Inhomogeneous cosmology in an anisotropic Universe
IInhomogeneous cosmology in ananisotropic Universe
Author:
Hayley J. Macpherson
Supervisors:
Ass. Prof. Daniel PriceDr. Paul Lasky
A thesis submitted in fulfillment of the requirementsfor the degree of Doctor of Philosophyin theMonash Centre for Astrophysics,School of Physics and Astronomy
October 30, 2019 a r X i v : . [ a s t r o - ph . C O ] O c t ii Copyright notice c (cid:13) Hayley J. Macpherson (2019)This thesis must be used only under the normal conditions of “fair deal-ing” under the Copyright Act. It should not be copied or closely para-phrased in whole or in part without the written consent of the author.Proper written acknowledgement should be made for any assistance ob-tained from this thesis. I certify that I have made all reasonable efforts tosecure copyright permissions for third-party content included in this thesisand have not knowingly added copyright content to my work without theowner’s permission.
Abstract
With the era of precision cosmology upon us, and upcoming surveys ex-pected to further improve the precision of our observations below the per-cent level, ensuring the accuracy of our theoretical cosmological model isof the utmost importance. Current tensions between our observations andpredictions from the standard cosmological model have sparked curiosityin extending the model to include new physics. Although, some sugges-tions include simply accounting for aspects of our Universe that are ignoredin the standard model. One example acknowledges the fact that our Uni-verse contains significant density contrasts on small scales; in the form ofgalaxies, galaxy clusters, filaments, and voids. This small-scale structureis smoothed out in the standard model, by assuming large-scale homogene-ity of the matter distribution, which could have a measurable effect dueto the nonlinearity of Einstein’s equations. This backreaction of small-scalestructures on the large-scale dynamics has been suggested to explain themeasured accelerating expansion rate of the Universe.Current standard cosmological simulations ignore the effects of GeneralRelativity by assuming purely Newtonian dynamics. In this thesis, we takethe first steps towards quantifying the backreaction of small-scale struc-tures by performing cosmological simulations that solve Einstein’s equa-tions directly. Simulations like these will allow us to quantify potentiallyimportant effects on our observations that could become measurable as theprecision of these observations increases into the future.We begin by testing our computational setup to ensure our results aretrustworthy. We then present simulations of a realistic matter distributionwith initial conditions inspired by the early moments of our own Universe.Analysing the averaged, large-scale evolution of an inhomogeneous uni-verse in full General Relativity, we find negligible difference from small-scale structures. However, we do find significant effects present on smallscales that could potentially influence future observations. While we sug-gest improvements to our computational framework to validate our results,we conclude that the backreaction of small-scale structures is unlikely to ex-plain the accelerating expansion of the Universe.Finally, we suggest future extensions to our analysis to improve thequantification of General-Relativistic effects on our cosmological observa-tions.ii
Publications list
1. H. J. Macpherson, P. D. Lasky, and D. J. Price (Mar. 2017). “Inhomo-geneous cosmology with numerical relativity”. In: Phys. Rev. D 95.6,064028, p. 064028.
DOI : . arXiv:
2. H. J. Macpherson, P. D. Lasky, and D. J. Price (Sept. 2018). “The Trou-ble with Hubble: Local versus Global Expansion Rates in Inhomo-geneous Cosmological Simulations with Numerical Relativity”. In:ApJ 865, L4, p. L4.
DOI :
10 . 3847 / 2041 - 8213 / aadf8c . arXiv:
3. H. J. Macpherson, D. J. Price, and P. D. Lasky (Mar. 2019). “Einstein’sUniverse: Cosmological structure formation in numerical relativity”.In: Phys. Rev. D 99.6, 063522, p. 063522.
DOI : . arXiv: x Declaration of Authorship
I, Hayley J. Macpherson, hereby declare that this thesis contains no ma-terial which has been accepted for the award of any other degree or diplomaat any university or equivalent institution and that, to the best of my knowl-edge and belief, this thesis contains no material previously published orwritten by another person, except where due reference is made in the textof the thesis.This thesis includes three original papers published in peer reviewedjournals in Chapters 3, 4, and 5. The core theme of the thesis is cosmologicalsimulations with numerical relativity. The ideas, development and writingup of all the papers in the thesis were the principal responsibility of myself,the student, working within the School of Physics and Astronomy underthe supervision of Ass. Prof. Daniel Price and Dr. Paul Lasky.The inclusion of co-authors reflects the fact that the work came from ac-tive collaboration between researchers and acknowledges input into team-based research. I have renumbered sections of submitted or published pa-pers in order to generate a consistent presentation within the thesis. In thecase of Chapters 3, 4, and 5, my contribution to the work in each case issummarised in the table on the following page.Student signature: Date: October 30, 2019I, Ass. Prof. Daniel Price, hereby certify that the above declaration cor-rectly reflects the nature and extent of the student’s and co-authors’ contri-butions to this work. In instances where I am not the responsible author Ihave consulted with the responsible author to agree on the respective con-tributions of the authors.Main Supervisor signature: Date: October 30, 2019 T h e s i sc h a p t e r Pub li c a ti o n titl e S t a t u s N a t u r ea nd % o f s t ud e n t c o n t r i bu ti o n C o - a u t h o r n a m e s , n a t u r ea nd % o f c o n t r i bu ti o n C o - a u t h o r s , M o n a s h s t ud e n t Y / N I nh o m o g e n e o u s c o s m o l o gy w i t hn u m e r i c a l r e l a t i v i t y P u b li s h e d ( P R D , M a r c h ) % – C o n c e p t , c o d i n g , a n a l y s i s o f r e s u l t s , m a n u s c r i p t a u t h o r s h i p P a u l L a s ky : % – C o n c e p t , c o d i n g , m a n u s c r i p t i n p u t . D a n i e l P r i c e : % – C o n c e p t , m a n u s c r i p t i n p u t . N E i n s t e i n ’ s U n i v e r s e : C o s m o l o g i c a l s t r u c t u r e f o r m a t i o n i nn u m e r i c a l r e l a t i v i t y P u b li s h e d ( P R D , M a r c h ) % – C o n c e p t , c o d i n g , a n a l y s i s o f r e s u l t s , m a n u s c r i p t a u t h o r s h i p D a n i e l P r i c e : % – C o n c e p t , c o d i n g , m a n u s c r i p t i n p u t . P a u l L a s ky : % – C o n c e p t , m a n u s c r i p t i n p u t . N T h e T r o u b l e w i t h H u bb l e : L o c a l v e r s u s G l o b a l E x p a n s i o n R a t e s i n I nh o m o g e n e o u s C o s m o l o g i c a l S i m u l a t i o n s w i t h N u m e r i c a l R e l a t i v i t y P u b li s h e d ( A p J L , S e p t e m b e r ) % – C o n c e p t , a n a l y s i s o f r e s u l t s , m a n u s c r i p t a u t h o r s h i p P a u l L a s ky : % – C o n c e p t , m a n u s c r i p t i n p u t . D a n i e l P r i c e : % – C o n c e p t , m a n u s c r i p t i n p u t . N i Acknowledgements
Firstly, I would like to thank my supervisors, Daniel Price and PaulLasky. From the beginning of this project back in 2015, you have both chal-lenged me and helped me grow, while always being plenty of fun to workwith. I wouldn’t be the scientist I am today if it weren’t for both of yourhelp, encouragement, and breadth of knowledge. I’m inspired by both ofyour curiosity, without which this work would never have happened. Iwould like to apologise for being such an expensive student, please acceptthis thesis as official reimbursement. It’s been a grand journey becomingcosmologists together, we made it.I would also like to thank my collaborators, especially Marco Bruni,Chris Blake, and Julian Adamek, for many valuable discussions and vis-its that helped me get to where I am today. I have been extremely luckyto travel so much during my PhD, and attend many conferences in someamazing places around the world. To everyone I met — and explored, ate,drank, and hiked with — along the way; thank you.Over my 7 years at Monash I have made many lifelong friendships. Aspecial thank you to Izzy for keeping me sane over those years, you willbe my best friend for life. I would also like to thank all of my office mates,especially Dave, Conrad, and Zac for priceless banter over the past 4 years.The majority of the postgrad cohort in the School of Physics and Astronomyhave contributed to my life at Monash in their own way, so to each of you;thank you. Perhaps my largest thanks goes to Jean Pettigrew, for doingbasically everything for everyone, all the time. I know I am not alone insaying that you keep us students afloat.Of course, I would like to thank my parents for their never-ending loveand support. Mum, your commitment and success have always inspiredme to be the best version of myself I can be, and the care you show forthose around you never ceases to amaze me. Hopefully this thesis willbe useful in explaining my research to your friends. Dad, your endlessencouragement and faith in me are a huge part of my achievements thusfar. Thanks for the science chats, South Park and tacos, the good (and thebad) surfs, and of course for being “a cool nerd... the Ultimate Person”(Mark Macpherson, 2014). A special thanks to my Grandma, not only forfeeding me and chatting MasterChef, but for your unconditional love andfor researching any place I’m travelling to better than I ever do. I love youall, thank you.Lastly, but certainly not least, I would like to thank my girls; Abbey,Meag, and Jess, for always sticking by me and being my best friends andsupport system since high school, and definitely for the rest of our livesto come. Also, a big thanks to Kym for helping me through the past fewmonths, I couldn’t have done it without you.This research was supported by an Australian Government ResearchTraining Program (RTP) Scholarship. I also acknowledge the AstronomicalSociety of Australia for their funding support that helped contribute to thiswork.iii
Contents Λ CDM . . . . . . . . . . . . . . . . . . . 16CMB power spectrum . . . . . . . . . . . . . . . . . . 16Hubble parameter . . . . . . . . . . . . . . . . . . . . 16Low-redshift Universe . . . . . . . . . . . . . . . . . . 17Proposed extensions . . . . . . . . . . . . . . . . . . . 171.3.4 Cosmological simulations . . . . . . . . . . . . . . . . 181.4 Cosmological perturbation theory . . . . . . . . . . . . . . . 191.4.1 The gauge problem . . . . . . . . . . . . . . . . . . . . 191.4.2 Linear perturbation theory . . . . . . . . . . . . . . . 201.4.3 Weak field approximation . . . . . . . . . . . . . . . . 211.5 Beyond the Standard Model . . . . . . . . . . . . . . . . . . . 221.5.1 General-Relativistic corrections . . . . . . . . . . . . . 221.5.2 Backreaction . . . . . . . . . . . . . . . . . . . . . . . . 24Averaging comoving domains . . . . . . . . . . . . . 24Averaging general foliations . . . . . . . . . . . . . . 27Improved general formalism . . . . . . . . . . . . . . 30Quantifying Q D . . . . . . . . . . . . . . . . . . . . . . 311.5.3 Exact inhomogeneous cosmology . . . . . . . . . . . 331.5.4 Numerical, General-Relativistic cosmology . . . . . . 34 INSTEIN T OOLKIT . . . . . . . . . . . . . . . . . . . . . 562.2.1 Base thorns . . . . . . . . . . . . . . . . . . . . . . . . 562.2.2
McLachlan . . . . . . . . . . . . . . . . . . . . . . . . 572.2.3
GRHydro . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2.4 Initial data and
FLRWSolver . . . . . . . . . . . . . . 58Linearly perturbed equations . . . . . . . . . . . . . . 60Linearly perturbed solutions . . . . . . . . . . . . . . 61Single-mode perturbation . . . . . . . . . . . . . . . . 62Multi-mode perturbations . . . . . . . . . . . . . . . . 632.2.5 Setup for this thesis . . . . . . . . . . . . . . . . . . . . 632.3 Post-processing analysis:
MESCALINE . . . . . . . . . . . . . 642.3.1 Key calculations . . . . . . . . . . . . . . . . . . . . . 64Ricci tensor and connection functions . . . . . . . . . 64Constraint violation . . . . . . . . . . . . . . . . . . . 66Expansion scalar . . . . . . . . . . . . . . . . . . . . . 67Shear tensor . . . . . . . . . . . . . . . . . . . . . . . . 68Other tensors . . . . . . . . . . . . . . . . . . . . . . . 68Effective scale factors . . . . . . . . . . . . . . . . . . . 692.3.2 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . 69Global averages . . . . . . . . . . . . . . . . . . . . . . 69Subdomain averaging . . . . . . . . . . . . . . . . . . 702.3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 70
ACTUS and FLRWSolver . . . . . . . . . . . . . . . . 934.2.2 Length unit . . . . . . . . . . . . . . . . . . . . . . . . 944.2.3 Redshifts . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.1 Linear Perturbations . . . . . . . . . . . . . . . . . . . 954.3.2 Cosmic Microwave Background fluctuations . . . . . 964.4 Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5 Averaging scheme . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.1 Cosmological parameters . . . . . . . . . . . . . . . . 1024.5.2 Post-simulation analysis . . . . . . . . . . . . . . . . . 1024.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.6.1 Global averages . . . . . . . . . . . . . . . . . . . . . . 1044.6.2 Local averages . . . . . . . . . . . . . . . . . . . . . . 104Cosmological parameters . . . . . . . . . . . . . . . . 104Scale factor . . . . . . . . . . . . . . . . . . . . . . . . 108Hubble parameter . . . . . . . . . . . . . . . . . . . . 1084.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.7.1 Global averages . . . . . . . . . . . . . . . . . . . . . . 1094.7.2 Local averages . . . . . . . . . . . . . . . . . . . . . . 1104.7.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A Newtonian Gauge 129B Averaging in the non-comoving gauge 133C Effective scale factors 135D Constraint violation 137E Convergence and errors 141 vi F Effective scale factors part 2 145Bibliography 151 vii
List of Abbreviations
GR G eneral R elativity LIGO L aser I nterferometer G ravitational-Wave O bservatory FLRW F riedmann- L emaître- R obertson- W alker SN1a S uper N ova Type osmic M icrowave B ackground Λ CDM Λ C old D ark M atter NR N umerical R elativity ADM A rnowitt D eser M isner BSSN B aumgarte- S hapiro- S hibata- N akamura CAMB C ode for A nisotropies in the M icrowave B ackground EOS E quation O f S tate LSST L arge S ynoptic S urvey T elescope SKA S quare K ilometre A rray SDSS S loan D igital S ky S urvey - d egree F ield G alaxy R edshift S urvey WMAP W ilkinson’s M icrowave A nisotropy P robe BAO B aryon A coustic O scillation SH0ES S upernova H for the E quation of S tate of dark energy RGTC R
IEMANNIAN G EOMETRY AND T ENSOR C ALCULUS
LTB L emaître T olman B ondi ET E
INSTEIN T OOLKIT
CCZ4 C onformal and C ovariant Z4TVD T otal V ariation D iminishing PPM P iecewise P arabolic M ethod ENO E ssentially N on- O scillatory HLLE H arten L ax van L eer E infeldt RK4 R unge- K utta th order MoL M ethod o f L inesix For every woman who thought she couldn’t
Chapter 1
Introduction and Background
Einstein’s general theory of relativity (GR) is the most accurate descrip-tion of gravity for the Universe. Rather than describing gravity as a forcebetween massive objects — as in Newton’s theory — gravity is a conse-quence of geometry. The presence of mass curves spacetime, the “fabric”of the Universe, which in turn affects the motion of passing matter. Ein-stein’s theory has proven to better describe dynamics in the local Universethan Newtonian gravity, successfully explaining the precession of the per-ihelion of Mercury (Einstein, 1916; Clemence, 1947) and the bending oflight around the Sun (Dyson, Eddington, and Davidson, 1920). More re-cently, the first detection of gravitational waves with the Laser Interferome-ter Gravitational-Wave Observatory (LIGO; Abbott et al., 2016) marked yetanother prediction of GR to be confirmed correct. While small-scale physicsin our Universe is described with great precision using either the weak- orstrong-field limit of GR, the applicability to large-scale physics remains tobe thoroughly tested.Early astronomical observations were limited to stars within our owngalaxy — which have small velocities — prompting scientists to believe thatthe Universe was static: neither expanding nor contracting. Einstein’s firstapplication of GR to cosmology thus required the cosmological constant, Λ ,to be added into his field equations to counteract the expansion; resulting ina static description of spacetime (Einstein, 1917). Aleksandr Friedmann andGeorges Lemaître independently derived expanding solutions to Einstein’sequations with no need for the cosmological constant (Friedmann, 1922;Friedmann, 1924; Lemaître, 1927). These solutions were largely dismisseduntil the first observational evidence for an expanding Universe emerged in1929. Edwin Hubble’s observations of extragalactic nebulae showed a pos-itive, linear trend between distance and radial velocity; suggesting that theUniverse is expanding (Hubble, 1929). Einstein later accepted the notionof an expanding Universe, deeming the cosmological constant unnecessary(Einstein, 1931; Einstein and de Sitter, 1932; Straumann, 2002).The basis for the current standard model of cosmology is the Friedmann-Lemaître-Robertson-Walker (FLRW) solution to Einstein’s equations; de-scribing an expanding, homogeneous, and isotropic spacetime. Applicationof the FLRW model to our cosmological observations revealed that thingswere not as expected; leading to an astounding discovery. Riess et al. (1998)and Perlmutter et al. (1999) discovered the accelerating expansion of theUniverse using observations of Type 1a supernovae (SN1a). These SN1aappear fainter than predicted by the FLRW model, implying the expansionof spacetime is accelerating. The return of the cosmological constant Λ wasimminent, deemed “dark energy”; a mysterious negative pressure forcing Chapter 1. Introduction and Background the expansion of the Universe to accelerate at late times.Early measurements of the first light after the Big Bang — the CosmicMicrowave Background (CMB) radiation — indicate that the Universe hasglobally flat geometry (Jaffe et al., 2001). Constraints that matter only ac-counted for ∼ of the total energy density of the Universe (Bennett etal., 2003) require a smoothly-distributed energy to reconcile this result withthat of a flat geometry. These measurements fit perfectly with the cosmo-logical constant, Λ .The existence of “dark matter” — a type of invisible matter that interactsonly gravitationally — was first inferred from rotation curves of galaxies(Freeman, 1970; Rubin and Ford, 1970; Rubin, Ford, and Thonnard, 1980).Since then, measurements of the weak gravitational lensing of light (e.g.Bacon, Refregier, and Ellis, 2000), and measurements of CMB anisotropy(e.g. Bennett et al., 2003; Hinshaw et al., 2013; Planck Collaboration et al.,2018) have strengthened the argument for the existence of dark matter (see,e.g., Bertone and Hooper, 2018, for a review).These discoveries each contributed to the current standard cosmologi-cal model; the Λ Cold Dark Matter ( Λ CDM) model, named after the mainconstituents of the Universe.To date, the Λ CDM model has successfully explained essentially all ofour cosmological observations. Notable successes include matching thepower spectrum of temperature fluctuations in the CMB (e.g. Planck Col-laboration et al., 2018), the location of the peak separation of large-scalestructures — the baryon acoustic oscillation (BAO) peak — (e.g. Blake etal., 2011), and the matter power spectrum of the large-scale structure at lowredshifts (e.g. Reid et al., 2010; Anderson et al., 2014).Aside from its successes, there are some tensions between the Λ CDMmodel and what we observe. Most notable are the lack of power at thelargest scales in the CMB power spectrum (Planck Collaboration et al., 2018);the low-multipole “bump” visible in the power spectrum data, and therecent . σ tension between local measurements of the Hubble parame-ter (Riess et al., 2018b; Riess et al., 2018a) compared to that inferred fromthe CMB (Planck Collaboration et al., 2018). These tensions may be due toinsufficient precision, systematic errors in the measurements, or — moreexcitingly — new physics (see Section 1.3.3).Existing physics that is currently neglected in the standard model couldexplain some of these tensions (e.g. Buchert et al., 2016). The basis of Λ CDMis the assumption of a homogeneous, isotropic background spacetime withexpansion described by the FLRW model. Current state-of-the-art cosmo-logical simulations model structures evolving under Newtonian gravity ontop of an a-priori assumed homogeneously expanding background space-time (e.g. Springel, 2005; Kim et al., 2011; Genel et al., 2014; Potter, Stadel,and Teyssier, 2017). These simulations are the primary comparison pointto our cosmological observations. Newtonian gravity has been shown tobe a good approximation for GR on small scales, however, it’s applicabil-ity to cosmological scales remains uncertain, if only because it is based oninstantaneous action-at-a-distance (Buchert et al., 2016). Newtonian cos-mological simulations thus ignore causality, which could have a significanteffect when considering cosmological scales.The main argument for the assumptions of homogeneity and isotropyunderlying the standard model is that our Universe is homogeneous and hapter 1. Introduction and Background smoothing over this small-scalestructure has a measurable effect on the large-scale dynamics of the Uni-verse. Buchert (2000) showed that the evolution of the average of an in-homogeneous universe does not coincide with the evolution of a homoge-neous universe, due to the non-commutation of averaging and time evolu-tion in fully nonlinear GR. Extra mathematical terms contribute to both theexpansion of spacetime and the acceleration of the expansion of spacetime,the significance of which has been the subject of much debate (e.g. Buchertet al., 2015; Green and Wald, 2016).Upcoming cosmological surveys using state-of-the-art telescopes suchas Euclid, the Large Synoptic Survey Telescope (LSST), and the Square Kilo-metre Array (SKA) are expected to reach percent-level precision (Ivezic,Tyson, and Abel, 2008; Maartens et al., 2015; Amendola, Appleby, andAvgoustidis, 2016), at which small differences between Newtonian grav-ity and GR on cosmological scales could be measurable. Drawing correctconclusions from our observations first requires accurate cosmological sim-ulations; simulations that include any potentially measurable GR effects.The magnitude of the backreaction of structures on the large-scale dynam-ics of the Universe can only be quantified with a full treatment of GR in acosmological simulation.Numerical relativity (NR) allows us to solve Einstein’s field equationsnumerically, often using a “3+1” decomposition. This involves splittingour four-dimensional spacetime into three space dimensions and one timedimension. The ADM formalism, named after its authors Arnowitt, Deser,and Misner (1959), casts Einstein’s equations into a weakly hyperbolic formfor numerical evolution. This weak hyperbolicity means the system is un-stable for long time evolutions. The Baumgarte-Shapiro-Shibata-Nakamura(BSSN) formalism improves on this by instead forming a strongly hyper-bolic system (Shibata and Nakamura, 1995; Baumgarte and Shapiro, 1999),which allows for arbitrarily long, stable time evolutions. Pretorius (2005a),Campanelli et al. (2006), and Baker et al. (2006) performed the first suc-cessful long-term evolutions of a binary black hole system, including themerger and emission of gravitational waves. Since then, the field of NRhas exploded and it is now widely used for simulations of binary mergersof compact objects such as black holes (e.g. Baker et al., 2006; Campanelliet al., 2006; Buonanno, Cook, and Pretorius, 2007; González et al., 2007;Hinder, Kidder, and Pfeiffer, 2018; Huerta et al., 2019), neutron stars (e.g.Baiotti, Giacomazzo, and Rezzolla, 2008; Paschalidis et al., 2011; Kastaunand Galeazzi, 2015; Chaurasia et al., 2018), stellar collapse and supernovaeexplosions (e.g. Duez, Shapiro, and Yo, 2004; Montero, Janka, and Müller,2012), and more recently, for cosmology (Giblin, Mertens, and Starkman,2016a; Giblin, Mertens, and Starkman, 2016b; Bentivegna and Bruni, 2016;Giblin et al., 2017; Macpherson, Lasky, and Price, 2017; Macpherson, Price,and Lasky, 2019; Macpherson, Lasky, and Price, 2018; East, Wojtak, andAbel, 2018; Daverio, Dirian, and Mitsou, 2019; Barrera-Hinojosa and Li,2019).It is now possible to investigate GR effects on our cosmological obser-vations, test the validity of the assumptions underlying the standard cos-mological model, and ensure our cosmological simulations are sufficiently
Chapter 1. Introduction and Background accurate for forthcoming precision cosmological surveys. These are the pri-mary motivations behind this thesis.We perform three-dimensional cosmological simulations of large-scalestructure formation that solve Einstein’s equations without approximationusing NR. Removing the assumptions underlying current cosmological sim-ulations allows us to fully understand GR’s role, and validity, on cosmolog-ical scales.In the remainder of this Chapter we describe the basis of modern cos-mology, including the mathematical background for the standard model,observational tests of Λ CDM, the current status of state-of-the-art cosmo-logical simulations, cosmological perturbation theory and its current use inthese simulations, current observational tensions with the standard modeland the resulting suggested extensions to Λ CDM.In Chapter 2 we outline the methods used to perform our simulationsand main analysis. We include an introduction to NR and derive the equa-tions of the BSSN formalism including a discussion of common coordinatechoices, a discussion of the workings of the E
INSTEIN T OOLKIT — the open-source NR code that we use to perform our simulations — and a descriptionof our post-processing analysis.In Chapter 3 we describe our computational framework in detail, andshow results of two code tests to demonstrate the accuracy of our results.We perform simulations of a homogeneous, isotropic FLRW spacetime andcompare to the analytic solution. Perturbing this FLRW spacetime withinitially small, simplified inhomogeneities, we show the growth of theseperturbations matches the analytic linear solution; but also surpasses thissolution into the nonlinear regime of growth where deriving analytic ex-pressions in full GR is not possible.In Chapter 4 we present NR cosmological simulations of a realistic mat-ter distribution. We generate initial conditions drawn from the temperatureanisotropies in the CMB, using the Code for Anisotropies in the MicrowaveBackground (CAMB; Seljak and Zaldarriaga, 1996). We then evolve theseinitially small inhomogeneities to redshift z ≈ , and from the resultingcosmic web we calculate averaged quantities and draw comparisons withan FLRW spacetime.In Chapter 5 we analyse the effect of local inhomogeneities on an ob-server’s measurement of the Hubble parameter (expansion rate) of the Uni-verse. Our calculations are presented to approximate the expected varianceon the local H measurement from SN1a in Riess et al. (2018b) due to anobserver’s physical location in an inhomogeneous, anisotropic Universe.A short summary is presented at the end of each chapter, and a fullsummary with details of future work is presented in Chapter 6.Throughout this thesis, we assume a metric signature ( − , + , + , +) un-less otherwise stated, and adopt the Einstein summation convention: im-plied summation over repeated indices. Greek indices represent spacetimeindices and take values (0 , , , , and Latin indices represent spatial in-dices and take values (1 , , . .1. Einstein’s field equations Einstein’s field equations of GR are G µν ≡ R µν − R g µν = 8 πGc T µν , (1.1)where G µν is the Einstein tensor, g µν is the metric tensor, G is the gravita-tional constant, c is the speed of light, and T µν is the stress-energy tensor ofmatter. The four-dimensional Riemann curvature tensor is R αµβν ≡ ∂ β (4) Γ αµν − ∂ ν (4) Γ αµβ + (4) Γ αλβ (4) Γ λµν − (4) Γ αλν (4) Γ λµβ , (1.2)where ∂ µ ≡ ∂/∂x µ is the partial derivative with respect to coordinate x µ ,and the four-dimensional Christoffel symbols (or connection functions) as-sociated with the metric g µν are defined as (4) Γ αµν ≡ g αβ ( ∂ µ g βν + ∂ ν g µβ − ∂ β g µν ) . (1.3)We distinguish these four-dimensional objects from their spatial counter-parts with (4) (see Section 2.1.1). The Ricci curvature tensor in Einstein’sequations (1.1) is then the contraction of the Riemann tensor R µν ≡ R αµαν ,and the Ricci scalar is the trace of the Ricci curvature tensor, R ≡ g µν R µν ,where the inverse metric g µν is defined such that g µν g να = δ µα , with δ µα theKronecker delta function. Indices of four-dimensional tensors are raisedand lowered using the metric tensor, i.e. A µν = g αµ A αν .The Einstein equations (1.1) satisfy the contracted Bianchi identities (Voss,1880), ∇ µ G µν = 0 , (1.4)which consequently implies the conservation of stress-energy, ∇ µ T µν = 0 , (1.5)from which the equations of General-Relativistic hydrodynamics are de-rived (see Section 2.1.6). In the above, ∇ µ is the covariant derivative associ-ated with the metric g µν , i.e. ∇ α g µν = ∂ α g µν + (4) Γ µαβ g βν + (4) Γ ναβ g µβ = 0 , (1.6)by construction.The set of 16 equations (1.1) — which reduces to 10 due to symmetries inthe metric tensor g µν — describe how matter, T µν , interacts with spacetime, G µν . Here, we approximate the matter content of the Universe as a perfectfluid, which has stress-energy tensor T µν = ρ hu µ u ν + P g µν , (1.7)where ρ and P are the rest-mass density and pressure, respectively, andthe specific enthalpy is h = c + (cid:15) + P/ρ , with (cid:15) the internal energy. Thedimensionless four velocity of the fluid is defined as u µ ≡ dx µ d ( cτ ) , (1.8) Chapter 1. Introduction and Background where x µ = ( ct, x i ) are the spacetime coordinates, and the four velocity isnormalised such that u µ u µ = − . The proper time, τ , is defined by c dτ ≡ − ds , (1.9)where ds is the infinitesimal distance between two nearby points in anyspacetime, ds ≡ g µν dx µ dx ν , (1.10)commonly referred to as the line element of a given spacetime.Often Einstein’s equations are written including the cosmological con-stant Λ , i.e. R µν − R g µν + Λ g µν = 8 πGc T µν , (1.11)commonly adopted to describe dark energy; the driving force of the appar-ent accelerating expansion of the Universe (see Section 1.3). Alternatively,the cosmological constant can be absorbed into the stress-energy tensor asan additional form of matter, using ρ R → ρ R + ρ Λ , and P → P + P Λ , with P Λ = − ρ Λ . Here, ρ R is the total mass-energy density as measured by anobserver comoving with the fluid, and is defined by projecting the stress-energy tensor into the rest frame of the fluid, i.e. ρ R c ≡ T µν u µ u ν . (1.12)Using (1.7), its relation to the rest-mass density ρ is therefore ρ R c = ρ hu µ u µ u ν u ν + P g µν u µ u ν , (1.13) = ρ c (cid:16) (cid:15)c (cid:17) . (1.14)For the remainder of this thesis, we assume ρ R contains all forms of energy-density — potentially including a cosmological constant — unless other-wise stated. Einstein’s field equations describe the relation between spacetime and mat-ter. In order to simulate the evolution of a four-dimensional spacetimeusing numerical relativity, we split that spacetime into a series of three-dimensional hypersurfaces (surfaces) that can be evolved in time. Choos-ing the way the spacetime is split is done via the lapse function, α , whichdescribes the spacing between subsequent spatial slices in time, and theshift vector, β i , which describes how the spatial coordinates are re-labelledbetween slices. These are known as gauge choices. The surfaces each havedimensionless normal vector n µ ≡ − α ∇ µ x , (1.15)which is normalised such that n µ n µ = − . We impose α > , so that n µ is time like. In terms of the lapse and shift functions, the covariant andcontravariant normals are n µ = ( − α, , , and n µ = (1 /α, − β i /α ) , re-spectively. From (1.15) we define the spatial metric tensor induced on the .2. Friedmann-Lemaître-Robertson-Walker spacetime γ µν ≡ g µν + n µ n ν , (1.16)which is used to express four-dimensional objects, such as the Riemanncurvature, in terms of purely spatial objects by projecting them onto thespatial surfaces.The normal vector describes the motion of coordinate observers, andthe fluid four velocity describes the motion of observers comoving with thefluid. The Lorentz factor between this pair of observers is W ≡ − n α u α = 1 (cid:112) − v i v i /c , (1.17)where v i is the fluid three velocity with respect to an Eulerian observer,which, in terms of the four velocity, lapse, and shift, is v i c = u i αu + β i α . (1.18)In this three-dimensional split, the components of the contravariant fourvelocity are u = dx d ( cτ ) = Wα , (1.19) u i = dx i d ( cτ ) = W (cid:18) v i c − β i α (cid:19) , (1.20)and the covariant components are u = W (cid:18) v i β i c − α (cid:19) , (1.21) u i = W v i c . (1.22)Indices of purely spatial objects are raised and lowered using the spatialmetric, i.e. v i = γ ij v j . For four-dimensional objects this is not the case, i.e.we have u i = g iα u α (cid:54) = g ij u j in general. Full details of the 3+1 foliation andthe resulting equations commonly used in numerical relativity are given inSection 2.1. The assumptions underlying the FLRW model are that the Universe is bothhomogeneous — the same everywhere — and isotropic — the same in alldirections. This is the cosmological principle .Applying the cosmological principle to Einstein’s equations in sphericalpolar coordinates results in the line element ds = − c dt + a ( t ) (cid:20) dr − kr + r (cid:0) dθ + sin θdφ (cid:1)(cid:21) , (1.23)where r is the comoving radial distance, and the curvature constant is k = − , , if the universe has an open, flat or closed geometry, respectively. Chapter 1. Introduction and Background
Assuming spatial flatness we can write the above line element in Cartesiancoordinates as ds = − c dt + a ( t ) δ ij dx i dx j . (1.24)In the above, a ( t ) is the spatially homogeneous scale factor describing thesize of the universe at any time. The coordinate time, t , in (1.24) is also known as the cosmic time ; the propertime for a clock with zero peculiar velocity in a vacuum FLRW spacetime.The proper time τ , defined in (1.9), represents the proper time of any clockmoving along a general path ds , whereas the cosmic time represents theproper time of a specific clock.The conformal time, η , explicitly describes the time it would take for aphoton to travel back to the Big Bang, if the expansion were to suddenlycease, at any time in the Universe’s history. For this reason, η is not a physi-cally meaningful time, however, the particle horizon , defined as cη , measuresthe maximum distance any information could have propagated since theBig Bang (in an FLRW spacetime). This horizon is useful in determiningcausality between different regions in the Universe, and identifying sim-ilarities between regions separated by distances larger than cη (e.g. the“horizon problem” in inflationary cosmology; see Dodelson, 2003). Theconformal time is widely used in cosmology, and is related to the cosmictime via a ( η ) dη = dt . The FLRW line element (1.24) in terms of conformaltime is therefore ds = a ( η ) (cid:0) − c dη + δ ij dx i dx j (cid:1) . (1.25) With the metric (1.25), the components of the Einstein tensor G µν are G = 3 c (cid:18) a (cid:48) a (cid:19) , (1.26) G ij = 1 c (cid:34)(cid:18) a (cid:48) a (cid:19) − a (cid:48)(cid:48) a (cid:35) δ ij , (1.27) G i = 0 , (1.28)where a = a ( η ) , and the (cid:48) represents a derivative with respect to conformaltime, i.e. a (cid:48) ≡ ∂ η a ≡ ∂a/∂η . The trace of (1.27) is therefore G ii = g ij G ij = 3 c (cid:34)(cid:18) a (cid:48) a (cid:19) − a (cid:48)(cid:48) a (cid:35) . (1.29)The assumption of isotropy underlying the FLRW solution implies the fluidis at rest with respect to the FLRW coordinates, i.e. u µ = (1 , , , , and theassumption of homogeneity implies the density and pressure are functionsof time only. Assuming a perfect fluid, the components of the stress-energy .2. Friedmann-Lemaître-Robertson-Walker spacetime T = ρ R c a , (1.30) T ij = P g ij , (1.31) T i = 0 , (1.32)and the trace of the spatial components is T ≡ T ii = 3 P. (1.33)Substituting the time-time components (1.26) and (1.30) into Einstein’sequations (1.1) gives the first Friedmann equation (cid:18) a (cid:48) a (cid:19) = 8 πGρ R a , (1.34)and the traces (1.29) and (1.33), also with (1.34), gives the acceleration equa-tion a (cid:48)(cid:48) a = − πGa (cid:18) ρ R + 3 Pc a (cid:19) . (1.35)From the time component of the conservation law for the stress-energytensor (1.5) ∇ µ T µ = ∂ µ T µ + (4) Γ µαµ T α + (4) Γ αµ T αµ = 0 , (1.36)and using the metric (1.25) to calculate the connection functions (see Sec-tion 2.3.1), we find ρ (cid:48) R = − a (cid:48) a (cid:18) ρ R + Pc (cid:19) , (1.37)which describes mass-energy conservation in an FLRW spacetime. Deriving the time evolution of the FLRW spacetime in most cases requiressolving (1.37) with (1.34), however, to close the system of equations wemust first specify an equation of state (EOS) relating the pressure, P , tothe mass-energy density, ρ R .The EOS we choose will depend on the cosmological era we want to de-scribe. At early times in the Universe’s history, around the recombinationera, radiation dominated the total energy density, with pressure P = ρ R c .Substituting this in (1.37) gives a density ρ R ∝ a − , meaning the densityof radiation dropped off rapidly with time. At later times, matter there-fore came to dominate the energy density, which is well approximated as a“dust” fluid with P = 0 . For matter domination the Friedmann equationsare H = 8 πGρ R a , (1.38a) a (cid:48)(cid:48) a = − πGρ R a , (1.38b)0 Chapter 1. Introduction and Background where we have defined the conformal Hubble parameter to be H ( η ) ≡ a (cid:48) a . (1.39)In some cosmologies the sign of the Hubble parameter H is ambiguous,e.g. in bouncing cosmologies (see e.g. Novello and Bergliaffa, 2008). Toderive the time evolution of the FLRW spacetime for these cases we mustinstead solve (1.37) with the acceleration equation (1.35). For the remainderof this thesis, we will only consider cosmologies with positive expansion,i.e. H > , for all time.Setting P = 0 in (1.37) gives ρ (cid:48) R ρ R = − a (cid:48) a , (1.40) ⇒ ρ R a = ρ ∗ (1.41)where ρ ∗ ≡ ρ R, init a is the conserved (constant) rest-frame energy-density, a init ≡ a ( η init ) and ρ R, init ≡ ρ R ( η init ) are the arbitrary initial values of thescale factor and density, respectively. Substituting (1.41) into the Friedmannequation (1.34) gives a (cid:48) √ a = (cid:114) πGρ ∗ , (1.42) ⇒ a ( ξ ) = a init ξ , (1.43)where we have defined the dimensionless scaled conformal time ξ ≡ (cid:114) πGρ ∗ a init η, (1.44)and we have set η init = 0 in deriving the above. Using (1.43) in (1.41) wefind ρ R ( ξ ) = ρ R, init ξ . (1.45)According to observations of SN1a, the expansion of the Universe is cur-rently accelerating. This implies a mysterious, negative pressure — knownas “dark energy” — now contributes ∼ of the total energy density, bestdescribed via the cosmological constant Λ (see Section 1.3). The evolutionof the energy density and scale factor in this case is found using an EOSwith a combination of dust and dark energy, i.e. P = − ρ Λ .The FLRW spacetime forms the basis of the current standard cosmo-logical model to describe the large-scale evolution of the Universe, and isadopted in almost all current simulations of cosmological structure forma-tion (see Section 1.3). The FLRW spacetime is also a common choice fora background cosmology in perturbation theory — a method commonlyused to study the growth of structures in the Universe (see Section 1.4) —and to estimate the size of General-Relativistic corrections to the standardcosmological model (see Section 1.5.1). .3. The Lambda Cold Dark Matter model The Λ CDM model became the widely-accepted cosmological model afterthe discovery of the accelerating expansion of the Universe (Riess et al.,1998; Perlmutter et al., 1999). This model is the adaptation of the FLRW so-lution of GR to our observable Universe, and is the concordance cosmologysupported by many cosmological observations. In this model, the Universehas a flat geometry, a perfect fluid content, and a non-zero cosmologicalconstant Λ to explain the accelerating expansion of the Universe. In addi-tion, the majority of matter in the Universe is observed to be a type of slowmoving (i.e. not relativistic) matter known as “cold dark matter”, whichonly interacts gravitationally and is therefore well approximated as “dust”. Considering matter, curvature, and dark energy, the Friedmann equation isderived from Einstein’s equations (1.11) with Λ (cid:54) = 0 and the metric (1.23),with curvature k (cid:54) = 0 in general, giving (cid:18) a (cid:48) a (cid:19) = 8 πGρ R a c − kc a . (1.46)With the definition of the conformal Hubble parameter, (1.39), we can writethis equation equivalently as m + Ω Λ + Ω k , (1.47)where the dimensionless cosmological parameters are Ω m ≡ πGρ R a H , (1.48a) Ω Λ ≡ Λ c H , (1.48b) Ω k ≡ − kc a H . (1.48c)These describe the contribution to the total energy-density of the Universefrom matter, dark energy, and curvature, respectively.Measuring the cosmological parameters using observations allows usto determine the main components of the Universe, under the assumptionthat the FLRW model is a valid description of our Universe on the scalesmeasured. We discuss the main observations constraining the values of thecosmological parameters in the following sections. The first discovery by Hubble (1929) that the Universe is expanding markedthe beginning of the standard cosmological model. Early work constrainingthe energy-density of matter to be Ω m < (e.g. Efstathiou, Sutherland,and Maddox, 1990; Efstathiou, Bond, and White, 1992; Kofman, Gnedin,and Bahcall, 1993; Ostriker and Steinhardt, 1995) paved the way for thediscovery of the accelerating expansion, implying Ω Λ > (Riess et al., 1998;Perlmutter et al., 1999).2 Chapter 1. Introduction and Background
The first detection of the CMB radiation revealed a homogeneous andisotropic background glow across the sky. Improvements in instrumenta-tion and better understanding of systematic errors led to the discovery thatthe CMB radiation was actually anisotropic , and even further improvementsnow allow us to constrain these anisotropies to the percent level. Surveys ofSN1a can now constrain cosmological parameters to within ∼ (Riess etal., 2018b), and forecasts of upcoming cosmological surveys predict betterthan percent-level precision (Ivezic, Tyson, and Abel, 2008; Maartens et al.,2015; Amendola, Appleby, and Avgoustidis, 2016; Square Kilometre ArrayCosmology Science Working Group et al., 2018; Zhan and Tyson, 2018).Currently, almost all of our cosmological observations are explained ex-tremely well under the standard model. Including the power spectrumof anisotropies in the CMB radiation, the large-scale clustering of galaxies,and the values of the cosmological parameters. Aside from its success, the Λ CDM model also has its fair share of “curiosities”, i.e. observations thatare not explained in the context of Λ CDM. These have driven explorationsinto whether they could be caused by modifications of GR, if the assump-tions underlying the standard model are flawed, or if some other exoticphysics could explain the discrepancies. We discuss some of these tensions,and some extensions to Λ CDM proposed to alleviate them, in Section 1.3.3.
