Production and evaporation of micro black holes as a link between mirror universes
aa r X i v : . [ a s t r o - ph . C O ] F e b Production and evaporation of micro black holes as a link between mirror universes
Viktor K. Dubrovich, ∗ Yury N. Eroshenko, † and Maxim Yu. Khlopov
3, 4, 5, 6, ‡ Special Astrophysical Observatory, St. Petersburg Branch,Russian Academy of Sciences, St. Petersburg, 196140 Russia Institute for Nuclear Research, Russian Academy of Sciences,pr. 60-letiya Oktyabrya 7a, Moscow, 117312 Russia Institute of Physics, Southern Federal University, Rostov on Don, Russia National Research Nuclear University ”MEPHI” (Moscow Engineering Physics Institute), 115409 Moscow, Russia Centre for Cosmoparticle Physics “Cosmion” 115409 Moscow, Russia Universit´e de Paris 7, CNRS, Astroparticule et Cosmologie, Paris, France (Dated: February 8, 2021)It is shown that the equalisation of temperatures between our and mirror sectors occurs duringone Hubble time due to microscopic black holes production and evaporation in particles collisionsif the temperature of the universe is near the multidimensional Plank mass. This effect excludesthe multidimensional Planck masses smaller then the reheating temperature of the universe ( ∼ GeV) in the mirror matter models, because the primordial nucleosynthesis theory requires thatthe temperature of the mirror world should be lower than the ours. In particular, the birth ofmicroscopic black holes on the LHC is impossible if the dark matter of our universe is representedby baryons of mirror matter. It excludes some of the possible co-existing options in particle physicsand cosmology.
I. INTRODUCTION
The mirror matter models was proposed by Lee andYang in [1] and developed in [2] (see review and bibliog-raphy in [3] and [4]). These models have many interest-ing consequences for cosmology and astrophysics [5–8],in particular, the mirror dark matter can form objects ofdifferent types [9–11], including domain structures [12].In several works the possibility was considered that ourand mirror words interact not only gravitationally, butalso through some exchange of energy and matter. Thematter can be transferred by the oscillations of chargedleptons [7], [8], [13, 14], neutrons [15] or neutrinos [16, 17]into the corresponding particles of mirror world and viceverse due to the high-order operators in the Lagrangian.Authors of [18] considered the hypothesis about the mix-ing of our and mirror photons by the following term inLagrangian εF µν F ′ µν , and the restriction ε < × − wasobtained from the primordial nucleosynthesis constraints.In more details the evolution of the temperatures in ourand in the mirror words due to the mixing of photonswas considered in [19], [20], where the constraints on themixing parameter ε were elaborated. The non-orientablewormholes provide another canal for the matter exchange[21].The formation of microscopic black holes (BH) in par-ticle collisions in the early universe was discussed earlierin articles [22–27]. In this paper, we consider the energyexchange between our and the mirror world by the birthand evaporation of microscopic BHs. As far as we know,previously this energy exchange channel with reference ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] to mirror matter was not considered. The connection ofthe worlds through microscopic BHs was considered inanother aspects in the works [28], [29], where it is shownthat microscopic black holes can provide bridges betweenclose branes.Primordial nucleosynthesis requires that the tempera-ture of the CMB in the mirror world be lower than in ours[30]. Otherwise, additional relativistic degrees of freedomappear, which change the dynamics of the primordial nu-cleosynthesis and change the yield of chemical elements.Can the temperatures of our and the mirror worlds inthe early Universe be leveled by the exchange of energybetween them? Aforementioned variant with the pho-tons mixing have already been considered. In the pres-ence of additional dimensions, multidimensional Planckmass M can be many orders of magnitude smaller thanthe usual 4-dimensional Planck mass, which reduces theenergy necessary for the BH production. This effect waswidely discussed in relation to the Large