Establishing the Dark Matter Relic Density in an Era of Particle Decays
Carlos Maldonado James Unwin

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    a]Carlos Maldonadob,c]and James Unwin[a]Departamento de Fisica, Universidad de Santiago de Chile, Santiago, Chile.[b]Department of Physics,  University of Illinois at Chicago, Chicago, Illinois, 60607, USA[c]Simons Center for Geometry and Physics, Stony Brook, New York, 11794, USAIf the early universe is dominated by an energy density which evolves other than radiation-like the normal Hubble-temperature relation H∝ T^2 is broken and dark matter relic density calculations in this era can be significantly different. We first highlight that with 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 the Boltzmann equations and impacts the dark matter relic abundance. We show that the form of the 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. In particular, a dark matter relic density set via freeze-in or non-thermal production is highly sensitive to the temperature dependance of the initial expansion rate.   This work generalises earlier studies which assumed initial expansion rates due to matter or kination domination.Establishing the Dark Matter Relic Density in an Era of Particle Decays
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        December 30, 2023
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    § INTRODUCTION
    
    
    
    
    
    
    The dark matter freeze-out paradigm, in particular the WIMP miracle, is prized for its simplicity and predictiveness. However, it is relatively straightforward to arrange for significant deviations in the predictions of freeze-out by changing either the particle physics model, or the cosmological history. For instance, standard freeze-out calculations typically assume that decoupling occurs whilst the energy density of the universe is dominated by radiation in which case the expansion rate of the universe is H∝ T^2. However, dark matter freeze-out could occur whilst the universe is dominated by some form of energy other than radiation in which case the usual Hubble-temperature relation H∝ T^2 is broken. In particular, one possibility which occurs quite naturally in many Standard Model extensions is the case of an early matter dominated period for which H∝ T^3/2, or an era of particle decays leading to significant entropy production in the thermal bath in which case H∝ T^4<cit.>. The case of dark matter freeze-out during an early period of matter domination was recently highlighted in <cit.> and freeze-out whilst H∝ T^4 was studied in <cit.>. 
    
    More generally the early universe could be dominated by the energy density of a population of states ϕ evolving as an arbitrary power of the scale factor ρ_ϕ(t)=ρ_ϕ(t_I)a^4+n for  t_I some initial time. This evolution  can also be expressed in terms of the ϕ equation of state ω_ϕ=(n+1)/3.  Indeed, observational constraints allow the equation of state for the inflaton to take values in ω_ϕ∈(-1,-1/3) between the end of the de Sitter inflationary period and onset the radiation domination <cit.> potentially allowing for n∈(-4,-2) during this 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 <cit.>, brane world cosmologies  <cit.>, and scalar-tensor theories <cit.>. An explicit example which allows general n was constructed in <cit.>, in which the energy density of the universe is dominated by the contributions from a real scalar field ϕ with a potential of the form
    
        V(ϕ)= 4-2n/(4+n)^2t_I^2  exp[(ϕ(t_I)-ϕ)√(n+4)] .
    
    Values of n∈(-4,2) imply positive potentials, corresponding to ω_ϕ∈(-1,1), however considering also negative potentials extends the range to n>2 (which was the focus of <cit.>).   
     
    
    Such early periods of non-standard cosmology can impact the dark matter evolution and thus it is interesting to explore the implications, while remaining agnostic regarding the form and origin of the evolution of the dominant energy density.
    
    Two recent studies by D'Eramo, Fernandez, & Profumo <cit.> considered the consequences for dark matter if the relic density is established during a period of fast expansion.  These papers focused on scenarios with n>0, in which case the expansion rate H∝ T^n/2+2 is faster than expected from a radiation dominated universe and the energy density in ϕ eventually redshifts to a negligible level. Note that the case n=2 corresponds to `kination domination', see e.g. <cit.>, in which the energy density of the universe is dominated by the kinetic energy of some scalar field  (the ϕ̇ term), for instance the inflaton. Conversely, for n<0, in order to recover the successes of standard cosmology, one requires that the state which dominates the energy density eventually decays. Specifically, the universe should be dominated by the Standard Model radiation bath at temperates around 10 MeV and below (until matter-radiation equality) so not to spoil the precision predictions of Big Bang nucleosynthesis (see e.g. <cit.>).
    
    Here we study the case of dark matter which freezes out or is produced during a period of particle production due to the decays of some (boson or fermion) state ϕ which dominates the energy density and under the assumption that ρ_ϕ∝ a^n+4 thus implying an initial expansion rate of the form H∝ T^n/2+2.  Notably, during this period of ϕ decays, entropy is no longer conserved in the thermal bath and this impacts the dark matter relic abundance calculation.  In particular, our work generalizes the earlier papers of  <cit.> which assume that the initial expansion rate corresponds to an early matter dominated phase. Additionally, this work can also be seen as an extension of dark matter freeze-out or production with a general expansion rate, as studied in <cit.>, to include decays of ϕ. 
    
