Decoupling of Asymmetric Dark Matter During an Early Matter Dominated Era
PPrepared for submission to JCAP
Decoupling of Asymmetric Dark Matter
During an Early Matter Dominated Era
Prolay Chanda and James Unwin
Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA
Abstract.
In models of Asymmetric Dark Matter (ADM) the relic density is set by a particleasymmetry in an analogous manner to the baryons. Here we explore the scenario in whichADM decouples from the Standard Model thermal bath during an early period of matterdomination. We first examine the phenomenology of this alternative cosmological scenarioin the context of an elegant SO(10) implementation of ADM in which the matter dominatedera is due to a long lived heavy right-handed neutrino. Subsequently, we present a modelindependent analysis for a generic ADM candidate with s-wave annihilation cross sectionwith more general assumptions regarding the origin of the early matter dominated period.We contrast our results to those from conventional ADM models which assume radiationdomination during decoupling, and discuss the prospects for superheavy ADM. a r X i v : . [ h e p - ph ] F e b ontents Asymmetric Dark Matter (ADM) [1–3] draws a direct analogy with the mechanism throughwhich the present-day density of baryons is set. The dark matter carries a conserved quantumnumber (analogous to baryon number B ), and there is an asymmetry in the number of darkmatter particles χ and its antiparticle χ . The asymmetry is typically defined as follows η χ ≡ n χ − n χ s , (1.1)where n χ is the number density, and s is the Standard Model entropy density. Provided thedark matter annihilation rate is sufficiently large, then the relic density is not determined bythe point of decoupling (as in freeze-out dark matter [4, 5]), but rather is set by the size ofthe asymmetry, such that Y χ (cid:39) η χ and Y χ (cid:28) Y χ (for reviews of ADM see [6, 7]).For dark matter that decouples in a matter dominated era, rather than during radiationdomination, there are two main consequences. First, the Hubble rate is different, and thisalters the dynamics of decoupling [8]. Secondly, entropy production can occur when radiationdomination is restored, which should occur prior to Big Bang Nucleosynthesis (BBN), thiseither dilutes the dark matter abundance or leads to dark matter production. Indeed, onecan see the difference in decoupling between the two scenarios by inspection of Figure 1.This figure shows the evolution of the dark matter abundance Y χ ≡ n χ /s and the anti-darkmatter abundance Y ¯ χ as a function of x = m χ /T where T is the temperature. This is shownfor both the case of freeze-out during radiation domination ( Y χ, ¯ χ ) RD and freeze-out assumingmatter domination ( Y χ, ¯ χ ) MD . Notably, for the same parameter values the relic density isasymmetric with ( Y ¯ χ ) RD (cid:28) ( Y χ ) RD for radiation dominated freeze-out, while for matterdominated freeze-out the final dark matter abundance consists of symmetric contributionsfrom the dark matter and anti-dark matter: ( Y ¯ χ ) MD (cid:39) ( Y χ ) MD .This paper is structured as follows; we first explore an elegant implementation of ADMin which the dark matter arises from an SO(10) unified theory, and the matter dominatedera is due to a long-lived heavy right-handed neutrino. We numerically calculate the relicdensity of ADM in this scenario and show that the assumption of dark matter decouplingduring matter domination makes the scenario significantly more attractive. Following this, in– 1 – MD ± Y RD + Y RD -
20 30 40 50 60 7010 - - - - - - - x Y ± η χ = - σ = T MD = GeV m χ =
10 GeV
Figure 1 . Evolution of the abundances of dark matter Y + and anti-dark matter Y − as functions of x = m χ /T , assuming radiation domination (RD) and matter dominated (MD), for an annihilationcross-section σ = 2 pb and an initial dark matter asymmetry η χ = 10 − . We assume that matterdomination occurs at T MD = 10 GeV for the MD curves. Observe that for the same particular choicesof particle physics parameters (masses, couplings), radiation dominated decoupling implies an ADMscenario with the late time abundance set by η χ , while the matter dominated case leads to symmetricdark matter. Thus cosmology can play a role in determining the late time dark matter scenario. Section 3, we adopt a model-independent perspective and obtain a semi-analytic solution tothe Boltzmann equation for the case of ADM with an s-wave annihilation cross-section whichdecouples during an early period in which the universe is matter dominated. In particular, wecalculate the abundances of the dark matter and anti-dark matter at the point of decoupling.Then, in Section 4, we discuss the importance of the entropy injection on the relic densityof ADM and thus identify regions of parameter space for which this scenario successfullyreproduces the observed dark matter relic abundance. In Section 5, we give some concludingremarks and discuss the prospect of realizing Superheavy ADM in the case that decouplingoccurs during a matter dominated era.
As a particularly interesting example (prior to the more general study), we first explorecandidates for asymmetric dark matter (ADM) within the context of non-supersymmetricSO(10) Grand Unified Theories. Specifically, an elegant connection between ADM modelsand SO(10) GUTs was recently highlighted in [9], in which the lightest member of the scalarmultiplets and , being a complex singlet with hypercharge zero, is identified as thedark matter state χ . A stabilizing Z symmetry emerges for χ from the intermediate scalebreaking of a subgroup by a dimensional representation [20–24]. As discussed in [9],alternatives potential dark matter candidates from the SO(10) representations (other thanthe singlet state χ ), either disrupt gauge coupling unification or, in the case of non-zerohypercharge states, encounter strong direct detection limits. For alternative discussions of ADM candidates within non-SUSY SO(10) models see [10–19]. In what follows we explore how the experimental limits on the singlet state can be relaxed due to anentropy injection from RH neutrino decays and, plausibly, these candidates might be similarly salvaged. – 2 – .1 Higgs Portal to Singlet Dark Matter from SO(10)
While in [9], it was concluded that the singlet state was only a viable ADM state for freeze-outvia resonant annihilation to Standard Model states, here we show that if the lightest right-handed (RH) neutrino N decays after dark matter freeze-out, then this significantly widensthe viable parameter space. This presents an excellent example of matter dominated freeze-out, with the RH neutrino dominating the energy density for a period of the early universe.The dark matter χ is assumed to have couplings such that it is initially in thermalequilibrium with the Standard Model and then subsequently freezes out non-relativistically.In models of ADM, one assumes a particle-antiparticle asymmetry between χ and χ † , whichperturbs the standard freeze-out calculation, as we shall discuss analytically in subsequentsections. Since χ is a complex singlet with a Z symmetry, it interacts with the StandardModel through renormalizable interactions involving the Higgs, the so called Higgs portal,with the following Lagrangian [25–28] L = L SM + 12 ( ∂ µ χ ) † ∂ µ χ − λ (cid:48) | χ | − µ (cid:48) | χ | − κ | χ | | H | , (2.1)where L SM is the Standard Model Lagrangian. After electroweak symmetry breaking, theLagrangian eq. (2.1) can be expanded around the Higgs VEV to obtain L = L SM + 12 ( ∂ µ χ ) † ∂ µ χ − λ (cid:48) | χ | − m χ | χ | − κ | χ | h − κvh | χ | . (2.2)The state χ acquires mass contributions from both the µ (cid:48) term and the cross-coupling term,such that the dark matter mass is m χ = (cid:112) µ (cid:48) + κv , where v is the Higgs VEV. For m χ > κ < . m χ (cid:39) µ (cid:48) to good approximation. This Lagrangian fixes the interactions between the dark matterand Standard Model and, in particular, one can calculate the annihilation cross-section of χ to Standard Model states. The dark matter will primarily annihilate to Standard Modelfermions, gauge bosons, or Higgs bosons depending on which are kinematically accessible(i.e. depending on m χ ), and we state the main annihilation cross-sections in Appendix A.1.For dark matter decoupling during a matter dominated era, the Hubble rate at freeze-out has a different temperature dependence compared to during a radiation dominated era[8]. It is helpful to define the fractional asymmetry F = Y χ † /Y χ , being the abundance ofanti-dark matter compared to the dark matter. After dark matter decouples, at the freeze-out temperature T f , the fractional asymmetry is constant F f ≡ F ( T f ) (in the absence of anysubsequent production of the dark matter). Thus F = 1 implies conventional symmetric darkmatter, while F < F ∼ .
