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

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

Modified Gravity and Cosmology

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.

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The Confrontation Between General Relativity and Experiment

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

According to general relativity, photons are deflected and delayed by the curvature of space-time produced by any mass. The bending and delay are proportional to gamma + 1, where the parameter gamma is unity in general relativity but zero in the newtonian model of gravity. The quantity gamma - 1 measures the degree to which gravity is not a purely geometric effect and is affected by other fields; such fields may have strongly influenced the early Universe, but would have now weakened so as to produce tiny--but still detectable--effects. Several experiments have confirmed to an accuracy of approximately 0.1% the predictions for the deflection and delay of photons produced by the Sun. Here we report a measurement of the frequency shift of radio photons to and from the Cassini spacecraft as they passed near the Sun. Our result, gamma = 1 + (2.1 +/- 2.3) x 10(-5), agrees with the predictions of standard general relativity with a sensitivity that approaches the level at which, theoretically, deviations are expected in some cosmological models.

A test of general relativity using radio links with the Cassini spacecraft

A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

Mach’s principle and the structure of dynamical theories

1. Dynamical Principles.- 2. The Gravitational Field of an Isolated Body.- 3. Planetary Rotation.- 4. Gravitational Torques and Tides.- 5. The Interior of the Earth.- 6. Planetary Magnetism.- 7. Atmospheres.- 8. Upper Atmospheres.- 9. The Sun and the Solar Wind.- 10. Magnetospheres.- 11. The Two-Body Problem.- 12. Perturbation Theory.- 13. The Three-Body Problem.- 14. The Planetary System.- 15. Dynamical Evolution of the Solar System.- 16. Origin of the Solar System.- 17. Relativistic Effects in the Solar System.- 18. Artificial Satellites.- 19. Telecommunications.- 20. Precise Measurements in Space.

Physics of the Solar System: Dynamics and Evolution, Space Physics, and Spacetime Structure

A structure of dynamical theories is proposed that implements Mach's ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein's general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

Mach's principle and the structure of dynamical theories

We have used precision Doppler tracking of the Cassini spacecraft during its 2001-2002 solar opposition to derive improved observational limits to an isotropic background of low-frequency gravitational waves. Using the Cassini multilink radio system and an advanced tropospheric calibration system, the effects of heretofore leading noises—plasma and tropospheric scintillation—were, respectively, removed and calibrated to levels lower than other noises. The resulting data were used to construct upper limits to the strength of an isotropic background in the 10-6 to 10-3 Hz band. Our results are summarized as limits on the strain spectrum Sh( f), the characteristic strain (hc = the square root of the product of the frequency and the one-sided spectrum of strain at that frequency), and the energy density (Ω = energy density in bandwidth equal to center frequency assuming a locally white energy density spectrum, divided by the critical density). Our best limits are Sh( f) < 6 × 10-27 Hz-1 at several frequencies in the millihertz band, hc < 2 × 10-15 at about 0.3 mHz, and Ω < 0.025 × h, where h75 is the Hubble constant in units of 75 km s-1 Mpc-1, at 1.2 × 10-6 Hz. These are the best observational limits in the low-frequency band, the bound on Ω, for example, being about 3 orders of magnitude better than previous constraints from Doppler tracking.

Bruno Bertotti

Papers

A test of general relativity using radio links with the Cassini spacecraft

Mach’s principle and the structure of dynamical theories

Physics of the Solar System: Dynamics and Evolution, Space Physics, and Spacetime Structure

Mach's principle and the structure of dynamical theories

Stochastic Gravitational Wave Background: Upper Limits in the 10–6 to 10–3 Hz Band