Dark matter amnesia in out-of-equilibrium scenarios
Joshua Berger, Djuna Croon, Sonia El Hedri, Karsten Jedamzik, Ashley Perko, Devin G. E. Walker
PPITT-PACC-1822
Dark matter amnesia in out-of-equilibrium scenarios
Joshua Berger, a Djuna Croon, b Sonia El Hedri, c,d
Karsten Jedamzik, e Ashley Perko, f Devin G. E. Walker f a Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA b TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, B.C. V6T2A3, Canada c Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands d Laboratoire Leprince Ringuet, École Polytechnique, 91120 Palaiseau, France e Laboratoire Univers et Particules de Montpellier, UMR5299-CNRS, Universite Montpellier II,34095 Montpellier, France f Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Models in which the dark matter is produced at extremely low rates from theannihilation of Standard Model particles in the early Universe allow us to explain the currentdark matter relic density while easily evading the traditional experimental constraints. Inscenarios where the dark matter interacts with the Standard Model via a new physicsmediator, the early Universe dynamics of the dark sector can be particularly complex,as the dark matter and the mediator could be in thermal and chemical equilibrium witheach other. This equilibration takes place via number-changing processes such as doubleCompton scattering and bremsstrahlung, whose amplitudes are cumbersome to calculate.In this paper, we show that in large regions of the parameter space, these equilibrationmechanisms do not significantly affect the final dark matter relic density. In particular, fora model with a light dark photon mediator, the relic density can be reasonably estimatedby considering that the dark matter is solely produced through the annihilation of StandardModel particles. This result considerably simplifies the treatment of a large class of darkmatter theories, facilitating in particular the superimposition of the relic density constraintson the current and future experimental bounds. a r X i v : . [ h e p - ph ] D ec ontents Understanding the nature of dark matter (DM) is one of the most pressing unresolvedproblems in particle physics and cosmology. An important class of dark matter theories aremodels where the dark sector has a gauge structure, the simplest of these scenarios beingextensions of the Standard Model (SM) by a dark U (1) (cid:48) gauge group. This symmetry isassociated with a dark photon A (cid:48) and can possibly be spontaneously broken by a new Higgsboson Φ . In the general case, A (cid:48) and Φ can mix with the SM hypercharge gauge bosonand with the SM Higgs respectively, thereby connecting the dark and visible sectors [3–7].In these so-called “portal models”, the dark matter can be produced from the annihilationof Standard Model and other dark sector particles in the early Universe, and can alsoannihilate into these particles at later times. This finding has led to the hypothesis thatthese production and annihilation processes completely determine the final dark matter relicdensity. Under this hypothesis, the couplings and the masses of the dark sector particleswould be strongly tied to this relic density, thus allowing for a determination of how theviable regions of the parameter space of existing models intersect with the current andfuture experimental constraints.Linking the relic density requirements for gauge and Higgs portal models to experi-mental constraints is all the more important, as the corresponding signatures are extremelydiverse. One particularly interesting avenue is to search not directly for the dark matter, butfor the mediator itself. New dark gauge bosons, in particular, could be directly produced atcolliders, affect the flavor observables, or be produced at beam dump experiments [8–28].More recently, however, indirect astrophysical and cosmological probes have been discussedfor models where the mediators are particularly light or couple extremely weakly to the SM.For particularly weak coupling, the dark mediators are sufficiently long lived as to decayaround the era of Big Bang Nucleosynthesis (BBN) or recombination [1, 2]. In this case,they are constrained by the observed light element yields in the Universe and the success of– 1 –he Λ CDM cosmology at predicting the Cosmic Microwave Background (CMB) spectrumrespectively. The sensitivity to CMB distortions will be greatly enhanced by a possiblyupcoming PIXIE experiment [35].Constraints from direct and indirect detection are also possible. Although the couplingof the dark matter to the SM is very weak, the mediator can be rather light in these sce-narios and spin independent direct detection constraints are strong [36–41]. On the otherhand, the lightness of the mediator works against the constraints on these models as therecoil momentum transfer becomes comparable to the mediator mass [42]. The lightnessof the mediator