Simulating Momentum Exchange in the Dark Sector
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 1–12 (2011) Printed 30 July 2018 (MN L A TEX style file v2.2)
Simulating Momentum Exchange in the Dark Sector
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.
30 July 2018
ABSTRACT
Low energy interactions between particles are often characterised by elastic scattering.Just as electrons undergo Thomson scattering with photons, dark matter particles mayexperience an analogous form of momentum exchange with dark energy. We investigate theinfluence such an interaction has on the formation of linear and nonlinear cosmic structure, byrunning for the first time a suite of N-body simulations with different dark energy equationsof state and scattering cross sections. In models where the linear matter power spectrumis suppressed by the scattering, we find that on nonlinear scales the power spectrum isstrongly enhanced. This is due to the friction term increasing the efficiency of gravitationalcollapse, which also leads to a scale-independent amplification of the concentration and massfunctions of halos. The opposite trend is found for models characterised by an increase ofthe linear matter power spectrum normalisation. More quantitatively, we find that powerspectrum deviations at nonlinear scales ( k ≈ h/ Mpc) are roughly ten times larger thantheir linear counterparts, exceeding for the largest value of the scattering cross sectionconsidered in the present work. Similarly, the concentration-mass relation and the halo massfunction show deviations up to and , respectively, over a wide range of masses.Therefore, we conclude that nonlinear probes of structure formation might provide muchtighter constraints on the scattering cross section between dark energy and dark matter ascompared to the present bounds based on linear observables.
Key words: dark energy – dark matter – cosmology: theory – galaxies: formation
Aside from their insensitivity to electromagnetism, the physicalcharacteristics of dark matter and dark energy remain highly un-certain. The dark matter particle may have a mass lying anywherein the range to eV, while the microphysical nature of darkenergy is even less clear. The observation that these two phenomenacurrently possess energy densities that are the same order of mag-nitude has led to speculation that they may forge a deeper connec-tion. Recent measurements of the geometry of the Universe pointtowards an expansion history consistent with dark energy takingthe form of a cosmological constant. The evolution of cosmologi-cal density perturbations provides a crucial independent test, whichhelps to break degeneracies between a cosmological constant andother theories of dark energy.Current observational data relating to density perturbationsin the low redshift Universe, such as weak gravitational lens-ing from CFHTLenS (Heymans et al. 2013), redshift space dis-tortions (Reid et al. 2012; Blake et al. 2011), and galaxy clusters(Vikhlinin et al. 2009), all indicate a slightly lower amplitude of clustering than has been inferred from the Cosmic MicrowaveBackground (CMB Planck Collaboration et al. 2013). This tensionhas already been the topic of statistical analysis (MacCrann et al.2014; Beutler et al. 2014), although the level of statistical signif-icance hinges on assumptions such as spatial flatness. It is worthstressing that – if confirmed – a retardation in the growth of largescale structure is not necessarily the work of a modified theory ofgravity, but may simply arise from an interaction between the twocomponents of the Universe which we know least about - dark mat-ter and dark energy Simpson et al. (2011).Models of coupled dark energy have been extensively studiedin the literature (see e.g. Wetterich 1995; Barrow & Clifton 2006;Amendola 2000; Baldi 2011b). However almost all proposed mod-els share a common feature which is that they predominantly in-volve energy exchange between the two fluids, with only a smalldegree of momentum exchange. (The distinction between energyexchange and momentum exchange depends on the frame of refer-ence, here we define these terms with respect to the rest frame of thedark energy fluid.) Models with energy exchange inevitably mod-ify both the expansion history and the growth of structure. In this c (cid:13) M. Baldi & F. Simpson work we elaborate on a model of momentum exchange presentedin Simpson (2010), by exploring for the first time the evolution ofnonlinear structure with the use of N-body simulations. Since theenergy exchange in this model is negligible, the expansion historyis the same as for the non-scattering case, and is determined solelyby the dark energy equation of state.Other forms of non-gravitational physics may also influencestructure formation. Scattering between dark matter particles leadsto a differential motion between baryons and dark matter. Thismay become apparent via the dissipation of substructure in clusters(Meneghetti et al. 2001), the formation of asymmetric tidal streams(Kesden & Kamionkowski 2006), and the sequestration of X-raygas (Harvey et al. 2014). Any interaction between dark matter anddark energy will also invoke this differential motion, and couldtherefore induce similar observational signatures.Cosmological simulations of models characterised by someform of interaction between dark energy and dark matter havebeen performed by several authors (see e.g. Macci`o et al. 2004;Baldi et al. 2010; Li & Barrow 2011; Carlesi et al. 2014a), also in-cluding the effects of hydrodynamical forces on the uncoupledbaryonic particles (as e.g. in Baldi 2011b; Baldi & Viel 2010;Carlesi et al. 2014b). Several numerical studies have been devotedto the investigation of various statistical properties of the large-scale structures in the presence of such interactions, as e.g. the halomass function (Baldi & Pettorino 2011; Cui et al. 2012), the clus-tering of galaxies in real and redshift space (Marulli et al. 2012;Moresco et al. 2013), the distribution of halo satellites (Baldi et al.2011) and of large cosmic voids (Li 2011; Sutter et al. 2014), aswell as weak lensing properties (Beynon et al. 2012; Giocoli et al.2013; Carbone et al. 2013; Pace et al. 2014). As mentioned above,all these investigations were carried out for interaction scenarioscharacterised by an energy-momentum exchange between dark en-ergy and dark matter, while our present study is the first – to ourknowledge – to focus on models of pure momentum exchange.The paper is organised as follows. In Section 2 we review themodification to the dynamics of dark matter particles in the eventthat they experience elastic scattering with the dark energy fluid.This framework is applied to N-body simulations in Section 3. Theresults of these simulations, in terms of the matter power spectrum,halo mass function, concentration-mass relation and halo velocitydispersion are presented in Section 4. Finally in Section 5 we drawour conclusions.
