Baryon asymmetry from primordial black holes

We propose a new scenario of the baryogenesis from primordial black holes (PBH). Assuming presence of microscopic baryon (or lepton) number violation and a CP violating operator such as $\partial_\alpha F(\mathcal{R_{....}} ) J^\alpha$, where $F(\mathcal{R_{....}})$ is a scalar function of the Riemann tensor, time evolution of an evaporating black hole generates baryonic (leptonic) chemical potential at the horizon; consequently PBH enumerates asymmetric Hawking radiation between baryons (leptons) and anti-baryons (leptons). Though the operator is higher dimensional and largely suppressed by a high mass scale $M_*$, we show that sufficient amount of asymmetry can be generated for a wide range of parameters of the PBH mass $M_{\rm PBH}$, its abundance $\Omega_{\rm PBH}$, and the scale $M_*$.


Introduction
The standard model of particle physics is completed by the discovery of the Higgs boson, and is surprisingly consistent with the experimental data up to 1 TeV scale. However, there still remain several unsolved questions, e.g., What is the dark matter in the universe? Why are baryons more abundant than anti-baryons?
In order to answer these questions, gravitational effects might play important roles. One of the interesting possibilities of gravitational effects will be primordial black holes (PBH) [1,2], which may be created in the early universe. PBHs could be formulated in the early universe by various processes such as large density fluctuations by inflation [3], preheating [4], in particular the tachyonic preheating [5], or bubble collisions [6] associated with first order phase transitions in the universe [7]. 1 A PBH evaporates by Hawking radiation until the present time if its mass M is lighter than M = 10 15 g. Consequently the abundance of the PBHs around M = 10 15 g is strongly constrained by observations of the cosmic gamma ray [10,11]. If the mass is between 10 9 g and 10 13 g, the Hawking radiation from the PBHs affect the big bang nucleosynthesis (BBN) and the abundance in this mass region is also strongly constrained (see e.g. [12]). PBHs with larger mass also play various important roles in cosmology. It may contribute to the dark matter in the universe for M 10 15 g although its abundance is severely constrained [13,14]. PBH may also explain the origin of the BHs with mass M = O(10 30 )g [15,16], whose binary mergers are observed in the recent detections of the gravitational waves by LIGO [17].
On the other hand, PBHs with smaller mass M < 10 8 g will play a different role. One of the important roles of lighter PBHs will be to give a stage for generating baryon asymmetry. Hawking [18], Carr [19] and Barrow [20] proposed a scenario of Baryogenesis in which GUT scale particle/right handed neutrino are created by the Hawking radiation and then decay in a C and CP violating manner (see also [21,22,23,24]). Recently Hook proposed a different scenario of Baryogenesis by using asymmetric Hawking radiation due to a dynamically generated baryonic chemical potential at the horizon [25]. There the CP violating interaction of the baryonic (or leptonic) current J α and the scalar curvature R is assumed, and the time evolution of the universe is used to generate the chemical potential µ =Ṙ/M 2 * for baryons. The same interaction is used in the gravitational baryogenesis [26]. Indeed, the mechanism [25] essentially utilizes the idea of the spontaneous baryogenesis [27,28] and gravitational baryogenesis [26] scenarios.
In this paper, we propose a new mechanism of the baryogenesis from evaporating PBHs. The mechanism is similar to that of [25], but instead of using the time evolution of the universe, we make use of the time evolution of the mass of the PBH itself for generating the Baryonic chemical potential 2 . This leads to a big difference between our mechanism and [25]. Since the scalar curvature around the PBH in vacuum is vanishing, we need to use higher dimensional operators such as where R µνρσ is the Riemann tensor 3 . The operator is largely suppressed by the scale M * and seems to be negligible but we show that it is sufficient to generate the desired asymmetry. It is due to the fact that the dynamically generated chemical potential µ, as well as the temperature of the Hawking radiation T H , is time-dependent through the mass of the BH. Indeed µ/T H is increasing as the PBH evaporates and becomes larger than 1 at the late stage of the evaporation. Consequently the asymmetry in the Hawking radiation becomes maximal after the PBH mass becomes smaller than a critical mass. The critical mass is determined by the scale M * . Once the chemical potential is generated at the horizon, Hawking radiation produces lepton/baryon asymmetry. Since the sphaleron process [31,32,33] violates B + L, it is necessary to generate B − L if the typical Hawking temperature is higher than 100 GeV. We thus assume violation of the B − L number in the underlining microscopic theories such as interactions with right-handed neutrinos or some effects related to the quantum gravity [34]. The CP symmetry is broken by the effective operator (2). The time dependence of the PBH mass due to Hawking radiation induces time dependent and position dependent chemical potential, which is apparently a non-equilibrium process. In this way, Sakharov's three conditions [35] are satisfied in the present model 4 .
2 A similar idea is proposed in [29]. Note that, by chemical potential, we here mean asymmetry of propagations between particles and anti-particles due to the interaction with the background geometry. If particles enter thermal equilibrium, the distributions become asymmetric. In the case of Hawking radiation, the radiation from black holes becomes asymmetric as if there is a chemical potential. Hence, in the present paper we call the term µJ 0 a chemical potential term. 3 Our investigation does not depend much on the specific form of the higher dimensional operators. We can instead use the Gauss-Bonne type which can be transformed to the scalaron picture [30], and is safe in view of the ghost modes. The first and the third terms in the Gauss-Bonne term vanish around the Schwartzschild BH, so it give the same effective CP-violating operator as the one we introduced. 4 It is often stated that the Sakharov's three conditions are not necessary in the spontaneous baryoge- The paper is organized as follows. In the next section, we explain the basic mechanism of the scenario, and estimate the order of the asymmetry. We show that in some region of the parameter space of the PBH mass M PBH and the scale M * , the desired asymmetry n B /s 8.7 × 10 −11 [36] can be generated. A possible origin of the higher dimensional operator (2) is given in Sec. 3, and we estimate the order of the scale M * . In Sec. 4 we show that washout of the generated lepton number outside the horizon does not occur for the typical interaction discussed in Sec. 3. Sec. 5 is devoted to the summary and discussions. In Appendix A and B, we discuss Hawking radiation with the chemical potential. In Appendix C, we give an analytical approximation of the function g n (X) used in Sec.2.

