Inflationary perturbations in anisotropic backgrounds and their imprint on the CMB (0707.4179v2)

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.

• The paper presents the extension of standard inflationary perturbation theory to a Bianchi I model that couples scalar and tensor modes.
• The analysis employs a Mukhanov-Sasaki framework to compute perturbations and reveal non-diagonal CMB multipole correlations.
• The study demonstrates that persisting early anisotropies may explain observed CMB anomalies and large-scale multipole alignments.

Inflationary Perturbations in Anisotropic Backgrounds and Their Imprint on the CMB

The paper investigates the extension of cosmological perturbation theories to include anisotropic, but homogeneous, universes. The primary focus is on the computation of perturbations within a Bianchi I model framework, which is a simple anisotropic model characterized by different scale factors along different spatial dimensions. The paper pays particular attention to the impact of such anisotropy on the Cosmic Microwave Background (CMB) anisotropies.

Theoretical Framework and Model Specification

In the context of modern cosmology, the standard model assumes a statistically isotropic universe—an assumption that is crucial both for the perturbation theory and the interpretation of CMB observations. The authors extend this paradigm by considering a background that is initially anisotropic but becomes isotropic with the onset of inflation. Specifically, the paper explores a Bianchi I model that retains isotropy in two spatial dimensions, simplifying both theoretical analysis and numerical computation. Anisotropic expansion rates are quantified by distinguishing two Hubble parameters, thereby creating a landscape where anisotropic inflation can be explicit.

Perturbation Modes and Methodology

In the isotropic case, perturbation modes are cleanly separated into scalar, vector, and tensor categories that evolve independently. However, anisotropy leads to inherent coupling between these modes. The authors identify that in an anisotropic Bianchi I universe, three physical modes arise: two tensors and one scalar, similar to the isotropic case but with essential coupling during the anisotropic phase.

The analysis employs an exhaustive mathematical approach to derive the canonical perturbative variables. This ensures the applicability of a Mukhanov-Sasaki-type framework even in anisotropic backgrounds. Perturbations are computed by fixing gauge freedoms in a manner that facilitates the integration of nondynamical modes, ultimately extracting the three physical modes from the eleven initial degrees of freedom in the metric and scalar field perturbations.

Impact on Cosmic Microwave Background

One of the significant outcomes of this paper is the exploration of how anisotropy affects CMB correlations. In an isotropic universe, CMB mode correlations vanish off the diagonal; however, anisotropy introduces coupling, leading to non-zero correlations between different multipoles. The correlation of anisotropic modes in the CMB spectrum is calculated to be of the form $$\langle a_{\ell m} a^*_{\ell' m'} \rangle \propto \delta_{\ell \ell'} \delta_{mm'}$$, implying new physics constraints on CMB data analysis.

The paper highlights that early universe anisotropies, if not fully erased by inflation, leave an imprint on large-scale structure, potentially explaining anomalies observed in the WMAP data, such as the alignment of large-scale multipoles and asymmetries in power between hemispheres.

Challenges and Future Work

The paper identifies significant challenges when attempting to set initial conditions for perturbations in the anisotropic regime. In particular, the requirement of an adiabatic vacuum—a cornerstone in perturbation theory—is complicated by anisotropy. The authors provide a detailed exploration of potential methods to resolve singular behaviors observed in perturbation spectra, such as by considering nonlinear effects or alternative early universe dynamics.

This work poses an intriguing possibility that deviations from isotropy in the early universe might not be completely obscured by subsequent inflation, aligning with certain anomalies currently observed in the CMB spectrum. The papel encourages further exploration into more complex anisotropic models or alternative mechanisms to achieve inflation in such frameworks, potentially involving additional fields or modifying gravitational constraints.

Conclusion

The paper ambitiously enhances our understanding of inflationary cosmology by examining non-standard pre-inflationary conditions. While providing detailed theoretical insights and computational methodologies, it underscores the pivotal role that initial anisotropies may play in shaping observable cosmic phenomena. Future developments in the observational sensitivity of CMB data analyses could either substantiate the proposed scenarios or motivate reconsiderations of early-universe dynamics within anisotropic frameworks.