aa r X i v : . [ h e p - t h ] J a n UG-FT-257/09CAFPE-127/09
Black hole gas in the early universe
M´onica Borunda, Manuel Masip CAFPE and Departamento de F´ısica Te´orica y del CosmosUniversidad de Granada, E-18071 Granada, Spain [email protected], [email protected]
Abstract
We consider the early universe at temperatures close to the fundamental scale ofgravity ( M D ≪ M P lanck ) in models with extra dimensions. At such temperatures asmall fraction of particles will experience transplanckian collisions that may result inmicroscopic black holes (BHs). BHs colder than the environment will gain mass, andas they grow their temperature drops further. We study the dynamics of a system (a black hole gas ) defined by radiation at a given temperature coupled to a distribution ofBHs of different mass. Our analysis includes the production of BHs in photon-photoncollisions, BH evaporation, the absorption of radiation, collisions of two BHs to give alarger one, and the effects of the expansion. We show that the system may follow twodifferent generic paths depending on the initial temperature of the plasma.
Introduction
Large (ADD) [1] or warped (RS) [2] extra dimensions open the possibility that the funda-mental scale of gravity M D is much lower than M P ≈ GeV. This could imply that the transplanckian regime [3] is at accesible energies. Collisions in that regime are very differentfrom what we have experienced so far in particle colliders: due to its spin 2 the gravitonbecomes strongly coupled and dominates at distances that increase with the center of massenergy √ s . In particular, at small impact parameters one expects that gravity bounds thesystem and the two particles collapse into a microscopic black hole (BH) of mass around √ s . Such BH would evaporate [4] very fast (the time scale is set by 1 /M D ) into final statesof high multiplicity, a possible LHC signature that has been extensively discussed in theliterature [5].These collisions, however, may have also occurred in the early universe if the temperaturewas ever close to M D . Consider a period of inflation produced by a field on the brane,followed by reheating into 4–dimensional (4-dim) species. If the reheating temperature is T rh ≥ . M D , particles in the tail of the Boltzmann distribution may collide with enoughenergy to form BHs. Hot BHs will evaporate, but if a mini-BH is colder than the environmentit absorbs more than it emits [6], growing and becoming colder, which in turn increases itsabsorption rate (see below). The fact that heavier BHs live longer distinguishes them frommassive particles or string excitations possibly produced at these temperatures, since thelifetime of the latter is inversely proportional to their mass. As a consequence, one expectsthat BHs are a critical ingredient at temperatures near the fundamental scale. Notice thatthese BHs are not the primordial ones formed from the gravitational collapse of densityfluctuations [7, 8, 9, 10] at lower temperature T .Although BHs would also be present in a 4-dim universe at T ∼ M P , they are mostrelevant in TeV gravity models. The reason is that the expansion of the universe is a long-distance process, so its rate is dictated by M P (not M D ) also in models of TeV gravity. If the bulk is basically empty [11, 12, 13] or thin (see below), then the expansion at T ≤ M D willbe negligible in terms of the fundamental time scale 1 /M D , and there is plenty of time forcollisions producing BHs to occur and for the BHs to grow. In contrast, in a 4dim universethe expansion rate at T ≈ M P is of order H − ≈ /M P , the temperature of the universedrops before BHs have grown, and once it goes below T BH they evaporate in the same timescale.In this article we explore the implications of having initial temperatures near the funda-mental scale of gravity. First we define a consistent set up for TeV gravity. Then we studythe behaviour of a single BH inside a thermal bath in an expanding universe. Finally we2onsider the generic case, a black hole gas , with radiation coupled to a distribution of BHsof different mass. We find remarkable that the effect of these mini-BHs in the early universehas been almost completely overlooked in the literature (the only analysis that we havefound is given by Conley and Wizansky in [14]), although many authors have consideredtemperatures close (and above) the Planck scale (see [15] and references therein). There are two generic frameworks that may imply unsuppressed gravity at M D ≈ M D < M P ). In the first one (ADD), n compact dimensions of length L < m c ≡ L − , couple very weakly ( ≈ √ s/M P ) to ordinary matter.However, the large number of effective gravitons ≈ ( √ s/m c ) n involved in a collision gives anamplitude of order one, sM P × √ sm c ! n ≈ , (1)at √ s = M D if m c = M D (cid:18) M D M P (cid:19) /n . (2)In contrast, in the second scenario (RS) the KK excitations of the graviton have unsuppressedcouplings to matter but large masses, right below M D , so a few KK modes suffice to definean order one gravitational interaction at that scale.The ADD set up has the basic cosmological problem pointed out in [12]. Essentially, attemperatures T ≫ m c KK gravitons will be abundantly produced in annihilations of braneparticles, due to the large multiplicity of final states. If the initial temperature is large thesegravitons will change the expansion rate at the time of primordial nucleosynthesis. Even ifthe initial temperature is as low as 1 MeV, their late decay will distort the diffuse gamma raybackground in an unacceptable way. Obviously, an initial temperature close to M D wouldbring too many massive gravitons.One solution would be to consider RS models of TeV gravity. There, at T ≈ M D bulk and brane species have similar abundances, but massive gravitons will decay fast once T < m c ≈ . M D , returning all the energy to the brane and defining acceptable 4-dimcosmological models.Even within the ADD framework, however, we can consider hybrid models where theconnection between m c and M D is not the one given in Eq. (2). This can be obtained, for3xample, with a warp factor [16]. The effect would be to push the KK modes towards the4dim brane, reducing the effective compact volume to V ≈ (1 /m c ) n while increasing theircoupling to matter, sM P → sM D (cid:18) m c M D (cid:19) n . (3)In this way, a smaller number of KK modes will imply an order one gravitational interactionat the same scale √ s = M D . If the free parameter m c takes the value in Eq. (2) we recoverADD, whereas for m c approaching M D we obtain RS. At distances below 1 /m c gravity wouldbe higher dimensional (similar to ADD) whereas at larger distances it becomes 4-dimensional(like in the usual RS scenario).In this framework the KK modes of the graviton are not produced at temperatures T < m c . Therefore, m c >
10 MeV avoids astrophysical bounds [17] for any number n ofextra dimensions [16] and M D = 1 TeV. On the other hand, as these KK gravitons havestronger couplings to matter, they can decay much faster than in the usual ADD modelor decouple at temperatures below their mass. By changing m c (with M D fixed) it seemseasy to obtain models with no gravitons at the time of primordial nucleosynthesis that areconsistent with all cosmological observations. Let us consider the hybrid framework described in the previous section with n extra dimen-sions, a fundamental scale M D , and an independent compactification mass m c ≥ M D and distances smaller than 1 /m c is strongly coupled andhigher dimensional (just like in ADD), so the radius and temperature of a BH of mass M are r H = a n M D (cid:18) MM D (cid:19) / ( n +1) ; T BH = n + 14 πr H , (4)with a n = n π ( n − / Γ (cid:16) n +32 (cid:17) n + 2 n +1 . (5)Once a BH reaches a radius r H = 1 /m c and fills up the whole compact space its (4-dim)size will not keep growing, since all the KK gravitons but the zero mode provide shortdistance interactions. The radius should start growing significantly only when the usual4-dim horizon (produced by the massless graviton) is of order 1 /m c , i.e. , for BH masses M ≈ M P /m c . Above this threshold BHs are basically 4-dim.4e will assume that the particles in the brane and the bulk (with g ∗ and g b degrees offreedom, respectively) are initially in thermal equilibrium at a temperature T = T . Noticethat reheating in just the brane after a period of inflation would not justify an empty