We are going to look at the growth
of
structure in simulations with differing cosmological parameters. The
simulations are the "Hubble Volume Simulations" and more information
can be found at http://www.mpa-garching.mpg.de/Virgo/hubble.html.
I have grabbed the cluster catalogs from two simulations: LCDM and tauCDM.
A description of these files can be found here --
look under "Cluster Catalog Files". The cosmological parameters of the
sims are given by:

Repeat the calculation just for the most massive clusters -- those with velocity dispersions > 600 km/s. Comment on any differences you see from the first plot.

You'll need to make use of astropy's cosmology routines for this, in order to work out the comoving volume in each of your redshift bins. You can get this by grabbing the total volume out to the edge of each of your bins, then doing a np.diff to get the volume within the bin; this shows how to do it for the LCDM cosmology:

- LCDM: H0=70 km/s/Mpc, OM=0.3, OL=0.7
- tauCDM: H0=50 km/s/Mpc, OM=1.0, OL=0.0

Repeat the calculation just for the most massive clusters -- those with velocity dispersions > 600 km/s. Comment on any differences you see from the first plot.

You'll need to make use of astropy's cosmology routines for this, in order to work out the comoving volume in each of your redshift bins. You can get this by grabbing the total volume out to the edge of each of your bins, then doing a np.diff to get the volume within the bin; this shows how to do it for the LCDM cosmology:

from
astropy import cosmology

LCDM=cosmology.LambdaCDM(H0=70, Om0=0.3, Ode0=0.7)

zbins=np.arange(0,2,dz) # THIS SETS UP THE EDGES OF YOUR BINS (dz is your redshift binwidth)

bincenters = (zbins[:-1] + zbins[1:]) / 2.0 # THIS CALCULATES THE CENTER OF THE BIN

dvol=LCDM.differential_comoving_volume(zbins).value # THIS GETS VOL PER UNIT REDSHIFT PER UNIT SOLID ANGLE

dvol=4.*np.pi * dz * dvol # THIS GETS THE TOTAL VOLUME WITHIN EACH BIN OF REDSHIFT

LCDM=cosmology.LambdaCDM(H0=70, Om0=0.3, Ode0=0.7)

zbins=np.arange(0,2,dz) # THIS SETS UP THE EDGES OF YOUR BINS (dz is your redshift binwidth)

bincenters = (zbins[:-1] + zbins[1:]) / 2.0 # THIS CALCULATES THE CENTER OF THE BIN

dvol=LCDM.differential_comoving_volume(zbins).value # THIS GETS VOL PER UNIT REDSHIFT PER UNIT SOLID ANGLE

dvol=4.*np.pi * dz * dvol # THIS GETS THE TOTAL VOLUME WITHIN EACH BIN OF REDSHIFT

2. The Galaxy Two Point Correlation Function

There is a "chunk of the Virgo
consortium universe" available for you here.
The data come from a massive simulation of a cube of the universe
measuring 140 Mpc on a side. Details of the simulation and the galaxy
creation can be found at http://www.mpa-garching.mpg.de/Virgo/data_download.html
The data give the x, y, and z coordinates in Mpc and star formation
rate in solar masses per year of 8384 simulated galaxies.

We are going to define subsets of galaxies as "late types" (ie Sb/Sc spirals) and "early types" (ellipticals and S0's) based on their star formation rates. Let's say late types are things with SFR's > 1 Msun/yr, and early types are things w/ SFR's < 0.1 Msun/yr. (Does this definition make sense?)

Calculating the 2ptCF: First, install astroml if you haven't already (see below). Then to calculate the 2ptCF of a sample of N galaxies with x,y,z coordinates, do the following:

Installing astroml:

We are going to define subsets of galaxies as "late types" (ie Sb/Sc spirals) and "early types" (ellipticals and S0's) based on their star formation rates. Let's say late types are things with SFR's > 1 Msun/yr, and early types are things w/ SFR's < 0.1 Msun/yr. (Does this definition make sense?)

- Plot up x vs y for all galaxies to give a feel for what the data look like (remember to set the aspect ratio properly on your plots!). Then do the same thing just for E's and just for S's. Describe any differences and explain whether or not it makes sense.
- Now calculate the two point correlation function for all the galaxies (see below for how to do this). Do this for seperations between 1 and 10 Mpc, then fit a power law and derive the clustering length, r0. How do your numbers compare to the numbers discussed in class? Also describe qualitatively what this plot is telling you about how galaxies are distributed.
- Now do the same thing just looking at 2 point correlation function for spirals only, and again for ellipticals only. Plot them up, fit power laws, and derive the clustering lengths. Describe the differences you find in the clustering lengths for each sample. What does this tell you about the clustering of different types of galaxies in the universe, and does this make sense qualitatively based on what you know about galaxies?

Calculating the 2ptCF: First, install astroml if you haven't already (see below). Then to calculate the 2ptCF of a sample of N galaxies with x,y,z coordinates, do the following:

from astroML.correlation import
two_point

pos=np.array([x,y,z]).T # YOU WANT AN ARRAY WITH SHAPE (N,3), NOT (3,N)

rbins=np.arange(1,12,1) # BINS OF SEPARATION (IN MPC)

rbincenters = (rbins[:-1] + rbins[1:]) / 2.0

corrfunc=two_point(pos,rbins)

pos=np.array([x,y,z]).T # YOU WANT AN ARRAY WITH SHAPE (N,3), NOT (3,N)

rbins=np.arange(1,12,1) # BINS OF SEPARATION (IN MPC)

rbincenters = (rbins[:-1] + rbins[1:]) / 2.0

corrfunc=two_point(pos,rbins)

Installing astroml:

In a jupyter notebook, Chris Carr reports that you should be able to simply say

- !python -m pip install --upgrade pip (if you need to update pip)
- !pip install astroml

If you don't use jupyter notebooks, you
can just do the same thing from the command line of a shell window,
just omit the exclamation points from the beginning of the lines.

Remember our discussion about the
turnaround and collapse of initial
overdensities in matter-only universes. If you have a little
overdense
bubble of the universe embedded in a flat, no-lambda universe, show
that the density of the overdensity at maximum size is given by

where R refers to the scale factor of the universe, R

I will review this and then we will sit down on Wed 11/28 for discussion and feedback for final version. Your oral in-class presentations will be done the last day of class (Dec 5). The draft review is to ensure your project is on track, that you haven't missed any critical details, and that your scientific emphasis is appropriate for the oral presentation.