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In this work, we examine the optical behaviors and thermodynamic phase structures using shadow analysis for four $\mathbb{S}\mathbb{T}\mathbb{U}$ black holes. The study is conducted for four cases of charge configurations on the parameter space $\mathcal{M}\left(q_1,q_2,q_3,q_4\right)$. As a matter of fact, both the electric charge as a parameter and the parameter space $\mathcal{M}\left(q_1,q_2,q_3,q_4\right)$ affect the geometry of the black hole shadow, particularly the size of the shadow. We also introduce a constraint on the charge of the black hole from the observational results of the M87$^\star$ {\color{black}and Sgr A$^\star$} shadow. Furthermore, we show that the electric charge and the parameter space $\mathcal{M}\left(q_1,q_2,q_3,q_4\right)$ have a non-trivial impact on the variation of the energy emission rate. Interestingly enough, we find novel scenarios in which the evaporation is slower, which causes the lifetime of the black holes to be considerably elongated. On the other side, the phase structure of four $\mathbb{S}\mathbb{T}\mathbb{U}$ black holes is explored for two cases of electric charge configuration. The findings show a perfect correlation between the shadow and event horizon radii. This correlation is, in fact, helpful in discovering the phase transition in terms of the shadow radius. In addition, the microstructure is being analyzed in terms of shadow analysis, providing similar behavior to the ordinary situation of the Ruppeiner formalism., Comment: 32 pages, 14 labeled figures, accepted for publication in Classical and Quantum Gravity

In this study, we explore the corrected thermodynamics of non-linear magnetic charged anti-de Sitter (AdS) black holes surrounded by quintessence, incorporating thermal fluctuations and deriving the corrected thermodynamic potentials. We analyze the effects of corrections due to thermal fluctuations on various thermodynamic potentials, including enthalpy, Helmholtz free energy, and Gibbs free energy. Our results show significant impacts on smaller black holes, with first-order corrections destabilizing them, while second-order corrections enhance stability with increasing parameter values. The specific heat analysis further elucidates the stability criteria, indicating that the large black holes ensure stability against phase transitions. However, the thermal fluctuations do not affect the physical limitation points as well as the second-order phase transition points of the black hole. Our findings highlight the intricate role of thermal fluctuations in black hole thermodynamics and their influence on stability, providing deeper insights into the behaviour of black holes under corrected thermodynamic conditions., Comment: 18 pages, 6 figures

This research paper presents a black hole solution with a rational non-linear electrodynamics source in the Rastall gravity framework. The paper analyzes the thermodynamic properties of the solution in normal phase space and explores its critical behavior. The phase structure is examined using the extended first law of thermodynamics, with the cosmological constant $\Lambda$ serving as pressure $P$. The isotherms exhibit van der Waals behavior at small values of horizon $r_+$. The paper also investigates the Gibbs free energy behavior and finds two critical points with two pressures where the reentrant phase transition occurs and disappears. On the other hand, we explore the prevalent microstructure of black holes in Ruppeiner geometry, uncovering significant deviations in the nature of particle interactions from conventional practice. Moreover, the thermodynamic geometry is analyzed using the Ruppeiner formalism, with the normalized Ricci scalar indicating possible point-phase transitions of the heat capacity, and the normalized extrinsic curvature having the same sign as the normalized Ricci scalar. The three-phase transitions of the heat capacity are those that we find for the normalized Ruppeiner curvatures. Thus, there is an absolute correspondence., Comment: 23 pages, 7 figures

We study a cosmological model in Rastall's theory of gravity in the framework of the flat FLRW metric. We formulate the value of the Hubble parameter, which contains two model parameters, $ \alpha $ and $ j $. Employing the Markov Chain Monte Carlo (MCMC) sampling technique, we determine the values of these model parameters along with their uncertainties. Moreover, we derive the equation of state (EoS) parameter, which converges around the quintessence region. We perform a dynamical system analysis using the linearization technique to validate the results independently. Also, we discuss various physical attributes of the model, highlighting the transition to acceleration and the violation of the strong energy condition (SEC) in the late stages of evolution. In conclusion, our model mimics the behavior of a dark matter fluid during the past epoch and transitions into a quintessence dark energy model in the future epoch., Comment: 18 pages, 17 figures

