Measurement of a Cosmographic Distance Ratio with Galaxy and Cosmic Microwave Background Lensing

We measure the gravitational lensing shear signal around dark matter halos hosting constant mass galaxies using light sources at $z\sim 1$ (background galaxies) and at the surface of last scattering at $z\sim 1100$ (the cosmic microwave background). The galaxy shear measurement uses data from the CFHTLenS survey, and the microwave background shear measurement uses data from the Planck satellite. The ratio of shears from these cross-correlations provides a purely geometric distance measurement across the longest possible cosmological lever arm. This is because the matter distribution around the halos, including uncertainties in galaxy bias and systematic errors such as miscentering, cancels in the ratio for halos in thin redshift slices. We measure this distance ratio in three different redshift slices of the constant mass (CMASS) sample and combine them to obtain a 17% measurement of the distance ratio, $r=0.390_{-0.062}^{+0.070}$, at an effective redshift of $z=0.53$. This is consistent with the predicted ratio from the Planck best-fit cold dark matter model with a cosmological constant cosmology of $r=0.419$.

Figure 1

Null test of optical lensing signal. The $R\sim 40h^{-1}\mathrm{Mpc}$ bins are consistently smaller than zero for all the redshift slices, and thus we do not use them. The $p$ value, or $P({\chi }^2>{\chi }_{\mathrm{obs}}^2)$, where $P$ is the ${\chi }^2$ distribution with degree of freedom of the 12 $R\lesssim 30h^{-1}\mathrm{Mpc}$ bins over the redshift slices, is 0.82, which is within a 95% C.L. region. We use these 12 data points for the distance ratio analysis.

Figure 2

Theory expectation of CMB tangential shear using an input $C_l^{\kappa {\delta }_g}$ curve generated with a linear matter power spectrum with a linear galaxy bias of 2. We do not use radial bins that have a mismatch between black crosses and red x’s (shaded regions) as that would make the optical and CMB analyses inconsistent. The green points show the shear from the data, and where those points deviate from the theory at small scales is where there is sensitivity to the one-halo term from the CMASS galaxy halos themselves.

Figure 3

CMB and optical shear around CMASS halos in the redshift range $0.43<z<0.7$. The dashed blue curve shows a theory fit to the optical data, which includes both the one-halo and two-halo terms. This red curve is given by scaling up the blue curve to the CMB source redshift.

Figure 4

Measured distance ratio for each radial bin and redshift slice of CMASS galaxies. Here the error bars are derived by Monte Carlo calculations of the covariance matrices for optical and CMB measurements, taking the ratio for each realization, and showing the 68% C.L. region around the mean ratio. The dashed line and error band show $r=0.390_{-0.062}^{+0.070}$, the best-fit value coadding all the radial bins and simultaneously fitting to the three redshift slices.

Figure 5

Comparison of the measured distance ratio with that predicted from different cosmological models. The black solid and dashed curves show the ratio for the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ and $w\mathrm{CDM}$ models, respectively, from the Planck $\mathrm{TT}+\text{lowP spectra}$ [45]. The thin solid curves show deviations from the best-fit Planck $\mathrm{\Lambda }\mathrm{CDM}$ model, as indicated.