Cosmic Degeneracies II: Structure formation in joint simulations of Warm Dark Matter and f(R) gravity
MMNRAS , 1–14 (2011) Preprint 2 October 2018 Compiled using MNRAS L A TEX style file v3.0
Cosmic Degeneracies II: Structure formation in jointsimulations of Warm Dark Matter and f ( R ) gravity Marco Baldi , , , Francisco Villaescusa-Navarro , Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Universit`a di Bologna, viale Berti Pichat, 6/2, I-40127 Bologna, Italy; INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy; INFN - Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy; INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34143, Trieste, Italy; INFN - Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy;
ABSTRACT
We present for the first time the outcomes of a cosmological N-body simulation thatsimultaneously implements a Warm Dark Matter (WDM) particle candidate and amodified gravitational interaction in the form of f ( R ) gravity, and compare its resultswith the individual effects of these two independent extensions of the standard ΛCDMscenario, and with the reference cosmology itself. We consider a rather extreme valueof the WDM particle mass ( m WDM = 0 . f ( R ) grav-ity with | ¯ f R | = 10 − , and we investigate the impact of these models and of theircombination on a wide range of cosmological observables with the aim to identifypossible observational degeneracies. In particular, we focus on the large-scale matterdistribution, as well as on the statistical and structural properties of collapsed halosand cosmic voids. Differently from the case of combining f ( R ) gravity with massiveneutrinos – previously investigated in Baldi et al. (2014) – we find that most of theconsidered observables do not show any significant degeneracy due to the fact thatWDM and f ( R ) gravity are characterised by individual observational footprints witha very different functional dependence on cosmic scales and halo masses. In particular,this is the case for the nonlinear matter power spectrum in real space, for the halo andsub-halo mass functions, for the halo density profiles and for the concentration-massrelation. However, other observables – like e.g. the halo bias – do show some levelof degeneracy between the two models, while a very strong degeneracy is observedfor the nonlinear matter power spectrum in redshift space, for the density profiles ofsmall cosmic voids – with radius below ≈ /h – and for the voids abundance asa function of the void core density. Key words: dark energy – dark matter – cosmology: theory – galaxies: formation
The presently accepted standard cosmological scenario –known as ΛCDM – has been proved as an extremely success-ful model capable to explain numerous cosmological obser-vations, from the statistical properties of the anisotropies inthe Cosmic Microwave Background (CMB, Ade et al. 2015)to the clustering properties of galaxies or cosmic neutralhydrogen on large-scales (Alam et al. 2016; Delubac et al.2015). This model assumes that the observed accelerated ex-pansion of the Universe (Riess et al. 1998; Perlmutter et al.1999; Schmidt et al. 1998) is driven by a cosmological con-stant Λ whose value is set by observations, and that the dark matter that drives the growth of cosmic structures is cold ,i.e. that it has negligible thermal velocities at all redshifts.Despite the spectacular success of the ΛCDM model,some of its predictions are in tension with observations onboth large and small scales. In particular, a persisting ten-sion has recently emerged between the best-fit cosmologicalparameters obtained for a ΛCDM cosmology from CMB ob-servations (Ade et al. 2015) and a number of large-scalestructure observations including weak gravitational lensing(Heymans et al. 2013; Hildebrandt et al. 2016), redshift-space distortions induced by the peculiar motion of galaxies(Blake et al. 2011; Reid et al. 2012; Simpson et al. 2016)or galaxy clusters (Vikhlinin et al. 2009), as well as from c (cid:13) a r X i v : . [ a s t r o - ph . C O ] A ug M. Baldi & F. Villaescusa-Navarro galaxy cluster counts (Planck Collaboration et al. 2015).Furthermore a number of longstanding observational ten-sions at small scales are still persisting. Among these thereare: i ) the cusp-core problem: N-body simulations predictthat the density profile of dark matter halos exhibits a cuspin their cores, while from observations we know that somegalaxies present a core in their density profile (Salucci et al.2007; Gilmore et al. 2007; van Eymeren et al. 2009; Kuzio deNaray et al. 2010; Walker & Pe˜narrubia 2011). ii ) The miss-ing satellite problem: the number of satellites around Milky-way dark matter halos from simulations is much larger thanthe number we observe (Moore et al. 1999; Klypin et al.1999). iii ) The too-big-to-fail problem: the kinematic prop-erties of the most massive subhalos around Milky-way ha-los from simulations are in strong tension with observations(Boylan-Kolchin et al. 2011, 2012).In order to overcome these tensions and to alleviatethe theoretical naturalness problems associated with theobserved energy scale of the cosmological constant, severalmodifications of the standard ΛCDM cosmology have beenproposed and investigated over the past decades. In partic-ular, in the present work we will be focusing on Warm DarkMatter (WDM) scenarios aiming to solve the small-scale is-sues of the standard model, and on Modified Gravity (MG)theories as possible alternative and more natural explana-tions for the observed accelerated cosmic expansion, alsopossibly providing new interpretations to the above men-tioned large-scale tensions.Both these alternative cosmological scenarios have beenwidely investigated in the literature and their effects on alarge number of observables have been tested and clarified.In particular, WDM models have been tested by numer-ous authors (see e.g. Col´ın et al. 2000; Bode et al. 2001;Avila-Reese et al. 2001; Yoshida et al. 2003; Viel et al.2012; Schneider et al. 2012; Lovell et al. 2012; Maccio et al.2012; Maio & Viel 2015; Carucci et al. 2015, and refer-ences therein) and their impact on the formation of struc-tures at small scales has been compared with observations(Narayanan et al. 2000; Viel et al. 2005; Miranda & Maccio2007; Markovic et al. 2011; Viel et al. 2013; Kennedy et al.2014) thereby placing constraints on the viable WDM par-ticle mass. Similarly, in more recent years MG cosmologieshave attracted significant interest for their expected impacton structure formation and on the evolution of collapsedstructures. In particular, the f ( R ) gravity theory that willbe discussed in the present work has been implemented insimulation codes of structure formation by several authors(Li et al. 2012b; Puchwein et al. 2013; Llinares et al. 2014)and a wide range of simulated observables within these MGscenarios have been obtained (just to metion some, see e.g.Li et al. 2011, 2012a; Jennings et al. 2012; Fontanot et al.2013; Arnold et al. 2013; Cai et al. 2014; Hellwing et al.2014; Arnold et al. 2015; Cai et al. 2015; Achitouv et al.2015).In the present work we aim to investigate the joint ef-fects of these two independent modifications of the standardcosmological model, testing for possible observational de-generacies and quantifying the deviations that each modelproduces on the expected signatures of the other. This typeof analysis has already been performed for the combinationof f ( R ) gravity theories with a non-negligible value of thetotal neutrino mass (Baldi et al. 2014) allowing to iden- tify a very strong degeneracy between these two classes ofmodels. We will therefore proceed along the lines of Baldiet al. (2014) and explore the joint effects of f ( R ) gravityand WDM particle candidates using high-resolution cosmo-logical simulations.The paper is organised as follows. In Section 2 we willprovide a brief overview on the two physical models consid-ered in this work, namely f ( R ) gravity in Section 2.1 andWDM in Section 2.2. In Section 3 we will describe the nume-rical setup adopted in the present work and the approachesemployed for the identification of collapsed halos and cos-mic voids. In Section 4 we will illustrate the outcomes ofour analysis on a wide range of cosmological observables.Finally, in Section 5 we will summarise our results and driveour conclusions. f ( R ) gravity For what concerns possible modifications of the theory ofgravity, as anticipated above, we will consider extensionsto standard General Relativity (GR) in the form of f ( R )gravity, which represents the most widely studied class ofModified Gravity models also down to their impact on linear(Pogosian & Silvestri 2008; Hu et al. 2016) and non-linearstructure formation (see e.g. Oyaizu et al. 2008; Schmidtet al. 2009; Li et al. 2012a; Puchwein et al. 2013; Llinareset al. 2014). f ( R ) gravity is characterised by the action S = (cid:90) d x √− g (cid:18) R + f ( R )16 πG + L m (cid:19) , (1)where the standard Einstein-Hilbert term R (with R be-ing the Ricci scalar curvature) is replaced by R + f ( R ). InEq. (1), G is Newton’s gravitational constant, g is the deter-minant of the metric tensor g µν , and L m is the Lagrangiandensity of all matter fields. The model can be described byan additional scalar degree of freedom associated with thequantity f R ≡ d f ( R ) / d R . In the weak-field and quasi-staticlimit this scalar field obeys an independent dynamic equa-tion (see again Hu & Sawicki 2007): ∇ f R = 13 ( δR − πGδρ ) , (2)where δR and δρ are the relative perturbations in the scalarcurvature and matter density, respectively.A popular choice within all possible forms of the func-tion f ( R ) was proposed by Hu & Sawicki (2007): f ( R ) = − m c (cid:0) Rm (cid:1) n c (cid:0) Rm (cid:1) n + 1 , (3)where m ≡ H Ω M is a mass scale while c , c , and n arenon-negative constant free parameters of the model. Thechoice of Eq. (3) has the appealing feature of allowing torecover with arbitrary precision the expansion history of aΛCDM cosmology by choosing c /c = 6Ω Λ / Ω M under the We work in units where the speed of light is set to unity, c = 1.MNRAS000
