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Special Issue "Modified Gravity: From Black Holes Entropy to Current Cosmology"

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Astrophysics and Cosmology".

Deadline for manuscript submissions: closed (20 September 2012)

Special Issue Editors

Guest Editor
Prof. Dr. Sergei D. Odintsov (Website)

ICREA, P. Lluis Companyas 23, 08010 Barcelona and Institute of Space Sciences (IEEC-CSIC), C. Can Magrans s/n, 08193 Barcelona, Spain
Phone: (34)935814368
Fax: +34 93 5814363
Interests: cosmology; dark energy and inflation; quantum gravity; modified gravity and beyond general relativity; quantum fields at external fields
Guest Editor
Prof. Dr. Kazuharu Bamba

Division of Human Support System, Faculty of Symbiotic Systems Science, Fukushima University, Fukushima 960-1296, Japan
Phone: +81-24-503-3263
Fax: +81-24-598-3187
Interests: modified theories of gravity; dark energy; cosmology; particle-theory and field-theory models of the early universe; electric and magnetic fields; quantum aspects of black holes, evaporation, thermodynamics

Special Issue Information

Dear Colleagues,

Recent cosmological observations strongly support that the current expansion of the universe is accelerating. The origin of such a cosmic acceleration mechanism is one of the most significant problems in modern cosmology. Indeed, this is shown by the fact that the Nobel Prize in Physics 2011 was presented to the finding of the current cosmic acceleration by means of the observations of the Type Ia supernovae.

There are two representative approaches to explain the current accelerated expansion of the universe. One is to introduce “dark energy” in the framework of general relativity. The other is to modify a gravitational theory, such as F(R) gravity, so that we can obtain so-called geometrical dark energy. It is believed that a modified gravitational theory must pass cosmological bounds and solar system tests because it corresponds to an alternative theory of gravitation to general relativity. As another meaningful touchstone of modified gravity, it is important to examine whether the second law of thermodynamics can be satisfied in the models of modified gravity.

The fundamental connection between gravitation and thermodynamics has been suggested by the discovery of black hole thermodynamics with black hole entropy and Hawking temperature. In addition, it was shown that the Einstein equation can be derived from the proportionality of the entropy to the horizon area together with the Clausius relation in thermodynamics. This consequence has been applied to various cosmological settings as well as modified gravitational theories. In particular, the connections between thermodynamics and modified gravity have recently been discussed extensively.

In this special issue, we discuss the application of thermodynamics to the test of a successful alternative gravitational theory to general relativity. Through this procedure, we can obtain a clue to resolve the dark energy problem “geometrically”. It is considered that any successful modified gravity theory should obey the second law of thermodynamics. If the second law is violated in certain universes in a model, it is more likely to be due to an incorrect generalization of the second law or some inherent inconsistency of the model itself. For the latter case, the model should be abandoned. It is strongly expected that the considerations of this special issue can produce our new physical understanding on entropy in the context of the relation between thermodynamics and gravitation and shed light on novel ingredients as well as insights on modern cosmology, in particular new properties of dark energy.

Prof. Dr. Sergei D. Odintsov
Dr. Kazuharu Bamba
Guest Editors

Keywords

  • Quantum aspects of black holes, evaporation, thermodynamics
  • Black hole entropy
  • Modified theories of gravity
  • Dark energy
  • Cosmology

Published Papers (17 papers)

