### Some Observable Effects of Modified Gravity in Cosmology and Astrophysics

#### Abstract

Among modified theories of gravitation, f (R) theories are possibly the most straightforward and “natural” purely geometric extension of GR. The first appearance of modified gravity theories dates back to the 1920’s [1, 2], although their relevance and popularity vastly increased about 40 years ago, when pioneering works [3–5] showed the possibility of generating the early inflationary period with quadratic theories, which arise naturally from quantum corrections in curved spacetime. After the discovery of the cosmic acceleration [6–9], new life was infused into f (R) theories, sparked by the early works [10–14]. These models were soon realised to suffer from severe instabilities [15, 16], because the additional scalar degree of freedom acquires an imaginary mass. During the following years, an impressive amount of work was directed to determining the cosmological viability conditions of f (R) models [17–26]. Clearly, despite the conceptual simplicity of f (R) theories, the additional dynamics and the higher-order equations make it difficult – and fun – to come up with “good” models to model dark energy. Furthermore, these models must be tested in a variety of cosmological and astrophysical situations, and may lead to important detectable signatures which could in principle be observed soon. As important as inventing new models is, finding new ways to constrain and even exclude them is no smaller task. It is believed that f (R) models can be considered as low-energy phenomenological limits of some more fundamental theory such as string theory, etc. [27–29]. Every step towards the “right” f (R) model may very well be a step towards the “right” theory of quantum gravity, so there is no overestimating the relevance of any result in this direction. This has been precisely the intent of my work. Chapter 1 is devoted to introducing the vacuum energy problem. After a brief review of the standard cosmological scenario and of the main observational indications for a vacuum energy component, I present a few of the most important theoretical models proposed to explain the cosmic acceleration, from both the modified matter (dark energy) and modified gravity standpoints. In Chapter 2, I study the radiation-dominated epoch in R + R2 gravity, discussing the modified curvature dynamics analytically and numerically. The curvature scalar exhibits fast oscillations around some power-law behaviour which may or may not correspond to the standard GR solution. These curvature oscillations are however damped due to gravitational particle production effects, so that eventually the solutions relax to the GR ones, but with an additional relic density of gravitationally produced particles which can in principle give some imprint on the cosmological evolution and perhaps even make up an effective mechanism to produce dark matter. In Chapter 3, I investigate the formation of curvature singularities inside astronomical contracting systems within the framework of two recently proposed f (R) models [30, 31], studying the problem analytically and comparing my estimates with exact numerical results. I show that such infinite-R, finite-r singularities can arise in a number of physically reasonable systems, and derived the time scales for this to happen. Naturally, as R approaches infinity, one expects high-curvature effects to come into play. In Chapter 4, I study the curvature evolution in the models [30, 31] with the addition of an R2 term (irrelevant for cosmology, but important for large R). I show that this term prevents the formation of the curvature singularity while still allowing R to reach very large values, and in turn may lead to strong particle production. I calculate the particle production rate, which depends on both physical properties of the system and on model parameters. These high-energy cosmic rays could in principle be detectable and, if observed, would represent a unique model-dependent signature. Some unexplained features in the cosmic ray spectrum, e.g. the so-called “ankle” [32–35], might find a fascinating explanation in this framework. In Chapter 5, I discuss another interesting and unexpected consequence of these high-R solutions, namely the possibility of gravitational repulsion in contracting systems. The modified Einstein equations lead to new solutions for the metric in spherically symmetric systems, and the new geodesic equation for a test particle essentially shows repulsive behaviour if curvature is large compared to its GR value. The phenomenology of such an antigravitational behaviour, which has not been fully explored yet, is probably rather interesting and may lead to additional interesting discoveries. riferimenti bibliografici [1] H. Weyl, Annalen Phys. 59, 101 (1919). [2] A. S. 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