The importance of using the correct kappa
It's important to use the correct scheme for combining kappas from different redshifts. When you choose a biased sample of lines of sight the result from the Keeton scheme varies significantly from the additive scheme. It's important that we use an approximation that is good enough for our purpose: 1% accuracy on time delay distances.
Fig 1. Reconstructions of (top) kappa_add and (middle) kappa_keeton compared with (bottom) Stefan's ray-traced kappa. This is shown for 1000 lines of sight that have between 80 and 87 (inclusive) galaxies within 45 arcseconds that have [i+z]/2 magnitudes between 18.5 and 24.5. This roughly corresponds to <N_45>*[2.0+/-0.1].
Fig 2. As figure one but shown for 1000 lines of sight that have between 38 and 45 (inclusive) galaxies within 45 arcseconds that have [i+z]/2 magnitudes between 18.5 and 24.5. This roughly corresponds to <N_45>*[1.0+/-0.1].
``[***
Bias in reconstructed kappa_add - raytraced kappa_hilbert.
When you assume mean kappa_reconstruction = 0, the bias in reconstruction-raytace is -0.00014 (this is the mean of stefans 10^8 lines of sight). There is a statistical error if you don't reconstruct infinite lines of sight, that goes as ~0.036/sqrt(N), although it's quite skewed until around N = 100.
Fig 3. A binned version of kappaReconstruction - kappaRay-trace vs kappaReconstruction. All of the bins are consistent with no bias - but clearly the reconstruction method doesn't do a good job for kappa greater than ~0.1.
Reconstruction Errors
Fig 4. Scatter plot of reconstructed kappa versus the 68 percent confidence interval on kappa_reconstruction. Green shows fixed halo mass, combined with a photoz error = 0.1(1+z). Blue shows fixed redshift, and stellar mass uncertainties given by a (log-gaussian) uncertainty of 0.15 dex multiplied with the GSMF of Fontana2006. Red shows photozerror = 0.1(1+z) and (log-gaussian) uncertainty of 0.45 dex multiplied with the GSMF - NOTE the stellar mass uncertainty is three times larger than the spec_z case.
Fig 5. A comparison of the reconstruction error with a SPEC-z (red points) versus the intrinsic error (black bars).]``