The presently accepted standard cosmological scenario –known as ΛCDM – has been proved as an extremely success-ful model capable to explain numerous cosmological obser-vations, from the statistical properties of the anisotropies inthe Cosmic Microwave Background (CMB, Ade et al. 2015)to the clustering properties of galaxies or cosmic neutralhydrogen on large-scales (Alam et al. 2016; Delubac et al.2015). This model assumes that the observed accelerated ex-pansion of the Universe (Riess et al. 1998; Perlmutter et al.1999; Schmidt et al. 1998) is driven by a cosmological con-stant Λ whose value is set by observations, and that the dark matter that drives the growth of cosmic structures is cold ,i.e. that it has negligible thermal velocities at all redshifts.Despite the spectacular success of the ΛCDM model,some of its predictions are in tension with observations onboth large and small scales. In particular, a persisting ten-sion has recently emerged between the best-fit cosmologicalparameters obtained for a ΛCDM cosmology from CMB ob-servations (Ade et al. 2015) and a number of large-scalestructure observations including weak gravitational lensing(Heymans et al. 2013; Hildebrandt et al. 2016), redshift-space distortions induced by the peculiar motion of galaxies(Blake et al. 2011; Reid et al. 2012; Simpson et al. 2016)or galaxy clusters (Vikhlinin et al. 2009), as well as from c (cid:13) a r X i v : . [ a s t r o - ph . C O ] A ug M. Baldi & F. Villaescusa-Navarro galaxy cluster counts (Planck Collaboration et al. 2015).Furthermore a number of longstanding observational ten-sions at small scales are still persisting. Among these thereare: i ) the cusp-core problem: N-body simulations predictthat the density profile of dark matter halos exhibits a cuspin their cores, while from observations we know that somegalaxies present a core in their density profile (Salucci et al.2007; Gilmore et al. 2007; van Eymeren et al. 2009; Kuzio deNaray et al. 2010; Walker & Pe˜narrubia 2011). ii ) The miss-ing satellite problem: the number of satellites around Milky-way dark matter halos from simulations is much larger thanthe number we observe (Moore et al. 1999; Klypin et al.1999). iii ) The too-big-to-fail problem: the kinematic prop-erties of the most massive subhalos around Milky-way ha-los from simulations are in strong tension with observations(Boylan-Kolchin et al. 2011, 2012).In order to overcome these tensions and to alleviatethe theoretical naturalness problems associated with theobserved energy scale of the cosmological constant, severalmodifications of the standard ΛCDM cosmology have beenproposed and investigated over the past decades. In partic-ular, in the present work we will be focusing on Warm DarkMatter (WDM) scenarios aiming to solve the small-scale is-sues of the standard model, and on Modified Gravity (MG)theories as possible alternative and more natural explana-tions for the observed accelerated cosmic expansion, alsopossibly providing new interpretations to the above men-tioned large-scale tensions.Both these alternative cosmological scenarios have beenwidely investigated in the literature and their effects on alarge number of observables have been tested and clarified.In particular, WDM models have been tested by numer-ous authors (see e.g. Col´ın et al. 2000; Bode et al. 2001;Avila-Reese et al. 2001; Yoshida et al. 2003; Viel et al.2012; Schneider et al. 2012; Lovell et al. 2012; Maccio et al.2012; Maio & Viel 2015; Carucci et al. 2015, and refer-ences therein) and their impact on the formation of struc-tures at small scales has been compared with observations(Narayanan et al. 2000; Viel et al. 2005; Miranda & Maccio2007; Markovic et al. 2011; Viel et al. 2013; Kennedy et al.2014) thereby placing constraints on the viable WDM par-ticle mass. Similarly, in more recent years MG cosmologieshave attracted significant interest for their expected impacton structure formation and on the evolution of collapsedstructures. In particular, the f ( R ) gravity theory that willbe discussed in the present work has been implemented insimulation codes of structure formation by several authors(Li et al. 2012b; Puchwein et al. 2013; Llinares et al. 2014)and a wide range of simulated observables within these MGscenarios have been obtained (just to metion some, see e.g.Li et al. 2011, 2012a; Jennings et al. 2012; Fontanot et al.2013; Arnold et al. 2013; Cai et al. 2014; Hellwing et al.2014; Arnold et al. 2015; Cai et al. 2015; Achitouv et al.2015).In the present work we aim to investigate the joint ef-fects of these two independent modifications of the standardcosmological model, testing for possible observational de-generacies and quantifying the deviations that each modelproduces on the expected signatures of the other. This typeof analysis has already been performed for the combinationof f ( R ) gravity theories with a non-negligible value of thetotal neutrino mass (Baldi et al. 2014) allowing to iden- tify a very strong degeneracy between these two classes ofmodels. We will therefore proceed along the lines of Baldiet al. (2014) and explore the joint effects of f ( R ) gravityand WDM particle candidates using high-resolution cosmo-logical simulations.The paper is organised as follows. In Section 2 we willprovide a brief overview on the two physical models consid-ered in this work, namely f ( R ) gravity in Section 2.1 andWDM in Section 2.2. In Section 3 we will describe the nume-rical setup adopted in the present work and the approachesemployed for the identification of collapsed halos and cos-mic voids. In Section 4 we will illustrate the outcomes ofour analysis on a wide range of cosmological observables.Finally, in Section 5 we will summarise our results and driveour conclusions. f ( R ) gravity For what concerns possible modifications of the theory ofgravity, as anticipated above, we will consider extensionsto standard General Relativity (GR) in the form of f ( R )gravity, which represents the most widely studied class ofModified Gravity models also down to their impact on linear(Pogosian & Silvestri 2008; Hu et al. 2016) and non-linearstructure formation (see e.g. Oyaizu et al. 2008; Schmidtet al. 2009; Li et al. 2012a; Puchwein et al. 2013; Llinareset al. 2014). f ( R ) gravity is characterised by the action S = (cid:90) d x √− g (cid:18) R + f ( R )16 πG + L m (cid:19) , (1)where the standard Einstein-Hilbert term R (with R be-ing the Ricci scalar curvature) is replaced by R + f ( R ). InEq. (1), G is Newton’s gravitational constant, g is the deter-minant of the metric tensor g µν , and L m is the Lagrangiandensity of all matter fields. The model can be described byan additional scalar degree of freedom associated with thequantity f R ≡ d f ( R ) / d R . In the weak-field and quasi-staticlimit this scalar field obeys an independent dynamic equa-tion (see again Hu & Sawicki 2007): ∇ f R = 13 ( δR − πGδρ ) , (2)where δR and δρ are the relative perturbations in the scalarcurvature and matter density, respectively.A popular choice within all possible forms of the func-tion f ( R ) was proposed by Hu & Sawicki (2007): f ( R ) = − m c (cid:0) Rm (cid:1) n c (cid:0) Rm (cid:1) n + 1 , (3)where m ≡ H Ω M is a mass scale while c , c , and n arenon-negative constant free parameters of the model. Thechoice of Eq. (3) has the appealing feature of allowing torecover with arbitrary precision the expansion history of aΛCDM cosmology by choosing c /c = 6Ω Λ / Ω M under the We work in units where the speed of light is set to unity, c = 1.MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM Parameter Value H − Mpc − Ω M DE b A s . × − n s Table 1.
