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Scalar-Scalar, Scalar-Tensor, and Tensor-Tensor Correlators from Anisotropic Inflation

A. E. Gumrukcuoglu, Burak Himmetoglu, Marco Peloso

Abstract

We compute the phenomenological signatures of a model (Watanabe et al' 09) of anisotropic inflation driven by a scalar and a vector field. The action for the vector is U(1) invariant, and the model is free of ghost instabilities. A suitable coupling of the scalar to the kinetic term of the vector allows for a slow roll evolution of the vector vev, and hence for a prolonged anisotropic expansion; this provides a counter example to the cosmic no hair conjecture. We compute the nonvanishing two point correlation functions between physical modes of the system, and express them in terms of power spectra with angular dependence. The anisotropy parameter g_* for the scalar-scalar spectrum (defined as in the Ackerman et al '07 parametrization) turns out to be negative in the simplest realization of the model, which, therefore, cannot account for the angular dependence emerged in some analyses of the WMAP data. A g_* of order -0.1 is achieved when the energy of the vector is about 6-7 orders of magnitude smaller than that of the scalar during inflation. For such values of the parameters, the scalar-tensor correlation (which is in principle a distinctive signature of anisotropic spaces) is smaller than the tensor-tensor correlation.

Scalar-Scalar, Scalar-Tensor, and Tensor-Tensor Correlators from Anisotropic Inflation

Abstract

We compute the phenomenological signatures of a model (Watanabe et al' 09) of anisotropic inflation driven by a scalar and a vector field. The action for the vector is U(1) invariant, and the model is free of ghost instabilities. A suitable coupling of the scalar to the kinetic term of the vector allows for a slow roll evolution of the vector vev, and hence for a prolonged anisotropic expansion; this provides a counter example to the cosmic no hair conjecture. We compute the nonvanishing two point correlation functions between physical modes of the system, and express them in terms of power spectra with angular dependence. The anisotropy parameter g_* for the scalar-scalar spectrum (defined as in the Ackerman et al '07 parametrization) turns out to be negative in the simplest realization of the model, which, therefore, cannot account for the angular dependence emerged in some analyses of the WMAP data. A g_* of order -0.1 is achieved when the energy of the vector is about 6-7 orders of magnitude smaller than that of the scalar during inflation. For such values of the parameters, the scalar-tensor correlation (which is in principle a distinctive signature of anisotropic spaces) is smaller than the tensor-tensor correlation.

Paper Structure

This paper contains 11 sections, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Model and the Background Evolution
  3. Perturbations
  4. Classification, gauge choice, and quadratic action for the dynamical modes
  5. Quantization and initial adiabatic vacuum
  6. Two point correlation functions
  7. Evolution of the perturbations and initial conditions
  8. Power spectra after isotropization
  9. Discussion
  10. Explicit quadratic action of the perturbations
  11. Gauge invariant perturbations in terms of perturbations in our gauge, and late time isotropic limit

Figures (4)

  • Figure 1: Evolution of the anisotropy factor $\dot{\alpha} / \dot{\sigma}$ for two different values of $c$ in the function (\ref{['Vf']}), as a function of the number of e-folds. $N=0$ corresponds to the end of inflation.
  • Figure 2: Time evolution for the power of a specific mode, on a nearly isotropic background ($c-1=10^{-5}$). The number of e-folds $\alpha$ is used as a "time" variable, and it is normalized to $0$ at the end of inflation. The mode shown leaves the horizon about $50$ e-folds before the end of inflation. See the main text for details.
  • Figure 3: Scalar-scalar, scalar-tensor, and tensor-tensor power spectra for a nearly isotropic background ($c-1=10^{-5}$). Notice that the scale on the $y$ axis is different for the different panels. For the scalar-scalar, and the tensor-tensor spectra, we also show the standard result for comparison (in the present model, the isotropic limit is reached for $c=1$).
  • Figure 4: Angular dependence (expressed through the ACW $g_*$ factor) of the scalar-scalar power spectrum for different choices of $c$.