thermal abundance of semirelativistic relics

Thermal abundance of semirelativistic relics

Manuel Drees,1,* Mitsuru Kakizaki,2,† and Suchita Kulkarni1,‡

1Physikalisches Institut and Bethe Center for Theoretical Physics, Universitat Bonn, D-53115 Bonn, Germany2LAPTH, Universite de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

(Received 23 April 2009; published 10 August 2009)

Approximate analytical solutions of the Boltzmann equation for particles that are either extremely

relativistic or nonrelativistic when they decouple from the thermal bath are well established. However, no

analytical formula for the relic density of particles that are semirelativistic at decoupling is yet known. We

propose a new ansatz for the thermal average of the annihilation cross sections for such particles, and find

a semianalytical treatment for calculating their relic densities. As examples, we consider Majorana- and

Dirac-type neutrinos. We show that such semirelativistic relics cannot be good cold dark matter

candidates. However, late decays of metastable semirelativistic relics might have released a large amount

of entropy, thereby diluting the density of other, unwanted relics.

DOI: 10.1103/PhysRevD.80.043505 PACS numbers: 98.80.Cq, 95.35.+d

I. INTRODUCTION

The accurate determination of cosmological parametersby up-to-date observations, most notably by the WilkinsonMicrowave Anisotropy Probe (WMAP) [1], increases theimportance of quantitative predictions. In particular, theestimate of the cosmological relic abundances of particlespecies is essential, for the history of the Universe dependson these quantities. One of the most important examples isthe cosmological abundance of dark matter [2,3], whosemass density is found to be [1]

DMh2 ¼ 0:1099 0:0062; (1)

where h ’ 0:7 is the scaled Hubble parameter in units of100 km s1 Mpc1 and is the mass density in units ofthe critical density.

The abundance of some particle species is determinedby solving the Boltzmann equation, which describes thechange of the particle number caused by particle reactionsas well as by the expansion of the Universe [2]. However,there is no analytical general solution of this nonlineardifferential equation, and therefore one needs to solve theequation numerically in many cases. In early studies, ap-proximate analytical formulas have been found for therelativistic [2,4] and nonrelativistic [5–7] regimes.

Stable or long-lived weakly interacting massive particles(WIMPs) with weak-scale masses are examples of coldrelic particles, which decouple from thermal equilibriumwhen they are nonrelativistic. In standard cosmology, de-coupling of WIMPs occurs in the radiation-dominated(RD) era after inflation, and analytic approximate formulasfor the WIMP relic abundance have been derived [5–7]. Inthe opposite limit, where decoupling of particles from thethermal background occurs when they are relativistic, the

relic abundance is approximated by its equilibrium value atthe decoupling temperature, and not sensitive to details ofits freeze-out [2]. The resulting relic density can thereforeeasily be computed analytically. On the other hand, noanalytical treatment to calculate the relic abundance inthe intermediate regime is known yet.In this paper, we revisit the relic density of particles

that decouple from the thermal bath when they were semi-relativistic, i.e. at freeze-out temperature TF m.

Assuming that the Maxwell-Boltzmann (MB) distributioncan be used for all participating particles, we introduce anexpression for the thermal average of the annihilationcross section which smoothly interpolates between theextremely relativistic and nonrelativistic regimes. It isshown that our new ansatz is capable of reproducing theexact thermally averaged annihilation cross section withaccuracy of a few percent. Given this approximated crosssection, we can define the freeze-out temperature by com-paring the annihilation rate to the expansion rate. Theassumption that the comoving density remains constantafter freeze-out turns out to be a good approximation forthe relic abundance of semirelativistic particles.We also discuss the roles such semirelativistic particles

could play in realistic cosmological scenarios. It should beemphasized that the abundance of semirelativistically de-coupled relics tends to be large because it is only verymildly Boltzmann suppressed. We point out that scenarioswhere semirelativistically decoupling particles form thedark matter have problems with structure formation, bigbang nucleosynthesis (BBN) and/or laboratory measure-ments. Nevertheless, such semirelativistic relics can beuseful for diluting the density of other, unwanted relicsby late-time out-of-equilibrium decay [8,9]. As an ex-ample, we investigate a scenario of decaying sterile neu-trino that is assumed to depart from thermal equilibriumwhen it is semirelativistic, in sharp contrast to nonthermalsterile neutrino scenarios [10]. Thermal equilibrium isattained by introducing some higher-dimensional operator.

*[email protected]†[email protected]‡[email protected]

PHYSICAL REVIEW D 80, 043505 (2009)

1550-7998=2009=80(4)=043505(10) 043505-1 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.043505

It is illustrated that an enormous amount of entropy can beproduced without spoiling the successful BBN predictionof the light element abundances.

This paper is organized as follows: In Sec. II we begin byreviewing briefly the method to calculate relic densities forrelativistic and nonrelativistic particles. In Sec. III weexplain the new formalism which is applicable for allfreeze-out temperatures in case of S- and P-wave crosssections. The way to calculate the freeze-out temperature isalso shown. In Sec. IV, the possibility for semirelativisticparticles to have the observed dark matter relic density ofDMh

2 ’ 0:1 is considered. Then, we discuss the amountof entropy produced by the decay of unstable semirelativ-istic species that decay in less than a second. Finally,Sec. VI is devoted to summary and conclusions. Someproperties of modified Bessel functions are described inAppendix A. In Appendix B we argue that the use of theMaxwell-Boltzmann distribution in the definition of thethermally averaged cross section only leads to a smallmistake in the final relic density.

