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Simple Types of Anisotropic Inflation

J. D. Barrow, S. Hervik

TL;DR

This work investigates gravity theories augmented by quadratic curvature corrections, focusing on Bianchi type I cosmologies. It derives a dynamical system with expansion-normalised variables from the action $S_G= rac{1}{2\kappa}\int d^4x\sqrt{|g|}\left(R+\alpha R^{2}+\beta R_{\mu\nu}R^{\mu\nu}-2\Lambda\right)$ and identifies two inflating fixed points: a de Sitter attractor and an anisotropically-inflating solution $\mathcal{A}(I)$, linked by a bifurcation. The slow decay of shear near the anisotropic fixed point allows a lasting imprint of anisotropy on the primordial fluctuations, yielding a non-scale-invariant quadrupolar CMB signature that is stronger on larger scales; these results extend to other Bianchi types, suggesting the anisotropic inflation phenomenon is generic in quadratic gravity and could relate to observed CMB anomalies.

Abstract

We display some simple cosmological solutions of gravity theories with quadratic Ricci curvature terms added to the Einstein-Hilbert lagrangian which exhibit anisotropic inflation. The Hubble expansion rates are constant and unequal in three orthogonal directions. We describe the evolution of the simplest of these homogeneous and anisotropic cosmological models from its natural initial state and evaluate the deviations they will create from statistical isotropy in the fluctuations produced during a period of anisotropic inflation. The anisotropic inflation is not a late-time attractor in these models but the rate of approach to a final isotropic de Sitter state is slow and is conducive to the creation of observable anisotropic statistical effects in the microwave background. The statistical anisotropy would not be scale invariant and the level of statistical anisotropy will grow with scale.

Simple Types of Anisotropic Inflation

TL;DR

This work investigates gravity theories augmented by quadratic curvature corrections, focusing on Bianchi type I cosmologies. It derives a dynamical system with expansion-normalised variables from the action and identifies two inflating fixed points: a de Sitter attractor and an anisotropically-inflating solution , linked by a bifurcation. The slow decay of shear near the anisotropic fixed point allows a lasting imprint of anisotropy on the primordial fluctuations, yielding a non-scale-invariant quadrupolar CMB signature that is stronger on larger scales; these results extend to other Bianchi types, suggesting the anisotropic inflation phenomenon is generic in quadratic gravity and could relate to observed CMB anomalies.

Abstract

We display some simple cosmological solutions of gravity theories with quadratic Ricci curvature terms added to the Einstein-Hilbert lagrangian which exhibit anisotropic inflation. The Hubble expansion rates are constant and unequal in three orthogonal directions. We describe the evolution of the simplest of these homogeneous and anisotropic cosmological models from its natural initial state and evaluate the deviations they will create from statistical isotropy in the fluctuations produced during a period of anisotropic inflation. The anisotropic inflation is not a late-time attractor in these models but the rate of approach to a final isotropic de Sitter state is slow and is conducive to the creation of observable anisotropic statistical effects in the microwave background. The statistical anisotropy would not be scale invariant and the level of statistical anisotropy will grow with scale.

Paper Structure

This paper contains 9 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Bianchi type I model
  3. The Inflating solutions
  4. The de Sitter solution, $\mathrm{dS}$:
  5. Anisotropically-inflating type I universes, $\mathcal{A}(I)$:
  6. Anisotropic inflation
  7. Consequences for the CMB
  8. Other Bianchi models
  9. Conclusions

Figures (2)

  • Figure 1: Inflation: The evolution of $Q$ and $\Sigma$ with time, $\tau$, for universes with three different initial values. Here, we set $\chi =6$.
  • Figure 2: The evolution of the dimensionless expansion-normalised shear, $\Sigma$, with time, $\tau$, for three universes starting away from the inflating solutions. They experience a transient period during which the universes inflate anisotropically. The evolution displayed requires some fine-tuning but the plot illustrates that there is a set of non-zero measure that gives the desired behaviour.