**Abstract**—In recent years, hypotheses about a cyclical Universe have been again actively considered. In these cosmological theories, the Universe, instead of a “one-time” infinite expansion, periodically shrinks to a certain volume, and then again experiences the Big Bang. One of the problems for the cyclic Universe will be its compatibility with a vast population of indestructible black holes that accumulate from cycle to cycle. The article considers a simple iterative model of the evolution of black holes in a cyclic Universe, independent of specific cosmological theories. The model has two free parameters that determine the iterative decrease in the number of black holes and the increase in their individual mass. It is shown that this model, with wide variations in the parameters, explains the observed number of supermassive black holes at the centers of galaxies, as well as the relationships between different classes of black holes. The mechanism of accumulation of relict black holes during repeated pulsations of the Universe may be responsible for the black hole population detected by LIGO observations and probably responsible for the dark matter phenomenon. The number of black holes of intermediate masses corresponds to the number of globular clusters and dwarf satellite galaxies. These results argue for models of the oscillating Universe, and at the same time impose substantial requirements on them. Models of a pulsating Universe should be characterized by a high level of relict gravitational radiation generated at the time of maximum compression of the Universe and mass mergers of black holes, as well as solve the problem of the existence of the largest black hole that is formed during this merger. It has been hypothesized that some neutron stars can survive from past cycles of the Universe and contribute to dark matter. These relict neutron stars will have a set of features by which they can be distinguished from neutron stars born in the current cycle of the birth of the Universe. The observational signs of relict neutron stars and the possibility of their search in different wavelength ranges are discussed.
*Editorial note.* This paper caused controversy in the expert community. Nevertheless, the Editorial Board decided to accept it and publish to raise a discussion on this actual field of research.

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## ACKNOWLEDGMENTS

The authors are grateful to A. Vasilkov, A. Bogomazov, J. Mather, as well as to anonymous reviewers for useful discussions and comments, and L.B. Potapova for help in preparing the work.

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Translated by T. Sokolova

## APPENDIX. OSCILLATING MODEL OF THE UNIVERSE WITH A VARIABLE GRAVITATIONAL MASS

### APPENDIX. OSCILLATING MODEL OF THE UNIVERSE WITH A VARIABLE GRAVITATIONAL MASS

The results of the LIGO group showed that about \(5\% \) of the mass of merging black holes is converted into gravitational radiation. The Nobel laureate Anderson (2018) noted that this requires the construction of a cosmological model with a variable gravitational mass. Indeed, there is an interpretation of GR that goes back to the works of Einstein, Eddington, and Schrodinger (see references in the article Gorkavyi and Vasilkov (2016)), according to which gravitational waves should not be included in the sources of the gravitational field, therefore, the transition of part of the mass of the Universe to gravitational radiation reduces its total gravitational mass. This is exactly what happens when the observed population of black holes merges during the compression of the Universe and dumps a significant part of its mass into gravitational waves, which contribute to the background of the relict gravitational radiation. Consider a model of a cyclic Universe with a variable gravitational mass. Kutschera (2003) obtained a modified Schwarzschild metric for the variable gravitational mass of a fireball in the weak field approximation:

where \(b\left( {t,r} \right) = 2GM{{\left( {{{t - r} \mathord{\left/ {\vphantom {{t - r} c}} \right. \kern-0em} c}} \right)} \mathord{\left/ {\vphantom {{\left( {{{t - r} \mathord{\left/ {\vphantom {{t - r} c}} \right. \kern-0em} c}} \right)} {(r{{c}^{2}})}}} \right. \kern-0em} {(r{{c}^{2}})}}\).

Kutschera (2003) concluded that a decrease in the gravitational mass of a fireball generates a monopole gravitational wave. Note that Birkhoff’s theorem, which is often referred to as an argument against the existence of monopole gravitational waves, is not applicable to systems with gravitational radiation that do not have spherical symmetry. In Gorkavyi and Vasilkov (2016), a system consisting of many gravitational wave emitters is considered, and a similar space-time metric for a variable mass is independently obtained. Consider a quasi-spherical system in which gravitational radiation is generated, for example, during the merger of black holes. For the weak external gravitational field of this system, we write: \({{g}_{{\mu \nu }}} = {{\eta }_{{\mu \nu }}} + {{h}_{{\mu \nu }}}\), where \({{\eta }_{{\mu \nu }}}\) is the Minkowski tensor and \({{\eta }_{{\mu \nu }}} \gg {{h}_{{\mu \nu }}}\). Let’s write down the Einstein equation for the weak field in the known form (see references in Gorkavyi and Vasilkov (2016)):

