aa r X i v : . [ a s t r o - ph ] D ec Free streaming in mixed dark matter
Daniel Boyanovsky ∗ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA (Dated: October 30, 2018)Free streaming in a mixture of collisionless non-relativistic dark matter (DM) particles is studiedby solving the linearized Vlasov equation implementing methods from the theory of multicomponentplasmas. The mixture includes Fermionic, condensed and non-condensed Bosonic particles decou-pling in equilibrium while relativistic, heavy thermal relics that decoupled when non-relativistic(WIMPs), and sterile neutrinos that decouple out of equilibrium when they are relativistic. The dif-ferent components interact via the self-consistent gravitational potential that they source. The free-streaming length λ fs is obtained from the marginal zero of the gravitational polarization function,which separates short wavelength Landau-damped from long wavelength Jeans-unstable collective modes. At redshift z we find λ fs ( z ) = z ) ˆ . ˜ P a ν a g d,a ( m a / keV) I a , where 0 ≤ ν a ≤ fractions of the respective DM components of mass m a that decouple when the effective numberof ultrarelativistic degrees of freedom is g d,a , and I a are dimensionless ratios of integrals of the dis-tribution functions which only depend on the microphysics at decoupling and are obtained explicitlyin all the cases considered. If sterile neutrinos produced either resonantly or non-resonantly thatdecouple near the QCD scale are the only DM component, we find λ fs (0) ≃ /m ) (non-resonant), λ fs (0) ≃ .
73 kpc (keV /m ) (resonant). If WIMPs with m wimp &
100 GeV decoupling at T d &
10 MeV are present in the mixture with ν wimp ≫ − , λ fs (0) . . × − pc is dominated by CDM. If a Bose Einstein condensate is a DM component its free streaming length is consistentwith CDM because of the infrared enhancement of the distribution function. I. INTRODUCTION
Candidate dark matter (DM) particles are broadly characterized as cold, hot or warm depending on their veloc-ity dispersions. The concordance
ΛCDM standard cosmological model emerging from CMB, large scale structureobservations and simulations favors the hypothesis that DM is composed of primordial particles which are cold andcollisionless[1]. Although this model is very successful in describing the large scale distribution of galaxies, recentobservations hint at possible discrepancies summarized as the “satellite” and “cuspy halo” problems. In the ΛCDMmodel the CDM power spectrum favors small scales, which become non-linear first and collapse in a hierarchical“bottom-up” manner and dense clumps survive the mergers in the form of “satellite” galaxies. Large-scale simula-tions within the ΛCDM paradigm lead to an overprediction of “satellite” galaxies [2], which is almost an order ofmagnitude larger than the number of satellites that have been observed in Milky-Way sized galaxies[2, 3, 4, 5, 6].Furthermore, large-scale N-body simulations of CDM clustering predict a density profile monotonically increasingtowards the center of the halos[2, 7, 8, 9, 10], with asymptotic behavior ρ ( r ) ∼ r − γ where 1 ≤ γ . . simple estimate of thefree-streaming length λ fs is obtained from the familiar Jeans’ length by replacing the speed of sound by the velocitydispersion of the particle. An equivalent estimate is obtained by computing the distance that the particle travelswithin a dynamical (Hubble) time[18] λ fs ∼ h ~V i /H .However, a thorough assessment of DM particles and structure formation requires a more detailed and reliabledetermination of the free-streaming length. The necessity for this has been recently highlighted by the recent resultsof ref.[19] that suggest that the first stars form in filaments of the order of the free streaming scale. ∗ Electronic address: [email protected]
Perturbations in a collisionless system of particles with gravitational interactions is fundamentally different fromfluid perturbations in the presence of gravity. The (perfect) fluid equations correspond to the limit of vanishing meanfree path. In a gravitating fluid pressure gradients tend to restore hydrostatic equilibrium with the speed of soundin the medium and short wavelength fluctuations are simple acoustic waves. For large wavelengths the propagationof pressure waves cannot halt gravitational collapse on a dynamical time scale. The dividing line is the Jeans length:perturbations with wavelengths shorter than this oscillate as sound waves, while perturbations with longer wavelengthundergo gravitational collapse.In a gas of collisionless particles with gravitational interaction the situation is different since the mean free path ismuch larger than the size of the system (Hubble radius) and the fluid description is not valid. Instead the Boltzmann-Vlasov equation for the distribution function must be solved to extract the dynamics of perturbations[20, 21]. Justas in the case of plasma physics, the linearized Boltzmann-Vlasov equation describes collective excitations[22]. In thecase of a collisionless gas with gravitational interactions these collective excitations describe particles free-streaming inand out of the gravitational potential wells of which they are the source. The damping of short wavelength collective excitations is akin to
Landau damping in plasmas [22] and is a result of the phenomenon of dephasing via phasemixing in which the particles are out of phase with the potential wells that they produce[23]. This situation is similarto Landau damping in plasmas where dephasing between the charged particles and the self-consistent electric fieldthat they produce lead to the collisionless damping of the collective modes[22]. For a thorough review of collectiveexcitations and their Landau damping in gravitational systems see ref.[24].Gilbert[25] studied the linearized Boltzmann-Vlasov equation in a matter dominated cosmology for non-relativisticparticles described by an (unperturbed) Maxwell-Boltzmann distribution function. In this reference the linearizedthis equation was cast as a Volterra integral equation of the second kind which was solved numerically. The result ofthe integration reveals a limiting value of the wavevector below which perturbations are Landau damped and abovewhich perturbations grow via gravitational instability. Eventually the redshift in the expanding cosmology makeswavevectors that are initially damped to enter the band of unstable modes and grow[25]. However, the dividingwavevector between damped and growing modes emerges at very early times during which the expansion can beneglected[25]. The results of the numerical study were consistent with replacing the speed of sound by the Maxwellianvelocity dispersion in the Jeans length (up to a normalization factor of O (1)). Gilbert’s equations were used alsoby Bond and Szalay[26] in their pioneering study of collisionless damping of density fluctuations in an expandingcosmology. These authors focused primarily on massive neutrinos and solved numerically Gilbert’s integral equationbut approximated the Fermi-Dirac distribution function and provided a fitting function for the numerical results.Gilbert’s equations were also solved numerically to study dissipationless clustering of neutrinos in ref.[27] but witha truncation of their Fermi-Dirac distribution function and an analytic fit to the numerical solution of the integralequation was provided. Bertschinger and Watts[21, 28] also studied numerically Gilbert’s equation within the contextof cosmological perturbations from cosmic strings and massive light neutrinos, and more recently similar integralequations were solved approximately for thermal neutrino relics in ref.