This paper investigates the contribution of the nonsingular Yukawa-modified potential in the context of four-dimensional Einstein-Gauss-Bonnet (EGB) gravity modeling by a static and spherically symmetric black hole solution. These Yukawa-type corrections are essentially described along two parameters, $\beta$ and $\lambda$, affecting Newton's law of gravity at large distances, and a deformation parameter $\ell_0$, which is essential at short distances. Primarily, the strongest effect is encoded in $\beta$, which alters the total mass of the black hole with additional mass proportional to $\beta\mathcal{M}$, imitating the effects of dark matter at large distances from the black hole. In contrast, the effect due to $\lambda$ is small for astrophysical values. On the other hand, the EGB gravity is ruled by the Gauss-Bonnet (GB) coupling constant $\alpha$, a fundamental parameter of the theory. We pay particular attention to thermodynamic stability, critical orbits, geodesics and quasinormal modes. The results demonstrate stability of the black hole solution for a range of values of the GB coupling constant $\alpha$. Furthermore, this study investigates the null geodesic motion, namely the shadow behavior, providing intriguing results in relation to the size of the black hole shadow., Comment: 30 pages, 8 tables, and 5 figures

In this paper, we have investigated some exact cosmological models in Myrzakulov gravity using a flat Friedmann-Lematre-Robertson-Walker (FLRW) spacetime metric. We have considered the modified Lagrangian function as $F(R,T)=R+\lambda T$, where $R, T$ are respectively the Ricci curvature scalar and the torsion scalar with respect to non-special connection, and $\lambda$ is a model parameter. We have obtained two exact solutions in two different situations for the scale factor $a(t)$. Using this scale factor, we have obtained various geometrical parameters to investigate cosmological properties of the universe. We have obtained the best fit values of model parameters through the MCMC analysis of two types latest observational datasets like $H(z)$ and Pantheon SNe Ia samples, with $1-\sigma, 2-\sigma$ \& $3-\sigma$ regions. We have performed a comparative and relativistic study of geometrical and cosmological parameters. In model-I, we have found that the effective equation of state (EoS) parameter $\omega_{eff}$ varies in the range $-1\le\omega_{eff}\le0$ while in the model-II, it varies as $-1.031\le\omega_{eff}\le0$. We have found that both models are transit phase (decelerating to accelerating) universe with transition redshift in the range $0.6

In the present paper, we investigate some exact cosmological models in Myrzakulov $F(R,T)$ gravity theory. We have considered the arbitrary function $F(R, T)=R+\lambda T$ where $\lambda$ is an arbitrary constant, $R, T$ are respectively, the Ricci-scalar curvature and the torsion. We have solved the field equations in a flat FLRW spacetime manifold for Hubble parameter and using the MCMC analysis, we have estimated the best fit values of model parameters with $1-\sigma, 2-\sigma, 3-\sigma$ regions, for two observational datasets like $H(z)$ and Pantheon SNe Ia datasets. Using these best fit values of model parameters, we have done the result analysis and discussion of the model. We have found a transit phase decelerating-accelerating universe model with transition redshifts $z_{t}=0.4438_{-0.790}^{+0.1008}, 0.3651_{-0.0904}^{+0.1644}$. The effective dark energy equation of state varies as $-1\le\omega_{de}\le-0.5176$ and the present age of the universe is found as $t_{0}=13.8486_{-0.0640}^{+0.1005}, 12.0135_{-0.2743}^{+0.6206}$ Gyrs, respectively for two datasets., Comment: 22 pages, 11 figures

In this paper we study the cosmological aspects of metric-affine $f(R)$ gravity with hyperfluid. The equations of motion of the theory are obtained by varying the action with respect to the metric and the independent connection. Subsequently, considering a Friedmann-Lema\^itre-Robertson-Walker background, we derive the modified Friedmann equations in the presence of a perfect cosmological hyperfluid. Especially, we focus on the particular case in which $f(R)=R+\beta R^2$, considering purely shear hypermomentum and finding exact solutions in the weak coupling limit., Comment: 26 pages

