April 2007
Constraints on the Very Early Universe from
Thermal WIMP Dark Matter
Manuel Dreesa, ∗, Hoernisa Iminniyaza,b, † and Mitsuru Kakizakia, ‡
aPhysikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany
bPhysics Dept., Univ. of Xinjiang, 830046 Urumqi, P.R. China
Abstract
We investigate the relic density nχ of non–relativistic long–lived or sta-
ble particles χ in non–standard cosmological scenarios. We calculate the
relic abundance starting from arbitrary initial temperatures of the radiation–
dominated epoch, and derive the lower bound on the initial temperature
T0 ≥ mχ/23, assuming that thermally produced χ particles account for the
dark matter energy density in the universe; this bound holds for all χ annihi-
lation cross sections. We also investigate cosmological scenarios with modified
expansion rate. Even in this case an approximate formula similar to the stan-
dard one is capable of predicting the final relic abundance correctly. Choosing
the χ annihilation cross section such that the observed cold dark matter abun-
dance is reproduced in standard cosmology, we constrain possible modifications
of the expansion rate at T ∼ mχ/20, well before Big Bang Nucleosynthesis.
∗drees@th.physik.uni-bonn.de
†hoernisa@th.physik.uni-bonn.de
‡kakizaki@th.physik.uni-bonn.de
http://arxiv.org/abs/0704.1590v2
1 Introduction
One of the most notable recent developments in cosmology is the precise determina-
tion of cosmological parameters from observations of the large–scale structure of the
universe, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP). In
particular, the accurate determination of the non–baryonic cold Dark Matter (DM)
density [1],
0.08 < ΩCDMh
2 < 0.12 (95% C.L.) , (1)
has great influence on particle physics models which possess dark matter candidates
[2, 3]. The requirement that the predicted DM density falls in the range (1) is a
powerful tool for discriminating between various models and for constraining the
parameter space of surviving models.
Many dark matter candidate particles have been proposed. In particular long–
lived or stable weakly interacting massive particles (WIMPs) with weak–scale masses
are excellent candidates. In standard cosmology WIMPs decoupled from the thermal
background during the radiation–dominated epoch after inflation. In this framework
convenient and accurate analytic approximate solutions for the relic abundance have
been derived [4, 5]. One of the best motivated candidates for WIMPs is the lightest
neutralino in supersymmetric (SUSY) models. Assuming that the neutralino is the
lightest supersymmetric particle (LSP) stabilized due to R–parity, its relic abundance
has been extensively discussed [3]. Similar analyses have also been performed for
other WIMPs whose existence is postulated in other extensions of the Standard
Model (SM) of particle physics. In many cases the cosmologically favored parameter
space of WIMP models can be directly tested at the CERN Large Hadron Collider
(LHC) in a few years [6]. The same parameter space often also leads to rates of
WIMP interactions with matter within the sensitivity of near–future direct DM
detection experiments.
This discussion shows that we are now entering an interesting time where the
standard cosmological scenario can be examined by experiments at high–energy col-
liders as well as DM searches [7]. In this respect we should emphasize that the
relic abundance of thermally produced WIMPs depends not only on their annihila-
tion cross section, which can be determined by particle physics experiments, but in
general is also very sensitive to cosmological parameters during the era of WIMP
production and annihilation. Of particular importance are the initial temperature
T0 at which WIMPs began to be thermally produced, and the expansion rate of the
universe H .
In the standard cosmological scenario, the expansion rate is uniquely determined
through the Friedmann equation of general relativity. In this scenario the density of
WIMPs with massmχ followed its equilibrium value until the freeze–out temperature
TF ≃ mχ/20. Below TF , interactions of WIMPs are decoupled, and thus the present
density is independent of T0 as long as T0 > TF .
It should be noted that in non–standard scenarios the relic density can be larger
or smaller than the value in the standard scenario. One example is the case where
T0 is smaller than or comparable to TF , which can be realized in inflationary models
with low reheat temperature. Since in many models the inflationary energy scale
must be much higher than mχ in order to correctly predict the density perturbations
[8], the standard assumption T0 > TF is not unreasonable. On the other hand, the
constraint on the reheat temperature from Big Bang Nucleosynthesis (BBN) is as low
as T0 >∼ MeV [9, 10]. From the purely phenomenological viewpoint, it is therefore
also interesting to investigate the production of WIMPs in low reheat temperature
scenarios [9, 11, 12, 13, 14].
The standard scenario also assumes that entropy per comoving volume is con-
served for all temperatures T ≤ TF . Late entropy production can dilute the predicted
relic density [15, 16]. The reason is that the usual calculation actually predicts the
ratio of the WIMP number density to the entropy density. On the other hand, if
late decays of a heavier particle non–thermally produce WIMPs in addition to the
usual thermal production mechanism, the resulting increase of the WIMP density
competes with the dilution caused by the decay of this particle into radiation, which
increases the entropy density [17, 18, 19, 20, 13, 21].
Another example of a non–standard cosmology changing the WIMP relic density
is a modified expansion rate of the universe. This might be induced by an anisotropic
expansion [16], by a modification of general relativity [16, 22], by additional contri-
butions to the total energy density from quintessence [23], by branes in a warped
geometry [24], or by a superstring dilaton [25].
These examples show that, once the WIMP annihilation cross section is fixed,
with the help of precise measurements of the cold dark matter density we can probe
the very early stage of the universe at temperatures of O(mχ/20) ∼ 10 GeV. This
is reminiscent of constraining the early evolution of the universe at T = O(100) keV
∗Note that TF can be formally defined in the standard way even if T0 < TF . In this case WIMPs
never were in full equilibrium, and correspondingly never “froze out”.
using the primordial abundances of the light elements produced by BBN.
The goal of this paper is to investigate to what extent the constraint (1) on
the WIMP relic abundance might allow us to derive quantitative constraints on
modifications of standard cosmology. So far the history of the universe has been
established by cosmological observations as far back as the BBN era. In this paper
we try to derive bounds on cosmological parameters relevant to the era before BBN.
Rather than studying specific extensions of the standard cosmological scenario, we
simply parameterize deviations from the standard scenario, and attempt to derive
constraints on these new parameters. Since we only have the single constraint (1),
for the most part we only allow a single quantity to differ from its standard value.
We expect that varying two quantities simultaneously will allow to get the right
relic density for almost any WIMP annihilation cross section. This has been shown
explicitly in [13] for the case that both late entropy production and non–thermal
WIMP production are considered, even if both originate from the late decay of a
single scalar field.
We first analyze the dependence of the WIMP abundance on the initial temper-
ature T0 of the conventional radiation–dominated epoch. We showed in [14] that for
fixed T0 the predicted WIMP relic density reaches a maximum as the annihilation
cross section is varied from very small to very large values. A small annihilation cross
section corresponds to a large TF > T0; in this case the relic density increases with
the annihilation cross section, since WIMP production from the thermal plasma is
more important than WIMP annihilation. On the other hand, increasing this cross
section reduces TF ; once TF < T0 a further increase of the cross section leads to
smaller relic densities since in this case WIMPs continue to annihilate even after the
temperature is too low for WIMP production. Here we turn this argument around,
and derive the lower bound on T0 ≥ mχ/23 under the assumption that all WIMPs
are produced thermally. Note that we do not need to know the WIMP annihilation
cross section to derive this bound.
We then examine the dependence of the predicted WIMP relic density on the
expansion rate in the epoch prior to BBN, where we allow the Hubble parameter to
depart from the standard value. The standard method of calculating the thermal
relic density [2, 4] is found to be still applicable in this case. Our working hypothesis
here is that the standard prediction for the Hubble expansion rate is essentially
correct, i.e. that the true expansion rate differs by at most a factor of a few from
the standard prediction. We then simply employ a generic Taylor expansion for the
temperature dependence of this modification factor; note that the success of standard
BBN indicates that this factor cannot deviate by more than ∼ 20% from unity at
low temperatures, T <∼ 1 MeV. Similarly, we assume that the WIMP annihilation
cross section has been determined (from experiments at particle colliders) to have
the value required in standard cosmology. Our approach is thus quite different from
that taken in [7], where present upper bounds on the fluxes of WIMP annihilation
products are used to place upper bounds on the Hubble expansion rate during WIMP
decoupling. The advantage of their approach is that no prior assumption on the
WIMP annihilation cross section needs to be made, whereas we assume a cross
section that reproduces the correct relic density in the standard scenario. On the
other hand, the bounds derived in refs.[7] are still quite weak, allowing the Hubble
parameter to exceed its standard prediction by a factor >∼ 30; moreover, no lower
bound on H can be derived in this fashion.
The remainder of this paper is organized as follows: In Sec. 2 we will briefly review
the calculation of the WIMP relic abundance assuming a conventional radiation–
dominated universe, and derive the lower bound on the initial temperature T0. In
Sec. 3 we discuss the case where the pre–BBN expansion rate is allowed to depart
from the standard one. Using approximate analytic formulae for the predicted WIMP
relic density for this scenario, we derive constraints on the early expansion parameter.
Finally, Sec. 4 is devoted to summary and conclusions.
2 Relic Abundance in the Radiation–Dominated
Universe
We start the discussion of the relic density nχ of stable or long–lived particles χ by
reviewing the structure of the Boltzmann equation which describes their creation and
annihilation. The goal of this Section is to find the lowest possible initial temperature
of the radiation–dominated universe, assuming that the present relic abundance of
cold dark matter is entirely due to thermally produced χ particles.
As usual, we will assume that χ is self–conjugate†, χ = χ̄, and that some symme-
try, for example R–parity, forbids decays of χ into SM particles; the same symmetry
then also forbids single production of χ from the thermal background. However,
†The case χ 6= χ̄ differs in a non–trivial way only in the presence of a χ− χ̄ asymmetry, i.e. if
nχ 6= nχ̄.
the creation and annihilation of χ pairs remains allowed. The time evolution of the
number density nχ of particles χ in the expanding universe is then described by the
Boltzmann equation [2],
+ 3Hnχ = −〈σv〉(n2χ − n2χ,eq) , (2)
where nχ,eq is the equilibrium number density of χ, and 〈σv〉 is the thermally averaged
annihilation cross section multiplied with the relative velocity of the two annihilating
χ particles. Finally, the Hubble parameter H = Ṙ/R is the expansion rate of the
universe, R being the scale factor in the Friedmann–Robertson–Walker metric. The
first (second) term on the right–hand side of Eq.(2) describes the decrease (increase)
of the number density due to annihilation into (production from) lighter particles.