[29].Most of the studies of the solutions of the linearized Boltzmann-Vlasov equation for collisionless particles addressedone single species of DM candidates and generally in terms equilibrium distribution functions or approximationsthereof, for example Maxwell-Boltzmann for relics that decouple non-relativistically or Fermi-Dirac (without chemicalpotential) (or truncations of this distribution) for neutrinos.However, it is likely that DM may be composed of several species, this possibility is suggested in ref.[30] and mostextensions of the standard model generally allow several possible candidates, from massive weakly interacting particles(WIMPs) to “sterile” neutrinos ( SU (2) singlets)[31].Furthermore, several possible WDM candidates may decouple out of local thermodynamic equilibrium (LTE) withdistribution functions which may be very different from the usual ones in LTE. This is the case for sterile neutrinosproduced non-resonantly via the Dodelson-Widrow (DW) mechanism[32] or through a lepton-driven MSW (Mikheyev-Smirnov-Wolfenstein) resonance[33]. A. Motivation and goals:
In this article we study free streaming of decoupled collisionless non-relativistic (DM)candidates focusing on two aspects: • A mixture of (DM) particles including CDM and WDM candidates: typical studies of structure formation invokeeither CDM or WDM, but it is likely that both candidates are present with different fractions ν of the totalDM component. In fact most particle physics extensions beyond the standard model have plenty of room for avariety of CDM, WDM or HDM candidates. Thus we allow all of these components, each one contributing an The exception being ref.[28] wherein massive neutrinos decoupled with a Fermi-Dirac distribution and a cosmic string source ∝ δ ( ~r )were studied, neglecting any radiation component. arbitrary fraction ν to the total (DM) content of the Universe. Although the DM candidates are collisionless anddo not interact directly with each other, they interact indirectly via the gravitational potential that they source.As a result the free-streaming length of the mixture is a non-trivial function of the individual free-streaminglengths. • Free streaming has mostly been studied in the above references with particles that decoupled either whenultrarelativistic (as is the case for neutrinos) or non-relativic as in the case of weakly interacting massive particles(WIMPs) but generally in local thermodynamic equilibrium (LTE), namely with Fermi-Dirac, Bose-Einstein orMaxwell-Boltzmann distributions respectively. We seek to obtain the corresponding free streaming lengths forparticles that decoupled in or out of LTE with arbitrary isotropic distribution functions, without any truncation.This aspect is important for sterile neutrinos either produced non-resonantly[32] or resonantly[31, 33] becausethese particles decoupled while relativistic but out of LTE. Therefore, we consider the most general mixture ofFermionic and Bosonic thermal relics that decouple when relativistic, including the possibility of a Bose-Einsteincondensate (BEC)[34], heavy non-relativistic thermal relics, as the case of WIMPs, and sterile neutrinos thatdecoupled out of LTE when ultrarelativistic.In order to carry out this program analytically we first neglect the cosmological expansion and solve the linearizedBoltzmann-Vlasov equation in the non-expanding case exactly by implementing methods from the theory of multi-component plasmas[22]. The neglect of the cosmological expansion is warranted by the detailed numerical study inrefs.[25, 26] wherein it was found that the dividing wavevector between the Landau damped modes and the modesthat grow under gravitational instability is insensitive to the expansion, just as in the case of the Jeans instabilitywhere the Jeans wavevector can be extracted in the non-expanding case include the redshift dependence of the density,speed of sound and scale factors a posteriori [18]. The analytic exact solution for the free-streaming wave-vector today k fs (0) = 2 π/λ fs (0) in terms of the full distribution functions without truncation, in or out of LTE is a main resultof this program, one which yields a reliable determination of free-streaming lengths for mixtures of (DM) componentsthat decoupled with arbitrary distribution functions.This program is carried out by implementing methods from the theory of multicomponent plasmas[22], in particularwe obtain the “gravitational polarization” function[23, 24] for a mixture of (DM) components akin to the dielectricresponse function of multicomponent plasmas[22]. The collective excitations are described by the zeroes of this functionin the complex frequency plane and the free-streaming wave-vector k fs is identified as that wavevector that separatesbetween the Landau-damped short wavelength modes and the gravitationally Jeans- unstable long-wavelength modes.Based on the Liouville evolution of the decoupled distribution functions and assuming that the expansion of theuniverse is slow enough so that it can be treated adiabatically, we provide a scaling argument that determines thefollowing dependence of the free streaming length on the redshift, λ fs ( z ) = λ fs (0) √ z . (1.1) B. Summary of Results:
Our main result for the comoving free streaming length λ fs ( z ) at redshift z of mixed(DM) is1 λ fs ( z ) = 1(1 + z ) h . i X species ( ν F g d,F (cid:16) m F keV (cid:17) I F [ u ] + ν s g d,s (cid:16) m s keV (cid:17) .
814 + ν B g d,B (cid:16) m B keV (cid:17) I B [ x d , u d ] +10 ν wimp g d,wimp (cid:16) m wimp
100 GeV (cid:17) (cid:16) T d
10 MeV (cid:17)) , (1.2)where ν a is the fraction of (DM) of each component with P a ν a = 1, g d,a is the effective number of ultrarelativisticdegrees of freedom at decoupling for each species ( a ) of mass m a , and the functions I F , I B are dimensionless ratiosof integrals of the distribution functions of the decoupled particles which are determined by the microphysics atdecoupling. Their explicit expressions in the cases considered are given in section (III). The label F refer to all possible Fermions with chemical potential µ decoupled in LTE at a temperature T d while ultrarelativistic, and sterileneutrinos produced non-resonantly via the (DW) mechanism[32] for which the chemical potential vanishes and I F [0] =2 ln(2) / ζ (3) = 0 . s refers solely to sterile neutrinos produced via a lepton-driven (MSW) resonancevia the mechanism described in ref.[33]. The label B corresponds to condensed or non-condensed Bosons of mass m and chemical potential µ that decoupled at temperature T d while ultrarelativistic. The function I B features aninfrared divergence in the limits µ/T d ; m/T d → µ = m for any value of the mass. This latter case correspondsto the case of a Bose-Einstein Condensate[34]. Thus Bosonic particles that decoupled while ultrarelativistic with µ/T d ≪ or that formed a BEC lead to small free-streaming lengths and behave as CDM. Finally, (WIMPs) areconsidered to be decoupled while non-relativistic with a Maxwell-Boltzmann distribution function.Eq.(1.2) clearly shows that (WIMPs) with m ∼
100 GeV that decoupled kinetically in LTE at T ∼
10 MeV[35]dominate all other contributions to k fs resulting in an extremely small free streaming length λ fs ∼ . × − pc unless their fractional abundance ν wimp . − .If (DM) is dominated by sterile neutrinos (produced either resonantly or non-resonantly) that decoupled nearthe QCD scale[32, 33] with m ∼ keV, we find that the typical free-streaming lengths today are λ fs ∼ − r c ∼ . r c ∼
10 kpc).