    This paper is structured as follows: In Section <ref> we discuss the formulation of the Boltzmann equation without entropy conservation, generalizing the derivations of <cit.>. Using these results, in Section <ref> we compare different scenarios for setting the dark matter relic abundance and discuss their dependance on the exponent n of the initial expansion rate. Concluding remarks are given in Section <ref>.
    
    
    
    
    
    § BOLTZMANN EQUATIONS WITHOUT ENTROPY CONSERVATION
    
    
    We start by deriving expressions for the evolution of the different particle populations in the case that the early universe is dominated by a state ϕ resulting in an initial expansion rate of H∝ T^n/2+2. We show that the expansion rate subsequently transitions to H∝ T^4 and derive an expression for the maximum temperature of the Standard Model radiation.
    
    The expressions derived reproduce the results of Giudice, Kolb, & Riotto <cit.> for matter domination prior to decays (n=-1) and Visinelli <cit.> with kination domination prior to decays (n=2).
    
    
    
     §.§ Boltzmann equations
    
    We start in familiar territory by defining the Hubble parameter
    
        H^2=8π/3M_ Pl^2(ρ_ϕ+ρ_R+ρ_X) ,
    
    where the terms indicate the energy density in ϕ, Standard Model radiation, and dark matter X, respectively. The evolution of quantities can be tracked relative to a dimensionless scale factor A defined as 
     
    A≡a/a_I  =aT_ RH .
    
    where a_I is an arbitrary initial reference point which is chosen  to be a_I=1/T_ RH and  T_ RH is the reheat temperature following ϕ decays.
    Emulating the analysis of <cit.>, we rewrite the relevant variables as dimensionless quantities, but where now the energy density of ϕ evolves as some arbitrary power a^4+n
    Φ   ≡ ρ_ϕ a^4+nT_ RH^n =ρ_ϕ A^4+n/T_ RH^4 ,
         
    R ≡ρ_Ra^4  =ρ_RA^4/T_ RH^4 , 
        
    X ≡ n_Xa^3 =n_XA^3/T_ RH^3 .
    
    With the assumption that ρ_X=⟨ E_X⟩ n_X, where n_X is the dark matter number density and ⟨ E_X⟩ is the expected energy of each dark matter state, one can express the Hubble rate in terms of these dimensionless variables to obtain
     
    H   = T_ RH^2/M_ PlA^2+n/2√(8π/3(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH)) .
    
    This recovers the form of H in <cit.> for n=-1.
    
    For an expansion rate of the form of eq. (<ref>), the Boltzmann equations which describe the evolution of the number densities can be expressed in a  manner highly reminiscent to  those studied in <cit.>
    
        ρ̇_̇ϕ̇+(4+n)Hρ_ϕ   =  -Γ_ϕρ_ϕ
        ρ̇_̇Ṙ+4Hρ_R    =Γ_ϕρ_ϕ +2⟨ E_X⟩(n_X^2-n_ eq^2)⟨σ v⟩
        ṅ_̇Ẋ+3Hn_X    = b/m_ϕΓ_ϕρ_ϕ - (n_X ^2 - n_ eq^2)⟨σ v⟩
    
    where m_ϕ and Γ_ϕ are the ϕ mass and decay rate to Standard Model. Dotted variables indicate differentiation with respect to time, the quantity b paramaterises the branching ratio of ϕ to dark matter, ⟨σ v⟩ is the thermally averaged dark matter annihilation cross section, and n_ eq denotes the equilibrium number density of dark matter which has its usual form. Furthermore, note that the ϕ decay rate can be expressed in terms of the reheat temperature after ϕ decays
    
        Γ_ϕ=√(4π^3g_*(T_ RH)/45)T_ RH^2/M_ Pl .
    
    
    
    
    We next re-express eqns. (<ref>)-(<ref>) in terms of the dimensionless units of eq. (<ref>).
    
    Taking first eq. (<ref>), simple substitution yields
       Ȧd/dA(Φ/(Aa_I)^4+nT_ RH^n)+(4+n)HΦ/(Aa_I)^4+nT_ RH^n=-Γ_ϕΦ/(Aa_I)^4+nT_ RH^n .
    
    After some manipulation,  and using that H=ȧ/a, this can be simplified to
    
    ȦΦ '=Γ_ϕΦ ,
    
    where the primed variable indicates differentiation with respect to A. Furthermore, using eqns. (<ref>) & (<ref>) we can express eq. (<ref>) in terms of Φ' as follows
    Φ '
       =-√(π^2g_*(T_ RH)/30)Φ A^1+n/2/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH) .
    
