Prepared for submission to JCAP

Establishing the Dark Matter RelicDensity in an Era of Particle Decays

Carlos Maldonadoa and James Unwinb,c

aDepartamento de Fisica, Universidad de Santiago de Chile, Santiago, Chile.bDepartment of Physics, University of Illinois at Chicago, Chicago, Illinois, 60607, USAcSimons Center for Geometry and Physics, Stony Brook, New York, 11794, USA

Abstract. If the early universe is dominated by an energy density which evolves other thanradiation-like the normal Hubble-temperature relation H ∝ T 2 is broken and dark matterrelic density calculations in this era can be significantly different. We first highlight thatwith a population of states φ sourcing an initial expansion rate of the form H ∝ T 2+n/2,for n ≥ −4, during the period of appreciable φ decays the evolution transitions to H ∝ T 4.The decays of φ imply a source of entropy production in the thermal bath which alters theBoltzmann equations and impacts the dark matter relic abundance. We show that the form ofthe initial expansion rate leaves a lasting imprint on relic densities established while H ∝ T 4

since the value of the exponent n changes the temperature evolution of the thermal bath. Inparticular, a dark matter relic density set via freeze-in or non-thermal production is highlysensitive to the temperature dependance of the initial expansion rate. This work generalisesearlier studies which assumed initial expansion rates due to matter or kination domination.

arX

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6v2

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Contents

1 Introduction 1

2 Boltzmann Equations without Entropy Conservation 22.1 Boltzmann equations 22.2 Radiation temperature maxima 42.3 Onset of radiation domination 7

3 Implications for Dark Matter 83.1 Freeze-in 83.2 Freeze-out during reheating 113.3 Non-thermal production 12

4 Concluding Remarks 13

1 Introduction

The dark matter freeze-out paradigm, in particular the WIMP miracle, is prized for its sim-plicity and predictiveness. However, it is relatively straightforward to arrange for significantdeviations in the predictions of freeze-out by changing either the particle physics model,or the cosmological history. For instance, standard freeze-out calculations typically assumethat decoupling occurs whilst the energy density of the universe is dominated by radiationin which case the expansion rate of the universe is H ∝ T 2. However, dark matter freeze-outcould occur whilst the universe is dominated by some form of energy other than radiationin which case the usual Hubble-temperature relation H ∝ T 2 is broken. In particular, onepossibility which occurs quite naturally in many Standard Model extensions is the case of anearly matter dominated period for which H ∝ T 3/2, or an era of particle decays leading tosignificant entropy production in the thermal bath in which case H ∝ T 4 [1]. The case ofdark matter freeze-out during an early period of matter domination was recently highlightedin [2] and freeze-out whilst H ∝ T 4 was studied in [3–6].

More generally the early universe could be dominated by the energy density of a pop-ulation of states φ evolving as an arbitrary power of the scale factor ρφ(t) = ρφ(tI)a

4+n

for tI some initial time. This evolution can also be expressed in terms of the φ equationof state ωφ = (n + 1)/3. Indeed, observational constraints allow the equation of state forthe inflaton to take values in ωφ ∈ (−1,−1/3) between the end of the de Sitter inflationaryperiod and onset the radiation domination [7] potentially allowing for n ∈ (−4,−2) duringthis transition period. Moreover, periods of non-standard expansion (with a range of rates)are independently motivated in a variety of contexts such as scalars with periodic potentials[8, 9], brane world cosmologies [10, 11], and scalar-tensor theories [12]. An explicit examplewhich allows general n was constructed in [13], in which the energy density of the universeis dominated by the contributions from a real scalar field φ with a potential of the form

V (φ) =4− 2n

(4 + n)2t2Iexp

[(φ(tI)− φ)

√n+ 4

].

Values of n ∈ (−4, 2) imply positive potentials, corresponding to ωφ ∈ (−1, 1), howeverconsidering also negative potentials extends the range to n > 2 (which was the focus of [13]).

– 1 –

Such early periods of non-standard cosmology can impact the dark matter evolution andthus it is interesting to explore the implications, while remaining agnostic regarding the formand origin of the evolution of the dominant energy density. Two recent studies by D’Eramo,Fernandez, & Profumo [13, 14] considered the consequences for dark matter if the relic densityis established during a period of fast expansion. These papers focused on scenarios withn > 0, in which case the expansion rate H ∝ Tn/2+2 is faster than expected from a radiationdominated universe and the energy density in φ eventually redshifts to a negligible level. Notethat the case n = 2 corresponds to ‘kination domination’, see e.g. [15–18], in which the energydensity of the universe is dominated by the kinetic energy of some scalar field (the φ term),for instance the inflaton. Conversely, for n < 0, in order to recover the successes of standardcosmology, one requires that the state which dominates the energy density eventually decays.Specifically, the universe should be dominated by the Standard Model radiation bath attemperates around 10 MeV and below (until matter-radiation equality) so not to spoil theprecision predictions of Big Bang nucleosynthesis (see e.g. [19]).

