Warm dark matter at small scales: peculiar velocities and phase space density
aa r X i v : . [ a s t r o - ph . C O ] M a y Warm dark matter at small scales:peculiar velocities and phase space density.
Daniel Boyanovsky ∗ Department of Physics and Astronomy,University of Pittsburgh, Pittsburgh, PA 15260 (Dated: June 9, 2018)
Abstract
We study the scale and redshift dependence of the power spectra for density perturbations andpeculiar velocities, and the evolution of a coarse grained phase space density for (WDM) particlesthat decoupled during the radiation dominated stage. The (WDM) corrections are obtained in aperturbative expansion valid in the range of redshifts at which N-body simulations set up initialconditions, and for a wide range of scales. The redshift dependence is determined by the kurtosis β of the distribution function at decoupling. At large redshift there is an enhancement of peculiarvelocities for β > β . For β > /
21 it growslogarithmically with the scale factor as a consequence of the suppression of statistical fluctuations.Two specific models for WDM are studied in detail. The (WDM) corrections relax the bounds onthe mass of the (WDM) particle candidate.
PACS numbers: 98.80.-k; 95.35.+d; 98.80.Bp ∗ Electronic address: [email protected] . INTRODUCTION The current paradigm of structure formation, the Λ
CDM standard cosmological model,describes large scale structure remarkably well. However, observational evidence has beenaccumulating suggesting that the cold dark matter (CDM) scenario of galaxy formation may have problems at small, galactic, scales.Large scale simulations seemingly yield an over-prediction of satellite galaxies[1] by almostan order of magnitude[1–5]. Simulations within the ΛCDM paradigm also yield a densityprofile in virialized (DM) halos that increases monotonically towards the center[1, 6–9] andfeatures a cusp, such as the Navarro-Frenk-White (NFW) profile[6] or more general centraldensity profiles ρ ( r ) ∼ r − β with 1 ≤ β . . small scale powerspectrum[23] providing an explanation of the smoother inner profiles of (dSphs) and fewersatellites.Furthermore recent numerical results hint to more evidence of possible small scale dis-crepancies with the Λ CDM scenario: another over-abundance problem, the “emptiness ofvoids” [24] and the spectrum of “mini-voids”[25], both of which may be explained by aWDM candidate. Constraints from the luminosity function of Milky Way satellites[26] sug-gest a lower limit for the mass of a WDM particle of a few keV, a result consistent withLyman- α [27–29], galaxy power spectrum[30] and lensing observations[31]. More recently,results from the Millenium-II simulation[32] suggest that the Λ CDM scenario overpredicts the abundance of massive & M ⊙ haloes, which is corrected with a WDM candidate of m ∼ Motivation and goals:
Redshift dependence of the power spectrum and peculiar velocities: recent N-body simu-lations of WDM[25, 26] set up initial conditions at z = 40[26] or z = 50[25] with a rescaledversion of the CDM power spectrum from a fit provided in ref.[29] that inputs a cutoff from2ree streaming, however, these simulations neglected the velocity dispersion of the WDMparticles in the initial conditions. We seek to understand both the redshift dependence ofthe matter and peculiar velocity power spectrum in this range of redshifts for a wide rangeof scales. Phase space density: in a seminal article Tremaine and Gunn[37] provided bounds onthe mass of the DM particle from phase space density considerations: whereas in absenceof self-gravity the fine-grained phase space density (or distribution function) is conservedafter the DM species decouples from the plasma, phase mixing theorems[38] assert that acoarse grained phase space density always diminishes as a result of phase mixing (violentrelaxation)[38, 39]. Therefore the microscopic phase space density provides an upper bound from which constraints on the mass can be extracted. These arguments were generalizedin refs.[10, 33, 40–43] to a coarse grained phase space density obtained from moments ofthe microscopic distribution function. In ref.[33, 41–43] this coarse grained phase spacedensity was combined with photometric observations of (dShps) to constrain the mass andthe number of relativistic degrees of freedom at decoupling.Although the microscopic phase space density, namely the distribution function, obeysthe collisionless Boltzmann equation, the evolution of the coarse grained phase space densityis not directly obtained from this equation (see discussion in ref.[39]). Although the proxy phase space density introduced in refs.[10, 40–42] is conserved after decoupling, its evolution does not include self-gravity. Therefore there remains the unexplored question of preciselywhat happens to the microscopic phase space density or its proxy introduced in refs.[10, 40–42] when gravitational perturbations are included in the Boltzmann equation. One aspectis clear: the perturbations of the distribution function (microscopic phase space density)feature two moments that grow under gravitational perturbations: the first moment (densityperturbations) and the second moment (velocity perturbations) which are actually relatedvia the continuity equation on sub-horizon scales. In this article we study the evolution ofthe coarse-grained phase space density introduced in refs.[10, 40–42] as a function of redshiftand scale for (WDM) particles in order to assess how the original arguments are modifiedby gravitational perturbations, again in the regime of redshifts at which N-body simulationsset up initial conditions.