    Analogously, eqns. (<ref>) & (<ref>) can be rewritten in a similar fashion to obtain
    
    R'=   √(π^2g_*(T_ RH)/30)Φ A^1-n/2/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH) 
    +√(3/8π)2⟨σ v⟩⟨ E_X⟩ M_ PlA^n/2-1(X^2-X_ eq^2)/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH)
    and
    
        X'=   √(π^2g_*(T_ RH)/30)b/m_ϕΦ T_ RHA^-n/2/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH) 
        -√(3/8π)⟨σ v⟩ M_ PlT_ RHA^n/2-2(X^2-X_ eq^2)/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH).
    
    Thus the dimensionless versions of  eqns. (<ref>)-(<ref>) are, respectively,   eqns. (<ref>)-(<ref>). 
    
    
    
     §.§ 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 initial energy density of ϕ can be written
    ρ_ϕ(a_I)=(3/8π)H_I^2 M_ Pl^2
    where H_I≡ H(a_I) is the initial expansion rate (throughout we will use the subscript I to mean the value of a give quantity  at a=a_I). In terms of  dimensionless variables this initial condition is
    
        Φ_I=3H_I^2 M_ Pl^2/8π T_ RH^4,  R_I=X_I=0,    A_I=1.
    
    One instance in which such initial conditions could arise, for instance, is immediately after inflation, as in the case of kination domination <cit.> corresponding to n=2.
    
    In  <cit.> the authors studied the evolution of the temperature of the Standard Model thermal 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 due to the decays of ϕ until it hits some maximum temperature T_ Max after which the expansion rate transitions to H∝ T^4. The bath then cools until the temperature T_ RH at H≃Γ_ϕ after which ϕ decays become negligible and the universe becomes radiation dominated with H∝ T^2.  We next generalise this analysis to the case that the early universe expansion rate sourced by ϕ is an arbitrary power of the temperature as in eq. (<ref>).
    
    
    The ϕ decays transfer energy to the Standard Model bath and one can derive the point A_ Max at which the maximum temperature of the radiation T_ Max occurs for the more general expansion rate.
    
    Since the dominant contribution comes from ϕ at early time, we can neglect the second term in eq. (<ref>), and using eq. (<ref>) we obtain
    
    
        R'=√(π^2g_*(T_ RH)/30)Φ_I^1/2A^1-n/2.
    
    Then integrating we  obtain the following 
    
    
        R=
        {[ √(π^2g_*(T_ RH)/30)√(Φ_I)(1/2-n/2)(A^2-n/2-1)                                     for   n<4;                √(π^2g_*(T_ RH)/30)√(Φ_I)ln(A)                                     for   n=4 ]. .
    
    The temperature is a measure of the radiation energy density, and thus we can obtain an expression for the evolution of T as a function of A from the expression 
    ρ_R=π^2g_*(T)/30T^4=R(T_ RH/A)^4 .
     
    Moreover, substituting eq. (<ref>) into eq. (<ref>) and rearranging, gives the evolution of the temperature (for n≠ 4)
    
    
    T   =   (45/4π^3g_*(T_ RH)/g_*^2(T))^1/8(H_IM_ PlT_ RH^2)^1/4[A^-(2+n/2)-A^-4/2-n/2]^1/4 .
    
    
    
    The critical point, with respect to A, of the factor in square brackets of eq. (<ref>) marks the maximum temperature and the value of the scale factor A_ Max at which the temperature stops increasing and begins to decrease. For  |n|<4 this is given  by
     
    A_ Max=(n+4/8)^2/(n-4) .
    
    For comparison, recall that A_I=1 corresponds to T=T_ RH and A>1 implies a> a_I. For A>A_ Max the A^-4 piece in eq. (<ref>) can be neglected and T∝ A^-(2+n/2). Observe that A_ Max is sensitive to the exponent n of the early universe expansion rate.
    
    The temperature extremum  T_ Max for |n|<4 is found at A=A_ Max given by
    
     T_ Max=
    (45g_*(T_ RH)/4π^3g_*^2(T_ Max))^1/8(M_ PlH_I T_ RH^2)^1/4(2/4-n)^1/4
    [(n+4/8)^4+n/4-n-(n+4/8)^8/4-n]^1/4.
    
