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Black holes are seen as thermodynamical systems. They can be characterized by thermodynamic intensive quantities such as temperature, pressure, and extensive quantities such as entropy and energy density. In this paper, we focus on the thermodynamical properties of (a) Schwarzschild, (b) Reissner–Nordström, (c) Kerr, and (d) Kerr–Newman black holes. These allow to estimate the corresponding equations-of-state (EoS) and the squared speed of sound for each of them. The latter helps to calculate the ratio of the bulk viscosity to the shear viscosity for each of the black holes. We conclude that the resulting EoS enhances when moving from Schwarzschild to Reissner–Nordström to Kerr, and to Kerr–Newman black holes.

We suggest that the incapability of the thermal models in reproducing the horn-like structure of the Kaon-to-pion ratios measured at AGS, SPS, and RHIC energies and then confirmed in the Beam Energy Scan programs is likely emerged from an ad hoc utilization of inappropriate types of statistics, either extensive additive Boltzmann-Gibbs (BG) or the Tsallis-type of nonextensivity. In this regard, we emphasize that any suggested statistical approach for the dynamical and likely nonequilibrium nonextensive particle production, for example, Kaon and pion yields, should not necessarily remain valid over the whole range of the collision energies. We claim that generic (non)extensive statistics with two equivalence classes (c,d) is the most suitable statistical approach to replace all those assumptions and conjectures. Accordingly, the statistical ensemble itself becomes autonomously capable to determine its degree of extensivity or nonextensivity, where these are not necessarily limited to BG orTsallis. The generic approach covers the entire (c,d)-space, in which both extensive BG and nonextensive Tsallis statistics are characterized by distinct equivalence classes, (1,1) and (0,q), respectively. The energy dependence of the light-, γq, and strange-quark occupation factor, γs, indicates that the produced particles are best described as a nonequilibrium ensemble, where the remarkable nonmonotonic behavior associating the horn-structure of K+/π+ ratio is qualitatively reproduced. On other hand, the resulting equivalence classes (c,d) refer to a generic nonextensivity associated with an extended exponential and a Lambert-W0 exponentially generating distribution function, which obviously emerge from free, short- and long-range correlations over the wide range of the collision energies. With this generic statistical approach, the hadron resonance gas model, ideal thermal model, excellently reproduces the K+/π+ ratio, including its nonmonotonic horn-like structure. Therefore, we conclude that the utilization of any specific type of extensive or nonextensive statistics must be strictly aligned to the statistical nature of the underlying ensemble, otherwise, the price to be paid becomes unexpectedly huge. Such a universal conclusion apparently goes beyond the reproduction of the horn-like structure of K+/π+ ratio or particle physics to all possible statistical implications either on stock markets or economy or sociology or biological evolution or weather prediction, etc. This brings about the great deal of the present paper.

In statistical physics, the dominance of Boltzmann--Gibbs distribution seems to endow an impression that this would be the only statistical approach. Indeed, there is a large class of statistical approaches referring to various types of nonextensivity. For instance, $\log$ and $\exp$ distributions play a crucial role in extensive and nonextensive statistical mechanics. Emerging in a physical system, Maxwell--Boltzmann statistics (MB) defines extensive entropy which relates the number of microstates to thermodynamic quantities. In this regard, the Boltzmann distribution is nothing but a probability distribution. The various types of nonextensive statistics categorically violate the fourth Shannon--Khinchen additivity. We compare between $\log$ and $\exp$ distributions in MB, Tsallis, and generic statistics and studied their mathematical properties. We conclude that their compatibility exclusively depends on the nonextensivity parameters. PACS numbers: 05.70.Ln, 05.65.+b