Results:
Armed with the results recently obtained in ref.[44] we obtain a perturbative expansionof the redshift corrections to the matter, peculiar velocity power spectra and evolution ofa coarse-grained phase space density. This expansion is valid in the regime z ≪ z eq for awide range of scales and is a distinct feature of (WDM) particles. These corrections dependon the kurtosis β of the unperturbed distribution function. Peculiar velocities contributeto the velocity dispersion and free streaming and lead to a suppression of the matter powerspectrum for β > z ≃ − grows logarithmically with the scale factor for β > /
21. Two specific models of (WDM) particles motivatedby particle physics are studied in detail. Implications on the bounds for the mass of the(WDM) particle are discussed. 3
I. PRELIMINARIES
We begin by establishing some notation and conventions that are used in the analysis.Since we focus on the region of redshift z ≫ H = ˙ a a = H (cid:20) Ω r a + Ω m a (cid:21) = H Ω m a [ a + a eq ] (II.1)where the dot stands for derivative with respect to conformal time ( η ), the scale factor isnormalized to a = 1 today, and a eq = Ω r Ω m ≃ . (II.2)Introducing ˜ a = aa eq , (II.3)it follows that d ˜ adη = (cid:20) H Ω m a eq (cid:21) [1 + ˜ a ] . (II.4)At matter-radiation equality we define k eq ≡ H eq a eq = √ (cid:20) H Ω m a eq (cid:21) = 9 . × − Mpc (II.5)corresponding to the comoving wavevector that enters the Hubble radius at matter-radiationequality, where we have used Ω m h = 0 . g = − a ( η ) h ψ ( ~x, η ) i (II.6) g ij = a ( η ) h − φ ( ~x, η ) i δ ij . (II.7)The perturbed distribution function is given by f ( p, ~x, η ) = f ( p ) + F ( p, ~x, η ) (II.8)where f ( p ) is the unperturbed distribution function, which after decoupling obeys the colli-sionless Boltzmann equation in absence of perturbations and ~p, ~x are comoving momentumand coordinates respectively. As discussed in ref.[41–43] the unperturbed distribution func-tion is of the form f ( p ) ≡ f ( y ; χ , χ , · · · ) (II.9)where y = pT ,d (II.10)4here p is the comoving momentum and T ,d is the decoupling temperature today , T ,d = (cid:16) g d (cid:17) T CMB , (II.11)with g d being the effective number of relativistic degrees of freedom at decoupling, T CMB =2 . × − eV is the temperature of the (CMB) today, and χ i are dimensionless couplingsor ratios of mass scales.We neglect stress anisotropies, in which case φ = ψ and introduce e F ( ~p, ~k, η ) = F ( ~p, ~k, η ) n ; e f ( p ) = f ( p ) n (II.12)where n = Z d p (2 π ) f ( p ) , (II.13)is the background density of (DM) today . Therefore δ ( ~k, η ) = Z d p (2 π ) e F ( ~k, η ) . (II.14)becomes δρ m /ρ m after the DM particle becomes non-relativistic.Introducing spatial Fourier transforms in terms of comoving momenta ~k (we keep thesame notation for the spatial Fourier transform of perturbations), and neglecting stressanisotropies the linearized Boltzmann equation for perturbations is given by[46–52]˙ e F ( ~k, ~p ; η ) + i k µ pǫ ( p, η ) e F ( ~k, ~p ; η ) + (cid:16) d e f ( p ) dp (cid:17)h p ˙ φ ( ~k, η ) − ik µ ǫ ( p, η ) φ ( ~k, η ) i = 0 (II.15)where dots stand for derivatives with respect to conformal time, µ = b k · b p , and ǫ ( p, η ) = p p + m a ( η ) is the conformal energy of the particle of mass m . The gravitational poten-tial is determined by Einstein’s equation[46, 47].As discussed in ref.[44] for a (WDM) particle with a mass in the ∼ keV range, there arethree stages of evolution: I) radiation domination and the DM particle is relativistic, II)radiation domination and the DM particle is non-relativistic and III) the matter dominatedstage, during which cold and warm DM particles are non-relativistic.During stages (I) and (II) the gravitational potential is completely determined by theradiation component and the Boltzmann equation for the distribution function of the WDMparticle is solved by integrating eqn. (II.15) with φ being determined by the radiationcomponent. During stage (III) the gravitational potential is determined by the mattercomponent and the Boltzmann equation becomes a self-consistent Vlasov-type equation.Since the Boltzmann equation is first order in time, the solution during stages (I) and(II) becomes the initial condition for the evolution during stage (III).In this article we focus on the evolution of peculiar velocities and phase space densityduring the matter dominated stage 10 ≤ z . z eq , corresponding to stage (III) during whichdark energy can be neglected. Typical N-body simulations setup initial conditions whichinput the matter power spectrum from linear perturbation theory at z ≃ − (cid:0) (cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11)(cid:12) (cid:13)(cid:14)(cid:15)(cid:16) (cid:17)(cid:18)(cid:19)(cid:20) (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" FIG. 1: u [ z ] for z ≤ z eq . In this stage the WDM is non-relativistic, hence p/ǫ ( p, η ) = p/m a ( η ), and the Bolzmannequation simplifies by introducing the variable s ( η ) = Z η dη ′ a ( η ′ ) ≡ √ u ( η ) k eq a eq (II.16)where the dimensionless variable u ( η ) = 12 ln " p a ( η ) − p a ( η ) + 1 ; u NR ≤ u ( η ) ≤ , (II.17)is normalized so that u ( ∞ ) = 0 and introduced[44] u NR = ln h √ ˜ a NR i ; ˜ a NR = h V ( t eq ) i (II.18)where ˜ a NR corresponds to the time when the particle becomes non-relativistic, and h V ( t eq ) i is the velocity dispersion of the DM particle at matter-radiation equality given by[44] h V ( t eq ) i ≃ . × − q y (cid:16) keV m (cid:17) (cid:16) g d (cid:17) . (II.19)In this expression g d is the number of relativistic degrees of freedom at decoupling and weintroduced the moments y n = R ∞ y n f ( y ) dy R ∞ y f ( y ) dy . (II.20)The function u [ z ] as function of redshift is displayed in fig. (1) and˜ a ( u ) = 1sinh [ u ] . (II.21)The solution of the Boltzmann equation during stage (III) is given in ref.