Melanopogenesis: Dark Matter of (almost) any Mass and Baryonic Matter from the Evaporation of Primordial Black Holes weighing a Ton (or less)
Logan Morrison Stefano Profumo Yan Yu
 
Sampling on the sphere from f(x)∝ x^TAx
    Richard Arnold
 
        School of Mathematics and Statistics

        Victoria University of Wellington

        Wellington, New Zealand
 
        richard.arnold@vuw.ac.nz
    December 30, 2023
================================================================================================================================================================================






§ INTRODUCTION



The nature of non-baryonic dark matter and the origin of the matter-antimatter asymmetry in the universe are two of the most pressing questions at the junction of particle physics and cosmology (see e.g. <cit.>).  The direct detection of gravitational waves <cit.> has triggered renewed interest in the role black holes might play in connection with these two outstanding open questions (see e.g. <cit.> and references therein).

The idea that black hole evaporation can lead to the production of particles in the early universe and possibly to the generation of a baryon asymmetry or of dark matter has a rather long history. In fact, it dates back to early work by Hawking <cit.> and Zel'dovich <cit.>, and to subsequent work by Carr <cit.>, where it was pointed out that evaporation might produce a baryon asymmetry because of intrinsic CP-violating effects, or because of accidental statistical excesses. Subsequent work invoked GUT-scale physics, which generically violates CP and baryon number, as a culprit for the generation of the matter-antimatter asymmetry, and specifically CP- and B-violating decays of GUT-scale particles produced in PBH evaporation. This was  first (to our knowledge) envisioned in Ref. <cit.>, and further elaborated upon in Ref. <cit.>.

More recently, studies have considered the possibility that primordial black holes be responsible for both the generation of a baryon asymmetry and of the dark matter. Specifically, Ref. <cit.> considers the concurrent generation of a baryon asymmetry from GUT boson decays in inflationary and in ekpyrotic/cyclic models, and the possible presence of dark matter in the form of Planck-scale relics from the evaporation of light primordial holes. The key assumption in Ref. <cit.> is that the black holes dominate the universe's energy budget by the time they decay. More recently, Ref. <cit.> dealt with the possibility of a generation of the matter asymmetry from asymmetric sterile neutrino decays, through leptogenesis, and the co-genesis of dark matter from Hawking evaporation. Here, the assumption is, again, that the energy density of the universe is dominated by the primordial black holes before they evaporate. Ref. <cit.> additionally allows for a possible entropy injection episode after the evaporation of the primordial holes. Ref. <cit.> and <cit.> recently focused on production of dark matter from black hole evaporation (the second specifically WIMP dark matter), while Ref. <cit.> and <cit.> considered scenarios technically similar to asymmetric dark matter production from black hole evaporation that we also entertain below, although the first study in the context of mirror matter and the second in the context of asymmetric Hawking radiation.

In the present study, we are not concerned with how primordial black holes were produced following inflation (any pre-existing population would be presumably inflated away). The conditions under which a cosmologically relevant population of primordial black holes are produced have been known for a long time, ever since the seminal work of Zel'dovich and Novikov from over a half century ago <cit.>. We will not review them here, referring the Reader to the extensive existing literature <cit.>. In fact, in what follows we will use the relative abundance of primordial black holes at formation as a free parameter.

We intend to study here the generic possibility that primordial black hole evaporation at very early times (much earlier than, say, the epoch when the synthesis of light elements occurs) plays a key role in the genesis of dark matter and/or the baryon asymmetry in the universe. We will be agnostic about the fate of black holes as their mass approaches the Planck scale, and consider both complete evaporation, and the possibility that evaporation stops around or below that scale, leaving Planck-scale relics. We  consider a variety of scenarios for the genesis of the matter-antimatter asymmetry, including GUT-scale baryogenesis, leptogenesis, and asymmetric co-genesis of dark and ordinary matter. Finally, we discuss possible tell-tale signals of this general framework.

The structure of this manuscript is as follows: in the following section <ref> we outline our framework; sec. <ref> and <ref>, respectively, discuss the generation of dark matter and of a baryon asymmetry from primordial black hole evaporation, after a general discussion of particle production from hole evaporation; sec. <ref> presents our results and a discussion thereof, and the final sec. <ref> concludes.



§ OUTLINE OF THE FRAMEWORK




In this study, we explore the possibility that primordial black hole evaporation produced both the dark matter and the baryon asymmetry. We entertain the possibility that the dark matter be a species χ, with mass m_χ belonging to a dark sector with a number of degrees of freedom g_χ that ranges, for definiteness, from 1 to 100. We do not make any assumption on the details of the dark sector spectrum. In principle, these details affect the temperature dependence of the number of degrees of freedom in the early universe; however, as we show below, the quantities relevant for us all depend quite weakly on that. We therefore neglect details of the dark sector spectrum and its impact on the number of degrees of freedom as a function of temperature, and assume the dark sector is all degenerate at the same mass scale m_χ.

We assume that the χ never attains thermal equilibrium after being produced (unless we specify otherwise, as is the case when we discuss the case of asymmetric dark matter), and that no processes exist that can freeze in any significant abundance of χ after black hole evaporation <cit.>. We also entertain both the possibility that black hole evaporation stops at a black hole mass f M_ Pl, with M_ Pl the reduced Planck mass, leading to a multi-component dark matter scenario consisting of the species χ produced by evaporation and of the Planck-scale relics, and the second possibility that evaporation leads to the complete disappearance of the primordial holes (f=0) and thus to a single-dark matter scenario.

As far as the production of the baryon asymmetry, we entertain three distinct possibilities:

    
  * That the baryon asymmetry is produced by the B-, L-, and CP-violating decays of a GUT boson X of mass 10^15≲ m_X/ GeV≲ 10^17;
    
  * That the baryon asymmetry stems from leptogenesis induced by the CP violating decays of heavy right-handed neutrinos with masses at the scale M_ν;
    
  * That both the baryon asymmetry and the dark matter are produced by the decay of heavy right-handed neutrinos (or some other massive species) into both dark matter/dark sector fields and standard model/visible sector fields, with the dark matter produced by evaporation annihilating away.


Unlike in previous studies, here we let the initial abundance β of primordial black holes at the epoch of formation, at a time t_i normalized to the radiation density,

    β≡ρ_ PBH(t_i)/ρ_ rad(t_i),

to be a free parameter. 

As mentioned above, we do consider the possibility that relics of mass f M_ Pl are left over from PBH evaporation. In this case the approximate constraint is <cit.>

    β(M_ PBH)≲ 2.8× 10^-28 f^-1(M_ PBH/M_ Pl)^3/2.