Eq.(2) assumes that χ is in kinetic equilibrium with standard model particles.
It is useful to rewrite Eq.(2) in terms of the scaled inverse temperature x = mχ/T
as well as the dimensionless quantities Yχ = nχ/s and Yχ,eq = nχ,eq/s. The entropy
density is given by s = (2π2/45)g∗sT
3, where
g∗s =
i=bosons
i=fermions
. (3)
Here gi denotes the number of intrinsic degrees of freedom for particle species i
(e.g. due to spin and color), and Ti is the temperature of species i. Assuming that
the universe expands adiabatically, the entropy per comoving volume, sR3, remains
constant, which implies
+ 3Hs = 0 . (4)
The time dependence of the temperature is then given by
. (5)
Therefore the Boltzmann equation (2) can be written as
= −〈σv〉s
(Y 2χ − Y 2χ,eq) . (6)
Thermal production of WIMPs takes place during the radiation–dominated epoch,
when the expansion rate is given by
, (7)
with MPl = 2.4× 1018 GeV being the reduced Planck mass and
i=bosons
i=fermions
. (8)
In the following we use Hst to denote the standard expansion rate (7). If the post–
inflationary reheat temperature was sufficiently high, WIMPs reached full thermal
equilibrium. This remains true for temperatures well below mχ. We can therefore
use the non–relativistic expression for the χ equilibrium number density,
nχ,eq = gχ
e−mχ/T . (9)
In the absence of non–thermal production mechanisms, nχ ≤ nχ,eq at early times.
The annihilation rate Γ = nχ〈σv〉 then depends exponentially on T , and thus drops
more rapidly with decreasing temperature than the expansion rate Hst of Eq.(7)
does. When the annihilation rate falls below the expansion rate, the number density
of WIMPs ceases to follow its equilibrium value and is frozen out.
For T ≪ mχ the annihilation cross section can often (but not always [5]) be
approximated by a non–relativistic expansion in powers of v2. Its thermal average
is then given by
〈σv〉 = a + b〈v2〉+O(〈v4〉) = a+ 6b
. (10)
In this standard scenario, the following approximate formula has been shown [4, 2, 5]
to accurately reproduce the exact (numerically calculated) relic density:
Yχ,∞ ≡ Yχ(x → ∞) ≃
1.3 mχMPl
g∗(xF )(a/xF + 3b/x
, (11)
with xF = mχ/TF , TF being the decoupling temperature. For WIMPs, xF ≃ 22.
Here we assume g∗ ≃ g∗s and dg∗/dx ≃ 0. It is useful to express the χ mass density
as Ωχ = ρχ/ρc, ρc = 3H
Pl being the critical density of the universe. The present
relic mass density is then given by ρχ = mχnχ,∞ = mχs0Yχ,∞; here s0 ≃ 2900 cm−3
is the present entropy density. Eq.(11) then leads to
2 = 2.7× 1010 Yχ,∞
100 GeV
≃ 8.5× 10
−11 xF GeV
g∗(xF )(a+ 3b/xF )
, (12)
where h ≃ 0.7 is the scaled Hubble constant in units of 100 km sec−1 Mpc−1. We
defer further discussions of this expression to Sec. 3, where scenarios with modified
expansion rate are analyzed. Note that in the standard scenario leading to Eq.(12),
the present χ relic density is inversely proportional to its annihilation cross section
and has no dependence on the reheat temperature. Recall that this result depends
on the assumption that the highest temperature in the post–inflationary radiation
dominated epoch, which we denote by T0, exceeded TF significantly.
On the other hand, if T0 was too low to fully thermalize WIMPs, the final result
for Ωχ will depend on T0. In particular, if WIMPs were thermally produced in a com-
pletely out–of–equilibrium manner starting from vanishing initial abundance during
the radiation–dominated era, such that WIMP annihilation remains negligible, the
present relic abundance is given by [14]
Y0(x → ∞) ≃ 0.014 g2χg−3/2∗ mχMPle−2x0x0
. (13)
Note that the final abundance depends exponentially on T0, and increases with
increasing cross section.
In in–between cases where WIMPs are not completely thermalized but WIMP
annihilation can no longer be neglected, we have shown [14] that re–summing the
first correction term δ enables us to reproduce the full temperature dependence of
the density of WIMPs:
1− δ/Y0
≡ Y1,r . (14)
Here δ < 0 describes the annihilation of WIMPs produced according to Eq.(13):
δ(x → ∞) ≃ −2.5 × 10−4 g4χg−5/2∗ m3χM3Ple−4x0x0
. (15)
Since δ is proportional to the third power of the cross section, the re–summed ex-
pression Y1,r is inversely proportional to the cross section for large cross section. In
ref.[14] we have shown that this feature allows the approximation (14) to be smoothly
matched to the standard result (12). Not surprisingly, as long as we only consider
thermal χ production, decreasing T0 can only reduce the final χ relic density.
With the help of these results, we can explore the dependence of the χ relic
density on T0 as well as on the annihilation cross section. Some results are shown in
Fig. 1, where we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0,
mχ = 100 GeV, gχ = 2 and g∗ = 90.
The results depicted in this Figure can be understood as follows. For small T0,
i.e. large x0, Eq.(13) is valid, leading to a very strong dependence of Ωχh
2 on x0.
10−10 10−9 10−8 10−7
a (GeV−2 )
b = 0
10−310−4
10−9 10−8 10−7 10−6
b (GeV−2 )
a = 0
10−310−4
Figure 1: Contour plots of the present relic abundance Ωχh2. Here we take (a) a 6= 0, b = 0, and
(b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2, g∗ = 90. The shaded region
corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh
2 < 0.12
(95% C.L.).
Recall that in this case the relic density is proportional to the cross section. In this
regime one can reproduce the relic density (1) with quite small annihilation cross
section, a + 6b/x0 <∼ 10−9 GeV−2, for some narrow range of initial temperature,
x0 <∼ 22.5. Note that this allows much smaller annihilation cross sections than the
standard result, at the cost of a very strong dependence of the final result on the
initial temperature T0.
In this Section we set out to derive a lower bound on T0. In this regard the
region of parameter space described by Eq.(13) is not optimal. Increasing the χ
annihilation cross section at first allows to obtain the correct relic density for larger
x0, i.e. smaller T0. However, the correction δ then quickly increases in size; as noted
earlier, once |δ| > Y0 a further increase of the cross section will lead to a decrease
of the final relic density. The lower bound on T0 is therefore saturated if Ωχh
2 as a
function of the cross section reaches a maximum. From Fig.1 we read off
T0 ≥ mχ/23 , (16)
if we require Ωχh
2 to fall in the range (1).
We just saw that in the regime where this bound is saturated, the final relic den-
sity is (to first order) independent of the annihilation cross section, ∂(Ωχh
2)/∂〈σv〉 =
0. If T0 is slightly above the absolute lower bound (16), the correct relic density can
therefore be obtained for a rather wide range of cross sections. For example, if
x0 = 22.5, the entire range 3 × 10−10 GeV−2 <∼ a <∼ 2 × 10−9 GeV
−2 is allowed. Of
course, the correct relic density can also be obtained in the standard scenario of
(arbitrarily) high T0, if a + 3b/22 falls within ∼ 20% of 2× 10−9 GeV−2.
3 Relic Abundance for Modified Expansion Rate
In this section we discuss the calculation of the WIMP relic density nχ in modi-
fied cosmological scenarios where the expansion parameter of the pre–BBN universe
differed from the standard value Hst of Eq.(7). For the most part we will assume
that WIMPs have been in full thermal equilibrium. Various cosmological models
predict a non–standard early expansion history [22, 23, 24, 25]. Here we analyze to
what extent the relic density of WIMP Dark Matter can be used to constrain the
Hubble parameter during the epoch of WIMP decoupling. As long as we assume
large T0 we can use a modification of the standard treatment [4, 2] to estimate the
relic density for given annihilation cross section and expansion rate. We will show
that the resulting approximate solutions again accurately reproduce the numerically
evaluated relic abundance.
Let us introduce the modification parameter A(x), which parameterizes the ratio
of the standard value Hst(x) to the assumed H(x):
A(x) =
Hst(x)
. (17)
Note that A > 1 means that the expansion rate is smaller than in standard cos-
mology. Allowing for this modified expansion rate, the Boltzmann equation (6) is
altered to
G(x)mχMPl
〈σv〉A(x)
Y 2χ − Y 2χ,eq
, (18)
where
G(x) =
. (19)
Following refs.[4, 2], we can obtain an approximate solution of this equation by
considering the differential equation for ∆ = Yχ − Yχ,eq. For temperatures higher
than the decoupling temperature, Yχ tracks Yχ,eq very closely and the ∆
2-term can
be ignored:
≃ −dYeq
− 4π√
mχMPl
G(x)〈σv〉A(x)
(2Yχ,eq∆) . (20)
Here dYχ,eq/dx ≃ −Yχ,eq for x ≫ 1. In order to keep |∆| small, the derivative d∆/dx
must also be small, which implies
∆ ≃ x
90)mχMPlG(x)〈σv〉A(x)
. (21)
This solution is used down to the freeze–out temperature TF , defined via
∆(xF ) = ξYχ,eq(xF ) , (22)
where ξ is a constant of order of unity. This leads to the following expression:
xF = ln
ξmχMPlgχ
〈σv〉A(x)
xg∗(x)
, (23)
which can e.g. be solved iteratively. In our numerical calculations we will choose
2− 1 [4, 2].
On the other hand, for low temperatures (T < TF ), the production term ∝ Y 2χ,eq
in Eq.(18) can be ignored. In this limit, Yχ ≃ ∆, and the solution of Eq.(18) is given
∆(xF )
∆(x → ∞)
mχMPlI(xF ) , (24)
where the annihilation integral is defined as
I(xF ) =
G(x)〈σv〉A(x)
. (25)
Assuming ∆(x → ∞) ≪ ∆(xF ), the final relic abundance is
Yχ,∞ ≡ Yχ(x → ∞) =
90)mχMPlI(xF )
. (26)
Plugging in numerical values for the Planck mass and for today’s entropy density,
the present relic density can thus be written as
8.5× 10−11
I(xF ) GeV
. (27)
The constraint (1) therefore corresponds to the allowed range for the annihilation
integral
7.1× 10−10 GeV−2 < I(xF ) < 1.1× 10−9 GeV−2 . (28)
The standard formula (12) for the final relic density is recovered if A(x) is set to
unity and G(x) is replaced by the constant
g∗(xF ).