CP-violating interactions
The scenario of the gravitational baryogenesis [26] assumes the CP-violating interaction (1) where M * is the scale of the underlying theory that generates such an interaction. In an expanding universe, the time derivative of the scalar curvatureṘ is non-zero and the interaction generates a chemical potential 5 µ =Ṙ/M 2 * . If a B-violating interaction is present 6 and the system is in thermal equilibrium, the distribution becomes asymmetric between baryons and anti-baryons. Then once the temperature drops below the freezing-out temperature of the B-violating interaction, the asymmetry remains in the later universe. The scenario is applied to the evaporating BH by the Hawking radiation [25]. The term µJ 0 is similarly generated by the evolution of the universe and the Hawking radiation becomes asymmetric, but the freezing-out scenario is different. Since the thermal radiation from black holes is created, not by thermal process of B-violating interaction, but by genuine quantum process, the condition of the thermal equilibrium and the freezing-out in [26] is not necessary to be introduced in the analysis of [25].
In this section, we generalize the idea of the gravitational baryogenesis from a PBH [25] by taking the direct effect of the decay of the PBH mass M (t). Since the scalar curvature R vanishes outside of the BH in vacuum 7 , we consider an nesis. In the present scenario, since the CP parity of F (R....) ∝ R µνλσ 2 is even, the higher dimensional operator breaks C and CP symmetries. 5 As explained in footnote 2, the energy spectrum becomes asymmetric between particles and antiparticles. Then, as long as typical time scale of the interaction is smaller than that of the expansion of the universe, particles enter in thermal equilibrium and µJ 0 term can be interpreted as a chemical potential. 6 In the present section 2.1, for simplicity, we use B(aryon) to represent the current J µ . It can be either baryons or leptons but a necessary condition is that it has non-vanishing B − L charge. 7 In [26] and [25], the radiation dominated universe with the trace anomaly of the energy momentum tensor is studied so as to make R non-vanishing. operator such as in (2). More generally, we can consider a class of higher dimensional operators 8 , a n M 4n * It can be further generalized to where F (R .... ) is any scalar function made of the curvature tensors. For the square of the Riemann curvature of the Schwartzschild BH is given by 9 a non-vanishing chemical potential µ = a n ∂ 0 (R µνρσ R µνρσ ) n /M 4n * is generated if the BH mass M is decaying. Here we have introduced the reduced Planck scale, where G is the Newton constant. Note that the chemical potential is dependent on time through M (t). It also changes with the distance r from the BH. Since the Hawking radiation is generated by the Bogoliubov transformation between the vacua of quantum fields near the horizon and at far-infinity from the BH, the chemical potential near the horizon is relevant to generate the asymmetry of the Hawking radiation. The propagations of baryon and anti-baryon become different in the vicinity of the horizon, which shift the energy between them. Accordingly the generated asymmetry is proportional to the chemical potential evaluated at the horizon r = r H . Here we note that even if we instead evaluate the chemical potential at, e.g., r 2r H , it does not change our conclusion very much (see the 2nd paragraph of Sec. 5).