We assume for simplicity that the mass function of primordial black holes (PBH) is monochromatic and centered at a mass M_ PBH (see e.g. <cit.> for a motivation to this assumption, and <cit.> for a recent study of the optimal mass function for dark matter in the form of PBH).  The range of viable black hole masses is constrained from below by the requirement that the black holes form when the Hubble rate is at or below the Hubble rate during inflation H_*. The latter is constrained by Planck observations <cit.> to be

    H_*/M_ Pl<2.7× 10^-5   ( 95% C.L.).

If, as we assume here, primordial black holes form during the radiation domination epoch, their initial mass is

    M_ PBH=γ4π/3ρ H^-3,

where γ∼0.2 <cit.>, and the Planck limit translates to

    M_ PBH/M_ Pl>4πγ/2.7× 10^-5≃ 9.1× 10^4   ( 95% C.L.).


Notice that the density of PBHs depends on the spectrum of primordial density perturbations δ≡δρ/ρ, which at a given epoch has a certain probability distribution P(δ), for instance of Gaussian form; the fraction of energy density of the universe that collapses to PBHs is then

    β(M_ PBH)=∫_δ_c^∞ P(δ)dδ,

with δ_c∼γ the critical overdensity; in the case of Gaussian density perturbations with mass variance σ at horizon crossing, β∼exp(-δ_c^2/(2σ^2); since generally δ_c≫σ, only a small fraction of the energy density of the universe thus collapses to form PBHs, explaining why β≪ 1.


The upper limit to the mass of the PBH, in our framework, derives from the requirement that evaporation happens well before Big Bang Nucleosynthesis. In principle this is not a hard requirement, but, should it not be satisfied, the resulting limits on β(M_ PBH) would rule out any significant production of either dark matter or the baryon asymmetry from PBH evaporation, defeating the point of our study.




We now quickly summarize PBH evaporation: upon integrating the mass loss rate of PBH from Hawking-Gibbons evaporation from a black body at temperature T_H≡ M_ Pl^2/M_ PBH, and neglecting grey-body factors <cit.>, the lifetime of a PBH of mass M_ PBH is

    τ=160/π gM_ PBH^3/M_ Pl^4,

where g are the number of degrees of freedom of the radiated particles (including the 7/8 factor for fermions), which, here, are all particles with mass smaller than T_H. The radiation bath temperature corresponding to the time τ is calculated in the standard way for a radiation-dominated cosmology, assuming instantaneous thermalization of the evaporation products: using Friedmann's equation <cit.>

    π^2/30g_* T_ evap^4=3M_ Pl^2 H^2_ evap≃ 3M_ Pl^2τ^-2

with g_* the number of effective relativistic degrees of freedom, we get

    T_ evap/M_ Pl≃ 0.77(g_*/100)^-1/4(g/100)^1/2(M_ Pl/M_ PBH)^3/2,

which is of course only valid if evaporation ends during radiation domination, which is always the case for us.

We list in Table <ref> masses, Hawking-Gibbons temperatures, lifetimes, and temperature corresponding to the evaporation time for a few illustrative black hole masses. The table shows that in order to avoid impacting the synthesis of light elements (at times of t∼𝒪(1  sec), the PBH under consideration here are required to be lighter than approximately 1,000 t, i.e. 10^9 grams.





§ PARTICLE PRODUCTION FROM PRIMORDIAL BLACK HOLE EVAPORATION

We intend to calculate the abundance of right-handed neutrinos or dark matter (generically, of any massive particle X) produced by the evaporation of PBH in an adiabatically expanding universe, if the relative density of PBH to radiation at PBH formation time t_i is β(t_i)=ρ_ PBH/ρ_ rad. Indicating with N_X the number of particles X produced in the evaporation of one single hole, the number-to-entropy density of particles X at the present epoch is

    n_X/s(t_ now)=N_Xn_ PBH/s(t_i)=N_X Y_i.

To calculate Y_i, we use the definition of β,

    β=M_ PBHn_ PBH(t_i)/ρ_ rad(t_i)=M_ PBHs(t_i)/ρ_ rad(t_i)Y_i=4/3M_ PBH/T_iY_i

where in the last equality we have assumed that at very large temperatures g≃ g_*, the latter indicating the entropic degrees of freedom. From Eq. (<ref>) we then get

    T_i/M_ Pl≃ 0.87(M_ Pl/M_ PBH)^1/2(g_*/100)^-1/4.

Now, substituting Eq. (<ref>) into Eq. (<ref>) and the expression for Y_i,

    Y_i=0.65 β (g_*/100)^-1/4(M_ Pl/M_ PBH)^3/2,

into Eq. (<ref>), we get

    n_X/s(t_ now) ≃ 0.65 β  N_X(g_*/100)^-1/4(M_ Pl/M_ PBH)^3/2.

The calculation of N_X follows Ref. <cit.>. Assume that M_X<T_H=M_ Pl^2/M_ PBH, and assume that evaporation does not stop at M_ Pl, i.e. f=0. In this case, 

    N_X≃g_X/g∫_M_ PBH^0-dM/3T=g_X/g∫_T_H^∞M_ Pl^2/3T^3dT=g_X/6g(M_ PBH/M_ Pl)^2.

In the first equality, we assumed that the number of radiated particle is given by the ratio of the radiation energy from PBH evaporation, -dM, divided by the mean energy of a black-body of temperature T, ⟨ E ⟩_T≃ 3T (we notice that this is an approximate result that assumes the particles produced at evaporation to be spin zero, as well as a trivial, constant absorption cross section; in more realistic setups, there is a complicated dependence of the factor in front of T on the spectrum and spin of the particles the PBH evaporates to, see e.g. the classical literature on this point, Ref. <cit.>). 

The ratio of the degrees of freedom g_X/g, where g=g_X+g_ SM, the latter g_ SM indicating the “standard model” degrees of freedom, corresponds to the approximate ratio of emitted X particles, neglecting the effects of charge and spin on the evaporation rate, and in the second equality we used the relation between Hawking-Gibbons temperature and black hole mass. In the case where evaporation stops when the black hole mass is equal to fM_ Pl, the equation above is modified as follows:

    N_X^f≃g_X/6g((M_ PBH/M_ Pl)^2-f^2),

and of course reduces to the result above if M_ PBH≫ f M_ Pl; given the constraint in Eq. (<ref>) above, unless f≳ 10^4, a range theoretically unmotivated for evaporation to stop, Eq. (<ref>) above is perfectly adequate, and we shall use it from now on.