The solution of the wave equation (12) is the retarded potential, which includes a variable gravitational mass. The zero component of the metric tensor will take the form (Kutschera (2003), Gorkavyi and Vasilkov (2016)):

In the paper Gorkavyi and Vasilkov (2016) for the modified Schwarzschild metric (11), (13), the gravitational acceleration was calculated and it was indicated that the mass merger of black holes at the last stage of the collapse of the Universe can generate a repulsive force that exceeds the gravitational attraction. We describe the variable mass of the Universe with the following function: \(M = {{M}_{0}}{{e}^{{ - \alpha (t - r/c)}}}\). Assuming a weak gravitational field and low velocities, we get an expression for the gravitational acceleration:

The first term characterizes the Newtonian attraction. For \(\alpha > 0\), the second term describes “anti-gravity”, and in the case of \(\alpha < 0\)—“hypergravity”. To clarify the physical meaning of equation (14), we can write the gravitational acceleration in terms of the quasi-Newtonian potential \(\varphi \): \(F = - \tfrac{{\partial \varphi }}{{\partial r}}\), where \(\varphi = - \tfrac{{GM({{t - r} \mathord{\left/ {\vphantom {{t - r} c}} \right. \kern-0em} c})}}{r}\) (this equation was published by Gorkavyi in 2003 (see the reference in Gorkavyi and Vasilkov (2016)). After differentiating this potential, the expression (14) is obtained. Attraction is analogous to the motion of the balls along the inclined surface of the potential funnel, and anti-gravity corresponds to the scattering of the balls if the potential forms a peak, and not a funnel. An analog of this antigravity has already been investigated earlier: a similar repulsive force is responsible for the escape of a black hole formed by the merger of black holes of unequal masses (Rezzolla et al., 2010). The resulting gravitational potential is not directed inwards, but outwards, and it throws the new hole away from the point of fusion, even if it was located in the center of gravity of the system. It is logical to assume that the second term of equation (14) is responsible for the Big Bang mechanism. Consider the conditions under which antigravity will be stronger than gravitational attraction. Let a system with mass \(M\) and radius \(R\) decrease its mass due to its transformation into gravitational waves. From equation (14), it is not difficult to obtain the condition for the dominance of antigravity over attraction:

where \(E\) is the energy of the gravitational matter. Although this condition was formally obtained for the case of weak fields, it can be expected that it will be satisfied for any fields, because a similar formula is obtained from quasi-Newtonian calculations, which are not subject to any restrictions (Gorkavyi and Vasilkov, 2016). To estimate the radiation of a system of gravitational waves, you can use the well-known expression for the radiation power of gravitational waves:

Here we have introduced the non-sphericity parameter \(S\), which is zero for a perfectly spherical system and of the order of \(1\) for a binary star system. Substituting (16) into (15), we obtain the condition under which antigravity begins to dominate:

It is easy to see that this condition coincides to the accuracy of the coefficient \(S\) with the condition of being inside a black hole:

Popławski (2016) develops an elegant model of a cyclical Universe pulsating inside a huge black hole. Within the framework of such a model, condition (18) should be satisfied, unless the introduced nonsphericity parameter \(S\) is abnormally small. As shown in many works, gravitational collapse leads to an increase in nonsphericity. This can also be formulated in Newtonian language: small fluctuations of the surface of the collapsing ball will be unstable due to stretching by tidal forces, which grow as \({1 \mathord{\left/ {\vphantom {1 {{{r}^{3}}}}} \right. \kern-0em} {{{r}^{3}}}}\), that is, faster than gravitational forces: \({1 \mathord{\left/ {\vphantom {1 {{{r}^{2}}}}} \right. \kern-0em} {{{r}^{2}}}}\). When the Universe is compressed to several light years, its radius is reduced by 10 orders of magnitude, increasing the right side of (18) by 40 orders of magnitude, so it can be assumed that at any realistic degree of nonsphericity, condition (18) will be satisfied. This removes the general singularity problem: any system within the Schwarzschild radius evaporates into gravitational waves and generates powerful repulsive forces before reaching the singularity. Note that this does not contradict the Hawking–Penrose theorems, which are not applicable to systems with antigravity and with a positive cosmological constant (Hawking and Penrose, 1970). Thus, the Big Bang is an expansion of the fireball of the compressed Universe, which got into the conditions of strong antigravity, which arose during the Big Compression.