Type 1a Supernovae
The luminosity distance d L of an object is related to the luminosity fluxreceived by an observer, F , and the absolute luminosity of the object, L , via F = L πd L . (1.49)The flux received by an observer is dependent on d L , which itself dependson the underlying cosmology; and therefore on the energy-density of darkenergy and curvature. Measurements of the luminosity distance are usedto determine the distance-redshift relation — i.e. the Hubble diagram —and can therefore also be used to determine the cosmological parametersdefined in (1.48).Phillips (1993) discovered that SN1a can be used as standardised mea-sures of distances, or “standard candles”, due to their predictable lightcurves. Combined with the fact that they have high intrinsic luminosities,and therefore can be observed at large distances, SN1a are perfect candi-dates for cosmological observation (see Leibundgut, 2000, for a review).The High-z Supernova Search Team (Schmidt et al., 1998; Riess et al., 1998)and the Supernova Cosmology Project (Perlmutter et al., 1999) took advan-tage of this and measured the Hubble diagram out to redshift z ∼ . forthe first time. The magnitudes of the distant SN1a they found were ∼ dimmer than anticipated by a flat FLRW model with zero cosmological con-stant, implying accelerating expansion with Λ > in the context of FLRW.Alternate explanations for this dimming include intrinsic evolution of theSN1a light curves as a function of redshift (Drell, Loredo, and Wasserman,2000) and extinction via extragalactic grey dust (Aguirre, 1999b; Aguirre,1999a). Further observations of even higher-redshift SN1a, with z > , .3. The Lambda Cold Dark Matter model deceleration , and therefore wasnot always dark-energy dominated (Riess et al., 2001; Riess et al., 2004;Riess et al., 2007).SN1a are now widely used as cosmological probes, and many subse-quent surveys align with the early results of Riess et al. (1998) and Perlmut-ter et al. (1999), including the new Supernova Cosmology Project (Knop etal., 2003), the Supernova Legacy Survey (Astier et al., 2006), the ESSENCEsupernova survey (Miknaitis, Wood-Vasey, and ESSENCE Team, 2006; Wood-Vasey et al., 2007), the Sloan Digital Sky Survey II, (SDSS Kessler et al.,2009), the Union2 compilation (Amanullah et al., 2010), the Supernova H for the Equation of State of dark energy (SH0ES) project (Riess et al., 2011;Riess et al., 2016), the Dark Energy Survey Supernova Program (Abbott etal., 2019), and combinations of different SN1a sample sets, for example, Su-percal (Scolnic et al., 2015) and the Pantheon Sample (Scolnic et al., 2018).If we wish to maintain the homogeneous, isotropic FLRW spacetime asthe base of the standard cosmological model, the dimming of SN1a can beexplained in one of two ways. First is that the Universe contains a sig-nificant amount of dark, non-baryonic matter with an effective negativepressure, called “dark energy”. This is often described as being smoothlydistributed through spacetime, as the cosmological constant Λ . Second isthat Einstein’s theory of GR breaks down on cosmological scales, and weneed an amended theory to explain our Universe on large scales (see Cliftonet al., 2012, for a review). The standard cosmological model assumes thatGR is the correct description of the Universe, and adopts the cosmologi-cal constant, with observations currently constraining Ω Λ ≈ . . Alterna-tively, abandoning the assumptions of homogeneity and isotropy has, insome cases, been able to explain the dimming without Λ (see Section 1.5).Observations of SN1a also allows precise measurement of the expansionrate locally via the Hubble parameter, H . Using the distance modulus rela-tion, and by calibrating the distances to SN1a using Cepheid variable starsin their host galaxies, the most up-to-date measurement of H from SN1ahas reached 2.3% precision, with H = 73 .
48 km s − Mpc − and a σ uncer-tainty of ± .
66 km s − Mpc − (Riess et al., 2018b). This measurement of thelocal expansion rate is in . σ tension with the model-dependent inferred value using the latest CMB data from the Planck Collaboration et al. (2018),with H = 67 . − Mpc − and a σ uncertainty of ± . − Mpc − We discuss this discrepancy more in Section 1.3.3, and investigate the roleof local inhomogeneities on the SN1a measurement in Chapter 5.
Cosmic Microwave Background
The discovery of the CMB radiation revealed a perfectly homogeneous andisotropic background glow across the sky with temperature ∼ . K (Pen-zias and Wilson, 1965), providing the first evidence in favour of Big-Bangcosmology (Dicke et al., 1965). Increasingly precise measurements of thebackground radiation revealed tiny anisotropies in temperature (Smoot etal., 1992), and recent measurements — using the Wilkinson MicrowaveAnisotropy Probe (WMAP) and the Planck satellite — have uncovered im-mense amounts of detail in the ∆ T / ¯ T ∼ − anisotropies around the mean4 Chapter 1. Introduction and Background ¯ T ≈ . K radiation (e.g. Komatsu et al., 2009; Planck Collaboration et al.,2016).The very early Universe was a dense, opaque plasma of baryons andphotons. Hot regions in this plasma expanded under high pressure andconsequently cooled. As they cooled, the outward pressure was eventu-ally overcome by the gravitational potential, causing these regions to re-collapse, again raising the temperature and pressure to the point at whichpressure dominated once again. This interplay between forces caused BAOs;acoustic waves in the dense plasma. As the Universe expanded, less andless Compton scattering occurred, and free electrons bound with protonsto form neutral hydrogen; this was the recombination era. These free elec-trons bound in their highest energy state, immediately dropping in energyand emitting the first photons that were able to freely travel through theUniverse; this was the photon decoupling era (see Durrer, 2008).The small-scale anisotropies we measure in the CMB radiation todayare dominated by the acoustic oscillations created during recombination.Measuring the location and amplitude of the resulting peaks in the angularpower spectrum allows us to constrain the geometry of the Universe, theenergy-density of baryons, and the energy-density of matter (Jungman etal., 1996). Temperature anisotropies themselves can be related to perturba-tions in the metric at recombination, consequently allowing us to constrainthe matter perturbations that gave rise to the large-scale structure we seetoday.Measurements of the CMB temperature power spectrum alone only con-strain a small subset of the cosmological parameters, and so Planck Collab-oration et al. (2018) combine the temperature power spectrum with mea-surements of the polarisation power spectrum and the lensing signatureof the CMB. Since the CMB photons measured have travelled the entirehistory of the Universe, they have passed by many massive objects alongtheir path, and therefore there will be a gravitational lensing signature inthe radiation we measure (Blanchard and Schneider, 1987). Measuring thislensing spectrum requires assumptions about the geometry of the Universe,the depth of gravitational potentials along the photon paths, the radial ex-tent of the potentials, and the average number of potentials any one photonmay pass through. These assumptions are made under linear perturbationtheory (see Section 1.4.2), since on the ∼ arcminute scales considered forCMB lensing studies, weak lensing should be a valid approximation (Lewisand Challinor, 2006). Large-Scale Structure
The initially small temperature anisotropies in the CMB gave rise to thelarge-scale distribution of galaxy clusters, filaments, and voids we see to-day. Signatures of the anisotropy in the CMB radiation are therefore presentin the measured clustering of galaxies, e.g. the sound horizon introduced inthe previous section (Eisenstein et al., 1998; Eisenstein and Hu, 1998). Thesound horizon is measured at low redshift as a peak in the matter powerspectrum, and the location of this peak (and hence, physical size of the sepa-ration) at different redshifts provides intuition into the method of structureformation, a geometrical distance measure, and insight into the expansion .3. The Lambda Cold Dark Matter model Λ CDM cosmology to simulate the growth of the large-scale structurefrom initially small perturbations. These simulations predict a certain dis-tribution and abundance of massive galaxy clusters at a given redshift (e.g.Warren et al., 2006). The formation of the dark-matter haloes surroundingthese clusters should be dependent only on the geometry of the Universeand the initial fluctuations that gave rise to the gravitational potentials to-day (Haiman, Mohr, and Holder, 2001). The abundance of these clusterswill also be dependent on the growth rate of structure, determined fromthe total energy density of matter, Ω m . Including analysis of the evolution of the abundance of galaxy clusters can improve the constraints on cosmo-logical parameters (Viana and Liddle, 1999). In addition, measuring theweak gravitational lensing of light as it passes by massive clusters allowsanother method of mapping the dark-matter distribution. This lensing dis-torts images of distant galaxies lying behind massive clusters along our lineof sight, and the degree of this lensing allows us to constrain the mass of thedark-matter halo of the cluster. As with the galaxy cluster abundances, thisallows constraints to be put on dark energy’s role in cosmological structureformation (Frieman, Turner, and Huterer, 2008).In a matter-dominated Universe, on scales where linear perturbationtheory is valid, the depth and size of the gravitational potentials are con-stant in time. When dark energy dominates at later times, the potentialsdecay due to the accelerated expansion, and photons are lensed as theytravel through these potentials. This is the late-time, integrated Sachs-Wolfeeffect (Rees and Sciama, 1968). Correlating the lensing signal of the CMBwith the position of super-voids and super-clusters ( ∼ Mpc in size) —as measured in the SDSS luminous red galaxies survey — supports a flat,dark-energy dominated Universe. However, comparisons with Λ CDM N-body simulations show a ∼ σ deviation from the observations (Granett,Neyrinck, and Szapudi, 2008). Concordance cosmology
Most cosmological observations only constrain certain parameters. For ex-ample, the CMB radiation does not strongly constrain dark energy, since atrecombination the amount of dark energy was negligible compared to mat-ter and radiation. Combinations of cosmological observations are thereforenecessary to build a complete picture of the cosmological model that bestdescribes our Universe. The Planck Collaboration et al. (2018) presented thelatest CMB data, making thorough comparisons to not only different kinds6
Chapter 1. Introduction and Background of CMB power spectra, but also to independent cosmological probes in-cluding BAOs, SN1a, and galaxy clusters, all of which agree on a spatially-flat Λ CDM cosmology with matter density Ω m = 0 . ± . , and dark-energy density Ω Λ = 0 . ± . , where the dark energy equation of stateis P = − wρ Λ with w = − . ± . (consistent with a cosmological con-stant). See Planck Collaboration et al. (2018) for the full parameter set de-scribing the currently favoured standard model. Λ CDM
The Λ CDM model has successfully predicted and explained most of ourcosmological observations. However, alongside its success are several ten-sions (see, e.g. Bull et al., 2016; Buchert et al., 2016, for reviews). These“curiosities” have sparked interest in investigating possible extensions oralterations to the standard model, most of which are also strongly moti-vated by the fact that inflation, dark matter, and dark energy — the maincomponents of the standard model — have largely eluded explanation todate.
CMB power spectrum
The Planck Collaboration et al. (2018) measurements of the CMB anisotropyconstrain many of the Λ CDM model parameters to better than percent-level precision, aligning with predictions of the lensing signal present inthe CMB radiation and the angular power spectrum at small scales. How-ever, at the largest angular scales measured by the Planck satellite, there isa suspicious “dip” in the power spectrum relative to the Λ CDM prediction,which is also seen in CMB data from WMAP (Bennett et al., 2003; Hinshawet al., 2013). The fact that both satellites independently measure the samedip at the same angular scale essentially rules out instrument systematicerrors or foreground structures as causes of the anomaly, instead pointingtowards a real feature in the CMB anisotropy (Planck Collaboration et al.,2014). Since small angular-scale data fits the Λ CDM prediction so well,there is less freedom to move away from the standard model. New physicsat the largest angular scales only could be required to explain this discrep-ancy, however, must maintain the match to observations at small scales.
Hubble parameter
Arguably, the most significant tension with the latest CMB measurementsis that of the Hubble expansion at redshift zero, i.e. H . The expansion rateinferred from the CMB power spectrum is H = 67 . ± . − Mpc − ,a highly model dependent result, explicitly assuming a base- Λ CDM model(Planck Collaboration et al., 2018). As discussed in Section 1.3.2, SN1a con-tribute to the cosmic distance ladder at low redshift, providing a model-independent measurement of the local expansion rate of H = 73 . ± .
66 km s − Mpc − (Riess et al., 2018b). This value is in . σ tension withthe inferred expansion rate from Planck data.The actual measured values of H have not changed significantly frompast measurements from SH0ES (Riess et al., 2011; Riess et al., 2016) andearlier Planck data (Planck Collaboration et al., 2014; Planck Collaboration .3. The Lambda Cold Dark Matter model H need to be improved (seeSection 6.2.2), we found the effect was below the percent-level, and there-fore not enough to explain the tension. Low-redshift Universe
Lithium levels measured in metal-poor stars in our own galaxy are − σ lower than predicted by Λ CDM, suggesting a lower primordial abundancethan predicted in Big-Bang nucleosynthesis (Cyburt, Fields, and Olive, 2008).This tension can be alleviated by assuming the existence of new particles(Cyburt et al., 2013), and in some cases supersymmetric particles — yetundetected — have also been shown to solve the problem (e.g. Jedamzik,2004). Low-redshift clustering of luminous red galaxies on ∼ Mpc scalesdiffers from Λ CDM predictions from cosmological simulations by up to σ (Wiegand, Buchert, and Ostermann, 2014), and predictions of the growthrate of structure — sensitive to Ω m — are significantly higher than thosemeasured with redshift space distortions (see Peacock et al., 2001). Whilethe standard model describes the early Universe extremely well, these dis-crepancies suggest we may need alternative explanations at low redshift.Suggested extensions include higher-order relativistic corrections to New-tonian dynamics (see Section 1.5.1), or considering that small-scale dynam-ics do contribute to the large-scale evolution of the Universe, and hence theassumptions of homogeneity and isotropy underlying the standard modelare not valid (see Section 1.5.2). Proposed extensions
Planck Collaboration et al. (2018) have investigated standard extensions tothe Λ CDM model, including primordial gravitational waves, non-zero spa-tial curvature, dynamical (evolving) dark energy models (see also DES Col-laboration et al., 2017), modifications to GR, and different neutrino massesor primordial element abundances. No significant evidence in favour ofany of these extensions, as opposed to Λ CDM, was found.8
Chapter 1. Introduction and Background
All of the tensions touched on here remain largely unexplained. Giventhe increasing precision of cosmological observations — implying tighterconstraints on our systematic errors — we may instead need to turn to in-herent flaws in the Λ CDM model, or new, exotic physics, to explain thesecuriosities.
As mentioned in the previous section, the small-amplitude anisotropies inthe CMB at recombination seeded the large-scale galaxy structure we seetoday. As these anisotropies grow over time under the influence of gravity,the dynamics of their evolution becomes increasingly more complicated.While the amplitudes are small (linear), the evolution can be predicted an-alytically (see Section 1.4). However, the nonlinear regime of structure for-mation is only accessible via numerical simulation.CMB measurements provide insight into the near-Gaussian anisotropiesat recombination, and large-scale galaxy clustering and BAOs measured atdifferent redshifts gives us an idea about the evolution of these perturba-tions. Ensuring the standard cosmological model correctly predicts the evo-lution of the large-scale structure, including the nonlinear dynamics at latetimes, requires cosmological simulations.These simulations generally adopt a background flat, FLRW spacetimethat expands according to the Friedmann equations (with Λ (cid:54) = 0 ) alongsidea purely-Newtonian description for gravity. Initially small perturbations tothe density field, based on measurements of the CMB, collapse over time toform a large-scale distribution of galaxy clusters, filaments, and voids thatcan then be compared to our observations.In the early days, matter dynamics were completely approximated by acollisionless, self-gravitating fluid using N-body particle methods — witheach particle of a certain mass representing a collection of physical darkmatter particles. While the majority of matter in the Universe is thought tobe cold dark matter, which is well approximated as dust, in order to mimicour cosmological observations we need to also consider gas dynamics (e.g.using smoothed particle hydrodynamics, see Monaghan, 1992).The first N-body cosmological simulation was performed by Peebles(1970), using 300 particles to simulate the formation of the Coma cluster.Since then, advancements in both supercomputing power and improvednumerical techniques have allowed the particle number of such simulationsto skyrocket to billions (e.g. Springel et al., 2005; Kim et al., 2009; Kim etal., 2011; Genel et al., 2014) and even trillions (e.g. Ishiyama et al., 2013;Skillman et al., 2014; Potter, Stadel, and Teyssier, 2017) over Gpc volumes.We need our cosmological simulations to sample both very large vol-umes (of order Gpc or more), while also sampling down to the scales ofindividual galaxies smaller than the Milky Way. Large-volume simulationsare appealing because of the recent rapid increase in sky area sampled bycosmological surveys, and so we need equivalently large simulations tocompare with these observations. In addition, to be able to study the preva-lence of very high-mass galaxy clusters in the Universe, we need to samplea large enough volume so as to gain a statistically significant sample ofthese rare objects. Sampling as small scales as possible, while still main-taining a large volume, will allow a complete sampling of the matter power .4. Cosmological perturbation theory can influence one another in Newtonian gravity, since in-formation propagates instantly. In GR, however, information travels at thespeed of light, and therefore causality becomes important on sufficientlylarge scales. In addition, the assumption of a homogeneously expandingbackground spacetime, that evolves independently of the nonlinear struc-ture formation taking place, is another potential issue with these simula-tions. In Section 1.5.2 we discuss the effects on the global expansion rateby small-scale inhomogeneities, the size of which can only be quantifiedusing cosmological simulations that solve Einstein’s equations directly (seeChapters 3, 4, and 5). The Universe is often approximated as being homogeneous and isotropic.However, the mere presence of stars, planets, galaxies, and galaxy clustersshows that in the early Universe there must have been small perturbationsthat coalesced over time into larger and larger structures. Investigatingwhere these initially small perturbations came from, and how they grewover time into the structure we see today, is a main goal of perturbationtheory in cosmology (Kodama and Sasaki, 1984). Perturbation theory usesa background cosmological model — for example, the FLRW model — andthen uses Einstein’s equations of General Relativity to describe small per-turbations around this background.
Due to the complete coordinate freedom of GR, perturbations themselvescan be dependent on the chosen coordinates. Therefore, solutions to theperturbed Einstein equations may include unphysical “gauge modes”, as inthe pioneering work of Lifshitz (1946) and Lifshitz and Khalatnikov (1963).This means the density perturbation itself is gauge dependent. A physi-cally meaningful perturbation should not be dependent on the coordinatesused, which is where the motivation for “gauge invariant” formulationsof perturbation theory originated. Bardeen (1980) wrote the perturbationequations in a completely gauge-invariant way, and analysed the physi-cal interpretation of the scalar, vector, and tensor perturbations. However,even in Bardeen’s formulation, the density perturbation remains dependenton the gauge, because it is defined as the difference between the real den-sity and the background density, i.e. δρ ≡ ρ − ¯ ρ . In this definition, wehave implicitly defined a particular mapping from the true Universe ( ρ ) tothe fictitious background spacetime ( ¯ ρ ), and therefore the perturbations areexplicitly dependent on the background spacetime, and hence the chosen0 Chapter 1. Introduction and Background gauge. The gauge transformation of an arbitrary tensor field perturbation δX α = X α − ¯ X α is (Stewart and Walker, 1974) δX (cid:48) α → δX α + L Y ¯ X α , (1.50)where L Y is the Lie derivative with respect to the vector field inducing thegauge transformation, Y (see Section 2.1.2). From (1.50) we can see that ifthe Lie derivative of the background quantity ¯ X α vanishes, the perturba-tion is gauge-invariant. Since the Lie derivative of the density in an FLRWspacetime may not vanish (i.e. the time derivative is nonzero), the per-turbation δρ is not gauge invariant. Ellis and Bruni (1989) address this bydescribing the density distribution instead in terms of the density gradi-ent , which is zero in the FLRW background, and therefore the variables aregauge invariant in a perturbed-FLRW Universe (see also Bruni, Dunsby,and Ellis, 1992). Early in the Universe’s history, perturbations to the curvature and the stress-energy tensor were small, allowing us to approximate their evolution usinglinear perturbation theory. Considering perturbations to the backgroundmetric tensor (indicated with an over bar) such that δg µν (cid:28) ¯ g µν , g µν = ¯ g µν + δg µν , (1.51)and taking the background cosmology as FLRW, the linearly-perturbed lineelement in the longitudinal gauge is ds = a ( η ) (cid:20) − (cid:18) ψc (cid:19) c dη + (cid:18) − φc (cid:19) δ ij dx i dx j (cid:21) , (1.52)where ψ and φ are the first-order scalar perturbations to the metric, relevantfor galaxy clustering and light propagation, respectively. Vector and tensorperturbations are subdominant in the linear regime, and so are usually ne-glected in the context of linear perturbation theory (but see Section 1.4.3).In this gauge, the scalar perturbations φ and ψ coincide with Bardeen’sgauge-invariant potentials Φ and Ψ , respectively (Bardeen, 1980). Thesesmall perturbations in the metric tensor are linked to perturbations in thestress-energy tensor via the perturbed Einstein equations ¯ G µν + δG µν = 8 πGc (cid:0) ¯ T µν + δT µν (cid:1) . (1.53)Solving these equations in linear perturbation theory involves neglectingterms second order or higher (see Section 2.2.4 for the derivation of theequations used for initial conditions in this thesis). This method is com-monly used to describe the high-redshift Universe — usually in generatinginitial conditions for cosmological simulations — and the low-redshift Uni-verse on sufficiently large scales such that fluctuations in the density fieldare small. .4. Cosmological perturbation theory On small enough scales we measure density fluctuations that are δ (cid:29) ,and therefore the assumption of linear perturbations is no longer appli-cable. Usually, Newtonian dynamics is used to describe the small-scalegrowth of structure, although cosmological N-body simulations adoptingNewtonian gravity also sample cosmological scales, at which point causal-ity can become important (Rigopoulos and Valkenburg, 2015). However,even though density contrasts may be large, the metric perturbation re-mains small, with amplitude φ/c ∼ − − − , from galactic up to cos-mological scales.Linear perturbation theory remains valid for large-scale perturbations(but see Section 1.5.2 for a discussion of how small-scale nonlinearities couldstill influence the large-scale dynamics), the dynamics of which are ex-tremely well understood. To bridge the gap between linear perturbationtheory on large scales and Newtonian dynamics on small scales, the weakfield approximation was developed for cosmology (Green and Wald, 2011;Green and Wald, 2012). Green and Wald consider a general backgroundmetric that describes the averaged behaviour of the spacetime, and differ-ences between the actual metric and the background metric are assumedto be small everywhere, neglecting the effects of relativistic objects suchas black holes and neutron stars. This assumption does not imply thatany derivatives of δg µν must be small. Placing no limit on second deriva-tives of the metric also implies that matter sources with δ (cid:29) are al-lowed within this framework, as is commonly observed on galactic andsub-galactic scales. The intention of this framework is to capture both small-scale nonlinear dynamics and the large-scale, averaged evolution of theUniverse.Adamek et al. (2013) applied the weak-field approximation to GEVOLU - TION ; a relativistic N-body code for simulations of cosmological structureformation, by adopting the perturbed metric in the Poisson gauge (here inunits with c = 1 ) ds = a ( η ) (cid:2) − (1 + 2 ψ ) dη − B i dx i dη + (1 − φ ) δ ij dx i dx j + h ij dx i dx j (cid:3) . (1.54)Here, B i and h ij are the vector and tensor perturbations, respectively, andare kept to first order only since they are, in general, small relative to thescalar perturbations φ and ψ . This is evident in the long-term success ofpredictions from Newtonian cosmological simulations — which neglect B i and h ij — and in the difficulty of observing these perturbations comparedto the scalar potentials (e.g. Everitt et al., 2011; Abbott et al., 2016).The scalar perturbations themselves are also only kept to linear order,except when their quadratic terms are multiplied with their spatial deriva-tives (Adamek et al., 2016b; Adamek et al., 2016a). Higher derivatives of thepotentials, corresponding to density fluctuations, are kept to all orders, andvelocities are kept to second order. These simulations capture significantlymore relativistic effects than Newtonian N-body simulations, however, theweak-field approximation still requires a background spacetime, and hencean explicit description for the averaged evolution of the Universe. Potentialissues with this assumption are discussed in Section 1.5.2.2 Chapter 1. Introduction and Background
According to the standard cosmological model, approximately 95% of theenergy-density of the Universe is made up of the so-called “dark compo-nents”, namely dark matter and dark energy. Neither of these have beendirectly observed, and the very nature of them both remains a mystery.Not only because of this, but also due to disagreement between some ob-servations and predictions from the standard model, many extensions arenow being explored. These include modifications of GR on cosmologi-cal scales (see Clifton et al., 2012, for a review), using relativistic pertur-bation theory to include effects that are neglected in the standard model(see Section 1.5.1), and others which question the validity of the underly-ing assumptions. An example of the latter is discussed in Section 1.5.2,where the assumptions of homogeneity and isotropy underlying the Λ CDMmodel are called into question. Other assumptions that have been investi-gated are the Gaussianity of the primordial fluctuations (see, e.g., Verdeet al., 2000) and the apparent fine-tuning of inflation (see, e.g., Branden-berger, 2011). While some “standard” extensions have been shown to bedisfavoured compared to Λ CDM (see, e.g. Planck Collaboration et al., 2018),there is still a huge amount to be explored.
Large-scale galaxy surveys are interpreted based on cosmological simula-tions that adopt purely Newtonian dynamics. The clustering of galaxies isexplained sufficiently in Newtonian gravity on small enough scales (Greenand Wald, 2012), however, there is skepticism regarding the applicability ofthe Newtonian limit on (or close to) the Hubble scale (e.g. Yoo, Fitzpatrick,and Zaldarriaga, 2009). Some have shown that the Newtonian limit pro-vides the correct dynamics even on large scales (discussed below; see alsoMatarrese and Terranova, 1996; Hwang and Noh, 2006; Chisari and Zal-darriaga, 2011; Oliynyk, 2014). Regardless of this, there are still quanti-ties in GR that simply do not exist in the Newtonian approximation (Bruni,Thomas, and Wands, 2014). In addition, relativistic corrections to our ob-servations arise because our observations take place on the past light cone,not over a spatial slice (Bertacca et al., 2015). Assessing the size of the GRcorrections to the Newtonian limit for large-scale cosmological simulationsis therefore important, especially with the increasing precision of cosmo-logical surveys.General-Relativistic effects could be important not only in the nonlin-ear evolution of the density field, but in the setting of initial conditionsfor simulations. Primordial Gaussian perturbations in the metric tensorare predicted from inflation (Maldacena, 2003), and N-body simulationsare usually initialised using a corresponding Gaussian distribution of den-sity perturbations. This relation is valid based on the Poisson equation,in which the density perturbation and gravitational potential are relatedlinearly. Bruni, Hidalgo, and Wands (2014) used the post-Friedmann ex-pansion (Milillo et al., 2015) to show that Gaussian fluctuations in the met-ric correspond to non-Gaussian density perturbations on large scales, dueto the nonlinearity of Einstein’s equations. Cosmological simulations sam-pling initial conditions above the causal horizon therefore must include this .5. Beyond the Standard Model not equivalent to the Poisson equa-tion in purely Newtonian gravity (see (2.132a) in Section 2.2.4). This sug-gests that N-body simulations are not solving the correct dynamics (Chis-ari and Zaldarriaga, 2011). Fidler et al. (2015) defined the
N-body gauge ,in which the density field calculated when counting particles in N-bodysimulations aligns with the comoving density field defined in Einstein’sequations. This gauge is therefore suggested as a useful gauge to interpretN-body simulations (see also Fidler et al., 2016).Aside from analysing the dynamic evolution in N-body simulations, andassessing their relevance in GR, some relativistic effects are exactly zeroin the Newtonian limit, and so require a full GR treatment. The frame-dragging potential, gravitational waves, and the difference between thetwo potentials φ and ψ in the metric (1.52) are all examples of relativisticeffects that we know exist, but are zero in Newtonian gravity. Gravitationalwaves have now been observed (Abbott et al., 2016), and the frame drag-ging effect is present in cosmological perturbation theory (Bardeen, 1980)and has also been measured in our own Solar System (Everitt et al., 2011).The “gravitational slip” | φ − ψ | (see Chapter 3) can be measured from theintegrated Sachs-Wolfe effect, weak gravitational lensing, and in the matterpower spectrum (Daniel et al., 2010). It is zero at first order in GR, how-ever becomes non-zero at higher orders in perturbation theory (and in somemodified gravity theories, see Daniel et al., 2008).Post-Friedmann expansion provides an approximation for GR that cap-tures both the small-scale nonlinear dynamics and the large-scale linear dy-namics (Milillo et al., 2015). In the Newtonian limit of this expansion, thereis a non-zero vector potential, in addition to the usual scalar potential, inthe metric tensor (Bruni, Thomas, and Wands, 2014). This encapsulatesthe frame-dragging effect, sourced by purely Newtonian terms, and there-fore can be calculated from nonlinear N-body simulations. Bruni, Thomas,and Wands (2014) performed the first calculation of the frame-dragging po-tential from a purely Newtonian simulation, showing it has small enoughmagnitude that N-body dynamics should be unaffected, however, could bemeasurable in weak-lensing cosmological surveys (see also Thomas, Bruni,and Wands, 2015b; Thomas, Bruni, and Wands, 2015a).4 Chapter 1. Introduction and Background
Our Universe is approximated as homogeneous and isotropic on large scales,however it is extremely inhomogeneous and anisotropic on small scales.The process of smoothing over these small-scale structures to achieve large-scale homogeneity is often referred to as the “averaging problem” (see Clark-son et al., 2011; Wiltshire, 2011). Due to the nonlinearity of Einstein’s equa-tions, when averaging an inhomogeneous fluid (e.g., the Universe on smallscales) over large scales, there are extra terms governing the evolution ofthe averaged fluid compared to a homogeneous fluid. The theory of back-reaction states that the globally-averaged expansion of an inhomogeneous,anisotropic Universe does not coincide with the expansion rate of the homo-geneous, isotropic model (Buchert, 2000). Extra terms appear in the equa-tion for the acceleration of the expansion rate of the Universe, and thereforehave been suggested as alternate explanations for dark energy.Quantifying the size of the backreaction effect ultimately requires sim-ulations that solve Einstein’s equations using numerical relativity. The for-malism for calculating cosmological averages is explicitly dependent on thechosen slicing conditions, since the averages themselves must be taken overa specified three-dimensional domain. The dependence of backreaction onslicing has been explored in Adamek et al. (2019), and can show up to a10% difference depending on which three-dimensional slices are chosen. Itis therefore important to specify a slicing condition that is physically inter-esting and best represents our measurements of averages in the Universe.
