Structure formation simulations with momentum exchange: alleviating tensions between high-redshift and low-redshift cosmological probes
MMNRAS , 1–14 (2011) Preprint 13 November 2018 Compiled using MNRAS L A TEX style file v3.0
Structure formation simulations with momentumexchange: alleviating tensions between high-redshift andlow-redshift cosmological probes
Marco Baldi , , , Fergus Simpson 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; ICC, University of Barcelona (UB-IEEC), Marti i Franques 1, 08028, Barcelona, Spain.
13 November 2018
ABSTRACT
Persisting tensions between the cosmological constraints derived from low-redshiftprobes and the ones obtained from temperature and polarisation anisotropies of theCosmic Microwave Background – although not yet providing compelling evidenceagainst the ΛCDM model – seem to consistently indicate a slower growth of densityperturbations as compared to the predictions of the standard cosmological scenario.Such behavior is not easily accommodated by the simplest extensions of GeneralRelativity, such as f ( R ) models, which generically predict an enhanced growth rate.In the present work we present the outcomes of a suite of large N-body simulationscarried out in the context of a cosmological model featuring a non-vanishing scatteringcross section between the dark matter and the dark energy fields, for two differentparameterisations of the dark energy equation of state. Our results indicate thatthese Dark Scattering models have very mild effects on many observables related tolarge-scale structures formation and evolution, while providing a significant suppres-sion of the amplitude of linear density perturbations and the abundance of massiveclusters. Our simulations therefore confirm that these models offer a promising routeto alleviate existing tensions between low-redshift measurements and those of theCMB. Key words: dark energy – dark matter – cosmology: theory – galaxies: formation
The physical characteristics of dark matter and dark energyremain poorly constrained. Proposed candidates for darkmatter include axions, WIMPs, and black holes, while darkenergy has been linked to a cosmological constant and scalarfields. And of course, there remains ample scope for eitherdark matter or dark energy to be described by a fundamen-tally new form of physics. One fairly well determined featureis their current energy density, and in that respect, these twophenomena appear comparable. This has fuelled speculationthat their relationship is not purely gravitational.A further hint of a connection between the universe’stwo dominant constituents stems from small but consistentdeviations between low redshift measurements of the ampli-tude of density perturbations, and the extrapolated valuebased on the amplitude of primary anisotropies in the Cos-mic Microwave Background (CMB Ade et al. 2015). These low redshift measurements include weak gravitational lens-ing from CFHTLenS (Heymans et al. 2013), redshift spacedistortions induced by the motions of galaxies (Reid et al.2012; Blake et al. 2011; Simpson et al. 2016), and galaxyclusters (Vikhlinin et al. 2009), all indicating a slightly loweramplitude of clustering than has been inferred from the Cos-mic Microwave Background. Additionally, even lensing ofthe CMB itself prefers lower values of the amplitude of lin-ear perturbations (Ade et al. 2015).If the growth of cosmological structure is confirmed todeviate from the theoretical predictions of the ΛCDM model,one interpretation of this result would be the discovery of anew regime of gravitational physics. However, many of themost popular modified gravity theories, such as f ( R ), Sym-metron, and nDGP models, generically lead to a strength-ening of the gravitational force. This naturally implies anenhanced growth of perturbations, in contrast with the ob-servations which favour a suppressed growth rate. In this c (cid:13) a r X i v : . [ a s t r o - ph . C O ] M a y M. Baldi & F. Simpson respect, it appears more plausible that a non-gravitationalinteraction is responsible for the anomalous behaviour. Fur-thermore, in light of the stringent constraints on GeneralRelativity derived from solar system measurements (Bertottiet al. 2003; Will 2005), and further restrictions derived fromthe propagation of gravitational waves (Lombriser & Lima2016), the latter options would also appear to offer a morenatural solution.Many models of coupled dark energy have been pro-posed in the literature (see e.g. Amendola 2000; Barrow &Clifton 2006; Baldi 2011). However the overwhelming major-ity focus on a specific form of energy-momentum exchangebetween the two fluids, in which the coupling current is time-like. In other words, the theoretical models predominantlytake the form of energy exchange rather than momentumexchange. Motivated by the tendency for low-energy inter-actions between Standard Model particles to result in elasticscattering, Simpson (2010) proposed a model which invokespure momentum exchange between the two fluids. Subse-quently, Pourtsidou et al. (2013) presented a comprehensiveclassification of interacting Dark Energy models where theyidentified a class of models (termed ‘Type 3’ ) which invokepure momentum exchange between dark matter and a scalarfield. The properties of these models were explored in greaterdetail in Skordis et al. (2015). In this work we will aim todevelop our understanding of the relationship between theType 3 models and the elastic scattering model, and exploretheir phenomenological effects.In Baldi & Simpson (2015) it was shown that if thedark matter particles experience a drag force as they passthrough a canonical (i.e. non-phantom) dark energy fluid,they leave two key observational signatures. First of all thematter power spectrum is suppressed in a scale-independentfashion on linear scales, and secondly it is scale-dependentlyenhanced on nonlinear scales. Here we present results froma suite of N-body simulations in which the dark energy fluidhas an evolving equation of state, as is to be expected fromdynamical models, and compare the resulting large scalestructure with those found in Baldi & Simpson (2015). Ouraim is then to test whether such a class of cosmologies mightprovide a way to reconcile cosmological constraints arisingfrom CMB data analysis and low-redshift measurements ofthe growth of structuresIn § § §
4, and theiroutputs are analysed in §
5. Our concluding remarks are pre-sented in § Pourtsidou et al. (2013) present three classifications of cou-pled dark energy models. Here we shall focus on the
Type 3 class of models, which generate an exchange of momentumbetween dark matter and dark energy. This is achieved byinvoking a coupling between the dark matter velocity field u µ and the covariant derivative of a scalar field φ , as follows w CPL-1CPL-2 HYP-1HYP-2HYP-3
Figure 1.
The equation of state evolution for the various param-eterisations considered in the present work. The black dashed andtriple-dot-dashed lines represent the constant values w = − . w = − .
1, respectively, that have been discussed in Baldi &Simpson (2015)Parameterisation w w a z t ξ σ ΛCDM -1 – – – 0.83 w09 -0.9 – – 10 ,
50 0 . , . w11 -1.1 – – 10 ,
50 0 . , . CPL-1 -1 0.2 – 50 0 . CPL-2 -1.1 0.3 – 50 0 . HYP-1 -1 0.2 2.5 50 0 . HYP-2 -1.1 0.3 2.5 50 0 . HYP-3 -1.05 0.25 2.5 50 0 . Table 1.