At sufficiently low energies the interactions between particles - ele-mentary or composite - can invariably be described by a process ofelastic scattering. Rayleigh scattering and Thomson scattering aretwo prominent examples.Consider the case of a particle traversing an isotropic fluidwhose stress energy tensor given by T ab = diag( ρ, wρ, wρ, wρ ) ,where w is the equation of state parameter of the fluid. Provided w = − , the particle observes a nonzero momentum flux ¯ T µ ,which imparts a force proportional to the scattering cross sec-tion σ c . It may be shown that the four-force g i takes the form(Padmanabhan 1997): g i = σ c h T ik u k − u i (cid:16) T ab u a u b (cid:17)i , (1)in order to satisfy the condition g i u i = 0 . These terms may be expressed as T ab u a u b = ργ (1 + v w ) ,T ab u b = ργ (1 , − vw ) , (2)where v is the velocity of the particle traversing the fluid and γ isthe Lorentz factor. This yields a force given by g i = ( γ f · v , γ f ) (3)where f = − (1 + w ) σ c γ ρ v . (4)One cosmological example of this scenario is in the early Universe,where non-relativistic electrons experienced a retardation from thebackground radiation, w γ = 1 / , given by the Thomson drag force F = − σ T vρ γ , (5)where σ T is the Thomson cross section, and ρ γ is the energy den-sity of the radiation.In this work we shall adopt the drag force given by (4) as theonly non-gravitational force acting between the two dominant flu-ids in the Universe, dark matter and dark energy. In particular, wewill consider the case of Cold Dark Matter (CDM) particles mov-ing through the Dark Energy (DE) fluid – such that w = w DE –with a DE-CDM scattering cross-section σ c . In linear perturbation theory the two fluids obey the coupled differ-ential equations (Simpson 2010): θ ′ Q = 2 Hθ Q − an D σ c ∆ θ + k φ + k δ Q w ,θ ′ c = − Hθ c + ρ Q ρ c (1 + w ) an D σ c ∆ θ + k φ , (6)where n D is the proper number density of dark matter particles, ∆ θ ≡ θ Q − θ c is the velocity contrast, ( θ i being the divergenceof the velocity perturbations for the field i ), and where we haveassumed the dark energy sound speed c s = 1 . The prime denotesa derivative with respect to time. The other perturbation equationsremain in their canonical form, i.e.: δ ′ Q = − (cid:20) (1 + w ) + 9 H k (cid:0) − w (cid:1)(cid:21) θ Q + 3(1 + w ) φ ′ − H (1 − w ) δ Q , (7) δ ′ c = − θ c + 3 ˙ φ. (8)The gravitational potential φ is sourced by the Poisson equa-tion. Baryons may be included as a third fluid with no couplingterm. This slightly weakens the modification to the growth rate incosmologies with coupled fluids Simpson et al. (2011), but other-wise leaves the functional form unchanged, so we shall neglect theirinfluence in this work. In the context of our N-body simulations, our equation of motionfor the dark matter fluid element is modified by the additional term c (cid:13)000
At sufficiently low energies the interactions between particles - ele-mentary or composite - can invariably be described by a process ofelastic scattering. Rayleigh scattering and Thomson scattering aretwo prominent examples.Consider the case of a particle traversing an isotropic fluidwhose stress energy tensor given by T ab = diag( ρ, wρ, wρ, wρ ) ,where w is the equation of state parameter of the fluid. Provided w = − , the particle observes a nonzero momentum flux ¯ T µ ,which imparts a force proportional to the scattering cross sec-tion σ c . It may be shown that the four-force g i takes the form(Padmanabhan 1997): g i = σ c h T ik u k − u i (cid:16) T ab u a u b (cid:17)i , (1)in order to satisfy the condition g i u i = 0 . These terms may be expressed as T ab u a u b = ργ (1 + v w ) ,T ab u b = ργ (1 , − vw ) , (2)where v is the velocity of the particle traversing the fluid and γ isthe Lorentz factor. This yields a force given by g i = ( γ f · v , γ f ) (3)where f = − (1 + w ) σ c γ ρ v . (4)One cosmological example of this scenario is in the early Universe,where non-relativistic electrons experienced a retardation from thebackground radiation, w γ = 1 / , given by the Thomson drag force F = − σ T vρ γ , (5)where σ T is the Thomson cross section, and ρ γ is the energy den-sity of the radiation.In this work we shall adopt the drag force given by (4) as theonly non-gravitational force acting between the two dominant flu-ids in the Universe, dark matter and dark energy. In particular, wewill consider the case of Cold Dark Matter (CDM) particles mov-ing through the Dark Energy (DE) fluid – such that w = w DE –with a DE-CDM scattering cross-section σ c . In linear perturbation theory the two fluids obey the coupled differ-ential equations (Simpson 2010): θ ′ Q = 2 Hθ Q − an D σ c ∆ θ + k φ + k δ Q w ,θ ′ c = − Hθ c + ρ Q ρ c (1 + w ) an D σ c ∆ θ + k φ , (6)where n D is the proper number density of dark matter particles, ∆ θ ≡ θ Q − θ c is the velocity contrast, ( θ i being the divergenceof the velocity perturbations for the field i ), and where we haveassumed the dark energy sound speed c s = 1 . The prime denotesa derivative with respect to time. The other perturbation equationsremain in their canonical form, i.e.: δ ′ Q = − (cid:20) (1 + w ) + 9 H k (cid:0) − w (cid:1)(cid:21) θ Q + 3(1 + w ) φ ′ − H (1 − w ) δ Q , (7) δ ′ c = − θ c + 3 ˙ φ. (8)The gravitational potential φ is sourced by the Poisson equa-tion. Baryons may be included as a third fluid with no couplingterm. This slightly weakens the modification to the growth rate incosmologies with coupled fluids Simpson et al. (2011), but other-wise leaves the functional form unchanged, so we shall neglect theirinfluence in this work. In the context of our N-body simulations, our equation of motionfor the dark matter fluid element is modified by the additional term c (cid:13)000 , 1–12 imulating Dark Scattering Parameter Value H − Mpc − Ω M Ω DE Ω b A s . × − n s Table 1.
The set of cosmological parameters adopted in the present work,consistent with the latest results of the Planck collaboration (Ade et al.2013). Here n s is the spectral index of primordial density perturbationswhile A s is the amplitude of primordial density perturbations. involving the CDM particle mass m CDM , its velocity v , the scat-tering cross section σ c , the dark energy equation of state w DE andthe dark energy density ρ DE , according to the equation: ˙ v = − (1 + w DE ) σ c vc ρ DE m CDM (9)where c is the speed of light.As we are assuming a DE sound speed c s = 1 , we expectDE perturbations to be dumped within the cosmic horizon so thatthe DE density and velocity field is approximately homogeneous.Therefore, we shall neglect the influence of dark energy perturba-tions within the simulation as we will concentrate on scales fallingwell within the horizon. We have numerically verified that this as-sumption has little impact on the derived growth rate even for largerscales. The net rate of change in the dark energy density may be expressedas a sum of the contributions from the dark matter and the adiabaticexpansion. dρ Q da = ρ Q [1 + w ( a )] (cid:20) σ c n v a H − a (cid:21) , (10)where n indicates the number density of dark matter particles atthe present day. Consider the evolution of ρ Q during the era ofradiation domination. The velocity dispersion v ≃ const . and H ∝ a − , thus for a constant equation of state w we find: ρ Q = ρ Q a − w ) e − wa σcn v eqHeq (11)This suggests that the dark energy grows extremely quickly inthe early universe, when the dark matter particles had a very highnumber density. The energy density reaches a maximum when ei-ther (a) the adiabatic decay takes over; or (b) the energy transferbecomes large enough to saturate the kinetic energy of dark matter;or (c) the equation of state transitions to w ≃ − . The nature of thisenergy transfer will be investigated in greater detail in future work.Here we shall focus on the consequences of momentum exchangeduring matter-domination, where the available kinetic energy den-sity is low and thus we can safely assume any change in the darkenergy density is negligible. In order to explore the nonlinear effects of the Dark Scattering sce-nario described above, we have run a series of CDM-only cosmo-logical N-body simulations by means of a suitably modified ver-sion of the widely-used TreePM code
GADGET-3 (Springel et al.
Run w DE ξ [bn · c / GeV] σ ( z = 0)Λ CDM − – 0.826w09-xi0 − . − . − . − . − . − . − . − . Table 2.
The suite of cosmological N-body simulations considered in thepresent work, with their main physical and numerical parameters. N particles in an expanding background the full accelerationequation experienced by the i -th particle then becomes: ˙ v i = − [1 + A ] H v i + X j = i Gm j r ij | r ij | (12)where r ij is the distance between the i -th and the j -th particle. Theextra scattering term A is defined as: A ≡ (1 + w DE ) σ c cm CDM DE πG H (13)and adds to the standard cosmological friction term.This extra drag force depends on three free quantities: the DEequation of state parameter w DE , the DE-CDM scattering crosssection σ c , and the CDM particle mass m CDM . While the formercould be in general an arbitrary function of redshift, w ( z ) , the lat-ter two quantities are more naturally modelled as dimensional con-stants. The overall magnitude of the drag force depends only ontheir ratio, so we can define the combined quantity ξ ≡ c · σ c m CDM (14)with dimensions of [bn · c / GeV] , as the characteristic parameterof our models, such that Eq. 13 becomes: A ≡ (1 + w ) 3Ω DE πG Hξ (15)In the present work, we restrict our investigation to the simplified(and rather unrealistic) case of a constant DE equation of state, w DE = const . This is done in order to reduce the number of freeparameters in this first numerical investigation so to focus on theeffects of the DE-CDM scattering using a relatively low numberof simulations. More realistic equations of state for the DE compo-nent will be discussed in a forthcoming paper, for which the presentsimplified analysis will serve as a guideline to select reasonablecombinations of possible w DE ( z ) and ξ .Under these assumptions, we consider three possible valuesof the DE equation of state parameter w DE = {− . , − , − . } ,and four possible values of the parameter ξ = { , , , } bn · c / GeV . As clearly shown by Eq. 15, for the case of w DE = − (corresponding to a cosmological constant and therefore having thesame expansion history as the standard Λ CDM cosmology) the ad-ditional drag force vanishes, irrespective of the value of ξ , and themodel fully recovers the standard Λ CDM scenario. We are there-fore left with 9 distinct models, summarised in Table 2, for eachof which we have performed a cosmological N-body simulationwithin a periodic box of 250 comoving Mpc /h a side, filled with c (cid:13) , 1–12 M. Baldi & F. Simpson + A z i = 99w = -0.9, ξ = 50w = -1.1, ξ = 50 Figure 1.