Notice that in principle massive particles X can be produced by PBH evaporation even if the initial Hawking-Gibbons temperature is lower than m_X. In that case, X production proceeds from the moment when t_H=M_X, thus

    N_X≃g_X/g∫_M_X^∞M_ Pl^2/3T^3dT=g_X/6g(M_X/M_ Pl)^-2.

In summary, given a sector with g_X degrees of freedom and a stable relic X of mass M_X, the cosmological abundance Ω_X=ρ_X/ρ_c, with ρ_c the critical energy density of the universe, is

    Ω_X   =   M_X/ρ_cn_X/s(t_ now)s_ now=M_X/ρ_cN_XY_i s_ now
       ≃   0.11 β M_X s_ now/ρ_c(g_*/100)^-1/4(g_X/g)(M_ PBH/M_ Pl)^1/2,     M_X<M_ Pl^2/M_ PBH,
       ≃   0.11 β M_X s_ now/ρ_c(g_*/100)^-1/4(g_X/g)(M_ Pl^7/M_ PBH^3 M_X^4)^1/2,     M_X>M_ Pl^2/M_ PBH.

Since <cit.>

    ρ_c=1.0537× 10^-5 h^2  GeV/ cm^3≃4.78× 10^-6(h/67.37)^2  GeV/ cm^3

and <cit.>

    s_ now=2,891.2(T_ CMB/2.7255)^2  cm^-3,

we can recast the equations above, for g_*=106.75+1 (which assumes g_ DM=1 for definiteness), as

    Ω_X   ≃   6.5× 10^7 β (M_X/ GeV) (g_X/g)(M_ PBH/M_ Pl)^1/2,     M_X<M_ Pl^2/M_ PBH,
       ≃   6.5× 10^7 β (M_X/ GeV)(g_X/g)(M_ Pl^7/M_ PBH^3 M_X^4)^1/2,     M_X>M_ Pl^2/M_ PBH.


Since the radiation energy density redshifts like a^-4, with a the scale factor, while pressure-less matter redshifts as a^-3, for sufficiently large β, the universe could become matter-dominated by PBH prior to evaporation. This condition can be expressed as 

    β_f=ρ_ PBH(T_ evap)/ρ_ rad(T_ evap)>1.

Assuming entropy conservation, we can relate the initial value of β to β_f in the equation above,

    β_f=βT_i/T_ evap≃ 1.1(g/100)^-1/2(M_ PBH/M_ Pl),

where we used Eq. (<ref>) and (<ref>) to express T_i and T_ evap as functions of M_ PBH. If PBH get to dominate the energy density of the universe prior to evaporation, the number density of particles produced by evaporation ceases to depend on β, as pointed out by Ref. <cit.>, as a result of the balance between the dilution of the number density of particles produced by evaporation and of the additional particles resulting from β_f>1. In all our results, we highlight the parameter space on the (M_ PBH,β) plane where β_f=1 (for g_χ=1 plus Standard Model degrees of freedom) with a thick purple line; for β values above that line, our results are β-independent.  



 §.§ Dark Matter from PBH evaporation


We consider three mechanisms for dark matter production from PBH evaporation: direct production from evaporation, Planck-scale relics from evaporation, and asymmetric dark matter production. We postpone the discussion of the latter to the next section, and Eq.(<ref>) directly gives the abundance of dark matter from PBH evaporation. 

If evaporation stops at a mass scale fM_ Pl, the cosmological abundance of Planck-scale relics, Ω_P=(f M_ Pln_ PBH(t_ now))/ρ_c is given by

    Ω_P   =   f M_ Pl/ρ_cY_i s_ now=0.65 β f M_ Pls_ now/ρ_c(g_*/100)^-1/4(M_ Pl/M_ PBH)^3/2,
       ≃   9.4× 10^26 β f (M_ Pl/M_ PBH)^3/2,

again with g_*=106.75+1 in the second equation.

Requiring that the dark matter in the universe have a density Ω_CDM≃ 0.21 <cit.> forces a condition across the model parameters g_χ, m_χ, M_ PBH, f and β (where we indicated with χ the dark matter from PBH evaporation) such that

    Ω_CDM=Ω_χ(g_χ,m_χ,M_ PBH,β)+Ω_P(f,M_ PBH,β).

Also, wherever Ω_P, Ω_χ>Ω_CDM, the corresponding region of parameter space is excluded as too much dark matter is produced by either evaporation or Planck-scale leftover relics, or both (of course, this assumes no episode of entropy injection that would dilute the relics' density, see e.g. <cit.>). Notice that regions with underabundant dark matter from either Planck relics or evaporation are not ruled out, since some other dark matter component might provide the remaining part of the observed cosmological dark matter density.

An important constraint on dark matter χ produced from PBH evaporation comes from the requirement that the dark matter be cold enough as to avoid disruption of small-scale structures via free-streaming. We follow here the discussion in Ref. <cit.>: The initial average energy of particles from the evaporation of a hole of mass M_ PBH is 6T_H=6M_ Pl^2/M_ PBH (see the derivation below in Eq. (<ref>)).  Because we assume the dark matter particles are never in kinetic or chemical equilibrium, the particle momentum today is simply the redshifted value of the momentum at production,

    p_ now=a_ evap/a_ nowp_ evap,

where a_ now=1. The energy of the dark matter particle at evaporation can be calculated as follows: the  average energy of particles radiated by a PBH with a Hawking-Gibbons radiation temperature T_H is 3T_H (with the caveats explained above – this is a simplifying approximation!). The total number of particles emitted by the PBH is approximately

    N=∫_0^N dn=∫_T_H^∞M_ Pl^2/3T^3dT=1/6(M_ Pl/T_H)^2.

The mean energy of the radiated particles is thus 

    E̅≃∫_0^N(3T)dn/N=6(T_H/M_ Pl)^2∫_T_H^∞M_ Pl^2/T^2dT=6(T_H/M_ Pl)^2(M^2_ Pl/T_H)=6T_H.