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Research

Jump to: Review

Open AccessArticle The Thermal Entropy Density of Spacetime
Entropy 2013, 15(1), 156-161; doi:10.3390/e15010156
Received: 28 November 2012 / Accepted: 25 December 2012 / Published: 8 January 2013
Cited by 2 | PDF Full-text (147 KB)
Abstract
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The [...] Read more.
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results confirm that there is a profound connection between gravity and thermodynamics. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Entropy and Gravity
Entropy 2012, 14(12), 2456-2477; doi:10.3390/e14122456
Received: 26 October 2012 / Revised: 22 November 2012 / Accepted: 23 November 2012 / Published: 4 December 2012
Cited by 3 | PDF Full-text (369 KB) | HTML Full-text | XML Full-text
Abstract
The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion. Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal [...] Read more.
The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion. Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal equilibrium. The difference between the evolution of the cosmic entropy in a co-moving volume in an inflationary era with repulsive gravity and a matter-dominated era with attractive gravity is discussed. The significance of conversion of gravitational energy to thermal energy in a process with gravitational clumping, in order that the entropy of the universe shall increase, is made clear. Entropy of black holes and cosmic horizons are considered. The contribution to the gravitational entropy according to the Weyl curvature hypothesis is discussed. The entropy history of the Universe is reviewed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Periodic Cosmological Evolutions of Equation of State for Dark Energy 
Entropy 2012, 14(11), 2351-2374; doi:10.3390/e14112351
Received: 27 August 2012 / Revised: 27 September 2012 / Accepted: 23 October 2012 / Published: 20 November 2012
Cited by 8 | PDF Full-text (195 KB) | HTML Full-text | XML Full-text
Abstract
We demonstrate two periodic or quasi-periodic generalizations of the Chaplygin gas (CG) type models to explain the origins of dark energy as well as dark matter by using the Weierstrass ξ(t), σ(t) and ζ (t) functions with two periods being infinite. If the [...] Read more.
We demonstrate two periodic or quasi-periodic generalizations of the Chaplygin gas (CG) type models to explain the origins of dark energy as well as dark matter by using the Weierstrass ξ(t), σ(t) and ζ (t) functions with two periods being infinite. If the universe can evolve periodically, a non-singular universe can be realized. Furthermore, we examine the cosmological evolution and nature of the equation of state (EoS) of dark energy in the Friedmann–Lemaître–Robertson–Walker cosmology. It is explicitly illustrated that there exist three type models in which the universe always stays in the non-phantom (quintessence) phase, whereas it always evolves in the phantom phase, or the crossing of the phantom divide can be realized. The scalar fields and the corresponding potentials are also analyzed for different types of models. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Viscosity in Modified Gravity
Entropy 2012, 14(11), 2302-2310; doi:10.3390/e14112302
Received: 19 October 2012 / Revised: 6 November 2012 / Accepted: 8 November 2012 / Published: 12 November 2012
Cited by 1 | PDF Full-text (193 KB) | HTML Full-text | XML Full-text
Abstract
A bulk viscosity is introduced in the formalism of modified gravity. It is shownthat, based on a natural scaling law for the viscosity, a simple solution can be found forquantities such as the Hubble parameter and the energy density. These solutions mayincorporate [...] Read more.
A bulk viscosity is introduced in the formalism of modified gravity. It is shownthat, based on a natural scaling law for the viscosity, a simple solution can be found forquantities such as the Hubble parameter and the energy density. These solutions mayincorporate a viscosity-induced Big Rip singularity. By introducing a phase transition inthe cosmic fluid, the future singularity can nevertheless in principle be avoided. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Dark Energy Problem, Physics of Early Universe and Some New Approaches in Gravity
Entropy 2012, 14(11), 2143-2156; doi:10.3390/e14112143
Received: 7 September 2012 / Revised: 19 October 2012 / Accepted: 29 October 2012 / Published: 2 November 2012
PDF Full-text (201 KB) | HTML Full-text | XML Full-text
Abstract
The dark energy problem is studied based on the approach associated with the cosmological term in General Relativity that is considered as a dynamic quantity. It is shown that a quantum field theory of the Early Universe (Planck scales) and its limiting [...] Read more.
The dark energy problem is studied based on the approach associated with the cosmological term in General Relativity that is considered as a dynamic quantity. It is shown that a quantum field theory of the Early Universe (Planck scales) and its limiting transition at low energy play a significant role. Connection of this problem with Verlinde’s new (entropic) approach to gravity is revealed within the frame of such statement as well as the Generalized Uncertainty Principle (GUP) and Extended Uncertainty Principle (EUP). The implications from the obtained results are presented, and a more rigorous statement of the Concordance Problem in cosmology is treated. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Accelerating Universe and the Scalar-Tensor Theory