([44]) e F ( ~k, ~p ; s ) = − φ ( ~k, s ) (cid:16) p d e f ( p ) dp (cid:17) + i m Z ss NR ds ′ a ( s ′ ) φ ( ~k, s ′ ) (cid:16) ~k · ~ ∇ p e f ( p ) (cid:17) e − i ~k · ~pm ( s − s ′ ) + e − i ~k · ~pm ( s − s NR ) h e F ( ~k, ~p ; η NR ) + φ ( ~k, η NR ) (cid:16) p d e fdp (cid:17)i . (II.22)6he term e F ( ~k, ~p ; η NR ) in the bracket in eqn. (II.22) is the solution of the Boltzmannequation at the beginning of stage (III) (end of stage (II)) its form is given in detail inref.[44] but is not necessary in the discussion that follows.After radiation-matter equality when the WDM particle is non-relativistic and (DM) per-turbations dominate the gravitational potential and for k ≫ k eq , the gravitational potential φ is determined by Poisson’s equation[47] φ ( k, η ) = − k eq k ˜ a δ ( ~k, s ) (II.23)For s > s eq , the integral in s ′ in (II.22) is split from s NR up to s eq and from s eq up to s . Inthe first integral the gravitational potential is determined by perturbations in the radiationfluid and in the second integral the gravitational potential is replaced by Poisson’s equation(II.23), leading to the result (valid for s > s eq )[44] e F ( ~k, ~p ; s ) = 34 k eq k ˜ a δ ( ~k, s ) (cid:16) p d e f ( p ) dp (cid:17) − i mk eq a eq k Z ss eq ds ′ ˜ a ( s ′ ) δ ( k, s ′ ) µ (cid:16) d e f ( p ) dp (cid:17) e − iµQ + F [ ~k, ~p ; s ] . (II.24)where Q = kpm ( s − s ′ ) ; µ = b k · b p , (II.25)and F [ ~k, ~p ; s ] is given by the second line in (II.22) plus the contribution from the integralbetween s NR and s eq (for details see ref.[44]).We are interested in the corrections to the power spectra in the regime of redshift 1 0, namely κ → 0, therefore all the WDM corrections are in termsof κ .In the (CDM) limit eqn.(II.27) reduces to Meszaro’s equation[53–55] for (CDM) pertur-bations in a radiation and matter dominated cosmology[44].The power spectrum of density perturbations is given by P δ ( k ) = Ak n s T ( k ) (II.33)where where n s = 0 . A is the am-plitude and T ( k ) is the transfer function. It is convenient to normalize the WDM powerspectrum and transfer function to CDM, namely P wdm ( k ) = P cdm ( k ) T ( k ) ; T ( k ) = T wdm ( k ; κ ) T cdm ( k ) (II.34)where P cdm ( k ) = Ak n s T cdm ( k ) (II.35)is the (CDM) power spectrum and the dependence on WDM is encoded in the κ dependenceof T wdm ( k ; κ ) so that T wdm ( k ; κ = 0) = T cdm ( k ). The dependence on κ describes the velocitydispersion and non-vanishing free streaming length of the WDM particle.In ref.[44] it is shown that eqn. (II.27) can be solved in a systematic Fredholm expansion,from which the transfer function of density perturbations at z = 0 is extracted. The leadingorder term is a Born-type approximation which provides a remarkably accurate approxima-tion to the transfer function and reproduces numerical results available in the literature inseveral cases (for discussion and comparison see[44]). The definition of the power spectrumand transfer function above are at z = 0. We seek to study the redshift dependence for z . − 40 at which N-body simulations set up initial conditions.Asymptotically during the matter dominated era as ˜ a → ∞ ( u → 0) it is found[44]that δ ( k, u ) → ˜ a ( u ) δ ( k, 0) + · · · where the dots stand for subleading terms. The leadingand subleading asymptotic behavior in the u → a → ∞ ) limit can be obtained fromeqn. (II.27). In this limit the inhomogeneity J [ k, 0] is a finite constant (see expressions in8ef.[44]), the integral term receives the largest contribution for u ′ ∼ u ∼ − α e Π (cid:2) α ( u − u ′ ) (cid:3) ≃ κ ( u − u ′ ) (cid:16) − β (cid:17) + · · · . (II.36)where β = y (cid:0) y (cid:1) (II.37)is the kurtosis of the distribution function of the decoupled particle, with the momentsdefined by eqn. (II.20), and the dots stand for terms that yield subleading corrections (seebelow).Since ˜ a ( u ) = 1sinh [ u ] ∼ u − 13 + O ( u ) , (II.38)we propose the asymptotic expansion δ ( k, u ) = δ ( k, u + δ ( k ) + δ ( k ) u ln[ − u ] + · · · (II.39)Introducing this expansion in eqn. (II.27) we find δ ( k ) = δ ( k, (cid:2) κ + 2 (cid:3) ; δ ( k ) = − δ ( k, κ (cid:16) − β (cid:17) , (II.40)where δ ( ~k, 0) is obtained from the asymptotic solution of the full eqn. (II.27). Therefore δ ( ~k, u ) = δ ( ~k, D [ k, u ] (II.41)where the wavevector dependent growth factor is found to be D [ k, u ] = D cdm [ u ] D [ k, u ] (II.42)with D cdm [ u ] being the CDM growth factor (for κ = 0) D cdm = 1 u " u · · · (II.43)and D [ k, u ] = " κ u ) κ u ) − ln[ − u ]) (cid:16) − β (cid:17) + · · · . (II.44)contains the WDM corrections as is manifest in the κ dependence.For u → D cdm [ u ] = 1 u + 13 ≃ ˜ a + 23which is recognized as the growing solution of Meszaros equation for CDM[53–55]. Further-more from (II.16) we recognize that − κ u = k l fs (cid:2) p h p i m , η , η (cid:3) (II.45)9here l fs (cid:2) √ h p i m , η , η (cid:3) is the comoving free streaming distance that a particle with (comov-ing) velocity p h p i /m travels between conformal time η and today η ≫ 1. We see thatup to logarithms, the expansion in powers of κ u is valid at late times for wavelengths