1, in which case the abundance of the anti-dark matter is depleted by order ofmagnitude only, or it could be highly asymmetric F (cid:28) F = F ( σ, m χ ) and a smaller fractional asymmetry requires a larger annihilation cross-section σ , potentially implying better detection prospects, but otherwise, F is a free parameter. Aswe derive later (or see [29]) the dark matter relic abundance can be expressed in terms of η χ and F f as follows Ω RelicDM = ζ (cid:18) s m χ ρ c (cid:19) (cid:18) η χ + 2 η χ F f − F f (cid:19) , (2.3)where ζ is a dilution factor due to the entropy injection from the decays of N states. For a discussion of the origins of this particle asymmetry in the context of the SO(10) models, see [9]. – 3 – .2 Right-Handed Neutrino Dominated Era
The RH neutrino states couple with the Higgs through the Yukawa coupling y , with thefollowing Lagrangian L ⊃ i ¯ N i /∂N i − (cid:2) y ik ¯ (cid:96) i HN k + M ik N i N k (cid:3) , (2.4)to generate masses for the left-handed neutrinos through the seesaw mechanism, where (cid:96) i = e, µ, τ represents the lepton doublets, H is the Higgs doublet, and N i stands for the RHneutrino states. The decay rate of the RH neutrinos is thusΓ N i = y i π M N i . (2.5)Since empirically, the lightest active neutrino can be arbitrarily light (or even massless) [30],the neutrino Yukawa matrix can exhibit substantially smaller couplings for one generation ofthe RH neutrinos N , compared to N and N . Since the mass of the lightest RH neutrino, M N , can be considerably lower, this can result in a long-lived heavy particle species. Thenthe population of N states can potentially evolve to dominate the energy density of theuniverse, leading to a matter dominated phase of the early universe until N decays [31]. InAppendix A.2 we confirm that for appropriate parameter choices that N can indeed lead toan early period of matter domination, for instance with κ ∼ − and M N ∼ GeV.Assuming that the N states decay simultaneously at H (cid:39) Γ N into Standard Modelradiation (the sudden decay approximation), the resulting reheat temperature T RH for theStandard Model thermal bath will be T RH = (cid:18) M π g (cid:63) (cid:19) / y M / N (cid:39)
20 GeV (cid:18) M π g (cid:63) (cid:19) / (cid:16) y − (cid:17) (cid:18) M N GeV (cid:19) / . (2.6)Moreover, the reheat temperature T RH is linked to the dilution factor ζ . Specifically,under the assumption that φ decays suddenly when the thermal bath has a temperature T Γ ,such that the reheating temperature of the Standard Model thermal bath following N decaysis T RH ∼ √ Γ M Pl , then the dilution factor ζ can be expressed as ζ = s before s after (cid:39) (cid:18) T Γ T RH (cid:19) . (2.7)In Section 4, we analytically derive the form of T Γ in terms of the decay rate Γ and the valueof the Hubble parameter at the point that the RH neutrinos become non-relativistic, i.e., M N , or more generally, for some arbitrary heavy decoupled state the Hubble parameter H (cid:63) at which evolution of this state becomes matter-like. Then the dilution of the relic abundancedue to an entropy injection from N decays to Standard Model states can be recast in termsof the reheat temperature T RH of the thermal bath following N decays. Taking the above together, in Figure 2 we present the parameter space for SO(10) singlet darkmatter χ which freezes-out during an early matter dominated era due to decoupled N stateswith mass M N ∼ GeV dominating the energy density of the universe. The contours– 4 – .0 1.5 2.0 2.5 3.0 3.5 4.0 - - - - - - - Log [ m χ / GeV ] Lo g κ T R H = G e V T R H = G e V T R H = G e V XENON1tLUX ν f l o o r Γ Inv, H Fermi F f = - - - - - - - - Log [ m χ / GeV ] Lo g κ T R H = G e V T R H = G e V T R H = G e V XENON1tLUX ν f l o o r Γ Inv, H Fermi F f = - Figure 2 . An entropy injection dilutes the dark matter after freeze-out introducing a new parameterfor calculating the dark matter relic density, the energy from the entropy injection sets the reheattemperature of the Standard Model radiation bath to T RH . We show contours of T RH for which F f = Y DM /Y DM = 10 − (left) and 10 − (right) evaluated at freeze-out, thus the dark matter isasymmetric and reproduces the observed relic density for couplings on or above a given contour. Themodel under consideration is a complex scalar dark matter annihilating through the Higgs portal, asmotivated by the SO(10) model discussed here. Experimental constraints are shown from XENON1T[32] (dashed red), LUX [33] (dashed orange), Fermi-LAT [34] (dashed purple) and the invisible Higgswidth [35, 36] (dashed brown). The grey shaded region indicates where the theoretical analysisbreaks down, corresponding to the assumption of a matter dominated universe. The neutrino flooris shown as the dashed green curve. We assume that the onset of matter domination occurs at T MD = M N / ∼ GeV. The white space in each panel is a viable parameter space in whichthe dark matter relic density is correct while evading experimental limits for an appropriate choice of T RH (it is, however, below the neutrino floor). show the couplings required to reduce the symmetric component of χχ † -pairs such that thefractional asymmetry at freeze-out is F f = 10 − (left panel) or 10 − (right panel) for differentvalues of the reheating temperature. The initial dark matter asymmetry is adjusted to matchthe observed relic abundance as m χ and T RH are varied, given that Ω ∝ ζm χ η χ (cid:12)(cid:12) initial . Thegrey region indicates regions in which our assumptions breakdown, namely in which darkmatter freeze-out occurs prior to N states dominating the energy density or near the pointof N decays around H ∼ Γ. To match observations the decays of N should produce a reheattemperature higher than the threshold of BBN at around 10 MeV [37]. For more discussionon these conditions, see [38] which presents similar plots and has additional details.We overlay Figure 2 with experimental constraints from XENON1T [32], LUX [33]Fermi-LAT [34] (dashed purple) and LHC determinations of the invisible Higgs width [35, 36](dashed brown), as well as the predicted neutrino floor. Observe that for the two scenariospresented which assume a fractional asymmetry of 10 − and 10 − there is viable parameterspace in which χ can be ADM and simultaneously satisfy the relic density requirement andexperimental bounds over a broad range of masses for reheat temperatures below 100 GeV.While it is unfortunate that the parameter space lies below the neutrino floor, these areperfectly acceptable dark matter scenarios which could be readily realised in nature if theUV completion of the Standard Model is a non-supersymmetric SO(10) GUT.– 5 – Boltzmann Analysis Assuming Matter Domination
Expanding on the specific example of Section 2, we now take a more model independentapproach in which the early matter dominated era is sourced by a general population ofmatter-like states and where we remain agnostic regarding the manner of dark matter freeze-out apart from the assumption that it is via s-wave annihilations. We start by outlining thedifferences between radiation dominated freeze-out and matter dominated freeze-out. Wethen quantify the impact on models of ADM by deriving the freeze-out abundance assumingthat decoupling occurs during a matter dominated era. Subsequently, in Section 4 we willdiscuss the entropy dilution of the freeze-out abundance due to the transition to radiationdomination, leading to the final relic density of dark matter.