Averaging comoving domains
The original formalism of Buchert (2000) is based on averaging over a spa-tial surface that is comoving with the fluid flow. This is called the syn-chronous, comoving gauge; a popular choice in relativistic cosmologicalperturbation theory, both due to its simplicity and the parallels that can bedrawn with Newtonian Lagrangian coordinates (Bruni, Dunsby, and Ellis,1992; Bruni et al., 2014). However, this spatial surface only exists if the fluidis vorticity free (Ehlers, 1993). In this gauge, the proper time measured by acomoving observer coincides with the coordinate time on the spatial slices— i.e. a lapse function α = 1 — and the coordinate observers follow thefluid flow — i.e. a shift vector β i = 0 . The normal vector orthogonal to thespatial slices in this case is therefore n µ = (1 , , , , here coinciding withthe four velocity of the fluid u µ .We want to study the averaged dynamics of inhomogeneous, anisotropicdust. The kinematical quantities describing the expansion rate, shear, andvorticity of this fluid are defined, respectively, by decomposing the fourvelocity of the fluid, Θ ≡ ∇ µ u µ , (1.55) σ µν ≡ b αµ b βν ∇ ( α u β ) −
13 Θ b µν , (1.56) w µν ≡ b αµ b βν ∇ [ α u β ] , (1.57)where in the comoving gauge w µν = 0 . The projection tensor b µν ≡ g µν + u µ u ν is purely spatial, and in this gauge is equivalent to the metric tensor .5. Beyond the Standard Model γ ij describing the spatial surfaces. In the above, we use rounded and squarebrackets around indices to denote the symmetric and antisymmetric partsof a tensor, respectively, i.e. A ( ij ) ≡
12 ( A ij + A ji ) , (1.58) A [ ij ] ≡
12 ( A ij − A ji ) . (1.59)When considering three-dimensional slices embedded in four-dimensionalspacetime, Einstein’s equations are split into the Hamiltonian and momen-tum constraint equations, and a system of evolution equations for the met-ric and extrinsic curvature of the slices (see Section 2.1.2). In terms of thekinematical quantities above, the Hamiltonian constraint equation can bewritten as (with Λ = 0 ) R + 13 Θ − σ = 8 πGc ρ, (1.60)where σ ≡ σ ij σ ij , and R is the Ricci scalar describing the intrinsic curva-ture of the surfaces. Here, the density ρ is the projection of the stress-energytensor into the spatial surfaces.Raychaudhuri’s equation governs the evolution of the expansion scalar, Θ , and is derived from the trace of the evolution equation for the extrinsiccurvature (Raychaudhuri, 1957; Matarrese and Terranova, 1996), ∂ Θ + 13 Θ + 2 σ + 4 πGc ρ = 0 , (1.61)where ∂ ≡ ∂/∂x = c − ∂ t , with t the coordinate time (which in the comov-ing gauge coincides with the proper time).The average of a scalar χ , which here is a function of Lagrangian (co-moving) coordinates and time, over some arbitrary domain D (lying on thespatial slice) is defined as (Buchert, 2000), (cid:104) χ ( t, x i ) (cid:105) b ≡ V b D (cid:90) D χ ( t, x i ) √ b d x, (1.62)where b is the determinant of the projection tensor b ij , the volume elementis dV = √ b d x , and the volume of the domain is defined as V b D ≡ (cid:90) D √ b d x. (1.63)The dimensionless, effective scale factor is defined from the volume, a b D ≡ (cid:18) V b D ( t ) V b D ( t init ) (cid:19) / , (1.64)and describes the expansion rate of the domain, where t init is the initialtime. We can then write the expansion scalar in terms of the effective scalefactor, (cid:104) Θ (cid:105) b = ∂ V b D V b D = 3 ∂ a b D a b D . (1.65)6 Chapter 1. Introduction and Background
To study the averaged dynamics of the fluid, we want to study the aver-aged Raychaudhuri equation and Hamiltonian constraint. First taking thetime derivative of (1.62), we get ∂ (cid:104) χ (cid:105) b = ∂ (cid:18) V b D (cid:19) (cid:90) D χ √ b d x + 1 V b D (cid:90) D (cid:16) √ b ∂ χ + χ∂ √ b (cid:17) d x, (1.66) = − ∂ V b D V b D (cid:104) χ (cid:105) b + (cid:104) ∂ χ (cid:105) b + 1 V b D (cid:90) D χ ∂ (cid:16) √ b (cid:17) d x. (1.67)Using the evolution equation for the projection tensor b ij (see Section 2.1.2),we can show (see Buchert, 2000), ∂ √ b = Θ √ b, (1.68)so (1.67) becomes ∂ (cid:104) χ (cid:105) b = − ∂ V b D V b D (cid:104) χ (cid:105) b + (cid:104) ∂ χ (cid:105) b + (cid:104) Θ χ (cid:105) b . (1.69)From this, we substitute (1.65) and find the commutation rule to be ∂ (cid:104) χ (cid:105) b − (cid:104) ∂ χ (cid:105) b = (cid:104) Θ χ (cid:105) b − (cid:104) Θ (cid:105) b (cid:104) χ (cid:105) b . (1.70)Now taking the average of Raychaudhuri’s equation (1.61) by averagingeach individual term, i.e. (cid:104) ∂ Θ (cid:105) b + 13 (cid:104) Θ (cid:105) b + 2 (cid:104) σ (cid:105) b + 4 πGc (cid:104) ρ (cid:105) b = 0 , (1.71)and using the commutation rule (1.70), we have (cid:104) ∂ Θ (cid:105) b = ∂ (cid:104) Θ (cid:105) b − (cid:104) Θ (cid:105) b + (cid:104) Θ (cid:105) b . (1.72)Substituting the above, along with ∂ (cid:104) Θ (cid:105) b = 3 ∂ (cid:18) ∂ a b D a b D (cid:19) , (1.73) = 3 ∂ a b D a b D − (cid:18) ∂ a b D a b D (cid:19) , (1.74)and (1.65), into (1.71) gives − (cid:104) Θ (cid:105) b + 23 (cid:104) Θ (cid:105) b + 3 ∂ a b D a b D + 2 (cid:104) σ (cid:105) b + 4 πGc (cid:104) ρ (cid:105) b = 0 , (1.75)which we rearrange to get the averaged Raychaudhuri equation, ∂ a b D a b D + 4 πGc (cid:104) ρ (cid:105) b = Q D . (1.76)Here, we have defined the kinematical backreaction term Q D ≡ (cid:16) (cid:104) Θ (cid:105) b − (cid:104) Θ (cid:105) b (cid:17) − (cid:104) σ (cid:105) b , (1.77) .5. Beyond the Standard Model (cid:104)R(cid:105) b + 13 (cid:104) Θ (cid:105) b − (cid:104) σ (cid:105) b = 8 πGc (cid:104) ρ (cid:105) b , (1.78)and substituting (1.77) and (1.65) quickly gives the averaged Hamiltonianconstraint, c (cid:18) ∂ t a b D a b D (cid:19) − πGc (cid:104) ρ (cid:105) b + 12 (cid:104)R(cid:105) b + 12 Q D = 0 . (1.79)We define the effective Hubble parameter in the domain D from theeffective scale factor, H D ≡ ∂ t a b D a b D , (1.80)and using this we can rewrite (1.79) to give Ω m + Ω R + Ω Q = 1 , (1.81)where we have defined the dimensionless cosmological parameters, Ω m ≡ πG (cid:104) ρ (cid:105) b H D , Ω R ≡ − (cid:104)R(cid:105) b c H D , Ω Q ≡ − Q D c H D , (1.82)describing, respectively, the matter, curvature, and backreaction content ofan averaged inhomogeneous Universe.This system is analogous to the FLRW model discussed in Section 1.2,however the cosmological parameters (1.82) are different to those in theFLRW model (1.48) since here they are derived in full GR, rather than underthe assumptions of homogeneity and isotropy. Averaging general foliations
While the synchronous, comoving gauge is a useful representation of ob-servers following the fluid flow, in practice it presents computational dif-ficulties (see Section 2.1.4). It is therefore useful to generalise the aboveaveraging formalism for any slicing condition, i.e. for any form of the lapsefunction or shift vector.Several generalisations of the Buchert (2000) averaging formalism to ar-bitrary coordinates have been proposed (Brown, Robbers, and Behrend,2009; Larena et al., 2009; Gasperini, Marozzi, and Veneziano, 2010), withdifferences in these formalisms stemming mainly from the definition of theHubble expansion, H D . In the previous section, we defined the Hubbleparameter from the expansion scalar Θ , which measures the divergenceof the four velocity of the fluid. In the specific case of the synchronous,comoving gauge, the four velocity of the fluid coincides with the normalvector orthogonal to the spatial surfaces, i.e. u µ = n µ . In the case ofa general foliation, this may not be the case, and we will generally have u µ (cid:54) = n µ . In Brown, Robbers, and Behrend (2009) and Gasperini, Marozzi,and Veneziano (2010), the Hubble parameter is defined as the divergenceof the normal vector, H D ∝ ∇ µ n µ , representing the expansion of coordinate Chapter 1. Introduction and Background observers . Similarly, the volume element in these works is defined using thespatial metric h µν ≡ g µν + n µ n ν , which is used to project four-dimensionalobjects onto the spatial slices orthogonal to n µ (in the general case, h µν isdistinct from b µν used in the previous section). Larena et al. (2009) definesthe Hubble parameter from the divergence of the fluid four velocity, henceanalysing a more physically interesting expansion rate, since we, as ob-servers, will measure the expansion rate of the fluid and not that of ourcoordinates. For this reason, here, we follow the generalised averaging for-malism of Larena et al. (2009).In general coordinates, the normal vector orthogonal to the spatial sur-faces is given by n µ = (1 /α, − β i /α ) . The four velocity of the fluid is againdecomposed into its expansion rate, shear, and vorticity, respectively θ ≡ h µν ∇ µ u ν , (1.83) σ µν ≡ h αµ h βν ∇ ( α u β ) − θh µν , (1.84) ω µν ≡ h αµ h βν ∇ [ α u β ] . (1.85)We also decompose the normal vector and the Eulerian velocity v i in a sim-ilar way, giving Σ µν ≡ h αµ h βν ∇ ( α n β ) + 13 Kh µν , (1.86) β µν ≡ h αµ h βν ∇ ( α v β ) − κh µν , (1.87) M µν ≡ h αµ h βν ∇ [ α v β ] , (1.88)where κ ≡ h αβ ∇ α v β , (1.89) K = − h αβ ∇ α n β . (1.90)Here, K is the trace of the extrinsic curvature of the spatial hypersurfaces(see Section 2.1.2).The Hamiltonian constraint can be written in terms of the above vari-ables, W R − σ − σ B + 23 ( θ + θ B ) − πGc W ρ = 0 , (1.91)where W is the Lorentz factor describing the motion between normal ob-servers and observers comoving with the fluid, and we have used the fol-lowing for convenience θ B ≡ − W κ − W B, (1.92) σ Bµν ≡ −
W β µν − W (cid:18) B ( µν ) − Bh µν (cid:19) , (1.93)where we have also defined σ ≡ σ ij σ ij , (1.94) σ B ≡ σ Bij σ ijB + σ ij σ ijB . (1.95) .5. Beyond the Standard Model B µν ≡ κ ( v µ n ν + v µ v ν ) + β αµ v α n ν + β αµ v α v ν + M αµ v α n ν + M αµ v α v ν , (1.96)and B = κv µ v µ + β µν v µ v ν is its trace.The averaging procedure for the non-comoving formalism is defined ina similar way, however we define the volume element here instead usingthe projection tensor h µν V h D ≡ (cid:90) D √ h d x, (1.97)where h ≡ det | h µν | , so that the average of a function χ is (cid:104) χ ( t, x i ) (cid:105) h ≡ V h D (cid:90) D χ ( t, x i ) √ h d x. (1.98)The effective Hubble parameter can then be defined from the expansionscalar (1.83) (Larena et al., 2009; Umeh, Larena, and Clarkson, 2011), H h D ≡ c (cid:104) αθ (cid:105) h , (1.99)which describes the expansion rate of the fluid as seen by an observer onthe hypersurface defined by h µν (Larena et al., 2009). The effective scalefactor defined in (1.64) describes the expansion of the volume element ofthe domain, V D . In the case of a comoving slice, u α = n α , this expansioncorresponds to the expansion of the fluid flow. In this case, we have ingeneral u α (cid:54) = n α and so the effective volume scale factor here is a V D ≡ (cid:18) V h D ( t ) V h D ( t init ) (cid:19) / , (1.100)which describes the expansion of the coordinate observers themselves. Wecan also define the effective fluid scale factor from the Hubble expansion,by setting H h D = ∂ t a h D a h D . (1.101)In this general formalism, using the evolution equation for the metric h ij (see Section 2.1.2), we can show √ h ∂ √ h = αW ( θ + θ B ) + D i β i , (1.102) We note an error in equation (30) in Larena et al. (2009) — corresponding to our equation(1.102) — and consequently (31) and (34) — our equations (1.104) and (1.105), respectively.In the paper, the author has − κ in place of θ B in our expressions. See Appendix F for details. Chapter 1. Introduction and Background which is equivalent to (1.68) in the case of a comoving formalism, i.e. α = 1 , W = θ B = β i = 0 . From (1.102), the rate of change of the volume is then ∂ V h D V h D = 3 ∂ a V D a V D , (1.103) = (cid:104) αW ( θ + θ B ) + D i β i (cid:105) h , (1.104)and using this with (1.101), we can show the relation between the two ef-fective scale factors is a h D = a V D exp (cid:18) − c (cid:90) tt init (cid:104) αW ( θ + θ B ) − αθ + D i β i (cid:105) h dt (cid:19) , (1.105)see Appendix C for more details.Averaging each term in (1.91), we find (cid:104) W R(cid:105) h − (cid:104) σ (cid:105) h − (cid:104) σ B (cid:105) h + 23 (cid:104) θ (cid:105) h + 43 (cid:104) θθ B (cid:105) h + 23 (cid:104) θ B (cid:105) h − πGc (cid:104) W ρ (cid:105) h = 0 . (1.106)The kinematical backreaction term adapted for this generalised foliation is Q h D ≡ (cid:16) (cid:104) θ (cid:105) h − (cid:104) θ (cid:105) h (cid:17) − (cid:104) σ (cid:105) h , (1.107)and the additional backreaction term due to the non-zero coordinate veloc-ity is L D ≡ (cid:104) σ B (cid:105) h − (cid:104) θ B (cid:105) h − (cid:104) θθ B (cid:105) h . (1.108)With these definitions in (1.106), we arrive at the averaged Hamiltonianconstraint Ω m + Ω R + Ω Q + Ω L = 1 . (1.109)Here, the cosmological parameters are Ω m ≡ πG (cid:104) W ρ (cid:105) h H h D , Ω R ≡ − (cid:104) W R(cid:105) h c H h D , (1.110) Ω Q ≡ − Q h D c H h D , Ω L ≡ L D c H h D , (1.111)which describe the content of an averaged, inhomogeneous universe as cal-culated by a general, non-comoving observer. Improved general formalism
The above formalism describes the averaged cosmological dynamics of thefluid, as seen by normal observers, by projecting properties of the fluid intothe hypersurfaces defined by h µν . However, the volume element (1.97) ispropagated along the normal vector to the hypersurfaces, rather than alongthe fluid four-velocity vector. This means that matter is free to flow into andout of the domain over time — implying mass is not conserved within the .5. Beyond the Standard Model comoving observers, rather than that seen by normal ob-servers as in Larena et al. (2009) (and see also Brown, Behrend, and Malik,2009; Gasperini, Marozzi, and Veneziano, 2010).The proper volume element comoving with the fluid is defined in thesame way as V b D in (1.63), and the averaging operator appears the same as(1.62), however the domain D in this case lies in the non-comoving hyper-surfaces, rather than the comoving hypersurfaces as in Buchert (2000). Theproper volume element is related to the Riemannian volume element V h D defined in (1.97), via V b D = (cid:104) W (cid:105) b V h D . (1.112)The averaged Hamiltonian constraint in this formalism is (for Λ = 0 ) H D c − πGc (cid:104) ˜ ρ (cid:105) b + 12 (cid:104) ˜ R b (cid:105) b + 12 ˜ Q D = 0 (1.113)where the effective Hubble parameter is defined in (1.80). The tilde repre-sents rescaled kinematic fluid variables, ˜ ρ ≡ α W ρ, ˜ R b ≡ α W R b , (1.114) ˜Θ ≡ α W Θ , ˜ σ ≡ α W σ , ˜ w ≡ α W w , (1.115)where w ≡ w ij w ij is the vorticity scalar, and the rescaled kinematic back-reaction term in (1.113) is defined as ˜ Q D ≡ (cid:16) (cid:104) ˜Θ (cid:105) b − (cid:104) ˜Θ (cid:105) b (cid:17) − (cid:104) ˜ σ (cid:105) b + 2 (cid:104) ˜ w (cid:105) b . (1.116)Here, R b is distinct from the Ricci scalar of the spatial hypersurfaces, R , andinstead represents the fluid rest-frame spatial curvature (Buchert, Mourier,and Roy, 2018). Quantifying Q D The basic result that averaged properties of an inhomogeneous fluid do notsatisfy Einstein’s equations has sparked many investigations into the sizeof the resultant effect. Some attempts have been able to completely explainthe accelerating expansion with
Λ = 0 , while others have shown that back-reaction itself is a small effect, however, still can be relevant for upcomingprecision cosmological surveys. In contrast, some argue that whether or not Q D provides acceleration is irrelevant, and actual connection to observables in the Universe is more important (Ishibashi and Wald, 2006; Green andWald, 2014). Because of these strikingly different results, the amplitude of Q D itself is still heavily debated (see, e.g. Buchert et al., 2015; Green andWald, 2015; Green and Wald, 2016).2 Chapter 1. Introduction and Background
Wiltshire’s “timescape” cosmology considers virialised objects to be spa-tially flat, and the void regions surrounding them to be negatively curved(see Wiltshire, 2007a; Wiltshire, 2007b; Wiltshire, 2008; Wiltshire, 2009). Inthe context of Buchert’s averaging scheme, observers located in the dense“walls” surrounding virialised objects measure an apparent cosmic expan-sion when the fraction of the total volume occupied by voids reaches ∼ . .The difference is largely due to the fact that the clocks of observers locatedin dense regions will tick differently to the globally-averaged clock. Thetimescape model has also been shown to fit SN1a light-curve data as wellas — or better than — Λ CDM in some cases (Dam, Heinesen, and Wiltshire,2017; Smale and Wiltshire, 2011).Backreaction as calculated from purely Newtonian simulations has alsobeen shown to be significant; in some cases explaining cosmic acceleration.Roukema (2018) used peculiar velocity gradients from Newtonian N-bodysimulations to calculate the backreaction parameter, and hence the differ-ential expansion due to structure formation (see also Räsänen, 2006a). Inthis model, an accelerating global expansion was found when considering ∼ . /h averaging regions in calculating Q D . Using a similar method,Rácz et al. (2017) calculated the local expansion rate of smoothed regionsfrom N-body simulations using the Friedmann equations (i.e., not consider-ing any relativistic quantities such as curvature or backreaction). With thismethod, again on a certain coarse-graining scale, the modified simulationsprovide an extremely close fit to the SN1a data, while also resolving thetension between the locally-measured Hubble expansion and that from theCMB. Both of these approaches have used purely Newtonian simulations tofind a global backreaction. However, the original Buchert and Ehlers (1997)averaging scheme clearly showed that there can be no global backreactioneffect in Newtonian simulations with periodic boundary conditions, since Q D itself manifests as a pure boundary term in Newtonian dynamics (seeBuchert, 2018; Kaiser, 2017, for direct comments on these works). Backre-action can still be studied in the context of these Newtonian simulationson sub-periodicity scales, however in this case the measurement is of cos-mic variance from peculiar velocities, rather than a pure GR effect (Buchert,Kerscher, and Sicka, 2000; Buchert and Räsänen, 2012).Studying backreaction in the context of perturbation theory is poten-tially problematic since it still requires a background spacetime. Regard-less, perturbation theory has still provided some constraints on the size ofthe backreaction effect in this context. High-order terms in the perturbativeseries get progressively smaller for the early Universe, however, at redshifts z (cid:46) these high-order terms have been shown to have similar magnitudes— i.e. the series does not converge — and therefore can contribute to ac-celerating expansion (Räsänen, 2004; Notari, 2006). Second-order pertur-bation theory has shown that super-horizon fluctuations generated at infla-tion could be responsible for the apparent accelerating expansion (Barausse,Matarrese, and Riotto, 2005; Kolb, Matarrese, and Riotto, 2006). However,these works have been criticised since the higher-order terms neglected inthe perturbation series can be shown to cancel these second-order effects(Hirata and Seljak, 2005).Second-order analysis of the scale dependence of backreaction showsthat averaged curvature effects can reach ∼ at 80 Mpc scales — justbelow the homogeneity scale of the Universe — and fall to order at .5. Beyond the Standard Model ∼ Mpc scales, with Q D becoming important inside ∼ Mpc scales (Liand Schwarz, 2007; Li and Schwarz, 2008). The weak-field approximationimproves on perturbation theory (as discussed in Section 1.4.3), in whichonly the metric perturbations are assumed small. In the weak-field limit,backreaction is small (Adamek et al., 2015; Adamek et al., 2019). How-ever, relativistic effects in the Hubble diagram (Adamek et al., 2018) andin redshift-space distortions on Gpc scales (Adamek, 2018) have been sug-gested to be important.Backreaction itself is an attractive explanation to the accelerating expan-sion without introducing any new, exotic physics, but simply by consider-ing Einstein’s GR in full. However, any suggestions that propose the stan-dard cosmological model — which has been accepted as correct for decades— as incorrect or flawed will be subject to a necessary amount of scrutiny.
Exact solutions to Einstein’s equations have proven to be extremely use-ful in analysing the behaviour and evolution of simple objects in astro-physics, for example, black holes using the Schwarzschild (1916) and Kerr(1963) solutions. For inhomogeneous cosmology, there are not many ex-act solutions to choose from (see, e.g. Bolejko et al., 2009; Bolejko, Célérier,and Krasi ´nski, 2011, for reviews). Commonly adopted are the Lemaître-Tolman-Bondi (LTB) model (Lemaître, 1933; Tolman, 1934; Bondi, 1947), aspherically-symmetric dust solution, the Szekeres model (Szekeres, 1975), ageneral non-symmetric dust solution, and “Swiss Cheese” models (e.g. Ein-stein and Straus, 1945), which are often groups of LTB or Szekeres solutionson a homogeneous background (Kai et al., 2007).The LTB model has been used to suggest that a nearby, large-scale inho-mogeneity is causing an apparent accelerating expansion. Measurementsby an observer located at the centre of such an inhomogeneity align withthe SN1a data without the need for dark energy (Célérier, 2000), while stilldescribing the position of the first peak in the angular power spectrum ofthe CMB anisotropy (Alnes, Amarzguioui, and Grøn, 2006) and observedBAO data (Garcia-Bellido and Haugbølle, 2008). Even though under-denseregions of the correct radius have been detected (Frith et al., 2003), the mea-sured density does not match the minimum requirement for acceleratingexpansion (Alexander et al., 2009).The LTB solution has proven useful as a toy model for inhomogeneouscosmology, however, its inherent symmetries call for the use of more gen-eral models to validate the results. Szekeres models are inhomogeneous,exact solutions to Einstein’s equations for a dust fluid containing no grav-itational radiation; a type of “silent” Universe (see Bruni, Matarrese, andPantano, 1995; Bolejko, 2018b). Ishak et al. (2008) used a Szekeres modelwith zero curvature at large distances from the observer and negative cur-vature nearby, alongside
Λ = 0 , to fit the SN1a data competitively with Λ CDM, while still satisfying the requirement of spatial flatness at CMBscales. Improving further on these models are “Swiss Cheese” models,considering multiple LTB (e.g. Biswas and Notari, 2008) or Szekeres (e.g.Bolejko, Célérier, and Krasi ´nski, 2011) holes in a homogeneous backgroundcheese. In some cases these inhomogeneities have been shown to produce apercent-level effect on our observations (Bolejko and Ferreira, 2012; Fleury,4
Chapter 1. Introduction and Background
Dupuy, and Uzan, 2013). Apparent accelerating expansion only arises if ob-servers are located in a large ∼ Mpc void (Marra et al., 2007; Alexanderet al., 2009; Bolejko, Célérier, and Krasi ´nski, 2011), which has been ruledout with CMB constraints in the case of Swiss cheese models (Valkenburg,2009). Moss, Zibin, and Scott (2011) found inconsistencies between LTBvoid models and observational data, finding the approximate models havevery low expansion rates, Universe ages inconsistent with observations,and much smaller local matter fluctuations than measured.Another family of exact inhomogeneous cosmological models commonlyused to address backreaction are black-hole lattices (Lindquist and Wheeler,1957). These models involve regular grids of Schwarzschild masses in anotherwise vacuum spacetime (see Bentivegna et al., 2018, for a review).Cosmological averaging in these spacetimes exhibits large backreaction ef-fects for small numbers of masses, however the global expansion approachesFLRW for large numbers of masses (e.g. Clifton, Rosquist, and Tavakol,2012; Bentivegna and Korzy ´nski, 2013). Regardless of their inherent large-scale homogeneity and global FLRW expansion, some optical propertiesmeasured in these spacetimes — such as the luminosity distance — do not match the prediction from the FLRW model (Bentivegna et al., 2017). Whileblack-hole lattices are useful toy models to study exact inhomogeneous cos-mology in the presence of strong-field objects, the mass distribution is ex-tremely simplified and the solutions themselves are static; missing the dy-namic aspect of inhomogeneous cosmology.
Significant progress has been made towards quantifying the backreactioneffect in inhomogeneous cosmologies. In some cases described above, theeffect has been significant enough to explain the accelerating expansion,and in other cases the effect is either percent level or completely negligi-ble. All the methods described above have their own respective drawbacks,be it due to simplifying assumptions or symmetries. In order to be able tofully quantify the effect of backreaction in our own Universe, we must solveEinstein’s equations in full for the complex, nonlinear matter distributionwe observe. Advances in numerical relativity and computational resourcesover the past two decades (see Section 2.1) now allow for the stable simula-tion of relativistic objects such as black holes and neutron stars. Applicationof numerical relativity to large-scale, inhomogeneous cosmological simu-lations has emerged over the past few years (Giblin, Mertens, and Stark-man, 2016a; Bentivegna and Bruni, 2016; Macpherson, Lasky, and Price,2017), and the field is rapidly advancing with new codes (Bentivegna, 2016;Mertens, Giblin, and Starkman, 2016; Daverio, Dirian, and Mitsou, 2017;East, Wojtak, and Abel, 2018), measurements of observables from fully rel-ativistic simulations (Giblin, Mertens, and Starkman, 2016b; Giblin et al.,2017), and the application of particle-based methods alongside numericalrelativity (as opposed to purely mesh-based codes; see Giblin et al., 2018;Daverio, Dirian, and Mitsou, 2019; Barrera-Hinojosa and Li, 2019). As ofyet, no significant backreaction effect has been measured in full numeri-cal relativity (Bentivegna and Bruni, 2016; Macpherson, Price, and Lasky,2019), however, a lack of virialisation and periodic boundary conditionsmay potentially explain these results (see Chapter 4). .5. Beyond the Standard Model
Chapter 2
Methods
In this Chapter, we outline the methods used for the simulations and anal-ysis presented in this thesis. We introduce a brief history of numerical rel-ativity, the formalism and system of equations, and how these are imple-mented in C
ACTUS and the E
INSTEIN T OOLKIT (ET); the numerical relativ-ity code used in this thesis. We derive the system of equations used to de-velop the initial conditions for our simulations, and the methods employedin
MESCALINE to perform post-simulation analysis.
Solving Einstein’s equations numerically allows the study of dynamics ofstrong-field gravitating objects, which are otherwise inaccessible with ana-lytic methods. Binary black holes and neutron stars — or black hole, neu-tron star pairs — and the resulting gravitational-wave emission and prop-agation, tidal disruption of a star by a black hole, accretion disks aroundspinning black holes, supernova explosions, and cosmological structureformation are all examples of the highly nonlinear, relativistic phenomenawhich can now be studied in detail due to the advancement of this field.The pioneers of numerical relativity knew the importance of using nu-merical techniques to study problems that were difficult to solve analyti-cally. Hahn and Lindquist (1964) numerically evolved the merger of twoends of a wormhole to study the two-body problem, Eppley (1977) createdinitial data for evolving source-free gravitational radiation, and Smarr andYork (1978a) studied coordinate choices in the numerical evolution of Ein-stein’s equations, relating these choices to different families of observers.In the decades following development of the ADM formalism (Arnowitt,Deser, and Misner, 1959, see the next section), the advancement of numerical-relativity simulations was hindered both by the lack of computational powerand stability issues with the form of the evolution equations themselves.Early works focused on the study of pure gravitational waves and stellarcollapse (Nakamura, Oohara, and Kojima, 1987), and inhomogeneous in-flationary cosmology (Laguna, Kurki-Suonio, and Matzner, 1991).The main obstacle standing in the way of long-term simulations of com-pact objects was an efficient method for dealing with singularities. Manysingularity-avoiding coordinates were proposed, for example “maximal slic-ing” (Smarr and York, 1978b, and see Section 2.1.4), in which time is stopped in the vicinity of the singularity, but continues moving forward in other re-gions. While this avoids issues in evolving the singularity itself, it induces8
Chapter 2. Methods strong gradients in the metric which generally cause the code to fail. Insome cases, these slices have been found to be successful in spherical sym-metry (e.g. Bona et al., 1995) and for evolving brief periods of black-holemergers (Anninos et al., 1995; Bruegmann, 1999). Alternatively, bound-ary conditions on the event-horizon edge can completely excise the singu-larity from the simulation, while still evolving all observable regions (e.g.Bardeen and Piran, 1983; Thornburg, 1987). This improved the situation,although still only short-lived simulations were possible (e.g. Seidel andSuen, 1992; Brandt et al., 2000; Alcubierre and Bruegmann, 2001; Thorn-burg, 2004).Simulations of the head-on collision of non-rotating black holes, includ-ing the extraction of gravitational waves, were limited to axially-symmetric,two-dimensional models (Smarr, 1977; Seidel and Suen, 1992; Anninos etal., 1993; Bernstein et al., 1994). As computing power and memory con-tinued to increase, parallelisation of codes became possible. Anninos etal. (1995) used excision to perform the first fully three-dimensional simu-lation in Cartesian coordinates of a single black hole, evolved for severallight-crossing times. The “puncture” method — i.e., placing the singular-ity away from the points on the computational grid where the variables areevaluated (Brandt and Bruegmann, 1997) — allowed for longer evolutionsof both distorted and colliding black holes without the need for excision(Alcubierre et al., 2003).Even with the improvement of stability via the BSSN formalism (Baum-garte and Shapiro, 1999; Shibata and Nakamura, 1995), at this time, all bi-nary black-hole and neutron-star (e.g. Oohara and Nakamura, 1999; Miller,Suen, and Tobias, 2001) simulations adopted some kind of symmetry, usu-ally to reduce the computational memory required. The first fully three-dimensional, non-axisymmetric binary black-hole simulations — includingthe merger and ring-down stages and the emission of gravitational waves— were performed by Pretorius (2005a), Campanelli et al. (2006), and Bakeret al. (2006). Since this ground-breaking work, the field of numerical rela-tivity has exploded.These early works paved the way for more recent advancement in nu-merical relativity, touching many aspects of relativistic astrophysics such asblack holes (e.g. Baker et al., 2006; Campanelli et al., 2006; Buonanno, Cook,and Pretorius, 2007; González et al., 2007; Hinder, Kidder, and Pfeiffer, 2018;Huerta et al., 2019), neutron stars (e.g. Baiotti, Giacomazzo, and Rezzolla,2008; Paschalidis et al., 2011; Kastaun and Galeazzi, 2015; Chaurasia et al.,2018), stellar collapse and supernovae explosions (e.g. Duez, Shapiro, andYo, 2004; Montero, Janka, and Müller, 2012), and more recently, for cos-mology (e.g. Giblin, Mertens, and Starkman, 2016a; Bentivegna and Bruni,2016; Macpherson, Lasky, and Price, 2017; East, Wojtak, and Abel, 2018;Daverio, Dirian, and Mitsou, 2019; Barrera-Hinojosa and Li, 2019).
In the 3+1 foliation of spacetime, Einstein’s four-dimensional equationsmay be written in terms of purely spatial objects constructed from the three-dimensional metric tensor of the embedded spatial hypersurfaces (surfaces),which are then evolved forward in time. Foliations of this type began withthe ADM formalism (Arnowitt, Deser, and Misner, 1959). .1. Numerical Relativity g µν and its firsttime derivative ∂ t g µν at every point on the initial space-like surface. To thenevolve these surfaces forwards in time we thus need to specify the secondtime derivatives ∂ t g µν , which, due to the symmetry of the metric tensor g µν ,implies we need ten equations in total to evolve the system. These secondtime derivatives will be present in some components of the Ricci tensor R µν in the field equations (and hence the Einstein tensor G µν ), and so we needto identify which components are relevant for time evolution.The contracted Bianchi identities give ∇ ν G µν = ∂ ν G µν + (4) Γ µνα G να + (4) Γ ννα G αµ = 0 , (2.1) ⇒ ∂ G µt = − ∂ i G µi − (4) Γ µνα G να − (4) Γ ννα G αµ , (2.2)where the right hand side contains only second time derivatives (in theRicci tensor R µν ). This implies that G µt cannot contain any second timederivatives itself, and the four components of Einstein’s equations G µt = 8 πGc T µt , (2.3)therefore cannot contribute to the evolution of the metric tensor, and in-stead act as constraint equations that must be satisfied on every surfaceduring the evolution. We now have only the remaining six components ofEinstein’s equations G ij = 8 πGc T ij . (2.4)Since we require ten equations in total to evolve the system, this leaves uswith four extra degrees of freedom for the evolution. The lapse function α describes how much proper time elapses between surfaces, and the shiftvector β i describes how the spatial coordinates x i transform from one sur-face at time t to the next at time t + dt . The remaining four degrees offreedom for the evolution are in the second time derivatives ∂ t α and ∂ t β i .The initial data we must specify in g µν and ∂ t g µν comprises in totaltwenty independent choices. Using the constraint equations (2.3) thesechoices are reduced to sixteen. Due to the coordinate invariance of GR,any metric which satisfies Einstein’s equations in some coordinate system x µ must also satisfy Einstein’s equations in some other coordinate system x µ (cid:48) . This means our choice of the coordinates x µ constrain the physical formof the metric g µν , and therefore eliminate four degrees of freedom from ourchoice of initial data. The evolution of our chosen coordinates is definedvia the lapse, α , and the shift, β i , including their first time derivatives ∂ t α and ∂ t β i . Since α and β i form part of the metric tensor itself (see (2.12)below), these eight gauge choices imply that overall, twelve of our totaltwenty degrees of freedom in the initial conditions are purely coordinate-based choices. Considering also the four constraint equations, we thereforeare left with only two physical degrees of freedom in the metric, and two inits first time derivative.We want to represent the four-dimensional Einstein equations in termsof purely spatial objects that exist on the chosen surfaces. This is done bydefining projection operators using the spatial metric (1.16) induced on the0 Chapter 2. Methods surfaces and the normal vector (1.15). Raising one index of the spatial met-ric, γ µν , is useful for projecting four-dimensional tensors into the spatialsurfaces, while the normal projector − n µ n ν extracts the time-like part of atensor. Every free index of the tensor being projected must be contractedwith one of these projection operators. This projection is also used to definethe covariant derivative with respect to the spatial metric, by projecting thefour-dimensional covariant derivative into the surfaces, e.g. for a scalar χ , D µ χ ≡ γ νµ ∇ ν χ, (2.5)for a vector (one-form) F ν , D µ F ν ≡ γ νβ γ αµ ∇ α F β , (2.6)and for a tensor P νβ , D µ P νβ ≡ γ (cid:15)β γ νδ γ αµ ∇ α P δ(cid:15) , (2.7)see Baumgarte and Shapiro (2010).The Riemann curvature tensor associated with the spatial metric, i.e. thecurvature of the spatial surfaces, is R kilj = ∂ l Γ kij − ∂ j Γ kil + Γ kml Γ mij − Γ kmj Γ mil , (2.8)where the spatial connection functions are Γ kij = 12 γ km ( ∂ i γ jm + ∂ j γ mi − ∂ m γ ij ) . (2.9)Both (2.8) and (2.9) can therefore be constructed entirely from the spatialmetric and its spatial derivatives. The spatial Ricci curvature tensor and itstrace are R ij ≡ R kikj and R ≡ R ii , respectively.The connection functions (2.9) differ from the spatial components of thefour-dimensional connection functions (1.3), denoted by (4) Γ kij . The con-travariant four-dimensional metric in the 3+1 foliation is (Baumgarte andShapiro, 2010) g µν = γ µν − n µ n ν (2.10) = (cid:20) − α − α − β i α − β j γ ij − α − β i β j (cid:21) , (2.11)and the covariant metric is g µν = (cid:20) − α + β k β k β i β j γ ij (cid:21) , (2.12)from which we can see that g ij = γ ij , however, in general g ij (cid:54) = γ ij . Thisimplies the spatial components of (1.3) will not be equivalent to the spatialconnection functions (2.9), except in the case of β i = 0 .The curvature (2.8) does not contain all of the information about thefour-dimensional spacetime curvature, since its four-dimensional relative(1.2) contains time derivatives, and R kilj can be constructed purely from .1. Numerical Relativity K ij , which measuresthe gradient of the normal vectors of the embedded surfaces, and hence de-scribes how these surfaces are placed in the four-dimensional manifold. Itis defined as the gradient of the normal vector projected onto the surfaces, K µν ≡ − γ αµ γ βν ∇ α n β , (2.13)and in the 3+1 decomposition can also be related to the time derivative ofthe spatial metric (see Section 2.1.2). The extrinsic curvature measures localdeviations in the direction of the normal vector, and thus describes how thesurfaces are deformed on each spatial slice. The line element, in Cartesian coordinates, in the 3+1 decomposition is ds = − α c dt + γ ij (cid:0) dx i + β i cdt (cid:1) (cid:0) dx j + β j cdt (cid:1) . (2.14)In the ADM formalism, Einstein’s equations are decomposed into four con-straint equations and a set of evolution equations for the spatial metric andthe extrinsic curvature. Constraint equations
The constraint equations are derived by relating the four-dimensional Rie-mann tensor R αµβν to its three-dimensional counterpart R kilj , which liveson the surfaces. This is done via the full spatial projection of the four-dimensional Riemann tensor, i.e. γ µa γ νb γ αc γ βd R µναβ . Contracting the re-sulting relation and eliminating the four-dimensional Riemann tensor us-ing Einstein’s equations (1.1) results in the Hamiltonian constraint, R + K − K ij K ij − πGc ρ = 0 , (2.15)where the mass-energy density measured by an observer moving along thenormal to the surfaces is defined as ρ c ≡ T µν n µ n ν . (2.16)Taking a spatial projection of the four-dimensional Riemann tensor withone index projected in the normal (time) direction, i.e. γ µa γ νb γ αc n β R µναβ ,which is related to spatial derivatives of the extrinsic curvature, and con-tracting the result gives the momentum constraint D j K ji − D i K − πGc S i = 0 , (2.17)where K = γ ij K ij is the trace of the extrinsic curvature, and the momentumdensity is defined as S i ≡ − γ iµ n ν T µν . (2.18)2 Chapter 2. Methods
For the full derivation of the constraint and evolution equations, see Baum-garte and Shapiro (2010).If the constraint equations are satisfied on this initial surface, and therelevant evolution equations are also satisfied, then the constraint equationswill be satisfied exactly on any future surface. Numerical evolution of theinitial surface forwards in time will result in nonzero constraint violationdue to truncation errors. We discuss this further in Section 2.2.
Evolution equations
The evolution equation for the spatial metric is derived from the definitionof the extrinsic curvature (2.13), giving ddt γ ij = − αK ij , (2.19)where the time derivative is defined as ddt ≡ ∂∂x − L β , (2.20)and L β is the Lie derivative associated with the shift vector β i . The Liederivative describes the change in a tensor, vector, or scalar along a vectorcongruence — in this case the shift vector β i — and is defined independentof the metric on the surface. For example, the Lie derivative of a covarianttensor X ij along β i is L β X ij ≡ β k ∂ k X ij + X ik ∂ j β k + X kj ∂ i β k , (2.21)and for a contravariant tensor is L β X ij ≡ β k ∂ k X ij − X ik ∂ k β j − X kj ∂ k β i . (2.22)So long as the connection functions associated with the spatial metric aresymmetric in their lower indices, i.e. Γ ijk = Γ i ( jk ) , then we can interchangethe partial derivatives in the Lie derivative with covariant derivatives toobtain a coordinate-free expression (see Baumgarte and Shapiro, 2010).A tensor is defined by its transformation law, however, if instead an ob-ject transforms slightly differently, specifically by picking up a power of theJacobian during the coordinate transform, it is instead a tensor density , with“weight” W equal to the power of the Jacobian in its coordinate transfor-mation (see Baumgarte and Shapiro, 2010). In the above, if X ij were insteada tensor density, we would have an extra term W X ij ∂ k β k added to the re-spective Lie derivative (this will be used in the next section).We can write the evolution equation for the contravariant spatial metric(which will be of use later), starting with the spatial metric γ ij γ ij = δ ii = 3 , (2.23) ⇒ ddt (cid:0) γ ij γ ij (cid:1) = 0 , (2.24) γ ij ddt γ ij = − γ ij ddt γ ij , (2.25) .1. Numerical Relativity ddt γ ij = 2 αK ij . (2.26)The evolution equation for the extrinsic curvature is derived from a pro-jection of the four-dimensional Riemann tensor, with two indices projectedalong the normal direction, i.e. n µ n α γ βa γ νb R µναβ , giving ddt K ij = α (cid:104) R ij − K ik K kj + KK ij (cid:105) − D i D j α − πGc α (cid:20) S ij − γ ij ( S − ρ c ) (cid:21) , (2.27)where we have defined the spatial stress and its trace to be, respectively, S ij ≡ γ iµ γ jν T µν , S ≡ γ ij S ij . (2.28)For a system of equations to be “well posed”, when treated as a Cauchyproblem, the solution to such a system must be bounded by an exponentialfunction that is independent of the initial data. The solution must not beable to grow unbounded. In terms of hyperbolic partial differential equa-tions, such as Einstein’s equations, the system would be said to be weaklyhyperbolic if it were not well posed (see Alcubierre, 2008, for more detail).Einstein’s equations in the ADM formalism are only weakly hyperbolic,and are therefore not expected to remain well behaved for long time evolu-tions (see Kidder, Scheel, and Teukolsky, 2001). The hyperbolicity is spoiledby the presence of mixed second derivatives in the spatial Ricci tensor inthe right-hand side of the evolution equation (2.27). Without these terms,the evolution equations could be written as a set of wave equations. Tostrengthen the hyperbolicity of the system these mixed derivative termscan be removed, as is done in the BSSN formalism (Baumgarte and Shapiro,1999; Shibata and Nakamura, 1995). There are other methods of stabilisingthe evolution of the ADM system, including abandoning the 3+1 foliationaltogether (see Baumgarte and Shapiro, 2010, and Sections 2.1.4 and 2.1.5for more details of some of these alternatives). The BSSN formalism (Shibata and Nakamura, 1995; Baumgarte and Shapiro,1999) re-casts the ADM equations into a form that is strongly hyperbolic,and hence allows for arbitrarily long, stable evolutions of Einstein’s equa-tions.