Basic properties of evaporating BH
We summarize some basic facts about an evaporating BH. For simplicity 10 we consider the Schwarzschild black hole with the metric, 8 In addition to the CP-violating interaction, we implicitly assume that the existence of B-violating operator since otherwise this term vanishes by performing an integration by parts. See Sec. 3 and App. A for the discussion of the physical meaning of this operator. 9 See, e.g., Ref. [37]. 10 In general, BH can have charge and angular momentum, but these would be quickly lost before most of BH mass disappears [38,39,40].
where M is the mass of the black hole. The radius of the horizon and Hawking temperature are given by Through the Hawking radiation, particles are emitted [41] from the BH with the rate where N, E are number and energy of emitted particles, ω is the frequency, and Γ is the absorption probability (or gray body factor), which is caused by gravitational scatterings of emitted particles outside the horizon. The absorption probability depends on particle species, especially on the spin of the emitted particles. At low frequency, ω → 0, the absorption cross sections σ := πω −2 Γ of spin 0 and 1/2 particles are constant while those of spin 1 and 2 particles are proportional to square and fourth power of frequency, respectively: const. for spin 0 and 1/2, ω 2 for spin 1, ω 4 for spin 2. .
As a result, most of the energy emitted from the PBH is carried by scalars and fermions [42]. We emphasize that the spectrum of the Hawking radiation is (almost) thermal, not because thermal plasma at temperature T H is realized due to sufficiently fast interactions between emitted particles, but simply because the quantum vacuum at the horizon behaves as if it is in the thermal equilibrium for an observer at far-infinity. In fact, even extremely weakly coupled particles (such as gravitons), that can be thermalized only at T H M P , are emitted according to the Hawking thermal spectrum.
Once we take into account the Hawking radiation, the spacetime is no longer stationary, and the metric (8) is no longer appropriate to describe the evaporating BH. Since the mass in (8) is the ADM mass which includes the energy of the emitted radiation it cannot correctly describe the mass of a decaying BH itself. A simplest alternative is the outgoing Vaidya metric [43], which is a solution of the Einstein equation describing outgoing null dust: where u = t − r * , r * = r + 2M log |(r − 2M )/2M |. The apparent horizon is located at r = r H = 2GM (u). The corresponding energy momentum tensor is given by which describes dust with energy density ρ = (−dM/du)/(4πr 2 ) moving with a four-velocity l µ . The mass M (u) represents the Bondi mass, which is nothing but the mass of the BH itself. In the following, we consider the time-evolution of the Bondi mass ∂ u M (u). Because of the (almost) thermal Hawking radiation, the black hole loses its energy following where α is a numerical coefficient [42] which can be determined by taking the effects of absorption cross section σ. As we discussed above, the dominant contribution to α in the standard model comes from fermions. In Ref. [42], it is shown that the contribution to α from ν e and ν µ is 1.575 × 10 −4 . Then, summing up all the fermionic degrees of freedom in the standard model, we obtain 11 Solving the Eq. (14), the time dependence of M is given by and the lifetime of the black hole τ becomes Here we take M = M PBH at the initial time u = u ini . From Eq. (16), we can see that the PBH completely evaporates until today if M PBH is smaller than 10 20 M P ∼ 10 15 g. 11 Here we assume that the value of α is the same to all the fermonic degrees of freedom, and the coefficients in the parenthesis are 1 for the SU (2) L singlet lepton, 2 for the doublet lepton, and 4 × 3 for up and down quarks with color degrees of freedom. The coefficient 3/2 is a transformation factor from the 2 generation calculation to 3.

Dynamically generated chemical potential
Since the square of the Riemann tensor outside a PBH is we have a chemical potential µ, if the CP-violating interaction Eq. (2) is present, By taking r = r H , the chemical potential evaluated at the horizon becomes Then the ratio of µ| r H to the Hawking temperature is T H = M 2 P /M is given by where we have defined the critical mass M cr by Note that the (absolute value of the) ratio is increasing as the BH mass M decreases. For M < M cr , the ratio exceeds 1 and the asymmetry of the radiation becomes maximal. It indicates that if the initial mass of the PBH is smaller than the critical mass, only baryons are emitted 12 .
Parametrizing the scale M * as M * = 10 x M P and the initial mass of the PBH as M PBH = 10 y M P , the condition of M PBH = M cr becomes We will see later that the relation plays an important role in generating the asymmetry.

Leptogenesis from a PBH
Now let us explicitly calculate the asymmetry produced by the evaporation of one PBH. Since we have in mind a model in which the CP-violating interaction is induced by the interactions with the right-handed neutrinos, we hereafter 12 Of course, particles without the baryon number are also emitted.
suppose that the chemical potential induced at the horizon is the leptonic one. Then, as we see that the temperature of the universe at the epoch of evaporation is much higher than the electroweak scale, sphaleron processes transmute the generated leptons into baryons. First note that the averaged energy per each emitted massless particle is where the subscripts L(L) and other represent the leptons (anti-leptons) and the other emitted particles (including only scalars and fermions) in the standard model respectively. n i is the number density for each species and given by where g i is the internal degrees of freedom. In the case of the standard model, g L = gL = 9 and g other = 76. n tot = n other + n L + nL is the total number. Li a (z) is the polylogarithmic function defined by Li a (z) := ∞ k=1 z k /k a . In the present convention, n L > nL since µ < 0. It is, of course, reversed if the sign of the coefficient of the CP violating operator is reversed. E i is the averaged energy for each species and the explicit expressions are given by Notice that E L behaves as E L ∼ T H for |µ| < T H , and E L ∼ |µ| for T H < |µ|. By using these formula, the lepton number asymmetry can be estimated as Here we introduced the lower cutoff M min for the mass of the PBH, under which the typical energy scale of the Hawking radiation becomes higher than M * and the present analysis becomes questionable. It is determined by the condition, either T H = M * or |µ| = M * , and given by For M < M min , either T H or |µ| is larger than the scale M * , and the present investigations are no longer valid 13 . Then changing the integration variable in The where the integrand is given by and Note that, from Eq. (22), X min is given by The function g(X) is numerically depicted in Fig. 1. One can see that it is peaked around X = O(1), which indicates that the dominant lepton number asymmetry is produced when the BH mass is comparable with the critical mass. The damping behavior of g(X) for large values of X means that when Since the generated lepton number is proportional to the integral of g(X), the asymmetry is efficiently produced around X ∼ 0.5, namely M ∼ 0.7M cr . For larger values of X = (M/M cr ) 2 , the chemical potential |µ| becomes negligibly small. For smaller values, the asymmetry is maximal but the number of emitted particles is reduced due to large |µ|.
the BH mass is larger than M cr , the radiation is almost symmetric and never contributes to the lepton number. On the other hand, the damping in small values of X implies that the chemical potential of leptons is too large, and the number of emitted leptons is significantly suppressed. See also App. C for the behavior and analytical approximations of f (X min , X 0 ) and g(X).