We consider the presence of a decoupled scalar field φ along with the visible sector anddark matter χ . Just after inflationary reheating, χ and φ behave like radiation until acharacteristic temperature ( T (cid:63) (cid:39) m φ ) when φ starts evolving matter-like. A long-lived φ results in a matter dominated early universe. If T (cid:63) (cid:29) T f (cid:29) T Γ , the dark matter freezes outin a matter dominated universe. The Hubble parameter for T (cid:63) > T > m χ can be defined asfollows [8] H = H (cid:63) (cid:20) r g ∗ g ∗ + g χ (cid:16) a (cid:63) a (cid:17) + (1 − r ) (cid:16) a (cid:63) a (cid:17) + r g χ g ∗ + g χ (cid:16) a (cid:63) a (cid:17) (cid:21) , (3.1)where the subscript ‘ (cid:63) ’ refers to the value of a physical quantity evaluated at T = T (cid:63) , H (cid:63) = (cid:112) π / h / i g / (cid:63) ( T (cid:63) /M Pl ), and g ∗ is the effective number of relativistic degrees offreedom of the Standard Model and g χ is the internal degrees of freedom of the DM, the factor h i = 1 , / i stands forthe bosons and the fermions respectively. The quantity r accounts for the fraction of theenergy density in radiation at a = a ∗ . Note also that provided the thermal bath entropy isconservation, the scale factor a ( T ) can bee related to the bath temperature T as follows a ( T (cid:63) ) a ( T ) (cid:39) (cid:18) g ∗ ( T ) g ∗ ( T (cid:63) ) (cid:19) / TT (cid:63) , (3.2)From eq. (3.1) and using eq. (3.2), it can be seen that for r (cid:39)
1, the Hubble parameter H ∝ T / , and therefore the universe is radiation dominated. On the other hand, for r (cid:28) H ∝ T ,corresponding to a matter dominated era at a = a (cid:63) . If theuniverse is radiation dominated while the φ field becomes non-relativistic at temperature T (cid:63) ,the φ field contribution to the energy density of the universe grows with time potentiallyleading to a matter dominated universe. We denote the temperature of the thermal bath atwhich the Universe becomes matter dominated as T MD (specifically, the point at which the φ field accounts for half of the total energy density of the Universe), this can be related to T (cid:63) via the following relationship T MD = (1 − r ) r (cid:18) g ∗ ( T (cid:63) ) g ∗ ( T MD ) (cid:19) / T (cid:63) . (3.3)For both matter dominated and the radiation dominated cosmology, H ( t ) scales as(time) − . This motivates us to adopt the following form for HH = νt . (3.4)– 6 –ith ν = 1 / φ decays. The conversecase leads to significant differences in the Boltzmann analysis, see [39]. We discuss for whatcases this assumption holds in Appendix A.4 (see [38] for a more detailed discussion). We next derive an expression for the evolution of the fractional asymmetry F , thus extendinga result of [29] to the case of ADM which decouples during a period of matter domination.Writing Y χ ≡ Y + and Y χ † ≡ Y − , the Boltzmann equation for dark matter and anti-darkmatter can be expressed as follows [40] (see also [41–43]) dY ± dt = −(cid:104) σv (cid:105) s ( T )( Y + Y − − Y +eq Y − eq ) , (3.5)where (cid:104) σv (cid:105) is dark matter self annihilation cross-section and s is the entropy density s ( T ) = π g ∗ ,s ( T ) T in terms of g ∗ ,s ( T ) (cid:39) g ∗ ( T ) the effective number of relativistic degrees of free-dom. Finally, the equilibrium abundances can be as follows Y ± eq = bx / e − x e ± µ/T , (3.6)where b = g √ π / g ∗ ,s and g is the dark matter internal degrees of freedom (e.g. g = 2for complex scalar or g = 4 for Dirac fermion). Here µ is the χ chemical potential whichcharacterizes the difference between dark matter and anti-dark matter, such that η ∝ sinh( µ ).The dark matter annihilation cross-section can be expanded in a power series around x = 0, (cid:104) σv (cid:105) = (cid:80) ∞ n =0 σ n x − n , expanding to second order (i.e. s-wave σ and p-wave σ terms)one has (cid:104) σv (cid:105) = σ + σ x − . Then from eqns. (3.1) and (3.4) we can rewrite (3.11) as follows dY ± dx = − (cid:88) n λ n x − n − g / (cid:0) Y + Y − − Y +eq Y − eq (cid:1) , (3.7)where the quantity λ n contains the cross section and is defined as λ n = νh − / i (cid:114) π m χ M Pl σ n . (3.8)We obtain numerical solutions to eq. (3.7) for a given dark matter annihilation cross-section σ (neglecting the higher-order contributions) and fixing the dark matter mass m χ ,the critical temperature T (cid:63) and the initial dark matter asymmetry η χ . An example evolutionwas given in Figure 1. Observe that in the scenario of dark matter decoupling during radiationdomination, the freeze-out dark matter abundance is determined by the initial asymmetry η χ for a particular choice of this parameter. In contrast, in the matter dominated freeze-outcase, with otherwise the same parameter values, the dark matter abundance at decouplingis comparable to the anti-dark matter abundance thus implying a symmetric dark matterscenario.Furthermore, it can be insightful to recast the evolution of the cosmic abundances darkmatter in terms of the fractional asymmetry