Reconstruction of lensing from the cosmic microwave background polarization

Abstract

Gravitational lensing of the cosmic microwave background (CMB) polarization field has been recognized as a potentially valuable probe of the cosmological density field. We apply likelihood-based techniques to the problem of lensing of CMB polarization and show that if the B-mode polarization is mapped, then likelihood-based techniques allow significantly better lensing reconstruction than is possible using the previous quadratic estimator approach. With this method the ultimate limit to lensing reconstruction is not set by the lensed CMB power spectrum. Second-order corrections are known to produce a curl component of the lensing deflection field that cannot be described by a potential; we show that this does not significantly affect the reconstruction at noise levels greater than 0.25 microK arcmin. The reduction of the mean squared error in the lensing reconstruction relative to the quadratic method can be as much as a factor of two at noise levels of 1.4 microK arcmin to a factor of ten at 0.25 microK arcmin, depending on the angular scale of interest.

Figures (8)

• Figure 1: CMB polarization power spectra for $E$-type and $B$-type polarization (upper and lower solid curves, respectively). The noise curves for the experiments of Table \ref{['tab:expt']} are shown as dashed lines; from top to bottom: WMAP, Planck, and Reference Experiments A, B, C, D, E, and F.
• Figure 2: (a) The convergence (upper curves) and field rotation (lower curves) power spectra in the fiducial cosmology. These are normalized to $\sigma_8^{\rm linear}=1.0$ (solid curves) and $\sigma_8^{\rm linear}=0.7$ (dashed curves). (b) The power spectra of $\Delta E$ (solid curves) and $\Delta B$ (long dashed) in "noise units" ($\mu$K arcmin). The short-dashed curves are the total $B$-mode power introduced by the convergence component. The upper curves are calculated for $\sigma_8^{\rm linear}=1.0$, the lower curves for $0.7$.
• Figure 3: The power spectrum of the error in the convergence reconstruction for Reference Expts. A--F. The top curve in each panel shows the overall convergence power spectrum $C^{\kappa\kappa}_l$. The middle curve shows the theoretical, i.e. from Eq. (\ref{['eq:qfisher']}) power spectrum of the convergence error $\hat{\kappa}-\kappa$ in the Wiener-filtered quadratic estimator Eq. (\ref{['eq:wfquad']}); the "$+$" data points indicate the power spectrum of this error as recovered from simulations. The error power spectrum for the iterative estimator Eq. (\ref{['eq:iter']}), again as recovered from simulations, is shown with the "$\times$" data points. The bottom curve shows the theoretical best performance if the Fisher matrix limit Eq. (\ref{['eq:fphi']}) can be achieved, i.e. if we had a truly optimal estimator and no curvature corrections. Note the more dramatic improvement provided by the iterative estimator when the noise is small. Field rotation was neglecting in the calculations for this figure.
• Figure 4: A simulated reconstruction of the lensing convergence using polarization and Reference Expt. C parameters. In the left panel, we display the realization of the convergence field $\kappa$ used to produce the simulated CMB. The reconstructions using the Wiener-filtered quadratic estimator and the iterative estimator are shown in the center and right panels, respectively. These frames are each $8^\circ 32'$ in angular width, corresponding to 1/16 of the simulated area; the scale ranges from black (diverging, $\kappa=-0.12$) through white (converging, $\kappa=+0.12$). Although all lensing multipoles up to $l=3600$ are simulated, we have only displayed the $l\le 1600$ modes in these figures for clarity. Field rotation was neglected in the calculations for this figure.
• Figure 5: The dependence of the mean squared error in lensing reconstruction, $\langle |\kappa_{\bf l}-\hat{\kappa}_{\bf l}|^2\rangle$, on the instrument parameters. The baseline is Ref. Expt. C, ${\cal N}_P=1.41$$\mu$K arcmin, $\theta_{FWHM}=4$ arcmin. The thick solid line is the raw power spectrum $C^{\kappa\kappa}_l$; the thin solid lines indicate the mean squared error for the lensing reconstruction using the iterative estimator. As described in the text, the iterative estimator is unusable for wide-beam experiments ($\ge$10 arcmin); we used the quadratic estimator for these cases (dot-dashed curves). The dashed lines indicate the ideal zero-noise reconstruction error from the quadratic estimator according to Eq. (\ref{['eq:qfisher']}) with polarization only (top) and temperature+polarization (bottom). (a) Changing ${\cal N}_P$; units are $\mu$K arcmin. (b) Changing $\theta_{FWHM}$; units are arcmin.