As shown in Gorkavyi and Vasilkov (2018), the problem of dark energy also finds its logical solution in the cosmological model with a variable gravitational mass. During the expansion phase of the Universe, the merging of black holes and the transition of their mass into gravitational radiation becomes a rare event, and black holes begin to grow, absorbing the background gravitational radiation. The biggest black hole (BBH) formed during the collapse of the Universe, grows faster than all. Let us study the influence of the gravitational field of the growing BBH on the expansion of the Universe. We derive the modified Friedman equations for a metric with variable mass in the comoving coordinates \({{x}_{ * }},{{y}_{ * }},{{z}_{ * }}\):

where \(b(t,r) = {{2GM(t,r)} \mathord{\left/ {\vphantom {{2GM(t,r)} {(r{{c}^{2}})}}} \right. \kern-0em} {(r{{c}^{2}})}}\) is a known function, and \(a(t,r)\) is an unknown scale factor. For \(\left| \alpha \right| \gg {c \mathord{\left/ {\vphantom {c r}} \right. \kern-0em} r}\), the dependence of \(a(t,r)\) on spatial coordinates is significantly weaker than that of the function *b*(*t*, *r*) (Gorkavyi and Vasilkov, 2018). The paper Gorkavyi and Vasilkov (2018) considered the more general case of metric (19), which takes into account the weak inhomogeneity of time, and it was shown that the modified Friedman equation includes perturbations \(b(t,r)\) only from the spatial part of the metric. For the case of a weak gravitational field and \(b(t,r) \ll 1\), we obtain the first Friedman equation in the form:

where the cosmological function \(\Lambda (t,r)\) is expressed as follows:

or

where \(x,y,z\) are the physical coordinates. From equation (22) for \(\left| \alpha \right| \gg {c \mathord{\left/ {\vphantom {c r}} \right. \kern-0em} r}\) we get

where \({{r}_{0}} = \tfrac{{2GM(t,r)}}{{r{{c}^{2}}}}\) is the Schwarzschild radius, \(T \approx 4 \times {{10}^{{17}}}\) s is the cosmological time. \(\Lambda (t,r)\) is equal to the observed value of the cosmological constant \(1.1 \times {{10}^{{ - 56}}}\mathop {\;{\text{cm}}}\nolimits^{ - 2} \) if \(\mathop {(\alpha T)}\nolimits^2 \tfrac{{{{r}_{0}}}}{r} = 1.6\) (Gorkavyi and Vasilkov, 2018). For example, the latter is true if the dimensionless parameter \(\frac{{{{r}_{0}}}}{r} = 0.016\) and \(\left| \alpha \right|T = 10\). The existence of relict gravitational waves is not in doubt (they are most intensively generated when black holes merge with each compression of the Universe), but the level of their energy is unknown. Their total energy was usually limited by the condition that they did not make a noticeable contribution to the total gravitational density of the Universe. From the point of view of Einstein–Eddington–Schrödinger, gravitational radiation does not contribute to the gravitational mass of the Universe, so this restriction becomes invalid. In the article Gorkavyi et al. (2018), the density of the medium of gravitational waves \({{\rho }_{{{\text{GW}}}}}\) was determined, with the absorption of which the BBH increases at a rate sufficient for the observed value of the cosmological constant:

Then we get

where \(\left| \alpha \right|T = 10\) and the BBH mass \(M = 6 \times {{10}^{{54}}}\) g (this BBH mass can be calculated from the condition \(\tfrac{{{{r}_{0}}}}{r} \approx 0.02\) (Gorkavyi and Vasilkov, 2018)). If we take \(\left| \alpha \right|T = 100\), then \(\tfrac{{{{r}_{0}}}}{r} \approx 2 \times {{10}^{{ - 4}}}\), the BBH mass \(M = 6 \times {{10}^{{52}}}\) g, and \({{\rho }_{{{\text{GW}}}}} \sim {{10}^{{ - 25}}}\) g cm\(^{{ - 3}}\). The perturbation of the gravitational field from BBH, which is now observed in the form of accelerated recession of galaxies, arose about 13 billion years ago, therefore, during this time, the energy density of the background of gravitational radiation should decrease by several orders of magnitude. Consider the case when a term with a cosmological function dominates over a term with an average density. For an exponential change in the BBH mass, we obtain \(\dot {\Lambda }(t,r) = - \alpha \Lambda (t,r)\). Then the second Friedman equation will take the form:

Hence it follows that the acceleration of the observable part of the Universe \(\tfrac{{\ddot {a}}}{a} > 0\) is caused by the effect of hypergravity at \(\alpha < 0\). Hypergravity from the growing BBH (or the population of the largest holes) stretches the observed set of galaxies so that, from the point of view of a local observer, the galaxies are moving away from each other with acceleration. A number of observations indicate an increase in the constant \({{H}_{0}}\) with time (Riess, 2020). This presents a problem for theories that tie the cosmological constant to the properties of the vacuum. In the model under consideration, the cosmological constant is a function of the BBH mass and the density of background gravitational waves \({{\rho }_{{{\text{GW}}}}}\), so the growth of \({{H}_{0}}\) is admissible.