II. RELIC ABUNDANCES IN THENONRELATIVISTIC AND RELATIVISTIC LIMITS

In this section we briefly review the standard analyticalapproximations for evaluating the relic abundance of hy-pothetical particles [2,5,7]. These are applicable to par-ticles that were either nonrelativistic or extremelyrelativistic at freeze-out.

The number density n is obtained by solving the cor-

responding Boltzmann equation [2]. For the moment, weassume that single production and decay are forbiddenby some symmetry or adequately suppressed. TheBoltzmann equation takes a simple form if one furtherassumes that the quantum statistics factors describing theBose enhancement or Fermi suppression of all final statescan be neglected; in Appendix B it is shown that this isessentially equivalent to assuming that the distributionfunctions of all relevant particles are proportional to theMaxwell-Boltzmann distribution. The Boltzmann equationfor n can then be written as [2]

dndt

þ 3Hn ¼ hviðn2 n2;eqÞ; (2)

where n;eq is the equilibrium number density, hvi isthe thermal average of the annihilation cross section multi-plied by the relative velocity between the two annihilating particles, and H is the Hubble expansion rate of theUniverse. The second term on the left-hand side describesthe dilution caused by the expansion of the Universe; thefirst (second) term on the right-hand side decreases (in-creases) the number density due to annihilation into (pro-duction from) other particles, which are assumed to be incomplete thermal equilibrium.

It is useful to express the above Boltzmann equation interms of the dimensionless quantities Y ¼ n=s and

Y;eq ¼ n;eq=s. The entropy density is given by s ¼ð22=45ÞgT3, with g being the effective number ofrelativistic degrees of freedom and T the temperature ofthe Universe. We also introduce the dimensionless ratio of mass m to the temperature, x ¼ m=T. Assuming an

adiabatic expansion of the Universe, Eq. (2) can then berewritten as [2]

dY

dx¼ hvis

HxðY2

Y2;eqÞ: (3)

The generic picture of decoupling from the thermalbath is as follows. After inflation, the Universe becomesradiation dominated with expansion rate

H ¼ T2

MPl

ffiffiffiffiffiffig90

r; (4)

where MPl ¼ 2:4 1018 GeV is the reduced Planck mass.The reheat temperate is assumed to be high enough for particles to reach full thermal (chemical as well as kinetic)equilibrium.1 Thermal equilibrium is maintained as long asthe interaction rate ¼ nhvi is larger than the Hubble

expansion rate H. As the temperature decreases, the inter-action rate decreases more rapidly than the expansion ratedoes. When the interaction rate falls below the expansionrate, is no longer kept in thermal equilibrium and thecomoving number density Y becomes essentially con-

stant. This transition temperature is referred to as thefreeze-out temperature TF.Analytical expressions for the resulting relic density

are known for the cases where decoupling happens when is nonrelativistic (xF m=TF 3) or relativistic (xF 3). We discuss these two limiting cases in the followingsections.

A. Relativistic case

First, consider the case where particles decouple whenthey are ultrarelativistic (xF 3). In this case the equilib-rium number density to entropy ratio Y;eqðxÞ depends onthe temperature only through the number of degrees offreedom g of the thermal bath. Therefore, the final relicabundance is to very good approximation equal to itsequilibrium value at the time of decoupling:

Y;1 Yðx ! 1Þ ¼ Y;eqðxFÞ ¼ 0:28ðgeff=gðxFÞÞ;(5)

where

geff ¼g ðfor bosonsÞ;3g=4 ðfor fermionsÞ; (6)

with g being the number of internal (e.g., spin or color)

degrees of freedom of . Following the conventional no-

1The case where the reheat temperature is too low for toattain chemical equilibrium has been discussed in [11,12].

MANUEL DREES, MITSURU KAKIZAKI, AND SUCHITA KULKARNI PHYSICAL REVIEW D 80, 043505 (2009)

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tation, we express the relic density as ¼ms0Y;1=c, where c ¼ 3H2

0M2Pl is the present critical

density of the Universe, and s0 ’ 2900 cm3 is the presententropy density. This yields

h2 ¼ 7:8 102 geff

gðxFÞm

1 eV

: (7)

It should be noted that the relic density is simply propor-tional to the mass of the particle in the relativistic case.

B. Nonrelativistic case

For the nonrelativistic case where xF 3, the relicabundance strongly depends on the freeze-out temperatureTF because the equilibrium abundance Y;eqðxÞ is exponen-tially suppressed as the temperature decreases. The tem-perature dependence of the thermal average of theannihilation cross section is obtained using the Taylorexpansion in powers of the velocity squared:

hvi ¼ aþ bhv2i þOðhv4iÞ ¼ aþ 6b

xþO

1

x2

: (8)

The numerically evaluated correct relic abundance is re-produced with an accuracy of a few percent using theapproximate analytic formula

Y;1 Yðx ! 1Þ

¼ 1

1:3mMPl

ffiffiffiffiffiffiffiffiffiffiffiffiffiffigðxFÞ

p ða=xF þ 3b=x2FÞ: (9)

For WIMPs with electroweak scale mass, freeze-out occursat xF ’ 20. The corresponding scaled relic density is thengiven by

h2 ¼ 2:7 108Y;1

m

1 GeV

¼ 8:5 1011xF GeV2ffiffiffiffiffiffiffiffiffiffiffiffiffiffigðxFÞ

p ðaþ 3b=xFÞ: (10)

Note that the relic density of a nonrelativistic particle isinversely proportional to its annihilation cross section, butdoes not depend explicitly on its mass.