F = Y − /Y + . Specifically, from eq. (1.1),– 7 –he equilibrium abundance eq. (3.6) follows from the geometric mean of the equilibriumabundance of the dark matter and anti dark matter in terms of F eq ≡ Y − eq Y +eq as Y eq ≡ (cid:113) Y − eq Y +eq = (cid:115) F eq η (1 − F eq ) . (3.9)This implies that the equilibrium fractional asymmetry evolves according to F eq = exp (cid:18) − − (cid:18) η Y eq (cid:19)(cid:19) . (3.10)Moreover, the evolution of evolution of dark matter and anti-dark matter Y ± can be expressedas follows dY ± dx = −(cid:104) σv (cid:105) dtdx s ( T ) (cid:32) η χ F (1 − F ) − Y +eq Y − eq (cid:33) , (3.11)where the dark matter asymmetry is defined as Y + − Y − = η χ . Thus in the early universe,much before dark matter decoupling occurs, it is expected that F (cid:39) F eq . On the other hand,at later times after decoupling (i.e. x (cid:29) x f ), the RHS of the eq. (3.11) will be dominated bythe first term since Y eq is exponentially dependent on x . Thus from eq. (3.7) it follows thatone can express the evolution of the fractional asymmetry as below dFdx = − η χ g / (cid:88) n λ n x − n − (cid:32) F − F eq (cid:18) − F − F eq (cid:19) (cid:33) , (3.12)where g eff is the effective number of relativistic degrees of freedom given by g / (cid:39) g / ∗ ( T (cid:63) ) (cid:18) T (cid:63) (1 − r ) xm χ + r (cid:19) (cid:18) T (cid:63) (1 − r ) xm χ + r (cid:19) − / . (3.13)Since we are interested in times much before BBN, we take g ∗ , to be constant and set it to beStandard Model value at temperatures above the top mass g ∗ ,s ( x ) (cid:39) g ∗ ( x ) (cid:39) g ∗ ( x (cid:63) ) (cid:39) . g / (cid:39) g / (cid:63) . At the point of dark matter decoupling, the two terms on the RHS of eq. (3.11) cease tobalance, however to good approximation d F d x ≈ d F eq d x , and it follows that F (cid:48) eq ≈ (cid:88) n δ n λ n η χ g / x − n − F eq , (3.14)where the prime indicates a derivative with respect to the x and δ n is a constant fixed byfitting the numerical solution, normally taken to be ( n + 1) [44].Next, we derive the temperature of dark matter decoupling x f . We use eq. (3.9), alongwith eq. (3.14), to re-express the condition of eq. (3.10) in the following manner (cid:18) − x f (cid:19) − F eq , f F eq , f ≈ − η χ g / (cid:88) n δ n λ n x n +2 f . (3.15)– 8 –estricting to the relevant limit in which η χ g / (cid:80) n λ n x − n − f (cid:28)
1, we can obtain an iterativeapproximate solution for the point of decoupling x f = ln( δ n g / ,f bλ n ) + 12 ln ln ( δ n g / ,f bλ n ) (cid:16) ln n +4 ( δ n g / ,f bλ n ) − ( δ n λ n η χ ) g eff ,f (cid:17) . (3.16)Using eqns. (3.13) and (3.8) with r ≤ .
99 (which implies that there is a non-zero contributionfrom the matter-like species φ to the energy density at T = T (cid:63) ) we obtain the followingexpression x f = ln M + 12 ln (cid:18) ln M ln n +4 M − N (cid:19) (3.17)in terms of M = √ g √ π ( n + 1) h − / i g − / (cid:63) (1 − r ) − / M Pl m / χ T − / (cid:63) x − / f σ n N = (cid:18)(cid:114) π
180 ( n + 1) h − / i (1 − r ) − / η χ M P l m / χ T − / (cid:63) x − / f σ n (cid:19) g ∗ ( T (cid:63) ) , (3.18)where the dependence of M on x f comes from the fact that g eff for the matter dominatedcase is a function of x f (3.13). Also, recall h i is the number of internal degrees of freedom ofthe dark matter and r is the fraction of the energy density contained in radiation at T = T (cid:63) . In the scenario in which the dark matter freezes out during matter domination in the presenceof the decoupled heavy scalar field φ , then φ must subsequently decay in order to restoreradiation domination prior to BBN to reproduce cosmological observables. The decay of φ dilutes the freeze-out dark matter abundance by a factor ζ , which is defined as the ratioof the entropy before decays s before and after decays s after , where we assume that φ decaysonly to Standard Model particles and not to the dark matter particles. This is interestingbecause the entropy injection from the φ decays allows the dark matter to be overabundant atfreeze-out and subsequently diluted in order to obtain the observed dark matter relic density.For ζ (cid:28)
1, smaller dark matter annihilation cross-sections or larger dark matter masses areallowed, therefore weakening search constraints and also unitarity limits m χ (cid:46)
