We develop a general framework for generating estimators of a given quantity which are unbiased to a given order in the difference between the true value of the underlying quantity and the fiducial position in theory space around which we expand the
likelihood. We apply this formalism to rederive the optimal quadratic estimator and show how the replacement of the second derivative matrix with the Fisher matrix is a generic way of creating an unbiased estimator (assuming choice of the fiducial model is independent of data). Next apply the approach to estimation of shear lensing, closely following the work of Bernstein and Armstrong (2014). Our first order estimator reduces to their estimator in the limit of zero shear, but it also naturally allows for the case of non-constant shear and the easy calculation of correlation functions or power spectra using standard methods. Both our first-order estimator and Bernstein and Armstrongs estimator exhibit a bias which is quadratic in true shear. Our third-order estimator is, at least in the realm of the toy problem of Bernstein and Armstrong, unbiased to 0.1% in relative shear errors $Delta g/|g|$ for shears up to $|g|=0.2$.
Updated constraints on dark matter cross section and mass are presented combining CMB power spectrum measurements from Planck, WMAP9, ACT, and SPT as well as several low-redshift datasets (BAO, HST, supernovae). For the CMB datasets, we combine WMAP9
temperature and polarization data for l <= 431 with Planck temperature data for 432 < l < 2500, ACT and SPT data for l > 2500, and Planck CMB four-point lensing measurements. We allow for redshift-dependent energy deposition from dark matter annihilation by using a `universal energy absorption curve. We also include an updated treatment of the excitation, heating, and ionization energy fractions, and provide updated deposition efficiency factors (f_eff) for 41 different dark matter models. Assuming perfect energy deposition (f_eff = 1) and a thermal cross section, dark matter masses below 26 GeV are excluded at the 2-sigma level. Assuming a more generic efficiency of f_eff = 0.2, thermal dark matter masses below 5 GeV are disfavored at the 2-sigma level. These limits are a factor of ~2 improvement over those from WMAP9 data alone. These current constraints probe, but do not exclude, dark matter as an explanation for reported anomalous indirect detection observations from AMS-02/PAMELA and the Fermi Gamma-ray Inner Galaxy data. They also probe relevant models that would explain anomalous direct detection events from CDMS, CRESST, CoGeNT, and DAMA, as originating from a generic thermal WIMP. Projected constraints from the full Planck release should improve the current limits by another factor of ~2, but will not definitely probe these signals. The proposed CMB Stage IV experiment will more decisively explore the relevant regions and improve upon the Planck constraints by another factor of ~2.