III. ABUNDANCE OF SEMIRELATIVISTICALLYDECOUPLING PARTICLES

In the previous section, we reviewed the known relativ-istic and nonrelativistic approximate formulas for the relicabundance. The main aim of this section is to find a simplemethod applicable between the two regimes: an analyticestimate of the relic abundance of semirelativistically de-coupling particles (xF 3).

One of the key quantities that determine the freeze-outtemperature is the thermally averaged cross section. In thenonrelativistic case, expressions for the equilibrium num-ber density as well as for the thermally averaged cross

section times velocity are rather simple. For a semirelativ-istic particle, however, the thermally averaged cross sec-tion involves multiple integrals and cannot be expandedwith respect to the velocity nor to the mass. Here wediscuss a method of approximating the thermally averagedcross section, by interpolating between its relativisticallyand nonrelativistically expanded expressions. We employthe Maxwell-Boltzmann distribution for the equilibriumnumber density [7]:

Y;eqðxÞ ¼ 45

44

ggðxÞ x

2K2ðxÞ: (11)

The thermal average of the cross section is then obtained as[7]

hvi ¼ 1

8m4TK

22ðm=TÞ

Z 1

4m2

dsðs 4m2Þ

ffiffiffis

pK1ð

ffiffiffis

p=TÞ; (12)

where K1ðxÞ and K2ðxÞ are the first and second modifiedBessel functions of the second kind; some properties ofthese functions are given in Appendix A.At first glance the use of the Maxwell-Boltzmann dis-

tribution seems improper for particles that are semirelativ-istic at decoupling, let alone for ultrarelativistic particles.However, we will argue in Appendix B that this should stillyield accurate results for the final relic density. This ispartly due to cancellations between the numerator anddenominator of Eq. (12), and partly due to the fact thatthe final result for becomes less sensitive to xF as xFdecreases.As examples of the annihilation of particles, we consider

the pair annihilation processes of Dirac and Majoranafermions (e.g. neutrinos) into a pair of massless fermions, ! f f [4,13,14]. It should, however, be noticed that thisassumption includes more general cases of any other spe-cies annihilating from S- or P-wave initial states. In arenormalizable model, the annihilation is mediated bysome heavy particle: for example, a Z boson with tinycoupling with , or a new spin-1 bosonU [15]. We assumethat the mass of this exchange particle is much larger thanm, so that the annihilation amplitude can be described

through an effective four-fermion interaction.The annihilation of two Dirac fermions proceeds from

an S wave, and the resulting cross section can be parame-trized as

Sv ¼ G2s

16; (13)

where s is the center-of-mass energy squared. G denotesthe coupling constant of the four-fermion interaction (e.g.the Fermi coupling constant, GF ¼ 1:17 105 GeV2).Finally, v is the relative velocity defined as

THERMAL ABUNDANCE OF SEMIRELATIVISTIC RELICS PHYSICAL REVIEW D 80, 043505 (2009)

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v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðpA pBÞ2 m4

qEAEB

; (14)

where pA;B and EA;B are the four-momenta and energies of

the two incident particles labeled A and B. The resultingthermally averaged cross section is given by

hSvi ¼ G2

256m4TK

22ðxÞ

Z 1

4m2

dss2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis 4m2

qK1ð

ffiffiffis

p=TÞ

¼ G2m2

x6K22ðxÞ

Z 1

0dtt2ðt2 þ x2Þ2K1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 þ x2

pÞ: (15)

Its relativistic and nonrelativistic limits read

hSviR ¼ G2m2

16x2ð12þ 5x2Þ; hSviNR ¼ G2m2

4;

(16)

respectively. A general expression for hSvi should repro-duce these results for x ! 0 and x ! 1, respectively. Asimple possibility is

hSviapp ¼G2m2

16

12

x2þ 5þ 4x

1þ x

: (17)

It should be noticed that this choice is not unique.Let us turn to the case of the annihilation from a P-wave

initial state, which is e.g. true if is a Majorana fermion.Equation (13) should then be replaced by

Pv ¼ G2s

16

1 4m2

s

: (18)

Thermal averaging leads to

hPvi ¼ G2

256m4TK

22ðxÞ

Z 1

4m2

dssðs 4m2Þ3=2K1ð

ffiffiffis

p=TÞ

¼ G2m2

x6K22ðxÞ

Z 1

0dtt4ðt2 þ x2ÞK1ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2 þ x2

pÞ: (19)

Following the same steps as in the S-wave case, we find thefollowing interpolation:

hPviapp ¼G2m2

16

12

x2þ 3þ 6x

ð1þ xÞ2: (20)

Figure 1 shows the ratio of the approximate to the exactcross section hviapp=hvi for the S- (solid line) and

P-wave (dashed line) cases. Note that this ratio dependsonly on x. We see that our approximate expressions repro-duce the exact ones with accuracy of better than 2% (0.5%)for annihilation from an S (P) wave, even in the semi-relativistic region (x 3).

