Primordial Black Holes

Primordial Black Holes

Maxim Yu. Khlopov Maxim.Khlopov@roma1.infn.it
Abstract

Primordial black holes (PBHs) are a profound signature of primordial cosmological structures and provide a theoretical tool to study nontrivial physics of the early Universe. The mechanisms of PBH formation are discussed and observational constraints on the PBH spectrum, or effects of PBH evaporation, are shown to restrict a wide range of particle physics models, predicting an enhancement of the ultraviolet part of the spectrum of density perturbations, early dust-like stages, first order phase transitions and stages of superheavy metastable particle dominance in the early Universe. The mechanism of closed wall contraction can lead, in the inflationary Universe, to a new approach to galaxy formation, involving primordial clouds of massive BHs created around the intermediate mass or supermassive BH and playing the role of galactic seeds.

Centre for Cosmoparticle Physics ”Cosmion”

Moscow, 125047, Miusskaya pl. 4

Moscow State Engineering Physics Institute,

Kashirskoe Sh., 31, Moscow 115409, Russia, and

APC laboratory 10, rue Alice Domon et Léonie Duquet

75205 Paris Cedex 13, France

1 Introduction

The convergence of the frontiers of our knowledge in micro- and macro- worlds leads to the wrong circle of problems, illustrated by the mystical Uhroboros (self-eating-snake). The Uhroboros puzzle may be formulated as follows: The theory of the Universe is based on the predictions of particle theory, that need cosmology for their test. Cosmoparticle physics [1, 2, 3, 4, 5, 6] offers the way out of this wrong circle. It studies the fundamental basis and mutual relationship between micro-and macro-worlds in the proper combination of physical, astrophysical and cosmological signatures. Some aspects of this relationship, which arise in the astrophysical problem of Primordial Black Holes (PBH) is the subject of this review.

In particle theory Noether’s theorem relates the exact symmetry to conservation of respective charge. Extensions of the standard model imply new symmetries and new particle states. The respective symmetry breaking induces new fundamental physical scales in particle theory. If the symmetry is strict, its existence implies new conserved charge. The lightest particle, bearing this charge, is stable. It gives rise to the deep relationship between dark matter candidates and particle symmetry beyond the Standard model.

The mechanism of spontaneous breaking of particle symmetry also has cosmological impact. Heating of condensed matter leads to restoration of its symmetry. When the heated matter cools down, phase transition to the phase of broken symmetry takes place. In the course of the phase transitions, corresponding to given type of symmetry breaking, topological defects can form. One can directly observe formation of such defects in liquid crystals or in superfluid He. In the same manner the mechanism of spontaneous breaking of particle symmetry implies restoration of the underlying symmetry. When temperature decreases in the course of cosmological expansion, transitions to the phase of broken symmetry can lead, depending on the symmetry breaking pattern, to formation of topological defects in very early Universe. Defects can represent new forms of stable particles (as it is in the case of magnetic monopoles [7, 8, 9, 10, 11, 12]), or extended structures, such as cosmic strings [13, 14] or cosmic walls [15].

In the old Big bang scenario cosmological expansion and its initial conditions were given a priori [16, 17]. In the modern cosmology expansion of Universe and its initial conditions are related to inflation [18, 19, 20, 21, 22], baryosynthesis and nonbaryonic dark matter (see review in [23, 24]). Physics, underlying inflation, baryosynthesis and dark matter, is referred to extensions of the standard model, and variety of such extensions makes the whole picture in general ambiguous. However, in a framework of each particular physical realization of inflationary model with baryosynthesis and dark matter the corresponding model dependent cosmological scenario can be specified in all details. In such scenario main stages of cosmological evolution, structure and physical content of the Universe reflect structure of the underlying physical model. The latter should include with necessity the standard model, describing properties of baryonic matter, and its extensions, responsible for inflation, baryosynthesis and dark matter. In no case cosmological impact of such extensions is reduced to reproduction of these three phenomena only. A nontrivial path of cosmological evolution, specific for each particular realization of inflational model with baryosynthesis and nonbaryonic dark matter, always contains some additional model dependent cosmologically viable predictions, which can be confronted with astrophysical data. Here we concentrate on Primordial Black Holes as profound signature of such phenomena.

It was probably Pierre-Simon Laplace [25] in the beginning of XIX century, who noted first that in very massive stars escape velocity can exceed the speed of light and light can not come from such stars. This conclusion made in the framework of Newton mechanics and Newton corpuscular theory of light has further transformed into the notion of ”black hole” in the framework of general relativity and electromagnetic theory. Any object of mass can become a black hole, being put within its gravitational radius At present time black holes (BH) can be created only by a gravitational collapse of compact objects with mass more than about three Solar mass [26, 27]. It can be a natural end of massive stars or can result from evolution of dense stellar clusters. However in the early Universe there were no limits on the mass of BH. Ya.B. Zeldovich and I.D. Novikov [28] noticed that if cosmological expansion stops in some region, black hole can be formed in this region within the cosmological horizon. It corresponds to strong deviation from general expansion and reflects strong inhomogeneity in the early Universe. There are several mechanisms for such strong inhomogeneity and we’ll trace their links to cosmological consequences of particle theory.

Primordial Black Holes (PBHs) are a very sensitive cosmological probe for physics phenomena occurring in the early Universe. They could be formed by many different mechanisms, e.g., initial density inhomogeneities [29, 30] and non-linear metric perturbations [31, 32, 33], blue spectra of density fluctuations [34, 35, 36, 37, 38, 39], a softening of the equation of state [40, 34, 35], development of gravitational instability on early dust-like stages of dominance of supermassive particles and scalar fields [41, 42, 43, 44] and evolution of gravitationally bound objects formed at these stages [45, 46], collapse of cosmic strings [47, 48, 49, 50, 51] and necklaces [52], a double inflation scenario [53, 54, 55, 56], first order phase transitions [57, 58, 59, 60, 61], a step in the power spectrum [62, 63], etc. (see [35, 3, 4, 64, 65] for a review).

Being formed, PBHs should retain in the Universe and, if survive to the present time, represent a specific form of dark matter [66, 67, 3, 4, 68, 69, 70, 65, 71]. Effect of PBH evaporation by S.W.Hawking [72] makes evaporating PBHs a source of fluxes of products of evaporation, particularly of radiation [73]. MiniPBHs with mass below  g evaporate completely and do not survive to the present time. However, effect of their evaporation should cause influence on physical processes in the early Universe, thus providing a test for their existence by methods of cosmoarcheology [74], studying cosmological imprints of new physics in astrophysical data. In a wide range of parameters the predicted effect of PBHs contradicts the data and it puts restrictions on mechanism of PBH formation and the underlying physics of very early Universe. On the other hand, at some fixed values of parameters, PBHs or effects of their evaporation can provide a nontrivial solution for astrophysical problems.

