Analyzing weak lensing of the cosmic microwave background using the likelihood function

TL;DR

This paper develops a likelihood-based framework for extracting CMB lensing information from temperature maps, focusing on the lensing potential $\Phi$ and its convergence $\kappa$. It shows that, in the weak-lensing limit, maximum-likelihood estimators reduce to the standard quadratic estimators, and it compares their performance via simulations for Planck-like and high-resolution instruments. The analysis demonstrates that for temperature data, the quadratic estimator is nearly optimal for Planck, with partial gains from a full nonlinear approach at high resolutions; it also outlines a path to jointly estimate the lensing power spectrum and cross-correlations with other observables. The work highlights the challenges and potential gains in using likelihood methods, while noting that polarization data may require more careful nonlinear treatment to fully exploit lensing information.

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.

Figures (5)

• Figure 1: The dimensionless nonlinearity parameter $R_2$, equal to the ratio of RMS bias in the quadratic lensing potential estimator [equation (\ref{['eq:c16']})] to the RMS value of the potential, is plotted here for several experiments as a function of multipole (wavenumber). Note that this quantity is less than unity for all of the experiments. See Table \ref{['tab:t6']} for experiment parameters.
• Figure 2: The solid line illustrates the model primary CMB temperature power spectrum $l(l+1)C_l^{\Theta\Theta}/2\pi$. The noise curves $l(l+1)C_l^{\epsilon\epsilon}/2\pi$ are shown for MAP 4-year data (top, long-dashed), Planck (center, short-dashed), and the high resolution reference experiment (bottom, dotted).
• Figure 3: The solid line illustrates the model convergence power spectrum $C_l^{\kappa\kappa}=l^2(l+1)^2C^{\Phi\Phi}/4$. The noise curves $l^2(l+1)^2(F_l^{\Phi\Phi})^{-1}/4$ are shown for (top to bottom) MAP 4-year data, Planck, and the high-resolution reference experiment, using curvature matrix elements $F_l$ computed from equation (\ref{['eq:c15']}).
• Figure 4: The ratio of the root mean squared error ($\sqrt{\rm MSE}$) for the nonlinear estimator, equation (\ref{['eq:e3']}), to that of the linear estimator, equation (\ref{['eq:c17']}), in bins of $\Delta l$=20. Results are obtained from a Monte Carlo simulation, which is responsible for the bumpiness of the graph. The solid line is for Planck parameters, and the dotted line is for the high-resolution experiment (see Table \ref{['tab:t6']}).
• Figure 5: The true convergence power spectrum, $C^{\kappa\kappa}_l$, is shown by the solid line. The points (with error bars) indicate estimated convergence power spectra from the linear (+ points) and nonlinear ($\times$ points) estimators (equations (\ref{['eq:d12']}) and (\ref{['eq:e5']}) respectively). To prevent the error bars from overlapping and causing confusion, we have displaced the data point for the linear estimator slightly to the left and the data point for the nonlinear estimator slightly to the right. The error bars are the $1\sigma$ Monte carlo error bars on the expectation value of the estimator. The estimated power spectra plotted are averages over $10$ trials of 0.14 sr solid angle each using Planck parameters, and thus shows the error bar on the power spectrum using data from a region of area $A(\Omega)=1.4$ sr.