We used blast wave model with Tsallis statistics to analyze the experimental data measured by ALICE Collaboration in proton-proton collisions at Large Hadron Collider and extracted the related parameters (kinetic freeze-out temperature, transverse flow velocity and kinetic freeze-out volume of emission source) from transverse momentum spectra of the particles. We found that the kinetic freeze-out temperature and kinetic freeze-out volume are mass dependent. The former increase while the latter decrease with the particle mass which is the evidence of a mass as well as volume differential kinetic freeze-out scenario. Furthermore we extracted the mean transverse momentum and initial temperature by an indirect method and observed that they increase with mass of the particles. All the above discussed parameters are observed to increase with energy. Triton ($t$), hyper-triton (${^3_{\bar\Lambda} H}$) and helion (${^3 He}$) and their anti-matter are observed to freeze-out at the same time due to isospin symmetry., Comment: 13 pages, 5 figures, 1 table

Based on statistical thermal approaches, the transverse momentum distribution of the well-identified produced particles, π+, π–, K+, K–, p, and $$\bar {p}$$, is studied. We aim at introducing a novel almost entirely empirical estimation for the inclusive chemical potential μ. From the partition function of a grand-canonical ensemble, we propose a generic expression for the dependence of μ on the rapidity y. Then, by fitting this expression with the experimental results of the most central p⊥ and d2N/2πp⊥dp⊥dy, at 7.7, 11.5, 19.6, 27, 39, 130, 200 GeV, we introduce a generic expression for the rapidity dependence of for all particle yields, at different energies, μ = a + by2. The resulting energy dependence reads $$\sqrt {{{s}_{{{\text{NN}}}}}} $$ = c[(μ – a)/b]d/2. As a validation test, we show that the proposed approach reproduces excellently the rapidity spectra of various particle yields measured, at different energies.

In this paper, artificial neural network (ANN) model is used to estimate the multiplicity per rapidity for charged pions and kaons observed in various high-energy experiments from central [Formula: see text] heavy-ion collisions with energies ranging from 2–200 GeV, and then compared to available experimental data, including RHIC-BRAHMS experiment, and covering the energy range of the future accelerator facilities at NICA and FAIR. We also used Landau hydrodynamical approach, which has a better description for the evolution of hot and dense matter produced in ultra-relativistic heavy-ion collisions. The approach is fitted to both results estimated from experiment and ANN simulation. We noted that the Landau model accurately reproduces the entire range of multiplicity per rapidity for all created particles at all energies. Also ANN model can reproduce the multiplicity per rapidity very well for all the considered particles. This encourages us to use ANN model to predict the multiplicity per rapidity for [Formula: see text] particles at energies [Formula: see text] and [Formula: see text] GeV.

The nature of fractality and correlations in relativistic heavy-ion interactions of $$^{16}$$ O–Ag/Br at collision energy 60 A GeV are investigated in the framework of the sandbox algorithm using the concept of isotropy and anisotropy of phase space. In this work, we have attempted to find the deterministic information in $$\eta -\phi $$ space using the concept of Hurst exponent (H) by means of precise determination of topological parameters from the complex network perspective. The experimental data have been compared with Monte Carlo (MC) generated events to extract non-statistical dynamics. It has been observed that the process under study is multifractal in nature. It is also observed that the process is ruled by a small number of large fluctuations and contributes to correlated stochastic long-term positive auto-correlations. The characteristics of the multiparticle production process under study are independent of the spatial distribution of $$\eta $$ and $$\phi $$ space, where $$\eta $$ is pseudorapidity and $$\phi $$ is the azimuthal angle, but the degree of multifractality depends on the spatial distribution of pions in anisotropic phase space during the multiparticle production process. It is also observed that for different Hurst exponents, the multifractality of the experimental data sets are significantly different from the MC-simulated events.