The full friction coefficient (1 + A ) as a function of redshift forthe two most extreme scenarios under consideration in the present work,namely the two cases of ξ = 50 bn · c / GeV for w DE = − . ( solid )and w DE = − . ( dashed ). As one can see in the plot, for these modelsthe effect of the DE-CDM scattering on the overall friction term in the per-turbations equations is negligible (below ) for z & , which makes ournumerical setup fully consistent. For more realistic scenarios such as EDEmodels we expect that this condition will no longer hold. particles with mass m c = 9 . × M ⊙ /h . The gravita-tional softening has been set to ǫ g = 12 kpc /h , corresponding toroughly 1/40 th of the mean inter-particle separation.All simulations started from the same initial conditions at z i = 99 . The initial conditions have been generated by ran-domly displacing particles from a homogeneous “glass” distri-bution (Davis et al. 1985) according to the Zel’dovich approxima-tion (Zel’dovich 1970) for a Λ CDM cosmology with parametersbased on the latest results of the Planck satellite mission (Ade et al.2013), which are summarised in Table 1. In doing so, we are dis-carding any possible effect of the DE-CDM scattering at redshiftshigher than z i . This approximation appears to be fully justified forthe models under investigation, since the effective drag force isproportional to the combination Ω DE H which rapidly vanishes athigh redshifts for our set of cosmologies. This is clearly shown inFig. 1 where we display the magnitude of the overall friction term (1 + A ) as a function of redshift for the two most extreme mod-els considered in the present paper. The effect of the scattering isfound to be negligible at z > z i . Clearly, for more realistic sce-narios as e.g. Early Dark Energy models (see e.g. Wetterich 2004;Doran & Robbers 2006), for which the DE fractional density Ω DE does not vanish at high redshifts, this approximation will have tobe dropped.Although starting from the same initial conditions, the vari-ous models will show a different evolution due to both the differentcosmological expansion history and the different impact of the DE-CDM scattering. Therefore, the amplitude of linear perturbations at z = 0 will be different, resulting in a different value of σ as shownin the last column of Table 2. We have tested that our modifiedcode correctly captures this different evolution of linear perturba-tions by comparing the linear growth factor extracted from the sim-ulations with the theoretical expectation obtained by numericallyintegrating Eqs. 6,8. The comparison is shown in Fig. 2, where weshow in the upper panel the ratio of the growth factor D + ( z ) to thestandard Λ CDM case both for the theoretical expectations in the w DE = − . and w DE = − . models (solid and dashed lines, D + / D + ( Λ CD M ) ξ = 50 ξ = 30 ξ = 10 ξ = 0 Λ CDMw=-0.9w=-1.11 101+z0.00.51.01.50.00.0 x | D + S i m - D + L i n | / D + L i n Figure 2.
Top : The ratio of the linear growth factor to the Λ CDM case forthe different models under investigation extracted from our suite of sim-ulations (open symbols) and computed by integrating the linear equations6,8 (solid and dashed lines).
Bottom : The accuracy of our N-body code incapturing the linear growth of the different models: the relative differencebetween the linear and the simulated growth functions never exceeds . . respectively) and for the simulations results (open symbols). As oncan see from the plot, the two relative deviations agree extremelywell from z = 45 (corresponding to the first snapshot of our simu-lations) to z = 0 . In the lower panel we display the absolute valueof the percent relative difference between the simulated and the ex-pected linear growth. The plot shows that the accuracy of the codeis at the ∼ level for all the models and redshifts. We now move to discuss the main results of our set of simulations,consisting of a series of basic large-scale structure statistics thatwill be systematically compared both with the standard Λ CDMcosmology and with the constant- w DE models with no DE-CDMscattering. We begin with the visual inspection of the CDM distribution in aslice of thickness . Mpc /h through the simulation box. In Fig. 3we show the projected CDM density field at z = 0 in the standard Λ CDM cosmology ( middle panel ) and in the two most extremedark scattering scenarios, namely the ξ = 50 [bn · c / GeV] modelswith w DE = − . and w DE = − . ( upper and lower panels,respectively). The size of the region shown in the figures is . Mpc /h a side (i.e. one fourth of the whole simulation box).By looking at the three images, one can notice how the globalshape of the large-scale cosmic web is the same in the three models,reflecting their common initial conditions. At the same time, fromthis visual inspection it is already possible to notice that the threemodels are characterised by a different level of evolution of thelarge-scale structures, with the w DE = − . clearly showing astronger clustering of the most prominent density peaks, while the w DE = − . cosmology has a lower clustering than the Λ CDMcase. This is consistent with the general picture of the DE-CDM c (cid:13) , 1–12 imulating Dark Scattering
10 Mpc/h w = -0.9 ξ = 50 [bn c / GeV] Λ CDM
10 Mpc/h w = -1.1 ξ = 50 [bn c / GeV] Figure 3.
Density slices in the reference Λ CDM cosmology ( middle ) and inthe two ξ = 50[bn · c / GeV] models with w = − . ( top ) and w = − . ( bottom ) scattering enhancing or suppressing the linear growth of structuresin the two cases. For all the simulations of our suite we extract the nonlinear mat-ter power spectrum through a Cloud-in-Cell mass assignment toa cubic cartesian grid with the same spacing of the PM meshused for the large-scale integration with
GADGET , i.e. gridnodes over the whole simulation box. This provides a measure-ment of the power spectrum up to the Nyquist frequency of thegrid k Ny = πN/L = 6 . Mpc /h . Beyond this frequency weestimate the power spectrum by means of the folding method ofJenkins et al. (1998); Colombi et al. (2009) and we smoothly inter-polate the two estimations around k Ny . The total power spectrumobtained in this way is then truncated at the frequency where theshot noise reaches 20% of the measured power.In Fig. 4 we display the ratio of the full nonlinear matterpower spectra of all the simulations with w DE = − to the modelwith identical expansion history and no scattering between DE andCDM (i.e. to the ξ = 0 case), and for three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). As the figure shows, onlinear scales the effect of the DE-CDM scattering is to suppress (en-hance) the power for w DE > − ( w DE < − ) cosmologies, witha maximum deviation from the non-scattering model of ≈ at z = 0 , ≈ at z = 0 . , and ≈ at z = 1 . This confirms theexpectations obtained through linear theory, as one should expectalso by looking at the test shown in Fig. 2.However, at nonlinear scales – which are evaluated here for thefirst time – the situation appears significantly different. First of all,the sign of the deviation is reversed with respect to linear scales,showing an enhancement of the nonlinear power for w DE > − and a suppression for w DE < − . The transition between the tworegimes lies in the range k ∼ . − h/ Mpc for the various mod-els and redshifts. Furthermore, the amplitude of the effect is muchlarger in the nonlinear regime than observed for the linear case,with a maximum deviation exceeding for the most extremescenarios at the smallest scales probed by our present resolution.Also, the deviation from the non-scattering case shows a strongscale-dependence in the nonlinear regime, suggesting that the am-plitude of the effect might keep increasing at even smaller scales.This result clearly shows how the nonlinear regime might providemuch tighter constraints on the scattering cross section σ c betweenDE and CDM (or more precisely on the ratio ξ between the crosssection and the CDM particle mass) as compared to what obtainedso far using only linear probes (Simpson 2010).Similarly to what displayed in Fig. 4, we show in Fig. 5 theratio of the full nonlinear matter power spectrum of the variouscosmologies to the Λ CDM case, thereby including in the compar-ison both the effect of the DE-CDM scattering and of the specificexpansion history associated with the two different values of theequation of state parameter w DE . As one can see from the figure,the overall effect is qualitatively similar to what observed in thecomparison with the ξ = 0 case for a fixed expansion history. Thisshows that the impact of the background cosmic evolution is sub-dominant with respect to the DE-CDM scattering, in particular athighly nonlinear scales, even for rather extreme choices on the DEequation of state as the ones adopted in this study. Therefore, thenonlinear effects of the DE-CDM scattering appear as a distinc-tive and prominent feature of the model, independently from ourignorance about the nature of the DE component and its dynam-ical evolution. Remarkably, while the amplitude of the deviation c (cid:13) , 1–12 M. Baldi & F. Simpson P ( k ) / P ( k ) ξ = w = -1.1w = -0.9 z = 0 ξ = 0 ξ = 10 ξ = 30 ξ = 50 0.1 1.0 10.0k [h/Mpc]0.51.01.52.0 P ( k ) / P ( k ) ξ = w = -1.1w = -0.9 z = 0.5 0.1 1.0 10.0k [h/Mpc]0.51.01.52.0 P ( k ) / P ( k ) ξ = w = -1.1w = -0.9 z = 1 Figure 4.