In this article, we study the expanding nature of universe in the contest of $f(R,L_m)$ gravity theory, here $ R $ represents the Ricci scalar and $ L_m $ is the matter Lagrangian density. With a specific form of $ f(R,L_m) $, we obtain the field equations for flat FLRW metric. We parametrize the deceleration parameter in terms of the Hubble parameter and from here we find four free parameters, which are constraints and estimated by using $H(z)$, $Pantheon$, and their joint data sets. Further, we investigate the evolution of the deceleration parameter which depicts a transition from the deceleration to acceleration phases of the universe. The evolution behaviour of energy density, pressure, and EoS parameters shows that the present model is an accelerated quintessence dark energy model. To compare our model with the $ \Lambda $CDM model we use some of the diagnostic techniques. Thus, we find that our model in $ f(R,L_m) $ gravity supports the recent standard observational studies and delineates the late-time cosmic acceleration., Comment: 14 pages, 11 figures

We consider some nonlinear models describing interactions of long and short (LS) waves. Such LS models have been derived and proposed with various motivations, which mainly come from fluid and plasma physics. In this paper, we study some of integrable LS models, namely, the Yajima-Oikawa equation, the Newell equation, the Ma equation, the Geng-Li equation and etc. In particular, the gauge equivalent counterparts of these integrable LS models (equations) are found. In fact, these gauge equivalents of the LS equations are integrable generalized Heisenberg ferromagnet equations (HFE) with self-consistent potentials (HFESCP). The associated Lax representations of these HFESCP are given. We also presented several spin-phonon equations which describe nonlinear interactions of spin and lattice subsystems in ferromagnetic materials., Comment: 20 pages

We extend the results of antecedent literature on quadratic Metric-Affine Gravity by studying a new quadratic gravity action in vacuum which, besides the usual (non-Riemannian) Einstein-Hilbert contribution, involves all the parity even quadratic terms in torsion and non-metricity plus a Lagrangian that is quadratic in the field-strengths of the torsion and non-metricity vector traces. The theory result to be equivalent, on-shell, to a Vector-Tensor theory. We also discuss the sub-cases in which the contribution to the Lagrangian quadratic in the field-strengths of the torsion and non-metricity vectors just exhibits one of the aforementioned quadratic terms. We then report on implications of our findings in the context of $F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity., Comment: 28 pages

We consider an extension of standard General Relativity in which the Hilbert-Einstein action is replaced by an arbitrary function of the Ricci scalar, nonmetricity, torsion, and the trace of the matter energy-momentum tensor. By construction, the action involves a non-minimal coupling between matter and geometry. The field equations of the model are obtained, and they lead to the nonconservation of the matter energy-momentum tensor. A thermodynamic interpretation of the nonconservation of the energy-momentum tensor is also developed in the framework of the thermodynamics of the irreversible processes in open systems. The Newtonian limit of the theory is considered, and the generalized Poisson equation is obtained in the low velocity and weak fields limits. The nonmetricity, the Weyl vector, and the matter couplings generate an effective gravitational coupling in the Poisson equation. We investigate the cosmological implications of the theory for two different choices of the gravitational action, corresponding to an additive and a multiplicative algebraic structure of the function $f$, respectively. We obtain the generalized Friedmann equations, and we compare the theoretical predictions with the observational data. \te{We find that the cosmological models can give a good descriptions of the observations up to a redshift of $z=2$, and, for some cases, up to a redshift of $z=3$., Comment: 19 pages, 6 figures, Matched with published version

We propose a cosmological scenario in which the universe undergoes through a non-singular bounce, and after the bounce, it decelerates having a matter-like dominated evolution during some regime of the deceleration era, and finally at the present epoch it evolves through an accelerating stage. Our aim is to study such evolution in the context of Chern-Simons corrected F(R) gravity theory and confront the model with various observational data. Using the reconstruction technique, and in addition by employing suitable boundary conditions, we determine the form of F(R) for the entire possible range of the cosmic time. The form of F(R) seems to unify a non-singular bounce with a dark energy epoch, in particular, from a non-singular bounce to a deceleration epoch and from a deceleration epoch to a late time acceleration era. It is important to mention that the bouncing scenario in the present context is an asymmetric bounce, in particular, the Hubble radius monotonically increases and asymptotically diverges at the late contracting era, while it seems to decrease with time at the present epoch. Such evolution of the Hubble radius leads to the primordial perturbation modes generate at the deep contracting era when all the perturbation modes lie within the horizon. We calculate the scalar and tensor power spectra, and as a result, the primordial observables are found to be in agreement with the latest Planck 2018 constraints. In this regard, the Chern-Simons term seems to have considerable effects on the tensor perturbation evolution, however keeping intact the scalar part of the perturbation with that of in the case of a vacuum F(R) model, and as a result, the Chern-Simons term proves to play an important role in making the observable quantities consistent with the Planck results. Furthermore the theoretical expectation of the dark energy observables are confronted with the Planck+SNe+BAO data., Comment: "Physics of the Dark Universe" accepted. Abstract is reduced due to arxiv limitation