The QCD phase diagram is studied, at finite magnetic field. Our calculations are based on the QCD effective model, the SU($3$) Polyakov linear sigma model (PLSM), in which the chiral symmetry is integrated in the hadron phase and in the parton phase, the up-, down- and strange-quark degrees of freedom are incorporated besides the inclusion of Polyakov loop potentials in the pure gauge limit, which are motivated by various underlying QCD symmetries. The Landau quantization and the magnetic catalysis are implemented. The response of the QCD matter to an external magnetic field such as magnetization, magnetic susceptibility and permeability has been estimated. We conclude that the parton phase has higher values of magnetization, magnetic susceptibility, and permeability relative to the hadron phase. Depending on the contributions to the Landau levels, we conclude that the chiral magnetic field enhances the chiral quark condensates and hence the chiral QCD phase diagram, i.e. the hadron-parton phase transition likely takes place, at lower critical temperatures and chemical potentials., Comment: 16 pages, 11 figures, the paper is accepted for publication in The European Physical Journal A (EPJA)

Based on quantum mechanics and general relativity, Karolyhazy proposed a generalization to the well-known Heisenberg uncertainty relation in which the energy density of quantum fluctuations of space-time plays a crucial role. Later on, various holographic DE models were suggested, in which the Hubble scale (size) and the age of the universe were assumed as measures for the largest infrared cutoff satisfying the holographic principle and energy bounds assuring applicability of quantum field theory. We review various models based on the holographic principle and the Karolyhazy relation and compare these to the space-time foam and superconducting DE models. We analyze their (in)stability against cosmological perturbation.

When minimal length uncertainty emerging from generalized uncertainty principle (GUP) is thoughtfully implemented, it is of great interest to consider its impacts on {\it "gravitational} Einstein field equations (gEFE) and to try to find out whether consequential modifications in metric manifesting properties of quantum geometry due to quantum gravity. GUP takes into account the gravitational impacts on the noncommutation relations of length (distance) and momentum operators or time and energy operators, etc. On the other hand, gEFE relates {\it classical geometry or general relativity gravity} to the energy-momentum tensors, i.e. proposing quantum equations of state. Despite the technical difficulties, we confront GUP to the metric tensor so that the line element and the geodesic equation in flat and curved space are accordingly modified. The latter apparently encompasses acceleration, jerk, and snap (jounce) of a particle in the {\it "quasi-quantized"} gravitational field. Finite higher-orders of acceleration apparently manifest phenomena such as accelerating expansion and transitions between different radii of curvature, etc., Comment: 4 pages, 0 figure, invited talk at 9th int. Workshop on Astronomy and Relativistic Astrophysics (IWARA2020 Video Conference). accepted for publication in Astronomical Note (AN)

The visibility graph (VG) and horizontal visibility graph (HVG) have been recently introduced as a mapping between the time series and the complex networks. Here, the relativistic nucleus–nucleus interaction data, $$^{24}$$ Mg-Ag/Br at 4.5 A GeV/c are analyzed and used to characterizing the multifractal properties by extending VG, HVG and the sandbox algorithm (SB). In this work, the HVG–SB techniques have been applied to confirm the scale-freeness and the presence of fractal properties in the process of multiplicity fluctuation. We try to find deterministic information by means of precise determination of topological parameter from a complex network perspective. We have analyzed the degree distributions which are associated with event-wise pseudorapidity distributions, typical of any multifractality measurement, different patterns of symmetry scaling, scale-freeness, correlation and clustering among the produced particles. We try to find deterministic information by means of precise determination of topological parameter from a complex network perspective. It has been also observed that the multifractal variables were derived by SB algorithm for the experimental data sets to some extent distinct from the respective Monte Carlo simulated results.

The rapidity distribution of well-identified particles such as pions, kaons, protons and their antiparticles measured in AGS, SPS, and BRAHMS experiments (Au+Au collisions), at various energies spanning from higher energies s NN = 200 down to lower energies 2 GeV, are compared with that obtained from huge statistical ensembles of 100, 000 events generated from the Cosmic Ray Monte Carlo (CRMC) EPOS event-generator. All these data are then compared to the results calculated in the frame work of Landau hydrodynamical model. The EPOS event-generator can describe the experimental data of the rapidity distribution for all mentioned particles at all considered energies. This leads to use the EPOS event-generator to produce the rapidity distribution for various particles at energies where there are no experimental data yet to compare with. Such a prediction might serve as an input for future experiments such as FAIR and NICA for instance. A successful comparison between the Landau hydrodynamical approach and the computational CRMC EPOS 1.99 model in describing the rapidity distribution for the mentioned particles is shown in the presence of the experimental data. We conclude that the Landau hydrodynamical model is capable of describing both experimental data and EPOS 1.99 event-generator results, and they are in a good agreement.

