This is a practical introduction (1 week crash course tutorial) to weak gravitational lensing
aimed at finishing undergraduates/starting PhD students wanting
to do a research project on weak lensing (e.g. cosmic shear).
Reading is the least stimulating way to learn something. I think
it is essential to try doing parts of the derivations yourself
(at very least write it out for yourself in your own handwriting),
and try out writing codes.
Don't let yourself get stuck, but add
to a list of specific questions you want to understand later and
plough on. Hassle some lensing people about your questions
over the coming days/years.
Preparation
Get a computer account with access to a matlab license.
It doesn't matter whether it is Linux or Windows (no doubt fine on a Mac too).
Optional: fire up matlab and type "demo" at the prompt to play
with the demos (esp the 1d, 2d, plotting demos).
Optional: Read the
introductory article by Peter Schneider
especially focussing on Sections 2.3, 3.1, 3.3, 4.3 (first bit of 4.3 only - ignore time delays) and 6.2.
Optional: derive the lensing bend angle from GR.
Day 1: Shear
Derive the shear map for a point mass lens, from the GR
bend angle equation (try to think about it before looking at
a worked answer pdf).
Consider a circular source galaxy.
If gamma_1~0.2, gamma_2=0: use the Jacobian
to work out what the
circle would look like on the sky, after being lensed.
Repeat for gamma_1=0, gamma_2~0.2.
Repeat for gamma_1~-0.2, gamma_2~0.
Repeat for gamma_1=0, gamma_2~-0.2.
Consider a circular source galaxy sitting on the positive
x axis on the sky (e.g. theta_1=3 arcmin, theta_2=0)
Roughly what might the values of gamma_1 and gamma_2 be (assume
|gamma|~0.2).
Consider circular source galaxies at a range of different
positions on the sky. For each galaxy think through roughly what their gamma_1 value will be. Repeat for gamma_2. Sketch a map of gamma_1 values around
the center of the lens. Ditto gamma_2.
Optional: Use the shear map for a point mass lens to derive
the shear map for an arbitrary lens, in the thin lens approximation
(try to think about it before looking at a worked answer
pdf).
Day 2: The Singular Isothermal Sphere
Derive the bend angle for a circularly symmetric lens
(pdf).
Derive the shear for a circularly symmetric lens.
Read esp the first two paragraphs of section 3.3 of the introductory Schneider article about the Singular Isothermal Sphere (SIS) lens profile.
This is the simplest model for a galaxy or cluster of galaxies (after
a point mass!).
Derive the bend angle as a function of angular radius (theta)
for a SIS from rho, in terms of the angle theta and the
SIS velocity dispersion sigma_v
(this should give some clues: pdf).
Derive the shear as a function of theta for a SIS.
For sigma_v=500 km/s and D_ds/D_s=1 (distant source approximation)
use matlab to make a 1d plot of
shear versus angular radius (e.g. theta=0.1:0.5:10; in arcminutes).
Make a 2d shear map for a SIS:
Read and run the sisdemo matlab script (/import/cyan/codeshare/sarah/gravlens/sisdemo.m) up to line 21.
Uncomment the imagesc(...gamma_1...) line and run it.
Compare with the gamma_1 mapyou sketched yesterday.
Ditto gamma_2.
Optional: Play around changing the parameters and try to understand
how the code works. Note the use of subroutines. Look at all the
subroutines. Make a list of questions.
Optional: Run and understand the rest of the sisdemo script.
Especially get some ideas about estimating parameters (no need to worry about the
details for now - see day 5).
Day 3: Shear Power Spectrum
Derive the shear power spectrum in terms of the matter
power spectrum:
Think about the effect of having source galaxies scattered at
various redshifts. This
pdf describes how it is conventionally done
(I think there is potential for improving this, but that would take a
bit of time!).
Think about dividing up the universe into slices
(pdf).
Show that the shear power spectrum is equal to the convergence
power spectrum (pdf).
Derive the convergence power spectrum
in terms of the matter power spectrum (pdf).
Use the lens_cls_demo matlab code
(/import/cyan/codeshare/sarah/gravlens/lens_cls_demo.m)
to reproduce the SNAP 3 paper plots.