Here we study the case of dark matter which freezes out or is produced during a period ofparticle production due to the decays of some (boson or fermion) state φ which dominates theenergy density and under the assumption that ρφ ∝ an+4 thus implying an initial expansionrate of the form H ∝ Tn/2+2. Notably, during this period of φ decays, entropy is no longerconserved in the thermal bath and this impacts the dark matter relic abundance calculation.In particular, our work generalizes the earlier papers of [3–6] which assume that the initialexpansion rate corresponds to an early matter dominated phase. Additionally, this workcan also be seen as an extension of dark matter freeze-out or production with a generalexpansion rate, as studied in [13, 14], to include decays of φ. This paper is structured asfollows: In Section 2 we discuss the formulation of the Boltzmann equation without entropyconservation, generalizing the derivations of [5]. Using these results, in Section 3 we comparedifferent scenarios for setting the dark matter relic abundance and discuss their dependanceon the exponent n of the initial expansion rate. Concluding remarks are given in Section 4.

2 Boltzmann Equations without Entropy Conservation

We start by deriving expressions for the evolution of the different particle populations in thecase that the early universe is dominated by a state φ resulting in an initial expansion rateof H ∝ Tn/2+2. We show that the expansion rate subsequently transitions to H ∝ T 4 andderive an expression for the maximum temperature of the Standard Model radiation. Theexpressions derived reproduce the results of Giudice, Kolb, & Riotto [5] for matter dominationprior to decays (n = −1) and Visinelli [20] with kination domination prior to decays (n = 2).

2.1 Boltzmann equations

We start in familiar territory by defining the Hubble parameter

H2 =8π

3M2Pl

(ρφ + ρR + ρX) , (2.1)

where the terms indicate the energy density in φ, Standard Model radiation, and dark matterX, respectively. The evolution of quantities can be tracked relative to a dimensionless scalefactor A defined as

A ≡ a

aI= aTRH . (2.2)

– 2 –

where aI is an arbitrary initial reference point which is chosen to be aI = 1/TRH and TRH

is the reheat temperature following φ decays. Emulating the analysis of [5], we rewrite therelevant variables as dimensionless quantities, but where now the energy density of φ evolvesas some arbitrary power a4+n

Φ ≡ ρφa4+nTnRH =

ρφA4+n

T 4RH

, R ≡ ρRa4 =ρRA

4

T 4RH

, X ≡ nXa3 =nXA

3

T 3RH

. (2.3)

With the assumption that ρX = 〈EX〉nX , where nX is the dark matter number density and〈EX〉 is the expected energy of each dark matter state, one can express the Hubble rate interms of these dimensionless variables to obtain

H =T 2RH

MPlA2+n/2

√8π

3

(Φ +RAn +

〈EX〉XAn+1

TRH

). (2.4)

This recovers the form of H in [5] for n = −1. For an expansion rate of the form ofeq. (2.4), the Boltzmann equations which describe the evolution of the number densities canbe expressed in a manner highly reminiscent to those studied in [5, 6]

ρφ + (4 + n)Hρφ = −Γφρφ (2.5)

ρR + 4HρR = Γφρφ + 2〈EX〉(n2X − n2eq)〈σv〉 (2.6)

˙nX + 3HnX =b

mφΓφρφ − (n2X − n2eq)〈σv〉 (2.7)

where mφ and Γφ are the φ mass and decay rate to Standard Model. Dotted variablesindicate differentiation with respect to time, the quantity b paramaterises the branchingratio of φ to dark matter, 〈σv〉 is the thermally averaged dark matter annihilation crosssection, and neq denotes the equilibrium number density of dark matter which has its usualform. Furthermore, note that the φ decay rate can be expressed in terms of the reheattemperature after φ decays

Γφ =

√4π3g∗(TRH)

45

T 2RH

MPl. (2.8)

We next re-express eqns. (2.5)-(2.7) in terms of the dimensionless units of eq. (2.3).Taking first eq. (2.5), simple substitution yields

Ad

dA

(Φ

(AaI)4+nTnRH

)+ (4 + n)H

Φ

(AaI)4+nTnRH

= −ΓφΦ

(AaI)4+nTnRH

. (2.9)

After some manipulation, and using that H = aa , this can be simplified to

AΦ′ = ΓφΦ , (2.10)

where the primed variable indicates differentiation with respect to A. Furthermore, usingeqns. (2.4) & (2.8) we can express eq. (2.10) in terms of Φ′ as follows

Φ′ = −√π2g∗(TRH)

30

ΦA1+n/2√Φ +RAn + 〈EX〉XAn+1

TRH

. (2.11)

– 3 –

Analogously, eqns. (2.6) & (2.7) can be rewritten in a similar fashion to obtain

R′ =

√π2g∗(TRH)

30

ΦA1−n/2√Φ +RAn + 〈EX〉XAn+1

TRH

+

√3

8π

2〈σv〉〈EX〉MPlAn/2−1(X2 −X2

eq)√Φ +RAn + 〈EX〉XAn+1

TRH

(2.12)and

X ′ =

√π2g∗(TRH)

30

b

mφ

ΦTRHA−n/2√

Φ +RAn + 〈EX〉XAn+1

TRH

−√

3

8π

〈σv〉MPlTRHAn/2−2(X2 −X2

eq)√Φ +RAn + 〈EX〉XAn+1

TRH

.

