Abstract

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity — such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

Highlights

  • General Relativity (GR) [225, 226] is widely accepted as a fundamental theory to describe the geometric properties of spacetime

  • This burst of activities is strongly motivated by the observational discovery of dark energy

  • The idea is that the gravitational law may be modified on cosmological scales to give rise to the late-time acceleration, while Newton’s gravity needs to be recovered on solar-system scales

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Summary

Introduction

General Relativity (GR) [225, 226] is widely accepted as a fundamental theory to describe the geometric properties of spacetime. While scalar-field models of inflation and dark energy correspond to a modification of the energy-momentum tensor in Einstein equations, there is another approach to explain the acceleration of the universe This corresponds to the modified gravity in which the gravitational theory is modified compared to GR. For large coupling models with |Q| = O(1) it is possible to design scalar-field potentials such that the chameleon mechanism works to reduce the effective matter coupling, while at the same time the field is sufficiently light to be responsible for the late-time cosmic acceleration This generalized BD theory leaves a number of interesting observational and experimental signatures [596]. The Greek indices μ and ν run from 0 to 3, whereas the Latin indices i and j run from 1 to 3 (spatial components)

Field Equations in the Metric Formalism
Equations of motion
Equivalence with Brans–Dicke theory
Conformal transformation
Inflationary dynamics
Dynamics in the Einstein frame
Reheating after inflation
Dynamical equations
Equation of state of dark energy
Local Gravity Constraints
Linear expansions of perturbations in the spherically symmetric background
Field profile of the chameleon field
Thin-shell solutions
Post Newtonian parameter
Experimental bounds from the violation of equivalence principle
Cosmological Perturbations
Perturbation equations
Gauge-invariant quantities
Perturbations Generated During Inflation
Curvature perturbations
Tensor perturbations
The power spectra in the Einstein frame
The Lagrangian for cosmological perturbations
Matter density perturbations
The impact on large-scale structure
Non-linear matter perturbations
Cosmic Microwave Background
Field equations
Background cosmological dynamics
Matter perturbations
10 Extension to Brans–Dicke Theory
10.2 Cosmological dynamics of dark energy models based on Brans– Dicke theory
10.3 Local gravity constraints
10.4 Evolution of matter density perturbations
11.1 Field equations
11.2 Constant density star
12 Gauss–Bonnet Gravity
12.1 Lovelock scalar invariants
12.2 Ghosts
12.3.2 Numerical analysis
12.3.3 Solar system constraints
12.3.4 Ghost conditions in the FLRW background
12.3.6 The speed of propagation in more general modifications of gravity
12.4 Gauss–Bonnet gravity coupled to a scalar field
13.1 Weak lensing
13.2 Thermodynamics and horizon entropy
13.5 Vainshtein mechanism
13.6 DGP model
13.7 Special symmetries
13.7.1 Noether symmetry on FLRW
13.7.2 Galileon symmetry
14 Conclusions
15 Acknowledgements
Full Text
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