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

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.