muchlarger than the free streaming distance that the particle would travel between that time andtoday.The identification (II.45) leads to a simple physical interpretation of the first term in D [ k, u ]: free streaming of collisionless particles suppresses the gravitational collapse of den-sity perturbations, the longer the time scale, the farther the free streaming particles cantravel away from the collapsing region erasing the perturbations. Therefore the first termreflects that at earlier times (larger values of u ) density perturbations are larger . The sec-ond term, however, has a more interesting interpretation. As it will be discussed below,it represents the peculiar velocity contribution to free streaming induced by gravitationalself-interaction (see discussion on peculiar velocity below). When β > increases the free streaming velocity leading to a suppression of power,which counterbalances the enhancement by the first term. Which term dominates dependson the scale k , the free streaming wavevector, a characteristic of the WDM particle, and theredshift. This will be analyzed in two specific models below.We emphasize that the expansion in (II.39) is valid at long time, in particular for κ u < − u ) only appear linearly in the logarithm but multiplied by higher powers of κ u , therefore for | κ u | < D is the leading logarithmic contribution, with higher contributions being of theform ( κ u ) n ln( − u ) ; n = 6 , · · · . This is an important observation: in particular within theregime of validity of the perturbative expansion | κu | < | κu ln( − u ) | ∼ . ≤ − u [ z ] ≤ . f or ≤ z ≤ , (II.46)for example in the region of redshifts where initial conditions for N-body simulations areset up, the WDM corrections to the growth factor are of O (10 − k & (1 − k fs which for a species with m ∼ keV decoupled with g d ∼ − 100 with y ∼ 10 correspondsto k & − 30 (Mpc) − .There is a caveat in this analysis of the reliability of the expansion, since it applies only inthe linear regime where the linearized Boltzmann equation describes the transfer function.It is conceivable that non-linear effects restrict further the regime of validity, but of coursethis cannot be assessed in the linear theory which is the focus of this discussion.Using Poisson’s equation (II.23), the asymptotic behavior δ ( ~k, u ) → δ ( ~k, 0) ˜ a ( u ) and thedefinition of the transfer function[47] T ( k ) φ ( k, ˜ a ≫ 1) = 910 φ i ( k ) T ( k ) (II.47)where φ i ( k ) is the primordial value of gravitational perturbations seeded by inflation. Itthen follows that δ ( ~k, 0) = − φ i ( ~k ) 6 k k eq T ( k ) . (II.48)10e emphasize that there are two different averages: i) the statistical average of a quantity O with the perturbed distribution function f + F to which we refer as hOi , ii) average overthe initial gravitational potential φ i which is a stochastic Gaussian field (we neglect possiblenon-Gaussianity) whose power spectrum is determined during the inflationary era φ i ( ~k ) φ i ( − ~k ′ ) = (2 π ) δ (3) ( ~k − ~k ′ ) P φ ( k ) (II.49)where the A B refers to averages with the primordial Gaussian distribution function for thegravitational potential . Therefore full expectation values correspond to averages both withthe perturbed distribution function and the Gaussian distribution function for the primordialgravitational potential, these are given by hOi , with the power spectrum of matter densityfluctuations δ ( ~k, δ ( − ~k ′ , 0) = (2 π ) δ (3) ( ~k − ~k ′ ) P δ ( k ) , (II.50)where P δ ( k ) is given by eqn. (II.33).Including the wavevector dependent growth factor D [ k, u ] (II.44) but keeping only the(WDM) ( κ = 0) corrections with redshift, the effective (WDM) power spectrum at z ≪ z eq is given by P δ [ k, z ] = P wdm ( k ) D [ k, z ] (II.51)where the scale and redshift dependent correction is given by (see eqn. (II.44)) D [ k, z ] = 1 + κ " z z eq − κ " z z eq ln " z eq z β − (cid:17) + · · · (II.52)For β > k > k fs " z eq )9 (1 + z )( β − 1) ln h (1+ z eq )(1+ z ) i (II.53)for β − ∼ O (1) and z ≃ − 50 one finds that the third term dominates over the secondfor k ∼ (1 − k fs . These are the scales beyond which the contribution from the peculiarvelocities to free streaming leads to a suppression of the power spectrum. Coincidentally thisis the scale at which the power spectrum displays (WDM) acoustic oscillations which arisefrom the competition between free streaming and gravitational collapse in the collisionless regime as described in ref.[44]. This definition should not be confused with that of the moments in eqn. (II.20) which refer to averageswith the unperturbed distribution function. The meaning of averages is unambiguously inferred from thecontext. II. PECULIAR VELOCITY AND PHASE SPACE DENSITY: Statistical averages of observables with the perturbed distribution function (II.8) in thelinearized theory (in terms of their spatial Fourier transform) are given by e O ( ~k ; η ) ≡ hO ( ~p, ~k, η ) i = R d p (2 π ) h f ( p ) + F ( ~p, ~k ; η ) i O ( ~p, ~k ; η ) R d p (2 π ) h f ( p ) + F ( ~p, ~k ; η ) i = e O ( ~k ; η ) + ∆ e O ( ~k ; η ) h δ ( ~k, η ) i (III.1)where e O ( ~k ; η ) = Z d p (2 π ) e f ( p ) O ( ~p, ~k ; η ) , (III.2)∆ e O ( ~k ; η ) = Z d p (2 π ) e F ( ~p, ~k ; η ) O ( ~p, ~k ; η ) . (III.3)where e F , e f are defined in eqn. (II.12). In the linearized approximation e O ( ~k ; η ) ≃ e O ( ~k ; η ) + (cid:16) ∆ e O ( ~k ; η ) − e O ( ~k ; η ) δ ( ~k, η ) (cid:17) . (III.4)With e F ( ~k, ~p ; s ) given by (II.26) and δ ( k, η ) given