Paper Draft

The original questions:

• How much of the total external convergence in a time delay lens system comes from visible galaxies? How does this change as a function of magnitude cut?
• Where are the most important galaxies? Do the massive galaxies dominate, or is it the the galaxies nearest the optical axis, or some combination of these? How many galaxies account for 50%, and 95% of the convergence on a typical line of sight?
• Can the Hilbert (ray-traced) convergence be recovered by halo model lightcone reconstruction? With what scatter and bias? Which recipes are the dominant sources of these uncertainties?

Paper Outline:

Introduction: What is the problem - reconstructing kappa. Why is this problem relevant? Time delay distances. Magnification. Previous work on environments: Auger, Momcheva, others. Hilbert et al.'s raytracing plus Fassnacht & Koopmans density, Suyu et al's kappa pdf. Why this approach may be insufficient - information provided in future surveys. Questions asked in this paper.

Theoretical Background:

• The mass sheet degeneracy - how kappa is degenerate in strong lens models (and magnifications). Weak lensing by NFW halos.
• Accounting for mass along the line of sight. Approximations to full ray tracing: combining kappas. Example ray traced kappa map from Stefan.

Modeling External Convergence:

• Halos N-body sims provide mass framework. MS halo catalog. Truncated spherical NFW approx based on virial masses. Stellar mass. Halo map corresponding to ray-traced kappa map, for illustration.
• Small-scale structure Not all DM is in halo catalog. Smooth component approximation.
• Voids How voids produce a negative kappa - filled beam distances require halos and voids to provide density contrast relative to mean. How we correct for voids. Comments on reasonable, but unphysical nature of this approach, tests to follow.

Reconstructing kappa with perfect knowledge of halo mass and redshift Raytracing through n-body particles as 'truth'. Motivation for assuming perfect knowledge of halo mass and z = testing model of previous section. Deterministic prediction of kappa.

• Is truncation choice important?
• How deep and wide does a reconstruction need to go?
• What is the irreducible uncertainty on kappa from this halo model? (Bias and scatter) Can use the ideal reconstruction to ask:
• Where does the kappa come from? Close to the LoS? Bright objects? How many objects are important?
• What error is made by considering subsets of available objects?

Reconstructing kappa with imperfect knowledge of halo mass and redshift Consider three sources of scatter in the halo model: zphot, Mstar|zphot, and Mhalo|Mstar,zphot. Mechanism for including scatter in model: draw samples from PDFs. Result is not one predicted kappa, but one P(kappa) for each lightcone. This is what is required by single-lens analyses. We can draw samples from this PDF and ask:

• How is the kappa inference offset and blurred by uncertainty on redshift, Mstar and Mhalo?
• What is the effect of using the wrong Mstar-Mhalo relation?
• How sensitive is reconstruction to zphot and Mstar error, by comparison?

Impact on Time Delay Distances Folding the kappa uncertainties into time delay distance uncertainties. Toy model: same zl and zs, allows simple product of pdfs to obtain estimate of P(D).

• What is the expected bias and scatter (centroid position and width) of P(D) as a function of N lenses?
• How does reconstruction compare with P(D) derived using P(kappa|N45) for each lightcone?
• How does reconstruction compare with P(D) derived assuming kappa=0 for each lightcone?

Discussion

• Why reconstruction works or doesn't work
• Major sources of uncertainty
• Ideas for improving the procedure