Entropy 2012, 14(10), 1997-2035; doi:10.3390/e14101997
Received: 18 September 2012 / Revised: 11 October 2012 / Accepted: 15 October 2012 / Published: 19 October 2012
PDF Full-text (587 KB) | HTML Full-text | XML Full-text
Abstract
To understand the accelerating universe discovered observationally in 1998, we develop the scalar-tensor theory of gravitation originally due to Jordan, extended only minimally. The unique role of the conformal transformation and frames is discussed particularly from a physical point of view. We [...] Read more.
To understand the accelerating universe discovered observationally in 1998, we develop the scalar-tensor theory of gravitation originally due to Jordan, extended only minimally. The unique role of the conformal transformation and frames is discussed particularly from a physical point of view. We show the theory to provide us with a simple and natural way of understanding the core of the measurements, Λobs ∼ t0−2 for the observed values of the cosmological constant and today’s age of the universe both expressed in the Planckian units. According to this scenario of a decaying cosmological constant, Λobs is this small only because we are old, not because we fine-tune the parameters. It also follows that the scalar field is simply the pseudo Nambu–Goldstone boson of broken global scale invariance, based on the way astronomers and astrophysicists measure the expansion of the universe in reference to the microscopic length units. A rather phenomenological trapping mechanism is assumed for the scalar field around the epoch of mini-inflation as observed, still maintaining the unmistakable behavior of the scenario stated above. Experimental searches for the scalar field, as light as ∼ 10−9 eV, as part of the dark energy, are also discussed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Exact Solution and Exotic Fluid in Cosmology
Entropy 2012, 14(9), 1771-1783; doi:10.3390/e14091771
Received: 21 August 2012 / Revised: 13 September 2012 / Accepted: 17 September 2012 / Published: 20 September 2012
Cited by 1 | PDF Full-text (538 KB) | HTML Full-text | XML Full-text
Abstract
We investigate cosmological consequences of nonlinear sigma model coupled with a cosmological fluid which satisfies the continuity equation. The target space action is of the de Sitter type and is composed of four scalar fields. The potential which is a function of only one of the scalar fields is also introduced. We perform a general analysis of the ensuing cosmological equations and give various critical points and their properties. Then, we show that the model exhibits an exact cosmological solution which yields a transition from matter domination into dark energy epoch and compare it with the Λ-CDM behavior. Especially, we calculate the age of the Universe and show that it is consistent with the observational value if the equation of the state ωf of the cosmological fluid is within the range of 0.13 < ωf < 0.22. Some implication of this result is also discussed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theories
Entropy 2012, 14(9), 1717-1770; doi:10.3390/e14091717
Received: 12 July 2012 / Revised: 14 August 2012 / Accepted: 24 August 2012 / Published: 18 September 2012
Cited by 83 | PDF Full-text (1458 KB) | HTML Full-text | XML Full-text
Abstract
Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are [...] Read more.
Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy–Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Cosmology of F(T) Gravity and k-Essence
Entropy 2012, 14(9), 1627-1651; doi:10.3390/e14091627
Received: 20 July 2012 / Revised: 17 August 2012 / Accepted: 21 August 2012 / Published: 4 September 2012
Cited by 20 | PDF Full-text (353 KB) | HTML Full-text | XML Full-text
Abstract
This a brief review on F(T) gravity and its relation with k-essence. Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough reconstruction analysis on the [...] Read more.
This a brief review on F(T) gravity and its relation with k-essence. Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough reconstruction analysis on the so-called F(T) models, where F(T) is some general function of the torsion term, and deduce the required conditions for the equivalence between of F(T) models with pure kinetic k-essence models. We present a new class of models of F(T)-gravity and k-essence. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip versus de Sitter
Entropy 2012, 14(8), 1578-1605; doi:10.3390/e14081578
Received: 21 July 2012 / Revised: 14 August 2012 / Accepted: 16 August 2012 / Published: 23 August 2012
Cited by 28 | PDF Full-text (314 KB) | HTML Full-text | XML Full-text
Abstract
We develop the general reconstruction scheme in two scalar model. The quintom-like theory which may describe (different) non-singular Little Rip or de Sitter cosmology is reconstructed. The number of scalar phantom dark energy models (with Little Rip cosmology or asymptotically de Sitter [...] Read more.