limits the constraints to the point where they are generally irrelevant tothe models considered below. Indirect detection does not face this issue, but the dominantannihilation channel for dark matter coupling to a light mediator is into mediators, ratherthan SM particles, softening the observed spectrum and weakening the constraints some-what [43]. The annihilation cross-section into mediators is rather large, though, leadingto some significant constraints if the dark matter coupling to the mediator makes up thetotality of the observed dark matter abundance in the universe.One crucial issue with cosmological experimental probes is that they constrain regions ofthe parameter space where the couplings of the dark matter to the SM are extremely small.In these regions, the dark matter can never be in equilibrium with the SM and thermaldark matter models will not be viable. Understanding how cosmological constraints on lightgauge and Higgs bosons affect dark matter models therefore seems to require a thoroughunderstanding of the dynamics of the dark matter when it is produced out of equilibrium.The associated scenarios can be extremely complex as the internal dynamics of the darksector could significantly affect the dark matter evolution. Taken at face value, resultantdark matter densities could differ by orders of magnitude depending on the degree of darksector equilibration.To illustrate this, let us consider two extreme cases: (a) full dark sector equilibrationand (b) complete absence of interactions between particles in the dark sector. In eithercase, dark sector particles are produced via freeze-in [29] of the mediator A (cid:48) and the darkmatter χ particles (having U (1) (cid:48) charges), with the former usually produced in much largernumbers. However, whereas in case (b) the dark matter abundance is simply given by itsfreeze-in value n fi χ , in case (a) production of dark sector particles via → processes andpair production of χ (cid:48) s via A (cid:48) annihilation lead to parametrically larger χ and A (cid:48) abundances.Here → processes favor the production of extra dark sector particles since the typicalenergy of a dark sector particle is approximately T , the visible sector temperature, whichis far too large for an equilibrated dark sector at temperature T (cid:48) (cid:28) T . Thus extra darkparticles have to be produced for equilibration to occur. A rough estimate of the equilibrated χ abundance can be obtained via the energy leaked into the dark sector, i.e. T n fi χ ∼ T (cid:48) and n eq χ ∼ T (cid:48) , leading to n eq χ n fi χ ∼ (cid:18) T n fi χ (cid:19) / , (1.1)which is of the order − in the parts of parameter space where the relic abundancesaturates Ω DM . Note that this ratio represents a serious underestimate as the bulk of the– 2 –isible energy is expected to be leaked into the mediator, an effect not taken into accountin the estimate. For such scenarios, the relic abundance calculation is thus plagued by largeuncertainties. Note that such scenarios have been studied in Ref. [30], and though thiswork has outlined important features of dark matter portal scenarios, it has not properlyaddressed these equilibration uncertainties.This paper will show that a class of dark sector models which are distinguished by a richthermal evolution are degenerate from a phenomenological perspective. In particular, wewill demonstrate that, in large regions of the parameter space, the dark matter relic densitydepends only weakly on the details of the chemical equilibration of the dark sector in theearly Universe. This result is due to the fact that, in general, the dark matter producedvia the annihilation of the dark mediator will have enough time to fully annihilate beforethe dark sector freezes out. Hence, the dark matter relic density in today’s Universe will befully determined by its production through the annihilation of SM particles and possiblyalso by its late time annihilation into dark sector particles. Since the cross-section for thelatter process is nearly constant at low temperature this annihilation rate depends onlyweakly on the degree of equilibration in the dark sector. For wide ranges of parameters, itis therefore possible to make order-of-magnitude estimates of the dark matter relic densitywithout any knowledge of the magnitude of the number-changing processes involving darksector particles. This result thus allows us to quickly and straightforwardly delineate theregions of parameter space most relevant in the quest of dark matter.The rest of this paper is outlined as follows. In section 2 we introduce the toy modelthat we use for our study, which is an extension of the SM with a fermionic dark mattersinglet and a new U (1) (cid:48) symmetry associated with a dark photon. We then describe thetwo extreme scenarios that we are going to consider: the case of full equilibrium withinthe dark sector, where the dark