Various aspects of PBH physics, mechanisms of their formation, evolution and effects are discussed in [75, 76, 77, 78, 35, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123] particularly specifying PBH formation and effects in braneworld cosmology [124, 125, 126, 127], on inflationary preheating [128], formation of PBHs in QCD phase transition [129, 130], properties of superhorizon BHs [131, 132], role of PBHs in baryosynthesis [133, 134, 135, 136, 137], effects of PBH evaporation in the early Universe and in modern cosmic ray, neutrino and gamma fluxes [138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169], in creation of hypothetical particles [170, 171, 172, 173], PBH clustering and creation of supermassive BHs [174, 175, 176, 177, 178, 179], effects in cosmic rays and colliders from PBHs in low scale gravity models [180, 181]. Here we outline the role of PBHs as a link in cosmoarcheoLOGICAL chain, connecting cosmological predictions of particle theory with observational data. We discuss the way, in which spectrum of PBHs reflects properties of superheavy metastable particles and of phase transitions on inflationary and post-inflationary stages. We briefly review possible cosmological reflections of particle physics (section 2), illustrate in section 3 some mechanisms of PBH formation on stage of dominance of superheavy particles and fields (subsection 3.1) and from second order phase transition on inflationary stage. Effective mechanism of BH formation during bubble nucleation provides a sensitive tool to probe existence of cosmological first order phase transitions by PBHs (section 4). Existence of stable remnants of PBH evaporation can strongly increase the sensitivity of such probe and we demonstrate this possibility in section 5 on an example of gravitino production in PBH evaporation. Being formed within cosmological horizon, PBHs seem to have masses much less than the mass of stars, constrained by small size of horizon in very early Universe. However, if phase transition takes place on inflationary stage, closed walls of practically any size can be formed (subsection 6.2) and their successive collapse can give rise to clouds of massive black holes, which can play the role of seeds for galaxies (section 6). The impact of constraints and cosmological scenarios, involving primordial black holes, is briefly discussed in section 7.

2 PBHs as cosmological reflection of new physics

The simplest primordial form of new physics is a gas of new stable massive particles, originated from early Universe. For particles with mass , at high temperature the equilibrium condition, is valid, if their annihilation cross section is sufficiently large to establish equilibrium. At such particles go out of equilibrium and their relative concentration freezes out. Weakly interacting species decouple from plasma and radiation at , when , i.e. at . This is the main idea of calculation of primordial abundance for WIMP-like dark matter candidates (see e.g. [3, 4, 74] for details). The maximal temperature, which is reached in inflationary Universe, is the reheating temperature, , after inflation. So, very weakly interacting particles with annihilation cross section , as well as very heavy particles with mass can not be in thermal equilibrium, and the detailed mechanism of their production should be considered to calculate their primordial abundance.

Decaying particles with lifetime , exceeding the age of the Universe, , , can be treated as stable. By definition, primordial stable particles survive to the present time and should be present in the modern Universe. The net effect of their existence is given by their contribution into the total cosmological density. They can dominate in the total density being the dominant form of cosmological dark matter, or they can represent its subdominant fraction. In the latter case more detailed analysis of their distribution in space, of their condensation in galaxies, of their capture by stars, Sun and Earth, as well as effects of their interaction with matter and of their annihilation provides more sensitive probes for their existence. In particular, hypothetical stable neutrinos of 4th generation with mass about 50 GeV are predicted to form the subdominant form of modern dark matter, contributing less than 0,1 % to the total density [186, 187]. However, direct experimental search for cosmic fluxes of weakly interacting massive particles (WIMPs) may be sensitive to existence of such component (see [182, 183, 184, 185] and references therein). It was shown in [188, 189, 190, 191] that annihilation of 4th neutrinos and their antineutrinos in the Galaxy can explain the galactic gamma-background, measured by EGRET in the range above 1 GeV, and that it can give some clue to explanation of cosmic positron anomaly, claimed to be found by HEAT. 4th neutrino annihilation inside the Earth should lead to the flux of underground monochromatic neutrinos of known types, which can be traced in the analysis of the already existing and future data of underground neutrino detectors [190, 192, 193, 194].

New particles with electric charge and/or strong interaction can form anomalous atoms and contain in the ordinary matter as anomalous isotopes. For example, if the lightest quark of 4th generation is stable, it can form stable charged hadrons, serving as nuclei of anomalous atoms of e.g. anomalous helium [195, 196, 197, 198, 199, 200].

Primordial unstable particles with lifetime, less than the age of the Universe, , can not survive to the present time. But, if their lifetime is sufficiently large to satisfy the condition , their existence in early Universe can lead to direct or indirect traces. Cosmological flux of decay products contributing into the cosmic and gamma ray backgrounds represents the direct trace of unstable particles. If the decay products do not survive to the present time their interaction with matter and radiation can cause indirect trace in the light element abundance or in the fluctuations of thermal radiation.

If particle lifetime is much less than s multi-step indirect traces are possible, provided that particles dominate in the Universe before their decay. On dust-like stage of their dominance black hole formation takes place, and spectrum of such primordial black holes traces particle properties (mass, frozen concentration, lifetime) [35]. Particle decay in the end of dust like stage influences the baryon asymmetry of the Universe. In any way cosmophenomenoLOGICAL chains link the predicted properties of even unstable new particles to the effects accessible in astronomical observations. Such effects may be important in analysis of the observational data.

Parameters of new stable and metastable particles are also determined by a pattern of particle symmetry breaking. This pattern is reflected in a succession of phase transitions in the early Universe. First order phase transitions proceed through bubble nucleation, which can result in black hole formation (see e.g. [59] and [65] for review and references). Phase transitions of the second order can lead to formation of topological defects, such as walls, string or monopoles. The observational data put severe constraints on magnetic monopole [9] and cosmic wall production [15], as well as on the parameters of cosmic strings [13, 14]. Structure of cosmological defects can be changed in succession of phase transitions. More complicated forms like walls-surrounded-by-strings can appear. Such structures can be unstable, but their existence can leave a trace in nonhomogeneous distribution of dark matter and give rise to large scale structures of nonhomogeneous dark matter like archioles [201, 202, 203]. Primordial Black Holes represent a profound signature of such structures.

3 PBHs from early dust-like stages

A possibility to form a black hole is highly improbable in homogeneous expanding Universe, since it implies metric fluctuations of order 1. For metric fluctuations distributed according to Gaussian law with dispersion

(1)

a probability for fluctuation of order 1 is determined by exponentially small tail of high amplitude part of this distribution. This probability can be even more suppressed in a case of non-Gaussian flutuations [31].

In the Universe with equation of state

(2)

with numerical factor being in the range

(3)

a probability to form black hole from fluctuation within cosmological horizon is given by (see e.g. [3, 4] for review and references)

(4)

It provides exponential sensitivity of PBH spectrum to softening of equation of state in early Universe () or to increase of ultraviolet part of spectrum of density fluctuations (). These phenomena can appear as cosmological consequence of particle theory.

3.1 Dominance of superheavy particles in early Universe

Superheavy particles can not be studied at accelerators directly. If they are stable, their existence can be probed by cosmological tests, but there is no direct link between astrophysical data and existence of superheavy metastable particles with lifetime . It was first noticed in [41] that dominance of such particles in the Universe before their decay at can result in formation of PBHs, retaining in Universe after the particles decay and keeping some information on particle properties in their spectrum. It provided though indirect but still a possibility to probe existence of such particles in astrophysical observations. Even the absence of observational evidences for PBHs is important. It puts restrictions on allowed properties of superheavy metastable particles, which might form such PBHs on a stage of particle dominance, and thus constrains parameters of models, predicting these particles.

