Cosmological evolution of warm dark matter fluctuations II: Solution from small to large scales and keV sterile neutrinos

0.71 MB·English

by  H. J. de Vega

Cosmological evolution of warm dark matter fluctuations II: Solution from small to
large scales and keV sterile neutrinos
H. J. de Vega (a,b)∗ and N. G. Sanchez (b)†
(a) LPTHE, Universit´e Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII),
Laboratoire Associ´e au CNRS UMR 7589, Tour 13-14, 4`eme. et 5`eme. ´etages,
Boite 126, 4, Place Jussieu, 75252 Paris, Cedex 05, France.
(b) Observatoire de Paris, LERMA, Laboratoire Associ´e au CNRS UMR 8112.
61, Avenue de l’Observatoire, 75014 Paris, France.
(Dated: July 4, 2012)
We solve the cosmological evolution of warm dark matter (WDM) density fluctuations within
2
the analytic framework of Volterra integral equations presented in the accompanying paper [1]. In
1
the absence of neutrinos, the anisotropic stress vanishes and the Volterra-type equations reduce to
0
a single integral equation. We solve numerically this single Volterra-type equation both for DM
2
fermions decoupling at thermal equilibrium and DM sterile neutrinos decoupling out of thermal

n equilibrium. We give the exact analytic solution for the density fluctuations and gravitational
a potential at zero wavenumber. We compute thedensity contrast as a function of thescale factor a
J
for a relevant range of wavenumbers k. At fixed a, the density contrast turns to grow with k for

0 k<k whileitdecreasesfork>k ,wherek 1.6/Mpc. Thedensitycontrastdependsonk anda
c c c
1 mainly through the product k a exhibiting a s≃elf-similar behavior. Our numerical density contrast
for small k gently approaches ouranalytic solution for k=0. Forfixedk<1/(60kpc),thedensity
] contrast generically grows with a while for k > 1/(60 kpc) it exhibits oscillations starting in the
O
radiationdominated(RD)erawhichbecomestrongeraskgrows. Wecomputethetransferfunction
C of the density contrast for thermal fermions and for sterile neutrinos decoupling out of equilibrium
. in two cases: the Dodelson-Widrow (DW) model and a model with sterile neutrinos produced by
h
a scalar particle decay. The transfer function grows with k for small k and then decreases after
p
reaching a maximum at k = k reflecting the time evolution of the density contrast. The integral
- c
o kernelsintheVolterraequationsarenonlocal intimeandtheirfalloff determinethememoryofthe
r past evolution since decoupling. We find that this falloff is faster when DM decouples at thermal
t
s equilibriumthanwhenitdecouplesoutofthermalequilibrium. Althoughneutrinosandphotonscan
a be neglected in the matter dominated (MD) era, they contribute to the Volterra integral equation
[ in the MD era through theirmemory from the RD era.

2
v
0
0
Contents
3
0
. I. Introduction and Summary of Results 2
1
1
1 II. The Volterra Integral Equations and Relevant physical scales 5
1 A. Density fluctuations and anisotropic stress fluctuations 5
: B. Relevant scales in the ultra-relativistic and non-relativistic DM regimes 7
v
i
X III. From the ultrarelativistic to the non-relativistic regime of the DM in the Volterra equations 9
r A. Transition Regime 10
a B. Non-relativistic Regime 10
C. The Gilbert equation from the Volterra equation in the MD era. 14
IV. Solving the Volterra equation for the DM density fluctuations (without anisotropic stress) 14
A. Numerical solution of the Volterra equation for a wide range of wavenumbers 15
B. Analytic solution of the Volterra equation at zero wavenumber 16
C. The transfer function for the density contrast 19
V. Fermions in thermal equilibrium and sterile neutrinos out of equilibrium 20
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
VI. Volterra integral equations for cold dark matter 22
Acknowledgments 23
A. The gravitational potential in the RD era 24
B. The free-streaming length in the different regimes. 25
References 27
I. INTRODUCTION AND SUMMARY OF RESULTS
Inanaccompanyingpaper[1]weprovidedaframeworktostudythecompletecosmologicalevolutionofdarkmatter
(DM) density fluctuations for DM particles that decoupled being ultrarelativisticduring the radiationdominated era
which is the case of keV scale warm DM (WDM). In this paper, we solve the evolution of DM density fluctuations
following the framework developed in ref. [1].