II. FREE STREAMING LENGTH FOR MULTICOMPONENT DARK MATTER
We study DM particles that decoupled in or out of equilibrium while relativistic or non-relativistic, but that are non-relativistic today . As argued above, Gilbert’s detailed numerical study[25] confirmed by Bond and Szalay[26] showsthat the value of k fs can be extracted from the marginal case between modes that grow under gravitational instabilityand those that are Landau damped. For this marginal case linear perturbations are stationary, and just as in the caseof the Jeans length this marginal value can be reliably extracted in a non-expanding cosmology, including a posteriori the scale factor dependence of the various quantities in the Jeans wavelength which separates the gravitationallystable and unstable modes[18]. In section (III A) we present arguments that determine the redshift dependence of thefree streaming length under a suitable approximation.Consider several species of DM candidates that are non-relativistic today with masses m a and distribution functions f a where the label a = 1 , · · · refers to the different components. Each component ( a ) obeys the collisionlessBoltzmann-Vlasov equation[20, 21, 23], ∂f a ( ~x, ~p ; t ) ∂t + ~pm a · ~ ∇ ~x f a ( ~x, ~p ; t ) − m a ~ ∇ ~x Φ( ~x ; t ) ~ ∇ ~p f a ( ~x, ~p ; t ) = 0 (2.1)where Φ is the total Newtonian potential which is the solution of the Poisson equation ∇ Φ( ~x ; t ) = 4 πG X a m a g a Z d p (2 π ) f a ( ~x, ~p ; t ) , (2.2)where g a is the number of internal degrees of freedom.We note that whereas each individual species obeys its own collisionless Boltzmann-Vlasov equation, the Newtoniangravitational potential is determined by all the components as indicated by the Poisson equation (2.2). Therefore,although the different DM components do not interact directly they interact indirectly via the self-consistent gravi-tational potential since all of the DM components act as source of this potential which enters in Boltzmann-Vlasovequation of each component.Linearizing the Boltzmann-Vlasov equation and writing f a ( ~x, ~p ; t ) = f (0) a ( p ) + f (1) a ( ~x, ~p ; t ) ; Φ( ~x ; t ) = Φ ( ~x ; t ) + Φ (1) ( ~x ; t ) (2.3)where f (0) a ( p ) are the distribution functions of the species that decoupled in or out of LTE , the only assumption isthat these are isotropic, namely only depend on p = | ~p | . In the non-expanding case the zeroth order equation requiresto invoke the usual “Jeans swindle” (see the textbooks[23]) whereas in the expanding case the zeroth order equationis solved in terms of the inhomogeneous gravitational potential which yields the expanding background (see[20]).It is convenient to perform a spatial Fourier transform of the perturbations in a volume Vf (1) a ( ~x, ~p ; t ) = 1 √ V X ~k F (1) a ( ~k, ~p ; t ) e i~k · ~x ; Φ (1) ( ~x ; t ) = 1 √ V X ~k e i~k · ~x ϕ (1) ( ~k ; t ) (2.4)in terms of which the Vlasov and Poisson equations for the perturbations become dF (1) a ( ~k, ~p ; t ) dt + i~k · ~pm a F (1) a ( ~k, ~p ; t ) − im a ~k · b ~p ϕ ( ~k ; t ) df (0) a ( p ) dp = 0 , (2.5) ϕ ( ~k ; t ) = − πGk X b m b g b Z d p (2 π ) F (1) b ( ~k, ~p ; t ) . (2.6)Inserting eqn. (2.6) into eqn. (2.5) makes explicit that all the DM components are actually interacting through theself-consistent gravitational perturbations which act as a dynamical self-consistent mean field, as discussed above.Decaying or growing perturbations must be treated as an initial value problem, and following the treatment ofLandau damping in plasmas[22] we introduce the Laplace transform of the perturbations e F (1) a ( ~k, ~p ; s ) = Z ∞ e − st F (1) a ( ~k, ~p ; t ) dt ; e ϕ (1) ( ~k ; s ) = Z ∞ e − st ϕ (1) ( ~k ; t ) dt . (2.7)The Laplace transform of the Bolztmann-Vlasov equation (2.5) leads to e F (1) a ( ~k, ~p ; s ) = e ϕ (1) ( ~k ; s ) im a ~k · b ~ps + i ~k · ~pm a df (0) a ( p ) dp + F (1) a ( ~k, ~p ; t = 0) s + i ~k · ~pm a . (2.8)Taking the Laplace transform of (2.6), multiplying (2.8) by ( − πGm a g a /k ), summing over a and integrating in p we are led to e ϕ (1) ( ~k ; s ) = i πGk ε ( k ; s ) X a m a g a Z d p (2 π ) F (1) a ( ~k, ~p ; t = 0) ~k · ~pm a − is (2.9)where the gravitational “polarization” function is given by ε ( k ; s ) = 1 + 4 πGk X a m a g a Z d p (2 π ) ~k · ˆ ~p ~k · ~pm a − is df (0) a ( p ) dp (2.10)The collective excitations of the collisionless self-gravitating system correspond to the poles of e ϕ (1) ( ~k ; s ) in the complexs-plane, these are the zeroes of ε ( k ; s ). The time dependence of the perturbation of the Newtonian gravitationalpotential is obtained by the inverse Laplace transform ϕ (1) ( ~k ; t ) = Z C ds πi e st e ϕ (1) ( ~k ; s ) (2.11)where C stands for the Bromwich contour parallel to the imaginary axis and to the right of all the singularities of e ϕ (1) ( ~k ; s ) in the complex s -plane.Using that f (0) a ( p ) is a function of p = | ~p | it is convenient to write ~k · ˆ ~p ~k · ~pm a − is = m a p is ~k · ~pm a − is (2.12)which allows to extract the s = 0 contribution. The resulting expression for ε ( k ; s ) becomes ε ( k ; s ) = ε ( k ; 0) + is πGk X a g a m a Z d p (2 π ) df (0) a ( p ) p dp~k · ~pm a − is (2.13)where ε ( k ; 0) = 1 + 4 πGk X a g a m a Z ∞ p df (0) a ( p ) dp dp π ≡ − k fs k . (2.14)In eqn. (2.14) we have integrated by parts and introduced the free streaming momentum k fs given by k fs = 4 πG X a ρ (0) a D ~V E a (2.15)where ρ (0) a = m a n (0) a ; n (0) a = g a Z d p (2 π ) f (0) a ( p ) (2.16)and D ~V E a = g a n (0) a Z d p (2 π ) m a p f (0) a ( p ) , (2.17)the free streaming length is obtained as λ fs = 2 πk fs . (2.18)The expression for k fs can be written as k fs = Ω DM h (Mpc) X a ν a D V ~V E a ; V = 122 . partial fractions are defined as ν a = Ω a Ω DM ; X a ν a = 1 . (2.20)The collective modes correspond to the zeroes of the “gravitational polarization” function ε ( k ; s ) eqn. (2.13) in thecomplex s − plane. These yield the time evolution for the gravitational perturbations ϕ (1) ( ~k ; t ) ∝ X p e s p ( k ) t (2.21)where s p ( k ) are the zeroes of ε ( k ; s ).It is illuminating to compare the expression (2.15) with that for the Jeans wavevector for a single fluid k J = 4 πGρ (0) c s , (2.22)where c s is the adiabatic speed of sound of the fluid. We see that for a single collisionless component we can obtainthe free streaming length from the Jeans length by the replacement c s ⇒ (cid:20)D ~V E(cid:21) − (2.23)which in general is different from replacing c s by the velocity dispersion q h ~V i . This difference becomes importantwhen the unperturbed distribution function favors small values of the momentum. This observation will becomecrucial when we study Bosonic particles and sterile neutrinos decoupled when relativistic but out of LTE . As it willbe seen in detail below, in these cases the distribution function favors the region of small momentum which leads todramatic consequences in the difference between 1 / h ~V i and h /~V i . A. Landau damping and Jeans instability