We develop the general reconstruction scheme in two scalar model. The quintom-like theory which may describe (different) non-singular Little Rip or de Sitter cosmology is reconstructed. The number of scalar phantom dark energy models (with Little Rip cosmology or asymptotically de Sitter evolution) is presented. Stability issue of such dark energy cosmologies as well as the flow to fixed points are studied. The stability of Little Rip universe which leads to dissolution of bound objects sometime in future indicates that no classical transition to de Sitter space occurs. The possibility of unification of inflation with Little Rip dark energy in two scalar theory is briefly mentioned. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle A Model of Nonsingular Universe
Entropy 2012, 14(7), 1296-1305; doi:10.3390/e14071296
Received: 19 April 2012 / Revised: 4 June 2012 / Accepted: 13 June 2012 / Published: 23 July 2012
Cited by 9 | PDF Full-text (112 KB) | HTML Full-text | XML Full-text
Abstract
In the background of Friedmann–Robertson–Walker Universe, there exists Hawking radiation which comes from the cosmic apparent horizon due to quantum effect. Although the Hawking radiation on the late time evolution of the universe could be safely neglected, it plays an important role [...] Read more.
In the background of Friedmann–Robertson–Walker Universe, there exists Hawking radiation which comes from the cosmic apparent horizon due to quantum effect. Although the Hawking radiation on the late time evolution of the universe could be safely neglected, it plays an important role in the very early stage of the universe. In view of this point, we identify the temperature in the scalar field potential with the Hawking temperature of cosmic apparent horizon. Then we find a nonsingular universe sourced by the temperature-dependent scalar field. We find that the universe could be created from a de Sitter phase which has the Planck energy density. Thus the Big-Bang singularity is avoided. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle The Dark Energy Properties of the Dirac–Born–Infeld Action
Entropy 2012, 14(7), 1203-1220; doi:10.3390/e14071203
Received: 1 May 2012 / Revised: 30 May 2012 / Accepted: 18 June 2012 / Published: 9 July 2012
PDF Full-text (309 KB) | HTML Full-text | XML Full-text
Abstract
Introducing a new potential, we deduce a general Lagrangian for Dirac–Born– Infeld (DBI) inflation, in which the determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into [...] Read more.
Introducing a new potential, we deduce a general Lagrangian for Dirac–Born– Infeld (DBI) inflation, in which the determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into each other at any same order, which is in agreement with the limit of classical physics. We also present a general sound speed in the evolutions of the universe, and the exact expressions of energy-momentum tensor, pressure and density. Furthermore, from the results we obtain the new equation of states. The analytic form of the action that is consistent with data turns out to be surprisingly simple and easy to categorize. Finally, we examine properties of the dark energy and introduce a novel mechanism for realizing either quintessence or phantom dark energy dominated phases within a string theoretical context. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle On Chirality of the Vorticity of the Universe
Entropy 2012, 14(5), 958-965; doi:10.3390/e14050958
Received: 16 April 2012 / Accepted: 11 May 2012 / Published: 16 May 2012
Cited by 4 | PDF Full-text (214 KB) | HTML Full-text | XML Full-text
Abstract
The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. [...] Read more.
The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. One can, consequently, expect the violation of the cosmic isotropy by the rotating Universe. We study chirality of the vorticity of the Universe within the Einstein–Cartan cosmology. The role of the spin of fermion species during the evolution of the Universe is studied by averaged spin densities and Einstein–Cartan equations. It is shown that spin density of the light Majorana neutrinos acts as a seed for vorticity at early stages of the evolution of the Universe. Its chirality can be evaluated in the vicinity of the spacelike infinity. It turns out that vorticity of the Universe has right-handed chirality. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle A Phase Space Diagram for Gravity
Entropy 2012, 14(5), 848-855; doi:10.3390/e14050848
Received: 20 March 2012 / Revised: 18 April 2012 / Accepted: 18 April 2012 / Published: 4 May 2012
Cited by 2 | PDF Full-text (108 KB) | HTML Full-text | XML Full-text
Abstract
In modified theories of gravity including a critical acceleration scale a0, a critical length scale rM = (GM/a0)1/2 will naturally arise with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger than rM. This adds a second critical scale to gravity, in addition to the one introduced by the criterion v < c of the Schwarzschild radius, rS = 2GM/c2. The distinct dependencies of the two above length scales give rise to non-trivial phenomenology in the (mass, length) plane for astrophysical structures, which we explore here. Surprisingly, extrapolation to atomic scales suggests gravity should be at the dark matter mimicking regime there. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