(2.13)

Thus the dimensionless versions of eqns. (2.5)-(2.7) are, respectively, eqns. (2.11)-(2.13).

2.2 Radiation temperature maxima

We assume that in the early universe the energy density is dominated by φ and, moreover,we further suppose that the initial radiation bath is negligible. This implies that the initialenergy density of φ can be written ρφ(aI) = (3/8π)H2

IM2Pl where HI ≡ H(aI) is the initial

expansion rate (throughout we will use the subscript I to mean the value of a give quantityat a = aI). In terms of dimensionless variables this initial condition is

ΦI =3H2

IM2Pl

8πT 4RH

, RI = XI = 0, AI = 1. (2.14)

One instance in which such initial conditions could arise, for instance, is immediately afterinflation, as in the case of kination domination [15–18] corresponding to n = 2.

In [5] the authors studied the evolution of the temperature of the Standard Modelthermal bath, assuming that prior to the decays of φ the radiation component is negligible.What was observed is that during the early matter dominated era the temperature rises dueto the decays of φ until it hits some maximum temperature TMax after which the expansionrate transitions to H ∝ T 4. The bath then cools until the temperature TRH at H ' Γφafter which φ decays become negligible and the universe becomes radiation dominated withH ∝ T 2. We next generalise this analysis to the case that the early universe expansion ratesourced by φ is an arbitrary power of the temperature as in eq. (2.1).

The φ decays transfer energy to the Standard Model bath and one can derive the pointAMax at which the maximum temperature of the radiation TMax occurs for the more generalexpansion rate. Since the dominant contribution comes from φ at early time, we can neglectthe second term in eq. (2.12), and using eq. (2.14) we obtain

R′ =

√π2g∗(TRH)

30Φ1/2I A1−n/2. (2.15)

Then integrating we obtain the following

R =

√

π2g∗(TRH)30

√ΦI

(1

2−n/2

)(A2−n/2 − 1) for n < 4√

π2g∗(TRH)30

√ΦI ln(A) for n = 4

. (2.16)

– 4 –

The temperature is a measure of the radiation energy density, and thus we can obtain anexpression for the evolution of T as a function of A from the expression

ρR =π2g∗(T )

30T 4 = R

(TRH

A

)4

. (2.17)

Moreover, substituting eq. (2.16) into eq. (2.17) and rearranging, gives the evolution of thetemperature (for n 6= 4)

T =

(45

4π3g∗(TRH)

g2∗(T )

)1/8 (HIMPlT

2RH

)1/4 [A−(2+n/2) −A−42− n/2

]1/4. (2.18)

The critical point, with respect to A, of the factor in square brackets of eq. (2.18) marksthe maximum temperature and the value of the scale factor AMax at which the temperaturestops increasing and begins to decrease. For |n| < 4 this is given by

AMax =

(n+ 4

8

)2/(n−4). (2.19)

For comparison, recall that AI = 1 corresponds to T = TRH and A > 1 implies a > aI . ForA > AMax the A−4 piece in eq. (2.18) can be neglected and T ∝ A−(2+n/2). Observe thatAMax is sensitive to the exponent n of the early universe expansion rate. The temperatureextremum TMax for |n| < 4 is found at A = AMax given by

TMax =

(45g∗(TRH)

4π3g2∗(TMax)

)1/8 (MPlHIT

2RH

)1/4( 2

4− n

)1/4[(

n+ 4

8

) 4+n4−n−(n+ 4

8

) 84−n]1/4

.

(2.20)As a reference, taking a few specific choices for n, the value of TMax can be approximated as

TMax '(MPlHIT

2RH

)1/4 ×

0.30 for n = −1

0.31 for n = −2

0.33 for n = −3

, (2.21)

where we take g∗(TRH) ≈ g∗(TMax) ≈ 100. For example, taking reasonable values for bothHI ∼ eV and TRH ∼ 1 TeV, then TMax ∼ 3 TeV for |n| ∼ O(1). Furthermore, we canre-express eq. (2.18) in terms of a normalised function f(AMax) = 1 as follows

T = TMaxf(A) (2.22)

for

f(A) ≡ κ(T )[A−(2+n/2) −A−4

]1/4(2.23)

with

κ(T ) =

[g∗(TMax)

g∗(T )

]1/4 [(4 + n

8

) 4+n4−n−(

4 + n

8

) 84−n]−1/4

. (2.24)

– 5 –

n=-4

n=-3

n=-2

n=0

n=-1

ΦI=10 10GeV | HI=1 eV, TRH=100 GeV

n=1

2 4 6 8 100

200

400

600

800

A

T(GeV

)

Figure 1. The figure shows the bath temperature T as function of A for different values of n,assuming that initially the expansion rate is H ∝ Tn/2+2 and that there is negligible energy in theradiation bath or dark matter R(aI) = X(aI) = 0. We fix ΦI = Φ(aI) = 1010 GeV, or equivalently(see eq. (2.14)) this corresponds to, for example, HI = 1 eV and TRH = 100 GeV. The curves followeq. (2.26). Of the cases shown n = −1 corresponds to a matter-like φ (blue, solid), n = 0 is radiation-like φ (dashed), and for n = 1 then φ redshifts faster than radiation (dotted). Note that the maximumtemperature drops (and occurs earlier) for increasing n following eq. (2.19).

