Abstract:
We consider the basic physical properties of matter forming a thin accretion disc in the static and spherically symmetric space-time metric of the vacuum $f(R)$ modified gravity models. The Lagrangian of the generalized gravity theory is also obtained in a parametric form, and the conditions of the viability of the model are discussed. The exact Schwarzschild type solution of the gravitational field equations in the $f(R)$ gravity contains a linearly increasing term, as well as a logarithmic correction, as compared to the standard Schwarzschild solution of general relativity, and it depends on four arbitrary integration constants. The energy flux and the emission spectrum from the accretion disk around the $f(R)$ gravity black holes are obtained, and they are compared to the general relativistic case. Particular signatures can appear in the electromagnetic spectrum, thus leading to the possibility of directly testing modified gravity models by using astrophysical observations of the emission spectra from accretion disks.

Abstract:
We generalize the virial theorem in f(R) modified gravity using the collisionless Boltzmann equation. We find supplementary geometric terms in the modified Einstein equation providing an effective contribution to the gravitational energy. The total virial mass is proportional to the effective mass associated with the new geometrical term, which may account for the well-known virial theorem mass discrepancy in clusters of galaxies. The model predicts that the geometric mass and its effects extend beyond the virial radius of the clusters. We also consider the behavior of the galaxy cluster velocity dispersion in f(R) models. Thus, the f(R) virial theorem can be an efficient tool in observationally testing the viability of this class of generalized gravity models.

Abstract:
This paper is devoted to study the energy conditions in F(R,T) gravity for FRW universe with perfect fluid, where $R$ is the Ricci scalar and $T$ is the torsion scalar. We construct the general energy conditions in this theory and reduce them in F(R) as well as F(T) theory of gravity. Further, we assume some viable models and investigate bounds on their constant parameters to satisfy the energy condition inequalities. We also plot some of the cases using present-day values of the cosmological parameters. It is interesting to mention here that the model F(R,T)=\mu R+ \nu T satisfies the energy conditions for different ranges of the parameters.

Abstract:
We discuss the validity of the energy conditions in a newly modified theory named as $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity, where $R$ and $T$ represent the scalar curvature and trace of the energy-momentum tensor. The corresponding energy conditions are derived which appear to be more general and can reduce to the familiar forms of these conditions in general relativity, $f(R)$ and $f(R,T)$ theories. The general inequalities are presented in terms of recent values of Hubble, deceleration, jerk and snap parameters. In particular, we use two specific models recently developed in literature to study concrete application of these conditions as well as Dolgov-Kawasaki instability. Finally, we explore $f(R,T)$ gravity as a specific case to this modified theory for exponential and power law models.

Abstract:
I present the junction conditions for F(R) theories of gravity and their implications: the generalized Israel conditions and equations. These junction conditions are necessary to construct global models of stars, galaxies, etc., where a vacuum region surrounds a finite body in equilibrium, as well as to describe shells of matter and braneworlds, and they are stricter than in General Relativity in both cases. For the latter case, I obtain the field equations for the energy-momentum tensor on the shell/brane, and they turn out to be, remarkably, the same as in General Relativity. An exceptional case for quadratic F(R), previously overlooked in the literature, is shown to arise allowing for a discontinuous R, and leading to an energy-momentum content on the shell with unexpected properties, such as non-vanishing components normal to the shell and a new term resembling classical dipole distributions. For the former case, they do not only require the agreement of the first and second fundamental forms on both sides of the matching hypersurface, but also that the scalar curvature R and its first derivative agree there too. I argue that, as a consequence, matched solutions in General Relativity are not solutions of F(R)-models generically. Several relevant examples are analyzed.

Abstract:
We introduce the Minimum Entropy Method, a simple statistical technique for constraining the Milky Way gravitational potential and simultaneously testing different gravity theories directly from 6D phase-space surveys and without adopting dynamical models. We demonstrate that orbital energy distributions that are separable (i.e. independent of position) have an associated entropy that increases under wrong assumptions about the gravitational potential and/or gravity theory. Of known objects, `cold' tidal streams from low-mass progenitors follow orbital distributions that most nearly satisfy the condition of separability. Although the orbits of tidally stripped stars are perturbed by the progenitor's self-gravity, systematic variations of the energy distribution can be quantified in terms of the cross-entropy of individual tails, giving further sensitivity to theoretical biases in the host potential. The feasibility of using the Minimum Entropy Method to test a wide range of gravity theories is illustrated by evolving restricted N-body models in a Newtonian potential and examining the changes in entropy introduced by Dirac, MONDian and f(R) gravity modifications.

Abstract:
In this paper on the basis of the generalized $f(R)$ gravity model with arbitrary coupling between geometry and matter, four classes of $f(R)$ gravity models with non minimal coupling between geometry and matter will be studied. By means of conditions of power law expansion and the equation of state of matter less than -1/3, the relationship among p, w and n, the conditions and the candidate for late time cosmic accelerated expansion will be discussed in the four classes of $f(R)$ gravity models with non minimal coupling. Furthermore, in order to keep considering models to be realistic ones, the Dolgov Kawasaki instability will be investigated in each of them.

Abstract:
The energy conditions are derived in the context of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor, which can reduce to the well-known conditions in $f(R)$ gravity and general relativity. We present the general inequalities set by the energy conditions in terms of Hubble, deceleration, jerk and snap parameters. In this study, we concentrate on two particular models of $f(R,T)$ gravity namely, $f(R)+\lambda{T}$ and $R+2f(T)$. The exact power-law solutions are obtained for these two cases in homogeneous and isotropic $f(R,T)$ cosmology. Finally, we find certain constraints which have to be satisfied to ensure that power law solutions may be stable and match the bounds prescribed by the energy conditions.

Abstract:
In this paper, we reconstruct cosmological models in the framework of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the stress-energy tensor. We show that the dust fluid reproduces $\Lambda $CDM, phantom-non-phantom era and the phantom cosmology. Further, we reconstruct different cosmological models including, Chaplygin gas, scalar field with some specific forms of $f(R,T)$. Our numerical simulation for Hubble parameter shows good agreement with the BAO observational data for low redshifts $z<2$.

Abstract:
Scalar modifications of gravity have an impact on the growth of structure. Baryon and Cold Dark Matter (CDM) perturbations grow anomalously for scales within the Compton wavelength of the scalar field. In the late time Universe when reionisation occurs, the spectrum of the 21cm brightness temperature is thus affected. We study this effect for chameleon-f(R) models, dilatons and symmetrons. Although the f(R) models are more tightly constrained by solar system bounds, and effects on dilaton models are negligible, we find that symmetrons where the phase transition occurs before z_* ~ 12 will be detectable for a scalar field range as low as 5 kpc. For all these models, the detection prospects of modified gravity effects are higher when considering modes parallel to the line of sight where very small scales can be probed. The study of the 21 cm spectrum thus offers a complementary approach to testing modified gravity with large scale structure surveys. Short scales, which would be highly non-linear in the very late time Universe when structure forms and where modified gravity effects are screened, appear in the linear spectrum of 21 cm physics, hence deviating from General Relativity in a maximal way.