Lepton asymmetry in the universe
In order to discuss the lepton asymmetry in the universe, we briefly discuss the cosmological history of the light PBHs. When the density perturbation becomes as large as the order one δρ/ρ ∼ 1, the PBH can be formed. The mass of PBH is determined by the energy within the Hubble horizon, namely, where γ is a numerical factor depending on details of the gravitational collapse.
For it is usually considered to be γ 0.2, we take γ = 0.2 for simplicity. Here H ini is the Hubble parameter at the time of the PBH formation. After it is formed, it emanates the Hawking radiation and when the Hubble parameter becomes the inverse of the PBH lifetime, the PBH evaporates completely.Thus the ratio of the Hubble parameters is given by If the universe continues to be in the radiation dominated phase, the Hubble parameter is related to the scale factor of the universe as H ∝ a −2 . Then the ratio of the scale factors a during the evaporation is given by Since the energy density of the universe changes as the ratio of the energy density of PBHs ρ PBH to the total energy density of the universe ρ rad increases as the universe expands. Of course, the evaporation transfers the energy from the PBH to the radiation component and the actual evolution is more complicated [44]. It is not further discussed in the present paper.
The temperature of radiation just after the PBH evaporation T eva can be estimated from Here g * is the effective degrees of freedom. Then we have Notice that the typical value of T eva is much higher than the electroweak scale for M PBH 10 11 M P ∼ 10 5 g, and the lepton asymmetry produced by the PBH evaporation can be converted into the baryon asymmetry by the sphaleron process.
We also note that the Hubble parameter H ini must be smaller than the Hubble parameter during inflation H inf ∼ 10 14 r/0.1 GeV, where r is the tensor to scalar ratio. Thus we have the lower bound on the PBH Only PBHs satisfying the condition can be created in our universe.
Having the above cosmological history in mind, we can estimate the lepton asymmetry after the PBH evaporation, where Ω P BH = ρ PBH /ρ tot is the ratio of the energy density of the PBHs to the total energy density at the epoch of evaporation which includes the radiation from PBHs. Assuming domination of the radiation after the evaporation, and using ρ tot /s = 3T eva /4, (29) and (39), we have which is conserved until now under the assumption that there is no other entropy production. From this rough estimation, we can see that the observed amount of the asymmetry, n L /s ∼ 10 −10 , can be successfully produced unless f (X min , X 0 ) is too small.
In the left panel of Fig. 2, we plot the lepton asymmetry generated in presence of the CP-violating interaction (2) for Ω PBH = 1 as a function of M * 14 . In order to generate the observed baryon asymmetry n B /s 8.7 × 10 −11 , the mass of the PBH (the vertical axis) and the scale M * suppressing the interaction must be on the line with ∼ 10 −10 . If the density ratio of PBHs is less than 1, i.e., Ω PBH = 10 −s with s > 0, the parameters (M PBH , M * ) must be on the line with a larger value ∼ 10 s−10 . There are 3 lines in the figure. On the dotted line (the lowest line), the initial mass M PBH of PBH is equal to the critical mass M cr . The other two lines represent |µ| = M * and T H = M * . The region below these lines is beyond the reach of this paper since the typical energy scale is larger than M * .
From Eq.(32), we see that the value M * ∼ 10 −3 M P corresponds to X min ∼ 1, and the M * dependence of asymmetry becomes different between the right region with X min 1 and the left region with X min 1. For X min 1, according to Fig. 1, f (an integral of g) becomes almost independent of X min . Since the asymmetry is most dominantly generated around the dotted line (X = 1) and the critical mass increases as M * decreases (see Eq.(22)), the asymmetry also increases when M PBH fixed. On the other hand, for X min 1, f strongly depends on the lower cutoff X min . It is shown in the first paragraph of Appendix C that the produced asymmetry becomes independent of M * for X min 1. This is the reason why the generated asymmetry becomes flat as a function of M * below M * ∼ 10 −3 M P . below which the typical energy scale becomes higher than the scale M * and the present calculation is no longer valid. The plot is drawn using the formulae in Appendix C. Below M * < 10 −3 M P , the produced asymmetry becomes flat and independent of M * . It is due to our prescription to cut off the X integral at X min (see the first paragraph of Appendix C and (32)), and the asymmetry in this region may be interpreted as a conservative estimation, as noted in footnote 13. A PBH with mass lower than M PBH ∼ 10 5 M P is not created in our universe as discussed in (40). The graph shows that the baryogenesis from the PBH works as far as Ω PBH > 10 −2 . The right panel shows the lepton asymmetry in the case of a generalized CP-violating operator for n = 10 discussed in section 2.6. We use Eq. (51) with a n = 1/n and Ω PBH = 1. For n ∼ 10, more asymmetry is efficiently generated compared to the n = 1 case in the left panel, and the density ratio of PBH can be as low as Ω PBH = 10 −6 . Below the dashed line, the asymmetry is suppressed because the function g n (X) significantly decreases for X 1. The magenta line (|µ| = M * ) almost coincides with the dashed line.. Unlike the left panel, the produced asymmetry does not become flat because, within the region of M * in the graph, X min < 1 is always satisfied and the integral is independent of the lower cutoff X min .
The shaded region is not allowed because below M PBH ∼ 10 5 g, PBH is not created in our universe (40). Therefore, Figure 2 shows that the baryogenesis from the PBH works as far as Ω PBH > 10 −2 for the simplest CP violating operator of dimension 8. It is based on our conservative assumption 15 that the lepton asymmetry produced in the region of T H > M * is not counted.

