R^2 Dark Matter

There is a non-trivial four-derivative extension of the gravitational spectrum that is free of ghosts and phenomenologically viable. It is the so called $R^2$-gravity since it is defined by the only addition of a term proportional to the square of the scalar curvature. Just the presence of this term does not improve the ultraviolet behaviour of Einstein gravity but introduces one additional scalar degree of freedom that can account for the dark matter of our Universe.Comment: 6 pages, 1 figure, to appear in the Proceedings of the sixth International Workshop on the Dark Side of the Universe (DSU2010) Leon, Guanajuato, Mexico 1-6 June 201

R2{\cal R}^2 Supergravity

We formulate $R^2$ pure supergravity as a scale invariant theory built only in terms of superfields describing the geometry of curved superspace. The standard supergravity duals are obtained in both "old" and "new" minimal formulations of auxiliary fields. These theories have massless fields in de Sitter space as they do in their non supersymmetric counterpart. Remarkably, the dual theory of $R^2$ supergravity in the new minimal formulation is an extension of the Freedman model, describing a massless gauge field and a massless chiral multiplet in de Sitter space, with inverse radius proportional to the Fayet-Iliopoulos term. This model can be interpreted as the "de-Higgsed" phase of the dual companion theory of $R+R^2$ supergravity.Comment: 12 page

R^2-corrections to Chaotic Inflation

Scalar density cosmological perturbations, spectral indices and reheating in a chaotic inflationary universe model, in which a higher derivative term is added, are investigated. This term is supposed to play an important role in the early evolution of the Universe, specifically at times closer to the Planck era.Comment: 14 pages, accepted for publication in MPL

Parabolic Anderson Model on R^2

For my thesis project we have been studying the analysis of the parabolic Anderson model in 2 spatial dimensions on the whole plane, performed by Hairer and Labbe in early 2015. This problem is a nice example as it requires renormalization to control the singularities and weighted spaces to control the divergence at infinity. After adding the necessary logarithmic counter term and posing the problem in the correct space we are then able to prove existence and uniqueness of the solution. Our main contribution is to offer a more explicit account than was previously available, and to correct some typos in the original work. This work is of importance because the parabolic Anderson model, which models a random walk driven by a random potential, can be used to study several topics such as spectral theory and some variational problems. Moreover, this analysis is of interest because it presents a particularly clean example, in that there is no need for any complicated (though more general) renormalization procedures. Rather, we use a trick from the analysis of smooth partial differential equations to identify the diverging terms and then add an appropriate counter term.Mathematic

Dark Matter from R^2-gravity

The modification of Einstein gravity at high energies is mandatory from a quantum approach. In this work, we point out that this modification will necessarily introduce new degrees of freedom. We analyze the possibility that these new gravitational states can provide the main contribution to the non-baryonic dark matter of the Universe. Unfortunately, the right ultraviolet completion of gravity is still unresolved. For this reason, we will illustrate this idea with the simplest high energy modification of the Einstein-Hilbert action: R^2-gravity.Comment: 5 pages, 2 figure