The Nature of Primordial Fluctuations from Anisotropic Inflation

TL;DR

This paper analyzes primordial fluctuations in an anisotropic inflation model where a vector field coupled to the inflaton induces small yet detectable anisotropy. Using a canonical gauge that generalizes the flat slicing, it reveals the coupling structure among curvature perturbations, vector waves, and gravitational waves, and identifies two sources of anisotropy, with the coupling-driven source dominating. Through numerical simulations and analytical in-in calculations, it predicts a curvature perturbation anisotropy $g_*\sim 24 I N^2(k)$, a smaller GW anisotropy $g_*^{GW}\sim 6 I \epsilon_H N^2(k)$, and a cross-correlation between scalar and tensor modes, all amplified by the e-folding number $N(k)$. The results imply potentially observable signatures in the CMB and future GW experiments, and motivate extensions to multi-vector or antisymmetric-field scenarios to explore the full phenomenology of anisotropic inflation.

Abstract

We study the statistical nature of primordial fluctuations from an anisotropic inflation which is realized by a vector field coupled to an inflaton. We find a suitable gauge, which we call the canonical gauge, for anisotropic inflation by generalizing the flat slicing gauge in conventional isotropic inflation. Using the canonical gauge, we reveal the structure of the couplings between curvature perturbations, vector waves, and gravitational waves. We identify two sources of anisotropy, i.e. the anisotropy due to the anisotropic expansion of the universe and that due to the anisotropic couplings among variables. It turns out that the latter effect is dominant. Since the coupling between the curvature perturbations and vector waves is the strongest one, the statistical anisotropy in the curvature perturbations is larger than that in gravitational waves. We find the cross correlation between the curvature perturbations and gravitational waves which never occurs in conventional inflation. We also find the linear polarization of gravitational waves. Finally, we discuss cosmological implication of our results.

Figures (4)

• Figure 1: Evolution of anisotropy of curvature perturbations. Here we depicted the anisotropy $P_{\delta\phi} ({\bf k}) |_{\theta = \pi/2} / P_{\delta\phi} ({\bf k}) |_{\theta = 0} - 1$ as a function of time. We set $e$-folding number to be zero at the time of horizon crossing of the given mode. The both axes are taken in log scale.
• Figure 2: Evolution of anisotropy in gravitational waves. Here we depicted the anisotropy $P_{X} ({\bf k})|_{\theta = \pi/2} /P_{X} ({\bf k})|_{\theta = 0} - 1$, where $X=\Gamma , G$, as a function of time. The axes are in log scale. As one can see, the difference between two modes is quite small.
• Figure 3: Evolution of cross correlation between curvature perturbations and the plus mode of gravitational waves. Here we depicted the value $-P_{\delta\phi G} ({\bf k}) |_{\theta = \pi/2} / P_{\delta\phi} ({\bf k}) |_{\theta = \pi/2}$ as a function of the time. The axes are in log scale.
• Figure 4: Evolution of the linear polarization of gravitational waves. Here we depicted the value $P_{\Gamma} ({\bf k}) |_{\theta = \pi/2}/ P_{G} ({\bf k}) |_{\theta = \pi/2} - 1$ as a function of time. The axes are in log scale.