Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology, and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories that include higher-order curvature invariants, and specifically the particular class of $f(R)$ theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. Here $f(R)$ theories of gravity are reviewed in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented---metric, Palatini, and metric affine---and the following topics are discussed: motivation; actions, field equations, and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; and astrophysical applications.

f ( R ) theories of gravity

https://physicstoday.scitation.org/doi/pdf/10.1063/1.1397398

Cosmological Inflation and Large-Scale Structure

For general relativistic spacetimes filled with irrotational ‘dust’ a generalized form of Friedmann's equations for an ‘effective’ expansion factor aD of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the ‘backreaction effect’ of inhomogeneities on the average expansion of the model. A universal relation between ‘backreaction’ and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to aD-2, the expansion law governing a generic domain can be found. However, as the general equations show, ‘backreaction’ acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.

http://cds.cern.ch/record/389354/files/arXiv:gr-qc_9906015.pdf

On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies

For general relativistic spacetimes filled with irrotational `dust' a generalized form of Friedmann's equations for an `effective' expansion factor $a_D (t)$ of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the `backreaction effect' of inhomogeneities on the average expansion of the model. A universal relation between `backreaction' and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to $a_D^{-2}$, the expansion law governing a generic domain can be found. However, as the general equations show, `backreaction' acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.

On average properties of inhomogeneous fluids in general relativity I: dust cosmologies

We study the existence and stability of cosmological scaling solutions of a nonminimally coupled scalar field evolving in either an exponential or inverse power law potential. We show that, for inverse power law potentials, there exist scaling solutions the stability of which does not depend on the coupling constant $\ensuremath{\xi}.$ We then study the more involved case of exponential potentials and show that the scalar field will asymptotically behave as a baryotropic fluid when $\ensuremath{\xi}\ensuremath{\ll}1.$ The general case $\ensuremath{\xi}\ensuremath{\ll}/1$ is then discussed and we illustrate these results by some numerical examples.

Cosmological scaling solutions of nonminimally coupled scalar fields

We derive the late-time behaviour of tilted Bianchi VII_0 cosmologies with an irrotational radiation fluid as source, and give the asymptotic form of the general solution as $t \to +\infty$, making comparisons with the dust-filled models. At first sight the radiation-filled models appear to approximate the flat FL model at late times, since the Hubble-normalized shear and the tilt tend to zero and the density parameter tends to one. The Hubble-normalized Weyl curvature diverges, however, indicating that physically significant anisotropy remains. We also discuss the influence of a cosmological constant on this phenomenon.

Tilted Bianchi VII_0 cosmologies -- the radiation bifurcation

We derive the late-time behaviour of tilted Bianchi VII0 cosmologies with an irrotational radiation fluid as source, and give the asymptotic form of the general solution as t → +∞, making comparisons with the dust-filled models. At first sight the radiation-filled models appear to approximate the flat FL model at late times, since the Hubble-normalized shear and the tilt tend to zero and the density parameter tends to 1. The Hubble-normalized Weyl curvature diverges, however, indicating that physically significant anisotropy remains. We also discuss the influence of a cosmological constant on this phenomenon.

Tilted Bianchi VII0 cosmologies—the radiation bifurcation

In this paper we consider second order perturbations of a flat Friedmann-Lemaitre universe whose stress-energy content is a single minimally coupled scalar field with an arbitrary potential. We der ...

The general solution at large scale for second order perturbations in a scalar field dominated universe

In this paper we consider second order perturbations of a flat Friedmann-Lemaitre universe whose stress-energy content is a single minimally coupled scalar field with an arbitrary potential. We derive the general solution of the perturbed Einstein equations in explicit form for this class of models when the perturbations are in the super-horizon regime. As a by-product we obtain a new conserved quantity for long wavelength perturbations of a single scalar field at second order.

In this paper we give, for the first time, a qualitative description of the asymptotic dynamics of a class of non-tilted spatially homogeneous (SH) cosmologies, the so-called exceptional Bianchi cosmologies, which are of Bianchi type VI$_{-1/9}$. This class is of interest for two reasons. Firstly, it is generic within the class of non-tilted SH cosmologies, being of the same generality as the models of Bianchi types VIII and IX. Secondly, it is the SH limit of a generic class of spatially inhomogeneous $G_{2}$ cosmologies. 
Using the orthonormal frame formalism and Hubble-normalized variables, we show that the exceptional Bianchi cosmologies differ from the non-exceptional Bianchi cosmologies of type VI$_{h}$ in two significant ways. Firstly, the models exhibit an oscillatory approach to the initial singularity and hence are not asymptotically self-similar. Secondly, at late times, although the models are asymptotically self-similar, the future attractor for the vacuum-dominated models is the so-called Robinson-Trautman SH model instead of the vacuum SH plane wave models.

John Wainwright

Papers

Tilted Bianchi VII_0 cosmologies -- the radiation bifurcation

Tilted Bianchi VII0 cosmologies—the radiation bifurcation

The general solution at large scale for second order perturbations in a scalar field dominated universe

The general solution at large scale for second order perturbations in a scalar field dominated universe

Asymptotic dynamics of the exceptional Bianchi cosmologies