More general CP-violating interactions
So far we have studied the CP-violating interaction of (2). We extend it to more general higher dimensional operators introduced in Eq. (4) 16 . The calculation is the same as the simplest case discussed so far. The chemical potential is dynamically generated at the horizon and the ratio to the Hawking temperature is given by where the critical mass is given by Notice that, compared to the n = 1 case in (22), the exponent of (M P /M * ) in √ C n is larger and accordingly the critical mass becomes larger. It is good for producing larger asymmetry since the asymmetry is produced when the BH mass is around the critical mass. Parametrizing M * as M * = 10 x M P and M PBH = 10 y M P , the condition of M PBH = M cr for n → ∞ with a n = 1/n becomes On the other hand, as shown in eq. (43), the chemical potential becomes too large when M becomes larger than M cr . It is bad for asymmetry generation because it enhances the typical energy E ∼ µ of emitted particles and consequently reduces number of particles dM/ E while the BH decreases its mass by dM . These two effects compete as n becomes large. 15 The assumption is partially based on the fact that the typical energy of the Hawking radiation is given by T H (> |µ|) and thus number of emanated particles is drastically reduced for very high T H . 16 A truncation of higher order terms is assumed in this subsection. In order to estimate the asymmetry starting from the ultraviolet theory such as the model in Sec. 3, it is necessary to compute the full propagator in curved background without relying on the derivative expansion. We want to come back to this problem in future publications.
By introducing a new integration variable X = (M/M cr ) 2 as before, we obtain the lepton number emitted from a single PBH as where f n is an integral f n (X min , X 0 ) = X 0 X min dXg n (X) over the PBH mass X. Here the function g n is given by and the lower bound of the integral is given by By using Eq. (44), it is found and, for n = 10, M * eq,n=10 ∼ 10 −28 M P . Thus, in the region of M * > 10 −8 M P , X min < 1 is always satisfied. The function f n (X min , X 0 ) behaves similarly to the previously defined function f (X min , X 0 ) = f 1 (X min , X 0 ), but, as shown in Fig.6, the function g n (the integrand of f n ) is more sharply peaked around X 0 ∼ 1. In the large n limit, Hence, for large n, the lepton asymmetry is found to be n L s 9.3 × 10 −7 106.75 g * 1/4 α 3.5 × 10 −3

1/2
× Ω PBH f n (X min , X 0 ) 10 5 M P M PBH 5/2 According to the behavior of a n , the asymptotic behavior of (na n ) 1/2n is given as for a n ∼ 1 n e n for a n ∼ 1 n! ∼ n −n e n .
As discussed in App. C, the function f n is suppressed by n as f n ∼ 1/n, and the asymmetry vanishes in the n → ∞ limit. It is due to the rapid increase of the chemical potential µ for a larger n at M < M cr . However, it is interesting to note that, for moderate values of n, the higher dimensional operators are more efficient to produce larger amount of the asymmetry thanks to the increasing of M cr . In order to see the competition of these two effects of taking large n, we plot n dependences of the function f n , the critical mass M 2 cr /M 2 P and the Lepton asymmetry n L /s = f n × M 2 cr /(2M 2 P ) in Fig. 3. It can be seen that f n behaves as 1/n for large n, as expected. On the contrary, M cr is a monotonically increasing function of n. As a result, n L /s has the peak around n = 10-20.
We now explicitly estimate the produced asymmetry created by the higher dimensional interactions, assuming (na n ) 1/2n ∼ 1. In the right panel of Fig. 2, we plot the produced asymmetry n L /s in Eq. (51) for Ω PBH = 1 and n = 10. We can see that the observed asymmetry, n B /s 8.7 × 10 −11 , is realized in a wide range of parameters as far as Ω PBH > 10 −6 .
We note that, if the mass of the PBH is less than 10 8 g [12], there are no observational constraints. Thus the mass region in which the present baryogenesis scenario works is all allowed. It is also interesting to note that the generalized higher dimensional interactions suppressed by a power of M * can generate larger baryon asymmetry. It is due to the fact that the critical mass M cr , around which the most lepton number is generated, becomes larger for larger n. But it also indicates that the present calculation of using the derivative expansions needs improvement. Actually, when the BH mass is reduced to the critical mass M cr , the scale of the curvature tensor becomes comparable to M * , and the Compton wave length of the underlying particles that induce the CP violating coupling to the background gravity becomes comparable to the horizon radius of the BH. It is important to study the asymmetric Hawking radiation without resorting to the derivative expansions adopted in the present paper. We hope to come back in future investigations. mechanism discussed in the previous section is not restricted to this particular possibility. But, in order to get a rough order estimation of the scale of M * , we study this possibility in this section 17 .
In order to generate these operators, the underlying physics need to violate CP symmetry and the lepton number. One of the simplest possibility is the Majorana mass terms of the right handed neutrinos. As usual, imaginary phases of the mass matrix break the CP symmetry. The Lagrangian we consider is where y N is the neutrino Yukawa coupling, and M N is the mass of the right handed neutrinos. Recently, Mcdonald and Shore [45,46] explicitly calculated loop diagrams containing the above interactions coupled with the gravity in the external lines. They took the conformally flat metric, g µν = (1 + h)η µν where the field h is treated as a background field, and then calculated the two-loop Feynman diagrams in Fig. 5 to obtain the higher dimensional operator used in the gravitational leptogenesis, Since the right handed neutrinos are integrated out, the mass scale of the higher dimensional interaction is essentially given by 18 The higher dimensional operators utilized in the present paper will also arise by calculating similar diagrams with more gravitons in the external lines 19 . To prove it, we need to consider a more generic form of the metric and further involved calculations are necessary, so we leave it for future investigations. From dimensional and diagramatic arguments, it can be expected that the mass scale M * is given by Let us now give a rough estimation for the scale M * using this formula. The see-saw mechanism indicates that the typical order of M N is given by M N ∼ 17 If the right-handed neutrinos are responsible for the CP-violating higher dimensional interactions, the ordinary leptogenesis can also occur and we need to take it into account to calculate the total abundance of baryon asymmetry. In the present paper, we consider only the abundance generated through the PBHs. 18 These authors further claimed possible appearance of an enhancement factor depending on the mass hierarchy of the right handed neutrinos. We do not discuss such effects here. 19 It is not evident what types of higher dimensional operators can be induced, but their specific forms are not very important in our discussions as commented in the footnote 3.
(y N v) 2 /m ν ∼ y 2 N 10 16 GeV, where v ∼ 100 GeV and m ν ∼ 10 −3 eV. The scale M * is proportional to M N but also dependent on the Yukawa coupling y N and the CP-violating parameter := Im((y N y † N ) 2 /|(y N y † N ) 2 |). If we take the Yukawa coupling as y N ∼ 1, smallness of makes the scale M * larger than 10 16 GeV. On the other hand, if we assume that y N is as small as, e.g. y N = 10 −5 and the CP-violating parameter is of order 1, the scale M * can be lowered to 10 11 GeV. Note that from recent arguments on the hierarchy problem of the electroweak scale, it is preferable to take the scale of the right-handed neutrinos relatively lower. Hence the scale M * may be expected to take a value in the region, 10 −7 M P < M * < M P .
We also emphasize that, other than the right handed neutrinos, if CP is broken in the UV theory including gravity, we can naturally expect appearance of the operators like Eq.(4). These are left for future studies.