Review

Jump to: Research

Open AccessReview Conformal Gravity: Dark Matter and Dark Energy
Entropy 2013, 15(1), 162-176; doi:10.3390/e15010162
Received: 28 August 2012 / Revised: 17 December 2012 / Accepted: 7 January 2013 / Published: 9 January 2013
Cited by 7 | PDF Full-text (216 KB) | HTML Full-text | XML Full-text
Abstract
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion [...] Read more.
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and gauge boson theory but modifies Einstein–Hilbert general relativity and the Higgs scalar field model, with no new physical fields. Subgalactic phenomenology is retained. Without invoking dark matter, conformal gravity and a conformal Higgs model fit empirical data on galactic rotational velocities, galactic halos, and Hubble expansion including dark energy. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessReview Conformal Relativity versus Brans–Dicke and Superstring Theories
Entropy 2012, 14(10), 1978-1996; doi:10.3390/e14101978
Received: 20 August 2012 / Revised: 23 September 2012 / Accepted: 27 September 2012 / Published: 18 October 2012
Cited by 3 | PDF Full-text (316 KB) | HTML Full-text | XML Full-text
Abstract
We show how conformal relativity is related to Brans–Dicke theory and to low-energy-effective superstring theory. Conformal relativity or the Hoyle–Narlikar theory is invariant with respect to conformal transformations of the metric. We show that the conformal relativity action is equivalent to the [...] Read more.
We show how conformal relativity is related to Brans–Dicke theory and to low-energy-effective superstring theory. Conformal relativity or the Hoyle–Narlikar theory is invariant with respect to conformal transformations of the metric. We show that the conformal relativity action is equivalent to the transformed Brans–Dicke action for ω = -3/2 (which is the border between standard scalar field and ghost) in contrast to the reduced (graviton-dilaton) low-energy-effective superstring action which corresponds to the Brans–Dicke action with ω = -1. We show that like in ekpyrotic/cyclic models, the transition through the singularity in conformal cosmology in the string frame takes place in the weak coupling regime. We also find interesting self-duality and duality relations for the graviton-dilaton actions. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessReview Exact Solutions in Modified Gravity Models
Entropy 2012, 14(7), 1140-1153; doi:10.3390/e14071140
Received: 24 April 2012 / Revised: 21 May 2012 / Accepted: 21 May 2012 / Published: 25 June 2012
Cited by 3 | PDF Full-text (267 KB) | HTML Full-text | XML Full-text
Abstract
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this [...] Read more.
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

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