The various DE parameterisations considered in thepresent work with their main parameters and the resulting valueof σ at z = 0. S φ = − (cid:90) d x √− g [ F ( Y ) + F ( Z ) + V ( φ )] , where the kinetic and velocity coupling terms are defined by Y ≡ ∇ µ φ ∇ µ φ ,Z ≡ u µ ∇ µ φ , Note that in defining a continuous velocity field, as isrequired to form Z , we necessarily introduce a smoothinglength over which the particle velocities are averaged. Thesmoothing length is assumed to be smaller than the cosmo-logical perturbations under consideration.The momentum flux S is given by Skordis et al. (2015): S = B δ DE + B θ DE + B θ c where δ and θ denote the density and velocity perturbations,and their three coefficients are MNRAS000
The various DE parameterisations considered in thepresent work with their main parameters and the resulting valueof σ at z = 0. S φ = − (cid:90) d x √− g [ F ( Y ) + F ( Z ) + V ( φ )] , where the kinetic and velocity coupling terms are defined by Y ≡ ∇ µ φ ∇ µ φ ,Z ≡ u µ ∇ µ φ , Note that in defining a continuous velocity field, as isrequired to form Z , we necessarily introduce a smoothinglength over which the particle velocities are averaged. Thesmoothing length is assumed to be smaller than the cosmo-logical perturbations under consideration.The momentum flux S is given by Skordis et al. (2015): S = B δ DE + B θ DE + B θ c where δ and θ denote the density and velocity perturbations,and their three coefficients are MNRAS000 , 1–14 (2011) ark Scattering beyond constant w B = 11 − ¯ ZF Z ρ c ¯ ZF Z c s w ,B = a − ¯ ZF Z ρ c (cid:20) ¯ X (cid:18) F Z F Y − Z (cid:19) + F Z (cid:18) µaF Z − F φ F Y (cid:19)(cid:21) ,B = − B + 3 H ¯ ZF Z c s − ¯ ZF Z ρ c . Here subscripts denote derivatives, for example F Z ≡ dF ( Z ) /dZ , and c s denotes the sound speed, while¯ X = 1 a (cid:104) ( ZF ZY − F ZZ ) ˙¯ Z − F Zφ ˙ φ − H F Z (cid:105) . In Skordis et al. (2015) the authors demonstrate thata formal equivalence cannot be drawn between the Type3 models and the elastic scattering case. This is under-standable given that the velocity coupling Z is associatedwith the gradient of the field, in contrast to the scatteringmodel where the interaction is associated with the local en-ergy density. However, for a particular subclass of Type 3models, the characteristic drag-like behaviour can be repro-duced. Provided the derivative of F ( Z ) is large, such that | F Z | (cid:29) | Z | , as could occur for a variety of functions suchas F ( Z ) ∝ exp( − Z ), then the expression simplifies consid-erably. There is only one contribution which has no explicitdependence on Z , which stems from the final term in equa-tion (2), so to leading order we have S = 3 H F Z ( θ c − θ DE ) + O ( Z )By comparison, in the elastic scattering model (Simpson2010) an expression is found which is also proportional tothe difference in the velocity perturbations of the two fluids S = − ρ DE (1 + w ) an D σ D ( θ c − θ DE )where n D ≡ n a − is the proper number density of darkmatter particles, w ≡ p/ρ is the dark energy equation ofstate, and σ D is the scattering cross-section. From the above,and utilising ρ DE (1+ w ) (cid:39) − ZF Z , we can define an effectivecross-section as follows σ eff ≡ − H a F Z n Z .
Furthermore, in the limit of weak coupling, we find thatall Type 3 models generate a scale-independent modifica-tion to the linear growth of cosmic structure. This is dueto the fact that, on scales smaller than the dark energysound horizon, the dark energy perturbations are driven bythe potential well associated with the dark matter pertur-bations. The density and velocity fields therefore all dis-play approximately the same spatial distributions: δ m ( x ) ∝ θ m ( x ) ∝ δ DE ( x ) ∝ θ DE ( x ) (see also eq 105 of Pourtsidouet al. (2013)). As a result, even in the most general formof Type 3 models ( S = B δ DE + B θ DE + B θ c ) are welldescribed on sub-horizon scales ( k (cid:29) k H ) by S ∝ θ m , The microphysical interpretation is that the dark matterparticles will experience a force directly proportional to, and(anti-)parallel with, their velocity vector. And it is this core phenomenological effect that we shall replicate within ournumerical simulations.
Motivated by the above outlined relation between modelsof elastic scattering in the dark sector and the particularsub-class of ‘Type 3’ coupled quintessence models proposedin Skordis et al. (2015), we briefly review in this Sectionthe main features of Dark Scattering cosmologies. We alsopresent the specific models under investigation in the presentwork, providing an overview of their background evolutionand of their main features related to linear and nonlinearstructure formation.
We will consider cosmological models characterised by an ex-change of momentum between Cold Dark Matter particlesand a Dark Energy field, modelled as a nearly-homogeneousfluid with a time-dependent equation of state parameter w ( a ). In the present work we will consider two possible pa-rameterisations of a freezing DE equation of state parameter w ( a ).The first one is the standard and widely employedChevalier-Polarski-Linder parameterisation (CPL hereafter,Chevallier & Polarski 2001; Linder 2003): w CPL ( a ) ≡ w + (1 − a ) w a , (1)where w is the value of the equation of state parameterat the present time and the parameter w a defines the low-redshift evolution of w CPL . It should be noticed here thatthe CPL parameterisation provides an evolution of w CPL ranging between w at z = 0 and ( w + w a ) for z → ∞ ,with a negative convexity.However, as it has been recently pointed out in Pantaziset al. (2016), freezing models with a transition between aconvex and concave shape might provide a less biased fit ofobservational data. Therefore, the second parameterisationthat we will consider in this work will span the same globalrange of our first model, but be characterised by a shallowerbehaviour at very low redshifts and a sharper transition tothe high- z asymptotic value taking place at some interme-diate redshift z t (that represents an additional free param-eter) which sets the point of inflection. Such behavior canbe modelled by a hyperbolic tangent function of the form : w HYP ( a ) ≡ w + w a (cid:18) a − z t (cid:19) . (2)The evolution of the equation of state parameter as afunction of redshift for these two parameterisations is shownin Fig. 1 for the parameters summarised in Table 1. In Fig. 1,like in all the figures of the present work, we also display forcomparison the behavior of the two constant- w models with A similar shape of the equation of state w ( z ) has been recentlyproposed also by Jaber & de la Macorra (Jaber & de la Macorra)using a different functional parameterisation.MNRAS , 1–14 (2011) M. Baldi & F. Simpson H / H Λ CD M CPL-1CPL-2HYP-1HYP-2HYP-3 w = -0.9w=-1.1
Figure 2.
The Hubble function ratio to the standard ΛCDM casefor the various parameterisations considered in the present work.The black solid and dashed lines represent the constant values w = − . w = − .
1, respectively, that have been discussedin Baldi & Simpson (2015) w = − . w = − . H ( z ): H ( z ) = H (cid:2) Ω M (1 + z ) + Ω r (1 + z ) + Ω K (1 + z ) +Ω DE e (cid:82) z w (˜ z ))1+˜ z d ˜ z (cid:21) (3)which is displayed in Fig. 2, along with its ratio to the stan-dard ΛCDM case. In Eq. 3, the dimensionless density param-eters Ω i ≡ ρ i /ρ crit refer to the components of matter ( M ),radition ( r ), curvature ( K ) and dark energy (DE), with thecritical density of the universe being ρ crit ≡ H / πG . Asone can see from the figure, all the models considered in thiswork do not deviate by more than 2.5% from the ΛCDM ex-pansion history, with a maximum deviation around z ≈ w models are found to be all closer toΛCDM compared to the two constant- w scenarios investi-gated in Baldi & Simpson (2015). Therefore, these modelsvery closely resemble the standard cosmological scenario atthe level of the background evolution. As already extensively discussed in Simpson (2010) andBaldi & Simpson (2015), the evolution of linear perturba-tions in the presence of a momentum exchange between darkenergy and CDM particles is described by the system of cou- + A z i = 99CPL-1, ξ =50CPL-2, ξ =50HYP-1, ξ =50HYP-2, ξ =50HYP-3, ξ =50w = -0.9, ξ ={10,50}w=-1.1, ξ ={10,50} Figure 3.