The further discussion is simplified if we use the normalized temperature z =
T/mχ ≡ 1/x, rather than x. Phenomenologically A(z) can be any function subject
to the condition that A(z) approaches unity at late times in order not to contradict
the successful predictions of BBN. We need to know A(z) only for the interval from
around the freeze-out to BBN: zBBN ∼ 10−5 − 10−4 <∼ z <∼ zF ∼ 1/20. This suggests
a parameterization of A(z) in terms of a power series in (z − zF,st):
A(z) = A(zF,st) + (z − zF,st)A′(zF,st) +
(z − zF,st)2A′′(zF,st) , (29)
where zF,st is the normalized freeze–out temperature in the standard scenario and
a prime denotes a derivative with respect to z. The ansatz (29) should be of quite
general validity, so long as the modification of the expansion rate is relatively modest.
This suits our purpose, since we wish to find out what constraints can be derived on
the expansion history if standard cosmology leads to the correct WIMP relic density.
We further introduce the variable
k = A(z → 0) = A(zF,st)− zF,stA′(zF,st) +
z2F,stA
′′(zF,st) , (30)
which describes the modification parameter at late times. Since zBBN is almost
zero, we treat k as the modification parameter at the era of BBN in this paper.‡
Deviations from k = 1 are conveniently discussed in terms of the equivalent number
of light neutrino degrees of freedom Nν . BBN permits that the number of neutrinos
differs from the standard model value Nν = 3 by δNν = 1.5 or so [26]. We therefore
take the uncertainty of k to be 20%. In the following we treat A(zF,st), A
′(zF,st) and
k as free parameters; A′′(zF,st) is then a derived quantity.
Note that we allow A(z) to cross unity, i.e. to switch from an expansion that
is faster than in standard cosmology to a slower expansion or vice versa. This is
illustrated in Fig. 2, which shows examples of possible evolutions of A(z) as function
of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each case
we consider scenarios with A(zF ) = 1.3 (slower expansion at TF than in standard
cosmology) as well as A(zF ) = 0.7 (faster expansion); moreover, we allow the change
of A at z = zF to be either positive or negative. However, we insist that H remains
positive at all times, i.e. A(z) must not cross zero. This excludes scenarios with
very large positive A′(zF,st), which would lead to A < 0 at some z < zF . Similarly,
‡Presumably the Hubble expansion rate has to approach the standard rate even more closely
for T < TBBN. However, since all WIMP annihilation effectively ceased well before the onset of
BBN, this epoch plays no role in our analysis.
 0  0.01  0.02  0.03  0.04  0.05
z = T/mχ
k = 1.2
A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
 0  0.01  0.02  0.03  0.04  0.05
z = T/mχ
k = 0.8A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
Figure 2: Examples of possible evolutions of the modification parameter A(z) as function of z
for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each frame we choose
A(zF ) = 1.3, A
′(zF ) = −3 (thick line), A(zF ) = 1.3, A′(zF ) = 9 (dashed), A(zF ) = 0.7, A′(zF ) =
−3 (dotted), A(zF ) = 0.7, A′(zF ) = 9 (dot–dashed).
demanding that our ansatz (29) remains valid for some range of temperatures above
TF excludes scenarios with very large negative A
′(zF,st). We will come back to this
point shortly.
Eq.(23) shows that zF 6= zF,st (xF 6= xF,st) if A(zF ) 6= 1. This is illustrated by
Fig. 3, which shows the difference between xF and xF,st in the (A(zF,st), A
′(zF,st))
plane. Here we take parameters such that Ωχh
2 = 0.099 in the standard cosmology,
which is recovered for A(zF,st) = 1, A
′(zF,st) = 0. Due to the logarithmic dependence
on A, xF (or zF ) differs by at most a few percent from its standard value if A(zF,st) is
O(1). Since TF only depends on the expansion rate at TF , it is essentially insensitive
to the derivative A′(zF,st).
In our treatment the modification of the expansion parameter affects the WIMP
relic density mostly via the annihilation integral (25). In terms of the normalized
temperature z, the latter can be rewritten as
I(zF ) =
dz G(z)〈σv〉A(z) . (31)
One advantage of the expansion (29) is that this integral can be evaluated analyti-
cally:
I(zF ) ≃ G(zF )
k(azF + 3bz
F ) + (A
′(zF,st)− zF,stA′′(zF,st))
z2F + 2bz
A′′(zF,st)
z3F +
. (32)
 0.6  0.8  1  1.2  1.4
A(zF,st )
xF − x F,st
a = 2.0*10−9  GeV −2
b = 0
k = 1
xF,st = 22.0
−0.4 −0.2 0 0.2 0.4
Figure 3: Contour plot of xF −xF,st in the (A(zF,st), A′(zF,st)) plane. Here we take a = 2.0×10−9
GeV−2, b = 0, mχ = 100 GeV, gχ = 2, g∗ = 90 (constant) and k = 1. This parameter set produces
xF,st = 22.0 and Ωχh
2 = 0.099 for the standard approximation.
 0.6  0.8  1  1.2  1.4
A(zF,st )
2(approx) / Ωχh
2(exact)
0.997
0.998
0.999
1.000
a = 2.0*10−9 GeV−2
b = 0
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st)
2(approx) / Ωχh
2(exact)
0.996
0.998
1.000
1.002
1.004 a = 0
b = 1.5*10−8 GeV−2
k = 1
Figure 4: Ratio of the analytic result of the relic density to the exact value in the (A(zF,st),
A′(zF,st)) plane for a = 2.0× 10−9 GeV−2, b = 0 (left frame) and for a = 0, b = 1.5× 10−8 GeV−2
(right). The other parameters are as in Fig. 3.
Here we have assumed that G(z) varies only slowly.
Before proceeding, we first have to convince ourselves that the analytic treatment
developed in this Section still works for A 6= 1. This is demonstrated by Fig. 4,
which shows the ratio of the analytic solution obtained from Eqs. (27) and (32)
to the exact one, obtained by numerically integrating the Boltzmann equation (18),
assuming constant g∗. We see that our analytical treatment is accurate to better
than 1%, and can thus safely be employed in the subsequent analysis.
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 2.0*10−9 GeV −2
b = 0
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1.2
 0.6  0.8  1  1.2  1.4
A(zF,st ) 
a = 2.0*10−9 GeV−2
b = 0
k = 0.8
Figure 5: Contour plots of the relic abundance in the (A(zF,st), A′(zF,st)) plane. Here we choose
(a) a = 2.0×10−9 GeV−2, b = 0, k = 1; (b) a = 0, b = 1.5×10−8 GeV−2, k = 1; (c) a = 2.0×10−9
GeV−2, b = 0, k = 1.2; (d) a = 2.0× 10−9 GeV−2, b = 0, k = 0.8. The other parameters are as in
Fig. 3.
We are now ready to analyze the impact of the modified expansion rate on the
WIMP relic density. In Fig. 5, we show contour plots of Ωχh
2 in the (A(zF,st),
A′(zF,st)) plane. Recall that large (small) values of A correspond to a small (large)
expansion rate. Since a smaller expansion rate allows the WIMPs more time to
annihilate, A > 1 leads to a reduced WIMP relic density, whereas A < 1 means
larger relic density, if the cross section is kept fixed.
However, unlike the freeze–out temperature, the annihilation integral is sensitive
to A(z) for all z ≤ zF . Note that A′(zF,st) > 0 implies A(z) < A(zF,st) for z <
zF,st ≃ zF . A positive first derivative, A′(zF,st) > 0, can therefore to some extent
compensate for A(zF,st) > 1; analogously, a negative first derivative can compensate
for A(zF,st) < 1. This explains the slopes of the curves in Fig. 5. Recall also that
A′(zF,st) = 0 does not imply a constant modification factor A(z); rather, the term
∝ A′′(zF,st) in Eq.(29) makes sure that A approaches k as z → 0. This explains why
a change of A by some given percentage leads to a smaller relative change of Ωχh
2, as
can be seen in the Figure. This also illustrates the importance of ensuring appropriate
(near–standard) expansion rate in the BBN era. Finally, since the expansion rate
at late times is given by Hst/k, bigger (smaller) values of k imply that the χ relic
density is reduced (enhanced).
Fig. 5 shows that we need additional physical constraints if we want to derive
bounds on A(zF,st) and/or A
′(zF,st). Once the annihilation cross section is known,
the requirement (1) will single out a region in the space spanned by our three new
parameters (including k) which describe the non–standard evolution of the universe,
but this region is not bounded. Such additional constraints can be derived from
the requirement that the Hubble parameter should remain positive throughout the
epoch we are considering. As noted earlier, requiring H > 0 for all T < TF,st leads
to an upper bound on A′(zF,st); explicitly,
A′(zF,st) <
A(zF,st) +
kA(zF,st)
zF,st
. (33)
On the other hand, a lower bound on A′(zF,st) is obtained from the condition that
the modified Hubble parameter is positive between the highest temperature Ti where
the ansatz (29) holds and TF,st:
A′(zF,st) > −
zi − zF,st
2− zi
zF,st
A(zF,st) + k
zF,st
, (34)
for (1− zF,st/zi)2k < A(zF,st), and
A′(zF,st) >
A(zF,st)−
kA(zF,st)
zF,st
, (35)
for A(zF,st) < (1− zF,st/zi)2k, where zi = Ti/mχ.