matter can be produced from dark photon annihilation,and the case where the DM and the dark photon never equilibrate. In section 3, we detailthe procedure we used to scan the parameter space for our example model. We then showthat, when the dark photon is much lighter than the dark matter both scenarios lead tosimilar relic densities. Finally, we conclude in section 4 that for a majority of the parameterspace of interest, the dark matter relic abundance can be determined using the simplifiedscenarios discussed in section 2 rather than the full out-of-equillibrium calculation. We consider a simple extension of the Standard Model with a broken U (1) (cid:48) gauge groupassociated with a dark photon A (cid:48) , and a fermionic dark matter candidate χ . The new U (1) (cid:48) gauge group kinetically mixes with hypercharge with strength (cid:15) . The model is describedby the following Lagrangian: L = −
14 ˆ B µν ˆ B µν −
14 ˆ F (cid:48) µν ˆ F (cid:48) µν − (cid:15) B µν ˆ F (cid:48) µν (2.1) + ¯ χ (cid:2) γ µ ( i∂ µ − g (cid:48) A (cid:48) µ ) − m χ (cid:3) χ + 12 m A (cid:48) ( A (cid:48) µ ) . (2.2)– 3 –ere, we do not introduce any dark scalars and hence we give A (cid:48) a Stueckelberg mass. Inwhat follows, we focus on the regime where the dark photon is light but can still decay toSM particles, that is, in the mass range m e < m A (cid:48) < GeV. The condition for thermalequilibrium between the dark sector and the Standard Model is roughly given by (cid:104) n f σ ff → χχ v (cid:105) (cid:38) H , (2.3)where n f is the SM fermion density and where (cid:104) σ ff → χχ v (cid:105) is the velocity-averaged cross-section for the processes connecting the DM to the SM, and is proportional to (cid:15) . Inthis paper we consider only very small values for (cid:15) , in the range − < (cid:15) < − , forwhich condition 2.3 is not satisfied. In this regime, instead of reaching its equilibriumvalue before freezing out at later times, the DM abundance initially grows at a very slowrate [29]. For small U (1) (cid:48) gauge couplings α (cid:48) = g (cid:48) / π , the DM annihilation processescan be neglected and the DM abundance steadily grows before “freezing-in” to a constantvalue. Conversely, when α (cid:48) is large, the exchange processes between the DM and A (cid:48) willbe sufficiently important for the whole dark sector to equilibrate at some temperature T (cid:48) different from the temperature of the SM sector. In this scenario, after the initial productionphase, the DM will be able to annihilate into dark photons with a significant rate beforethe dark sector freezes out. We can therefore identify two main regimes:A. Pure freeze-in: in the small α (cid:48) limit, communication between the DM and A (cid:48) is toolimited to impact the DM number density evolution. The latter therefore dependsonly on the DM production rate and steadily increases with time until it “freezes in”when it can no longer be produced due to kinematic reasons, i.e. when T < m χ .B. Reannihilation: when α (cid:48) is large, χ and A (cid:48) are in equilibrium in the early Universeand the dark sector has a temperature T (cid:48) . The DM is first produced through theannihilation of SM fermions and dark photons and later annihilates into dark photons.Hence, the relic density first steadily grows, then diminishes again until the χχ → A (cid:48) A (cid:48) processes freeze out.The two regimes in fact correspond to two extreme limits of the dark matter-dark photoncoupling α (cid:48) . For intermediate values of α (cid:48) and especially when the dark photon is massive,partial equilibrium can occur. Equilibration is ensured by number-changing processes suchas χ A (cid:48) → χ A (cid:48) A (cid:48) (followed by A (cid:48) A (cid:48) → χ ¯ χ ), that can occur either through double Comptonscattering or bremsstrahlung. The associated tree-level amplitudes are however infrared di-vergent, involving large logarithms that need to be resummed. Determining to what extentthe dark sector is in equilibrium therefore requires particularly involved computations. Forstudies of dark matter phenomenology, however, the knowledge of the full thermal evolutionof the dark sector is often not necessary, and approximate estimates of the final relic densityare often enough to derive meaningful theoretical constraints on our models. In this paper,we therefore evaluate the impact of dark sector equilibration in the early Universe on theDM relic density by considering the following two extreme scenarios: We expect other contributing processes, such as production of A (cid:48) by inverse decay, to be subdominant. – 4 –. The dark sector is in equilibrium in the early Universe, at a temperature T (cid:48) . Forsizable α (cid:48) , dark photon production via inverse decay of SM particles and its subsequentannihilation into pairs of χ can therefore contribute to the DM production in theearly Universe. At