Hadron Collider.If a BH borns when two particles of our world collide,the BH evaporates both in our and in mirror particles.Thus, there is a transfer of energy from our to the mirrorworld. The reverse flow of energy will be less, becausethe temperature of the mirror world is lower. Due to theenergy exchange, the temperatures of our and the mirrorworld can be equalized. This contradicts the primordialnucleosynthesis constraint and implies a lower bound onthe Planck mass in multidimensional models. The mainresult of this work is the restriction on the multidimen-sional Planck mass. Namely, it was obtained that themultidimensional Planck mass should exceed the reheat-ing temperature ∼ GeV .With the temperature decrease the universe evolvesfrom multidimensional to our 4D state. The transitiontakes place near T ∼ M . We will show that the temper-ature equalisation between our and mirror worlds occursduring one Hubble time. This means that to state thefact of the temperature equalisation one has no need toconsider the temperatures at T > M and the complicatedmultidimensional physics. It’s enough to consider onlythe T ∼ M epoch. In this epoch the usual 4-dimensionalphysics are at place. Therefore, our consideration will bein much respects independent on the particular modelsof the multidimensional word at T > M . II. PRELIMINARY ESTIMATES
Consider the cosmological model where our dark mat-ter is a mirror substance (mirror barion, mirror lep-tons etc), and the temperature of our world is different(higher) from the temperature of the mirror world. Attemperatures T ≥ M , BH will be born in the particle col-lisions, and in a world with a higher temperature, theirbirth is more efficient. During the quantum evaporationand decay of the BH, particles of both our and the mir-ror universe are equally likely to be born. Thus, therewill be a flow of energy from our hotter universe to thecolder mirror universe. Below, we will evaluate how ef-fective this process is, and whether the temperatures ofour world and the mirror world will equalize.With the additional dimensions present, the multidi-mensional Planck mass M can be less than the usual4-dimensional Planck mass M Pl ≃ . × GeV. If M ∼
10 TeV the creation of microscopic BHs at theLarge Hadron Collider is possible [31–33]. The cross sec-tion for the production of BHs in pp collisions in the firstapproximation is written in the form [31, 34], σ ≃ πR s = 1 M " M BH M (cid:0) n +32 (cid:1) n + 2 ! / ( n +1) , (1)where n is the number of additional dimensions (4 + n intotal), R s is the Schwarzschild radius of the multidimen-sional BH with mass M BH , and the units ~ = c = 1 arein use here and further. The Boltzmann constant is alsoassumed to be equals 1.Let us begin from the simple estimates. Assume thatfor T ∼ M all BHs born with masses M BH ∼ M . Inreality, the mass spectrum should be formed. Assumealso that the BHs decay immediately after birth withouta stable remnants (or Plankions). In reality, the Hawkingevaporation takes some time, and there is a time delayfrom the moment of birth to the moment of final decay.Taking into account the 1st assumption, the birth cross-section (1) is written as σ ∼ M . (2)Let one and only one BH appears with the cross-section(2) when any two particles collide at T ≥ M , and thistakes place for each effective degree of freedom (total of g ∗ ∼ M appears only inthe BH birth cross-section, and the usual 4-dimensionalPlanck mass M Pl is used for the universe evolution. To-tal number density of particles in the universe at theradiation-dominated stage [35] n ≃ g ∗ ρ T , (3)where ρ = π g ∗ T . (4)The number of BHs born in a volume V per unit time is˙ N ∼ σn V. (5)The products of BH quantum evaporation immediatelythermalize, going into the cosmic plasma. Relative rateof energy transfer to the mirror universe˙ ρ BH ρ ∼ ρ M ˙ N V , (6)where the multiplier 1 / H = ˙ aa = T M ∗ Pl , (7)where [35] M ∗ Pl = M Pl / (1 . √ g ∗ ),( ˙ ρ BH /ρ ) /H ∼ × (cid:18) M Pl M (cid:19) ≫ . (8)Thus, the multidimensional Planck mass M must belarger than the reheating temperature ∼ GeV. Oth-erwise, the temperatures of our world and the mirrorworld equalise, and the universe becomes symmetrical,violating the primordial nucleosynthesis constraints.