It is clear from (2.13) that there is a pole in the Laplace transform of the Newtonian perturbation for the marginalvalue s = 0 ; k = k fs . (2.24)This is akin to the marginal value k = k J in a fluid where k J is the Jeans wave vector, in a fluid for k > k J pressure gradients hinder gravitational collapse and the perturbations are simple acoustic oscillations, for k < k J pressure gradients cannot prevent the collapse and the self-gravitational fluid undergoes the Jeans instability towardsgravitational collapse. We can study the dynamics of collective excitations in region k ≈ k fs searching for zeroes in ε ( k ; s ) for s ≈
0. The second term in (2.13) can be evaluated by performing the angular integral and using that f (0) a only depends on p = | ~p | , namely Z − d (cos θ ) kp cos θm a − is = m a kp ln " kpm a − is − kpm a − is . (2.25)For s ≈ Re ( s ) >
0, which isprecisely recognized as the Landau prescription for the evaluation of the integrals[22]. We find ε ( k ; s ) = 1 − k fs k + is Gπk X a g a m a Z ∞ " − df (0) a ( p ) dp ln " kpm a + is kpm a − is − iπ ) dp (2.26)Because for small s the logarithm in (2.26) is linear in s , the last term in the bracket ( − iπ ) contributes to theleading order in k − k fs . To lowest order in s for k ≈ k fs , the condition ε ( k ; s ) = 0 yields s ( k ) = C (cid:2) k fs − k (cid:3) ; C = kG P a g a m a f (0) a (0) > . (2.27)From the time evolution of the gravitational perturbation eqn. (2.21) we find s ( k ) < k > k fs ⇒ Landau damping s ( k ) > k < k fs ⇒ Jeans instability . (2.28)The long wavelength limit k → ~k · ~pm − is ) − in the integrand in eqn. (2.13) in powers of ~k · ~p/ms . In the resulting expression only the odd powers survive the angular integration. Keeping up to ( ~k · ~p/ms ) the long wavelength limit of ε ( k ; s ) is found to be ε ( k ; s ) = 1 − πGs X a ρ (0) a (cid:20) − (cid:10) V (cid:11) a k s + · · · (cid:21) (2.29)and the zeroes of ε ( k ; s ) in the long-wavelength limit are found to be s ± ( k ) = ± h Ω J − V k i + · · · (2.30)where Ω J = 4 πG X a ρ (0) a ; V = X a ν a (cid:10) ~V (cid:11) a (2.31)and ν a are the partial fractions. The Jeans frequency Ω J is the same as that for single component fluids, howeverthe relationship between the free streaming wavevector k fs (2.15) and the Jeans frequency Ω J is different from thatof the Jeans wavevector and the Jeans frequency in a single component fluid. For a single collisionless component wefind Ω J = "D ~V E − k fs (2.32)whereas for a single fluid one finds[20] Ω J = c s k J . (2.33) B. An example: the Maxwell-Boltzmann distribution
In general the momentum integral in (2.10) cannot be found in closed form without approximating the distributionfunction. However the Maxwell-Boltzmann distribution provides an example for which a closed form expression for(2.10) can be found. This distribution function is relevant for the description of WIMPs which are heavy relics thatdecoupled in LTE while non-relativistic[18]. In this case f (0) ( ~p ) = N e − ~p mTd (2.34)where the normalization N is obtained from the solution of the kinetic equation for the distribution function, whichcan be found in section (5.2) in ref.[18]. The particle density is ρ = mg Z d p (2 π ) f (0) ( ~p ) = N mg h mT d π i . (2.35)The momentum integrals in eqn. (2.10) can be carried out straightforwardly because f (0) ( ~p ) is a function of ~p . Thisis achieved by splitting the vector ~p into components parallel and perpendicular to ~k . With d p = dp k d p ⊥ theintegrals along the parallel and perpendicular directions can be done straightforwardly. We obtain the result ε ( k ; s ) = 1 − k fs k + sk g N m e δ " − √ π Z δ e − t dt ; δ = sk h m T d i . (2.36)It is straightforward to analytically continue this function to Re( s ) < f (0) ( ~
0) = N we find for the marginal case s = 0 the solution k = k fs , where k fs = 4 πGρ D ~V E , (2.37)and ρ is given by eqn. (2.35). Furthermore it is a simple exercise to confirm that near the marginal case the pole inthe function ε ( k ; s ) is given by eqn. (2.27). For the Maxwell-Boltzmann distribution it follows that D ~V E = 3 Tm ; D ~V E = mT = 1 D ~V E . (2.38)The long-wavelength limit can be obtained by expanding the integral in (2.36) for δ ≫
1, namely " − √ π Z δ e − t dt = e − δ δ √ π h − δ + 34 δ + · · · i (2.39)which after tedious but straightforward algebra leads to the expressions for the poles given by eqn. (2.31) with ν = 1and Ω J = 4 πGρ where ρ is given by (2.35) and (cid:10) ~V (cid:11) given by (2.38). III. FREE STREAMING LENGTHS FOR DECOUPLED PARTICLES
The distribution function of decoupled particles in a homogeneous and isotropic cosmological background in absenceof gravitational perturbations is constant along geodesics and obey the Liouville or collisionless Boltzmann equationin terms of an affine parameter λ [18, 34, 36] ddλ f [ P f ; t ] = 0 ⇒ df [ P f ; t ] dt = 0 (3.1)where P f = p c /a ( t ) is the physical momentum, and p c the time independent comoving momentum. Taking P f as anindependent variable this equation leads to the familiar form[18, 34, 36] ∂f [ P f ; t ] ∂t − HP f ∂f [ P f ; t ] ∂P f = 0 , (3.2)where H = ˙ a/a is the Hubble parameter and P f a = p c = constant is a characteristic of the equation. Obviously asolution of this equation is f [ P f ; t ] ≡ f d [ a ( t ) P f ] = f d [ p c ] . (3.3)If a particle of mass m has been in LTE but it decoupled from the plasma with decoupling temperature T d itsdistribution function is f d ( p c ) = 1 e √ m p c − µdTd ± − ) respectively allowing for a chemical potential.Since the distribution function is dimensionless, without loss of generality we can always write for a decoupled particle[34] f d ( p c ) = f d (cid:18) p c T d ; mT d ; α i (cid:19) (3.5)where α i are a collection of dimensionless constants determined by the microphysics, for example dimensionlesscouplings or ratios between T d and particle physics scales or in equilibrium µ d /T d etc. To simplify notation in whatfollows we will not include explicitly the set of dimensionless constants m/T d ; α i , etc, in the argument of f d , but theseare implicit in generic distribution functions. If the particle decouples when it is still relativistic m/T d → y = p c T d = P f T d ( t ) ; T d ( t ) = T d a ( t ) ; x d = mT d . (3.6)We emphasize that the distribution functions (3.5) are general and not necessarily describing particles decoupledwhile in local thermal equilibrium. When the particle becomes non-relativistic, its contribution to the energy densityis ρ = m n ( t ) (3.7)where[34] n ( t ) = g T d ( t )2 π Z ∞ y f d ( y ) dy (3.8)and g is the number of internal degrees of freedom. From entropy conservation[18, 36], the decoupling temperatureat redshift z is related to the temperature of the CMB today by T d ( z ) = (1 + z ) (cid:18) g d (cid:19) T cmb = (cid:18) g d (cid:19) . × − (1 + z ) eV , (3.9)where g d is the number of effective ultrarelativistic degrees of freedom at decoupling. For a given species ( a ) ofparticles with g a internal degrees of freedom that decouples when the effective number of ultrarelativistic degrees offreedom is g d,a , the relic abundance today ( z = 0) is given by[34]Ω a h = (cid:16) m a .