The modified friction term (1 + A ) for the various pa-rameterisations considered in the present work. As on can see inthe figure, the variable- w models determine a stronger modifica-tion at high redshifts and a weaker modification at low redshiftscompared with the constant- w case. Colors and linestyles are thesame as in Fig. 2 pled equations in Fourier space : θ (cid:48) DE = 2 Hθ DE − an CDM σ D ∆ θ + k φ + k δ DE w ,θ (cid:48) CDM = − Hθ CDM + ρ DE ρ CDM (1 + w ) an CDM σ D ∆ θ + k φ , (4)where n CDM is the proper number density of CDM particles,∆ θ ≡ θ DE − θ CDM is the velocity contrast, ( θ i being thedivergence of the velocity perturbations for the field i ), φ isthe gravitational potential sourced by the Poisson equation k φ = 4 πG ( δ CDM + δ DE ), and a prime denotes a derivativewith respect to cosmic time.By assuming a DE sound speed c s = 1, which is pre-dicted by most DE models based on light scalar fields, wecan expect DE perturbations to be damped within the cos-mic horizon so that the DE density and velocity fields areapproximately homogeneous (i.e. δ DE = θ DE = 0), as wasconfirmed numerically in the previous paper. Therefore, aswe will concentrate on sub-horizon scales, we shall neglectthe influence of dark energy perturbations within the simu-lation and safely approximate ∆ θ ≈ − θ CDM . With such ap-proximation the linear Euler equation for CDM becomes: θ (cid:48) CDM = − Hθ CDM [1 + A ] + k φ , (5)where the second term in the brackets is the additional fric-tion associated with the momentum exchange, defined as: A ≡ ρ DE Hρ CDM (1 + w ) n CDM σ D =(1 + w ) σ D m CDM DE πG H . (6)This extra drag force depends on three free quantities: the In the present paper – if not stated otherwise – we will alwaysassume units in which the speed of light is unity, c = 1.MNRAS000
The modified friction term (1 + A ) for the various pa-rameterisations considered in the present work. As on can see inthe figure, the variable- w models determine a stronger modifica-tion at high redshifts and a weaker modification at low redshiftscompared with the constant- w case. Colors and linestyles are thesame as in Fig. 2 pled equations in Fourier space : θ (cid:48) DE = 2 Hθ DE − an CDM σ D ∆ θ + k φ + k δ DE w ,θ (cid:48) CDM = − Hθ CDM + ρ DE ρ CDM (1 + w ) an CDM σ D ∆ θ + k φ , (4)where n CDM is the proper number density of CDM particles,∆ θ ≡ θ DE − θ CDM is the velocity contrast, ( θ i being thedivergence of the velocity perturbations for the field i ), φ isthe gravitational potential sourced by the Poisson equation k φ = 4 πG ( δ CDM + δ DE ), and a prime denotes a derivativewith respect to cosmic time.By assuming a DE sound speed c s = 1, which is pre-dicted by most DE models based on light scalar fields, wecan expect DE perturbations to be damped within the cos-mic horizon so that the DE density and velocity fields areapproximately homogeneous (i.e. δ DE = θ DE = 0), as wasconfirmed numerically in the previous paper. Therefore, aswe will concentrate on sub-horizon scales, we shall neglectthe influence of dark energy perturbations within the simu-lation and safely approximate ∆ θ ≈ − θ CDM . With such ap-proximation the linear Euler equation for CDM becomes: θ (cid:48) CDM = − Hθ CDM [1 + A ] + k φ , (5)where the second term in the brackets is the additional fric-tion associated with the momentum exchange, defined as: A ≡ ρ DE Hρ CDM (1 + w ) n CDM σ D =(1 + w ) σ D m CDM DE πG H . (6)This extra drag force depends on three free quantities: the In the present paper – if not stated otherwise – we will alwaysassume units in which the speed of light is unity, c = 1.MNRAS000 , 1–14 (2011) ark Scattering beyond constant w Parameter Value H − Mpc − Ω M DE b A s . × − n s Table 2.
A summary of the cosmological parameters adopted forall the simulations discussed in the present work.
DE equation of state w ( z ), the DE-CDM scattering crosssection σ D , and the CDM particle mass m CDM . In particular,the overall magnitude of the drag force depends on the lattertwo parameters only through their ratio, so that we candefine the combined quantity ξ ≡ σ D m CDM (7)with dimensions of [bn / GeV], as the main characteristicparameter of our models, such that Eq. 6 becomes: A ≡ (1 + w ) 3Ω DE πG Hξ . (8)From Eqs. 6,8, we notice that the additional term A canbe both positive or negative (i.e. acting as a friction or asa dragging force) for values of the DE equation of state w above or below the cosmological constant value w = − w = {− . , − . } for differentvalues of the parameter ξ , in the present work we aim to gobeyond such rather unrealistic assumption and test DarkScattering scenarios with variable w ( z ), focusing on the fewmodels described in Table 1, while keeping fixed the valueof ξ .A new friction term, analogous to the Thomson dragforce experienced by electrons, is introduced to the equationof motion for individual CDM particles.˙ v i = − [1 + A ] H v i + (cid:88) j (cid:54) = i Gm j r ij | r ij | (9)where r ij is the distance between the i -th and the j -thparticle. The evolution of the factor (1 + A ) is shown inFig. 3 for the different variable- w models under investigationwith a value of ξ = 50 [bn / GeV], while the dashed and dot-dashed lines enclosing the dark-grey and light-grey shadedareas correspond to the case of a constant w with ξ = 10and 50 [bn / GeV], respectively. As one can see by comparingFigs. 2 and 3, the DE parameterisations considered in thiswork have a weaker impact on the background expansionhistory and on low-redshift structure formation as comparedto the constant- w models investigated in Baldi & Simpson(2015), while they are expected to have a stronger effect onthe growth of structures at high redshifts. As we will seelater in the paper, this will imply an overall weaker impacton most cosmological observables while retaining interestingand non-trivial effects on the abundance of massive clustersand on the expected weak lensing signal. For all the models summarised in Table 1, and for a referenceΛCDM cosmology, we have performed a set of intermediate-size simulations with the modified version of
GADGET (Springel 2005) described in Baldi & Simpson (2015), whichself-consistently implements the drag force associated withthe DE-CDM momentum exchange. These simulations havea box size of 250 Mpc /h a side and follow the evolution of512 CDM particles in a periodic cosmological volume from z i = 99 down to z = 0. The mass resolution is m = 1 × M (cid:12) /h and the gravitational softening is (cid:15) ≈
12 kpc /h . Allthe simulations share the same initial conditions (since theeffect of the momentum exchange is negligible at z (cid:38) w models on two basic cosmologicalobservables: the nonlinear matter power spectrum and thehalo mass function (see 5.1). These preliminary results al-lowed us to identify the most relevant sets of parametersfor both the CPL and the HYP parameterisations to be in-vestigated more extensively with a set of larger simulations.The latter are cosmological boxes of 1 Gpc /h a side filledwith 1024 CDM particles, and have therefore a poorer mass( m = 8 × M (cid:12) /h ) and space ( (cid:15) ≈
24 kpc /h ) resolutioncompared to the smaller runs, but significantly improve thestatistics of massive clusters and of cosmic voids (see 5.2),thereby allowing a more significant assessment of the impactof the momentum exchange on the statistical and structuralproperties of these classes of objects.For all the simulations initial conditions have been gen-erated by displacing particle positions from a “glass” ho-mogeneous distribution (Davis et al. 1985) according tothe Zeldovich approximation (Zel’dovich 1970) to set up arandom-phase realization of the power spectrum predictedby CAMB (Lewis et al. 2000) for a ΛCDM cosmology withthe chosen cosmological parameters. Therefore, all the dif-ferences that will be identified among the simulations out-puts at low redshifts can be unambiguously ascribed to theeffects of the different cosmological models. Furthermore,as the initial conditions are identical the comparison of thevarious models will not be affected by sample variance, andstatistical uncertainties will be only due to Poisson noise. In the present section we will discuss the main out-comes of our simulations, starting with the results of theintermediate-scale runs and then moving to the large-scalerealisations for the selected subset of models.