Using Eqs. (17) or (20) instead of the exact expressions(15) or (19) greatly reduces the numerical effort required tosolve the Boltzmann equation (3). At the cost of somefurther loss of accuracy, an even faster estimate of the relicdensity can be obtained by using an approximate solution

of the Boltzmann equation instead of the accurate evalu-ation of the relic abundance by solving the Boltzmannequation numerically. The determination of the tempera-ture where some interaction decouples plays an importantrole in the analytical prediction of the relic abundance.Indeed, for nonrelativistically decoupling particles the relicabundance is sensitive to the freeze-out temperature xFbecause of the Boltzmann suppression. In this case, a roughestimate of xF obtained by equalizing the interaction rate and the Hubble expansion rate H is not sufficient to makean accurate prediction. Here we show that for semirelativ-istically decoupling particles a simple comparison of theinteraction rate and the Hubble expansion rate still gives areasonably accurate result for the final relic density.We define the freeze-out temperature by equalizing the

interaction rate and the Hubble expansion rate,2

ðxFÞ n;eqhviðxFÞ ¼ HðxFÞ: (21)

We then simply assume that Y does not change after

decoupling from the thermal bath, so that

Y;1 ¼ Y;eqðxFÞ: (22)

Let us see to what extent this method can reproduce thecorrect relic abundance. As an example, we consider thepair annihilation of neutrinolike particles via the mediationof the weak standard model (SM) gauge bosons.3 In the leftframe of Fig. 2 we plot the relic abundance h

2 as

function of m. The solid curves show predictions for the

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

0.1 1 10

<σv

>ap

p /<

σv>

x

S wave

P wave

FIG. 1. Ratio of the approximate to the exact thermally aver-aged annihilation cross sections hviapp=hvi as a function of x

for annihilation from an S- (solid line) and P-wave (dashed)initial state.

2This definition of the freeze-out temperature should not beused for the prediction of the relic abundance through Eqs. (9)and (10).

3Our primary concern here is to test our approximation for therelic abundance, and thus, as an illustration, we take such anunrealistic setup.


043505-4

relic abundance obtained by solving the Boltzmann equa-tion (3) numerically, while the dashed curves have beenobtained using the analytic approximation describedabove. The upper curves are for Majorana fermions anni-hilating from a P wave, and the lower curves are for theDirac fermions case annihilating from an S wave. We takeG ¼ GF ¼ 1:17 105 GeV2, g ¼ 10, and g ¼ 2.4

The right frame shows the corresponding values of xF.This figure shows that our very simple analytical treat-

ment reproduces the correct relic density with an error of atmost 20% (5%) for semirelativistically decoupling parti-cles annihilating from an S (P) wave. Not surprisingly, ourtreatment becomes exact for particles that are relativistic atdecoupling.5 In the Dirac case, our approximation coin-cides with the exact relic abundance for m ¼ 1 GeV,

corresponding to xF ’ 16; however, the deviation becomeslarger again for larger m. Therefore, our method is not

applicable for the entire region of cold relics. Instead, onecould switch to the usual nonrelativistic treatment de-scribed in Sec. II B at the crossover value, i.e. at xF ¼16. In the P-wave case the crossover already occurs atm ¼ 30 MeV, corresponding to xF ’ 4:5. The dotted

curve shows that the nonrelativistic approximation is al-ready quite reliable at this point. We can thus smoothlymatch our approximation to the usual nonrelativistic treat-ment for both S- and P-wave annihilation.

IV. SEMIRELATIVISTIC DARK MATTER?

As a first application, let us analyze whether semirela-tivistically decoupled particles (xF ’ 3) can be a darkmatter candidate, whose cosmological abundance shouldbeh

2 & 0:1. The final number density of such a particle

is of the order of Y;eqðx ’ 3Þ 102. Combining this

value with the observed amount of dark matter, the upperbound of the semirelativistic particle turns out to be m &

100 eV, which would thus decouple at temperatures of afew dozen eV. These particles would therefore still beultrarelativistic during the formation of 4He. Moreover,the effective coupling would have to be very large, G103 GeV2. Such a scenario is therefore tightlyconstrained.Note that a particle with m 100 eV could only

annihilate into light neutrinos.6 Moreover, the exchangeparticle would also need to be quite light, with mass &30 MeV if all couplings are 1.The simplest case is that of a real scalar . Since it only

adds a single degree of freedom, its presence would not bein serious conflict with current BBN constraints [16]. In arenormalizable theory, ! could then proceed eitherthrough exchange of a fermion in the t or u channel, orthrough boson exchange in the s channel.

103

104

105

10−4

10−3

10−2

10−1

100

Ωχh

2

mχ (GeV)

P wave

S wave

FIG. 2 (color online). Scaled freeze-out temperature xF (right panel) and scaled relic abundance h2 (left panel) as a function of

m. In the right (left) frame the lower (upper) curves are for Majorana fermions, and the upper (lower) curves for Dirac fermions. In the

left frame the solid curves show exact solutions of Eq. (3), and the dashed curves our analytic approximations. The dotted curve showsthe nonrelativistic approximation, Eq. (10), for P-wave annihilation. Here we take G¼GF¼1:17105 GeV2, g ¼10, and g ¼ 2.