Evidently the lower bound on A′(zF,st) depends on zi, i.e. on the maximal tem-
perature where we assume our ansatz (29) to be valid. In ref.[14] we have shown that
in standard cosmology (A ≡ 1) essentially full thermalization is already achieved for
xi <∼ xF − 0.5, even if nχ(xi) = 0. However, it seems reasonable to demand that H
should remain positive at least up to xi = xF−(a few). In Fig. 6 we therefore show
 0  1  2  3  4  5  6  7
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1
xF,st − x i  = 4
xF,st − x i  = 10
 0  1  2  3  4  5  6  7
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
xF,st − x i  = 4
xF,st − x i  = 10
Figure 6: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), A′(zF,st)) plane. The
dashed line corresponds to the upper bound on A′(zF,st). The dotted lines correspond to the lower
bounds calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and
a = 0, b = 1.5× 10−8 GeV−2 (right frame). The other parameters are as in Fig 3.
the physical constraints on the modification parameter A(z) for xF,st − xi = 4, 10
and k = 1. The dashed and dotted lines correspond to the upper and lower bounds
on A′(zF,st), described by Eq.(33) and Eqs.(34), (35), respectively. We see that when
xF,st − xi = 4 the allowed region is 0.4 <∼ A(zF,st) <∼ 6.5 with −60 <∼ A′(zF,st) <∼ 400
for b = 0 (left frame), and 0.4 <∼ A(zF,st) <∼ 4.5 with −60 <∼ A′(zF,st) <∼ 300 for a = 0
(right frame). When xF,st − xi = 10, the lower bounds are altered to 0.6 <∼ A(zF,st),
−10 <∼ A′(zF,st) for b = 0 (left frame), and 0.6 <∼ A(zF,st), −20 <∼ A′(zF,st) for a = 0
(right frame). Note that the lower bounds on A(zF,st), which depend only weakly on
xi so long as it is not very close to xF , are almost the same in both cases, which also
lead to very similar relic densities in standard cosmology. However, the two upper
bounds differ significantly. The reason is that large values of A(zF,st), i.e. a strongly
suppressed Hubble expansion, require some degree of finetuning: One also has to
take large positive A′(zF,st), so that A becomes smaller than one for some range of
z values below zF , leading to an annihilation integral of similar size as in standard
cosmology. Since the b−terms show different zF dependence in the annihilation in-
tegral (32), the required tuning between A(zF,st) and A
′(zF,st) is somewhat different
than for the a−terms, leading to a steeper slope of the allowed region. This allowed
region therefore saturates the upper bound (33) on the slope for somewhat smaller
A(zF,st).
The effect of this tuning can be seen by analyzing the special case where A′′(zF,st) =
 0  0.5  1  1.5  2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
 4 0.08
 0  0.5  1  1.5  2
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
Figure 7: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), k) plane for A′′(zF,st) = 0.
The dotted lines correspond to the lower bounds of A(zF,st), calculated for xF,st − xi = 4, 10. We
take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5 × 10−8 GeV−2 (right frame).
The other parameters are as in Fig 3.
0. The modification parameter then reads
A(z) =
A(zF,st)− k
zF,st
z + k . (36)
Note that A is now a monotonic function of z, making large cancellations in the
annihilation integral impossible. Imposing that A(z) remains positive for zF,st ≤
z ≤ zi leads to the lower limit
A(zF,st) >
zF,st
k . (37)
There is no upper bound, since A(z) is now automatically positive for all z ∈ [0, zF,st]
if A(zF,st) and A(0) ≡ k are both positive. Fig. 7 shows constraints on the relic
abundance in the (A(zF,st), k) plane for A
′′(zF,st) = 0. The dotted lines correspond
to the lower bounds (37) on A(zF,st) for xF,st − xi = 4, 10. As noted earlier, k
is constrained by the BBN bound. This leads to the bounds 0.5 <∼ A(zF,st) <∼ 1.8
for b = 0 (left frame), and 0.65 <∼ A(zF,st) <∼ 1.6 for a = 0 (right frame), when
xF,st − xi = 10. Evidently the constraints now only depend weakly on whether the
a− or b−term dominates in the annihilation cross section. As the initial temperature
is lowered, the impact of the constraint (37) disappears.
So far we have assumed in this Section that the reheat temperature was high
enough for WIMPs to have attained full thermal equilibrium. If this was not the
case, the initial temperature as well as the suppression parameter affects the final
 0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1, A’’(zF,st ) = 0
0.080.12
Figure 8: Contour plot of the relic abundance Ωχh2 in the (A(zF,st), x0) plane. Here we choose
a = 2.0 × 10−9 GeV−2, b = 0, k = 1, A′′(zF,st) = 0. The other parameters are as in Fig. 3.
The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 <
ΩCDMh
2 < 0.12 (95% C.L.).
relic abundance. Here we show that the lower bound on the reheat temperature
derived in the previous Section survives even in scenarios with altered expansion
history as long as WIMPs were only produced thermally.
This can be understood from the observation that the Boltzmann equation with
modified expansion rate is obtained by replacing 〈σv〉 in the radiation–dominated
case by 〈σv〉A. Increasing (decreasing) A therefore has the same effect as an in-
crease (decrease) of the annihilation cross section. Since the lower bound on T0 was
independent of σ (more exactly: we quoted the absolute minimum, for the optimal
choice of σ), we expect it to survive even if A(z) 6= 1 is introduced.
This is borne out by Fig. 8, which shows the relic abundance Ωχh
2 in the (A(zF,st),
x0) plane for the simplified case A
′′(zF,st) = 0; similar results can be obtained for
the more general ansatz (29). The shaded region corresponds to the bound (1) on
the cold dark matter abundance. As expected, this figure looks similar to Fig. 1 if
the annihilation cross section in Fig. 1 is replaced by A(zF,st). The maximal value of
x0 consistent with the WMAP data remains around 23 even in these scenarios with
modified expansion rate. Fig. 8 also shows that A(zF,st) ≪ 1 is allowed for some
narrow range of initial temperature T0 < TF . This is analogous to the low cross
section branch in Fig. 1.
4 Summary and Conclusions
In this paper we have investigated the relic abundance of WIMPs χ, which are non-
relativistic long–lived or stable particles, in non–standard cosmological scenarios.
One motivation for studying such scenarios is that they allow to reproduce the ob-
served Dark Matter density for a large range of WIMP annihilation cross sections.
Our motivation was the opposite: we wanted to quantify the constraints that can
be obtained on parameters describing the early universe, under the assumption that
thermally produced WIMPs form all Dark Matter. Wherever necessary, we fixed
particle physics quantities such that standard cosmology yields the correct relic den-
sity.
Specifically, we first considered scenarios with low post–inflationary reheat tem-
perature, while keeping all other features of standard cosmology (known particle
content and Hubble expansion parameter during WIMP decoupling; no late entropy
production; no non–thermal WIMP production channels). If the temperature was so
low that WIMPs could not achieve full thermal equilibrium, the dependence of the
abundance on the mass and annihilation cross section of the WIMPs is completely
different from that in the standard thermal WIMP scenario. In particular, if the
maximal temperature T0 is much less than the decoupling temperature TF , nχ re-
mains exponentially suppressed. By applying the observed cosmological amount of
cold dark matter to the predicted WIMP abundance, we therefore found the lower
bound of the initial temperature T0 >∼ mχ/23. One might naively think that this
bound could be evaded by choosing a sufficiently large WIMP production (or anni-
hilation) cross section. However, increasing this cross section also reduces TF . For
sufficiently large cross section one therefore has TF ≤ T0 again; in this regime the
relic density drops with increasing cross section. Our lower bound is the minimal
T0 required for any cross section; once the latter is known, the bound on T0 might
be slightly sharpened. As a by–product, we also noted that the final relic density
depends only weakly on the annihilation cross section if T0 is slightly above this
lower bound.
We also investigated the effect of a non–standard expansion rate of the universe
on the WIMP relic abundance. In general the abundance of thermal relics depends
on the ratio of the annihilation cross section to the expansion rate; the latter is
determined unambiguously in standard cosmology. We found that even for non–
standard Hubble parameter the relic abundance can be calculated accurately in
terms of an annihilation integral, very similar to the case of standard cosmology. We
assumed that the WIMP annihilation cross section is such that the standard scenario
yields the observed relic density, and parameterized the modification of the Hubble
parameter as a quadratic function of the temperature. In this analysis it is crucial to
make sure that at low temperatures the Hubble parameter approaches its standard
value to within ∼ 20%, as required for the success of Big Bang Nucleosynthesis
(BBN).
Keeping the annihilation cross section fixed and allowing a 20% variation in the
relic density, roughly corresponding to the present “2σ” band, we found that the
expansion of the universe at T = TF might have been more than two times faster, or
more than six times slower, than in standard cosmology. These large variations of
H(TF ) can only be realized by finetuning of the parameters describing H(T < TF ).
However, even if we forbid such finetuning by choosing a linear parameterization for
the modification of the expansion rate, a 20% variation of Ωχh
2 allows a difference
between H(TF ) and its standard expectation of more than 50%. This relatively weak
sensitivity of the predicted Ωχh
2 on H(TF ) is due to the fact that the relic density
depends on all H(T < TF ); as stressed above, we have to require that H(T ≪ TF )
approaches its standard value to within ∼ 20%. The fact that determining Ωχh2 will
yield relatively poor bounds on H(TF ) remains true even if the annihilation cross
section is such that a non–standard behavior of H(T ) is required for obtaining the
correct χ relic density. Finally, we showed that the absolute lower bound on the
temperature required for thermal χ production is unaltered by allowing H(T ) to
differ from its standard value.
Of course, in order to draw the conclusions derived in this article, we need to
convince ourselves that WIMPs do indeed form (nearly) all Dark Matter. This
requires not only the detection of WIMPs, e.g. in direct search experiments; we
also need to show that their density is in accord with the local Dark Matter density
derived from astronomical observations. To that end, the cross sections appearing in
the calculation of the detection rate need to be known independently. This can only
be done in the framework of a definite theory, using data from collider experiments.
For example, in order to determine the cross section for the direct detection of
supersymmetric WIMPs, one needs to know the parameters of the supersymmetric
neutralino, Higgs and squark sectors [3]. We also saw that inferences about H(TF )
can only be made if the WIMP annihilation cross section is known. This again
requires highly non–trivial analyses of collider data [27], as well as a consistent
overall theory. We thus see that the interplay of accurate cosmological data with
results obtained from dark matter detections and collider experiments can give us
insight into the pre–BBN universe, which to date remains unexplored territory.
Acknowledgments
This work was partially supported by the Marie Curie Training Research Network
“UniverseNet” under contract no. MRTN-CT-2006-035863, as well as by the Euro-
pean Network of Theoretical Astroparticle Physics ENTApP ILIAS/N6 under con-
tract no. RII3-CT-2004-506222.