After reheating, at

(5)

particles with mass and relative abundance (where is frozen out concentration of particles and is concentration of relativistic species) must dominate in the Universe before their decay. Dominance of these nonrelativistic particles at , where

(6)

corresponds to dust like stage with equation of state at which particle density fluctuations grow as

(7)

and development of gravitational instability results in formation of gravitationally bound systems, which decouple at

(8)

from general cosmological expansion, when for fluctuations, entering horizon at with amplitude .

Formation of these systems can result in black hole formation either immediately after the system decouples from expansion or in result of evolution of initially formed nonrelativistic gravitationally bound system.

If density fluctuation is especially homogeneous and isotropic, it directly collapses to BH as soon as the amplitude of fluctuation grows to 1 and the system decouples from expansion. A probability for direct BH formation in collapse of such homogeneous and isotropic configurations gives minimal estimation of BH formation on dust-like stage.

This probability was calculated in [41] with the use of the following arguments. In the period , when fluctuation decouples from expansion, its configuration is defined by averaged density , size , deviation from sphericity and by inhomogeneity of internal density distribution within the fluctuation. Having decoupled from expansion, the configuration contracts and the minimal size to which it can contract is

(9)

being determined by a deviation from sphericity

(10)

where , and define a deformation of configuration along its three main orthogonal axes. It was first noticed in [41] that to form a black hole in result of such contraction it is sufficient that configuration returns to the size

(11)

which had the initial fluctuation , when it entered horizon at cosmological time . If

(12)

configuration is sufficiently isotropic to concentrate its mass in the course of collapse within its gravitational radius, but such concentration also implies sufficient homogeneity of configuration. Density gradients can result in gradients of pressure, which can prevent collapse to BH. This effect does not take place for contracting collisionless gas of weakly interacting massive particles, but due to inhomogeneity of collapse the particles, which have already passed the caustics can free stream beyond the gravitational radius, before the whole mass is concentrated within it. Collapse of nearly spherically symmetric dust configuration is described by Tolmen solution. It’s analysis [42, 43, 204, 35] has provided a constraint on the inhomogeneity within the configuration. It was shown that both for collisionless and interacting particles the condition

(13)

is sufficient for configuration to contract within its gravitational radius.

A probability for direct BH formation is then determined by a product of probability for sufficient initial sphericity and homogeneity of configuration, which is determined by the phase space for such configurations. In a calculation of one should take into account that the condition (12) implies 5 conditions for independent components of tensor of deformation before its diagonalization (2 conditions for three diagonal components to be close to each other and 3 conditions for nondiagonal components to be small). Therefore, the probability of sufficient sphericity is given by [41, 42, 43, 204, 35]

(14)

and together with the probability for sufficient homogeneity

(15)

results in the strong power-law suppression of probability for direct BH formation

(16)

Though this calculation was originally done in [41, 42, 43, 204, 35] for Gaussian distribution of fluctuations, it does not imply specific form of high amplitude tail of this distribution and thus should not change strongly in a case of non-Gaussian fluctuations [31].

The mechanism [41, 42, 43, 204, 35, 3, 4] is effective for formation of PBHs with mass in an interval

(17)

The minimal mass corresponds to the mass within cosmological horizon in the period when particles start to dominate in the Universe and it is equal to [41, 42, 43, 204, 35, 3, 4]

(18)

The maximal mass is indirectly determined by the condition

(19)

that fluctuation in the considered scale , entering the horizon at with an amplitude can manage to grow up to nonlinear stage, decouple and collapse before particles decay at For scale invariant spectrum the maximal mass is given by [65]

(20)

The probability, given by Eq.(16), is also appropriate for formation of PBHs on dust-like preheating stage after inflation [44, 3, 4]. The simplest example of such stage can be given with the use of a model of homogeneous massive scalar field [3, 4]. Slow rolling of the field in the period (where is the mass of field) provides chaotic inflation scenario, while at the field oscillates with period . Coherent oscillations of the field correspond to an averaged over period of oscillations dust-like equation of state at which gravitational instability can develop. The minimal mass in this case corresponds to the Jeans mass of scalar field, while the maximal mass is also determined by a condition that fluctuation grows and collapses before the scalar field decays and reheats the Universe.

The probability determines the fraction of total density

(21)

corresponding to PBHs with mass . For this fraction, given by Eq.(16), is small. It means that the bulk of particles do not collapse directly in black holes, but form gravitationally bound systems. Evolution of these systems can give much larger amount of PBHs, but it strongly depends on particle properties.

Superweakly interacting particles form gravitationally bound systems of collisionless gas, which remind modern galaxies with collisionless gas of stars. Such system can finally collapse to black hole, but energy dissipation in it and consequently its evolution is a relatively slow process [205, 3, 4]. The evolution of these systems is dominantly determined by evaporation of particles, which gain velocities, exceeding the parabolic velocity of system. In the case of binary collisions the evolution timescale can be roughly estimated [205, 3, 4] as

(22)

for gravitationally bound system of particles, where the free fall time for system with density is This time scale can be shorter due to collective effects in collisionless gas [206] and be at large of the order of

(23)

However, since the free fall time scale for gravitationally bound systems of collisionless gas is of the order of cosmological time for the period, when these systems are formed, even in the latter case the particles should be very long living to form black holes in such slow evolutional process.

The evolutional time scale is much smaller for gravitationally bound systems of superheavy particles, interacting with light relativistic particles and radiation. Such systems have analogy with stars, in which evolution time scale is defined by energy loss by radiation. An example of such particles give superheavy color octet fermions of asymptotically free SU(5) model [45] or magnetic monopoles of GUT models. Having decoupled from expansion, frozen out particles and antiparticles can annihilate in gravitationally bound systems, but detailed numerical simulation [46] has shown that annihilation can not prevent collapse of the most of mass and the timescale of collapse does not exceed the cosmological time of the period, when the systems are formed.

3.2 Spikes from phase transitions on inflationary stage

Scale non-invariant spectrum of fluctuations, in which amplitude of small scale fluctuations is enhanced, can be another factor, increasing the probability of PBH formation. The simplest functional form of such spectrum is represented by a blue spectrum with a power law dispersion

(24)

with amplitude of fluctuations growing at to small . The realistic account for existence of other scalar fields together with inflaton in the period of inflation can give rise to spectra with distinguished scales, determined by parameters of considered fields and their interaction.

In chaotic inflation scenario interaction of a Higgs field with inflaton can give rise to phase transitions on inflationary stage, if this interaction induces positive mass term . When in the course of slow rolling the amplitude of inflaton decreases below a certain critical value the mass term in Higgs potential

(25)

changes sign and phase transition takes place. Such phase transitions on inflationary stage lead to the appearance of a characteristic spikes in the spectrum of initial density perturbations. These spike–like perturbations re-enter the horizon during the radiation or dust like era and could in principle collapse to form primordial black holes. The possibility of such spikes in chaotic inflation scenario was first pointed out in [207] and realized in [62] as a mechanism of of PBH formation for the model of horizontal unification [208, 209, 210, 211].

For vacuum expectation value of a Higgs field

(26)

and the amplitude of spike in spectrum of density fluctuations, generated in phase transition on inflationary stage is given by [62]

(27)

with

(28)

where

If phase transition takes place at –folding before the end of inflation and the spike re-enters horizon on radiation dominance (RD) stage, it forms Black hole of mass

(29)

where is the Hubble constant in the period of inflation.