Effect of washout outside the BH
If the CP-violating interaction such as Eq.(4) is present, decay of the PBH mass by Hawking radiation breaks time-reversal symmetry and generates chemical potential at the horizon. Then the radiation becomes asymmetric. As already mentioned before, the distribution looks thermal but it does not mean that the lepton number violating interaction is in the thermal equilibrium. The Hawking radiation is emitted because of the quantum mechanical effect, and reflects the fact that the quantum vacuum near the horizon is different from the Minkowski vacuum. In this respect, the mechanism of baryogenesis is very different from the gravitational baryogenesis [26], in which decoupling of the baryon number violating interaction is necessary to fix the final amount of asymmetry.
It is, however, necessary to check whether the generated asymmetry is washed out by the baryon (or lepton) number violating interactions outside of the black hole horizon. As discussed in the previous section, we assumed that the microscopic process violates the lepton number through the interaction with the right-handed neutrinos. By integrating out them, we have the following dimension-5 operator, After the Higgs acquires the vev H = v 246 GeV, it gives the mass to the neutrinos. Hence Λ is determined to be where m ν is the neutrino mass and we put m ν = 10 −3 eV.
The cross section is estimated by σ ∼ 1/Λ 2 at low energy scale. The particle number density is given by n ∼ T 3 H near the horizon where the particles are created. Then the scattering rate of the lepton number violating interaction at the horizon is given by But the rate is much slower than the typical time scale of the created particle to move away from the BH, namely Γ 0 / L r −1 H ∼ (M P /M )M P for the BH mass M 10 5 M P . Hence the particle density is quickly diluted by the factor (r H /r) 2 , where r is the distance from the BH, and the interaction rate is reduced to Γ / L = Γ 0 Since the Hubble parameter of the universe at the epoch of evaporation in (34) is estimated to be H eva ∼ 10 −14 (10 5 M P /M ) 3 M P , the dilution factor (r H /r) 2 instantly makes Γ / L much smaller than H eva . Hence the interaction is not in the chemical equilibrium at the time of evaporation with the Hubble H eva .
Next we check the condition of washout when the particles are at higher energy than M N . The infinite blue shift near the BH horizon enhances the energy of the particle by the factor (1 − r H /r) −1/2 . Then, near the horizon, the cross section of lepton number violating interaction is replaced by its high energy counterpart σ ∼ y 2 N /s ∼ y 2 N (1 − r H /r)/T 2 H . Taking the dilution factor (r H /r) 2 into account, we have The maximal value of Γ / L is obtained by at the position r ∼ 1.5 r H . Again this is much smaller than r −1 H by a factor 10 −2 y 2 N , and the emitted particles move away quickly from the BH, so the scattering rate is reduced to Γ / L ∼ (r H /r) 2 Γ max . Compared with the Hubble H eva in (34), when the particle moves to r ∼ 10 3 r H , Γ / L < H eva is satisfied. Hence, if y N < 0.1 the time that the particle moves to that position is sufficiently short for the lepton number violating interaction to occur, and the generated lepton number is never washed out.