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	Introduction
	Relic Abundance in the Radiation–Dominated Universe
	Relic Abundance for Modified Expansion Rate
	Summary and Conclusions

  We investigate the relic density n_\chi of non-relativistic long-lived or
stable particles \chi in non-standard cosmological scenarios. We calculate the
relic abundance starting from arbitrary initial temperatures of the
radiation-dominated epoch, and derive the lower bound on the initial
temperature T_0 \geq m_\chi/23, assuming that thermally produced \chi particles
account for the dark matter energy density in the universe; this bound holds
for all \chi annihilation cross sections. We also investigate cosmological
scenarios with modified expansion rate. Even in this case an approximate
formula similar to the standard one is capable of predicting the final relic
abundance correctly. Choosing the \chi annihilation cross section such that the
observed cold dark matter abundance is reproduced in standard cosmology, we
constrain possible modifications of the expansion rate at T \sim m_\chi/20,
well before Big Bang nucleosynthesis.

Introduction
One of the most notable recent developments in cosmology is the precise determina-
tion of cosmological parameters from observations of the large–scale structure of the
universe, most notably by the Wilkinson Microwave Anisotropy Probe (WMAP). In
particular, the accurate determination of the non–baryonic cold Dark Matter (DM)
density [1],
0.08 < ΩCDMh
2 < 0.12 (95% C.L.) , (1)
has great influence on particle physics models which possess dark matter candidates
[2, 3]. The requirement that the predicted DM density falls in the range (1) is a
powerful tool for discriminating between various models and for constraining the
parameter space of surviving models.
Many dark matter candidate particles have been proposed. In particular long–
lived or stable weakly interacting massive particles (WIMPs) with weak–scale masses
are excellent candidates. In standard cosmology WIMPs decoupled from the thermal
background during the radiation–dominated epoch after inflation. In this framework
convenient and accurate analytic approximate solutions for the relic abundance have
been derived [4, 5]. One of the best motivated candidates for WIMPs is the lightest
neutralino in supersymmetric (SUSY) models. Assuming that the neutralino is the
lightest supersymmetric particle (LSP) stabilized due to R–parity, its relic abundance
has been extensively discussed [3]. Similar analyses have also been performed for
other WIMPs whose existence is postulated in other extensions of the Standard
Model (SM) of particle physics. In many cases the cosmologically favored parameter
space of WIMP models can be directly tested at the CERN Large Hadron Collider
(LHC) in a few years [6]. The same parameter space often also leads to rates of
WIMP interactions with matter within the sensitivity of near–future direct DM
detection experiments.
This discussion shows that we are now entering an interesting time where the
standard cosmological scenario can be examined by experiments at high–energy col-
liders as well as DM searches [7]. In this respect we should emphasize that the
relic abundance of thermally produced WIMPs depends not only on their annihila-
tion cross section, which can be determined by particle physics experiments, but in
general is also very sensitive to cosmological parameters during the era of WIMP
production and annihilation. Of particular importance are the initial temperature
T0 at which WIMPs began to be thermally produced, and the expansion rate of the
universe H .
In the standard cosmological scenario, the expansion rate is uniquely determined
through the Friedmann equation of general relativity. In this scenario the density of
WIMPs with massmχ followed its equilibrium value until the freeze–out temperature
TF ≃ mχ/20. Below TF , interactions of WIMPs are decoupled, and thus the present
density is independent of T0 as long as T0 > TF .
It should be noted that in non–standard scenarios the relic density can be larger
or smaller than the value in the standard scenario. One example is the case where
T0 is smaller than or comparable to TF , which can be realized in inflationary models
with low reheat temperature. Since in many models the inflationary energy scale
must be much higher than mχ in order to correctly predict the density perturbations
[8], the standard assumption T0 > TF is not unreasonable. On the other hand, the
constraint on the reheat temperature from Big Bang Nucleosynthesis (BBN) is as low
as T0 >∼ MeV [9, 10]. From the purely phenomenological viewpoint, it is therefore
also interesting to investigate the production of WIMPs in low reheat temperature
scenarios [9, 11, 12, 13, 14].
The standard scenario also assumes that entropy per comoving volume is con-
served for all temperatures T ≤ TF . Late entropy production can dilute the predicted
relic density [15, 16]. The reason is that the usual calculation actually predicts the
ratio of the WIMP number density to the entropy density. On the other hand, if
late decays of a heavier particle non–thermally produce WIMPs in addition to the
usual thermal production mechanism, the resulting increase of the WIMP density
competes with the dilution caused by the decay of this particle into radiation, which
increases the entropy density [17, 18, 19, 20, 13, 21].
Another example of a non–standard cosmology changing the WIMP relic density
is a modified expansion rate of the universe. This might be induced by an anisotropic
expansion [16], by a modification of general relativity [16, 22], by additional contri-
butions to the total energy density from quintessence [23], by branes in a warped
geometry [24], or by a superstring dilaton [25].
These examples show that, once the WIMP annihilation cross section is fixed,
with the help of precise measurements of the cold dark matter density we can probe
the very early stage of the universe at temperatures of O(mχ/20) ∼ 10 GeV. This
is reminiscent of constraining the early evolution of the universe at T = O(100) keV
∗Note that TF can be formally defined in the standard way even if T0 < TF . In this case WIMPs
never were in full equilibrium, and correspondingly never “froze out”.
using the primordial abundances of the light elements produced by BBN.
The goal of this paper is to investigate to what extent the constraint (1) on
the WIMP relic abundance might allow us to derive quantitative constraints on
modifications of standard cosmology. So far the history of the universe has been
established by cosmological observations as far back as the BBN era. In this paper
we try to derive bounds on cosmological parameters relevant to the era before BBN.
Rather than studying specific extensions of the standard cosmological scenario, we
simply parameterize deviations from the standard scenario, and attempt to derive
constraints on these new parameters. Since we only have the single constraint (1),
for the most part we only allow a single quantity to differ from its standard value.
We expect that varying two quantities simultaneously will allow to get the right
relic density for almost any WIMP annihilation cross section. This has been shown
explicitly in [13] for the case that both late entropy production and non–thermal
WIMP production are considered, even if both originate from the late decay of a
single scalar field.
We first analyze the dependence of the WIMP abundance on the initial temper-
ature T0 of the conventional radiation–dominated epoch. We showed in [14] that for
fixed T0 the predicted WIMP relic density reaches a maximum as the annihilation
cross section is varied from very small to very large values. A small annihilation cross
section corresponds to a large TF > T0; in this case the relic density increases with
the annihilation cross section, since WIMP production from the thermal plasma is
more important than WIMP annihilation. On the other hand, increasing this cross
section reduces TF ; once TF < T0 a further increase of the cross section leads to
smaller relic densities since in this case WIMPs continue to annihilate even after the
temperature is too low for WIMP production. Here we turn this argument around,
and derive the lower bound on T0 ≥ mχ/23 under the assumption that all WIMPs
are produced thermally. Note that we do not need to know the WIMP annihilation
cross section to derive this bound.
We then examine the dependence of the predicted WIMP relic density on the
expansion rate in the epoch prior to BBN, where we allow the Hubble parameter to
depart from the standard value. The standard method of calculating the thermal
relic density [2, 4] is found to be still applicable in this case. Our working hypothesis
here is that the standard prediction for the Hubble expansion rate is essentially
correct, i.e. that the true expansion rate differs by at most a factor of a few from
the standard prediction. We then simply employ a generic Taylor expansion for the
temperature dependence of this modification factor; note that the success of standard
BBN indicates that this factor cannot deviate by more than ∼ 20% from unity at
low temperatures, T <∼ 1 MeV. Similarly, we assume that the WIMP annihilation
cross section has been determined (from experiments at particle colliders) to have
the value required in standard cosmology. Our approach is thus quite different from
that taken in [7], where present upper bounds on the fluxes of WIMP annihilation
products are used to place upper bounds on the Hubble expansion rate during WIMP
decoupling. The advantage of their approach is that no prior assumption on the
WIMP annihilation cross section needs to be made, whereas we assume a cross
section that reproduces the correct relic density in the standard scenario. On the
other hand, the bounds derived in refs.[7] are still quite weak, allowing the Hubble
parameter to exceed its standard prediction by a factor >∼ 30; moreover, no lower
bound on H can be derived in this fashion.
The remainder of this paper is organized as follows: In Sec. 2 we will briefly review
the calculation of the WIMP relic abundance assuming a conventional radiation–
dominated universe, and derive the lower bound on the initial temperature T0. In
Sec. 3 we discuss the case where the pre–BBN expansion rate is allowed to depart
from the standard one. Using approximate analytic formulae for the predicted WIMP
relic density for this scenario, we derive constraints on the early expansion parameter.
Finally, Sec. 4 is devoted to summary and conclusions.
2 Relic Abundance in the Radiation–Dominated
Universe
We start the discussion of the relic density nχ of stable or long–lived particles χ by
reviewing the structure of the Boltzmann equation which describes their creation and
annihilation. The goal of this Section is to find the lowest possible initial temperature
of the radiation–dominated universe, assuming that the present relic abundance of
cold dark matter is entirely due to thermally produced χ particles.
As usual, we will assume that χ is self–conjugate†, χ = χ̄, and that some symme-
try, for example R–parity, forbids decays of χ into SM particles; the same symmetry
then also forbids single production of χ from the thermal background. However,
†The case χ 6= χ̄ differs in a non–trivial way only in the presence of a χ− χ̄ asymmetry, i.e. if
nχ 6= nχ̄.
the creation and annihilation of χ pairs remains allowed. The time evolution of the
number density nχ of particles χ in the expanding universe is then described by the
Boltzmann equation [2],
+ 3Hnχ = −〈σv〉(n2χ − n2χ,eq) , (2)
where nχ,eq is the equilibrium number density of χ, and 〈σv〉 is the thermally averaged
annihilation cross section multiplied with the relative velocity of the two annihilating
χ particles. Finally, the Hubble parameter H = Ṙ/R is the expansion rate of the
universe, R being the scale factor in the Friedmann–Robertson–Walker metric. The
first (second) term on the right–hand side of Eq.(2) describes the decrease (increase)
of the number density due to annihilation into (production from) lighter particles.
Eq.(2) assumes that χ is in kinetic equilibrium with standard model particles.
