# Non-linear metric perturbations and production of primordial black holes

###### Abstract

We consider the simple inflationary model with peculiarity in the form of ”plateau” in the inflaton potential. We use the formalism of coarse-grained field in order to describe the production of metric perturbations of an arbitrary amplitude, and obtain non-Gaussian probability function for such metric perturbations. We associate the spatial regions having large perturbations with the regions going to primordial black holes after inflation. We show that in our model the non-linear effects can lead to overproduction of the primordial black holes.

PACS number(s): 98.80.Cq, 97.60.Lf, 98.70. Vc, 98.80.Hw

## I Introduction

Starting from pioneering works by Zel’dovitch and Novikov [1], and also by Hawking [2], the primordial black holes (hereafter PBH’s) were subject of extensive ivestigations. The presence of PBH’s may significantly influence on physical processes and effects in the Universe (such as nucleosynthesis, CMBR spectral distortions, or distortions of -ray background radiation) due to Hawking effect [3], PBH’s may be a component of dark matter (see e.g. [4], [5]). The formation of PBH’s is determined by small scale, but large amplitude inhomogeneities in the Early Universe, and the processes of PBH’s formation, evolution and decay link the physical conditions of Early Universe with conditions in the radiation-dominated epoch and present-day cosmology. Even the very absence of PBH’s may significantly constraint the models of the beginning of cosmological evolution.

Usually the processes of PBH’s formation are associated with production of the scalar mode of perturbations during inflation (see e.g. [5-9]) or phase transitions in the Early Universe [10]. In this paper we are going to discuss the first possibility, which allows to use the powerful and well-elaborated theory of instability of the expanding Universe for analysis of conditions, under which PBH’s can form.

The theory of generation of adiabatic perturbations during inflation started from pioneering papers [11-13]. It was established that the RMS-amplitude of metric perturbations is connected with the parameters of inflationary theory by means of relation

where is the Hubble parameter, is the velocity of the field, evolving during inflation. To get PBH’s abundance in an observable amount, one should have (see, e. g. [14]).On the other hand COBE CMBR data, as well as analysis of Large-Scale Structure formation constraint the amplitude of perturbations at super-large scales. Therefore to get PBH’s one should increase the amplitude of the perturbations by a factor at small scales. Unfortunately this cannot be reached in the simplest inflationary models, since in these models logarithmically grows with increase of scale, and one should use nonstandard models having additional power at small scales to obtain significant PBH’s amount.

Recently, several models of such type were proposed. For instance, Carr and Lidsey [6] proposed toy model having blue type spectrum (the spectrum , where is the wavenumber, and is the spectral index), and investigated the constraint on the spectral index associated with possible PBH’s formation in such model. Linde [15] has shown that blue type spectra can be naturally obtained in the two-field model of so-called hybrid inflation.

Another type of model having a spike in the power spectrum at some scale
was proposed by Ivanov, Naselsky and Novikov ([5], hereafter INN)
^{1}^{1}1See also the papers by Hodges and Blumenthal, Hodges
et
all[16] and Kates
et
all [17], who employed similar models in contest of Large-Scale Structure
formation theory .
They considered one-field inflationary model with
inflaton and assumed that
the potential has
a ”plateau” region at some scale , and has a
standard form (say, power-law form) outside the ”plateau” region.
The field slows down in the ”plateau” region giving increase of
the spectrum of perturbations at the scale
according to eqn. . One can
adjust the parameters of ”plateau” region to obtain the desired increase of
the spectrum,
and consequently the desired PBH’s amount. Garcia-Bellido et all [8]
and also Randall et all [9] considered more realistic two field models
having a saddle point in two-dimensional form of potential .
Like the one-field model, the evolution of the system
of fields slows down near the saddle point giving an increase of the spectrum
power. Randall et all pointed out that such models solve several
fine-tuning problems of the standard inflation, and therefore look
very natural from the point of view of high energy physics. Garcia-Bellido et
all carefully investigated the process of PBH’s formation in such models
(see also recent work by Yokoyama, [18]).

If the primordial black holes are not super-large, they probably collapse during the radiation dominated epoch of the evolution of the Universe. This means that the amplitude of the metric inhomogeneities inside the regions going to PBH’s should be of order of unity to overcome the strong pressure forces during collapse of the perturbed region [14]. These large amplitude metric inhomogeneities are assumed to be generated during inflation as rare events in the random field of the metric perturbations. Since the amplitude of the inhomogeneities is rather large, the natural question appears: to what extent we can rely on the linear theory of perturbations which usually gives Gaussian probability distribution of PBH’s formation?