For reference, a selection of specific values for κ are

κ(T ) ≈[g∗(TMax)

g∗(T )

]1/4

(88

33·55

)1/20for n = −1(

44

33

)1/12for n = −2(

166

77

)1/28for n = −3

. (2.25)

Starting from a = aI the temperature increases from a negligible value to TMax, andthen subsequently decreases according to eq. (2.25), as illustrated in Figure 1. The evolutionof the bath temperature in Figure 1 assumes that φ dominates the energy density, and theevolution will be altered once the energy in radiation becomes comparable to φ, we denotethis A×, as we discuss in the next section. Thus for AMax < A < A× one can approximatethe temperature evolution as follows

T ∼ κTMaxA−(2+n/2)/4 . (2.26)

Between the time when TMax is reached and the point of radiation domination, at the earlierof TRH or T×, the φ field energy density scales as ρφ = ΦIT

4RH/A

4+n. Since the dominantcontribution to the Hubble parameter at early times comes from ρφ, it follows from eq. (2.4)and eq. (2.14) that for AMax < A < A× then

H2 ' 8π

3MPl

ΦIT4RH

A4+n=(HIA

−(4+n)/2)2

. (2.27)

Moreover, using eq. (2.20) and eq. (2.26), we can express A in terms of the temperature

A−(4+n)/2 =

(4π3(2− n/2)2g2∗(T )

45g∗(TRH)

)1/2T 4

HIMPlT2RH

. (2.28)

– 6 –

Substituting eq. (2.28) into (2.27) gives an expression for H for AMax < A < A×

H = |4− n|(

π3g2∗(T )

45g∗(TRH)

)1/2T 4

MPlT2RH

. (2.29)

Thus the point AMax indicates the A at which the evolution transitions to H ∝ T 4. Inter-estingly, the form of H at this stage is independent of the preceding expansion rate apartfrom the prefactor, however because the values of AMax and TMax differ the evolution ofcosmological abundances still changes for different values of n.

2.3 Onset of radiation domination

Since φ is decaying eventually radiation will come to dominate the energy density of theuniverse, indeed this is desirable to match early universe cosmology such as Big Bang nucle-osynthesis observations. The φ energy density changes as described by eq. (2.11) and at earlytime (where X and R are negligible) can be expressed as follows via separation of variables

dΦ′√Φ

= −dA

√π2g∗(TRH)

30A1+n/2 . (2.30)

Evaluating this integral from AI = 1 we find

Φ =

ΦI · exp

[−√

π2g∗(TRH)30

12+n/2(A2+n/2 − 1)

]for n 6= −4

ΦI ·A−√π2g∗(TRH)/30 for n = −4

. (2.31)

Since the energy density in φ is falling quickly, whilst the radiation component grows grad-ually, at some point (which we denote A×) the contributions from radiation and φ becomecomparable i.e. Φ(A×) ' R(A×). Shortly after A× the universe transitions to radiationdomination and the expansion rate transitions to H ∝ T 2. Importantly, for A & A× theneq. (2.18) (and Figure 1) no longer well describe the evolution, since it is not reasonable toneglect R in the derivation.1 To find A× we numerically solve the coupled differential equa-tions eqns. (2.11) & (2.12) with the initial conditions R(aI) = X(aI) = 0. In Figure 2 (left)we show the values of A× for different values of n (i.e. initial expansion rates), and whereΦ(aI) ≡ ΦI is treated as a free parameter. Fitting to A× we find the form A× = cnΦmn

I

where mn and cn are constants, for instance, for n = −1 then m−1 ≈ 0.20 and c−1 ≈ 0.68.By inspection of Figure 2 (left) it is seen that for ΦI & 108 then eq. (2.31) is valid up to

A & 10, which is the range of Figure 1. The approximation remains good for lower ΦI andhigher A for larger values of n. Moreover, AMax marks the point of transition from increasingto decreasing bath temperature, thus we should check that AMax A×. Inspecting eq. (2.19)we note that for n ≥ −3 then AMax ≤ 2 and AMax A× for reasonable values of ΦI andn. In particular, for ΦI & 1 TeV (typically the range of interest) the bath evolves into thedecreasing temperature regime prior to A×.

For a given TRH one can translate A× into the corresponding temperature T× at whichthe radiation and φ components become comparable, as shown in Figure 2 (center). Theapproximate value of TMax ' 0.3× (MPlHIT

2RH)1/4 is shown as the black dashed line and the

1This approximation also breaks down if the value of X grows too large, but since the growth of X dependson the small free parameter b/mφ, we continue to neglect X in deriving A×.