If the spike re-enters horizon on matter dominance (MD) stage it should form black holes of mass

(30)

4 First order phase transitions as a source of black holes in the early Universe

First order phase transition go through bubble nucleation. Remind the common example of boiling water. The simplest way to describe first order phase transitions with bubble creation in early Universe is based on a scalar field theory with two non degenerated vacuum states. Being stable at a classical level, the false vacuum state decays due to quantum effects, leading to a nucleation of bubbles of true vacuum and their subsequent expansion [212]. The potential energy of the false vacuum is converted into a kinetic energy of bubble walls thus making them highly relativistic in a short time. The bubble expands till it collides with another one. As it was shown in [57, 213] a black hole may be created in a collision of several bubbles. The probability for collision of two bubbles is much higher. The opinion of the BH absence in such processes was based on strict conservation of the original O(2,1) symmetry. As it was shown in [59, 60, 61] there are ways to break it. Firstly, radiation of scalar waves indicates the entropy increasing and hence the permanent breaking of the symmetry during the bubble collision. Secondly, the vacuum decay due to thermal fluctuation does not possess this symmetry from the beginning. The investigations [59, 60, 61] have shown that BH can be created as well with a probability of order unity in collisions of only two bubbles. It initiates an enormous production of BH that leads to essential cosmological consequences discussed below.

In subsection 4.1 the evolution of the field configuration in the collisions of bubbles is discussed. The BH mass distribution is obtained in subsection 4.2. In subsection 4.3 cosmological consequences of BH production in bubble collisions at the end of inflation are considered.

4.1 Evolution of field configuration in collisions of vacuum bubbles

Consider a theory where a probability of false vacuum decay equals and difference of energy density between the false and true vacuum outside equals . Initially bubbles are produced at rest however walls of the bubbles quickly increase their velocity up to the speed of light because a conversion of the false vacuum energy into its kinetic ones is energetically favorable.

Let us discuss dynamics of collision of two true vacuum bubbles that have been nucleated in points and which are expanding into false vacuum. Following papers [57, 214] let us assume for simplicity that the horizon size is much greater than the distance between the bubbles. Just after collision mutual penetration of the walls up to the distance comparable with its width is accompanied by a significant potential energy increase [215]. Then the walls reflect and accelerate backwards. The space between them is filled by the field in the false vacuum state converting the kinetic energy of the wall back to the energy of the false vacuum state and slowdown the velocity of the walls. Meanwhile the outer area of the false vacuum is absorbed by the outer wall, which expands and accelerates outwards. Evidently, there is an instant when the central region of the false vacuum is separated. Let us note this false vacuum bag (FVB) does not possess spherical symmetry at the moment of its separation from outer walls but wall tension restores the symmetry during the first oscillation of FVB. As it was shown in [214], the further evolution of FVB consists of several stages:

1) FVB grows up to the definite size until the kinetic energy of its wall becomes zero;

2) After this moment the false vacuum bag begins to shrink up to a minimal size ;

3) Secondary oscillation of the false vacuum bag occurs.

The process of periodical expansions and contractions leads to energy losses of FVB in the form of quanta of scalar field. It has been shown in the [214, 216] that only several oscillations take place. On the other hand, important note is that the secondary oscillations might occur only if the minimal size of the FVB would be larger than its gravitational radius, . Then oscillating solutions of ”quasilumps” can be realized [217]. The opposite case ( ) leads to a BH creation with the mass about the mass of the FVB. As it was shown in [59, 60, 61] the probability of BH formation is almost unity in a wide range of parameters of theories with first order phase transitions.

4.2 Gravitational collapse of FVB and BH creation

Consider following [59, 60, 61, 65, 4] in more details the conditions of converting FVB into BH. The mass of FVB can be calculated in a framework of a specific theory and can be estimated in a coordinate system where the colliding bubbles are nucleated simultaneously. The radius of each bubble in this system equals to half of their initial coordinate distance at first moment of collision. Apparently the maximum size of the FVB is of the same order as the size of the bubble, since this is the only parameter of necessary dimension on such a scale: . The parameter is obtained by numerical calculations in the framework of each theory, but its exact numerical value does not affect significantly conclusions.

One can find the mass of FVB that arises at the collision of two bubbles of radius:

(31)

This mass is contained in the shrinking area of false vacuum. Suppose for estimations that the minimal size of FVB is of order of wall width . The BH is created if minimal size of FVB is smaller than its gravitational radius. It means that at least at the condition

(32)

the FVB can be converted into BH (where G is the gravitational constant).

As an example consider a simple model with Lagrangian

(33)

In the thin wall approximation the width of the bubble wall can be expressed as . Using (32) one can easily derive that at least FVB with mass

(34)

should be converted into BH of mass M. The last condition is valid only in case when FVB is completely contained within the cosmological horizon, namely where the mass of the cosmological horizon at the moment of phase transition is given by . Thus for the potential (33) at the condition a BH is formed. This condition is valid for any realistic set of parameters of theory.

The mass and velocity distribution of FVBs, supposing its mass is large enough to satisfy the inequality (32), has been found in [59, 60, 61]. This distribution can be written in the terms of dimensionless mass :

(35)

The numerical integration of (35) revealed that the distribution is rather narrow. For example the number of BH with mass 30 times greater than the average one is suppressed by factor . Average value of the non dimensional mass is equal to . It allows to relate the average mass of BH and volume containing the BH at the moment of the phase transition:

(36)

4.3 First order phase transitions in the early Universe

Inflation models ended by a first order phase transition hold a dignified position in the modern cosmology of early Universe (see for example [218, 219, 220, 221, 222, 223, 224]). The interest to these models is due to, that such models are able to generate the observed large-scale voids as remnants of the primordial bubbles for which the characteristic wavelengths are several tens of Mpc. [223, 224]. A detailed analysis of a first order phase transition in the context of extended inflation can be found in [225]. Hereafter we will be interested only in a final stage of inflation when the phase transition is completed. Remind that a first order phase transition is considered as completed immediately after establishing of true vacuum percolation regime. Such regime is established approximately when at least one bubble per unit Hubble volume is nucleated. Accurate computation [225] shows that first order phase transition is successful if the following condition is valid:

(37)

Here is the bubble nucleation rate. In the framework of first order inflation models the filling of all space by true vacuum takes place due to bubble collisions, nucleated at the final moment of exponential expansion. The collisions between such bubbles occur when they have comoving spatial dimension less or equal to the effective Hubble horizon at the transition epoch. If we take in Universe the comoving size of these bubbles is approximately . In the standard approach it believes that such bubbles are rapidly thermalized without leaving a trace in the distribution of matter and radiation. However, in the previous subsection it has been shown that for any realistic parameters of theory, the collision between only two bubble leads to BH creation with the probability closely to 100% . The mass of this BH is given by (see (36))

(38)

where and is the mass that could be contained in the bubble volume at the epoch of collision in the condition of a full thermalization of bubbles. The discovered mechanism leads to a new direct possibility of PBH creation at the epoch of reheating in first order inflation models. In standard picture PBHs are formed in the early Universe if density perturbations are sufficiently large, and the probability of PBHs formation from small post- inflation initial perturbations is suppressed (see Section 3). Completely different situation takes place at final epoch of first order inflation stage; namely collision between bubbles of Hubble size in percolation regime leads to copious PBH formation with masses

(39)

where is the mass of Hubble horizon at the end of inflation. According to (36) the initial mass fraction of this PBHs is given by . For example, for typical value of the initial mass fraction is contained in PBHs with mass .