67 eV (cid:17) g a R ∞ y f d,a ( y ) dy g d,a ζ (3) . (3.10)If this decoupled species contributes a fraction ν a to dark matter, with Ω a = ν a Ω DM and taking Ω DM h = 0 . ν a = (cid:16) m a .
227 eV (cid:17) g a g d,a Z ∞ y f d,a ( y ) dy . (3.11)The constraint on the fractional abundance 0 ≤ ν a ≤ V /c ) ≪
1. The momentum that enters in the Boltzmann-Vlasov equationis the physical momentum, which in the non-relativistic limit is related to the velocity as ~V = ~P f m . (3.12)The unperturbed distribution functions for decoupled particles that enter in the linearized Boltzmann-Vlasov eqn.(2.5) are the solutions of the unperturbed collisionless Boltzmann equation, namely f (0) ( p ) = f (0) d ( p c ) (3.13)where f d ( p c ) are given by eqn. (3.5). Restoring the speed of light c , the average D ~V E a = (cid:18) m a T d,a ( z ) c (cid:19) I a , (3.14)where T d,a ( z ) is given by eqn. (3.9) for the species a , and we have introduced I a ≡ " R ∞ f (0) d,a ( y ) dy R ∞ y f (0) d,a ( y ) dy . (3.15)This dimensionless ratio of integrals only depends on the ratios m a /T d , µ a /T d and dimensionless couplings from themicrophysics at decoupling. Using eqn. (3.9) we obtain D V ~V E a = 1 . z ) (cid:16) m a eV (cid:17) g d,a I a . (3.16)Inserting this result into the expression (2.19) and using Ω DM h = 0 . today , for z = 0 k fs = h . i X a ν a g d,a (cid:16) m a keV (cid:17) I a . (3.17)For light particles that decouple while they are ultrarelativistic the distribution function f d,a ( y ) does not dependon the ratio m a /T d , however, for particles that decouple when they are non-relativistic, their distribution function istypically a Maxwell-Boltzmann distribution which does depend on this ratio.Inserting the result (3.11) for the fractions (3.11) and (3.16) into eqn. (3.17) leads to the alternative form k fs = h . i X a g a g d,a (cid:16) m a keV (cid:17) Z ∞ f (0) d,a ( y ) dy (3.18)where g a are the internal degrees of freedom of the particle of species ( a ). The simple expressions (3.17,3.18) are someof the main results of this article. A. Redshift dependence
The Boltzmann-Vlasov equation in a non-expanding cosmology (2.1) is obtained by setting the scale factor a ≡ today . In an expanding cosmology the decay or growth of perturbations is no longerexponential but typically a power of the scale factor[20, 21]. However, here we are not concerned directly with themanner in which linear perturbations grow or decay, but with the marginal wave-vector k fs that determines thecrossover of behavior from growth to damping of collective excitations. If we assume that the expansion is sufficientlyslow that it can be treated adiabatically we can obtain the redshift behavior of the free-streaming length λ fs ( z ) byreplacing the densities, velocities and wavevectors with the corresponding scale factors given by eqns. (3.8,3.9,3.16),namely: ρ → ρ ( z ) = ρ (0)(1 + z ) D ~V E → D ~V ( z ) E = D ~V (0) E (1 + z ) − k → k (1 + z ) , (3.19)1where z = 0 refers to today . Defining the comoving free-streaming wavevector k fs ( z ) = 2 π/λ fs ( z ) upon rescaling bythe corresponding scale factor k fs → k fs ( z ) (1 + z ) . (3.20)Assuming the validity of this adiabatic scaling in eqn (2.15) we obtain, k fs ( z ) = 4 πG (1 + z ) X a ρ (0) ( z ) D ~V ( z ) E . (3.21)Since the DM density and velocity dispersions for all components scale as in eqn. (3.19) we find the following redshiftdependence of the comoving free-streaming wavevector k fs ( z ) = 4 πG (1 + z ) X a ρ (0) (0) D ~V (0) E . (3.22)This result is similar to the expression for the Jeans’s wavevector in a Newtonian fluid[20, 21] upon replacing (cid:10) /~V ( z ) (cid:11) → c s ( z ) where c s ( z ) is the (adiabatic) speed of sound in the medium as a function of redshift. In fact thevalidity of this assumption is confirmed not only by the similarity with the familiar Jeans’ result for Newtonian fluidsin an expanding cosmology, but also by the exact solution obtained in ref.[29] for the case of neutrinos decoupled inLTE. Therefore we identify k fs given by eqn. (3.17) as the comoving free streaming wave-vector. The scaling behaviorof the comoving free streaming length λ fs ( z ) = λ fs (0) p (1 + z ) leads to the free-streaming mass M fs ( z ) = 4 π X a ρ (0) a ( z ) λ fs ( z )1 + z ! = M fs (0)(1 + z ) , (3.23)a relation similar to the that of the Jeans’ mass in the non-relativistic regime. Therefore, under the validity of theadiabatic assumption, the simple re-scaling of the free-streaming wave-vector and length given by eqn. (3.22) indicatesthat we can simply obtain these quantities today ( z = 0) and extrapolate to an arbitrary redshift z via eqn. (3.22)provided the redshift is still small enough that the species are non-relativistic.The validity of the adiabatic assumption relies on the fact that in the non-relativistic regime with h ~V /c i ≪
1, thefree-streaming length is much smaller than the Hubble radius, which is found below to be a consistent assumption,or alternatively k fs /H ≫