by (II.39,II.40) we can now obtain anystatistical average by expanding O ( ~p, ~k ; η ) ≡ O ( p, k, µ ; η ) in Legendre polynomials in µ and carrying out the integrals in p, µ leading to an expansion in spherical Bessel functions.However, here we focus on obtaining the leading asymptotic expansion of these averages for z ≪ z eq , namely for u ≪ 1. This is readily achieved by using the asymptotic expansion(II.39) with the coefficients given by (II.40), expandingexp[ − iµQ ] ≃ − iµQ − µ Q + i µ Q + · · · and integrating over µ and p term by term in the expansion. A. Peculiar velocity Writing the comoving peculiar velocity in terms of the longitudinal and transverse com-ponents ~v ( ~k, η ) = h ~pm i ≡ ~v T + b k v L , ~k · ~v T = 0 (III.5)where v L = pm µ ; µ = b k · b p (III.6)and p is the comoving momentum. In the linearized approximation, the expectation valueof k v L is given by k v L ( ~k, η ) = Z d p (2 π ) e F ( ~p, ~k ; η ) ~k · ~pm . (III.7)12urthermore, e F ( ~p, ~k ; η ) is a function of k and ~k · ~p leading to ~v T = 0 in the linearizedapproximation. Since the gravitational potential is only a function of k , the first term onthe right hand side of (II.22) does not contribute and we find k v L ( ~k, η ) = i dds Z d p (2 π ) " e F ( ~p, ~k ; s ) + φ ( k, s ) (cid:16) p d e f ( p ) dp (cid:17) = i dds " δ ( ~k, s ) − φ ( k, s ) (III.8)Using d/ds = ad/dη equation (III.8) becomes dδdη − dφdη + i ka v L = 0 (III.9)which is recognized as the continuity equation in presence of the gravitational potential[47] for the comoving longitudinal velocity. For ˜ a ≫ k ≫ k eq the second term in thecontinuity eqn. (III.9) can be safely neglected, leading to v L ( ~k, u ) = i k eq a eq √ k dδ ( ~k, u ) du . (III.10)As a function of redshift we find v L ( ~k, z ) = i k eq a eq √ k δ ( k, − u [ z ]) V wdm [ k, z ] (III.11)where we used the asymptotic expansion (II.39,II.40) and introduced V wdm [ k, z ] = " κ h z z eq i ln h z eq z i (cid:16) β − (cid:17) + · · · . (III.12)In the CDM limit κ → u − ≃ ˜ a which is recognized as the growthof comoving peculiar velocity in a matter dominated cosmology, in this limit T ( k ) → V wdm [ k, u ] → 1. The function V wdm [ k, u ] encodes the corrections to the peculiar velocityat small scales. It is clear that as compared to the CDM case, when the kurtosis β > κ & larger at higher redshift. Comparing eqn. (III.12)with the third term in eqn. (II.52) confirms the interpretation of the suppression of thepower spectrum at small scales and high redshift as a consequence of the peculiar velocitycontribution to free streaming. B. Statistical fluctuations and correlation functions: In the linearized approximation (and with adiabatic perturbations only), the perturbationin the distribution function e F ( ~k, ~p ; u ) is linear in the primordial gravitational potential φ i ( k )which is a Gaussian variable determined by the power spectrum of perturbations during theinflationary stage (here we neglect possible non-gaussianities). Therefore as discussed in the Note that the Newtonian potential in eqn. (II.7) features a minus sign with respect to the definition in[47]. statistical average with the perturbeddistribution function f + F , ii) with the initial Gaussian probability distribution of P φ ( k )in eqn. (II.49). Statistical fluctuations are contained in the variance of the various quantities calculatedwith the perturbed distribution e F . These are linear in δ ( k, φ i , thereforethey feature Gaussian fluctuations with the probability distribution function P φ ( k ), but withnon-gaussian statistical variances.As an example of a statistical fluctuation consider∆ v L = 6 h p m i Z u du ′ ˜ a ( u ′ ) δ ( k, u ′ ) " ( u − u ′ ) − κ β ( u − u ′ ) · · · du ′ (III.13)Using the asymptotic expansions (II.38,II.39),II.40) we find up to leading logarithmic order Z u ˜ a ( u ′ ) δ ( k, u ′ )( u − u ′ ) du ′ = δ ( k, u " − κ u ln[ − u ] + · · · (III.14) Z u ˜ a ( u ′ ) δ ( k, u ′ )( u − u ′ ) du ′ = − δ ( k, 0) ln[ − u ] + · · · (III.15)leading to similar statistical fluctuations for the total velocity dispersion h p /m i and thetransverse component ~v T , namely (all quantities are comoving)∆ h v L i = h p i m δ ( k, u " − κ u ln[ − u ] (cid:16) − β (cid:17) + · · · (III.16)∆ h p m i = 53 h p m i δ ( k, u " − κ u ln[ − u ] (cid:16) − β (cid:17) + · · · (III.17)∆ h v T i = 23 h p i m δ ( k, u " − κ u ln[ − u ] (cid:16) − β (cid:17) + · · · (III.18)Restoring units, writing h p i = y T ,d and expressing these expressions in terms of redshift,we find the following statistical fluctuations ∆ h v L i ≃ . (cid:16) kmsec (cid:17) (cid:16) keV m (cid:17) y δ ( k, h z eq z i " − κ h z z eq i ln h z eq z i(cid:16) β − (cid:17) + · · · (III.19)∆ h p m i ≃ . (cid:16) kmsec (cid:17) (cid:16) keV m (cid:17) y δ ( k, h z eq z i " − κ h z z eq i ln h z eq z i(cid:16) β − (cid:17) + · · · (III.20)∆ h v T i ≃ . (cid:16) kmsec (cid:17) (cid:16) keV m (cid:17) y δ ( k, h z eq z i " − κ h z z eq i ln h z eq z i(cid:16) β − (cid:17) + · · · (III.21)14hese expressions also show that the WDM corrections (proportional to κ ) suppress thestatistical fluctuations at small scales k & k fs where the peculiar velocity contribution tofree streaming becomes important (we will see below that at least for the (WDM) candidatesconsidered here β > ξ ij ( ~x, ~x ′ ; u ) = Z d k (2 π ) e i~k · ~x Z d k ′ (2 π ) e − i~k ′ · ~x ′ v i ( ~k, u ) v ∗ j ( − ~k ′ , u ) (III.22)using (II.50) and (III.10) we find ξ ij ( ~r ; z ) = k eq a eq u Z d k (2 π ) e