In this work, we present a new scheme to study the tachyon-dark-energy model using dynamical systems analysis by considering parametrization of the equation of state (EOS) of the dark energy. Both the canonical and phantom field dynamics are investigated. In our method, we do not require any explicit form of the tachyon potential. Instead of the potential, we start with an approximate form of the EOS of the tachyon field. This EOS is phenomenologically motivated and contains some dimensionless parameters. Using our method, we can construct the dynamical system which gives rise to the time evolution of the Universe. We have considered two different parametrizations of the EOS and studied the phase space dynamics in detail. Our analysis shows Taylor series parametrization of the EOS has serious cosmological limitations. We have also provided an example of how this method can be applied to coupled tachyon models with a specific form of interaction. Our proposal is generic in nature and can be applied to other scalar field dark energy models.

We propose a new framework for studying the cosmology of $f(R)$ gravity which completely avoids using the reconstruction program. This allows us to easily obtain a qualitative feel of how much the $\mathrm{\Lambda }\mathrm{CDM}$ model differs from other $f(R)$ theories of gravity at the level of linear perturbation theory for theories that share the same background dynamics. This is achieved by using the standard model independent cosmographic parameters to develop a new dynamical system formulation of $f(R)$ gravity which is free from the limitation of having to first specify the functional form of $f(R)$. By considering a set of representative trajectories, which are indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$, we use purely qualitative arguments to determine the extent to which these models deviate from the standard model by including an analysis of the linear growth rate of density fluctuations and also whether or not they suffer from the Dolgov-Kawasaki instability. We find that if one demands that a late time $f(R)$ cosmology is observationally close to the $\mathrm{\Lambda }\mathrm{CDM}$ model, there is a higher risk that it suffers from a Dolgov-Kawasaki instability. Conversely, the more one tries to construct a physically viable late time $f(R)$ cosmology, the more likely it is observationally different from the $\mathrm{\Lambda }\mathrm{CDM}$ model.

We present the reconstruction method of $f(R)$ gravity for the homogeneous and anisotropic Bianchi-I spacetime, which was previously formulated only for homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker spacetime. We argue in this paper that for anisotropic spacetimes, the total anisotropy behaves as an independent metric degree of freedom (d.o.f.) on top of the average scale factor in $f(R)$ gravity. This is not like general relativity, where specifying the form of the average scale factor as a function of time also specifies the total anisotropy as a function of time uniquely. We link this peculiar fact to an interesting intertwining between the definition of Ricci scalar for anisotropic metric and anisotropy evolution equation in $f(R)$ gravity. Consequently, specifying an anisotropic solution of $f(R)$ gravity implies specifying both the average scale factor and the total anisotropy as functions of time. The reconstruction method hence formulated is applied to two scenarios where anisotropy suppression is important, namely, a quasi-de-Sitter expansion as required in inflation, and a power law contraction as required in ekpyrotic bounce models.

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 not only help 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.