Abstract. In this paper, we examine observational constraints on the power law cosmology; essentially dependent on two parameters H 0 (Hubble constant) and q (deceleration parameter). We investigate the constraints on these parameters using the latest 28 points of H(z) data and 580 points of Union2.1 compilation data and, compare the results with the results of ΛCDM. We also forecast constraints using a simulated data set for the future JDEM, supernovae survey. Our studies give better insight into power law cosmology than the earlier done analysis by Kumar [arXiv:1109.6924] indicating it tuning well with Union2.1 compilation data but not with H(z) data. However, the constraints obtained on < H 0 > and < q > i.e. H 0 average and q average using the simulated data set for the future JDEM, supernovae survey are found to be inconsistent with the values obtained from the H(z) and Union2.1 compilation data. We also perform the statefinder analysis and find that the power-law cosmological models approach the standard ΛCDM model as q → −1. Finally, we observe that although the power law cosmology explains several prominent features of evolution of the Universe, it fails in details.

A modified Mimetic gravity (MMG) is proposed as a generalization of general relativity. The model contains a physical metric which is the function of an auxiliary (unphysical) metric and a Lyra's metric. We construct different kinds of conformally invariant models in different levels of the expansion parameter λ. This model phenomenologically has been extended to higher-order forms. Cosmology of a certain class of such models has been investigated in detail. A cosmological solution has been proposed in inhomogeneous form of scalar field. For homogeneous case, energy conditions are widely investigated. We have shown that the system evaluated at intervals shorter than a certain time Tc meets all the energy conditions.

We consider the thermodynamics and Geometrothermodynamics of the Myers-Perry black holes in five dimensions for three different cases, depending on the values of the angular momenta. We follow Davies approach to study the thermodynamics of black holes and find a nontrivial thermodynamic structure in all cases, which is fully reproduced by the analysis performed with the techniques of Geometrothermodynamics. Moreover, we observe that in the cases when only one angular momentum is present or the two angular momenta are fixed to be equal, i.e. when the thermodynamic system is two dimensional, there is a complete agreement between the divergences of the generalized susceptibilities and the singularities of the equilibrium manifold, whereas when the two angular momenta are fully independent, that is, when the thermodynamic system is three dimensional, additional singularities in the curvature appear. However, we prove that such singularities are due to the changing from a stable phase to an unstable one.

We study thermodynamic of strange quark matter (SQM) using the analytic expressions of free and internal energies. We investigate two regimes of the high density and low density separately. As a vital program, in the case of a massless gluon and massless quarks at finite temperature, we also present a geometry of thermodynamics for the gluon and Bosons using a Legendre invariance metric, it is so called as geometrothermodynamic (GTD) to better understanding of the phase transition.The GTD metric and its second order scalar invariant have been obtained, and we clarify the phase transition by study the singularities of the scalar curvature of this Riemannian metric. This method is ensemble dependence and to complete the phase transition. Meanwhile, we also investigate enthalpy and entropy and internal energy representations. Our work exposes new pictures of the nature of phase transitions in SQM.

In this work, we have studied accretion of dark energy (DE) onto MorrisThorne wormhole with three different forms, namely, holographic dark energy, holographic Ricci dark energy and modified holographic Ricci dark energy . Considering the scale factor in power-law form we have observed that as the holographic dark energy accretes onto wormhole, the mass of the wormhole is decreasing. In the next phase we considered three parameterization schemes that are able to get hold of quintessence as well as phantom phases.Without any choice of scale factor we reconstructed Hubble parameter from conservation equation and dark energy densities and subsequently got the mass of the wormhole separately for accretion of the three dark energy candidates. It was observed that if these dark energies accrete onto the wormhole, then for quintessence stage, wormhole mass decreases up to a certain finite value and then again increases to aggressively during phantom phase of the universe.

The work presented in this paper aims to study the inflationary cosmology of [Formula: see text] gravity where [Formula: see text] by two models of Dark Energy (DE), namely Variable Generalized Chaplygin Gas (VGCG) and the Nojiri–Odintsov Holographic Dark Energy. The reconstruction has been carried out in the absence as well as presence of viscosity. The viscous pressure is considered to be [Formula: see text], where [Formula: see text] and [Formula: see text] is the Hubble parameter. In viscous and nonviscous scenarios, the reconstruction has been carried out for power law scale factor and bounce scale factor. For the specific range of cosmic time [Formula: see text], the quintessence and quintom behavior are observed for the equation of state parameters. The slow roll parameters have been studied for both the cases in nonviscous scenario and for the case, namely bounce scale factor for VGCG of nonviscous scenario, we have found that there is a scope of exit from inflation. Furthermore, constraints have been deduced to study the presence of singularity. Finally, the [Formula: see text] gravity is reconstructed as a function of cosmic time [Formula: see text] and is found to stay at positive level for a range of cosmic time [Formula: see text].

It is shown that the inflationary model is the result of the symmetry of the generalized F(R,T,X,φ)-cosmological model using the Noether symmetry. It leads to a solution, a particular case of which is Starobinsky’s cosmological model. It is shown that even in the more particular case of cosmological models F(R,X,φ) and F(T,X,φ) the Monge–Ampère equation is still obtained, one of the solutions including the Starobinsky model. For these models, it is shown that one can obtain both power-law and exponential solutions for the scale factor from the Euler–Lagrange equations. In this case, the scalar field φ has similar time dependences, exponential and exponential. The resulting form of the Lagrangian of the model allows us to consider it as a model with R2 or X2. However, it is also shown that previously less studied models with a non-minimal relationship between R and X are important, as one of the possible models. It is shown that in this case the power-law model can have a limited evolutionary period with a negative value of the kinetic term.

One of the actual problems of mathematical physics is to relate differential geometry and nonlinear differential equation. Research in this direction is very important, as the results are a theoretical and practical application. In this paper, we investigate the Camassa-Holm equation. It is well known that the integrable nonlinear Camassa-Holm equation play an important role in the study of wave propagation. We present the relationship between Camassa-Holm equation and soliton surfaces. The rst and second fundamental forms, surface area and curvature for Camassa-Holm equation are found.