It is useful to rewrite Eq.(2) in terms of the scaled inverse temperature x = mχ/T
as well as the dimensionless quantities Yχ = nχ/s and Yχ,eq = nχ,eq/s. The entropy
density is given by s = (2π2/45)g∗sT
3, where
g∗s =
i=bosons
i=fermions
. (3)
Here gi denotes the number of intrinsic degrees of freedom for particle species i
(e.g. due to spin and color), and Ti is the temperature of species i. Assuming that
the universe expands adiabatically, the entropy per comoving volume, sR3, remains
constant, which implies
+ 3Hs = 0 . (4)
The time dependence of the temperature is then given by
. (5)
Therefore the Boltzmann equation (2) can be written as
= −〈σv〉s
(Y 2χ − Y 2χ,eq) . (6)
Thermal production of WIMPs takes place during the radiation–dominated epoch,
when the expansion rate is given by
, (7)
with MPl = 2.4× 1018 GeV being the reduced Planck mass and
i=bosons
i=fermions
. (8)
In the following we use Hst to denote the standard expansion rate (7). If the post–
inflationary reheat temperature was sufficiently high, WIMPs reached full thermal
equilibrium. This remains true for temperatures well below mχ. We can therefore
use the non–relativistic expression for the χ equilibrium number density,
nχ,eq = gχ
e−mχ/T . (9)
In the absence of non–thermal production mechanisms, nχ ≤ nχ,eq at early times.
The annihilation rate Γ = nχ〈σv〉 then depends exponentially on T , and thus drops
more rapidly with decreasing temperature than the expansion rate Hst of Eq.(7)
does. When the annihilation rate falls below the expansion rate, the number density
of WIMPs ceases to follow its equilibrium value and is frozen out.
For T ≪ mχ the annihilation cross section can often (but not always [5]) be
approximated by a non–relativistic expansion in powers of v2. Its thermal average
is then given by
〈σv〉 = a + b〈v2〉+O(〈v4〉) = a+ 6b
. (10)
In this standard scenario, the following approximate formula has been shown [4, 2, 5]
to accurately reproduce the exact (numerically calculated) relic density:
Yχ,∞ ≡ Yχ(x → ∞) ≃
1.3 mχMPl
g∗(xF )(a/xF + 3b/x
, (11)
with xF = mχ/TF , TF being the decoupling temperature. For WIMPs, xF ≃ 22.
Here we assume g∗ ≃ g∗s and dg∗/dx ≃ 0. It is useful to express the χ mass density
as Ωχ = ρχ/ρc, ρc = 3H
Pl being the critical density of the universe. The present
relic mass density is then given by ρχ = mχnχ,∞ = mχs0Yχ,∞; here s0 ≃ 2900 cm−3
is the present entropy density. Eq.(11) then leads to
2 = 2.7× 1010 Yχ,∞
100 GeV
≃ 8.5× 10
−11 xF GeV
g∗(xF )(a+ 3b/xF )
, (12)
where h ≃ 0.7 is the scaled Hubble constant in units of 100 km sec−1 Mpc−1. We
defer further discussions of this expression to Sec. 3, where scenarios with modified
expansion rate are analyzed. Note that in the standard scenario leading to Eq.(12),
the present χ relic density is inversely proportional to its annihilation cross section
and has no dependence on the reheat temperature. Recall that this result depends
on the assumption that the highest temperature in the post–inflationary radiation
dominated epoch, which we denote by T0, exceeded TF significantly.
On the other hand, if T0 was too low to fully thermalize WIMPs, the final result
for Ωχ will depend on T0. In particular, if WIMPs were thermally produced in a com-
pletely out–of–equilibrium manner starting from vanishing initial abundance during
the radiation–dominated era, such that WIMP annihilation remains negligible, the
present relic abundance is given by [14]
Y0(x → ∞) ≃ 0.014 g2χg−3/2∗ mχMPle−2x0x0
. (13)
Note that the final abundance depends exponentially on T0, and increases with
increasing cross section.
In in–between cases where WIMPs are not completely thermalized but WIMP
annihilation can no longer be neglected, we have shown [14] that re–summing the
first correction term δ enables us to reproduce the full temperature dependence of
the density of WIMPs:
1− δ/Y0
≡ Y1,r . (14)
Here δ < 0 describes the annihilation of WIMPs produced according to Eq.(13):
δ(x → ∞) ≃ −2.5 × 10−4 g4χg−5/2∗ m3χM3Ple−4x0x0
. (15)
Since δ is proportional to the third power of the cross section, the re–summed ex-
pression Y1,r is inversely proportional to the cross section for large cross section. In
ref.[14] we have shown that this feature allows the approximation (14) to be smoothly
matched to the standard result (12). Not surprisingly, as long as we only consider
thermal χ production, decreasing T0 can only reduce the final χ relic density.
With the help of these results, we can explore the dependence of the χ relic
density on T0 as well as on the annihilation cross section. Some results are shown in
Fig. 1, where we take (a) a 6= 0, b = 0, and (b) a = 0, b 6= 0. We choose Yχ(x0) = 0,
mχ = 100 GeV, gχ = 2 and g∗ = 90.
The results depicted in this Figure can be understood as follows. For small T0,
i.e. large x0, Eq.(13) is valid, leading to a very strong dependence of Ωχh
2 on x0.
10−10 10−9 10−8 10−7
a (GeV−2 )
b = 0
10−310−4
10−9 10−8 10−7 10−6
b (GeV−2 )
a = 0
10−310−4
Figure 1: Contour plots of the present relic abundance Ωχh2. Here we take (a) a 6= 0, b = 0, and
(b) a = 0, b 6= 0. We choose Yχ(x0) = 0, mχ = 100 GeV, gχ = 2, g∗ = 90. The shaded region
corresponds to the WMAP bound on the cold dark matter abundance, 0.08 < ΩCDMh
2 < 0.12
(95% C.L.).
Recall that in this case the relic density is proportional to the cross section. In this
regime one can reproduce the relic density (1) with quite small annihilation cross
section, a + 6b/x0 <∼ 10−9 GeV−2, for some narrow range of initial temperature,
x0 <∼ 22.5. Note that this allows much smaller annihilation cross sections than the
standard result, at the cost of a very strong dependence of the final result on the
initial temperature T0.
In this Section we set out to derive a lower bound on T0. In this regard the
region of parameter space described by Eq.(13) is not optimal. Increasing the χ
annihilation cross section at first allows to obtain the correct relic density for larger
x0, i.e. smaller T0. However, the correction δ then quickly increases in size; as noted
earlier, once |δ| > Y0 a further increase of the cross section will lead to a decrease
of the final relic density. The lower bound on T0 is therefore saturated if Ωχh
2 as a
function of the cross section reaches a maximum. From Fig.1 we read off
T0 ≥ mχ/23 , (16)
if we require Ωχh
2 to fall in the range (1).
We just saw that in the regime where this bound is saturated, the final relic den-
sity is (to first order) independent of the annihilation cross section, ∂(Ωχh
2)/∂〈σv〉 =
0. If T0 is slightly above the absolute lower bound (16), the correct relic density can
therefore be obtained for a rather wide range of cross sections. For example, if
x0 = 22.5, the entire range 3 × 10−10 GeV−2 <∼ a <∼ 2 × 10−9 GeV
−2 is allowed. Of
course, the correct relic density can also be obtained in the standard scenario of
(arbitrarily) high T0, if a + 3b/22 falls within ∼ 20% of 2× 10−9 GeV−2.
3 Relic Abundance for Modified Expansion Rate
In this section we discuss the calculation of the WIMP relic density nχ in modi-
fied cosmological scenarios where the expansion parameter of the pre–BBN universe
differed from the standard value Hst of Eq.(7). For the most part we will assume
that WIMPs have been in full thermal equilibrium. Various cosmological models
predict a non–standard early expansion history [22, 23, 24, 25]. Here we analyze to
what extent the relic density of WIMP Dark Matter can be used to constrain the
Hubble parameter during the epoch of WIMP decoupling. As long as we assume
large T0 we can use a modification of the standard treatment [4, 2] to estimate the
relic density for given annihilation cross section and expansion rate. We will show
that the resulting approximate solutions again accurately reproduce the numerically
evaluated relic abundance.
Let us introduce the modification parameter A(x), which parameterizes the ratio
of the standard value Hst(x) to the assumed H(x):
A(x) =
Hst(x)
. (17)
Note that A > 1 means that the expansion rate is smaller than in standard cos-
mology. Allowing for this modified expansion rate, the Boltzmann equation (6) is
altered to
G(x)mχMPl
〈σv〉A(x)
Y 2χ − Y 2χ,eq
, (18)
where
G(x) =
. (19)
Following refs.[4, 2], we can obtain an approximate solution of this equation by
considering the differential equation for ∆ = Yχ − Yχ,eq. For temperatures higher
than the decoupling temperature, Yχ tracks Yχ,eq very closely and the ∆
2-term can
be ignored:
≃ −dYeq
− 4π√
mχMPl
G(x)〈σv〉A(x)
(2Yχ,eq∆) . (20)
Here dYχ,eq/dx ≃ −Yχ,eq for x ≫ 1. In order to keep |∆| small, the derivative d∆/dx
must also be small, which implies
∆ ≃ x
90)mχMPlG(x)〈σv〉A(x)
. (21)
This solution is used down to the freeze–out temperature TF , defined via
∆(xF ) = ξYχ,eq(xF ) , (22)
where ξ is a constant of order of unity. This leads to the following expression:
xF = ln
ξmχMPlgχ
〈σv〉A(x)
xg∗(x)
, (23)
which can e.g. be solved iteratively. In our numerical calculations we will choose
2− 1 [4, 2].
On the other hand, for low temperatures (T < TF ), the production term ∝ Y 2χ,eq
in Eq.(18) can be ignored. In this limit, Yχ ≃ ∆, and the solution of Eq.(18) is given
∆(xF )
∆(x → ∞)
mχMPlI(xF ) , (24)
where the annihilation integral is defined as
I(xF ) =
G(x)〈σv〉A(x)
. (25)
Assuming ∆(x → ∞) ≪ ∆(xF ), the final relic abundance is
Yχ,∞ ≡ Yχ(x → ∞) =
90)mχMPlI(xF )
. (26)
Plugging in numerical values for the Planck mass and for today’s entropy density,
the present relic density can thus be written as
8.5× 10−11
I(xF ) GeV
. (27)
The constraint (1) therefore corresponds to the allowed range for the annihilation
integral
7.1× 10−10 GeV−2 < I(xF ) < 1.1× 10−9 GeV−2 . (28)
The standard formula (12) for the final relic density is recovered if A(x) is set to
unity and G(x) is replaced by the constant
g∗(xF ).