later times, the DM can annihilate into pairs of dark photons.Additionally, we assume that the dark photon number density follows its equilibriumvalue as long as it communicates with the dark matter.2. χ ¯ χ ↔ A (cid:48) A (cid:48) equilibrium is not realized. The relic density of χ particles is hence fullydetermined by the freeze-in of ¯ f f → χ ¯ χ production and, for large α (cid:48) , the freeze-outof χ ¯ χ → A (cid:48) A (cid:48) annihilation.The two scenarios correspond to extreme cases of equilibration. In the first one, the chemicalpotentials of the dark matter particle and anti-particle are equal to each other in the earlyUniverse and the chemical potential of the dark photon is always zero. In the second one,the dark sector is always out of equilibrium and the chemical potentials are large. Figure 1shows the evolution of the DM comoving number density Y χ in these two scenarios for agiven parameter point of our model. For x = m χ /T < . , the DM density in scenario 1 ismuch larger than in scenario 2, since the A (cid:48) A (cid:48) ↔ χχ equilibrium favors DM production. For x (cid:38) . , however, this number density sharply drops and closely tracks the one obtained inscenario 2. Ultimately, the resulting relic densities differ by no more than a factor of two,the early Universe dynamics of the dark matter having been almost completely washed out.In the rest of this work, we compare the values of the DM relic density obtained inscenarios 1 and 2 in order to determine whether the behavior observed in figure 1 holds forother choices of parameters. Given our assumptions for these two scenarios, it is safe toassume that if both lead to similar DM relic densities, intermediate scenarios, with partialequilibrium in the dark sector, will lead to similar results . We first detail the evolutionequations corresponding to scenarios 1 and 2 and discuss interesting limit cases. In this section we consider the case 1, where the dark sector equilibrates at a temperature T χ = T A (cid:48) = T (cid:48) (cid:54) = T . Additionally, we consider that the dark photon comoving numberdensity always tracks its equilibrium value Y A (cid:48) ( T A (cid:48) ) ≈ Y A (cid:48) ,eq ( T A (cid:48) ) = 1 s g A (cid:48) (2 π ) (cid:90) ∞ f A (cid:48) ( p, T A (cid:48) ) d p , (2.4)where g A (cid:48) = 3 is the number of degrees of freedom for a massive spin-1 boson, and f A (cid:48) ( p, T A (cid:48) ) is the Bose-Einstein distribution of the A (cid:48) particles at temperature T A (cid:48) , which is differentfrom the temperature T of the SM sector. Here, Y i = n i /s is the ratio of the number densityof n i for a given particle i to the total entropy in the Universe s . This assumption (2.4) isvalid when A (cid:48) is much lighter than the dark matter. We therefore expect this scenario tobe most relevant to that region of parameter space. For the couplings we are interested in,we always have T (cid:48) (cid:28) T . We will therefore consider that the dark sector contributions tothe total energy density, the Hubble parameter, and the entropy are negligible, and we can– 5 – x Y Figure 1 : Comoving number density of the dark matter Y χ as a function of x = m χ /T for m χ = 800 MeV, m A (cid:48) = 20 MeV, (cid:15) = 4 . × − , and α (cid:48) = 0 . . The blue and red solid linesrepresent the values of Y χ in scenarios 1 and 2 respectively. The orange and green dashed linesshow Y QSE (defined in equation 2.14) in the following two limit cases: the dark sector has anequilibrium temperature T (cid:48) and the dark sector has zero temperature. Finally, the black dot-dashed line shows the value of Y χ when the dark sector is at equilibrium at a temperature T (cid:48) . Here,we started integrating the evolution equations only when the DM departs from equilibrium and,before that time, approximated its density by its value in its various equilibrium states, hence thesharp turn for the blue curve around x = 0 . . write these quantities as ρ = ρ SM = π g eff, SM ( T ) T (2.5) H = 1 M P (cid:114) πρ (2.6) s = s SM = 2 π h eff, SM ( T ) T , (2.7)where g eff and h eff are the statistical weights for energy and entropy, respectively. In ournumerical study we will take the effective numbers of degrees of freedom (as a function oftemperature) for the visible sector from the tables in micrOMEGAs [32].We compute the dark sector temperature T (cid:48) following the procedure described in [30],using the total energy density of the dark sector ρ (cid:48) = ρ χ + ρ A (cid:48) , defined as ρ (cid:48) ( T (cid:48) ) = g χ π (cid:90) ∞ E χ p d pe E χ /T (cid:48) + 1 + g A (cid:48) π (cid:90) ∞ E A (cid:48) p d pe E (cid:48) A /T (cid:48) − , (2.8)where we again impose µ χ = µ A (cid:48) = 0 . Here, since χ is a Dirac fermion, its number of degreesof freedom is g χ = 4 . Since the dark photon is always massive in our study, g A (cid:48) = 3 . For– 6 – (cid:48) (cid:28) T , as expected