– 7 –

=-

=-

=-

=-

104 106 108 1010

2

5

10

20

50

ΦI (GeV)

A×

TRH = 1 TeV

n=-3

n=-2

n=-1

n=0

n=+1

TMax

107 1010 1013 1016

500

1000

5000

1×104

5×104

ΦI(GeV)T×(GeV

)

TRH = 1 TeV, ΦI = 1010 GeV

TMaxn=-3

TMaxn=-2

T×n=-3

T×n=-2

5 10 15 20 25

4000

5000

6000

7000

8000

A

T(GeV

)

Figure 2. (Left). The point A× at which R(a×) = Φ(a×) as ΦI is varied and for different n, foundby solving the coupled differential eqns. (2.11) & (2.12) with the initial conditions of eq. (2.14). Theline styles match Figure 1, with n = 0 is dashed and n = 1 dotted. The point A× signifies thebreakdown of eq. (2.31), which underlies Figure 1. (Center). For a given value of TRH (the reheatingtemperature after φ decay) A× is associated to a specific temperature T× via eq. (2.18). Here we showT× as a function of ΦI for TRH = 1 TeV. The black dashed curve indicates the maximum temperatureTMax ∼ 0.3×(MPlHIT

2RH)1/4. Changes in TRH simply scales the y-axis and the relative orientations of

the lines are unchanged. (Right). We illustrate the temperature evolution for two cases with n = −2and n = −3, taking TRH = 1 TeV and ΦI = 1010 GeV, and we highlight where TMax and T× occur.

separation between TMax and T× gives an indication of the length of the transition from theinitial expansion rate to radiation domination. Figure 2 (right) illustrates the temperatureevolution and the points at which TMax and T× occur for two specific cases. Once thetemperature drops below either T× or TRH the system transitions to radiation dominationwith H ∝ T 2. The scenario of interest here is the case in which the dark matter relic densityis set prior to the onset of radiation domination, as we discuss in the next section.

3 Implications for Dark Matter

In the preceding section we studied the behavior of the temperature and the expansion ratefor the case of a period in the early universe in which the expansion follows some generalpower law and while φ is decaying. We consider next the implications for dark matter, inparticular, how the predicted dark matter relic density depends on the exponent n in theinitial expansion rate H ∝ Tn/2+2. We will break the discussion into the following cases:

§3.1: Freeze-in: Thermal production without chemical equilibrium.

§3.2: Freeze-out during reheating: Thermal production without chemical equilibrium.

§3.3: Non-thermal production.

3.1 Freeze-in

First we assume that dark matter is always non-relativistic and does not reach chemicalequilibrium (X Xeq). The case of thermal production of non-relativistic dark matterwithout reaching chemical equilibrium with the thermal radiation bath is an instance of darkmatter freeze-in formulated more generally in [21] and developed in e.g. [22–27]. We considerthe evolution of X following eq. (2.13), for now taking b = 0, in which case

X ′ =

√3

8π〈σv〉MPlTRHΦ

−1/2I An/2−2X2

eq , (3.1)

– 8 –

in terms of the equilibrium distribution given by

Xeq ≡ a3neqX =A3

T 3RH

g

(MXT

2π

)3/2

e−MXT , (3.2)

where g is the number of internal degrees of freedom of the dark matter state. Using eq. (2.26)and substituting eq. (3.2) into eq. (3.1) we obtain

X ′ = 〈σv〉g2M3

Xκ3T 3

Max

8π3HIT 3RH

A(20+n)/8e− 2MXA

(4+n)/8

κTMax . (3.3)

Expressing the cross section in terms of the s and p-wave pieces 〈σv〉 = αs/M2X + αpT/M

3X ,

and integrating (neglecting the temperature dependence in κ, i.e. κ = κRH ≡ κ(TRH)) gives

X∞ =2−(n+28)/(n+4)

n+ 4g2

(κRHTMax)(4n+40)/(n+4)

π3HIT 3RHM

24/n+4X

Γ

(n+ 28

n+ 4

)(αs +

n+ 4

12αp

), (3.4)

where here Γ indicates the gamma function. Assuming that the dark matter is non-relativisticand does not enter chemical equilibrium for b ≈ 0 (more precisely provided that the first termof the Boltzmann equation for X, eq. (2.13), can be safely neglected) then

ρX(TRH) = MXnX(TRH) = MXX∞T 3RH

A3RH

. (3.5)

Furthermore, it is known that at the point of reheating H ' Γφ the energy density forradiation is

ρR(TRH) =π2g∗(TRH)

30T 4RH (3.6)

and thus comparing the ratio of energy densities now and at reheating we have

ρX(Tnow)

ρR(Tnow)=TRH

Tnow

ρX(TRH)

ρR(TRH)=

MX

Tnow

30

A3RHπ

2g∗(TRH)X∞ . (3.7)

Substituting eq. (3.4) into eq. (3.7) it follows that

ρX(Tnow)

ρR(Tnow)=

30× 2−n+28n+4 MX

A3RHπ

5g∗(TRH)Tnow(n+ 4)

g2 (κRHTMax)4n+40n+4

HIT 3RHM

24n+4

X

Γ

(n+ 28

n+ 4

)(αs +

n+ 4

12αp

).