The set of cosmological parameters adopted in thepresent work, consistent with the latest results of the Planck col-laboration (Ade et al. 2013). Here n s is the spectral index ofprimordial density perturbations while A s is the amplitude ofscalar perturbations at the redshift of the CMB. condition c ( R/m ) n (cid:29)
1, so that the scalar field f R takesthe approximate form: f R ≈ − n c c (cid:18) m R (cid:19) n +1 . (4)In this work we will only consider models with n = 1,which leaves c as the only free parameter of the model.The latter can be also expressed in terms of the associatedvalue of the mean scalar degree of freedom at the presentepoch, ¯ f R . As this convention has become the standardone in studies of f ( R ) gravity models, we will also specifyour cosmologies by their ¯ f R value.In f ( R ) models, the dynamical gravitational potentialΦ satisfies (Hu & Sawicki 2007): ∇ Φ = 16 πG δρ − δR , (5)so that the total gravitational force is governed by a modifiedpotential Φ = Φ N − δR/ MG-GADGET code (Puchwein et al. 2013) that consistently includes the ef-fects of the modified potential and its associated
Chameleon screening mechanism (Khoury & Weltman 2004). The
MG-GADGET code features a Newton-Gauss-Seidl iterativescheme to solve Eq. (2) for a generic density field producedby a set of discrete particles, and computes the total forceexperienced by each particle through Eq. (5) by including inthe gravitational source term the curvature perturbation δR derived according to Eq. (2). We refer the interested readerto the MG-GADGET code paper for a more thorough pre-sentation of the numerical implementation.
A possible way to solve both the cusp-core and missing satel-lite problems, while preserving the success of CDM on largescales, is to enable the possibility that dark matter has non-negligible thermal velocities. This is what is referred to bythe term Warm Dark Matter (WDM). In WDM models, theformation of halos/subhalos on scales smaller than the darkmatter free-streaming length will be extremely suppressed,since only processes as halo fragmentation may produce suchobjects. On the other hand, if dark matter has a non-zerotemperature (i.e. thermal velocities are non negligible), thedark matter phase-space is finite, which implies that the density profile of halos can not be cuspy, but it should ex-hibit an inner core. Thus, both the missing satellite and thecusp-core problems would be naturally alleviated by invok-ing non-negligible dark matter thermal velocities.In terms of matter power spectrum, models with WDMwill exhibit a cut-off on small scales, produced by the darkmatter thermal velocities, that inhibits the matter clusteringon scales smaller than the free-streaming length. The cut-offin the primordial density power spectrum can be expressedas a transfer function with the form (Bode et al. 2001): T ( k ) ≡ P WDM ( k ) /P ΛCDM ( k ) = (1 + ( α k ) ν ) − /ν ,α ( m WDM ) = 0 . (cid:18) m WDM (cid:19) . (cid:18) Ω WDM . (cid:19) . (cid:18) h . (cid:19) . (6)with ν = 1 .
2. Other forms of the linear power suppressionhave been found in the literature (see e.g. Hansen et al. 2002)with a similar qualitative behaviour.This signature can be used to put constraints on themass (or magnitude of the thermal velocities) of the WDMparticles. The tightest constraints to date arise from mea-surements of the amplitude and shape of the Ly α powerspectrum, resulting in a 2 σ limit of m WDM (cid:62) . m WDM > . We perform four high-resolution simulations for four differ-ent cosmological models characterised by different laws ofgravity and dark matter particle mass.As a reference model, we consider a standard ΛCDMscenario, i.e. a model where gravity is governed by stan-dard General Relativity and the dark matter particle massis formally infinite, in the sense that thermal velocities arenegligible and the linear matter power spectrum does notexhibit any cut-off on small scales.For the same dark matter particle mass we investigatealso the case where gravity is described by the f ( R ) theorydescribed above in Section 2.1 for a scalar amplitude at z = 0of | ¯ f R | = 10 − .In addition, we will consider two cosmological modelscharacterised by these two different theories of gravity (GRand | ¯ f R | = 10 − ) and by a dark matter particle mass of0 . MNRAS , 1–14 (2011)
M. Baldi & F. Villaescusa-Navarro observables we will consider in the present work, as the ther-mal cut-off for such low mass is already significant at scalesthat are more easily resolved by our simulations. Therefore,our setup will provide a very prominent example of how thedegeneracy with the underlying theory of gravity might af-fect some observational features that are usually employedto derive constraints on the dark matter particle mass. Fur-thermore, this value corresponds to the most extreme valueconsidered in the recent work by Yang et al. (2015) aboutthe impact of WDM on the properties of cosmic voids towhich we aim comparing some of our results.Our simulations evolve a set of 512 particles in a peri-odic cosmological box of 100 comoving Mpc /h aside, with amass resolution of m ≈ . × M (cid:12) /h and a gravitationalsoftening of (cid:15) g = 6 kpc /h . All simulations share the samecosmological parameters (summarised in Table 1, consistentwith Planck cosmological constraints, Ade et al. 2015) andthe same background expansion history since both WDMand f ( R ) gravity (for the Hu & Sawicki setup adopted inthe present work) do not appreciably affect the backgroundevolution of the universe.For the WDM simulations we imprint the thermal cut-off described by Eq. (6) onto the primordial power spectrumwhen generating the initial conditions, which are producedby displacing particles from a regular cartesian grid accord-ing to Zel’dovich approximation (Zel’dovich 1970) at a start-ing redshift of z = 127. On top of the peculiar velocities, weadd to the particles thermal velocities whose magnitude israndomly drawn from the WDM momentum distribution.The orientation of the thermal velocity vector is taken ran-domly within the sphere.It is a well known fact that standard N-body simulationsof WDM suffer of numerical artifacts such as the forma-tion of spurious halos along the filaments of the cosmic web(Wang & White 2007). Such artificial fragmentation then re-sults in a population of small halos that do not correspond toreal physical objects. Numerical approaches to overcome thisproblem have been discussed e.g. by Angulo et al. (2013). Wedo not employ such alternative approaches as we are mostlyinterested in the degeneracy with the underlying theory ofgravity and a possible increase of the abundance of low masshalos due to artificial fragmentation would simply make ourresults more conservative.Due to limitations of computational resources and thehighly demanding integration of the combined simulation westop our runs at z = 0 .
25 and we will perform all our analysisat this redshift. This is not expected to affect significantlythe main conclusions of our investigation.