Conformal decomposition
First, we define the conformal metric ¯ γ ij , defined by decomposing the spatialmetric into two parts using a conformal factor γ ij = e φ ¯ γ ij , or , γ ij = e − φ ¯ γ ij . (2.29)In general, conformal decompositions of the metric involve an arbitrary fac-tor ψ . The BSSN formalism corresponds to setting ψ = e φ , which turns out4 Chapter 2. Methods to be a convenient choice when deriving the evolution equations. Quanti-ties written with an over bar are associated with the conformal metric ¯ γ ij ,and those associated with the spatial metric γ ij are written without an overbar. Indices of conformal quantities are raised and lowered with the confor-mal metric. In Cartesian coordinates, we choose e φ = det( γ ij ) / ≡ γ / ,so that ¯ γ = 1 . This convenient choice makes the conformal factor φ a tensordensity of weight W = 1 / , implying the covariant conformal metric ¯ γ ij isa tensor density of weight W = − / .The traceless part of the extrinsic curvature is defined as A ij ≡ K ij − γ ij K, (2.30)and the conformal traceless part of the extrinsic curvature is A ij = e φ ¯ A ij , A ij = e − φ ¯ A ij . (2.31)Substituting the conformal metric (2.29) into the definition of the spatialconnection functions (2.9) gives their conformal transformation Γ ijk = ¯Γ ijk + 2 (cid:16) δ ij ∂ k φ + δ ik ∂ j φ − ¯ γ jk ¯ γ il ∂ l φ (cid:17) , (2.32)which we substitute into the contraction of (2.8) to get the conformal trans-formation of the spatial Ricci tensor ¯ R ij = R ij + 2 (cid:16) ¯ D i ¯ D j φ + ¯ γ ij ¯ γ lm ¯ D l ¯ D m φ (cid:17) − (cid:104) ¯ D i ( φ ) ¯ D j ( φ ) − ¯ γ ij ¯ γ lm ¯ D l ( φ ) ¯ D m ( φ ) (cid:105) , (2.33)or ¯ R ij = R ij − ¯ R φij , (2.34)where ¯ R φij is the part of (2.33) that only depends on the conformal factor φ .From (2.33) we can write the conformal transformation of the Ricci scalar tobe ¯ R ≡ ¯ γ ij ¯ R ij = e φ R + 8 e − φ ¯ D e φ , (2.35)where we have used ¯ D e φ = e φ (cid:2) ¯ γ ij ¯ D i ¯ D j φ + ¯ γ ij ¯ D i ( φ ) ¯ D j ( φ ) (cid:3) , (2.36)and ¯ D ≡ ¯ γ ij ¯ D i ¯ D j is the conformal, spatial, covariant Laplacian. Constraint equations
In the following section, we will slightly alter the form of the ADM evolu-tion equations to strengthen the hyperbolicity of the system. We do notneed to do this for the constraint equations since they are not evolved.However, see Section 2.1.5 for some formalisms that do evolve the con-straint equations to restrict the violations in numerical-relativity simula-tions. Here, we write the constraint equations in terms of the conformalvariables. For the Hamiltonian constraint (2.15) we first use (2.30), which .1. Numerical Relativity K ij K ij = A ij A ij + 13 K = ¯ A ij ¯ A ij + 13 K , (2.37)and using the conformal decomposition of the spatial Ricci scalar (2.35), wecan write (2.15) as e φ R − ¯ D e φ − e φ A ij ¯ A ij + e φ K − πGc e φ ρ = 0 . (2.38)The first term in the momentum constraint (2.17), using (2.30), is D j K ji = D j ¯ A ji + 13 ¯ D i K, (2.39)since A ji = ¯ A ji , D i K = ¯ D i K , and ¯ K = K . We now need to relate thecovariant derivative D i in the above to its conformal counterpart ¯ D i , whichwe do using the relation between the conformal and non-conformal spatialconnection functions. First we expand the spatial covariant derivative D j ¯ A ji = ∂ j ¯ A ji + Γ jkj ¯ A ki − Γ kji ¯ A jk , (2.40)and now using (2.32) for the two right-most terms above, we find D j ¯ A ji = ¯ D j ¯ A ji + 6 ¯ A ji ¯ D j φ. (2.41)Substituting this expression into (2.39), and then the result into (2.17) givesthe conformal momentum constraint e − φ ¯ D k (cid:16) e φ ¯ A ki (cid:17) −
23 ¯ D i K − πGc S i = 0 . (2.42) Evolution equations
Introducing the conformal factor φ in the BSSN formalism means we nowrequire an additional equation. We derive this by first substituting the con-formal metric (2.29) into (2.19), giving ddt ¯ γ ij + 4¯ γ ij ddt φ = − α ¯ K ij , (2.43)taking the trace gives ¯ γ ij ddt ¯ γ ij + 12 ddt φ = − αK. (2.44)We now use the identity (see Carroll, 1997) ddt ln γ / = 12 γ ij ddt γ ij , (2.45)which for the conformal metric is, ddt ln¯ γ / = 12 ¯ γ ij ddt ¯ γ ij = 0 , (2.46)6 Chapter 2. Methods since ¯ γ = 1 in Cartesian coordinates, and so (2.44) becomes ddt φ = − αK. (2.47)The evolution equation for the trace of the extrinsic curvature is found byfirst taking the trace of (2.27), which results in γ ij ddt K ij = α (cid:0) R − K ij K ij + K (cid:1) − D α + 4 πGc α (cid:0) S − ρc (cid:1) , (2.48)where D ≡ γ ij D i D j is the covariant Laplacian associated with the spatialmetric. We then eliminate R using the Hamiltonian constraint (2.15), giving γ ij ddt K ij = − αK ij K ij − D α + 4 πGc α (cid:0) S + ρc (cid:1) . (2.49)We can then expand the derivative of the trace of the extrinsic curvature togive ddt K = ddt ( γ ij K ij ) , (2.50) = γ ij ddt K ij + 2 αK ij K ij , (2.51)where we have used (2.19). Substituting this into (2.49), and using (2.30),we find ddt K = α (cid:18) ¯ A ij ¯ A ij + 13 K (cid:19) − D α + 4 πGc α (cid:0) S + ρ c (cid:1) . (2.52)We can find the traceless part of the evolution equations by subtracting(2.47) and (2.52) from the ADM evolution equations (2.19) and (2.27). Theseare ddt ¯ γ ij = − α ¯ A ij , (2.53)and ddt ¯ A ij = e − φ (cid:20) − ( D i D j α ) TF + α (cid:18) R TF ij − πGc S TF ij (cid:19)(cid:21) + α (cid:16) K ¯ A ij − A ik ¯ A kj (cid:17) , (2.54)where the superscript TF represents the trace-free part of a tensor. That is,we define, S TF ij ≡ S ij − γ ij S, (2.55) R TF ij ≡ R ij − γ ij R , (2.56) ( D i D j α ) TF ≡ D i D j α − γ ij D α. (2.57)We can use the conformal decomposition of the Ricci tensor, (2.34), and theRicci scalar, (2.35), to relate the trace-free Ricci tensor (2.56) to its conformal .1. Numerical Relativity ¯ R T Fij ≡ ¯ R ij −
13 ¯ γ ij ¯ R . (2.58)We find R T Fij = ¯ R T Fij + ¯ R φij + 83 ¯ γ ij e − φ ¯ D e φ , (2.59)where ¯ R φij is defined from (2.33).Computing the Ricci tensor R ij in (2.54) would again introduce similarmixed derivative terms which spoil the hyperbolicity of the ADM equa-tions. To avoid this, we introduce the contracted conformal connectionfunctions ¯Γ i ≡ ¯ γ jk ¯Γ ijk , (2.60)where the ¯Γ ijk are the connection functions associated with the conformalmetric. The conformal Ricci tensor in terms of these conformal connectionfunctions is then ¯ R ij = −
12 ¯ γ lm ∂ m ∂ l ¯ γ ij + ¯ γ k ( i ∂ j ) ¯Γ k + ¯Γ k ¯Γ ( ij ) k + ¯ γ lm (cid:16) kl ( i ¯Γ j ) km + ¯Γ kim ¯Γ klj (cid:17) , (2.61)where Γ ijk ≡ γ im Γ mjk for both conformal and non-conformal connectionfunctions, and round brackets around indices denote the symmetric partsof a tensor, defined in (1.58).There are several ways to ensure the mixed derivatives in R ij are elim-inated in the evolution equations. One of these is to make (2.60) a gaugechoice, and choose ¯Γ i = 0 so that the mixed derivative terms vanish com-pletely; the “Gamma-driver” condition (see Section 2.1.4). However, thisreduces the gauge freedom of the system, and may lead to undesirable co-ordinates that could form coordinate singularities. Another method — usedin the BSSN formalism — is to evolve (2.60) as a new variable; which botheliminates the mixed derivative terms and retains the gauge freedom of thesystem via the lapse and the shift vector.In Cartesian coordinates, where we have chosen ¯ γ = 1 , we can write(2.60) as ¯Γ i = − ∂ j ¯ γ ij , (2.62)using the identity (2.46). Taking the time derivative of the above gives ∂ ¯Γ i = − ∂ j ∂ ¯ γ ij . (2.63)Expanding the Lie derivative in (2.53) gives α ¯ A ij = ∂ ¯ γ ij − L β ¯ γ ij , (2.64) ⇒ ∂ ¯ γ ij = 2 α ¯ A ij + β k ∂ k ¯ γ ij − γ k ( i ∂ k β j ) + 23 ¯ γ ij ∂ k β k , (2.65)and then substituting this expression into (2.63) we find ∂ ¯Γ i = − A ij ∂ j α − α∂ j ¯ A ij + β k ∂ k ¯Γ i − ¯Γ k ∂ k β i + 13 ¯ γ ij ∂ j ∂ k β k + ¯ γ kj ∂ k ∂ j β i + 23 ¯Γ i ∂ k β k , (2.66)where we have used the definition (2.60) to simplify the expression. Wecan eliminate the derivative term ∂ j ¯ A ij in the above with the momentum8 Chapter 2. Methods constraint (2.17), which along with (2.30) gives ∂ j A ij = 8 πGc S i + 23 γ ij ∂ j K − Γ ijk A kj − Γ jjk A ik . (2.67)We now substitute the conformal trace-free extrinsic curvature ¯ A ij and con-formal metric ¯ γ ij into the above expression, and after some simplificationwe find ∂ j ¯ A ij = 8 πGc ¯ γ ij S j + 4 ¯ A ij ∂ j φ − (cid:16) Γ ijk ¯ A kj + Γ jjk ¯ A ik (cid:17) + 23 ¯ γ ij ∂ j K. (2.68)The only non-conformal objects remaining in this expression are the con-nection functions, which we can relate to their conformal counterparts us-ing (2.32), so that the connection function terms in (2.68) become Γ ijk ¯ A kj + Γ jjk ¯ A ik = ¯Γ ijk ¯ A kj + 10 ¯ A ij ∂ j φ + ¯Γ jjk ¯ A ik , (2.69)and using (2.45) we can show that ¯Γ jjk = 0 . Substituting this expressionback into (2.68) gives ∂ j ¯ A ij = 8 πGc ¯ γ ij S j − A ij ∂ j φ − ¯Γ ijk ¯ A kj + 23 ¯ γ ij ∂ j K. (2.70)Finally, to arrive at the evolution equation for the contracted connectioncoefficients we substitute the above expression for ∂ j ¯ A ij into (2.66), whichgives ddt ¯Γ i = − A ij ∂ j α + 2 α (cid:18) ¯Γ ijk ¯ A kj −
23 ¯ γ ij ∂ j K − πGc ¯ γ ij S j + 6 ¯ A ij ∂ j φ (cid:19) + 23 ¯Γ i ∂ j β j + 13 ¯ γ li ∂ l ∂ j β j + ¯ γ lj ∂ j ∂ l β i . (2.71)This equation, together with equations (2.47), (2.52), (2.53), and (2.54), formthe full BSSN system of evolution equations, summarised in Table 2.1. Evolv-ing the contracted connection functions as independent functions meansthat the definition (2.60) acts as a new constraint equation together with theHamiltonian (2.38) and momentum constraints (2.42).The matter source terms ρ, S i , and S ij appearing in both the ADM andBSSN formalisms above are projections of the stress-energy tensor into the spatial surfaces . These quantities are distinct from those measured in the restframe of the fluid. From (2.16) we can relate ρ to the mass-energy densitymeasured in the fluid rest frame, ρ R , ρc ≡ T µν n µ n ν , (2.72) = ρ hu µ n µ u ν n ν + P g µν n µ n ν , (2.73) = ρ R c W + P (cid:0) W − (cid:1) , (2.74)where we have used n µ n µ = − and the definition of the Lorentz factor(1.17). .1. Numerical Relativity As mentioned in Section 2.1.1, there exist four degrees of freedom along-side the evolution and constraint equations derived in Section 2.1.2 and2.1.3. These freedoms are encompassed in choosing the lapse function andthe shift vector. The lapse function α describes the time slicing; the relationbetween proper time and coordinate time between, and across, spatial sur-faces. The shift vector β i describes the spatial gauge; describing how thespatial coordinates are translated from one surface to the next. These func-tions are traditionally chosen in ways that avoid (or prevent) coordinate orphysical singularities, with some coordinate choices made purely for sim-plicity, and others developed to avoid numerical issues with these simplerchoices, a few of which we will discuss briefly below. Geodesic slicing
Geodesic slicing is the simplest choice for the spatial gauge and time slicing,in which we have α = 1 and β i = 0 . The choice of zero shift implies thatcoordinate observers coincide with normal observers, and the choice of α =1 implies that these observer’s proper time coincides with the coordinatetime of the surfaces. While this choice simplifies the evolution equations, italso can form coordinate singularities when evolving nonlinear problems.To show this, we can take the trace of (2.27), which gives (see Smarr andYork, 1978a), ddt K = − γ ij D i D j α + α (cid:20) K ij K ij + 4 πGc ( ρc + S ) (cid:21) , (2.75)and substituting the conditions for geodesic slicing, α = 1 , β i = 0 , gives ∂ K = K ij K ij + 4 πGc ( ρc + S ) , (2.76)and for a perfect fluid (using (1.7)) we can write the trace of the spatial stressas S ≡ γ ij S ij = γ ij γ iµ γ jν T µν (2.77) = γ jµ γ jν u µ u ν ( ρc + P ) + γ jµ γ jν g µν P (2.78) = ( ρc + P ) u j u j + 3 P (2.79)and in geodesic slicing we have u j = n j = 0 , so we have ∂ K = K ij K ij + 4 πGc ( ρc + 3 P ) . (2.80)So long as the strong energy condition for a perfect fluid is satisfied, ρc + 3 P ≥ , (2.81)then both terms on the right hand side are positive, implying that K willincrease without limit. From (2.13), we can write K = −∇ µ n µ , (2.82)0 Chapter 2. Methods which shows that in geodesic slicing the normal vectors will therefore con-verge and create caustics, and hence coordinate singularities. This is ex-pected as these choices correspond to observers freely falling along geodesics(i.e. with no acceleration), and geodesics will converge during gravitationalcollapse. So long as we are not simulating nonlinear gravitational collapse,e.g. simulating evolution of an FLRW spacetime, geodesic slicing is wellsuited. However, in simulations of large-scale cosmological structure for-mation, geodesics will cross once nonlinear structures begin to form.
Maximal slicing
Maximal slicing provides a condition that prevents the convergence of co-ordinate observers seen in geodesic slicing (Smarr and York, 1978a). From(2.82), an obvious choice to stop the focusing of normal observers is K = 0 ,and choosing also ∂ t K = 0 ensures this will be true on all subsequent timeslices. Equation (2.52) can then be simplified into an elliptic equation forthe lapse D α = αK ij K ij + 4 πGc α (cid:0) S + ρ c (cid:1) . (2.83)In practice, the maximal slicing condition will only be satisfied approxi-mately, i.e. K (cid:54) = 0 , due to truncation errors in the simulation. Even if ∂ t K = 0 is enforced throughout, the maximal slicing condition will still beviolated. Instead, a condition is specified to drive K back towards zero, ∂ K = − mK, (2.84)where m is a positive constant with dimensions of inverse length. Againusing (2.52) we find a new elliptic equation for the lapse, D α = α (cid:20) K ij K ij + 4 πGc (cid:0) S + ρ c (cid:1)(cid:21) + β i ∂ i K + mK, (2.85)which is essentially correcting (2.83) for violations of the maximal slicingcondition. The lapse function that satisfies (2.85) describes the maximalslices. Elliptic equations are computationally expensive to invert in threedimensions, so to make this equation cheaper to solve numerically, we con-vert it into a parabolic form using a derivative of the lapse (i.e. similarto relaxation methods for solving elliptic equations; Press, Flannery, andTeukolsky, 1986). We introduce an arbitrary time coordinate λ = (cid:15)t , where (cid:15) is some constant with dimension L . We then set ∂ λ α equal to (2.85), whichgives a parabolic equation for the lapse ∂ α = (cid:15)D α − (cid:15)α (cid:20) K ij K ij + 4 πGc (cid:0) S + ρ c (cid:1)(cid:21) − (cid:15)β i ∂ i K − (cid:15)mK, (2.86)or, more simply ∂ α = − (cid:15) ( ∂ K + mK ) , (2.87)which is referred to as a “K-driver” condition (Balakrishna et al., 1996). Themaximal slicing condition is then satisfied in the limit (cid:15) → ∞ , however avery large (cid:15) would require an extremely small time step, which can signifi-cantly increase the computational cost of the simulation. .1. Numerical Relativity Harmonic coordinates and slicing
Writing Einstein’s equations in harmonic coordinates was first done to avoidsome of the computational issues associated with the ADM formalism (e.g.Fischer and Marsden, 1973), by abandoning the 3+1 decomposition alto-gether and keeping Einstein’s equations in a four-dimensional form. Wefirst define the four-dimensional contracted connection functions, similarto in Section 2.1.3, (4) Γ α ≡ g µν (4) Γ αµν . (2.88)If we choose for these contracted connection functions to vanish, (4) Γ α = 0 , (2.89)then the coordinates themselves satisfy the wave equation, ∇ α ∇ α x µ = 0 ,and are therefore harmonic functions (e.g. York, 1979). As discussed in Sec-tion 2.1.3, defining the contracted connection functions simplifies the formof the Riemann — and hence Ricci — tensor by absorbing second deriva-tives of the metric into first derivatives of the functions (2.88). The harmoniccoordinate choice (2.89) will obviously simplify this even further, and re-duces Einstein’s equations to a set of nonlinear wave equations, which pro-vides a mathematical advantage since the behaviour of equations of thistype is extremely well understood (Baumgarte and Shapiro, 2010).An issue with harmonic coordinates arises because there is no strict re-quirement for the time coordinate to remain time-like throughout the sim-ulation, and this can cause numerical problems (Garfinkle, 2002). A wayaround this is to introduce source functions to the wave equation for the co-ordinates, i.e. ∇ α ∇ α x µ = H µ . The function H µ can then be used to controlthe behaviour of the coordinates in the simulation, known as “generalisedharmonic coordinates” (Pretorius, 2005b).Using completely harmonic coordinates is not normally done in numer-ical simulations because the choices of initial data and coordinates are notas clear as in a 3+1 decomposition. Usually, the coordinates are chosen viathe lapse and shift, which both have a clear geometric interpretation in re-lation to the spatial surfaces. In harmonic coordinates, H µ encompasses thecoordinate choices, but does not have a clear geometric interpretation, mak-ing it difficult to form a desirable coordinate system (Pretorius, 2005b). Be-cause the spacetime is not split into a series of spatial surfaces when usingharmonic coordinates, this also makes the generation of initial data moredifficult. The regular 3+1 constraint equations, along with the evolutionequations, can instead be used to generate the initial data necessary for theharmonic evolution equations; specifying the four-metric g µν and its timederivative at some initial instant in time.Harmonic slicing of the lapse is much more common in numerical sim-ulations, and involves setting only the time component of the contractedconnection functions to be zero, i.e. (4) Γ = 0 . Alongside a zero shift vector,this gives the evolution equation for the lapse to be ∂ α = − α K, (2.90)2 Chapter 2. Methods which can be integrated by first substituting a contraction of the ADM evo-lution equation (2.19), namely, αK = D i β i − ∂ ln γ / , (2.91)again with zero shift, which gives ∂ ln α = − ∂ ln γ / . (2.92)Solving the above gives the general form of the lapse function in harmonicslicing to be α = F ( x i ) γ / (2.93)where F ( x i ) is an arbitrary, dimensionless, purely spatial function. Generalised slicing form
The appeal of the evolution of the lapse function using (2.93) is that thereis no need to invert an elliptic equation at each time step, as with maximalslicing. However, the singularity avoidance of harmonic slicing is not asstrong as in maximal slicing (Shibata and Nakamura, 1995).The Bona-Masso family of slicing conditions (Bona et al., 1995) are ageneralisation of the evolution equation for the lapse (2.90), ∂ α = − α f ( α ) K, (2.94)where f ( α ) is a positive, dimensionless, arbitrary function (which may bea function of the lapse, or a constant). Choosing f = 1 reduces the slicingcondition to harmonic slicing, and f = 0 reduces it to geodesic slicing.Another popular choice is f = 2 /α , which results in the lapse α = 1 + ln γ, (2.95)known as “1+log” slicing, which has been proven to have better singularityavoidance than maximal slicing (Alcubierre, 2008). Gamma-driver condition
The Gamma-driver condition is related to the minimal distortion shift con-ditions (see Baumgarte and Shapiro, 2010), which was developed to limitthe time evolution of the conformal metric in order to reduce spurious co-ordinate modes in ¯ γ ij (Smarr and York, 1978a). We can limit the confor-mal metric’s evolution by setting the time derivative of the conformal, con-tracted connection functions to zero, ∂ ¯Γ i = 0 , (2.96) .1. Numerical Relativity β j ∂ j ¯Γ i − ¯Γ j ∂ j β i + 23 ¯Γ i ∂ j β j + 13 ¯ γ li ∂ l ∂ j β j + ¯ γ lj ∂ j ∂ l β i = 2 ¯ A ij ∂ j α − α (cid:18) ¯Γ ijk ¯ A kj −
23 ¯ γ ij ∂ j K − πGc ¯ γ ij S j + 6 ¯ A ij ∂ j φ (cid:19) . (2.97)As in Section 2.1.4, we want to drive away any potential violations of (2.96),so we instead choose ∂ ¯Γ i = − η ¯Γ i . (2.98)We then convert (2.97) into parabolic form in the same way as in maximalslicing, giving the “Gamma-driver” condition for the shift (Alcubierre andBruegmann, 2001; Duez et al., 2003) ∂ β i = k (cid:0) ∂ ¯Γ i + η ¯Γ i (cid:1) , (2.99)where k and η are arbitrary constants with dimensions length and inverselength, respectively. The constraint equations (2.15) and (2.17) are zero analytically, however, fi-nite differencing errors introduce a non-zero constraint violation into theevolution. In the ADM formalism, since the constraint equations are notevolved, any local constraint violation remains where it is and can grow,which may become unstable. In deriving the evolution equations for theBSSN formalism, specifically the evolution equation for the contracted con-nection functions (2.71), we brought the momentum constraint into the evo-lution equations. This means that local violation of the momentum con-straint can now propagate and move off the grid (if coupled with suitableboundary conditions). This stabilises the evolution and damps the con-straint violation in the simulation.
Z4 Formulation
In the original ADM formalism, the 3+1 decomposition of spacetime splitsEinstein’s equations into separate evolution and constraint equations, thelatter of which are only enforced on the initial data, and not constrainedduring the simulation. Any resulting constraint violations are not invariantunder coordinate transforms, breaking the general covariance of the sys-tem. The Z4 formulation (Bona et al., 2003) is a covariant extension to Ein-stein’s equations, which involves introducing a new four-vector Z µ , suchthat the field equations become G µν + ∇ µ Z ν + ∇ ν Z µ = 8 πGc T µν , (2.100)with the constraint Z µ = 0 now acting as a numerical check on the accu-racy of a simulation. A 3+1 decomposition of (2.100) results in a system ofequations with no static constraints, with Z µ evolved as a part of the sys-tem, maintaining general covariance. The regular 3+1 constraint equations4 Chapter 2. Methods (2.15) and (2.17) can be solved to generate initial data for the Z4 system,since the two formulations are equivalent in the case Z µ = 0 . The extent towhich the solution is matching Einstein’s equations can then be monitoredvia Z µ (Bona et al., 2003). The addition of this new four-vector is analogousto the addition of the contracted connection functions in harmonic coordi-nates, which also results in a general covariant system, however the twosystems share similar drawbacks, as discussed in Section 2.1.4 (Garfinkle,2002; Pretorius, 2005b). Damped Z4 system
The λ − system is a method of constraint-violation damping by adding dy-namical variables into the system, which act as time derivatives of eachconstraint expression (Brodbeck et al., 1999). Additional terms are broughtinto the evolution equations of these extra variables such that the variablesthemselves, and hence the constraints, are damped during the evolution.The Z4 system is a λ − system with no damping terms; the Z µ four-vectoracts as the additional dynamical variables related to the constraints. The“damped Z4 system” (Gundlach et al., 2005) takes advantage of this byadding damping terms to (2.100), G µν + ∇ µ Z ν + ∇ ν Z µ + κ [ n µ Z ν + n ν Z µ − (1 + κ ) g µν n σ Z σ ] = 8 πGc T µν , (2.101)where κ , κ are free parameters to control the level of damping. Conformal and covariant Z4 system
The conformal and covariant Z4 system (CCZ4; Alic et al., 2012) is an ex-tension to the damped Z4 system, by performing a 3+1 decomposition of(2.101) and casting the equations into conformal trace-free form, as in theBSSN formalism. The aim is to combine the benefits of the BSSN and gen-eralised harmonic formalisms. The BSSN formalism is appealing numer-ically because of its gauge freedom in the form of the lapse function andthe shift vector, which allow for singularity avoidance without the need tocompletely excise a region of spacetime from the simulation. In addition,the conformal decomposition allows for potentially singular terms to be ab-sorbed into the conformal factor, rather than the metric. The appeal of thegeneralised harmonic formalism is the evolution of the constraints is in-cluded, meaning initial data that satisfies the constraints will satisfy themat all times (Pretorius, 2005b). In the case of initially small, inhomogeneousconstraint violations, the CCZ4 system will constrain the growth of theseviolations during the simulation with appropriately chosen values for κ i (see Gundlach et al., 2005). Many astrophysical (and general hydrodynamical) phenomena produce shockwaves, and the numerical modelling of these nonlinear waves poses sev-eral issues. High resolution shock-capturing methods cast the evolution .1. Numerical Relativity primitive variables; mass density,pressure, velocity, and internal energy, while the conserved variables are theones used for evolution. In addition, the primitive variables themselvesmay be required to calculate the source terms to evolve the conserved vari-ables. Solving for the primitive variables from the conserved variables canbe difficult for particular systems, since a simple analytic relation betweenthe two may not exist. This must be performed at every time step, addingsignificant computational time to the calculation.The equations of General-Relativistic hydrodynamics are derived fromthe conservation of rest-mass and energy-momentum (see Banyuls et al.,1997) ∇ µ ( ρ u µ ) = 0 , ∇ µ T µν = 0 . (2.102)These equations can be written in flux-conservative form as ∂ U + ∂ i F i = S , (2.103)where U = [ D, S cj , τ c ] are the conserved variables, here defined in Euleriancoordinates as (Wilson, 1972; Font, 2008) D ≡ √ γρ W, (2.104a) S ci ≡ √ γρ hW v i c , (2.104b) τ c ≡ √ γ (cid:0) ρ hW − P (cid:1) − Dc . (2.104c)Here, h = c + (cid:15) + P/ρ is the specific enthalpy, and (cid:15) is the specific internalenergy. The Lorentz factor, W , and the fluid three velocity, v i , are definedin (1.17) and (1.18), respectively.The vectors F i and S in (2.103) are the fluxes and source terms, respec-tively, defined by F i = (cid:2) αD ˜ v i , α (cid:0) S cj ˜ v i + δ ij P (cid:1) , α (cid:0) τ c ˜ v i + P v i (cid:1)(cid:3) , (2.105a) S = (cid:104) , T µν (cid:16) ∂ µ g νj − (4) Γ λµν g λj (cid:17) , α (cid:16) T µ ∂ µ ln α − T µν (4) Γ µν (cid:17)(cid:105) , (2.105b)where ˜ v i ≡ v i c − β i α . (2.106)Defining an EOS for the fluid in question — i.e. a description of the pressureof the fluid in terms of rest-mass density and internal energy — closes thesystem.While the conserved variables D, S cj , and τ c are the variables actuallyevolved in the simulation, the primitive variables ρ , P, v i , and (cid:15) are re-quired to calculate the source terms S for the evolution of the conservedvariables (via the stress-energy tensor). From (2.104) we can see that con-verting primitive to conservative variables is straightforward analytically,but the reverse is not. We discuss briefly a few different methods for con-verting conservative to primitive variables in Section 2.2.3.6 Chapter 2. Methods
INSTEIN T OOLKIT
The C
ACTUS code was first written in 1992 by Ed Seidel and his group atthe Max Planck Institute for Gravitational Physics (Albert Einstein Insti-tute), initially to allow for a collaborative, parallel platform for numerical-relativity simulations. The C
ACTUS framework consists of a central core,“flesh”, and application modules, “thorns”, which communicate with eachother via the C
ACTUS flesh, allowing for thorns to be developed and main-tained independently from one another. The C
ACTUS framework was latergeneralised for other computational scientists requiring large-scale collab-orative computing (Goodale et al., 2003), and the numerical-relativity capa-bilities of the C
ACTUS code were collected into the E
INSTEIN T OOLKIT (ET)(Löffler et al., 2012).The ET itself consists of about 100 thorns used for relativistic astro-physics, including vacuum spacetime solvers (e.g. McLachlan ), General-Relativistic hydrodynamics (e.g.
GRHydro ), adaptive mesh refinement, anal-ysis thorns, and thorns for different initial conditions. We briefly introducethe main thorns used in this thesis, and discuss the relevant equations beingsolved.
The main appeal of the ET’s structure is the ability to be used collabora-tively, and for different parts of the code to be used in different ways. Mainevolution thorns (e.g. for the hydrodynamics and spacetime) are writtenin a way such that supplementary thorns can be easily substituted in theirplace. A large part of this structure being able to work is through the use ofseveral “base” thorns, which store the sets of variables common amongstdifferent methods for numerical evolution of particular systems, and there-fore the variables that are common among different thorns. For example,hydrodynamic evolution thorns communicate directly with the base thorn
HydroBase , which stores the primitive hydrodynamical variables (see Sec-tion 2.1.6), and spacetime evolution thorns communicate with
ADMBase ,which stores the variables evolved using a 3+1 decomposition of spacetime(as discussed in Section 2.1.2 and 2.1.3). The base thorns
HydroBase and
ADMBase also act as an interface to specify initial conditions, and for per-forming analyses (see Zilhão and Löffler, 2013, for a detailed discussion ofC
ACTUS and ET structure).Spacetime evolution thorns in the ET evolve only the left hand side ofEinstein’s equations and therefore must be sourced by the stress-energy ten-sor if matter is present. However, calculating the stress-energy tensor (1.7)requires both matter and spacetime variables. The thorn
TmunuBase buildsthe stress-energy tensor T µν by communicating separately with the hydro-dynamic and spacetime thorns, and can then feed it back into the relevantspacetime evolution thorn. This means that the evolution of the hydrody-namics and spacetime are completely independent, and so different thornscan be easily substituted. .2. The E INSTEIN T OOLKIT McLachlan
The
McLachlan group of thorns is a code for solving the left hand sideof Einstein equations using a 3+1 conformal decomposition (Brown et al.,2009). The code itself is generated by Kranc — a M ATHEMATICA pro-gram that converts a system of partial differential equations into C
ACTUS code. While the code itself solves the vacuum Einstein equations, it can belinked to seperate thorns that solve the hydrodynamical system coupled tothe spacetime, such as
GRHydro , explained in the next section. The space-time variables are discretised on a grid, with options for adaptive meshrefinement via the
Carpet driver thorn, which also handles memory, par-allelisation, input and output, and time evolution (Schnetter, Hawley, andHawke, 2004).The
McLachlan code implements both the conformal trace-free BSSN(as described in Section 2.1.3) and CCZ4 (as described in Section 2.1.5) for-malisms, via two thorns
ML_BSSN and
ML_CCZ4 . The variables evolvedin
ML_BSSN are the conformal factor φ , the trace of the extrinsic curvature K , the conformal metric ¯ γ ij , the conformal trace-free extrinsic curvature ¯ A ij , and the contracted conformal connection functions ¯Γ i (see Table 2.1 forthe full system). ML_CCZ4 extends this system with additional dampingterms related to the four-vector Z µ , as in (2.101), which are controlled usingthe parameters κ , κ , and also by evolving the quantity ˆ¯Γ i ≡ ¯Γ i + 2¯ γ ij Z j ,along with the projection of Z µ along the normal direction; n µ Z µ (see Alicet al., 2012). Both thorns adopt a generalised Bona-Masso slicing of thelapse function, using (2.94), and evolve the shift vector under the “Gamma-driver” condition, using (2.99). GRHydro
GRHydro is the main hydrodynamical evolution thorn in the ET, evolvingthe equations of ideal General-Relativistic hydrodynamics (or magnetohy-drodynamics), and was built from the public
Whisky code (Baiotti et al.,2005; Hawke, Löffler, and Nerozzi, 2005; Giacomazzo and Rezzolla, 2007;Baiotti, Giacomazzo, and Rezzolla, 2008; Mösta et al., 2014). The methodof lines thorn MoL is used for time evolution, implementing a numericalmethod for solving partial differential equations, in which spatial deriva-tives are discretised and time derivatives are left continuous. This then al-lows use of a regular numerical method for ordinary differential equations.Discretising a fluid on a computational grid results in artificial disconti-nuities in the fluid across cell boundaries. This is often dealt with by averag-ing the primitive variables across a cell boundary in order to calculate theflux of the fluid across the boundary, which in itself requires reconstruct-ing the primitive variables between neighbouring cells. Performing this re-construction at high accuracy can result in spurious numerical oscillationswhen near a shock (according to Godunov’s theorem). To achieve mono-tonicity, reconstruction methods such as total variation diminishing (TVD),the piecewise parabolic method (PPM; Colella and Woodward, 1984), andessentially non-oscillatory (ENO; Harten et al., 1987) methods are required; http://kranccode.org Chapter 2. Methods each of which are implemented in
GRHydro . Once the primitive variableshave been reconstructed on the cell boundaries, they are used as initialconditions for the chosen Riemann solver in
GRHydro , which may be theHarten-Lax-van Leer-Einfeldt (HLLE; Harten, Lax, and Van Leer, 1983; Ein-feldt, 1988), Roe (Roe, 1981), or Marquina (Marquina et al., 1992) solver.The EOS of the fluid is specified and handled separately in the thorn
EOS_Omni , in which “Polytype” EOS, P = P ( ρ ) (including a polytropicEOS), or general EOS, P = P ( ρ , (cid:15) ) (including a gamma-law and hybridEOS), are implemented (see Löffler et al., 2012, for full details of all thoseavailable).To simplify the calculations performed in GRHydro , time derivativesof the spatial metric in the source terms S in (2.105b) are eliminated usingthe evolution equation (2.19). Time components of the four-dimensionalconnection functions are written in terms of spatial and time derivativesof the spatial metric, lapse, and shift. The time derivatives of the lapse andshift are specified in the chosen gauge, and explicit spatial derivatives of thespatial metric are eliminated using its spatial covariant derivative, which iszero by construction, i.e. D i γ jk = ∂ i γ jk + 2 γ lk Γ jil = 0 . (2.107)Hence, spatial derivatives of γ ij can be written in terms of the connectionfunctions.The conversion from primitive to conservative variables is simple ana-lytically, as can be seen in (2.104), however the reverse is not as straightfor-ward. In GRHydro this conversion is performed using a Newton-Raphsoniteration, however, the specific method is dependent on the user-chosenEOS. In the case of a general EOS, i.e. P = P ( ρ, (cid:15) ) , the root of the func-tion f = ¯ P − P ( ¯ ρ, ¯ (cid:15) ) is found using approximate guesses for ¯ P , ¯ ρ, ¯ (cid:15) . In thiscase the pressure is a function of both the density and internal energy, sothe derivatives dP/dρ and dP/d(cid:15) (required to find the root) are suppliedfrom the relevant EOS thorn. In the case of a “Polytype” EOS, the root ofthe function f = ¯ ρ ¯ W − D/ √ γ is found using a similar method (see Löffleret al., 2012, and the GRHydro documentation).
