Impact of cosmological signatures in two-point statistics beyond the linear regime
Dante V. Gomez-Navarro, Alexander Mead, Alejandro Aviles, Axel de la Macorra
MMNRAS , 1–14 (2020) Preprint 30 September 2020 Compiled using MNRAS L A TEX style file v3.0
Impact of cosmological signatures in two-point statistics beyond thelinear regime
D. V. Gomez-Navarro , A. J. Mead , A. Aviles , (cid:63) and A. de la Macorra , Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México C.P. 04510, México Institut de Ciències del Cosmos, Universitat de Barcelona, Martí Franquès 1, E08028 Barcelona, Spain Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582, Colonia Crédito Constructor, Del. Benito Jurez, 03940, Ciudad de México, México. Departamento de Física, Instituto Nacional de Investigaciones Nucleares, Apartado Postal 18-1027, Col. Escandón, Ciudad de México,11801, México.
30 September 2020
ABSTRACT
Some beyond Λ CDM cosmological models have dark-sector energy densities that suffer phasetransitions. Fluctuations entering the horizon during such a transition can receive enhance-ments that ultimately show up as a distinctive bump in the power spectrum relative to a modelwith no phase transition. In this work, we study the non-linear evolution of such signatures inthe matter power spectrum and correlation function using N -body simulations, perturbationtheory and HMCODE - a halo-model based method. We focus on modelling the response, com-puted as the ratio of statistics between a model containing a bump and one without it, ratherthan in the statistics themselves. Instead of working with a specific theoretical model, we in-ject a parametric family of Gaussian bumps into otherwise standard Λ CDM spectra. We findthat even when the primordial bump is located at linear scales, non-linearities tend to producea second bump at smaller scales. This effect is understood within the halo model due to amore efficient halo formation. In redshift space these nonlinear signatures are partially erasedbecause of the damping along the line-of-sight direction produced by non-coherent motionsof particles at small scales. In configuration space, the bump modulates the correlation func-tion reflecting as oscillations in the response, as it is clear in linear Eulerian theory; however,they become damped because large scale coherent flows have some tendency to occupy re-gions more depleted of particles. This mechanism is explained within Lagrangian PerturbationTheory and well captured by our simulations.
Key words: dark energy – large-scale structure of Universe – astroparticle physics
An understanding of the Universe within the context of the Λ CDMmodel is by now well established. High-precision measurements,such as those of the CMB from
Planck (Planck Collaboration2018) and large-scale structure in SDSS-IV collaboration (2020),including Baryon Acoustic Oscillations (BAO) and Lyman-Alphaforest observations, can individually be well understood within Λ CDM. However, there exist some observational problems withinthe model: there is a well-known tension in the estimated value ofthe rate expansion of the Universe, H , between early- and late-time observations (e.g., Verde et al. 2019), which may indicate theneed to extend the Λ CDM model; and weak gravitational lensingstudies (e.g., Abbott et al. 2018; Hikage et al. 2019; Heymans et al.2020) seem to prefer a lower fluctuation amplitude than would beexpected from CMB observations. Furthermore, from a theoreticalperspective, there is no comprehensive understanding of the nature (cid:63)
E-mail: [email protected] of dark matter and dark energy that Λ CDM requires to together ac-count for (cid:39)
95 per cent of the energy density of the Universe today.Extensions of the standard model of particle physics, such asin grand-unification theories (Zyla et al. 2020), may account fordark matter and dark energy. For example, the recently proposedBound Dark Energy (BDE) cosmology (de la Macorra & Almaraz2018; Almaraz & de la Macorra 2019) introduces a dark gaugegroup SU(3) similar to the strong QCD force in the standard model.The particles in this gauge group are massless and their cosmicabundance decays like radiation at early times. However, the un-derlying non-perturbative dynamics cause a phase transition to takeplace, at a scale factor a c , and the elementary particles form mas-sive bound states (equivalent to mesons and baryons in the stan-dard model of particle physics) and the lightest scalar field φ cor-responds to dark energy. The energy density of BDE dilutes as ρ ∼ a − at the phase transition scale factor a c and remains sub-dominant for a long period of time. Eventually BDE reappears dy-namically close to present time, thus accelerating the expansion ofthe universe. The transition leaves imprints on large-scale structurestatistics Almaraz et al. (2020) – for a model independent analy- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p sis see Jaber-Bravo et al. (2020) – and a signature is generated at ascale k T entering the horizon about the phase-transition time, whichshows up as a ‘bump’ in the matter power spectrum. We call mod-els that generate features through phase transitions Rapid DilutedEnergy Density (RDED) de la Macorra & et al. (2020), and BDEis one such example. Locating unusual features in the power spec-trum may help in elucidating the nature of dark energy, dark matteror physics beyond the standard model.In this work we are motivated by RDED effects in the lin-ear matter power spectrum. However, we will work in a model-independent way by introducing a parametrized bump to the linearmatter power spectrum that will vary in position and width. Westudy the signatures imprinted in the linear matter power spectrumand we follow them beyond the linear regime using different, butcomplementary, tools.Higher-order Perturbation theory (e.g., Bernardeau et al.2002) (PT) approaches can successfully describe the intermediate,quasi-linear scales of the matter clustering, with different perturba-tive schemes having different advantages. For instance, LagrangianPerturbation Theory (LPT) is very accurate in modelling the two-point correlation function, particularly the smearing and shift of theBaryon Acoustic Oscillation (BAO) peak, which although locatedat a large scale ( ∼ h − Mpc), is not well captured by the lineartheory. On the other hand, Standard (Eulerian) Perturbation The-ory (SPT) is more successful in describing the broadband powerspectrum, but poorly models the BAO (e.g., Tassev 2014; Baldaufet al. 2015). As soon as non-linearities become dominant, field ex-pansions become meaningless and perturbative approaches breakdown. At this stage, the dynamics of the self-gravitating dark mattersystem can be tracked accurately by N -body simulations, althoughthese have the disadvantage of being computationally expensive. The non-linear dark matter statistics, however, can be described viaso-called halo models (e.g., Seljak 2000; Peacock & Smith 2000;Cooray & Sheth 2002).How to accurately model the power spectrum for standard Λ CDM cosmologies, with standard linear spectra, has been studiedin detail: PT provides high accuracy at large scales or high redshiftsBernardeau et al. (2002), whereas at more deeply non-linear scalesone can either use fitting functions (e.g., Smith et al. 2003; Taka-hashi et al. 2012), halo-model based methods (e.g.,
HMCODE : Meadet al. 2015, 2016; Mead et al. 2020b) or emulators (e.g., Lawrenceet al. 2010, 2017), all of which have been tuned to reproduce thepower spectra measured in accurate, high-resolution simulations.Recently, attention has been focused on modelling the power spec-trum ‘response’, which is the ratio of two power spectra, with thenumerator typically the cosmology of interest and the denomina-tor typically a cosmology whose power spectrum is well known.The response has the virtue of being both easier to simulate, requir-ing only moderate resolution simulations, and easier to model. Ithas been shown that the response can be accurately modelled fordark-energy cosmologies (Casarini et al. 2016; Mead 2017), modi-fied gravity (Cataneo et al. 2019), massive neutrinos (Cataneo et al.2020) and even for the effects of baryonic feedback (Mead et al.2020a). Once an accurate model for the power-spectrum responsehas been developed, this can simply be converted into an accuratemodel for the power spectrum by multiplying by an accurate modelfor the power spectrum of the cosmology that appeared in the de-nominator when creating the response. This could be from a high-resolution simulation, fitting function,