    As a reference, taking a few specific choices for n, the value of T_ Max can be approximated as
    
    T_ Max≃(M_ PlH_I T_ RH^2)^1/4×{[       0.30 for   n=-1;       0.31 for   n=-2;       0.33 for   n=-3;            ]. ,
    
    where we take  g_*(T_ RH)≈ g_*(T_ Max)≈100. For example, taking reasonable values for both H_I∼ eV and T_ RH∼ 1 TeV, then T_ Max∼ 3 TeV for |n|∼𝒪(1). 
    
    Furthermore, we can re-express eq. (<ref>) in terms of a normalised function f(A_ Max)=1 as follows
    
    T= T_ Max f(A)
    
    for 
    
    f(A)≡κ(T)[A^-(2+n/2)-A^-4]^1/4
     with
    κ(T)=[g_*(T_ Max)/g_*(T)]^1/4[(4 + n/8)^4 + n/4 - n-(4 + n/8)^8/4 - n]^-1/4 .
    
    For reference, a selection of specific values for κ are
    κ(T)≈[g_*(T_ Max)/g_*(T)]^1/4{[ (8^8/3^3·5^5)^1/20         for   n=-1;     (4^4/3^3)^1/12         for   n=-2;    (16^6/7^7)^1/28         for   n=-3 ]. .
    
    
    Starting from a=a_I the temperature increases from a negligible value to T_ Max, and then subsequently decreases according to eq. (<ref>), as illustrated in Figure <ref>. The evolution of the bath temperature in  Figure <ref> assumes that ϕ dominates the energy density, and the evolution will be altered once the energy in radiation becomes comparable to ϕ, we denote this A_×, as we discuss in the next section.
    
    Thus for A_ Max<A<A_× one can  approximate the temperature evolution as follows
    
    T∼κ T_ Max A^-(2+n/2)/4 .
    
    Between the time when T_ Max is reached and the point of radiation domination, at the earlier of T_ RH or T_×, the ϕ field energy density scales as ρ_ϕ=Φ_IT_ RH^4/A^4+n.  Since the dominant contribution to the Hubble parameter at early times comes from ρ_ϕ, it follows from eq. (<ref>) and eq. (<ref>) that for A_ Max<A<A_× then
    
    
    H^2   ≃   8π/3M_ PlΦ_IT_ RH^4/A^4+n=(H_IA^-(4+n)/2)^2 .
    
    Moreover, using eq. (<ref>) and eq. (<ref>), we can express A in terms of the temperature
    
        A^-(4+n)/2=(4π^3(2-n/2)^2g^2_*(T)/45g_*(T_ RH))^1/2T^4/H_IM_ PlT_ RH^2 .
    
    Substituting eq. (<ref>) into (<ref>) gives an expression for H for  A_ Max<A<A_×
    
        H=|4-n|(π^3g^2_*(T)/45g_*(T_ RH))^1/2T^4/M_ PlT_ RH^2 .
    
    Thus the point  A_ Max  indicates the A at which the evolution transitions to H∝ T^4. Interestingly,  the form of H at this stage is independent of the preceding expansion rate apart from the prefactor, however because the values of A_ Max and T_ Max differ the evolution of cosmological abundances still changes for different values of n. 
    
    
    
     §.§ Onset of radiation domination
    
    
    Since ϕ is decaying eventually radiation will come to dominate the energy density of the universe, indeed this is desirable to match early universe cosmology such as Big Bang nucleosynthesis  observations. The ϕ energy density changes as described by  eq. (<ref>) and at early time (where X and R are negligible) can be expressed as follows  via separation of variables
     d Φ'/√(Φ)   =-
      d A 
    √(π^2g_*(T_ RH)/30)  A^1+n/2 .
    
    Evaluating this integral from A_I=1  we find
    
        Φ=
        {[ Φ_I· exp[-√(π^2g_*(T_ RH)/30)1/2+n/2(A^2+n/2-1)]                                       for   n≠-4;                     Φ_I · A^-√(π^2g_*(T_ RH)/30)                                       for   n=-4 ]. .
    
    Since the energy density in ϕ is  falling quickly, whilst the radiation component  grows gradually, at some point (which we denote A_×) the contributions from radiation and ϕ become comparable i.e. Φ(A_×)≃ R(A_×). Shortly after A_× the universe transitions to radiation domination and the expansion rate transitions to H∝ T^2.
    
    Importantly, for A≳ A_× then eq. (<ref>) (and Figure <ref>) no longer well describe the evolution, since it is not reasonable to neglect R in the derivation.[This approximation also breaks down if the value of X grows too large, but since the growth of X depends on the small free parameter b/m_ϕ, we continue to neglect X in deriving A_×.]
    To find  A_× we numerically solve the coupled differential equations  eqns. (<ref>) & (<ref>) with the initial conditions R(a_I)=X(a_I)=0. In  Figure <ref> (left) we show the values of A_× for different values of n (i.e. initial expansion rates), and where   Φ(a_I)≡Φ_I is treated as a free parameter. Fitting to A_× we find the form A_×=c_n Φ_I^m_n where m_n and c_n are constants, for instance, for n=-1 then m_-1≈0.20 and c_-1≈ 0.68.
    