(3.8)Using the form of ARH from eq. (2.26) we can re-express the dark matter relic abundance as

ΩXh2

ΩRh2=

30× 2−n+28n+4 g2

π5(n+ 4)HIg∗(TRH)TnowΓ

(n+ 28

n+ 4

)(κRHTMax)4

T3(n−4)(n+4)

RH M20−nn+4

X

(αs +

n+ 4

12αp

), (3.9)

in terms of the observed fractional energy densities ΩR,X for radiation and dark matter.One could further rewrite eq. (3.9) in terms of TRH by substituting the form of TMax fromeq. (2.20). For n = −1 this scenario is studied in ‘Case A’ of [5], and eq. (3.9) generalisesthis to other values of the exponent n, reproducing the earlier result for n = −1.

– 9 –

-

- -

= -

=-

5.0 5.5 6.0 6.5 7.0 7.5 8.02

3

4

5

6

7

Log[MX /GeV]

Log[TRH/GeV

]×>* ( ) ∝

*>

Φ<=-α =

-

= -

Ω = Ω

0 100000 200000 300000 400000 5000000

500

1000

1500

2000

Mx [GeV]

TRH[GeV

]Figure 3. (Left). Shaded areas indicate regions of the TRH-MX plane for which AMax < A∗ < A× forn = −2 and different values of the initial expansion rate HI and we also require ΦI(HI , TRH) < MPl.(Right). As an example we fix n = −2 and HI = 10−4 GeV and plot the TRH which gives the observedrelic density ΩObs by freeze-in as MX is varied assuming αs = 10−18 and αp = 0, as described ineq. (3.9). We highlight the region AMax < A∗ < A× and ΦI < MPl for HI = 10−4 GeV (matching theleft panel), outside of this region the relic density curve is unreliable and we indicate this by dashingthe line. Note that TRH ∼ 1 TeV and HI ∼ 10−4 GeV corresponds to ΦI ∼ 1017 GeV.

Additionally, note that the point of peak dark matter production A∗ can be found bylooking at where the derivative in eq. (3.3) vanishes. For the s-wave case this occurs for2

A∗ =

[(20 + n

4 + n

)κTMax

2MX

] 84+n

' (ΦI)1

4+n

[0.3 · TRH

2MX

(20 + n

4 + n

)] 84+n

, (3.10)

where in the final equation we have used eqns. (2.14) & (2.25). Observe that A∗ dependsstrongly on TRH/MX but is relatively insensitive to ΦI .

For the relic density to be described by eq. (3.9), i.e. while H ∝ T 4, it is required thatAMax < A∗ < A×. Recall from Figure 2 (left) that A× ∼ O(10), thus dark matter productionoccurs prior to the onset of radiation domination for A∗ < A× ∼ O(10). Typically it canbe arranged that A∗ ∼ O(10) < A× by choosing an appropriate ΦI , although requiring thatΦI < MPl introduces an additional restriction. In Figure 3 (left) we highlight parametervalues for which AMax < A∗ < A× for the case of n = −2. Note that if ΦI is relatively largethen the dark matter mass must be fairly heavy to reproduce the observed relic density.

In Figure 3 (right) we show the parameter region AMax < A∗ < A× along with curvefor which the observed dark matter relic density is reproduced (ΩX = ΩObs) for a specificexample taking n = −2 and HI = 10−4 with couplings αs = 10−18 and αp = 0. We highlightthat generally for the relic density curve to align with the region AMax < A∗ < A× diminutivecouplings are needed α 1, however this is actually fortuitous since such feeble couplingstrengths are required in freeze-in models in order to ensure that the dark matter remainsout of equilibrium [21] (as assumed for this case). In the next subsection we shall derive thecondition on α under which X < Xeq at all times.

2Note taking n = −1 we find that the prefactor is 19/3 which is slightly different from derived in [5] whichgive the prefactor as 17/2. We believe the authors use different criteria and approximations.

– 10 –

3.2 Freeze-out during reheating

Next we consider the case in which the dark matter reaches chemical equilibrium and thenfreezes out while H ∝ T 4. The point of freeze-out can be defined implicitly by

neqX (TF )〈σv〉 = H(TF ) . (3.11)

Using that neqX = XeqA−3T 3RH and eq. (3.2) we can rewrite the lefthand side of the above

equation in terms of TF and for the righthand side we substitute eq. (2.29) to obtain

g√8π3〈σv〉(MXTF )3/2 exp

(−MX

TF

)= |4− n|

(π3g2∗(TF )

45g∗(TRH)

)1/2T 4F

MPlT2RH

. (3.12)

Thus the calculation is largely unchanged from earlier studies, differing only in the factor|4− n|, and the freeze-out temperature is analogous to as derived in [5], given by

xF = ln

(3gMPlT

2RHg∗(TRH)1/2

|4− n|√

5 · 8π3g∗(TF )M3X

(αsx

5/2F + αpx

3/2F

)). (3.13)

Note that, as usual, because of the insensitivity of the logarithm dependences for a largerange of reasonable parameter values xF ∼ O(10). Comparing TF ∼ MX/O(10) to T× inFigure 2 (centre) one finds that T× ∼ (O(100) − O(1000)) GeV therefore for MX & 1 TeVthen typically TF > T×. Thus for a large range of parameters dark matter freeze-out canoccur well before the transition to radiation domination, while H is described by eq. (2.29).