For all our intermediate-size simulations we extract the non-linear matter power spectrum through a Cloud-in-Cell mass , 1–14 (2011) M. Baldi & F. Simpson P ( k ) / P ( k ) Λ CD M z = 0 w=-0.9, ξ =10, 50w=-1.1, ξ =10, 50CPL-1, ξ = 50CPL-2, ξ = 50HYP-1, ξ = 50HYP-2, ξ = 50HYP-3, ξ = 50 P ( k ) / P ( k ) Λ CD M z = 0.5 P ( k ) / P ( k ) Λ CD M z = 1 Figure 4.
The nonlinear matter power spectrum ratio to the reference ΛCDM model at three different redshifts z = 0 ( left ), z = 0 . middle ), and z = 1 ( right ) for the various models investigated with our intermediate-size simulations. The colours and linestyles are thesame as in Fig. 1 ∆ w=-0.9, ξ =10, 50w=-1.1, ξ =10, 50CPL-1, ξ = 50CPL-2, ξ = 50HYP-1, ξ = 50HYP-2, ξ = 50HYP-3, ξ = 50 Figure 5.
The relative difference between the nonlinear (i.e.measured at k = 10 h/ Mpc) and the linear (i.e. measured at k =0 . h/ Mpc) effects on the matter power spectrum for the variousmodels of momentum exchange. assignment to a cubic cartesian grid having the same spac-ing of the mesh used for the large-scale N-body integration,i.e. 512 grid nodes. This allows to measure the power spec-trum from the fundamental mode k ≈ . h/ Mpc up to theNyquist frequency of the grid k Ny ≈ . h /Mpc. In orderto extend this range to smaller scales we employ the foldingmethod of Jenkins et al. (1998); Colombi et al. (2009) andwe smoothly interpolate the two estimations around k Ny .Then, the combined power spectrum obtained in this way istruncated at the scale where the shot noise reaches 20% ofthe measured power. We apply this procedure to the simu-lation snapshots corresponding to three different redshifts z = { , . , } . The results are displayed in Fig. 4 where weshow the ratio of the measured power of each simulation tothe corresponding ΛCDM result. All the variable- w cosmolo-gies (solid coloured lines with symbols) are characterised bya value of ξ = 50 [bn / GeV] while for the two constant- w models used as a reference (dashed and dot-dashed blacklines) we consider both ξ = 10 [bn / GeV] (thin lines) and ξ = 50 [bn / GeV] (thick lines). As one can see in the plots, and most evidently in the z = 0 panel, the variability of the DE equation of state in-troduces non-trivial features in the behavior of the matterpower as a function of scale. For constant- w (as already dis-cussed in Baldi & Simpson 2015) the effect of the DE-CDMmomentum exchange on the power spectrum shows a scale-independent suppression (enhancement) of power at largescales and a transition to a scale-dependent enhancement(suppression) at small scales for w > − w < − k fordecreasing redshift. Also, there is a direct correspondencebetween the magnitude of the linear and nonlinear effects,with a larger effect at linear scales being always associatedwith a larger effect also at nonlinear scales.For the variable- w case the effects appear more diverse,with a wide range of behaviours and of possible linear-nonlinear transitions, as well as a less direct correspondencebetween the magnitude of the effect at linear scales andits nonlinear counterpart. In this respect, it is interestingto consider the relation between the power spectrum ratioat large scales and that at smaller scales as the formerwill have mostly an impact on the statistical propertiesof large-scale structures while the latter will have directconsequences on the structural properties of collapsedobjects. This comparison is shown in Fig. 5, where wedisplay the relative difference of the observed power en-hancement at k = 0 . h/ Mpc and at k = 10 h/ Mpc definedas ∆ ≡ [ P ( k ) /P ( k ) ΛCDM ] k =10 / [ P ( k ) /P ( k ) ΛCDM ] k =0 . − w models have a weaker impact at nonlinearscales compared to the constant- w models with the samevalue of the ξ parameter ( ξ = 50 [bn / GeV], thick lines),and two of them (the CPL-2 and the HYP-1 models) evenshow a smaller nonlinear impact at z = 0 compared to theconstant- w models with the lower value of ξ = 10 [bn / GeV](thin lines). This provides us with a useful guideline to selectrelevant combinations of parameters, since we are interestedin identifying models that produce a sizeable suppres-sion of the large-scale structures growth without changingtoo dramatically the structural properties of collapsed halos.A similar analysis can be performed using the abun-dance of halos as a test observable. To this end, we haveidentified particle groups in our simulations by means of a
MNRAS000
MNRAS000 , 1–14 (2011) ark Scattering beyond constant w M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M w=-0.9, ξ =10, 50w=-1.1, ξ =10, 50CPL-1, ξ = 50CPL-2, ξ = 50HYP-1, ξ = 50HYP-2, ξ = 50HYP-3, ξ = 50 z = 0 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M z = 0.5 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M z = 1 Figure 6.
The mass function ratio to the ΛCDM case for all the models under investigation in the present work. The three panels referto the same redshifts considered above ( z = 0 , . Friends-of-Friends algorithm with linking length 0 . SUBFIND algorithm (Springel et al. 2001) inorder to select gravitationally bound substructures. Withthese catalogs at hand, we have computed the cumulativehalo mass function for all the models by binning the ha-los in mass bins according to their M mass defined as themass contained in a sphere centered on the most bound par-ticle of each main substructure with a radius R enclosinga mean density 200 times larger than the critical density ofthe universe. We have then compared these mass functionsto the outcome of the ΛCDM simulation, and the results aredisplayed in Fig. 6.As one can see in the figure, also in this case wefind that the variable- w models predict a milder impacton the abundance of halos at all masses compared totheir constant- w counterparts with the same value of ξ = 50 [bn / GeV], especially at z = 0. Furthermore, consis-tently with the previous findings shown in Figs. 5 and 4,the two models CPL-2 and HYP-1 appear to be the closestmatch to the ΛCDM result at low masses, while showingsome significant suppression of the abundance of halos atthe high-mass end of the available catalogs, differently fromall the other models that are found to determine largedeviations in the abundance also for galaxy-sized halos.This is a consequence of the suppression of large-scale linearclustering, since the halo mass function is exponentiallysensitive to the value of σ , and to the relatively weakimpact of the interaction at highly nonlinear scales. There-fore, these models represent promising candidates to easethe persisting tensions between both the observed weaklensing amplitude (Heymans et al. 2013) and abundance ofPlanck SZ clusters (Planck Collaboration et al. 2015) onone side and their predicted values based on the maximumlikelihood Planck 2015 cosmological parameters estimation(Ade et al. 2015).Based on these preliminary checks, we have then se-lected the two models CPL-2 and HYP-1 to be investigatedin more detail through larger simulations. along with twoconstant- w models with ξ = 10 [bn / GeV] and a referenceΛCDM cosmology. The outcomes of these larger simulations,that represent the core results of the present work, are dis-cussed in the following Section.