    
    	
    
     By inspection of Figure <ref> (left) it is seen that for Φ_I≳ 10^8 then eq. (<ref>) is valid up to A≳10, which is the range of Figure <ref>. The approximation remains good for lower Φ_I and higher A for larger values of n. Moreover, A_ Max marks the point of transition from increasing to decreasing bath temperature, thus we should check that A_ Max≪ A_×. Inspecting eq. (<ref>) we note that for n≥-3 then A_ Max≤2 and A_ Max≪ A_× for reasonable values of Φ_I and n.  In particular, for Φ_I≳1 TeV (typically the range of interest) the bath evolves into the decreasing temperature regime prior to A_×. 
    
    For a given T_ RH one can   translate A_× into the corresponding temperature  T_× at which the radiation and ϕ components become comparable, as shown in Figure <ref> (center). The approximate value of T_ Max≃0.3× (M_ PlH_I T_ RH^2)^1/4 is shown as the  black dashed line and the separation between T_ Max and T_× gives an indication of the length of the transition from the initial expansion rate to radiation domination.
    
     Figure <ref> (right) illustrates the temperature evolution and the points at which T_ Max and T_× occur for two specific cases. Once the temperature drops below either  T_× or T_ RH the system transitions to radiation domination with H∝ T^2. The scenario of interest here is the case in which the dark matter relic density is set prior to the onset of radiation domination, as we discuss in the next section.
    
    
    
    
    
    
    
    § IMPLICATIONS FOR DARK MATTER 
    
    
    
    In the preceding section we studied the behavior of the temperature and the expansion rate for the case of a period in the early universe in which the expansion follows some general power law and while ϕ is decaying.  We consider next the implications for dark matter, in particular, how the predicted dark matter relic density depends on the exponent n in the initial expansion rate H∝ T^n/2+2. We will break the discussion into the following cases:
    
      <ref>: Freeze-in: Thermal production without chemical equilibrium.
    
    
    
      <ref>: Freeze-out during reheating: Thermal production without chemical equilibrium.
    
    
    
      <ref>:  Non-thermal production.
    
    
    
    
     §.§ Freeze-in
    
    First we assume that dark matter is always non-relativistic and does not reach chemical equilibrium   (X≪ X_ eq).  The case of thermal production of non-relativistic dark matter  without reaching chemical equilibrium with the thermal radiation bath is an instance of  dark matter freeze-in  formulated more generally  in <cit.> and developed in e.g. <cit.>.  
    
    We consider the evolution of X following eq. (<ref>), for now taking b=0, in which case
    
    
        X'=√(3/8π)⟨σ v⟩ M_ PlT_ RHΦ_I^-1/2A^n/2-2X_ eq^2 ,
    
    in terms of the equilibrium distribution given by
    
    
        X_ eq≡ a^3n_X^ eq=A^3/T_ RH^3g(M_X T/2π)^3/2e^-M_X/T ,
    
    where g is the number of internal degrees of freedom of the dark matter state.
    Using eq. (<ref>) and substituting eq. (<ref>) into eq. (<ref>) we obtain
    
    
    X' =⟨σ v⟩g^2M_X^3κ^3T_ Max^3/8π^3H_IT_ RH^3A^(20+n)/8e^-2M_X A^(4+n)/8/κ T_ Max .
    
    Expressing the cross section in terms of the s and p-wave pieces ⟨σ v⟩=α_s/M_X^2+ α_pT/M_X^3, and integrating (neglecting the temperature dependence in κ, i.e. κ=κ_ RH≡κ(T_ RH))  gives
    
        X_∞=2^-(n+28)/(n+4)/n+4g^2(κ_ RH T_ Max)^(4n+40)/(n+4)/π^3H_IT_ RH^3M_X^24/n+4Γ(n+28/n+4)(α_s+n+4/12α_p) ,
    
     where here Γ indicates the gamma function.  Assuming that the dark matter is non-relativistic and does not enter  chemical equilibrium for b≈0 (more precisely provided that the first term of the Boltzmann equation for X, eq. (<ref>), can be safely neglected) then
    
    
        ρ_X(T_ RH)=M_Xn_X(T_ RH)=M_XX_∞T_ RH^3/A_ RH^3 .
    