Based on quantum mechanical framework for the minimal length uncertainty, we demonstrate that the generalized uncertainty principle (GUP) parameter could be best constrained by recent gravitational waves observations on one hand. On other hand this suggests modified dispersion relations (MDRs) enabling an estimation for the difference between the group velocity of gravitons and that of photons. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant (apparently manifesting gravitational influences on the vacuum energy density), we suggest a possible solution for the cosmological constant problem., 5 pages, 0 figures, an invited talk given at the int. Workshop on Astronomy and Relativistic Astrophysics (IWARA $2020$ Video Conference), accepted for publication in Astronomische Nachrichten

We compare results from the Polyakov linear-sigma model (PLSM) in optimized perturbation theory (OPT) with the mean-field approximation (MFA). At finite temperatures and chemical potentials, the chiral condensates and the decofinement order parameters, the thermodynamic pressure, the pseudo-critical temperatures, the subtracted condensates, the second- and high-order moments of various conserved charges (cumulants) obtained in MFA are compared with OPT and also confronted to available lattice QCD simulations. We conclude that when moving from lower- to higher-order moments of various quantum charges, OPT becomes more closer to QCD., 15 pages, 8 figures, 1 table, submitted to PRC

The proposed correspondence between the Hawking-Unruh radiation mechanism in rotating, electrically-charged and electrically-charged-rotating black holes and the hadronization in high-energy physics is assumed. This allows us to determine the well-profound freezeout parameters of the heavy-ion collisions. Furthermore, black holes thermodynamics is found analogical to that of the high-energy collisions. We also introduce a relation expressing the dependence of the angular momentum and the angular velocity deduced from rotating black holes on the chemical potential. The novel phase diagram for rotating, electrically-charged and electrically-charged-rotating black holes are found in an excellent agreement with the phase diagrams drawn for electrically-charged black holes and also with the ones mapped out from the statistical thermal models and the high-energy experiments. Moreover, our estimations for the freezeout conditions $\langle E\rangle/\langle N\rangle$ and $s/T^3$ are in excellent good agreement with the ones determined from the hadronization process, especially at $\mu\leq 0.3$ GeV., Comment: 15 pages, 2 figures, submitted IJMPD

The transverse momentum spectra of the well-identified produced particles, $\pi^+$, $\pi^-$, $K^+$, $K^-$, $p$, $\bar{p}$, $K_s^0$, $\Lambda$, $\bar{\Lambda}$, $\Xi^-$, and $\Xi^+$ are analyzed in a statistical approach. From the partition function of grand-canonical ensemble, we propose a generic expression for the dependence of the generic chemical potential $\mu$ on the rapidity $y$. Then, by fitting this expression to the experimental results on the most central $p_{\perp}$ and $d^2 N/2 \pi p_{\bot} dp_{\bot} dy$, at energies ranging from $7.7$ to $200~$GeV, we have introduced a generic expression for the rapidity dependence of $\mu$, at different energies, namely $\mu=a+b y^2$, where $a$ and $b$ is constants. We find that the resulting energy dependence $\sqrt{s_{\mathtt{NN}}}=c[(\mu-a)/b]^{d/2}$ agrees well with the statistical thermal models. We also present precise estimations of various types of chemical potentials, $\mu_B$, $\mu_S$, and $\mu_Q$, the baryon, the strangeness, and the charge chemical potential, respectively., Comment: 15 pages, 3 eps figures, 13 tables, submitted to INJP