bulk, asin a time of order M D /T most of the energy would escape into KK modes. Here, however,the bulk may be much thinner and emptier than in the usual ADD model, implying a slowexpansion in terms of the fundamental time 1 /M D (see below).It is easy to see that if the plasma temperature T and 1 /r H are larger than m c a BH atrest in the brane will change its mass according to [14]d M d t ≈ σ A ( T − T BH ) + σ n A n ( T n − T nBH ) ≈ π (cid:18) n + 1 T BH (cid:19) g ∗ (cid:16) T − T BH (cid:17) + g b c n T n T nBH T − T BH !! , (6)where σ = g ∗ π / c n is an order 1 coefficient that depends on the geometry of thecompact space, and we have neglected gray-body factors [18]. The expression above assumesa thermalized photon (in the brane) and graviton (in the bulk) plasma, so it requires thatchanges occur slowly . In particular, if the BH grows very fast one has to make sure that thegain in mass is always smaller than the total energy of the plasma in causal contact withthe hole: d M d t < π t T g ∗ + g b c n T n m nc ! , (7)where we have assumed t > /m c .As the BH grows it enters a new phase when its mass reaches M ≈ M D (cid:18) M D a n m c (cid:19) n +1 , (8)which corresponds to a radius r H = 1 /m c filling up the whole extra volume. For M > M the BH keeps gaining mass as far as the plasma temperature T is above T BH ≈ m c . However,as explained above, KK fields do not reach distances beyond r H ≈ /m c , so the BH radius(and its temperature) will stay basically constant.Finally, masses above M ≈ M p /m c turn the BH into a purely 4 dimensional object: itsradius r H = 2 MM P (9)grows larger than 1 /r c and the BH becomes too cold to emit massive gravitons. In thisregime, if T is larger than m c the BH changes its mass according tod M d t ≈ π g ∗ + g b c n T n m nc ! T T BH , (10)5hereas at lower plasma temperature it goes asd M d t ≈ π g ∗ T T BH − T BH ! . (11)Eq.(6) implies that lighter (hotter) BHs evaporate with an approximate lifetime τ ≈ M D (cid:18) MM D (cid:19) n n , (12)and that BHs colder than the plasma will gain mass: heat flows from the hot plasma to thecold BH, but the effect is to cool the BH further and increase T − T BH . This is, indeed, avery peculiar two-component thermodynamical system.On the other hand, the expansion of the universe is also affected by the presence ofbulk species. We will assume that the extra dimensions are frozen (do not expand) andwill integrate the matter content in the bulk. The large values of m c that we will considerimply large enough couplings with brane photons, so that the (equilibrium) abundance ofKK modes of mass larger than the plasma temperature will be negligible. Therefore, at T > m c we have a universe with a radiation density ρ rad ≈ π T g ∗ + g b c n T n m nc ! , (13)and an expansion rate ˙ R R = 8 πG ρ rad , (14) i.e. , ρ rad = ρ rad ( R /R ) . Notice that the second term in Eq.(13) will slow down the changein the plasma temperature T ( t ) due to the expansion. In particular, if the bulk energydominates then T ∝ t / (4+ n ) for times larger than a Hubble time. At temperatures below m c this term vanishes exponentially and all the bulk energy is transferred to the brane.To ilustrate the orders of magnitude involved, let us discuss a toy model with M D = 1TeV, n = 1 ( c = 0 . g ∗ = 2) and gravitons ( g b = 5) at T = 100 GeV.A BH of mass M < M crit = 12 TeV would be hotter than the environment, and it wouldevaporate in a time of order τ ≈ − GeV − = 6 . × − s. The Hubble time of a universeat this temperature is H − = R ˙ R = 1 . × GeV − = 9 . × − s . (15)A BH of initial mass M = 100 TeV will have a starting temperature of 8.7 GeV. In a timeof order 17 GeV − its mass is already around 4 . × GeV and its radius as large as the size6 [ G e V ] .. t [GeV − ] 10 Figure 1: Evolution of a single BH of mass M = 100 TeV for M D = 1 TeV and n = 1 in anexpanding universe at T = 100 (lower line) and 200 GeV (upper line).of the extra dimension (0 . − ). Then the BH keeps growing at approximately constantrate, since its size and temperature ( T BH ≈ . M ≈ . × GeV. At later times the expansion coolsthe plasma, which slows the growth of the BH. Its maximum mass, M ≈ × GeV, isachieved when T = T BH at times