when (cid:15) is small, ρ (cid:48) obeys the following energy transfer equation: d ( ρ (cid:48) /ρ ) dT = − HT ρ (cid:88) f (cid:34) g f π (cid:90) ds σ ff → χχ ( s )( s − m f ) sT K (cid:18) √ sT (cid:19) + 2 α EM ( (cid:15) cos θ W q f ) π m A (cid:48) (cid:32) m f m A (cid:48) (cid:33) (cid:115) − m f m A (cid:48) T K (cid:16) m A (cid:48) T (cid:17) , (2.9)where we sum over all species of SM fermions f and where the first and second terms onthe right-hand-side account for energy deposition into the dark sector via Standard Modelannihilation into dark fermions and inverse decay into dark photons, respectively.The Boltzmann equations in the non-relativistic limit are dY χ dx = s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH ( Y χ,eq ( T ) − Y χ ) + s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:32) Y χ,eq ( T (cid:48) ) Y A (cid:48) ,eq ( T (cid:48) ) Y A (cid:48) − Y χ (cid:33) dY A (cid:48) dx = Γ A (cid:48) → ¯ ff xH ( Y A (cid:48) ,eq ( T ) − Y A (cid:48) ) − s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:32) Y χ,eq ( T (cid:48) ) Y A (cid:48) ,eq ( T (cid:48) ) Y A (cid:48) − Y χ (cid:33) . (2.10)The subscript for the velocity-averaged cross sections indicates the temperature at whichthey should be evaluated. Note, in particular, that the χχ → A (cid:48) A (cid:48) cross section is evaluatedat T (cid:48) and not at T . For small (cid:15) , we can approximate these equations by dY χ dx ≈ s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) + s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:32) Y χ,eq ( T (cid:48) ) Y A (cid:48) ,eq ( T (cid:48) ) Y A (cid:48) − Y χ (cid:33) dY A (cid:48) dx ≈ Γ A (cid:48) → ¯ ff xH Y A (cid:48) ,eq ( T ) − s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:32) Y χ,eq ( T (cid:48) ) Y A (cid:48) ,eq ( T (cid:48) ) Y A (cid:48) − Y χ (cid:33) . (2.11)Assuming that A (cid:48) is fully thermalized, we can finally write the dark matter evolutionequation as dY χ dx ≈ s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) + s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:0) Y χ,eq ( T (cid:48) ) − Y χ (cid:1) . (2.12)When α (cid:48) is large, the last term in this equation dominates over the f ¯ f → χχ productionterm for an extended period of time in the early Universe and the DM density Y χ is nearlyequal to its equilibrium value Y χ,eq ( T (cid:48) ) . As detailed in [30], the dark matter-dark photonexchange processes will ultimately freeze out, when the A (cid:48) A (cid:48) → χχ production term shutsoff for T (cid:48) (cid:46) m χ . This will lead to the DM density either stabilizing to its final value orevolving further following the equation dY χ dx ≈ s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) − s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH Y χ . (2.13)In practice, the f ¯ f → χχ production term and the χχ → A (cid:48) A (cid:48) annihilation term will oftenbalance each other, leading to a new temporary equilibrium, dubbed the “Quasi-Static– 7 –quilibrium” (QSE) in [30]. In this case, the relic density of the DM is approximately givenby Y χ ≈ (cid:115) (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) Y χ,eq ( T ) ≡ Y QSE ( T ) (2.14)before freeze-out. The simplified scenario in this paragraph is therefore fully defined bythe analytic expression (2.14) and the freeze-out temperature. A quasi-static equilibriumbehavior can for example be observed in figure 1 for . (cid:46) x (cid:46) . In the simplified scenario 2, we assume that the DM and the dark photon never equilibrateand, in particular, that dark matter production from dark photon annihilation can beneglected. The dark matter number density will therefore be set first by DM productionfrom SM fermion annihilation and later, when the U (1) (cid:48) structure constant α (cid:48) is large, byDM annihilation into dark photons. In this regime, the Boltzmann equation for χ in thenon-relativistic limit is given by dY χ dx = s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) − s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH Y χ . (2.15)As before, we assumed that Y χ ( T ) (cid:28) Y χ,eq ( T ) , since the (cid:15) coupling connecting the DM andthe SM is very small. Since the DM annihilation into dark photons becomes important onlyat late times and therefore low temperatures, and occurs dominantly in the s -wave, we cansafely approximate the χχ → A (cid:48) A (cid:48) cross-section by its value at zero temperature, i.e. (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) ≈ (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) . (2.16)The final simplified Boltzmann equation then is dY χ dx = s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) − s (cid:104) σ ¯ χχ → A (cid:48) A (cid:48) v (cid:105) xH Y χ . (2.17)When α (cid:48) is small, this equation can be approximated by a “pure freeze-in” equation ofthe form dY χ dx = s (cid:104) σ ¯ χχ → ¯ ff v (cid:105) T xH Y χ,eq ( T ) . (2.18)In this regime the interactions A (cid:48) A (cid:48) ↔ ¯ χχ play no significant role in determining the relicdensity of either dark sector particle. The simplified evolution equation can be solvedanalytically, taking the late time limit. The result is given by [29] Y ( t → ∞ ) ≈ (cid:113) e M p (cid:15) π √ g eff h eff m χ . (2.19)Conversely, when α (cid:48) is large, the last term of equation 2.17 can balance out the DMproduction term. In this case, just as in the equilibrated case discussed in section 2.1,– 8 –he DM reaches a quasi-static equilibrium until the f ¯ f → χχ production term shuts offand the annihilation ¯ χχ → A (cid:48) A (cid:48) process freezes out. The associated relic density willalso be given by equation 2.14, evaluated at the freeze-out temperature T f and taking T (cid:48) = 0 . For parameter points where a QSE occurs, we can therefore already predict thatthe only discrepancy in the final DM relic density between scenarios 1 and 2 will arise fromcomputing the A (cid:48) A (cid:48) → χχ cross-sections at T (cid:48) and zero respectively. As can be seen infigure 1, this difference is not expected to be significant since temperatures in the darksector are usually low at late times. Finally, for intermediate α (cid:48) values, the equation 2.17has to be solved numerically but this procedure is particularly simple as it does not involvecomputing chemical potentials and temperatures in the dark sector.In the following section, we will perform a scan over the different parameters of ourmodel to compute the DM relic densities in the equilibrated scenario 1 and the out-of-equilibrium scenario 2 in order to assess the impact of the internal dark sector dynamics inthe early Universe on the final DM relic density. In this section, we determine in what region of the parameter space the dark matter relicdensity is nearly independent of the internal dark sector dynamics at high temperatures.To this end, we perform a numerical scan over the different parameters of our model andcompare the relic density obtained in the fully out-of-equilibrium scenario 2, via integrationof Eq. 2.17, to the one obtained in scenario 1 where either A (cid:48) or the whole dark sector isfully equilibrated, via integration of Eq. 2.12. The true relic density is expected to lie inbetween those limiting cases. Details on how we evaluate the relic densities numerically aregiven in Appendix B.Our model involves four parameters, (cid:15) , α DM , m DM , and m A (cid:48) , that can vary over ordersof magnitude. We therfore perform a scan over the following ranges: m DM ∈ [0 . , , step size . m A (cid:48) ∈ [2 MeV , min { m DM , } ] , step size . (cid:15) ∈ [10 − , − ] , step size . α DM ∈ [10 − , − ] , step size . . (3.1)The dimensionless step sizes above are logarithmic, so, for a given step size δ , the value ofa parameter p with minimum value p min at step n will be p = p min × nδ . (3.2)We express the discrepancy between the (partially) equilibrated scenario 1 and the out-of-equilibrium scenario 2 by E = Ω scenario2 Ω scenario1 . (3.3)Fig. 2 shows the color-coded values of E for our parameter scan in the ( α (cid:48) , m χ ) spacefor fixed values of the portal coupling (cid:15) and the mediator mass m A (cid:48) . The black line in each– 9 – igure 2 : Relative error E versus α (cid:48) and m χ , where masses are in GeV. We study values of (cid:15) that are small enough that the dark sector is out of thermal equilibrium with the visible sector,but large enough to produce the observed relic abundance. The dashed line is the approximatetransition between the freeze-in regime (below the line) and the re-annihilation regime (above theline), the solid line indicates where the relic abundance calculated using the equilibrium calculation(scenario 1) matches the observed relic abundance, and the dotted line indicates where the relicabundance calculated using the simplified calculation (scenario 2) matches the observed abundance. panel shows the approximate transition region between pure freeze-in (regime A.) and re-annihilation (regime B.). Here, we chose values of (cid:15) sufficiently large to obtain the observedDM relic density in some regions of the parameter space. We note that, although we choseto show our results only for specific values of (cid:15) and m A (cid:48) for the sake of clarity, the behaviorsand numerical results shown here are representative of the ones we obtained in the rest ofthe parameter space.From Fig. 2 we first observe that, in most of the parameter space, assuming the out-of-equilibrium scenario 2 leads to values of the DM relic density that are within a factor of– 10 –wo of the ones obtained in the equilibrated scenario 1. The only region in which scenario 2significantly differs from scenario 1 is the m A (cid:48) > − m χ region, where since A (cid:48) is heavy,equilibrium is unlikely to occur and