Dark Energy Constraints from Weak Lensing Cross-Correlation Cosmography

TL;DR

The paper develops cross-correlation cosmography for weak lensing, using foreground mass templates from photometric redshifts to derive a geometry-driven observable that isolates angular-diameter-distance ratios. By cross-correlating background galaxy shapes with these templates, the method eliminates dependence on the uncertain mass power spectrum and reduces susceptibility to certain systematics, while leveraging non-linear mass structures. A Fisher-matrix formalism is constructed, incorporating nuisance cross-correlations and priors, and applied to forecast dark energy constraints for SNAP, LSST, and CFHLS, yielding projected uncertainties on $w_0$ and $w_a$ that can be competitive with other probes. Achieving the promised precision requires unprecedented control of distortion calibration and photometric redshift biases, motivating space-based observations and extensive spectroscopic calibration campaigns; with these advances, cross-correlation cosmography could provide powerful, near-future insights into the dark-energy equation of state.

Abstract

We present a method to implement the idea of Jain & Taylor to constrain cosmological parameters with weak gravitational lensing. Photometric redshift information on foreground galaxies is used to produce templates of the mass structure at foreground slices z_\ell, and the predicted distortion field is cross-correlated with the measured shapes of sources at redshift z_s. The variation of the cross-correlation with z_s depends purely on ratios of angular diameter distances. We propose a formalism for such an analysis that makes use of all foreground-background redshift pairs, and derive the Fisher uncertainties on the dark energy parameters that would result from such a survey. Surveys from the proposed SNAP satellite or the LSST observatory could constrain the dark energy equation of state to 0.01 f_sky^{-1/2} in w_0 and 0.035 f_sky^{-1/2} in w_a after application of a practical prior on Ω_m. Advantages of this method over power-spectrum measurements are that it is unaffected by residual PSF distortions, is not limited by sample-variance, and can use non-linear mass structures to constrain cosmology. The signal is, however, very small, amounting to a change of a few parts in 10^3 of the lensing distortion. The calibration of lensing distortion must be independent of redshift to comparable levels, and photometric redshifts must be similarly free of bias. Both of these tasks require substantial advance over the present state of the art, but we discuss how such accurate calibrations might be achieved using internal consistency tests. Elimination of redshift bias would require spectroscopic redshifts of 10^4-10^5 high redshift galaxies.

Figures (2)

• Figure 1: The top panel shows the fractional change in the geometric factor $g_{\ell s}=(\chi_s-\chi_\ell)/\chi_s$ when we shift from a pure $\Lambda$CDM Universe to one with $w_a=0.2$; this should be discernible at $1\sigma$ significance in the SNAP survey. The horizontal axis is $z_s$ and each line corresponds to a different $z_\ell$. The triangle at the end of each line marks $z_\ell$. Because the mass normalization in each lens slice is free to vary, vertical shifts of each line carry no cosmological information, so we align them all to be unity at $z_\ell$. The cosmological information is carried in the departures of each line from horizontal; these departures are small, amounting to only a few parts in $10^{-3}$ change in the induced background distortion. The smallness of this signal implies that the calibration of the distortion measurement and the source redshifts must be accurate to roughly a part in $10^3$. The lower panel plots the assumed source redshift distribution (dotted line) and the expected distortion variance per unit redshift (solid line) using estimated non-linear power spectra. The dashed line shows the relative contribution of different lens planes to the constraint on $w_a$.
• Figure 2: Fisher uncertainty ellipses for dark energy parameters derived from three candidate weak lensing surveys are plotted. All ellipses are 68% confidence two-dimensional regions ($\Delta\chi^2=2.3$) after application of a Gaussian prior on $\Omega_m$ with $\sigma=0.03$ and marginalization over $\Omega_m$. The solid, dashed, and dotted contours are for SNAP, LSST, and CFHLS surveys, respectively. Survey parameters are listed in Table \ref{['surveys']} and a flat Universe is assumed. Fiducial models for both $\Lambda$CDM and a supergravity-inspired model are plotted as stars, and the shaded regions are the expected constraints from the SNAP Type Ia supernova measurement plus $\Omega_m$ prior (E. Linder, private communication). Unlike the weak lensing contours, the SN contours include the estimated effects of the dominant systematic errors.