of order t ≈ GeV − . Finally, the BH will evaporateafter τ ≈ GeV − ≈ T = 200 GeV. In this second case the expansion rate is faster (the Hubble time H − ≈ . × GeV − is shorter), but the higher radiation density provides larger BH masses.Lower (higher) values of m c ( M D ) would also imply larger BH masses.In none of the two cases described above the BH becomes purely 4 dimensional, i.e., witha mass M > GeV. These large values of M can be obtained increasing the number n ofextra dimensions and/or the ratio M D /m c . A BH radius larger than 1 /m c , however, wouldnot stop its growth, on the contrary, it would enhance the absorption rate of radiation bythe BH. What stops the growth of a BH in any TeV-gravity framework is just the drop inthe temperature of the plasma due to the expansion.It is also important to emphasize that the qualitative features of the process describedabove do not depend much on the details of the compactification (the length or the shape)7f the extra dimensions. They just depend on the fact that the fundamental scale of gravity M D is low and that the initial temperature of the radiation is close to it. In particular, thesequence of events would be similar in RS models, although there m c is close to M D and theBHs are smaller ( r H ≈ /m c , and the absortion rate grows with the BH area).Finally, notice that the usual cosmological constraints [12, 13] on the initial (reheating)temperature of the early universe or the astrophysical bounds on M D are actually a probeof 1 /L = m c , so they are avoided if m c is above 1 GeV. Let us now deduce the equations that describe the production and evolution of BHs insidea thermal bath of temperature T . If T < M D < M the BHs will be produced in collisionsbetween radiation in the high-energy tail of the Boltzmann distribution f γ ( ~k ). Once formedthese BHs will be non-relativistic, with a kinetic energy K ≈ T negligible versus the mass M and a velocity v ≈ q T /M . We will then assume that the two components of the system arewell described by the temperature T ( t ) of the plasma and the distribution f ( M, t ) expressingthe number of BHs of mass M per unit mass and volume at time t . The total energy densityis ρ ( t ) = ρ rad ( t ) + ρ BH ( t ) ≈ π T g ∗ + g b c n T n m nc ! + Z d M M f ( M, t ) . (16)We identify the following processes changing the number density of BHs of mass M . • BH production in photon-photon collisions. Photons will collide with a cross section σ ≈ πr H to form a BH of mass M ≈ √ s : ∂f ( M, t ) ∂t ! γγ → M = Z d k (2 π ) d k (2 π ) f γ ( ~k ) f γ ( ~k ) σ ( M ) | ~v − ~v | δ ( q ( k µ + k µ ) − M ) , (17)with v i = 1. If T ≪ M this expression can be approximated in terms of a modifiedBessel function, ∂f ( M, t ) ∂t ! γγ → M ≈ g ∗ a n π T M (cid:18) MM D (cid:19) n n K ( M/T ) . (18)To simplify our analysis we will neglect BH production in collisions of bulk particles.Although KK modes dominate the energy density, their cross section to form a BH issmaller (their wave function is diluted within the bulk), so this would be an order onecontribution. 8 The collision of two BHs, of mass M and M , to form a BH of mass M = M + M : ∂f ( M, t ) ∂t ! M M → M = Z d M d M f ( M , t ) f ( M , t ) σ ( M , M ) v δ ( M + M − M ) , (19)where the BH velocity is v i = q T /M i and v = h| ~v − ~v |i . We will take a BH–BHcross section of σ ( M , M ) = π ( r H + r H ) . (20)Notice also that for a minimum BH mass of M D , this contribution to f ( M, t ) is nonzeroonly if M ≥ M D . • A BH of mass M may collide with any other BH, which would reduce f ( M, t ): ∂f ( M, t ) ∂t ! MM → M = − Z d M f ( M, t ) f ( M , t ) σ ( M, M ) v , (21) • We can describe the absorption and emision of radiation using d M/ d t in (6). The BHsof mass M will have a mass M + d M at t + d t , i.e. , f ( M, t ) = f ( M + d M, t + d t ) . (22)This implies ∂f ( M, t ) ∂t ! abs/em = ∂f ( M, t ) ∂M d M d t . (23) • Finally, we have to add the effect of the expansion. The 4-dim scale factor growsaccording to ˙ R R = 8 πG ρ rad + ρ BH ) . (24)This dilutes the number of BHs at a rate ∂f ( M, t ) ∂t ! exp = − f ( M, t ) ˙