FLRWSolver
The ET contains several thorns for initialising different setups in both space-time and matter. These include binary neutron stars, binary black holes,magnetised neutron stars, Minkowski and Kasner spacetimes, Kerr andSchwarzschild spacetimes (in several coordinate systems), and linear grav-itational waves. The ET is not used extensively for cosmology, with onlya few tests of exact cosmological spacetimes having being previously per-formed (Vulcanov and Alcubierre, 2002).We developed an initial-condition thorn for linearly-perturbed FLRWspacetimes;
FLRWSolver (see Macpherson, Lasky, and Price, 2017). Aroundthe same time, a group of thorns CTT
HORNS (C OSMO T OOLKIT ) was re-leased and added to the public release of the ET, to both initialise cosmo-logical spacetimes and evolve them with a new hydrodynamic evolutionthorn for dust (see Bentivegna and Bruni, 2016). .2. The E INSTEIN T OOLKIT
FLRWSolver communicates directly with thebase thorns
HydroBase and
ADMBase , by filling the initial data for theprimitive hydrodynamic variables — the rest-mass density, pressure, veloc-ity, and internal energy — and the spacetime variables — the spatial metric,extrinsic curvature, lapse, and shift.In
FLRWSolver , we currently only consider small perturbations aroundthe flat, dust FLRW model, under the assumption that linear perturbationtheory is valid, i.e., | φ | c , | ψ | c , | δ | , | v i | c (cid:28) , (2.108)for the metric, density, and velocity perturbations, respectively.From the linearly perturbed Einstein equations (1.53), we have ¯ G µν = 8 πGc ¯ T µν , (2.109)and δG µν = 8 πGc δT µν , (2.110)solving (2.109) results in the Friedmann equations derived in Section 1.2.2.In the following derivation, we solve (2.110) to find analytic evolution equa-tions for the metric, density, and velocity perturbations. We neglect anyterms that are second order or higher. We use the resulting equations togenerate initial conditions for the cosmological simulations in Chapters 3,4, and 5.We use the Riemannian Geometry and Tensor Calculus (RGTC) pack-age for M ATHEMATICA to calculate the components of the Einstein tensor, G µν , for the metric (1.52). The time-time and time-space components are, tolinear order, G = 3 H c + 2 ∂ φc − H φ (cid:48) c , (2.111) G i = 2 c (cid:18) H ∂ i ψc + ∂ i φ (cid:48) c (cid:19) , (2.112)where ∂ ≡ δ ij ∂ i ∂ j . The spatial components are G ij = (cid:20) c (cid:18) H − a (cid:48)(cid:48) a (cid:19) (cid:18) − ψc − φc (cid:19) + 2 H c (cid:18) ψ (cid:48) c + 2 φ (cid:48) c (cid:19)(cid:21) δ ij + (cid:18) φ (cid:48)(cid:48) c + ∂ ψc − ∂ φc (cid:19) δ ij − ∂ i ∂ j ψc + ∂ i ∂ j φc . (2.113)For dust, i.e. using (1.7) with P = 0 , and zero shift, the time-time compo-nent of the stress-energy tensor is T = ρ R c W α , = ρ R c W a (cid:18) ψc (cid:19) , (2.114) Written by Sotirios Bonanos, see: http://library.wolfram.com/infocenter/MathSource/4484/ Chapter 2. Methods where we have used (1.21). The time-space components are T i = − ρ R c W v i c α, = − ρ R c W v i c a (cid:18) ψc (cid:19) , (2.115)where we have used (1.22) and the linear approximation α = a (cid:112) ψ/c ≈ a (1 + ψ/c ) . The spatial components are T ij = ρ R W v i v j . (2.116)Expanding the density and velocity in terms of the background and thelinear perturbations gives ρ R = ¯ ρ R (1 + δ ) , (2.117) v i = ¯ v i + δv i , (2.118)where we have introduced the fractional density perturbation δ ≡ δρ/ ¯ ρ R =( ρ R − ¯ ρ R ) / ¯ ρ R . For FLRW, ¯ v i = 0 , so from here on we denote v i = δv i . TheLorentz factor is W ≈ to linear order, and with the above perturbationsthe components of the stress-energy tensor become T = ¯ ρ R c a + ¯ ρ R c a (cid:18) δ + 2 ψc (cid:19) , (2.119) T i = − ¯ ρ R c a v i c , (2.120) T ij = 0 . (2.121) Linearly perturbed equations
The time-time component of (2.110) is, using (2.119) and (2.111), ∂ φc − H φ (cid:48) c = 8 πGc (cid:20) ¯ ρ R c a (cid:18) δ + 2 ψc (cid:19)(cid:21) , (2.122)which gives ∂ φ − H (cid:18) φ (cid:48) c + H ψc (cid:19) = 4 πG ¯ ρ R δa . (2.123)The time-space components of (2.110) are, using (2.120) and (2.112), H ∂ i ψ + ∂ i φ (cid:48) = − πG ¯ ρ R av i . (2.124)Now considering the spatial components, we first take the trace of (2.113) G kk = 3 a c (cid:18) H − a (cid:48)(cid:48) a (cid:19) (cid:18) − ψc − φc (cid:19) + 6 H a c (cid:18) ψ (cid:48) c + 2 φ (cid:48) c (cid:19) + 6 φ (cid:48)(cid:48) a c + 2 a (cid:18) ∂ ψc − ∂ φc (cid:19) , (2.125) .2. The E INSTEIN T OOLKIT T = g ij T ij = 0 ,we find φ (cid:48)(cid:48) + H ( ψ (cid:48) + 2 φ (cid:48) ) + (cid:18) a (cid:48)(cid:48) a − H (cid:19) ( φ + ψ ) + c ∂ ( ψ − φ ) = 0 . (2.126)Next, we consider the trace-free part of the spatial components by subtract-ing the spatial trace (2.125) from the full spatial components (2.113), i.e. bydefining G TF ij ≡ G ij − g ij G kk = G ij − a δ ij G kk , (2.127) T TF ij ≡ T ij − g ij T = 0 , (2.128)which hold for an FLRW background, and solving G TF ij = 8 πGc T TF ij . (2.129)This gives G TF ij = (cid:20)(cid:18) ∂ i ∂ j − δ ij ∂ (cid:19) ( φ − ψ ) (cid:21) = 0 , (2.130) ⇒ ∂ (cid:104) i ∂ j (cid:105) ( φ − ψ ) = 0 , (2.131)where ∂ (cid:104) i ∂ j (cid:105) ≡ ∂ i ∂ j − / δ ij ∂ . Equation (2.131) implies, in the linearregime, the temporal and spatial perturbations of the metric are equal, i.e. φ = ψ . Our full system of equations therefore simplifies to ∂ φ − H (cid:18) φ (cid:48) c + H φc (cid:19) = 4 πG ¯ ρ R δa , (2.132a) H ∂ i φ + ∂ i φ (cid:48) = − πG ¯ ρ R av i , (2.132b) φ (cid:48)(cid:48) + 3 H φ (cid:48) = 0 , (2.132c)where we have used a (cid:48)(cid:48) /a − H = 0 , which can be shown using the Fried-mann equations (1.35) and (1.34) with P = 0 . Linearly perturbed solutions
For a flat, matter-dominated FLRW universe, the expansion rate follows(1.43), giving the analytic form of H . This means we can solve (2.132c) toarrive at an analytic expression for the metric perturbation, φ ( ξ ) = f ( x i ) − g ( x i )5 ξ , (2.133)where f, g are arbitrary, time-independent functions, and ξ is the scaledconformal time defined in (1.44). We now substitute (2.133) into the Hamil-tonian constraint (2.132a) to derive the analytic form for the fractional den-sity perturbation, δ . We find δ ( ξ ) = C ξ ∂ f − c f − C ξ − ∂ g − c ξ − g, (2.134)2 Chapter 2. Methods where we have used the Friedmann equation (1.38a), and the solutions(1.45) and (1.43), for the background density, ¯ ρ R , and scale factor, a , respec-tively. We also define C ≡ a init / (4 πGρ ∗ ) , as in Macpherson, Lasky, andPrice (2017).We now substitute (2.133) into (2.132b) to derive the analytic form of thevelocity perturbation v i = C a init ξ − ∂ i f + 3 C a init ξ − ∂ i g, (2.135)where C ≡ − (cid:112) a init / (6 πGρ ∗ ) .The system of equations (2.133), (2.134), and (2.135) describes the evo-lution of linear perturbations to the FLRW metric, so long as our initialassumptions about the magnitude of the perturbations themselves remainvalid.The general evolution equation for the fractional density perturbation(2.134) contains both growing and decaying modes. Since we are interestedin analysing the growth of structure in the Universe, we choose g = 0 andextract only the growing mode. The analytic solutions governing the evo-lution of the metric, density, and velocity perturbations in the linear regimethen become φ = f, (2.136a) δ = C ξ ∂ f − c f, (2.136b) v i = C a init ξ − ∂ i f. (2.136c)We therefore have φ (cid:48) = 0 , i.e. the spatial distribution of the metric pertur-bation is constant in the linear regime. These analytic solutions provide theinitial conditions in FLRWSolver , for different choices of f .The fluid three velocity with respect to the Eulerian observer (2.136c),defined in HydroBase via (1.18), implies here (for zero shift) dx i dt = αv i . (2.137)The velocity v i decays over time, as per (2.136c), whereas the coordinatevelocity dx i /dt ∝ ξ in the linear regime, when α ≈ a . Single-mode perturbation
We can choose a simple form of φ to be a single-mode, sinusoidal perturba-tion with dimensionless amplitude φ (cid:28) , φc = φ (cid:88) i =1 sin (cid:18) πx i L (cid:19) , (2.138)where L is the wavelength of the perturbation with dimension of length.With this form of the metric perturbation, the corresponding density and .2. The E INSTEIN T OOLKIT ξ = 1 ) are then δ = − (cid:18) π c C L + 2 (cid:19) φ (cid:88) i =1 sin (cid:18) πx i L (cid:19) , (2.139) v i c = 2 πc C L a init φ cos (cid:18) πx i L (cid:19) . (2.140)We evolve these initial perturbations in the ET in Chapter 3, Section 3.4, andcompare the numerical growth to the analytic solutions (2.136). Multi-mode perturbations
We can analyse the growth of perturbations similar to those in our ownUniverse by drawing our initial conditions from the spectrum of perturba-tions in the CMB. The Code for Anisotropies in the Microwave Background(CAMB; Seljak and Zaldarriaga, 1996; Lewis and Bridle, 2002) is a cosmol-ogy code primarily used for generating linear power spectra of the fluctua-tions in the CMB. CAMB is written in the synchronous (comoving) gauge,i.e. α = 1 , ∂ t α = 0 and β i = 0 , which differs to the longitudinal gauge,as used in FLRWSolver . However, for the scales we currently sample, thematter power spectra in the synchronous and longitudinal gauges are al-most identical, and so we expect negligible difference in the generation ofinitial conditions on (cid:46) − Gpc scales.To generate the CMB-like initial conditions for
FLRWSolver we useparameters consistent with Planck Collaboration et al. (2016). We use thematter power spectrum output from CAMB as the spectrum of initial fluc-tuations in the density perturbation δ . We invert (2.136b) in Fourier space tosolve for the corresponding metric perturbation φ , from which we specifythe velocity field v i . The full method of generating the initial conditions isdiscussed in Chapter 4, Section 4.3.2. For the work presented in this thesis, we use our thorn
FLRWSolver toinitialise several cosmological spacetimes, described in detail in Chapters 3and 4. Aside from this thorn, we use
McLachlan , specifically
ML_BSSN , toevolve the spacetime variables using the BSSN formalism, with shift vector β i = 0 and the generalised slicing condition (2.94) with f ( α ) = 1 / , i.e., ∂ α = − α K, (2.141)which for FLRW gives the conformal time parameterisation, i.e. α = a .For the evolution of the hydrodynamical variables we use GRHydro , withPPM reconstruction (Colella and Woodward, 1984) and the HLLE Riemannsolver (Harten, Lax, and Van Leer, 1983; Einfeldt, 1988). In
EOS_Omni weuse the “Polytype” equation of state, specifically a polytrope with pressuredefined by P = K poly ρ , (2.142) https://camb.info Chapter 2. Methods since
GRHydro currently cannot handle P = 0 (dust). We choose K poly =0 . , which sufficiently satisfies P (cid:28) ρ . For this EOS, the internal energy isnot evolved, and is instead set directly from the rest-mass density, i.e. (cid:15) = ρ . We use periodic boundary conditions on a regular Cartesian mesh (i.e.,no adaptive mesh refinement), and use Runge-Kutta fourth order (RK4)time integration in the thorn MoL , with the condition ∆ t = 0 . x , where ∆ t, ∆ x are the time step and grid spacing, respectively. We expect second-order convergence of our solutions, since the spatial order of GRHydro issecond order. The full system of equations solved in our simulations issummarised in Table 2.1.Using initial conditions describing a flat, dust, FLRW spacetime, andseperate initial conditions describing small perturbations to this spacetime(see Section 1.4), in
FLRWSolver and evolving with the above setup, wematched the analytic evolution for the homogeneous scale factor and den-sity to within − , and linear perturbations to the density, velocity, andmetric to within − . We tested the convergence of our errors with increas-ing resolution, including the Hamiltonian and momentum constraints, andsaw the expected fourth-order convergence for FLRW (only time deriva-tives) and second-order convergence for the linear perturbations (both timeand space derivatives). For more details of the computational tests per-formed for this setup, see Chapter 3 (and Chapter 4 for more complex per-turbations). MESCALINE
The ET has some built in thorns specifically for analysis, which calculatequantities such as the trace of the extrinsic curvature, the Ricci scalar, thedeterminant of the spatial metric, the metric and extrinsic curvature in dif-ferent coordinates, while some thorns include routines for locating blackhole horizons and calculating constraint violation (Löffler et al., 2012; Zil-hão and Löffler, 2013). While these thorns are useful for generating out-put of physically interesting quantities while the simulation is running,to analyse General-Relativistic effects in inhomogeneous cosmology, thereare many more quantities that we are interested in. M
ESCALINE is a post-processing analysis code to read in three-dimensional C
ACTUS data in HDF5format and calculate quantities such as the spatial Ricci tensor and its trace,the trace of the extrinsic curvature, the expansion rate and shear of thefluid, and spatially-averaged quantities over both the entire volume and insub-domains within the volume. In
MESCALINE we adopt geometric units, G = c = 1 , and so for this section we present all equations in these units. Ricci tensor and connection functions
The spatial Ricci tensor is the contraction of the spatial Riemann curvaturetensor of the spatial surfaces, (2.8), R ij = R kikj = ∂ k Γ kij − ∂ j Γ kik + Γ klk Γ lij − Γ kjl Γ lik . (2.143) .3. Post-processing analysis: MESCALINE BSSN Equations Eq. number ddt φ = − αK (2.47) ddt K = α (cid:18) ¯ A ij ¯ A ij + 13 K (cid:19) − D α + 4 πGc α (cid:0) S + ρ c (cid:1) (2.52) ddt ¯ γ ij = − α ¯ A ij (2.53) ddt ¯ A ij = e − φ (cid:20) − ( D i D j α ) TF + α (cid:18) R TF ij − πGc S TF ij (cid:19)(cid:21) + α (cid:16) K ¯ A ij − A ik ¯ A kj (cid:17) (2.54) R TF ij = ¯ R ij −
13 ¯ γ ij ¯ R + ¯ R φij + 83 ¯ γ ij e − φ ¯ D e φ (2.59) ¯ R ij = −
12 ¯ γ lm ∂ m ∂ l ¯ γ ij + ¯ γ lm (cid:16) kl ( i ¯Γ j ) km + ¯Γ kim ¯Γ klj (cid:17) + ¯ γ k ( i ∂ j ) ¯Γ k + ¯Γ k ¯Γ ( ij ) k (2.61) ddt ¯Γ i = − A ij ∂ j α + 2 α (cid:18) ¯Γ ijk ¯ A kj −
23 ¯ γ ij ∂ j K (cid:19) + 23 ¯Γ i ∂ j β j − α (cid:18) πGc ¯ γ ij S j + 6 ¯ A ij ∂ j φ (cid:19) + 13 ¯ γ li ∂ l ∂ j β j + ¯ γ lj ∂ j ∂ l β i (2.71) Gauge conditions Eq. number β i = 0 N/A ∂ α = 13 α K (2.94) Equations of Hydrodynamics Eq. number ∂ D + ∂ i (cid:0) αD ˜ v i (cid:1) = 0 (2.104a), (2.105) ∂ S cj + ∂ i (cid:2) α (cid:0) S cj ˜ v i + δ ij P (cid:1)(cid:3) = T µν (cid:16) ∂ µ g νj − (4) Γ λµν g λj (cid:17) (2.104b), (2.105) ∂ τ c + ∂ i (cid:2) α (cid:0) τ c ˜ v i + P ˜ v i (cid:1)(cid:3) = α (cid:16) T µ ∂ µ ln α − T µν (4) Γ µν (cid:17) (2.104c), (2.105) T ABLE
INSTEIN T OOLKIT . Chapter 2. Methods In MESCALINE we calculate R ij directly from ET output using the spatialmetric and its spatial derivatives — via the spatial connection functions —with either a second-order or fourth-order approximation of the derivative.The Ricci scalar is then the trace of the Ricci tensor, R ≡ γ ij R ij . We cal-culate the trace of the extrinsic curvature K ≡ γ ij K ij using direct outputof K ij from the ET. When calculating the trace, we assume the rank-2, co-variant tensor in question is symmetric in its indices; true for all cases in MESCALINE .It is useful to write the time components of the four-dimensional con-nection functions in terms of purely spatial objects, some of which we useto calculate the expansion scalar θ . These are (4) Γ = − αK, (4) Γ i = 1 α ∂ i α, (4) Γ ij = − α K ij , (2.144a) (4) Γ i = αγ ij ∂ j α, (4) Γ i k = − γ ij αK kj , (2.144b)and the spatial components of the four-dimensional connection functionsare here equal to the spatial connection functions, since g ij = γ ij (always),and g i = 0 , g i = 0 , and g ij = γ ij for β i = 0 . Constraint violation
We calculate the violation of the Hamiltonian and momentum constraintequations via (2.15) and (2.17), respectively. The violation in the Hamilto-nian constraint is H ≡ R + K − K ij K ij − πρ, (2.145)where ρ is the total mass-energy density projected into the normal frame,which we can relate to the rest-mass density (as output from the ET, andtherefore read into MESCALINE ) via (2.74), which for dust gives, ρ = ρ R W = ρ (1 + (cid:15) ) W , (2.146)where we have used (1.14). For our chosen EOS we have (cid:15) = ρ (see Sec-tion 2.2.5), and so ρ = ρ W + ρ W , (2.147) ≈ ρ W , (2.148)since (in code units) we set ρ ≈ × − , and so ρ ≈ − (see Chapter 3).The violation in the momentum constraint is M i ≡ D j K ji − D i K − πS i , (2.149)where the momentum density is defined in (2.18), which we can write as S i = − γ ij n T j , (2.150)since n i = 0 and γ i = 0 . Using (1.7) with P = 0 , and four velocity (1.19)and (1.20), we find S i = γ ij v j W ρ R , (2.151)where ρ R is the projection of the stress-energy tensor into the rest frameof the fluid (1.12), and is related to the rest-mass density via (1.14). The .3. Post-processing analysis: MESCALINE M ≡ γ ij M i M j , andfor both violations we calculate the L error at each time, L ( H ) = 1 N N (cid:88) a =1 | H a | , (2.152) L ( M ) = 1 N N (cid:88) a =1 | M a | , (2.153)where N is the total number of grid cells, and H a , M a are the Hamiltonianviolation and magnitude of the momentum violation at grid cell a , respec-tively. The above L errors quantify the raw constraint violation in eachcase, but to calculate the relative violation we define the “energy scales”as the sum of the squares of the individual terms in each violation (as inMertens, Giblin, and Starkman, 2016), i.e. [ H ] ≡ (cid:113) R + ( K ) + ( K ij K ij ) + (16 πρ ) , (2.154) [ M ] ≡ (cid:114)(cid:16) D j K ji (cid:17) (cid:16) D j K ji (cid:17) + D i ( K ) D i ( K ) + (8 π ) S i S i , (2.155)and calculate the relative L violations as L ( H/ [ H ]) = N (cid:80) Na =1 | H a | N (cid:80) Na =1 [ H ] a , (2.156)and L ( M/ [ M ]) = N (cid:80) Na =1 | M a | N (cid:80) Na =1 [ M ] a . (2.157) Expansion scalar
We implement the generalised averaging scheme described in Section 1.5.2(Larena et al., 2009; Umeh, Larena, and Clarkson, 2011). To calculate thebackreaction terms Q h D (1.107) and L D (1.108), and hence the cosmologicalparameters, we first must calculate the expansion scalar, θ , the shear tensor σ µν , as well as the additional scalars and tensors related to the divergenceof the peculiar velocity field v i , at every coordinate point in the domain.The expansion scalar is defined as the divergence of the fluid four ve-locity projected into the surface defined by h µν , θ ≡ h µν ∇ µ u ν , (2.158) = h ij ∇ i u j , (2.159)since h µν is purely spatial and so h = 0 and h i = 0 . Expanding thecovariant derivative gives θ = h ij (cid:16) ∂ i u j − (4) Γ αij u α (cid:17) (2.160) = h ij (cid:16) ∂ i u j − (4) Γ ij u − Γ kij u k (cid:17) (2.161)8 Chapter 2. Methods since ∇ i is the spatial component of the covariant derivative associatedwith the metric g µν , the connection functions involved are still the four-dimensional connection functions, and hence we have a term involving thetime component (4) Γ ij , which we substitute from (2.144a). The expansionscalar is then θ = h ij ∂ i u j − W K − h ij Γ kij u k , (2.162)where we have also used (1.21). Shear tensor
The definition of the shear tensor is σ µν ≡ h αµ h βν ∇ ( α u β ) − θh µν , (2.163) ⇒ σ ij = ∇ ( i u j ) − θh ij , (2.164)i.e., it is purely spatial. Expanding the covariant derivative, and again using(2.144a), we find σ ij = ∂ ( i u j ) − W K ij − Γ kij u k − θh ij , (2.165)where we have again substituted (4) Γ ij from (2.144a). We then calculate therate of shear as σ ≡ σ ij σ ij . Other tensors
To be able to form the cosmological parameters we also need to calculate thescalars σ B (1.95) and θ B (1.92). The scalar θ B is built from the divergence ofthe peculiar velocity, κ (1.89), and the trace of the tensor B ij (1.96), whichitself depends on the tensor β ij (1.87). From their definitions, we can writethese quantities as κ = h ij ∂ i v j − h ij Γ kij v k , (2.166) β ij = ∂ ( i v j ) − Γ kij v k − κh ij , (2.167) B = 13 κv i v i + β ij v i v j , (2.168)where we have used the fact the peculiar velocity is purely spatial, i.e. v =0 . The scalar θ B is then θ B = − W κ − W B. (2.169)To calculate σ B we first must calculate the tensor σ Bij using (1.93), whichwe can write as σ Bij = − W β ij − W (cid:18) B ( ij ) − Bh ij (cid:19) (2.170)where we have B ij = 13 κv i v j + β ki v k v j + ∂ ( k v i ) v k v j . (2.171) .3. Post-processing analysis: MESCALINE Effective scale factors
We calculate the volume scale factor a V D using (1.100) and the definition ofthe volume element (1.97). To find the scale factor describing the expan-sion of the fluid, a h D , we follow Larena (2009) and use the expressions forthe Hubble parameter (1.101) and (1.99), along with the evolution of thevolume element (1.104), to get ∂ t a V D a V D − ∂ t a h D a h D = 13 (cid:104) A (cid:105) h , (2.172)where A ≡ αW ( θ − κ ) − αθ for simplicity . This implies ∂ t ln (cid:18) a V D a h D (cid:19) = 13 (cid:104) A (cid:105) h . (2.173)Using a second-order approximation for the time derivative, we have ln a V D a h D (cid:12)(cid:12)(cid:12) n − ln a V D a h D (cid:12)(cid:12)(cid:12) n − t ≈ (cid:104) A (cid:105) h | n − (2.174)where n , n − , and n − represent times t , t − ∆ t , and t − t , respectively.We therefore calculate the fluid scale factor from the volume scale factorusing a h D a V D (cid:12)(cid:12)(cid:12)(cid:12) n ≈ exp (cid:18) − t (cid:104) A (cid:105) h | n − (cid:19) a h D a V D (cid:12)(cid:12)(cid:12)(cid:12) n − . (2.175) As discussed previously, we calculate the expansion scalar and shear ten-sor at every cell in the computational domain. However, to investigate theeffect of inhomogeneities on the global evolution of the expansion rate andcosmological parameters, we must calculate averages of these quantities.The averaging domain D is entirely arbitrary, and choosing it depends onthe physical problem we are interested in. See Chapter 4 to see the resultsof averaging an inhomogeneous cosmological simulation with numericalrelativity using MESCALINE . Global averages
The simplest choice of the averaging domain is the entire computationalgrid. This is useful to study the large-scale (global) effects of backreaction,and to look into the potential for these effects to explain the acceleratingexpansion of the Universe; i.e. to explain dark energy (see Section 1.5.2).Global averages, however, can be susceptible to boundary problems sincethe boundary is included in the averaging process. It has not yet beenstudied how the use of periodic boundary conditions affects the size of the We note an error in Larena (2009) in this relation, which is corrected with − κ → θ B in A . We have not corrected it here since we use this form in our analysis, however, seeAppendix F for a re-analysis with the error fixed. Chapter 2. Methods backreaction effect measured globally in full GR simulations. In Newto-nian simulations with periodic boundary conditions, the global backreac-tion vanishes identically (see Buchert and Ehlers, 1997; Buchert, 2018). Wediscuss this more in Chapter 6.
Subdomain averaging
Rather than studying the global dynamics, we can analyse the averageproperties on smaller scales using subdomains located within the compu-tational domain. This will largely remove any spurious boundary effects,simply by not including the boundary in the averaging. In
MESCALINE we perform this subdomain averaging using an arbitrary number, N , ofrandomly placed spherical domains of an arbitrary radius, r D , within theglobal domain. The number of spheres and each sphere’s radius (whereall N spheres are given the same radius) are specified by the user beforecompiling. From these, we randomly generate N sets of x i = ( x, y, z ) co-ordinates lying within the computational grid, representing the origins ofeach individual sphere. These origins are generated such that the edge ofthe outermost sphere is not allowed to exit the computational domain, andthe spheres are allowed to overlap with one another.An issue with the averaging formalism used here, briefly discussed inSection 1.5.2, is that the domain D does not conserve mass during evolu-tion. That is, if we choose to calculate averages within subdomains as afunction of time, the mass contained within these domains is free to moveinto and out of the sphere itself. This issue is present in the averaging in MESCALINE because the coordinate positions of the sphere’s origins stayfixed during the evolution, and are not propagated along with the fluidflow. In order to address this, not only the origin of the sphere but the edgesof the sphere also need to be propagated along the fluid four-velocity vec-tor, i.e., we would need to allow the sphere to deform as the fluid evolves.The computational overhead of this propagation is outside the scope of thisthesis, and so long as the velocities in the simulation are v i (cid:28) c (as is thecase here), the approximation of stationary origins is sufficiently valid. In writing
MESCALINE we have adopted a few key assumptions to simplifythe code, namely:1. Regular Cartesian grid,2. Zero shift vector,3. Periodic boundary conditions,4. Matter-dominated fluid, i.e. P = 0 ,5. Geometric units, i.e. G = c = 1 ,each of which are common choices in cosmological simulations with nu-merical relativity (see e.g. Bentivegna and Bruni, 2016; Giblin, Mertens, andStarkman, 2016a; Macpherson, Lasky, and Price, 2017). The assumption .3. Post-processing analysis: MESCALINE ∂f∂x i ≈ − f ( x i + 2∆ x i ) + 8 f ( x i + ∆ x i ) − f ( x i − ∆ x i ) + f ( x i − x i )12∆ x i (2.176)where we set ∆ x i = ∆ x = ∆ y = ∆ z .Assuming a zero shift vector, i.e. β i = 0 , simplifies the expressions forthe time components of the four-dimensional connection functions, and thetime derivative (2.20), since for β i = 0 we have L β = 0 and hence ddt = ∂ t (both used in calculating the backreaction terms). It also simplifies (1.20)relating the four velocity and the Eulerian three velocity, the latter of whichis output from the ET. In addition, adopting a zero shift affects the waywe raise and lower indices of four-dimensional objects, i.e. for the spatialcomponents of the four velocity, u i = g iµ u µ , (2.177) = g i u + g ij u j , (2.178) = g ij u j . (2.179)Periodic boundary conditions are implemented in our approximation ofspatial derivatives only, and are the most reasonable choice for cosmologi-cal simulations without needing to simulate the entire past light cone of anobserver.In MESCALINE we assume the matter content is dust ( P = 0 ) in relatingdensities ρ and ρ measured in the normal frame and fluid rest-frame, re-spectively, and in calculating the momentum density, S i , for the momentumconstraint violation. However, since GRHydro is not equipped for P = 0 ,we instead use P (cid:28) ρ in the simulations themselves (see Section 2.2.5),which we found to be sufficient to match the dust evolution for FLRW andsmall perturbations to this background (see Chapter 3).3 Chapter 3
Inhomogeneous cosmologywith numerical relativity
Published in:Macpherson, Lasky, and Price (2017). Physical Review D, . Abstract
We perform three-dimensional numerical relativity simulations of homo-geneous and inhomogeneous expanding spacetimes, with a view towardsquantifying non-linear effects from cosmological inhomogeneities. We demon-strate fourth-order convergence with errors less than one part in inevolving a flat, dust Friedmann-Lemaître-Roberston-Walker (FLRW) space-time using the E INSTEIN T OOLKIT within the C
ACTUS framework. We alsodemonstrate agreement to within one part in between the numericalrelativity solution and the linear solution for density, velocity and metricperturbations in the Hubble flow over a factor of ∼ change in scalefactor (redshift). We simulate the growth of linear perturbations into thenon-linear regime, where effects such as gravitational slip and tensor per-turbations appear. We therefore show that numerical relativity is a viabletool for investigating nonlinear effects in cosmology. A note on notation
We have altered the notation throughout this chapter, including Appendix A,to be consistent with Chapters 1 and 2, unless explicitly stated otherwise.For these exceptions, we maintain the notation of the publication for consis-tency with figures in their published form. Aside from these changes, thischapter is consistent with the accepted version of Macpherson, Lasky, andPrice (2017).4
Chapter 3. Inhomogeneous cosmology with numerical relativity
Modern cosmology relies on the cosmological principle — that the Universeis sufficiently homogeneous and isotropic on large scales to be described bya Friedmann-Lemaître-Robertson-Walker (FLRW) model. Cosmological N-body simulations (e.g. Genel et al., 2014; Springel et al., 2005; Kim et al.,2011) encode these assumptions by prescribing the expansion to be that ofthe FLRW model, governed by the Friedmann equations, while employinga Newtonian approximation for gravity.The transition to cosmic homogeneity begins on scales ∼ h − Mpc e.g.Yadav, Bagla, and Khandai, 2010; Scrimgeour et al., 2012, but is inhomoge-neous and anisotropic on smaller scales. Upcoming cosmological surveysutilising Euclid, the Square Kilometre Array (SKA) and the Large Synop-tic Survey Telescope (LSST) (Amendola, Appleby, and Avgoustidis, 2016;Maartens et al., 2015; Ivezic, Tyson, and Abel, 2008) will reach a precisionat which nonlinear General-Relativistic effects from these inhomogeneitiescould be important. A more extreme hypothesis (Räsänen, 2004; Kolb etal., 2005; Kolb, Matarrese, and Riotto, 2006; Notari, 2006; Räsänen, 2006b;Räsänen, 2006a; Li and Schwarz, 2007; Li and Schwarz, 2008; Larena et al.,2009; Buchert et al., 2015; Green and Wald, 2016; Bolejko and Lasky, 2008) isthat such inhomogeneities may provide an alternative explanation for theaccelerating expansion of the Universe, via backreaction (see Buchert, 2008;Buchert and Räsänen, 2012, for a review), replacing the role assigned todark energy in the standard Λ CDM model (Riess et al., 1998; Perlmutteret al., 1999; Parkinson et al., 2012; Samushia et al., 2013).Quantifying the General-Relativistic effects associated with nonlinearstructures ultimately requires solving Einstein’s equations. Post-Newtonianapproximations are a worthwhile approach (Matarrese and Terranova, 1996;Räsänen, 2010; Green and Wald, 2011; Green and Wald, 2012; Adamek et al.,2013; Adamek et al., 2016b; Adamek et al., 2016a; Sanghai and Clifton, 2015;Oliynyk, 2014; Noh and Hwang, 2004), however the validity of these mustbe checked against a more precise solution since the density perturbationsthemselves are highly nonlinear.An alternative approach is to use numerical relativity, which has en-joyed tremendous success over the past decade (Pretorius, 2005a; Campan-elli et al., 2006; Baker et al., 2006).Cosmological modelling with numerical relativity began with evolu-tions of planar and spherically symmetric spacetimes using the Arnowitt-Deser-Misner (ADM) formalism (Arnowitt, Deser, and Misner, 1959), in-cluding Kasner and matter-filled spacetimes (Centrella and Matzner, 1979),the propagation and collision of gravitational wave perturbations (Cen-trella, 1980; Centrella and Matzner, 1982) and linearised perturbations to ahomogeneous spacetime (Centrella and Wilson, 1983; Centrella and Wilson,1984). More recent work has continued to include symmetries to simplifythe numerical calculations (e.g. Rekier, Cordero-Carrión, and Füzfa, 2015;Torres et al., 2014).Simulations free of these symmetries have only emerged within the lastyear. Giblin, Mertens, and Starkman (2016a) studied the evolution of smallperturbations to an FLRW spacetime, exploring observational implicationsin (Giblin, Mertens, and Starkman, 2016b). Bentivegna and Bruni (2016) .2. Numerical Method
INSTEIN T OOLKIT , based on the C
ACTUS infrastructure (Löffler et al., 2012; Zilhãoand Löffler, 2013). We benchmark our three-dimensional numerical imple-mentation on two analytic solutions of Einstein’s equations relevant to cos-mology: FLRW spacetime and the growth of linear perturbations. We alsopresent the growth of perturbations into the nonlinear regime, and analysethe resulting gravitational slip (Daniel et al., 2008; Daniel et al., 2009) andtensor perturbations.In Section 3.2 we describe our numerical methods, including choices ofgauge (3.2.1) and an overview of the derivations of the linearly perturbedEinstein equations used for our initial conditions (3.2.2). In Section 3.3 wedescribe the setup (3.3.1) and results (3.3.2) of our evolutions of a flat, dustFLRW universe. The derivation of initial conditions for linear perturba-tions to the FLRW model are described in 3.4.1, with results presented in3.4.2. The growth of the perturbations to nonlinear amplitude is presentedin 3.5, with analysis of results and higher order effects in 3.5.1 and 3.5.2 re-spectively. In this chapter, we adopt geometric units with G = c = 1 , Greekindices run from 0 to 3 while Latin indices run from 1 to 3, with repeatedindices implying summation. We integrate Einstein’s equations with the E
INSTEIN T OOLKIT , a free, open-source code for numerical relativity (Löffler et al., 2012). This utilises theC
ACTUS infrastructure, consisting of a central core, or “flesh”, with appli-cation modules called “thorns” that communicate with this flesh (Goodaleet al., 2003). The E
INSTEIN T OOLKIT is a collection of thorns for com-putational relativity, used extensively for simulations of binary neutronstar and black hole mergers (e.g Kastaun and Galeazzi, 2015; Radice, Rez-zolla, and Galeazzi, 2015; Baiotti et al., 2005). Numerical cosmology withthe E
INSTEIN T OOLKIT is a new field (Bentivegna and Bruni, 2016). Weuse the
McLachlan code (Brown et al., 2009) to evolve spacetime usingthe Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism (Shibata andNakamura, 1995; Baumgarte and Shapiro, 1999), and the
GRHydro codeto evolve the hydrodynamical system (Baiotti et al., 2005; Giacomazzo andRezzolla, 2007; Mösta et al., 2014); a new setup for cosmology with the E IN - STEIN T OOLKIT .We use the fourth-order Runge-Kutta method, adopt the Marquina Rie-mann solver and use the piecewise parabolic method for reconstructionon cell interfaces.
GRHydro is globally second order in space due to the6
Chapter 3. Inhomogeneous cosmology with numerical relativity coupling of hydrodynamics to the spacetime (Hawke, Löffler, and Nerozzi,2005; Mösta et al., 2014). We therefore expect fourth-order convergence ofour numerical solutions for the spatially homogeneous FLRW model. Onceperturbations are introduced to this model we expect our solutions to besecond-order accurate.We have developed a new thorn,
FLRWSolver , to initialise an FLRWcosmological setup with optional linear perturbations. We evolve our sim-ulations in a cubic domain on a uniform grid with periodic boundary con-ditions with x i in [-240,240]. Our domain sizes are , and , respec-tively using 70 (8 cores), 380 (8 cores) and 790 (16 cores) CPU hours. The gauge choice corresponds to a choice of the lapse function, α , and shiftvector, β i . The metric written in the (3 + 1) formalism is ds = − α dt + γ ij ( dx i + β i dt )( dx j + β j dt ) , (3.1)where γ ij is the spatial metric. Previous cosmological simulations withnumerical relativity adopt the synchronous gauge, corresponding to α =1 , β i = 0 (Giblin, Mertens, and Starkman, 2016a; Bentivegna and Bruni,2016). We instead utilise the general spacetime foliation of Bona et al. (1995), ∂ t α = − α f ( α ) K, (3.2)where f ( α ) > is an arbitrary function, and K = γ ij K ij . We set the shiftvector β i = 0 . Harmonic slicing uses f = const., while f = 1 /α correspondsto the “1+log” slicing common in black hole binary simulations. We choose f = 0 . to maintain the stability of our evolutions, as in Torres et al. (2014).This allows for longer evolutions for the same computational time, com-pared to “1+log” slicing, due to the increased rate of change of the lapse.We adopt this gauge for numerical convenience, and acknowledge possiblealternative methods include using synchronous gauge with adaptive time-stepping.We use (3.2) for evolution only. We scale to the gauge described in thenext section for analysis. Bardeen’s formalism of cosmological perturbations (Bardeen, 1980) was de-veloped with the intention to connect metric perturbations to physical per-turbations in the Universe. This connection is made clear by defining theperturbations as gauge-invariant quantities in the longitudinal gauge.The general line element of a perturbed, flat FLRW universe, includingscalar ( Φ , Ψ ), vector ( B i ) and tensor ( h ij ) perturbations takes the form ds = a ( η )[ − (1 + 2Ψ) dη − B i dx i dη + (1 − δ ij dx i dx j + h ij dx i dx j ] , (3.3)where η is conformal time, a ( η ) is the FLRW scale factor and δ ij is the iden-tity matrix. We derive initial conditions from the linearly perturbed Ein-stein equations, implying negligible vector and tensor perturbations (Adamek .2. Numerical Method ds = a ( η )[ − (1 + 2Ψ) dη + (1 − δ ij dx i dx j ] , (3.4)where Φ and Ψ here coincide with Bardeen’s gauge-invariant scalar poten-tials (Bardeen, 1980). Here we see that Ψ , the Newtonian potential, willlargely influence the motion of non-relativistic particles; where the time-time component of the metric dominates the motion. The Newtonian po-tential plays the dominant role in galaxy clustering. Relativistic particleswill also be affected by the curvature potential Φ , and so both potentialsinfluence effects such as gravitational lensing (Bertschinger, 2011; Bardeen,1980).The metric perturbations are coupled to perturbations in the matter dis-tribution via the stress-energy tensor. We approximate the homogeneousand isotropic background as a perfect fluid in thermodynamic equilibrium,giving T µν = ( ρ + P ) u µ u ν + P g µν , (3.5)where ρ is the total energy density , P is the pressure and u µ is the four-velocity of the fluid. We assume a dust universe, implying negligible pres-sure ( P (cid:28) ρ ), and we solve the perturbed Einstein equations, δG µν = 8 π δT µν , (3.6)using linear perturbation theory. From the time-time, time-space, trace andtrace free components of (3.6), we obtain the following system of equations(Sachs and Wolfe, 1967; Adamek et al., 2013) ∂ Φ − H (cid:0) Φ (cid:48) + H Ψ (cid:1) = 4 π ¯ ρ δa , (3.7a) H ∂ i Ψ + ∂ i Φ (cid:48) = − π ¯ ρ a δ ij δv j , (3.7b) Φ (cid:48)(cid:48) + H (cid:0) Ψ (cid:48) + 2Φ (cid:48) (cid:1) = 13 ∂ (Φ − Ψ) , (3.7c) ∂ (cid:104) i ∂ j (cid:105) (Φ − Ψ) = 0 . (3.7d)Here H ≡ a (cid:48) /a is the Hubble parameter, ∂ i ≡ ∂/∂x i , ∂ ≡ ∂ i ∂ i , ∂ (cid:104) i ∂ j (cid:105) ≡ ∂ i ∂ j − / δ ij ∂ , and (cid:48) represents a derivative with respect to conformal time, η . The quantity | Φ − Ψ | is known as the gravitational slip (Daniel et al., 2008;Daniel et al., 2009; Bertschinger, 2011), which is zero in the linear regimeand in the absence of anisotropic stress. At higher orders in perturbationtheory, the gravitational slip is non-zero, and Φ (cid:54) = Ψ (see e.g. Ballesteroset al., 2012). These perturbations are equivalent to φ and ψ used in Chapter 1 and Section 2.2.4. The energy-density ρ used throughout this Chapter is the total rest-frame energy-density,and is equivalent to ρ R used in Chapters 1 and 2. Chapter 3. Inhomogeneous cosmology with numerical relativity
We perturb the density and coordinate three velocity by making thesubstitutions ρ = ¯ ρ (1 + δ ) , (3.8a) v i = δv i , (3.8b)where ¯ ρ represents the background FLRW density, and ¯ v i = 0 . We de-rive the relativistic fluid equations from the components of the energy-momentum conservation law, ∇ α T αµ = 0 , (3.9)where ∇ α is the covariant derivative associated with the 4-metric. The re-sulting continuity and Euler equations are, ∂ t δ = 3 ∂ t Φ − ∂ i v i , (3.10a) H v i + ∂ t v i = − ∂ i Ψ . (3.10b) a / a i n it Time ( η ) − − − − a / a F L R W − − − − ρ / ρ i n it ExactNumerical Time ( η ) − − − − ρ / ρ F L R W − F IGURE a (left)and the density, ρ (right), relative to their initial values a init and ρ init , as a function of conformal time η . Bottom: errorsin the FLRW scale factor (left) and density (right) at domainsizes , and . We test our thorn
FLRWSolver together with the E
INSTEIN T OOLKIT ontwo analytic solutions to Einstein’s equations relevant to cosmology. Ourfirst and simplest test is the flat, dust FLRW model. Here we initialisea homogeneous and isotropic matter distribution and spatial metric, andevolve in the gauge outlined in Section 3.2.1. While the E
INSTEIN T OOLKIT The velocity used throughout this Chapter is v i ≡ dx i /dt , which differs from the veloc-ity used in Chapter 2 via (2.137). .3. FLRW spacetime
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Resolution ( N ) L e rr o r N a
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Resolution ( N ) N r
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Resolution ( N ) N H c F IGURE L error as a function of resolution for thescale factor (left), density (middle), and Hamiltonian con-straint (right). N refers to the number of grid points alongone spatial dimension. Filled circles indicate data pointsfrom our simulations, dashed lines join these points, andblack solid lines indicate the expected N − convergence. has been previously tested on FLRW and Kasner cosmologies (Löffler etal., 2012; Vulcanov and Alcubierre, 2002), this is an important first test of FLRWSolver and its interaction with the evolution thorns.
The line element for a spatially homogeneous and isotropic FLRW space-time is given by ds = − dt + a ( t ) (cid:20) dr − kr + r (cid:0) dθ + sin θdφ (cid:1)(cid:21) , (3.11)where k = − , , if the universe is open, flat or closed respectively. As-suming homogeneity and isotropy Einstein’s equations reduce to the Fried-mann equations (Friedmann, 1922; Friedmann, 1924), (cid:18) a (cid:48) a (cid:19) = 8 πρ a − k, (3.12a) ρ (cid:48) = − a (cid:48) a ( ρ + P ) . (3.12b)In the remainder of this chapter we assume a flat spatial geometry, sup-ported by combined Planck and Baryon Acoustic Oscillation data (PlanckCollaboration et al., 2016). The flat ( k = 0 ), dust ( P (cid:28) ρ ) solution to (3.12)is aa init = ξ , ρρ init = ξ − , (3.13)where a init , ρ init are the values of a, ρ at η = 0 respectively, and we haveintroduced the scaled conformal time coordinate ξ ≡ (cid:114) πρ ∗ a init η, (3.14)0 Chapter 3. Inhomogeneous cosmology with numerical relativity where ρ ∗ = ρ a is the conserved (constant) comoving density for an FLRWuniverse. The familiar ˆ t / solution for the scale factor arises in the Newto-nian gauge with ds = − d ˆ t + γ ij dx i dx j (for a flat spacetime; see AppendixA). We initialise a homogeneous and isotropic matter distribution by speci-fying constant density ρ init = 10 − and zero velocity in FLRWSolver , with a init = 1 . The E INSTEIN T OOLKIT then initialises the stress-energy ten-sor, coupled to our homogeneous and isotropic spacetime, characterisedby the spatial metric, γ ij = a ( η ) δ ij , and extrinsic curvature, also set in FLRWSolver . We define the extrinsic curvature via the relation ddt γ ij = − αK ij , (3.15)where d/dt ≡ ∂/∂t − L β , and L β is the Lie derivative with respect to theshift vector. Since we choose β i = 0 , we have d/dt = ∂/∂t . The extrinsiccurvature for our FLRW setup is therefore K ij = − ∂ t ( a ) aα δ ij . (3.16)We evolve the system until the domain volume has increased by one mil-lion, corresponding to a change in redshift of ∼ .To analyse our results we scale the time from the metric (3.1) to the lon-gitudinal gauge (3.4) using the coordinate transform t = t ( η ) . This gives dtdη = a ( η ) α ( t ) , (3.17)which we integrate to find the scaled conformal time in terms of t to be ξ ( t ) = (cid:18)(cid:112) πρ init (cid:90) α ( t ) dt + 1 (cid:19) / , (3.18)where we numerically integrate the lapse function α using the trapezoidalrule. This coordinate transformation allows us to simulate longer evolu-tions for less computational time, while still performing our analysis in thelongitudinal gauge to extract physically meaningful results. Figure 3.1 compares our numerical relativity solutions with the exact so-lutions to the Friedmann equations. The top panels show the time evo-lution of a and ρ (dashed magenta curves) relative to their initial values,which may be compared to the exact solutions, a FLRW and ρ FLRW (blacksolid curves). The bottom panels show the residuals in our numerical so-lutions at resolutions of , and . The error can be seen to decreasewhen the spatial resolution is increased. The increase in spatial resolutioncauses the time step to decrease via the Courant condition. To quantify this,we compute the L error, given by (e.g. for the scale factor) L ( a ) = 1 n n (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12) aa F LRW − (cid:12)(cid:12)(cid:12)(cid:12) , (3.19) .4. Linear Perturbations n is the total number of time steps. As outlined in Section 3.2, weexpect fourth-order convergence due to the spatial homogeneity. Figure 3.2demonstrates this is true for the scale factor (left), density (middle) and theHamiltonian constraint (right), H ≡ R + K − K ij K ij − πρ = 0 , (3.20)where R is the 3-Riemann scalar and K = γ ij K ij . For the FLRW modelthis reduces to the first Friedmann equation (3.12a).The results of this test demonstrate that the E INSTEIN T OOLKIT , in con-junction with our initial-condition thorn
FLRWSolver , produces agreementwith the exact solution for a flat, dust FLRW spacetime, with relative errorsless than − , even at low spatial resolution ( ). − − − δ ρ / ¯ ρ Time ( η ) − − − δ / δ ex a c t − − δ v ExactNumerical Time ( η ) − − − δ v / δ v e x ac t − F IGURE η ) evo-lution of the fractional density perturbation (top left) andthe velocity perturbation (top right) computed from one-dimensional slices along the x axis of our domain. Bottom:relative errors for calculations at , and . For our second test we introduce small perturbations to the FLRW model.The evolution of these perturbations in the linear regime can be found bysolving the system of equations (3.7). We use these solutions (derived be-low) to set the initial conditions.