HMCODE or an emulator.In this work we study the non-linear behavior of signaturesin two-point statistics that can be approximated by a bump in thelinear power spectrum, with our bump cosmology parametrized as in equation (9). We are mostly interested in the response of thebump cosmology compared to a standard Λ CDM cosmology withno bump. Henceforth, our response functions are constructed sim-ply by taking the ratios of non-linear statistics of cosmologies withbumps to a standard Λ CDM cosmology with no bump. We usecomplementary approaches to model non-linearities, for the realspace power spectrum, we use 1-loop SPT, the HMCODE model ,and low-resolution N -body simulations. We further consider the ef-fect of redshift space distortions in the power spectrum using theTNS model (Taruya et al. 2010) and the model of Scoccimarro(2004). The non-linear correlation function is obtained throughthe Convolution-LPT (CLPT) of Carlson et al. (2012); Vlah et al.(2015).This paper is organized as follows. In Section 2 we reviewhow cosmic phase transitions lead to different cosmological fea-tures; this section can be safely skipped by the reader interested inthe more phenomenological aspects of our work. In Section 3 weintroduce the parametric bump cosmology to be used in the rest ofthe work and we present specifications of our N -body simulationssuite employed to test the analytical methods. We also review dif-ferent analytical models to describe the redshift and real space mat-ter power spectrum, as well as the correlation function. In Section4, we present the numerical and analytical results for the responsefunctions. Finally we conclude in Section 5. Models beyond Λ CDM may leave different detectable features inthe matter power spectrum P ( k , a ) . These features can have dif-ferent origins. For instance, in modifications of General Relativity,as in Hordenski models Bellini et al. (2016); Pogosian & Silvestri(2016); Bayarsaikhan et al. (2020); Shi et al. (2020) or changes inthe evolution of the energy density, sound speed c s and anisotropicstress of the dark matter or dark energy Linton et al. (2018); de Put-ter et al. (2010); Koivisto & Mota (2006); Garcia-Arroyo et al.(2020); Mastache & de la Macorra (2019); Chandrachani Devi et al.(2019); Almaraz et al. (2020),as well having non-adiabatic termswhich may arise if we have different rest frames of these dark flu-ids. Besides the effect of structure formation due to the dark sector,the amount of baryonic matter GÃl’nova-Santos et al. (2015); Grif-fiths et al. (2001), deviations in the primordial power spectrum P s Kumazaki et al. (2011) and particle content beyond Λ CDM plays asignificant role in the determination of the matter power spectrum.Deviations from Λ CDM cosmologies impact the evolution ofthe energy density evolution ρ ( t , x ) at both the level of the back-ground and level of the density perturbations. The background evo-lution has a direct effect on cosmological distances, changing forexample the acoustic scale r s ( a (cid:63) ) or angular distances D A ( a ) orthe growth of structure δ ρ ( t , x ) leaving important signatures in thematter power spectrum as for example in Perenon et al. (2015);Gallego Cadavid et al. (2016); Batista & Pace (2013); Lee (2014);Gil-Marin et al. (2012).Another possibility considers the addition of extra radiationat early epochs, before recombination, to reconcile the sound hori-zon size with the current value of the Hubble constant. In this typeof extension, as a consequence of the additional extra relativisticspecies, the expansion rate and the growth of matter over-densities Recently, it has been shown that features in the power spectrum can beindeed well modeled within PT (Chen et al. 2020). https://github.com/alexander-mead/HMcode MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies are affected, leaving detectable signals in the matter power spec-trum. This extra density, henceforth ρ ex , may be considered in a cer-tain context as extra radiation or have the generic name early darkenergy (EDE) Linder & Robbers (2008); Calabrese et al. (2011).However, there is no precise definition of EDE models and theymay differ significantly on the evolution of the equation of state(EoS) w ex = p ex / ρ ex . For example, earlier efforts of EDE had anEoS close to w = − w = − z = O ( − ) Bartelmann et al.(2006); Grossi & Springel (2009); Fontanot et al. (2012); Franciset al. (2008), while more recently EDE models have w = − z < We are interested in the cosmological consequences, and particu-larly imprints on the matter power spectrum, generated by an ex-tra energy density ρ ex , beyond the standard Λ CDM, for a ≤ a c ,which dilutes rapidly after a transition takes place at the scale fac-tor a c , and with a mode k c = a c H ( a c ) crossing the horizon at such atime. For definiteness, we will assume that such component, hence-forth called Rapid Diluted Energy Density (RDED) de la Macorra& et al. (2020), tracks the leading background energy density, in ei-ther radiation, matter or DE domination epochs and we study thesedifferent cases.Let us consider a Λ CDM cosmology with and additional en-ergy density ρ ex , and cosmic abundance Ω ex ≡ ρ ex ρ sm + ρ ex = − H sm ( a ) H smx ( a ) , (1)where H sm and H smx are the Hubble parameters for the model withand without the RDED component, respectively.We consider thecase for which Ω ex is a constant for a ≤ a c , i.e. ρ ex tracks the lead-ing background energy density with w ex = w sm at the time of thetransition ( w sm = / w sm = ρ ex takes place at a c . At this time theEoS w ex ≡ p ex / ρ ex suffers a transition from w c ≡ w ex ( a c ) = w sm for a ≤ a c to w f ≡ w ( a f ) > w c for a f > a c . The value of ∆ w ≡ w f − w c > ∆ a ≡ ( a f − a c ) / a c set the steepnessof the transition of the energy density of ρ ex ( a c ) , i.e. how fast itdilutes. For a > a c we have ρ ex ρ sm = (cid:18) aa o (cid:19) − ∆ w , (2)with ∆ w ≡ w f − w c and for w c = / , w f = ∆ w = / Ω ex will decrease. Notice thatwe would expect a massive particle to go from being relativistic for T / m (cid:29) w c = / T / m (cid:28) w f = ∆ w = w f − w c = − / Ω ex .This property is not accomplished by an RDED component,hence it is beyond the standard model. However, it can be imple-mented using a scalar field φ , with energy density ρ φ = E k + V andpressure p φ = E k − V , where E k is the kinetic term and V the poten-tial energy (for a review see Copeland et al. 2006). The evolutionof the homogeneous scalar field φ ( t ) is given by the Klein Gordonequation ¨ φ + H ˙ φ + dVd φ = . (3)Widely used scalar potentials are either exponential or inverse power laws Steinhardt et al. (1999); de la Macorra & Stephan-Otto(2002), e.g. V ( φ ) = Λ e − αφ , V = Λ + n φ − n (4)with n a dimensionless constant while Λ , α − have mass dimen-sion. These potentials V have different behaviours depending onthe value of the parameters and the initial conditions of the scalarfield. The energy density ρ φ can track the background evolution(named tracker fields Steinhardt et al. (1999); Armendariz-Piconet al. (2001)) either in radiation or matter domination with w i = w i = / w f = V (cid:29) E k ) for a long periodof time, diluting ρ φ and generating a rapid dilution ρ ex situation.For example, for inverse power low (IPL) scalar field poten-tial V = Λ + n φ − n de la Macorra & Stephan-Otto (2002) the evolu-tion of φ goes through an epoch of kinetic-term domination with w =
1, before transitioning to potential-term domination regionwith w = − w ≈ − . n ∼ n = / n > w > − . n < n = / a c ∼ − . Inthis case the EoS is w c = / a c it leads to w f = w ∼ − z ∼ w ∼ − .
93 at present time. The complete evo-lution of w ( φ ) is a consequence of the dynamics of the equationof motion and is not set by hand. The BDE example satisfies thecondition ( ∆ w = w f − w c >
0) of RDED twice. The first is duringradiation domination at a c ∼ − with a k c = .
94 and the sec-ond takes place close to present time ( z ∼ . Matter perturbations are affected by RDED, leaving distinctive fea-tures in cosmological observables such as the matter perturbations.The amplitude of these perturbations is increased in RDED cos-mology compared to a standard Λ CDM cosmology, and a bumpfeature is produced in the ratio of power spectrum P smx / P sm . Thebump is generated for modes k > k c , with k c ≡ a c H ( a c ) , enteringthe horizon at a scale factor a < a c .We will sketch how this bump is generated and we will studyin detail the properties of these bumps in both the linear and non-linear approach. Let us explore here the linear regime of matterdensity fluctuations. In this case, the evolution of CDM perturba-tions obey the equation of motion: δ (cid:48)(cid:48) c + H δ (cid:48) c − H ∑ i Ω i δ i ( c s , i + ) = , (5)where the sum runs over all the fluids with sound speed c s , i = δ P i / δ ρ i and the prime denotes derivatives with respect to confor-mal time, with H = a (cid:48) / a . We see that the ρ ex has a twofold effect MNRAS000