To answer this question we can apply the formalism of coarse-grained fields (introduced by Starobinsky [19]) as an alternative approach to the linear theory that can describe large amplitude deviations of the field and the metric from the background quantities. According to this approach, the spatially inhomogeneous field is divided into two parts: the large-scale part , which consists of the modes with physical wavelengths greater than some characteristic scale , and the small scale part which consists of modes with . During inflation, the physical wavelengths are stretched, and new perturbations are added to . This effect may be considered as a new random force in the equation of motion of the field , and usually the dynamics of is described in terms of diffusion equation for probability density . This equation was subject of a number of works in connection with problems of Quantum Gravity and Large-Scale Structure formation. Recently, it was pointed out, that this equation can be employed for calculations of the probability to find large amplitude peaks in the random distribution of field , and it was mentioned that such approach can be applied to the problem of PBH’s formation [20].

Here we would like to note that when studying the effects originating after the end of inflation, such as PBH’s formation, one should use the large scale part of metric instead of large scale part of field. Contrary to the field , the large scale part of the metric, namely the ”inhomogeneous scale factor ” (see eqns. for exact definition) is the quantity conserving during the evolution outside the horizon, and this property allows to connect the physical conditions during inflation with the physical conditions during radiation-dominated epoch, when PBH’s are formed. Moreover, the criterion for PBH’s formation can be directly formulated in terms of ( Refs. [21], [22]). Therefore, the calculation of gives a tool to describe quantitatively the generation of non-linear metric perturbations, and the evolution of these perturbations into PBH’s.

In this paper we calculate the probability distribution function in the model with almost flat region in the inflaton potential. The main idea of our calculations has already been applied in the models of so-called stochastic inflation (see, e. g. [23] and references therein), and is very simple. When the field evolves inside the plateau region it slows down, and the random kicks (described by the force ) significantly influence on its evolution. So, the trajectory of the field inside the plateau region becomes stochastic, and the time that the field spends on the plateau, depends on the realization of the stochastic process. The total increase of the scale factor during the field evolution on the plateau, is obviously determined by : . Since different regions of the Universe separated by distances greater than evolve independently, the increase of corresponding to different regions is determined by different realizations of the random process. Thus the scale factor varies from one region to another after the field passes the plateau, that is the quantum effects generate the coordinate dependence of the scale factor. The shape of function is conserved during the subsequent evolution of the Universe until the scale of inhomogeneity crosses horizon at the second time. At that time, in the regions with significant contrast of the primordial black holes are formed.

Using the approach described above we calculate the probability distribution function
. With help of a simple criterion of PBH’s formation we
relate to the probability of PBH’s formation. We show that
in our case the non-linear effects over-produce PBH’s
^{2}^{2}2Note that this result differs from that obtained in Ref. [20]..
Although this result is very important qualitatively, it does not significantly
change the estimate based on the linear theory.

We use the simple one-field model, proposed by INN (see also Refs. [24], [25]). Due to simplicity of this model the bulk of our results are obtained analytically. We hope that our approach provides a reasonable approximation to the case of more complicated two-field models. We are going to discuss these models in our future work.

The paper is organized as following. We introduce our model and discuss the classical dynamics of the metric and field in Section 2. In Section 3 we obtain an expression for . We consider the role of non-linear effects on the statistics of PBH’s production in Section 4. We summarize our conclusions and discuss applicability of our approach in Section 5.

## Ii The dynamics of classical model

In this Section we consider the classical dynamics of spatially homogeneous parts of metric and field in the simplest inflationary model with a single scalar field (inflaton) and with the peculiarity in the inflaton potential. In this case the system of dynamical equations contains only two dynamical variables -scale factor and spatially homogeneous part of the field , and reduces to the Hamiltonian constraint equation

and to the equation of motion for field

where , and another symbols have their usual meaning. We hereafter use the natural system of units.

We assume that the effective potential has a small almost flat region (’plateau’) between some characteristic values of field and . The potential is also assumed to be proportional to outside the ’plateau’ region

at

at , and

at . Here , , so we can set . At sufficiently large values of the kinetic term in the equation is negligible in comparison with the potential term . As we will see below the size of the flat region is very small

and the equation reduces to an algebraical relation between and (so-called slow-roll approximation)

From the equation it follows that the Universe expands quasi-exponentially ( and ) at .