i~k · ~r b k i b k j P δ ( k ) k V wdm [ k, z ] ; ~r = ~x − ~x ′ . (III.23)Since there is only one vector ~r we write ξ ij ( ~r, z ) = P ⊥ ij ( b r ) ξ ⊥ ( r ; z ) + P k ij ( b r ) ξ k ( r ; z ) (III.24)where P ⊥ ij ( b r ) = δ ij − b r i b r j ; P k ij ( b r ) = b r i b r j . (III.25)are the projectors on directions parallel and perpendicular to r .We find ξ k ( r ; z ) = k eq a eq π ( u [ z ]) Z dk P δ ( k ) V wdm [ k, z ] h j ( kr ) − j ( kr ) i (III.26) ξ ⊥ ( r ; z ) = k eq a eq π ( u [ z ]) Z dk P δ ( k ) V wdm [ k, z ] h j ( kr ) + j ( kr ) i (III.27)where j , are spherical Bessel functions.Thus we see that the effective (WDM) power spectrum for peculiar velocities is P δ ( k ) V wdm [ k, z ].From expression (III.12) it is clear that for β > enhance thepeculiar velocity autocorrelation function for z ≃ − 50. This enhancement of the velocitycorrelation function is in concordance with the suppression of the power spectrum, since thelarger velocity dispersion induced by self-gravity leads to a larger free streaming velocityand a further suppression of the power spectrum. C. Phase space density: In their seminal article Tremaine and Gunn [37] argued that the coarse grained phasespace density is always smaller than or equal to the maximum of the (fine grained) micro-scopic phase space density, namely, the distribution function, allowing to establish boundson the mass of the DM particle.Such argument relies on a theorem [38, 39] that states that collisionless phase mixingor violent relaxation by gravitational dynamics (mergers or accretion) can only diminishthe coarse grained phase space density. A similar argument was presented in refs. [10, 33,40–42] where a proxy for a coarse-grained phase space density in absence of gravitationalperturbations was introduced. 15owever, whereas the distribution function obeys the collisionless Boltzmann equation,Dehnen[39] clarifies that the coarse grained phase space density does not necessarily evolvewith the collisionless Boltzmann equation, and introduces an excess mass function which isargued to always diminish upon gravitational phase space mixing.Numerical simulations confirm the evolution of a coarse grained phase space densitytowards smaller values during violent relaxation events such as encounters, mergers andaccretion of haloes[56, 57]. In the simulations in ref.[56] a phase space density Q is obtainedby averaging ρ, σ over a determined volume, and its evolution with redshift is followed from z = 10 until z = 0 diminishing by a factor ≃ 40 during this interval.However, to the best of our knowledge, a consistent study of the evolution of the micro-scopic phase space density including gravitational effects even in the linearized approxima-tion has not yet been provided.In linearized theory, the corrections to the distribution function F , or rather the nor-malized perturbation e F ( ~p, ~k, η ) defined by eqn. (II.12) obeys the collisionless Boltzmannequation (II.15), whose solution in the regime when the DM is non-relativistic is given byeqn. (II.22). Thus the time evolution of the microscopic phase space density is completelydetermined. Two aspects of this solution invite further scrutiny: i) density perturbations grow from self-gravity effects, ii) peculiar velocities also grow , a direct consequence of thecontinuity equation (III.9) and explicitly shown by eqn. (III.11). That both quantities grow upon gravitational collapse suggests an examination of the phase space evolution in thelinearized regime.In principle one could perform the Fourier transform back to (comoving) spatial co-ordinates and obtain e F ( ~p, ~r ; η ), however, δ ( ~k, η ) is a stochastic variable with a Gaussianprobability distribution determined by the power spectrum of the primordial gravitationalpotential. Therefore, the linear correction to the microscopic phase space density itselfbecomes a stochastic variable as discussed above.Rather than pursuing the Fourier transform, which in the linearized approximation canbe performed at any state in the calculation, we follow refs.[10, 33, 40–43] and define thecoarse grained (dimensionless) primordial phase space density D ≡ n ( t ) (cid:10) ~P f (cid:11) , (III.28)where ~P f = ~p/a ( t ) is the physical momentum . In absence of gravitational perturbations, the(unperturbed) distribution function of the decoupled species is frozen and n ( t ) = n /a ( t ),therefore it is clear that D is a constant , namely a Liouville invariant. In absence of self-gravity it is given by D = g π " R ∞ y f ( y ) dy " R ∞ y f ( y ) dy , (III.29)where f ( y ) is the decoupled distribution function, and g the number of internal degrees offreedom of the WDM particle. 16hen the particle becomes non-relativistic ρ ( t ) = m n ( t ) and (cid:10) ~V (cid:11) = (cid:10) ~P f m (cid:11) , therefore, D = ρm (cid:10) ~V (cid:11) = Q DH m (III.30)where Q DH = ρ/ (cid:10) ~V (cid:11) is the phase-space density introduced in refs. [10, 40].In the non-relativistic regime D is related to the coarse grained phase space density Q T G introduced by Tremaine and Gunn [37] Q T G = ρm (2 π σ ) = (cid:18) π (cid:19) D . (III.31)where σ is the one-dimensional velocity dispersion. The observationally accessible quantityis the phase space density ρ/σ , therefore, using ρ = mn for a decoupled particle that isnon-relativistic today and eq.(III.30), we define the primordial phase space density ρ DM σ DM = 3 m D ≡ . × D h m keV i M ⊙ / kpc (cid:0) km / s (cid:1) . (III.32)In refs.