1. And as mentioned above the result for the free streaming length obtained from thisadiabatic hypothesis is similar to the usual result for the Jeans’ length[20, 21] and is confirmed in ref.[29] for the caseof a neutrino thermal relic.A more detailed analysis of Gilbert’s equation for mixtures of DM components with arbitrary (but isotropic)distribution functions in the adiabatic approximation will be provided elsewhere[38].We now gather the above results to give the general expression for the free streaming length of an arbitrary mixture of non-relativistic species that decoupled in or out of LTE either ultrarelativistic or non-relativistic, in terms of the partial fraction ( ν ) that each contributes to the (DM) content and the dimensionless ratios I a λ fs ( z ) = 1(1 + z ) h . i X a ν a g d,a (cid:16) m a keV (cid:17) I a , (3.24)where I a = " R ∞ f (0) d,a ( y ) dy R ∞ y f (0) d,a ( y ) dy (3.25)in terms of the general distribution functions (3.5) which only depend on the ratios m a /T d , µ a /T d and dimensionlesscouplings and are completely determined by the microphysics at decoupling.We now proceed to obtain the contributions to the free streaming wave-vector today ( z = 0) from the variouscomponents: thermal relics that decoupled either relativistic or non-relativistic in LTE and non-thermal relics thatdecoupled while relativistic but out of LTE.2 B. Thermal relics
Let us consider ultrarelativistic Fermionic or Bosonic particles decoupled in LTE with chemical potentials and with m a /T d,a ≪ • Ultrarelativistic Fermions:
Neglecting m/T d in the ultrarelativistic limit, but keeping the chemical potential µ , the distribution function is f (0) d ( y ) = 1 e ( y − u ) + 1 ; u = µT d (3.26)where we have neglected m/T d ≪
1. For this distribution Z ∞ f (0) d ( y ) dy = ln[1 + e u ] (3.27)Combining this result with eqn. (3.18) we note that larger chemical potentials lead to shorter free streamingscales.Denote I F [ u ] the ratio I a , eqn. (3.15) for an ultrarelativistic Fermionic thermal relic with chemical potential µ .It is depicted in fig. (1) as a function of u = µ/T d . For µ = 0 we find I F [0] = 2 ln(2)3 ζ (3) = 0 . . (3.28) (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. 1: I F [ u ] vs u = µ/T d for f d ( y ) = 1 / ( e ( y − u ) + 1). I F [0] = 0 . For µ = 0 and taking ν = 1 for this Fermionic species and vanishing abundance for all others, the result for k fs given by eqn. (3.17) with I a = 2 ln(2) / ζ (3) agrees with that found in ref.[29]. Therefore, a Fermionic thermalrelic as a unique DM component yields a free-streaming length today ( z = 0) λ fs (0) = 14 g d p I F [ u ] (cid:16) keV m (cid:17) kpc . (3.29)For example a neutrino with m ∼ eV decoupling at T d ∼ g d ∼
10, yields a free streaming lengthtoday λ fs (0) ∼
10 Mpc , (3.30)which is the usual estimate for free-streaming lengths for HDM candidates. • Ultrarelativistic Bosons:
The distribution function is f (0) d ( y ) = 1 e ( y − u d ) − u d = µT d (3.31)3where we have neglected m/T d ≪ µ ≤ Z ∞ f (0) d ( y ) dy = − ln[1 − e −| u d | ] (3.32)This expression clearly reveals that for m = 0 , µ d = 0 the distribution function diverges logarithmically at p = 0. In an expanding cosmology with a particle horizon the smallest wavevector that can describe causalmicroscopic physics is determined by the Hubble scale. Since we are considering decoupling in LTE, it isconsistent to assume an infrared momentum cutoff of order H d ∼ √ g d T d /M P l , the Hubble parameter at thetime of decoupling. Therefore assuming such an infrared cutoff, keeping m/T d ≪ Z ∞ f (0) d ( y ) dy ∼ ln " T d H d + p H d + m . (3.33)In the denominator in eqn. (3.15) we can set x d = 0 in the ultrarelativistic limit and for vanishing chemicalpotential we find I B [ x d , ∼ ζ (3) ln " T d H d + p H d + m . (3.34)Keeping the mass and the chemical potential the distribution function becomes f (0) d ( y ) = 1 e √ y + x d − u d − x d = mT d ; u d = µT d (3.35)for which we find Z ∞ f (0) d ( y ) dy = x d ∞ X l =1 e − l | u d | K (cid:2) l x d (cid:3) , (3.36)Where K is a Hankel function. For m/T d , µ/T d ≪ m/T d , µ/T d ≪ I B [ x d ; u d ] ≃ x d ζ (3) ∞ X l =1 e − l | u d | K (cid:2) l x d (cid:3) . (3.37)This function features an infrared divergence in the limit m/T d , µ/T d → Bose-Einstein Condensation : If chemical freeze out occurs before kinetic decoupling, it is possible that bosonicparticles undergo Bose-Einstein condensation[34]. Under this circumstance the homogeneous condensate ( ~p = 0)contributes to the dynamics of the scale factor and inhomogeneous perturbations only modify the distributionfunction of the particles outside of the condensate. The linearized Boltzmann-Vlasov equation therefore appliesto the non-condensate part, since the dynamics of the condensate cannot be described by the incoherent Boltz-mann equation but by the equation of motion of the coherent and homogeneous condensate, which in turn iscoupled to the Friedmann equations for the scale factor.In the case of Bose-Einstein condensation the chemical potential attains its maximum value µ = m in which casethe contribution from the particles outside the condensate is I B [ x d , x d ] which has a strong infrared divergence for any value of the mass . This is a consequence of the fact that for ultrarelativistic Bose-Condensed particlesthe distribution function diverges at p = 0 for any value of the mass .However, as discussed above this infrared divergence associated with a Bose-Einstein condensate must be care-fully assessed in an expanding cosmology because infrared modes with wavelengths larger than the size of the4horizon will not be in LTE since no causal processes can establish thermalization for superhorizon wavelengths.Hence we conjecture that the integrals of the Bose-Einstein distribution function must be cutoff in the infraredat a momentum of the order of the Hubble scale at decoupling, H d ∼ √ g d T d /M P l . Such a cutoff leads to theestimate I BEC ∼ m M P l √ g d T d ∼ √ g d (cid:16) m keV (cid:17) (cid:16) GeV T d (cid:17) . (3.38)If this were the only DM component, the resulting comoving free-streaming length at z = 0 ( today ) is given by λ fs (0) ∼ .