The nonlinear matter power spectrum ratio to the ξ = 0 case for models with both w DE = − . ( solid lines ) and w DE = − . ( dashed lines ) atthree different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). The different colours and open symbols refer to the different values of the parameter ξ . As the figures show, the DE-CDM scattering results in two opposite effects at linear and nonlinear scales: while at the largest scales we observe a weakand scale-independent suppression (enhancement) of the spectrum normalisation for w DE = − . ( w DE = − . ), corresponding to the different expectedvalues of σ for the different models, at smaller scales we find a strong scale-dependent enhancement (suppression) of power for the same classes of models,respectively. The transition scale between the two behaviours lies in the range . − h/ Mpc depending on the model and on the redshift. P ( k ) / P ( k ) Λ CD M w = -1.1w = -0.9 z = 0 Λ CDM ξ = 0 ξ = 10 ξ = 30 ξ = 50 0.1 1.0 10.0k [h/Mpc]0.51.01.52.02.5 P ( k ) / P ( k ) Λ CD M w = -1.1w = -0.9 z = 0.5 0.1 1.0 10.0k [h/Mpc]0.51.01.52.02.5 P ( k ) / P ( k ) Λ CD M w = -1.1w = -0.9 z = 1 Figure 5.
The nonlinear matter power spectrum ratio to the reference Λ CDM model at three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). The line styles, colours, and symbols are the same as in Fig. 4. Also in this case it is possible to observe a transition between the linear and the nonlinearimpact of the DE-CDM scattering on the measured power. at the smallest scales seems to be roughly insensitive to the back-ground expansion history assumed as a reference, the linear devia-tion appears larger when comparing to the expected Λ CDM result,showing how in the linear regime the effects of a w DE = const . expansion history and of the associated DE-CDM scattering mod-ify the growth of perturbations in the same direction. This wouldfurther increase the significance of this signature when comparingto real observational data.These modification to the power spectrum can be readily in-terpreted as the effect of the energy-momentum dissipation due tothe DE-CDM scattering. This is expected to alter the collapse andthe virialisation process of gravitationally bound structures at smallscales. More specifically, while in the linear regime the velocityfield is always aligned with the spatial gradient of the gravitationalpotential, such that the drag term associated with the DE-CDMscattering always acts in the same direction of the gravitational ac-celeration, this is no longer true in the nonlinear regime (i.e. aftershell crossing) when the collapsing structures start to gain angularmomentum. Therefore, while in the linear regime an extra friction(drag) will necessarily suppress (enhance) the growth of structures,in the nonlinear regime the main effect would be to lower (increase)the kinetic energy of bound structures thereby altering their virialequilibrium and causing them to contract, in the case of a friction,or to expand in the case of a drag.This explains the opposite trend of the nonlinear matter power spectrum in the different models and the transition scale betweensuppression and enhancement coinciding with the transition be-tween linear and nonlinear scales. Interestingly, an analogous ef-fect has already been discussed in the context of interacting DEcosmologies by Baldi et al. (2010); Baldi (2011a) where a similartype of drag term appears in the perturbations equations as a conse-quence of momentum conservation. However, while in interactingDE cosmologies such an extra drag is only one amongst severalother modifications of the dynamical equations, and its impact istherefore mitigated by other competing effects, in the dark scatter-ing models under investigation in the present work it represents theonly effect at play, thereby maximally displaying its impact in thelarge-scale matter distribution. As we will see below, the nonlinearbehaviour of the extra drag characterising dark scattering cosmolo-gies will have a significant impact also on the structural propertiesof CDM halos such as their concentration and their velocity disper-sion. This result illustrates how the nonlinear regime of structureformation might be used to place much tighter constraints on theratio ξ between the DE-CDM scattering cross section σ c and theCDM particle mass M CDM as compared with the bounds that canbe derived using only linear observables. c (cid:13)000
The nonlinear matter power spectrum ratio to the reference Λ CDM model at three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). The line styles, colours, and symbols are the same as in Fig. 4. Also in this case it is possible to observe a transition between the linear and the nonlinearimpact of the DE-CDM scattering on the measured power. at the smallest scales seems to be roughly insensitive to the back-ground expansion history assumed as a reference, the linear devia-tion appears larger when comparing to the expected Λ CDM result,showing how in the linear regime the effects of a w DE = const . expansion history and of the associated DE-CDM scattering mod-ify the growth of perturbations in the same direction. This wouldfurther increase the significance of this signature when comparingto real observational data.These modification to the power spectrum can be readily in-terpreted as the effect of the energy-momentum dissipation due tothe DE-CDM scattering. This is expected to alter the collapse andthe virialisation process of gravitationally bound structures at smallscales. More specifically, while in the linear regime the velocityfield is always aligned with the spatial gradient of the gravitationalpotential, such that the drag term associated with the DE-CDMscattering always acts in the same direction of the gravitational ac-celeration, this is no longer true in the nonlinear regime (i.e. aftershell crossing) when the collapsing structures start to gain angularmomentum. Therefore, while in the linear regime an extra friction(drag) will necessarily suppress (enhance) the growth of structures,in the nonlinear regime the main effect would be to lower (increase)the kinetic energy of bound structures thereby altering their virialequilibrium and causing them to contract, in the case of a friction,or to expand in the case of a drag.This explains the opposite trend of the nonlinear matter power spectrum in the different models and the transition scale betweensuppression and enhancement coinciding with the transition be-tween linear and nonlinear scales. Interestingly, an analogous ef-fect has already been discussed in the context of interacting DEcosmologies by Baldi et al. (2010); Baldi (2011a) where a similartype of drag term appears in the perturbations equations as a conse-quence of momentum conservation. However, while in interactingDE cosmologies such an extra drag is only one amongst severalother modifications of the dynamical equations, and its impact istherefore mitigated by other competing effects, in the dark scatter-ing models under investigation in the present work it represents theonly effect at play, thereby maximally displaying its impact in thelarge-scale matter distribution. As we will see below, the nonlinearbehaviour of the extra drag characterising dark scattering cosmolo-gies will have a significant impact also on the structural propertiesof CDM halos such as their concentration and their velocity disper-sion. This result illustrates how the nonlinear regime of structureformation might be used to place much tighter constraints on theratio ξ between the DE-CDM scattering cross section σ c and theCDM particle mass M CDM as compared with the bounds that canbe derived using only linear observables. c (cid:13)000 , 1–12 imulating Dark Scattering M [h -1 M O • ]24681012 c * m ean ξ = 0 ξ = 10 ξ = 30 ξ = 50z=0w = -0.9w = -1.1 10 M [h -1 M O • ]24681012 c * m ean ξ = 0 ξ = 10 ξ = 30 ξ = 50z=0.5w = -0.9w = -1.1 10 M [h -1 M O • ]24681012 c * m ean ξ = 0 ξ = 10 ξ = 30 ξ = 50z=1w = -0.9w = -1.1 Figure 6.