For all our four simulations we identify particle groupsthrough a Friends-of-Friends algorithm (Davis et al. 1985)with a linking length of 0 . SUBFIND algorithm(Springel et al. 2001) to identify gravitationally bound sub-strucures of each FoF group, and store halos with at least 20particles for which we compute spherical overdensity quan-tities as e.g. the virial mass M and virial radius R referred to the critical density of the universe, defined by the relation: 4 π R ∆ ρ crit = M (7)with ∆ = 200 and ρ crit = 3 H / (8 πG ).For our analysis we will only use halos whose main sub-structure has a virial mass M in the range (cid:2) , (cid:3) M (cid:12) /h : for higher halo masses the number of halos is too lowto provide robust statistical conclusions while lower massesare poorly resolved. With such catalogues at hand we willinvestigate the effects of WDM and of its combination with f ( R ) gravity on a number of statistical and structural prop-erties of the halo populations arising in the various cosmolo-gies, as described below in Section 4.2 We identify cosmic voids in our simulations suite by means ofthe publicly available void finder
VIDE (Sutter et al. 2015)by running it on a random subsample of the dark matterparticle distribution extracted from the z = 0 .
25 snapshotof the various runs.
VIDE is based on the
ZOBOV algorithm(Neyrinck 2008), which allows to identify wells in the densityfield produced by a set of points. This is done by means of aa Voronoi tessellation scheme that associates a cell to eachtracer of the density field and subsequently identifies localdensity minima among these cells by looking for cells with alarger Voronoi volume than all surrounding cells. By joiningtogether the Voronoi cells around a local density minimumbased on the Watershed Transform algorithm (see Platenet al. 2007), a hierarchy in the structures of the identifiedvoids is naturally obtained. Finally, the
VIDE toolkit elabo-rates on the void catalog obtained by
ZOBOV by performingvarious possible selections of the void sample as e.g. differ-ent cuts on the void density contrast or on the void centraloverdensity.For our analysis we will consider only voids with a cen-tral density below 20% of the mean density of the universeand with a density contrast between the most underdenseparticle of the void and the void boundary no lower than1 .
57, corresponding to a probability that the void arises fromPoisson noise below ∼
5% (see Neyrinck 2008).A standard quantity to characterise cosmic voids andstudy their relative abundance is given by their effectiveradius R eff , defined from the Voronoi volume of the void asthe radius of a sphere having the same volume as the void: V VOID ≡ N (cid:88) i =1 V pi = 43 πR . (8)However, following the discussion presented in Yang et al.(2015), for the analysis presented in this work it is moreconvenient to characterise voids based on their core density,defined as the density of the most underdense Voronoi cellof each void. Such quantity will be employed in Section 4.3.1below to characterise the abundance of voids in the differentcosmologies under investigation. We now present the main outcomes of our simulations. Thevarious observables that will be discussed in this section have
MNRAS000
MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM P ( k ) / P ( k ) Λ CD M z = 0.25 Viel et al. 2012GR CDMGR 0.4 keVfR5 CDMfR5 0.4 kev
Figure 1.
The ratio of the non-linear matter power spectrum tothe reference model for the different cosmologies under investiga-tion. been extracted from the simulations snapshots using stan-dard and well-tested numerical pipelines that are routinelyemployed for the analysis of cosmological simulations, andare presented in a very standard form. Therefore, we willnot describe very extensively these procedures, the noveltyof the present work being in the cosmological models inves-tigated rather than on the analysis performed. We will fo-cus on the statistical and structural properties of both darkmatter halos and cosmic voids.
For all the simulations presented above we have computedthe matter power spectrum at z = 0 .
25 through a Cloud-in-Cell mass assignment to a cartesian grid with 768 nodes,thereby spanning the range of Fourier modes between k =0 . h/ Mpc and the Nyquist frequency of k Ny ≈ h/ Mpc.In Figure 1 we show the ratio of the nonlinear matter powerspectrum at z = 0 .
25 to the standard cosmological sce-nario in the various models under investigation. As onecan see from the plot, the WDM simulation within stan-dard GR (solid red curve with open squares) shows the ex-pected suppression of power at small scales, with deviationsfrom the standard ΛCDM case starting at the cut-off scale k c ≈ h/ Mpc. As a reference, we overplot as a dashed linethe result of the nonlinear fitting function provided by Vielet al. (2012): T ( k ) ≡ P WDM ( k ) /P ΛCDM ( k ) = (1 + ( α k ) νl ) − s/ν ,α ( m WDM , z ) = 0 . (cid:18) m WDM (cid:19) . (cid:18) z (cid:19) . , (9)where we have used the values ν = 3, l = 0 . s = 0 . f ( R ) simulation for the case of CDM particles(solid green curve with triangles) gives rise to the expectedscale-dependent power enhancement due to the action of thefifth-force associated with the extra scalar degree of free-dom f R . This is consistent with a number of previous worksand in particular with the outcomes of the code comparisonproject for f ( R ) gravity cosmologies presented in Wintheret al. (2015).When we consider the combined simulation for a WDMparticle with mass m WDM = 0 . f ( R ) gravity (solidblue curve with diamonds) we find (quite expectedly) thatat large scales no significant difference appears with respectto the CDM- f ( R ) case, while at scales below the cut-off k c the power is suppressed compared to the CDM case. This isqualitatively expected, but the present work provides thefirst quantitative determination of the power suppressiondue to WDM in the context of a modified gravity cosmo-logical model. In particular, it is remarkable to notice thatthe nonlinear suppression relative to the CDM model within f ( R ) gravity shows the same shape and amplitude that isfound for the GR case: the blue dashed curve in Figure 1 –which represents the suppression obtained by applying thebest-fit suppression function from Eq. (9) for the GR caseto the f ( R )-CDM simulation – very well captures the powerspectrum ratio computed from the simulations.We have also computed the monopole of the matterpower spectrum in redshift-space at z = 0 .
25 for the fourdifferent models we investigate in this work. We move theparticles from real-space to redshift-space using the distantobserver approximation. We obtain three different realiza-tions in redshift-space by using the peculiar velocities ofthe particles along the three different cartesian axes. Themonopole in redshift-space is obtained by averaging over theresults from each realization. In Fig. 2 we show the results.The upper panel of Fig. 2 displays the matter monopolein redshift-space, normalized by the monopole of the ΛCDMmodel. Differences among all models are much smaller inredshift-space than in real-space. The model with modifiedgravity and CDM is degenerate with ΛCDM at the ∼ k → P s ( k ) P r ( k ) = 1 + 23 β + 15 β , (10)where the redshift-space distortion parameter is β = f ( z ) /b ,with f ( z ) being the linear growth factor and b is the bias(in this case b = 1). Unfortunately, our simulation boxes aretoo small to reach the above Kaiser limit. MNRAS , 1–14 (2011)
M. Baldi & F. Villaescusa-Navarro
Figure 2.
Upper panel : Monopole of the matter power spec-trum in redshift-space normalized by the monopole of the ΛCDMmodel.
Lower panel : Ratio between the monopoles in redshift-and real-space for the four different models. All results are shownat z = 0 . We have also investigated the clustering properties of darkmatter halos in the different models using the halo biasas our cosmological statistics. We have computed for eachmodel the halo bias as the ratio between the halo-mattercross-power spectrum to the matter auto-power spectrum.We have used this definition of halo bias to avoid stochas-ticity. Results are shown in Fig. 3.Unfortunately, our simulation boxes are not largeenough to allow us to explore the linear bias and whetherany of these models exhibit a scale-dependent bias on largescales. We find that models with WDM present a higheramplitude of the halo bias than their CDM counterparts,irrespective of the underlying gravity model. This is the ex-pected effect of WDM (Maccio et al. 2012; Dunstan et al.2011): suppressing the abundance of low mass halos andtherefore enhancing the clustering of halos of all masses.The halo bias of the modified gravity models is sys-tematically lower than the one of the standard GR mod-els by ∼ b ( k ) = P h m ( k ) / P mm ( k ) z = 0 . GR CDMGR 0.4 keVfR5 CDMfR5 0.4 keV -1 k [ h Mpc − ] R a t i o Figure 3.
Halo bias at z = 0 .