014 pc g d (cid:16) keV m (cid:17) (cid:16) T d GeV (cid:17) (3.39)It is clear from the discussion above that the microphysics of decoupling of light bosonic particles, with orwithout a Bose-Einstein condensate requires a thorough assessment of the infrared behavior of the distributionfunction. The primordial velocity dispersion is very sensitive to this cutoff whose origin lies in the causal aspectsof decoupling. This important physical aspect must be studied in deeper detail, a task that is certainly beyondthe realm of this article, but we can nevertheless conclude that a light bosonic particle decoupled in LTE caneffectively act as CDM as a consequence of the infrared sensitivity of the moment h /~V i for Bosonic thermalrelics either condensed or not. • Non-relativistic particles:
The distribution function after freeze out in LTE is the Maxwell-Boltzmanndistribution f (0) d ( p c ) = n d (cid:20) mT d π (cid:21) − e − ~p c mTd = n d (cid:20) mT d π (cid:21) − e − y Td m (3.40)where n d is the number of particles per comoving volume at freeze-out[18] n d = 2 π g d T d Y ∞ (3.41)and Y ∞ is obtained from the solution of the kinetic equation and is a function of the annihilation cross section(see section 5.2 in ref.[18]). We find Z ∞ f (0) d ( y ) dy = 4 π g d Y ∞ x d (3.42)and for the integral I a eqn. (3.15) denoted by I NR for the non-relativistic (Maxwell-Boltzmann) distributionwe obtain I NR = T d m . (3.43)In the case of (WIMPs) with[35], m ∼
100 GeV ; T d ∼
10 MeV as candidates for cold dark matter, x d ∼ − .If this is the only DM candidate with ν = 1 with vanishing abundance of the other WDM or HDM candidates,the comoving free streaming length at z = 0 is given by λ fs (0) ∼ .
014 pc g d (cid:16)
100 GeV m (cid:17) (cid:16)
10 MeV T d (cid:17) , (3.44)from which it follows that for WIMPs with m ∼
100 GeV that decouple kinetically at T d ∼
10 MeV[35] when g d ∼ λ fs (0) ∼ . × − pc . (3.45)5 C. Decoupling out of LTE
In ref.[34] the following distribution function for particles that decouple out of LTE and that effectively modelsseveral cases of cosmological relevance was introduced, f d ( y ) = f f eq (cid:16) yη (cid:17) θ ( y − y ) , (3.46)where f eq (cid:0) p c ηT d (cid:1) is the equilibrium distribution function for a relativistic particle at an effective temperature ηT d . Thisform is motivated by detailed studies of production[39] and thermalization process that proceeds by energy transferfrom long to short wavelengths via a cascade with a front that moves towards the ultraviolet[40]. If the interactionrate for mode mixing becomes smaller than the expansion rate the advance of this front is interrupted at a fixedvalue of the momentum, identified here to be p c = y T d where T d is the temperature of the environmental degrees offreedom that are in LTE at the time of decoupling[34]. The amplitude f and effective temperature ηT d ≤ T d reflectan incomplete thermalization behind the front of the cascade and determine the average number of particles in its wake [40]. The non-equilibrium distribution function (3.46) yields a fairly accurate description of these processes andthe decoupling out of LTE.Remarkably, this non-LTE distribution function also describes[34] sterile neutrinos produced non-resonantly via theDodelson-Widrow[32] (DW) mechanism or resonantly via a lepton-driven MSW resonance[33].For the general form (3.46) of the distribution function we find I a = 1 η H h p c η T d i ; H [ s ] = R s f eq ( y ) dy R s y f eq ( y ) dy . (3.47)The function H ( s ) for f eq ( y ) = 1 / ( e y + 1) is a monotonically decreasing function of s with limiting behavior H ( s ) ∼ /s for s → H ( s ) → / ζ (3) for s → ∞ , it is displayed in fig. (2) in the interval 0 . ≤ s ≤ ’()* +,- ./0 123 456789: ;<=>?@ABCDE FIG. 2: H ( s ) vs s for f eq ( y ) = 1 / ( e y + 1) . Although a more detailed assessment of the production and pathway towards thermalization of sterile neutrinos isstill being debated[41, 42] we will adopt the semi-phenomenological results of refs.[32, 33] as guiding models for thedistribution functions of sterile neutrinos decoupled out of LTE. • Sterile neutrinos produced via the (DW) mechanism [32]: for this case η = 1; s → ∞ ; f ∼ . /m ][32, 34]. The integral I a does not depend on the amplitude f (only the fractional abundance ν is affected by f ) and is therefore given by the value for an ultrarelativistic fermion with vanishing chemicalpotential decoupled in LTE[32], namely I DW = 2 ln(2) / ζ (3) ∼ . If these sterile neutrinos are the only DM component, namely ν = 1, the free streaming length today is given by the same form as for a Fermionicthermal relic eqn. (3.29) but with vanishing chemical potential, λ fs (0) = 22 . g d (cid:16) keV m (cid:17) kpc . (3.48)6In the Dodelson-Widrow scenario[32] the sterile neutrino production rate peaks at T ∼
130 MeV which is nearthe QCD scale. Taking the decoupling temperature in this range results in g d ∼ λ fs (0) ≃ (cid:16) keV m (cid:17) . (3.49)Therefore m ∼ keV sterile neutrinos produced via the (DW) mechanism yield free streaming lengths today ofthe order of λ fs ∼ g d because near the QCDphase transition there is an abrupt change in the effective number of relativistic degrees of freedom. However,because the cube-root of g d enters in λ fs the free streaming length is not very sensitive to this ambiguity. • Lepton-driven resonantly produced sterile neutrinos [33]: for this case η = 1 , s ∼ . f = 1[33, 34], andwe find I a = H (0 .
7) = 6 . only DM component we find the free-streaming scale today λ fs (0) = 5 . g d (cid:16) keV m (cid:17) kpc . (3.50)The production rate in the resonant case also seems to peak near the QCD scale[33] at which g d ∼
30. Takingthis value for the decoupling temperature we obtain the estimate λ fs (0) = 1 .
73 kpc (cid:16) keV m (cid:17) . (3.51)Hence m ∼ keV sterile neutrinos produced via the lepton-driven resonant mechanism[33] mechanism yield freestreaming lengths today of the order of λ fs ∼ ∼ I a and consequently the shortening of the free streaming scale in the case of sterile neutrinosproduced resonantly out of LTE follows from the fact that the distribution function resulting from the resonantproduction mechanism favors low momenta. Therefore for the same value of the contribution of sterile neutrinosproduced resonantly or non-resonantly out of LTE, the free streaming length in the case of resonance productionis ∼ smaller than either a thermal Fermionic relic with vanishing chemical potential or a sterile neutrinoproduced non-resonantly via the (DW) mechanism. Just as in the (DW) scenario, there is an ambiguity in theprecise determination of g d near the QCD scale, but again the free streaming length is not very sensitive to thisambiguity because of the power 1 / mixture of Fermionic andBosonic thermal relics, including a possible BEC, WIMPs and sterile neutrinos produced non-resonantly or resonantlyassumed to be described by the respective distribution functions quoted in refs.[32, 33] is given by1 λ fs ( z ) = 1(1 + z ) h . i X species ( ν F g d,F (cid:16) m F keV (cid:17) I F [ u ] + ν s g d,s (cid:16) m s keV (cid:17) .
