Black Hole Genesis of Dark Matter
Olivier Lennon, John March-Russell, Rudin Petrossian-Byrne, Hannah Tillim
PPrepared for submission to JCAP
Black Hole Genesis of Dark Matter
Olivier Lennon, John March-Russell, Rudin Petrossian-Byrne, andHannah Tillim
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, OxfordOX1 3NP, United KingdomE-mail: [email protected], [email protected],[email protected], [email protected]
Abstract.
We present a purely gravitational infra-red-calculable production mechanism fordark matter (DM). The source of both the DM relic abundance and the hot Standard Model(SM) plasma is a primordial density of micro black holes (BHs), which evaporate via Hawkingemission into both the dark and SM sectors. The mechanism has four qualitatively differentregimes depending upon whether the BH evaporation is ‘fast’ or ‘slow’ relative to the initialHubble rate, and whether the mass of the DM particle is ‘light’ or ‘heavy’ compared to theinitial BH temperature. For each of these regimes we calculate the DM yield, Y , as a functionof the initial state and DM mass and spin. In the ‘slow’ regime Y depends on only the initialBH mass over a wide range of initial conditions, including scenarios where the BHs are a smallfraction of the initial energy density. The DM is produced with a highly non-thermal energyspectrum, leading in the ‘light’ DM mass regime ( ∼
260 eV and above depending on DMspin) to a strong constraint from free-streaming, but also possible observational signaturesin structure formation in the spin 3/2 and 2 cases. The ‘heavy’ regime ( ∼ . × GeVto M Pl depending on spin) is free of these constraints and provides new possibilities for DMdetection. In all cases there is a dark radiation component predicted. a r X i v : . [ h e p - ph ] F e b ontents It is now widely accepted that ∼
85% of the matter content of the Universe is made up of anon-luminous component un-accounted for by Standard Model (SM) physics [1]. While thenature and origin of this dark matter (DM) remains obscure, some properties are known – aDM candidate must be cosmologically long-lived, cold (or possibly warm), with a dominantcomponent(s) having mass(es) in the range: 10 − eV (cid:46) µ (cid:46) eV. The self- and DM-SM-interaction strengths are bounded above and consistent with being purely gravitational.The most studied production mechanism for a relic abundance of particle-DM, thermalfreeze-out [2, 3], assumes that the DM in the early Universe was in equilibrium with thehot SM plasma due to much-stronger-than-gravitational DM-SM interactions, but later fellout of equilibrium when annihilation rates failed to keep up with Hubble expansion. Thismechanism depends on, at minimum, the mass and couplings of the DM, and the numberof degrees of freedom. For suitably chosen values, often taken at the weak scale, freeze-outreproduces the relic abundance of the DM. This mechanism, however, is now under strainfrom a combination of DM direct- and indirect-detection searches, and LHC bounds on newphysics. Thus it is of great importance to investigate other possible calculable productionmechanisms that do not rely on close-to-weak-scale interactions with the SM, such as freeze-inproduction of feebly-interacting DM from an initial state with negligible DM density [4].Here we present a calculable, infra-red mechanism for DM production utilising only themandatory coupling of DM to gravity . Specifically, we show that both the DM abundanceand the hot-Big-Bang SM plasma can result from an initial density of primordial black holes(pBHs).It has long been thought that processes in the early Universe could have formed pBHs [5].By Hawking’s famous result [6], these pBHs emit all states that couple to gravity and, if theyare sufficiently small, can completely evaporate in the early Universe, contributing to theprimordial radiation density. As a subcomponent of this Hawking radiation will necessarilybe the DM particles, pBHs can source some, or all, of the DM relic abundance. For earlierwork in this direction, see [7–10].We will argue that a simple set of assumptions results in un-elaborate (and often largelyinitial-state independent) expressions for the final DM yield, Y ≡ n DM /s tot , and thus forΩ DM h . (Here s tot is the total SM entropy density of the Universe.) Our starting hypothesesare: 1) There exists a population of micro pBHs with mean initial number density, n , withoverlaid fluctuations. We take the pBH masses to be narrowly peaked around an initialmass, M (the generalisation of our results to somewhat broader initial mass distributionsis straightforward). Here we do not seek to justify or provide a detailed mechanism for the– 1 –xistence of this population. 2) On large scales the initial energy density of BHs, ρ BH = M n ,inherits the approximately scale-invariant spectrum of density fluctuations, δρ/ρ (cid:39) − , seenby cosmic microwave background and large-scale structure observations. On tiny scales therecan be significant departures from this fluctuation spectrum, e.g., due to Poisson variationsin the number of BHs per initial Hubble patch. 