Notice that this average energy is different from the average energy at the beginning of evaporation, since it averages over all temperatures from that initial temperature to infinity. Now, since m_χ<T_H in order for the dark matter to be produced, E̅≃p̅=p_ evap. The last ingredient to calculate the dark matter velocity today is a_ evap. Fixing a_ now=1, the scale factor at matter-radiation equality a_ eq=Ω_r/Ω_m. Then, using Friedman's equation, and the fact that  ρ∼ a^4 in radiation domination, we have

    a_ evap=a_ eq(ρ_ eq/ρ_ evap)^1/4=a_ eq(ρ_c/a_ eq^3/3M_ Pl^2/t_ evap^2)^1/4≃ 7× 10^-32(M_ PBH/M_ Pl)^3/2

where we used Eq. (<ref>) in the next-to-last equality. The present velocity of dark matter produced from the evaporation of a PBH of mass M_ PBH is thus

    v_χ=p_ now/m_χ≃ 4× 10^-31(m_χ/M_ Pl)^-1(M_ PBH/M_ Pl)^1/2.

Assuming that only redshift contributes to setting the current dark matter velocity, using the constraint of Ref. <cit.> on the velocity of thermal relics today, 

    v_χ≲ 4.9× 10^-7,

we have

    m_χ/1  GeV≳ 2× 10^-6(M_ PBH/M_ Pl)^1/2.


The constraint above, together with the minimal primordial black hole mass allowed by CMB results, Eq. (<ref>),  sets the minimal possible dark matter mass, if produced from evaporation,

    m_χ≳ 0.6  MeV.

Notice that the constraint in Eq. (<ref>) applies only if a substantial fraction of the dark matter is produced by evaporation from PBH's of mass M_ PBH. Assuming that such fraction is, say, 10% of the dark matter in the universe, given that the maximal density of dark matter from evaporation corresponds to the dark matter dominating the number of degrees of freedom PBH evaporate to, our constraint applies to (for instance for m_χ<M_ Pl^2/M_ PBH)

    β≳ 0.1 ρ_c/m_χ s_ now(g_*/100)^1/4(g_χ/g)(M_ PBH/M_ Pl)^-1/2.

Notice that this is likely a fairly conservative constraint, as the limit in Eq. (<ref>) assumes 100% of the dark matter has the quoted velocity.










 §.§ Baryon Asymmetry from PBH evaporation

Here, we consider three classes of models for baryogenesis via PBH evaporation: baryogenesis via non-thermal leptogenesis,  baryogenesis via the decay of grand unification gauge bosons (GUT baryogenesis) and, finally, we entertain the possibility that the dark matter is produced in conjunction with an asymmetry in the baryon sector.

In all cases, we determine whether PBH evaporation can lead to  the observed baryon asymmetry, with a baryon-number-to-entropy density of <cit.>

    n_B/s≈ 8.8× 10^-11.



  §.§.§ Baryogenesis via leptogenesis

In the case of baryogenesis via leptogenesis,

    n_B/s=N_ν ε κ Y_i≃ 0.65 N_ν ε κ β(g_*/100)^-1/4(M_ Pl/M_ PBH)^3/2,

with 

    N_ν≃g_ν/g∫_M_ PBH^0-dM/3T=g_ν/g∫_T_0^∞M_ Pl^2/3T^3dT=g_ν/6g(M_ PBH/M_ Pl)^2,   M_ν<T_H=M_ Pl^2/M_ PBH


    N_ν≃g_ν/g∫_M_ν^∞M_ Pl^2/3T^3dT=g_ν/6g(M_ν/M_ Pl)^-2,   M_ν>T_H,

and where M_ν is the right-handed neutrino mass scale (for simplicity we assume all right-handed neutrinos to be close-to-degenerate in mass), with ε the CP asymmetry factor of the right-handed neutrino decays, and with κ≈ 0.35 the conversion ratio of leptons to baryons <cit.>. An important constraint for the baryogenesis-via-leptogenesis scenario is that the inverse-decay of right handed neutrinos be out of equilibrium. This is guaranteed if the temperature of the universe at PBH evaporation is smaller than M_ν, i.e. if

    M_ν>T_ evap≃  1.9× 10^18  GeV(M_ Pl/M_ PBH)^3/2,

  where we assumed g_*≃ g≃ 100.  
  In the baryogenesis-via-leptogenesis scenario we also require that the evaporation temperature be larger than the electroweak scale, under which sphaleron rates are highly suppressed, thus enforcing
  
    T_ evap≳100  GeV⇒ M_ PBH≲ 7.1× 10^10 M_ Pl≃ 1.4× 10^6  grams.


In a model-independent way, the parameter ε is a priori unconstrained and of 𝒪 (1). However, in specific model realizations, ε can be bounded from above. For instance, for type I seesaw models, barring tuned right-handed neutrino Yukawa textures <cit.>, one has <cit.>
  
    ε<3M_ν m_ max/16π v^2≃ 240(M_ν/M_ Pl)(m_ max/0.05  eV),

  with v the electroweak vacuum expectation value, and m_ max the mass of the heaviest left-handed neutrino. In light of that and of cosmological constraints on m_ max <cit.>, and in this specific model context, only for M_ν≃ 10^14 GeV could ε∼𝒪(1), but not for smaller right-handed neutrino masses. However, larger phases are generically possible, see e.g. fig. 4 of Ref. <cit.>. In what follows we consider a model independent scenario, and in order to show the maximal possible range of viable parameters, we set here  ε=0.5.
  
  Requiring a baryon asymmetry yield matching observations, and assuming M_ν>T_H and g_ν=6, we get the following relation between the right-handed neutrino mass scale, the PBH mass and β 
  
    β≃2.3× 10^-9/ε κ(M_ Pl^7/M_ν^4M_ PBH^3)^-1/2.

  

  §.§.§ GUT baryogenesis

In the scenario where baryogenesis originates from the CP and B-number violating decays of a GUT boson X, carrying g_X degrees of freedom, the produced baryon asymmetry depends on the CP violating parameter <cit.>

    γ≡∑_i B_i Γ(X→ f_i)-Γ(X̅→f̅_i)/Γ_X,

where B_i is the baryon number of the particular final state f_i, and Γ_X the X decay width. The expression for the resulting baryon asymmetry is then simply

    n_B/s=N_X γ Y_i≃ 0.65 N_X γβ (g_*/100)^-1/4(M_ Pl/M_ PBH)^3/2,

with, just as above,

    N_X≃g_X/6g(M_ PBH/M_ Pl)^2,   M_X<T_H=M_ Pl^2/M_ PBH
               ≃g_X/6g(M_X/M_ Pl)^-2,   M_X>T_H,

We consider a fairly generous range for the mass scale M_X of the GUT gauge bosons X whose decay is responsible for the generation of the baryon asymmetry, 

    10^15≲ M_X/ GeV≲ 10^17;
 a variety of mechanisms can shift the precise energy scale of gauge coupling unficiation, and even when that scale is fixed, M_X is not exactly determined (see e.g. Ref. <cit.> and references therein). In the plots, we use g_X=25 and γ=0.1.