In the absence of anisotropic stress we have
Ψ = Φ . Equation (3.7c) thenbecomes purely a function of Φ and the FLRW scale factor a . Solving this The density in (3.20) is technically the density observed in the normal frame, equivalentto ρ in Chapter 2. However, for dust in FLRW and linear perturbation theory, we have ρ = ρ R , see (2.74). Chapter 3. Inhomogeneous cosmology with numerical relativity
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Resolution ( N ) − − L e rr o r N − δ
20 30 40 50 60 70 80
Resolution ( N ) − N − δ v F IGURE L errors in the density (left)and velocity perturbations (right). N refers to the numberof grid points along one spatial dimension. Filled circlesindicate data points from our simulations, dashed lines jointhese points, and black solid lines indicate the expected N − convergence. gives Φ = f ( x i ) − g ( x i )5 ξ , (3.21)where f, g are functions of only the spatial coordinates. We substitute (3.21)into the Hamiltonian constraint, Equation (3.7a), to give the fractional den-sity perturbation δ ≡ δρ/ ¯ ρ , in the form δ = C ξ ∂ f ( x i ) − f ( x i ) − C ξ − ∂ g ( x i ) − ξ − g ( x i ) , (3.22)where we have defined C ≡ a init πρ ∗ , C ≡ a init πρ ∗ . (3.23)Using the momentum constraint, Equation (3.7b), the velocity perturbation δv i is therefore δv i = C ξ ∂ i f ( x i ) + 310 C ξ − ∂ i g ( x i ) (3.24)where we have C ≡ − (cid:114) a init πρ ∗ . (3.25)Equation (3.22) demonstrates both a growing and decaying mode for thedensity perturbation (Bardeen, 1980; Mukhanov, Feldman, and Branden-berger, 1992). We set g ( x i ) = 0 to extract only the growing mode, giving .4. Linear Perturbations Φ = f ( x i ) , (3.26a) δ = C ξ ∂ f ( x i ) − f ( x i ) , (3.26b) δv i = C ξ ∂ i f ( x i ) , (3.26c)from which we set our initial conditions. We choose Φ = Φ (cid:88) i =1 sin (cid:18) πx i L (cid:19) , (3.27)where L is the length of one side of our computational domain. We requirethe amplitude Φ (cid:28) so that our assumptions of linearity are valid, and sowe set Φ = 10 − . This choice then sets the form of our density and velocityperturbations, as per (3.26b) and (3.26c). At η = 0 ( ξ = 1 ) these are, δ = − (cid:34)(cid:18) πL (cid:19) C + 2 (cid:35) Φ (cid:88) i =1 sin (cid:18) πx i L (cid:19) , (3.28) δv i = 2 πC L Φ cos (cid:18) πx i L (cid:19) , (3.29)and the choice of Φ results in amplitudes of δ ∼ − and δv i ∼ − .We set these matter perturbations in FLRWSolver , implementing negligi-ble pressure and again using (3.15) to define the extrinsic curvature. For alinearly perturbed FLRW spacetime with
Ψ = Φ and Φ (cid:48) = 0 we have K ij = − ∂ t ( a ) aα (1 − δ ij . (3.30)We evolve these perturbations in the harmonic gauge until the volume ofthe domain has increased by 125 million, (∆ a ) ∼ . × , correspondingto a factor of 500 change in redshift. Dashed magenta curves in Figure 3.3 show the conformal time evolutionof the fractional density perturbation, δ ≡ δρ/ ¯ ρ (top left), and the velocityperturbation, δv (top right). Solid black curves show the solutions (3.26b)for δ exact and (3.26c) for δv exact . Bottom panels show the relative errorsfor three different resolutions. Figure 3.4 shows the L error as a functionof resolution, demonstrating the expected second-order convergence. Fig-ure 3.5 shows the Hamiltonian (top), and momentum (bottom) constraintsas a function of conformal time at our three chosen resolutions. The Hamil-tonian constraint was defined in Equation (3.20). For our linearly perturbedFLRW spacetime this reduces to Equation (3.7a). The momentum constraintis M i ≡ D j K ji − D i K − S i = 0 , (3.31)where D j is the covariant derivative associated with the 3-metric, and thematter source S i = − γ iα n β T αβ , with n β the normal vector (Baumgarte andShapiro, 1999). For linear perturbations this constraint reduces to Equation(3.7b).4 Chapter 3. Inhomogeneous cosmology with numerical relativity Time ( η ) − − − − H a m ilt on i a n Time ( η ) − − − M o m e n t u m F IGURE η at resolutions , and . Figure 3.5 shows a better preservation of the Hamiltonian constraintwith increasing resolution. The momentum constraint shows the opposite.We attribute this to the momentum constraint being preserved to of orderthe roundoff error, which will become larger with an increase in resolution.Even at the highest resolution the momentum constraint is preserved towithin − .This second test has demonstrated a match to within ∼ − of our nu-merical relativity solutions to the exact solutions for the linear growth ofperturbations, while exhibiting the expected second-order convergence. In order to evolve our perturbations to nonlinear amplitude in a reason-able computational time, we increase the size of our initial perturbationsto Φ = 10 − , which in turn gives δ ∼ − and δv i ∼ − . The linearapproximation remains valid.We choose the starting redshift to be that of the cosmic microwave back-ground (CMB). That is, we set z = 1000 , such that our initial density pertur-bation is roughly consistent with the amplitude of temperature fluctuationsin the CMB ( ∼ − ) (Bennett et al., 2013). We emphasise that this redshift,and all redshifts shown in figures, should not be taken literally; its purpose .5. Nonlinear evolution
100 0 100 200 d r / ¯ r ( ⇥ ) h = z =
100 0 100 200 h = . ⇥ z ⇡
100 0 100 200 h = . ⇥ z ⇡
100 0 100 200 h = . ⇥ z ⇡
100 0 100 200 x d v ( ⇥ )
100 0 100 200 x
100 0 100 200 x
100 0 100 200 x F IGURE domainduring the conformal time ( η ) evolution. All quantities areshown in code units. Dashed magenta curves show ournumerical solutions, and black solid curves show the ex-act solutions for the linear regime. Initial data ( η = 0 ; firstcolumn) and η = 1 . × (second column) match lin-ear theory. We see a clear deviation from linear theory at η = 6 . × (third column) and η = 1 . × (fourth col-umn). Simulation redshifts are shown as an indicator of the change in redshift.F IGURE η = 0 and η = 1 . × . All quantities are shown in code units forour simulation. Grey dashed lines indicate the positionof the one-dimensional slices shown in Figure 3.6. is to assign an approximate change in redshift, calculated directly from theFLRW scale factor. Figure 3.6 shows a series of one-dimensional slices through the origin of the y and z axes at four different times. Dashed magenta curves show solutionsfor the density (top row) and velocity (bottom row) perturbations, which6 Chapter 3. Inhomogeneous cosmology with numerical relativity Time ( η ) − − δ ρ / ¯ ρ Exact (linear)Numerical 10 Time ( η ) − − δ v F IGURE η (in code units), while black solid curves showthe analytic solutions for linear growth. Here we show thesimulation with domain size . Blue circles represent thetimes of the ( η > ) panels shown in Figure 3.6. may be compared to the black solid curves showing the analytic solutionsfor linear perturbations. At η = 0 and η = 1 . × (first and secondcolumns respectively) the solutions are linear, while at η = 6 . × (thirdcolumn) both the density and velocity perturbations deviate from lineartheory. The perturbations are nonlinear at η = 1 . × (fourth column)where matter collapses towards the overdensity, indicated by the shift inthe maximum velocity.The final column shows an apparent decrease in the average density.This is simply an artefact of taking a one-dimensional slice through a three-dimensional box. Figure 3.7 shows the column-density perturbation, δ col ,computed by integrating the density perturbation along the z axis. Panelsshow η = 0 and η = 1 . × respectively. The right panel shows an increaseof ∼ times in the column-density perturbation at x, y ≈ − , . Acorresponding void can be seen in the lower right of Figure 3.7, explainingthe underdensity along the y axis seen in the final column of Figure 3.6.Figure 3.8 shows the maximum value of the density (left) and velocity(right) perturbations as a function of time. Dashed magenta curves showthe numerical solutions, which may be compared to the black curves show-ing the linear analytic solutions. Perturbations can be seen to deviate fromthe linear approximation at η ≈ × , when δρ/ ¯ ρ ≈ . . At η ≈ ,the maximum of the density and velocity perturbations have respectivelygrown 25 and 2 times larger than the linear solutions. Gravitational slip is defined as the difference between the two potentials Φ and Ψ (Daniel et al., 2008; Daniel et al., 2009), which is zero in the linearregime, see equation (3.7d), but nonzero in the nonlinear regime (see e.g.Ballesteros et al., 2012).We reconstruct Φ and Ψ from the metric components, although we notethe interpretation of these potentials becomes unclear in the nonlinear regime. .5. Nonlinear evolution γ ij = a [(1 − δ ij + h ij ] , (3.32)and we adopt the traceless gauge condition δ ij h ij = 0 (Green and Wald,2012; Adamek et al., 2013). The potential Φ is then Φ = 12 (cid:18) − δ ij γ ij a (cid:19) , (3.33)which holds for all times the metric (3.3) applies. The potential Ψ is morecomplicated: our gauge choice implies lapse evolution according to (3.2),where we have set f ( α ) = 1 / , and K = − ∂ t aaα , (3.34)in the linear regime, which gives ∂ t α = 34 ∂ t ( a ) αa . (3.35)Integrating this results in a lapse evolution of αα init = D ( x i ) (cid:18) aa init (cid:19) / , (3.36)where D ( x i ) is a function of our spatial coordinates. According to the met-ric (3.4), and with α init = a init = 1 this implies α = √ a / , (3.37)from which we reconstruct the potential Ψ to be Ψ = 12 (cid:20)(cid:16) αa / (cid:17) − (cid:21) , (3.38)valid in the linear regime. Our gauge choice β i = 0 implies that in thenonlinear regime we expect additional modes to be present in this recon-struction of Ψ .We use an FLRW simulation for the scale factor a in (3.33) and (3.38),from which we calculate the gravitational slip | Φ − Ψ | . This is potentiallyproblematic once the perturbations become nonlinear, as the gauges of thetwo simulations will differ. Figure 3.9 shows one-dimensional slices of thegravitational slip at the same times as was shown in Figure 3.6. Dashedcurves show the numerical results, with black lines showing the linear solu-tion; zero gravitational slip. In the fourth panel ( η = 1 . × ) we see a posi-tive shift of the gravitational slip to ≈ × − for this one-dimensional slice,with the maximum value in the three-dimensional domain being . × − at this time. The Newtonian potential Ψ has a positive average value at η = 1 . × , due to the majority of the domain being underdense (seeFigure 3.7), and the potential Φ takes a negative average value. This can beinterpreted as an overall positive contribution to the expansion, from themetric (3.3).8 Chapter 3. Inhomogeneous cosmology with numerical relativity
100 0 100 200 x | F Y | ( ⇥ ) h = z =
100 0 100 200 x h = . ⇥ z ⇡
100 0 100 200 x h = . ⇥ z ⇡
100 0 100 200 x h = . ⇥ z ⇡ F IGURE domain, while black solid lines showthe solution for linear perturbation theory; zero. The po-tentials Φ and Ψ are reconstructed according to (3.33) and(3.38) respectively. Initial data ( η = 0 ) is shown in the leftcolumn, and time increases towards the right as indicatedby timestamps. We show the simulation redshift as an in-dicator of the approximate change in redshift only, and allquantities here are shown in code units. Relativistic corrections to one-dimensional N-body simulations in (Adameket al., 2013) resulted in a gravitational slip of × − . We show a gravi-tational slip of the same amplitude, including the full effects of GeneralRelativity in a three-dimensional simulation, for a time when our densityperturbation is comparable in size to that of Adamek et al. (2013). Gravita-tional slip is a measurable effect that can be quantified by combining weakgravitational lensing and galaxy clustering (Bertschinger, 2011). Our sim-ulations show tentative evidence for the importance of gravitational slipdue to nonlinear gravitational effects. However, robust predictions requirehigher resolution simulations with more realistic initial conditions.In our initial conditions we neglected vector and tensor perturbationsin the perturbed FLRW metric (3.3), since in the linear regime the scalarperturbations dominate. These higher order perturbations appear in thenonlinear regime. The tensor perturbation can be extracted from the off-diagonal, spatial components of the metric, γ ij = a h ij for i (cid:54) = j, (3.39)however, details of these tensor modes may be dependent on the choice ofgauge. We calculate h ij using the value of a as per the scalar perturbations.Figure 3.10 shows a two-dimensional cross-section of the xy componentof the tensor perturbation h ij . All other components are identical. Thecross-section is shown at η = 1 . × , corresponding to the right panel ofFigures 3.6, 3.7 and 3.9. While the maximum amplitude of the tensor pertur-bation is small ( ∼ × − ), an asymmetry develops in h xy , correspondingto the location of the overdensity in Figure 3.7. We also see a diffusion ofthe tensor perturbation in the void, indicating the beginning of growth ofhigher order perturbations. .6. Discussion and Conclusions
100 0 100 200 x y . . . . . . . . . h xy ( ⇥ ) F IGURE xy component ofthe tensor perturbation h ij at η = 1 . × . We use (3.39) tocalculate h xy using the off-diagonal metric component g xy .All quantities are shown in code units for our simula-tion. We have demonstrated the feasibility of inhomogeneous cosmological sim-ulations in full General Relativity using the E
INSTEIN T OOLKIT . The overallapproach is similar to other recent attempts (Giblin, Mertens, and Stark-man, 2016a; Bentivegna and Bruni, 2016), with the main difference beingin the construction of initial conditions which allows us to simulate a puregrowing mode, instead of a mix of growing and decaying modes (see Dav-erio, Dirian, and Mitsou, 2017). We also use a different code to (Giblin,Mertens, and Starkman, 2016a), allowing for independent verification. Aswith the other studies we were able to demonstrate the evolution of a den-sity perturbation into the nonlinear regime.As this is a preliminary study, we have focused on the numerical ac-curacy and convergence, rather than a detailed investigation of physicaleffects such as backreaction. Our main conclusions are:1. We demonstrate fourth-order convergence of the numerical solutionto the exact solution for a flat, dust FLRW universe with errors ∼ − even at low spatial resolution ( ).2. We demonstrate second-order convergence of the numerical solutionsfor the growth of linear perturbations, matching the analytic solutionsfor the cosmic evolution of density, velocity and metric perturbationsto within ∼ − .0 Chapter 3. Inhomogeneous cosmology with numerical relativity
3. We show that numerical relativity can successfully be used to followthe formation of cosmological structures into the nonlinear regime.We demonstrate the appearance of non-zero gravitational slip andtensor modes once perturbations are nonlinear with amplitudes of ∼ × − and ∼ × − respectively.The main limitation to our study is that we have employed only low-resolution simulations compared to current Newtonian N-body cosmologi-cal simulations (e.g. Genel et al., 2014; Springel et al., 2005; Kim et al., 2011),and used only simple initial conditions rather than a more realistic spec-trum of perturbations (but see Giblin, Mertens, and Starkman, 2016a). Rep-resenting the density field on a grid means our simulations are limited bythe formation of shell-crossing singularities. The relative computational ex-pense means that General-Relativistic simulations are unlikely to replacethe Newtonian approach in the near future. However, they are an impor-tant check on the accuracy of the approximations employed.1 Chapter 4
Einstein’s Universe:Cosmological structureformation in numericalrelativity
Published in:Macpherson, Price, and Lasky (2019). Physical Review D, . Abstract
We perform large-scale cosmological simulations that solve Einstein’s equa-tions directly via numerical relativity. Starting with initial conditions sam-pled from the cosmic microwave background, we track the emergence of acosmic web without the need for a background cosmology. We measure thebackreaction of large-scale structure on the evolution of averaged quantitiesin a matter-dominated universe. Although our results are preliminary, wefind the global backreaction energy density is of order − compared to theenergy density of matter in our simulations, and is thus unlikely to explainaccelerating expansion under our assumptions. Sampling scales above thehomogeneity scale of the Universe ( − h − Mpc), in our chosen gauge,we find − variations in local spatial curvature. A note on notation
We have altered the notation throughout this chapter, including Appen-dices B, C, D, and E, to be consistent with Chapters 1 and 2, unless ex-plicitly stated otherwise. For these exceptions, we maintain the notation ofthe publication for consistency with figures in their published form. Asidefrom these changes, this chapter is consistent with the accepted version ofMacpherson, Price, and Lasky (2019).2
Chapter 4. Einstein’s Universe
Modern cosmology derives from the Friedmann-Lemaître-Robertson-Walker(FLRW) metric — an exact solution to Einstein’s equations that assumeshomogeneity and isotropy. The formation of cosmological structure meansthat the Universe is neither homogeneous nor isotropic on small scales. TheLambda Cold Dark Matter ( Λ CDM) model assumes the FLRW metric, andhas been the leading cosmological model since the discovery of the acceler-ating expansion of the Universe (Riess et al., 1998; Perlmutter et al., 1999).Since then it has had many successful predictions, including the location ofthe baryon acoustic peak (e.g. Kovac et al., 2002; Eisenstein et al., 2005; Coleet al., 2005; Blake et al., 2011; Ata et al., 2018), the polarisation of the cosmicmicrowave background (CMB) (Planck Collaboration et al., 2016; Hinshawet al., 2013), galaxy clustering, and gravitational lensing (e.g. Bonvin et al.,2017; Hildebrandt et al., 2017; DES Collaboration et al., 2017). Despite thesesuccesses, tensions with observations have arisen. Most notable is the re-cent . σ tension between measurements of the Hubble parameter, H , lo-cally (Riess et al., 2018a) and the value inferred from the CMB under Λ CDM(Planck Collaboration et al., 2016).The assumptions underlying the standard cosmological model are basedon observations that our Universe is, on average , homogeneous and isotropic.However, the averaged evolution of an inhomogeneous universe does notcoincide with the evolution of a homogeneous universe (Buchert and Ehlers,1997; Buchert, 2000). Additional “backreaction” terms exist, but their sig-nificance has been debated (e.g. Räsänen, 2006b; Räsänen, 2006a; Li andSchwarz, 2007; Li and Schwarz, 2008; Larena et al., 2009; Clarkson andUmeh, 2011; Wiltshire, 2011; Wiegand and Schwarz, 2012; Green and Wald,2012; Buchert and Räsänen, 2012; Green and Wald, 2014; Buchert et al.,2015; Green and Wald, 2016; Bolejko and Korzy ´nski, 2017; Roukema, 2018;Kaiser, 2017; Buchert, 2018).State-of-the-art cosmological simulations currently employ the FLRWsolution coupled with a Newtonian approximation for gravity (Springelet al., 2005; Kim et al., 2011; Genel et al., 2014). These simulations haveproven extremely valuable to furthering our understanding of the Uni-verse. However, General-Relativistic effects on our observations cannot befully studied when the formation of large-scale structure has no effect onthe surrounding spacetime. Whether or not these effects are significant canonly be tested with numerical relativity, which allows us to fully removethe assumptions of homogeneity and isotropy. Initial works have shownemerging relativistic effects such as differential expansion (Bentivegna andBruni, 2016), variations in proper length and luminosity distance relative toFLRW (Giblin, Mertens, and Starkman, 2016a; Giblin, Mertens, and Stark-man, 2016b), and the emergence of tensor modes and gravitational slip(Macpherson, Lasky, and Price, 2017). A comparison between Newtonianand fully General-Relativistic simulations found sub-percent differences inthe weak-field regime (East, Wojtak, and Abel, 2018), in agreement withpost-Friedmannian N-body calculations (Adamek et al., 2013; Adamek, Dur-rer, and Kunz, 2014).In this work, we present cosmological simulations with numerical rela-tivity, using realistic initial conditions, evolved over the entire history of theUniverse. Here we use a fluid approximation for dark matter, however, this .2. Computational Setup G = c = 1 , where G is the gravitational constant and c is the speed oflight. Greek indices take values 0 to 3, and Latin indices from 1 to 3, withrepeated indices implying summation. ACTUS and FLRWSolver
To evolve a fully General-Relativistic cosmology we use the open-sourceE
INSTEIN T OOLKIT (Löffler et al., 2012), a collection of codes based on theC
ACTUS framework (Goodale et al., 2003). Within this toolkit we use the
ML_BSSN thorn (Brown et al., 2009) for evolution of the spacetime variablesusing the BSSN formalism (Shibata and Nakamura, 1995; Baumgarte andShapiro, 1999), and the
GRHydro thorn for evolution of the hydrodynamics(Baiotti et al., 2005; Giacomazzo and Rezzolla, 2007; Mösta et al., 2014). Inaddition, we use our initial-condition thorn,
FLRWSolver (Macpherson,Lasky, and Price, 2017), to initialise linearly-perturbed FLRW spacetimeswith perturbations of either single-mode or CMB-like distributions.We assume a dust universe, implying pressure P = 0 , however GRHydro currently has no way to implement zero pressure for hydrodynamical evo-lution. Instead we set P (cid:28) ρ , with a polytropic equation of state, P = K poly ρ , (4.1)where K poly is the polytropic constant, which we set K poly = 0 . in codeunits. We have found this to be sufficient to match the evolution of a homo-geneous, isotropic, matter-dominated universe. Deviations from the exactsolution for the scale factor evolution, at resolution, are within − (seeMacpherson, Lasky, and Price, 2017).We perform a series of simulations with varying resolutions, , ,and , and comoving physical domain sizes, L = 100 Mpc, 500 Mpc,and 1 Gpc, to study different physical scales. We simulate all three domainsizes at and resolution, and only the L = 1 Gpc domain size at resolution due to computational constraints. During the evolution we donot assume a cosmological background, and for convenience, since we have4 Chapter 4. Einstein’s Universe not yet implemented a cosmological constant in the E
INSTEIN T OOLKIT , weassume
Λ = 0 .Post-processing analysis is performed using the
MESCALINE code, whichwe introduce and describe in Section 4.5.2.
We choose the comoving length unit of our simulation domain to be 1 Mpc,implying a domain of L = 100 in code units is equivalently L = 100 Mpc.In geometric units c = 1 , and so we can relate our length unit, l = 1 Mpc, and our time unit, t c , via the speed of light (in physical units) t c = lc . (4.2)To find our background FLRW density we use H ( z = 0) = H , with unitsof s − . This implies H , code × t c = H , phys , (4.3)where H , code and H , phys = 100 h km s − Mpc − are the Hubble parameterexpressed in code units and physical units, respectively. We use (4.2) to-gether with (4.3) and the Friedmann equation for a flat, matter-dominatedmodel H = ˙ aa = (cid:114) π ¯ ρ , (4.4)where an overdot represents a derivative with respect to proper time, ¯ ρ isthe homogeneous density, a is the FLRW scale factor, and we have G = 1 incode units. We find the background FLRW density, evaluated at z = 0 , incode units, to be ¯ ρ , code = 1 . × − h . (4.5)For computational reasons we adopt the initial FLRW scale factor a init = a ( z = 1100) = 1 , whilst the usual convention in cosmology is to set a = a ( z = 0) = 1 . The density (4.5) was calculated using the Hubble parameter H , phys evaluated with a = 1 . The comoving (constant) FLRW densityis ρ ∗ = ¯ ρ a = ¯ ρ a , and so (4.5) is the comoving density ρ ∗ . We choose h = 0 . , and our choice a init = 1 implies our initial background densityis the comoving FLRW density. Simulations are initiated at z = 1100 and evolve to z = 0 . We quote red-shifts computed from the value of the FLRW scale factor at a particularconformal time, a ( η ) = z cmb + 1 z ( η ) + 1 , (4.6)where z cmb = 1100 . Since we set a init = 1 , we have a = 1101 . The evolutionof the FLRW scale factor in conformal time is a ( η ) = a init ξ , (4.7)where ξ is the scaled conformal time defined in Section 4.3.1. Importantly,the redshifts presented throughout this chapter are indicative only of the .3. Initial conditions We solve the linearly-perturbed Einstein equations to generate our initialconditions. Assuming only scalar perturbations, the linearly-perturbed FLRWmetric in the longitudinal gauge is ds = − a ( η ) (1 + 2 ψ ) dη + a ( η ) (1 − φ ) δ ij dx i dx j . (4.8)In this gauge the metric perturbations φ and ψ are the Bardeen potentials(Bardeen, 1980). These are related to perturbations in the matter distribu-tion via the linearly perturbed Einstein equations ¯ G µν + δG µν = 8 π (cid:0) ¯ T µν + δT µν (cid:1) , (4.9)where an over-bar represents a background quantity, and δX represents asmall perturbation in the quantity X , with δX (cid:28) X . A matter-dominated(dust) universe has stress-energy tensor T µν = ρ u µ u ν , (4.10)where ρ is the mass-energy density , u µ = dx µ /dτ is the four-velocity of thefluid, and τ is the proper time. Assuming small perturbations to the matterwe have ρ = ¯ ρ + δρ = ¯ ρ (1 + δ ) , (4.11) v i = δv i , (4.12)where the fractional density perturbation is δ ≡ δρ/ ¯ ρ , and v i = dx i /dt is thethree-velocity , with t the coordinate time.Solutions to (4.9) are found by taking the time-time, time-space, traceand trace-free components, given by ∂ φ − H (cid:0) φ (cid:48) + H ψ (cid:1) = 4 π ¯ ρ δa , (4.13a) H ∂ i ψ + ∂ i φ (cid:48) = − π ¯ ρ a δ ij v j , (4.13b) φ (cid:48)(cid:48) + H (cid:0) ψ (cid:48) + 2 φ (cid:48) (cid:1) = 13 ∂ ( φ − ψ ) , (4.13c) ∂ (cid:104) i ∂ j (cid:105) ( φ − ψ ) = 0 , (4.13d)respectively, where we have assumed all perturbations are small such thatsecond-order (and higher) terms can be neglected. Here, ∂ i ≡ ∂/∂x i , ∂ = ∂ i ∂ i , ∂ (cid:104) i ∂ j (cid:105) ≡ ∂ i ∂ j − / δ ij ∂ , a (cid:48) represents a derivative with respect toconformal time, and H ≡ a (cid:48) /a is the conformal Hubble parameter. Solving The density ρ used throughout this Chapter is the total rest-frame mass-energy density,and is equivalent to ρ R used in Chapters 1 and 2. The velocity used throughout this Chapter differs from the velocity used in Chapter 2,they are related via (2.137). Chapter 4. Einstein’s Universe these equations, we find ψ = φ = f ( x i ) − g ( x i )5 ξ , (4.14a) δ = C ξ ∂ f ( x i ) − f ( x i ) − C ξ − ∂ g ( x i ) − ξ − g ( x i ) , (4.14b) v i = C ξ ∂ i f ( x i ) + 310 C ξ − ∂ i g ( x i ) , (4.14c)where f, g are arbitrary functions of spatial position, we introduce the scaledconformal time coordinate ξ ≡ (cid:114) πρ ∗ a init η, (4.15)and we have defined C ≡ a init πρ ∗ , C ≡ a init πρ ∗ , C ≡ − (cid:114) a init πρ ∗ . (4.16)Equations (4.14) contain both a growing and decaying mode for the densityand velocity perturbations. We choose g = 0 to extract only the growingmode of the density perturbation, and our solutions become ψ = φ = f ( x i ) , (4.17a) δ = C ξ ∂ f ( x i ) − f ( x i ) , (4.17b) v i = C ξ ∂ i f ( x i ) , (4.17c)implying φ (cid:48) = 0 in the linear regime. We use (4.14) along with the Code for Anisotropies in the Microwave Back-ground (CAMB; Lewis and Bridle, 2002) to generate the matter power spec-trum at z = 1100 , with parameters consistent with Planck Collaborationet al. (2016) as input. Figure 4.1 shows the matter power spectrum fromCAMB (grey curve), as a function of wavenumber | k | = (cid:113) k x + k y + k z . Weuse the Python module c2raytools to generate a 3-dimensional Gaus-sian random field drawn from the CAMB power spectrum. This providesthe initial density perturbation. The magenta curve in Figure 4.1 showsthe region of the matter power spectrum sampled in our highest resolu-tion ( ), largest domain size ( L = 1 Gpc) simulation. The smallest k component sampled represents the largest wavelength of perturbations —approximately the length of the box, L — and the largest k componentsampled represents the smallest wavelength of perturbations — two gridpoints. To relate the initial density perturbation to the corresponding ve-locity and metric perturbations, we transform (4.14) into Fourier space. Ini-tially, ξ = 1 which gives a density perturbation of the form δ ( k ) = − (cid:0) C | k | + 2 (cid:1) φ ( k ) , (4.18) https://github.com/hjens/c2raytools .3. Initial conditions -4 -2 | k | [Mpc − ] -2 -4 -6 -8 -10 -12 -14 P ( | k | ) [ M p c ] CAMB256 , F IGURE | k | = (cid:113) k x + k y + k z . The magenta curveshows the section of the power spectrum we sample whenusing a domain size of L = 1 Gpc with resolution . where k = ( k x , k y , k z ) , so we can define an arbitrary function δ ( k ) , andconstruct the metric perturbation and velocity, respectively, using φ ( k ) = − δ ( k ) C | k | + 2 , (4.19a) v ( k ) = C i k φ ( k ) , (4.19b)where i = − . With the Fourier transform of the Gaussian random fieldas δ ( k ) , we calculate the velocity and metric perturbations in Fourier spaceusing (4.19), and then use an inverse Fourier transform to convert the per-turbations to real space. The density perturbation δ is already dimension-less, and we normalise by the speed of light, c , to convert v i and φ to codeunits. Figure 4.2 shows initial conditions at resolution for box sizes L = 1 Gpc, 500 Mpc, and 100 Mpc in the left to right columns, respectively.The top row shows the density perturbation δ , the middle row shows thenormalised metric perturbation φ/c , and the bottom row shows the mag-nitude of the velocity perturbation normalised to the speed of light | v | /c .These initial conditions are sufficient to describe a linearly-perturbed FLRWspacetime in FLRWSolver .We assume a flat FLRW cosmology for the initial instance only. Simula-tions begin with small perturbations at the CMB, and so the assumption ofa linearly-perturbed FLRW spacetime is sufficiently accurate.8
Chapter 4. Einstein’s Universe F IGURE L = 1 Gpc, 500 Mpc, and 100 Mpc. We show a two-dimensional slice through the midplane of each domain.All initial conditions shown here are at resolution, andall quantities are shown in code units – normalised by thespeed of light for the metric and velocity perturbations. Themagnitude of the velocity is | v | = (cid:113) v x + v y + v z . .4. Gauge The (3+1) decomposition of Einstein’s equations (Arnowitt, Deser, and Mis-ner, 1959) results in the metric ds = − α dt + γ ij (cid:0) dx i + β i dt (cid:1) (cid:0) dx j + β j dt (cid:1) , (4.20)where γ ij is the spatial metric, α is the lapse function, β i is the shift vector,and x i are the spatial coordinates. The lapse function determines the rela-tionship between proper time and coordinate time from one spatial slice tothe next, while the shift vector determines how spatial points are relabelledbetween slices. In cosmological simulations with numerical relativity thecomoving synchronous gauge (geodesic slicing) is a popular choice (e.g.Bentivegna and Bruni, 2016; Giblin, Mertens, and Starkman, 2016a; Giblin,Mertens, and Starkman, 2016b; Giblin, Mertens, and Starkman, 2017; Gib-lin et al., 2017), which involves fixing α = 1 , β i = 0 , and u µ = (1 , , , ,or u µ = (1 /a, , , for conformal time, throughout the simulation. Thisgauge choice can become problematic at low redshifts when geodesics be-gin to cross, and can form singularities. We choose β i = 0 and evolve thelapse according to the general spacetime foliation ∂ t α = − f ( α ) α K, (4.21)where f ( α ) is a positive and arbitrary function, and K = γ ij K ij is the traceof the extrinsic curvature. We choose f = 1 / , and use the relation from the(3+1) ADM equations (Shibata and Nakamura, 1995) ∂ t ln( γ / ) = − αK, (4.22)where γ is the determinant of the spatial metric. Integrating (4.21) gives α = C ( x i ) γ / , (4.23)where C ( x i ) is an arbitrary function of spatial position.For our initial conditions we have γ ij = a (1 − φ ) δ ij , implying γ / = a √ − φ . We therefore choose C ( x i ) = √ ψ √ − φ , (4.24)on the initial hypersurface, so that α = a √ ψ , as in the metric (4.8). We adopt the averaging scheme of Buchert (2000) generalised for an arbi-trary coordinate system (Larena, 2009; Brown, Robbers, and Behrend, 2009;Brown, Behrend, and Malik, 2009; Clarkson, Ananda, and Larena, 2009;Gasperini, Marozzi, and Veneziano, 2010; Umeh, Larena, and Clarkson,2011) . The average of a scalar quantity ψ ( x i , t ) is defined as During the review of this paper, Buchert, Mourier, and Roy (2018) raised some concernsregarding the averaging formalism of Larena (2009). We aim to investigate the proposedalterations in a later work (see Sections 1.5.2 and 6.2.1). Chapter 4. Einstein’s Universe (cid:104) ψ (cid:105) = 1 V D (cid:90) D ψ √ γ d x, (4.25)where the average is taken over some domain D lying within the chosen hy-persurface, and V D = (cid:82) D √ γd x is the volume of that domain. The normalvector to our averaging hypersurface is n µ = ( − α, , , , corresponding tothe four-velocity of observers within our simulations. These observers arenot comoving with the fluid, implying n µ (cid:54) = u µ , and the tilt between thesetwo vectors results in additional backreaction terms due to nonzero pecu-liar velocity v i . As in Larena (2009), Clarkson, Ananda, and Larena (2009),and Brown, Latta, and Coley (2013), we define the Hubble expansion of adomain D to be associated with the expansion of the fluid, θ , H D ≡ (cid:104) θ (cid:105) , (4.26)where θ ≡ h αβ ∇ α u β , (4.27)is the projection of the fluid expansion onto the three-surface of averaging,with the projection tensor h αβ ≡ g αβ + n α n β . In our case, this represents theexpansion of the fluid as observed in the gravitational rest frame (Umeh,Larena, and Clarkson, 2011).Averaging Einstein’s equations in this frame, with P = Λ = 0 , gives theaveraged Hamiltonian constraint H D = 16 π (cid:104) W ρ (cid:105) − R D − Q D + L D , (4.28)where W is the Lorentz factor, R D is the averaged Ricci curvature scalar, Q D is the kinematical backreaction term, and L D is the additional backreactionterm due to nonzero peculiar velocities in our gauge. For definitions ofthese terms, see Appendix B.We define the effective scale factor, a D , describing the expansion of thefluid, via the Hubble parameter H D = a (cid:48)D a D . (4.29)This is related to the effective scale factor describing the expansion of thecoordinate grid (volume) a V D ≡ V (cid:48)D V D = (cid:18) V D ( η ) V D ( η init ) (cid:19) / , (4.30)via a D = a V D exp (cid:18) − (cid:90) (cid:104) αW ( θ − κ ) − α θ (cid:105) dη (cid:19) . (4.31)See Appendix C for details . The fluid scale factor a D here is equivalent to a h D used in Chapter 1 and Section 2.3. After the publication of this paper, we noticed an error in (4.31) from Larena (2009). InAppendix F we show this error makes negligible difference to our results. .5. Averaging scheme z = 1099 z = 9 . z = 2 . z = 1 . z = 0 . z = 0 .
500 Mpc F IGURE simulation, in an L = 1 Gpc domain. This simulation has evolved from the cos-mic microwave background ( z = 1100 ; top left) until today( z = 0 ; bottom right). Each panel shows a two-dimensionalslice of the density perturbation in the midplane of the do-main. We can see the familiar web structure of moderncosmological N-body simulations using Newtonian gravity,however this cosmic web contains all of the correspondingGeneral-Relativistic information. The standard deviationsof the fractional density perturbation δ for each panel (pro-gressing in time) are σ δ = 0 . , . , . , . , . , and . , respectively. Chapter 4. Einstein’s Universe z = 0 . z = 0 . z = 0 .
050 Mpc F IGURE (leftto right) with domain sizes L = 1 Gpc, 500 Mpc, and 100Mpc, respectively. All snapshots show a two-dimensionaldensity slice in the midplane of the simulation domain atredshift z = 0 . The dimensionless cosmological parameters describe the content of the Uni-verse. From (4.28) we define Ω m = 8 π (cid:104) W ρ (cid:105) H D , Ω R = − R D H D , (4.32a) Ω Q = − Q D H D , Ω L = L D H D , (4.32b)giving the Hamiltonian constraint in the form Ω m + Ω R + Ω Q + Ω L = 1 . (4.33)We require this to be satisfied at all times. Here, Ω m is the matter energydensity, Ω R is the curvature energy density, Ω Q + Ω L is the energy den-sity associated with the backreaction terms; a purely General-Relativisticeffect. For a standard Λ CDM cosmology, these cosmological parametersare Ω m = 0 . ± . , | Ω R | = | Ω k | < . , Ω Q = 0 , and Ω L = 0 (PlanckCollaboration et al., 2016). The Universe is measured to be homogeneous and isotropic on scales largerthan ∼ − h − Mpc
Scrimgeour et al., 2012. Above these scales it is un-clear whether the evolution of the average of our inhomogeneous Universecoincides with the FLRW (or Λ CDM) equivalent. In attempt to address this,we calculate averages over our entire simulation domain, but also over sub-domains within the simulation to sample a variety of physical scales. Wemeasure averages over spheres of varying radius r D embedded in the totalvolume, from which we calculate the dimensionless cosmological param-eters (4.32), the Hubble parameter (4.26), and consequently the effectivematter expansion a D . .6. Results
500 Mpc θ/ (3 H all )
40 20 0 20 40 60 80 100 R/ (6 H ) log £ σ / (3 H ) ⁄ F IGURE θ , the spatial Ricci curvature R , andthe shear σ , respectively, each relative to the global Hubbleexpansion H all . Each panel shows a two-dimensional sliceat z = 0 through the midplane of the L = 1 Gpc domain at resolution. The spatial Ricci tensor R ij is the contraction of the Riemann tensor. Wecalculate this directly from the metric using R ij = ∂ k Γ kij − ∂ j Γ kik + Γ klk Γ lij − Γ kjl Γ lik , (4.34)where the spatial connection coefficients are Γ kij ≡ γ kl ( ∂ i γ jl + ∂ j γ li − ∂ l γ ij ) . (4.35)We use our analysis code MESCALINE , written to analyse three-dimensionalHDF5 data output from our simulations. The code reads in the spatial met-ric γ ij , the lapse α , the extrinsic curvature K ij , the density ρ , and the veloc-ity v i from the E INSTEIN T OOLKIT three-dimensional output. From thesequantities we calculate the spatial Ricci tensor R ij from the spatial metric,and hence the Ricci scalar via R = γ ij R ij . We take the trace of the extrinsiccurvature K = γ ij K ij and with the set of equations defined in Appendix Bwe calculate averages and the resulting backreaction terms. We also use MESCALINE to calculate the Hamiltonian and momentum constraint viola-tion, discussed in Appendix D. We compute derivatives using centred finitedifference operators, giving second order accuracy in both space and time,the same order as the E
INSTEIN T OOLKIT ’s spatial discretisation.