94 and the sec-ond takes place close to present time ( z ∼ . Matter perturbations are affected by RDED, leaving distinctive fea-tures in cosmological observables such as the matter perturbations.The amplitude of these perturbations is increased in RDED cos-mology compared to a standard Λ CDM cosmology, and a bumpfeature is produced in the ratio of power spectrum P smx / P sm . Thebump is generated for modes k > k c , with k c ≡ a c H ( a c ) , enteringthe horizon at a scale factor a < a c .We will sketch how this bump is generated and we will studyin detail the properties of these bumps in both the linear and non-linear approach. Let us explore here the linear regime of matterdensity fluctuations. In this case, the evolution of CDM perturba-tions obey the equation of motion: δ (cid:48)(cid:48) c + H δ (cid:48) c − H ∑ i Ω i δ i ( c s , i + ) = , (5)where the sum runs over all the fluids with sound speed c s , i = δ P i / δ ρ i and the prime denotes derivatives with respect to confor-mal time, with H = a (cid:48) / a . We see that the ρ ex has a twofold effect MNRAS000 , 1–14 (2020) on matter perturbations by modifying first the expansion rate H ,and second by the adding of an extra source term proportional to Ω ex δ ex .Initially, for modes outside the horizon, the matter perturba-tions do not evolve over time, leaving the ratio Q m ≡ δ smx m / δ sm m fixed. However, matter perturbations in the model with ρ ex are ini-tially suppressed compared to a Λ CDM model because the ini-tial amplitude of the matter perturbations, through the gravita-tional potential Ψ , depends on the fraction of relativistic particles R ν = ρ e f f / ( ρ e f f + ρ γ ) , with ρ γ the density of photons, ρ sm e f f = ρ ν and ρ smx e f f = ρ ν + ρ ex giving an initial suppression factor Q ini =( + ( / ) R sm ν ) / ( + ( / ) R smx ν ) , which depends solely on ρ ex .Modes start evolving once they cross the horizon a h , definedimplicitly in terms of the Hubble radius by k = a h H ( a h ) . However,there is a marked difference depending on whether they cross be-fore a c , with a corresponding mode k c ≡ a c H ( a c ) , or after the extraenergy density ρ ex has diluted. Small modes k > k c are further sup-pressed with respect to Λ CDM because the crossing time in thiscase is delayed by the presence of the ρ ex . Comparing the same k h = ( a h H ( a h )) (cid:12)(cid:12) smx = ( a h H ( a h )) (cid:12)(cid:12) sm we have a smx h a sm h = H sm H smx = √ − Ω ex , (6)and therefore small modes k > k c cross the horizon earlier in Λ CDM than in Λ CDMex. This is reflected in an early suppressionin ∆ δ m = δ ρ smxm / δ ρ smm , supplemented by a smaller amplitude athorizon crossing. However, after the initial suppression at horizoncrossing, matter perturbations in the smx model have a higher grow-ing rate than in the Λ CDM model that not only compensates butalso reverses the initial suppression (de la Macorra & et al. 2020).In radiation domination the matter perturbations evolve as δ m ∝ ln a with the growth function f = d ln δ m / d ln a ∝ / δ m , andsince δ m ( smx ) < δ m ( sm ) and f ( smx ) > f ( sm ) and Q m increases.This increase is boosted by the rapid dilution of ρ ex , where the EoSleaps abruptly from w i = / w f = a c affecting only themodes crossing the horizon before a c , i.e. k ≥ k c . This is the char-acteristic signature of RDED and was presented in Almaraz & de laMacorra (2019) and Jaber-Bravo et al. (2020). However, to fullyasses the growth of structure in these RDED cosmologies we mustgo beyond the linear regime, which is the main goal of this work.To gain physical intuition on how the rapid diluted energydensity affects matter-density fluctuations well inside the radiationdominated epoch we can analyze a simplified version of the equa-tions where the gravitational potentials are subdominant: We canestimate this increase in radiation domination, where the amplitude δ m = δ ρ m / ρ m has a logarithmic growth δ m ( a ) = δ mi [ ln ( a / a h ) + / ] , (7)with a h and H h the scale factor and Hubble parameter at horizoncrossing. We compare the evolution of the same mode k smx = k sm crossing the horizon before a c and using equations (7) and (6)we find the ratio ∆ δ m = δ smxm / δ smm = ( δ smxmi / δ smmi )([ ln ( a / a smxh ) + / ] / [ ln ( a / a smh ) + / ]) . The evolution of ∆ δ m after the RDEDtransition takes place at a c is given by de la Macorra & et al. (2020) ∆ δ m = δ smxmi δ smmi (cid:104)(cid:16) H smx + H smx − (cid:17) ln (cid:16) aa c (cid:17) + ln (cid:16) a smh a smxh (cid:17) + ln (cid:16) a c a smh (cid:17) + (cid:105) ln (cid:16) aa c (cid:17) + ln (cid:16) a c a smh (cid:17) + , (8)where we have taken into account in Λ CDMx the contribution of ρ ex in H smx + ( a c ) for a < a c and ρ ex = H smx − ( a c ) for a > a c , giving the ratio H smx − / H smx + = a smx h / a sm h = √ − Ω ex (c.f. equa-tion (6)). The matter power spectrum therefore shows a bump formodes ∆ δ m for a (cid:29) a c for modes k > k c entering the horizon before a c when we compare RDED and Λ CDM cosmologies.
To encompass a range of theoretical models, we choose to workwith a parametrization that we refer throughout as the ‘bump cos-mology’, where the linear power spectrum is given by a modifica-tion to that of a standard Λ CDM cosmology: P bump ( k , z ) = (cid:2) + F ( k ) (cid:3) P Λ CDM ( k , z ) , (9)with F ( k ) = A exp (cid:18) − [ ln ( k / k T )] σ (cid:19) . (10) A , k T and σ are the amplitude, scale, and width of the bump, re-spectively. We considered other choices for the function F ( k ) , suchas a Gaussian in k -space, and found qualitatively similar results.We consider nine different bump cosmologies, in each case wefix the amplitude A = .
15. We choose this amplitude motivated byBDE, where the energy density of the dark sector Ω BDE ( a c ) = . a c = . × − due to a phase transition of theunderlying physics and rendering a Ω BDE (cid:28) a > a c .Meanwhile, the width of the bump corresponds to how fast therapid diluted energy density takes place, for which we investigatethree different values: σ = .
0, 0 .
3, and 0 .
1, and place the bump atthree different scales: k T = .
05, 0 .
1, and 1 h Mpc − (see Table 1).We study structure formation in these bump cosmologies at the red-shifts z =
0, 0 .
5, 1, 2, 3, and 4. The cosmological parameters used togenerate the Λ CDM power spectrum are reported below in Section3.1 and are the same in both the bump and standard cosmologies,so that the only difference between the models is the presence ofthe bump.We investigate the response of the real-space matter powerspectra and correlation functions together with the redshift-spacemultipole power spectra. We construct the response as the ratio ofthe measurement or prediction between a bump and Λ CDM cos-mology. We consider different approaches for the non-linear the-ory: moderate-resolution N -body simulations, and HMCODE . In ad-dition, for the real-space power spectra we also consider one-loopSPT, for the redshift-space multipole power spectra we use the TNSand Scoccimarro models and finally, for the real-space correlationfunction we use CLPT.
We ran 12 N -body simulations using the cosmological simulationcode GADGET -2 (Springel 2005), one each for the cosmologies de-tailed in Table 1. We assume a background Λ CDM cosmology withtotal matter density Ω m = .
3, baryon density Ω b = .
05, dark en-ergy density Ω Λ = .
7, amplitude of the matter power spectrum σ = .
8, spectral index n s = .
96, and dimensionless Hubble con-stant h = .
7. Initial conditions were generated at z =
99 using
NGENIC , which implements the Zeldovich approximation to dis-place particles from an initially Cartesian grid and assigns themvelocities based on a ballistic trajectory. We chose to run simula-tions with different box sizes for the different k T values, to ensure , 1–14 (2020) on-linear evolution in bump cosmologies Name A σ k T [ h Mpc − ] L box [ h − Mpc ] FATBUMP - K .
15 1 . . MEDBUMP - K .
15 0 . . THINBUMP - K .
15 0 . . Λ CDM- K − − − FATBUMP - K P .
15 1 . . MEDBUMP - K P .
15 0 . . THINBUMP - K P .
15 0 . . Λ CDM- K P − − − FATBUMP - K P
05 0 .
15 1 . .
05 1024
MEDBUMP - K P
05 0 .
15 0 . .
05 1024
THINBUMP - K P
05 0 .
15 0 . .
05 1024 Λ CDM- K P − − − Table 1.
Specifications of our N -body simulation suite. The background cosmological parameters are the same for all the simulations: Ω m = . Ω b = . Ω Λ = . Ω ν = h = . n s = . σ = .
8. Each simulation uses 512 particles distributed over N grid = cells. We consider the redshifts z = , . , , , , that there was always a good sampling of modes around k T . Thebox sizes of the simulations are L box = h − Mpc,for k T = .
05, 0 . h Mpc − respectively. Each simulation uses512 particles to approximate the density field, which is quite mod-est compared to modern simulation standards. One may worry thatmeasurements from these simulations would be systematically bi-ased as a result of the low mass resolution. However, we checkedfor convergence with respect to low-resolution 256 particle simu-lations and found that our results for the power spectrum response were only significantly affected ( > HMCODE
We look at predictions for the non-linear matter power spectrumof our cosmological models using the
HMCODE (Mead et al. 2015)model.
HMCODE is an augmented version of the traditional halo-model calculation for the non-linear power spectrum (e.g., Seljak2000; Peacock & Smith 2000; Cooray & Sheth 2002) where theaugmentations account for defects and replace ingredients that aremissing from the standard halo-model calculation. This improvesthe accuracy of the calculation from ∼
30 per cent to ∼ HMCODE takes asinput the linear power spectrum of the cosmology in question, andthen uses some information about the background cosmological pa-rameters and power-spectrum shape and amplitude in order to makeits predictions. Although
HMCODE was not calibrated on the bumpcosmologies investigated in this paper, the grounding of the modelin physical reality means that we can hope that it will make reason- able predictions.