The new framework presented in ref. [1] and here is generic for any type of DM and applies in particular to cold
DM (CDM) too. The collisionless and linearized Boltzmann-Vlasov equations (B-V) for WDM and neutrinos in the
presenceofphotonsandcoupledtothelinearizedEinsteinequationsarestudiedindetailinthepresenceofanisotropic
stress with the Newtonian potential generically different from the spatial curvature perturbations.
In ref. [1] the full system of B-V equations for DM and neutrinos is recasted as a system of coupled Volterra
integral equations. (Ref. [18] has recently considered this issue in absence of anisotropic stress). These Volterra-type
equations are valid both in the radiation dominated (RD) and matter dominated (MD) eras during which the WDM
particles are ultrarelativistic and then nonrelativistic. This generalizes the so-called Gilbert integral equation only
valid for nonrelativistic particles in the MD era.
WesucceedtoreducethesystemoffourVolterraintegralequationsforthedensityandanisotropicstressfluctuations
of DM and neutrinos into a system of only two coupled Volterra equations.
Insummary,thepairofpartialdifferentialBoltzmann-VlasovequationsinsevenvariablesforDMandforneutrinos
become a system of four Volterra linear integral equations on the density fluctuations ∆ (η,~k), ∆ (η,~k) and
dm ν
anisotropic stress Σ (η,~k),Σ (η,~k) for DM and neutrinos, respectively.
dm ν
In addition, because we deal with linear fluctuations evolving on an homogeneous and isotropic cosmology, the
Volterra kernel turns to be isotropic, independent of the ~k directions. As stated above, the kˇ dependence factorizes
out and we arrive to a final system of two Volterra integral equations in two variables: the modulus k and the time
that we choose to be as
y a(η)/a 3200a(η). (1.1)
eq
≡ ≃
We have thus considerably simplified the original problem: we reduce a pair of partial differential B-V equations on
seven variables η, ~q, ~x into a pair of Volterra integral equations on two variables: η, k.
The customary DM density contrast δ(η,~k) is connected with the density fluctuations ∆ (η,~k) by [3]
dm
∆ (η,~k) 1
δ(η,~k)= dm , a , (1.2)
eq
ρ [a +a(η)] ≃ 3200
dm eq
where ρ is the averageDM density today.
dm
It is convenient to define dimensionless variables as
k l 2 T Idm
α fs , l = d 4 ,
fs
≡ I4dm H0 m saeq Ωdm
where l stands for the free-streaming lenpgth [9, 10, 14], T is the comoving DM decoupling temperature and Idm is
fs d 4
the dimensionless square velocity dispersion given by
∞
Idm = Qn fdm(Q)dQ , whilefdm(Q)is normalizedby Idm =1. (1.3)
n 0 0 2
Z0
3
Q is the dimensionless momentum Q q/T whose typical values are of order one.
d
≡
A relevantdimensionless rate emerges: the ratiobetween the DM particle mass m andthe decoupling temperature
at equilibration,
maeq m gd 31
ξ =4900 ,
dm
≡ T keV 100
d
(cid:16) (cid:17)
g being the effective number of UR degrees of freedom at the DM decoupling. Therefore, ξ is a large number
d dm
provided the DM is non-relativistic at equilibration. For m in the keV scale we have ξ 5000.
dm
∼
DM particles and the lightest neutrino become non-relativistic by a redshift
z +1 m 1.57 107 m gd 13 for DM particles , zν =34 mν for the lightest neutrino.
trans ≡ T ≃ × keV 100 trans 0.05eV
d
(cid:16) (cid:17) (1.4)
z denoting the transition redshift from ultrarelativistic regime to the nonrelativistic regime of the DM particles.
trans
The final pair of dimensionless Volterra integral equations take the form
y
∆˘(y,α)=C(y,α)+B (y)φ¯(y,α)+ dy′ G (y,y′)φ¯(y′,α)+Gσ(y,y′)σ¯(y′,α) , (1.5)
ξ α α
Z0
y (cid:2) (cid:3)
σ¯(y,α)=Cσ(y,α)+ dy′ Iσ(y,y′)σ¯(y′,α)+I (y,y′)φ¯(y′,α) , (1.6)
α α
Z0
(cid:2) (cid:3)
withinitialconditions∆˘(0,α)=1 , σ¯(0,α)= 2 I .ThispairofVolterraequationsiscoupledwiththelinearized
5 ξ
Einstein equations.