Notice that GUT baryogenesis requires a source of B-L violation to prevent sphaleron washout of the produced baryon asymmetry for models where evaporation happens before the electroweak phase transition, here for masses M_ PBH≲ 10^6 g. We postulate in this case the mechanism outlined in Ref. <cit.>, which posits the existence of heavy right-handed neutrinos interacting with the Standard Model Higgs doublet via an effective dimension five operator; as long as the induced lepton-number violating reaction is fast compared to the sphaleron rate (which is generically the case at high enough temperatures) then the Δ L component of the generated lepton-baryon asymmetry is erased, leaving a net Δ B which is unaffected by sphaleron washout.



  §.§.§ Asymmetric Dark Matter

Finally, we consider a simple incarnation of asymmetric dark matter, inspired by the scenario detailed in Ref. <cit.>. Schematically, the Standard Model is augmented with a dark-sector scalar field ϕ and a Dirac fermion χ coupled to right-handed neutrinos N_i, with  Lagrangian density

    -ℒ=-ℒ_ SM+1/2M_iN_i^2+Y_iαN_iL_α H+λ_iN_iχϕ+ h.c.

plus mass terms for the ϕ and χ. χ has lepton number +1, and χ and ϕ are charged under a discrete Z_2 symmetry that ensures the stability of the lightest dark sector state; we assume m_χ<m_ϕ, so χ is the stable species[Note that lepton number conservation forces χ to be a Dirac fermion, and to get mass from another fermion χ̃ with opposite lepton number.]. We also need to assume fast, lepton-conserving interactions that thermalize leptons l, the Higgs, and the dark sector fields, annihilating away the symmetric components l+l̅ and χ+χ̅ (including, here, those non-thermally produced by PBH evaporation). Since the symmetric component of the dark matter must annihilate away by hypothesis in this scenario, the dark matter will generically reach kinetic equilibrium , thus reducing its velocity. As a result, the limit in Eq. (<ref>) does not apply here.



The N_i decays are CP-violating, and the resulting decay asymmetries are defined summing upon Standard Model generations α=1...3, ϵ_L=∑_αϵ_L_α, where

    ϵ_χ=∑_iΓ(N_i→χϕ)-Γ(N_i→χ̅ϕ^†)/Γ_N_i,  ϵ_L=∑_iΓ(N_i→ l h)-Γ(N_i→l̅h^†)/Γ_N_i.

The final asymmetry in each sector does not only depend on the decay asymmetries above, but also by  on the details of the models and on washout and transfer effects, which, following Ref. <cit.>, we parameterize with the quantities η_L and η_χ in the two sectors, respectively. Finally, the asymptotic asymmetries must satisfy <cit.>

    Y^∞_Δ L   =   ϵ_Lη_L N_ν Y_i=(n_B/s)37/12≃ 2.7× 10^-10
    Y^∞_Δχ   =   ϵ_χη_χ N_ν Y_i≃ 4.4× 10^-10(1  GeV/m_χ),

where Y_i is the same as what given in Eq. (<ref>), N_ν is as given in Eq. (<ref>) and, again as above, we assume the N_i to be out of equilibrium and produced from PBH evaporation (thus with a mass satisfying the constraints of Eq. (<ref>)).

In the case, for instance, where M_ν<T_H, and thus N_ν is independent of M_ν, we find that 

    N_ν Y_i≃ 0.04β(M_ PBH/M_ Pl)^1/2

and thus, given a value for ϵ_Lη_L, there is one value of β that satisfies Eq. (<ref>), namely

    β≃6.8× 10^-9/ϵ_Lη_L(M_ PBH/M_ Pl)^-1/2.

In turn, given N_ν Y_i as in Eq. (<ref>), there is a one-to-one correspondence between ϵ_χη_χ and m_χ via Eq. (<ref>). Specifically,

    ϵ_Lη_L/ϵ_χη_χ≡ r_Lχ≃ 0.61 (m_χ/1  GeV).




§ RESULTS

We discuss in this section all of our numerical results for the framework described above. Sec. <ref> assumes complete PBH evaporation and no Planck-scale relics (thus, f=0, where f indicates the mass of PBH relics from evaporation in units of the reduced Planck mass); we show results for both the baryogenesis via leptogenesis (see sec. <ref>) and for the GUT baryogenesis (sec. <ref>) scenarios, for a variety of dark matter masses; the following sec. <ref> assumes f≠ 0, and thus the existence of Planck-scale relics contributing to the global cosmological dark matter density, again for both baryogenesis scenarios, and again for a variety of dark matter masses; finally, in sec. <ref> we show results for asymmetric dark matter, for two different values of the right-handed neutrino mass scale. 



 §.§ Baryogenesis and Dark Matter from (complete) PBH evaporation

As outlined above, we assume exclusive non-thermal dark matter production from PBH evaporation, and we also assume that the dark matter never thermalizes. We intend to address two questions:


(1) What is the range of viable dark matter masses?


(2) Can dark matter and baryogenesis both be accounted for from PBH evaporation? If so, for which PBH masses?


We outlined above general constraints on the dark matter mass: the lower limits stems from Eq. (<ref>), while the upper limit corresponds to the maximal mass that can be produced from the evaporation of a PBH of mass M_ PBH; the upper limit lies in the regime where M_X>T_H=M_ Pl^2/M_ PBH (for M_X<T_H, M_X<M_ Pl/(9.1× 10^4), because of Eq. (<ref>)), and is given by the requirement that N_X>1; The maximal possible N_X corresponds to g_X, g→∞ and thus to M_X<M_ Pl/√(6)≃ 10^18  GeV.




We present our results in Fig. <ref>. All plots in our study utilize the same parameter space: the (M_ PBH,β) plane (as a reminder, β is the relative energy density of primordial black holes at the time of their genesis). In the plots, we shade in yellow the region at low PBH masses ruled out by the CMB limit of Eq. (<ref>) from the lowest possible Hubble rate during inflation; we shade in blue the region excluded by the current dark matter velocity limit, Eq. (<ref>); finally, we shade in grey the region where PBH evaporation ends below the electroweak scale, and thus baryogenesis via leptogenesis is not effective because of suppressed sphaleron rates, Eq. (<ref>).

We indicate with a thick purple line the region where β_f=1 for g_χ=1 plus Standard Model degrees of freedom, i.e. for β above that line, PBH eventually dominate the energy density of the universe prior to evaporation, and particle production becomes β-independent (hence the lines corresponding to successful baryogenesis and dark matter production become vertical).

