A study of perturbations in scalar–tensor theory using 1 + 3 covariant approach

Abstract

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 ...

Selected topics in scalar–tensor theories and beyond

“…Scalar-tensor theories of gravity 41,64,65,[175][176][177][178][179][180][181][182][183][184][185][186][187] are a generalization of the Brans-Dicke theory to allow the BD coupling to be a function of the scalar field: ω BD → ω(φ), i. e., to be a varying parameter:…”

“…where Λ is an integration constant. Hence, the Friedmann equation in (184) reads: 3Ĥ 2 = Λ, and sinceφ = 0, the relationship (185) implies that in the Jordan frame we will have also a GR-de Sitter solution H =Ĥ = Λ/3. From the relationship (185) it follows, besides, that there can be other de Sitter solutions in the Jordan frame (H = H 0 ), that would require an evolving scalar field ϕ = ϕ(t) which compensates the time evolution of the EF Hubble parameter:…”

1 + 3 covariant perturbations in power-law f(R) gravity

“…The 1 + 3 covariant approach has been applied to study scalar perturbations in f (R) gravity [9,[19][20][21]. It is also studied in [22], where the approach was used to explore the equivalence between 0123456789(). : V,-vol scalar-tensor theory and f (R) gravity but they used a different definition of the scalar field.…”

“…Linear perturbations have severally been discussed in f (R) models in different studies using different approach [19,[21][22][23][24][25][26]. In these studies, the investigation between dark energy and f (R) models was done by analysing the behavior of scalar perturbations around FLRW background.…”

Scalar perturbations in f(T) gravity using the $$1 + 3$$ covariant approach

“…In the 1 + 3 covariant formalism, the perturbations defined describe true physical degrees of freedom and no physical gauge modes exist. In recent years, there has been active research on cosmological perturbations theory for both GR [38][39][40][41] and different extended gravity theories [34,35,42] using the 1+3 covariant formalism.…”