The kernels and the inhomogeneous terms in eqs.(1.5)-(1.6) are given explicitly by eqs.(2.12)-(2.15), (2.16)-(2.18)
and (2.7)-(2.11). The arguments of these functions contain the dimensionless free-streaming distance l(y,Q),
y dy′
l(y,Q)= . (1.7)
Z0 [1+y′] y′2+(Q/ξ )2
dm
r
h i
The coupled Volterra integral equations (1.5)-(1.6) are easily amenable to a numerical treatment.
During the RD era the gravitationalpotential is dominated by the radiation fluctuations (photons and neutrinos).
The photons canbe describedin the hydrodynamicalapproximation(their anisotropicstressis negligible). The tight
coupling of the photons to the electron/protons in the plasma suppresses before recombination all photon multipoles
except Θ and Θ . (The Θ stem from the Legendre polynomial expansion of the photon temperature fluctuations
0 1 l
Θ(η,~q,~k) [2]).
Θ and Θ obey the hydrodynamical equations [2]
0 1
dΘ dφ
0
+k Θ (η,α~)= , (1.8)
1
dη dη
dΘ k k
1
Θ (η,α~)= φ(η,α~) . (1.9)
0
dη − 3 3
This is a good approximation for the purposes of following the DM evolution [2].
The photons gravitationalpotential is given in the RD and MD eras by (ref. [2] and Appendix A)
√3 κy a 1
φ(η,~k)=ψ(η,~k)=3ψ(0,~k) j , κ=k η∗ , η∗ eq =143Mpc, (1.10)
1
κy (cid:18)√3(cid:19) ≡rΩM H0
where j (x) is the spherical Bessel function of order one.
1
For redshift z <30000 the kernel G (y,y′) in eq.(1.5) simplifies as
α
G (y,y′)y,y=′>0.1 ξdm κ y y′ Π[α (s(y) s(y′))] where,
α 2I √1+y′ −
ξ
4
∞ 1
Π(x)= QdQfdm(Q) sin(Qx) and s(y)= ArgSinh . (1.11)
0 − √y
Z0 (cid:18) (cid:19)
In this regime z <30000, y >0.1, the anisotropic stress σ¯(y,α) turns to be negligible and eqs.(1.5)-(1.6) becomes a
single Volterra integral equation. In the MD era this equation takes the form
∆˘(y,α) 6 s(y)
=g(y,α)+ ds′ Π[α(s(y) s′)] ∆˘(y(s′),α) , y 1 , (1.12)
y α − ≥
Zs(1)
where the inhomogeneoustermg(y,α) containsthe memoryfromthe previoustimes y <1 ofthe RD era. When DM
isnon-relativisticthememoryfromtheregimewhereDMwasultrarelativisticturnsouttofadeoutas1/ξ 0.0002
dm
∼
compared to the recent memory where DM is non-relativistic.
The falloff of the kernel Π[α(s s′)] determines the memory in the regime where DM is non-relativistic. We
−
find that this falloff is faster when DM decouples at thermal equilibrium than when it decouples out of thermal
equilibrium (see fig. 5). This can be explained by the general mechanism of thermalization [25]: in the out of
equilibrium situation the momentum cascade towards the ultraviolet is incomplete and there is larger occupation at
low momenta and smaller occupation at large momenta than in the equilibrium distribution. Therefore, the out of
equilibrium kernel Π(x) which is the Fourier transform eq.(1.11) of the freezed out momentum distribution exhibits
a longer tail than the equilibrium kernel.
Neutrinos and photons can be neglected in the matter dominated era. However, they contribute to the Volterra
integral equation in the MD era through the memory integrals over 0<y′ <1, namely the memory of the RD era.
Whenthe anisotropicstressσ¯(y,α) isnegligible,eqs.(1.5)-(1.6)reduce to asingleVolterraintegralequationforthe
DMdensityfluctuations∆˘ (y,α)whentheanisotropicstressσ¯(y,α)isnegligible. Wefindthesolutionofthissingle
dm
Volterra equation for a broad range of wavenumbers 0.1/Mpc<k <1/5kpc.
At zero wavenumber k=0 the kernelof this Volterra equation vanishes and the DM fluctuations can be expressed
explicitly in terms of the gravitational potential φ. The gravitational potential at k = 0 follows solely from the
hydrodynamic equations for the radiation combined with the regularity requirement at k = 0 of the first linearized
Einsteinequation. Namely, the gravitationalpotential φis solelyobtainedfromthe radiationwithoutspecifying the
sources of the DM and radiation fluctuations. Using this explicit and well known form of φ, (see e. g. ref. [2]) the
DM fluctuations are obtained at α=0. The fact that the Einstein equations constraintheir sourceswas first noticed
in ref. [16] in a completely different context.