    Furthermore, it is known that at the point of reheating H≃Γ_ϕ the energy density for radiation is
    
    
        ρ_R(T_ RH)=π^2g_*(T_ RH)/30T_ RH^4
    
    and thus comparing the ratio of energy densities now and at reheating we have
    
    
        ρ_X(T_ now)/ρ_R(T_ now)=T_ RH/T_ nowρ_X(T_ RH)/ρ_R(T_ RH)=M_X/T_ now30/A_ RH^3π^2g_*(T_ RH)X_∞ .
    
    Substituting  eq. (<ref>) into eq. (<ref>) it follows that 
    
    
        ρ_X(T_ now)/ρ_R(T_ now)
        = 30× 2^-n+28/n+4 M_X/A_ RH^3π^5g_*(T_ RH)T_ now(n+4)g^2(κ_ RH T_ Max)^4n+40/n+4/H_IT_ RH^3M_X^24/n+4Γ(n+28/n+4)(α_s+n+4/12α_p) .
    
    Using the form of A_ RH from eq. (<ref>) we can re-express the dark matter relic abundance as
    
        Ω_Xh^2/Ω_Rh^2
        =30× 2^-n+28/n+4g^2/π^5(n+4)H_Ig_*(T_ RH)T_ nowΓ(n+28/n+4)
         (κ_ RHT_ Max)^4/T_ RH^3(n-4)/(n+4)M_X^20-n/n+4(α_s+n+4/12α_p) ,
    
    in terms of the observed fractional energy densities Ω_R,X for radiation and dark matter. One could further rewrite eq. (<ref>) in terms of T_ RH by substituting the form of T_ Max from eq. (<ref>). For n=-1 this scenario is studied in `Case A' of  <cit.>,  and eq. (<ref>) generalises this to other values of the exponent n, reproducing the earlier result for n=-1. 
    
    
    	
    	
    
    	
    	
    Additionally, note that the point of peak dark matter production A_* can be found by looking at where the derivative in  eq. (<ref>) vanishes. 
    
    For the s-wave case this occurs for[Note taking n=-1 we find that the prefactor is 19/3 which is slightly different from derived in <cit.> which give the prefactor as 17/2. We believe the authors use different criteria and approximations.]
    
        A_*=[(20+n/4+n)κ T_ Max/2 M_X]^8/4+n≃(Φ_I)^1/4+n[0.3·T_ RH/2M_X(20+n/4+n)]^8/4+n,
    
    where in the final equation we have used eqns. (<ref>) & (<ref>). Observe that A_* depends strongly on T_ RH/M_X but is relatively insensitive to Φ_I. 
    
    For the relic density to be described by eq. (<ref>), i.e. while H∝ T^4, it is required that A_ Max<A_*<A_×. Recall from Figure <ref> (left) that A_×∼𝒪(10), thus dark matter production  occurs prior to the onset of radiation domination for A_*<A_×∼𝒪(10).  Typically it can be arranged that  A_*∼𝒪(10)<A_× by choosing an appropriate Φ_I, although requiring that Φ_I<M_ Pl introduces an additional restriction.  In Figure <ref> (left) we highlight parameter values for which A_ Max<A_*<A_× for the case of n=-2.  Note that if Φ_I is relatively large then the dark matter mass must be fairly heavy to reproduce the observed relic density. 
     
    
     
    In Figure <ref> (right) we show the parameter region  A_ Max<A_*<A_× along with curve for which the observed dark matter relic density is reproduced (Ω_X=Ω_ Obs) for a specific example taking n=-2 and H_I=10^-4 with couplings α_s=10^-18 and α_p=0. We highlight that generally for the relic density curve to align with the region A_ Max<A_*<A_× diminutive couplings are needed α≪1, however this is actually fortuitous since such feeble coupling strengths are required in freeze-in models in order to ensure that the dark matter remains out of equilibrium <cit.> (as assumed for this case). In the next subsection we shall derive the condition on α under which X< X_ eq at all times.
    
    
    
    
     §.§ Freeze-out during reheating
    
    
    
    
    Next we consider the case in which the dark matter reaches chemical equilibrium and then freezes out while H∝ T^4. The point of freeze-out can be defined implicitly by 
    
        n_X^ eq(T_F)⟨σ v⟩=H(T_F) .
    
    Using that n_X^ eq=X^ eqA^-3T_ RH^3 and eq. (<ref>) we can rewrite the lefthand  side of the above equation in terms of T_F and for the righthand side we substitute eq. (<ref>) to obtain
    
        g/√(8π^3)⟨σ v⟩ (M_X T_F)^3/2exp(-M_X/T_F)=
        |4-n|(π^3g^2_*(T_F)/45g_*(T_ RH))^1/2T_F^4/M_ PlT_ RH^2 .
    
