Mauro Francaviglia

Extended Theories of Gravity and their Cosmological and Astrophysical Applications

Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large scale structure and cosmic dynamics. The emerging picture is a spatially flat, homogeneous Universe undergoing the today observed accelerated phase. Despite of the good quality of astrophysical surveys, commonly addressed as Precision Cosmology, the nature and the nurture of dark energy and dark matter, which should constitute the bulk of cosmological matter-energy, are still unknown. Furthermore, up to now, no experimental evidence has been found, at fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of General Relativity in its simplest formulation (a linear theory in the Ricci scalar R, minimally coupled to the standard perfect fluid matter) and claiming for the “correct” theory of gravity as that derived by matching the largest number of observational data, without imposing any theory a priori. As a working hypothesis, accelerating behavior of cosmic fluid, large scale structure, potential of galaxy clusters, rotation curves of spiral galaxies could be reproduced by means of extending the standard theory of General Relativity. In other words, gravity could acts in different ways at different scales and the above “shortcomings” could be due to incorrect extrapolations of the Einstein gravity, actually tested at short scales and low energy regimes. After a survey of what is intended for Extended Theories of Gravity in the so called “metric” and “Palatini” approaches, we discuss some cosmological and astrophysical applications where the issues related to the dark components are addressed by enlarging the Einstein theory to more general f (R) Lagrangians, where f (R) is a generic function of Ricci scalar R, not assumed simply linear. Obviously, this is not the final answer to the problem of “dark-components” but it can be considered as an operative scheme whose aim is to avoid the addition of unknown exotic ingredients to the cosmic pie.

Dark energy dominance and cosmic acceleration in first order formalism

The current accelerated universe could be produced by modified gravitational dynamics as it can be seen, in particular, in its Palatini formulation. We analyze here a specific nonlinear gravity-scalar system in the first-order Palatini formalism which leads to a Friedmann-Robertson-Walker cosmology different from the purely metric one. It is shown that the emerging Friedmann-Robertson-Walker cosmology may lead either to an effective quintessence phase (cosmic speed-up) or to an effective phantom phase. Moreover, the already known gravity assisted dark energy dominance occurs also in the first-order formalism. Finally, it is shown that a dynamical theory able to resolve the cosmological constant problem exists also in this formalism, in close parallel with the standard metric formulation.

Accelerated cosmological models in Ricci squared gravity

Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of the Universe at present time. In this paper we proceed further in this analysis of cosmological applications of alternative gravitational theories depending on (other) curvature invariants. We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini formalism. It is striking that both Ricci scalar and Ricci squared theories are described in the same mathematical framework and both the generalized Einstein equations and generalized Friedmann equations have the same structure. In the framework of the cosmological principle, without the introduction of exotic forms of dark energy, we thus obtain modified equations providing values of

Nonlinear gravitational Lagrangians

{w}_{\mathrm{e}\mathrm{f}\mathrm{f}}l\ensuremath{-}1

Variational formulation of general relativity from 1915 to 1925 “Palatini's method” discovered by Einstein in 1925

in accordance with the experimental data. The spacetime bi-metric structure plays a fundamental role in the physical interpretation of results and gives them a clear and very rich geometrical interpretation.

Natural and Gauge Natural Formalism for Classical Field Theorie

Abstract“Alternative gravitational theories” based on Lagrangian densities that depend in a nonlinear way on the Ricci tensor of a metric are considered. It is shown that, provided certain weak regularity conditions are met, any such theory is equivalent, from the Hamiltonian point of view, to the standard Einstein theory for a new metric (which, roughly speaking, coincides with the momentum canonically conjugated to the original metric), interacting with external matterfields whose nature depends on the original Lagrangian density.

Accelerated cosmological models in first-order nonlinear gravity

Among the three basic variational approaches to general relativity, the metric-affine variational principle, according to which the metric and the affine connection are varied independently, is commonly known as the “Palatini method.” In this paper we revisit the history of the “golden age” of general relativity, through a discussion of the papers involving a variational formulation of the field problem. In particular we find that the original Palatini paper of 1919 was rather far from what is usually meant by “Palatinis method,” which was instead formulated, to our knowledge, by Einstein in 1925.

Post-newtonian parameters from alternative theories of gravity

I The Geometric Setting Introduction.- 1. Fiber Bundles.- 2. Jet Bundles.- 3. Principal Bundles and Connections.- 4. Natural Bundles.- 5. Gauge Natural Bundles.- II The Variational Structure of Field Theories Introduction.- 6. The Lagrangian Formalism.- 7. Natural Theories.- 8. Gauge Natural Theories.- III Spinor Fields Introduction.- 9. Spin Structures and Spin Frames.- 10. Spinor Theories.- Final Word.- References.- Analitic Index.

Universality of the Einstein equations for Ricci squared Lagrangians

The evidence of the acceleration of universe at present time has led to investigate modified theories of gravity and alternative theories of gravity, which are able to explain acceleration from a theoretical viewpoint without the need of introducing dark energy. In this paper we study alternative gravitational theories defined by Lagrangians which depend on general functions of the Ricci scalar invariant in minimal interaction with matter, in view of their possible cosmological applications. Structural equations for the spacetimes described by such theories are solved and the corresponding field equations are investigated in the Palatini formalism, which prevents instability problems. Particular examples of these theories are also shown to provide, under suitable hypotheses, a coherent theoretical explanation of earlier results concerning the present acceleration of the universe and cosmological inflation. We suggest moreover a new possible Lagrangian, depending on the inverse of

Conformal aspects of the Palatini approach in Extended Theories of Gravity

\mathrm{sinh}(R),