3) Any remaining initial energy densitynot in the form of pBHs is in the form of a SM radiation component. Similarly to freeze-inproduction, we assume an initially negligible number density of DM particles. 4) The DMis in the form of cosmologically long-lived massive particles with sub-Planckian mass andspin ≤ DM h can be parametrically large). 5) Stable Planckian-massBH relics do not exist. From these assumptions a successful mechanism of DM productionfollows, with noteworthy observational signatures in some cases.In detail, in Section II we calculate the DM yield and SM ‘reheat’ temperature, T RH ,at the end of pBH evaporation . We present analytic expressions in the four qualitativelydifferent regimes: ‘fast’ or ‘slow’ decay of the pBHs (with the general case being solved numer-ically) in both ‘light’ and ‘heavy’ DM scenarios. The observational constraints are applied inSection III, allowing us to find the acceptable parameter ranges of our mechanism. In SectionIV, we consider variations of our basic mechanism with additional possible signatures anddiscuss future directions.Finally, during manuscript preparation Ref. [12] appeared. This work, although havingsome overlap, differs considerably in both detail and general setup from ours. We first review the Hawking evaporation of a single, Schwarzschild BH, motivated by the factthat BHs with charge and angular momentum very quickly radiate these away (e.g., [13]).The instantaneous temperature of a BH of mass M is given by T BH = M / (8 πM ) (we workin units where (cid:126) = c = k B = 1) [6]. As the BH evaporates, T BH scans all temperaturesfrom an initial T = M / (8 πM ) up to T ∼ M Pl where Hawking’s calculation breaks down.In what follows we always assume that T > v EW = 246GeV, corresponding to an initialmass of M < M Pl . In this limit, all states in the SM are created ultra-relativisticallythroughout the BH’s lifetime and so we need not consider mass thresholds or the physics ofcrossing through the weak or QCD scales. Thus, during the entire evaporation process, theSM states can be well approximated as massless. (Our results generalise naturally to thecase where T is less than some SM masses.) Taking, then, effectively massless final states,the rate of BH mass-loss is [14]d M d t = − M M (cid:88) s,i e s,i g s,i ≡ − e T M M , (2.1)where i labels a state of spin s , g s,i is the associated number of degrees of freedom (dof), e s,i are the dimensionless grey-body ‘power-factors’, given in Table 1, and e T is the total emissioncoefficient, which does not depend upon the mass of the BH. The grey-body factors appeardue to a spin-dependent (and mass-dependent for particles of mass m (cid:54)(cid:28) T ) potential barrier This is similar to a scenario discussed in [11], wherein the early Universe is reheated via Hawking emissionof BHs post-inflation. – 2 –utside of the horizon, which causes back-scattering into the BH. These factors significantlyalter the emission rates and must be taken into account for a correct evaluation of the DMyield. For Hawking emission dominated by radiation into effectively massless SM states plusgravitons, e T ≈ e T , SM (cid:39) . × − . (This is a good approximation up to small correctionsdue the existence of the DM sector with small total dof g DM (cid:28) g SM (cid:39) .) Thus the BHmass evolves as M ( t ) = M − e T M ( t − t ), where M is the BH mass at initial time, t .We consider two broad cases with regards to the production of DM particles by Hawkingradiation: the first, where the initial temperature of the BH is greater than all mass scales–the‘light’ DM case–and the second, where it is below the DM mass, µ . In the latter ‘heavy’ case,creation of DM particles effectively only takes place after the BH has decayed sufficientlythat T > ∼ µ .Concretely, the emission rate of a species i of spin s , and mass µ , and with energy in arange ( ω, ω + d ω ) is [14]d (cid:18) d N s,i d t (cid:19) = (cid:88) (cid:96),h (2 (cid:96) + 1)2 π Γ i,s,(cid:96),h ( ω )exp( ω/T ( t )) + ( − (2 s +1) d ω (2.2)where (cid:96) is the spherical harmonic, h the helicity or polarisation of the emitted particle,and Γ i,s,(cid:96),h ( ω ) is the absorption probability for that mode, which encodes the grey-bodyfactor. The total rate of emission, integrated over all final state energies ω = (cid:112) p + µ , with p ∈ [0 , ∞ ], is, for our purposes, well approximated byd N s,i d t ≈ M M f s,i g s,i Θ (cid:18) d s M πM − µ (cid:19) (2.3)where the coefficients f s,i are dimensionless greybody ‘rate-factors’, with values given inTable 1, Θ is the Heaviside step-function, and d s is a spin-dependent dimensionless coefficientwe have found from numerical and semi-analytic solutions of the exact Hawking emissionequations. To a sufficient approximation d bosons (cid:39) .