The particle production in relativistic heavy-ion collisions seems to be created in a dynamically disordered system which can be best described by an extended exponential entropy. In distinguishing between the applicability of this and Boltzmann-Gibbs (BG) in generating various particle-ratios, generic (non)extensive statistics (GNS) is introduced to the hadron resonance gas model. Accordingly, the degree of (non)extensivity is determined by the possible modifications in the phase space. Both BG extensivity and Tsallis nonextensivity are included as very special cases defined by specific values of the equivalence classes $(c, d)$. We found that the particle ratios at energies ranging between $3.8$ and $2760~$GeV are best reproduced by nonextensive statistics, where $c$ and $d$ range between $\sim0.9$ and $\sim1$. The present work aims at illustrating that the proposed approach is well capable to manifest the statistical nature of the system on interest. We don't aim at highlighting deeper physical insights. In other words, while the resulting nonextensivity is neither BG nor Tsallis, the freezeout parameters are found very compatible with BG and accordingly with the well-known freezeout phase-diagram, which is in an excellent agreement with recent lattice calculations. We conclude that the particle production is nonextensive but should not necessarily be accompanied by a radical change in the intensive or extensive thermodynamic quantities, such as internal energy and temperature. Only, the two critical exponents defining the equivalence classes $(c, d)$ are the physical parameters characterizing the (non)extensivity., 14 pages, 4 figures, 1 table, accepted for publication in INJP

The current status of implementing Tsallis (nonextensive) statistics on high-energy physics is briefly reviewed. The remarkably low freezeout-temperature, which apparently fails to reproduce the first-principle lattice QCD thermodynamics and the measured particle ratios, etc. is discussed. The present work suggests a novel interpretation for the so-called {\it "Tsallis-temperature"}. It is proposed that the low Tsallis-temperature is due to incomplete implementation of Tsallis algebra though exponential and logarithmic functions to the high-energy particle-production. Substituting Tsallis algebra into grand-canonical partition-function of the hadron resonance gas model seems not assuring full incorporation of nonextensivity or correlations in that model. The statistics describing the phase-space volume, the number of states and the possible changes in the elementary cells should be rather modified due to interacting correlated subsystems, of which the phase-space is consisting. Alternatively, two asymptotic properties, each is associated with a scaling function, are utilized to classify a generalized entropy for such a system with large ensemble (produced particles) and strong correlations. Both scaling exponents define equivalence classes for all interacting and noninteracting systems and unambiguously characterize any statistical system in its thermodynamic limit. We conclude that the nature of lattice QCD simulations is apparently extensive and accordingly the Boltzmann-Gibbs statistics is fully fulfilled. Furthermore, we found that the ratios of various particle yields at extreme high and extreme low energies of RHIC-BES is likely nonextensive but not necessarily of Tsallis type., 25 pages, 3 figures, accepted for publication in PEPAN

It was argued that the Raychaudhuri equation with a quantum correction term seems to avoid the Big Bang singularity and to characterize an everlasting Universe [PLB741,276(2015)]. Critical comments on both conclusions and on the correctness of the key expressions of this work were discussed in literature [MPLA31(2016)1650044]. In the present work, we have analyzed the perturbative (in)stability conditions in the inflationary era of the early Universe. We conclude that both unstable and stable modes are incompatible with the corresponding ones obtained in the standard FLRW Universe. We have shown that unstable modes do exist at small (an)isotropic perturbation and for different equations of state. Inequalities for both unstable and stable solutions with the standard FLRW space were derived. They reveal that in the FLRW flat Universe both perturbative instability and stability are likely. While negative stability modes have been obtained for radiation- and matter-dominated eras, merely, instability modes exist in case of a finite cosmological constant and also if the vacuum energy dominates the cosmic background geometry., 14 pages, 1 figure, accepted for publication in INJP