    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 <cit.>, given by
    
    x_F   =   ln(3gM_ PlT_ RH^2g_*(T_ RH)^1/2/|4-n|√(5)· 8π^3g_*(T_F)M_X^3(α_sx_F^5/2+α_px_F^3/2)) .
    
    Note that, as usual, because of the insensitivity of the logarithm dependences for a large range of reasonable parameter values x_F∼𝒪(10). Comparing T_F∼ M_X/𝒪(10) to T_× in Figure <ref> (centre) one finds that T_×∼(𝒪(100)-𝒪(1000)) GeV therefore for M_X≳ 1 TeV then typically T_F > T_×. Thus for a large range of parameters dark matter freeze-out can occur well before the transition to radiation domination, while H is described  by eq. (<ref>).
    
    The dark matter abundance remains constant after the point of reheating at H≃Γ_ϕ. To ascertain the abundance of dark matter at reheating it is necessary to evolve the freeze-out abundance from T_F to T_ RH as follows
    
        ρ_X(T_ RH) = (a(T_ RH)/a(T_F))^-3ρ_X(T_F)
         = (g_*(T_ RH)/g_*(T_F))^2 (T_ RH/T_F)^8 ρ_X(T_F) ,
    
    where we use that the ratio of FRW scale factors can be replaced by the ratio of dimensionless A factors. 
    It follows that the dark matter relic density is given by 
    
        Ω_Xh^2/Ω_Rh^2 =|4-n|5√(5)/4√(π)√(g_*(T_ RH))/g_*(T_F)T_ RH^3/T_ nowM_XM_ Pl1/α_sx_F^-4+α_px_F^-5/5 .
    
    Notably, the abundance is essentially insensitive to the expansion rate prior to reheating in the 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 matter enters equilibrium. Specifically,  for X_∞≲ X_ eq(T_*), where T_* is the temperature of dominant particle production given by eq. (<ref>),  the dark matter will remain out of equilibrium at all times and the production rate  sets the relic density (the freeze-in scenario). Thus there is a critical value of the coupling α^ (crit) above which  the inequality X_∞≲ X_ eq(T_*) is violated.
    
    From eqns. (<ref>) & (<ref>) we have that A_* corresponds to a temperatures T_* which for s-wave is
    
        T_*≃ M_X(8+2n/20+n) .
    
    Applying the criteria that for α=α^ (crit) then X_∞= X_ eq(T_*), it follows that for the case with ⟨σ v⟩≃α_s/M_X^2 the critical coupling α_s^ (crit) is given by
    α_s^ (crit)=2π^3 M_X^3(20+n)^3(12-n)/2(n+4)(4+n)^5n-28/2(n+4)|4-n|/√(45)gΓ(28+n/4+n)e^20+n/2(n+4)M_ PlT_ RH^2g_*(T_*)/g_*(T_ RH)^1/2.
    
    Thus for α_s<α_s^ (crit) the relic density is set by freeze-in, as in Section <ref>, whereas for  α_s>α_s^ (crit) freeze-out dynamics determines the dark matter relic density.
    
    
    
    
    
     §.§ Non-thermal production
    
    
    
     The scenario of non-thermal production is important for  b≉0, in which case a significant (possibly dominant) population of dark matter may be produced directly from ϕ decays <cit.>. The case of non-thermal production without chemical equilibrium is described by eq. (<ref>) with the second (b-independent)  term neglected
    
    X'=   √(π^2g_*(T_ RH)/30)b/m_ϕΦ T_ RHA^-n/2/√(Φ+RA^n+⟨ E_X⟩ X A^n+1/T_ RH) .
    
    Integrating (for n≠2) from A_I to A_ RH and applying the boundary conditions of eq. (<ref>) we find the total population of dark matter produced due to ϕ decays is[For  n=2 then rather X(T_ RH)∝ln(A_ RH), this case was studied in <cit.> and we will not discuss it further.]
    
        X_ RH≡ X(T_ RH)≃-2η/n-2√(π g_*(T_ RH)/80)H_IM_ Pl/T_ RH(A_ RH^1-n/2-1) ,
    
    where we define η≡ b/m_ϕ which parmaterises the ϕ-dark matter branching fraction. 
    
    Using the above equation and  eq. (<ref>) & (<ref>) it follows that
    
    ρ_X_b(T_ now)/ρ_R(T_ now)≃15M_XH_IM_ Plη/√(5)(2-n)π^3/2g_*(√(T_ RH)) T_ RH^4 T_ now(A_ RH^2-n/2-1)T_ RH^3/A_ RH^3 .
    
