Abstract: Recent constrains on the sum of neutrino masses inferred by analyzing cosmological data, show that detecting a non-zero neutrino mass is within reach of forthcoming cosmological surveys. Such a measurement will imply a direct determination of the absolute neutrino mass scale. Physically, the measurement relies on constraining the shape of the matter power spectrum below the neutrino free streaming scale: massive neutrinos erase power at these scales. However, detection of a lack of small-scale power from cosmological data could also be due to a host of other effects. It is therefore of paramount importance to validate neutrinos as the source of power suppression at small scales. We show that, independent on hierarchy, neutrinos always show a footprint on large, linear scales; the exact location and properties are fully specified by the measured power suppression (an astrophysical measurement) and atmospheric neutrinos mass splitting (a neutrino oscillation experiment measurement). This feature cannot be easily mimicked by systematic uncertainties in the cosmological data analysis or modifications in the cosmological model. Therefore the measurement of such a feature, up to 1% relative change in the power spectrum for extreme differences in the mass eigenstates mass ratios, is a smoking gun for confirming the determination of the absolute neutrino mass scale from cosmological observations. It also demonstrates the synergy between astrophysics and particle physics experiments.

Abstract: General Relativity has shown an outstanding observational success in the scales where it has been directly tested. However, modifications have been intensively explored in the regimes where it seems either incomplete or signals its own limit of validity. In particular, the breakdown of unitarity near the Planck scale strongly suggests that General Relativity needs to be modified at high energies and quantum gravity effects are expected to be important. This is related to the existence of spacetime singularities when the solutions of General Relativity are extrapolated to regimes where curvatures are large. In this sense, Born-Infeld inspired modifications of gravity have shown an extraordinary ability to regularise the gravitational dynamics, leading to non-singular cosmologies and regular black hole spacetimes in a very robust manner and without resorting to quantum gravity effects. This has boosted the interest in these theories in applications to stellar structure, compact objects, inflationary scenarios, cosmological singularities, and black hole and wormhole physics, among others. We review the motivations, various formulations, and main results achieved within these theories, including their observational viability, and provide an overview of current open problems and future research opportunities.