2, while d fermions (cid:39) . s = 0 , , s = 1 / , / N s,i (cid:39) f s,i g s,i e T (cid:18) M M Pl (cid:19) (cid:26) T > µ/d s ) d s T /µ ( T < µ/d s ) . (2.4) Light Dark Matter:
Consider, first, the case where the DM mass is less than theinitial effective BH temperature µ < d s T . The evolution equations of the energy densitiesof the pBHs, ρ BH = M ( t ) n ( t ), and the SM radiation, ρ rad , in the early Universe are:d ρ BH d t + 3 Hρ BH = − e T M M n ; (2.5)d ρ rad d t + 4 Hρ rad = + e T M M n . (2.6)We solve these equations together with Eq. (2.1) and the Friedmann equation, H ( t ) =8 π ( ρ BH + ρ rad ) / M , subject to M > M Pl and ρ BH (0) = M n < M . We also assumethat the process begins in a pBH-matter-dominated universe with ρ rad (0) ≈ e s f s . × − . × − / . × − . × − . × − . × − / . × − . × − . × − . × − Table 1 . The numerical power, e s , and rate, f s , greybody factors per degree of freedom for neutralparticles of given spin s , extracted from [14–16]. These have been summed over the dominant angularmomentum emission modes [17]. For spin 3 / e s and f s if particles are charged. we discuss the (in)sensitivity of our results to the relaxation of this assumption. It shouldbe noted that a number of studies into formation process for pBHs give the number of BHsper initial Hubble patch, 4 πn / H ∼ τ dec ( M ( t )), and the Hubble time, t H ( t ). We define B ( t ) ≡ t H /τ dec , with initial value( B ) = e π M n (cid:18) M Pl M (cid:19) . (2.7)When B (cid:29) B (cid:28) ρ rad thermalises quickly, while, as statedin the introduction (assumption 4), the DM does not interact significantly with either itselfor the SM, so that no other processes affect the DM yield. The two important calculablequantities from these analytic cases are the temperature of the SM radiation bath at the endpBH decay – the ‘reheat’ temperature, T RH – and the yield of the DM. We will minimallyimpose that T RH > µY = 0 .
43 eV, ensuring, respectively, that BBNis standard and Ω DM h = 0 .
11 today, as observed.From our analytic solution we find, in the case of slow initial decay, B (cid:28) T slowRH (cid:39) . e / g / ∗ M Pl (cid:18) M Pl M (cid:19) / , (2.8)where g ∗ is the effective number of SM dof at T RH . From the ensuing total entropy density, s tot (well approximated by that in the SM radiation bath in our limit g DM (cid:28) g SM ), and usingEq. (2.4), the yield of species i with spin s at the end of the decay process is then given by Y slow ≡ n s,i s tot (cid:39) . f s,i g s,i g / ∗ e / (cid:18) M Pl M (cid:19) / . (2.9)Notice that both of these expressions are independent of the initial pBH number density.– 4 – igure 1 . Schematic behaviour of Ω DM h as a function of DM mass µ , here given in arbitrary units,with initial pBH properties M and n held fixed. The orange dashed line represents the observedvalue of Ω DM h . The green dashed lines denote the ‘light’ ( µ l ) and ‘heavy’ ( µ h ) DM masses that areconsistent with the observed relic abundance. In the fast case, B (cid:29)
1, the pBHs effectively dump all of their energy into radiationinstantaneously without significant Hubble red-shifting. Thus, T fastRH (cid:39) (cid:18) n M π g ∗ (cid:19) / , (2.10)with corresponding particle yield Y fast (cid:39) . f s,i g s,i g / ∗ e T (cid:18) n M (cid:19) / (cid:18) M M Pl (cid:19) / . (2.11) Heavy Dark Matter:
We now consider the case µ > d s T . The reheat temperatureremains, in both slow and fast cases, of the form given in Eqs. (2.8) and (2.10), up to smallcorrections of O ( g DM /g SM ). From Eq. (2.4), we can then write the yield in this scenario as Y h = Y l T µ h d s , (2.12)where the subscripts h and l refer, respectively, to the heavy and light cases. In Fig. 1, we seethat the observable quantity, µY (equivalently Ω DM h ), exhibits two contrasting behavioursas a function of the DM mass µ , namely linearly rising for small µ , while decreasing as 1 /µ forlarge values. Thus, for any value of µY lower than the maximum (we find that µY = 0 .