We present here the results of our suite of large simulationsfocusing on the effects that the momentum exchange be-tween dark energy and CDM particles has on a series of stan-dard cosmological observables. As our analysis will show, thetwo selected models CPL-2 and HYP-1 will result in a cos-mological evolution of structures that closely resembles thatof ΛCDM for most of the observables, with the noticeableexception of the abundance of very massive objects and theoverall normalisation of the linear matter power spectrum,possibly easing tensions between CMB constraints and localmeasurements of large-scale structures.
As a first diagnostics of the effects of the momentum ex-change in the two selected variable- w models we computethe projected density field of a slice of thickness 30 Mpc /h through the simulation box. We assign the mass of particlesin the slice to a 4096 cartesian grid trough a Cloud-In-Cell(CIC) mass assignment scheme based on their projected po-sitions in the x − y plane and we compute the logarithmof the density contrast in the grid. In Fig. 7 we show thedensity field at z = 0 in a region of side 500 Mpc /h cen-tered on the most massive halo identified in the simulation.In the inset displayed in the bottom-right part of the mapswe show a zoom on the central halo with side 50 Mpc /h .This preliminary visual inspection shows a very similarshape of the large-scale matter distribution, with no signifi-cant difference in the location, shape, and size of overdenseregions and voids. The closer look at the region around amassive halo displayed in the zoomed inset highlights an al-most identical geometry of the cosmic web of filaments con-verging onto the central structure, even though some differ-ences appear in the relative position of the most prominentsubstructures in the vicinity of the central halo.Therefore, the overall density field seems very mildlyaffected by the momentum exchange with some appreciableeffect showing up only in the vicinity of the most overdenseregions of the simulated volume. In Fig. 8 we show the nonlinear mater power spectrum ratioto the ΛCDM reference simulation for all the models simu-
MNRAS , 1–14 (2011)
M. Baldi & F. Simpson
Figure 7.
The density field at z = 0 in a slice of size 500 Mpc /h and thinkness 30 Mpc /h for the reference ΛCDM simulation (cen-tral panel) and for the selected CPL and HYP parameterisations(top and bottom panels, respectively). The slices have been cen-tered on the most massive structure identified in the simulationswhich is displayed in the zoomed inset. lated in the large box. The plots are the same as in Fig. 4 al-though covering a different range of scales due to the largersize of the simulated box. As the plots clearly show, theeffect of the momentum exchange at large linear scales inthe CPL-2 and HYP-1 models is twice as large as for theconstant- w model with w = − . ξ = 10 [bn / GeV] athigh redshift, while it becomes comparable to the latter at z = 0. This is consistent with the variable- w models having amore efficient momentum transfer at high z due to the largervalue of w (see Fig. 3). At the same time, it is interesting tonotice that also the scale dependence of the effect at smallnonlinear scales is much more pronounced in the variable- w models than in the constant- w case at high redshift while theopposite occurs at z = 0. This suggests that at high redshiftour selected models of dark scattering might be character-ized by significantly overconcentrated collapsed structuresembedded in a less evolved large-scale matter distribution,as will be explicitly verified below (see Section 5.2.5). Suchprediction could be verified by combining weak and stronglensing observations at high z . For all our large box simulations we have identified cosmicvoids using the
VIDE public toolkit (Sutter et al. 2015) inboth a random subsampling of the CDM particle distribu-tion with a tracer density of 0 .
02 particles per cubic Mpc /h and in the distribution of halos of our FoF sample, with aminimum halo mass M FoF, min ( z = 0) ≈ . × M (cid:12) /h , forthe same three redshifts investigated above z = { , . , } .The voids are identified using a Voronoi tessellation proce-dure and a watershed algorithm to join underdense Voronoicells until a border to a neighboring underdense region isreached. Then, an effective radius R eff ≡ [3 · V void / (4 π )] / is associated to the void volume V void assuming sphericity(see Sutter et al. 2015, for a detailed description of the al-gorithm implemented in the VIDE code).Starting from the void catalogue produced by VIDE weidentify the main voids (i.e. those voids that are not em-bedded within larger voids) and remove pathological voidsfollowing the procedure described in Pollina et al. (2016).In this way, we ensure that the final void catalogue containsonly disjoint voids with a central overdensity δ min < . δ c > .
57. With such catalogue of selectedvoids we compute the differential void size distribution, i.e.the number density of voids as a function of their effectiveradius R eff . The comparison of the void size distributionfunctions of the different models is displayed in Fig. 9 wherethe upper plot refers to the voids in the subsampled CDMfield while the lower plot to the voids in the FoF halos cata-logs. In each plot the top panel shows the void size distribu-tion function while the bottom panel presents the relativedifference with respect to the ΛCDM reference in units of thestatistical significance computed by propagating the Poissonnoise in each bin of effective radius to the relative difference.As one can see from the plots, the variable- w modelsCPL-2 and HYP-1 show a slight suppression of the abun-dance of large voids in the CDM density field compared to https://bitbucket.org/cosmicvoids/vide publicMNRAS000
57. With such catalogue of selectedvoids we compute the differential void size distribution, i.e.the number density of voids as a function of their effectiveradius R eff . The comparison of the void size distributionfunctions of the different models is displayed in Fig. 9 wherethe upper plot refers to the voids in the subsampled CDMfield while the lower plot to the voids in the FoF halos cata-logs. In each plot the top panel shows the void size distribu-tion function while the bottom panel presents the relativedifference with respect to the ΛCDM reference in units of thestatistical significance computed by propagating the Poissonnoise in each bin of effective radius to the relative difference.As one can see from the plots, the variable- w modelsCPL-2 and HYP-1 show a slight suppression of the abun-dance of large voids in the CDM density field compared to https://bitbucket.org/cosmicvoids/vide publicMNRAS000 , 1–14 (2011) ark Scattering beyond constant w P ( k ) / P ( k ) Λ CD M z = 0 w=-0.9, ξ =10w=-1.1, ξ =10CPL-2, ξ = 50HYP-1, ξ = 50 P ( k ) / P ( k ) Λ CD M z = 0.5 P ( k ) / P ( k ) Λ CD M z = 1 Figure 8.
The nonlinear matter power spectrum ratio to the reference ΛCDM model at three different redshifts z = 0 ( left ), z = 0 . middle ), and z = 1 ( right ) for the selected models of parameterised w ( z ) and for the two constant- w cosmologies already investigatedin Baldi & Simpson (2015). The color coding and linestyles are the same as in all previous figures. d N / d R [ h / M p c ]
10 20 30 40 50R eff [Mpc/h]-3-2-10123 ( N / N Λ CD M - ) / σ Λ CDMCPL-2, ξ =50HYP-1, ξ =50w09, ξ =10w11, ξ =10 CDM, 0.02 h /Mpc z = 0.00
10 20 30 40R eff [Mpc/h] z = 0.50 eff [Mpc/h] z = 1.00 d N / d R [ h / M p c ]
20 40 60 80R eff [Mpc/h]-4-2024 ( N / N Λ CD M - ) / σ Λ CDMCPL-2, ξ =50HYP-1, ξ =50w09, ξ =10w11, ξ =10 All halosz = 0.00
20 40 60 80R eff [Mpc/h] z = 0.50
20 40 60 80R eff [Mpc/h] z = 1.00
Figure 9.
