Narrow Results

The equivalence between $f(R)$ and scalar–tensor theories is revisited, we consequently explored different $f(R)$ models. After consideration of specific definition of the scalar field, we derived the potentials $V(\varphi )$ for each $f(R)$ model focusing on the early Universe, mostly the inflation epoch. For a given potential, we applied the slow-roll approximation approach to each $f(R)$ model and obtained the expressions for the spectral index $n_s$ and tensor-to-scalar ratio $r$. We determined the corresponding numerical values associated with each of the $f(R)$ models. Our results showed that for certain choice of parameter space, the values of $n_s$ and $r$ are consistent with the Planck survey results and others produce numerical values that are in the same range as suggested by Planck data. We further constructed the Klein–Gordon equations (KGEs) of each $f(R)$ model. We found numerical solutions to each KGE considering different values of free parameters and initial conditions of each $f(R)$ model. All models showed that the scalar field decreases as time increases, an indication that there is less content of the scalar field in the late Universe.

We study $f(R)$ gravity models in the language of scalar–tensor (ST) theories. The correspondence between $f(R)$ gravity and ST theories is revisited since $f(R)$ gravity is a subclass of Brans–Dicke models, with a vanishing coupling constant ($\omega =0$). In this treatment, four $f(R)$ toy models are used to analyze the early-universe cosmology, when the scalar field $\varphi$ dominates over standard matter. We have obtained solutions to the Klein–Gordon equation for those models. It is found that for the first model $(f(R)=\beta R^n)$, as time increases the scalar field decreases and decays asymptotically. For the second model $(f(R)=\alpha R+\beta R^n)$, it was found that the function $\varphi (t)$ crosses the $t$-axis at different values for different values of $\beta$. For the third model $\left(\right.f(R)=R-\frac{{\nu }^4}{R}\left)\right.$, when the value of $\nu$ is small, the potential $V(\varphi )$ behaves like the standard inflationary potential. For the fourth model $(f(R)=R-(1-m){\nu }^2\left(\right.\frac{R}{{\nu }^2}{\left)\right.}^m-2\mathrm{\Lambda })$, 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 $n_s$ and the tensor-to-scalar ratio $r$.

In this paper, we explore the equivalence between two theories, namely $f(R)$ and scalar–tensor theories of gravity. We use this equivalence to explore several $f(R)$ toy models focusing on the inflation epoch of the early universe. The study is done based on the definition of the scalar field in terms of the first derivative of $f(R)$ model. We have applied the slow-roll approximations during inflationary parameters consideration. The comparison of the numerically computed inflationary parameters with the observations is done. We have inspected that some of the $f(R)$ models produce numerical values of $n_s$ that are in the same range as the suggested values from observations. But for the case of the tensor-to-scalar ratio $r$, we realized that some of the considered $f(R)$ models suffer to produce a value which is in agreement with the observed values for different considered space parameter.

We present a reconstruction technique for models of $f(R)$ gravity from the Chaplygin scalar field in flat de Sitter spacetimes. Exploiting the equivalence between $f(R)$ gravity and scalar–tensor (ST) theories, and treating the Chaplygin gas (CG) as a scalar field model in a universe without conventional matter forms, the Lagrangian densities for the $f(R)$ action are derived. Exact $f(R)$ models and corresponding scalar field potentials are obtained for asymptotically de Sitter spacetimes in early and late cosmological expansion histories. It is shown that the reconstructed $f(R)$ models all have General Relativity (GR) as a limiting solution.

This work discusses scalar–tensor theories of gravity, with a focus on the Brans–Dicke sub-class, 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) spacetime. 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 matter-dominated era, it is shown that the dust energy density perturbations grow exponentially, a result which agrees with those already existing in the literatures. 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.

In this paper, the scalar–tensor theory is applied to the study of perturbations in a multifluid universe, using the $1+3$ covariant approach. Both scalar and harmonic decompositions are instituted on the perturbation equations. In particular, as an application, we study perturbations on a background Friedmann-Robertson-Walker (FRW) cosmology consisting of both radiation and dust in the presence of a scalar field. We consider both radiation-dominated and dust-dominated epochs, respectively, and study the results. During the analysis, quasi-static approximation is instituted. It is observed that the fluctuations of the energy density decrease with increasing redshift, for different values of $n$ of a power-law $R^n$ model.