The c ∗ − M relation for all the models with w DE = − considered in the present work, at three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). Solid lines refer to the w DE = − . cosmologies while dashed lines refer to w DE = − . models, while the different coloursand open symbols refer to the different values of the parameter ξ . As the plots show, w DE = − . gives rise to higher concentrations for increasing values of ξ , while lower concentrations are obtained for w DE = − . . M [h -1 M O • ]0.00.51.01.52.02.5 c * m ean / c * m ean ( ξ = ) ξ = 0 ξ = 10 ξ = 30 ξ = 50z = 0w = -0.9w = -1.1 10 M [h -1 M O • ]0.00.51.01.52.02.5 c * m ean / c * m ean ( ξ = ) ξ = 0 ξ = 10 ξ = 30 ξ = 50z = 0.5w = -0.9w = -1.1 10 M [h -1 M O • ]0.00.51.01.52.02.5 c * m ean / c * m ean ( ξ = ) ξ = 0 ξ = 10 ξ = 30 ξ = 50z = 1w = -0.9w = -1.1 Figure 7.
The ratio of the c ∗ − M relation to the ξ = 0 case for both values of the equation of state at three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). Line styles, colours, and symbols are the same as in Fig. 6 For all the simulations of our suite we have identified particlegroups by means of a Friends-of-Friends (FoF) algorithm with link-ing length ℓ = 0 . × ¯ d , where ¯ d is the mean inter-particle separa-tion. Furthermore, for each FoF group we have identified gravita-tionally bound substructures by means of the SUBFIND algorithm(Springel et al. 2001) and we associated to the main substructureof each FoF halo a spherical overdensity mass M defined as themass enclosed in a sphere of radius R centred on the particlewith the minimum gravitational potential such that the mean den-sity within R corresponds to times the critical density of theuniverse, ρ crit ≡ H / πG .For each halo in our sample we then compute the halo concen-trations c ∗ following the method devised in Springel et al. (2008)as: c ∗ ln(1 + c ∗ ) − c ∗ / (1 + c ∗ ) = 7 . δ V (16)with δ V defined as: δ V = 2 (cid:18) V max H r max (cid:19) (17)where V max and r max are the maximum rotational velocity of thehalo and the radius at which this velocity peak is located, respec-tively.In Fig. 6 we show the average concentrations obtained withthis method within five logarithmically equispaced mass bins as afunction of the bin mass M for all the models under investiga- tion, and at the usual three different redshifts. The grey shaded areain each panel indicates the statistical Poissonian error based on theabundance of halos in each bin of the reference simulation.As the plot clearly shows, the DE-CDM scattering determinesa significant enhancement (suppression) of the normalisation of the c ∗ − M relation for w DE > − ( w DE < − ). The effect ismaximum at low redshifts and for the largest value of ξ , reaching afactor of ≈ in both directions for ξ = 50 [bn · c / GeV]. This ismore evident by looking at Fig. 7, where we display the ratio of theaverage concentration c ∗ mean in each mass bin to the reference caseof no DE-CDM scattering ( c ∗ mean ( ξ = 0) ) with fixed backgroundexpansion history. Also in this case, as for the comparison of thehalo mass function, the deviation from the reference case does notshow any clear dependence on the halo mass. This is therefore a fur-ther peculiar feature of dark scattering models, which also predicta significant impact on the average structural properties of CDMhalos besides the above mentioned effects on the large-scale matterdistribution.Since the drag force experienced by the dark matter particlesscales with their velocity, particles within larger halos experience agreater force. However over a given period of time, the fractionalreduction in velocity ∆ v/v is independent of v , and therefore eachhalo is subject to the same proportional change in concentration.These results are consistent with the effects observed on thenonlinear matter power spectrum and with the physical interpre-tation provided above for the transition between the linear and thenonlinear behavior of an extra-drag term in the dynamical evolutionof CDM particles. More specifically, our findings on the halo con- c (cid:13) , 1–12 M. Baldi & F. Simpson centrations, and in particular their evolution with redshift, clearlyshow how the virial equilibrium of collapsed halos is continuouslyaltered at low redshifts by the dissipation (for w DE > − ) or theinjection (for w DE < − ) of kinetic energy from (or into) the sys-tems, resulting in or expansion of the halos, respectively, with mat-ter moving towards (or out of) the halo core. With our halo catalogs at hand we have computed – for each cos-mology – the halo mass function as the number of halos withvirial mass M lying within a series of logarithmically eq-uispaced mass bins in the range − M ⊙ /h . This al-lows us to have a statistically significant sample of halos for eachmass bin, with a minimum number of halos for the poorest binused in our analysis (corresponding to the most massive bin in the w = − . , ξ = 50 [bn · c / GeV] model).Similarly to what was shown above for the nonlinear matterpower spectrum, in Figs. 8 and 9 we display the ratio of the halomass function of all the models under investigation to the ξ = 0 case (for fixed expansion history), and to the standard Λ CDM cos-mology, respectively. In both figures the grey shaded regions repre-sent the Poissonian error on the ratio based on the number of objectsincluded in each bin of the reference simulation, which for Fig. 8is taken to be the w DE = − . , ξ = 50 [bn · c / GeV] model.Also in this case, the two comparisons are qualitatively similar,although some differences between the two appear at the largestmasses as a consequence of the exponential dependence of the halomass function on the linear perturbations amplitude. This changesfrom model to model as a consequence of the different expansionhistory associated with the three possible different values of w DE .However, in both cases we observe a significant enhancement (sup-pression) of the abundance of halos at all masses within our massrange for the w DE > − ( w DE < − ) models, with the magnitudeof the effect increasing with the value of ξ , and with decreasingredshift.This turns out to be another highly distinctive feature of theDE-CDM scattering. In fact, while most non-standard cosmologies,including e.g. DE (see e.g. Courtin et al. 2011; Cui et al. 2012)and Modified Gravity (see e.g. Baldi et al. 2013; Lombriser et al.2013) scenarios, as well as primordial non-Gaussianity (see e.g.Grossi et al. 2007; Wagner et al. 2010), massive neutrinos (see e.g.Castorina et al. 2013) and Warm Dark Matter (see e.g. Angulo et al.2013) models, all affect the halo mass function with a specific massdependence – in most cases having a stronger impact on the high-mass tail – the effect of the DE-CDM scattering appears to have avery weak dependence on the halo mass resulting in a roughly con-stant enhancement (or suppression) of the halo abundance over awide mass range, at least at low redshifts. Some more pronouncedmass dependence appears at higher redshifts where the most mas-sive halos are less affected by the scattering as compared to thelow mass ones. The maximum relative deviation from the Λ CDMmodel is obtained for the two ξ = 50 [bn · c / GeV] models at z = 0 and ranges between 20 and 25%.This is the result of the superposition of two distinct effects.On one side, the weak modulation of the linear perturbations am-plitude is expected to slightly suppress (enhance) the abundanceof halos at large masses for w DE > − ( w DE < − ) as a con-sequence of the exponential dependence of the high-mass tail ofthe halo mass function on σ . On the other side, the strong distor-tion of the matter power spectrum at nonlinear scales is expectedto increase (reduce) the abundance of objects of all masses for w DE > − ( w DE < − ) by changing the value of M as aconsequence of the dynamical change of the halo concentrations,as discussed above. Remarkably, the overall effect appears to bealmost mass-independent, at least at low redshifts, while at higherredshifts the low-mass end of the halo mass function is more sig-nificantly affected. As a further statistics of the cosmic structures properties in the pres-ence of a DE-CDM scattering we investigate the relation betweenthe one-dimensional velocity dispersion σ and M for all thehalos in our catalogs, and compare the results to the case of noscattering ξ = 0 . This comparison is shown in Fig. 10 for modelswith fixed expansion history at the usual three different redshifts z = 0 , z = 0 . , and z = 1 . In the upper panels the coloureddots represent a random subsample of all the halos in the catalogswhile the solid and dashed lines trace the mean value of σ within10 logarithmically equispaced mass bins for the w DE = − . and the w DE = − . models, respectively. In the bottom panelswe display the ratio of the binned average 1-D velocity disper-sion of each