It can also be easily shown that outside the plateau region the field moves with large friction at , so

The friction dominated condition helps to simplify the integration of the system . Integrating the eqns. with help of inequalities , at , we have

and

where and are some initial values of the field and scale factor.

is the number of e-folds of the scale factor during the field rolling down starting from some initial value of and down to the field . The similar formulae hold at

where and , and is the number of e-folds up to the end of inflation: , where we assume that inflation ends at standard (for theory) value of . Note, that should be rather large. For example, to get a feature in the spectrum at scales, corresponding to the solar mass, we should have . Therefore, the value of should be greater than unity ( for ).

Now let us consider the dynamics of inflaton in the ”plateau” region . In this region the equation is simplified to

where . The solution of eqn. can be written as

and for the field velocity we have

where

Our model has two possible limiting variants depending on the relation between the time of the crossing of plateau region by the field () and . If the field crosses the plateau mainly due to inertia. In this case the parameter determines the number of e-folds during plateau crossing , and therefore the width of produced bump in the spectrum remains small and fixed. The model of similar type was discussed by INN. Here we consider another possible case , where the field spends some time on the plateau, evolving in the friction-dominated approximation. In this case the width of the spectrum is determined by the value of , which is the free parameter of our model. Instead of we will parameterize our model by the quantity -the ratio of wave numbers, corresponding to the fields , , respectively, . The parameter cannot be too small and we take in the estimations. If is not extremely large , the size of plateau is of order of typical size . The typical relative size of plateau is very small

Thus, the correction due to the presence of plateau practically does not influence on the dynamics of the field outside plateau region and we can set . On the other hand, the size of plateau is much greater than - the typical size of quantum fluctuations, .

Typically, the estimate holds for arbitrary power-low potentials provided power is not very large. However the opposite limiting case is also possible. For example, Bullock and Primack [20] proposed the potential of the form

where the constant is chosen to normalize the large-scale part of spectrum to the RMS-amplitude . The flat region in this potential starts from and ends at , and inflation ends itself at . It was mentioned by Bullock Primack that this potential leads to strongly non-Gaussian statistics of field perturbations.

## Iii Non-linear metric perturbations from the quantum dynamics of coarse-grained field

It is well known that there are two equivalent ways to describe inhomogeneous Universe. The first way is to consider inhomogeneities as a small corrections to the homogeneous space-time and study them in the frameworks of linear theory of perturbations. Another approach splits the metric and the field into large-scale part (coarse-grained over some scale greater than horizon scale), and small-scale part. During inflation, the dynamical equations for coarse-grained field and coarse-grained scale-factor are equivalent to eqns. provided the quantum effects are switched off. The quantum effects continuously produce new inhomogeneities of random amplitude with scales greater than the scale of coarse-graining. These inhomogeneities should be added to and and effectively this leads to the presence of stochastic force term in the equations of motion. Therefore, the dynamics of coarse-grained variables can be described in terms of the distribution functions of and , and in principal these distribution functions can provide the same information as the power spectrum of perturbations, and furthermore the coarse-grained formalism gives a tool for description of the metric perturbations with amplitude, greater than 1.

The effective dynamical equation for the field has the form
[19]
^{3}^{3}3See also recent papers [26] and references therein.

where , and is delta-correlated random force, . The equation for coarse-grained scale factor remains unchanged

The solution of the set of eqns. is extremely difficult problem, and can be done under some additional simplifying assumptions. For example if we choose the featureless potential, and consider the friction-dominated solutions of the eqn. , we can obtain the solutions describing self-reproduced inflationary Universe (provided the stochastic term in dominates over potential term, see for example Linde [23]). In our case we cannot use the friction-dominated condition in the beginning of the field evolution inside the plateau region. However we can adopt another simplifying assumptions: first we can set inside and near the plateau region, and second, we can omit the stochastic term in the eqn. outside the plateau region, assuming the field moves along the classical trajectory there. Under these assumptions the statistics of the scale factor is totally determined by the time that field spends in the plateau region

