Cosmic Microwave Background Statistics for a Direction-Dependent Primordial Power Spectrum

TL;DR

This paper develops a framework to test departures from statistical isotropy in the primordial perturbations by allowing a direction-dependent power spectrum $P(\mathbf{k})$. It introduces two complementary tools: (i) power multipole moments $b_{LM}$ that probe anisotropies in the CMB variance in a model-independent way, and (ii) minimum-variance estimators for the anisotropy coefficients $g_{LM}(k)$ that optimally constrain directional dependence; these yield observable off-diagonal correlations in the CMB multipoles through $D^{LM}_{ll'}$. Applying the formalism to a quadrupolar anisotropy in the primordial power, the authors show Planck's TT data could detect $|g_{2M}|$ down to roughly $2\%$, illustrating the method’s sensitivity. The framework generalizes to temperature-polarization data and offers a path to combine with galaxy surveys and future 21-cm probes for comprehensive tests of statistical isotropy in the early Universe.

Abstract

Statistical isotropy of primordial perturbations is a common assumption in cosmology, but it is an assumption that should be tested. To this end, we develop cosmic microwave background statistics for a primordial power spectrum that depends on the direction, as well as the magnitude, of the Fourier wavevector. We first consider a simple estimator that searches in a model-independent way for anisotropy in the square of the temperature (and/or polarization) fluctuation. We then construct the minimum-variance estimators for the coefficients of a spherical-harmonic expansion of the direction-dependence of the primordial power spectrum. To illustrate, we apply these statistics to an inflation model with a quadrupole dependence of the primordial power spectrum on direction and find that a power quadrupole as small as 2.0% can be detected with the Planck satellite.