# Lecture Notes

**[Lucas' Notes for Lecture 7](/SummerCourseRiemannSphere2.pdf)**

# Sphere Pixelation

Somewhat related to the contents of this lecture and our discussions afterwards, "good" pixelations of the sphere are an ongoing topic of math/numerical research with applications to astrophysics data analysis. Pretty pictures:

![This is the most popular current scheme, developed at NASA/JPL (http://healpix.jpl.nasa.gov/)](/ClassJuly20/healpixGridRefinement.jpg)

![This is an icosahedron scheme by Max Tegmark (http://space.mit.edu/home/tegmark/icosahedron.html)](/icosahedron.gif)

One of the most important metrics calculated from CMB (cosmic microwave background) skymaps (like those from WMAP, or this [super pretty new map](http://4.bp.blogspot.com/_GVA115I1I8Y/TDLWxv-FiDI/AAAAAAAAAww/P_cC-oE67uQ/s1600/PLANCK_FSM_03_Black.jpg) from Plank) is the power spectrum: This is essentially the Fourier transform of the surface of the sphere using spherical harmonics (indexed by $\ell$ instead of $n$; familiar from physical chemistry and the quantum mechanics of the hydrogen atom). The power spectrum of the skymap, which basically describes the relative populations of lumps in the skymap, is the established meeting ground between early universe theorists (who calculate what the power spectrum would look like under different cosmological models, such as inflationary, open/closed/flat, cosmic strings, quantum loop gravity, cold-dark-matter-dominated, and other pseudo-nonsense wink-wink) and observational scientists who try to constrain the spectrum with better and better measurements. 

![Power spectrum image from NASA/WMAP team](/ClassJuly20cmbpower.jpg)

Just recently the number crunching of large observational datasets has become an obstacle to progress: we need better algorithms ([like the "the fastest Boltzmann code in the west"](http://arxiv.org/abs/astro-ph/9603033)) for dealing with these skymaps, particularly when trying to measure "non-Gaussianities" in the distribution of bumps. I have a feeling that some of the techniques we use in this course are applicable!

More background (heh):

- http://backreaction.blogspot.com/2007/12/cmb-power-spectrum.html
- http://space.mit.edu/home/tegmark/cmb/pipeline.html
- http://space.mit.edu/home/tegmark/strategy.html
- http://space.mit.edu/home/tegmark/karhunen.html
- http://www.astro.ucla.edu/~wright/cosmo_01.htm