In this work we present some cosmologically relevant solutions using the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime in metric f (R) gravity where the form of the gravitational Lagrangian is given by 1 α e αR . In the low curvature limit this theory reduces to ordinary Einstein-Hilbert Lagrangian together with a cosmological constant term. Precisely because of this cosmological constant term this theory of gravity is able to support nonsingular bouncing solutions in both matter and vacuum background. Since for this theory of gravity f and f is always positive, this is free of both ghost instability and tachyonic instability. Moreover, because of the existence of the cosmological constant term, this gravity theory also admits a de-Sitter solution. Lastly we hint towards the possibility of a new type of cosmological solution that is possible only in higher derivative theories of gravity like this one. *
We investigate the dynamics of f (R) gravity in Jordan and Einstein frames. First, we perform a phase-space singularities analysis in both frames. We show that, typically, anisotropic singularities are absent in the Einstein frame, whereas they may appear in the Jordan frame. We conciliate this apparent inconsistency by showing that the necessary conditions for the existence of the Einstein frame are namely the same ones assuring the absence of the anisotropic singularities in the Jordan frame. In other words, we show that, at least in the context of Bianchi I cosmologies, the Einstein frame is available only when the original formulation in the Jordan frame is free of anisotropic singularities. Furthermore, we present a novel dynamical system formulation for anisotropic cosmologies in which both frames, provided they exist, will be manifestly equivalent from the dynamical point of view, even though they fail to be diffeomorphic in general. Our results could help not only the construction of viable (free of anisotropic singularities) f (R) cosmological models, but also contribute to the still active debate on the physical interpretation of the two frames.
We present a dynamical analysis in terms of new expansion-normalized variables for homogeneous and anisotropic Bianchi-I spacetimes in f (R) gravity in the presence of anisotropic matter. With a suitable choice of the evolution parameter, the Einstein's equations are reduced to an autonomous 5dimensional system of ordinary differential equations for the new variables. Further restrictions lead to considerable simplifications. For instance, we show that for a large class of functions f (R), which includes several cases commonly considered in the literature, all the fixed points are polynomial roots, and hence they can be determined with good accuracy and classified for stability. Moreover, typically for these cases, any fixed point corresponding to isotropic solutions in the presence of anisotropic matter will be unstable. The assumption of a perfect fluid as source and or the vacuum cases imply some dimensional reductions and even more simplifications. In particular, we find that the vacuum solutions of f (R) = R 1+δ , with δ a constant, are governed by an effective bi-dimensional phase space which can be analytically constructed, leading to an exactly soluble dynamics. Finally, we demonstrate that several results already reported in the literature can be re-obtained in a more direct and easy way by exploring our dynamical formulation.
The present work is related to anisotropic cosmological evolution in metric f (R) theory of gravity. The initial part of the paper develops the general cosmological dynamics of homogeneous anisotropic Bianchi-I spacetime in f (R) cosmology. The anisotropic spacetime is pervaded by a barotropic fluid which has isotropic pressure. The paper predicts nonlinear growth of anisotropy in such spacetimes. In the later part of the paper we display the predictive power of the nonlinear differential equation responsible for the cosmological anisotropy growth in various relevant cases. We present the exact solutions of anisotropy growth in Starobinsky inflation driven by quadratic gravity and exponential gravity theory. Semi-analytical results are presented for the contraction phase in quadratic gravity bounce. The various examples of anisotropy growth in Bianchi-I model universe shows the complex nature of the problem at hand. *
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