A measurement of gravitational lensing of the microwave background using South Pole Telescope data

TL;DR

This work presents a high-significance detection of gravitational lensing in the CMB using South Pole Telescope data from 2008–2009 and measures the lensing potential power spectrum. Employing quadratic estimators, the authors reconstruct deflection maps and evaluate the lensing signal with both all-$l$ and $l$-split approaches, rigorously testing for systematics through curl estimators and extensive simulations. They report a lensing amplitude $A_{ m lens}$ around $0.90$ (marginalized over LCDM) and demonstrate substantial improvements in cosmological parameters when the lensing information is combined with WMAP7 data, notably tightening constraints on curvature, $\sigma_8$, neutrino masses, and the dark energy equation of state $w$. The results affirm LCDM predictions for lensing and illustrate CMB lensing as a powerful probe of growth and geometry, with future gains expected from larger surveys and polarization data.

Abstract

We use South Pole Telescope data from 2008 and 2009 to detect the non-Gaussian signature in the cosmic microwave background (CMB) produced by gravitational lensing and to measure the power spectrum of the projected gravitational potential. We constrain the ratio of the measured amplitude of the lensing signal to that expected in a fiducial LCDM cosmological model to be 0.86 +/- 0.16, with no lensing disfavored at 6.3 sigma. Marginalizing over LCDM cosmological models allowed by the WMAP7 results in a measurement of A_lens=0.90+/-0.19, indicating that the amplitude of matter fluctuations over the redshift range 0.5 <~ z <~ 5 probed by CMB lensing is in good agreement with predictions. We present the results of several consistency checks. These include a clear detection of the lensing signature in CMB maps filtered to have no overlap in Fourier space, as well as a "curl" diagnostic that is consistent with the signal expected for LCDM. We perform a detailed study of bias in the measurement due to noise, foregrounds, and other effects and determine that these contributions are relatively small compared to the statistical uncertainty in the measurement. We combine this lensing measurement with results from WMAP7 to improve constraints on cosmological parameters when compared to those from WMAP7 alone: we find a factor of 3.9 improvement in the measurement of the spatial curvature of the Universe, Omega_k=-0.0014+/-0.0172; a 10% improvement in the amplitude of matter fluctuations within LCDM, sigma_8=0.810+/ 0.026; and a 5% improvement in the dark energy equation of state, w=-1.04+/-0.40. When compared with the measurement of w provided by the combination of WMAP7 and external constraints on the Hubble parameter, the addition of the lensing data improve the measurement of w by 15% to give w=-1.087+/-0.096.

Figures (12)

• Figure 1: Impact of apodization: (left) reconstruction of lensing deflection for one of the SPT fields (ra5h30dec-55); (middle) mean estimated deflection for 100 simulations, indicating the mean apodization feature; (right) resulting estimate of the deflection in the SPT field after subtracting the estimated apodization feature. All maps have the same greyscale ($\pm$0.005).
• Figure 2: Theoretical lensing reconstruction noise curves $A_{\mathbf{L}}$ (Eq. \ref{['eq:weighting']}) for SPT map filtering and noise levels, showing slices along the $L_x$-direction (dotted), along the 45-degree line (dashed) and in the $L_y$-direction (dot-dashed). Due to the anisotropic noise statistics and filtering, the lensing data are $\sim 4$ times noisier in the $L_y$-direction than in the $L_x$-direction. The azimuthally-averaged mean theoretical noise curve, given by the thick solid line, is the noise bias which must be subtracted from the lensing power spectrum estimate. Note that the variance of the bandpowers in the lensing power spectrum estimate will not rise as quickly with $L$, as the number of lensing modes to average over is $\propto L$. The fiducial lensing power spectrum $L^4 C_L^{\phi\phi}$ is given by the grey line.
• Figure 3: Individual all-$l$ raw power spectra for each field for the main lensing signal (the divergence; bottom, black, points in each panel) and the curl component (top, red, points in each panel). Curves show the results of the lensed and unlensed simulations; i.e., the lower curves show the Gaussian noise biases estimated from simulations and the upper curves show the sum of the noise bias and the expected lensing signal in our fiducial cosmological model. The extra ticks on the error bars show the impact of the correlated covariance arising from the uncertainty in the Gaussian noise bias subtraction.
• Figure 4: Ratio of the power excess measured in the SPT data compared to the power excess from lensed simulations. The left panel uses all-$l$ maps, while the right panel uses the $l$-split method of using disjoint annuli in $l$ space to avoid a noise bias. The horizontal lines indicate lensing amplitudes of 0 and 1. Each field is shown as a different color, offset in $L$ for clarity: the ra5h30dec-55 field in green; the ra23h30dec-55 field in blue; the ra3h30dec-60 field in magenta; and the ra21hdec-60 field in orange. The heavy black points show the combined best-fit estimate of the lensing amplitude. Note the expanded scale in the right panel; the $l$-split method has less statistical power. No lensing is excluded at 6.3$\,\sigma$ (left) and $3.9\,\sigma$ (right)
• Figure 5: Change in $2 \ln L$ compared to best fit for the SPT lensing power spectrum, when the fiducial lensing power spectrum is multiplied by a lensing scale factor ${A}_{\rm lens}^{0}$. A strong detection is evidenced for both the less-sensitive $l$-split method (blue, long dashed line) and the more-sensitive all-$l$ technique (black, solid line). Using the curl signal in the data (red, short dashed), lensing is also tentatively detected.