On the joint analysis of CMB temperature and lensing-reconstruction
  power spectra
  Gravitational lensing provides a significant source of cosmological
information in modern CMB parameter analyses. It is measured in both the power
spectrum and trispectrum of the temperature fluctuations. These observables are
often treated as independent, although as they are both determined from the
same map this is impossible. In this paper, we perform a rigorous analysis of
the covariance between lensing power spectrum and trispectrum analyses. We find
two dominant contributions coming from: (i) correlations between the
disconnected noise bias in the trispectrum measurement and sample variance in
the temperature power spectrum; and (ii) sample variance of the lenses
themselves. The former is naturally removed when the dominant N0 Gaussian bias
in the reconstructed deflection spectrum is dealt with via a partially
data-dependent correction, as advocated elsewhere for other reasons. The
remaining lens-cosmic-variance contribution is easily modeled but can safely be
ignored for a Planck-like experiment, justifying treating the two observable
spectra as independent. We also test simple likelihood approximations for the
deflection power spectrum, finding that a Gaussian with a parameter-independent
covariance performs well.