RR . (25)The total change in f ( M, t ) per unit time will result from the addition of the 5 contributionsabove. This fixes ˙ ρ BH : ˙ ρ BH = Z d M M ∂f ( M, t ) ∂t . (26)On the other hand, to obtain the change in T (or ρ rad ( T )) we impose energy conservation,d( ρ rad R ) + d( ρ BH R ) = − ρ rad d R , (27)9here we have neglected the pressure of the BHs (they behave like non-relativistic matter).The equation above implies ˙ ρ rad = − ρ rad ˙ RR − ˙ ρ BH − ρ BH ˙ RR . (28)Notice that if the radiation and the BHs where decoupled, then one would have d( ρ BH R ) =0, i.e. , ˙ ρ rad = − ρ rad ( ˙ R/R ). This, however, is not the case since there is energy exchangebetween the two components. In particular, a variation in ρ rad changes T , implying a changein the absorption/emission rate of the BHs and in ρ BH . It is easy to see thatd( ρ BH R ) = R α d ρ rad , (29)where we define α as α ≡ ∂ρ BH ∂ρ rad ! R = cons. . (30)Substituting this equation in (27) we obtain R (1 + α ) d ρ rad = − ρ rad d R (31)The equation above expresses that BH evaporation can slow down the cooling of theradiation due to the expansion: as the universe expands, T drops, this leaves some BHshotter than the plasma, so they evaporate and reheat the environment. Taking an intervalwhere α is constant, eq. (31) can be integrated to ρ rad ≈ ρ rad (cid:18) R R (cid:19) α . (32)If α ≫ stop the change in the density and the temperature of theradiation due to the expansion. We will now apply these equations to an initial configuration with only radiation (no BHs)at a given temperature T (0) = T . We find two generic scenarios depending on the valueof T . Values closer to M D produce a larger number of BHs, that grow and absorb all theradiation before a Hubble time. Lower values of T imply a smaller number of BHs, thatgrow at basically constant temperature up to times of order H − . In this section we willfocus on the first case. We will describe the sequence of events using the toy model with n = 1, g ∗ = 2, g b = 5, m c = 10 GeV, M D = 1 TeV and an initial temperature of T = 20010 t f ( M , t ) [ G e V ] .. M [GeV]5000 15000 25000 35000 4500010 − − − − − Figure 2: Distribution f ( M, t ) for t i = 0 . , − and T = 200 GeV. Dashes describe theevolution neglecting BH–BH collisions.GeV. The critical BH mass (corresponding to T BH = T ) is M crit = 3 . f ( M, t ) for two values of t . We have included the distributions withand without the effect of collisions of two BHs to form a larger one.We can distinguish four phases in the evolution of the BH gas.1. In a first phase BH’s of M > M crit are produced at constant rate (see Fig. 3) andabsorb radiation of temperature T ≈ T . As the number of BHs grows (see Fig. 2),collisions between two BHs to produce a larger one become important. This reducesthe number of BHs and increses their average mass (in Fig. 3). At times around t ≈ − BHs start dominating the energy density and the temperature of the radiationdrops (see Fig. 4).2. The drop in T stops the production of BHs in photon-photon collisions. In addition,the lighter BHs become hotter than the plasma, so they evaporate and feed ρ rad . Theevaporation reduces the number of BHs per unit volume, but the average BH massgrows: there is a continuous transfer of energy from the radiation and from the lighter(evaporating) BHs to the larger BHs of lower temperature. Once the BHs get a massaround 5 × GeV their radius stops growing.3. At t ≈ GeV − the temperature of the radiation and of the BHs is similar, around11 B H [ G e V ] .. t [GeV − ] 10001001010 . M [ G e V ] .. t [GeV − ] 10001001010 . Figure 3: Number of BHs per unit volume and average BH mass as a function of time for T = 200 GeV.1 . ρ = ρ rad + ρ BH is matter (BH) dominated.4. At times of order H − ≈ . × GeV − the expansion cools the radiation. The BHs,of mass around 10 GeV, decay fast ( τ ≈ × GeV − ) and the universe becomesradiation dominated. The lightest KK modes also decay fast ( τ KK ≈ M nD /m nc ≈ GeV − ), so only 4dim photons survive below T ≈ T below 0.1 GeV (twotypes of matter, baryons and BHs, could coexist in models with lower values of m c /M D ).Second, this generic high T case, with M D > natural to obtain a plasmadominated by the standard particles at T ≈ m c ≫ Λ QCD . The predictions for primordialnucleosynthesis