We derive the full set of field equations for the Metric-Affine version of the Myrzakulov gravity model and also extend this family of theories to a broader one. More specifically, we consider theories whose gravitational Lagrangian is given by $F(R,T,Q, {\cal T},{\cal D})$ where $T$, $Q$ are the torsion and non-metricity scalars, ${\cal T}$ is the trace of the energy-momentum tensor and ${\cal D}$ the divergence of the dilation current. We then consider the linear case of the aforementioned theory and assuming a cosmological setup we obtain the modified Friedmann equations. In addition, focusing on the vanishing non-metricity sector and considering matter coupled to torsion we obtain the complete set of equations describing the cosmological behaviour of this model along with solutions., Comment: 11 pages, no figures

Slow-roll inflation is analyzed in the context of modified gravity within the Palatini formalism. As shown in the literature, inflation in this framework requires the presence of non-traceless matter, otherwise it does not occur just as a consequence of the non-linear gravitational terms of the action. Nevertheless, by including a single scalar field that plays the role of the inflaton, slow-roll inflation can be performed in these theories, where the equations lead to an effective potential that modifies the dynamics. We obtain the general slow-roll parameters and analyze a simple model to illustrate the differences introduced by the gravitational terms under the Palatini approach, and the modifications on the spectral index and the tensor to scalar ratio predicted by the model., Comment: 11 pages, version published

In this paper, two inhomogeneous single fluid models for the Universe, which are able to naturally solve the $H_{0}$ tension problem, are discussed. The analysis is based on a Bayesian Machine Learning approach that uses a generative process. The adopted method allows to constrain the free parameters of each model by using the model itself, only. The observable is taken to be the Hubble parameter, obtained from the generative process. Using the full advantages of our method, the models are constrained for two redshift ranges. Namely, first this is done with mock $H(z)$ data over $z\in [0,2.5]$, thus covering known $H(z)$ observational data, which are most helpful to validate the fit results. Then, aiming to extend to redshift ranges to be covered by the most recent ongoing and future planned missions, the models are constrained for the range $z\in[0,5]$, too. Full validation of the results for this extended redshift range will have to wait for the near future, when higher redshift $H(z)$ data become available. This makes our models fully falsifiable. Finally, our second model here is able to explain the BOSS reported value for $H(z)$ at $z=2.34$., Comment: 13 pages, 4 figures

These results continue our studies of integrable generalized Heisenberg ferromagnet-type equations (GHFE) and their equivalent counterparts. We consider the GHFE which is the spin equivalent of the Zakharov-Ito equation (ZIE). We have established that these equations are gauge and geometrical equivalent to each other. The integrable motion of space curves induced by the ZIE is constructed. The 1-soliton solution of the GHFE is obtained from the seed solution of the ZIE., Comment: 12 pages

We study the integrability and equivalence of a generalized Heisenberg ferromagnet-type equation (GHFE). The different forms of this equation as well as its reduction are presented. The Lax representation (LR) of the equation is obtained. We observe that the geometrical and gauge equivalent counterpart of the GHFE is the modified Camassa-Holm equation (mCHE) with an arbitrary parameter $\kappa$. Finally, the 1-soliton solution of the GHFE is obtained., Comment: 16 pages

In this paper, we study the generalized Heisenberg ferromagnet equation, namely, the M-CVI equation. This equation is integrable. The integrable motion of the space curves induced by the M-CVI equation is presented. Using this result, the Lakshmanan (geometrical) equivalence between the M-CVI equation and the two-component Camassa-Holm equation is established. Note that these equations are gauge equivalent each to other., Comment: 11 pages. arXiv admin note: text overlap with arXiv:nlin/0501028 by other authors