Inspired by Lifshitz theory for quantum critical phenomena in condensed matter, Horava proposed a theory for quantum gravity with an anisotropic scaling in ultraviolet. In Horava-Lifshitz gravity (HLG), we have studied the impacts of six types of equations of state on the evolution of various cosmological parameters such as Hubble parameters and scale factor. From the comparison of the general relativity gravity with the HLG with detailed and without with non-detailed balance conditions, remarkable differences are found. Also, a noticeable dependence of singular and non-singular Big Bang on the equations of state is observed. We conclude that HLG explains various epochs in the early universe and might be able to reproduce the entire cosmic history with and without singular Big Bang., Comment: 20 pages, 3 figures, accepted for publication in Int. J. Theor. Phys

We report the first measurement of rapidity-odd directed flow (v_{1}) for D^{0} and D^{0}[over ¯] mesons at midrapidity (|y

The particle ratios $k^+/\pi^+$, $\pi^-/K^-$, $\bar{p}/\pi^-$, $\Lambda/\pi^-$, $\Omega/\pi^-$, $p/\pi^+$, $\pi^-/\pi^+$, $K^-/K^+$, $\bar{p}/p$, $\bar{\Lambda}/\Lambda$, $\bar{\Sigma}/\Sigma$, $ \bar{\Omega}/\Omega$ measured at AGS, SPS and RHIC energies are compared with large statistical ensembles of $100,000$ events deduced from the CRMC EPOS $1.99$ and the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) hybrid model. In the UrQMD hybrid model two types of phase transitions are taken into account. All these are then confronted to the Hadron Resonance Gas Model. The two types of phase transitions are apparently indistinguishable. Apart from $k^+/\pi^+$, $k^-/\pi^-$, $\Omega/\pi^-$, $\bar{p}/\pi^+$, and $\bar{\Omega}/\Omega$, the UrQMD hybrid model agrees well with the CRMC EPOS $1.99$. Also, we conclude that the CRMC EPOS $1.99$ seems to largely underestimate $k^+/\pi^+$, $k^-/\pi^-$, $\Omega/\pi^-$, and $\bar{p}/\pi^+$., Comment: 8 pages, 2 figures, submitted to JETP

The SU($3$) Polyakov linear-sigma model (PLSM) in mean-field approximation is utilized in analyzing the chiral condensates $\sigma_u$, $\sigma_d$, $\sigma_s$ and the deconfinement order parameters $\phi$, $\bar{\phi}$, at finite isospin asymmetry. The bulk thermodynamics including pressure density, interaction measure, susceptibility, and second-order correlations with baryon, strange and electric charge quantum numbers are studied in thermal and dense medium. The PLSM results are confronted to the available lattice QCD calculations. The excellent agreement obtained strengthens the reliability of fixing the PLSM parameters and therefore supports further predictions even beyond the scope of the lattice QCD numerical applicability. From the QCD phase structure at finite isospin chemical potential ($\mu_I$), a novel expression for the explicit symmetry breaking term $h_3$ is introduced, we find that the pseudo-critical temperatures decrease with the increase in $\mu_I$. We conclude that the QCD phase structure in ($T_\chi$-$\mu_I$) plane seems to extend the one in ($T_\chi$-$\mu_B$) plane., Comment: 24 pages, 6 figures and 1 table, accepted for publication in IJMPA

Various thermodynamic quantities for baryon-free matter are calculated by combining the most reliable non-perturbative and perturbative calculations, especially the most recent ones including as many quark flavors as possible. We extend these calculations by including other degrees of freedom (dof), such as photons, neutrinos, leptons, electroweak particles, and Higgs bosons, that allows us to consider the temperatures up to the TeV-scale. The calculations show that similar to QCD, the EW phase transition is also a crossover. We have found that while the equation of state for the hadronic matter is linear, $p/\rho\simeq 0.2$, the one for higher temperatures is rather complex; it exhibits two crossover-type phase transitions, corresponding to strong and EW matter. At even larger energy densities, the deduced EoS becomes linear again and close to ideal gas. The combined equation of state can be used for modeling the expansion of the Universe from very early times and through the EW and QCD era., Comment: 17 pages, 4 figures, submitted to PRD