We depict in figs. 2 the normalized density contrast vs. y (the scale factor divided by a ) for thermal fermions
eq
and sterile neutrinos in the Dodelson-Widrow (DW) model [5] (both models yield identical density fluctuations for a
given value of ξ ). Similar curves are obtained in the χ model where sterile neutrinos are produced by the decay of
dm
a real scalar [21].
Atfixedywefindthatthedensitycontrastgrowswithkfork <k whileitdecreasesfork >k ,wherek 1.6/Mpc.
c c c
≃
Wefindthatthedensitycontrastdependsonαandymainlythroughtheproductαyexhibitingaself-similarbehavior.
Thedensitycontrastcurvescomputednumericallyforsmallαgentlyapproachintheupperfig. 2ouranalyticsolution
forα=0. Forfixedα<1,thedensitycontrastgenericallygrowswithywhileforα>1itexhibitsoscillationsstarting
in the RD era which become strongeras α grows(see fig. 2). The density contrastbecomes proportionalto y (to the
scale factor) at sufficiently late times. The larger is α, the later starts δ(y,α) to grow proportional to y (see fig. 2).
Also, the larger is α>1, the later the oscillations remain.
We depict in fig. 3 the transferfunction for thermalfermions and sterile neutrinos in the DW modeland for sterile
neutrinos decoupling out of equilibrium in the χ model. The transfer function grows with k for small k and then
decreases after reaching a maximum at k =k .
c
WeanalyzeinsectionIIIthesystemoftwoVolterraintegralequationsintheregimeswhereDMisinthetransition
from UR to NR and when DM is nonrelativistic. In sec. IIIC we take the nonrelativistic limit of our system of
VolterraintegralequationsintheMDera. ThisyieldstheGilbertequation(plusextraterms). Wefindextramemory
terms and different inhomogeneities arising from our system of Volterra equations.
InsectionIVweconsiderthe zeroanisotropicstresscasewherethesystemofVolterraequationsreducesto asingle
Volterraequation. The numericalsolutionfor the DMfluctuations ina broadrangeofwavenumbersis presentedand
discussed, as well as the transfer function and the analytic solution for zero wavenumber.
Wepresentinsec. Vthedistributionfunctions,mainparametersandintegralkernelsforsterileneutrinosdecoupling
out of equilibrium and compare them to fermions decoupling with a Fermi-Dirac distribution. Finally, we present in
sec. VI the generalization of the Volterra integral equation for cold dark matter.
5
Inthe RDerawhereradiationfluctuationsdominatesthe gravitationalpotentialwederiveinAppendix Aasecond
order differential equation for the gravitational potential. We show that the solution of this differential equation is
well approximated by the Bessel function of order one eq.(1.10).
We provide in Appendix B explicit and useful expressions for the free-streaming distance l(y,Q) [see eq.(1.7)] in
the main relevant regimes.
II. THE VOLTERRA INTEGRAL EQUATIONS AND RELEVANT PHYSICAL SCALES
We recall here the pair of coupled Volterra integral equations derived in the accompanying paper [1] from the
Boltzmann-Vlasov equations for DM and for neutrinos.