    Further, using. eq. (<ref>) to replace A_ RH this can be rewritten to obtain an expression for the dark matter relic abundance
    Ω_X/Ω_R   ≃   15M_XH_IM_ Plη/√(5)(2-n)g_*(√(T_ RH))π^3/2T_ RH^3T_ now([κ T_ Max/T_ RH]^4(2-n)/n+4-1)(T_ RH/κ T_ Max)^24/n+4.
    
    Similar to previously, the above result generalises expressions in Case 3 of <cit.> from n=-1 to general n.
    
    
    Note that the preceeding calculation assumes the ordering A_ Max<A_ RH<A_× where A_ RH≡ A(T_ RH). We can obtain an expression for A_ RH from eqns. (<ref>) & (<ref>) as follows A_ RH∼  (√(Γ_ϕ M_ Pl)/ κ T_ Max)^-8/(4+n). Since A_ RH depends on Γ_ϕ and the other quantities do not depend on Γ_ϕ, the above inequality involving A_ RH is typically not constraining. Although we highlight here that Big Bang nucleosynthesis constraints <cit.> imply a limit T_ RH≳ 10 MeV.
    
    	
    
    
    
    In Figure <ref>  we illustrate some example parameter ranges which reproduce the observed dark matter relic density for the case of  non-thermal production without chemical equilibrium for n=0,1,-1.  In particular, we highlight that changes in n have a modest impact on the appropriate T_ RH required to reproduce the relic density, however there is a great degree of freedom in  η≡ b/m_ϕ which can lead to substantially larger impacts on the required value of T_ RH 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-thermal production is reduced due to dark matter annihilations. The production of dark matter due to ϕ decays can maintain the dark matter at an equilibrium abundance past T∼ M_X and dark matter only freezes out at T∼ T_ RH, when dark matter production ceases. Thus the period of freeze-out occurs during the era of radiation domination. As argued in <cit.> (Case 4), this leads to a scaling of the radiation dominated relic density due to the difference in the entropy density between T_ RH and the radiation domination freeze-out temperature T_ FO. Specifically, the dark matter relic abundance will be Ω_X∼(T_ FO/T_ RH)Ω_ RD where Ω_ RD is the 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 be largely insensitive to the temperature dependance of the initial expansion rate.
    
    
    
    
    
    § CONCLUDING REMARKS
    
    
    
    A myriad of scenarios exists in which the early universe is not immediately radiation dominated but goes through periods with expansion rates different to the commonly assumed relationship H∝ T^2. In this work we have focused on a previously unstudied case in which the early universe is dominated by some state ϕ which leads to a general expansion rate of the form H∼ T^2+n/2, but due to the fact that  ϕ is decaying there is a subsequent transition to H∼ T^4. Notably, the form of the initial expansion rate leaves a lasting imprint on relic densities established while H∝ T^4, because the value of the exponent n changes the temperature evolution of the Standard Model thermal bath
    
    The prospect of the dark matter relic abundance being established during a period of entropy injection following an early matter dominated era was originally studied in influential papers  of  Giudice, Kolb & Riotto <cit.> and Gelmini & Gondolo <cit.>. This was later  adapted to the case of a period of decays following kination domination by Visinelli <cit.>. In this work we have further generalised to the case in which the initial epoch has a general expansion rate  of the form H∼ T^2+n/2. We have highlighted how the choice of n propagates into the cosmology of the era of significant entropy injection from ϕ decays and into calculations of the dark matter relic density. While freeze-out during reheating (<ref>) is largely insensitive to the initial expansion rate, the abundances of dark matter produced via freeze-in (<ref>) or non-thermal production without equilibrium (<ref>) are sensitive to the value of the exponent n. 
    
    Notably, the temperature dependance of the initial expansion rate can significantly impact the form of the  dark matter relic density. Since such variant cosmologies can alter the predicted dark matter relic density  they have previously been used to adjust the freeze-out abundance or evade experimental constraints, see e.g. <cit.>. It would be interesting 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, in future work we plan to explore to what extent the temperature dependance of the early expansion rate imprints on cosmological parameters and observables, such as deviations in the matter power spectrum <cit.>, and whether there is a degeneracy between the initial temperature dependence of H (the value of n) and the magnitude of the initial expansion rate H_I. 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 of thermalisation on the dark matter abundance, generalising the work of <cit.> to different initial expansion rates. 
    
    
    
    
    
     Acknowledgments.
    CM is funded by FONDECYT (project 1161150), CONICYT-PCHA/Doctorado
    Nacional/2018-21180309 and thanks the University of Illinois at Chicago for hospitality. JU is grateful to New College, Oxford and the Simons Center for Geometry and Physics (Program: Geometry & Physics of Hitchin Systems) for hospitality and support.
    
    
    
    
    
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