HMCODE models the power spectrum as a sumof (almost) linear theory and a one-halo term, which accounts forsmall-scale, deeply non-linear power under the assumption that allsuch power originates from the clustering within haloes. The bumpcosmologies will therefore affect the
HMCODE predictions in twoways. The first, is trivially that the
HMCODE prediction at largescales is essentially linear theory and because linear theory con-tains the bump then so will
HMCODE . The second, is that within
HMCODE the one-halo power is determined by the halo mass func-tion, which itself depends on the linear power spectrum via the vari-ance in the power spectrum as a function of scale. The bump willtherefore affect the halo mass function and we expect that it willboost the predicted numbers of haloes in certain mass ranges.Before running
HMCODE , we can make the prediction that itshould generally boost power in the one-halo term for a bump cos-mology compared to a cosmological model that lacks a bump. Thetraditional halo model calculation has a problem in the transitionregion between the two- and one-halo terms ( k ∼ . h Mpc − at z = HMCODE is a smoothing of the transition re-gion. Based on this discussion, we could predict that the
HMCODE predictions for the bump cosmologies may be better in the deeplyone-halo regime ( k > h Mpc − ), and that they may be less im-pressive in the transition region. In the linear region they should beperfect, given that the HMCODE prediction is identical to linear the-ory at large scales. In the quasi-linear regime ( k ∼ . h Mpc − at z =
0) we would expect perturbation theory to perform better than
HMCODE since the latter lacks any formal grounding in analyticalperturbation theory.
In this paper we consider two variations of perturbation theory,the first, ‘standard’ perturbation theory is constructed in Eulerianspace, where we consider the evolution of the density field at
MNRAS000
MNRAS000 , 1–14 (2020) fixed positions. We consider the phenomenological, but physicallywell motivated, redshift-space models of Scoccimarro (2004) andTaruya et al. (2010) to describe the redshift-space power spectrum,which improve the description of redshift-space distortion (RSD)effects on the power spectrum compared to Kaiser linear theoryKaiser (1987). In the following we shall refer to these models asSc04 and TNS, respectively.The anisotropic redshift-space clustering originates from thepeculiar velocities v of matter, or more generally any tracer of it,such that an object located at a real space position r is observedto be located at an apparent redshift-space position s . The relationbetween coordinates system is inferred via the Doppler effect tobe s = r + ˆ n v (cid:107) ( aH ) − , where ˆ n is a the line-of-sight direction ofthe point-process sample, and v (cid:107) = v · ˆ n . That is, we are using theplane-parallel approximation on which the observer is located at adistant position of the sample of objects over which we perform thestatistics. The redshift-space power spectrum, (cid:104)| δ s ( k ) | (cid:105) is givenby P s ( k ) = (cid:90) d r e i k · r (cid:68) e ik µ ∆ v (cid:107) / ( aH ) (cid:18) δ ( x ) − aH ∇ (cid:107) v (cid:107) ( x ) (cid:19) × (cid:18) δ ( x (cid:48) ) − aH ∇ (cid:107) v (cid:107) ( x (cid:48) ) (cid:19)(cid:69) (11)where r = x − x (cid:48) and ∆ v (cid:107) = v (cid:107) ( x ) − v (cid:107) ( x (cid:48) ) and µ = ˆ n · ˆ k is the anglebetween the wave vector and the line-of-sight direction.The RSD correction at linear order, known as the Kaiserformula, is given by δ sL ( k ) = ( + f µ ) δ L ( k ) , where f = d log D + ( a ) / d log a ( t ) and D + ( t ) the linear growth function. Theredshift-space power spectrum at linear order becomes P sK ( k , µ ) = ( + f µ ) P L ( k ) . (12)To move beyond linear theory it is common to define the di-mensionless velocity divergence θ = − ∇ · v / ( aH f ) , such that at lin-ear order we have θ = δ .The exponential oscillatory factor inside the correlator inequation (11) is due to virialized, non-coherent random motionsof dark-matter particles along the line-of-sight direction, hence it isin essence non-perturbative. In (Scoccimarro 2004) this factor is re-placed by a phenomenological damping function that accounts forthe velocity dispersion σ v = (cid:104) θ (cid:105) . By rotational symmetry aroundthe line-of-sight direction one obtains a simple prescription P s Sc04 ( k , µ ) = exp (cid:2) − k µ f σ v (cid:3) × (cid:104) P δδ ( k ) + f µ P δθ ( k ) + f µ P θθ ( k ) (cid:105) , (13)where P δδ , P θθ and P δθ are respectively the non-linear matter den-sity, velocity divergence, and density-velocity divergence power-spectra. In linear theory equation (13) reduces to the Kaiser powerspectrum (equation 12) times the damping factor. Several otherfunctional forms for this damping factor have been used in the liter-ature. In this work, we opt for the most common – Gaussian damp-ing. The velocity dispersion is physically motivated to be given byPT as σ v = π (cid:90) d p p P θθ ( p ) , (14)but due to its non-perturbative origin it is commonly replaced by afree parameter σ , especially for parameter estimation. Howeverin this work, we regard σ v as the linear velocity dispersion, whichis obtained from the above equation by replacing P θθ by its linearvalue P L . This prescription has shown to give good results whencomparing theory to simulations (e.g., Taruya et al. 2010). The TNS formalism, on the other hand, expands in cumulantsthe correlator in equation (11), and then replaces a residual expo-nential factor of the form exp (cid:2) (cid:104) e ik µ ∆ v (cid:107) / ( aH ) (cid:105) c (cid:3) , by a position inde-pendent, phenomenological damping factor that can be brought outof the integral. The standard formula for TNS is P s TNS ( k , µ ) = exp ( − k µ f σ v ) (cid:2) P δδ ( k ) + f µ P δθ ( k )+ f µ P θθ ( k ) + A ( k , µ ) + B ( k , µ )] , (15)where the new correction terms, A ( k , µ ) and B ( k , µ ) , are given by A ( k , µ ) = k µ f (cid:90) d p ( π ) p · ˆ n p B σ ( p , k − p , − k ) , (16) B ( k , µ ) = ( k µ f ) (cid:90) d p ( π ) F ( p ) F ( k − p ) , (17)with the bispectrum ( π ) δ D ( k + k + k ) B σ ( k , k , k ) = (cid:68) θ ( k ) × (cid:34) δ ( k ) + f ( k · ˆ n ) k θ ( k ) (cid:35)(cid:34) δ ( k ) + f ( k · ˆ n ) k θ ( k ) (cid:35)(cid:69) , (18)and F ( p ) = p · ˆ n p (cid:18) P δθ ( p ) + f ( p · ˆ n ) p P θθ ( p ) (cid:19) . (19)Hence, the TNS model partially accounts for the interaction be-tween the Kaiser effect and the non-linear random motion of parti-cles, reflected in the two extra functions A and B . On the other hand,the Sc04 model factorizes the linear and Finger-of-God effects, soeach can be treated separately.The real-space matter power spectrum is given by P δδ ( k ) = (cid:104)| δ ( k ) | (cid:105) , which can be obtained from both of the above redshift-space formalisms by considering only the perpendicular to line-of-sight components (i.e., µ = P SPT1-loop ( k , t ) = P L ( k ) + P ( k ) + P ( k ) , (20)where P and P are the usual 1-loop corrections (e.g.,Bernardeau et al. 2002). The linear power spectrum is the dominantterm at large scales. However, at smaller scales, the loop correctionscontribute with similar magnitude to the linear power, therefore sig-nificantly contributing to the total power spectrum. In contrast to the Eulerian scheme, in a Lagrangian picture we fol-low the trajectories of individual particles with initial position q and current position x . The Lagrangian coordinates q are related tothe Eulerian coordinates x through the coordinates transformation x ( q , t ) = q + Ψ ( q , t ) , (21)where Ψ ( q , t ) is the Lagrangian displacement and Ψ ( q , t = t ini ) =
0, meaning that at a sufficiently early time t ini both coordinatessystem coincide. Furthermore, mass conservation implies the rela-tion between Lagrangian displacements and overdensities (Taylor& Hamilton 1996)1 + δ ( x , t ) = (cid:90) d q δ D [ x − q − Ψ ] , (22)from which the LPT correlation function is obtained as1 + ξ ( r ) = (cid:90) d k ( π ) (cid:90) d qe i k · ( r − q ) (cid:104) e − i k · ∆ (cid:105) , (23) MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies where ∆ i = Ψ i ( q ) − Ψ i ( q ) are the Lagrangian displacement dif-ferences at two positions q and q separated by a distance q = | q − q | . The idea behind CLPT is to expand the correlator ineq. (23) in cumulants and thereafter to expand out of the exponen-tial all but linear terms in the Lagrangian displacement, such thatthe k -integral can be performed analytically using standard Gaus-sian integration techniques (see Carlson et al. (2012)), by which wearrive at1 + ξ CLPT ( r ) = (cid:90) d q ( π ) det [ A Li j ] / e A Lij ( r i − q i )( r j − q j ) × (cid:20) − G i j A loop i j + Γ i jk W loop i jk + ··· (cid:21) , (24)with cumulants A i j = (cid:104) ∆ i ∆ j (cid:105) c and W i jk = (cid:104) ∆ i ∆ j ∆ k (cid:105) c , and tensors G i j = A − Li j − g i g j , Γ i jk = A − Li j g k + A − L jk g i + A − Lki g j − g i g j g k , and g i j = A − Li j ( r j − q j ) . The label ‘ L ’ in the A function denotes thelinear piece and ‘loop’ the pure 1-loop piece, such that A i j = A Li j + A loop i j . Notice that the ‘1’ in the squared brackets of the aboveequation corresponds to the Zeldovich correlation function and theother two terms yield the next-to-leading order, 1-loop contribu-tions.To solve numerically the CLPT correlation function of equa-tion (24) we use the code MGPT (Aviles et al. 2018), which alsoaccounts for the exact kernels for a Λ CDM background cosmology,instead of the most commonly used Einstein-de Sitter ( Ω m = . z = We use the different approaches discussed in the previous Sectionto study the evolution, dilution and shift of the bump as seen in theresponse functions R ( k ) = P bump ( k ) P Λ CDM ( k ) . (25)That is, we use our N -body simulations, HMCODE and PT methodsas complementary tools. The results are contrasted with the lin-ear theory, for which the response in the power spectrum is simply R L ( k ) = + F ( k ) at all z because linear growth is scale indepen-dent. The power spectra data are extracted from the simulations usinga cloud-in-cell mass-assignment scheme, on a grid with resolu-tion of N grid = cells. They are binned in 100 evenly log-spaced k -points over a range [ k min , k Ny ] , where k Ny = N grid π / L isthe Nyquist frequency, and k min = π / L . L is the size of the box,which is given in Table 1 for the different simulations. Traditionallypower spectra measured in this way are considered to be accurateup to half of the Nyquist frequency, with the power at smaller scalespotentially corrupted by aliasing from smaller scales. In our plotswe show all measured scales, up to k Ny , and the reader should keepthis in mind. https://github.com/cosmoinin/MGPT Figs. 1, 2, and 3 show the responses using the bump cosmol-ogy from equation (9) located at the scales k T = .