model to the non scattering case with identical expan-sion history, again shown as solid (dashed) lines for w DE = − . ( w DE = − . ).Also for these figures, the grey shaded region indicates thePoissonian error associated with the abundance of halos in the dif-ferent bins of the reference simulation. As one can see from theplots, the models with w DE = − . show a systematic enhance-ment of the 1-D velocity dispersion with respect to the ξ = 0 caseover the whole mass range covered by our sample, with a weakmass dependence giving rise to a slightly stronger effect for thelargest masses probed by our catalogs. The enhancement increasesfor increasing values of ξ and for later times, reaching a maximumvalue of ≈ − for the most massive halos in the ξ = 50 [bn · c / GeV] model at z = 0 . This effect can be again interpreted asa consequence of the DE-CDM scattering on the structural prop-erties of the halos: when a drag term is acting on the dark matterparticles, the halos are more concentrated and therefore our defini-tion of R moves inwards. Therefore, for a fixed halo mass, thepotential well at R is deeper, which corresponds to an enhancedvelocity dispersion.Interestingly, we do not find a similar effect in the opposite di-rection for the models with w DE = − . , as it was always the casefor all the other observables investigated in this work. On the con-trary, all the w DE = − . cosmologies show very little deviationsfrom their reference model, never exceeding ≈ − even forthe largest value of the ξ parameter. Such different efficiency of thescattering in changing the halo velocity dispersion for quintessence and phantom expansion histories might be related to the later onsetof the extra-scattering term occurring for the latter models, whichis clearly visible in Fig. 1. A detailed investigation of this effectwould require a larger statistical sample and a higher resolutionthan is allowed by our present simulations, and is left for futurework. It is nonetheless interesting to notice here that the absence ofany impact of the DE-CDM scattering on the halo velocity disper-sion for w DE < − , if confirmed by future investigations, mightprovide a way to break possible degeneracies present in other ob-servables between the signature of the scattering and the effects ofthe background expansion history. c (cid:13)000
The ratio of the c ∗ − M relation to the ξ = 0 case for both values of the equation of state at three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). Line styles, colours, and symbols are the same as in Fig. 6 For all the simulations of our suite we have identified particlegroups by means of a Friends-of-Friends (FoF) algorithm with link-ing length ℓ = 0 . × ¯ d , where ¯ d is the mean inter-particle separa-tion. Furthermore, for each FoF group we have identified gravita-tionally bound substructures by means of the SUBFIND algorithm(Springel et al. 2001) and we associated to the main substructureof each FoF halo a spherical overdensity mass M defined as themass enclosed in a sphere of radius R centred on the particlewith the minimum gravitational potential such that the mean den-sity within R corresponds to times the critical density of theuniverse, ρ crit ≡ H / πG .For each halo in our sample we then compute the halo concen-trations c ∗ following the method devised in Springel et al. (2008)as: c ∗ ln(1 + c ∗ ) − c ∗ / (1 + c ∗ ) = 7 . δ V (16)with δ V defined as: δ V = 2 (cid:18) V max H r max (cid:19) (17)where V max and r max are the maximum rotational velocity of thehalo and the radius at which this velocity peak is located, respec-tively.In Fig. 6 we show the average concentrations obtained withthis method within five logarithmically equispaced mass bins as afunction of the bin mass M for all the models under investiga- tion, and at the usual three different redshifts. The grey shaded areain each panel indicates the statistical Poissonian error based on theabundance of halos in each bin of the reference simulation.As the plot clearly shows, the DE-CDM scattering determinesa significant enhancement (suppression) of the normalisation of the c ∗ − M relation for w DE > − ( w DE < − ). The effect ismaximum at low redshifts and for the largest value of ξ , reaching afactor of ≈ in both directions for ξ = 50 [bn · c / GeV]. This ismore evident by looking at Fig. 7, where we display the ratio of theaverage concentration c ∗ mean in each mass bin to the reference caseof no DE-CDM scattering ( c ∗ mean ( ξ = 0) ) with fixed backgroundexpansion history. Also in this case, as for the comparison of thehalo mass function, the deviation from the reference case does notshow any clear dependence on the halo mass. This is therefore a fur-ther peculiar feature of dark scattering models, which also predicta significant impact on the average structural properties of CDMhalos besides the above mentioned effects on the large-scale matterdistribution.Since the drag force experienced by the dark matter particlesscales with their velocity, particles within larger halos experience agreater force. However over a given period of time, the fractionalreduction in velocity ∆ v/v is independent of v , and therefore eachhalo is subject to the same proportional change in concentration.These results are consistent with the effects observed on thenonlinear matter power spectrum and with the physical interpre-tation provided above for the transition between the linear and thenonlinear behavior of an extra-drag term in the dynamical evolutionof CDM particles. More specifically, our findings on the halo con- c (cid:13) , 1–12 M. Baldi & F. Simpson centrations, and in particular their evolution with redshift, clearlyshow how the virial equilibrium of collapsed halos is continuouslyaltered at low redshifts by the dissipation (for w DE > − ) or theinjection (for w DE < − ) of kinetic energy from (or into) the sys-tems, resulting in or expansion of the halos, respectively, with mat-ter moving towards (or out of) the halo core. With our halo catalogs at hand we have computed – for each cos-mology – the halo mass function as the number of halos withvirial mass M lying within a series of logarithmically eq-uispaced mass bins in the range − M ⊙ /h . This al-lows us to have a statistically significant sample of halos for eachmass bin, with a minimum number of halos for the poorest binused in our analysis (corresponding to the most massive bin in the w = − . , ξ = 50 [bn · c / GeV] model).Similarly to what was shown above for the nonlinear matterpower spectrum, in Figs. 8 and 9 we display the ratio of the halomass function of all the models under investigation to the ξ = 0 case (for fixed expansion history), and to the standard Λ CDM cos-mology, respectively. In both figures the grey shaded regions repre-sent the Poissonian error on the ratio based on the number of objectsincluded in each bin of the reference simulation, which for Fig. 8is taken to be the w DE = − . , ξ = 50 [bn · c / GeV] model.Also in this case, the two comparisons are qualitatively similar,although some differences between the two appear at the largestmasses as a consequence of the exponential dependence of the halomass function on the linear perturbations amplitude. This changesfrom model to model as a consequence of the different expansionhistory associated with the three possible different values of w DE .However, in both cases we observe a significant enhancement (sup-pression) of the abundance of halos at all masses within our massrange for the w DE > − ( w DE < − ) models, with the magnitudeof the effect increasing with the value of ξ , and with decreasingredshift.This turns out to be another highly distinctive feature of theDE-CDM scattering. In fact, while most non-standard cosmologies,including e.g. DE (see e.g. Courtin et al. 2011; Cui et al. 2012)and Modified Gravity (see e.g. Baldi et al. 2013; Lombriser et al.2013) scenarios, as well as primordial non-Gaussianity (see e.g.Grossi et al. 2007; Wagner et al. 2010), massive neutrinos (see e.g.Castorina et al. 2013) and Warm Dark Matter (see e.g. Angulo et al.2013) models, all affect the halo mass function with a specific massdependence – in most cases having a stronger impact on the high-mass tail – the effect of the DE-CDM scattering appears to have avery weak dependence on the halo mass resulting in a roughly con-stant enhancement (or suppression) of the halo abundance over awide mass range, at least at low redshifts. Some more pronouncedmass dependence appears at higher redshifts where the most mas-sive halos are less affected by the scattering as compared to thelow mass ones. The maximum relative deviation from the Λ CDMmodel is obtained for the two ξ = 50 [bn · c / GeV] models at z = 0 and ranges between 20 and 25%.This is the result of the superposition of two distinct effects.On one side, the weak modulation of the linear perturbations am-plitude