Lightcone PDFs
We want to estimate the effect that the uncertain convergence has on the time delay distance measurement, and so we need to make a (prior) PDF for kappa that we can marginalise over. We do this by a Monte Carlo method: we make lots of individual realisations for each lightcone, drawing galaxy masses from various scatter-full scaling relations.
We replace the MS stellar mass, with one drawn from the Behroozi Mstar-Mhalo relation for central galaxies. This gives something that is a) more realistic than the SAM-generated MS Mstars, and b) something that we should be able to recover accurately (ie it gives us a functional test).
First, we take the true halo mass from the catalogues, and generate a stellar mass, according to the Behroozi relation (which has scatter tabulated in his paper). This stellar mass is then given an 'observed' stellar mass which is drawn from a log(Gaussian) centred on the true stellar mass, and width 0.15 dex (as appropriate for galaxies with known redshift). These observed stellar masses are the starting point for our kappa PDF estimation.
Fig. 1a. The Behroozi relation, Pr(Mstar|Mhalo). Lines mark the mean, and then enclose 68% and 99% of the conditional probability. We use this to simulate observed stellar masses.
We now invert the Monte Carlo process to generate a plausible Pr(kappa|data).
From the observed stellar masses, we draw a halo mass, using the Inverted Behroozi relation. This relation is derived straightforwardly from the MS catalog, essentially transforming PDFs by Monte Carlo integration. Using the true Mstar and Mhalos from the MS catalog, we can estimate P(Mhalo|Mstar)~ P(Mstar|Mhalo)*P(Mhalo). Note: P(Mhalo) is the halo mass function, and is naturally encoded in the MS catalogue. (Both the Behroozi and Inverted Behroozi relations are evaluated at the redshift of the object. The halo mass function is binned up into dz = 0.2, and the halo mass function in each redshift bin is fitted as a powerlaw.) The Inverted Behroozi relation is shown in Fig. 1b.
Fig. 1b. The Inverted Behroozi relation, Pr(Mhalo|Mstar). Lines mark the mean, and then enclose 68% and 99% of the conditional probability. This is derived by multiplying the P(Mstar|Mhalo) by P(Mh) (or in practice, drawing halos from an N-body simulation catalog) and renormalizing along the Mstar direction.
Using the method outlined above, we repeatedly draw stellar masses and halo masses, for the same lightcone, and generate a PDF for kappa. This kappa is then convolved with the typical reconstruction error on kappa_truth-kappa_hilbert (this accounts for the difference between n-body ray tracing, and our N-body NFW halo approach. This error is typically larger than the error from mass reconstruction, but it varies between lightcones - some are much easier to reconstruct than others. [more on this later] )
A randomly selected lightcone is shown in Fig 2. and its PDF is shown in Fig 3.
Fig. 2. The mass distribution near a randomly selected lightcone. Every halo in the the lightcone is shown, with their centres shown in red (centre defined as r < the NFW scale radius), and the outer regions shown in blue (out to r = 3*R_vir, where we truncate). Any halo that is individually contributing a kappa>0.0001 has a black dot at its centre.
Fig. 3. Pr(kappa) for the lightcone shown in Fig. 2, following the Monte Carlo method laid out above. This PDF assumes no redshift errors. [Interestingly, this PDF looks fairly similar to the pdf from lots of randomly selected lightcones. But it is definitely the right PDF!]
But was that lightcone a typical lightcone? How do the PDFs look for lots of lightcones? They are typically non-Gaussian, having a long tail of realizations where the mass of a nearby halo is overestimated, so we summarize the widths of the PDFs here as half of the 68% confidence level.
Fig. 3. The distribution of the width of the PDF Pr(kappa), for different lines of sight. The PDF width is defined as half of the interval containing the central 68% of the samples. One lightcone Pr(kappa) with a width of 0.17 is not shown here.
Clearly some lines of sight are harder to reconstruct than others. But are there observable properties, that can tell us if the LoS will be hard to reconstruct or not?
Fig. 4. The widths of the kappa PDFs vs the mean of the PDFs. For low kappa, the width is dominated by the intrinsic scatter term added for the scatter on kappa_hilbert-kappa_reconstruct_truth, whereas high kappa LoS show larger widths - they have more mass on them so they are harder to reconstruct.
Are the reconstructions biased?
Fig. 5. The difference between mean reconstructed kappa and kappa hilbert (both y-axes) plotted against kappa_hilbert (left x-axis) and mean_kappa_reconstruct (right x-axis). There is clearly a bias compared to kappa_hilbert (but this isn't an observable), but the reconstructed mean does not appear biased. The mean value of kappa_hilbert-mean_kappa_reconstruct is 0.0016 - perfectly reasonable given 1000 samples and kappa_reconstruct having typical widths of ~0.03.
I (TC) take the results of Fig 4. to mean that kappa_hilbert still includes some physics that we aren't including, but that the reconstruction process isn't fundamentally biased.
-
If we can get spectroscopic redshifts for those few halos, then we can reconstruct fairly tight pdfs. (But remember, not many halos are important, so we don't need too many spectra. And we can pre-estimate the important halos from their colours and photo-z.) Next, I'll soon be looking at how well we can do with photo-zs only.
-
Large kappa lines of sight are much harder to accurately reconstruct than small kappa LoS. If you are very close to a massive halo, then small uncertainties in its halo mass become large uncertainties in the reconstructed kappa.
These results do not include any redshift uncertainties so far - the Mstar is assumed measured to 0.15 dex. We also assumed that all halos with mass over 10^10 follow the Behroozi relation. (I then trimmed out the halos with Mstar <10^7, since I felt this was pushing the Behroozi relation too far, so it's really a limit on Mhalo of about 10^10.5 - halos below this were not reconstructed. [possible room for improvement?])
TC 2012-30-06
Note. We've altered the method for calculating the smooth component - we have set mean_kappa to 0, calculated from the kappa for 1000 different reconstructed lightcones.