Inflation with stable anisotropic hair: is it cosmologically viable?

TL;DR

This paper analyzes inflation with a vector-field–driven anisotropic hair, showing a stable anisotropic fix-point that violates the cosmic no-hair theorem. By extending the dynamics to axisymmetric spacetimes BI, BII, BIII and KS, the authors construct an autonomous system in state-space and show the anisotropic attractor is reached after a few e-folds for broad initial data when the coupling Q is large. They find that curvature at the end of inflation scales as exp(-2N) while the residual shear remains at the percent level, followed by isotropization over about 2N/3 e-folds and a tracking phase in which curvature grows relative to shear, with curvature dominating about N e-folds after inflation. Using SN Ia isotropy bounds they derive N_min in the interval (21,48), indicating the model is cosmologically viable for the restricted class of spacetimes, with KS either collapsing or following the same attractor behavior.

Abstract

Recently an inflationary model with a vector field coupled to the inflaton was proposed and the phenomenology studied for the Bianchi type I spacetime. It was found that the model demonstrates a counter-example to the cosmic no-hair theorem since there exists a stable anisotropically inflationary fix-point. One of the great triumphs of inflation, however, is that it explains the observed flatness and isotropy of the universe today without requiring special initial conditions. Any acceptable model for inflation should thus explain these observations in a satisfactory way. To check whether the model meets this requirement, we introduce curvature to the background geometry and consider axisymmetric spacetimes of Bianchi type II,III and the Kantowski-Sachs metric. We show that the anisotropic Bianchi type I fix-point is an attractor for the entire family of such spacetimes. The model is predictive in the sense that the universe gets close to this fix-point after a few e-folds for a wide range of initial conditions. If inflation lasts for N e-folds, the curvature at the end of inflation is typically of order exp(-2N). The anisotropy in the expansion rate at the end of inflation, on the other hand, while being small on the one-percent level, is highly significant. We show that after the end of inflation there will be a period of isotropization lasting for about 2N/3 e-folds. After that the shear scales as the curvature and becomes dominant around N e-folds after the end of inflation. For plausible bounds on the reheat temperature the minimum number of e-folds during inflation, required for consistency with the isotropy of the supernova Ia data, lays in the interval (21,48). Thus the results obtained for our restricted class of spacetimes indicates that inflation with anisotropic hair is cosmologically viable.

Figures (3)

• Figure 1: The phase flow of $X$, $Z$ and $\Omega_K$ with $\lambda=0.1$ and $Q=50$. The black, green, red and blue curves, respectively, represents simulations with BI, BII, BIII and KS initial conditions close to fixpoint (a). All trajectories converge towards a common point which is the type BI anisotropic fix point (a).
• Figure 2: The phase flow of $X$, $Y$ and $Z$ for KS initial conditions with $\lambda=0.1$ and $Q=50$. Initial conditions and direction of flow is indicated by the arrow. The fixpoint (a), (b), (c) and (d) are indicated by the colored points. Initial conditions are carefully chosen such that the solution "rides" on both KS-saddles (d) and (c) before ending up in the BI anisotropic attractor (a).
• Figure 3: Simulation of BIII after inflation with equation of state $\omega=1/3$ for the perfect fluid. The initial conditions comes from an inflation model with $\lambda=0.1$$Q=50$ and number of e-folds $N=60$.