therefore assuming the fully equilibrated scenario 1 leadsto overestimating the relic density. Note that, while the relic density contours correspondingto the PLANCK value coincide in scenarios 1 and 2 for m DM (cid:29) m m A (cid:48) , they start stronglydiverging as the dark matter mass becomes lower. The m DM ∼ m A (cid:48) region thereforedeserves a more in-depth treatment.Since the abundances predicted in scenarios 1 and 2 bracket the true relic abundance wecan conclude the following: as long as the dark photon is much lighter than the dark matter,the degree of equilibration in the dark sector has an extremely limited influence on thevalue of the DM relic density. It is therefore possible to considerably simplify the treatmentof a wide range of dark matter models with a light vector mediator. At first sight, ourresults may appear very surprising. We have argued in the introduction that thermalizationuncertainties in the considered dark matter scenario may potentially indroduce orders-of-magnitude uncertainties in the determination of relic abundances. However, the closeness ofpredicted abundances in scenarios 1 and 2 maybe understood from the following observation.These two scenarios distinguish themselves by the A (cid:48) A (cid:48) → χχ production term and theevaluation of the χχ → A (cid:48) A (cid:48) cross section at either T (cid:48) (cid:54) = 0 or T (cid:48) = 0 . We have alreadynoted that the latter difference is minimal for s -wave annihilation at χ freeze-out. Darkmatter χ production via A (cid:48) self-annihilation may become important for large A (cid:48) densities(i.e. large (cid:15) and large T (cid:48) ) as well as large A (cid:48) self-annihilation cross section (i.e. large α (cid:48) ).In this case one could expect for scenario 1 to give a much larger abundance than scenario2. However, large (cid:15) and large α (cid:48) also implies the likely re-annihilation of the produceddark matter, with the end effect that though in the early Universe there may be orders ofmagnitude differences in the χ abundance (cf. Fig 1), the freeze-out value is essentially thesame. It is not clear whether our results are generic to other dark sector Lagrangians; weleave this to future work. In this work, we have demonstrated that, in many cases, the complicated intermediatestages for the production of dark matter via the freeze-in mechanism can be skipped inthe calculation of the final relic abundance. This implies, for the great majority of pa-rameter space, the relic abundance can be approximated well through an analytic or asimplified numeric calculation. This is a great simplification and in particular circumventsthe great uncertainties associated with the relic abundance calculation, in particular fromnumber-changing processes. Specifically, we demonstrated the close resemblance of the fullyequilibrated and simplified scenarios with a direct numerical comparison, shown in Fig. 2. Itis seen that the approximation holds best for light mediators, i.e. large mass gap m χ /m A (cid:48) .It is interesting to note that for part of the parameter space considered in this work,the relic abundance is the only distinguishing feature of the dark sector. Direct detectionconstraints are weakened for light mediators [42], and indirect detection constraints areweakened as the dominant annihilation channel is into dark mediators [43]. The mediator– 11 –ixing (cid:15) is generally too large in the scenarios considered here to be probed by CMB orBBN constraints, with the mediator decaying just before nucleosynthesis starts. Constraintsdue to decays during BBN exist in the part of parameter space with smaller mixing [1].Assuming no hidden sector decays, dark photons that are lighter than the electron mass arestable, dark matter matter candidates [44]. Other constraints do not apply in the parameterspace considered here as the gauge boson mass is too large. For example, strong constraintson the massless A (cid:48) scenario come from milli-charged particles, and from the effects of long-range interactions on DM haloes (see for example [33, 34]). Such interactions would deformthe DM halo profiles in elliptic galaxies, which gives a strong constraint which essentiallyrules out the re-annihilation scenario in this case. Acknowledgements
The authors thank C. Sun, R. Caldwell and C. Smith for valuable discussions. This workwas performed in part at the Aspen Center for Physics, which is supported by NationalScience Foundation grant PHY-1607611. The work of JB is supported in part by U.S.Department of Energy grant no. de-sc0007914 and in part by PITT PACC. TRIUMFreceives federal funding via a contribution agreement with the National Research Councilof Canada and the Natural Science and Engineering Research Council of Canada. DGEWis supported by a Burke faculty fellowship. SEH has been supported by the NWO Vidigrant “Self-interacting asymmetric dark matter”.