In this paper, we study one of generalized Heisenberg ferromagnet equations with self-consistent sources, namely, the so-called M-CIV equation with self-consistent sources (M-CIVESCS). The Lax representation of the M-CIVESCS is presented. We have shown that the M-CIVESCS and the CH equation with self-consistent sources (CHESCS) is geometrically equivalent each to other. The gauge equivalence between these equations is proved. Soliton (peakon) and pseudo-spherical surfaces induced by these equations are considered. The one peakon solution of the M-CIVESCS is presented., Comment: 13 pages

In this paper, we provide the geometric formulation to the two-component Camassa-Holm equation (2-mCHE). We also study the relation between the 2-mCHE and the M-CV equation. We have shown that these equations arise from the invariant space curve flows in three-dimensional Euclidean geometry. Using this approach we have established the geometrical equivalence between the 2-mCHE and the M-CV equation. The gauge equivalence between these equations is also considered., Comment: 11 pages

In the present paper, we investigate some geometrical properties of the Camass-Holm equation (CHE). We establish the geometrical equivalence between the CHE and the M-CIV equation using a link with the motion of curves. We also show that these two equations are gauge equivalent each to other., Comment: 12 pages

This paper is a continuation of our previous work in which we studied a dispersionless limits of some integrable spin systems (magnetic equations). Now, we shall present dispersionless limits of some integrable generalized Heisenberg ferromagnet equations., Comment: 10 pages

This is a write-up of the lectures on dispersionless equations in 1+1, 2+1 and 3+1 dimensions presented by one of us (RM) at "Eurasian International Center for Theoretical Physics" (EICTP). We provide pedagogical introduction to the subject and summarize well-known results and some recent developments in theory of integrable dispersionless equations. Almost all results presented are available in the literatures. We just add some new results related to dispersionless limits of integrable magnetic equations. On the contrary, some of the basic tools of the integrable dispersionless systems and related theoretical techniques are not available in a pedagogical format. For this reason we think that it seems worthwhile to present basics of theory of dispersionless equations here for the benefit of beginner students. We begin with a detailed exposition of the well-known dispersionless systems like dKdVE, dNLSE, dKPE, dDSE and other classical soliton equations. We present in detail some new dispersionless systems. Next we develop the dispersionless limits of known magnetic equations. Lastly, we discuss in full detail the Lax representation formulation of some presented dispersionless equations. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field of integrable dispersionless equations and related topics., Comment: 18 pages

We propose a new integrable coupled dispersionless equation with self-consistent sources (CDESCS). We obtain the Lax pair and the equivalent generalized Heisenberg ferromagnet equation (GHFE), demonstrating its integrability. Specifically, we explore the geometry of these equations. Last, we consider the relation between the motion of curves/surfaces and the CDESCS and the GHFE., Comment: 21 pages

In the present paper, we study the Myrzakulov-XIII (M-XIII) equation geometrically. From the geometric point of view, we establish a link of the M-XIII equation with the motion of space curves in the 3-dimensional space $R^{3}$. We also show that the complex coupled dispersionless (CCD) equation can be derived from the geometrical formalism such that their curve flows are formulated. Finally, the gauge equivalence between the M-XIII equation and the CCD equation is established., Comment: 13 pages

In the present paper, we study the integrable 2-layer generalized Heisenberg ferromagnet equation (HFE). The relation between this generalized HFE and differential geometry of curves is established. Using this relation we found the geometrical equivalent counterpart of the 2-layer spin system which is the 2-component KdV equation. Finally, the gauge equivalence between these equations is established., Comment: 12 pages

In this paper, we analyze the early-time inflation in a scalar-tensor theory of gravity where the scalar field is minimally coupled with the Gauss-Bonnet four dimensional topological invariant. The theory belongs to a class of Horndeski models where the field equations are at the second order like in General Relativity. A viable inflationary scenario must correctly reproduce the last Plank satellite data. By starting from some simple assumptions on the field and on the coupling function between the field and the Gauss-Bonnet term, we derive the spectral index and the tensor-to-scalar ratio of the model. Once the model is viable, it is finally possible to fully reconstruct its Lagrangian., Comment: 11 pages, accepted in EPJ Plus

In this paper, we consider a cosmological model of the flat and homogeneous universe for the Starobinsky model $F(R)=\alpha R + \beta R^{2}$, which is non-minimally coupled with f-essence. For this model we obtained the field equations and considered particular solutions of the coupling and fermionic field functions. It is shown that the fermionic field can describe a nature of the universe., Comment: 6 pages