would then be consistent with observations.12 [ G e V ] .. t [GeV − ] 10001001010 . Figure 4: Temperature T of the radiation as a function of t . At larger values of t (up to aHubble time ≈ GeV − ) T ≈ . Let us now discuss the scenario with a low initial temperature and ρ BH ≪ ρ rad at any t . Wetake n = 1, g ∗ = 2, g b = 5, m c = 10 GeV, M D = 1 TeV and T = 100 GeV. The critical BHmass (corresponding to T BH = T ) is M crit = 12 TeV, higher than in the previous case. Asa consequence, the production rate of BHs colder than the plasma is much smaller. We canseparate three phases in the evolution of this BH gas.1. At times below H − = 1 . × GeV − BHs are produced at constant rate in photon-photon collisions, with T ≈ T . The number of BHs is so small that BH collisions canbe neglected. All these BHs grow like the one discussed in Section 2.2. When the expansion cools the photons, BH production drops exponentially, whereasBH growth slows down. In Fig. 5 we plot the BH distribution f ( M, t ) after one andfour Hubble times. The BHs reach a maximum mass around M = 10 GeV (thereis just a 10% mass difference between 99% of the BHs) and T BH ≈ . t ≈ GeV − the photon gas becomescolder than the BHs.3. In the last phase all the BHs evaporate in a time scale τ ≈ GeV − .13 t f ( M , t ) [ G e V ] .. M [GeV]5 · · · · − − − − − − Figure 5: Distribution f ( M, t ) for t = H − = 1 . × GeV − and t = 4 H − with T = 100 GeV.This generic scenario is in principle consistent with primordial nucleosynthesis, sinceBHs are just a small fraction of the total energy density and do not change the expansionrate. In the particular case that we have discussed they decay when the plasma temperatureis T ≈ .
01 GeV, but increasing the ratio M D /m c one can obtain BHs that become 4dimensional and with a much longer lifetime. Their late decay could introduce distortions inthe diffuse gamma ray background. In addition, in a more complete set up including baryonsand structure formation they might work as seeds for macroscopic (primordial) BHs. A transplanckian regime at accessible energies would have new and peculiar implicationsin collider physics and cosmology. What makes this regime special is that as the energygrows, softer physics dominate. For example, the production of a regular heavy particle atthe LHC would provide an event with very energetic (hard) jets from its decay. In contrast,the production of a mini-BH would be seen as a high multiplicity event, with dozens of lessenergetic (soft) jets. Analogously, the production of massive particles or mini-BHs in theearly universe would have very different consequences. The heavier the elementary particle,the shorter its lifetime, which tends to decouple these particles from low temperatures. Formini-BHs is just the opposite, heavier BHs are colder and live longer. In addition, if an14pproximate symmetry makes a massive relic long lived, its late decay will produce veryenergetic particles. BHs would imply a much softer spectrum of secondaries [19], withdifferent cosmological consequences.In this paper we have analyzed the dynamics of a two-component gas with photons andBHs in an expanding universe. The system is characterized by the temperature T ( t ) ofthe radiation and the distribution f ( M, t ) of BHs. Our equations take into account BHproduction, absorption of photons, BH evaporation, and collisions of two BHs. We havediscussed the two generic scenarios that may result from an initial temperature close to thefundamental scale of gravity M D . In the first scenario BHs empty of radiation the universeand dominate ρ before a Hubble time, whereas in the second case there are few BHs thatgrow at constant T up to t ≈ H − .We think that the work presented here is a necessary first step in the search for observableeffects from these BHs. It could also lead us to a better understanding of the early universeat the highest temperatures. Acknowledgments
We would like to thank Mar Bastero, Thomas Hahn and Iacopo Mastromatteo for usefuldiscussions. This work has been supported by MEC of Spain (FIS2007-63364 and FPA2006-05294) and by Junta de Andaluc´ıa (FQM-101 and FQM-437). MB acknowledges a Juan dela Cierva fellowship from MEC of Spain.