For temperatures $T$ ranging from a few MeV up to TeV and energy density $\rho$ up to $10^{16}~$GeV/fm$^3$, the bulk viscosity $\zeta$ is calculated in non-perturbation (up, down, strange, charm, and bottom) and perturbation theories with up, down, strange, charm, bottom, and top quark flavors, at vanishing baryon-chemical potential. To these calculations, results deduced from the effective QCD-like model, the Polyakov linear-sigma model (PLSM), are also integrated in. The PLSM merely comes up with essential contributions for the vacuum and thermal condensations of the gluons and the quarks (up, down, strange, and charm flavors). Furthermore, the thermal contributions of the photons, neutrinos, charged leptons, electroweak particles, and scalar Higgs boson, are found very significant along the entire range of $T$ and $\rho$ and therefore could be well integrated in. We present the dimensionless quantity $9 \omega_0 \zeta/Ts$, where $\omega_0$ is a perturbative scale and $s$ is the entropy density and conclude that $9 \omega_0 \zeta/Ts$ exponentially decreases with increasing $T$. We also conclude that the resulting $\zeta$ with the non-perturbative and perturbative QCD contributions non-monotonically increases with increasing $\rho$. But with nearly-entire standard model contributions considered in the present study, $\zeta$ almost-linearly increases with increasing $\rho$. Apparently, these results offer a great deal to explore in astrophysics, cosmology, and nuclear collisions., Comment: 21 pages, 5 figures

It is conjectured that in cosmological applications the particle current is not modified but finite heat or energy flow. Therefore, comoving Eckart frame is a suitable choice, as it merely ceases the charge and particle diffusion and conserves charges and particles. The cosmic evolution of viscous hadron and parton epochs in casual and non-casual Eckart frame is analyzed. By proposing equations of state deduced from recent lattice QCD simulations including pressure p, energy density ρ, and temperature T, the Friedmann equations are solved. We introduce expressions for the temporal evolution of the Hubble parameter H˙, the cosmic energy density ρ˙, and the share η˙ and the bulk viscous coefficient ζ˙. We also suggest how the bulk viscous pressure Π could be related to H. We conclude that the relativistic theory of fluids, the Eckart frame, and the finite viscous coefficients play essential roles in the cosmic evolution, especially in the hadron and parton epochs.

Visibility graphs (VGs), horizontal visibility graphs (HVGs) and the sandbox algorithm (SB) are applied for multifractal characterization of complex network systems that are converted from time series measurements, are used to characterize the fluctuations in pseudorapidity densities of singly charged particles produced in [Formula: see text]O–AgBr interactions at 60[Formula: see text]A[Formula: see text]GeV. The work presents the analysis of ring-like and jet-like events in terms of multifractality characterization of [Formula: see text]O–AgBr interactions at 60[Formula: see text]A[Formula: see text]GeV. We systematically compared the experimental events of both ring- and jet-like events with Monte Carlo (MC) simulated events. The investigation reveals that the multifractal spectrum for the jet-like events and ring-like events is different and can be distinguished. The ring-like events have its parameters slightly higher than that of jet-like events. Further analysis shows that the strength of the nonstatistical fluctuations is larger for ring-like events than those of jet-like events. The SB method presented here appears to be more useful than the conventional methods used for multifractal characterization.

We argue that the modified Landau–Raychaudhuri equations should first be analysed in a large class of spacetimes and in dependence on various equations of states, before endorsing any conclusion about (non)singular Big Bang. From the corrected entropy-area law in a large class of metrics, the generalized uncertainty principle (GUP) and the modified dispersion relation (MDR) approaches, and various equations of states, the modified Friedmann equations are derived. They are applied on Landau–Raychaudhuri equations in emergence of cosmic space framework from fixed point method. We show that any conclusion about (non)singular Big Bang is simply badly model-dependent, especially when utilizing GUP and MDR approaches, which can not replace a good theory for quantum gravity. We conclude that the various quantum gravity approaches, metrics and equations of state lead to different modifications in Friedmann and Landau–Raychaudhuri equations and thus to different (non)singular solutions for Big Bang theory.