We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Dynamics of dark energy

On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

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

Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models

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.

f(R) theories

We discuss a class of phantom ($p < - \varrho$) cosmological models Except for phantom we admit various forms of standard types of matter and discuss the problem of singularities for these cosmologies The singularities are different from those of standard matter cosmology since they appear for infinite values of the scale factor We also find an interesting relation between the phantom models and standard matter models which is like the duality symmetry of string cosmology

Phantom cosmologies

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and Einstein relativity. Because of that, in this paper we discuss the rules of conformal transformations for geometric quantities in general relativity. In particular, we discuss the conformal transformations of the matter energy-momentum tensor. We thoroughly discuss the latter and show the subtlety of the conservation law (i.e., the geometrical Bianchi identity) imposed in one of the conformal frames in reference to the other. The subtlety refers to the fact that conformal transformation ``creates'' an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is ``created'' due to work done by the conformal transformation to bend the spacetime which was originally flat. We also discuss how to construct the conformally invariant gravity which, in the simplest version, is a special case of the Brans-Dicke theory -- the one with the Brans-Dicke parameter $\omega = -3/2$. Conformal transformations are also used to investigate higher-order gravity theories. Motivated by this, we find the conformal transformation rules for the curvature invariants $R^2$, $R_{ab}R^{ab}$, $R_{abcd}R^{abcd}$ and, as a consequence, for the Gauss-Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present all already obtained rules of the conformal transformations in the fashion of the duality transformation of the superstring theory. In such a case the transitions between conformal frames can just be obtained by a simple change of the sign of the quantity $\om = \ln{\Om}$, where $\Om$ is the conformal factor.

Conformal transformations and conformal invariance in gravitation

Phantom Friedmann cosmologies and higher-order characteristics of expansion

We find that sudden future singularities of pressure may also appear in spatially inhomogeneous Stephani models of the universe. They are temporal pressure singularities and may appear independently of the spatial finite density singularities already known to exist in these models. It is shown that the main advantage of the homogeneous sudden future singularities which is the fulfillment of the strong and weak energy conditions may not be the case in the inhomogeneous case.

Inhomogenized sudden future singularities

We show that homogeneous G\"odel spacetimes need not contain closed timelike curves in low-energy-effective string theories. We find exact solutions for the G\"odel metric in string theory for the full $O({\ensuremath{\alpha}}^{\ensuremath{'}})$ action including both dilaton and axion fields. The results are valid for bosonic, heterotic and super-strings. To first order in the inverse string tension ${\ensuremath{\alpha}}^{\ensuremath{'}},$ these solutions display a simple relation between the angular velocity of the G\"odel universe, \ensuremath{\Omega}, and the inverse string tension of the form ${\ensuremath{\alpha}}^{\ensuremath{'}}=1/{\ensuremath{\Omega}}^{2}$ in the absence of the axion field. The generalization of this relationship is also found when the axion field is present.

Mariusz P. Dabrowski

Papers

Phantom cosmologies

Conformal transformations and conformal invariance in gravitation

Phantom Friedmann cosmologies and higher-order characteristics of expansion

Inhomogenized sudden future singularities

Godel Universes in String Theory