Let us consider the problem of the transition of the expanding Universe to contraction. The fact that the observed local average density of matter is less than the critical density required to close the Universe does not mean that the Universe is open. If BBH grows, absorbing not only ordinary matter and radiation, but also background gravitational waves, then the total gravitational mass of the Universe will also increase. BBH can expand and include all observed galaxies and stars. For a resting external observer, the signature of the Schwarzschild metric inside a black hole “deteriorates”, which is usually corrected by a change (redesignation) of the spatial and temporal axes, leading to exotic dynamics of bodies inside black holes. On the one hand, this topic has not been studied enough, and other options for correcting the signature are possible. On the other hand, the dynamics of a freely falling observer obeys the principle of equivalence, according to which such a local observer cannot distinguish his fall from rest. Therefore, a freely falling observer easily crosses the surface of a black hole in its own reference frame (from the point of view of an external stationary observer, this is impossible, because time stops at the boundary of a black hole). From the point of view of local observers, all their galaxies will freely cross the surface of a large black hole and will exist and move inside it, following the laws of General Relativity. This permits the consideration of cosmology inside a black hole, as, for example, Popławski does (Popławski, 2016). Thus, the growth of BBH and the absorption of all matter in the Universe by it solves the problem of stopping the Universe and its subsequent collapse. It is well known that the average density required to close the Universe is the same as the density of a black hole of size of the Universe. If the Universe is closed, then space should have a slight positive curvature. Indeed, from the data from the WMAP and Planck satellites, the positive curvature of the Universe space is established with a confidence of more than \(99\% \) (Di Valentino et al. 2020). Consider the entropy growth problem, which is difficult for all cyclic cosmologies. Black holes have huge entropy (Egan and Lineweaver, 2010):

where \(k\) is the Boltzmann constant. It is known that the main entropy of the Universe is contained in the largest black holes (Egan and Lineweaver, 2010). From this formula, we obtain the following estimate of the entropy value for the population of black holes: \({\text{Entropy}} \sim {{10}^{{77}}}{{({{M}_{ \odot }})}^{2}}N[k]\). Entropy for a population of black holes with the same hole mass of 30 solar masses (assuming they make up all dark matter): \(M = 30{{M}_{ \odot }};\) \(N = {{10}^{{22}}}\); \({\text{Entropy}} \sim {{10}^{{102}}}[k]\). The entropy of one BBH is 20 orders of magnitude higher: \(M = 5 \times {{10}^{{22}}}{{M}_{ \odot }}\); \(N = 1;\,\,{\text{Entropy}} \sim 2.5 \times {{10}^{{122}}}[k]\). Thus, practically all the entropy of the Universe will be contained in the BBH that arose in this cycle, so the entropy problem can be formulated as follows: what to do with the BBH that arises at each cycle of the Universe? Obviously, the only way to get rid of BBH is to get into it and start a new cosmological cycle inside, which is what our scenario suggests. In this case, the entropy problem is solved automatically: the Bekenstein–Hawking entropy is determined only for an outside observer who measures a certain thermal temperature of a black hole. This temperature, and hence the entropy of the black hole, is not determined for the internal observer. Thus, a set of galaxies, getting inside a black hole, find themselves in a system whose entropy is twenty orders of magnitude less than the entropy of the Universe at the stage of expansion. This sharp decrease in entropy also manifests itself in the dynamics of matter and radiation: at the stage of expansion of the Universe, the density of matter dropped, as did the temperature of the background radiation, and after entering the big black hole, the density of matter begins to grow, as does the temperature of radiation. The surface of the big black hole, having absorbed all matter and background radiation of the Universe, goes to the periphery, and inside this huge hole the observed part of the Universe begins a new cycle under conditions of reduced entropy. Formally, the entropy of the Universe as a whole, taking into account the surface of the outer outer shell of a black hole (slowly increasing or “multilayered”), grows with each cycle, but in practice, the observed set of galaxies enters a new cycle, dropping entropy to a minimum. Thus, the Universe turns out to be a closed system or an object with variable gravitational mass, pulsating inside a huge black hole. A similar model has been considered not only by Popławski (2016), but also by other authors (see references in Gorkavyi and Vasilkov (2018)). This cosmological model, in addition to baryons, gravitational and electromagnetic radiation, contains an extensive population of black holes, the evolution of which is the subject of study in the main text of this article.

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Gorkavyi, N.N., Tyul’bashev, S.A. Black Holes and Neutron Stars in an Oscillating Universe.
*Astrophys. Bull.* **76, **229–247 (2021). https://doi.org/10.1134/S199034132103007X

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### Keywords:

- cosmology: dark matter—stars: black holes—stars: neutron