where is the value of scale factor at the time of entrance of the field in the plateau region, and corresponds to the moment , when the field leaves the plateau region. To see that let us consider the evolution of the scale factor in the comoving coordinate system. Outside the horizon the hypersurfaces of constant comoving time practically coincide with hypersurfaces of constant energy density . On the other hand, the field evolves slowly during inflation and hypersurfaces of constant energy density are close to hypersurfaces , and therefore we can put . After the field passes the plateau region, the evolution of can be described by the standard expression , so we have

where is nearly constant inside of coarse-grained regions with comoving scale , but changes from one region to another. Thus, the metric outside horizon has the quasi-isotropic form

where we represent the scale factor as a multiplication of two factors: and . Here and are determined by the classical equations , , and the spatial coordinates are coarse-grained over the regions with scale . To estimate the change of metric from one region to another quantitatively, we introduce the definition of non-linear metric perturbation

(remind, that is the time which the field spends in the plateau region moving along the classical trajectory when the stochastic term in is switched off). Note, that in the limit of small , the metric assumes the form

and the definition is reduced to the standard expression for growing mode of adiabatic perturbation outside the horizon. Namely, in this case reduces to gauge independent quantities, introduced by a number of authors [11-13], [27] up to a constant factor. The variables do not depend on time outside the horizon. Therefore, using of these variables is very convenient to match the perturbations, generated during inflation with the perturbations, crossing horizon at the normal stage of the Universe evolution. As one can see from the metric perturbations are determined by stochastic variable and the distribution of must follow from the solution of eqn. . Note, that the definition of non-linear metric perturbations should be taken with a caution. In principal, one can use another definition relating to by some non-linear transformation, and having the same limit in the case of small . For example, Bond and Salopek [28] used the quantity to define non-linear metric perturbations. However, the criterion for PBH’s formation can be directly expressed in terms of the quantity (see next Section), and therefore this quantity is the most natural variable for our purposes.

Although the assumption of constant greatly simplifies the
problem it still remains rather complicated for a simple
analytical treatment
^{4}^{4}4In this case our problem is reduced to the first-passage problem
for one-dimensional Fokker-Plank (Kramers) equation, associated with
eqn. [29]. The general solution of this problem demands too much formalism
[30], and is not considered here. Note, however that the simple asymptotic estimates are
still possible in this case [30], [31].

For further progress we have to make some additional assumptions. We will consider below the plateau region of sufficiently large size. For this case the field approaches to the end of plateau in the friction-dominated approximation, which greatly simplifies the treatment of diffusion processes. To estimate the relevance of friction-dominated approximation we should compare the time and the time of the decay of the inertial term in the eqns. . If and therefore , the inertial term in these equations can be neglected at . In this regime the solution of the classical equation of motion has the form

and the equation becomes

where , and we introduce the dimensionless time , and is associated with simple diffusion type equation, describing the evolution of positions probability distribution . The stochastic equation

Now we assume that the distribution is not spread out sufficiently before and take -distributed function at the moment as the initial condition for our problem

where is the value of field corresponding
to the beginning of ”friction-dominated” part of plateau region
^{5}^{5}5The estimates show that the characteristic width of
is of order of and much less than the size
of ”friction-dominated” region ..

Together with initial condition we should specify the boundary condition at . This condition depends on the form of the transition layer between the plateau region and the part of potential with steep slope . We assume this transition to be sharp, and therefore set the condition of absorbing wall at the downstream point

Note, that this boundary condition was used by Aryal and Vilenkin [32] for analysis of stochastic inflation in the theory with top-hat potentials. In that paper it was shown that the more reasonable smooth transitions between the flat and steep regions of the potential are unlikely to modify significantly the resulting distribution.

In our case the probability density of time relates to the solution of eqn. as

where we define by the probability current . The conservation of the probability current allows to estimate the correction term to eqn. due to nonzero . Assuming that field moves along the classical trajectory after , we have . Therefore the correction to the expression is times smaller than the leading term.

The conditions determine the solution of eqn. . This solution can be found by standard methods of the theory of diffusion equations (see, e. g. Ref. [33]), and in our case has the form

Substituting to the equation we find the explicit expression for

The expression for probability distribution of metric can be readily obtained from . Using the eqn. to express the time in terms of , taking into account the eqn and the definitions of , , and assuming , we obtain

where is the standard metric amplitude calculated for the plateau parameters, and

are the numbers of e-folds for the classical path and for a random path , which start at and end at .