Figure 4.3 shows time evolution of a two-dimensional slice of the density ρ through the midplane of the L = 1 Gpc domain at resolution. Weshow the growth of structures from z = 1100 (top left) through to z = 0 (bottom right). The σ variance in δ evolves from σ δ = 0 . (top left) to σ δ = 3 . (bottom right).Figure 4.4 shows two-dimensional slices through the midplane of three resolution simulations with domain size L = 1 Gpc, 500 Mpc, and 10004
Chapter 4. Einstein’s Universe
Mpc (left to right), at redshift z = 0 . As we sample smaller scales we see amore prominent web structure forming. Our fluid treatment of dark matterimplies over-dense regions continue to collapse towards infinite density,rather than forming virialised structures. This should, in general, yield ahigher density contrast on small scales than we expect in the Universe.Figure 4.5 shows (left to right) the matter expansion rate θ , the spatialRicci curvature R , and the shear σ , respectively, at z = 0 . Each quantity isnormalised to the global Hubble expansion H all . The curvature and shearpanels are normalised to correspond to the respective density parameters: Ω R defined in (4.32), and Ω σ = (cid:104) σ (cid:105) / (3 H ) defined in Montanari and Räsä-nen (2017). We calculate θ using (4.27), σ using (B.6) and (B.5), and R usingthe definitions (4.34) and (4.35). Each panel shows a two-dimensional slicethrough the midpoint of the L = 1 Gpc domain at resolution. Ourrelativistic quantities can be seen to closely correlate with the density dis-tribution at the same time, shown in the bottom right panel of Figure 4.3. Figure 4.6 shows the global evolution of the effective scale factor, a D . Theblue curve shows a D calculated over the whole L = 1 Gpc, resolutiondomain with (4.31). The purple dashed curve in the top panel shows thecorresponding FLRW solution for the scale factor, a FLRW , found by solv-ing the Hamiltonian constraint for a flat, matter-dominated, homogeneous,isotropic Universe in the longitudinal gauge, a (cid:48) a = (cid:114) πG ¯ ρ a , (4.36)giving the solution (4.7). The bottom panel of Figure 4.6 shows the residualerror between the two solutions, which remains below − for the evolu-tion to z = 0 .Analysing the cosmological parameters as an average over the entiresimulation domain we find agreement with the corresponding FLRW modelin our chosen gauge. Globally, at z = 0 , we find Ω m ≈ , Ω R ≈ − , and Ω Q + Ω L ≈ − . Systematic errors on these values are discussed in Ap-pendix E. Cosmological parameters
Figure 4.7 shows cosmological parameters calculated within spheres of var-ious averaging radii, r D , within an L = 1 Gpc domain at resolution.Left to right panels correspond to increasing time (decreasing z ), showing z = 9 . , . , and , respectively. Black points show the mean value over1000 spheres at the corresponding averaging radius, showing filled circlesfor Ω m , filled squares for Ω R , and crosses for Ω Q + Ω L . Over these 1000spheres we also show the 68%, 95%, and 99.7% confidence intervals for Ω m and Ω R as progressively lighter blue and purple shaded regions, respec-tively. The same confidence intervals for the contribution from the backre-action terms, Ω m + Ω L , are shown as dashed, dot-dashed, and dotted lines .6. Results a D a FLRW -1 z -8 -6 -4 a D /a FLRW − F IGURE a D , calculated over the entire L = 1 Gpcdomain. The dashed magenta curve shows the equivalentFLRW solution (with Ω m = 1 ), as a function of redshift. Thebottom panel shows the residual error for this resolu-tion calculation.
20 40 60 80 100 r D (Mpc) z = 9 . m Ω R Ω Q + Ω L
20 40 60 80 100 r D (Mpc) z = 1 .
20 40 60 80 100 r D (Mpc) z = 0 . F IGURE r D , randomly placedwithin an L = 1 Gpc domain at resolution. Blackpoints show mean values over 1000 spheres at each radius,progressively lighter blue and purple shaded regions showthe 68%, 95%, and 99.7% confidence intervals for Ω m and Ω R , respectively. Crosses show the mean contribution frombackreaction terms Ω Q + Ω L , while dashed, dot-dashed,and dotted lines show the 68%, 95%, and 99.7% confidenceintervals, respectively. Left to right panels are redshifts z = 9 . , . , and . , respectively. Chapter 4. Einstein’s Universe
50 100 150 200 250 r D (Mpc)
0. 50. 00. 51. 01. 5 Ω m Ω R Ω Q + Ω L F IGURE r D = 250 Mpc.Black points show the mean Ω m , Ω R , and Ω Q + Ω L over the1000 spheres at each radius. Progressively lighter blue andpurple shaded regions show the 68%, 95%, and 99.7% con-fidence intervals for Ω m and Ω R , while dashed, dot-dashed,and dotted lines show these for Ω Q + Ω L . .6. Results
0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 a D , all ( × )
0. 00. 20. 40. 60. 81. 01. 2 a D ( × ) δ ≥ . δ − . a D , all F IGURE a D calculated inspheres of radius 100 Mpc as a function of global expansion a D , all . We calculate a D in an L = 500 Mpc simulation at resolution. Blue curves show overdense regions with δ ≥ . , while purple curves show underdense regions with δ ≤ − . . The black dashed line shows the mean expansionover the whole domain. respectively. Figure 4.8 shows the same calculation of the cosmological pa-rameters at z = 0 , extending averaging radii to r D = 250 Mpc.At redshift z = 0 , considering averaging radii corresponding to theapproximate homogeneity scale of the Universe (Scrimgeour et al., 2012), < r D < h − Mpc, we find Ω m = 1 . ± . , Ω R = − . ± . ,and Ω Q + Ω L = − . ± . . These are the mean values over all sphereswith r D = 80 − h − Mpc; 3000 spheres in total. Variations are the 68%confidence intervals of the distribution.Below the measured homogeneity scale, with r D < h − Mpc, we use13 individual radii each with a sample of 1000 spheres. We find Ω m =1 . +0 . − . , Ω R = − . +0 . − . , and Ω Q + Ω L = − . +0 . − . .Considering scales above this homogeneity scale, we use < r D < h − Mpc with a total of 11 radii sampled and 1000 spheres each. Onthese scales we find Ω m = 0 . ± . , Ω R = 0 . ± . , and Ω Q + Ω L =0 . ± . .Systematic errors in all quoted cosmological parameters here are dis-cussed in Appendix E.08 Chapter 4. Einstein’s Universe δ H D / H all δ linear r D =20 Mpcslope: -3.30 0.8 0.6 0.4 0.2 0.0 0.2 0.4 δ H D / H all r D =40 Mpcslope: -3.07 0.8 0.6 0.4 0.2 0.0 0.2 0.4 δ H D / H all r D =80 Mpcslope: -2.97 F IGURE δ , and the deviation in the Hubble parameter, δ H D / H all , for averaging radii r D = 20 , , and 80 Mpc (leftto right), respectively. Points in each panel represent indi-vidual spheres of 1000 sampled at each radius, and the solidline of the same colour is the best-fit linear relation, withslope indicated in each panel. The black dashed line is theprediction from linear theory. Scale factor
Figure 4.9 shows the evolution of the effective scale factor calculated withinspheres of r D = 100 Mpc, relative to the global value a D , all , which we useas a proxy for time. The dashed line shows the global average, blue curvesshow a D for overdense regions with δ ≥ . , and purple curves for un-derdense regions with δ ≤ − . . In total, we sample 1000 spheres withrandomly placed (fixed) origins within an L = 1 Gpc, resolution sim-ulation. Underdense regions with δ ≤ − . expand − % faster than themean at z = 0 , while overdense regions with δ ≥ . expand − % slower. Hubble parameter
Figure 4.10 shows the relation between the density, δ , of a spherical do-main and the corresponding deviation in the Hubble parameter δH D / ¯ H D =( H D − ¯ H D ) / ¯ H D ; the expansion rate of that sphere. We show the density andvariation in the Hubble parameter for averaging radii r D = 20 , , and 80Mpc, (left to right) respectively. Points in each panel show individual mea-surements within 1000 randomly placed spheres of the same radius. Thesolid line of the same colour in each panel is the linear best-fit for the datapoints, with slope indicated in each panel.Linear perturbation theory predicts the relation between the averagedensity, (cid:104) δ (cid:105) , of a spherical perturbation and the deviation from the Hubbleflow of that spherical region, δ H D / H all , to be (Lahav et al., 1991) (cid:104) δ (cid:105) = − F δ H D H all , (4.37)where F = Ω . m is the growth rate of matter (Linder, 2005), which for ourglobal average Ω m ≈ is F = 1 . This in turn implies that the growth rateof structures in our simulations is larger than in the Λ CDM Universe where Ω m ≈ . (e.g. DES Collaboration et al., 2017; Bonvin et al., 2017; PlanckCollaboration et al., 2016; Bennett et al., 2013). The black dashed line ineach panel of Figure 4.10 is the relation (4.37), a slope of -3. On 20 Mpc .7. Discussion We have presented simulations of nonlinear structure formation with nu-merical relativity, beginning with initial conditions drawn from the CMBmatter power spectrum. These simulations allow us to analyse the effectsof large density contrasts on the surrounding spacetime, and consequentlyon cosmological parameters. We calculate the cosmological parameters Ω m , Ω R , Ω Q , and Ω L , together describing the content of the Universe, for spher-ical subdomains embedded within a 256 resolution, L = 1 Gpc simulation.We vary the averaging radius between ≤ r D ≤ Mpc, representingscales both below and above the measured homogeneity scale of the Uni-verse.Our results were obtained using simulations sampling the matter powerspectrum down to scales of two grid points. Quantifying the errors in sucha calculation is difficult because structure formation occurs fastest on smallscales, implying different physical structures at different resolutions. Thisis a known problem in cosmological simulations, not unique to GeneralRelativistic cosmology (see e.g. Schneider et al., 2016). To correctly quantifysuch errors, we must maintain the same density gradients between sev-eral simulations at different computational resolution. This becomes diffi-cult when the perturbations themselves are nonlinear. Even with identicalinitial conditions, we see a different distribution of structures at redshift z = 0 when sampling nonlinear scales at different resolutions. To approx-imate the errors on our main results, we instead analyse a set of test sim-ulations in which we simulate a fixed amount of large-scale structure (seeAppendix E). This allows for a reliable Richardson extrapolation of the so-lution to approximate the error in our main results at redshift z = 0 .Regardless of this, the main result of this chapter is that we find Ω m ≈ in all simulations we analyse here. Any unquantified errors are unlikelyto significantly shift this result, and all global effects of backreaction andcurvature are likely to remain small with an improved sampling of smallscales. We find global cosmological parameters consistent with a matter-dominated,flat, homogeneous, isotropic universe, and therefore no global backreac-tion. The evolution of the effective scale factor a D , evaluated over the wholedomain, coincides with the corresponding FLRW model, as shown in Fig-ure 4.6. The < − discrepancy between the two solutions does not cor-relate with the onset of nonlinear structure formation, indicating that thisdifference is most likely computational error.We find a globally flat geometry in our simulations with Ω R ≈ − .This could be a result of our treatment of the matter as a fluid. We can-not create virialised objects and so any “clusters” will continue to collapsetowards infinite density. In reality, a dark matter halo or galaxy clusterwould form, be supported by velocity dispersion, and stop collapsing. The10 Chapter 4. Einstein’s Universe surrounding voids would continue to expand and potentially contribute toa globally negative curvature (see e.g. Bolejko, 2017; Bolejko, 2018b). With-out a particle description for dark matter alongside numerical relativity wecannot properly capture this effect.Any contribution from backreaction, Q D or L D , is due to variance in theexpansion rate and shear. The left panel of Figure 4.5 shows the matter ex-pansion rate θ , where collapsing regions (yellow, orange, and red) balancethe expanding regions (green) due to our treatment of matter. While we seespatial variance in θ , there is no global contribution from backreaction un-der our assumptions. Therefore, in our chosen gauge and under the caveatsdescribed in Section 4.7.3 below, backreaction from structure formation isunlikely to explain dark energy. We find strong positive curvature on scales below the homogeneity scaleof the Universe. Variations in measured cosmological parameters are upto 31% based purely on location in an inhomogeneous matter distribution.Our result is similar to that of Bolejko (2017) on small scales, but with largervariance in Ω R because of increased small-scale density fluctuations due toour fluid treatment of dark matter.On the approximate homogeneity scale of the Universe we find meancosmological parameters consistent with the corresponding FLRW modelto ∼ . Aside from this, we find the parameters can deviate from thesemean values by 4-9% depending on physical location in the simulation do-main. This implies that, although on average these coincide with a flat,homogeneous, isotropic Universe, an observers interpretation may differby up to 9% based purely on her position in space.As we approach larger averaging radii within a 1 Gpc volume, we be-gin to move away from independent spheres, and each sphere begins tooverlap with others; effectively sampling the same volume. Due to this, theconfidence intervals contract, and eventually at r D ≈ Mpc most spheresbecome indistinguishable from the mean. The beginning of this is evidentin Figure 4.8 as we approach r D = 250 Mpc. This transition appears to bedue to overlapping spheres, although could in part be due to the statisticalhomogeneity of the matter distribution at these scales.Local observations of type 1a supernovae generally probe scales of − h − Mpc (Wu and Huterer, 2017). Nearby objects are excluded from thedata in an effort to minimise cosmic variance on the result (Riess et al., 2016;Riess et al., 2018b; Riess et al., 2018a). In this work, we cannot meaningfullysample scales above 250 Mpc because our maximum domain size is only 1Gpc . In order to sample all scales used in nearby SN1a surveys, we wouldneed a domain size of L (cid:38) h − Gpc, with a resolution up to . Cur-rent computational constraints, and the overhead of numerical relativity,currently restrict us to domain sizes and resolutions used in this work. Toaddress scales as similar as possible to those used in local surveys, we con-sider < r D < h − Mpc. On these scales we find Ω m = 1 . ± . , Ω R = 0 . ± . , and Ω Q + Ω L = 0 . ± . , where variances are the68% confidence intervals due to local inhomogeneity. This implies based onan observers physical location, measured deviations from homogeneity onthese scales could be up to 6%. We expect this variance to decrease when .7. Discussion h − Mpc.We investigate this further, including extrapolation to larger scales, in ourcompanion paper Macpherson, Lasky, and Price (2018).While the global effective scale factor demonstrates pure FLRW evolu-tion, we find inhomogeneous expansion within spheres of 100 Mpc radius.Figure 4.9 shows the expansion rate differs by − depending on therelative density of the region sampled. These differences agree with lin-ear perturbation theory, to within 1%, on (cid:38) Mpc scales, with smallerscales showing differences of up to 10%. These differences are most likelydue to the nonlinearity of the density field on these scales, although, inaddition, could involve General-Relativistic corrections. To properly testthis we would require an equivalent Newtonian cosmological simulationto compare this relation at nonlinear scales, which we leave to future work.
1. Our treatment of dark matter as a fluid is the main limitation of thiswork. Under this assumption, we are unable to form bound struc-tures supported from collapse by velocity dispersions. In cosmologi-cal N-body simulations, particle methods are adopted so as to capturethe formation of galaxy haloes, and local groups of galaxies as boundstructures. Adopting a fully General-Relativistic framework in addi-tion to particle methods would allow us to adopt a proper treatmentof dark matter in parallel with inhomogeneous expansion.2. We take averages over purely spatial volumes. In reality, an observerwould measure her past light cone, and hence the evolving Universe.Our results can thus be considered an upper limit on the variance dueto inhomogeneities, since any structures located in the past light conewill be more smoothed out.3. Our results are explicitly dependent on the chosen averaging hyper-surface. The result of averaging across different hypersurfaces hasbeen investigated (Adamek et al., 2019; Giblin et al., 2018), and the re-sults can show significant differences. It is clear the physical choice ofhypersurface can be important for quantifying the backreaction effect.4. We assume
Λ = 0 , and begin our simulations assuming a flat, matterdominated background cosmology with small perturbations. Through-out the evolution, on a global scale, we find the average Ω m ≈ ;consistent with this model. It is extremely well constrained that ourUniverse is best described by a matter content Ω m ≈ . (e.g. DES Col-laboration et al., 2017; Bonvin et al., 2017; Planck Collaboration et al.,2016; Bennett et al., 2013). The growth rate of cosmological structuresin our simulations will therefore be amplified relative to the Λ CDMUniverse.5. Given our limited spatial resolution, we underestimate the amount ofstructure compared to the real Universe. In addition, we resolve struc-tures down to scales of two grid points, which means these structuresmay be under resolved.12
Chapter 4. Einstein’s Universe
We summarise our findings as follows:1. We find no global backreaction under our assumptions. Over the en-tire simulation domain we have Ω m ≈ , Ω R ≈ − , and Ω Q + Ω L ≈ − , in our chosen gauge; consistent with a matter-dominated, flat,homogeneous, isotropic universe.2. We find strong deviation from homogeneity and isotropy on smallscales. Below the measured homogeneity scale of the Universe ( r D (cid:46) h − Mpc) we find deviations in cosmological parameters of − based purely on an observers physical location.3. Above the homogeneity scale of the universe ( < r D < h − Mpc)we find mean cosmological parameters coincide with the correspond-ing FLRW model, with potential − deviations due to inhomo-geneity.4. We find agreement with linear perturbation theory within 1% on ≥ Mpc scales for the relation between the density of a spherical regionand its corresponding deviation from the Hubble flow. However,these few percent deviations on smaller scales may prove importantin forthcoming cosmological surveys.While we find no global backreaction in our cosmological simulations, ournumerical relativity calculations show significant contributions from cur-vature and other nonlinear effects on small scales.13
Chapter 5
The Trouble with Hubble:Local versus Global ExpansionRates in InhomogeneousCosmological Simulations withNumerical Relativity
Published in:Macpherson, Lasky, and Price (2018). The Astrophysical Journal Letters . Abstract
In a fully inhomogeneous, anisotropic cosmological simulation performedby solving Einstein’s equations with numerical relativity, we find a localmeasurement of the effective Hubble parameter differs by less than 1%compared to the global value. This variance is consistent with predictionsfrom Newtonian gravity. We analyse the averaged local expansion rate onscales comparable to Type 1a supernova surveys, and find that local vari-ance cannot resolve the tension between the Riess et al. (2018a) and PlanckCollaboration et al. (2018) measurements.
A note on notation
We have altered the notation throughout this chapter to be consistent withChapters 1 and 2, unless explicitly stated otherwise. For these exceptions,we maintain the notation of the publication for consistency with figures intheir published form. Aside from these changes, this chapter is consistentwith the accepted version of Macpherson, Lasky, and Price (2018).14
Chapter 5. The Trouble with Hubble
Recently, the tension in the locally measured value of the Hubble parame-ter, H (Riess et al., 2011; Riess et al., 2016) and that inferred from the cos-mic microwave background (CMB; Planck Collaboration et al., 2018) hasreached . σ (Riess et al., 2018b; Riess et al., 2018a). This tension has bothmotivated the search for extensions to the standard cosmological model,and for the improvement of our understanding of systematic uncertainties(e.g. Efstathiou, 2014; Addison et al., 2016; Dhawan, Jha, and Leibundgut,2018). The higher local expansion rate (Riess et al., 2018b; Riess et al., 2018a)suggests we may live in a void (Cusin, Pitrou, and Uzan, 2017; Sundell,Mörtsell, and Vilja, 2015), consistent with local ∼ − underdensitesthat have been found in the supernovae Type 1a (SN1a) data (Zehavi et al.,1998; Jha, Riess, and Kirshner, 2007; Hoscheit and Barger, 2018).In an attempt to address this tension, we perform cosmological simu-lations of nonlinear structure formation that solve Einstein’s equations di-rectly with numerical relativity. In this letter we quantify local fluctuationsin the Hubble parameter based purely on physical location in an inhomo-geneous, anisotropic universe. Further details of our simulations are givenin Macpherson, Price, and Lasky (2019, see Chapter 4), including a quantifi-cation of backreaction of inhomogeneities on globally averaged quantities.Local fluctuations in the expansion rate due to inhomogeneities havebeen analysed using Newtonian and post-Friedmannian N-body cosmolog-ical simulations (e.g. Shi and Turner, 1998; Wojtak et al., 2014; Odderskov,Hannestad, and Haugbølle, 2014; Odderskov, Koksbang, and Hannestad,2016; Adamek et al., 2019), second-order perturbation theory (Ben-Dayan etal., 2014), and exact inhomogeneous models (e.g. Marra et al., 2013). Theseapproaches predict local fluctuations in the Hubble parameter of up to afew percent. Inhomogeneities have also been proposed to have an effect onthe globally measured expansion rate (e.g. Buchert et al., 2015; Roy et al.,2011), with analytical approaches showing this can contribute to an acceler-ated expansion (e.g. Räsänen, 2006a; Räsänen, 2008; Ostrowski, Roukema,and Buchert, 2013). Under the “silent universe” approximation, a globally,non-flat geometry has been shown to fully alleviate the Hubble tension(Bolejko, 2017; Bolejko, 2018a). These works are important steps towardsfully quantifying the effects of inhomogeneities on the Hubble expansion,although simplifying assumptions about the inhomogeneities themselveslimit the ability to make a strong statement.Considering a fully inhomogeneous, anisotropic matter distribution inGeneral Relativity allows us to analyse the effects of inhomogeneities with-out simplifying the structure of the Universe. Simulations of large-scalestructure formation with numerical relativity have been shown to be a vi-able way to study inhomogeneities (Giblin, Mertens, and Starkman, 2016a;Bentivegna and Bruni, 2016; Macpherson, Lasky, and Price, 2017; Giblin,Mertens, and Starkman, 2017; East, Wojtak, and Abel, 2018), although fluc-tuations in the Hubble parameter have not yet been considered. In thiswork we attempt to quantify the discrepancy between local and global ex-pansion rates using cosmological simulations performed without approxi-mating gravity or geometry.We present our computational and analysis methods in Section 5.2, andoutline our method for calculating the Hubble parameter in Section 5.2.2. .2. Method G = c = 1 , unless otherwise stated. Greek indices run from 0 to 3, andLatin indices run from 1 to 3, with repeated indices implying summation. We have simulated the growth of large-scale cosmological structures us-ing numerical relativity. Our initial conditions were drawn from temper-ature fluctuations in the CMB radiation, using the Code for Anisotropiesin the Microwave Background (CAMB; Lewis and Bridle, 2002). The ini-tial density perturbation is a Gaussian random field drawn from the matterpower spectrum of the CMB , and the corresponding velocity and space-time perturbations were found using linear perturbation theory. We use thefree, open-source E INSTEIN T OOLKIT along with our thorn
FLRWSolver (Macpherson, Lasky, and Price, 2017) for defining initial perturbations. Ina previous paper we benchmarked our computational setup for homoge-neous and linearly perturbed cosmological solutions to Einstein’s equa-tions, achieving precision within ∼ − (see Macpherson, Lasky, and Price,2017). We refer the reader to Chapter 4 for full details of our computationalmethods (see also Section 2.1), including generation of initial conditionsand derivations of the appropriate equations (see also Section 2.2.4), anddetails of gauge conditions (see also Section 2.2.5).We evolve Einstein’s equations in full, with no assumed backgroundcosmology, beginning in the longitudinal gauge from z = 1100 , through to z = 0 . Since we have not yet implemented a cosmological constant in theE INSTEIN T OOLKIT , we assume
Λ = 0 , and a matter-dominated ( P (cid:28) ρ )universe. This implies the age of our model universe will differ from theUniverse where Λ (cid:54) = 0 . We simulate a range of resolutions and domainsizes, detailed in Macpherson, Price, and Lasky (2019). Here we analysea resolution, L = 1 Gpc simulation, where the total volume is L .Length scales are quoted under the assumption h = 0 . (see Macpherson,Price, and Lasky, 2019), and we use periodic boundary conditions in allsimulations. The right panel of Figure 5.1 shows the density distributionat z = 0 , showing a two-dimensional slice through the midplane of thedomain, normalised to the global average density, (cid:104) ρ (cid:105) all . We evolve thematter distribution on a grid, treating dark matter as a fluid. This implieswe cannot form virialised structures, and any dense regions will continueto collapse towards infinite density. This is a current limitation of any fullyGeneral-Relativistic cosmological simulation, since numerical relativity N-body codes for cosmology currently do not exist . To create a Gaussian random field following a particular power spectrum, we use thePython module c2raytools : https://github.com/hjens/c2raytools The energy-density ρ used throughout this Chapter is the total rest-frame energy-density,and is equivalent to ρ R used in Chapters 1 and 2. Since the publication of this Letter, several codes for N-body numerical-relativity cos-mology have been developed. See Giblin et al. (2018), Daverio, Dirian, and Mitsou (2019),and Barrera-Hinojosa and Li (2019) Chapter 5. The Trouble with Hubble F IGURE H all .Right panel shows the density distribution relative to theglobal average, (cid:104) ρ (cid:105) all . Both panels show a slice through themidplane of a resolution simulation with L = 1 Gpc.
It is common to compare the evolution of global averages in an inhomoge-neous, anisotropic universe (Buchert and Ehlers, 1997; Buchert, Kerscher,and Sicka, 2000) to the evolution of a homogeneous, isotropic universe.However, the correct choice of averaging time-slice remains ambiguousdue to the presence of nonlinearities. We adopt the averaging scheme ofBuchert, Kerscher, and Sicka (2000), generalised to any hypersurface of av-eraging (Larena, 2009; Brown, Robbers, and Behrend, 2009; Brown, Behrend,and Malik, 2009; Clarkson, Ananda, and Larena, 2009; Gasperini, Marozzi,and Veneziano, 2010; Umeh, Larena, and Clarkson, 2011). The average of ascalar function ψ over a domain D , located within the chosen hypersurface,is (cid:104) ψ (cid:105) = 1 V D (cid:90) D ψ √ γ d x, (5.1)where V D = (cid:82) D √ γ d X is the volume of the domain, with γ the determi-nant of the spatial metric γ ij . We define our averaging hypersurfaces byobservers with four-velocity n µ = ( − α, , , , where α is the lapse func-tion, and we set the shift vector β i = 0 . The four-velocity of these observersdiffers from the four-velocity of the fluid u µ ≡ dx µ /dτ , where τ is the propertime. The local expansion rate of the fluid projected onto our averaging hyper-surface is θ ≡ γ µν ∇ µ u ν , (5.2) .3. Results γ µν ≡ g µν + n µ n ν , and ∇ µ is the covariant derivative associated withthe metric tensor g µν . We define the effective Hubble parameter in a domain D to be H D ≡ (cid:104) θ (cid:105) . (5.3)In a Friedmann-Lemaître-Robertson-Walker spacetime, (5.3) reduces to theusual conformal Hubble parameter H = a (cid:48) /a , where (cid:48) represents a deriva-tive with respect to conformal time.The local expansion rate is not necessarily what the observer measures.Observations of SN1a (Riess et al., 2018b; Riess et al., 2018a) measure thedistance-redshift relation, and it is unclear how this relates to the local ex-pansion rate. Recreating what an observer measures in an inhomogeneousUniverse ultimately requires ray tracing (see Giblin, Mertens, and Stark-man, 2016b; East, Wojtak, and Abel, 2018), which we leave to future work. In order to quantify H D on different physical scales, we calculate averagesover spherical subdomains placed randomly within the volume shown inFigure 5.1. This allows us to analyse the effect of inhomogeneities indepen-dent of boundary effects. We calculate θ for each grid cell, and calculate H D by averaging over subdomains of various radii r D .Observations of SN1a in the local universe span a redshift range of . (cid:46) z (cid:46) . (Riess et al., 2011; Riess et al., 2016; Riess et al., 2018b;Riess et al., 2018a), corresponding to distances of (cid:46) r D (cid:46) h − Mpc(Wu and Huterer, 2017; Odderskov, Hannestad, and Haugbølle, 2014). Lo-cal SN1a with z (cid:46) . are excluded from the analysis in attempt to min-imise cosmic variance; their inclusion results in a 3% higher H , suggestingwe are located in a void (Jha, Riess, and Kirshner, 2007).We approximate a measurement of the Hubble expansion using SN1aby calculating the average local expansion rate over a variety of scales. Wesample spherical regions with radii up to r D = 250 Mpc to ensure indi-vidual spheres are sufficiently independent within our L = 1 Gpc domain.We therefore calculate H D on scales < r D < h − Mpc, correspond-ing to an effective survey range of . (cid:46) z (cid:46) . . The reduced rangeis due to the computational overhead of numerical relativity currently lim-iting us to domain sizes and resolutions of this order. We extrapolate to r D = 450 h − Mpc to estimate the variance over the full range adopted inRiess et al. (2018b) and Riess et al. (2018a). We perform this extrapolationby fitting a function of the form δ H D / H all ∝ /r D using our calculated vari-ance at r D ≥ Mpc, to minimise the effect of small-scale fluctuations (seelower panel of Figure 5.2). To properly test the full range of observations, alarger simulation volume and resolution would be required.
The left panel of Figure 5.1 shows deviations in the Hubble parameter, rel-ative to the global mean H all , at z = 0 . We show a two-dimensional slicethrough the midplane of the L = 1 Gpc domain. Green regions are ex-panding ( θ > ), while yellow to red regions are collapsing ( θ < ). Thisexpansion is strongly correlated with the density field shown in the right18 Chapter 5. The Trouble with Hubble
50 100 150 200 2500. 150. 100. 050. 000. 050. 100. 15 δ H D / H a ll
100 200 300 400 500 600 r D (Mpc)
0. 030. 000. 03 δ H D / H a ll extrapolated variance F IGURE H D at z = 0 . Top panel is the fractional deviation measured inany one sphere from the average over the whole domain, H all , as a function of averaging radius r D . Progressivelylighter blue shaded regions are the 68%, 95% and 99.7% con-fidence intervals, respectively. The red line is the measure-ment from Riess et al. (2018a), and the shaded region rep-resents the 1 σ uncertainty. Dashed curves represent 68%,95%, and 99.7% confidence intervals for the same sampleof spheres weighted as a function of redshift in accordancewith the SN1a sample used in Riess et al. (2018b) and Riesset al. (2018a) (Wu and Huterer, 2017; Camarena and Marra,2018). Bottom panel shows the variance extrapolated to thefull sample range (Riess et al., 2018b; Riess et al., 2018a).Progressively lighter blue curves are the extension of the68%, 95%, and 99.7% confidence intervals, respectively. .3. Results r D . Crosses represent the radii at whichour calculations were done, and progressively lighter blue shaded regionsrepresent the 65%, 98%, and 99.7% confidence intervals over 1000 randomlyplaced spheres with the corresponding radius r D . The red line and shadedregion show the mean and σ deviation of the Riess et al. (2018a) measure-ment from the Planck Collaboration et al. (2018) measurement, respectively.The bottom panel of Figure 5.2 shows the 68%, 95%,and 99.7% confidencecontours (dark to light blue curves, respectively) extrapolated to the fullredshift range used in Riess et al. (2018b) and Riess et al. (2018a).Considering our averaging spheres as a survey volume including SN1aat redshifts . (cid:46) z (cid:46) . , and assuming an isotropic distribution ofobjects across the sky with equal numbers of SN1a at all redshift, we es-timate the expected variance in a local H measurement due to inhomo-geneities as the variance in H D . We calculate the ± σ variance in a mea-surement as the th and th percentiles of the full distribution of spheressampled over the effective survey range, and similarly for the − σ vari-ance. Sampling all scales in the top panel of Figure 5.2, including local SN1awith z (cid:46) . , results in a σ variance of ± . . Excluding these localSN1a the variance drops to (+1.2,-1.1)%. We extrapolate to the full surveyrange . (cid:46) z (cid:46) . (bottom panel of Figure 5.2) by fitting a function δ H D / H all ∝ /r D to each confidence contour in Figure 5.2. While not in-tended to be a precise measure of the variance at large scales, we estimate a σ variance of (+0.8,-0.4)%.The blue distribution in Figure 5.3 shows the local deviation in the Hub-ble parameter relative to the global mean, versus the fraction of total sphereswith that deviation, N sph /N tot . We show the full sample of spheres in therange . (cid:46) z (cid:46) . , with the corresponding σ variations shown asdashed lines. The blue line and shaded region represent the Planck Collab-oration et al. (2018) measurement and σ uncertainties, respectively, whilethe red line and shaded region shows the Riess et al. (2018a) measurementand the σ uncertainties, respectively.The Supercal SN1a compilation (Scolnic et al., 2015), used by Riess etal. (2016), does not contain equal numbers of SN1a at all redshifts; a largernumber of objects are sampled at low redshifts. Weighting our results inline with the redshift distribution of the sample (as shown in Wu and Huterer,2017; Camarena and Marra, 2018) we find the variance in the Hubble pa-rameter increases to (+1 . , − . over our reduced redshift range. Dashedcurves in the top panel of Figure 5.2 show the variance as a function of av-eraging radius for the weighted sample. We proceed using the weightedsample for further analysis.Extending to the σ variance over . (cid:46) z (cid:46) . we find a localHubble constant can be up to 6.2% larger than the mean. Taking the PlanckCollaboration et al. (2018) measurement of . ± . km s − Mpc − as theglobal mean expansion rate, this implies that if an observers position in thecosmic web is relatively underdense, she may measure a Hubble parameterup to . km s − Mpc − larger. Hence a local measurement using SN1a20 Chapter 5. The Trouble with Hubble H D − H all (km s − Mpc − )
0. 00. 10. 20. 30. 40. 50. 6 N s ph / N t o t Riess et. al σ All spheresPlanck σ F IGURE < r D < h − Mpc in blue.The dashed blue lines represent the σ deviation of the in-homogeneous distribution. The blue shaded region repre-sents the σ uncertainties on the Planck Collaboration et al.(2016) measurement, while the solid red line and shaded re-gion represent the mean and σ deviation in the Riess et al.(2018a) measurement, respectively. could reach H = 71 . ± . km s − Mpc − , assuming the same statisticaluncertainties as Riess et al. (2018a). This measurement would then be in . σ tension with Planck Collaboration et al. (2018).In order to completely resolve the tension between a local measurementand the global value, we must restrict our sample range to < r D < h − Mpc, or . (cid:46) z (cid:46) . . Over these scales, our σ variance in theHubble parameter implies a local H measurement could be up to 8.7%,or . km s − Mpc − , larger than the global expansion. Again taking thePlanck Collaboration et al. (2018) value as the global expansion, a local mea-surement could reach H = 73 . ± . km s − Mpc − purely based on theobservers location in an inhomogeneous universe. This is consistent withthe Riess et al. (2018a) measurement within σ . The variance in the effective Hubble parameter shown in Figure 5.2 cannotresolve the tension between the Planck Collaboration et al. (2018) and Riesset al. (2018a) measurements. Excluding local SN1a with z (cid:46) . we findthe variance in the Hubble parameter due to inhomogeneities is (+1.5,-1.6)%over a reduced redshift range. We find an observer can only measure a localHubble parameter up to 8.7% higher than the global value when further reducing the survey range to . (cid:46) z (cid:46) . . The restricted range requiredfor such a measurement emphasises that it is unlikely to completely resolvethe tension by local variance in expansion rate. Extrapolating our results to .4. Discussion r D = 50 , and h − Mpc we find variations of ± . , ± . , and (+1 . , − . , respectively. These are consistent with Newtonian pre-dictions, also sampling observers randomly located in space, from Wojtak etal. (2014) and Odderskov, Koksbang, and Hannestad (2016) to within (cid:46) .However, to address whether this difference is due to General-Relativisticeffects or computational differences, we ultimately require a particle treat-ment of dark matter alongside numerical relativity.Our results may be considered an upper limit for the variance in theHubble parameter over the scales we sample for several reasons. We as-sume averages over a purely spatial volume, when in reality an observerwould measure their past light cone. As we look back in time, structuresare more smoothed out, which would reduce the overall variance. In ad-dition, we evolve our simulations assuming Λ = 0 ; a matter-dominateduniverse at the initial instance. We do not fix Ω m = 1 over the course ofthe simulation, however, globally we find Ω m = 1 to within computationalerror for all time (Macpherson, Price, and Lasky, 2019). This implies thegrowth rate, f , of structures in our simulation will be larger than in Λ CDM,since f = Ω . m (Linder, 2005), resulting in a larger density contrast in gen-eral. This will also increase our variance in the Hubble parameter relativeto that measured in the Universe where Ω m ≈ . is well constrained (e.g.DES Collaboration et al., 2017; Bonvin et al., 2017; Planck Collaboration etal., 2018; Bennett et al., 2013).The effects of inhomogeneities can be dependent on the choice of ob-servers. Adamek et al. (2019) used weak-field relativistic N-body simu-lations to study variance in the Hubble parameter in the comoving syn-chronous gauge and the Poisson gauge. In the comoving gauge the vari-ance in the Hubble parameter reached 10% at z = 0 , while the Poissongauge remained below 0.01%. A direct comparison to this work is not pos-sible due to different definitions of the local expansion, however it outlinesthe importance of carefully choosing the averaging hypersurface. The co-moving gauge is often used to represent observers on Earth, however thisgauge breaks down at low redshifts due to shell crossings, and so it hasbeen suggested the Poisson gauge — similar to the gauge used here — isbetter suited to study the effects of inhomogeneities in the nonlinear regimewith simulations (Adamek et al., 2019).22 Chapter 5. The Trouble with Hubble
We have investigated the effects of inhomogeneities on local measurementsof the Hubble parameter. Using numerical relativity we have simulated thegrowth of density fluctuations drawn from the CMB through to z = 0 . Wehave calculated the expansion rate of dark matter within randomly placedspheres of various radii from a resolution simulation with domain size L = 1 Gpc. Our conclusions are:1. We measure a (+1.5,-1.6)% variance in the local expansion rate due toinhomogeneities over . (cid:46) z (cid:46) . with a weighted sample ofaveraging spheres.2. Estimating an extension to our results over . (cid:46) z (cid:46) . reducesthe variance to (+0.8,-0.4)%. This is consistent with predictions fromNewtonian N-body simulations.3. Our σ variance in the Hubble parameter of 6.2%, over . (cid:46) z (cid:46) . , could reduce the tension between a local and global measure-ment to . σ .4. When restricting the survey range to include more nearby SN1a, thetension is resolved. Over scales . (cid:46) z (cid:46) . , a local calculation of H D can be up to 8.7% larger than the global value. However, since theRiess et al. (2018b) and Riess et al. (2018a) measurement considers asignificantly wider survey range, we conclude that the tension cannotbe explained by local inhomogeneities under our assumptions.23 Chapter 6
Conclusions
In this thesis, we have presented simulations of cosmological structure for-mation that solve Einstein’s equations of GR directly. In Chapter 1 we out-lined the current status of cosmological theory and observations, includingthe standard cosmological model; the Λ CDM model, cosmological pertur-bation theory, and proposed extensions to Λ CDM based on some currenttensions with observational data. In Chapter 2 we derived the 3+1 foli-ation of Einstein’s equations, specifically the BSSN formalism, for evolv-ing arbitrary spacetimes numerically. We also discussed several commoncoordinate choices and an improvement to the BSSN formalism in termsof constraint violation management via the CCZ4 formalism. We then de-scribed the E
INSTEIN T OOLKIT ; the computational framework used for thenumerical-relativity simulations presented in this thesis, along with ourinitial-condition thorn
FLRWSolver , and detailed
MESCALINE ; the post-processing analysis code used to extract our results from these simulations.In this final chapter we summarise the main findings from Chapters 3,4, and 5, and suggest directions for future work to further investigate andbuild on these results.