The further discussion is simplified if we use the normalized temperature z =
T/mχ ≡ 1/x, rather than x. Phenomenologically A(z) can be any function subject
to the condition that A(z) approaches unity at late times in order not to contradict
the successful predictions of BBN. We need to know A(z) only for the interval from
around the freeze-out to BBN: zBBN ∼ 10−5 − 10−4 <∼ z <∼ zF ∼ 1/20. This suggests
a parameterization of A(z) in terms of a power series in (z − zF,st):
A(z) = A(zF,st) + (z − zF,st)A′(zF,st) +
(z − zF,st)2A′′(zF,st) , (29)
where zF,st is the normalized freeze–out temperature in the standard scenario and
a prime denotes a derivative with respect to z. The ansatz (29) should be of quite
general validity, so long as the modification of the expansion rate is relatively modest.
This suits our purpose, since we wish to find out what constraints can be derived on
the expansion history if standard cosmology leads to the correct WIMP relic density.
We further introduce the variable
k = A(z → 0) = A(zF,st)− zF,stA′(zF,st) +
z2F,stA
′′(zF,st) , (30)
which describes the modification parameter at late times. Since zBBN is almost
zero, we treat k as the modification parameter at the era of BBN in this paper.‡
Deviations from k = 1 are conveniently discussed in terms of the equivalent number
of light neutrino degrees of freedom Nν . BBN permits that the number of neutrinos
differs from the standard model value Nν = 3 by δNν = 1.5 or so [26]. We therefore
take the uncertainty of k to be 20%. In the following we treat A(zF,st), A
′(zF,st) and
k as free parameters; A′′(zF,st) is then a derived quantity.
Note that we allow A(z) to cross unity, i.e. to switch from an expansion that
is faster than in standard cosmology to a slower expansion or vice versa. This is
illustrated in Fig. 2, which shows examples of possible evolutions of A(z) as function
of z for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each case
we consider scenarios with A(zF ) = 1.3 (slower expansion at TF than in standard
cosmology) as well as A(zF ) = 0.7 (faster expansion); moreover, we allow the change
of A at z = zF to be either positive or negative. However, we insist that H remains
positive at all times, i.e. A(z) must not cross zero. This excludes scenarios with
very large positive A′(zF,st), which would lead to A < 0 at some z < zF . Similarly,
‡Presumably the Hubble expansion rate has to approach the standard rate even more closely
for T < TBBN. However, since all WIMP annihilation effectively ceased well before the onset of
BBN, this epoch plays no role in our analysis.
 0  0.01  0.02  0.03  0.04  0.05
z = T/mχ
k = 1.2
A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
 0  0.01  0.02  0.03  0.04  0.05
z = T/mχ
k = 0.8A=1.3, A’=−3
A=1.3, A’= 9
A=0.7, A’=−3
A=0.7, A’= 9
Figure 2: Examples of possible evolutions of the modification parameter A(z) as function of z
for zF = 0.05. Here we take k = 1.2 (left frame) and k = 0.8 (right). In each frame we choose
A(zF ) = 1.3, A
′(zF ) = −3 (thick line), A(zF ) = 1.3, A′(zF ) = 9 (dashed), A(zF ) = 0.7, A′(zF ) =
−3 (dotted), A(zF ) = 0.7, A′(zF ) = 9 (dot–dashed).
demanding that our ansatz (29) remains valid for some range of temperatures above
TF excludes scenarios with very large negative A
′(zF,st). We will come back to this
point shortly.
Eq.(23) shows that zF 6= zF,st (xF 6= xF,st) if A(zF ) 6= 1. This is illustrated by
Fig. 3, which shows the difference between xF and xF,st in the (A(zF,st), A
′(zF,st))
plane. Here we take parameters such that Ωχh
2 = 0.099 in the standard cosmology,
which is recovered for A(zF,st) = 1, A
′(zF,st) = 0. Due to the logarithmic dependence
on A, xF (or zF ) differs by at most a few percent from its standard value if A(zF,st) is
O(1). Since TF only depends on the expansion rate at TF , it is essentially insensitive
to the derivative A′(zF,st).
In our treatment the modification of the expansion parameter affects the WIMP
relic density mostly via the annihilation integral (25). In terms of the normalized
temperature z, the latter can be rewritten as
I(zF ) =
dz G(z)〈σv〉A(z) . (31)
One advantage of the expansion (29) is that this integral can be evaluated analyti-
cally:
I(zF ) ≃ G(zF )
k(azF + 3bz
F ) + (A
′(zF,st)− zF,stA′′(zF,st))
z2F + 2bz
A′′(zF,st)
z3F +
. (32)
 0.6  0.8  1  1.2  1.4
A(zF,st )
xF − x F,st
a = 2.0*10−9  GeV −2
b = 0
k = 1
xF,st = 22.0
−0.4 −0.2 0 0.2 0.4
Figure 3: Contour plot of xF −xF,st in the (A(zF,st), A′(zF,st)) plane. Here we take a = 2.0×10−9
GeV−2, b = 0, mχ = 100 GeV, gχ = 2, g∗ = 90 (constant) and k = 1. This parameter set produces
xF,st = 22.0 and Ωχh
2 = 0.099 for the standard approximation.
 0.6  0.8  1  1.2  1.4
A(zF,st )
2(approx) / Ωχh
2(exact)
0.997
0.998
0.999
1.000
a = 2.0*10−9 GeV−2
b = 0
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st)
2(approx) / Ωχh
2(exact)
0.996
0.998
1.000
1.002
1.004 a = 0
b = 1.5*10−8 GeV−2
k = 1
Figure 4: Ratio of the analytic result of the relic density to the exact value in the (A(zF,st),
A′(zF,st)) plane for a = 2.0× 10−9 GeV−2, b = 0 (left frame) and for a = 0, b = 1.5× 10−8 GeV−2
(right). The other parameters are as in Fig. 3.
Here we have assumed that G(z) varies only slowly.
Before proceeding, we first have to convince ourselves that the analytic treatment
developed in this Section still works for A 6= 1. This is demonstrated by Fig. 4,
which shows the ratio of the analytic solution obtained from Eqs. (27) and (32)
to the exact one, obtained by numerically integrating the Boltzmann equation (18),
assuming constant g∗. We see that our analytical treatment is accurate to better
than 1%, and can thus safely be employed in the subsequent analysis.
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 2.0*10−9 GeV −2
b = 0
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
 0.6  0.8  1  1.2  1.4
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1.2
 0.6  0.8  1  1.2  1.4
A(zF,st ) 
a = 2.0*10−9 GeV−2
b = 0
k = 0.8
Figure 5: Contour plots of the relic abundance in the (A(zF,st), A′(zF,st)) plane. Here we choose
(a) a = 2.0×10−9 GeV−2, b = 0, k = 1; (b) a = 0, b = 1.5×10−8 GeV−2, k = 1; (c) a = 2.0×10−9
GeV−2, b = 0, k = 1.2; (d) a = 2.0× 10−9 GeV−2, b = 0, k = 0.8. The other parameters are as in
Fig. 3.
We are now ready to analyze the impact of the modified expansion rate on the
WIMP relic density. In Fig. 5, we show contour plots of Ωχh
2 in the (A(zF,st),
A′(zF,st)) plane. Recall that large (small) values of A correspond to a small (large)
expansion rate. Since a smaller expansion rate allows the WIMPs more time to
annihilate, A > 1 leads to a reduced WIMP relic density, whereas A < 1 means
larger relic density, if the cross section is kept fixed.
However, unlike the freeze–out temperature, the annihilation integral is sensitive
to A(z) for all z ≤ zF . Note that A′(zF,st) > 0 implies A(z) < A(zF,st) for z <
zF,st ≃ zF . A positive first derivative, A′(zF,st) > 0, can therefore to some extent
compensate for A(zF,st) > 1; analogously, a negative first derivative can compensate
for A(zF,st) < 1. This explains the slopes of the curves in Fig. 5. Recall also that
A′(zF,st) = 0 does not imply a constant modification factor A(z); rather, the term
∝ A′′(zF,st) in Eq.(29) makes sure that A approaches k as z → 0. This explains why
a change of A by some given percentage leads to a smaller relative change of Ωχh
2, as
can be seen in the Figure. This also illustrates the importance of ensuring appropriate
(near–standard) expansion rate in the BBN era. Finally, since the expansion rate
at late times is given by Hst/k, bigger (smaller) values of k imply that the χ relic
density is reduced (enhanced).
Fig. 5 shows that we need additional physical constraints if we want to derive
bounds on A(zF,st) and/or A
′(zF,st). Once the annihilation cross section is known,
the requirement (1) will single out a region in the space spanned by our three new
parameters (including k) which describe the non–standard evolution of the universe,
but this region is not bounded. Such additional constraints can be derived from
the requirement that the Hubble parameter should remain positive throughout the
epoch we are considering. As noted earlier, requiring H > 0 for all T < TF,st leads
to an upper bound on A′(zF,st); explicitly,
A′(zF,st) <
A(zF,st) +
kA(zF,st)
zF,st
. (33)
On the other hand, a lower bound on A′(zF,st) is obtained from the condition that
the modified Hubble parameter is positive between the highest temperature Ti where
the ansatz (29) holds and TF,st:
A′(zF,st) > −
zi − zF,st
2− zi
zF,st
A(zF,st) + k
zF,st
, (34)
for (1− zF,st/zi)2k < A(zF,st), and
A′(zF,st) >
A(zF,st)−
kA(zF,st)
zF,st
, (35)
for A(zF,st) < (1− zF,st/zi)2k, where zi = Ti/mχ.
