  Future experiments will produce high-resolution temperature maps of the
cosmic microwave background (CMB) and are expected to reveal the signature of
gravitational lensing by intervening large-scale structures. We construct
all-sky maximum-likelihood estimators that use the lensing effect to estimate
the projected density (convergence) of these structures, its power spectrum,
and cross-correlation with other observables. This contrasts with earlier
quadratic-estimator approaches that Taylor-expanded the observed CMB
temperature to linear order in the lensing deflection angle; these approaches
gave estimators for the temperature-convergence correlation in terms of the CMB
three-point correlation function and for the convergence power spectrum in
terms of the CMB four-point correlation function, which can be biased and
non-optimal due to terms beyond the linear order. We show that for sufficiently
weak lensing, the maximum-likelihood estimator reduces to the computationally
less demanding quadratic estimator. The maximum likelihood and quadratic
approaches are compared by evaluating the root-mean-square (RMS) error and bias
in the reconstructed convergence map in a numerical simulation; it is found
that both the RMS errors and bias are of order 1 percent for the case of Planck
and of order 10--20 percent for a 1 arcminute beam experiment. We conclude that
for recovering lensing information from temperature data acquired by these
experiments, the quadratic estimator is close to optimal, but further work will
be required to determine whether this is also the case for lensing of the CMB
polarization field.