III. THE RATE OF MICRO BLACK HOLESPRODUCTION IN THE EARLY UNIVERSE
Let us consider the micro black hole production ratemore exactly. The rate of particle interactions (numberof events per time interval dt inside the volume element dV ) is expressed through the invariant cross-section σ [36] dνdtdV = σ p ( p µ p µ ) − m m E E n n , (9)where n and n are the number densities of the col-liding particles. The generalization of this equation forthe physical situation in the early universe requires theintegration over particles distributions and the heat pro-duction looks as δQdtdV = 12 X i,j Z d p Z d p σ ∆ E p ( p µ p µ ) − m m E E × π ) (cid:2) e ( E − µ ) /T + i (cid:3) (cid:2) e ( E − µ ) /T + j (cid:3) , (10)where ∆ E is the energy transferred into BH, indexes i, j are equal to 1 in the case of fermion particles, and − i = j = 1, i = j = − i = − j = 1, and i = − j = −
1. The sum-mation goes over all possible degrees of freedom (bosonicand fermionic). The factor 1 / T ∼ M . In this sense, using the exactexpressions for Fermi-Dirac and Bose-Einstein distribu-tions seems redundant. However, it may be justified bythe following reasons. At T ≪ M the microscopic BH arenot born in the collisions of the most particles. But thedistributions continue towards higher energies, and rareparticles with energies E ∼ M from these distributionscan produce microscopic BH in collisions. It cannot beexcluded in advance that even a small fraction of all par-ticles with energies E ∼ M will lead to the temperatureequalization. Therefore, we keep the distributions in ourcalculations.For the approximate calculation of (10) we do the fol-lowing simplifications. First of all, we consider suffi-ciently high temperatures T and ultrarelativistic case byneglecting m and m in the further equations. In thiscase E = | ~p | , E = | ~p | and p µ p µ = | ~p || ~p | (1 − cos θ ) , (11)where θ is the angle between ~p and ~p . Let us introducethe dimensionless variables u = | ~p | T , u = | ~p | T , (12)then ∆ E = T ( u + u ) . (13)The chemical potential µ = µ = 0 because of fast ther-malization in hot plasma. Really, the thermalization ofthe evaporated radiation proceeds very fast. Sunyaev andZeldovich have shown in [37] that if the energy injectiontakes place prior to the epoch of the e + e − pairs annihi-lation no observable distortions are expected in the spec-trum of primordial radiation. It was obtained in [38] thateven significant energy release at the red-shifts z ≥ would be completely thermalized. Therefore, we use thethermal distributions for bosons and fermions. We con-sider only the process χ + χ → BH neglecting the pos-sible additional canals of the type χ + χ → BH + some-thing else. In particular, we suppose, that gravitational Ξ min J ij Figure 1. The functions J ij at n = 1 in the cases (from up todown) i = j = − i = − j = 1, and i = j = 1. waves generated during the particles collisions carry outenergy of the order or less than ∆ E . Under this condi-tion our calculations are valid at the order of magnitudeat least. The center of mass energy squared is s = ( p + p ) = 2 T u u (1 − cos θ ) = M ≥ γ M , (14)where the factor γ in the last inequality follows fromthe entropy arguments, and γ ∼ u u ≥ ξ min = γ M T . (15)The invariant cross-section σ is calculated in the lab-oratory system where one of the particle is at rest. Notehowever that during the transition to the center of masssystem the σ does not change and is given by (1) be-cause this geometrical cross-section represents the trans-verse direction under the Lorentz transformations. Notein addition that dtdV in (9) is invariant. For the masslessparticles the aforementioned laboratory system shouldbe considered in the limit m →
