We study the reconstruction of the lensing potential power spectrum from CMB temperature data, with an eye to the Planck experiment. We work with the optimal quadratic estimator of Okamoto and Hu, which we characterize thoroughly in an application to the reconstruction of the lensing power spectrum. We find that at multipoles $L<250$, our current understanding of this estimator is biased at the 15% level by beyond-gradient terms in the Taylor expansion of lensing effects. We present the full lensed trispectrum to fourth order in the lensing potential to explain this effect. We show that the low-$L$ bias, as well as a previously known bias at high $L$, is relevant to the determination of cosmology and must be corrected for in order to avoid significant parameter errors. We also investigate the covariance of the reconstructed power, finding broad correlations of $\approx 0.1\%$. Finally, we discuss several small improvements which may be made to the optimal estimator to mitigate these problems.

• Received 26 August 2010

DOI:https://doi.org/10.1103/PhysRevD.83.043005