When the perturbations are small , the distribution has the standard Gaussian form

and in the opposite case of very large metric perturbations and the distribution deviates sharply from the Gaussian law and has the power-law form

As seen from eqns. , the non-Gaussian effects over-produce the metric perturbations of high amplitude in our model. To understand this fact, let us discuss the origin of non-Gaussian effects in our model. There are two sources for such effects. First, note that the ”effective dispersion” in eqn. depends itself on the value of the stochastic variable . Qualitatively, it can be explained as follows. In the linear theory the dispersion is proportional to the time spent by the classical background field on the plateau. In non-linear theory the coarse-grained field plays the role of background field, and therefore the distribution of the family of neighboring to paths should be described in terms of the probability distribution with dispersion , which is proportional to the time spent by field on the plateau. Second, the amplitude of large metric perturbations depends on exponentially (), so order of magnitude increase of leads to exponential increase of . Obviously, these two effects increase the probability of large amplitude metric perturbations.

## Iv Probability of black holes formation

Although the distribution provides very important information about the geometry of spatial part of metric outside horizon, it cannot be directly applied to the estimates of PBH’s formation. Indeed, the distribution is formed by the field inhomogeneities with wave-numbers in the range . The process of PBH formation is determined mainly by the field modes with wave-numbers , where is the typical PBH wavenumber. The modes with compose the large-scale background part of metric at the moment of PBH formation, and do not influence on the formation of PBH’s significantly. The modes with lead to high-frequency modulation of the perturbation with , which is also unimportant, provided the mode with crosses the horizon second time at the radiation-dominated epoch. Therefore, in order to obtain the probability of PBH’s formation, we should subtract the contribution of the large-scale and small-scale metric perturbations.

In general it is very difficult to separate the perturbations of a given
scale in the frameworks of non-linear approach. However, we can estimate
the probability density of the perturbations, corresponding
to the smallest scale
^{6}^{6}6In this connection, let us note that the black holes of smallest
mass should give the major contribution to the present fraction of
black holes, provided PBH’s spectrum is flat (Carr, 1975 [14])..
For that we simply
put in eqns. , assuming that the random process
starts when the mode with wavenumber crosses horizon.
This procedure automatically subtracts the large-scale contribution
of modes with . The small-scale contribution is also absent
due to our absorbing boundary condition. We have

from the eqn. , where , and in the limit of small we obtain again the Gaussian distribution

The distribution has nonzero first momentum (the lower limit of integration should be -1, since the metric perturbations with are cut off). The contribution of should be added to the background part of metric, and further we will use the renormalized metric perturbation instead of . The probability to find the metric perturbations with amplitude greater than some threshold value can be estimated as

where , and we assume . The same quantity, but calculated for the Gaussian distribution takes the well-known form

The observed quantities (such as, e.g. the matter density of PBH’s in different cosmological epochs) can be easily expressed in terms of the probability , provided the mass of PBH’s and some criterion for PBH’s formation are fixed. In our case the criterion for PBH’s formation should give the information about the threshold value . Since this criterion plays very important role, let us discuss it in some details. First let us note that PBH’s are formed from high amplitude peaks in the density distribution which are approximately spherically-symmetric (see e.g. Ref. [34]). It can also be easily shown that the maxima in the matter density correspond to the maxima in the function . The form of totally specifies the number of regions going to PBH’s as well as dynamics of the collapsing regions. Therefore we formulate the criterion of PBH’s formation in terms of conditions imposed on the function .

The first criterion was formulated by Carr in his seminal paper [14]. It was shown that an over-dense region forms PBH if the density contrast at the horizon scale lies approximately within the limits the NNP criterion reads . The first part of this inequality tells that the over-dense region should stop expansion before the scale of the region crosses the sound horizon. The second part requires that the over-dense region does not collapse before crossing the causal horizon, and consequently the perturbation does not produce a closed world separated from the rest of the Universe. Then the criterion for PBH’s formation was improved by Nadegin, Novikov and Polnarev [21] (hereafter NNP), and also by Biknell and Henriksen [22] with help of numerical computations. The initial condition used by NNP was chosen as a non-linear metric perturbation having the form of a part of the closed Friedman Universe matched with the spatially flat Universe through an intermediate layer of negative density perturbation. The conditions for PBH’s formation depend on the size of this part (i.e. the amplitude of the perturbation), as well as on the size of the matching layer. The smaller matching layer is, the larger the pressure gradients needed to prevent collapse will be. Therefore, the amplitude of the perturbation forming PBH must be greater in the case of narrow intermediate layer. In terms of our function

where is the value of at the maximum of the perturbation
and is the same quantity outside the perturbed region
^{7}^{7}7In the linear theory
the density perturbation at horizon scale relates to the metric perturbation
by is in agreement with Carr’s result. (see, e.g. [27]).
Therefore the estimate .
The first number on the right hand side of corresponds to the matching
layer of size comparable with size of the over-dense region, and the second
number corresponds to the narrow matching layer. Assuming
the matching layer to be sufficiently large we take as a
criterion of PBH’s formation.

