Primordial Power Spectra from Anisotropic Inflation

TL;DR

This work studies primordial perturbations in an anisotropic inflation model where a U(1) gauge field couples to the inflaton via f(φ)F_{μν}F^{μν}, enabling a persistent anisotropy on a Bianchi I background. By formulating the quadratic action, diagonalizing the kinetic terms, and applying the in-in formalism to both odd and even sectors, the authors derive the dominant direction-dependent corrections to the tensor and scalar power spectra. They show that small anisotropies can produce dramatic directional signatures, with scalar perturbations more strongly affected than tensors, and derive a consistency relation connecting gravitational-wave and scalar anisotropies; observational bounds on g_* constrain the vector field energy density during inflation. The results highlight potential CMB signatures of primordial anisotropy and place stringent limits on ρ̂_A/ε, with implications for the viability and detectability of anisotropic inflation scenarios in current and future data.

Abstract

We examine cosmological perturbations in a dynamical theory of inflation in which an Abelian gauge field couples directly to the inflaton, breaking conformal invariance. When the coupling between the gauge field and the inflaton takes a specific form, inflation becomes anisotropic and anisotropy can persist throughout inflation, avoiding Wald's no-hair theorem. After discussing scenarios in which anisotropy can persist during inflation, we calculate the dominant effects of a small persistent anisotropy on the primordial gravitational wave and curvature perturbation power spectra using the "in-in" formalism of perturbation theory. We find that the primordial power spectra of cosmological perturbations gain significant direction dependence and that the fractional direction dependence of the tensor power spectrum is suppressed in comparison to that of the scalar power spectrum.

Figures (2)

• Figure 1: Log plot of $\Sigma$ and $\epsilon$ as a function of $e$-foldings ($\Delta \alpha = \alpha - \alpha_0$) during inflation. The plot was generated with the potential $V = {1 \over 2} m^2 \phi^2$ and coupling function $f(\phi) = \exp\left[{\kappa^2 \phi^2 \over 2}\right]$. The initial conditions were $\phi_0 = 17.5/\kappa$, $\phi_0' = 0$, $\alpha_0 = -75$, $\beta_0 = 0$ and $\beta_0' = 0$. The constants $m$ and $p_A$ were chosen so that initially $\rho_A/\rho_\phi \approx 10^{-6}$. Notice that $\Sigma$ very quickly settles to a value that is somewhat smaller than the square of the slow-roll parameter $\epsilon$.
• Figure 2: The function $e^z \tilde{I}(-e^z,-e^{-z_*})$ on a linear scale. The axes cross at the point $\{ 0,0 \}$. For $0 < z < z_*$ the function is well approximated by $- {2 \over 3} e^{2 z_*}$. The frequency of oscillation for $z < 0$ does not vary much as $z_*$ increases---only the amplitude changes. The plot above was generated using $z_* = 15$.