Estimators for CMB Statistical Anisotropy

TL;DR

This paper develops a quadratic maximum-likelihood framework to constrain Gaussian but anisotropic CMB fluctuations and applies it to WMAP data to test several anisotropy models. The method yields optimal, model-agnostic estimators for modulation fields and primordial power anisotropy, with mean-field corrections, Fisher forecasts, and robust significance testing. The results show a detectable large-scale modulation at low multipoles that does not persist at smaller scales, while claims of a primordial quadrupole are found to be dominated by beam and scan-systematics; a local modulation estimator is proposed for future robust constraints. The work provides a scalable, systematic pipeline for isolating true CMB anisotropy signals and diagnosing instrumental effects in high-precision cosmology.

Abstract

We use quadratic maximum-likelihood (QML) estimators to constrain models with Gaussian but statistically anisotropic Cosmic Microwave Background (CMB) fluctuations, using CMB maps with realistic sky-coverage and instrumental noise. This approach is optimal when the anisotropy is small, or when checking for consistency with isotropy. We demonstrate the power of the QML approach by applying it to the WMAP data to constrain several models which modulate the observed CMB fluctuations to produce a statistically anisotropic sky. We first constrain an empirically motivated spatial modulation of the observed CMB fluctuations, reproducing marginal evidence for a dipolar modulation pattern with amplitude 7% at L < 60, but demonstrate that the effect decreases at higher multipoles and is 1% at L~500. We also look for evidence of a direction-dependent primordial power spectrum, finding a very statistically significant quadrupole signal nearly aligned with the ecliptic plane; however we argue this anisotropy is largely contaminated by observational systematics. Finally, we constrain the anisotropy due to a spatial modulation of adiabatic and isocurvature primordial perturbations, and discuss the close relationship between anisotropy and non-Gaussianity estimators.

Figures (10)

• Figure 1: Pseudo-$C_{l}$ of the $f_{l m}$ reconstruction for the WMAP V-band foreground-reduced data, with KQ85 mask and $l_{\rm max}=64$ (black solid). The full-sky, isotropic normalization was used rather than the actual inverse Fisher matrix. The $[25,75]\%$ (dark gray) and $[5,95]\%$ (light gray) confidence intervals measured from Monte-Carlo simulations are overlaid.
• Figure 2: Reconstructed maps of $f(\hat{\mathbf{n}})$ for three values of $l_{\rm max}$, smoothed with a ten degree beam. We have used the isotropic normalization for simplicity, which is invalid close to the sky-cut, but works well otherwise. The '$+$' symbols mark the peak of the QML dipole. The '$\times$' symbol and ring in the $l_{\rm max}=64$ plot marks the M-L dipole and error found by Hoftuft:2009rq, which agrees well with our result.
• Figure 3: Summary of modulation dipole results for the foreground-reduced WMAP data. Solid lines correspond to KQ85 masking, and dashed lines use the KQ75 mask. Upper panel: Dipole amplitudes $|\mathbf{A}|$ as a function of the maximum multipole used in the reconstruction. The black dashed line gives the expected value for a cosmic-variance limited experiment, which is non-zero due to the estimator noise. The dotted lines give the reconstruction noise spectra measured from the simulations. They separate into two groups for KQ85 and KQ75 masking and are well described as $f_{\rm sky}^{-1}$ times the ideal result for $l_{\rm max}<300$, but decrease more slowly at higher-$l$ as the instrumental noise becomes non-negligible. Lower panel: $\chi^2$ significances of the reconstructions in the isotropic model.
• Figure 4: Sensitivity of modulation results to various tests, similar to Fig. \ref{['fig:modulation_ml_QVW_std_AS']}. All data are for WMAP V-band. Blue solid is the V-band foreground-reduced data, with KQ85 mask. Magenta is with the KQ85+CS5 mask. Green is for reconstructions with $l_{\rm min}=20$. Red lines are for raw maps, without template cleaning (solid/dashed correspond to KQ85/KQ75 masking respectively.)
• Figure 5: $\sqrt{l(l+1)l_2(l_2+1)}C_{l l _2}/2\pi$ for $a(k)=1$, with $l_2=l$ (thick solid), $l_2=l+2$ (dashed), $l_2=l+4$ (dot-dashed) and $l_2=l+20$ (dotted).