is expected to slightly suppress (enhance) the abundanceof halos at large masses for w DE > − ( w DE < − ) as a con-sequence of the exponential dependence of the high-mass tail ofthe halo mass function on σ . On the other side, the strong distor-tion of the matter power spectrum at nonlinear scales is expectedto increase (reduce) the abundance of objects of all masses for w DE > − ( w DE < − ) by changing the value of M as aconsequence of the dynamical change of the halo concentrations,as discussed above. Remarkably, the overall effect appears to bealmost mass-independent, at least at low redshifts, while at higherredshifts the low-mass end of the halo mass function is more sig-nificantly affected. As a further statistics of the cosmic structures properties in the pres-ence of a DE-CDM scattering we investigate the relation betweenthe one-dimensional velocity dispersion σ and M for all thehalos in our catalogs, and compare the results to the case of noscattering ξ = 0 . This comparison is shown in Fig. 10 for modelswith fixed expansion history at the usual three different redshifts z = 0 , z = 0 . , and z = 1 . In the upper panels the coloureddots represent a random subsample of all the halos in the catalogswhile the solid and dashed lines trace the mean value of σ within10 logarithmically equispaced mass bins for the w DE = − . and the w DE = − . models, respectively. In the bottom panelswe display the ratio of the binned average 1-D velocity disper-sion of each model to the non scattering case with identical expan-sion history, again shown as solid (dashed) lines for w DE = − . ( w DE = − . ).Also for these figures, the grey shaded region indicates thePoissonian error associated with the abundance of halos in the dif-ferent bins of the reference simulation. As one can see from theplots, the models with w DE = − . show a systematic enhance-ment of the 1-D velocity dispersion with respect to the ξ = 0 caseover the whole mass range covered by our sample, with a weakmass dependence giving rise to a slightly stronger effect for thelargest masses probed by our catalogs. The enhancement increasesfor increasing values of ξ and for later times, reaching a maximumvalue of ≈ − for the most massive halos in the ξ = 50 [bn · c / GeV] model at z = 0 . This effect can be again interpreted asa consequence of the DE-CDM scattering on the structural prop-erties of the halos: when a drag term is acting on the dark matterparticles, the halos are more concentrated and therefore our defini-tion of R moves inwards. Therefore, for a fixed halo mass, thepotential well at R is deeper, which corresponds to an enhancedvelocity dispersion.Interestingly, we do not find a similar effect in the opposite di-rection for the models with w DE = − . , as it was always the casefor all the other observables investigated in this work. On the con-trary, all the w DE = − . cosmologies show very little deviationsfrom their reference model, never exceeding ≈ − even forthe largest value of the ξ parameter. Such different efficiency of thescattering in changing the halo velocity dispersion for quintessence and phantom expansion histories might be related to the later onsetof the extra-scattering term occurring for the latter models, whichis clearly visible in Fig. 1. A detailed investigation of this effectwould require a larger statistical sample and a higher resolutionthan is allowed by our present simulations, and is left for futurework. It is nonetheless interesting to notice here that the absence ofany impact of the DE-CDM scattering on the halo velocity disper-sion for w DE < − , if confirmed by future investigations, mightprovide a way to break possible degeneracies present in other ob-servables between the signature of the scattering and the effects ofthe background expansion history. c (cid:13)000 , 1–12 imulating Dark Scattering M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) ξ = ξ = 0 ξ = 10 ξ = 30 ξ = 50w = -0.9w = -1.1z = 0 10 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) ξ = w = -0.9w = -1.1z = 0.5 10 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) ξ = w = -0.9w = -1.1z = 1 Figure 8.
The mass function ratio to the ξ = 0 case for all the models with w DE = − at three different redshifts z = 0 ( left panel ), z = 0 . ( middle panel ),and z = 1 ( right panel ). Solid lines refer to the w DE = − . case while dashed lines refer to the w DE = − . models, while the different colours and opensymbols refer to the different values of the parameter ξ . The effect of enhancement or suppression of the halo abundance is almost mass-independent for allthe models at z = 0 , while at higher redshifts the large masses appear less affected than the low masses. M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M Λ CDM ξ = 0 ξ = 10 ξ = 30 ξ = 50w = -0.9w = -1.1z = 0 10 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M w = -0.9w = -1.1z = 0.5 10 M [M O • /h]0.60.81.01.21.41.6 N ( M ) / N ( M ) Λ CD M w = -0.9w = -1.1z = 1 Figure 9.
The mass function ratio to the Λ CDM case for all the models under investigation in the present work. The three panels, colours, symbols, and linestyles are the same as displayed in Fig. 8. Also in this case the effect of enhancement or suppression appears almost scale-independent for all the models at z = 0 while for higher redshift the mass dependence is more pronounced than in the comparison among models with the same expansion history. At largemasses at high redshifts it is also possible to observe a suppression of the abundance of large-mass halos for w = − . models which did not occur at fixedexpansion history. We conclude our analysis by investigating the halo bias in our setof dark scattering cosmologies. We compute the bias by takingthe ratio of the halo-matter cross-power spectrum P hm ( k ) to thematter-matter power spectrum P mm ( k ) for halos with mass above × M ⊙ /h . In Fig. 11 we show the ratio of the halo bias ofall the scattering DE models to the corresponding non-scatteringreference model with identical expansion history in the range ofscales . k · h/ Mpc . Interestingly, also in this case the w DE = − . models show a significant deviation with respect tothe non-scattering scenario, with an increase of the halo bias a allscales, while the w DE = − . models have much smaller devia-tions. The increase of the bias for the w DE = − . models alsoshows a clear enhancement at the crossing between linear and non-linear scales, reflecting the sharp transition displayed in Figure 4for the matter power spectrum. Also in this case, we argue that thedifferent behaviour of the w DE > − and w DE < − modelsmight be due to the later onset of the extra-scattering term for a phantom expansion history, as shown in Fig. 1. We leave for futurework a more detailed investigation of how the redshift evolution ofthe extra-force (which is in turn dictated by the evolution of the DEdensity) is related to the impact of the scattering on both the halovelocity dispersion and the bias. In particular, for models of EDEwhere the DE density never vanishes at high redshifts we would ex-pect – for a fixed scattering parameter ξ – a direct relation between the amount of EDE and the efficiency of the scattering in alteringboth the halo velocity dispersion and the bias. In the present work we have investigated for the first time the obser-vational effects of an elastic scattering between Cold Dark Matterparticles and a perfect fluid associated with the cosmic Dark En-ergy in the nonlinear regime of structure formation by means ofa suite of intermediate-resolution N-body simulations. As a firstnumerical investigation of these scenarios, we have restricted ouranalysis to the simplified case of a constant Dark Energy equationof state parameter taking both a canonical value w DE = − . anda phantom-like value w DE = − . and for each of these cases wehave run four simulations with different values of the dimensionalparameter ξ associated with the ratio between the Dark Energy-Cold Dark Matter scattering cross section σ c and the CDM particlemass m CDM , namely ξ = { , , , } [bn · c / GeV]. A fur-ther simulation for the standard Λ CDM cosmology has been runas reference. All simulations shared the same initial conditions,thereby discarding possible effects of the scattering before the start-ing redshift of the runs z i = 99 , which we verified to be a safeapproximation for the specific models under investigation.We analysed our simulations suite through a series of basic c (cid:13) , 1–12 M. Baldi & F. Simpson σ [ k m / s ] M [h -1 M O • ]0.900.951.001.051.10 σ / σ ( ξ = ) w=1.1w=-0.9 z = 0 ξ = 0 ξ = 10 ξ = 30 ξ = 50 10 σ [ k m / s ] M [h -1 M O • ]0.900.951.001.051.10 σ / σ ( ξ = ) w=1.1w=-0.9 z = 0.5 ξ = 0 ξ = 10 ξ = 30 ξ = 50 10 σ [ k m / s ] M [h -1 M O • ]0.900.951.001.051.10 σ / σ ( ξ = ) w=1.1w=-0.9 z = 1.0 ξ = 0 ξ = 10 ξ = 30 ξ = 50 Figure 10.