25 in real-space. Upper panelsshows the halo bias, computed as the ratio of the halo-mattercross-power spectrum over the matter auto-power spectrum forthe different models, as indicated in the legend. The bottom paneldisplays the halo bias normalized by the results of the ΛCDMcosmology. tive differences among the various models are almost scale-independent.
We compute the differential halo mass function of all ourcosmologies as the number of halos identified by the
SUB-FIND algorithm having a virial mass M (see above) fallingwithin 12 logarithmically equispaced mass bins ranging be-tween 10 and 10 M (cid:12) /h . In Fig. 4 we display the ratioof the abundance computed in each bin to the fiducial caserepresented by the ΛCDM cosmology.As expected, the WDM model (shown as red opensquares in the plot) results in a strong suppression of theabundance of small halos with a reduction of ≈
40% of ob-jects at the smallest mass side of our considered interval. Atlarger masses the halo abundance of the WDM cosmologyrecovers the ΛCDM expectation within statistical errors, dis-played as the grey-shaded area in Fig. 4 and correspondingto the Poissonian error in each bin.The f ( R ) cosmology with CDM particles (blue open di-amonds) shows on the contrary a mass-dependent enhance-ment of the abundance of halos that becomes more pro-nounced for larger halo masses, reaching a ≈
35% increasedhalo abundance at the largest mass available to our halosample. This is also an expected result and fully consistentwith the outcomes of several previous works (as e.g., Liet al. 2012b; Puchwein et al. 2013; Lombriser et al. 2013;Winther et al. 2015; Achitouv et al. 2015, just to mentionsome).
MNRAS000
MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM M [M O • /h]0.60.81.01.21.4 N ( M ) / N ( M ) Λ CD M fR5 0.4 kevfR5 CDMGR 0.4 keVGR CDM z = 0.25 Figure 4.
The halo mass function for the three cosmologies underinvestigation. The grey-shaded region represent the Poissonianerror propagated to the counts ratio based on the number of halosin each bin.
By comparing these first two simulations it appears al-ready clear that no degeneracy between WDM and f ( R )gravity is to be expected for an observable as the halomass function, since the two scenarios give rise to the op-posite mass-dependence of the deviation from the standardcosmological scenario. This is confirmed by directly check-ing the combined simulation (green open triangles) whichshows how the small-mass suppression and the large-massenhancement both persist in the combined halo mass func-tion. Therefore, the combination of WDM and f ( R ) gravityis expected to enhance the signal of a mismatch between thedetected abundance of halos at large and small masses. We also compute – for all the cosmologies under investiga-tion – the subhalo mass function, defined as the abundanceof substructures of mass M sub that are gravitationally boundto a main halo of virial mass M , as a function of the massratio M sub /M . The results are displayed in Fig. 5 with thesame color- and symbol-coding of the previous figures.As one can see from the plot, the WDM case showsthe expected suppression of the abundance of substructures,with about 25 −
40% fewer subhalos compared to ΛCDMfor a mass fraction M sub /M below 10 − . The effect issignificant when compared to the statistical error due toPoisson noise, which is represented in the plot by the grey-shaded region.This suppression does not appear to be significantlymodified when moving from GR to f ( R ) as the underlyingtheory of gravity. In fact, the combined simulation showsapproximately the same behavior as the GR-WDM run. Onthe contrary, the f ( R )-CDM simulation – for which no sup-pression of the abundance of substructures is expected – isfound to be consistent with the standard scenario withinstatistical errors.Therefore, also for the case of the abundance of bound dn ( M s ub / M ) / dLog ( M s ub / M ) -3.0 -2.5 -2.0 -1.5 -1.0 Log[M sub /M ]0.500.751.001.251.50 dn / dn Λ CD M z = 0.25fR5 0.4 kevfR5 CDMGR 0.4 keVGR CDM Figure 5.
The subhalo mass function for the three cosmologiesunder investigation. The grey-shaded region represent the Pois-sonian error propagated to the counts ratio based on the numberof subhalos in each bin. substructures within virialised halos no significant degener-acy is found between the effects of a WDM particle and amodified theory of gravity in the form of f ( R ). For each of the 12 mass bins employed to compute the dif-ferential halo mass function discussed above we have com-puted the average density profiles of up to 100 halos. Thismeans that we have computed the mass density in a setof 30 logarithmic radial shells centered on the most boundparticle of a halo for a random sample of halos belongingto each mass bin, and then stacked these density profilesafter rescaling the individual radial coordinates in units ofthe halo virial radius R to obtain an average profile foreach bin. For those bins not reaching 100 members (nor-mally the last three) we have performed the stacking usingall the available members.All the profiles, independently on the mass bin and onthe cosmological model, have then been normalised to unityat the virial radius and we show the ratio of these profilesto the ΛCDM cosmology in Fig. 6 for four out of the twelveavailable mass bins. We restrict the analysis to masses above ≈ × M (cid:12) /h as the lower mass halos are poorly re-solved and the resulting profiles are noisy. The grey-shadedareas in the plot represent the 2- σ confidence limits basedon the standard deviation of the mean density profiles com-puted through a bootstrap resampling technique with 1000re-samples of the 100 individual profiles.As one can see in the plots, the WDM realisation for astandard GR theory of gravity (red open squares) always re-sults in shallower density profiles with a significant suppres-sion of the central overdensity and a lower profile slope, eventhough the effect is – as expected – more pronounced for thelowest mass bin and progressively decreases for higher-masshalos. In particular, for the largest masses available to oursample the profile is marginally consistent with the standard MNRAS , 1–14 (2011)
M. Baldi & F. Villaescusa-Navarro [ ρ / ρ ] / [ ρ Λ CD M / ρ Λ CD M ]
The ratio of the stacked halo density profiles to the fiducial GR+CDM cosmology for about 100 halos within 4 out of the 12mass bins considered for the halo mass function and concentration analysis. The grey shaded area represents a 2 − σ uncertainty on thestacked profile based on a bootstrap procedure. cosmological result at ≈ σ . This is the very well known ef-fect of the thermal cut-off in the primordial density powerspectrum, which determines the formation of cored densityprofiles for low mass halos and of progressively steeper pro-files for increasing halo mass (Villaescusa-Navarro & Dalal2011; Maccio et al. 2012).On the contrary, the CDM realisation for f ( R ) gravity(green open triangles) shows a more complicated modula-tion of the deviation with respect to the standard modelas a function of the halo mass. More specifically, while theprofiles are only mildly affected at the lowest mass bin andcompletely unaffected at the highest mass bin displayed inthe plots, the effect is more pronounced for intermediatemasses and results in a significant steepening of the profileswith up to a 40% enhancement of the central overdensity forhalos of mass ≈ − × M (cid:12) /h . This behaviour is consis-tent with the effects of the Chameleon screening mechanismthat characterises f ( R ) gravity, with the most massive halosbeing fully screened due to their deep potential wells, andthe lowest mass halos having formed and virialised at earlierepochs when the higher average cosmic density significantlysuppresses the scalar fifth force.It is particularly interesting then to test the outcomes ofthe combined WDM- f ( R ) simulation (open blue diamonds)where the interplay between these opposite effects and theirdifferent dependence on the host halo mass might give riseto characteristic observational footprints. In particular, wefind (quite surprisingly) that the shallowing of the densityprofiles for the low-mass halos presented in Figure 6 is sig-nificantly enhanced in f ( R ) gravity compared to standardGR. On the contrary, for intermediate masses (top-right andbottom-left plots in the Figure) the effect of the fifth-forceseems to overcome the suppression due to WDM and the resulting density profiles are steeper than in the standardcosmological scenario, even though with a weaker enhance-ment of the central overdensity as compared to the CDMcase. Finally, for the most massive halos considered in ouranalysis the effect of the fifth-force seems to be completelyabsent (due to the Chameleon screening, as discussed above)and the combined model appears fully consistent with theWDM result in standard GR.