05, 0 .
1, and,1 h Mpc − , respectively. The non-linear power is computed usingour different methods and then divided by their counterparts in Λ CDM to create the response function. This analysis compareshow the bump cosmologies are modified by non-linearities withinthe different approaches.As expected, at higher redshifts non-linear effects are smallerand the responses of all methods are very similar. In the σ = z decreases we see two things in the simulated measure-ments: First, at the small-scale edge of the bump we see the powergrow above the bump, an effect that is captured extremely well bySPT and less well by HMCODE . Second, we see the generation ofa second bump at much smaller scales, with a peak k ∼ h Mpc − .This second bump is well modelled by HMCODE but not capturedat all by SPT. The failure of SPT at these scales (which even pre-dicts a decrease in the response at higher z ) is not surprising giventhat these scales are far beyond its reach. In Fig. 1, correspond-ing to k T = . h Mpc − , this non-linear ‘second bump’ peaks at10 times smaller scale, k ≈ . h Mpc − , and reaches a maximum at z =
0, peaking in amplitude at ∼
10, 2, and 1 per cent for the σ = .
3, and 0 . k T = .
1, shown in Fig. 2, thissecond non-linear bump is even clearer, peaking at k ≈ h Mpc − for z (cid:39)
0, but at smaller scales for higher redshifts. For some of ourcosmologies, this second bump is even larger than the primordialbump. For example, it is larger for σ = z = . z = .
5, contributing ≈
18 and 16 per cent to the whole re-sponse respectively. We note that as the width of primordial bumpdecreases, the second, non-linear bump amplitude decreases. In allcases, the location and amplitude of the non-linear bump as seen inthe simulation response is in remarkable agreement with the predic-tions from
HMCODE . For the cases of k T = .
05 and 0 . h Mpc − ,shown in Figs. 1 and 2, HMCODE and the simulated data providesimilar results at lower redshifts and in the non-linear regime. Con-versely, 1-loop SPT gives results closer to those obtained fromsimulations at higher redshifts and in the mildly non-linear regime( k (cid:46) . h Mpc − ).The non-linear bumps in the response functions are a con-sequence of the interaction between the primordial bump and theone-halo term, being highly enhanced for the wider bumps simplybecause these provide a greater enhancement of linear power. Phys-ically, this can be understood within the halo model (and thereforewithin HMCODE ) as follows: the one-halo term is given by the in-tegral of the halo mass function multiplied by the squared Fourier-space halo profile. The halo mass function itself is related to thestandard-deviation in the density field when smoothed over the La-grangian radius of the halo, σ R . Our bump cosmology increasesthe power over a certain range of scales, such that σ R will also in-crease, and therefore so will the mass function. Hence, one effectof the linear bump is to accelerate halo formation relative to a cos-mology with no bump. A different way to think about this is via aPress & Schechter (1974) type argument, where the increased am-plitude of some modes, given here by our bump cosmology, is help-ing more small scale fluctuations to cross over the critical thresh-old to collapse. This occurs even when the bump is at very linearscales as long as the width is sufficiently large, because those longwavelength modes exist underneath the smaller-scale fluctuationsenhancing the collapse to form the actual haloes. We remark thatthis is a highly non-linear effect, such that the second bump is notwell captured by PT, where the main non-linear effect is the spread-ing and enhancement of the primordial bump.In Fig. 3 we show the k T = h Mpc − bump cosmologies. The MNRAS000
05 and 0 . h Mpc − ,shown in Figs. 1 and 2, HMCODE and the simulated data providesimilar results at lower redshifts and in the non-linear regime. Con-versely, 1-loop SPT gives results closer to those obtained fromsimulations at higher redshifts and in the mildly non-linear regime( k (cid:46) . h Mpc − ).The non-linear bumps in the response functions are a con-sequence of the interaction between the primordial bump and theone-halo term, being highly enhanced for the wider bumps simplybecause these provide a greater enhancement of linear power. Phys-ically, this can be understood within the halo model (and thereforewithin HMCODE ) as follows: the one-halo term is given by the in-tegral of the halo mass function multiplied by the squared Fourier-space halo profile. The halo mass function itself is related to thestandard-deviation in the density field when smoothed over the La-grangian radius of the halo, σ R . Our bump cosmology increasesthe power over a certain range of scales, such that σ R will also in-crease, and therefore so will the mass function. Hence, one effectof the linear bump is to accelerate halo formation relative to a cos-mology with no bump. A different way to think about this is via aPress & Schechter (1974) type argument, where the increased am-plitude of some modes, given here by our bump cosmology, is help-ing more small scale fluctuations to cross over the critical thresh-old to collapse. This occurs even when the bump is at very linearscales as long as the width is sufficiently large, because those longwavelength modes exist underneath the smaller-scale fluctuationsenhancing the collapse to form the actual haloes. We remark thatthis is a highly non-linear effect, such that the second bump is notwell captured by PT, where the main non-linear effect is the spread-ing and enhancement of the primordial bump.In Fig. 3 we show the k T = h Mpc − bump cosmologies. The MNRAS000 , 1–14 (2020) k [ h Mpc ]0.80.91.01.11.21.01.11.21.01.11.21.01.11.21.01.11.21.01.11.2 P / P C D M k T = 0.05, = 1.0 z=4.0z=3.0z=2.0z=1.0z=0.5z=0.0 10 k [ h Mpc ] k T = 0.05, = 0.3 Linear1-loopHM codeN-body sims 10 k [ h Mpc ] k T = 0.05, = 0.1 Figure 1.
Response functions for the bump cosmologies at k T = . h Mpc − . From top to bottom, yellow curves are for redshift z =
4; magenta for z = z =
2; red for z =
1; green for z = .
5; and blue for z =
0. The left panel shows the bump cosmology for σ =
1; middle panel for σ = .
3; and rightpanel for σ = .
1. Dash-dotted (black) curves are for the linear theory; dashed (color) are for 1-loop SPT; solid for
HMCODE model; and crosses are for themeasurement from N -body simulations. k [ h Mpc ]0.80.91.01.11.21.01.11.21.01.11.21.01.11.21.01.11.21.01.11.2 P / P C D M k T = 0.1, = 1.0 z=4.0z=3.0z=2.0z=1.0z=0.5z=0.0 10 k [ h Mpc ] k T = 0.1, = 0.3 Linear1-loopHM codeN-body sims 10 k [ h Mpc ] k T = 0.1, = 0.1 Figure 2.
Same as Fig. 1 but for the bump cosmologies with k T = . h Mpc − . MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies k [ h Mpc ]0.80.91.01.11.21.01.11.21.01.11.21.01.11.21.01.11.21.01.11.2 P / P C D M k T = 1.0, = 1.0 z=4.0z=3.0z=2.0z=1.0z=0.5z=0.0 10 k [ h Mpc ] k T = 1.0, = 0.3 Linear1-loopHM codeN-body sims 10 k [ h Mpc ] k T = 1.0, = 0.1 Figure 3.