References [1] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B (1998)263 [arXiv:hep-ph/9803315]; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos andG. R. Dvali, Phys. Lett. B (1998) 257 [arXiv:hep-ph/9804398].[2] L. Randall and R. Sundrum, Phys. Rev. Lett. (1999) 3370 [arXiv:hep-ph/9905221].[3] T. Banks and W. Fischler, “A model for high energy scattering in quantumgravity,” arXiv:hep-th/9906038; R. Emparan, Phys. Rev. D (2001) 024025[arXiv:hep-th/0104009]; S. B. Giddings and S. D. Thomas, Phys. Rev. D (2002)056010 [arXiv:hep-ph/0106219]; D. M. Eardley and S. B. Giddings, Phys. Rev. D (2002) 293 [arXiv:hep-ph/0112161].[4] S. W. Hawking, Nature (1974) 30; S. W. Hawking, Commun. Math. Phys. (1975) 199 [Erratum-ibid. (1976) 206]; D. N. Page, Phys. Rev. D (1976) 198.[5] S. Dimopoulos and G. L. Landsberg, Phys. Rev. Lett. (2001) 161602[arXiv:hep-ph/0106295].[6] A. S. Majumdar, Phys. Rev. Lett. (2003) 031303 [arXiv:astro-ph/0208048];R. Guedens, D. Clancy and A. R. Liddle, Phys. Rev. D (2002) 083509[arXiv:astro-ph/0208299].[7] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. (1974) 399.[8] J. D. Barrow, E. J. Copeland and A. R. Liddle, Mon. Not. Roy. Astron. Soc. (1991)675; J. D. Barrow, E. J. Copeland and A. R. Liddle, Phys. Rev. D (1992) 645.[9] P. C. Argyres, S. Dimopoulos and J. March-Russell, Phys. Lett. B (1998) 96[arXiv:hep-th/9808138].[10] A. S. Majumdar and N. Mukherjee, Int. J. Mod. Phys. D (2005) 1095[arXiv:astro-ph/0503473].[11] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D (1999) 086004[arXiv:hep-ph/9807344].[12] S. Hannestad, Phys. Rev. D (2001) 023515 [arXiv:hep-ph/0102290]; L. J. Hall andD. Tucker-Smith, Phys. Rev. D (1999) 085008 [arXiv:hep-ph/9904267].[13] G. D. Starkman, D. Stojkovic and M. Trodden, Phys. Rev. D (2001) 103511[arXiv:hep-th/0012226].[14] J. A. Conley and T. Wizansky, Phys. Rev. D (2007) 044006 [arXiv:hep-ph/0611091].[15] R. H. Brandenberger and C. Vafa, Nucl. Phys. B (1989) 391; M. Gasperiniand G. Veneziano, Phys. Rept. (2003) 1 [arXiv:hep-th/0207130]; B. A. Bas-sett, M. Borunda, M. Serone and S. Tsujikawa, Phys. Rev. D (2003) 123506[arXiv:hep-th/0301180]; M. Borunda and L. Boubekeur, JCAP (2006) 002[arXiv:hep-th/0604085].[16] G. F. Giudice, T. Plehn and A. Strumia, Nucl. Phys. B (2005) 455[arXiv:hep-ph/0408320]. 1617] S. Hannestad and G. G. Raffelt, Phys. Rev. D (2003) 125008 [Erratum-ibid. D (2004) 029901] [arXiv:hep-ph/0304029].[18] D. N. Page, Phys. Rev. D (1976) 198.[19] P. Draggiotis, M. Masip and I. Mastromatteo, JCAP (2008) 014 [arXiv:0805.1344[hep-ph]]; M. Masip and I. Mastromatteo, JCAP0812