43 eValways satisfies this), two solutions for µ exist – one in each regime. The two solutions satisfy µ h µ l = T d s . Numerics:
We have numerically solved the full set of coupled equations (Eqs. (2.1), (2.5), (2.6),and the Friedmann equation). Our results, shown in Fig. 2, confirm our limiting approxima-tions are excellent over the majority of parameter space. The line B = 1 is also confirmedto be a good discriminant, marking a quick but smooth transition between the two regimes.The plots also make obvious the different behaviour of µ l and µ h over the parameter space,i.e. whether they increase or decrease towards the origin in the upper left corner where M is small and n large. – 5 – igure 2 . Contours of DM mass µ giving observed Ω DM h , for the ‘light’ and ‘heavy’ DM cases, asa function of log ( M /M Pl ) and log ( n /M ) across the a priori allowed parameter space beforeconstraints are imposed . The red line is B = 1. Figure 3 . Constraints in the log ( M /M Pl )–log ( n /M ) plane for ‘light’ and ‘heavy’ DM cases(left and right panels), for a single real s = 0 dof (other spins display precisely the same features,differing only in numerical values) before the additional, strong constraint coming from free-streamingis imposed. (In the case of ‘light’ DM this strongly DM-spin-dependent constraint excludes DM spin s ≤
1, marginally excludes s = 3 /
2, and allows a region of the spin s = 2 case to survive. There areno significant constraints on the ‘heavy’ DM case from free-streaming. See text for more details.) Thered line ( B = 1) marks the boundary between the ‘fast’ and ‘slow’ regimes; the blue line correspondsto one pBH per initial Hubble patch with increasing pBH number density below. The blue shadedregions are disallowed as T RH < T RH = 200 GeV. The upperpurple, and in the ‘heavy’ case khaki, shaded regions are excluded as ρ BH (0) > M and µ > M Pl ,respectively. – 6 – pin g s µ/ GeV (slow, light) µ/ GeV (slow, heavy) µ/ GeV (fast, light) µ/ GeV (fast, heavy)0 1 (cid:2) . × − , . (cid:3) (cid:2) . × , M Pl (cid:3) (cid:2) . × − , . × (cid:3) (cid:2) . × , M Pl (cid:3) / (cid:2) . × − , . (cid:3) (cid:2) . × , M Pl (cid:3) (cid:2) . × − , . × (cid:3) (cid:2) . × , M Pl (cid:3) (cid:2) . × − , . (cid:3) (cid:2) . × , M Pl (cid:3) (cid:2) . × − , . × (cid:3) (cid:2) . × , M Pl (cid:3) / (cid:2) × − , (cid:3) (cid:2) × , M Pl (cid:3) (cid:2) × − , × (cid:3) (cid:2) × , M Pl (cid:3) (cid:2) . × − , (cid:3) (cid:2) . × , M Pl (cid:3) (cid:2) . × − , . × (cid:3) (cid:2) . × , M Pl (cid:3) Table 2 . Limits on DM mass giving the correct relic abundance in the four regimes of ‘light’ and‘heavy’ DM mass and ‘slow’ and ‘fast’ BH decay, which satisfy our observational and theoreticalconstraints (except free-streaming in ‘light’ case – see text). A single DM particle of given spin isassumed. In particular, given the large masses allowed above, this mechanism provides a new way forproducing so-called ‘WIMPzillas’ [22] in the early Universe.
In order for our model to follow standard cosmological results from BBN onwards, we min-imally require that T RH > M > M Pl and ρ BH (0) = M n < M , is shown in Fig. 3 in the ‘light’ DM mass (left) and ‘heavy’ DM mass(right) cases. We show only constraint plots for a single real s = 0 dof as other spins displayprecisely the same features, differing only in numerical values. These, together with the DMmass contour plots given in Fig. 2, lead to the maximally allowed ranges for the DM mass inTable 2 in each of the four regimes, and for DM spins 0 ≤ s ≤ before imposition of the additional, strong constraint coming from free-streaming).In the slow regime, T RH and Y (Eqs. (2.8) and (2.9)) are independent of n , and iso-contours of T RH , µ giving the observed Ω DM h and M will all be parallel. Thus, constrainingthe reheat temperature and the initial BH mass will provide constraints on the DM mass.Since, in the light (respectively, heavy) case, the DM mass contours increase (decrease) withincreasing M , we find that T RH (cid:46) µ , while M > M Pl sets a lower (upper) bound. However, in the heavy case, we must exclude the region where µ > M Pl . This then leads to an upper bound on the reheat temperature in the heavy case ranging from T RH (cid:46) . s = 0 up to T RH (cid:46) . s = 2.In the fast regime, the mass contours are of differing gradients for the light and heavycases, setting different locations for the bounds on the DM mass. For the light (heavy) case,the crossover point from the fast to the slow regime at low M and high n provides a lower(upper) bound on the DM mass. As this point is a crossover between regimes, by continuity,we must find that the bounds are parametrically similar. This can indeed be seen in Table 2.Excluding super-Planckian DM masses we again obtain a maximum T RH in the heavy case,which, by continuity, is the same for the slow regime. From the slope of the mass contours,we see that we set an upper bound in the light case for low M and n , deep in the fastregion. Similarly, in the heavy case, we set a bound at the crossover for low n and high M .If electroweak baryogenesis is to be accommodated in our model, then we further requirethat T RH > ∼
