Smooth component: solved?

We believe we've solved the problem of the smooth component!

The formalism is as follows:

0. Apply any selection cuts to the catalogue (e.g. a magnitude cut)
1. Bin the universe into redshift slices.
2. Calculate all the mass in each slice (neglecting any mass in halos that failed selection cuts). This requires a truncation radius to picked, which must be the same as the truncation radius used in the kappa reconstructions. (We looked at truncation last time)
3. Calculate kappa_slice according to (TotalMass_slice/area_slice)/sigmacrit_slice
4. Add kappa_slices according to the keeton prescription, to give kappa_smooth.
5. Subtract kappa_smooth from the total kappa_keeton of each line of sight.

The kappa_smooth is a function of truncation radius and magnitude cut. It can be calculated in ~20 seconds for a 1 square degree patch of Stefan's catalogues. You only need to calculate it once, unless you change the truncation radius or the selection cuts (i.e. magnitude). This correction doesn't account for the finite nature of our apperture, but if the apperture >> the truncation radius, this should be a small correction.

Fig 1. A 'smooth corrected' histogram of kappa_keeton, kappa_hilbert and the difference between the two (1000 sightlines). The equation at the top is the bias and Gaussian scatter on kappa_keeton-kappa_hilbert.

Fig 1a. As Figure 1, but with kappa_TC in place of kappa_keeton. kappa_TC is given by kappa_TC = (1-beta)*kappa_deflector. This is the simplest possible rescaling of kappa, and removes the shear dependence of kappa_keeton.

[recall that kappa_keeton is (kappa_TC+(beta^2-beta)(kappa_d^2-gamma_d^2))/((1-(betakappa_d)^2)-(betagamma_d)^2)]

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Can we minimize the scatter/bias on the kappa_TC-kappa_Hilbert?

Fig 2. The scatter on kappa_TC-kappa_Hilbert as a function of the Halo Truncation Radius. The solid line shows truncations as a multiple of the virial radius, whilst the dotted line shows truncations as a function of the NFW scale radius. Clearly it doesn't make a significant difference, so long as kappa_truncation >~ 2 R_Vir. All the following plots use a truncation at 3 R_Vir.

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Fig 3, The mean, median and 68% error bars on kappa_TC-kappa_Hilbert as a function of the reconstruction aperture (by which I mean, the maximum distance from the LoS to which the halos are reconstructed). Clearly it is rare for a halo to be important beyond 3 arminutes.

Note that for the median LoS we overestimate kappa, but the mean is fairly close to 0. This is because kappa_TC and kappa_hilbert show different skews. When reconstructing kappa symmetric errors are a poor approximation. It's likely that the different skews come from:

a) Halos that aren't reconstructed by kappa_TC (mass that isn't in the halo catalogue)

b) Incorrect placement of voids along the line of sight (which obviously don't show up in a catalogue of light!).

c) Scatter in the mass profiles of halos. (Either concentration, or deviations from NFW)

d) Offsets between light, and the halo centres.

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Fig 4 Reconstruction depth. A plot of kappa_TC-kappa_Hilbert as a function of the reconstruction depth (only reconstruct kappa if the galaxy is brighter than a threshold value).

Unsurprisingly, as you increase the magnitude limit, the reconstruction improves, but the catalogue stops at i=25, which makes it hard to draw conclusions from this plot. How deep do you need to reconstruct???

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A reminder of our 3 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?