Choose your own sets of cosmological and source galaxy parameters
and make a new plot, labeling it etc.
Day 4: A more realistic lens model: NFW
Read section 6.2 of the introductory Schneider article about
the NFW lens. This is the profile of galaxies and clusters of
galaxies predicted by simple n-body simulations.
Run the nfwdemo matlab script
(/import/cyan/codeshare/sarah/gravlens/nfwdemo.m)
up to line 36 to see some pretty pictures.
Change some of the parameters at the top of the
code to see how it affects the plots (tip: type "figure" at the
matlab prompt so that the new plots come up in a new figure window;
you can then compare with your previous results by bringing up the
earlier figures.)
Understand better how nfwdemo is making a shear map
from the nfw parameters.
e.g. Find the equations for getting a shear map Osaka?-Wright & Brainerd
and match them up with those in the code.
Optional: Run and understand the rest of the nfwdemo script.
Day 5: Essentials of research
Use matlab to simulate some data from a straight
line and find the best fit straight line parameters
(pdf).
Read through nfwdemo and understand how it is finding the
best fit mass profile parameters and the errors on them.
Work out how you are going to organise your computer
files and hand written notes.
I strongly recommend you always have a text editor (e.g. emacs or matlab editor) open and
each day enter the date and all the useful commands you use plus
comments about what you are learning.
Each time you finish something, move all the text to the bottom
of another file. In this way you have a record of everything you have
ever done, so you can easily repeat things, check what you did on
a given day, check what your matlab scripts do, by reverse searching
on the file. Unfortunately the matlab editor doesnt seem to have a
reverse search facility so use emacs (or e.g. WinEdt) for that.
To understand better what I mean, please take a look at my files here
/cyan/codeshare/sarah/gravlens/mstuff/gravlens_allnotes.m
/cyan/codeshare/sarah/gravlens/mstuff/gravlens_notes.m
i.e. yes I actually do this!
Please e.g. reverse search my above notes files to learn things.
Plan now how you are going to backup your files.
I strongly recommend separating your essential files
(that you would cry if you lost) from your large data files (e.g.
that you could at worst reproduce from your essential files, or
that you could re-download from somewhere/someone).
I suggest making e.g. a "work" directory inside which you will keep
all your essential files (e.g. matlab files, notes files, latex files
e.g. /home/username/work),
and e.g. a "data" directory which could go on a disk which is not
backed up (e.g. /import/cyan/username/data e.g. for any plots or tables of
numbers that you make, and for any data you download).
Do your first backup:
rsync -azv -e ssh --delete /home/username/work/ /import/cyan/username/work/
ssh cyan
rsync -azv -e ssh --delete /cyan/username/data/ /cyan2/username/data/
This will protect you from any disks failing, or accidental overwrites.
Ideally also copy your essential files to another computer
or computer account in a different location.
Check the literature
to make sure your research project hasn't already been done!
Check for new cosmology papers daily or weekly. This should give you
ideas for future research projects and also you should be checking
whether someone else has just published the research you are working on!
Optional: Learn latex so you can write reports with equations
easily inserted, and research papers. Practice by latexing some of
the handwritten notes above and sending me the result(!)
Optional: Read Sivia book on statistics.
Day 6+: Further Reading
Read the Refregier ARAA review article
Work through my
(Catania lensing school Nov 2006)
lectures on cosmic shear tomography
and photometric redshifts
ppt
| pdf.
If you use the ppt version on slideshow mode then you can test your
understanding.
The advanced lensing article by Peter Schneider.
Other lensing review articles:
Munshi, Valageas, Van Waerbeke, Heavens
Mellier
Bartelmann & Schneider
Schneider, Ehlers & Falco book
Optional: Derive the errors on the shear power spectrum assuming
fluctuations are Gaussian.
Optional: derive the shear power spectrum using the halo model.
Your comments and corrections on the above are very welcome.
Thanks very much for helpful comments and corrections to
Keith Biner, Lisa Voigt, Eduardo Cypriano, Alex
Abate, David Sutton, Geraldine Marien.
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