200 GeV. We include a contour for this reheat temperature in Fig. 3. This contouris particularly constraining in the heavy case, yielding, for example, a revised range for theDM mass of 5 . × GeV (cid:46) µ (cid:46) M Pl for a real scalar. Free-Streaming:
The ‘light’ DM mass case is significantly constrained by the freestreaming of the ultra-relativistically-emitted DM particles: we now calculate the DM mo-mentum distribution function F ( p, t ) after the end of pBH decay. Here, p denotes themagnitude of the 3-momentum. The number of particles emitted by a BH per dof, perunit time, with energy in the range ( ω, ω + d ω ) is given by Eq. (2.2), where the grey-– 7 – igure 4 . DM momentum distribution function resulting at end of pBH decay. The green curveshows the distribution resulting from ‘fast’ decay which exhibits a significant high energy tail comparedto a thermal spectrum. The blue curve is an example of the distortion arising from a ‘slow’ decay(same T but larger n ) when the interim expansion of the universe is significant. body factor, Γ i,s,(cid:96),h ( ω ), is related to the absorption cross-section of the BH by σ i,s,h ( ω ) = π (cid:80) l (2 l + 1)Γ i,s,(cid:96),h ( ω ) /ω . As the form of F ( p, t ) is dominantly determined by the ever-increasing BH temperature, T ( t ), and the subsequent red-shifting of the DM momentum,and not the grey-body factors to a first approximation, we may use σ i,s,h ( ω ) = 27 πM /M -the geometrical optics limit. Notice that, since this ignores small spin-dependent low- ω sup-pressions of the spectrum [14] the result will be a slight underestimate of the total portionof relativistic (high ω ) particles. The instantaneous distribution of emitted momenta is thusd ˙ N d p ( p, t ) = 27 M ( t ) πM p e p/T ( t ) ± . (3.1)Where we are justified in taking the ultrarelativistic limit µ → F ( p, t ) we are left with at some time t after the BH has decayed away ( t end )is a superposition of all the above instantaneous distributions, each red-shifted appropriatelyfrom its time of emission F ( p, t ) = (cid:90) t end t d τ d ˙ N d p (cid:18) p R ( t ) R ( τ ) , T ( τ ) (cid:19) R ( t ) R ( τ ) . (3.2)Here we are interested in finding this expression at the end of the BH decay, F ( p, t end ).In the ‘fast’ regime, where the scale factor is approximately constant, this takes the formof M T − ˜ F ( p/T ), with ˜ F dimensionless, and can be evaluated analytically in terms ofpoly-logarithms. We have numerically integrated the general case. In Fig. 4 we show theexact solution for the bosonic ‘fast’ case and a typical effect on the distribution of a ‘slower’decay. We find that at low energies the non red-shifted distribution can be fitted by aneffective temperature T eff ≈ . T (which is higher than that of the SM thermal bath) andan additional pseudo-exponential high-energy tail. Note that T eff does not characterise thetotal energy density of the DM, or its number density, which is very far below that of theSM particles. A ‘slower’ decay distorts the lower energy spectrum towards zero as particlesproduced early are Hubble red-shifted during the BH lifetime. The tail is unaffected by this– 8 –s its major contributors are produced in the final explosive moments of the BH life, a processthat is intrinsically ‘fast’.From the BH decay time onwards these distributions are simply redshifted by Hubbleexpansion as p ∼ /R . From this we deduce that, as the universe cools to some temperature T SM , the number of particles that are still relativistic is (cid:90) ∞ p min d p F ( p, t end ) , p min = µ T RH T SM (cid:18) g ∗ S ( T RH ) g ∗ S ( T SM ) (cid:19) / . (3.3)Explicitly, in our two separate regimes (and taking T SM = 1 keV, see below) we have p min ≈ (cid:18) g ∗ S ( T RH )3 . (cid:19) / e T f s g s T (cid:26) . . , (3.4)where the spin dependence is made manifest. By inspecting Table 1 and Fig. 4 it is clearthat the lower the spin, the higher the proportion of relativistic particles.It is possible to numerically evaluate the above integral expression and, as a rough-and-ready criterion for successful structure formation we impose that when T SM = 1 keV(at which stage the horizon mass is ∼ M (cid:12) ) less than 10% of the DM is relativistic (see,e.g. [23] for a relevant discussion of free streaming constraints). We find that for the ‘light’DM case almost the entire parameter space is excluded. In particular we find that for the‘light’ DM mass solution the distribution of DM momenta at T SM = 1 keV is almost entirelyrelativistic for spins s ≤