4Since we concentrate on the ratio of the exact to the approxi-mate relic densities, we discard the temperature dependence ofg, which is basically an overall factor.

5We will see in Appendix B that one should use the correctFermi-Dirac or Bose-Einstein expression for n;eq in Eq. (22) inorder to accurately predict the relic density for xF & 1.

6The neutrinos themselves are not in thermal equilibrium withthe photons any more at T & 100 eV. However, as long as T m they would still have a thermal distribution. The Boltzmannequation (2) should therefore still be applicable, if T is taken tobe the temperature of the neutrinos, which is somewhat lowerthan the photon temperature.


043505-5

In the former case, the exchange particle would have tobe an SUð2Þ doublet, if the low-energy theory only con-tains left-handed, SUð2Þ doublet neutrinos. The presenceof such a light SUð2Þ doublet fermion is excluded by LEPdata. In principle the light neutrinos might also be Diracparticles, allowing ! R R annihilation via exchangeof a singlet fermion, possibly R itself. However, onewould then need R to also be in thermal equilibrium,increasing the number of additional degrees of freedompresent during BBN to an unacceptable level.

For a real scalar , s-channel exchange could onlyproceed through another scalar . However, a scalar can only couple to L R or its Hermitian conjugate. Thisscenario would therefore again require R to have been inequilibrium. We therefore conclude that such a light particle cannot be a real scalar.7

If is a complex field, one needs g ¼ 2, which is only

marginally compatible with BBN [16]. In principle, ! could then proceed through t- or u-channel exchange ofan SUð2Þ singlet. However, then either or this lightexchange particle would have to carry hypercharge, sothat it would have been produced copiously in Z decays.The argument against s-channel exchange of a scalar is thesame as for real .

However, a complex , either a scalar or a Weyl fer-mion,8 could couple to a new gauge bosonU, which in turncould couple to L L. This new boson would contributethree additional bosonic degrees of freedom at T * mU.Since we already added 2 degrees of freedom in , con-sistency with BBN would require mU * 1 MeV. This inturn would require the coupling of U to left-handed neu-trinos to exceed 0.01, assuming its coupling to is pertur-bative. By SUð2Þ invariance, U would have to couple withequal strength to a left-handed charged lepton. This com-bination of U boson mass and coupling is excluded, by alarge margin, for both the electron and muon family bymeasurements of the respective magnetic moments [15].No analogous measurement exists for the third generation,so a U boson with few MeV mass coupling exclusively tothird generation leptons and particles might still becompatible with laboratory data. However, given that and neutrinos are known to mix strongly [18], a gaugeinvariant model where U couples to but does not coupleto muons is difficult, if not impossible, to construct.

Finally, such a light particle would form hot, or at leastwarm, dark matter. This possibility is strongly constrainedby observations of early structures in the Universe, inparticular, the ‘‘Lyman- forest’’ [19]. Such a particlecould thus at best form a subdominant component of thetotal dark matter.

In combination, these arguments strongly indicate thatsemirelativistically decoupling particles should not be ab-solutely stable. In the next section we will show that suchparticles may nevertheless have a role to play in the historyof the Universe, if they are meta-stable.

V. ENTROPY PRODUCTION BY DECAYINGPARTICLES

In this section we demonstrate that semirelativisticallydecoupling particles can be useful for producing a largeamount of entropy, which could dilute the density of otherrelics to an acceptable level. Examples of such relics aredecaying gravitinos, which can lead to problems with bigbang nucleosynthesis [20], or supersymmetric neutralinos,whose relic density often exceeds the required dark matterdensity by 1 or 2 orders of magnitude [21]. The density ofsuch relics will be diluted only if the entropy is releasedafter they decouple from the thermal bath. This will simul-taneously dilute any preexisting baryon asymmetry. Onethus either has to increase the efficiency of early baryo-genesis, or introduce late baryogenesis after the release ofthe additional entropy. Both possibilities can be realized inthe framework of Affleck-Dine baryogenesis [22].Generally [8,9], out-of-equilibrium decays of long-lived

particles can only produce a significant amount of entropyif the decaying particle dominates the energy density of theUniverse prior to its decay. The abundance of nonrelativ-istically decoupling particles is suppressed by a factorexF , hence their contribution to the energy density is smallat decoupling. However, after decoupling their energydensity only drops like R3 / T3, while that of the domi-nant radiation component decreases like T4 as the Universecools off. Thermally produced particles can thereforedominate the energy density of the Universe only at tem-perature T exFTF. Significant entropy production bythe late decay of nonrelativistically decoupling particles istherefore only possible if they are simultaneously verymassive and quite long lived. For semirelativistic particles,on the other hand, the abundance at decoupling is large andthus a significant amount of entropy can be produced evenif their mass is small, since their density will becomedominant quite soon after decoupling.Let us consider the out-of-equilibrium decay of long-

lived particles which semirelativistically decoupled fromthe thermal background. For simplicity we work in theinstantaneous decay approximation, i.e. we assume thatall particles decay at time td ¼ , where is the

lifetime of . While this approximation does not describethe time dependence of the entropy (or temperature) fort very well, it does reproduce the entropy enhance-

ment factor, i.e. the entropy at t , quite accurately. We

assume that particles were in full thermal equilibrium forsufficiently high temperatures in the RD epoch. When thetemperature decreased to T ¼ TF ’ m, the number

density n froze out. At decoupling, particles contributed

7Of course, this argument does not exclude the possibility thata much heavier real singlet scalar could be cold dark matter[17].

