Estimators for CMB statistical anisotropy

We use quadratic maximum-likelihood (QML) estimators to constrain models with Gaussian but statistically anisotropic 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 l60, but demonstrate that the effect decreases at higher multipoles and is 1% at l500. 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.