The dark matter abundance remains constant after the point of reheating at H ' Γφ.To ascertain the abundance of dark matter at reheating it is necessary to evolve the freeze-outabundance from TF to TRH as follows

ρX(TRH) =

(a(TRH)

a(TF )

)−3ρX(TF ) =

(g∗(TRH)

g∗(TF )

)2(TRH

TF

)8

ρX(TF ) , (3.14)

where we use that the ratio of FRW scale factors can be replaced by the ratio of dimensionlessA factors. It follows that the dark matter relic density is given by

ΩXh2

ΩRh2= |4− n| 5

√5

4√π

√g∗(TRH)

g∗(TF )

T 3RH

TnowMXMPl

1

αsx−4F + αpx

−5F /5

. (3.15)

Notably, the abundance is essentially insensitive to the expansion rate prior to reheating inthe case that dark matter freeze-out is non-relativistic and in thermal equilibrium.

Whether the relic density is set via freeze-out or freeze-in depends on if the dark matterenters equilibrium. Specifically, for X∞ . Xeq(T∗), where T∗ is the temperature of dominantparticle production given by eq. (3.16), the dark matter will remain out of equilibrium at alltimes and the production rate sets the relic density (the freeze-in scenario). Thus there isa critical value of the coupling α(crit) above which the inequality X∞ . Xeq(T∗) is violated.From eqns. (2.26) & (3.10) we have that A∗ corresponds to a temperatures T∗ which fors-wave is

T∗ 'MX

(8 + 2n

20 + n

). (3.16)

– 11 –

Applying the criteria that for α = α(crit) then X∞ = Xeq(T∗), it follows that for the case

with 〈σv〉 ' αs/M2X the critical coupling α

(crit)s is given by

α(crit)s =

2π3M3X (20 + n)

3(12−n)2(n+4) (4 + n)

5n−282(n+4) |4− n|

√45gΓ

(28+n4+n

)e

20+n2(n+4)MPlT

2RH

g∗(T∗)

g∗(TRH)1/2. (3.17)

Thus for αs < α(crit)s the relic density is set by freeze-in, as in Section 3.1, whereas for

αs > α(crit)s freeze-out dynamics determines the dark matter relic density.

3.3 Non-thermal production

The scenario of non-thermal production is important for b 6≈ 0, in which case a significant(possibly dominant) population of dark matter may be produced directly from φ decays [6].The case of non-thermal production without chemical equilibrium is described by eq. (2.13)with the second (b-independent) term neglected

X ′ =

√π2g∗(TRH)

30

b

mφ

ΦTRHA−n/2√

Φ +RAn + 〈EX〉XAn+1

TRH

. (3.18)

Integrating (for n 6= 2) from AI to ARH and applying the boundary conditions of eq. (2.14)we find the total population of dark matter produced due to φ decays is3

XRH ≡ X(TRH) ' − 2η

n− 2

√πg∗(TRH)

80

HIMPl

TRH

(A

1−n/2RH − 1

), (3.19)

where we define η ≡ b/mφ which parmaterises the φ-dark matter branching fraction.Using the above equation and eq. (3.5) & (3.6) it follows that

ρXb(Tnow)

ρR(Tnow)' 15MXHIMPlη√

5(2− n)π3/2g∗(√TRH)T 4

RHTnow

(A

2−n2

RH − 1

)T 3RH

A3RH

. (3.20)

Further, using. eq. (2.26) to replace ARH this can be rewritten to obtain an expression forthe dark matter relic abundance

ΩX

ΩR' 15MXHIMPlη√

5(2− n)g∗(√TRH)π3/2T 3

RHTnow

[κTMax

TRH

] 4(2−n)n+4

− 1

( TRH

κTMax

) 24n+4

. (3.21)

Similar to previously, the above result generalises expressions in Case 3 of [6] from n = −1to general n.

Note that the preceeding calculation assumes the ordering AMax < ARH < A× whereARH ≡ A(TRH). We can obtain an expression for ARH from eqns. (2.8) & (3.16) as followsARH ∼ (

√ΓφMPl/κTMax)−8/(4+n). Since ARH depends on Γφ and the other quantities do not

depend on Γφ, the above inequality involving ARH is typically not constraining. Although wehighlight here that Big Bang nucleosynthesis constraints [19] imply a limit TRH & 10 MeV.