A. Density fluctuations and anisotropic stress fluctuations
In the companion paper [1] we defined dimensionless density fluctuations ∆¯ (y,α) and ∆¯ (y,α) and dimension-
dm ν
less anisotropic stress fluctuations σ¯(y,α) factoring out the initial gravitational potential ψ(0,~k) in order to obtain
quantities independent of the~k direction. These relevant quantities are expressed as
d3Q Ψ (y,Q~,~κ) d3Q Ψ (y,Q~,~κ)
∆¯ (y,α)= ε(y,Q)fdm(Q) dm , ∆¯ (y,α)= Qfν(Q) ν ,
dm 4π 0 ψ(0,~κ) ν 4π 0 ψ(0,~κ)
Z Z
φ(y,~κ)=ψ(0,~k)φ¯(y,α) , ψ(y,~κ)=ψ(0,~k)ψ˘(y,α) and ψ˘(0,α)=1,
σ(y,~κ)=ψ(0,~k)σ¯(y,α) , σ¯(y,α)=φ¯(y,α) ψ˘(y,α) , σ¯(0,α)=φ¯(0,α) 1. (2.1)
− −
We then introduce in ref. [1] the combined density fluctuation ∆˘(y,α)
1 1 R (y) Idm
∆˘(y,α)= ∆¯ (y,α)+ ν ∆¯ (y,α) , I = 3 +R (0) R (0)=0.727 , ∆˘(0,α)=1,
−2I ξ dm Iν ν ξ ξ ν ≃ ν
ξ (cid:20) dm 3 (cid:21) dm
(2.2)
where ξ is the ratio between the DM particle mass m and the physical decoupling temperature at equilibration
dm
redshift z +1 3200,
eq
≃
maeq m gd 13 m 43 1
ξdm = =4900 =5520 (gdm Ndm)3 . (2.3)
T keV 100 keV
d
(cid:16) (cid:17) (cid:16) (cid:17)
We use here the dimensionless wavenumbers [1, 14]
2 2 T a 1
κ k η∗ and α κ= d k where η∗ eq =143Mpc. (2.4)
≡ ≡ ξdm H0 m aeq Ωdm ≡rΩM H0
Using ∆˘(y,α) and σ¯(y,α) in ref. [1] allowed to rpeduce the system of four Volterra integral equations into a the
following pair of Volterra integral equations:
y
∆˘(y,α)=C(y,α)+B (y)φ¯(y,α)+ dy′ G (y,y′)φ¯(y′,α)+Gσ(y,y′)σ¯(y′,α) , (2.5)
ξ α α
Z0
(cid:2) (cid:3)
y
σ¯(y,α)=Cσ(y,α)+ dy′ Iσ(y,y′)σ¯(y′,α)+I (y,y′)φ¯(y′,α) , (2.6)
α α
Z0
(cid:2) (cid:3)
with the initial conditions [1]
2 2
∆˘(0,α)=1 , σ¯(0,α)= I R (0) .
ξ ν
5 ≃ 5
We have in eqs.(2.5)-(2.6)
1 a(y,α) R (y) aσ(y,α) R (y)
C(y,α)= + ν aur(y,α) , Cσ(y,α) + ν aurσ(y,α), (2.7)
−2I ξ Iν ≡ ξ Iν
ξ (cid:20) dm 3 (cid:21) dm 3
6
1
B (y)= [y b (y)+4R (y)] ,
ξ dm ν
−2I
ξ
κ 1 R (y)
G (y,y′)= N (y,y′)+ ν Nur(y,y′) , (2.8)
α −2I √1+y′ ξ α Iν α
ξ (cid:20) dm 3 (cid:21)
κ 1 R (y)
Gσ(y,y′)= Nσ(y,y′) ν Nur(y,y′) , (2.9)
α −2I √1+y′ ξ α − 2Iν α
ξ (cid:20) dm 3 (cid:21)
κ 1 R (y)
I (y,y′)= U (y,y′)+ ν Uur(y,y′) , (2.10)
α √1+y′ ξ α Iν α
(cid:20) dm 3 (cid:21)
κ 1 R (y)
Iσ(y,y′)= Uσ(y,y′) ν Uur(y,y′) . (2.11)
α √1+y′ ξ α − 2Iν α
(cid:20) dm 3 (cid:21)
In eqs.(2.5)-(2.6) we can use I R (0). The DM integral kernels and inhomogeneity functions in eqs. (2.7)-(2.11)
ξ ν
≃
are given by
∞ dfdm α
a(y,α)= Q2 dQε(y,Q) fdm(Q)c¯0 (Q)+φ¯(0) 0 j Ql(y,Q) , (2.12)
0 dm dlnQ 0 2
Z0 (cid:20) (cid:21) h i
∞ Q2 dQ
y ξ b (y)= fdm(Q) 4Q2+3(ξ y)2 , (2.13)
dm dm ε(y,Q) 0 dm
Z0
(cid:2) (cid:3)
∞ dfdm Q2
N (y,y′)= Q2 dQε(y,Q) 0 j [αl (y,y′)] ε(y′,Q)+ , (2.14)
α dQ 1 Q ε(y′,Q)
Z0 (cid:20) (cid:21)
∞ dfdm
Nσ(y,y′)= Q2 dQ 0 j [αl (y,y′)] ε(y,Q)ε(y′,Q). (2.15)
α − dQ 1 Q
Z0
3 ∞ Q4 dQ dfdm α
aσ(y,α)= fdm(Q)c¯0 (Q)+φ¯(0) 0 j Ql(y,Q) , (2.16)
κ2 y2 ε(y,Q) 0 dm dlnQ 2 2
Z0 (cid:20) (cid:21) h i
3 ∞ Q4 dQ dfdm Q2
U (y,y′)= 0 ε(y′,Q)+ 2j [αl (y,y′)] 3j [αl (y,y′)] , (2.17)
α −5κ2 y2 ε(y,Q) dQ ε(y′,Q) { 1 Q − 3 Q }
Z0 (cid:20) (cid:21)
3 ∞ Q4 dQ dfdm
Uσ(y,y′)= 0 ε(y′,Q) 2j [αl (y,y′)] 3j [αl (y,y′)] . (2.18)
α 5κ2 y2 ε(y,Q) dQ { 1 Q − 3 Q }
Z0
The function c¯0 (Q) determines the intial conditions. We have for thermal initial conditions (TIC) and for thermal
dm
initial conditions (TIC) [1]
1 dlnfdm 1 dlnfν
0 for thermal initial conditions (TIC), 0 for TIC,
c¯0dm(Q)= 2 dlnQ c¯0ν(Q)= 2 dlnQ (2.19)
 2 for Gilbert initial conditions (GIC) .  2 for GIC .
