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

List of contributors Preface 1 The geometry of cosmological models G F R Ellis, S T C Siklos and J Wainwright 2 Friedmann-Lemaitre universes G F R Ellis and J Wainwright 3 Cosmological observations G F R Ellis and J Wainwright 4 Introduction to dynamical systems R Tavakol 5 Qualitative analysis of Bianchi cosmologies G F R Ellis, C Uggla and J Wainwright 6 Bianchi cosmologies: non-tilted class A models J Wainwright 7 Bianchi cosmologies: non-tilted class B models C G Hewitt and J Wainwright 8 Bianchi cosmologies: extending the scope C G Hewitt, C Uggla and J Wainwright 9 Exact Bianchi cosmologies and state space C G Hewitt, S T C Siklos, C Uggla and J Wainwright 10 Hamiltonian cosmology C Uggla 11 Deterministic chaos and Bianchi cosmologies D W Hobill 12 G2 cosmologies C G Hewitt and J Wainwright 13 Silent universes M Bruni, S Matarrese and O Pantano 14 Cosmological density perturbations P K S Dansby 15 Overview G F R Ellis and J Wainwright References Index

Dynamical systems in cosmology

The authors write the Einstein field equations for the orthogonal class B Bianchi cosmologies as an autonomous DE in terms of expansion-normalized dimensionless variables. The theory of dynamical systems is then used to give a qualitative description of the evolution of the models. Comparisons are made with other work on this problem.

https://iopscience.iop.org/article/10.1088/0264-9381/6/10/011/pdf

A dynamical systems approach to Bianchi cosmologies: Orthogonal models of class A

We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\v{\i}, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models

Past attractor in inhomogeneous cosmology

This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=μ, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.

John Wainwright

Papers

Dynamical systems in cosmology

A dynamical systems approach to Bianchi cosmologies: Orthogonal models of class A

Past attractor in inhomogeneous cosmology

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=μ

Dynamical systems in cosmology. Overview.