0. We don’t considerthe possible exponential suppression of the geometricalcross-section which was proposed in [39] and initiateddiscussion in several works.The integration over the angle θ in (10) can be doneanalytically. After this the (10) takes the form δQdtdV = Φ T (7 n +9) / ( n +1) (16)whereΦ = J n + 12 n + 3 2 / ( n +1) (2 π ) M (2 n +4) / ( n +1) (cid:0) n +32 (cid:1) n + 2 ! / ( n +1) , (17) J = X J ij , (18) J ij = ∞ Z du ∞ Z du ( u + u ) n ( u u ) (2 n +3) / ( n +1) − ξ (2 n +3) / ( n +1)min o θ H ( u u − ξ min )( e u + i )( e u + j ) , (19)where θ H is the Heaviside step function. The examplesof these functions are shown at Fig. 1. It’s easy to seethat J ij → const at ξ min →
0, i.e. at T ≫ M . IV. TEMPERATURE EVOLUTIONA. General equations
Let us consider the evolution in time of the densities (ortemperatures) of radiation in our and mirror universes.The values related to our world are marked by lower in-dex of “1”, and the values related to the mirror worldare marked by “2”. The first necessary equation is oneof the Friedmann equations12 (cid:18) dadt (cid:19) − πG a ( ε + ε ) = − k , (20)where the densities enter in the form of a simple sumaccording to the summation of energy-momentum ten-sors in Einstein’s equations, and further we consider aflat model with k = 0. The energy densities of our andmirror worlds are expressed through their temperatures ε = g ∗ ( T ) π ~ c T , ε = g ∗ ( T ) π ~ c T , (21)where g ∗ ( T ) is the effective number of degrees of freedom.Note that the multidimensional Planck mass M en-ters only the BH production cross section, and the usualfour-dimensional Planck mass M Pl is used in the cosmo-logical evolution equations, because the Einstein equa-tions contain the already reduced gravitational constantat T ≪ M .Let us write down the first law of thermodynamics forthe matter of our world: δQ = p dV + dE , (22)where δQ is the energy change in the volume V due tothe energy transfer to the mirror world and due to thereverse energy flow, E = ε V , pressure p = ε /
3. Wewrite the cubic volume element in the form V = a r .For a fixed comoving volume ( r = const ) one has δQ V dt = 3 ˙ aa ( p + ε ) + ˙ ε = 4 ˙ aa ε + ˙ ε . (23)And the similar relationship holds for the mirror world δQ V dt = 3 ˙ aa ( p + ε ) + ˙ ε = 4 ˙ aa ε + ˙ ε , (24) with δQ = − δQ . Here we neglect the energy that isstored in black holes at any time prior to evaporation,assuming that the evaporation takes place very quickly.Summing (23) and (24), we obtain for the total values ε = ε + ε and p = p + p the usual relation dεdt = − aa ( p + ε ) (25)with known solutions [35] ε = 3 c πGt , a ( t ) ∝ t / . (26)In the general case, the effective number of degrees offreedom g ∗ depends on the temperature. However, weconsider temperatures T ≥ g ∗ = const because the new degrees of freedom are notexcited with the temperature increase. Limiting values athigh temperatures are g ∗ = 106 .
75 and g ∗ = 228 .
75, re-spectively, in the Standard Model and in MSSM [40]. Inany case, we assume that g ∗ = const in the finite temper-ature range. This is true if the temperature equalizationoccurs fairly quickly at times on the order of the Hubbletime, so that g ∗ = const is a good approximation. Let usdenote θ = T and θ = T . In this case, the equations(23) and (24) take the form˙ θ + 4 ˙ aa θ = Φ2 α ( θ (7 n +9) / n +1)2 − θ (7 n +9) / n +1)1 ) . (27)˙ θ + 4 ˙ aa θ = Φ2 α ( θ (7 n +9) / n +1)1 − θ (7 n +9) / n +1)2 ) , (28)where α = g ∗ π / (30 ~ c ). B. Finite lifetime
The important question is the finite lifetime of BH inthe Hawking evaporation. Upper we considered the in-stantaneous decay of BHs. Now we discuss the influenceof the time-delay. The BHs lifetime is estimated as [32] τ ∼ ~ M c (cid:18) M BH M (cid:19) ( n +3) / ( n +1) . (29)For the moving BH the additional Lorentz-factor Γ ∼ E/M BH arises in the lifetime (29). But typically E ∼ T , M BH ∼ T , and Γ ∼