Summary and discussions
In this paper, we have proposed a new scenario of baryogenesis from evaporating PBHs. The key element is the CP violating operator Eq. (2), which generates the splitting of energy spectrum between particles and anti-particles if the BH is decaying. The mechanism is similar in spirit to the gravitational baryogenesis [26] or the mechanism by Hook [25] who applied the gravitational baryogenesis to the PBH, but an essential difference is that we make use of the time-evolution of the BH itself [29], not the cosmological evolution, to generate the chemical potential. Because of this, the ratio of the chemical potential to the Hawking temperature µ/T H becomes a function of the decaying mass of evaporating BHs and, when the mass becomes under the critical mass, the ratio |µ/T H | exceeds 1. After this epoch, maximal asymmetry can be generated. Due to such efficiency of generation mechanism, even though the CP-violating operator is largely suppressed by high energy scale M * , we show that sufficient amount of baryon asymmetry n B /s 8.7 × 10 −11 can be obtained in wide range of parameter space if the density ratio of PBHs at the epoch of evaporation is Ω PBH > 10 −6 . If the scenario really explains the baryon asymmetry of the universe, it constrains model buildings beyond the standard model because the PBH can radiate heavy particles that may decay later, e.g. during the BBN. Thus the present scenario favors simpler model buildings such as [47].
The estimation of the generated asymmetry and the requirement for Ω PBH in the present paper will change if we take other effects into account. In our analysis we evaluated the chemical potential at the horizon r = r H . There is, however, a discussion [48] [49] that the Hawking radiation originates in the larger-region r = cr H > r H . According to [49], c = 3 √ 3/2 ∼ 2.6 is given for high frequency modes. But the effect can be always absorbed in the definition of the scale M * . Indeed if we instead evaluate µ at r = cr H , the chemical potential is reduced by c −6n and the critical mass M cr is reduced by the factor c −6n/(4n+2) . From the definition of the critical mass, it is equivalent to increasing the effective M * by c 6n/(4n+2) . For n = 1 and n = 10 with c = 3 √ 3/2, the numerical factors are 2.6 and 3.9 respectively.
In the analysis we used the adiabatic approximation, namely we have implicitly assumed that the time scale characterizing the Hawking radiation is shorter than that of the change of mass of the PBH. Here we confirm the validity of this assumption. The typical time scale of Hawking radiation is estimated from the uncertainty relation between time and energy, The assumption of the adiabaticity is justified if the change of the PBH mass is negligibly small during ∆t. Using Eq. which is much smaller than M for M M P . Therefore, we can safely treat the system adiabatically.
If PBHs are responsible for the baryon asymmetry in the universe, they can also be responsible for gravitational waves. PBHs emit gravitational waves either by Hawking radiation or by a formation of PBH binaries, but the Hawking radiation would provide the strongest signal [50]. In Ref. [50], they estimated the peak frequency f (peak) and the peak amplitude h 2 0 Ω GW as if PBH dominates the universe. Note that the peak amplitude does not depend on M PBH . In Fig. 4, the peak frequency of the gravitational waves from the PBH is plotted as a function of its mass. It is amusing to consider a cosmological history in which PBHs dominate in the early universe. But the expected gravitational waves seem to be difficult to observe since its frequencies are too high for the near future experiments. The final comment is the formation mechanism of the PBHs. There are various possibilities to create PBHs in the early universe as summarized in the introduction. The bubble collisions [6] associated with the first order phase transition in the universe [7] is becoming more interesting recently since the discovery of the Higgs boson and strong constraints on the TeV scale physics have stimulated reconsideration of our view of the cosmological history. In particular, revival of radiative symmetry breaking via the Coleman Weinberg mechanism [51,52,53] suggests that the phase transition will be a strong first order type. In such models, bubble collisions of the true vacua can generate strong gravitational waves [54], topological objects such as monopoles [55], or PBHs [6]. It is interesting to pursue further cosmological consequences of the strong first order phase transitions, in relation to the particles physics models beyond the SM.

A Is chemical potential physical?
If the (lepton number) current is conserved, the operator of the type in Eq. (5) seems to vanish by integrating by parts or by rotating the phase of leptons as ψ as ψ → exp(iF (R .... ))ψ. It implies that the lepton number violation is necessary to justify the microscopic origin of such CP-violating effective operators. Indeed McDonald and Shore [46] obtained such a term by integrating the right-handed neutrinos whose interactions violate the CP and the lepton number conservation microscopically.
But it is still paradoxical why we are not allowed to do such phase rotation to remove the effect of lepton asymmetry. First note that if the interaction is replace by µJ 0 where µ =Ḟ is assumed to be a constant, the phase rotation becomes ψ → exp(iµt)ψ. This is nothing but the shift of the energy ω → ω − µ for leptons and ω → ω + µ anti-leptons. Since the vacuum state is defined so as to fill all negative energy levels, the phase rotations simply changes the definitions of the vacuum. Thus such phase rotation should be correlated with the definition of the vacuum state, and with the definition of the lepton number of the vacuum.
In the case we are discussing, the current is coupled to the total derivative ∂ α F (R) and F (R) is a smooth function in the spacetime. Especially it vanishes at r = ∞ and takes a nonvanishing value at the BH horizon r = r H . Therefore, if we assume that F (R) is a slowly changing function in time and can be expanded as F (t, r) = F 0 (r) + µ(r)t + · · · , the phase rotation changes the energy levels of leptons ω → ω − µ(r) as a function of the position r. In this sense, the situation is similar to the discussion of chiral anomaly as the spectral flow in the Hamiltonian formulation. Now the question is that how we can define the vacuum of quantum field at r = ∞ and r = r H separately. And this is nothing but the issue of the Hawking radiation. At r = ∞, F (R) vanishes and there is no ambiguity in defining the vacuum. At r = r H , we first define the appropriate vacuum state (such as the Unruh vacuum) so that an infalling observer does not encounter any divergences, and then calculate the effective action. Once the effective operator is induced as performed in [46,45], then we can no longer shift the energy level (see also [56] and [57].) This is the reason why we should not rotate the basis after we calculate the effective interaction, and the CPviolating operator and the resulting chemical potential derived from Eq. (53) has a physical meaning. 20 It will be interesting to obtain the asymmetry in spectrum of the Hawking radiation without resorting to the calculation of the effective interaction, i.e. by explicitly calculating the lepton wave function in the eikonal approximation with the coupling to the right-handed neutrinos included. We want to come back to this problem in future. (See also Appendix B.)

