First measurement of the cross-correlation of CMB lensing and galaxy lensing

We measure the cross-correlation of cosmic microwave background (CMB) lensing convergence maps derived from Atacama Cosmology Telescope data with galaxy lensing convergence maps as measured by the Canada-France-Hawaii Telescope Stripe 82 Survey. The CMB-galaxy lensing cross power spectrum is measured for the first time with a significance of $4.2\sigma$, which corresponds to a 12% constraint on the amplitude of density fluctuations at redshifts $\sim 0.9$. With upcoming improved lensing data, this novel type of measurement will become a powerful cosmological probe, providing a precise measurement of the mass distribution at intermediate redshifts and serving as a calibrator for systematic biases in weak lensing measurements.

Figure 1

The lensing kernel $W^{{\kappa }_{\mathrm{gal}}}$ (solid) for the CS82 redshift distribution of source galaxies [as given in Eq. (6)] and normalized to a unit maximum. For comparison, the kernel for CMB lensing [Eq. (3)] is shown as dashed, also normalized to a unit maximum.

Figure 2

The overlapping sky coverage of the ACT and CS82 data used in this work. Regions excluded as part of the CS82 mask are shown in grey, while unmasked regions, totaling 121 square degrees in area, are shown in white.

Figure 3

Redshift distribution of CS82 source galaxies. Left: Redshift distribution for a matched sample of galaxies from the COSMOS survey. The blue solid line indicates our fit to the COSMOS matched sample. Right: We test how the amplitude of the theoretical lensing-lensing cross power spectrum changes when we vary the peak of the $dn/dz$ (red, dashed lines) and the high-redshift tail of the distribution (orange, dash-dotted lines). These variations in the $dn/dz$ lead to changes of order 10%–20% in the amplitude of the theoretical lensing-lensing cross power spectrum. Accurate estimates of $dn/dz$ will be crucial for this kind of cross-correlation in the future.

Figure 4

The ratio of the auto power spectrum of simulated lensing convergence maps measured using our analysis pipeline $C_{\ell }^{\text{meas}}$ to the true, input auto spectrum $C_{\ell }^{\text{true}}$. The analysis pipeline described in Sec. 4 properly recovers the input spectrum from simulations.

Figure 5

The CMB lensing-galaxy lensing convergence cross power spectrum (red points), measured using ACT and CS82 data. Error bars are computed using Monte Carlo methods (see text), and the significance of the measurement is $4.2\sigma$. The solid and dashed black lines show the expected power spectra assuming the $\text{Planck}+\text{lensing}+\mathrm{WP}+\text{high}\mathrm{L}$ and $\mathrm{WMAP}9+\mathrm{eCMB}$ cosmological models, respectively. The theoretical spectra shown correspond to $A=1$, and relative to these models, the best-fit amplitudes to our data are $A^{\text{Planck}}=0.78\pm 0.18$ and $A^{\mathrm{WMAP}}=0.92\pm 0.22$.

Figure 6

Two successful null tests, both consistent with zero. Top: The mean correlation between 500 randomized galaxy lensing maps and the true ACT data. Bottom: The mean correlation between the true CS82 data and 58 ACT “shuffled” maps, constructed by shifting the data in intervals of $0.75°$ along the right ascension direction. The probabilities to exceed the measured ${\chi }^2$ for these tests are 63% and 34%, respectively. Note that the scaling of the $y$-axis here is an order of magnitude smaller than the $y$-axis of Fig. 5.

Figure 7

Two tests of the CS82 galaxy lensing data, both consistent with zero. The data points show the correlations between the ACT data and the CS82 B-mode ellipticity data (top) and the PSF ellipticity data (bottom). The probabilities to exceed the measured ${\chi }^2$ for these tests are 55% and 13%, respectively. Note that the quantity ${\ell }^2{C}_{\ell }$ is plotted in the bottom panel in order to increase the dynamic range of the plot.