Spin-Torsion in Chaotic Inflation

The role of spin-torsion coupling to gravity is analyzed in the context of a model of chaotic inflation. The system of equations constructed from the Einstein-Cartan and inflaton field equations are studied and it is shown that spin-torsion interactions are effective only at the very first e-folds of inflation, becoming quickly negligible and, therefore, not affecting the standard inflationary scenario nor the density perturbations spectrum predictions. PACS number(s): 98.80 Cq E-mail: [email protected] E-mail: [email protected] Inflation, in its many different implementations, has become one of the most important cosmological paradigm today [for reviews, see for instance, [1]]. The underlying idea of inflation, of a period of accelerated expansion of the scale factor, when the energy density is dominated by a vacuum energy density, is able to provide in a simple way a solution to the cosmological horizon and flatness problems and at the same time provides a model for density perturbations in the early Universe. Earlier studies by Gasperini [2] have shown that inflation could be driven by a spin density dominated epoch in the early Universe, even in the absence of vacuum dominant contributions to the energy density, showing that a spin-torsion interaction acts like a source of repulsive gravity. This then poses us with the question whether primordial spin-torsion interactions are able to support inflation in standard inflaton driven inflationary scenarios, by, e.g., easing the conditions for slow-roll of the inflaton field. Previous works on spin/torsion effects in inflation that we are aware of [3] have not detailed or elucidated the real role of spin-torsion in an inflationary epoch. Torsion makes an important role in very different physical models [4, 5]. In particular, torsion is natural to many models of higher dimensional theory, as in Einstein-Kalb-Ramond models and string theory [6] and in gauge theories of the Poincarè group [7]. Therefore, it is natural to expect that torsion may be particularly important in pre-inflationary models, where quantum gravity effects may be introduced, from the geometrical aspects of the space-time, by a torsion interaction term. This may be the case in chaotic inflationary scenarios, where the inflaton initial conditions are taken around the Planck era and, then, quantum gravity effects may become important to determine the initial conditions prior to inflation. Based on the above motivations, in this letter, by considering the spin-spin interactions of matter as described by the Einstein-Cartan theory (see, e.g., Ref. [7]), we study the role of spin-torsion in the simplest model of chaotic inflation, which is that of an inflaton with a quadratic potential. We do not expect that more general models of chaotic inflation will lead to results much different to this simple model, when regarding the effects of spin-torsion, which is introduced through a generalization of the gravity equations. In the Einstein-Cartan theory, the gravity equations are modified such that the Friedman equation (we are assuming a spatially flat Friedman-Robertson-Walker metric) and the acceleration equations read [2], respectively, H = 8πG 3 (ρφ + ρs) (1) and ä a = − 4πG 3 (ρφ + 3pφ − 8πGρs) , (2) where H = ȧ/a is the Hubble parameter and G = 1/M pl, with Mpl the Planck mass. In the above equations we have also defined ρs, as ρs = 〈SμναS 〉/2, the average of the square of the spin density tensor Sμνα. The spins are taken as randomly oriented (from not polarized spinning matter fields) [2], so the average value of S is zero. The torsion