In general the Hawking evaporation of mini BHs could give rise to a variety possible end states. It is generally assumed, that evaporation proceeds until the PBH vanishes completely [226], but there are various arguments against this proposal (see e.g. [227, 75, 228, 229]). If one supposes that BH evaporation leaves a stable relic, then it is naturally to assume that it has a mass of order , where . We can investigate the consequences of PBH forming at the percolation epoch after first order inflation, supposing that the stable relic is a result of its evaporation. As it follows from the above consideration the PBHs are preferentially formed with a typical mass at a single time . Hence the total density at this time is

(40)

where denotes the fraction of the total density, corresponding to PBHs in the period of their formation . The evaporation time scale can be written in the following form

(41)

where is the number of effective massless degrees of freedom.

Let us derive the density of PBH relics. There are two distinct possibilities to consider.

The Universe is still radiation dominated (RD) at . This situation will be hold if the following condition is valid . It is possible to rewrite this condition in terms of Hubble constant at the end of inflation

(42)

Taking the present radiation density fraction of the Universe to be ( being the Hubble constant in the units of ), and using the standard values for the present time and time when the density of matter and radiation become equal, we find the contemporary densities fraction of relics

(43)

It is easily to see that relics overclose the Universe () for any reasonable and .

The second case takes place if the Universe becomes PBHs dominated at period . This situation is realized under the condition , which can be rewritten in the form

(44)

The present day relics density fraction takes the form

(45)

Thus the Universe is not overclosed by relics only if the following condition is valid

(46)

This condition implies that the masses of PBHs created at the end of inflation have to be larger than

(47)

From the other hand there are a number of well–known cosmological and astrophysical limits [230, 138, 231, 232, 233, 234, 235] which prohibit the creation of PBHs in the mass range (47) with initial fraction of mass density close to .

So one have to conclude that the effect of the false vacuum bag mechanism of PBH formation makes impossible the coexistence of stable remnants of PBH evaporation with the first order phase transitions at the end of inflation.

5 Gravitino production by PBH evaporation and constraints on the inhomogeneity of the early Universe

Presently there are no observational evidences, proving existence of PBHs. However, even the absence of PBHs provides a very sensitive theoretical tool to study physics of early Universe. PBHs represent nonrelativistic form of matter and their density decreases with scale factor as , while the total density is in the period of radiation dominance (RD). Being formed within horizon, PBH of mass , can be formed not earlier than at

(48)

If they are formed on RD stage, the smaller are the masses of PBHs, the larger becomes their relative contribution to the total density on the modern MD stage. Therefore, even the modest constraint for PBHs of mass on their density

(49)

in units of critical density from the condition that their contribution into the the total density

(50)

for does not exceed the density of dark matter

(51)

converts into a severe constraint on this contribution

(52)

in the period of their formation. If formed on RD stage at , given by (48), which corresponds to the temperature , PBHs contribute into the total density in the end of RD stage at , corresponding to , by factor larger, than in the period of their formation. The constraint on , following from Eq.(51) is then given by

(53)

The possibility of PBH evaporation, revealed by S. Hawking [72], strongly influences effects of PBHs. In the strong gravitational field near gravitational radius of PBH quantum effect of creation of particles with momentum is possible. Due to this effect PBH turns to be a black body source of particles with temperature (in the units )

(54)

The evaporation timescale BH is (see Eq.(41) and discussion in previous section) and at  g is less, than the age of the Universe. Such PBHs can not survive to the present time and the magnitude Eq.(51) for them should be re-defined and has the meaning of contribution to the total density in the moment of PBH evaporation. For PBHs formed on RD stage and evaporated on RD stage at the relationship Eq.(53) between and is given by [236, 35]

(55)

The relationship between and has more complicated form, if PBHs are formed on early dust-like stages [43, 35, 78, 3], or such stages take place after PBH formation[78, 3]. Relative contribution of PBHs to total density does not grow on dust-like stage and the relationship between and depends on details of a considered model. Minimal model independent factor follows from the account for enhancement, taking place only during RD stage between the first second of expansion and the end of RD stage at , since radiation dominance in this period is supported by observations of light element abundance and spectrum of CMB [43, 35, 78, 3].

Effects of PBH evaporation make astrophysical data much more sensitive to existence of PBHs. Constraining the abundance of primordial black holes can lead to invaluable information on cosmological processes, particularly as they are probably the only viable probe for the power spectrum on very small scales which remain far from the Cosmological Microwave Background (CMB) and Large Scale Structures (LSS) sensitivity ranges. To date, only PBHs with initial masses between  g and  g have led to stringent limits (see e.g. [75, 76, 77, 35]) from consideration of the entropy per baryon, the deuterium destruction, the He destruction and the cosmic-rays currently emitted by the Hawking process [72]. The existence of light PBHs should lead to important observable constraints, either through the direct effects of the evaporated particles (for initial masses between  g and  g) or through the indirect effects of their interaction with matter and radiation in the early Universe (for PBH masses between  g and  g). In these constraints, the effects taken into account are those related with known particles. However, since the evaporation products are created by the gravitational field, any quantum with a mass lower than the black hole temperature should be emitted, independently of the strength of its interaction. This could provide a copious production of superweakly interacting particles that cannot not be in equilibrium with the hot plasma of the very early Universe. It makes evaporating PBHs a unique source of all the species, which can exist in the Universe.

Following [3, 4, 78, 66] and [171, 172] (but in a different framework and using more stringent constraints), limits on the mass fraction of black holes at the time of their formation () were derived in [237] using the production of gravitinos during the evaporation process. Depending on whether gravitinos are expected to be stable or metastable, the limits are obtained using the requirement that they do not overclose the Universe and that the formation of light nuclei by the interactions of He nuclei with nonequilibrium flux of D,T,He and He does not contradict the observations. This approach is more constraining than the usual study of photo-dissociation induced by photons-photinos pairs emitted by decaying gravitinos. It opened a new window for the upper limits on below  g. The cosmological consequences of the limits, obtained in [237], are briefly reviewed in the framework of three different scenarios: a blue power spectrum, a step in the power spectrum and first order phase transitions.

5.1 Limits on the PBH density

Several constraints on the density of PBHs have been derived in different mass ranges assuming the evaporation of only standard model particles : for the entropy per baryon at nucleosynthesis was used [138] to obtain , for the production of pairs at nucleosynthesis was used [233] to obtain , for deuterium destruction was used [232] to obtain , for spallation of He was used [238, 78] to obtain , for the gamma-rays and cosmic-rays were used [235, 167] to obtain . Slightly more stringent limits were obtained in [239], leading to for masses between and and in [168], leading to for . Gamma-rays and antiprotons were also recently re-analyzed in [156] and [153], improving a little the previous estimates. Such constraints, related to phenomena occurring after the nucleosynthesis, apply only for black holes with initial masses above  g. Below this value, the only limits are the very weak entropy constraint (related with the photon-to-baryon ratio) and the constraint, assuming stable remnants of black holes forming at the end of the evaporation mechanism as described in the previous Section.