8A massive two-component Weyl fermion can equivalently bedescribed by a four-component Majorana fermion.


043505-6

a few percent to the total energy density of the Universe;however, as noted earlier, the ratio of the radiation and

energy densities decreased by a factor Td=TF ¼ffiffiffiffiffiffiffiffiffiffiffiffitF=

qbetween decoupling and decay of ; here Td refers to thetemperature at time t ¼ , just prior to decay. If tF, the energy density at the time of the decay is wellapproximated by ;d ¼ mn;d, and dominated over the

radiation. In this case, the ratio of the final entropy densitysf after the decay to the initial entropy density si before the

decay is given by [8]

sfsi

¼ 0:82g1=4mY;d

1=2

M1=2Pl

; (23)

for si sf. Here Y;d ¼ n;d=si is proportional to the

abundance just prior to its decay.In light of the BBN prediction of the primordial abun-

dances of the light elements, the lifetime is constrainedas & 1 s [23]. Equations (10) and (23) show that the

entropy ratio is proportional to the relic density h2 that

would have if it were stable. We saw in Fig. 2 that forfixed coupling G this quantity is maximal if TF m;

more accurately, the maximum of h2 is achieved for

xF ¼ 1:8ð2:1Þ if particles annihilate from an S- (P-)waveinitial state. Entropy production by late decays is thusmost efficient when the particles decoupled semirelativ-istically, with their lifetime fixed to the maximal value of1 s.

We can construct a feasible scenario that fulfills theseconditions by introducing a sterile neutrino which mixeswith an ordinary neutrino. Here we treat both m and the

mixing angle as free parameters. In sharp contrast toconventional cosmological scenarios with sterile neutrinos[10], the sterile neutrino is assumed to be in thermalequilibrium in the early universe. In ordinary sterileneutrino models, thermal equilibrium is not reached be-cause the Yukawa coupling of the sterile neutrino withSM particles is tiny. One possible method for the pairproduction and annihilation to reach thermal equilibriumis to extend sterile neutrino models by adding anotherhypothetical boson Z0. Let it have coupling gZ0f with the

SM fermion pair f f, and gZ0 with the sterile neutrino

pair. If the Z0-boson mass mZ0 is larger than the energy,the annihilation cross section has the form of Eq. (18)with G ¼ gZ0gZ0f=m

2Z0 . Although gZ0f and mZ0 are con-

strained by high energy experiments, gZ0 can be as

large as unity. Therefore, annihilation can be in thermalequilibrium before its semirelativistic decoupling.

Decoupling occurred at T m if m’1GeVð3109 GeV2=GÞ2=3g1=6 .

In order to estimate the amount of entropy released bythe decay of sterile neutrinos in this setup, we have tocalculate their lifetime. For simplicity we ignore propaga-tor effects. When the sterile neutrino mass is smaller than

theW-boson mass mW ¼ 80 GeV, it decays into three SMfermions, with decay width

¼27 16sin2W þ 80

3sin4W

G2

Fm5

1923sin2; (24)

where sin2W ¼ 0:23 is the weak mixing angle. When thesterile neutrino mass is larger than the Z-boson massmZ ¼91 GeV, the sterile neutrino predominantly decays into aSM gauge boson and a lepton. Its decay width is thenproportional to m3

, and given by

¼ GFm3

8ffiffiffi2

p

2

1m2

W

m2

21þ 2m2

W

m2

þ1 m2

Z

m2

21þ 2m2

Z

m2

sin2: (25)

In the in-between case where mW <m <mZ, we obtain

¼ 2GFm

3

8ffiffiffi2

p

1m2

W

m2

21þ 2m2

W

m2

sin2

þ11 20sin2W þ 80

3sin4W

G2

Fm5

1923sin2: (26)

Figure 3 shows contours of the entropy increase sf=sidue to sterile neutrino decay in the ð1= ffiffiffiffi

Gp

; sinÞ plane. Weset the freeze-out temperature to xF ¼ 2:1, which max-imizes mY;i; this can be achieved by choosing the mass

m appropriately. The thick line indicates the BBN limit on

the sterile neutrino lifetime, ¼ 1 sec . Equation (23)

shows that for given neutrino mass, the released entropywill be maximal if is chosen such that reaches this

upper limit.

10−14

10−13

10−12

10−11

10−10

10−9

10−8

104 105 106 107

sin

θ

G−1/2 (GeV)

τχ=1sec

10

102

103

104

FIG. 3. Contour of the entropy increase sf=si caused by semi-relativistic sterile neutrino decay in the ð1= ffiffiffiffi

Gp

; sinÞ plane. Wechoose m such that xF ¼ 2:1. The solid line indicates the BBN

limit on the sterile neutrino lifetime ¼ 1 s.