1, and so this regime is definitively excluded for these spins (atleast for the basic DM genesis mechanism we discuss in this Section). For s = 3 / M - n plane we are situated, and so this case is marginal at very best giventhe criterion for successful structure formation that we have employed . In the case of‘light’ s = 2 DM we find that there are substantial regions where 10% or less of the DM isrelativistic at T SM = 1 keV, so this case naively survives and provides possibly observableand even useful modifications to the galaxy structure power spectrum. The fraction of DMparticles that are still relativistic at T SM = 1keV in these two cases, s = 3 / , / T SM = 1 keV, and thus we have in this case a successful newmechanism of DM production in both the ‘slow’ and ‘fast’ regimes of BH decay. Dark Radiation:
The change, compared to the SM, in the effective number of rela-tivistic dof contributing to the energy density at the surface of last scattering is bounded fromPLANCK2015 data [1] by ∆ N eff < . ± . N eff , SM (cid:39) . e (cid:39) . × − (cid:28) e T , SM (cid:39) . × − (for related earlier work see [24]). Specifically at the end of pBH decay ∆ ρ grav /ρ rad =2 e /e T , SM (cid:39) . × − , and this translates to ∆ N eff by using the appropriate changes in For earlier work on the spin 3/2 case, see [9]. – 9 – igure 5 . Free-streaming constraints in the ‘light’ DM case for spin 3/2 and 2 (left and right panels),where colour shading shows fraction of DM particles that are still relativistic at T SM = 1 keV, and wehave at every point imposed a ‘light’ solution DM mass such that the correct Ω DM h is reproduced.Note the differing colour scales in the two cases with the spin 3/2 case having more than ∼
40% ofparticles relativistic over the entire plane, while the spin 2 case has substantial regions where less than ∼
10% of DM particles are relativistic. Red line ( B = 1) marks the boundary between the ‘fast’ and‘slow’ regimes. the number of effective SM relativistic dof from T RH down. In our case we find that theadditional ∆ N eff due to gravitons depends upon the reheat temperature: for T RH ∼ N eff , grav (cid:39) . × − while for T RH ∼ N eff , grav (cid:39) . × − . As ∆ N eff , grav further decreases for increasing T RH , this does not place a bound on our mechanism. Never-theless, the larger figure may provide an accessible target for planned future high-precisiondeterminations of N eff . In Section IV we consider motivated variations of our basic modelwith additional dark radiation contributions. Density Perturbations:
Limits on the spectrum and form of density perturbationsprovide another potential source of constraint. We have here taken as a basic hypothesisthat on moderate-to-large scales (but not on small scales as we soon quantify) the initial BHnumber density, and thus the initial energy density, ρ BH , inherits the spectrum of Gaussian,almost scale-invariant density perturbations that are the result of, say, an early epoch ofinflation. (We have nothing new here to say about the origin of these perturbations. Note thatwe do not necessarily require inflation: any mechanism or dynamics in the very early Universethat gives the approximately scale-invariant spectrum of observed density perturbations issuitable as long as there is a population of micro pBHs produced/present at the end satisfyingthe conditions set out in the introduction, and discussed further in Section IV.) One significantpoint is that because the evaporation of our pBHs sources all forms of matter (both SM anddark), the perturbations at the end of pBH decay are naturally adiabatic and not isocurvature.Nevertheless, many production mechanisms for pBHs, for instance their production inultra-relativistic bubble collisions following, say, a quantum phase transition (e.g., of the typediscussed in [25]), lead to substantial random variation in the number of pBHs produced perinitial Hubble volume, and thus to a significant extra source of density perturbations on small scales of order H − . To quantify the potential later effect of this on structure formation,consider BHs that are initially distributed at t with density variations between initial Hubblevolumes V H . This implies at later times δρ/ρ ( t ) ≈ / √ N = 1 / (cid:112) K ( t ) n V H ( t ), where K is– 10 –he number of V H patches within the horizon at t and N is therefore the total number of BHswithin that horizon. If we assume a Poisson distribution of BHs, this approximation is veryaccurate at later times when K is very large. Assuming matter domination during BH decay,and radiation domination thereafter, we require that δρ/ρ at matter-radiation equality be atmost of order 10 − . We find that this constraint is very weak and the the parameter regionsthus excluded lie well within the section of the plane already forbidden by the requirementthat T RH ≥ Finally, we make some comments, and briefly discuss some motivated variations of our basicmechanism, as well as directions for future work.