The differential void size distribution in the CDM field (top) and in the halos distribution (bottom) as computed using theVIDE void finding toolkit. As one can see in the figures, the slight reduction in the abundance of large voids found for the CDM field ismostly erased in the distribution of halos.MNRAS , 1–14 (2011) M. Baldi & F. Simpson ρ / ρ m ean z = 0R eff = (10 - 20) Mpc/h Λ CDMCPL-2, ξ =50HYP-1, ξ =50w09, ξ =10w11, ξ =10 eff -0.2-0.10.00.10.2 ( ρ / ρ Λ CD M - ) z = 0R eff = (30 - 40) Mpc/h Λ CDMCPL-2, ξ =50HYP-1, ξ =50w09, ξ =10w11, ξ =10 eff Figure 10.
The stacked density profiles for voids identified in the CDM distribution of the various large simulations. The stacking hasbeen performed using 100 randomly selected voids with effective radius in the range 10 −
20 Mpc /h ( left plot ) and 30 −
40 Mpc /h ( rightplot ). The bottom panels display the ratio of the density profiles to the reference ΛCDM model, and the grey-shaded region representsthe 1- σ statistical significance according to a bootstrap estimation. As one can see in the figure, the mommentum exchange determinesslightly but significantly shallower void profiles. M [M O • /h]0.40.60.81.01.21.4 N ( M ) / N ( M ) Λ CD M w = -1.1, ξ = 10w = -0.9, ξ = 10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDM z = 0 10 M [M O • /h]0.40.60.81.01.21.4 N ( M ) / N ( M ) Λ CD M w = -1.1, ξ = 10w = -0.9, ξ = 10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDM z = 0.5 10 M [M O • /h]0.40.60.81.01.21.4 N ( M ) / N ( M ) Λ CD M w = -1.1, ξ = 10w = -0.9, ξ = 10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDM z = 1
Figure 11.
The mass function ratio to the ΛCDM case for the models included in our suite of large-svale simulations. The three redshiftsdisplayed in the different panels, as well as colours, symbols, and line styles are the same as displayed in Fig. 8. As one can see in thefigures, the variable- w models – differently from the constant- w cases – determine a very significant suppression of the abundance ofcluster-sized halos, thereby alleviating current persisting tensions between low- z observational data and best-fit cosmological parametersderived from primary CMB anisotropies. the reference ΛCDM simulation. The effect is not too dra-matic, with a significance level of ≈ − σ , and is consistentwith the suppression of large scales perturbations alreadyshown by the power spectrum comparison. Interestingly, alsofor this class of models – as it has already been shown tooccur for modified gravity (Cai et al. 2014; Achitouv et al.2015), massive neutrinos (Massara et al. 2015) and inter-acting dark energy (Pollina et al. 2016) cosmologies – thiseffect is strongly suppressed when looking at the distributionof voids in the halo catalogs, where no significant deviationcan be observed besides statistical oscillations around thereference model, at all redshifts.We also compared the structural properties of voidsin the different models by computing the stacked radialdensity profiles around the voids macrocenters identifiedin the ΛCDM simulation for 100 randomly selected voidswithin two bins of void effective radius, namely R eff ∈{ − , − } Mpc /h . The results are shown in Fig. 10and clearly show how the variable- w models of momentum exchange result in shallower profiles for voids of both bins.Therefore, voids appear to be less empty in Dark Scatter-ing cosmologies, which might result in a lower weak lensingsignal at low multipoles. As already done in Fig. 6 for the intermediate-scale simula-tions, in Fig. 11 we display the ratio of the differential halomass function to the ΛCDM case for the models that weresimulated in the larger boxes. This allows to increase thestatistics of massive halos and extend the range of the com-puted mass function to larger masses, thereby investigatingthe impact of the momentum exchange on the abundance ofmassive clusters of galaxies as a function of redshift.This more extended mass range allows to see that boththe variable- w cosmologies have a very significant impacton the abundance of very massive objects, suppressing thenumber density of clusters with mass around 10 h − M (cid:12) by MNRAS000
The mass function ratio to the ΛCDM case for the models included in our suite of large-svale simulations. The three redshiftsdisplayed in the different panels, as well as colours, symbols, and line styles are the same as displayed in Fig. 8. As one can see in thefigures, the variable- w models – differently from the constant- w cases – determine a very significant suppression of the abundance ofcluster-sized halos, thereby alleviating current persisting tensions between low- z observational data and best-fit cosmological parametersderived from primary CMB anisotropies. the reference ΛCDM simulation. The effect is not too dra-matic, with a significance level of ≈ − σ , and is consistentwith the suppression of large scales perturbations alreadyshown by the power spectrum comparison. Interestingly, alsofor this class of models – as it has already been shown tooccur for modified gravity (Cai et al. 2014; Achitouv et al.2015), massive neutrinos (Massara et al. 2015) and inter-acting dark energy (Pollina et al. 2016) cosmologies – thiseffect is strongly suppressed when looking at the distributionof voids in the halo catalogs, where no significant deviationcan be observed besides statistical oscillations around thereference model, at all redshifts.We also compared the structural properties of voidsin the different models by computing the stacked radialdensity profiles around the voids macrocenters identifiedin the ΛCDM simulation for 100 randomly selected voidswithin two bins of void effective radius, namely R eff ∈{ − , − } Mpc /h . The results are shown in Fig. 10and clearly show how the variable- w models of momentum exchange result in shallower profiles for voids of both bins.Therefore, voids appear to be less empty in Dark Scatter-ing cosmologies, which might result in a lower weak lensingsignal at low multipoles. As already done in Fig. 6 for the intermediate-scale simula-tions, in Fig. 11 we display the ratio of the differential halomass function to the ΛCDM case for the models that weresimulated in the larger boxes. This allows to increase thestatistics of massive halos and extend the range of the com-puted mass function to larger masses, thereby investigatingthe impact of the momentum exchange on the abundance ofmassive clusters of galaxies as a function of redshift.This more extended mass range allows to see that boththe variable- w cosmologies have a very significant impacton the abundance of very massive objects, suppressing thenumber density of clusters with mass around 10 h − M (cid:12) by MNRAS000 , 1–14 (2011) ark Scattering beyond constant w M [h -1 M O • ]0.60.70.80.91.01.11.21.3 c * m ean / c * m ean ( Λ CD M ) Λ CDMCPL, w = -1.1, ξ = 50HYP, w = -1, ξ = 50w = -0.9, ξ = 10w = -1.1, ξ = 10z=0 M [h -1 M O • ]0.60.70.80.91.01.11.21.3 c * m ean / c * m ean ( Λ CD M ) Λ CDMCPL, w = -1.1, ξ = 50HYP, w = -1, ξ = 50w = -0.9, ξ = 10w = -1.1, ξ = 10z=0.5 M [h -1 M O • ]0.60.70.80.91.01.11.21.3 c * m ean / c * m ean ( Λ CD M ) Λ CDMCPL, w = -1.1, ξ = 50HYP, w = -1, ξ = 50w = -0.9, ξ = 10w = -1.1, ξ = 10z=1 Figure 12.