− −
The neutrino integral kernels in eqs.(2.7)-(2.11) and inhomogeneity functions are given by
Nur(y,y′)= 8Iν j [κr(y,y′)] ,
α − 3 1
24Iν
Uur(y,y′)= 3 2j [κr(y,y′)] 3j [κr(y,y′)] , (2.20)
α 5κ2 y2 { 1 − 3 }
7
aur(y,α)= 2Iν 1+2φ¯(0) j [κr(y,0)] , (2.21)
− 3 0
(cid:2) (cid:3)
j [κr(y,0)]
aurσ(y,α)= 6Iν 1+2φ¯(0) 2 . (2.22)
− 3 κ2 y2
(cid:2) (cid:3)
The Volterra integral equations (2.5)-(2.6) are coupled with the linearized Einstein equations derived in the accom-
panying paper [1]
d 1 1 R (y)
(1+ (y)) +1 + (κy)2 φ¯(y,α)=[1+ (y)]σ¯(y,α) ∆¯ (y,α) ν ∆¯ (y,α) 2R (y)Θ¯ (y,α).
R0 dy 3 R0 −2ξ dm − 2Iν ν − γ 0
(cid:20) (cid:18) (cid:19) (cid:21) dm 3
(2.23)
Here,
ρ (y) ∞ Q2 ρ
(y) dm = Q2 dQ y2+ fdm(Q) , ρ (y)= r and (2.24)
R0 ≡ ρr(y) Z0 s ξd2m 0 r a4(y)
Idm
3 1+ ξ2 y2 , ξ y .1,
ξ O dm dm
(y)= dm (2.25)
R0  y+ 2(cid:2)ξI42dmy (cid:0)+O ξ(cid:1)4(cid:3)1y3 , ξdm y &5.
dm (cid:18) dm (cid:19)

B. Relevant scales in the ultra-relativistic and non-relativistic DM regimes
The evolution of the DM fluctuations presented here is valid generically for DM particles that decouple at redshift
z , being ultrarelativistic in the RD era and become non-relativistic in the same RD era. That is, the evolution
d
presentedhere is valid as long as ξ 1 which is the case fromeq.(2.3) providedDM decouples ultrarelativistically
dm
≫
deep enough in the RD era.
The frameworkpresentedin this paperis general,validfor anyDMparticle,notnecessarilyinthe keVscale. More
precisely, the treatment presented here is valid for ξ 1 and:
dm
≫
m m ξ
dm
1 = =3200
≫ T T z z
dphys d d d
which implies z 3200ξ . The redshift at decoupling turns to be
d dm
≫
z +1= Tdphys =1.571015 Tdphys gd 31 . (2.26)
d
T 100GeV 100
d
(cid:16) (cid:17)
where we used T =(2/g )1/3 T and T =0.2348 meV.
d d cmb cmb
DM particles are ultra-relativistic (UR) for z & z , z being the redshift at the transition from ultra-
trans trans
relativistic to non-relativistic DM particles
z +1 m 1.57 107 m gd 31 . (2.27)
trans
≡ T ≃ × keV 100
d
(cid:16) (cid:17)
Then, they become non-relativistic (NR) for z . z . In terms of the variable y [eq.(1.1)] the transition from UR
trans
to NR DM particles takes place around y y while decoupling happens well before y by y y :
trans trans d
∼ ∼
y =1/ξ 0.0002 , y =3200/z 2 10−12 .
trans dm d d
≃ ≃ ×
Notice that modes that reenter the horizon by the UR-NR transition y y , have from eqs. (2.30) and (2.40) of
trans
∼
the accompanying paper ref. [1], wavenumbers
1 ξ 2 Idm 1
k dm = 4 .
