The Initial State of a Primordial Anisotropic Stage of Inflation

Abstract

We investigate the possibility that the inflationary period in the early universe was preceded by a primordial stage of strong anisotropy. In particular we focus on the simplest model of this kind, where the spacetime is described by a non-singular Kasner solution that quickly evolves into an isotropic de Sitter space, the so-called Kasner-de Sitter solution. The initial Big Bang singularity is replaced, in this case, by a horizon. We show that the extension of this metric to the region behind the horizon contains a timelike singularity which will be visible by cosmological observers. This makes it impossible to have a reliable prediction of the quantum state of the cosmological perturbations in the region of interest. In this paper we consider the possibility that this Kasner-de Sitter universe is obtained as a result of a quantum tunneling process effectively substituting the region behind the horizon by an anisotropic parent vacuum state, namely a $1+1$ dimensional spacetime compactified over an internal flat torus, $T_2$, which we take it to be of the form ${de Sitter}_2 \times T_2$ or ${Minkowski}_2 \times T_2$. As a first approximation to understand the effects of this anisotropic initial state, we compute the power spectrum of a massless scalar field in these backgrounds. In both cases, the spectrum converges at small scales to the isotropic scale invariant form and only present important deviations from it at the largest possible scales. We find that the decompactification scenario from $M_2 \times T_2$ leads to a suppressed and slightly anisotropic power spectrum at large scales which could be related to some of the anomalies present in the current CMB data. On the other hand, the spectrum of the universe with a $dS_2 \times T_2$ parent vacuum presents an enhancement in power at large scales not consistent with observations.