Inflationary perturbations in anisotropic backgrounds and their imprint on the CMB

TL;DR

The paper tackles inflationary perturbations in a homogeneous but anisotropic background, focusing on a Bianchi I model with a residual 2d isotropy that isotropizes at inflation onset. It generalizes the Mukhanov-Sasaki formalism to anisotropic spacetimes, revealing that three physical perturbations are linearly coupled during the anisotropic phase, which induces off-diagonal correlations in the CMB multipoles. The authors derive the axisymmetric CMB covariance, analyze background dynamics, and compute the curvature power spectrum, finding isotropy at small scales but potential large-scale anisotropic signatures, though linear theory breaks down for the most sensitive configurations without including nonlinear effects. The work highlights a framework to test departures from statistical isotropy in the CMB and points to the need for more sophisticated anisotropic models or nonlinear analyses to make robust predictions for observations.

Abstract

We extend the standard theory of cosmological perturbations to homogeneous but anisotropic universes. We present an exhaustive computation for the case of a Bianchi I model, with a residual isotropy between two spatial dimensions, which is undergoing complete isotropization at the onset of inflation; we also show how the computation can be further extended to more general backgrounds. In presence of a single inflaton field, there are three physical perturbations (precisely as in the isotropic case), which are obtained (i) by removing gauge and nondynamical degrees of freedom, and (ii) by finding the combinations of the remaining modes in terms of which the quadratic action of the perturbations is canonical. The three perturbations, which later in the isotropic regime become a scalar mode and two tensor polarizations (gravitational wave), are coupled to each other already at the linearized level during the anisotropic phase. This generates nonvanishing correlations between different modes of the CMB anisotropies, which can be particularly relevant at large scales (and, potentially, be related to the large scale anomalies in the WMAP data). As an example, we compute the spectrum of the perturbations in this Bianchi I geometry, assuming that the inflaton is in a slow roll regime also in the anisotropic phase. For this simple set-up, fixing the initial conditions for the perturbations appears more difficult than in the standard case, and additional assumptions seem to be needed to provide predictions for the CMB anisotropies.

Figures (3)

• Figure 1: Time evolution of the two expansion rates $H_a$ (red-solid) and $H_b$ (green-dashed), in units of $m$, and of the inflaton (blue-dotted), in units of $M_p$, for a choice of chaotic inflaton potential $V=m^2\phi^2/2$. The left panel refers to the positive branch, while the right one to the negative branch. $V_0$ denotes the initial potential of the inflaton, with $\phi_0 = 16\, M_p$ in this example. Notice that for the negative branch $H_a$ starts from negative values, indicating that this direction is initially contracting.
• Figure 2: Power spectrum of the comoving curvature perturbation ${\cal R}$ in the inflationary model $V = m^2 \,\phi^2 /2$. The inflaton field starts with $\phi = 16 \,M_p$, and is evolved until the moment shown in the figure, when $\phi = M_p$. Modes with $k>k_{\rm iso}$ leave horizon during the later isotropic stage of inflation. The standard result is recovered for these modes.
• Figure 3: Sections of the power spectrum (same evolution as in the previous figure) at fix values of $\xi \,$. We see that the power spectrum becomes isotropic (no $\xi$ dependence) at large momenta; at lower momenta it presents some oscillations, whose amplitude increases as $\xi$ decreases (this leads to the $1/\xi$ divergency mentioned in the main text). Finally, the power spectrum is suppressed as $k \rightarrow 0 \,$.