To derive a limit in the initial mass range  g, gravitinos emitted by black holes were considered in [237]. Gravitinos are expected to be present in all local supersymmetric models, which are regarded as the more natural extensions of the standard model of high energy physics (see, e.g., [240] for an introductory review). In the framework of minimal Supergravity (mSUGRA), the gravitino mass is, by construction, expected to lie around the electroweak scale, i.e. in the 100 GeV range. In this case, the gravitino is metastable and decays after nucleosynthesis, leading to important modifications of the nucleosynthesis paradigm. Instead of using the usual photon-photino decay channel, the study of [237] relied on the more sensitive gluon-gluino channel. Based on [241, 242, 243, 244, 245], the antiprotons produced by the fragmentation of gluons emitted by decaying gravitinos were considered as a source of nonequilibrium light nuclei resulting from collisions of those antiprotons on equilibrium nuclei. Then, Li, Li and Be nuclei production by the interactions of the nonequilibrium nuclear flux with He equilibrium nuclei was taken into account and compared with data (this approach is supported by several recent analysis [246, 247] which lead to similar results). The resulting Monte-Carlo estimates [244] lead to the following constraint on the concentration of gravitinos: , where is the gravitino mass in GeV. This constraint has been successfully used to derive an upper limit on the reheating temperature of the order [244]:  GeV. The consequences of this limit on cosmic-rays emitted by PBHs was considered, e.g., in [169]. In the approach of [237] this stringent constraint on the gravitino abundance was related to the density of PBHs through the direct gravitino emission. The usual Hawking formula [72] was used for the number of particles of type emitted per unit of time and per unit of energy . Introducing the temperature defined by Eq. (54) taking the relativistic approximation for , and integrating over time and energy, the total number of quanta of type can be estimated as:

(56)

where is in GeV,  g, , is the particle mass and accounts for the number of degrees of freedom through where is the black hole mass. Once the PBH temperature is higher than the gravitino mass, gravitinos will be emitted with a weight related with their number of degrees of freedom. Computing the number of emitted gravitinos as a function of the PBH initial mass and matching it with the limit on the gravitino density imposed by nonequilibrium nucleosynthesis of light elements leads to an upper limit on the PBH number density. If PBHs are formed during a radiation dominated stage, this limit can easily be converted into an upper limit on by evaluating the energy density of the radiation at the formation epoch. The resulting limit is shown on Fig. 1 and leads to an important improvement over previous limits, nearly independently of the gravitino mass in the interesting range. This opens a new window on the very small scales in the early Universe.

Figure 1: Constraints of [237] on the fraction of the Universe going into PBHs (adapted from [75, 76, 77, 35]). The two curves obtained with gravitinos emission in mSUGRA correspond to = 100 GeV (lower curve in the high mass range) and = 1 TeV (upper curve in the high mass range)

It is also possible to consider limits arising in Gauge Mediated Susy Breaking (GMSB) models [248]. Those alternative scenarios, incorporating a natural suppression of the rate of flavor-changing neutral-current due to the low energy scale, predict the gravitino to be the Lightest Supersymmetric Particle (LSP). The LSP is stable if R-parity is conserved. In this case, the limit was obtained [237] by requiring , i.e. by requiring that the current gravitino density does not exceed the matter density. It can easily be derived from the previous method, by taking into account the dilution of gravitinos in the period of PBH evaporation and conservation of gravitino to specific entropy ratio, that [237]:

(57)

where is the total number of gravitinos emitted by a PBH with initial mass , is the end of RD stage and when a non-trivial equation of state for the period of PBH formation is considered, e.g. a dust-like phase which ends at [43]. The limit (57) does not imply thermal equilibrium of relativistic plasma in the period before PBH evaporation and is valid even for low reheating temperatures provided that the equation of state on the preheating stage is close to relativistic. With the present matter density [249] this leads to the limit shown in Fig. 2 for  GeV. Following (57) this limit scales with gravitino mass as . Models of gravitino dark matter with , corresponding to the case of equality in the above formula, were recently considered in [250, 251].

Figure 2: Constraints of [237] on the fraction of the Universe going into PBHs (adapted from [75, 76, 77, 35]). The curve obtained with gravitinos emission in GMSB correspond to GeV and scales with gravitino mass as .

5.2 Cosmological consequences

Upper limits on the fraction of the Universe in primordial black holes can be converted into cosmological constraints on models with significant power on small scales [237].

The easiest way to illustrate the importance of such limits is to consider a blue power spectrum and to derive a related upper value on the spectral index of scalar fluctuations (). It has recently been shown by WMAP [249] that the spectrum is nearly of the Harrison-Zel’dovich type, i.e. scale invariant with . However this measure was obtained for scales between and times larger that those probed by PBHs and it remains very important to probe the power available on small scales. The limit on given in [237] must therefore be understood as a way to constrain at small scales rather than a way to measure its derivative at large scales : it is complementary to CMB measurements. Using the usual relations between the mass variance at the PBH formation time and the same quantity today [87],

(58)

where is the Hubble mass at time and is the equilibrium time, it is possible to set an upper value on which can be expressed as

(59)

where is the minimum density contrast required to form a PBH. The limit derived in the previous subsection leads to in the mSUGRA case whereas the usually derived limits range between 1.23 and 1.31 [87, 94, 252]. In the GMSB case, it remains at the same level for  GeV and is slightly relaxed for smaller masses of gravitino. This improvement is due to the much more important range of masses probed by the method [237].

In the standard cosmological paradigm of inflation, the primordial power spectrum is expected to be nearly –but not exactly– scale invariant [253]. The sign of the running can, in principle, be either positive or negative. It has been recently shown that models with a positive running , defined as

(60)

are very promising in the framework of supergravity inflation (see, e.g., [254]). The analysis [237] strongly limits a positive running, setting the upper bound at a tiny value . This result is more stringent than the upper limit obtained through a combined analysis of Ly forest, SDSS and WMAP data [255], , as it deals with scales very far from those probed by usual cosmological observations. The order of magnitude of the running naturally expected in most models –either inflationary ones (see, e.g., [256]) or alternative ones (see, e.g., [257])– being of a few times our upper bound should help to distinguish between different scenarios.

In the case of an early dust-like stage in the cosmological evolution [41, 35, 3, 4], the PBH formation probability is increased to where is the density contrast for the considered small scales (see subsection 3.1). The associated limit on is strengthened to .

Following [87], it is also interesting to consider primordial density perturbation spectra with both a tilt and a step. Such a feature can arise from underlying physical processes [258] and allows investigation of a wider class of inflaton potentials. If the amplitude of the step is defined so that the power on small scales is times higher than the power on large scales, the maximum allowed value for the spectral index can be computed as a function of . Figure 3, taken from [237], shows those limits, which become extremely stringent when is small enough, for both the radiation-dominated and the dust-like cases.

Figure 3: Upper limit from [237] on the spectral index of the power spectrum as a function of the amplitude of the step.

Another important consequence of limits [237] concerns PBH relics dark matter (see also discussion in subsection 4.3). The idea, introduced in [259], that relics possibly formed at the end of the evaporation process could account for the cold dark matter has been extensively studied. The amplitude of the power boost required on small scales has been derived, e.g., in [173] as a function of the relic mass and of the expected density. The main point was that the ”step” (or whatever structure in the power spectrum) should occur at low masses to avoid the constraints available between  g and  g. The limit on derived in [237] closes this dark matter issue except within a small window below  g.