Sensitivity to Initial Conditions:
The reader might be concerned that we have sofar assumed that the initial condition is one of pure pBH matter domination. However, wehave found that our DM yield prediction is insensitive to changes in this assumption overa wide range of parameter space. In the fast BH decay regime, provided any initial SMradiation component satisfies ρ rad , SM (0) < ∼ . ρ BH (0) the predicted SM reheat temperatureis unaffected up to O (10%) corrections or less. Regarding the DM yield, any primordial DMnumber density ∆ n DM (0) is unimportant if it is substantially less than that produced by thepBH decay. Given the expression for the total number of DM particles emitted, Eq. (2.4),this translates into the requirement∆ n DM (0) n (cid:28) × − (cid:18) M n (cid:19) / , (4.1)where for concreteness we have inserted typical values for scalar or spin-1/2 fermion DM inthe ‘light’ case, and used the condition B ∼ n for a given value of M . Since overmuch of the allowed parameter space n /M can be as small as 10 − Eq.(4.1) is not aparticularly restrictive requirement.The situation in the slow BH decay regime is even better, as any initial SM radiationcomponent is now substantially red-shifted away during the long period of BH decay. Specif-ically, we find from our analytic slow-regime solution that, at end of pBH decay, any initialSM radiation has been diluted by a factor∆ ρ rad , SM ( t end )∆ ρ rad , SM ( t ) (cid:39) (cid:18) B . (cid:19) / (cid:28) B < B (cid:28)
1. This is to be compared to the SM radiationcomponent produced by pBH emission, which is ρ rad , SM ( t end ) (cid:39) . B ρ BH ( t ). Thus aslong as ρ rad , SM ( t ) < ∼ . B − / ρ BH ( t ) (4.3)both the SM reheat temperature and the DM yield are substantially unaffected, up to O (10%)corrections or less, by the presence of the early SM radiation component. Since, consistentwith constraints, we find that in the slow regime B − / can be as large as B − / ∼ . M Pl /T RH ) / (cid:29) , (4.4)– 11 – dominant initial SM radiation component, as big as ∼ times larger than the initial pBHenergy density can be present without affecting our results . (This ratio is for T RH = 3 MeV.For T RH = 1 TeV, a ratio as large as ∼ is still allowed.) It is clear that the slow region isparticularly insensitive to initial conditions, and thus attractive in this regard. In additionthe region of parameter space where there is of order one BH per initial horizon patch, andso possibly favoured by BH production mechanisms, is contained within the slow regime. Other DM Interactions:
Although our mechanism requires that the traditionalfreeze-out and recent freeze-in mechanisms of DM production give sub-dominant effects,this doesn’t rule out some small self- or DM-SM-interactions with possibly important ob-servational consequences. We thus now briefly turn to the possibilities for detection of DMthrough non-gravitational signatures.One natural possibility is the existence of operators which allow the DM to slowlydecay to the SM sector. (See, e.g., [26–33] for a selection of earlier work on the topicof decaying heavy DM.) This is well motivated if the stabilising symmetry of the DM isglobal, as, by general arguments, all theories incorporating gravity are widely believed topossess no exact global symmetries. (It is not known if the strength of this violation issuppressed by some large mass scale, such as the Planck mass, or is non-perturbatively smallin some dimensionless coupling.) Since our DM, especially in the ‘heavy’ case, is surprisinglymassive, even exponentionally tiny symmetry-violating effects can lead to cosmologicallyrelevant decay times. As a rule of thumb, lifetimes of order 10 − s are on the allowedborder for decaying DM, depending on exact details on the decay. Since for our mechanismthe DM is often very massive, the energy of the SM decay products can be very large,thus leading to ultra-high-energy cosmic rays. If we parameterise the effective dimensionlesscoupling constant that violates the symmetry as λ ∼ exp( − π /g ), as might be suitable fora non-perturbative violation, then for a DM mass of order the GUT scale, ∼ GeV, the10 s lifetime constraint translates into g < .
8, a not unreasonable number for a fundamentaldimensionless coupling. Of course, such a non-perturbatively determined lifetime is verysensitive to the underlying and unknown coupling g so an observationally interesting lifetimeis in no way predicted. An alternative, and less sensitive, possibility exists for a DM mass inthe region of 100 TeV, for then dimension-six Planck-suppressed operators lead to a lifetimeof order 10 s with only mild polynomial dependence on the mass.The DM can also have self-interactions as long as they don’t lead to a freeze-out pro-cess. For example, in the case of a single real massive scalar a quartic interaction will notallow physical, kinematically allowed number-changing interactions, but will lead to elasticscattering which can modify the DM momentum distribution.For our heavier mass regions the DM can also have substantial interactions with SMmatter. This is because for DM mass µ (cid:29) T RH the freeze-in process is exponentially sup-pressed [4] in addition to being down by small couplings, while any potential freeze-outreprocessing via, say, self-annihilations of our DM into SM states is suppressed by the ex-tremely tiny number density of DM that applies in the case that µ (cid:29) M Pl , and, moreover, is operative for states of spin s ≥ /
2, motivates the intriguing possibility that either GUT, or even higher-spin string Reggeexcitations, if they are sufficiently stable (say due to a combination of J PC conservation,– 12 –inematics, and maybe discrete gauge symmetry quantum numbers), might be the DM! Additional Dark Radiation:
If there are new very light dof in addition to the gravitonand SM neutrinos then these will also be produced during the decay of the pBHs and willbe an additional dark radiation component beyond that of the gravitons. A particularlywell-motivated almost massless dof is the axion, whether the QCD axion [34–36] or multipleaxion-like particles as in the string axiverse scenario [37, 38]. If the number of light axions, N a , is much less than the number of dof at T RH , then the effective change in the numberof dof is: for T RH ∼ N eff = 0 . N a ; for T RH ∼ N eff = 4 . × − N a .For N a ≤
3, both of these are consistent with Planck measurements – however, for thelower reheat temperature, this could be discovered or ruled out by future experiments (see,e.g., [39, 40]).