∼ η∗ y ∼ η∗ l ∼ l
trans pfs fs
Thatis,whenDMparticlesbecomenonrelativisticthefree-streaminglengthl isoftheorderofthecomovinghorizon
fs
[18].
8
a z +1 3200
UniverseEvent redshift z y= = eq
aeq z+1 ≃ z+1
DM decoupling z 1.61015 Tdp gd 13 y 2 10−12
d ∼ 100GeV 100 d ≃ ×
(cid:0) (cid:1)
neutrinodecoupling zν 6 109 yν 0.5 10−6
d ≃ × d ≃ ×
1 1
DMparticles transition from UR to NR z 1.6 107 keV gd 3 y = 0.0002
trans≃ × m (cid:0)100(cid:1) trans ξdm ≃
10−6 <y<0.01
Transition from theRD to theMD era z 3200 y =1
eq eq
≃
m 0.05eV
The lightest neutrinobecomes NR zν =95 ν yν =34
trans 0.05eV trans m
ν
Today z =0 y 3200
0 0
≃
TABLE I: Main eventsin theDM, neutrinos and universeevolution.
At decoupling, the covariant neutrino temperature, decoupling neutrino redshift and y variable are,
Tν =0.1710−3 eV , zν 6 109 and yν 0.5 10−6 .
d d ≃ × d ≃ ×
The lightest neutrinos become non-relativistic at a redshift
m 0.05eV
zν =95 ν and yν =34 .
trans 0.05eV trans m
ν
Namely,neutrinosbecomenon-relativisticintheMDerawhentheirdensityaswellastheirfluctuationsarenegligible.
Thus, we can treat the neutrinos as ultra-relativistic or neglect them.
The neutrino and photon fractions of the energy density are defined in general as
ρ (η) Ω ρ (η) Ω
ν ν γ γ
R (η) = , R (η) =
ν γ
≡ ρ(η) Ω +a(η)Ω ≡ ρ(η) Ω +a(η)Ω
r M r M
where ρ (η), ρ (η) and ρ(η) stand for the neutrino, photon and total energy density, respectively. In the radiation
ν γ
dominated eraΩ a(η)Ω and R (η)+R (η)=1. The neutrino fractionchangesafter neutrino decoupling when
r M ν γ
the cosmic temper≫ature crosses the e+ e− threshold, that is [2],
−
0.727 , 4 109 .z .6 109
× ×
Rν(η)= 0.41 , 3200.z .4 109 . (2.28)
×
 0 , 0 z .3200
≤
The quantity I defined by eq.(2.2) is dominated by the neutrino piece R (0) and takes the value
ξ ν
I R (0)=0.727. (2.29)
ξ ν
≃
In the MD dominated era both R (η) and R (η) become very small and can be neglected.
ν γ
9
Range of Validity ε(y,Q) l(y,Q)
URDM particles
ξ y 1
ξ y 1 Q dm
dm ≪ Q 1+ I3dm
q ξdm
0<y<10−6
Transition regime from UR
ξ y
to NRDM particles Q2+(ξ )2 y2 ArgSinh dm
dm (cid:18) Q (cid:19)
p
10−6 <y<0.01
NRDM particles
1 8ξ 1 Q 1 Q 2
0.01<y<3200 ξ y 2ArgSinh +log dm + 3y 1+y+y+2
dm − (cid:18)√y(cid:19) (cid:18) Q (cid:19) 2 ξdm − 8 (cid:18)yξdm(cid:19) h p i
ξ y 1
dm
≫
MD era
2 8ξ 1 Q
y 1 ξ y +log dm +
≫ dm −√y (cid:18) Q (cid:19) 2 ξdm
NRDM particles
TABLE II: The different regimes ultra-relativistic (UR), transition and non-relativistic (NR) of the free-streaming distance
l(y,Q). Notice that the second (third) formula for l(y,Q) is also valid in the first (fourth) formula for 0<y <10−6 (y 1).
≫
Inaddition,thethirdformulaofl(y,Q)fory 1matchesforξ y 1withtheasymptoticbehaviourofthesecondformula
dm
≪ ≫
for l(y,Q). The precise behaviours of l(y,Q) are derived in Appendix B and given by eqs.(B3)-(B8). When DM is UR l(y,Q)
growsasthecomovinghorizonη∗yandthusfree-streamingefficientlyerasesfluctuations. WhenDMbecomesNRl(y,Q)grows
much slower and free-streaming is inefficient to erase fluctuations.
