On implications of modified gravity models of dark energy

Resumen

Modern cosmology aims to model the large scale structure and dynamics of the universe, and its origin and fate. It was born as a quantitative science after the advent of Einstein’s theory of General Relativity (GR) in 1915. Soon after, GR found its first empirical verification in its successful prediction of two solar system phenomena: the perihelion precession of Mercury and the deflection of light by the Sun. Since then, GR has become an essential tool in the study of astrophysical and cosmological phenomena. According to GR, the energy and momentum of matter are the source of the gravitational field, and at large scales determine the geometry of the universe. In particular, Einstein studied solutions of the GR field equations for a homogeneous distribution of matter, as a model of the large scale structure of the universe. He discovered soon that his theory, in its simplest form, does not support a static universe, but instead predicts an expanding or a contracting one. By contrast, the small velocities of the stars observed at that time suggested that the universe is stationary. In order to allow for a steady-state solution, Einstein had to add what came to be called the cosmological constant to the field equations of GR. The dynamics of a homogeneous and isotropic universe according to GR was subsequently studied more systematically by Alexander Friedmann in 1922, again finding expanding or contracting solutions. In parallel, Edwin Hubble discovered the existence of galaxies outside of the Milky Way, which until then was believed to comprise the entire universe. His subsequent observations showed, in 1929, that the recessional velocity of galaxies increases with their distance from the Earth, implying that the universe is expanding. In consequence, Einstein dropped the cosmological constant from the field equations, famously regretting his ad hoc hypothesis of the universe being static as his “greatest blunder.” However, from a quantum field theory perspective, the cosmological constant in fact arises naturally as the energy density of the vacuum; due to the quantum uncertainty principle, the ground state energy of all matter fields is nonzero, and contributes to the vacuum energy density, acting like a cosmological constant for the gravitational field. Thus dropping the cosmological constant from the field equations turned out to beg a theoretical justification. Nevertheless, until the 1990’s the standard cosmological paradigm was described by a homogeneous and isotropic matter-dominated expanding universe, with no cosmological constant. A great turnover happened in 1998 when observations of type Ia Supernovae (SN Ia) revealed for the first time a very recent trend in the expansion of the universe: at a redshift z < 1 the expansion of the universe started accelerating. This observation has been subsequently corroborated by other independent cosmological observations: the Cosmic Microwave Background (CMB), the Large Scale Structure (LLS), the Baryonic Acoustic Oscillations (BAO) and the estimated age of the universe. The fundamental nature of such acceleration remains unknown and constitutes one of the major problems to be solved in cosmology. The geometry of spacetime is determined by the energy content of the universe. But according to the current cosmological models, based on Einstein’s field equations, the gravitational interaction of the standard matter-energy density components is attractive, and can only slow down the expansion. Thus, in the GR framework, a vacuum energy density component with negative pressure, is thought to be responsible for the accelerated expansion. Such vacuum energy, which effectively acts as “anti-gravity”, has been called dark energy. Observations of the universe’s expansion require the dark energy component to account for about 70% of the total energy density in the universe. The remaining 30% is accounted for by matter; a ? 25% accounted for by a dark matter component and only about 5% accounted for by ordinary matter. Furthermore, the energy density of the dark energy component appears to be constant, playing the role of a cosmological constant in Einstein’s field equations. This fact, supported by the most accurate observations over the past fifteen years, have led to the current standard cosmological model, in which the accelerated expansion is due to a positive cosmological constant, whose corresponding energy density is currently dominating the expansion history. This is the so-called ? Cold Dark Matter (?CDM) model. Naturally, the observed constant vacuum energy density was attributed to the ground state energy of matter fields. However, reasonable estimates from quantum field theory of this zero-point energy, yield values of the cosmological constant which are many orders of magnitude larger than the value necessary to account for the ob- served acceleration of the universe; this discrepancy, or the extreme fine-tuning which is necessary to avoid it, is referred to as the cosmological constant problem. Another fundamental question arises from the fact that the present value of the vacuum energy density is of the same order of magnitude as the present matter energy