1. Let us compare this lifetime withcosmological (Hubble) time in the early universe in thecase n = 1 τt ∼ . × − (cid:18) TM (cid:19) (cid:18) T GeV (cid:19) − . (30)Therefore, the condition τ < t requires M > × − T (cid:18) T GeV (cid:19) − / , (31)and in this case the rough condition for the BHs pro-duction T ≥ M can be satisfied only in the temperaturerange T = (2 . × − − × GeV . (32)Under the conditions (31) and (32) the BHs decay typ-ically during one Hubble time and the energy transferbetween our and mirror universes can be considered asinstantaneous. In this case we can use the expression(16) for the energy transfer. For n > M . The time-delay makes the energy trans-fer even more effective, because the radiation energy ofthe evaporated BH is red-shifted and diluted as 1 /a ( t )during the universe expansion. But the energy stored inthe non-relativistic part of the BHs spectrum is rarefiedslowly as 1 /a ( t ). Therefore, neglecting the finite lifetimewe will obtain the lower limit for energy transfer whichis enough for our purposes.We assume also that the BHs evaporate without stableremnants (Planckions). C. Case n = 1 In the n = 1 case the exact analytical solution can befound. The equations (27) and (28) have the form˙ θ + 4 ˙ aa θ = Φ2 α ( θ − θ ) . (33)˙ θ + 4 ˙ aa θ = Φ2 α ( θ − θ ) . (34)Now we take the difference of these equations. The right-hand-side can be decomposed θ − θ = ( θ − θ )( θ + θ )and the general expression (26) can be used for the sum θ + θ . The resultant equation for θ − θ have the simpleexact solution. Let us also denote δ = θ − θ θ + θ . (35)At the time of reheating δ = δ i ≤
1, and the maximum δ i = 1 corresponds to a completely cold or empty mirroruniverse. With the initial condition δ ( t i ) = δ i one hasthe solution δ ( t ) = δ i exp (cid:26) c Φ32 πGα (cid:18) t − t i (cid:19)(cid:27) . (36)We require that at δ i ∼ t ≫ t i the situation δ ( t ) ≪ c Φ32 πGα t i ≃ (cid:18) T i GeV (cid:19) (cid:18) M GeV (cid:19) − . (37) Let us consider the temperatures T i ∼ M . We see that(37) is less then 1 for M > × GeV. Otherwise thetemperatures equalization take place during one Hubbletime. In other words, the mass M cannot be less thanthe reheating temperature as long as the reheating tem-perature of the universe is less than ∼ GeV.
D. General case
For n ≥
2, the exact solution of the equations (27) and(28) can not be found, but, nevertheless, one can obtaina sufficiently strong lower bound on M . Taking again thedifference (27) and (28), we obtain the equation ddt ( θ − θ )+ 2 t ( θ − θ ) = − Φ α ( θ (7 n +9) / n +1)1 − θ (7 n +9) / n +1)2 ) . (38)If we replace the right-hand side of (38) by a smallerquantity (by absolute value), then the resulting equationwill describe the process of energy transfer with lowerefficiency than the original equation, and from the prop-erties of its solution we obtain a lower bound on M . Wewill consider the equation (38) in the bounded tempera-ture range T i > T > T f , where T f will be chosen later.Note, that β = 7 n + 94( n + 1) = 1 + 34 + 12( n + 1) . (39)Note also that the function φ ( x ) = x β at β > φ ( x ) − φ ( x ) > ( x − x ) φ ′ ( x ) = ( x − x ) βx β − (40)at x > x . As x , we take x , = T , /M >
1. In the case x > / [2( n + 1)] is thrown out the exponent β −
1. Therefore,we replace the equation (38) by the following ddt ( θ − θ )+ 2 t ( θ − θ ) = − M α ( θ − θ ) (cid:18) T f M (cid:19) , (41)where ˜Φ = 4 . × − J is obtained after the minimizationof (17). Solving (41), we find for the relative change δ ( t ) = δ i exp ( − M α (cid:18) T f M (cid:19) ( t − t i ) ) . (42)With the effective temperature equalization near T ∼ T f , we have the situation T ∼ T . Let us consider thetemperature variation during one Hubble time after t i ,i.e. we set again M ∼ T f ∼ T i . Also by order of magni-tude J ∼ g ∗ . Under these conditions, the exponent in(42) at t = t f ∼ t i is7 ˜Φ M t i α ≃ × (cid:18) M GeV (cid:19) − . (43)We see that in this case the temperature equalisationoccurs at all masses M during one Hubble time. The onlyway to avoid it is to suppose that M is larger then themaximum temperature in the history of the hot universe,i.e. the reheating temperature. Therefore the effect ofthe microscopic BH production exclude the masses M V. CONCLUSION In this paper the implications of micro black holesformation in high-energy