100 TeV [46].We first derive the evolution of the scale factor at critical temperature a (cid:63) to the point a Γ (at which the φ states decay simultaneously), using H (cid:39) Γ in eq. (3.1) we obtain (cid:18) a (cid:63) a Γ (cid:19) ≈ (cid:18) Γ H (cid:63) (1 − r ) / (cid:19) / . (4.1)Taking this with T RH ∼ √ Γ M Pl , we can express T Γ in terms of T RH and T (cid:63) T Γ (cid:39) (cid:18) π g (cid:63) (1 − r ) T T (cid:63) (cid:19) / . (4.2) In principle, φ decays could produce dark matter and anti-dark matter. In particular, this could occurvia loops of Standard Model particles, as discussed in [38, 45]. While one might be concerned that even if theproduction from decays of dark matter is negligible, anti-dark matter production might alter the fractionalasymmetry F , in Appendix A.3 we argue that this is not the case in the parameter regions of interest. – 9 –oreover, using eq. (2.7) we can obtain a formula which relates ζ to T RH and T (cid:63) ζ (cid:39) π T RH (1 − r ) T (cid:63) g (cid:63) , (4.3) In Section 3.3 we derived an expression for the point of dark matter decoupling, eq. (3.17), wenow use this result to derive the dark matter relic abundance. We start from an approximatedanalytic solution to eq. (3.7) (which we derive in Appendix A.5), specifically, for x (cid:29) x f theanti-dark matter abundance is given by Y − f (cid:39) η χ exp (cid:18) η χ (cid:80) n λ n Φ n ( ∞ , m χ ) (cid:19) − , (4.4)where Φ n ( ∞ , m χ ) is defined asΦ n ( x, m χ ) = (cid:90) xx f dx (cid:48) x (cid:48)− n − g / . (4.5)It then follows that the total dark matter density at decoupling can be expressed as followsΩ DM , f h = s m χ ρ c h (cid:16) Y + f + Y − f (cid:17) , (4.6)where ρ c = 3 H M / (8 π ) with M Pl = 1 . × GeV and s ≈ − .Provided dark matter production after the freeze-out is negligible, the dark matter relicabundance corresponds to the abundance at decoupling diluted by a factor ζ due to theentropy injection of φ decays. The ζ factors is given in eq. (1.1) and this lead to the followingexpression for the relic density of the dark matterΩ FO h = ζ s m χ ρ c h (cid:16) Y − f + η χ (cid:17) . (4.7)Therefore, using eq. (4.4) the dark matter relic density after decoupling during a matterdominated era and subsequent entropy dilution is given byΩ RelicDM h (cid:39) . × ζη χ m χ GeV − exp (cid:32)(cid:112) π h − / i g / (cid:63) η χ √ − r m χ M Pl (cid:80) n σ n x − ( n + 32 ) f ( n + ) x / (cid:63) (cid:33) − . (cid:16) m χ GeV (cid:17) ζ (cid:16) η χ − (cid:17) . (4.8)For η χ (cid:28) ( (cid:80) n λ n Φ n ( ∞ , m χ )) − the expression above can be simplified to obtainΩ RelicDM h (cid:39) ζ . × h / i g / (cid:63) M Pl (cid:88) n (1 − r ) − / σ n GeV( n + 3 / x n +3 / f x − / (cid:63) − + 2 . (cid:16) m χ GeV (cid:17) ζ (cid:16) η χ − (cid:17) . (4.9)Note that the first term on the RHS corresponds to the asymmetric contribution to the relicdensity and the second term represents the symmetric component.The dark matter relic density today is set by both the symmetric and the asymmetriccontributions, the first and the second term of eq. (4.9). For a certain choice of parameter– 10 –pace, the symmetric part annihilates away, and the dark matter density today is set bythe asymmetric part. The final asymmetry, as we observe today, is the initial asymmetryfollowed by the entropy injection η final χ = ζη initial χ . (4.10)Notably, the entropy injection also dilutes the initial baryon asymmetry η initial B to the finalbaryon asymmetry η final B in a similar fashion. If we suppose that the initial asymmetry canbe no larger than η initial B ∼ O (1) (as argued in [47]) then the observed baryon asymmetrytoday η B ∼ − implies a weak bound on the magnitude of ζ . From eq. (4.6), the relic density of dark matter is the decoupling abundance multiplied bythe dilution factor ζ which can be expressed in terms of F f as followsΩ RelicDM h = ζ s m χ ρ c h (cid:18) η χ + 2 η χ F f − F f (cid:19) . (4.11)To proceed, we re-express the evolution of the fractional asymmetry, as given in eq. (4.12),in terms of Φ and F eq as follows F = F eq , f exp (cid:32) − (cid:88) n λ n η χ Φ n ( x, m χ ) (cid:33) . (4.12)In the case of symmetric dark matter ( F f →
1) the relic density however should dependstrongly on the cross-section, which can be seen by the fact that fractional asymmetryeq. (4.12) depends strongly on the cross-section σ n and in the symmetric dark matter limitthe fractional asymmetry can be approximated as F f ≈ − (cid:88) n λ n η χ Φ n , (4.13)where the parameter λ n is directly proportional to the cross-section σ n . Moreover, in the casewhere the initial asymmetry is sufficiently small such that Y + f (cid:39) Y − f (cid:29) η χ , and therefore, F eq , f ≈
1, then the fractional asymmetry simplifies to F f (cid:39) exp (cid:32) − (cid:88) n λ n η χ Φ n ( ∞ , m χ ) (cid:33) . (4.14)However, as suggested in [29], the above form remains a valid approximation for strongerinitial asymmetries provided F f (cid:38) − .Furthermore, the relic density of baryons today can be written as followsΩ Relic B h = ζ s m p ρ c h η B , (4.15)where m p is the proton mass and η B is the initial baryon asymmetry. It follows that theratio of the dark matter to the baryon relic density can be expressed in the form [29]Ω RelicDM Ω Relic B = (cid:18) F f − F f (cid:19) m χ η χ m p η B . (4.16)– 11 – - - - - - - [ m χ / GeV ] Log [ η χ i n i t i a l ] ζ = � ζ = �� - � ζ = �� - � ζ = �� - � ζ = �� - � ζ = �� - �� ��� � * = �� � ��� κ = ��� - - - - - - [ m χ / GeV ] Log [ η χ i n i t i a l ] ζ = � ζ = �� - � ζ = �� - � ζ = �� - � ζ = �� - � ζ = �� - �� ��� � * = �� � ��� κ = ��� Figure 3 . Contours for which the observed relic density Ω relic h ≈ .