[33, 41–43] the phase mixing theorem was invoked to argue that the observed phasespace density is smaller than the primordial value (III.32) with D replaced by D given byeqn. (III.29), leading to lower bound on the mass of the (WDM) particle. However, asemphasized in ref.[39] the phase mixing theorem[38] does not directly address the evolutionof D , nor has there yet been an analysis of its evolution in the linearized regime.The results obtained above allow us to directly calculate the corrections to D from self-gravity in the linearized theory. Using the identity (III.3) for linearized statistical averages,it is given by D = n h δ ( k, u ) i " h p i + ∆ h p i ≃ D " δ ( ~k, κ ln h z eq z i (cid:16) β − (cid:17) . (III.33)were we have used eqns.(II.39,II.40) and (III.17). In the (CDM) limit κ → increases with thelogarithm of the scale factor when β > . 19. We will see below that this the case at leastfor two examples of (WDM) candidates supported by particle physics models. The reasonfor the increase in the coarse grained phase space density can be tracked to the suppressionof statistical fluctuations: the leading term in δ ( ~k, u ) = δ ( ~k, /u + · · · cancels againstthe leading term proportional to δ ( ~k, /u in the statistical fluctuation (III.17), these arethe only contributions that remain in the (CDM) limit, however the (WDM) contribution keV (cid:0) km / s (cid:1) = 1 . M ⊙ kpc . increase of thecoarse grained phase space density as a consequence of the suppression of the statisticalfluctuations (statistical variance) of the velocity dispersion in the (WDM) case. IV. TWO SPECIFIC EXAMPLES. We now focus on two specific examples of WDM candidates: sterile neutrinos producedvia the Dodelson Widrow (DW) mechanism[58] for which f dw ( y ) = ξe y + 1 (IV.1)where the constant ξ is a function of the active-sterile mixing angle[58], and sterile neutri-nos produced near the electroweak scale via the decay of scalar or vector bosons (BD) forwhich[43, 59] f bd ( y ) = λ √ y ∞ X n =1 e − ny n (IV.2)where λ ∼ − .We implement the Born approximation to the matter power spectrum presented in ref.[44]to obtain the corrected power spectrum normalized to (CDM) [ T ( k ) D [ k, z ]] . As discussedin ref.[44] the Born approximation yields excellent agreement with the power spectrumobtained in ref.[29] for (DW) sterile neutrinos. (DW) sterile neutrinos: For the distribution function (IV.1) we find: y = 12 . 939 ; β = 2 . 367 ; k fs = 5 . (cid:16) m keV (cid:17) (cid:16) g d . (cid:17) (cid:16) Mpc (cid:17) − (IV.3)The (DW) case is displayed in fig.(2): the fig. for [ D [ k, z ]] for the “standard” value g d = 10 . m = 1 keV (the value used in the figure) k fs = 5 . 44 (Mpc) − , and the figureclearly shows that the crossover from enhancement to suppression occurs at k ≈ − k fs for 30 ≤ z ≤ 50 . The corrections from D [ k, z ] are not resolved in the log-log scale, howevera linear-linear display of the region k & k fs reveals the 10 − 15 % suppression of the powerspectrum. This range of small scales is where the power spectrum develops the oscillatorybehavior associated with the (WDM) acoustic oscillations discussed in ref.[44]. (BD) sterile neutrinos: Sterile neutrinos produced by the decay of scalar or vector bosons at the electroweakscale[43, 59] are colder for two reasons, i) their decoupling occurs when g d ∼ 100 and theydo not reheat when the entropy from other degrees of freedom is given off to the thermalplasma, ii) their distribution function (IV.2) is more enhanced at small momentum therebyyielding smaller velocity dispersion. For this species y = 8 . 509 ; β = 2 . 890 ; k fs = 14 . (cid:16) m keV (cid:17) (cid:16) g d (cid:17) (cid:16) Mpc (cid:17) − (IV.4)18 >?@ABCDE F G H IJ KLMNOPQRSTU VWXYZ[\]^_‘abcdefghi jklmnopqrstuvwx yz{|}~(cid:127)(cid:128) (cid:129) (cid:130)(cid:131)(cid:132) (cid:133)(cid:134)(cid:135) (cid:136) (cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147) (cid:148)(cid:149)(cid:150) (cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:159)(cid:160)¡¢£⁄¥ƒ §¤'“«‹›fifl(cid:176)–†‡·(cid:181)¶•‚„”»…‰(cid:190)¿(cid:192)`´ˆ˜¯˘˙¨(cid:201) ˚¸(cid:204) ˝˛ˇ —(cid:209)(cid:210)(cid:211) (cid:212)(cid:213)(cid:214)(cid:215)(cid:216)(cid:217)(cid:218)(cid:219)(cid:220) (cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226)ª(cid:228)(cid:229) (cid:230)(cid:231) ŁØ Œº (cid:236)(cid:237) (cid:238)(cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß(cid:252) (cid:253)(cid:254)(cid:255)(cid:2)(cid:0)(cid:1)(cid:6)(cid:3)(cid:4)(cid:5)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !" FIG. 2: DW, m = 1 keV , g d = 10 . 75: upper left panel D [ k, z ] and [ T ( k ) D ( k, z )] , lower panel:small scale region of [ T ( k ) D ( k, z )] and V wdm [ k, z ] all for z = 30 , , This case is displayed in fig.(3): the fig. for [ D [ k, z ]] for g d = 100 (corresponding tofreeze-out at the electroweak scale) also shows the crossover from an early enhancement as aconsequence of free streaming to a later suppression of the power spectrum as a consequenceof the extra contribution to free streaming from peculiar velocity at small scales. For m =1 keV (the value used in the figure) k fs = 14 . 107 (Mpc) − , and the figure clearly showsthat, again, the crossover from enhancement to suppression occurs at k ≈ − k fs for30 ≤ z ≤ 50 . The corrections from D [ k, z ] are not resolved in the log-log scale of thepower spectrum, however a linear-linear display of the region k & k fs reveals the 10 − 15 %suppression of the power spectrum. In this region the figure displays a hint of the (WDM)acoustic oscillations discussed in ref.