Paper Figures and Conclusions
Here are the figures that are in the current draft of the paper.
##Figures:
Reconstructions with perfect knowledge of halo mass and redshift
Figure 1. The standard deviation in kappa_reconstruction minus kappa_raytrace as a function of halo truncation radius in virial radii, using the truncated NFW profile of Baultz, Marshall and Oguri. In our models we truncate our halos at 5R_200
Figure 2. kappa_reconstruction minus kappa_raytrace as a function of kappa_reconstruction, for 10000 reconstructed lines of sight. The points are put into bins of width 0.01 in kappa_reconstruction, except the last bin containing all reconstructions with kappa_reconstruction > 0.14. The errorbars show the population standard deviation for each bin. Since all of the errorbars are consistant with zero, there is no evidence for kappa_reconstruction being a biased estimator of kappa_raytrace
Figure 3. The 16, 50 and 84% confidence intervals on kappa_reconstruction minus kappa_raytrace as a function of the limiting i band depth of the halo reconstruction - there is no significant improvement from including small halos with magnitudes less than i = 24
Figure 4. The 16, 50 and 84% confidence intervals on kappa_reconstruction minus kappa_raytrace as a function of the limiting radius (in arcminutes) beyond which halos are not reconstructed. The majority of the convergence comes from halos centred within an arcminute of the line of sight
Reconstructions with imperfect knowledge of halo mass and redshift
Figure 5. 68 % and 95 % contours of the joint distribution P(kappa_raytrace, Sum kappa_halos) given a mock reconstruction of our 40000 calibration lines of sight. Sum kappa_halos is the median of P(Sum kappa_halo) given a spectroscopic redshift for each halo. kappa_raytrace and Sum kappa_halos are strongly correlated, despite the blurring effect of uncertain halo masses.
Figure 6. Widths of P(kappa_ext) after applying our reconstruction of all halos down to i = 26 and within 5 arcminutes of the line of sight. Results for 10000 lines of sight are shown. Green is for a reconstruction given perfect knowledge of halo mass and redshift, blue is for a reconstruction given a spectroscopic redshift for every halo, and red is for a reconstruction with photometric redshifts only. The dashed black line shows the width of the global P(kappa_ext) distribution; the reconstruction process provides roughly a factor of 2 improvement for the majority of lines of sight. Spectroscopic redshifts improve the reconstruction, but at signifcant observational cost.
Figure 7. Widths of P(kappa_ext) after a photometric reconstruction of all halos within different fields. The different field sizes and depths are given in the legend. As the field becomes smaller or shallower, fewer halos are being reconstructed and so the width of P(kappa_ext) grows.
In this work we have investigated a simple halo model prescription for reconstructing all the mass along a line of sight to an intermediate redshift source. We have used the raytraced lensing convergence along lines of sight through the Millenium Simulation to test this approach, and to calibrate estimates of the total convergence along a line of sight to an observed distant galaxy made by summing the convergences due to each object in a photometric catalogue. Having found that the reconstruction process is effective given perfect knowledge of halo mass and redshift, we investigated the effects of reasonable uncertainties in the stellar mass and redshift of each halo, and propogated these uncertainties into a P(kappa_ext) for each line of sight. We draw the following conclusions:
• Our model uses a truncated spherical NFW profile for each dark matter halo and neglects voids, but despite the model’s simplicity the reconstructed P(Sum kappa_halo) is a good tracer of kappa_ext. We found that with perfect knowledge of the halo mass and redshift (from the Millenium Simulation catalogs), the reconstruction gives an unbiased estimate of P(kappa_ext) for a single line of sight that is almost almost a factor of 2 less broad than the global P(kappa_ext). • For the most overdense lines of sight, the reconstruction produces a very broad PDF, but since the reconstruction can be performed before follow-up time is invested it will be possible to discard the most uncertain lines of sight and prevent the waste of follow-up time. • With complete spectroscopic redshift coverage and just an empirical stellar mass to halo mass relation, we find that the median of P(Sum kappa_halo) is a useful indicator for generating an estimate of P(·ext) from the ensemble of simulated lines of sight. The resulting PDF tends to be around 10% broader than it would have been given perfect knowledge of both halo mass and redshift; given only photometric redshifts (which in turn give rise to much less certain stellar masss estimates) causes another ∼10% increase to the width of P(kappa_ext).
• It is very rare for halos further than 2 arcminutes to make a significant contribution to kappa_ext. We also found that including halos whose host galaxy is less luminous than i = 24 does not significantly improve our reconstruction proceedure. A photometric survey to this depth of a 4×4 arcminute patch around the lens would approach the limiting uncertainties of our simple reconstruction recipe, and yield a P(kappa_xt) that has, on average, a width of 0.0163 – 50% less broad than the global P(kappa_ext).
• We find that the lines of sight with the sharpest P(kappa_ext) are typically under-dense. With a photometric reconstruction of all lens fields, and following up only the lenses with the most constraining P(kappa_ext), the width of the dominant statistical uncertainty in time-delay cosmography can be halved, whilst at the same time decreasing any potential for a systematic error due to lenses being biased in kappa_ext.
Whilst halos contribute a positive convergence to ·ext, voids contribute a negative convergence.When a raybundle passes through a region of space that is less dense than rho_cr(z) the rays are de-focused.Whilst the full raytraced solution takes into account the effect of voids, our reconstruction cannot – the halo catalogue does not include voids. In principle the absence of a halo could be taken to infer a void, but the under-density of such a void is hard to infer. Neglecting voids would lead to a heavily biased estimate of kappa_ext. We account for the voids statistically; we reconstruct the kappa_ext contribution of halos for an ensemble of calibration sightlines and form the joint distribution P(kappa_raytrace, P(Sum kappa_halo). The calibration can then be accounted for by marginalizing over all lines of sight with a similar P(Sum kappa_halo). We will refer to this calibration proceedure as the ”void correction”, although it also accounts for all of the physics that our simple spherical halo prescription neglects in a statistical manner.