We summarize in Table I the ranges of the redshift z and the variable y (the scale factor normalized to unity at
equilibration) for the main events in the DM, neutrinos and the universe evolution.
The free-streamingdistance l(y,Q)isexpressedby eq. (1.7). l(y,Q)canbe expressedingeneralinterms ofelliptic
integrals. In the present case where ξ 5000 we find in appendix B excellent approximations to l(y,Q) in terms
dm
∼
of simple elementary functions. We display the free-streamingdistance l(y,Q)and the particle energy ε(y,Q) for the
different regimes in Table II. It must be stressed that each of the four formulas displayed in Table II match with its
neighboring expression as discussed in appendix B.
From eqs.(2.3) and (2.4) we obtain for the dimensionless variable α,
1
keV 100 3
α=58.37 k kpc. (2.30)
m g
(cid:18) d (cid:19)
In terms of α, the primordial gravitationalpotential eq.(3.17) in the accompanying paper [1] becomes,
1.848 α 12(ns−1) keV 3 100
ψ(0,α~)= (kpc)3 G(α~), (2.31)
α32 (cid:18)α0(cid:19) (cid:18) m (cid:19) gd
1
where α =1.16710−4 (keV/m) (100/g )3 and
0 d
<G(α~)G∗(α~′)>=δ(α~ α~′).
−
III. FROM THE ULTRARELATIVISTIC TO THE NON-RELATIVISTIC REGIME OF THE DM IN
THE VOLTERRA EQUATIONS
We investigate here the system of Volterra integral equations (2.5)-(2.6) first in the transition regime for DM and
then in the non-relativistic DM regime.
10
14
l(y,Q=0.1)/Q vs. log y
10
l(y,Q=1)/Q vs. log y
10
12 l(y,Q=10)/Q vs. log y
10
10
8
6
4
2
0
-5 -4 -3 -2 -1 0 1 2 3 4
FIG. 1: The free-streaming length in dimensionless variables l(y,Q) divided by Q vs. log y for Q = 0.1, 1 and 10. [Recall
10
that λ = (η∗/ξ ) Q l(y,Q) [1]]. We explicitly compute l(y,Q) in appendix B. l(y,Q) is given in the different regimes by
FS dm
eqs.(B2), (B3), (B8) and (B9). Notice that log y=0 corresponds to equilibration. We choose here ξ =5000.
10 dm
A. Transition Regime
We consider here the coupled Volterra integral equations (2.5-(2.6)) in the transition regime from ultrarelativistic
to non-relativistic DM particles 0.5 10−6 < y < 0.01 well inside the RD era where the neutrinos are ultrarelativistic
and they have already decoupled.
The second entry of Table II the one-particle energy ε(y,Q) = (ξ )2 y2+Q2 and the free-streaming length
dm
l(y,Q) applies now. Therefore, we have from eq.(B3),
p
3 Q 2 ξ y 1 3 Q 2 Q
l(y,Q)= 1 ArgSinh dm 1 y y2+ + (y3).
" − 16 (cid:18)ξdm(cid:19) # (cid:18) Q (cid:19)− 2 (cid:18) − 8 (cid:19)s (cid:18)ξdm(cid:19) − ξdm O
 
2 2
Q 3 Q ξ y Q 3 Q
l (y,y′)= 1 ArgSinh dm 1 y y2+ y y′ .
Q
2 " − 16 (cid:18)ξdm(cid:19) # (cid:18) Q (cid:19)− 4 (cid:18) − 8 (cid:19)s (cid:18)ξdm(cid:19) −{ ⇒ }
These formulas are to be inserted in eqs.(2.12)-(2.18) for a(y,α), aσ(y,α), N (y,y′), Nσ(y,y′), U (y,y′) and
α α α
Uσ(y,y′).
α
B. Non-relativistic Regime
We write here the Volterra integral equation in the non-relativistic regime 3200>y >0.01.
The thirdentryofTableII forthe one-particleenergyε(y,Q) ξ y andthe free-streaminglengthl(y,Q)applies
dm
≃
in this case. Notice that the difference of the free-streaming lengths which appears in the integrand of the kernel
N (y,y′) eq.(2.14) is now Q-independent because the DM particles are non-relativistic:
α
Q
l (y,y′)= [l(y,Q) l(y′,Q)]=Q [s(y) s(y′)] ,
Q
2 − −