TITLE:
Analyzing weak lensing of the cosmic microwave background using the likelihood function.
AUTHOR(S):
Christopher M. Hirata, Uros Seljak (Princeton University)
DATE:
2002 Sep 24 (arXiv, posted, v1);
2002 Sep 24 (Phys. Rev. D, submitted);
2002 Nov 26 (revised);
2002 Dec 09 (Phys. Rev. D, accepted);
2003 Feb 12 (Phys. Rev. D, published).
AVAILABILITY:
arXiv astro-ph/0209489 (free);
APS (requires subscription).
PUBLICATION INFORMATION:
Physical Review D, 67, 043001 (2003), 19 pages
ABSTRACT:
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.
ADS BIBLIOGRAPHIC CODE:
2003PhRvD..67d3001H
COMMENTS:
It has been known for some time that the cosmic microwave background radiation (CMB) should be gravitationally lensed.
In the several years preceding this paper, several methods had been proposed for analyzing CMB
data and extracting a gravitational lensing signal (see the references), all based on perturbation theory analyses of
gravitational lensing. This paper goes beyond the perturbation theory to (nearly) a full treatment of the lensing of
CMB anistropies and their statistical recovery from observables. It was found that the perturbative methods made nearly
optimal use of the temperature anisotropy data on angular scales larger than an arcminute. (We later discovered that the same is not true for the polarization data.) For the first attempt at CMB lensing reconstruction on
actual data, see our paper on correlating SDSS and WMAP.