In Chapter 3 we presented two important code tests to ensure the validityof our computational setup. We initialised a homogeneous, isotropic FLRWmetric with
FLRWSolver , and evolved the flat, dust spacetime with nu-merical relativity using the E
INSTEIN T OOLKIT . We evolved over a changein the scale factor (and hence redshift) of ∆ a ≈ , and matched the an-alytic solutions to the Friedmann equations for the scale factor (1.43) anddensity (1.45) to within − in a simulation with grid cells. We demon-strated the expected fourth-order convergence, for the time integrator, ofthe L error in our solutions, in addition to the violation in the Hamilto-nian constraint. Next, we initialised small perturbations to the backgroundFLRW spacetime in the density, velocity, and metric. We chose a single-mode, sinusoidal form for the metric perturbation, and related this to thecorresponding density and velocity perturbations using the solutions foundin linear perturbation theory, (2.136b) and (2.136c), respectively. We chose Φ = 10 − (cid:28) , with corresponding amplitudes of the density and veloc-ity perturbations of − and − , respectively, so that our assumption oflinear perturbations was valid for the initial conditions. We evolved over achange in scale factor of ∆ a ≈ , and found agreement with linear theoryto within − for the growth of the density and velocity perturbations. We24 Chapter 6. Conclusions demonstrated the expected second-order convergence, for the spatial inte-grator, of the L error of these solutions. Beginning with slightly larger per-turbations, Φ = 10 − , we simulated the growth of these perturbations intothe nonlinear regime, and found the gravitational slip — the difference be-tween the temporal and spatial metric perturbations — was nonzero, withan amplitude of ∼ × − at z ≈ . We found the tensor perturbation,zero in linear theory, grew in the over-dense region of the simulation andwas smoothed out in the under-dense region. This work was an impor-tant proof-of-concept test that cosmological simulations into the nonlinearregime of structure formation — inaccessible with analytic methods — ispossible using numerical relativity, specifically with the open-source E IN - STEIN T OOLKIT .The main aim of this thesis was to complete the first steps towards afull investigation of GR effects in our own Universe. In Chapter 4 we there-fore extended the work performed in Chapter 3 to a more realistic mat-ter distribution. We drew a spectrum of Gaussian density perturbationsfrom the anisotropies in the CMB using the matter power spectrum fromCAMB (Seljak and Zaldarriaga, 1996), and found the corresponding metricand velocity perturbations using linear perturbation theory. We generatedthese initial conditions at a number of different resolutions and physicalbox sizes, with the main results presented for a 1 Gpc domain with grid points. The simulations were each evolved over a total change in scalefactor of ∆ a ≈ , i.e. from the CMB to z ≈ . We used Buchert averag-ing for a general foliation (Larena et al., 2009) in this simulation to computethe global contributions to the total energy-density from matter, curvature,and backreaction, and compared with the equivalent FLRW model. Wealso calculated these contributions within sub-domains in the simulation toassess the affects from curvature and backreaction on small scales. Glob-ally, we found the effect from curvature and backreaction to be negligible,with Ω R ≈ − and | Ω Q + Ω L | ≈ − , respectively, with the contributionfrom matter dominating; a match to the equivalent FLRW model. On smallscales, we found the contribution from backreaction and curvature couldbe significant, anywhere between ∼ on ≥ Mpc scales, up to ∼ on (cid:46) Mpc scales. These results showed that, while backreaction fromstructures is unlikely to explain the accelerating expansion rate (under ourassumptions), percent-level effects are possible, and could be relevant inupcoming precision cosmological surveys.In Chapter 5 we used the simulations presented in Chapter 4 to inves-tigate the effect of local inhomogeneities on a measurement of the Hubbleparameter. Our aim was to address the recent tension between locally mea-sured values of H using SN1a (Riess et al., 2018a), and that inferred fromthe CMB (Planck Collaboration et al., 2018), which has been suggested tobe caused by local inhomogeneities (see Section 1.3.3). We defined the ef-fective Hubble parameter within a chosen domain to be the averaged ex-pansion rate of the fluid within that domain, and calculated the varianceas a function of domain size. From this we estimated the expected vari-ance on a local measurement of H using SN1a within a particular red-shift interval, purely due to an observers physical location in an inhomoge-neous universe. Due to computational limits on resolution, and thereforethe physical box size of our simulations, we could not sample the full red-shift range, . (cid:46) z (cid:46) . , used for the local SN1a measurement in Riess .2. Future work (+0 . , − . ; not sufficient to ex-plain the current . σ tension. It is therefore unlikely that the tension canbe explained purely due to local inhomogeneities. This result is subject toseveral caveats — and the importance of these remains to be investigated —including periodic boundary conditions, the fluid treatment of dark matter,and the validity of using averaging to approximate a measurement of theHubble parameter. Even small variances on our measurements could beimportant in upcoming precision survey data, and therefore investigationinto the effect of these caveats is necessary. Understanding the role of GR on our observations is imperative as we moveinto the era of precision cosmology. Upcoming surveys are expected to pro-duce data at percent-level precision, and to ensure we correctly interpretthese data we must first validate the accuracy of the underlying assump-tions of our cosmological model. The work presented in this thesis wasa step towards this goal, providing essential tests of the required compu-tational framework and early quantifications of the effect of structure for-mation on the large-scale evolution of the Universe. This truly is only thebeginning of this field, and many aspects of the work presented here can beimproved on to solidify and extend our results.
Recently, Buchert, Mourier, and Roy (2018) pointed out some potential is-sues in the averaging formalism of Larena et al. (2009), which we usedto analyse our simulations in Chapters 4 and 5. Similar issues are alsopresent in the averaging procedures of Brown, Behrend, and Malik (2009)and Gasperini, Marozzi, and Veneziano (2010) (see also Umeh, Larena, andClarkson, 2011).The main issue with these formalisms is that the domain of averagingis non-conservative; the fluid is free to flow into and out of the domainof averaging because it is propagated along the normal to the hypersur-faces, rather than the fluid normal. These averaging schemes are thereforebased on an “extrinsic approach” of studying the averaged fluid quantitiesas seen by observers located in the spatial hypersurfaces defined by the nor-mal vector (which does not coincide with the fluid four velocity). Instead,Buchert, Mourier, and Roy (2018, and in a forthcoming publication) de-rive a coordinate-independent averaging formalism for general foliationsof spacetime to study average fluid quantities as seen by fluid observers;an “intrinsic approach”, which we outlined in Section 1.5.2. The domainsof averaging in this formalism are mass conserving, since they are propa-gated along the fluid four velocity vector.Including the lapse function in the definition of H h D , e.g. in (1.99), istechnically arbitrary (see Umeh, Larena, and Clarkson, 2011), however, in-cluding it ensures a covariant expression for the Hubble parameter. Sincethe effective fluid scale factor is subsequently defined from this, a transfor-mation of time will therefore give a vastly different scale factor if α is not26 Chapter 6. Conclusions included in the definition (recently pointed out by Buchert, Mourier, andRoy, 2018, and in a forthcoming publication via private communication).We include the lapse in the relation between the fluid and volume scale fac-tors for the analysis presented in Chapter 4. However, we do not includeit when calculating the Hubble parameter or the cosmological parameters.We do not expect this to significantly affect the magnitude of the cosmologi-cal parameters, although the inhomogeneous nature of the lapse could havean effect on the kinematic backreaction term itself.It is important to verify the effect of including the lapse in all averages on the cosmological parameters, kinematic backreaction, and average cur-vature. We leave this investigation, along with a general improvement ofour averaging scheme to the newly suggested intrinsic approach, to futurework.
While the averaging procedures outlined in Section 1.5.2 are useful for study-ing the large-scale evolution of our Universe compared to the homoge-neous, isotropic equivalent, they are explicitly dependent on the chosenaveraging domain. Adamek et al. (2019) showed backreaction can differby 3-5 orders of magnitude depending on the spatial hypersurface chosenfor averaging. However, whether the authors are actually measuring back-reaction here, as opposed to cosmic variance , is a point of contention (seeBuchert, Mourier, and Roy, 2018).Connecting the results of averaging to our observations is not so clear,since for cosmology, the fact that our observations are made along our pastlight cone — and not on a purely spatial hypersurface — can become impor-tant (see e.g. Buchert and Räsänen, 2012). While the global expansion ratewill affect the redshift and distance of objects, we really must study the pastlight cone in an inhomogeneous Universe to determine the full, measurableeffect.Light propagation in inhomogeneous cosmology is not a new field (e.g.Zel’dovich, 1964; Tomita, 1998; Rose, 2001; Kostov, 2010; Bolejko, 2011;Fleury, Dupuy, and Uzan, 2013), however, application of this work to nu-merical simulations has only recently emerged. Giblin, Mertens, and Stark-man (2016b) studied the effects on the Hubble diagram in their inhomoge-neous cosmological simulations (see Mertens, Giblin, and Starkman, 2016,for details), and found overall agreement with FLRW. The structures con-sidered in Giblin, Mertens, and Starkman (2016b) are of large wavelengthsonly (with small amplitudes), and therefore the lensing effects are wellapproximated by perturbation theory, with no significant deviations fromFLRW. Extending this work to more realistic matter distributions, i.e. in-cluding small-scale structures that can have large amplitudes — and there-fore result in significant lensing — is essential to determine the expectedscatter on the Hubble diagram we measure. This scatter has been esti-mated in the context of N-body simulations adopting the weak-field ap-proximation (Adamek et al., 2013; Adamek et al., 2016b), and was foundto be small. In addition, a percent-level bias was found on the measuredcurvature parameter due to relativistic effects (Adamek et al., 2018). Thesesimulations are however dependent on the weak-field approximation, andperhaps more importantly on the assumption of a background, flat FLRW .2. Future work
Throughout this thesis we have adopted a fluid approximation for the mat-ter content of the Universe, as in most numerical-relativity simulations.This approximation usually presents no issue in the context of most as-trophysical phenomena in GR (see, e.g. Font, 2008), however, it can beproblematic for cosmology. The reason for this is the common adoptionof the comoving gauge in theoretical relativistic cosmology, which is nowused in numerical-relativity simulations (Giblin, Mertens, and Starkman,2016a; Bentivegna and Bruni, 2016). Simulations of nonlinear structure for-mation in this gauge will fail because shell-crossing singularities form asthe structures collapse; since the coordinates coincide with the fluid flowlines (see Section 2.1.4). These singularities can be avoided with the ad-dition of a small amount of pressure (e.g. Bolejko and Lasky, 2008), or bysimply adopting a different gauge for cosmological simulations (Macpher-son, Lasky, and Price, 2017; East, Wojtak, and Abel, 2018). However, theassumption of a continuous fluid itself also means we cannot properly cap-ture the process of structure formation. As a galaxy or galaxy cluster col-lapses, it will reach a point at which its internal velocity dispersion preventsfurther collapse; and the structure becomes virialised . This is an importantaspect of structure formation in the Universe which cannot be captured nat-urally in the case of a continuous fluid.An effective virialisation technique was implemented in Bolejko (2018b),where regions were no longer evolved once they began to collapse, i.e.when they had Θ < . In this case, voids continue expanding and an over-all, global negative curvature arises, which has been shown to explain thedimming of SN1a observations and the tension in the Hubble parameter(Bolejko, 2018a). This suggests that virialisation is an important aspect ofinhomogeneous cosmology (see also Roukema, 2018).Discretising the fluid with particles, as is done in Newtonian cosmo-logical simulations (see Section 1.3.4), allows virialised objects to form nat-urally. Early work adopting collisionless particles in numerical-relativitysimulations focused on stellar collapse and black-hole or singularity for-mation (Shapiro and Teukolsky, 1985; Shapiro and Teukolsky, 1986; Shapiroand Teukolsky, 1991; Shibata, 1999). More recent applications include black-hole formation from gravitational waves (Pretorius and East, 2018) andnon-spherical gravitational collapse (Yoo, Harada, and Okawa, 2017).In a cosmological context, a particle description of matter has been adoptedin the weak-field limit (Adamek et al., 2013; Adamek, Durrer, and Kunz,2014), and also recently incorporated into numerical-relativity simulations(Giblin et al., 2018; Daverio, Dirian, and Mitsou, 2019; Barrera-Hinojosa and28 Chapter 6. Conclusions
Li, 2019, with each of these codes in the active development and testingphase). Ensuring we have a collection of independent codes with which toperform comparisons and validate results is essential for the advancementof this field. Extending already widely-used, open-source software, such asthe E
INSTEIN T OOLKIT , for this purpose will allow for contribution to thisfield from the wider community.
In this work we used the BSSN formalism of Einstein’s equations for ourcosmological simulations. As discussed in Section 2.1.5, there are alterna-tive formalisms that have been developed to damp the growth of constraintviolation by evolving the constraint variables themselves. Any constraintviolating modes can therefore be propagated off the grid. The conformaland covariant Z4 system (CCZ4; Alic et al., 2012) is an example we dis-cussed in Section 2.1.5. This has been implemented in the numerical rela-tivity formalism of Giblin et al. (2018) used for cosmological simulations,and in the GRC
HOMBO numerical relativity package (Clough et al., 2015);used for simulations of inflationary cosmology (see, e.g. Clough et al., 2017).Comparing the BSSN formalism and the CCZ4 system for the simulationspresented here is beyond the scope of this thesis. However, future workcomparing the amplitude and evolution of constraint violations in the E IN - STEIN T OOLKIT for both of these formalism is important in choosing thebest evolution system for cosmological simulations.The use of periodic boundary conditions is common in cosmologicalsimulations in Newtonian gravity (e.g. Boylan-Kolchin et al., 2009; Genelet al., 2014; Potter, Stadel, and Teyssier, 2017), and in the case of numer-ical relativity (e.g. Giblin, Mertens, and Starkman, 2016a; Bentivegna andBruni, 2016; Macpherson, Lasky, and Price, 2017; Daverio, Dirian, and Mit-sou, 2017; Barrera-Hinojosa and Li, 2019). Choosing boundary conditionsinherently defines the topology of the domain, with periodic boundary con-ditions corresponding to a three-torus in Cartesian coordinates. In 2+1 di-mensional GR the chosen topology sets the averaged curvature evolution,and in the case of periodic boundaries the averaged curvature will alwaystend towards zero; and hence towards a homogeneous, FLRW expansion.However, in 3+1 dimensional GR the connection between topology and cur-vature remains unclear (see, e.g. Buchert and Räsänen, 2012). The globallyzero curvature and homogeneous expansion seen in Chapter 4 could bea consequence of our chosen topology. To properly test this, a numericalrelativity code utilising a different coordinate system, e.g. spherical coordi-nates, must be used.29
Appendix A
Newtonian Gauge
Throughout Chapter 3 we work in the longitudinal gauge. For complete-ness, we show here the equivalent background and perturbation equationsin the Newtonian gauge. In this Appendix we use geometric units with G = c = 1 . The flat FLRW metric is ds = − d ˆ t + a (ˆ t ) dx i dx j δ ij , (A.1)which gives the Friedmann equations for a dust ( P (cid:28) ρ ) universe to be (cid:18) ˙ aa (cid:19) = 8 πρ , (A.2a) ˙ ρ = − aa ρ, (A.2b)where a dot represents d/d ˆ t . Solutions to these equations give the familiartime dependence of the scale factor, aa init = s / , ρρ init = s − , (A.3)where s ≡ (cid:112) πρ ∗ ˆ t. (A.4)We match our numerical evolution to this alternative set of solutions byinstead making the coordinate transform t = t (ˆ t ) . With this we see the ex-pected fourth-order convergence and maximum errors in the scale factorand density of ∼ − for our highest resolution ( ) simulation. Fig-ure A.1 shows the convergence of the scale factor (left), density (middle)and Hamiltonian constraint (right) for analysis performed in this gauge.The linearly perturbed FLRW metric in this gauge, including only scalarperturbations, is ds = − (1 + 2 ψ N ) d ˆ t + a (ˆ t )(1 − φ N ) δ ij dx i dx j , (A.5)where ψ N , φ N are not the usual gauge-invariant Bardeen potentials (whichare defined in the longitudinal gauge). Solving the perturbed Einstein equa-tions (3.6) in this gauge using the time-time, time-space, trace and trace free As in Chapter 3, the density ρ here is the total rest-frame energy-density, and is equiva-lent to ρ R used in Chapters 1 and 2. Appendix A. Newtonian Gauge
20 30 40 50 60 70 80
Resolution ( N ) L e rr o r N a
20 30 40 50 60 70 80
Resolution ( N ) N r
20 30 40 50 60 70 80
Resolution ( N ) N H c F IGURE
A.1: Fourth-order convergence of the FLRW solu-tions analysed in the Newtonian gauge. We show L erroras a function of resolution for the scale factor (left), den-sity (middle), and Hamiltonian constraint (right). N refersto the number of grid points along one spatial dimension.Filled circles indicate data points from the simulations, andblack solid lines indicate the expected N − convergence. components gives ∂ φ N − a ˙ a (cid:18) ˙ φ N + ˙ aa ψ N (cid:19) = 4 π ¯ ρ δa , (A.6a) ˙ aa ∂ i ψ N + ∂ i ˙ φ N = − π ¯ ρ a δ ij δ ˆ v j , (A.6b) ¨ φ N + ˙ aa (cid:16) ˙ ψ N + 3 ˙ φ N (cid:17) = 13 a ∂ ( φ N − ψ N ) , (A.6c) ∂ (cid:104) i ∂ j (cid:105) ( φ N − ψ N ) = 0 , (A.6d)in the linear regime. Solving these equations we find the form of the poten-tial φ N to be φ N = f ( x i ) − s − / g ( x i ) , (A.7)where f, g are functions of the spatial coordinates. From this we find thedensity and velocity perturbations to be, respectively, δ = C s / ∂ f ( x i ) − f ( x i ) (A.8a) + 3 C s − ∂ g ( x i ) − a s − / g ( x i ) ,δ ˆ v i = C s − / ∂ i f ( x i ) + 3 C s − ∂ i g ( x i ) , (A.8b)where the C i were defined in (3.23) and (3.25). We set g ( x i ) = 0 to extractonly the growing mode of the density perturbation, giving exact solutionsto be φ N = f ( x i ) , (A.9a) δ = C s / ∂ f ( x i ) − f ( x i ) , (A.9b) δ ˆ v i = C s − / ∂ i f ( x i ) . (A.9c)We note that these solutions give equivalent initial conditions to those foundin Section 3.4.1 since, initially, s = ξ = 1 . The velocity here is ˆ v i ≡ dx i /d ˆ t , which differs from the velocity used in Chapter 3. ppendix A. Newtonian Gauge t = t (ˆ t ) .We find the expected second-order convergence with maximum errors inthe density and velocity perturbations of ∼ − for our highest resolution( ) simulation. Figure A.2 shows the convergence of the density (left) andvelocity (right) perturbations when analysed in the Newtonian gauge.
20 30 40 50 60 70 80
Resolution ( N ) − − L e rr o r N − δ
20 30 40 50 60 70 80
Resolution ( N ) − − − N − δ v F IGURE
A.2: Second-order convergence of the numericalsolutions for a linearly perturbed FLRW spacetime, anal-ysed in the Newtonian gauge. We show L errors in thedensity (left) and velocity perturbations (right). N refersto the number of grid points along one spatial dimension.Filled circles indicate data points from our simulations, andblack solid lines indicate the expected N − convergence. Appendix B
Averaging in the non-comovinggauge
Averaging Einstein’s equations in a non-comoving gauge results in the av-eraged Hamiltonian constraint H D = 16 π (cid:104) W ρ (cid:105) − R D − Q D + L D , (B.1)where we define R D ≡ (cid:104) W R(cid:105) , (B.2) Q D ≡ (cid:0) (cid:104) θ (cid:105) − (cid:104) θ (cid:105) (cid:1) − (cid:104) σ (cid:105) , (B.3) L D ≡ (cid:104) σ B (cid:105) − (cid:104) θ B (cid:105) − (cid:104) θθ B (cid:105) . (B.4)Here, W = 1 / (cid:112) − v i v i is the Lorentz factor, R ≡ γ ij R ij is the three-dimensional Ricci curvature of the averaging hypersurfaces, with R ij thespatial Ricci tensor. In this Appendix we work in geometric units with G = c = 1 . Here σ = 12 σ ij σ ji , (B.5)where σ ij is the shear tensor, defined as σ µν ≡ h αµ h βν ∇ ( α u β ) − θh µν . (B.6)As in (Umeh, Larena, and Clarkson, 2011), we introduce for simplification σ B = 12 σ iBj σ jBi + σ ij σ ijB (B.7a) σ Bij ≡ −
W β ij − W (cid:18) B ( ij ) − Bh ij (cid:19) (B.7b) θ B ≡ − W κ − W B (B.7c) β µν ≡ h αµ h βν ∇ ( α v β ) − κh µν (B.7d) B µν ≡ κ ( v µ n ν + v µ v ν ) + β αµ v α n ν + β αµ v α v ν (B.7e) + M αµ v α n ν + M αµ v α v ν , (B.7f)34 Appendix B. Averaging in the non-comoving gauge where we also define κ ≡ h αβ ∇ α v β , M µν ≡ h αµ h βν ∇ [ α v β ] , (B.8a) B = 13 κv α v α + β µν v µ v ν . (B.8b)For a given tensor A µν we adopt the notation A ( µν ) = ( A µν + A νµ ) and A [ µν ] = ( A µν − A νµ ) .35 Appendix C
Effective scale factors
The effective expansion of an inhomogeneous domain can be defined by ∂ η a V D a V D ≡ ∂ η V D V D , (C.1) ⇒ a V D = (cid:18) V D ( η ) V D ( η init ) (cid:19) / , (C.2)where V D ( η ) is the volume of the domain D at a given conformal time. Thephysical interpretation of this scale factor depends on the chosen hyper-surface of averaging. If we choose the averaging surface to be comovingwith the fluid; a surface with normal u µ , then the scale factor a V D describesthe effective expansion of the fluid averaged over the domain. We definethe averaging surface to be comoving with a set of observers with normal n µ ; not coinciding with u µ . In this case, a V D describes the expansion of thevolume element, not of the fluid itself.We define the Hubble parameter as the expansion of the fluid projectedinto the gravitational rest frame; the frame of observers with normal n µ .From this we define the effective scale factor of the fluid, a D in (4.29). Wecan relate the two scale factors by first considering the rate of change of thevolume (with β i = 0 ) in the (3+1) formalism (Larena, 2009) , ∂ η V D V D = (cid:104) αW ( θ − κ ) (cid:105) . (C.3)Now, with ∂ η a D /a D = ∂ η ln( a D ) , we can write ∂ η ln( a D ) = 13 (cid:104) αθ (cid:105) , (C.4) ∂ η ln( a V D ) = 13 (cid:104) αW ( θ − κ ) (cid:105) , (C.5)subtracting (C.5) from (C.4) we arrive at the relation a D = a V D exp (cid:18) − (cid:90) (cid:104) αW ( θ − κ ) − αθ (cid:105) dη (cid:19) . (C.6)Here, a V D is found by calculating the volume of the domain relative to theinitial volume. Figure 4.6 shows the evolution of (C.6) (blue solid curve) The fluid scale factor a D here is equivalent to a h D used in Chapter 1 and Section 2.3. After the publication of this paper, we noticed an error in (C.3), (C.5), and (C.6) fromLarena (2009). In Appendix F we show this error has negligible effect on our results. Appendix C. Effective scale factors as a function of redshift for a simulation over an L = 1 Gpc domain,relative to the equivalent FLRW solution (purple dashed curve).37
Appendix D
Constraint violation
In numerical relativity, the error can be quantified by analysing the viola-tion in the Hamiltonian and momentum constraint equations, defined by H ≡ R + K − K ij K ij − πρ = 0 , (D.1a) M i ≡ D j K ji − D i K − πS i = 0 , (D.1b)respectively, where S i ≡ − γ iα n β T αβ and D i represents the covariant deriva-tive associated with the three-metric γ ij . We define the magnitude of themomentum constraint to be M = (cid:112) M i M i . In this Appendix we adopt ge-ometric units with G = c = 1 . An exact solution to Einstein’s equationswill identically satisfy (D.1). Since we are solving Einstein’s equations nu-merically, we expect some non-zero violation in the constraints. We use the MESCALINE code, described in Section 2.3 and 4.5.2, to calculate the con-straint violation as a function of time.For the simulations we present in this work, we do not expect (in gen-eral) to see convergence of the constraint violation. At each different resolu-tion we are sampling a different section of the power spectrum, and hencea different physical problem. In order to see convergence of the constraintsat the correct order, we must analyse a controlled case in which the gradi-ents are kept constant between resolutions. We perform three simulationsat resolutions , , and inside an L = 1 Gpc domain. We gener-ate the initial conditions for the simulation using CAMB; restricting theminimum sampling wavelength to be λ min = 10∆ x = 312 . Mpc. We uselinear interpolation to generate the same initial conditions at and .Figure D.1 shows the violation in the Hamiltonian (top panels) and mo-mentum (bottom panels) constraints, for the set of simulations with a con-trolled number of physical modes, as a function of effective redshift. Leftpanels show the raw L error for the violation, which for the Hamiltonianconstraint we define as L ( H ) = 1 N N (cid:88) a =1 | H a | , (D.2)where N is the total number of grid cells, and H a is the Hamiltonian con-straint violation at grid cell a , and similarly for the momentum constraint.To quantify the "smallness" of this violation, we normalise the constraintviolations to their relative "energy scales". Similar to Mertens, Giblin, and38 Appendix D. Constraint violation -18 -16 -14 -12 L ( H ) N = 32 N = 64 N = 128 -2 -1 z -22 -20 -18 -16 -14 L ( M ) -6 -4 -2 L ( H / [ H ] ) -2 -1 z -2 -1 L ( M / [ M ] ) F IGURE
D.1: Relative (right panels) and raw (left panels)constraint violation as a function of effective redshift calcu-lated using (D.4) and (D.2), respectively. We show viola-tions for the simulations with a controlled number of phys-ical modes. Top panels show the Hamiltonian constraint vi-olation, and bottom panels show the momentum constraintviolation. Colours show different resolutions as indicatedby the legend. ppendix D. Constraint violation N -12 L e rr o r N − H N -15 -14 L e rr o r N − M z = 1100 N -18 -17 L e rr o r N − H N -22 -21 L e rr o r N − M z = 0 . F IGURE
D.2: Second order convergence of the Hamiltonianand momentum constraints for the set of simulations with afixed number of physical modes. We show the L error cal-culated using (D.2) for both constraints. The top two panelsshow the L error for the initial conditions, at z = 1100 , andthe bottom two panels show the L error for z = 0 . Appendix D. Constraint violation
Starkman (2016) and Giblin, Mertens, and Starkman (2017), we define [ H ] ≡ (cid:113) R + ( K ) + ( K ij K ij ) + (16 πρ ) , (D.3a) [ M ] ≡ (cid:113) ( D j K ji )( D k K ki ) + ( D i K )( D i K ) + (8 π ) S i S i . (D.3b)Right panels in Figure D.1 show the relative L error for each constraintviolation, which we define as L ( H/ [ H ]) = N (cid:80) Na =1 | H a | N (cid:80) Na =1 [ H ] a , (D.4)where [ H ] a is the energy scale calculated at grid cell a , and similarly for themomentum constraint. We take the positive root of both [ H ] a and [ M ] a .Figure D.2 shows the raw L error for the same set of simulations, forthe initial data ( z = 1100 ; top two panels) and for the data at z = 0 (bot-tom two panels). The left panels show the L error for the Hamiltonianconstraint, and the right panels show the L error for the momentum con-straint. We see the expected second order convergence for the E INSTEIN T OOLKIT , with the exception of the N = 128 simulation’s violation in themomentum constraint. Our initial speculation was that this was roundofferror, given the smallness of the quantities involved. The top panels of Fig-ure D.2 show that this issue is not due to the non-convergence of our initialdata. Whether or not this is roundoff error remains unclear, and cannot beclarified without re-performing our simulations in quad precision.For the simulations with a controlled number of modes, at z = 0 for res-olution N = 128 the relative Hamiltonian constraint violation is L ( H/ [ H ]) =1 . × − , the momentum constraint is L ( M/ [ M ]) = 2 . × − . For thesimulations with full power spectrum sampling, at z = 0 and resolution N = 256 we find L ( H/ [ H ]) = 4 . × − , and L ( M/ [ M ]) = 5 . × − .41 Appendix E
Convergence and errors
In Appendix D we discussed the convergence of the constraint violation,which for the main simulations presented in Chapter 4 we do not expect toreduce with resolution, since the physical problem is changing. Regardlessof this, we expect the power spectrum of fractional density fluctuations, δ ,to be converged in these simulations. Coloured curves in Figure E.1 showthe z = 0 power spectrum of the fractional density fluctuations for resolu-tions , , and , each within an L = 1 Gpc domain. The dashedcurve shows the linear power spectrum at z = 0 calculated with CAMB.The pink shaded region shows one-dimensional scales of − Mpc,roughly corresponding to scales which are converged. Below these scaleswe therefore underestimate the growth of structures, and hence underesti-mate the contribution from curvature and backreaction in our calculations.Figure E.2 shows the global cosmological parameters as a function ofredshift for simulations sampling the full power spectrum (left panel), andfor those with a restricted sampling of the power spectrum; our controlledcase discussed in the previous section (right panel). Dotted, dashed, andsolid curves show different resolutions as indicated in each seperate leg-end. Blue curves show the density parameter Ω m , green curves show thecurvature parameter | Ω R | , and purple curves show the backreaction param-eters | Ω Q + Ω L | . As resolution increases in the left panel — as we add moresmall-scale structure — the contributions from curvature and backreactionincrease, however still remain negligible relative to the matter content, Ω m ,and are unlikely to grow large enough to be significant when reaching a re-alistic resolution. In the right panel, we have kept the physical problem con-stant and varied only the computational resolution, and so we see all globalparameters converged towards a single value. Comparing the left panel tothe right panel, the value of the curvature and backreaction parameters dif-fer by almost an order of magnitude. This is due to the restricted powerspectrum sampling for the simulations in the right panel, in which we onlysample structures down to λ min = 10∆ x = 312 . Mpc. These simulationsshould therefore not be considered the most realistic representation of ourUniverse. From this comparison we see that adding more structure results(in general) in a larger contribution from curvature and backreaction.We calculate the errors in the cosmological parameters using a Richard-son extrapolation, which requires the gradients between resolutions to re-main the same. This is not the case for the simulations with full powerspectrum sampling, however it is the case for the controlled simulationswith a restricted mode sampling. We therefore use the controlled simula-tions to approximate the errors for our main calculations. Figure E.3 showsthe values of the globally averaged cosmological parameters at z = 0 for the42 Appendix E. Convergence and errors -2 -1 k [Mpc − ] P ( k ) [ M p c ] linear P ( k ) N = 64 N = 128 N = 256100 −
150 Mpc F IGURE
E.1: Power spectrum of fractional density fluctu-ations δ at z = 0 . Solid coloured curves show P ( k ) forthree simulations with , , , each for an L = 1 Gpc domain, and the dashed curve shows the linear powerspectrum at z = 0 . The pink shaded region represents one-dimensional scales of − Mpc.
0. 00. 20. 40. 60. 81. 0 z ( × ) -12 -10 -8 -6 -4 -2 Full power spectrum sampling Ω m | Ω R || Ω Q + Ω L |
0. 00. 20. 40. 60. 81. 0 z ( × ) -12 -10 -8 -6 -4 -2 Restricted power spectrum sampling Ω m | Ω R || Ω Q + Ω L | F IGURE
E.2: Global cosmological parameters as a func-tion of effective redshift, for simulations with full powerspectrum sampling (left panel) and a controlled number ofphysical modes (right panel). Dotted, dashed, and solidcurves show resolutions as indicated in each seperate leg-end. Blue curves show Ω m , green curves show | Ω R | , andpurple curves show | Ω Q + Ω L | . ppendix E. Convergence and errors N Ω m N | Ω R | ( × . − N z = 0.0 | Ω Q + Ω L | ( × . − F IGURE
E.3: Cosmological parameters measured at z =0 for simulations with a controlled number of physicalmodes. Coloured points show Ω m (left), | Ω R | (middle),and | Ω Q + Ω L | (right) for resolutions N = 32 , , and .Dashed curves are the convergence fit for each parameter,detailed in the text. We use these curves for a Richardsonextrapolation to calculate the true value of the parametersand hence the errors on our measurements.
80 100 120 140 160 180 200 220 240 r D (Mpc) -4 -3 -2 -1 % e rr o r N = 128 Ω m Ω R Ω Q + Ω L F IGURE
E.4: Richardson extrapolated errors for cosmolog-ical parameters calculated within subdomains, for the resolution controlled simulation, as a function of averagingradius r D . Blue points show the percentage error for Ω m ,green points for Ω R , and purple points for Ω Q + Ω L . Appendix E. Convergence and errors controlled simulations. Coloured points show Ω m (left panel), | Ω R | (middlepanel), and | Ω Q + Ω L | (right panel) at resolutions N = 32 , , and . Weuse the function curve_fit as a part of the SciPy Python package to fiteach set of points with a curve of the form Ω i ( N ) = Ω inf + E × N − , where Ω inf is the value of the relevant cosmological parameter at N → ∞ , and E is a constant. Black dashed curves in each panel of Figure E.3 show thebest-fit curves.The best-fit value for Ω inf provides an approximation of the correct valueof each cosmological parameter for this set of test simulations. The residualbetween our calculations and Ω inf gives the error in our calculations. Forthe controlled simulation with resolution, the errors in the global cos-mological parameters are − , × − , and × − for Ω m , Ω R , and Ω Q + Ω L , respectively. Expressed as a percentage error, these are − % ,0.27%, and 1.9%.We follow the same procedure to estimate the errors on the cosmologicalparameters calculated within subdomains. Figure E.4 shows the percentageerror in each parameter as a function of averaging radius of the subdomain, r D , for the controlled simulation with resolution. Blue points show theerror for Ω m , green points show Ω R , and purple points show Ω Q + Ω L . Thejump in errors evident at ∼ Mpc in Ω R and Ω Q + Ω L is due to a changein sign of the curvature and backreaction parameters. https://scipy.org Appendix F
Effective scale factors part 2:investigation into an error
In Section 1.5.2 we introduced the averaging formalism of Larena (2009) forgeneral foliations of spacetime. Here, we detail an error in Larena’s equa-tion (30) for the rate of change of √ h , subsequently carried through intoequation (31) for the rate of change of the volume, and equation (34) relat-ing the fluid and volume scale factors — used for our analysis in Chapter 4.Here we re-analyse the simulations presented in Chapter 4 and find the er-ror makes a difference of ≈ − globally, and ≈ − on 100 Mpc scales,both at z ≈ . In this Appendix, we work in units with c = 1 .Expressing the evolution equation for the spatial metric (2.19) in termsof the fluid variables defined in Section 1.5.2, we arrive at equation (24) inLarena (2009), Wα ∂ t h ij = 23 ( θ + θ B ) h ij + 2 ( σ ij + σ Bij ) + 2
Wα D ( i β j ) , (F.1)and taking the trace of this results in an evolution equation for √ h √ h ∂ t √ h = αW ( θ + θ B ) + D i β i , (F.2)where we have used (2.45) and the fact that both σ ij and σ Bij are traceless.The equivalent to equation (F.2) in Larena’s paper, equation (30), reads √ h ∂ t √ h = αW ( θ − κ ) + D i β i . (F.3)Comparing (F.2) and (F.3) we can see there is a difference of θ B → − κ . InLarena (2009), this error is propagated into equation (31) for the evolutionof the volume, and subsequently into equation (34) for the relation betweenthe effective volume and fluid scale factors, which reads a V D = a h D exp (cid:18)(cid:90) tt init (cid:104) αW ( θ − κ ) − αθ + D i β i (cid:105) h dt (cid:19) . (F.4)We find this relation, instead using the evolution of the volume derivedfrom (F.2), to be a V D = a h D exp (cid:18) (cid:90) tt init (cid:104) αW ( θ + θ B ) − αθ + D i β i (cid:105) h dt (cid:19) , (F.5)see Section 1.5.2 for the derivation.46 Appendix F. Effective scale factors part 2
We used (F.4) for our analysis in Chapter 4, however, only the calculationof the fluid scale factor is affected, and all other analysis remains valid.We also note the missing factor of / in the exponential in (F.4) that was corrected in our analysis in Chapter 4.Starting from the definition of θ B in (1.92), and using the trace of B µν (1.96) as B = κv i v i + β ij v i v j , we find θ B = − W κ − W κv i v i − W β ij v i v j , (F.6) = − W κ − W v i v j ∇ i v j , (F.7)where we have substituted β ij from (1.87). Even for small velocities, with W ≈ , it is not clear that θ B ≈ − κ since the gradient of the velocity maynot be small, and so the second term in (F.7) is not obviously negligible.It is therefore important to verify whether this makes a difference to thecalculation of the fluid scale factor.We perform the same analysis outlined in Chapter 4, on the same sim-ulations, but instead using (F.5) to calculate the effective fluid scale factor, a h D , from the volume scale factor, a V D . We average over the whole domainas well as within subdomains, i.e. we reproduce the calculations in Fig-ures 4.6 and 4.9, respectively, and compare the evolution with and withoutthe error.The top panel of Figure F.1 shows a h D calculated over the entire domainas a function of effective redshift, for a h D ( − κ ) (error present; solid blackcurve), and a h D ( θ B ) (error corrected; dashed red curve). The bottom panelshows the relative difference, i.e. a h D ( − κ ) /a h D ( θ B ) − , (F.8)which remains below − , even at z ≈ .Each curve in Figure F.2 shows the relative difference (F.8) for an indi-vidual sphere of radius r D = 100 Mpc as a function of effective redshift. Wehave coloured the curves depending on their averaged density contrast at z ≈ , with over-dense regions in blue and under-dense regions in purple.The curves differ from one another because each sphere contains differ-ent structures, and therefore will contain different velocity gradients. Fig-ure F.3 shows the relative difference (F.8) for all 1000 spheres sampled with r D = 100 Mpc at three different redshifts z = 0 . , . , and . , as a functionof the averaged density contrast. At high redshift, we see the spread in δ isless, implying smoother structures and therefore smaller velocity gradients.At z ≈ the maximum difference still remains below × − .We have detailed an error we found in the derivation of Larena’s gen-eralised averaging formalism, which was used in our calculation of thefluid scale factor. We found maximum differences between the globally-averaged and subdomain-averaged fluid scale factors, with and withoutthe error fixed, of − and × − , respectively. These differences arenegligible compared with our estimated errors (see Appendix E). We note that in Chapter 4 we denote the fluid scale factor by a D , equivalent to a h D inthis Appendix, and in Chapter 1. ppendix F. Effective scale factors part 2 a h D ( − ) a h D ( θ B ) -1 z -12 -10 -8 Relative difference F IGURE
F.1: Globally-averaged, effective fluid scale fac-tors with the error described in this section present, a h D ( − κ ) ;black solid curve, and with the error corrected, a h D ( θ B ) ; reddashed curve. Top panel shows the evolution as a func-tion of effective redshift, and the bottom panel shows therelative difference (F.8), which remains below − for theentire simulation. Appendix F. Effective scale factors part 2 -2 -1 z R e l a ti v e d i ff e r e n ce ( × − ) δ − . δ ≥ . F IGURE
F.2: Relative difference (F.8) between effective fluidscale factors (with and without the error corrected) calcu-lated in spheres of radius r D = 100 Mpc. Each curveshows the difference for an individual sphere, either over-or under-dense as indicated by the legend, as a function ofeffective redshift. The difference remains of order ≈ − throughout the simulation. ppendix F. Effective scale factors part 2
0. 1 0. 0 0. 1 0. 2 0. 3 δ R e l a ti v e d i ff e r e n ce ( × − ) z = 0 . z = 2 . z = 4 . F IGURE
F.3: Relative difference (F.8) between effective fluidscale factors (with and without the error corrected) cal-culated in 1000 spheres of radius r D = 100 Mpc. Eachpoint shows an individual sphere measured at redshifts z = 0 . , . , . as indicated by the legend. As density con-trasts grow, so too does the difference between the effec-tive scale factors. The difference remains of order ≈ − throughout the simulation. Bibliography
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