Same as Fig. 1 but for the bump cosmologies with k T = h Mpc − . simulated data show that the primordial bump become erased by thenon-linear evolution. Such effect is well captured by the HMCODE –PT simply does not capture the non-linearities of the evolution ofthe bump since it is far out of the reach of its regime of validity,although at the highest redshifts it works moderately well. Interest-ing is the non presence of the second bump, because the primordialbump and the 1-halo term are located about the same scale.Figs. 1, 2 and 3 demonstrate that perturbation theory providesan accurate model for the power spectrum response at large scales,with the accuracy extending to smaller scales at higher redshift. Onthe other hand,
HMCODE provides a reasonable (though less per-fect) model for the response at the smaller scales, those typicallyassociated with halo formation. It does not take a great leap of theimagination to consider combining these two approaches to pro-vide an accurate model for the response that would be valid acrossa wider range of scales. This approach has been explored in the lit-erature previously (e.g., Mohammed & Seljak 2014; Seljak & Vlah2015; Philcox et al. 2020), but has never been applied to the bumpcosmology specifically. It may be possible to directly add perturba-tion theory to the halo-model, perhaps replacing the two-halo termwith some perturbative expansion, or else to use the response fromperturbation theory at one extreme and
HMCODE at the other, inter-polating between the two around some ‘non-linear’ wavenumber,perhaps defined via σ ( a / k nl ) = α where σ is the standard devia-tion in the density field when smoothed on a given physical scaleand a and α are fitted constants of order unity. In this Section, we study the responses of the redshift-spacemonopole and quadrupole power spectra using the non-linear mod-els Sc04 and TNS. For comparison we also use Kaiser linear the-ory. The multipole power spectra have been extracted from the N - body simulations using a triangular-shaped-cloud mass-assignmentfunction, on a grid with N grid = cells, using the N - BODYKIT package of Hand et al. (2018). The response functions are definedfor each multipole, such that R (cid:96) ( k ) = P bump (cid:96) ( k ) / P Λ CDM (cid:96) ( k ) .In the top panel of Fig. 4 we show the response functions inthe monopole P at redshifts z = z = k T = . h Mpc − . We notice, as expected, thatnon-linear models perform closer to the simulated data than thelinear theory in the quasi-linear regime, in the sense that the re-sponses are closer to those from N -body simulations. However, for k > . h Mpc − both theoretical models make deviations from thesimulated data. In the bottom panel of Fig. 4 we show the responsefunctions for the bump cosmologies located at k T = . h Mpc − .The behaviour of the PT methods is qualitatively similar to thecase of k T = . h Mpc − , but the non-linear features of the bumpare less well captured when comparing to the simulated data. Thismust be since the scale of the bump is located at the edge of theperturbative regime of validity. This has the consequence that thelinear theory response provides a better match to the simulationsfor some non-linear scales, obviously the good performance of lin-ear theory here is only a lucky coincidence. As in real space, thesimulated data show the appearance of the non-linear second bumpat ≈ h Mpc − , which is not present in the k T = . h Mpc − case,even though in real space this second non-linear bump was visiblein both cases. We suggest that the reason for this is the dampingalong the line-of-sight direction of the redshift-space power spec-trum, which ultimately comes from the highly oscillatory behaviorof the correlator inside the integral of equation (11) at large k . Thisredshift-space effect damps the multipoles in all bump cosmolo-gies, but since the second (real space) bump is larger for the case of https://nbodykit.readthedocs.ioMNRAS000
HMCODE at the other, inter-polating between the two around some ‘non-linear’ wavenumber,perhaps defined via σ ( a / k nl ) = α where σ is the standard devia-tion in the density field when smoothed on a given physical scaleand a and α are fitted constants of order unity. In this Section, we study the responses of the redshift-spacemonopole and quadrupole power spectra using the non-linear mod-els Sc04 and TNS. For comparison we also use Kaiser linear the-ory. The multipole power spectra have been extracted from the N - body simulations using a triangular-shaped-cloud mass-assignmentfunction, on a grid with N grid = cells, using the N - BODYKIT package of Hand et al. (2018). The response functions are definedfor each multipole, such that R (cid:96) ( k ) = P bump (cid:96) ( k ) / P Λ CDM (cid:96) ( k ) .In the top panel of Fig. 4 we show the response functions inthe monopole P at redshifts z = z = k T = . h Mpc − . We notice, as expected, thatnon-linear models perform closer to the simulated data than thelinear theory in the quasi-linear regime, in the sense that the re-sponses are closer to those from N -body simulations. However, for k > . h Mpc − both theoretical models make deviations from thesimulated data. In the bottom panel of Fig. 4 we show the responsefunctions for the bump cosmologies located at k T = . h Mpc − .The behaviour of the PT methods is qualitatively similar to thecase of k T = . h Mpc − , but the non-linear features of the bumpare less well captured when comparing to the simulated data. Thismust be since the scale of the bump is located at the edge of theperturbative regime of validity. This has the consequence that thelinear theory response provides a better match to the simulationsfor some non-linear scales, obviously the good performance of lin-ear theory here is only a lucky coincidence. As in real space, thesimulated data show the appearance of the non-linear second bumpat ≈ h Mpc − , which is not present in the k T = . h Mpc − case,even though in real space this second non-linear bump was visiblein both cases. We suggest that the reason for this is the dampingalong the line-of-sight direction of the redshift-space power spec-trum, which ultimately comes from the highly oscillatory behaviorof the correlator inside the integral of equation (11) at large k . Thisredshift-space effect damps the multipoles in all bump cosmolo-gies, but since the second (real space) bump is larger for the case of https://nbodykit.readthedocs.ioMNRAS000 , 1–14 (2020) k [ h Mpc ]1.01.041.081.121.161.01.041.081.121.161.2 P / P , C D M k T = 0.05= 1.0 z = 1.0 z = 0.0 KaiserTNSSc04N-body sims 10 k [ h Mpc ] k T = 0.05= 0.3 z = 1.0 z = 0.0 k [ h Mpc ] k T = 0.05= 0.1 z = 1.0 z = 0.0 k [ h Mpc ]1.01.041.081.121.161.01.041.081.121.161.2 P / P , C D M k T = 0.1= 1.0 z = 1.0 z = 0.0 k [ h Mpc ] k T = 0.1= 0.3 z = 1.0 z = 0.0 KaiserTNSSc04N-body sims 10 k [ h Mpc ] k T = 0.1= 0.1 z = 1.0 z = 0.0 Figure 4.
Responses of the monopole power spectrum at z = k T = .
05 (top panel) and 0 . h Mpc − (bottom panel), withdifferent widths σ = , . , . k T = . h Mpc − , it can overcome the damping and it still showsup in the redshift-space responses.We also show, in Fig. 5, the quadrupole redshift-space powerspectra for k T = . h Mpc − and k T = . h Mpc − . Althoughthe simulated quadrupole measurements are noisier than for themonopole, we note a similar trend to that predicted by the analyti-cal approaches, most obviously for the case of k T = . h Mpc − at z =
1. We also observe that the non-linear, second bumps do notappear in the quadrupole. We suggest that this is because this mul-tipole gives maximum weight to the line-of-sight direction wherethe damping effect occurs, while the monopole gives equal weightto all directions.
In this sub-section we compare the measured real-space correla-tion function responses, defined as ξ bump ( r ) / ξ Λ CDM ( r ) , from boththe simulated data and the analytical correlation function calculatedaccording to the CLPT method. All simulated correlation functionshave been measured by employing the N - BODYKIT code, using 60linearly spaced bins in the range 1–121 h − Mpc for the bump cos-mologies at k T = . h Mpc − , 30 bins in the range 1–61 h − Mpcfor models at k T = .
01, and 15 bins in the range 1–31 h − Mpc formodels at k T = . h Mpc − .Fig. 6 shows the response functions for the bump at k T = . h Mpc − for four different redshifts z =
2, 1, 0 .
5, and 0 andthe bump cosmologies with widths σ =
1, 0 .
3, and 0 .
1. We find adip in the response around 85 h − Mpc, which is due to a long wave- length modulation of the whole correlation function because of thelocalized bump feature in the corresponding power spectrum. Byconsidering equation (9), the linear correlation function responsebecomes R L ( r ) = + ξ Λ CDM L ( r ) (cid:90) ∞ d r (cid:48) ξ Λ CDM L ( r (cid:48) ) ˜ F ( | r − r (cid:48) | ) , (26)with˜ F ( r ) = (cid:90) ∞ dk π k F ( k ) j ( kr ) , (27)where j is the zero-order spherical Bessel function. Hencefor scales r (cid:29) k − T we have that ˜ F ( r ) → j ( kr ) ; while as r →
0, ˜ F ( r ) → ( π ) − (cid:82) dkk F ( k ) , a constant that when computed turns out tobe much smaller than the correlation function at those small scales;such that the linear response goes to unity at both large scales andsmall scales limits. On the other hand, k F ( k ) reaches its maximumat scales k max > k T , the larger the width σ , the bigger are k max andthe maximum values k max F ( k max ) ; however for larger widths, thefunction k F ( k ) overlaps with more oscillations and ˜ F ( r ) is sup-pressed. The inverse transformed bump feature, ˜ F ( r ) , turns out tobe a sinc-like function in real space with amplitude proportional tothe σ and first trough around r T (cid:46) ( / ) π / k T . Moreover, the afore-mentioned effects compete and this becomes more pronounced forthe width σ = . k T = . h − Mpc, in the correlation function, and both reinforce eachother to produce the large dip observed in Fig. 6.
MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies k [ h Mpc ]1.01.041.081.121.161.01.041.081.121.161.2 P / P , C D M k T = 0.05= 1.0 z = 1.0 z = 0.0 k [ h Mpc ] k T = 0.05= 0.3 z = 1.0 z = 0.0 KaiserTNSSc04N-body sims 10 k [ h Mpc ] k T = 0.05= 0.1 z = 1.0 z = 0.0 k [ h Mpc ]1.01.041.081.121.161.01.041.081.121.161.2 P / P , C D M k T = 0.1= 1.0 z = 1.0 z = 0.0 k [ h Mpc ] k T = 0.1= 0.3 z = 1.0 z = 0.0 KaiserTNSSc04N-body sims 10 k [ h Mpc ] k T = 0.1= 0.1 z = 1.0 z = 0.0 Figure 5.
As Fig. 4 but for the quadrupole response.
More generally, for the different choices of k T the changes inthe correlation-function response will follow a similar pattern, butthe wave-length modulations will be related to the characteristicscales: For larger k T and smaller σ , the oscillations in the corre-lation function have higher frequency and are damped at smallerscales. This can be seen in Figs. 7 and 8, for the cases k T = . h Mpc − , respectively. The main qualitative difference comparedto the k T = . h Mpc − case is that the reinforcement with theBAO characteristic dip is not present in these cases.The peaks and troughs are more pronounced in the lin-ear theory since the correlation function scales simply with thescale-independent linear growth function. On the other hand forCLPT, even in the Zeldovich approximation, overdense regionsare partially depleted, while underdense regions populated, dueto the free-streaming of coherent matter flows over a scale set-tled by the Lagrangian displacements 1-dimensional variance σ Ψ = (cid:82) ∞ dkP L ( k ) / ( π ) . This effect is captured very well by our numer-ical results, and the damping in the responses become very simi-lar for simulated data and CLPT. More remarkably for the casesof k T = h Mpc − with σ = . .
1, corresponding to middleand right panels of fig. 8, where the oscillations in the power spec-trum are practically erased. The reason for this is, of course, thatthe peaks and troughs are more closer to each other and the free-streaming of particles can cover such distances. Indeed, particlestravel, on average, a distance settled by the standard deviation ofthe displacement field σ Ψ ( z ) ∼ × D + ( z ) h − Mpc with D + ( z ) the This effect has the same origin as the smearing of the BAO peak observedin LPT and simulations (see, e.g. Tassev 2014). linear growth function, so the process of particles moving out frommore dense regions becomes very efficient.
Phase transitions in the dark sector are common in theories of cos-mology beyond Λ CDM, and these leave fingerprints that are poten-tially detectable by current and future surveys. One of these sig-natures is the creation of enhanced features in the power spectrumat scales where otherwise the power would be smooth. The gen-eration of these can be understood since all k -modes entering thehorizon during the time elapsed by the phase transition suffer anenhancement on their amplitude since adding an extra ρ ex increasethe growth rate of the linear matter perturbations impacting modes k ≥ k c entering the horizon for a < a c . In this work we have fo-cused on bumps generated in the power spectrum, motivated by therecently proposed BDE model of de la Macorra & Almaraz (2018)and SEOS Jaber-Bravo et al. (2020).We have studied non-linear evolution of parametric bump cos-mologies . We have chosen to be as model independent as possi-ble, instead of considering bumps generated by any specific BDEmodel, since we are interested in a wider range of theoretical mod-els. We have also fixed the background cosmology to be Λ CDM toallow us to investigate the phenomenology of bumps in isolation. Inorder to do so, we have run modest-resolution N -body simulations,which are complemented by perturbation theory models and non-linear halo-model calculations from the HMCODE model of Meadet al. (2015). We expect the different methods used to work overdifferent ranges of scales, and this complementarity is important,
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20 40 60 80 100 120 r [ h Mpc]-0.50.00.51.01.52.01.01.52.01.01.52.01.01.5 / C D M k T = 0.05, = 1.0 z=2.0z=1.0z=0.5z=0.0 20 40 60 80 100 120 r [ h Mpc] k T = 0.05, = 0.3 LinearCLPTN512 20 40 60 80 100 120 r [ h Mpc] k T = 0.05, = 0.1 Figure 6.
The real-space correlation function response: We plot ξ / ξ Λ CDM for bump cosmologies at k T = .
05 in units h Mpc − for different redshifts ( z = . , . , . , . z =
2; red for z =
1; green for z = .
5; and blue for z =
0. Right: Bumps with σ = .
0. Middle:Bumps with σ = .
3. Left: Bumps with σ = .
1. The CLPT correlations are represented by the solid lines. The plus markers denote the N -body simulations.The black dash-dotted lines denote the linear correlation function.
20 40 60 r [ h Mpc]0.751.01.251.01.251.01.251.01.25 / C D M k T = 0.1, = 1.0 z=2.0z=1.0z=0.5z=0.0 20 40 60 r [ h Mpc] k T = 0.1, = 0.3 LinearCLPTN-body sims 20 40 60 r [ h Mpc] k T = 0.1, = 0.1 Figure 7.
As Fig. 6 but for the bump cosmologies with k T = . h Mpc − . MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies
10 20 30 r [ h Mpc]0.981.01.021.01.021.01.021.01.021.0 / C D M k T = 1.0, = 1.0 z=2.0z=1.0z=0.5z=0.0 10 20 30 r [ h Mpc] k T = 1.0, = 0.3 Linear1-loopN-body sims 10 20 30 r [ h Mpc] k T = 1.0, = 0.1 Figure 8.
As Fig. 1 but for the bump cosmologies with k T = h Mpc − . since although bumps can be localized at a given scale, these arenaturally spread by non-linear evolution, typically covering scalesthat may be outside the range of validity of some particular method.Bearing in mind that non-linear Λ CDM is well studied, we haveput attention to the power spectrum response, constructed as theratio of the power in a bump cosmology to a cosmology with nobump, instead than on the power spectrum itself. Once an accuratemodel for the response is at hand, this can be converted into an ac-curate model for the power spectrum by multiplying by an accuratemodel for the Λ CDM non-linear power spectrum. We have stud-ied the non-linearities in both real and redshift space for the powerspectrum and how these fingerprints are translated to configurationspace in ¢athe correlation function.Much of the non-linear physics is understood within the HM - CODE method in the real space power spectrum. Of particular im-portance is the appearance of a second bump feature in the responsegenerated at smaller scales than the first, primordial bump. Thereason for this is a non-linear coupling of the bump and one-haloterm in the following simple mechanism: long wave-length den-sity fluctuations are enhanced to form the bump, but at the sametime, small-scale fluctuations in regions located inside these over-densities, corresponding to 1-halo regions in halo models, are fur-ther amplified and can cross-over the threshold density for collapsemore easily, leading to a more efficient halo formation than in amodel without a bump. PT, on the other hand successfully followthe data at quasilinear scales, though it fails to model the second,non-linear bump, which is out of its reach. In redshift space thissecond, non-linear bump is partially erased because of the dampingalong the line-of-sight direction that is provided by the random mo-tion of virialized regions that generate the "Fingers-of-God". Sucheffect is more clear in the quadrupole, for which the second bumpis almost completely erased, since this multipole gives more weightto the line-of-sight direction. The monopole, on the other hand, still shows the second bump since it gives equal weight to all directions.This redshift-space effect, being highly non-linear, is not capturedby perturbation theory, however at quasi-linear scales the simulateddata and theory predicted by using the two popular methods ofTaruya et al. (2010) and Scoccimarro (2004) behave similarly.A localized bump in the power spectrum corresponds to os-cillations in the correlation function with amplitude proportional toits width and a frequency governed by its position. The effect is tomodulate the response about unity: higher wavenumbers at whichthe bump is located translate to higher oscillation frequencies; andwider bumps enhance the modulation, but are also more rapidlydamped. This basic picture is explained well within linear Euleriantheory. By moving to Lagrangian space, we find that the signaturesin the correlation function become even more damped since coher-ent flows have a finite probability to leave overdense regions andpopulate underdense regions. This effect has the same origin as thesmearing of the BAO peak, that is well captured in LPT; and in thebump cosmology is much more evident for large- k located bumpswith small widths, since in these cases linear theory shows up rapidoscillations, and the displacement fields sizes, typically given bytheir standard deviations, are large enough, such that particles findthe time to deplete the overpopulated regions.In the future it would be interesting to investigate the phe-nomenology of the bump cosmology for different bump ampli-tudes, which is fixed at 0 .
