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

Increasingly accurate observations are driving theoretical cosmology towards the use of more sophisticated descriptions of matter and the study of nonlinear perturbations of Friedmann–Lemaitre cosmologies, whose governing equations are notoriously complicated. Our goal in this paper is to formulate the governing equations for linear perturbation theory in a particularly simple and concise form in order to facilitate the extension to nonlinear perturbations. Our approach has several novel features. We show that the use of so-called intrinsic gauge invariants has two advantages. It naturally leads to (i) a physically motivated choice of a gauge invariant associated with the matter density, and (ii) two distinct and complementary ways of formulating the evolution equations for scalar perturbations, associated with the work of Bardeen and of Kodama and Sasaki. In the first case, the perturbed Einstein tensor gives rise to a second-order (in time) linear differential operator, and in the second case to a pair of coupled first-order (in time) linear differential operators. These operators are of fundamental importance in cosmological perturbation theory, since they provide the leading order terms in the governing equations for nonlinear perturbations.

https://iopscience.iop.org/article/10.1088/0264-9381/28/17/175017/pdf

Cosmological perturbation theory revisited

In this paper we investigate the energy-momentum tensor of the most general second-order vector-tensor theory of gravitation and electromagnetism which has field equations which are (i) derivable from a variational principle, (ii) consistent with the notion of conservation of charge, and (iii) compatible with Maxwell's equations in a flat space. This energy-momentum tensor turns out to be quadratic in the first partial derivatives of the electromagnetic field tensor and depends upon the curvature tensor. The asymptotic behavior of this energy-momentum tensor is examined for solutions to Maxwell's equations in Minkowski space, and it is demonstrated that this energy-momentum tensor predicts regions of negative energy density in the vicinity of point sources.

Energy-momentum tensor of the electromagnetic field

We present simple expressions for the relativistic first and second order fractional density perturbations for FL cosmologies with dust, in four different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. We include a cosmological constant and arbitrary spatial curvature in the background. A distinctive feature of our approach is our description of the spatial dependence of the perturbations using a canonical set of quadratic differential expressions involving an arbitrary spatial function that arises as a conserved quantity. This enables us to unify, simplify and extend previous seemingly disparate results. We use the primordial matter and metric perturbations that emerge at the end of the inflationary epoch to determine the additional arbitrary spatial function that arises when integrating the second order perturbation equations. This introduces a non-Gaussianity parameter into the expressions for the second order density perturbation. In the special case of zero spatial curvature we show that the time evolution simplifies significantly, and requires the use of only two non-elementary functions, the so-called growth supression factor at the linear level, and one new function at the second order level. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

Second order density perturbations for dust cosmologies

In this paper we give five gauge-invariant systems of governing equations for first and second order scalar perturbations of flat Friedmann-Lemaitre universes that are minimal in the sense that the ...

Dynamics of cosmological perturbations at first and second order

In this paper we present four simple expressions for the relativistic first and second order fractional density perturbations for ΛCDM cosmologies in different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. A distinctive feature of our approach is the use of a canonical set of quadratic differential expressions involving an arbitrary spatial function, the so-called comoving curvature perturbation, to describe the spatial dependence, which enables us to unify, simplify and extend previous seemingly disparate results. The simple structure of the expressions makes the evolution of the density perturbations completely transparent and clearly displays the effect of the cosmological constant on the dynamics, namely that it stabilizes the perturbations. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

John Wainwright

Papers

Cosmological perturbation theory revisited

Energy-momentum tensor of the electromagnetic field

Second order density perturbations for dust cosmologies

Dynamics of cosmological perturbations at first and second order

Simple expressions for second order density perturbations in standard cosmology