[44]. As discussed in ref.[44] the smaller amplitude ofthe (WDM) acoustic oscillations as compared to the (DW) case are a reflection of the factthat (BD) sterile neutrinos are colder as explained above.In both these cases, we see that there is a suppression of the power spectrum for z ∼ − k ≃ (1 − k fs and an enhancement of the peculiar velocity in thesame region, both effects are at the 10 − 15% level and clearly correlated: the larger peculiarvelocity adds to free streaming depressing the power spectrum. Although these effects areat the level of few percent, it is conceivable that they may be magnified by the inherent non-19 (cid:140)(cid:141)(cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147) (cid:148)(cid:149) (cid:150)(cid:151) (cid:152)(cid:153) (cid:154)(cid:155) (cid:156)(cid:157)(cid:158)(cid:159)(cid:160)¡¢£⁄¥ƒ §¤'“«‹›fifl(cid:176)–†‡·(cid:181)¶•‚„”»…‰(cid:190)¿(cid:192)`´ ˆ˜¯ ˘˙¨(cid:201)˚¸(cid:204) ˝ ˛ ˇ —(cid:209)(cid:210) (cid:211)(cid:212)(cid:213)(cid:214)(cid:215)(cid:216)(cid:217)(cid:218)(cid:219)(cid:220)(cid:221)(cid:222) (cid:223) (cid:224)Æ(cid:226)ª(cid:228)(cid:229)(cid:230)(cid:231)ŁØŒ º(cid:236)(cid:237) (cid:238)(cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß(cid:252)(cid:253) (cid:254)(cid:255)(cid:1)(cid:0)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) (cid:17)(cid:18)(cid:19) (cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25) (cid:26) (cid:27) (cid:28) (cid:29) (cid:30)(cid:31) !" FIG. 3: BD, m = 1 keV , g d = 100: upper left panel D [ k, z ] and [ T ( k ) D ( k, z )] , lower panel: smallscale region of [ T ( k ) D ( k, z )] and V wdm [ k, z ] all for z = 30 , , linearities in the process of gravitational collapse, perhaps leading to important consequencesfor galaxy formation in N-body simulations. V. CONCLUSIONS Motivated by recent and forthcoming N-body simulations of galaxy formation in (WDM)scenarios, we set out to study the redshift corrections to the matter and peculiar velocitypower spectra and corrections to the phase space density from gravitational perturbationsin the region 30 ≤ z ≤ 50. This is the region in redshift where N-body simulations set upinitial conditions and the dark energy component can be safely neglected.Drawing from results in ref.[44], we implemented a perturbative expansion for the redshiftand scale dependence of the distribution function, matter density perturbations and coarsegrained phase space density valid for z/z eq ≪ β , the kurtosis of theunperturbed distribution function after freeze-out, with an enhancement of of the peculiarvelocity power spectrum and autocorrelation function at larger redshift for β > 1. This en-20ancement in the peculiar velocity hastens free streaming and leads to a further suppressionof the matter power spectrum for k > (1 − k fs , where k fs is the free streaming wavevector.For (WDM) gravitational perturbations lead to a suppression of the statistical fluctuationsof velocities when β > / β > . − 15 % forscales k ≃ (1 − k fs and redshifts z ≃ − Impact on the bounds on the mass: The scale and redshift dependence ofthe power spectra are encoded in the effective matter and velocity power spectra P δ ( k ) D [ k, z ] ; P δ ( k ) V [ k, z ] with D [ k, z ] ; V [ k, z ] given by eqns. (II.52,III.12) respectively.To assess the impact of the above results on the bounds on the mass of the (WDM)particle consider two N-body simulations with a particle of the same mass both setting upinitial conditions at the same z ≃ − 50, one with the matter and peculiar velocity powerspectra at z ∼ β > suppression and the peculiar velocity power spectrum features the (WDM) enhancement for k & k fs found above. These effects at small scales are akin to the suppression of densityfluctuations and enhancement of velocity dispersion associated with a lighter particle forthe un-corrected power spectra. This is because a lighter particle features a smaller k fs and a larger velocity dispersion. Therefore these corrections allow larger masses to describethe same large scale structure output from the N-body simulations as compared to theun-corrected power spectra. Thus one aspect of the corrections is to allow larger mass(WDM) particles, thereby relaxing the bound on the mass, at least for those models forwhich β > 1. However, this is not all there is to the corrections, because the coarse-grainedphase space density increases , which would correspond to a colder particle with smallervelocity dispersion. Thus the net effect of the corrections cannot be simply characterized asbeing described by an increase or decrease of the mass of the particle and ultimately mustbe understood via a full N-body simulation.Although these corrections are relatively small, non-linearities arising from gravitationalcollapse may result in a substantial amplification of these effects, if this is the case, and onlylarge scale N-body simulations with the corrected power spectra can assess this possibility,then it is conceivable (and expected) that the bounds on the mass of the (WDM) particlemay need substantial revision.The results obtained here suggest a breakdown of perturbation theory either at largeredshift and or small scales k ≫ k fs , this is clearly an artifact of the expansion, the integralin (II.27) which yields the logarithmic contribution is bounded and well behaved both inthe small scale and u → u NR limits[44]. 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