Phil, you have push authority, so you could even do this -- I am assigning to you. Hogg
Comments on Hogg's "Weak Lensing" data analysis problem outline
In principle the positions of the lensed sources contain information about
the mass distribution lying in front of them. In the z direction, the shear
applied to the galaxy shape gets stronger the more distant the source is
(enabling "tomographic" inference of the mass map in 3D). In the plane of the
sky, sources appear deflected away from the intervening mass concentrations,
while fainter sources are magnified into view. Whether this magnification
effect results in an apparent overdensity or underdensity of sources depends
on the slope of the source luminosity function. In any case, the shear map is
not only sourced by a three-dimensional mass map, it looks different depending
on how far way the sources are! These considerations imply that it is the 3D
mass map that is the correct thing to marginalise over, with the predicted
shear of each source calculable from it.
The comparison with CMB maps is interesting. You write: "In the case of weak
lensing, there is no way to hide or implicitly marginalize, because the
cosmological parameters interact with the map and the map interacts with the
data with so many nonlinearities and complications." But I think its true that
the phases of the CMB map are conceptually equivalent to the actual large
scale structures causing cosmic shear: both influence the data heavily, but
neither are interesting to cosmologists. And in fact the standard cosmic shear
analysis involves taking catalogs of noisy galaxy shapes, and forming
correlation functions and so on that can be compared directly with similar
functions predicted from the cosmological parameters - this is, I think, the
same implicit marginalisation that the CMB analysts carry out. It just maybe
that the approximations made by the WL analysts are not as accurate or
well-motivated as those made by the CMB analysts.
To continue on the CMB theme: CMB telescopes do not make maps of temperature
fluctuations, they return samples of the local intensity field taken in time
series. A map has to be made from these raw data, and this map is then
used as input to a power spectrum estimation routine (which may or may not be
inferential). And then these "Cl's" are then fitted with predictions given
cosmological parameters. One could argue that WL analysis is easier than
this, because no shear, or mass, map is ever made. However, is it possible
that one could improve the WL analysis accuracy by including information about
the mass map before doing the marginalisation explicitly, rather than ignoring
what we know about gravitational collapse and doing it implicitly?
The best source of information we have about galaxy shapes is all the
imaging of the sky that has been taken to date. I write this because it may
not be the case that the bst source of information about the LSST galaxy
shapes is the LSST survey data: surely we have learned a lot about galaxy
shapes from the HST surveys that have been carried out? What needs to be done
with care is joint analysis of ground based and space-based data on the same
galaxies, in order to get a complete picture. HST is sensitive to small
features (star forming regions), whereas deep ground-based imaging can be more
sensitive to the low surface brightness outer regions. It would be good to use
all this information to inform the PDF of galaxy shape parameters, in addition
to any new survey data we take.
The best source of additional information we have about the 3D mass map of
the Universe is probably the many cosmological simulations that have been made
of it. Suppose we were going to try and reconstruct the 3D mass map, and then
marginalise over its parameters: we can imagine using large ensembles of
numerical simulations to generate large numbers of sample mass maps, and
distill their statistical properties into a prior PDF. Maximum Entropy is
unlikely to be a good prior: mass clusters by gravity, no matter what the
cosmology is, so that the pixels of a mass map are going to be correlated. The
pixel histogram will not be Gaussian: I would expect it to be skewed, and
peakier than Gaussian. The sparsity of this distribution would be something
easy to learn from a simulation suite - Jean Luc Stark may have investigated
this already, following his interest in sparsity in image reconstruction.
I am interested in this project but it is a giant one. (It's name is
"cosmology with weak lensing.") There is a student of David Bacon's at
Portsmouth that is trying to improve on my MaxEnt mass mapping code by
including power spectrum information in the prior. I think this is a good
idea, but he might fall into the trap of assuming Gaussian statistics for a
non-Gaussian field. Still, he is a possible implementer to work with, he seems
good. Rafal somebody. Joe Zuntz is also someone who might be trickable into
working on all this stuff.
Phil, you have push authority, so you could even do this -- I am assigning to you. Hogg
Comments on Hogg's "Weak Lensing" data analysis problem outline