Opening the ‘Light’ DM Window:
Another variation of our basic mechanism is toallow the number of dof in the SM sector to be much increased beyond the 106.75 of the SM –for instance, due to TeV-scale (R-parity violating) supersymmetry or strong dynamics. Thissituation can possibly resurrect some of the low-spin, s ≤ e T .Thus, all other things kept fixed, the DM must be heavier to satisfy the Ω DM h observationand so becomes non-relativistic earlier. Early Matter Domination & BH Mergers:
The minimal
DM genesis mechanismthat we have presented so far has an early period of matter domination due to the pBHs.(As explained above during the discussion of insensitivity to initial conditions, it is possiblethat a much dominant SM radiation component is initially present, thus making the periodof matter domination by pBHs very short. Note, the period of matter domination in the fastregime is effectively automatically non-existent, due to the decay of the BHs into radiationwithin a Hubble time.) It is therefore possible that density perturbations re-entering thehorizon could gravitationally collapse into bound structures comprised of pBHs. The pBHscould then coalesce into larger BHs before they evaporate and so survive either down tothe present epoch, or, more dangerously, decay post-BBN and lead to observable changes inBBN or CMBR predictions, or to constrained gamma ray or ultra-high-energy cosmic raybackgrounds. In principle the general characteristics of this competition between decay andcoalescence is determined by the ratio of the Hawking evaporation time τ decay = M / M e T to the coalescence time τ coalesce (cid:39) − × M a /M predicted by gravity wave emission (here a is the BH binary separation, and we take quasi-circular orbits). Thus evaporation occursbefore coalescence if M < ∼ . M Pl a ) / M Pl .To evaluate the evaporation vs coalescence condition we need to know the distributionfunction of orbital radii a . It is natural to assume that typical values of a are set by the averagedistance ¯ d between pBHs in our scenario, which is growing due to Hubble expansion untilthe binary forms and the BHs separate from the Hubble flow. The maximally restrictiveassumption is that some BH binaries form early when ¯ d ∼ (3 / πn ) / , leading to thecondition M /M Pl < ∼ . n /M ) − / . This cuts off the upper right corner of the slowregion, and interestingly also requires more than one pBH per initial Hubble volume, butstill allows a significant part of the slow regime parameter space to survive, as well as all ofthe fast BH decay regime. However, we have ignored some complicating issues. First, rareclose binaries may form with much shorter coalescence times. Second, binaries with orbits of– 13 –ery high eccentricity, e ∼
1, also have much shorter coalescence times τ coalesce ∝ (1 − e ) / .Thus our discussion here is highly provisional, though it seems clear that much, and likely all,of the fast regime parameter space, and possibly much of the slow regime, survive unaffectedby BH mergers. We comment, though, that such mergers would, if they are frequent enough,substantially increase the gravitational dark radiation component beyond that calculatedin Section III. Moreover the result of such a merger is a spinning Kerr BH, and for suchBHs the Hawking emission of higher-spin particle states is enhanced, further increasing thegravitational dark radiation (and also possibly the DM yield in the case s > T RH ∼ ∼ M (cid:12) which, though (cid:28) M dwarf , is a possibly intriguing number asregards the seeding of supermassive galactic BHs. Summary:
We intend to return in future work to the physics of BH mergers in our sce-nario, as well as the other aforementioned topics, including a more precise characterisation ofthe free-streaming bounds. It is our belief, however, that our calculations and discussion herepresented have made a convincing case that the mechanism of DM production by Hawkingevaporation is both viable and behaviourally rich, and highly worthy of further exploration.
Acknowledgments
We gratefully thank Isabel Garcia Garcia and Kazunori Kohri for useful and stimulatingdiscussions. OL and HT are supported by the Science and Technology Facilities Council(STFC). RPB is supported by a Clarendon Scholarship from the University of Oxford.
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