particles collisions for the mir-ror matter cosmologies are considered. MultidimensionalPlanck mass M can be less than the usual 4-dimensionalPlanck mass easing the micro black holes production.Consider the model of the universe with two sectors: ourusual sector and the mirror one. The temperature of ourworld should be higher then the temperature of the mir-ror one due to the primordial nucleosynthesis constraints.The production of the microscopic black holes is more ef-ficient in a world with a higher temperature. During thequantum evaporation of black holes, particles of both ourand mirror universes will be emitted with equal probabil-ity. Thus, there will be a flow of energy from our hotteruniverse to the colder mirror world, and the equalizationof their temperatures is possible. This effect allows to ob-tain the constraints on the multidimensional Planck mass M in the mirror matter model. Namely, M should belarger than the reheating temperature ∼ GeV – themaximum temperature in the hot universe. Otherwise,the temperatures would be equalized and the primordialnucleosynthesis constraint would be violated. The equal-isation of temperatures between our and mirror worldsoccurs during one Hubble time near T ∼ M (even if ithas not occurred early). Therefore, the physics of themultidimensional universe at T ≪ M is not very impor-tant. We can use ordinary 4D physics near T ∼ M forestimates.Let us conclude with a few comments.1. Entropy transfer. As it was noted in [41], althoughthere is a balance of energy, the total entropy increases,because δQ = − δQ , but at the different temperatures δQ /T = − δQ /T . The increase of entropy occurs inthe same way as the entropy increases when the temper-atures of two bodies initially having different tempera-tures are equalized. In [18–20] the mixing of our photonsand mirror worlds photon was considered. With this mix-ing, the entropy in the intermediate states is not delayed.Our variant with BHs is more interesting in this respect, since it is known, that BHs themselves carry entropy, andthe BH entropy is expressed through its horizon area byknown formulas. Therefore, it is interesting to considerthe two questions: how much entropy a BH transfers be-tween words in comparison with own BH’s entropy andhow much entropy is enclosed in BHs at every cosmolog-ical instant of time. The last question has sense becausethe BHs evaporate not instantaneously, but have a cer-tain lifetimes. One should take into account that blackholes can born with relativistic velocities, therefore theirenergy M c / p − v /c can exceed the rest energy M c .However, the BH motion does not affect its entropy as inthe case of the moving medium [42].2. Planckeons. The remnants of primordial BHs wereconsidered in many works in different aspects. In partic-ular, the remnants can help solving the information lossparadox [43]. The remnants of the micro BHs can format the particles collisions (not primordial) in the earlyuniverse [22, 23], [25], [27]. In the case the black holesleave stable remnants (Planckeon) the fate of the multi-dimensional universe would be dramatic not only in themirror matter models but even for single particle sectorbecause the universe goes into the dust-like stage veryearly.3. Primordial BHs. The evaporation of primordial BHscan also be considered as a canal between our and mirrorworlds (especially the region of their masses < g).Equalization of the temperatures in this case providesnew constraints on the primordial black holes at smallmass region. One can assume that in the early epochthe primordial BHs begin to dominate in density, thenevaporate, and all was thermalized. In ordinary cosmol-ogy, this would have consequences for entropy generation[44]. In models with mirror matter, due to the evapora-tion of primordial BHs, the temperature asymmetry be-tween our and the mirror world will be destroyed. Thus,it is possible to obtain new constraints on the primordialBHs in models with mirror matter in comparison withthe known entropy bounds on primordial BHs [44]. Mi-croscopic primordial BHs may from the preheating insta-bility, subsequently dominate the content of the universe,and their evaporation may be the source of reheating [45–47]. ACKNOWLEDGEMENTS Authors are grateful to Z. Berezhiani and A. Gazizovfor useful discussions. The research by M.K.was finan-cially supported by Southern Federal University, 2020Project VnGr/2020-03-IF. [1] T.D. Lee and C.N. Yang, Physical Review , 254(1956).