043505-7

The behavior of the contours in Fig. 3 is easy to under-stand from Eq. (23). In the relevant limit 1 and keep-ing g constant, we have / 2m5

ð2m3 Þ for

m < ð>ÞmW . The entropy ratio thus scales as 1m3=2

/1Gð1m1=2

/ 1G1=3Þ for m < ð>ÞmW . Along the

¼ 1 s contour, the entropy release increases propor-

tional to m / G2=3 both for m <mW and for m >

mW . Figure 3 can be extended to even smaller G, i.e. largerZ0 masses, so long as m is smaller than the reheat tem-

perature after inflation, so that was in thermal equilib-rium in the RD epoch. If at the same time is decreased sothat ¼ 1 s remains constant, very large entropy dilution

factors could be realized,

sfsi

103

G1=2

106 GeV

4=3

: (27)

This result is only valid if the mixing-induced interac-tions of are not in thermal equilibrium for T & m. Since

these interactions are also responsible for decay, thisassumption is satisfied whenever tF; we saw in the

discussion of Eq. (23) that this strong inequality is in anycase a condition for significant entropy release from decay.

We finally note that for givenm the entropy released in

decays is maximal if G is so small that was ultrarela-tivistic at decoupling, since this maximizes Y;eqðxFÞ.Again setting ¼ 1 s by appropriate choice of , this

yields

sfsi

104 m

103 GeV: (28)

VI. CONCLUSION

In this paper we have developed an approximate analyticmethod for calculating the thermally averaged annihilationcross section of semirelativistically decoupling particlesand for estimating their relic density. We have shown thatthis approximate solution can be smoothly matched to thewell-known nonrelativistic approximation at the point ofintersection. We have argued that such relics cannot formthe observed cosmological dark matter. However, wepointed out that the late decay of metastable semirelativ-istically decoupling relics can be an efficient source ofentropy production. As an example of this entropy produc-tion mechanism we discussed a scenario with a sterileneutrino, and illustrated to what extent entropy can beincreased.

ACKNOWLEDGMENTS

This work was partially supported by the Marie CurieTraining Research Network ‘‘UniverseNet’’ underContract No. MRTN-CT-2006-035863, and by theEuropean Network of Theoretical Astroparticle Physics

ENTApP ILIAS/N6 under Contract No. RII3-CT-2004-506222. The work of M.K. was also supported by theMarie Curie Training Research Network ‘‘HEPTools’’under Contract No. MRTN-CT-2006-035505.

APPENDIX A: MODIFIED BESSEL FUNCTIONS

In this Appendix, we summarize some properties of themodified Bessel function. Using an integral representation,the modified Bessel function of the second kind is definedby

KðzÞ ¼ffiffiffiffi

p ðz=2Þðþ 1=2Þ

Z 1

1dteztðt2 1Þ1=2;

ReðÞ> 1

2; ReðzÞ> 0:

(A1)

In particular, the calculation of the relic abundance in-volves K1ðzÞ and K2ðzÞ,

K1ðzÞ ¼ zZ 1

1dteztðt2 1Þ1=2; ReðzÞ> 0;

K2ðzÞ ¼ z2

3

Z 1

1dteztðt2 1Þ3=2; ReðzÞ> 0:

(A2)

The lower order terms of the series expansion of K1ðzÞ andK2ðzÞ are given by

K1ðzÞ ¼ 1

zþ ; K2ðzÞ ¼ 2

z2 1

2þ : (A3)

The asymptotic expansion of KðzÞ is given by

KðzÞ ffiffiffiffiffi

2z

rez

1þ 42 1

8zþ

: (A4)

APPENDIX B: VALIDITY OF THEMAXWELL-BOLTZMANN DISTRIBUTION

In the calculations of this paper we used the Maxwell-Boltzmann (MB) distribution also for particles that weresemirelativistic at decoupling; this assumption is e.g. im-plicit in Eq. (12). At first sight this seems quite dangerous.For example, at T ¼ m, i.e. x ¼ 1, the MB result for n;eqoverestimates the Fermi-Dirac distribution by about 7%,and underestimates the Bose-Einstein distribution by about10%. Since annihilation always involves two particles,one might assume that the total error associated with theuse of the MB distribution is about twice as large. In thisAppendix we show that the MB distribution can indeed beused to compute the thermally averaged cross section andthe decoupling temperature as long as xF * 1. For smallerxF, one has to use the proper Fermi-Dirac or Bose-Einsteindistribution only in the very last step, when calculatingY;1.


043505-8

We begin by expanding the true distribution function,

f;eqðEÞ ¼ 1

eE=T 1’ eE=Tð1 eE=TÞ; (B1)

where the upper (lower) sign is for fermionic (bosonic) particles. Note that the correction term in parentheses hasexactly the same form as the ‘‘statistics factors’’ appearingin the collision term of the full Boltzmann equation [2]. Forconsistency these statistics factors therefore also have to beincluded. Up to first order in these correction factors, thetemperature dependent terms in the integrand defining thecollision term for $ f f processes then read for fermi-onic f:

I ¼ eðE1þE2

Þ=T ½c2ð1 eE1=T eE2

=T

eEf=T eE f=TÞ ð1 eEf=T eE f=T ce

E1=T ce

E2=TÞ

¼ eðE1þE2

Þ=T ½ðc2 1Þð1 eE1=T eE2

=T

eEf=T eE f=TÞ ðc 1ÞðeE1=T þ eE2

=TÞ;(B2)

here c ¼ f=f;eq is independent of energy as long as is

in kinetic equilibrium (through elastic scattering on SMparticles); in that case we can equivalently write c ¼n=n;eq. In order to derive the full collision term, I has

to be multiplied with the squared matrix element andintegrated over phase space [2].