density; whereas in principle there is no correlation between these two quantities. This is the so-called coincidence problem. Furthermore, there is no evidence indicating that the vacuum energy has been constant over the past history of the universe, when radiation or matter where the dominant components, nor even if it is really constant at the present. Thus, the true nature of the dark energy component, and hence the underlying cause of the accelerated expansion of the universe, is still an unsettled fundamental question which constitutes a major open problem in cosmology, suggesting that there is new physics missing from our standard cosmological paradigm. With the aim of understanding the fundamental nature of the accelerated expansion, many alternative theories to GR with a cosmological constant have been proposed. One set of alternatives replace the cosmological constant with a dynamic cosmic scalar field, the so-called quintessence, that varies with time and space, slowly approaching its ground state. However, quintessence models are no better than the cosmological constant scenario in avoiding the fine-tuning problem; they do not provide an explanation (e.g. via a symmetry principle) for the tiny value of the potential at its ground state, which plays the role of an effective cosmological constant. A more attractive alternative is to modify gravity itself. Modifications of Einstein’s gravitational field equations are not unexpected from an effective 4-dimensional description of higher dimensional theories in high-energy physics. Various modified gravity theories have been studied in the context of the accelerated cosmic expansion. Some proposed modified gravity models involve extra spatial dimensions, while others modify the Hilbert-Einstein action, e.g., to higher derivative theories, scalar-tensor theories, etc. In particular, f(R) models modify GR by generalizing the Hilbert-Einstein action, replacing the scalar curvature R in its Lagrangian density with a nonlinear function of it. f(R) models are particularly simple phenomenological models which allow to search for small deviations from the ?CDM scenario. Although these modifications of gravity, in principle, do not solve the cosmological constant problem, they intend to shed light in the interpretation of the dark energy phenomenon. In recent years, f(R) modified gravity theories have been considered as a possible explanation of the accelerated expansion of the universe. Some specific types of f(R) models have been proposed and studied in the literature. In order to be cosmologically viable, these models must reproduce the GR expectations at small scales (e.g. in the solar system) while giving rise to accelerated expansion at large, cosmic scales. Effectively, viable f(R) models contain small corrections to the Hilbert-Einstein action. In this thesis, the goal is to analyze phenomenological f(R) modified gravity models and their empirical viability against cosmological and solar system constraints. We begin this thesis by introducing the standard cosmological framework and describing the observations which indicate that the expansion of the universe is currently in an accelerated phase in Chapter 2. Then in Chapter 3, we introduce the f(R) modified gravity models as a possible alternative to explain the dark energy phenomenon. We derive the modified Friedmann equations, governing the expansion of the homogeneous universe, as well as the linear perturbation analysis describing the evolution of small inhomogeneities. We also give a general discussion of the conditions under which f(R) models would not violate the constraints imposed by solar system tests. In Chapter 4, we describe a dynamical systems approach which allows to classify f(R) models according to their qualitative cosmological viability in a homogeneous universe; we summarize under which conditions particular f(R) models can reproduce an acceptable expansion history according to observations. Five different types of f(R) models are shown to have a qualitatively acceptable expansion history. Considering these five types of models, in Chapter 5 we perform a quantitative analysis which includes the perturbative analysis of cosmic inhomogeneities up to the linear order; here we confront the models’ expansion history and the growth of structure formation with various experimental datasets. Specifically, we confront the expansion history of these f(R) models with SN Ia data, measurements on the CMB shift parameter, BAO acoustic parameter measurements, and data on the Hubble parameter derived from galaxy ages. We furthermore fit the linear growth of structure in these f(R) models to the growth information derived from observations of redshift space distortions. We show that the combination of geometrical probes with data on the linear growth of structure constitutes a robust approach to rule out f(R) models. Finally, for the types of models which result to be consistent with all the cosmological datasets exploited in this thesis, we analyze their slow-motion, weak-field limits in the parameter ranges in which they are viable at cosmological scales. Using this analysis we determine which models are compatible with the tight constraints imposed by solar system tests.