814 + ν B g d,B (cid:16) m B keV (cid:17) I B [ x d , u d ] +10 ν wimp g d,wimp (cid:16) m wimp
100 GeV (cid:17) (cid:16) T d
10 MeV (cid:17)) , (3.52)where the index F refers to thermal Fermions and sterile neutrinos produced non-resonantly via the (DW)mechanism[32] for which I F [0] = 2 ln(2) / ζ (3) = 0 . s refers solely to sterile neutrinos producedresonantly via the mechanism in ref.[33]. In each case g d,a is the effective number of ultrarelativistic degrees of freedomwhen the corresponding species decouples. For sterile neutrinos produced by either mechanism g d ∼
30 correspondingto a decoupling temperature near the QCD scale.7
IV. CONCLUSIONS AND FURTHER QUESTIONS
In this article we have implemented methods from the theory of multicomponent plasmas to study free streamingof a mixture of non-relativistic DM candidates that include Fermionic and Bosonic particles that decouple in LTEwhile relativistic, including the possibility of a Bose-Einstein condensate, heavy thermal relics that decoupled inwhen non-relativistic (WIMPs), and sterile neutrinos that decouple out of LTE when they are relativistic. We solve exactly the Boltzmann-Vlasov equation for the gravitational perturbations in a non-expanding cosmology and obtainthe “gravitational polarization function” whose zeroes determine the dispersion relations of the collective excitationsof the self-gravitating collisionless gas of particles. The free-streaming wave vector is obtained from the marginalsolution that separates Landau damped short wavelength perturbations from unstable collective modes. We obtainthe free-streaming length for arbitrary (but isotropic) distributions of the particles that decoupled in or out of LTE solely in terms of the fractional abundance of the different species and integrals of their distribution functions whichdepend on the microphysics at decoupling.Because all of the components are non-relativistic and assuming that the expansion is slow we provide an adiabaticityargument that allows us to implement a simple rescaling of densities, velocities dispersion and wavelengths to extractthe redshift dependence of the free-streaming length. The validity of this adiabatic approximation is confirmed bythe similarity of the result to the Jeans’ length for Newtonian perturbations in an expanding cosmology and by theexplicit result for the free-streaming wavevector for thermal neutrinos obtained in ref.[29].The main result for the free streaming length as a function of redshift for an arbitrary mixture of DM componentsis 1 λ fs ( z ) = 1(1 + z ) h . i X a ν a g d,a (cid:16) m a keV (cid:17) I a , (4.1)where I a = " R ∞ f (0) d,a ( y ) dy R ∞ y f (0) d,a ( y ) dy (4.2)is a ratio of integrals of the distribution functions that only depends on the m a /T d , µ a /T d and dimensionless couplingsand is completely determined by the microphysics at decoupling. Evaluating these integrals for thermal Fermionic andBosonic relics (with or without condensation), WIMPs and sterile neutrinos decoupled out of LTE either resonantlyor non-resonantly with the distribution functions obtained in refs.[32, 33] respectively, we find the general result1 λ fs ( z ) = 1(1 + z ) h . i X species ( ν F g d,F (cid:16) m F keV (cid:17) I F [ u ] + ν s g d,s (cid:16) m s keV (cid:17) .
814 + ν B g d,B (cid:16) m B keV (cid:17) I B [ x d , u d ] +10 ν wimp g d,wimp (cid:16) m wimp
100 GeV (cid:17) (cid:16) T d
10 MeV (cid:17)) , (4.3)where ν a is the partial fraction of each component with P a ν a = 1, the sum over Fermionic species ( F ) includes thermalrelics and sterile neutrinos produced non-resonantly via the (DW) mechanism[32], for which the chemical potentialvanishes and I F [0] = 0 . s is for sterile neutrinos produced via a lepton-driven (MSW) resonance asdescribed in ref.[33], and the label B stands for condensed or non-condensed Bosonic thermal relics.This expression features several important consequences relevant for large scale structure formation: • A non-negligible ν wimp ≫ − , for a CDM candidate with m ∼
100 GeV which decoupled at T d ∼
10 MeV[35]overwhelms all other components (but for a possible BEC) and leads to small free streaming lengths consistentwith CDM regardless of the presence of WDM or HDM components. For ν wimp ≫ − the free-streaminglength of mixed DM is completely dominated by WIMPs and is given today by λ fs (0) ∼ .
014 pc g d (cid:16)
100 GeV m (cid:17) (cid:16)
10 MeV T d (cid:17) , (4.4)from which it follows that for WIMPs with m ∼
100 GeV that decouple kinetically at T d ∼
10 MeV[35] when g d ∼ λ fs (0) ∼ . × − pc . (4.5)This cut-off scale might well be related to the smallest non-linear structures found in[10] unless there is somesubstantial violent relaxation and merging.8 • For vanishing chemical potential u d = 0, non-BEC Bosonic ultrarelativistic relics feature an infrared enhance-ment in I B [ x d ,
0] which must be regulated by assuming that the integrals of the distribution function are cutoff oforder of the Hubble scale at decoupling H d ∼ √ g d T d /M P l . If this is the only
DM component, the free-streaminglength today is λ fs (0) ∼ g d p I B [ x d , (cid:16) keV m (cid:17) kpc ; I B [ x d , ∼ ζ (3) ln " T d H d + p H d + m . (4.6)For BEC Bosons x d = u d the function I B [ x d , x d ] is divergent as a consequence of the infrared divergence inthe numerator of I a (eq. 3.15). However, we highlighted that the physics of BEC formation in an expandingcosmology must be assessed in greater detail in order to understand the behavior of superhorizon modes. Thisobservation also holds for the non-condensed Bose gas decoupling when relativistic with µ/T d ≪ I a of the order of theHubble scale, since superhorizon modes cannot establish thermal equilibrium via causal processes as discussedabove. If a BEC is the only DM component, its free-streaming length today is approximately given by λ fs (0) ∼ .
014 pc g d (cid:16) keV m (cid:17) (cid:16) T d GeV (cid:17) . (4.7)Therefore even when these relics decoupled when they were ultrarelativistic, they could effectively act as CDMcomponents. This is a consequence of the fact that the distribution functions favor small momenta. • If sterile neutrinos that decouple out of LTE near the QCD scale produced either non-resonantly via the (DW)[32]mechanism or via a lepton-driven MSW resonance[33] near the QCD scale are the only
DM components we findthe following free-streaming lengths today λ fs (0) ≃ (cid:16) keV m (cid:17) non − resonant (4.8) λ fs (0) ≃ .
73 kpc (cid:16) keV m (cid:17) resonant . (4.9)The smaller values of the free streaming length are compatible with those in ref.[19] where it is found that firststars form in filamentary structures with length scales of the order of the free streaming scale and within afactor ∼ − ∼ . ∼ −
10 kpc.We believe that these results lead to a significant advance in the understanding of collisionless (DM) becausetrying to obtain a reliable estimate of the free-streaming lengths via the numerical integration of Gilbert’s equationsfor a combination of arbitrary distribution functions corresponding to particles that decoupled in or out of LTE isundoubtedly a daunting task.The sensitivity of the free-streaming scale to the details of the distribution function at low momentum and theimportance of a reliable determination of the free-streaming length as a measure of the cutoff of the power spectrum oflinearized cosmological perturbations require a fundamentally sound understanding of the microphysics of productionand decoupling of sterile neutrinos, a program currently underway[41].
Acknowledgments
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