The ratio of the binned average concentration to the ΛCDM reference at the same three different redshifts considered in theprevious figures. As one can see in the plots, the variable- w models do not show a strong impact on the concentrations of halos at verylow redshifts, while at higher redshifts the effect appears to be somewhat enhanced. ≈ −
50% at z = 0. The effect is milder at higher redshifts,but still significant with a suppression of the abundance of10 h − M (cid:12) halos by ≈ −
30% at z = 1.This represents one of the most prominent features ofthe cosmological models under investigation and might pro-vide a way to reconcile the low abundance of SZ clustersdetected by Planck (Planck Collaboration et al. 2015) withtheir expected number based on the cosmological constraintsarising from the angular power spectrum of temperatureand polarisation anisotropies (Ade et al. 2015). It is alsoremarkable that the impact on the abundance of smaller ob-jects – down to Milky-Way sized halos of ∼ M (cid:12) /h – isvery mild, thereby leaving unaffected the expected numberof galaxies. For each halo in our sample we have computed the con-centrations c ∗ following the approach described in Springelet al. (2008) as:2003 c ∗ ln(1 + c ∗ ) − c ∗ / (1 + c ∗ ) = 7 . δ V (10)where δ V is defined as: δ V = 2 (cid:18) V max H r max (cid:19) (11)with V max and r max being the maximum rotational velocityof the halo and the radius at which this velocity peak islocated, respectively.In Fig. 12 we show the ratio of these average concen-trations at the usual three different redshifts z = { , . , } as a function of the halo mass M for a set of logarith-mically equispaced mass bins. The grey shaded area showsthe Poissonian error based on the number of halos in eachbin for the reference ΛCDM run. As the figures show, at z = 0 the variable- w models have a significantly weaker im-pact on the halo concentrations than the constant- w oneseven for a higher value of the interaction parameter ξ . Bothmodels determine an increase of concentrations below 7%over the whole mass range covered by our halo catalogs.This implies that no dramatic effect in low-redshift stronglensing observations is expected for the models under inves-tigation. On the other hand, as anticipated in Section 5.2.2,the situation changes at higher redshifts where the models with a variable equation of state show a more significant in-crease of halo concentrations. They are comparable to (at z = 0 .
5) or even larger than (at z = 1) the constant- w case(even though the latter has a weaker interaction parameter ξ ). Therefore, these models predict a somewhat enhancedstrong lensing efficiency at high redshift despite the overallreduction of large-scale power which is expected to result ina lower weak lensing signal. As a final probe of the momentum exchange between darkenergy and dark matter particles we investigate the abun-dance and spatial distribution of substructures within mas-sive collapsed halos.First, we compute the subhalo mass function, definedas the number of substructures with a given fractional masswith respect to the virial mass of their host main halo( M sub /M ) as a function of the fractional mass itself. Wecompute this quantity by binning in logarithmic fractionalmass bins the whole sample of substructures belonging tohost halos with virial mass above a minimum threshold of M min = 10 M (cid:12) /h , thereby restricting this analysis to mas-sive galaxy cluster halos (due to the limited resolution of thesimulations). In the left panel of Fig. 13 we show the sub-halo mass function at z = 0 for the various models and inthe bottom plot we display their ratio to the fiducial ΛCDMcosmology.Then, we also compute the subhalo radial distribution(shown in the Right panel of Fig. 13), defined as the frac-tional number density of substructures in a series of logarith-mically equispaced radial bins in units of the virial radius R of the host halo.Both these observables show very little deviationsamong all the models and the reference ΛCDM cosmology,thereby showing that the momentum exchange does not sig-nificantly alter the distribution of substructures at smallscales. For all the halos of each simulation sample we compute theone-dimensional velocity dispersion σ and compare it to thecorresponding behavior of the ΛCDM run. The results are MNRAS , 1–14 (2011) M. Baldi & F. Simpson dn ( M s ub / M ) / dLog ( M s ub / M ) -3.0 -2.5 -2.0 -1.5 -1.0 Log[M sub /M ]0.11.010.0 dn / dn Λ CD M z = 0.0w11, ξ =10w09, ξ =10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDM -0.6 -0.4 -0.2 0.0 0.2 Log[R/R ]01234 ( d N / N ) /[ π / ( R e3 - R i ) / R ] z = 0.0 Λ CDMCPL, w = -1.1, ξ = 50HYP, w = -1, ξ = 50w09, ξ =10w11, ξ =10 Figure 13.
The sub halo mass function ( left ) and the sub halo radial distribution ( right ) for the various models considered in our suiteof large-scale simulations at z = 0. As one can see in the plots, the momentum exchange does not significantly alter the abundance andthe radial distribution of substructures identified in the mass range allowed by the resolution of our simulations. σ [ k m / s ] M [h -1 M O • ]0.900.951.001.051.10 σ / σ ( Λ CD M ) z = 0 w = -1.1, ξ = 10w = -0.9, ξ = 10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDM
Figure 14.
The velocity dispersion of halos at z = 0 as a func-tion of halo mass for the various models under investigation. Theoverall deviation from the reference ΛCDM cosmology does notexceed a few percent over the whole mass range accessible to oursimulations. shown in Fig. 14 for the present epoch ( z = 0), across 10 log-arithmically equispaced mass bins for the different models.In the upper panel we display as coloured dots a randomsubsample of all the halos in the catalogs, while the linestrace the mean value of σ . In the bottom panels we plotthe ratio of the binned average 1-D velocity dispersion tothe ΛCDM case, and the grey shaded region indicates thePoissonian error associated with number counts of halos ineach bin of the reference simulation.As one can see from the plots, the Dark Scattering mod-els generate a very mild enhancement (only ≈ − b / b Λ CD M w = -1.1, ξ = 10w = -0.9, ξ = 10HYP, w = -1, ξ = 50CPL, w = -1.1, ξ = 50 Λ CDMw = -0.9w = -1.1z = 0
Figure 15.