In the usual treatment of WIMP decoupling, all theexponential terms in the square parentheses are neglected,so that the collision term becomes proportional to n2 n2;eq times the thermally averaged cross section defined in

Eq. (12). Unfortunately the full correction term introducesadditional dependence on the final state energies Ef and

E f. In order to keep the numerics manageable, we assume

that they can be replaced by E1and E2

, respectively. This

is certainly true (by energy conservation) for the sum Ef þE f; this has already been used in deriving Eq. (B2). Note

furthermore that we will need the collision term for tem-peratures * TF, where jc 1j 1, so that we can ap-

proximate c 1 ’ ðc2 1Þ=2. These approximations

yield

I ’ðc21ÞeðE1þE2

Þ=T½1ðeE1=TþeE2

=TÞ; (B3)

where ¼ 1=2 (3=2) for bosonic (fermionic) particles.In the following we will assume to be fermionic, whichaccording to Eq. (B3) should lead to larger deviations fromthe MB result.

Inserting this corrected collision term into theBoltzmann equation, and following the formalism of [7],finally yields a modified thermally averaged cross sectiontimes initial state velocity:

hvi ¼ 1

n2;eq

g2

8ð2Þ4Z

dEþdEdsð FÞðsÞeEþ=T

½1 ðeðEþþEÞ=ð2TÞ þ eðEþEÞ=ð2TÞÞ; (B4)

with F ¼ 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4m2

=sq

, Eþ ¼ E1þ E2

, and E ¼E1

E2. This reduces to Eq. (12) if the expression in

1 10x

0.96

0.98

1

1.02

1.04

ratio

<v σ>MB

/ <v σ><v σ>

par / <v σ>

MB

<v σ>par

/ <v σ>

FIG. 4 (color online). Various approximations for the ther-mally averaged cross section as a function of the scaled inversetemperature x ¼ m=T for fermionic particles annihilating from

an S wave. The solid (black) curve shows the ratio of thecorrected cross section (B4) to the Maxwell-Boltzmann (MB)result (12), while the dashed (red) curve shows this ratio if Eq.(12) is replaced by our approximation (17). The dotted (blue)curve is the same as the solid curve in Fig. 1.

0.001 0.01 0.1 1 10

mχ [GeV]

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

Rat

io

Ωχ,MB / Ωχ

Ωχ,mix / Ωχ

FIG. 5 (color online). Effect of using the Maxwell-Boltzmanndistribution on the predicted relic density, calculated using theapproximation Y;1 ¼ Y;eqðxFÞ, for fermionic particles anni-

hilating from an S-wave initial state. The dashed (red) curveshows the ratio of the prediction using the Maxwell-Boltzmanndistribution everywhere to the corrected prediction based onEqs. (B1) and (B4). The solid (black) curve shows the analogousratio, where correct Fermi-Dirac distribution has been used toevaluate Y;eqðxFÞ, but hvi and xF have still been obtained

using the MB distribution. Parameters are as in Fig. 2.


043505-9

square brackets is simply replaced by 1. In the followingwe assume that particles annihilate from an S wave.P-wave annihilation would favor larger energies, wherethe correction terms in Eq. (B4) are smaller. Note that wealso have to use the expanded form (B1) of the distributionfunction when calculating n;eq in Eq. (B4); otherwise the

solution of the Boltzmann equation will not yield n ’n;eq, including the correction terms, at T TF.

The size of the correction terms in Eq. (B4) is illustratedby the solid (black) curve in Fig. 4. We see that thecorrection amounts to less than 2% for all x * 1. This isdue to a strong cancellation between the corrections in theintegrand of Eq. (B4) and those in the overall factor1=n2;eq. The dashed (red) curve shows that for x 2 the

errors due to the use of the MB distribution and due to oursimple parametrization (17) add up, leading to a total errorof about 2.7% at most. The Fermi-Dirac corrections to the

thermally averaged cross section begin to be significant forx & 0:5. However, here one enters the ultrarelativisticregime, where the final relic density is no longer sensitiveto the decoupling temperature. We therefore expect theeffect of using the MB distribution in Eq. (12) on the finalprediction of the relic density to be quite small throughout.This is illustrated in Fig. 5, where the relic density has

been calculated using the simple assumption Y;1 ¼Y;eqðxFÞ; we have used the same parameters as in Fig. 2.

The dashed (red) curve shows that using the MB distribu-tion everywhere will overestimate the relic density form & 5 MeV, i.e. for xF & 1. However, the black curve

shows that this can easily be corrected by using the Fermi-Dirac distribution only in the final step, i.e. when calculat-ing Y;eqðxFÞ; hvi and xF can still been calculated using

the MB distribution. This validates our treatment in themain text.

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043505-10

thermal abundance of semirelativistic relics

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