Abstract
We study the planar regime of curvature perturbations for single field inflationary models in an axially symmetric Bianchi I background. In a theory with standard scalar field action, the power spectrum for such modes has a pole as the planarity parameter goes to zero. We show that constraints from back reaction lead to a strong lower bound on the planarity parameter for high-momentum planar modes and use this bound to calculate the signal-to-noise ratio of the anisotropic power spectrum in the CMB, which in turn places an upper bound on the Hubble scale during inflation allowed in our model. We find that non-Gaussianities for these planar modes are enhanced for the flattened triangle and the squeezed triangle configurations, but show that the estimated values of the fNL parameters remain well below the experimental bounds from the CMB for generic planar modes (other, more promising signatures are also discussed). For a standard action, fNL from the squeezed configuration turns out to be larger compared to that from the flattened triangle configuration in the planar regime. However, in a theory with higher derivative operators, non-Gaussianities from the flattened triangle can become larger than the squeezed configuration in a certain limit of the planarity parameter.
Highlights
Solution with the de Sitter solution leads to a description of the late time dynamics of the curvature perturbation in terms of an excited state on the Bunch-Davies vacuum
We review the field theory results for the case of single field inflation with standard action and higher derivative operators, respectively, in a Kasner-de Sitter background
The geometry in question is of Kasner type at early times and isotropizes to a de Sitter phase at late times
Summary
A Bianchi I geometry of the Kasner-de Sitter type appears naturally in a theory of Einstein gravity with a minimally coupled single scalar field. The metric in the above action (which we shall refer to as the background metric) is chosen to be an axially symmetric version of the Bianchi I metric ds2 = −dt2 + e2ρ(dx)2 + e2β(dy2 + dz2) The evolution of such a scalar field in this anisotropic geometry has been discussed in much detail in earlier works [2, 3, 17, 18, 19]. It turns out that only for the positive branch, one can impose initial conditions on the cosmological perturbations at early times via the usual WKB approximation [19] This is the class of backgrounds we shall focus on for the rest of the paper. All common inflaton potentials obey this condition and the above solution (2.9) is valid even for a non-constant potential in the t → 0+ In this case, we have Hφ2 ≈ t → 0, so that V is essentially constant at early times
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