12 is obtained as functions ofthe initial asymmetry η initial χ and the mass of dark matter m χ for different entropy dilution factor ζ as indicated. With σ as in eq. (4.17), the above plots assumes two coupling constants κ χ = 0 . m χ > × T (cid:63) since m χ /
25 is the characteristic freeze-out point and thus beyond this the dark matterfreezes out in a radiation domination regime prior to φ matter domination. We will consider the simplest case of dark matter with an s-wave annihilation channel toStandard Model states which we parameterise in a model-independent manner as follows σ ≈ κ m χ . (4.17)Thus the dark matter relic density eq. (4.9) is a function of the initial dark matter asymmetry η initial χ , the entropy dilution ζ , the dark matter mass m χ , the single coupling constant κ andin the matter dominated scenario, the critical temperature T (cid:63) . We illustrate this in Figure 3which presents contours of the observed dark matter relic density as a function of the initialasymmetry η initial χ and the dark matter mass m χ for different entropy dilution factors ζ , fortwo choices of the coupling κ χ = 0 . T (cid:63) .Conversely, we can look to ascertain the annihilation cross section appropriate to repro-duce the correct relic density. Again, we emulate the treatment of [29] and give our results interms of the thermal WIMP cross section σ , WIMP which is appropriate for giving the correctrelic density for symmetric s-wave ( n = 0) freeze-out in a radiation dominated era. We writethis in terms of the ratio Ω B / Ω χ (cid:39) . , WIMP = x − f g / (cid:63) (following from eq. (4.5)) σ , WIMP = (cid:114) π h / i Φ − , WIMP m p η B M Pl Ω Relic B Ω FO , (4.18)Since we can neglect the dependence of x f on σ , because of the logarithmic nature of x f , wetake x f (cid:39)
25 which implies that the symmetric radiation dominated freeze-out cross sectionis of order σ , WIMP (cid:39) . × − GeV − . In Figure 4 we show the final fractional asymmetry F f as a function of cross-section σ relative to the traditional WIMP cross section σ , WIMP .– 12 – * = � × �� � ���� η χ = �� - �� � * = � × �� � ���� η χ = �� - � � * = � × �� � ���� η χ = �� - �� � * = � × �� � ���� η χ = �� - � � * = � × �� � ���� η χ = �� - �� � * = � × �� � ���� η χ = �� - � σ σ F ∞ m χ =
10 GeV � * = � × �� � ��� � * = � × �� � ��� � * = � × �� � ��� � * = � × �� � ��� � * = � × �� � ��� � * = � × �� � ��� σ σ F ∞ η χ = - Figure 4 . The present fractional asymmetry F f as function of the cross-section σ relative to thetraditional WIMP cross section σ , WIMP . One can observe variations of the late time fractionalasymmetry F f with σ /σ , WIMP , for a fixed mass (left panel) and for a fixed asymmetry η χ (rightpannel) for two different dark matter masses m χ = 1 . × GeV (solid line) and m χ = 1 . × GeV(dashed line). The case of decoupling in the radiation dominated era is shown as the black curve.
When the initial dark matter asymmetry η χ is relatively small such that F f (cid:46)
1, thedark matter relic density cannot be determined only with the asymmetry as the second termof the LHS of eq. (4.11) gives a significant contribution. Using eqns. (4.16) and (4.18) onehas the following expression involving the fractional asymmetry [29] F f = exp (cid:18) − σ σ , WIMP (cid:18) − F f F f (cid:19) Φ Φ , WIMP (cid:19) . (4.19)Then the fractional asymmetry F f can be expressed as a function of the s-wave cross-section σ , for both the matter and radiation dominated dark matter freeze-out scenarios as follows F f = exp (cid:32) − (1 − r ) − / (cid:18) x (cid:63) x f (cid:19) / σ σ , WIMP (cid:18) − F f F f (cid:19)(cid:33) . (4.20)Notably, the strong dependence of F f on the s-wave cross-section σ , can be seen in Figure 4. The conventional motivation for ADM is that it offers an explanation for the observed O (1)coincidence of the present day baryon and dark matter abundances: Ω χ ≈ B . Specifically,if one assumes that η χ ∼ η B by linking the baryon and dark matter asymmetries, and thedark matter mass is similar to the proton mass m χ ∼ m p ≈ χ Ω B (cid:39) η χ m χ η B m p ∼ O (1) . (5.1)However, this relationship also holds away from the assumptions η χ /η B ∼ m χ /m p ∼ et al. for [48] one possible realization). Notably,from inspection of Figure 3 this framework of matter domination decoupling of ADM providesmotivated examples of this less studied ADM scenario with m χ (cid:29) m p .– 13 –he first model of Superheavy Asymmetric Dark Matter was introduced in [49]. Whilethe work of [49] provides a proof of principle that Superheavy ADM can give the correctrelic density, it introduces a strong assumption regarding the ordering of events and thusdoes not fully explore the potential parameter space. Specifically, it was assumed that darkmatter freeze-out occurred during an era of radiation domination. However, the entropyproduction, which is assumed to follow dark matter freeze-out, must be sourced by someother significant energy density in the early universe, and this can potentially significantlyalter the cosmological history. For instance, if the entropy injection is due to the decays ofsome heavy decoupled particle species, then over a large swathe of parameter space, the darkmatter will freeze-out in a matter dominated universe. Since this case was neglected in thisinitial study [49], this work completes this interesting picture by exploring the parameterspace of Superheavy ADM which decouples during matter domination.The central obstruction to realizing Superheavy ADM is that for the relic density to beasymmetric, the symmetric component of pairs of dark matter and anti-dark matter statesmust be removed via annihilations. For heavier dark matter, a smaller asymmetry is requiredsince fewer particles are needed at late time to make up the observed relic abundance. Forinstance, for PeV mass ADM one requires η today χ ∼ × − and in accordance with eq. (5.1)this scales as follows Ω Relic χ Ω Relic B (cid:39) × (cid:16) m χ (cid:17) (cid:18) η now χ × − (cid:19) . (5.2)Since the asymmetry is smaller, a larger annihilation rate is needed for Y χ (cid:28) Y χ to ensurethat the final relic density is asymmetric. It follows that in calculating the annihilation crosssection needed to deplete the symmetric component of dark matter for increasing mass, leadsto a unitarity limit analogous to the classic Griest and Kamionkowski unitarity bound [46].Assuming perturbative annihilations, the ADM mass bound is roughly m χ (cid:46)
100 TeV [50].As seen in Section 4 this ADM unitarity bound can be evaded via the introductionof an entropy injection event in the early universe. Specifically, perturbative unitarity [46]constraints the s-wave annihilation cross-section to satisfy σ (cid:46) πm χ , (5.3)For a specific critical temperature T (cid:63) = 10 GeV, the upper bound on the dark matter massfrom the unitarity limit is deduced from eq. (4.9) to be m χ (cid:46) GeV.Notably, Superheavy ADM models are also partially motivated by a second, distinct,compelling observation, namely that these scenarios can have significant impacts on astro-physical bodies. Notably, significant quantities of ADM can accumulate in objects such asmain sequence stars, white dwarfs, and neutron stars [51–56]. The reason ADM can have alarger impact than conventional ‘symmetric’ dark matter is due to the fact that the present-day abundance of anti-dark matter Y χ is typically negligible in ADM and therefore darkmatter pair annihilations are rare even in dense environments. In particular, it has beensuggested that ADM in the mass range 0.1-100 PeV could be responsible for the collapseof pulsars near the Galactic Center [57] and ignition of type-Ia supernovae [58–60], both ofwhich are open problems in astrophysics. Acknowledgements.