This result can be re-formulated in a more general way. If the nature of cosmological dark matter is related with superweakly interacting particles, which can not be present in equilibrium in early Universe and for which nonequilibrium processes of production e.g. in reheating are suppressed, the early Universe should be sufficiently homogeneous on small scales to exclude copious creation of these species in miniPBH vaporation.

Finally, the limits [237] also completely exclude the possibility of a copious PBH formation process in bubble wall collisions [59, 60, 61], considered in the previous Section. This has important consequences for the related constraints on first order phase transitions in the early Universe and on symmetry breaking pattern of particle theory.

6 Massive Primordial Black Holes from collapse of closed walls

A wide class of particle models possesses a symmetry breaking pattern, which can be effectively described by pseudo-Nambu–Goldstone (PNG) field and which corresponds to formation of unstable topological defect structure in the early Universe (see [65] for review and references). The Nambu–Goldstone nature in such an effective description reflects the spontaneous breaking of global U(1) symmetry, resulting in continuous degeneracy of vacua. The explicit symmetry breaking at smaller energy scale changes this continuous degeneracy by discrete vacuum degeneracy. The character of formed structures is different for phase transitions, taking place on post-inflationary and inflationary stages.

6.1 Structures from succession of U(1) phase transitions

At high temperatures such a symmetry breaking pattern implies the succession of second order phase transitions. In the first transition, continuous degeneracy of vacua leads, at scales exceeding the correlation length, to the formation of topological defects in the form of a string network; in the second phase transition, continuous transitions in space between degenerated vacua form surfaces: domain walls surrounded by strings. This last structure is unstable, but, as was shown in the example of the invisible axion [201, 202, 203], it is reflected in the large scale inhomogeneity of distribution of energy density of coherent PNG (axion) field oscillations. This energy density is proportional to the initial value of phase, which acquires dynamical meaning of amplitude of axion field, when axion mass is switched on in the result of the second phase transition.

The value of phase changes by around string. This strong nonhomogeneity of phase, leading to corresponding nonhomogeneity of energy density of coherent PNG (axion) field oscillations, is usually considered (see e.g. [260, 261] and references therein) only on scales, corresponduing to mean distance between strings. This distance is small, being of the order of the scale of cosmological horizon in the period, when PNG field oscillations start. However, since the nonhomogeneity of phase follows the pattern of axion string network this argument misses large scale correlations in the distribution of oscillations’ energy density.

Indeed, numerical analysis of string network (see review in [262]) indicates that large string loops are strongly suppressed and the fraction of about 80% of string length, corresponding to long loops, remains virtually the same in all large scales. This property is the other side of the well known scale invariant character of string network. Therefore the correlations of energy density should persist on large scales, as it was revealed in [201, 202, 203].

The large scale correlations in topological defects and their imprints in primordial inhomogeneities is the indirect effect of inflation, if phase transitions take place after reheating of the Universe. Inflation provides in this case equal conditions for phase transition, taking place in causally disconnected regions.

If phase transitions take place on inflational stage new forms of primordial large scale correlations appear. The value of phase after the first phase transition is inflated over the region corresponding to the period of inflation, while fluctuations of this phase change in the course of inflation its initial value within the regions of smaller size. Owing to such fluctuations, for the fixed value of in the period of inflation with e-folding corresponding to the part of the Universe within the modern cosmological horizon, strong deviations from this value appear at smaller scales, corresponding to later periods of inflation with . If , the fluctuations can move the value of to in some regions of the Universe. After reheating in the result of the second phase transition these regions correspond to vacuum with , being surrounded by the bulk of the volume with vacuum . As a result massive walls are formed at the border between the two vacua. Since regions with are confined, the domain walls are closed. After their size equals the horizon, closed walls can collapse into black holes.

This mechanism can lead to formation of primordial black holes of a whatever large mass (up to the mass of AGNs [263], see for latest review [264]). Such black holes appear in the form of primordial black hole clusters, exhibiting fractal distribution in space [267, 268, 65]. It can shed new light on the problem of galaxy formation [65, 265, 266].

6.2 Formation of closed walls in inflationary Universe

To describe a mechanism for the appearance of massive walls of a size essentially greater than the horizon at the end of inflation, let us consider a complex scalar field with the potential[263, 267, 268, 65]

(61)

where . This field coexists with an inflaton field which drives the Hubble constant during the inflational stage. The term

(62)

reflecting the contribution of instanton effects to the Lagrangian renormalization (see for example [269]), is negligible on the inflational stage and during some period in the FRW expansion. The omitted term (62) becomes significant, when temperature falls down the values . The mass of radial field component is assumed to be sufficiently large with respect to , which means that the complex field is in the ground state even before the end of inflation. Since the term (62) is negligible during inflation, the field has the form , the quantity acquiring the meaning of a massless field.

At the same time, the well established behavior of quantum field fluctuations on the de Sitter background [18] implies that the wavelength of a vacuum fluctuation of every scalar field grows exponentially, having a fixed amplitude. Namely, when the wavelength of a particular fluctuation, in the inflating Universe, becomes greater than , the average amplitude of this fluctuation freezes out at some non-zero value because of the large friction term in the equation of motion of the scalar field, whereas its wavelength grows exponentially. Such a frozen fluctuation is equivalent to the appearance of a classical field that does not vanish after averaging over macroscopic space intervals. Because the vacuum must contain fluctuations of every wavelength, inflation leads to the creation of more and more new regions containing a classical field of different amplitudes with scale greater than . In the case of an effectively massless Nambu–Goldstone field considered here, the averaged amplitude of phase fluctuations generated during each e-fold (time interval ) is given by

(63)

Let us assume that the part of the Universe observed inside the contemporary horizon Mpc was inflating, over e-folds, out of a single causally connected domain of size , which contains some average value of phase over it. When inflation begins in this region, after one e-fold, the volume of the Universe increases by a factor . The typical wavelength of the fluctuation generated during every e-fold is equal to . Thus, the whole domain , containing , after the first e-fold effectively becomes divided into separate, causally disconnected domains of size . Each domain contains almost homogeneous phase value . Thereby, more and more domains appear with time, in which the phase differs significantly from the initial value . A principally important point is the appearance of domains with phase . Appearing only after a certain period of time during which the Universe exhibited exponential expansion, these domains turn out to be surrounded by a space with phase . The coexistence of domains with phases and leads, in the following, to formation of a large-scale structure of topological defects.

The potential (61) possesses a symmetry, which is spontaneously broken, at least, after some period of inflation. Note that the phase fluctuations during the first e-folds may, generally speaking, transform eventually into fluctuations of the cosmic microwave radiation, which will lead to imposing restrictions on the scaling parameter . This difficulty can be avoided by taking into account the interaction of the field with the inflaton field (i.e. by making parameter a variable [65]). This spontaneous breakdown is holding by the condition on the radial mass, . At the same time the condition

(64)

on the angular mass provides the freezing out of the phase distribution until some moment of the FRW epoch. After the violation of condition (64) the term (62) contributes significantly to the potential (61) and explicitly breaks the continuous symmetry along the angular direction. Thus, potential (61) eventually has a number of discrete degenerate minima in the angular direction at the points .

