The Problem of the Smooth Component

The halos in the catalogs (either simulated or reconstructed from photometric galaxy catalogs) do not contain all the mass in the Universe. The rest is presumably distributed in smaller clumps, along filaments and not in voids, but in any case hard to predict. Can this missing matter be modelled as a uniform density distribution, the "smooth component"? What does this smooth component do to the final observed kappa? Is it positive or negative?

What do we need to do in order to predict kappa accurately? The Keeton approximation is obviously wrong; kappakeeton always adds! Is there a way to recover from this, such that the curves of growth asymptote to a constant value, and so that kappa_hilbert is accurately recovered in the MS fields?

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The Mass Budget

The mass budget is an important concept: however we assign mass, the global mean density at redshift z must end up being Omegam(z)*rhocrit(z). But the local mean density can vary: it's this that might be related to the number and mass of halos present in the locality. And then, we need to convert density to over/under-density, so that distances calculated in the uniform density assumption (FRW dust model) are still approximately right.

The amount of matter in between galaxies is very hard to measure. So how do we know if our line of sight is under- or over-dense, and by how much? Following Chris and Leon's work, we can compare reconstructed lens lightcones with reconstructed non-lens lightcones in the simulations (to check we are using an accurate recipe) and then again with observed fields. Note that in each case we need to assume an Omegam to compare the mass in halos with the total mass in the cone (which on average should be \int Omegam(z)*rhocrit(z)*dv), so checking against simulations with different cosmological parameters could be important. We will also need a large number of uncorrelated non-lens lightcones to make sure we not limited by sample variance.

As usual, the hypothesis is that reconstructing in 3 dimensions will allow a more accurate kappa estimate - so the budget balancing would need to be done in a number of thin redshift bins to capture the detected structures and the voids in between them. Note that we will be relating both halo number and mass to an estimate of the density of the smooth component in each slice of the cone, rather than just number - and also that the result will be a procedure that can be applied to any dataset, simulated or observed, as long as many lightcones are sampled.

Where to put the smooth component?

If we want to balance the mass budget by adding matter to the reconstructed lightcones there are (at least) two places we can put it: a) In the outskirts of halos, to account for the unresolved mass not found by the halo finder. b) In between halos, according to some statistical prescription.

Option a) is actually not crazy: it corresponds to an assumption of "ultimate clustering", that all matter not associated with halos can be represented as belonging to a halo, and that the voids are literally empty. It's possible that by choosing the halo truncation radius
wisely, we could recover the correct convergence.

Option b) seems more likely to be needed, but it's difficult to inform it. Presumably, there is more matter near a huge halo, than floating around in a void. But how much more? and where statistically is it? Note that scheme b) can be seen as a generalisation of a) - sticking more mass to the outer parts of halos is just one statistical prescription.

Given these desiderata, our basic plan for laying in a smooth component is to slice up the lightcone in redshift, snap the halos to this grid (which will speed up distance calculations by allowing fast look-up instead of slow re-calculation), and then compute densities in the resulting disks based on our understanding of density in halos compared to the local and global mean, extracted from a large number of non-lens lightcones. The required smooth component will then be added in as a sheet of mass with given surface density and zero shear, that is propagated via the Keeton formula.

Proposed Recipe

0. Derive the PDF for the smooth component density rhoS at redshift z given the density in halos there, rhoH, from N non-lens lightcones: Pr(rhoS|rhoH,z)

1. Start with an empty universe.

2. Add in the halos in the catalog, with truncation radius Rt = a*Rvir, at their given sky positions and redshift slices

3. In each redshift slice:

   • Compute density in halos rhoH
   • Draw a smooth density rhoS from Pr(rhoS|rhoH,z)
   • Add in a mass sheet with the corresponding surface density SigmaS
   • Subtract off the global mean surface density

4. Compute kappa for lightcone from the sum of each halo and sheet's contributions.

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