We are grateful to Jakub Scholtz for helpful comments. Here ‘superheavy’ is taken to mean dark matter with mass 1 PeV (cid:46) m χ (cid:46) M Pl (the Planck scale). – 14 – Appendices
A.1 Higgs Portal Annihilation Cross Sections
In this appendix we give the relevant annihilation cross sections for Higgs Portal dark matter.As can be inferred from eq. (2.2), scalar dark matter can annihilate to Standard Modelparticles via three main routes, (i). dark matter annihilations to Standard Model fermions χχ → f ¯ f ; (ii). dark matter to Standard Model vector bosons χχ → V ¯ V ; (iii). dark matter toHiggs bosons χχ → hh . Accordingly, the partial cross-sections in terms of the Mandelstamvariable s are respectively σ χχ → f ¯ f = N c c χ κ m f πs (cid:113) − m f s (cid:113) − m χ s s − m f ( s − m h ) + m h Γ h σ χχ → V ¯ V = δ V c χ κ m V πs (cid:113) − m V s (cid:113) − m χ s (cid:18) s − m V ) m V (cid:19)(cid:2) ( s − m h ) + ( m h Γ h ) (cid:3) σ χχ → hh = κ c χ πs (cid:113) − m h s (cid:113) − m χ s , (A.1)where N c is the number of colors, V = Z, W ± with δ Z = and δ W + ,W − = 1, and the Higgsboson mass and width are m h and Γ h ≈
13 MeV [30].Taking the non-relativistic limit, we expand the partial thermally averaged cross-sections (cid:104) σv (cid:105) i to order v (in terms of the relative velocity v ≈ v i / v i the velocity of a darkmatter particle during freeze-out) leading to (cid:104) σv (cid:105) = σ + σ v + · · · (A.2)Thus we obtain the following partial annihilation cross sections [38, 61–63]) (cid:104) σv (cid:105) χχ → f ¯ f = N c c χ κ r f πm χ (cid:16) − r f (cid:17) / (cid:0) − r h (cid:1) + r v r f + r h (1 − r f ) + 3 r f ( r h + r ) − (cid:16) − r f (cid:17) (cid:104)(cid:0) − r h (cid:1) + r (cid:105) (cid:104) σv (cid:105) χχ → V ¯ V = δ V c χ κ πm χ (cid:113) − r V (1 − r V + 12 r V ) (cid:0) − r h (cid:1) + r (cid:34) v K − (cid:0) − r V + 12 r V (cid:1) (cid:2) (1 − r h ) + r (cid:3) (cid:35) (cid:104) σv (cid:105) χχ → hh = κ c χ πm χ (cid:113) − r h (cid:20) v r h − − r h (cid:21) , (A.3)in terms of the ratios r i = m i / (2 m χ ) and r Γ = m h Γ h / (4 m χ ), and where F is given by K = 14 r V − r V + 168 r V − r h (12 r V − r V + 240 r V ) + ( r h + r )(1 − r V − r V − r V ) . By replacing, v = 6 /x f the expressions above can be re-written in terms of the mass scaledinverse temperature, x . – 15 – .2 Early Matter Domination from RH Neutrinos In this section we confirm that for suitable parameters the lightest RH neutrino N can leadto an early period of matter domination. Specifically, we outline a scenario in which the RHneutrinos are not ever coupled to the thermal bath for coupling of order ∼ − . If the RHneutrinos, N , never thermalises with the thermal bath, for T > M N the abundance of the N states, Y N (cid:28) Y eq , where Y eq ∼ T < M N , due to the inclusion of the mass the equilibrium number density, n eq , will be Boltzmann suppressed. Thus to prevent the N number density, n N , becomingcomparable to n eq , the decoupling temperature for the N states, T N ,f , is needed to muchlarger compared to its mass [64].Let us also suppose that the right-handed neutrinos N are in thermal equilibrium withsome temperature T , which may be different to the temperature T of the Standard Modelbath and that T < T . We shall assume that the Standard Model states were in thermalequilibrium since T ∼ GeV, it is less obvious whether the Standard Model was ever inequilibrium with a temperature in excess of this. A simple manner to ensure that the N thermalise is to have a state φ with modest couplings to the N . Subsequently the φ VEVcan be identified with mass (cid:104) φ (cid:105) = M of N . It then follows that T /T (cid:39) ( ρ N /ρ SM ) / where ρ N and ρ SM are the energy densities of N and the Standard Model, respectively. Theseenergy densities are set by inflationary reheating according to the branching ratios of theinflaton. We will take ξ ≡ T /T to be a free parameter with ξ < T (cid:29) M N , the N interaction rate isΓ ∼ n N (cid:104) σv (cid:105) ∼ κ T π , (A.4)where the N number density n N ∼ T /π and the cross-section (cid:104) σv (cid:105) ∼ κ /T , and κ is thecoupling strength. Therefore, we can express the ratio between the interaction rate and theHubble expansion rate as Γ /H = (cid:114) π M Pl g (cid:63) κ ξT . (A.5)For a choice of κ ∼ − and g (cid:63) ∼ .
75 this suggests that the interaction rate Γ is alwaysless than the Hubble expansion rate H , as the universe cools down from T ∼ GeV to atemperature comparable to the N mass scale.Moreover, at later times, when T (cid:28) M N , the interaction rate of the N states isΓ ∼ n N (cid:104) σv (cid:105) ∼ κ (2 π ) / (cid:18) T M N (cid:19) / exp ( − M N /T ) , (A.6)where the N number density n N ∼ ( M N T / π ) / exp ( − M N /T ) and the cross-section (cid:104) σv (cid:105) ∼ κ /M N . Again, the ratio between the interaction rate and the Hubble rateΓ /H ∼ (cid:114) π M Pl g (cid:63) κ ( M N T ) − / ξ / exp (cid:18) − M N ξT (cid:19) (A.7)this implies that with the same chosen coupling strength κ ∼ − , the N states again arefound to remain decoupled. – 16 –n the case where the N states freeze-in through renormalisable Yukawa interactionsat the early times T (cid:29) M N , the abundance of the N ’s given at a temperature T is [65] Y N ( T ) ∼ κ M Pl M N T , (A.8)where κ is the strength of the Yukawa coupling. Thus, for κ ∼ − , and M N ∼ GeV,at any given temperature,
T > M N , the abundance is Y N < − (cid:28)
1. When the N states decouple from the thermal bath H ( T N ,f ) ∼ (cid:104) σv (cid:105) ζ (3) T N ,f π , (A.9)where the thermally averaged cross-section (cid:104) σv (cid:105) can be approximated as ∼ κ /M N . Follow-ing the treatment of [64], the requirement that T N ,f (cid:29) M N can be quantified as κ (cid:28) (cid:18) π g (cid:63) (cid:19) / π (cid:112) ζ (3) M Pl M / N ∼ − (cid:18) M N . (cid:19) / , (A.10)which suggests that for κ ∼ − , the requirement, T N ,f (cid:28) M N , is well satisfied. A.3 Dark Matter Production in the Transition from Matter Domination toRadiation Domination