B Hawking radiation with a chemical potential
Here we briefly explain why the chemical potential modifies the spectrum of Hawking radiation. We start from the following action in the Schwartzschild black hole geometry.
The action has the same form as by identifying C ∂ µ R αβγδ R αβγδ /M 4 * with the gauge potential A µ . Thus the coupling to the background gravity is nothing but the pure gauge configuration (but it cannot be gauged away as discussed in the previous Appendix). It is now instructive to briefly sketch the derivation of Hawking radiation in the case of the charged black hole following Refs. [58,59]. At the outer event horizon, the action of the charged fermion becomes Here we omit the contribution from angular components for simplicity, and Q is the charge of the black hole. In order to obtain the outgoing flow of the energy at the future infinity, we usually impose the ingoing boundary condition for the current at the horizon. But since the above scalar potential diverges A U ∝ A t /U at the horizon U = 0 in the Kruskal coordinate, we need to take the gauge in which A t = 0 at the horizon (see discussions e.g. [58,59]) This shifts the energy level of the charged particles and the spectrum of the Hawking radiation is obtained by replacing ω by ω − µ = eQ/r H . Thus eQ/r H indeed plays the role of the chemical potential in the thermal radiation. The same procedure is applicable to the current setup. In our case, the "gauge transformation" to regularize the action at the horizon in the Kruskal coordinate is given by Thus becomes the chemical potential to describe the thermal radiation from the PBH.

C Analytical approximation
In this appendix, we present an analytical approximation of the function g n (X) which is defined in Eq. (47) and appears in the calculation of the asymmetry. We investigate the large n behavior of the integral. For large X, it can be expanded with respect to 1/X as g n (X) 40g L 7 ((2g L + g other )π 2 ) 1 X 2n+1 + 40g L (46g L − 7g other ) 49(2g L + g other ) 2 π 4 1 X 2n+1 3 + · · · (71) For X → ∞, the first term gives a good approximation for g n (X) and we have an approximated formula for the integral; for 1 ≤ X min ≤ X max . Then, from Eq.(46), we have δN L ∝ M 2 cr X −2n min . In this region (X > 1), T H > µ and X min is given by X min = (M 2 P /M cr M * ) 2 . By using the result for the critical mass M cr in (44), it turns out that δN L is independent of the mass scale M * . Since the entropy s is determined by the life time of the PBH, namely M PBH , and independent of M * , n L /s also becomes independent of M * . The region 1 < X min is given by the region where the line of M PBH = M cr is below the line of T H (M PBH ) = M * , and corresponds to the region with M * 10 −3 M P in the left panel of Fig. 2. This is the reason why the generated asymmetry is constant as a function of M * for M * 10 −3 M P . But we notice that the behavior is given by our prescription to cut the integral at X min where the typical energy of the Hawking radiation becomes higher than M * and the present investigations become questionable. Though it is beyond our approximation to estimate δN L for X X min , larger asymmetry may be produced in this region. In this sense, the estimation in this paper would give a conservative value for the lepton asymmetry.
On the other hand, in the region X ∼ 0, we have g app (X) = − 40g L X 2n+1 1 + π 2 (X 2n+1 ) 2 30g L + 60g L π 2 (X 2n+1 ) 2 + 7(2g L + g other )π 4 (X 2n+1 ) 4 , where C := 30g L /(7π 4 (2g L + g other )). Surprisingly, as shown in Fig. 6, this expression gives a good approximation even for X > 1 as well as for X ∼ 0. Since Eq. (73) is a rational function of X, its integral can be performed  Figure 6: The solid orange and dashed black lines correspond to g(X) and g app (X), respectively. One can see that g app (X) is good approximation for all the region of X, and the integrand has strong peak around X ∼ 1. Note that the position of the peak for n = 10 is closer to X = 1 than for n = 1.

(74)
Here we have defined 2n + (1/2) 2n + 1 ± 1 2 ; 1 where 2 F 1 is the hypergeometric function. In the numerical calculation for drawing the figures, we used the expression of Eq. (74). Finally let us examine the large n behavior of f app (X 0 , X min ). We concen-trate on the region X 0 > 1 for simplicity. The behavior of X min is given by X min → 1 − 1 8n (10 log 2 + log 3 + 4 log π) , and thus Then the hypergeometric function is approximated for large n as for Y = X min .