[2] I.Yu. Kobzarev, L.B. Okun, and I.Ia. Pomeranchuk,Yadernaia Fizika , 1154 (1966). [3] L.B. Okun, Phys. Usp. , 380 (2007).[4] M. Khlopov, Fundamentals of Cosmic particle Physics (CISP-Springer: Cambridge, UK, 2012). [5] Z. Berezhiani, D. Comelli, and F.L. Villante, Phys. Lett.B , 362 (2001).[6] Z. Berezhiani et al., Int. J. Mod. Phys. D , 107 (2005).[7] L. Bento and Z. Berezhiani, Phys. Rev. Lett. , 231304(2001).[8] Z. Berezhiani, Int. J. Mod. Phys. A, , 3775 (2004).[9] S.I. Blinnikov and M.Y. Khlopov, Sov. J. Nucl. Phys. ,472 (1982).[10] S.I. Blinnikov and M.Y. Khlopov, Soviet Astronomy ,37 (1983); Translation from Astronomicheskii Zhurnal , 632 (1983).[11] M.Y. Khlopov et al., Soviet Astronomy , 21 (1991).[12] V.K. Dubrovich and M.Y. Khlopov, Soviet Astronomy , 116 (1989).[13] L. Bento and Z. Berezhiani, “Baryogenesis: The Lep-ton Leaking Mechanism”, Talk given (LB) at the XIInternational School: Particles and Cosmology, BaksanValley, Kabardino-Balkaria, Russia, 18-24 April, 2001;arXiv:hep-ph/0111116.[14] Z. Berezhiani, “Through the Looking-Glass: Alice’s Ad-ventures in Mirror World”, Published in Ian KoganMemorial Collection ”From Fields to Strings: Circum-navigating Theoretical Physics”, Eds. M. Shifman etal., World Scientific, Singapore, vol. 3, pp. 2147-2195;arXiv:hep-ph/0508233.[15] Z. Berezhiani and A. Gazizov, Eur. Phys. J. C , 2111(2012).[16] V. Berezinsky and A. Vilenkin, Phys. Rev. D , 083512(2000).[17] V. Berezinsky, M. Narayan, and F. Vissani, NuclearPhysics B , 254 (2003).[18] E.D. Carlson and S.L. Glashow, Physics Letters B ,168 (1987).[19] P. Ciarcelluti and R. Foot, Phys. Lett. B , 278 (2009).[20] R. Foot and S. Vagnozzi, Phys. Rev. D , 023512 (2015).[21] V.I. Dokuchaev and Yu.N. Eroshenko, Phys. Rev. D ,024056 (2014).[22] A. Barrau, C. Feron, and J. Grain, Astrophys. J. ,1015 (2005).[23] J.A. Conley and T. Wizansky, Phys. Rev. D , 044006(2007). [24] M. Borunda and M. Masip, JCAP , 027 (2010).[25] T. Nakama and J. Yokoyama, Phys. Rev. D , 061303(2019).[26] A. Saini and D. Stojkovic, JCAP , 071 (2018).[27] A. Barrau, K. Martineau, F. Moulin, and J.-F. Ngono,Phys. Rev. D , 123505 (2019).[28] G. Dvali and O. Pujolas, Phys. Rev. D , 064032 (2009).[29] G. Dvali and M. Redi, Phys. Rev. D , 055001 (2009).[30] Z.G. Berezhiani, A.D. Dolgov, and R.N. Mohapatra,Phys. Lett. B , 26 (1996).[31] S. Dimopoulos and G. Landsberg, Phys. Rev. Lett. ,161602 (2001).[32] K. Cheung, Phys. Rev. Lett. , 221602 (2002).[33] CMS Collaboration, Physics Letters B , 434 (2011).[34] T. Banks and W. Fischler, arXiv:hep-th/9906038.[35] D.S. Gorbunov and V.A. Rubakov, Introduction to theTheory of the Early Universe: Hot Big Bang Theory (World Scientific Publishing Company, 2011).[36] L.D. Landau and E.M. Lifshitz, The Classical Theory ofFields (Pergamon, Oxford, 1975).[37] R.A. Sunyaev and Ya.B. Zeldovich, Astrophysics andSpace Science , 20 (1970).[38] A.F. Illarionov and R.A. Siuniaev, Soviet Astronomy ,691 (1975); Translation from Astronomicheskii Zhurnal , 1162 (1974).[39] M.B. Voloshin, Physics Letters B , 137 (2001).[40] D.J. Schwarz, Annalen Phys. , 220 (2003).[41] R. Foot, Int. J. Mod. Phys. A , 1430013 (2014).[42] L.D. Landau, E.M. Lifshitz, Statistical Physics , Vol. 5,3rd ed. (Butterworth-Heinemann, 1980).[43] P. Chen, Y.C. Ong, and D. Yeom, Physics Reports ,1 (2015).[44] Ya.B. Zel’dovich and A.A. Starobinskii, JETP Letters , 571 (1976).[45] J. Martin, T. Papanikolaou, and V. Vennin, JCAP ,024 (2020).[46] J. Martin, T. Papanikolaou, L. Pinol, and V. Vennin,JCAP05