Once the criterion is specified, we can link the desired PBH’s abundance with the parameters of our model. For instance, consider the model having the matter density of PBH’s equal to the critical one (the density parameter ). In this model we have [3], [6]

Equating the expression to the probability function , we have the equation determining the amplitude required for PBH’s abundance as a function of

and equating the expressions and we obtain the analogous equation for determining the reference amplitude when the non-Gaussian effects are switched off. The solution of these equations is given in Fig. 1.

One can see from this Fig. that the quantities and increase with increasing of and is always smaller than . It means that non-Gaussian effects over-produce PBH’s in our model (at least when the simple criterion is used), and the slope of potential can be steeper than that required in the Gaussian case. Typically, the ratio is about . Say, for the case of , we have and . We plot the probability function for in Fig. 2.

In this Fig., we also plot the Gaussian probability function for (dashed line) and the same quantity for (dotted line). Comparing the curves that correspond to the same PBH’s abundance, we see that the non-Gaussian curve is flatter having larger values of at large . The values of the Gaussian curve with the same plateau parameter is smaller by many orders of magnitude than the values of the non-Gaussian curve in the case of large .

Finally, let us note, that the non-Gaussian effects does not modify significantly the estimates based on the Gaussian theory. As we have seen, the ambiguity in the choice of the plateau slope due to these effects is about . This ambiguity seems to be less than the ambiguity in other parameters and can be obviously absorbed by a small change of the potential slope.

## V Discussion

We have demonstrated that the non-Gaussian effects related to the dynamics of the coarse-grained field (inflaton) and to the evolution of the large-scale part of metric over-produce large-amplitude inhomogeneities of the metric compared to the prediction of the Gaussian (linear) theory of perturbations. We have derived an analytical expression for non-Gaussian probability distribution for non-linear metric perturbations, and estimated the influence of non-linear effects on the probability of primordial black holes formation. We used the simple single field inflationary model with peculiarity in form of the flat region in inflaton potential , and power-law slope of the potential outside the peculiarity region. The key point of our approach is in the using of inhomogeneous coarse-grained metric function instead of the coarse-grained field as a basic quantity. This allows to match the physical condition of production of inhomogeneities during inflation with the ”observable” quantities.

Our results can be considered as semi-qualitative only. The uncertainties come fromthe phenomenological character of our inflationary model as well as from the oversimplified treatment of the process of PBH’s formation. The uncertainties related to the choice of parameters of inflationary model are mainly due to unknown form of the potential between the steep and flat regions, and also due to our friction-dominated assumption in the consideration of the stochastic process. These uncertainties can be eliminatedwith help of numerical simulations of stochastic process in more realistic models of inflation. The ambiguities concerning the criterion of PBH’s formation are mainly due to the one-point treatment of this process. Actually, PBH formation is nonlocal, and dynamics of collapsing region depends strongly on the form of the spatial profile of the density perturbation (see e.g. Refs. [22], [35] for discussion of this point). The form of the spatial profile can be studied by means of n-point correlation functions of the coarse-grained metric and field. Unfortunately, the formalism of n-point correlation functions is still not elaborated (see, however the Ref. [35] for the first discussion). Note, that probably the influence of the spatial profile of the collapsing region may be taken into account by a redefinition of the threshold value , and this value might be effectively less. In this case the role of the non-linear effects would be damped.

Finally we would like to note that the form of the distribution does not depend explicitly on the specific parameters of our model. This allows to suppose that similar distributions can be obtained in more complicated models, say, in two-field models proposed in Refs. [8], [9]. We are going to check this very interesting assumption in our future work.

Acknowledgments

The author acknowledges P. Naselsky and I. Novikov for many valuable discussions, and also A. Beloborodov, A. Dolgov and D. Markovic for useful comments. This work was supported in part by the Danish Research Foundation through its establishment of the Theoretical Astrophysic

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