We study f (R) gravity models in the language of scalar-tensor theories. The correspondence between f (R) gravity and scalar-tensor theories is revisited since f (R) gravity is a subclass of Brans-Dicke models, with a vanishing coupling constant (ω = 0). In this treatment, four f (R) toy models are used to analyze the early-universe cosmology, when the scalar field φ dominates over standard matter. We have obtained solutions to the Klein-Gordon equation for those models. It is found that for the first model (f (R) = βR n ), as time increases the scalar field decreases and decays asymptotically. For the second model (f (R) = αR + βR n ) it was found that the function φ(t) crosses the t-axis at different values for different values of β. For the third model f (R) = R − ν 4 R , when the value of ν is small the potential V (φ) behaves like the standard inflationary potential. For the fourth model f (R) = R − (1 − m)ν 2 R ν 2 m − 2Λ , we show that there is a transition between 1.5 < m < 1.55. The behavior of the potentials with m < 1.5 is totally different from those with m > 1.55. The slow-roll approximation is applied to each of the four f (R) models and we obtain the respective expressions for the spectral index ns and the tensor-to-scalar ratio r.

The cosmological scalar perturbations of standard matter are investigated in the context of extended teleparallel f(T) gravity theories using the $$1+3$$1+3 covariant formalism. After a review of the background gravitational field equations of f(T) gravity and the introduction of the covariant perturbation variables, the usual scalar and harmonic decomposition have been performed, and the analysis of the growth of the density contrasts in the quasi-static approximation for two non-interacting fluids scenarios, namely torsion-dust and torsion-radiation mixtures is presented for the generic f(T) gravity theory. Special applications to two classes of f(T) gravity toy models, namely $$f(T) = \mu T_0 \Big (\frac{T}{T_0}\Big )^n$$f(T)=μT0(TT0)n and $$f(T) = T+ \mu T_0 \Big (-\frac{T}{T_0}\Big )^n$$f(T)=T+μT0(-TT0)n, have then been made within the observationally viable regions of their respective parameter spaces, and the growth of the matter density contrast for both torsion-dust and torsion-radiation epochs of the Universe has been examined. The exact solutions of the dust perturbations, with growing amplitudes in cosmic time, are obtained for some limiting cases of n. Similarly, the long- and short-wavelength modes in the torsion-radiation case are treated, with the amplitudes either oscillating or monotonically growing with time. Overall, it is noted that f(T) models contain a richer set of observationally viable structure growth scenarios that can be tested against up-and-coming observational data and can accommodate currently known features of the large-scale structure power spectrum in the general relativistic and $$\varLambda CDM$$ΛCDM limits.

This work discusses scalar-tensor theories of gravity, with a focus on the Brans-Dicke subclass, and one that also takes note of the latter's equivalence with f (R) gravitation theories. A 1+3 covariant formalism is used in this case to discuss covariant perturbations on a background Friedmann-Laimaître-Robertson-Walker (FLRW) space-time. Linear perturbation equations are developed, based on gauge-invariant gradient variables. Both scalar and harmonic decompositions are applied to obtain second-order equations. These equations can then be used for further analysis of the behavior of the perturbation quantities in such a scalar-tensor theory of gravitation. Energy density perturbations are studied for two systems, namely for a scalar fluid-radiation system and for a scalar fluid-dust system, for R n models. For the matterdominated era, it is shown that the dust energy density perturbations grow exponentially, a result which agrees with those already in existing literature. In the radiation-dominated era, it is found that the behavior of the radiation energy-density perturbations is oscillatory, with growing amplitudes for n > 1, and with decaying amplitudes for 0 < n < 1. This is a new result.keywords : f (R) gravity -scalar-tensor -scalar field -cosmology -covariant perturbation. P ACSnumbers : 04.50. Kd, 95.36.+x, 98.80.Cq; M SCnumbers : 83Dxx, 83Fxx, 83Cxx arXiv:1801.01758v1 [gr-qc] 5 Jan 2018to study linear perturbation in General Relativity [15]. The same studies has been done in f (R) gravity at linear order and results have been obtained [17,18]. In scalar-tensor theory, the 1 + 3 covariant linear perturbation has been studied [19,20], but it was limited to vacuum case.The assumption that the hyper-surfaces are with constant scalar field was made in that study. Those hyper-surfaces are perpendicular to a vector field.In this paper, we consider hyper-surfaces with constant curvature. They have been considered for f (R) covariant perturbations [18,17,21]. This consideration is taken due to the equivalence between metric f (R) theory of gravity and Brans-Dicke scalar-tensor theory. The equivalence between f (R) gravity and scalar-tensor theory in five dimensions have been considered in [7,22], where Jordan frame was given much attention and bulk consideration which resulted in hyper-surfaces of four dimensional space-times. In our study, the hyper-surfaces are constructed from 1 + 3 covariant decomposition of spacetime such that they have three dimensions. The work presented in this paper is a follow-up of the work previously done by the authors [11], where the equivalence between f (R) theory and scalar-tensor theory has been explored. Here, this equivalence is extended to covariant linear perturbations for two fluid system with the consideration that scalar field behaves like a fluid (scalar fluid). The two fluid system (radiation-scalar field or dust-scalar field) are considered with the motivation that towards the end of a scalar field driven inflation the universe experienced a mixture of scalar field ...