In the plots, the colorful solid lines correspond to different numbers of dark-sector degrees of freedom: the upper blue line corresponds to g_χ=1, the orange line to 10 and the green to 100. The dot-dashed line shows the parameter space compatible with GUT baryogenesis for M_X=10^15 GeV. Finally, the dotted lines correspond to baryogenesis via leptogenesis with right-handed neutrino mass scales M_ν=10^14 GeV (upper line), M_ν=10^11 GeV (middle line) and M_ν=10^6 GeV (lower line). We truncate the dotted lines in this plot and in the following plots at PBH masses such that the corresponding non-thermally produced neutrinos would thermalize, thus violating Eq. (<ref>). Notice that for intermediate values of M_ν, the envelope giving the lowest possible β for successful leptogenesis is uninterrupted, and that values of M_ν<10^6 GeV are also possible. Once again, viable leptogenesis occurs in the region encompassed by the dotted lines. 

We start with a dark matter mass of 1 MeV in the upper left panel of the top-four plots in figure <ref>. This mass is only slightly above the limit in Eq. (<ref>) (incidentally, we note that slightly lower masses, between 0.6 and 1 MeV, are possible, but correspond to very narrow viable parameter space in M_ PBH). For such light masses, the constraint on the present dark matter velocity, from Eq. (<ref>), is quite stringent, and pushes against the constraint on M_ PBH in Eq. (<ref>). Notice that despite the fact that lighter black holes have a larger temperature, the earlier evaporation time means the produced dark matter has more time to cool by redshifting.

Light dark matter particles means, via Eq. (<ref>), that larger values of β are needed at a given M_ PBH. In turn, this makes it easier to combine the generation of dark matter from evaporation and of the observed baryon asymmetry. Fig. <ref> shows that for dark matter masses at around 1 MeV, both leptogenesis and GUT baryogenesis work, with the former suitable for a large number of degrees of freedom, and the latter for a low number of degrees of freedom. PBH dominance of the universe's energy density forces g_χ>1 in this case. The PBH mass needs to be right around 1 gram for a dark matter of 1 MeV. For sub-MeV dark matter masses we find that the only viable scenario is GUT baryogenesis, for PBH mass slightly below 1 gram, and again g_χ∼ 10.

The upper right plot shows m_ DM∼ 10 MeV. GUT baryogenesis is now no longer possible, while there is substantial overlap with leptogenesis across a fairly extended range of PBH masses from 0.5 to around 100 grams. The same applies to 100 MeV dark matter masses, although here the viable PBH mass range is extended to larger values, up to the limit from the current dark matter velocity, which, for a dark matter mass of 100 MeV, is around a few tens of kg. Finally, for DM masses at the GeV (lower right panel) or more, the parameter space keeps enlarging as the constraint on the dark matter velocity weakens (for 1 GeV up to ton-scale PBH); leptogenesis remains viable as long as the right-handed neutrino mass is sufficiently low.

In the lower four panels, we show a different regime, where the dark matter mass is very heavy (10^13 to 10^18 GeV), and is produced by PBH whose initial temperature is lower than the dark matter mass. In this regime, the dependence with M_ PBH is no longer Ω_χ∼ M_ PBH^1/2 but is instead Ω_χ∼ M_ PBH^-3/2 (see Eq.(<ref>)). The top two panels illustrate that the dark matter mass must be at least a few × 10^13 to be viable, with generally very low right-handed neutrino masses. Notice that the constraint on the PBH mass from evaporation ending prior to the EW phase transition, forces the lowest dark matter mass to be heavier than a few times 10^12 GeV. Also, notice that GUT baryogenesis is never an option for very heavy dark matter masses.




 §.§ Baryogenesis and Dark Matter from PBH evaporation and PBH relics

Here we discuss the possibility that evaporation stops at a mass f M_ Pl, leaving the dark matter produced by the PBH evaporation together with a second population of stable Planck-scale relics of mass M_ relic=f× M_ Pl; we explore this two-component dark matter scenario on the same parameter space as before, taking into consideration the over-closure constraint from the PBH relics (the corresponding excluded region of parameter space is shaded in dark red, and is at the top left of the plots).

In the top four panels of fig. <ref> we show the case where f=10^-7. This is, admittedly, a very low mass scale for PBH evaporation relics, but given ignorance about how evaporation might stop due to quantum gravity effects, it cannot be a priori ruled out. For such light PBH relics, we find that successful baryogenesis (via leptogenesis) plus two-component dark matter is possible for masses between roughly 10 MeV and a few GeV in the regime where m_χ<T_H, and is possible again for very heavy dark matter m_χ>10^13 GeV (in the figure, we show m_χ=10^14 GeV), but in this case the contribution of Planck relics is very sub-dominant. The right-handed neutrino mass needs to be between 10^7 and 10^12 GeV, and the PBH mass between 1 g and around a ton for this scenario to be successful in the low-dark matter mass regime; the heavy dark matter, as before, demand low right-handed neutrino masses, around 10^6 GeV or so. For dark matter larger than or around 10 GeV but lighter than around 10^13 GeV, dark matter and the baryon asymmetry cannot be jointly produced; also, we find that GUT baryogenesis never works if Planck relics are around (the corresponding region of parameter space is ruled out by overclosure from the density of Planck relics, unless f→0).


The lower four panels show the case where evaporation stops at the Planck scale, i.e. f=1. In this case the two-component dark matter is viable for a broad range of masses; demanding successful baryogenesis via leptogenesis forces the dark matter mass to be at the GeV scale (top left panel) and right handed neutrinos to be around 10 TeV; lighter dark matter particles in the MeV range are ruled out for f∼1, as are heavier masses (see top left panel showing m_χ=100 GeV). The bottom, left panel, with m_χ=10^13 GeV, shows (at around M_ PBH∼ 1 g) the turnover of the regime when m_χ∼ T_H; we do not find, however, regions of successful baryogenesis and dark matter; slightly heavier dark matter masses again make it possible to have successful leptogenesis, for sufficiently low right-handed neutrino masses (see bottom right panel, with m_χ=10^16 GeV).
