For each halo in our sample we have computed the concen-tration c ∗ following the approach described in Springel et al.(2008) as: 2003 c ∗ ln(1 + c ∗ ) − c ∗ / (1 + c ∗ ) = 7 . δ V (11)where δ V is defined as: δ V = 2 (cid:18) V max H r max (cid:19) (12)with V max and r max being the maximum rotational veloc-ity of the halo and the radius at which this velocity peak islocated, respectively. Both quantities have been computedin all models based only on the gravitational potential asso-ciated with the matter density profile, therefore neglectingthe effect of the additional fifth-force. This approach to com-pute halo concentrations has already been employed in thecontext of f ( R ) modified gravity theories by Arnold et al.(2016) for a small set of very well resolved Milky-Way-sizedhalos (by means of a set of zoomed high-resolution simula-tions) and compared with the results obtained from fittingthe individual density profiles with a Navarro-Frenk-White MNRAS000
For each halo in our sample we have computed the concen-tration c ∗ following the approach described in Springel et al.(2008) as: 2003 c ∗ ln(1 + c ∗ ) − c ∗ / (1 + c ∗ ) = 7 . δ V (11)where δ V is defined as: δ V = 2 (cid:18) V max H r max (cid:19) (12)with V max and r max being the maximum rotational veloc-ity of the halo and the radius at which this velocity peak islocated, respectively. Both quantities have been computedin all models based only on the gravitational potential asso-ciated with the matter density profile, therefore neglectingthe effect of the additional fifth-force. This approach to com-pute halo concentrations has already been employed in thecontext of f ( R ) modified gravity theories by Arnold et al.(2016) for a small set of very well resolved Milky-Way-sizedhalos (by means of a set of zoomed high-resolution simula-tions) and compared with the results obtained from fittingthe individual density profiles with a Navarro-Frenk-White MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM M [h -1 M O • ]4681012 c * m ean z=0.25 M [h -1 M O • ]0.40.60.81.01.21.41.6 c * m ean / c * m ean ( Λ CD M ) fR5 0.4 kevfR5 CDMGR 0.4 keVGR CDMz=0.25 Figure 7.
The concentration-mass relation (
Left ) and its ratio to the standard cosmological model (
Right ) for the three cosmologiesunder investigation. The grey-shaded regions represent the Possonian errors on both the concentrations ratio and the mean concentrationvalues due to the number of halos used in each mass bin. The dotted lines in the
Left plot indicate the spread of 68% of the halos ineach bin. (NFW Navarro et al. 1997) shape. In our setup we have amuch poorer resolution than in the simulations of Arnoldet al. but a significantly larger statistics. We have there-fore repeated the comparison by performing a 1-parameterfitting of an NFW shape for the individual profiles of allthe randomly-selected halos used to compute the stackeddensity profiles discussed in the previous section, and com-pared these determinations of the average concentration ineach of the 12 mass bins adopted for both the halo massfunction and the halo density profiles with the one obtainedfrom Eq. (11). We found consistent results on the relativeeffects of the different models between these two differentways to compute concentrations for the 5 most massive bins(for which the density profiles are reasonably well resolvedand the fitting procedure is not exceedingly affected by theradial fitting range). For lower mass halos the agreement issignificantly worse, but this is to be expected due to the poorresolution of the individual profiles and the large scatter inthe fitted parameters. In the following we will then assumethat the concentration values provided through Eq. (11) arereliable over the whole mass range covered by our sampleand we will show only these results in the following. Thisassumption would clearly require a more detailed scrutinythough a sample of simulated halos with comparable statis-tics but higher resolution than allowed by our simulations.In Figure 7 we show the concentration-mass relation( left panel) and its ratio to the standard cosmological model( right panel) for all the cosmologies under investigation. Thegrey-shaded areas represent the 2- σ confidence regions basedon the Poisson errors on the mean for each mass bin, whilethe dotted curves in the left panel show the spread of 68%of halos around the mean in each bin.As one can see from the two plots, the WDM model instandard GR shows the expected behaviour of suppressinghalo concentrations at low masses and progressively recov-ering the standard ΛCDM concentrations at larger masses.A maximum suppression of ≈
30% is found for the lowest masses available to our sample, while at the four largestmass bins the model is again consistent with ΛCDM at 2- σ .The CDM model within f ( R ) gravity, instead, shows asignificant increase of halo concentrations for intermediatemasses and recovers the standard value of the concentration-mass relation only at the highest mass bin. Interestingly,the lowest mass halos with virial masses M (cid:46) × M (cid:12) /h show a lower concentration compared to the standardmodel. This is qualitatively consistent with the behaviour ofthe density profiles shown in Figure 6 above. Such suppres-sion of halo concentrations has been found for a fraction ofthe considered halos with masses around 2 × M (cid:12) /h alsoin the higher-resolution study of Arnold et al. (2016) and fora lower value of the f ( R ) scalar amplitude ( ¯ f R = − − ),while the majority of halos did show an increased concen-tration. Again, a resolution comparable to Arnold et al. fora halo sample with similar statistics to our work would berequired to robustly assess the impact of f ( R ) gravity onthe concentrations of halos below such mass value.Finally, we investigated the outcomes of the combinedWDM- f ( R ) cosmology which shows a very interesting dis-tortion of the concentration-mass relation, again fully con-sistent with the results obtained for the density profiles: aweaker enhancement of halo concentrations compared to theCDM realisation of f ( R ) gravity for intermediate masseswith a steeper and stronger suppression of halo concentra-tions at low masses even when compared with the WDMcosmology for standard GR.The suppression of the inner halo density – and conse-quently of the halo concentration – for low mass halos thatwe find for the f ( R ) cosmologies for both CDM and WDMcompared to the standard GR case (Figures 6 and 7) is noteasily understandable in terms of the different regimes of ac-tion of the Chameleon screening mechanism and would de-serve further investigation, in particular – as already statedabove – a more detailed analysis based on higher-resolutionsimulations, which we defer to future work.
MNRAS , 1–14 (2011) M. Baldi & F. Villaescusa-Navarro ρ c /< ρ >10 -5 -4 n v [ h / M p c ] GR CDMGR 0.4 keVfR5 CDMfR5 0.4 kev
Figure 8.
The differential distribution function of cosmic voidsas a function of their core density (i.e. the number density n v ofvoids in each core density bin) for the various cosmologies. Theerror bars on the reference ΛCDM model correspond to the 2- σ Poissonian error on the number counts in each bin.
Cosmic voids have attracted significant interest in the pastfew years as possible complementary cosmological probes tothe standard large-scale structure observables. In particu-lar, previous investigation of the statistical and structuralproperties of cosmic voids in modified gravity theories havebeen performed by e.g. Cai et al. (2015), while a study ofthe impact of WDM on cosmic voids has been presented inYang et al. (2015).In the present work we investigate for the first time thejoint effects of these two modifications of the standard cos-mological model on cosmic voids, focusing in particular onthe abundance and on the density profiles of relatively smallvoids. Although a realistic determination of the expected ob-servational properties of cosmic voids should properly takeinto account the effect of the bias of the visible tracers (suchas galaxies and clusters) employed to identify the voids inlarge cosmological surveys (see e.g. Pollina et al. 2016; Na-dathur & Hotchkiss 2015), a clear understanding of the prop-erties of cosmic voids in the underlying matter density fieldrepresents an essential step to employ voids as cosmologicalprobes.