While one could be concerned that even in the case that there is negligible production fromdecays of dark matter, even a small amount of anti-dark matter production might alter thefractional asymmetry F . We next argue that this is not the case in the parameter regionsof interest Emulating the treatment of [45], the ratio of the contribution to the dark matterrelic density from the φ decay, Ω χ, decay , to that produced from freeze-out, Ω relic , is derived asΩ χ, decay Ω relic ∼ . × − (cid:18) B χ × − (cid:19) (cid:18) GeV m φ (cid:19) (cid:18) T RH (cid:19) (cid:16) m χ (cid:17) , (A.11)which can be calculated using the branching ratio derived in [38] B χ ∼ × − (cid:16) κ . (cid:17) . (A.12)It follows that, for the selected parameter space above, Ω χ, decay is obtained as ∼ − , whichis much smaller compared to the density of dark matter and anti-dark matter producedthrough the freeze-out which are normalized, such to agree with the observed relic density. A.4 Entropy Production in the Thermal Bath During Decoupling
While the assumption that φ states decay simultaneously at H (cid:39) Γ φ provides a usefulsimplification, the natural expectation is a exponential decay law for the φ particles [66]compared to the instantaneous decay of φ at some t ∼ Γ − . However, the approximationof sudden decay of the φ particles remains valid until the entropy violation caused by theradiation due to the decay of the φ particles can no longer be neglected, such that the oldradiation becomes comparable to the new radiation produced by φ decay [66].– 17 –ne can define a temperature T EV [38], so that for T (cid:38) T EV the universe remains matterdominated H ∝ T / and T ∝ a − as the φ decays are unimportant, whereas for T (cid:46) T EV the φ decays become significant such that H ∝ T and T ∝ a − / as in [66]. This temperaturethreshold at which entropy violation in the bath is non-negligible is given by T EV (cid:39) T (cid:63) (cid:18) (cid:18) r − r vΓ t (cid:63) (cid:19)(cid:19) (1 − v) / v , (A.13)where v ≡ / ω ) − + 1, with ω = 0 and 1/3 implies matter domination and radiationdomination respectively, and t (cid:63) is defined as the time when φ starts behaving matter-like( t (cid:63) ∼ H (cid:63) ). Thus, for dark matter freeze-out to occur during matter domination, it is requiredthat T f (cid:38) T EV . If, however, the φ particle decays become non-negligible before the darkmatter decouples from the thermal bath, the dark matter freeze-out calculations should beperformed similar to the classic paper of Giudice, Kolb, & Riotto [39] and we do not considerthis case here. A.5 Derivation of the Asymmetric Yield
Here an analytic solution of Boltzmann equation eq. (3.7) is obtained with the similar pro-cedure as described in [40] to arrive at eq. (4.4) in the main body. Defining ∆ − = Y − − Y − eq the anti-DM abundance is given by eq. (3.7) dY − dx = − (cid:88) n λ n x − n − g / (∆ − (∆ − + 2 Y − eq ) + η χ ∆ − ) d ∆ − dx = − dY − eq dx − (cid:88) n λ n x − n − g / (∆ − (∆ − + 2 Y − eq ) + η χ ∆ − ) (A.14)where we have used Y +eq − Y − eq = η χ . Much before the freeze out, when the temperature washigh (1 < x (cid:28) x f ), Y ± tracks Y ± eq very closely. During that period of time, ∆ − and d ∆ − dx become very small. Neglecting d ∆ − dx and higher order terms of ∆ − in eq. (A.14) we obtain dY − eq dx (cid:39) − (cid:88) n λ n x − n − g / (2∆ − Y − eq + η χ ∆ − ) . (A.15)From eq. (3.7) it should be noted that dY − eq dx vanishes in equilibrium and thus with eq. (3.9)leads to Y − eq ( Y − eq + η χ ) − Y = 0 Y − eq = − η χ (cid:115) η χ Y (A.16)Inserting the eq. (A.16) in eq. (A.15) we get12 (cid:18) η Y (cid:19) − / dY dx = − (cid:88) n λ n x − n − g / ∆ − (cid:115) η χ Y , (A.17)and thus using eq. (3.6) we obtain b (3 x − x ) e − x = − (cid:88) n λ n x − n − g / ∆ − (cid:32) η χ Y (cid:33) . (A.18)– 18 –n the epoch of interest (1 < x (cid:28) x f ) and therefore the x term can be neglected as comparedto the x term b ( − x ) e − x (cid:39) − (cid:88) n λ n x − n − g / ∆ − (cid:32) η χ Y (cid:33) , (A.19)which leads us to the following solution∆ − (cid:39) Y g / (cid:80) n λ n x − n − (cid:16) η χ + Y (cid:17) . (A.20)At later times much after the freeze out, the temperature will be sufficiently low ( x (cid:29) x f ) such that Y − tracks Y − eq very poorly. As a result, the abundance Y − becomes comparableto ∆ − (∆ − (cid:39) Y − (cid:28) Y − eq ). This allows us to neglect terms in eq. (A.14), which depends on dY − eq dx and Y − eq , leading to the expression d ∆ − dx (cid:39) − (cid:88) n λ n g / x − n − (∆ − + η χ ∆ − ) . (A.21)From the above we can obtain ∆ − by integrating in the interval [ x f , ∞ ] (cid:90) ∞ x f d ∆ − (∆ − + η χ ∆ − ) = − (cid:90) ∞ x f dx (cid:88) n λ n g / x − n − (A.22)and partially evaluating yields1 η χ ln ∆ − ∆ − + η χ (cid:12)(cid:12)(cid:12) ∞ x f = − (cid:88) n (cid:90) ∞ ¯ x f dx λ n g / x − n − . (A.23)For x (cid:29) x f , one can assume ∆ − ( x f ) (cid:29) ∆ −∞ , which leads us to the following expression1 η χ ln (cid:18) ∆ −∞ + η χ ∆ −∞ (cid:19) (cid:39) (cid:88) n (cid:90) ∞ x f dx λ n g / x − n − . (A.24)It follows that the anti-dark matter abundance set by the decoupling is Y −∞ (cid:39) ∆ −∞ = η χ exp (cid:32) η χ (cid:80) n ∞ (cid:82) x f dx λ n g / x − n − (cid:33) − , (A.25)Furthermore, since the dark matter mass m χ is independent of the bath temperature T ,the parameter λ n can be pulled out of the integration and the anti-dark matter abundancecan be further simplified to obtain Y −∞ (cid:39) η χ exp (cid:18) η χ (cid:80) n λ n Φ n ( ∞ , m χ ) (cid:19) − , (A.26)with the definition Φ n ( x, m χ ) = (cid:90) xx f dx (cid:48) x (cid:48)− n − g / . (A.27)– 19 – eferences [1] S. Nussinov, Technocosmology: Could A Technibaryon Excess Provide A ’Natural’ Missing MassCandidate?,
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