 §.§ Asymmetric Baryogenesis and Dark Matter from PBH evaporation



In the asymmetric dark matter scenario, in addition to the plots' parameter space, i.e. the (M_ PBH,β) plane, the framework we consider has four additional parameters: the CP-asymmetry-washout-factor products ϵ_χη_χ and ϵ_Lη_L, the dark matter mass m_χ, and the right-handed neutrino mass M_ν. We consider two representative right-handed neutrino mass scales, M_ν=10^11 GeV  in fig. <ref> and M_ν=10^13 GeV in fig. <ref>. In each of the top-four panels of each figure we fix the dark matter mass m_χ and show, on the (M_ PBH,β) plane the necessary values for ϵ_χη_χ to reproduce the universe's observed dark matter abundance, superimposed with regions where the baryon asymmetry can be produced for a given range of ϵ_Lη_L. Specifically, for definiteness we shade in light green the region corresponding to 

    10^-8<ϵ_Lη_L<10^-2.

(Notice that a broader range is theoretically possible). For a given dark matter mass, we find that there is ample parameter space to produce the observed dark matter density via PBH evaporation and subsequent asymmetric right-handed neutrino decay. Of course, from Eq. (<ref>) it follows that the lower the dark matter mass, the larger the needed ϵ_χη_χ.

In fig. <ref>, the right-handed neutrino mass scale is low enough that for the relevant PBH mass range both T_H>M_ν and  T_H<M_ν are possible (the latter at masses larger than around 100 grams, the former for lighter masses), hence the shape of the green-shaded regions. The figure illustrates that a broad range of dark matter masses are possible, depending on model details fixing the ϵ_χη_χ and ϵ_Lη_L products.

In the bottom four panels, we fix the product ϵ_χη_χ to several different values, namely 10^-6, 10^-5, 10^-4, 10^-3, and show lines corresponding to values of the dark matter mass that, in turn, would produce the observed dark matter density. Again, a wide range of values for the dark matter mass is possible, depending on the value of the parameter ϵ_χη_χ. The lower ϵ_χη_χ, the heavier the possible range of masses where the asymmetric dark matter and baryon asymmetry generation is possible.

We note that the limits on the current dark matter velocity are here different than before. First, the dark matter originates from right-handed neutrino decays rather than directly from evaporation. The right-handed neutrino lifetime is always much shorter than the PBH evaporation time scale; hence, effectively, the right-handed neutrino has no time to redshift, and the dark matter is produced by immediate subsequent decay. Assuming for simplicity isotropic decays in the rest frame of the neutrino, as we show in the Appendix the average dark matter velocity in this case is a factor 2 smaller than in the case of direct production from evaporation. As a result, the constraints on the dark matter mass are generically a factor 2 weaker (see Appendix <ref>).

However, since in the asymmetric dark matter production scenario we posit that processes exist that deplete the symmetric χ̅χ component produced by evaporation, the dark matter velocity might, and will, be affected by such processes. For instance, should the depletion of the symmetric component proceed via annihilation with the visible sector, i.e. χ̅χ→ SM, then the dark matter would be presumably brought in kinetic equilibrium and thus cool to the visible sector temperature, weakening the limit discussed above; if, on the other hand, the depletion occurs via 2n→2 “cannibal” processes, with n>1, such as χ̅χχ̅χ→χ̅χ,  then effectively the dark matter would heat itself up, strengthening the limits discussed above. 

For reference, in the figures we leave a vertical thick blue line in the top-left panel, corresponding to the heating from sterile neutrino decay, with the understanding that such limits are model-dependent and could be stronger or weaker than what the lines indicates. In practice, however, these constraints are largely outside the region of parameter space of interest.

Notice that a similar discussion to what we treat in the Appendix would be in order if the dark sector particles contained particles with masses largely different from the dark matter mass they eventually decay into. As mentioned above, here we make the simplifying assumption that the dark sector spectrum is trivially degenerate at a mass scale close to m_χ.




§ DISCUSSION AND CONCLUSIONS


We studied the joint production of the observed matter-antimatter asymmetry and of the cosmological dark matter from the evaporation of light primordial black holes (PBH) in the very early universe, at times t_ evap≪ 1 sec. The parameters of the model we considered include a universal mass for the primordial black holes, and their relative abundance at generation. We assumed that the dark matter belongs to a “dark sector” with a certain number of dark degrees of freedom. We also considered a “mixed dark matter scenario”, where the dark matter is both produced by PBH evaporation and consists of PBH relics from the end of evaporation at around, or below, the Planck scale. Finally, we considered three scenarios for the generation of the matter-antimatter asymmetry: (i) CP- and B-violating decays of GUT gauge bosons, (ii) baryogenesis through (non-thermal) leptogenesis via out-of-equilibrium CP- and L- violating decays of heavy right-handed neutrinos, and (iii) asymmetric dark matter and baryogenesis, again via decays of heavy right handed neutrinos.

The parameter space under consideration is constrained by a variety of considerations, from limits on the dark matter velocity inherited from the large Hawking-Gibbons temperature scales at which the dark matter was produced, to limits on the PBH mass from the Hubble rate during inflation, to an excessive density of relic PBH from the end of evaporation at the Planck scale.

Unlike in previous studies that focused on scenarios where PBH dominate the energy density of the universe at production (see e.g. Ref. <cit.>), here PBH can be a subdominant component to the early universe's energy density, with generally different conclusions (although our results correspond to those of Ref. <cit.> for values of β such that prior to evaporation PBH dominate the energy density of the universe).

If evaporation does not stop, and PBH vanish completely, both GUT baryogenesis and leptogenesis can be successful in conjunction with dark matter production from evaporation. GUT baryogenesis only works if the dark matter is between 1 and 10 MeV, while leptogenesis works either for dark matter masses between 1 MeV and a few GeV, or for super-heavy masses from 10^13 to 10^18 GeV, the maximal possible dark matter in this scenario. The needed PBH masses range from a few grams (for light dark matter particle masses), to around a ton for super-heavy dark matter.

If PBH evaporation does stop at some scale f M_ Pl, GUT baryogenesis is ruled out entirely, and leptogenesis works only for masses up to a few GeV or, again, for very heavy dark matter masses. In the former case, right handed neutrino masses must be large (10^11 GeV or so), in the latter, they must be much lighter (10^6 GeV or less).

Asymmetric dark matter and baryogenesis is successful, in this framework, for a broad range of effective CP times washout factors ϵη for the visible and dark sector. The larger the dark sector value of the product ϵ_χη_χ, the lighter the viable range of dark matter masses. The dark matter, in this scenario, must have a mass between a fraction of a GeV and 10 TeV or so.

If the scenario discussed here is indeed the backdrop for the generation of visible and dark matter, the detection outlook is relatively daunting. Searches for relic Planck-scale objects are possible, and in some cases might set some limits on this scenario, especially if the relic PBH are a substantial fraction of the dark matter, and/or if the relic are charged <cit.>. Directly or indirectly detecting the dark matter produced in PBH evaporation in the present scenario is problematic: since we assume no thermal equilibrium at any temperature, the indirect detection rates generically are highly suppressed, and so are the direct detection rates. 