Following the discussion presented in Yang et al. (2015) wehave computed the abundance of cosmic voids as a functionof their core density within 30 bins spanning the range ofcore densities covered by our void sample – after perform-ing the selection procedures described above in Section 3.2 –which goes from 0 to ≈ . σ errors computed as the Poissonian uncertainties associatedwith the number counts in each bin.As one can see in the plot, consistently with the out-comes of Yang et al. (2015), we do find that our WDM modelwithin standard GR (red open squares, having a WDM massof 0 . f ( R ) gravity (open green tri-angles), shows exactly the opposite behaviour, with a signif-icant enhancement of the most empty voids and a suppres-sion of the large core-density tail of the distribution.Quite remarkably, finally, the core density distributionfunction that results from the WDM model within the f ( R )gravity simulation shows that the effect associated with thethermal cut-off due to the low WDM particle mass is almostperfectly counterbalanced by the modified gravitational evo-lution, so that the resulting abundance of voids is againconsistent with the standard model within statistical uncer-tainties. This result provides the first evidence of an intrinsicobservational degeneracy between WDM and f ( R ) gravityat the level of the statistical properties of the population ofsmall cosmic voids.Even though the degeneracy between these two modifi-cations of the standard cosmological model is easily brokenby a number of other observables (as we have extensivelyshown in the previous sections) it is interesting to notice thatfor objects that are only mildly nonlinear (such as cosmicvoids) but still at relatively small scales the two models maskeach other almost perfectly. This outcome, as we will showin the next section, is not a peculiar coincidence occurringonly for the core density distribution function, but appearsto characterise also the structural properties of small voids,while for larger voids (arising from larger modes of the pri-mordial density spectrum) the effect of WDM becomes lesspronounced so that the expected imprint of f ( R ) gravitystems out prominently. As a final test of our set of cosmological models we havecomputed the average void density profiles for voids witheffective radius R eff below 5 Mpc /h and between 5 and 10Mpc /h , by stacking the individual spherically-averaged den-sity profiles of 100 randomly selected voids for each of thesetwo bins of R eff . The resulting mean profiles are displayedin the left and right panels of Fig. 9, respectively. In the twoplots, the top panels show the density profiles, with the errorbars on the standard ΛCDM curves representing the Pois-sonian errors on the mean, while the bottom panels showthe relative difference with respect to the standard model,with the grey-shaded regions indicating the 2- σ confidenceregion based on a bootstrap resampling technique with 1000 MNRAS000
Following the discussion presented in Yang et al. (2015) wehave computed the abundance of cosmic voids as a functionof their core density within 30 bins spanning the range ofcore densities covered by our void sample – after perform-ing the selection procedures described above in Section 3.2 –which goes from 0 to ≈ . σ errors computed as the Poissonian uncertainties associatedwith the number counts in each bin.As one can see in the plot, consistently with the out-comes of Yang et al. (2015), we do find that our WDM modelwithin standard GR (red open squares, having a WDM massof 0 . f ( R ) gravity (open green tri-angles), shows exactly the opposite behaviour, with a signif-icant enhancement of the most empty voids and a suppres-sion of the large core-density tail of the distribution.Quite remarkably, finally, the core density distributionfunction that results from the WDM model within the f ( R )gravity simulation shows that the effect associated with thethermal cut-off due to the low WDM particle mass is almostperfectly counterbalanced by the modified gravitational evo-lution, so that the resulting abundance of voids is againconsistent with the standard model within statistical uncer-tainties. This result provides the first evidence of an intrinsicobservational degeneracy between WDM and f ( R ) gravityat the level of the statistical properties of the population ofsmall cosmic voids.Even though the degeneracy between these two modifi-cations of the standard cosmological model is easily brokenby a number of other observables (as we have extensivelyshown in the previous sections) it is interesting to notice thatfor objects that are only mildly nonlinear (such as cosmicvoids) but still at relatively small scales the two models maskeach other almost perfectly. This outcome, as we will showin the next section, is not a peculiar coincidence occurringonly for the core density distribution function, but appearsto characterise also the structural properties of small voids,while for larger voids (arising from larger modes of the pri-mordial density spectrum) the effect of WDM becomes lesspronounced so that the expected imprint of f ( R ) gravitystems out prominently. As a final test of our set of cosmological models we havecomputed the average void density profiles for voids witheffective radius R eff below 5 Mpc /h and between 5 and 10Mpc /h , by stacking the individual spherically-averaged den-sity profiles of 100 randomly selected voids for each of thesetwo bins of R eff . The resulting mean profiles are displayedin the left and right panels of Fig. 9, respectively. In the twoplots, the top panels show the density profiles, with the errorbars on the standard ΛCDM curves representing the Pois-sonian errors on the mean, while the bottom panels showthe relative difference with respect to the standard model,with the grey-shaded regions indicating the 2- σ confidenceregion based on a bootstrap resampling technique with 1000 MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM ρ / ρ m ean z = 0.25 R eff = 0-5 Mpc/hGR CDMGR 0.4 keVfR5 CDMfR5 0.4 keV eff -0.2-0.100.10.2 ρ / ρ G R + CD M - z = 0.25 R eff = 5-10 Mpc/hGR CDMGR 0.4 keVfR5 CDMfR5 0.4 keV eff Figure 9.
The stacked void density profiles in two different ranges of effective radius for the various cosmologies. The upper plotsshow the mean spherically-averaged density profiles of 100 randomly selected voids, with the (barely visible) error bars representing thePoissonian error on the mean. The lower plots display the relative difference with respect to the standard ΛCDM cosmology with thegrey-shaded regions representing the 2- σ confidence region based on a bootstrap computation of the standard deviation of the averageprofiles. re-samples of the 100 individual profiles, as already done forthe halo density profiles discussed in Section 4.2.3 above.Consistently with Yang et al. (2015) and with the pre-vious results on the core density distribution, we find thatWDM within standard GR results in shallower density pro-files for the smallest voids, while no statistically significanteffect appears for somewhat larger voids.On the contrary, the effects of f ( R ) gravity for a stan-dard CDM particle candidate are present for both small andlarger voids, resulting in a steeper profile with a lower cen-tral density. This is consistent with previous studies on thestructural properties of cosmic voids in modified gravity cos-mologies (see again Cai et al. 2015).As a result, the density profiles for the smallest voids inthe combined WDM- f ( R ) cosmology are again fully consis-tent with the standard ΛCDM scenario, thereby confirmingthe degeneracy between these two models in the propertiesof cosmic voids already identified in the previous section forthe core-density distribution. For the larger voids, quite ex-pectedly, the degeneracy is much weaker due to the lowerimpact of the WDM cut-off and the still significant effect ofthe scalar fifth-force of the f ( R ) modified gravity model. We have performed an extended analysis of two cosmologicalN-body simulations featuring either a Warm Dark Matterparticle candidate or an f ( R ) modified gravity theory andcompared their outcomes to the standard ΛCDM cosmology. This analysis has provided fully consistent results with thelong series of previous works investigating the same models.Additionally, we have presented for the first time theresults of a cosmological simulation jointly including theeffects of both these deviations from the ΛCDM cosmol-ogy, and tested possible degeneracies of the two indepen-dent modifications of the standard model on a wide rangeof large-scale structure statistics.In particular, we have investigated the properties of thelarge-scale matter distribution by extracting from all oursimulations the nonlinear matter power spectrum both inreal and redshift space, the statistics of the halo populationswithin the different models, and the structural properties ofcollapsed structures over a wide range of masses, as wellas the properties of cosmic voids arising in these differentcosmological scenarios. Our results have shown that mostobservables do not show a significant degeneracy betweenthe effects of the WDM particle and those of a modified lawof gravity, while a few other observables are indeed charac-terised by a strong degeneracy.More specifically, our main results can be summarisedas follows. (cid:63) The matter power spectrum P ( k ) in configura-tion space shows the well known features for the individ-ual WDM and f ( R ) cosmologies, i.e. a scale-dependent sup-pression of power below a cut-off scale k c ≈ h/ Mpc forthe former and an scale-dependent enhancement of powerfor the latter, both with an amplitude increasing with scale.These effects have been widely discussed in the literatureand are well understood as resulting from the primordial