The halo bias as a function of scale at z = 0 forthe models considered in the large-scale simulations. As one cansee in the figure, the momentum exchange determines a slightincrease of the bias with a characteristic scale-dependence of theenhancement. As a final test of our models we compute the halo bias bytaking the ratio of the halo-matter cross-power spectrum P hm ( k ) to the matter-matter power spectrum P mm ( k ) forhalos with mass above 5 × M (cid:12) /h . In Fig. 15 we displaythe ratio of the halo bias of the various models to the ΛCDMcase in the range of scales 0 . (cid:54) k · h/ Mpc (cid:54)
1. As alreadyshown in Baldi & Simpson (2015), the constant- w modelsdetermine a ≈ −
7% increase (reduction) of the halo biaswith a slight scale-dependence for the interaction parameter ξ = 10 adopted in our simulations for w = − . w = − . w models under investiga-tion, despite the higher interaction parameter ξ = 50, arefound to yield only a marginal increase of the large scalebias. The effect at the smallest scales under consideration MNRAS000
7% increase (reduction) of the halo biaswith a slight scale-dependence for the interaction parameter ξ = 10 adopted in our simulations for w = − . w = − . w models under investiga-tion, despite the higher interaction parameter ξ = 50, arefound to yield only a marginal increase of the large scalebias. The effect at the smallest scales under consideration MNRAS000 , 1–14 (2011) ark Scattering beyond constant w is moderately stronger than before, reaching a ≈ − The physical nature of dark matter and dark energyremains highly active topic of theoretical and experimentalinvestigation. In this work we began by highlighting aconnection between the phenomenologically motivatedelastic scattering model, and a subclass of coupled scalarfield models. Furthermore we showed that, quite generically,the prediction of models which invoke momentum exchangebetween dark energy and dark matter is one where thelinear growth rate is modulated in a scale-independentmanner. In order to investigate further consequences of thisinteraction, we performed a series of N-body simulations,building on the work of Baldi & Simpson (2015), but nowincorporating a more realistic trajectory for the dark energyequation of state. In this context, the key consequenceof an evolving equation of state - specifically freezingmodels which tend to mimic a cosmological constant atlate times - is that the coupling is naturally weakened inthe late universe, when the bulk of the non-linear structureformation takes place. As a result, the amplification of thenon-linear matter power spectrum which had been foundin the earlier simulations of Baldi & Simpson (2015) issignificantly suppressed.We began by considering two different realisations ofa freezing Dark Energy equation of state w ( z ) with eithernegative or positive convexity near the present epoch,namely the widely-used CPL parameterisation and a novelstep-like parameterisation based on hyperbolic functions,respectively. We then explored the relevant parameter spaceof these two forms of freezing Dark Energy models with asuite of intermediate-size simulations, assuming a relativelylarge value of the interaction parameter ( ξ = 50 bn / GeV).With these simulations at hand, we identified those specificmodels that appear to affect some basic statistics of thelarge-scale matter distribution (such as the nonlinearmatter power spectrum and the halo mass function) inthe direction of a suppressed growth of perturbations,without impacting too dramatically on the highly nonlinearregime at very small scales. For such models, we performedsimulations on larger scales to improve the statistics of ourresults.Finally, by means of these larger simulations, we havebeen able to verify that a number of observable quanti-ties are not significantly perturbed by the interaction. Morespecifically: (cid:63)
The density field of the Dark Scattering models showsthe same shape of the large-scale cosmic web as comparedto the standard ΛCDM realisation, with tiny differences inthe position of prominent substructures around very massivesystems appearing only at very small scales; (cid:63)
The abundance of galaxy-sized and group-sized halos(up to 10 M (cid:12) /h ) is mildly affected by the interaction inthe redshift range 0 (cid:54) z (cid:54) (cid:63) The abundance of cosmic voids identified in the dis-tribution of collapsed halos does not show any significantdeviation compared to ΛCDM; (cid:63)
The relative abundance of substructures encoded by thesubhalo mass function as well as the radial distribution ofsubstructures around their host halo is basically unchanged,with differences appearing consistent with statistical uncer-tainties. (cid:63)
The 1-dimensional velocity dispersion of halos is onlyvery slightly enhanced in Dark Scattering models as com-pared to ΛCDM, with an effect not exceeding ≈ −
2% at z = 0; (cid:63) The halo bias is weakly increased (in a scale-dependentfashion), with a deviation from the ΛCDM reference rangingbetween ≈
5% at large scales ( k ≈ − h/ Mpc) and ≈ k ≈ h/ Mpc); more specifically, the biasat k ∼ . − . h/ Mpc is enhanced from the value b ≈ .
08 in ΛCDM to b ≈ .
15 for both the CPL and HYPparameterisations.On the other hand, we identified a set of characteristic ob-servational footprints of the model that might alleviate thepersisting tensions between high-redshift and low-redshiftcosmological constraints. In particular: (cid:63)
The matter power spectrum is suppressed at linearscales by up to ∼
10% at z = 0 for the specific parame-ters considered in our analysis, thereby providing a weakercontribution to the overall weak lensing signal and a lowerdetermination of σ from low-redshift probes; (cid:63) The abundance of cluster-sized halos is very signifi-cantly reduced, with a suppression reaching ∼
60% for halosof ∼ M (cid:12) /h at z = 0; (cid:63) The concentration-mass relation, which appears verymildly affected at the present epoch (with a mass-independent increase of the normalisation below 7 %) showsa somewhat larger deviation from ΛCDM at higher redshifts,with an enhancement up to ∼
20% at 0 . (cid:54) z (cid:54)
1, whichmight result in a higher efficiency for strong lensing at theseredshifts; (cid:63)
The abundance of cosmic voids in the CDM distribu-tion shows a statistically significant suppression (at the levelof about 1 − σ ) for voids with radius (cid:38)
20 Mpc /h , whilethe spherically-averaged stacked density profiles of voids arefound to be shallower, with a ≈
10% higher central over-density compared to ΛCDM, which is also consistent with alower expected weak lensing efficiency;To summarise, the cosmological models featuringelastic scattering between CDM particles and a DarkEnergy field with variable equation of state w ( z ) definedas in Eqs.1 and 2 are challenging to distinguish from thestandard ΛCDM cosmology at the level of the backgroundexpansion history, while providing a growth of structuresthat results in a lower amplitude of linear density pertur-bations (i.e. lower weak lensing signal, lower σ ) and astrongly suppressed abundance of very massive clusters atredshifts below 1. A similar conclusion has been recentlyreached – based on linear structure formation only – forsome specific realisations of Type 3 interacting Dark Energymodels by Pourtsidou & Tram (2016). In this respect, thesemodels might alleviate current observational tensions and
MNRAS , 1–14 (2011) M. Baldi & F. Simpson deserve further investigations.As a final note, let us briefly consider the relativemerits of interpreting abnormal structure formation aseither signs of coupled dark energy or as an indicationof modified gravity. A priori, before any experiments areconducted, we might consider both scenarios equally likely.But over the past few decades, a huge swathe of themodified gravity landscape has been erased. Following highprecision measurements of local gravitational potentialswithin the solar system, and cosmological potentials in thecosmic microwave background, only a small fraction of (thevery large) parameter space remains viable. A further nullresult was recently derived from the velocity of gravitationalwaves in the LIGO detection of merging black holes Abbottet al. (2016). So there have been numerous opportunitiesfor modified gravity to show its face elsewhere, but ithas failed to. In contrast, coupled models do not directlyalter the motions of planets, the gravitational potentials,nor the speed of gravitational waves. Crucially then, fromthe perspective of Bayesian model selection, couplingmodels are favoured because they are more predictive.They induce a pure change in cosmological structureformation, while naturally satisfying these complementarytests. Furthermore, a broad spectrum of modified gravitymodels generically predict an enhancement of structureformation, which would aggravate rather than alleviate theaforementioned tensions in observational data.Ultimately these two distinct explanations are obser-vationally distinguishable, due to the coupled model pro-ducing a small segregation of the baryonic and dark mattermotions. This would represent a highly challenging but notinsurmountable task for future cosmological surveys.
ACKNOWLEDGMENTS
We are deeply thankful to Alkistis Pourtsidou for usefuldiscussions on the connection between the Dark Scatteringmodels and the Type 3 class of coupled dark energy. FSwould like to thank Baojiu Li for helpful comments. MBacknowledges support from the Italian Ministry for Edu-cation, University and Research (MIUR) through the SIRindividual grant SIMCODE, project number RBSI14P4IH.FS acknowledges support by the European Research Coun-cil under the European Community’s Seventh FrameworkProgramme FP7-IDEAS-Phys.LSS 240117. The numericalsimulations presented in this work have been performed andanalysed on the Hydra cluster at the RZG supercomputingcentre in Garching.
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