Evidently the lower bound on A′(zF,st) depends on zi, i.e. on the maximal tem-
perature where we assume our ansatz (29) to be valid. In ref.[14] we have shown that
in standard cosmology (A ≡ 1) essentially full thermalization is already achieved for
xi <∼ xF − 0.5, even if nχ(xi) = 0. However, it seems reasonable to demand that H
should remain positive at least up to xi = xF−(a few). In Fig. 6 we therefore show
 0  1  2  3  4  5  6  7
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1
xF,st − x i  = 4
xF,st − x i  = 10
 0  1  2  3  4  5  6  7
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
k = 1
xF,st − x i  = 4
xF,st − x i  = 10
Figure 6: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), A′(zF,st)) plane. The
dashed line corresponds to the upper bound on A′(zF,st). The dotted lines correspond to the lower
bounds calculated for xF,st − xi = 4, 10. We take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and
a = 0, b = 1.5× 10−8 GeV−2 (right frame). The other parameters are as in Fig 3.
the physical constraints on the modification parameter A(z) for xF,st − xi = 4, 10
and k = 1. The dashed and dotted lines correspond to the upper and lower bounds
on A′(zF,st), described by Eq.(33) and Eqs.(34), (35), respectively. We see that when
xF,st − xi = 4 the allowed region is 0.4 <∼ A(zF,st) <∼ 6.5 with −60 <∼ A′(zF,st) <∼ 400
for b = 0 (left frame), and 0.4 <∼ A(zF,st) <∼ 4.5 with −60 <∼ A′(zF,st) <∼ 300 for a = 0
(right frame). When xF,st − xi = 10, the lower bounds are altered to 0.6 <∼ A(zF,st),
−10 <∼ A′(zF,st) for b = 0 (left frame), and 0.6 <∼ A(zF,st), −20 <∼ A′(zF,st) for a = 0
(right frame). Note that the lower bounds on A(zF,st), which depend only weakly on
xi so long as it is not very close to xF , are almost the same in both cases, which also
lead to very similar relic densities in standard cosmology. However, the two upper
bounds differ significantly. The reason is that large values of A(zF,st), i.e. a strongly
suppressed Hubble expansion, require some degree of finetuning: One also has to
take large positive A′(zF,st), so that A becomes smaller than one for some range of
z values below zF , leading to an annihilation integral of similar size as in standard
cosmology. Since the b−terms show different zF dependence in the annihilation in-
tegral (32), the required tuning between A(zF,st) and A
′(zF,st) is somewhat different
than for the a−terms, leading to a steeper slope of the allowed region. This allowed
region therefore saturates the upper bound (33) on the slope for somewhat smaller
A(zF,st).
The effect of this tuning can be seen by analyzing the special case where A′′(zF,st) =
 0  0.5  1  1.5  2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
 4 0.08
 0  0.5  1  1.5  2
A(zF,st )
a = 0
b = 1.5*10−8 GeV−2
Figure 7: Contour plots of the relic abundance Ωχh2 in the (A(zF,st), k) plane for A′′(zF,st) = 0.
The dotted lines correspond to the lower bounds of A(zF,st), calculated for xF,st − xi = 4, 10. We
take a = 2.0 × 10−9 GeV−2, b = 0 (left frame) and a = 0, b = 1.5 × 10−8 GeV−2 (right frame).
The other parameters are as in Fig 3.
0. The modification parameter then reads
A(z) =
A(zF,st)− k
zF,st
z + k . (36)
Note that A is now a monotonic function of z, making large cancellations in the
annihilation integral impossible. Imposing that A(z) remains positive for zF,st ≤
z ≤ zi leads to the lower limit
A(zF,st) >
zF,st
k . (37)
There is no upper bound, since A(z) is now automatically positive for all z ∈ [0, zF,st]
if A(zF,st) and A(0) ≡ k are both positive. Fig. 7 shows constraints on the relic
abundance in the (A(zF,st), k) plane for A
′′(zF,st) = 0. The dotted lines correspond
to the lower bounds (37) on A(zF,st) for xF,st − xi = 4, 10. As noted earlier, k
is constrained by the BBN bound. This leads to the bounds 0.5 <∼ A(zF,st) <∼ 1.8
for b = 0 (left frame), and 0.65 <∼ A(zF,st) <∼ 1.6 for a = 0 (right frame), when
xF,st − xi = 10. Evidently the constraints now only depend weakly on whether the
a− or b−term dominates in the annihilation cross section. As the initial temperature
is lowered, the impact of the constraint (37) disappears.
So far we have assumed in this Section that the reheat temperature was high
enough for WIMPs to have attained full thermal equilibrium. If this was not the
case, the initial temperature as well as the suppression parameter affects the final
 0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2
A(zF,st )
a = 2.0*10−9 GeV−2
b = 0
k = 1, A’’(zF,st ) = 0
0.080.12
Figure 8: Contour plot of the relic abundance Ωχh2 in the (A(zF,st), x0) plane. Here we choose
a = 2.0 × 10−9 GeV−2, b = 0, k = 1, A′′(zF,st) = 0. The other parameters are as in Fig. 3.
The shaded region corresponds to the WMAP bound on the cold dark matter abundance, 0.08 <
ΩCDMh
2 < 0.12 (95% C.L.).
relic abundance. Here we show that the lower bound on the reheat temperature
derived in the previous Section survives even in scenarios with altered expansion
history as long as WIMPs were only produced thermally.
This can be understood from the observation that the Boltzmann equation with
modified expansion rate is obtained by replacing 〈σv〉 in the radiation–dominated
case by 〈σv〉A. Increasing (decreasing) A therefore has the same effect as an in-
crease (decrease) of the annihilation cross section. Since the lower bound on T0 was
independent of σ (more exactly: we quoted the absolute minimum, for the optimal
choice of σ), we expect it to survive even if A(z) 6= 1 is introduced.
This is borne out by Fig. 8, which shows the relic abundance Ωχh
2 in the (A(zF,st),
x0) plane for the simplified case A
′′(zF,st) = 0; similar results can be obtained for
the more general ansatz (29). The shaded region corresponds to the bound (1) on
the cold dark matter abundance. As expected, this figure looks similar to Fig. 1 if
the annihilation cross section in Fig. 1 is replaced by A(zF,st). The maximal value of
x0 consistent with the WMAP data remains around 23 even in these scenarios with
modified expansion rate. Fig. 8 also shows that A(zF,st) ≪ 1 is allowed for some
narrow range of initial temperature T0 < TF . This is analogous to the low cross
section branch in Fig. 1.
4 Summary and Conclusions
In this paper we have investigated the relic abundance of WIMPs χ, which are non-
relativistic long–lived or stable particles, in non–standard cosmological scenarios.
One motivation for studying such scenarios is that they allow to reproduce the ob-
served Dark Matter density for a large range of WIMP annihilation cross sections.
Our motivation was the opposite: we wanted to quantify the constraints that can
be obtained on parameters describing the early universe, under the assumption that
thermally produced WIMPs form all Dark Matter. Wherever necessary, we fixed
particle physics quantities such that standard cosmology yields the correct relic den-
sity.
Specifically, we first considered scenarios with low post–inflationary reheat tem-
perature, while keeping all other features of standard cosmology (known particle
content and Hubble expansion parameter during WIMP decoupling; no late entropy
production; no non–thermal WIMP production channels). If the temperature was so
low that WIMPs could not achieve full thermal equilibrium, the dependence of the
abundance on the mass and annihilation cross section of the WIMPs is completely
different from that in the standard thermal WIMP scenario. In particular, if the
maximal temperature T0 is much less than the decoupling temperature TF , nχ re-
mains exponentially suppressed. By applying the observed cosmological amount of
cold dark matter to the predicted WIMP abundance, we therefore found the lower
bound of the initial temperature T0 >∼ mχ/23. One might naively think that this
bound could be evaded by choosing a sufficiently large WIMP production (or anni-
hilation) cross section. However, increasing this cross section also reduces TF . For
sufficiently large cross section one therefore has TF ≤ T0 again; in this regime the
relic density drops with increasing cross section. Our lower bound is the minimal
T0 required for any cross section; once the latter is known, the bound on T0 might
be slightly sharpened. As a by–product, we also noted that the final relic density
depends only weakly on the annihilation cross section if T0 is slightly above this
lower bound.
We also investigated the effect of a non–standard expansion rate of the universe
on the WIMP relic abundance. In general the abundance of thermal relics depends
on the ratio of the annihilation cross section to the expansion rate; the latter is
determined unambiguously in standard cosmology. We found that even for non–
standard Hubble parameter the relic abundance can be calculated accurately in
terms of an annihilation integral, very similar to the case of standard cosmology. We
assumed that the WIMP annihilation cross section is such that the standard scenario
yields the observed relic density, and parameterized the modification of the Hubble
parameter as a quadratic function of the temperature. In this analysis it is crucial to
make sure that at low temperatures the Hubble parameter approaches its standard
value to within ∼ 20%, as required for the success of Big Bang Nucleosynthesis
(BBN).
Keeping the annihilation cross section fixed and allowing a 20% variation in the
relic density, roughly corresponding to the present “2σ” band, we found that the
expansion of the universe at T = TF might have been more than two times faster, or
more than six times slower, than in standard cosmology. These large variations of
H(TF ) can only be realized by finetuning of the parameters describing H(T < TF ).
However, even if we forbid such finetuning by choosing a linear parameterization for
the modification of the expansion rate, a 20% variation of Ωχh
2 allows a difference
between H(TF ) and its standard expectation of more than 50%. This relatively weak
sensitivity of the predicted Ωχh
2 on H(TF ) is due to the fact that the relic density
depends on all H(T < TF ); as stressed above, we have to require that H(T ≪ TF )
approaches its standard value to within ∼ 20%. The fact that determining Ωχh2 will
yield relatively poor bounds on H(TF ) remains true even if the annihilation cross
section is such that a non–standard behavior of H(T ) is required for obtaining the
correct χ relic density. Finally, we showed that the absolute lower bound on the
temperature required for thermal χ production is unaltered by allowing H(T ) to
differ from its standard value.
Of course, in order to draw the conclusions derived in this article, we need to
convince ourselves that WIMPs do indeed form (nearly) all Dark Matter. This
requires not only the detection of WIMPs, e.g. in direct search experiments; we
also need to show that their density is in accord with the local Dark Matter density
derived from astronomical observations. To that end, the cross sections appearing in
the calculation of the detection rate need to be known independently. This can only
be done in the framework of a definite theory, using data from collider experiments.
For example, in order to determine the cross section for the direct detection of
supersymmetric WIMPs, one needs to know the parameters of the supersymmetric
neutralino, Higgs and squark sectors [3]. We also saw that inferences about H(TF )
can only be made if the WIMP annihilation cross section is known. This again
requires highly non–trivial analyses of collider data [27], as well as a consistent
overall theory. We thus see that the interplay of accurate cosmological data with
results obtained from dark matter detections and collider experiments can give us
insight into the pre–BBN universe, which to date remains unexplored territory.
Acknowledgments
This work was partially supported by the Marie Curie Training Research Network
“UniverseNet” under contract no. MRTN-CT-2006-035863, as well as by the Euro-
pean Network of Theoretical Astroparticle Physics ENTApP ILIAS/N6 under con-
tract no. RII3-CT-2004-506222.
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	Introduction
	Relic Abundance in the Radiation–Dominated Universe
	Relic Abundance for Modified Expansion Rate
	Summary and Conclusions