As soon as the angular mass is of the order of the Hubble rate, the phase starts oscillating about the potential minimum, initial values being different in various space domains. Moreover, in the domains with the initial phase , the oscillations proceed around the potential minimum at , whereas the phase in the surrounding space tends to a minimum at the point . Upon ceasing of the decaying phase oscillations, the system contains domains characterized by the phase surrounded by space with . Apparently, on moving in any direction from inside to outside of the domain, we will unavoidably pass through a point where because the phase varies continuously. This implies that a closed surface characterized by the phase must exist. The size of this surface depends on the moment of domain formation in the inflation period, while the shape of the surface may be arbitrary. The principal point for the subsequent considerations is that the surface is closed. After reheating of the Universe, the evolution of domains with the phase proceeds on the background of the Friedman expansion and is described by the relativistic equation of state. When the temperature falls down to , an equilibrium state between the ”vacuum” phase inside the domain and the phase outside it is established. Since the equation of motion corresponding to potential (62) admits a kink-like solution (see [262] and references therein), which interpolates between two adjacent vacua and , a closed wall corresponding to the transition region at is formed. The surface energy density of a wall of width is of the order of 111The existence of such domain walls in theory of the invisible axion was first pointed out in [270]..

Note that if the coherent phase oscillations do not decay for a long time, their energy density can play the role of CDM. This is the case, for example, in the cosmology of the invisible axion (see [260, 261] and references therein).

It is clear that immediately after the end of inflation, the size of domains which contains a phase essentially exceeds the horizon size. This situation is replicated in the size distribution of vacuum walls, which appear at the temperature whence the angular mass starts to build up. Those walls, which are larger than the cosmological horizon, still follow the general FRW expansion until the moment when they get causally connected as a whole; this happens as soon as the size of a wall becomes equal to the horizon size . Evidently, internal stresses developed in the wall after crossing the horizon initiate processes tending to minimize the wall surface. This implies that the wall tends, first, to acquire a spherical shape and, second, to contract toward the centre. For simplicity, we will consider below the motion of closed spherical walls 222The motion of closed vacuum walls has been driven analytically in [271, 272]..

The wall energy is proportional to its area at the instant of crossing the horizon. At the moment of maximum contraction, this energy is almost completely converted into kinetic energy [273]. Should the wall at the same moment be localized within the gravitational radius, a PBH is formed.

Detailed consideration of BH formation was performed in [263]. The results of these calculations are sensitive to changes in the parameter and the initial phase . As the value decreases to GeV, still greater PBHs appear with masses of up to g. A change in the initial phase leads to sharp variations in the total number of black holes.As was shown above, each domain generates a family of subdomains in the close vicinity. The total mass of such a cluster is only 1.5–2 times that of the largest initial black hole in this space region. Thus, the calculations confirm the possibility of formation of clusters of massive PBHs ( and above) in the pregalactic stages of the evolution of the Universe. These clusters represent stable energy density fluctuations around which increased baryonic (and cold dark matter) density may concentrate in the subsequent stages, followed by the evolution into galaxies.

It should be noted that additional energy density is supplied by closed walls of small sizes. Indeed, because the smallness of their gravitational radius, they do not collapse into BHs. After several oscillations such walls disappear, leaving coherent fluctuations of the PNG field. These fluctuations contribute to a local energy density excess, thus facilitating the formation of galaxies.

The mass range of formed BHs is constrained by fundamental parameters of the model and . The maximal BH mass is determined by the condition that the wall does not dominate locally before it enters the cosmological horizon. Otherwise, local wall dominance leads to a superluminal expansion for the corresponding region, separating it from the other part of the Universe. This condition corresponds to the mass [65]

(65)

The minimal mass follows from the condition that the gravitational radius of BH exceeds the width of wall and it is equal to[267, 65]

(66)

Closed wall collapse leads to primordial GW spectrum, peaked at

(67)

with energy density up to

(68)

At GeV this primordial gravitational wave background can reach For the physically reasonable values of

(69)

the maximum of spectrum corresponds to

(70)

Another profound signature of the considered scenario are gravitational wave signals from merging of BHs in PBH cluster. These effects can provide test of the considered approach in LISA experiment.

7 Discussion

For long time scenarios with Primordial Black holes belonged dominantly to cosmological anti-Utopias, to ”fantasies”, which provided restrictions on physics of very early Universe from contradiction of their predictions with observational data. Even this ”negative” type of information makes PBHs an important theoretical tool. Being formed in the very early Universe as initially nonrelativistic form of matter, PBHs should have increased their contribution to the total density during RD stage of expansion, while effect of PBH evaporation should have strongly increased the sensitivity of astrophysical data to their presence. It links astrophysical constraints on hypothetical sources of cosmic rays or gamma background, on hypothetical factors, causing influence on light element abundance and spectrum of CMB, to restrictions on superheavy particles in early Universe and on first and second order phase transitions, thus making a sensitive astrophysical probe to particle symmetry structure and pattern of its breaking at superhigh energy scales.

Gravitational mechanism of particle creation in PBH evaporation makes evaporating PBH an unique source of any species of particles, which can exist in our space-time. At least theoretically, PBHs can be treated as source of such particles, which are strongly suppressed in any other astrophysical mechanism of particle production, either due to a very large mass of these species, or owing to their superweak interaction with ordinary matter.

By construction astrophysical constraint excludes effect, predicted to be larger, than observed. At the edge such constraint converts into an alternative mechanism for the observed phenomenon. At some fixed values of parameters, PBH spectrum can play a positive role and shed new light on the old astrophysical problems.

The common sense is to think that PBHs should have small sub-stellar mass. Formation of PBHs within cosmological horizon, which was very small in very early Universe, seem to argue for this viewpoint. However, phase transitions on inflationary stage can provide spikes in spectrum of fluctuations at any scale, or provide formation of closed massive domain walls of any size.

In the latter case primordial clouds of massive black holes around intermediate mass or supermassive black hole is possible. Such clouds have a fractal spatial distribution. A development of this approach gives ground for a principally new scenario of the galaxy formation in the model of the Big Bang Universe. Traditionally, Big Bang model assumes a homogeneous distribution of matter on all scales, whereas the appearance of observed inhomogeneities is related to the growth of small initial density perturbations. However, the analysis of the cosmological consequences of the particle theory indicates the possible existence of strongly inhomogeneous primordial structures in the distribution of both the dark matter and baryons. These primordial structures represent a new factor in galaxy formation theory. Topological defects such as the cosmological walls and filaments, primordial black holes, archioles in the models of axionic CDM, and essentially inhomogeneous baryosynthesis (leading to the formation of antimatter domains in the baryon-asymmetric Universe [274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 3, 65, 4]) offer by no means a complete list of possible primary inhomogeneities inferred from the existing elementary particle models.

Observational cosmology offers strong evidences favoring the existence of processes, determined by new physics, and the experimental physics approaches to their investigation. Cosmoparticle physics [1, 2, 3, 4], studying the physical, astrophysical and cosmological impact of new laws of Nature, explores the new forms of matter and their physical properties. Its development offers the great challenge for theoretical and experimental research. Physics of Primordial Black holes can play important role in this process.

Acknowledgement

I express my gratitude to J.A. de Freitas Pacheco for inviting me to write this review and I am grateful to A. Barrau, S.G.Rubin and A.S.Sakharov for discussions and help in preparation of manuscript.

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