15 in this paper, somewhat arbitrarily.One could also investigate more physical examples of ‘bump’ cos-mology, such as that generated by the physical BDE model, wherethe background expansion is also changed relative to Λ CDM andwhere the bump shape will not necessarily be Gaussian. Since thiswas our first investigation, in this paper we focused on modellingstatistics of the matter field, which unfortunately are not direct ob-servables. In future, it would be fruitful to consider how the statis-tics of biased tracers of the density, such as haloes or galaxies, were
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HMCODE is publicly available at https://github.com/alexander-mead/HMcode. The perturbation theory code
MGPT is publiclyavailable at https://github.com/cosmoinin/MGPT. The simulateddata and theory curves generated as part of this work will be sharedon reasonable request to the corresponding author.
ACKNOWLEDGEMENTS
DG and AM acknowledge partial support from Project IN103518PAPIIT–UNAM, DG thanks support from a CONACyT PhD fel-lowship and AM from PASPA–DGAPA, UNAM and CONA-CyT. AJM has received funding from the Horizon 2020 researchand innovation programme of the European Union under MarieSkłodowska-Curie grant agreements No. 702971. AA acknowl-edges partial support from Conacyt Grant No. 283151.
REFERENCES
Abbott T. M. C., et al., 2018, Phys. Rev. D, 98, 043526Almaraz E., de la Macorra A., 2019, Phys. Rev. D, 99, 103504Almaraz E., Li B., de la Macorra A., 2020, JCAP, 03, 016Armendariz-Picon C., Mukhanov V. F., Steinhardt P. J., 2001, Phys. Rev. D,63, 103510Aviles A., Rodriguez-Meza M. A., De-Santiago J., Cervantes-Cota J. L.,2018, JCAP, 11, 013Baldauf T., Mirbabayi M., SimonoviÄ ˘G M., Zaldarriaga M., 2015, PhysicalReview D, 92Bartelmann M., Doran M., Wetterich C., 2006, Astron. Astrophys., 454, 27Batista R., Pace F., 2013, JCAP, 06, 044Baumann D., Nicolis A., Senatore L., Zaldarriaga M., 2012, JCAP, 07, 051Bayarsaikhan B., Koh S., Tsedenbaljir E., Tumurtushaa G., 2020, arXiv e-prints, p. arXiv:2005.11171Bellini E., Cuesta A. J., Jimenez R., Verde L., 2016, JCAP, 02, 053Bernardeau F., Colombi S., Gaztaôsaga E., Scoccimarro R., 2002, PhysicsReports, 367, 1â ˘A¸S248Calabrese E., Huterer D., Linder E. V., Melchiorri A., Pagano L., 2011,Phys. Rev. D, 83, 123504Carlson J., Reid B., White M., 2012, Monthly Notices of the Royal Astro-nomical Society, 429, 1674â ˘A¸S1685Casarini L., Bonometto S. A., Tessarotto E., Corasaniti P.-S., 2016, J. Cos-mology Astropart. Phys., 8, 008 Cataneo M., Lombriser L., Heymans C., Mead A. J., Barreira A., Bose S.,Li B., 2019, MNRAS, 488, 2121Cataneo M., Emberson J. D., Inman D., Harnois-Déraps J., Heymans C.,2020, MNRAS, 491, 3101Chandrachani Devi N., Jaber-Bravo M., Aguilar-Argüello G., Valen-zuela O., de la Macorra A., Velázquez H., 2019, arXiv e-prints, p.arXiv:1911.02402Chen S.-F., Vlah Z., White M., 2020Chevallier M., Polarski D., 2001, International Journal of Modern PhysicsD, 10, 213Cooray A., Sheth R., 2002, Physics Reports, 372, 1Copeland E. J., Sami M., Tsujikawa S., 2006, Int. J. Mod. Phys., D15, 1753Fontanot F., Springel V., Angulo R. E., Henriques B., 2012, Mon. Not. Roy.Astron. Soc., 426, 2335Francis M. J., Lewis G. F., Linder E. V., 2008, Mon. Not. Roy. Astron. Soc.,394, 605Gallego Cadavid A., Romano A. E., Gariazzo S., 2016, Eur. Phys. J. C, 76,385Garcia-Arroyo G., Cervantes-Cota J. L., Nucamendi U., Aviles A., 2020,Phys. Dark Univ., 30, 100668Gil-Marin H., Wagner C., Verde L., Porciani C., Jimenez R., 2012, JCAP,11, 029Griffiths L. M., Melchiorri A., Silk J., 2001, Astrophys. J. Lett., 553, L5Grossi M., Springel V., 2009, Mon. Not. Roy. Astron. Soc., 394, 1559GÃl’nova-Santos R., Atrio-Barandela F., Kitaura F., MÃijcket J., 2015, As-trophys. J., 806, 113Hand N., Feng Y., Beutler F., Li Y., Modi C., Seljak U., Slepian Z., 2018,Astron. J., 156, 160Heymans C., et al., 2020, arXiv e-prints, p. arXiv:2007.15632Hikage C., et al., 2019, PASJ, 71, 43Jaber-Bravo M., Almaraz E., de la Macorra A., 2020, Astropart. Phys., 115,102388Kaiser N., 1987, Monthly Notices of the Royal Astronomical Society, 227,1Klypin A., et al., 2020, arXiv e-prints, p. arXiv:2006.14910Koivisto T., Mota D. F., 2006, Phys. Rev. D, 73, 083502Kumazaki K., Yokoyama S., Sugiyama N., 2011, JCAP, 12, 008Lawrence E., Heitmann K., White M., Higdon D., Wagner C., Habib S.,Williams B., 2010, ApJ, 713, 1322Lawrence E., et al., 2017, ApJ, 847, 50Lee S., 2014, Eur. Phys. J. C, 74, 3146Linder E. V., 2003, Phys. Rev. Lett., 90, 091301Linder E. V., Robbers G., 2008, JCAP, 06, 004Linton M. S., Pourtsidou A., Crittenden R., Maartens R., 2018, JCAP, 04,043Mastache J., de la Macorra A., 2019, arXiv e-prints, p. arXiv:1909.05132Mead A. J., 2017, MNRAS, 464, 1282Mead A. J., Peacock J. A., Heymans C., Joudaki S., Heavens A. F., 2015,MNRAS, 454, 1958Mead A. J., Heymans C., Lombriser L., Peacock J. A., Steele O. I., WintherH. A., 2016, MNRAS, 459, 1468Mead A. J., Tröster T., Heymans C., Van Waerbeke L., McCarthy I. G.,2020a, arXiv e-prints, p. arXiv:2005.00009Mead A., Brieden S., Tröster T., Heymans C., 2020b, arXiv e-prints, p.arXiv:2009.01858Mohammed I., Seljak U., 2014, MNRAS, 445, 3382Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144Perenon L., Piazza F., Marinoni C., Hui L., 2015, JCAP, 11, 029Philcox O. H., Spergel D. N., Villaescusa-Navarro F., 2020, Phys. Rev. D,101, 123520Planck Collaboration 2018, arXiv e-prints, p. arXiv:1807.06209Pogosian L., Silvestri A., 2016, Phys. Rev. D, 94, 104014Poulin V., Smith T. L., Karwal T., Kamionkowski M., 2019, Phys. Rev.Lett., 122, 221301Press W. H., Schechter P., 1974, ApJ, 187, 425Scoccimarro R., 2004, Phys. Rev. D, 70, 083007Seljak U., 2000, MNRAS, 318, 203Seljak U., Vlah Z., 2015, Phys. Rev. D, 91, 123516MNRAS , 1–14 (2020) on-linear evolution in bump cosmologies Shi J., Katsuragawa T., Qiu T., 2020, Phys. Rev. D, 101, 024046Simpson F., James J. B., Heavens A. F., Heymans C., 2011, Phys. Rev. Lett.,107, 271301Smith R. E., et al., 2003, MNRAS, 341, 1311Smith R. E., Scoccimarro R., Sheth R. K., 2007, Phys. Rev. D, 75, 063512Springel V., 2005, MNRAS, 364, 1105Steinhardt P. J., Wang L.-M., Zlatev I., 1999, Phys. Rev. D, 59, 123504Takahashi R., Sato M., Nishimichi T., Taruya A., Oguri M., 2012, ApJ, 761,152Taruya A., Nishimichi T., Saito S., 2010, Phys. Rev. D, 82, 063522Tassev S., 2014, JCAP, 06, 008Taylor A., Hamilton A., 1996, Mon. Not. Roy. Astron. Soc., 282, 767Verde L., Treu T., Riess A. G., 2019, Nature Astronomy, 3, 891Vlah Z., White M., Aviles A., 2015, Journal of Cosmology and Astroparti-cle Physics, 2015, 014â ˘A¸S014White M., 2016, J. Cosmology Astropart. Phys., 2016, 057Zyla et al. 2020, Prog. Theor. Exp. Phys., 083C01collaboration S., 2020, ApJS, 249, 3de Putter R., Huterer D., Linder E. V., 2010, Physical Review D, 81de la Macorra A., Almaraz E., 2018, Phys. Rev. Lett., 121, 161303de la Macorra A., Piccinelli G., 2000, Phys. Rev. D, 61, 123503de la Macorra A., Stephan-Otto C., 2002, Phys. Rev. D, 65, 083520de la Macorra A., et al. 2020, In preparationMNRAS000