Traversable wormholes, tunnel like structures introduced by Morris & Thorne [1], have a significant role in connection of two different space-times or two different parts of the same space-time. The characteristics of these wormholes depend upon the redshift and shape functions which are defined in terms of radial coordinate. In literature, several shape functions are defined and wormholes are studied in f (R) gravity with respect to these shape functions [55,57,60]. In this paper, two shape functions (i) b(r) = r 0 log(r + 1) log(r 0 + 1) and (ii) b(r) = r 0 ( r r 0 ) γ , 0 < γ < 1 are considered. The first shape function is newly defined, however the second one is collected from the literature [77]. The wormholes are investigated for each type of shape function in f (R) gravity with f (R) = R + αR m − βR −n , where m, n, α, and β are real constants. Varying parameters α or β, f (R) model is studied in five subcases for each type of shape function. In each case, the energy density, radial & tangential pressures, energy conditions that include null energy condition, weak energy condition, strong energy condition & dominated energy condition, and anisotropic parameter are computed. The energy density is found to be positive and all energy conditions are obtained to be violated which supports the existence of wormholes. Also, the equation of state parameter is obtained to possess values less than -1, that shows the presence of the phantom fluid and leads towards the expansion of the universe.

In this work, wormholes, tunnel like structures introduced by Morris & Thorne [1], are explored within the framework of f (R) gravity. Using the shape function b(r) = r 0where 0 < γ < 1, and the equation of state p r = ωρ, the f (R) function is derived and the field equations are solved. Then null, weak, strong and dominated energy conditions are analyzed and spherical regions satisfying these energy conditions are determined. Furthermore, we calculated the range of the radius of the throat of the wormhole, where the energy conditions are satisfied.

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We compare the gravitational collapse of homogeneous perfect fluid with various equations of state in the framework of General Relativity and in R 2 gravity. We make our calculations using dimensionless time with characteristic timescale tg ∼ (Gρ) −1/2 where ρ is a density of collapsing matter. The cases of matter, radiation and stiff matter are considered. We also account the possible existence of vacuum energy and its influence on gravitational collapse. In a case of R 2 gravity we have additional degree of freedom for initial conditions of collapse. For barotropic equation of state p = wρ the result depends from the value of parameter w: for w > 1/3 the collapse occurs slowly in comparison with General Relativity while for w < 1/3 we have opposite situation. Vacuum energy as expected slows down the rate of collapse and for some critical density gravitational contraction may change to expansion. It is interesting to note that for General Relativity such expansion is impossible. We also consider the collapse in the presence of so-called phantom energy. For description of phantom energy we use Lagrangian in the form −X − V (where X and V are the kinetic and potential energy of the field respectively) and consider the corresponding Klein-Gordon equation for phantom scalar field.

Modified gravity theories have received increased attention lately to understand the late time acceleration of the universe. This viewpoint essentially modifies the geometric components of the universe. Among numerous extension to Einstein's theory of gravity, theories which include higher order curvature invariant, and specifically the class of f (R) theories, have received several acknowledgments. In our current work we try to understand the late time acceleration of the universe by modifying the geometry of the space and using dynamical system analysis. The use of this technique allows to understand the behavior of the universe under several circumstances. Apart from that we study the stability properties of the critical point and acceleration phase of the universe which could then be analyzed with observational data. We consider a particular model f (R) = R − µR c (R/R c ) p with 0 < p < 1, µ, R c > 0 for the study. As a first case we consider the matter and radiation component of the universe with an assumption of no interaction between them. Later, as a second case we take matter, radiation and dark energy (cosmological constant) where study on effects of linear, non-linear and no interaction between matter and dark energy is considered and results have been discussed in detail.

In this work, the study of traversable wormholes in f (R) gravity with the function f (R) = R + αR n , where α and n are arbitrary constants, is taken into account. The shape function b(r) = r exp(r−r 0 ) , proposed by Samanta et al. [71], is considered. The energy conditions with respect to both constant and variable redshift functions are discussed and the existence of wormhole solutions without presence of exotic matter is investigated.

We propose a novel shape function, on which the metric that models traversable wormholes is dependent. With this shape function, the energy conditions, equation of state and anisotropy parameter are analyzed in f (R) gravity, f (R, T ) gravity and general relativity. Furthermore, the consequences obtained with respect to these theories are compared. In addition, the existence of wormhole geometries is investigated.

Morris & Thorne [12] proposed geometrical objects called traversable wormholes that act as bridges in connecting two spacetimes or two different points of the same spacetime. The geometrical properties of these wormholes depend upon the choice of the shape function. In literature, these are studied in modified gravities for different types of shape functions. In this paper, the traversable wormholes having shape function b(r) = r 0 tanh(r) tanh(r 0 ) are explored in f (R) gravity with f (R) = R + αR m − βR −n , where α, β, m and n are real constants.For different values of constants in function f (R), the analysis is done in various cases. In each case, the energy conditions, equation of state parameter and anisotropic parameter are determined.