A Velocity-averaged cross sections
Here, we follow the procedure described in [31] to calculate the velocity-averaged crosssections needed to evaluate the condition in equation 2.3. We consider W ff = [ σsv ] ¯ χχ → ¯ ff ,defined for the photon channel as: [ σsv ] ¯ χχ → ¯ ff [ s, m χ , m f ] = N cf πα EM (cid:15) α (cid:48) q f s (cid:113) s − m f ( s + 2 m f )( s + 2 m χ ) , (A.1)Now, using equation (65) from [31], the velocity-averaged cross section can be written as (cid:104) σ eff v (cid:105) = (cid:82) ∞ p min,f d p p W ( p, m ) K ( √ s/T ) m T K ( x ) , (A.2)with x = m/T and p min ,f = (cid:114) max (cid:110) , m − m f (cid:111) . For x < , we define our integrationvariable as z = p/T , which gives (cid:104) σ eff v (cid:105) = T (cid:82) ∞ z min,f d z z W ( z, x ) K (2 √ z + x ) m K ( x ) . (A.3)To properly compute the Boltzmann equations, it makes more sense to define directly (cid:104) σ eff v (cid:105) T , which is equal to (cid:104) σ eff v (cid:105) T = (cid:82) ∞ z min,f d z z W ( z, x ) K (2 √ z + x ) x K ( x ) . (A.4)– 12 –n particular, when x goes to , we have (cid:104) σ eff v (cid:105) T = 14 (cid:90) ∞ d z z W ( z, K (2 z ) . (A.5)In practice, in the code we choose the upper limit of integration to be z max = z min,f + 10 (so a different one for each fermion species).Now, when x > , we use z = p/m as an integration variable and we obtain (cid:104) σ eff v (cid:105) T = (cid:82) ∞ z min,f d z z W ( z, K (2 x √ z + 1) xK ( x ) . (A.6)We use a similar approach to evaluate the velocity-averaged cross section for χχ → A (cid:48) A (cid:48) . B Details on the numerical evaluation of relic abundances
As the equations to solve numerically may be quite stiff, we evaluate them in the followingway. For each parameter point, we estimate how far the dark sector is from equilibrium.The equilibrium condition, derived from Eq. 2.12 with d Y χ / d x → d Y χ,eq ( T (cid:48) ) /dx , can bewritten as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y χ Y χ,eq ( T (cid:48) ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s (cid:104) σ χχ → ff v (cid:105) T xH Y χ,eq ( T ) − dY χ,eq ( T (cid:48) ) dx Y χ,eq ( T (cid:48) ) s (cid:104) σ χχ → A (cid:48) A (cid:48) v (cid:105) T (cid:48) xH (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . (B.1)We consider that the parameter points that verify this condition are also in the rean-nihilation regime. In order to simplify our analysis, instead of directly integrating equa-tion 2.12, we first check whether the quasi-static equilibrium (QSE) mentioned in section 2.1is reached. For this, we use a condition analogous to equation B.1, replacing Y χ,eq ( T (cid:48) ) by Y QSE ( T ) . If this QSE is reached, then the late time evolution of the dark matter den-sity will depend on its early Universe dynamics only through the value of the dark sectortemperature at the time when the A (cid:48) A (cid:48) ↔ χχ processes freeze out. Thus, the discrepancybetween the values of the DM relic density in scenarios 1 and 2 would entirely stem fromevaluating the χχ → A (cid:48) A (cid:48) cross-section at different temperatures in the two regimes. Giventhe weak sensitivity of said cross-section in the temperature at late times, this discrepancyis expected to remain extremely moderate. In our numerical scan, we therefore considerthe following three configurations:• Equilibrium within the dark sector and
QSE: here, the DM production from A (cid:48) anni-hilation will only weakly affect the value of the DM relic density, as explained above.This case can therefore be well described by the simplified scenario 2 from the in-troduction. To quantify the (small) error arising from the T (cid:48) = 0 approximation, weintegrate the Boltzmann equations 2.12 and 2.17 starting from the time of departurefrom QSE and compare the resulting relic densities Ω scenario1 and Ω scenario2 .• Equilibrium within the dark sector without
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