The one-dimensional velocity dispersion σ as a function of halo mass M and its ratio to the ξ = 0 model at the usual three different redshifts z = 0 ( left ), z = 0 . ( middle ), and z = 1 ( right ). In the upper panels we display as coloured dots a random subsample of the halos of each model and as solidand dashed lines the binned mean of the data points for the w DE = − . and w DE = − . , respectively. In the lower panels we show the binned ratio to thenon scattering model with identical expansion history. b / b ξ = ξ =50 ξ =30 ξ =10 ξ =0w = -0.9w = -1.1z = 0 Figure 11.
The ratio of the halo bias to the non-scattering reference modelwith identical expansion history for both w DE = − . ( solid lines ) and w = − . ( dashed lines ). statistical properties of the large-scale matter distribution and struc-tural properties of collapsed halos. More specifically, the main re-sults of our work can be summarised as follows: • Matter Power Spectrum – By comparing the full nonlinearmatter power spectrum extracted from our simulations both to thestandard Λ CDM reference and to the non-scattering scenario withidentical expansion history (i.e. to the ξ = 0 run with the same w DE for each model) we have shown that the power spectrumis affected by the scattering process in a radically differentway between the linear and the nonlinear regime of structureformation. In the former, a scale-independent shift in the poweramplitude appears at linear scales, resulting in a suppression or anenhancement of the linear perturbations amplitude for w DE > − or w DE > − , respectively, with a maximum relative differenceof ≈ for the most extreme scenario. In the latter, instead, astrongly scale-dependent deviation with a much larger amplitudeand an opposite sign as compared to the linear regime is found forscales below k ∼ . − h/ Mpc depending on the model and onthe redshift. Such behaviour can be interpreted as a consequenceof the injection or dissipation of kinetic energy within virialized (or virializing) structures, thereby altering the efficiency of thegravitational collapse. • Halo Concentrations – We computed the concentrationof all the main substructures of our halo catalogs and comparedthe concentration-mass relation of all the models over the massrange ≈ − M ⊙ /h . We then compared this relation tothe corresponding one for the case of no scattering ( ξ = 0 ) andidentical expansion history. As a general trend, we found thatthe scattering between Cold Dark Matter particles and the DarkEnergy fluid induces an increase of the normalisation of the c − M relation for w DE > − and correspondingly a decrease of thenormalisation for w DE < − . The effect is largest at z = 0 andrapidly decreases with redshift, with a maximum deviation of afactor ≈ for the most extreme models ( ξ = 50 bn · c / GeV) at z = 0 . Such evolution can also be ascribed to the nonlinear effectof the extra-drag term associated with the elastic scattering, whichdetermines an injection or a dissipation of kinetic energy frombound structures for w < − and w > − , respectively, with aconsequent expansion or contraction of the halos resulting in alower or higher concentration for a fixed halo mass. • Halo Mass Function – For all our simulations we comparedthe abundance of halos in the mass range − M ⊙ /h toboth the non scattering model with the identical expansion historyand the standard Λ CDM cosmology. In both cases the scatteringresults in a significant increase of the halo abundance over thewhole mass range for w > − and a corresponding decrease ofthe abundance for w < − . Also in this case, the effect is largestat z = 0 and rapidly decreases with redshift, with a maximumrelative deviation of about at z = 0 for the most extremerealisation of elastic scattering. In the comparison with the Λ CDMcosmology, we observe the superposition of the effects due tothe scattering and to the different expansion history, especially athigher redshifts when the impact of the scattering term is signif-icantly weaker than at later epochs, and the high-mass tail of thehalo mass function shows differences from model to model that aremainly related to the exponential dependence of the multiplicityfunction on the underlying value of the σ normalisation. • Velocity dispersions – We investigated the relation betweenthe one-dimensional velocity dispersion of the halos and their massin the various models under study, and compared the obtainedrelation to the case of no scattering with identical cosmological c (cid:13) , 1–12 imulating Dark Scattering expansion history. Our results indicate a systematic increase of thevelocity dispersion at fixed mass for increasing values of the scat-tering parameter ξ in the w DE > − scenarios. The effect is againlargest at low redshifts and shows a slightly positive correlationwith the halo mass, reaching a maximum value of ≈ for thelargest masses in the most extreme model. Interestingly, the samefeature does not appear for the w < − models which show nosignificant deviations from the non-scattering case even for largevalues of the scattering parameter ξ . • Halo Bias – We compared the halo bias of all the dark scatter-ing models under investigation to the corresponding non-scatteringmodel with identical expansion history, finding a systematic in-crease of the bias at all scales for the w DE = − . cosmolo-gies, for increasing values of the scattering parameter ξ , with aclear additional enhancement occurring at the transition betweenlinear and nonlinear scales reflecting the transition observed in thematter power spectrum. Also in this case, the w DE = − . scenar-ios show much smaller deviations, similarly to what we found forthe halo velocity dispersion. This starkly different behaviour of the“canonical” and “phantom” Dark Energy scenarios in the presenceof a scattering with the Cold Dark Matter particles deserves furtherinvestigations, and will be addressed in future works.To conclude, we have presented the first investigation of non-linear structure formation in the context of cosmological modelsfeaturing a non-vanishing scattering cross section σ c between ColdDark Matter particles and a Dark Energy fluid characterised by aconstant equation of state parameter with either a “canonical” or a“phantom” behaviour. Our results, based on the first N-body simu-lations of this class of models ever performed, indicate that the non-linear evolution of cosmic structures might provide significantlytighter constraints on a possible process of elastic scattering in thedark sector as compared to the present bounds based on linear ob-servables only. A more realistic investigation of the effects of thescattering in the presence of a time-dependent Dark Energy equa-tion of state, as for e.g. Early Dark Energy cosmologies, is presentlyongoing and will be discussed in an upcoming paper. ACKNOWLEDGMENTS
We are deeply thankful to Licia Verde and Raul Jimenez for theirhelpful comments on the draft. MB is supported by the MarieCurie Intra European Fellowship “SIDUN” within the 7th Frame-work Programme of the European Commission. FS acknowledgessupport from the European Research Council under the SeventhFramework Programme FP7-IDEAS-Phys.LSS 240117. The nu-merical simulations presented in this work have been performedand analysed on the Hydra cluster at the RZG supercomputing cen-tre in Garching.
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