Testing a direction-dependent primordial power spectrum with observations of the Cosmic Microwave Background

TL;DR

This paper develops quadratic estimators to test for a direction-dependent primordial power spectrum using CMB temperature and polarization data. It models a quadrupole modulation (L=2) with scale dependence and derives the induced off-diagonal covariances in the CMB, quantified via the Fisher matrix. Planck forecasts show that the temperature data can constrain each quadrupole component to ~0.01 (2σ) for a scale-invariant case, with polarization offering a strong, largely independent cross-check at ~0.03 under an extended observing program, and potential constraint of the scale-dependence parameter q to Δq ≈ 0.3 (1σ) if the asymmetry is sufficiently large. The framework also accommodates axisymmetric models, enabling a directional constraint of a few degrees, and underscores the method's applicability to future analyses involving anisotropic noise and masking.

Abstract

Statistical isotropy is often assumed in cosmology and should be tested rigorously against observational data. We construct simple quadratic estimators to reconstruct asymmetry in the primordial power spectrum from CMB temperature and polarization data and verify their accuracy using simulations with quadrupole power asymmetry. We show that the Planck mission, with its millions of signal-dominated modes of the temperature anisotropy, should be able to constrain the amplitude of any spherical multipole of a scale-invariant quadrupole asymmetry at the $0.01$ level ($2σ$). Almost independent constraints can be obtained from polarization at the $0.03$ level after four full-sky surveys, providing an important consistency test. If the amplitude of the asymmetry is large enough, constraining its scale-dependence should become possible. In scale-free quadrupole models with $1\%$ asymmetry, consistent with the current limits from WMAP temperature data (after correction for beam asymmetries), Planck should constrain the spectral index $q$ of power-law departures from asymmetry to $Δq = 0.3$. Finally, we show how to constrain models with axisymmetry in the same framework. For scale-free quadrupole models, Planck should constrain the direction of the asymmetry to a $1σ$ accuracy of about $2$ degrees using one year of temperature data.

Figures (5)

• Figure 1: Noise-free simulation of a model with scale-invariant quadrupole asymmetry in the primordial power with $g_{LM} = 0.1 \delta_{L2}\delta_{M0}$. The isotropic component of the temperature map is shown on the left, and the anisotropic component on the right. The colour scales are in $\mu \mathrm{K}$.
• Figure 2: Anisotropic components of the $Q$ (left) and $U$ (right) polarization maps in a noise-free simulation of the model in Fig. \ref{['mapT']}. The maps have been smoothed here with a Gaussian beam of FWHM $3^\circ$ to enhance the imprint of the preferred axis in the $Q$ map. The colour scales are in $\mu \mathrm{K}$.
• Figure 3: Estimates of the $G_{20}$ (left) and $G_{2\,-2}$ (right) anisotropy parameters (shown with points) and their (one-sigma) Fisher errors ([Red] solid lines) as a function of $l_{\mathrm{max}}$ from five simulations of the model in Fig. \ref{['mapT']} for one year of Planck data. The input parameters $G_{20}=0.1$ and $G_{2\,-2}$ are shown with horizontal [Blue] solid lines. From top to bottom we analyse temperature only, $E$-mode polarization only and temperature plus polarization.
• Figure 4: Fisher errors for $G_{20}$ from temperature, $E$-mode polarization, and temperature plus polarization in models with power-asymmetry spectral indices $q=0$ (left), $q=1$ (middle) and $q=2$ (right). For $q=0$ and $q=1$ we show results for one and two years of observations; for $q=2$ we show only the one-year errors since they improve very little with further observing time. Note that the one- and two-year errors from temperature alone are indistinguishable when $q\geq 0$.
• Figure 5: Marginal distributions for the direction (left) and amplitude ($g_{*2}$, right) from a simulation of the nominal (one-year) Planck survey for a model with an axisymmetric quadrupole asymmetry aligned with the polar axis ($g_{LM}=0.1 \delta_{L2} \delta_{M0}$). We parameterise the direction with the equal-area projection $x=2\sin(\beta/2)\cos(\alpha)$ and $y=2\sin(\beta/2)\sin(\alpha)$ and show in solid lines the 68% [Red], 95% middle [Blue] and 99% outer [Green] contours from the temperature alone; dashed-line contours are from the $E$-mode polarization alone. For the amplitude (right), we plot the marginal distributions from $T$ alone (solid line) and $E$ alone (dashed line).