One possible route to test this scenario (and in fact any scenario involving light PBH) is to look for gravitational wave emission from evaporation <cit.>: while all evaporation products quickly thermalize in our scenario, gravitons do not, leaving an imprint that is in principle detectable. There is a one-to-one correspondence between the frequency ν of gravity waves at the present time and the corresponding  frequency at emission ν_*, emission which assume here to happen at the PBH evaporation time:

    ν≃0.34 ν_* T_0/T_*(100/g_*(T_ evap))^1/3,

where the temperature of the PBH evaporation T_ evap is given in  Eq. (<ref>) and T_0 corresponds to the CMB frequency, around 160.4 GHz. As a result, we have

    ν≃7.1× 10^10  Hz(ν_*/M_ Pl)(100/g_*(T_ evap))^1/12(g/100)^-1/2(M_ PBH/M_ Pl)^3/2.

The maximal value for ν_* is around a few times the Hawking-Gibbons temperature, T_H=M_ Pl^2/M_ PBH. Using Eq. (<ref>), we get that the maximal frequency of gravity waves today is around 10^16 Hz. Generally, the spectrum peaks at ν_*∼ 2.8T_H <cit.>, therefore producing a signal at frequencies much higher than current gravity wave detectors. As we explain below, detection is however possible through the inverse Gertsenshtein effect <cit.>.

We estimate here the strain corresponding to the predicted gravity wave signal from PBH evaporation. Ref. <cit.> calculates that the energy density of gravity waves from PBH evaporation integrated over frequencies, and accounting for our assumption that PBH do not dominate the energy density of the universe, but rather constitute a fraction β of it at production, is approximately

    Ω_ GW(t_ evap)≃ 0.006(g_G/100)^2β,

with g_G=2 the number of graviton degrees of freedom. The equation above also assumes graviton production to happen instantaneously at the evaporation time (see also Ref. <cit.>). The red-shifted gravitational wave density today is

    Ω_ GWh^2(t_ now)≃ 1.67× 10^-5(100/g_*(T_ evap))^1/3Ω_ GW(t_ evap)≃ 10^-11(g_G/10^-2)^2β.

The corresponding strain is then

    h∼ 10^-36β^1/2.

The frequencies for the gravity wave emitted from PBH evaporation are well beyond currently operating or future interferometers; however they could be detectable via the so-called inverse Gertsenshtein effect <cit.>: The passage of gravity waves in a static magnetic field sources electromagnetic waves. As long as the induced signal “beats” thermal noise, a signal can be detected <cit.>.

In conclusion, early evaporation of light primordial black holes can lead to co-genesis of a baryon asymmetry and of the dark matter. We demonstrated that several possible baryogenesis scenarios are viable (GUT baryogenesis, leptogenesis, asymmetric dark matter), for a broad range of dark matter masses and of primordial black hole masses. The dark matter itself can originate entirely from evaporation, or from decay of particles produced by PBH evaporation, or it can be a mix of particles from evaporation, and Planck-scale relics of the evaporation process. Detection prospects for the dark matter are discouraging, but this scenario would leave an imprint of very high-frequency gravitational waves, of calculable spectrum and intensity, possibly detectable via the inverse Gertsenshtein effect.



§ ACKNOWLEDGEMENTS

We are grateful to Nicolas Fernandez, Benjamin V. Lehmann and John Tamanas for helpful comments; SP is grateful to his former undergraduate student Michael Shamma for pointing out a possible connection to asymmetric dark matter in this context. SP acknowledges gracious lessons in modern Greek neologisms from Vasiliki Pavlidou and John Tamanas. This work is partly supported by the U.S. Department of Energy grant number de-sc0010107.



§ APPENDIX: DM VELOCITY CONSTRAINT FOR INTERMEDIATE DECAY THROUGH A HEAVY NEUTRINO


Here we compute the average dark matter (DM) momentum from decay of a heavy neutrino N to a DM fermion χ and a scalar ϕ. We will assume a flat matrix element for this process for simplicity. In the rest frame (RF) of the heavy neutrino, the four-momentum is given by

    P^μ_N,RF = (M_N, 0, 0, 0)

Consider the heavy neutrino in a frame boosted along the z-axis. Let's call this frame the lab-frame (LF). In this boosted frame, the four-momentum for the heavy neutrino is:

    P^μ_N,LF = (E_N, 0, 0, p_N)

where 

    E_N   = γ M_N    p_N   = γ v_N M_N

with γ and v_N being the relativistic boost factor and the velocity of the heavy neutrino, respectively. In the rest frame of the heavy neutrino, the DM four-momentum can be considered to be in the xz-plane, given by:

    P^μ_χ, RF   = (E_χ, p_χsinθ, 0, p_χcosθ)

In the LF, the DM four-momentum becomes

    P^μ_χ, LF   = (γ E_χ + v_Nγ  p_χcosθ, p_χsinθ, 0, γ p_χcosθ + v_Nγ E_χ)

In the case where both the DM and the scalar are ultra-relativistic, we have that 

    E_χ = p_χ = M_N / 2

In this case, the four-momentum of the DM in the LF simplifies to:

    P^μ_χ, LF   = M_N/2(γ + v_Nγcosθ, sinθ, 0, γcosθ + v_Nγ)

The magnitude of the DM three-momentum is given by:

    |P⃗_χ,LF| = M_N/2√(sin^2θ + γ^2(cosθ + v_N)^2)
    	= γM_N/2√((1-β^2)sin^2θ + (cosθ + v_N)^2)

Expanding out the terms inside the square root, we find

    (1-v_N^2)sin^2θ + (cosθ + v_N)^2   = (1+v_Ncosθ)^2

where we used γ = (1-v_N^2)^-1/2. Therefore:

    |P⃗_χ,LF|
    	= γM_N/2(1+v_Ncosθ)

Averaging this expression over θ from 0 to π, we find 

    ⟨|P⃗_χ,LF|⟩ = 1π∫_0^π|P⃗_χ,LF|dθ =  γM_N/2

If the neutrino energy is given by 6T_H, then γ = 6T_H/M_N and 

    ⟨|P⃗_χ,LF|⟩ = 3T_H

Combining this result with Eq. (<ref>), Eq. (<ref>), and  Eq. (<ref>), we find that 

    m_χ1  GeV > 1×10^-6(M_PBHM_Pl)^1/2



JHEP