In Figure 4 we illustrate some example parameter ranges which reproduce the observeddark matter relic density for the case of non-thermal production without chemical equilibrium

3For n = 2 then rather X(TRH) ∝ ln(ARH), this case was studied in [20] and we will not discuss it further.

– 12 –

= -

η=-

η=-

η=

107 108 109 1010

1

10

100

1000

Mx [GeV]

TRH[GeV

]

η=

=-

=

=

1×109 2×109 5×109 1×1010

100

200

300

400

500

Mx [GeV]

TRH[GeV

]

Figure 4. Plot shows lines in the TRH-MX plane for which the dark matter relic density is reproducedvia non-thermal production, following eq. (3.21). Taking three different exponents of the initialexpansion rate n = 1 (dotted), n = 0 (dashed) and n = −1 (solid) and pameterising the φ-dark matterbranching fraction in terms of η ≡ b · GeV/mφ we show three different values η = 0.5, 10−4, 10−7.The plot fixes the initial Hubble rate to be HI =eV. The right panel shows an enlargement of thedashed rectangle of the left panel and illustrates the difference between n = 0, 1, and −1.

for n = 0, 1,−1. In particular, we highlight that changes in n have a modest impact on theappropriate TRH required to reproduce the relic density, however there is a great degree offreedom in η ≡ b/mφ which can lead to substantially larger impacts on the required value ofTRH needed to reproduce the observed dark matter relic density.

Note that if the branching fraction of φ decays (controlled by b) is sufficiently large,the dark matter will enter equilibrium in which case the contribution from non-thermalproduction is reduced due to dark matter annihilations. The production of dark matter dueto φ decays can maintain the dark matter at an equilibrium abundance past T ∼ MX anddark matter only freezes out at T ∼ TRH, when dark matter production ceases. Thus theperiod of freeze-out occurs during the era of radiation domination. As argued in [6] (Case4), this leads to a scaling of the radiation dominated relic density due to the difference inthe entropy density between TRH and the radiation domination freeze-out temperature TFO.Specifically, the dark matter relic abundance will be ΩX ∼ (TFO/TRH)ΩRD where ΩRD isthe dark matter abundance expected from thermal freeze-out during radiation domination.Since in this case freeze-out occurs during radiation domination, the relic abundance will belargely insensitive to the temperature dependance of the initial expansion rate.

4 Concluding Remarks

A myriad of scenarios exists in which the early universe is not immediately radiation dom-inated but goes through periods with expansion rates different to the commonly assumedrelationship H ∝ T 2. In this work we have focused on a previously unstudied case in whichthe early universe is dominated by some state φ which leads to a general expansion rate ofthe form H ∼ T 2+n/2, but due to the fact that φ is decaying there is a subsequent transi-tion to H ∼ T 4. Notably, the form of the initial expansion rate leaves a lasting imprint onrelic densities established while H ∝ T 4, because the value of the exponent n changes thetemperature evolution of the Standard Model thermal bath

– 13 –

The prospect of the dark matter relic abundance being established during a period ofentropy injection following an early matter dominated era was originally studied in influentialpapers of Giudice, Kolb & Riotto [5] and Gelmini & Gondolo [6]. This was later adapted tothe case of a period of decays following kination domination by Visinelli [20]. In this workwe have further generalised to the case in which the initial epoch has a general expansionrate of the form H ∼ T 2+n/2. We have highlighted how the choice of n propagates into thecosmology of the era of significant entropy injection from φ decays and into calculations of thedark matter relic density. While freeze-out during reheating (§3.2) is largely insensitive to theinitial expansion rate, the abundances of dark matter produced via freeze-in (§3.1) or non-thermal production without equilibrium (§3.1) are sensitive to the value of the exponent n.

Notably, the temperature dependance of the initial expansion rate can significantlyimpact the form of the dark matter relic density. Since such variant cosmologies can alterthe predicted dark matter relic density they have previously been used to adjust the freeze-out abundance or evade experimental constraints, see e.g. [6, 17, 18, 28–47]. It would beinteresting to examine how these specific particle physics models (such as the bino, neutralino,and Higgs portal) vary in the context of the generalise scenario outlined here. Moreover, infuture work we plan to explore to what extent the temperature dependance of the earlyexpansion rate imprints on cosmological parameters and observables, such as deviations inthe matter power spectrum [48–50], and whether there is a degeneracy between the initialtemperature dependence of H (the value of n) and the magnitude of the initial expansionrate HI . Also, while the results presented here assume that thermalisation occurs quickly,it would be interesting to consider the converse case, and to track the potential impact ofthermalisation on the dark matter abundance, generalising the work of [51–53] to differentinitial expansion rates.

Acknowledgments. CM is funded by FONDECYT (project 1161150), CONICYT-PCHA/Doctorado Nacional/2018-21180309 and thanks the University of Illinois at Chicagofor hospitality. JU is grateful to New College, Oxford and the Simons Center for Geometryand Physics (Program: Geometry & Physics of Hitchin Systems) for hospitality and support.

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