MNRAS , 1–14 (2011) M. Baldi & F. Villaescusa-Navarro cut-off in the density power spectrum and from the effect ofthe fifth-force associated with the scalar degree of freedom f R , respectively. In particular, for the WDM simulation wefind that the cut-off is well captured by the fitting formulaprovided by Viel et al. (2012), which provides a consistencycheck for our numerical integration. The combined modelshows an identical behaviour as the f ( R ) cosmology downto the cut-off scale k c , followed by a suppression of the mod-ified gravity enhancement. Quite remarkably, we find thatthe suppression induced by WDM with respect to the CDMscenario within f ( R ) gravity follows the same best-fit shapeobtained by the fitting formula of Viel et al. (2012) for thestandard GR simulation. Therefore, our results provides thefirst validation of this widely-used fitting formula for non-GR cosmologies. (cid:63) The matter power spectrum P s ( k ) in redshiftspace exhibits differences among models of a maximum of ∼ (cid:63) The halo bias b ( k ) shows that models with WDM havea larger amplitude, on all scales, with respect to the modelswith CDM, independently of the underlying gravity theory.We find that on scales k (cid:46) . h Mpc − relative differencesamong models are scale-independent and amount to ∼ ∼ (cid:63) The halo mass function shows the well known sup-pression of the abundance of small halos for the WDM cos-mology, and the expected mass-dependent enhancement ofthe abundance of halos for the f ( R ) gravity model. In thiscase, however, differently from the situation encountered forthe matter power spectrum, the two effects have the oppo-site dependence on the halo mass, with the former increasingfor progressively lower masses and the latter increasing forprogressively larger masses. This different mass dependenceensures that no degeneracy appears when the two models arejointly at work. In fact, the low-mass range of the resultinghalo mass function follows the same suppression found forthe WDM model while the high-mass range closely followsthe f ( R ) mass function. The transition between these tworegimes is found to occur at ≈ × M (cid:12) /h . (cid:63) The subhalo mass function shows a strong suppres-sion of the abundance of substructures, in particular for sub-halos with mass ratio to their host main structure below10 − , while no statistically significant change in the abun-dance of substructures is observed for the f ( R ) gravity sce-nario. Then, quite expectedly, the combination of the twomodels substantially follows the behaviour of the WDM cos-mology in standard GR with a comparable reduction of thenumber of small subhalos. (cid:63) The halo density profiles , compared among the dif-ferent models through a stacking procedure of 100 randomlyselected halos for several different halo mass bins, shows a significant shallowing of the density profiles for the WDMmodel at small halo masses, which is progressively reducedfor higher mass bins. On the contrary, the f ( R ) model showsa more complex modulation of the deviation from the stan-dard ΛCDM profiles, due to the different impact of the Chameleon screening mechanism for different overdensityenvironments, with very little deviations observed at thesmallest and largest mass bins, and a significant steepen-ing of the profiles for intermediate masses. Interestingly, thecombined model is found to enhance the WDM suppressionof the inner halo overdensities at small masses and to reducethe f ( R ) increase of the inner halo overdensities at interme-diate masses. (cid:63) The concentration-mass relation shows an interest-ing behaviour, fully consistent with the results obtained forthe halo density profiles in the different mass ranges. On onehand, the WDM scenario within standard GR displays thewell known suppression of the halo concentrations for lowmass halos, while the standard c − M relation is recovered forlarger masses; on the other hand, the two f ( R ) cosmologiesfor both CDM and WDM are found to provide an enhance-ment of halo concentrations at intermediate masses, with apeak of the deviation from ΛCDM at ≈ M (cid:12) /h , and asuppression of concentration at the lowest masses covered byour sample, which is more pronounced for the WDM realisa-tion as compared to the CDM case. While this suppressionmay be expected in the former case, the deviation foundfor CDM in f ( R ) gravity represents an interesting featuredeserving further investigation with higher resolution simu-lations; (cid:63) The abundance of cosmic voids is found to be one ofthe few observables (together with the matter power spec-trum in redshift space discussed above and with the voidprofiles summarised below) for which WDM and f ( R ) grav-ity show a significant degeneracy. In particular, as alreadyfound by Yang et al. (2015), WDM determines a suppressionof the abundance of small voids with core density below 5%of the mean cosmic density, and a corresponding enhance-ment of the abundance of voids above this threshols. Thiscorresponds to voids being less underdense in WDM com-pared to CDM. On the other hand, f ( R ) gravity is foundto determine the opposite effect on the population of smallcosmic voids, with an enhancement of the abundance of themost underdense voids and a corresponding suppression ofthe less underdense ones. As a result, the combined cosmol-ogy featuring both WDM and f ( R ) gravity shows a verystrong degeneracy with the standard ΛCDM result, withthe core-density distribution function being consistent withthe standard model at 2- σ confidence level. (cid:63) The density profiles of cosmic voids show a differ-ent range of effects for the different cosmologies dependingon the void size. For small voids ( R eff (cid:54) /h ) WDM isfound to make the void profiles shallower, with an inner den-sity about 20% larger than in the standard ΛCDM model,consistently with the previous results of Yang et al. (2015),while for larger voids (5 Mpc /h < R eff (cid:54)
10 Mpc /h ) theimpact of WDM is very mild, with the density profiles con-sistent at 2- σ with the standard scenario. On the other hand, f ( R ) gravity has the opposite effect of steepening the voidprofiles on both ranges of void size, even though the am-plitude of the suppression of the inner void density is morepronounced for the smallest voids. As a result, the com- MNRAS000
10 Mpc /h ) theimpact of WDM is very mild, with the density profiles con-sistent at 2- σ with the standard scenario. On the other hand, f ( R ) gravity has the opposite effect of steepening the voidprofiles on both ranges of void size, even though the am-plitude of the suppression of the inner void density is morepronounced for the smallest voids. As a result, the com- MNRAS000 , 1–14 (2011) osmic Degeneracies II: structure formation in f ( R ) -WDM bined model featuring at the same time WDM and f ( R )gravity is strongly degenerate with ΛCDM for the small-est voids while resulting marginally distinguishable from thestandard model for the larger ones. In this respect, we haveextended the results of Yang et al. (2015) to the case of anon-standard theory of gravity, finding evidence of a strongdegeneracy between the two models in the observables wherethe footprints of WDM first identified by Yang et al. (2015)are more pronounced.To summarise, we have performed and presented in thispaper – for the first time – the results of a cosmologicalsimulation of structure formation for a Warm Dark Matterparticle candidate evolving under the effect of a modifiedtheory of gravity, and compared its results to the referenceΛCDM cosmology and to the separate effects of these twomodifications of the standard cosmology.We have performed an extensive analysis of the simu-lated matter distribution and of the properties of the result-ing linear and non-linear structures, including halos, sub-halos, and voids, and identified possible observational de-generacies between the models. We found that – differentlyfrom the case of cosmologies featuring simultaneously mod-ified gravity and massive neutrinos discussed in Baldi et al.(2014) – most of the standard statistics do not show any sig-nificant degeneracy due to the fact that Warm Dark Matterand f ( R ) gravity impact these statistics with very differentfunctional dependencies on cosmic scales and halo masses.However, we also found that a clear degeneracy existsin the observational properties of small cosmic voids, withboth their abundance and their density profiles being hardlydistinguishable in the combined WDM- f ( R ) simulation fromthe corresponding results in the standard ΛCDM cosmology,even if the two models show – individually – a clear anddistinctive footprint in these observables. ACKNOWLEDGMENTS
MB acknowledges support from the Italian Ministry for Ed-ucation, University and Research (MIUR) through the SIRindividual grant SIMCODE, project number RBSI14P4IH.The numerical simulations presented in this work have beenperformed and analysed on the Hydra cluster at the RZGsupercomputing centre in Garching.
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