Astr 328/428 Homework #3 -- Due Nov 17th

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To very close accuracy, under ANY cosmology regardless of OmegaM and Lambda, the early universe was essentially flat (ie Omega=1). Let's show this.

Start with the Friedmann equation, and the definition of critical density, and show that

Then explain what happens to these terms in the early universe, and why that implies that ANY universe was essentially flat early on.

In the early universe, the energy density of radiation exceeded that of matter. We're going to calculate how the expansion factor of the universe changed with time during this period. Let's do this in parts.

 Start (as always) with the Friedmann equation. Write down the terms and think about the following:

• if radiation dominates the energy density of the Universe, how and why does rho change with R?
• in the early universe, how can we handle the term involving the cosmological constant?
• in the early universe, what can we say about the spatial curvature of  the universe?
  

Given your answers to those questions, show that R ~ t^(1/2) in the early universe.

Calculate the overdensity (delta) of the following:

• the globular cluster M13
• the Milky Way galaxy
  
• the Coma Cluster
• the Bootes void

In each case, explain your reasoning, describe whether (and why) your number is an upper limit, a lower limit, or reasonably close, and detail your information sources.

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".

Make a plot of the number of clusters as a function of z in the two simulations, as well as the ratio N(z|LCDM)/N(z|tauCDM). Describe why the shapes of the plots look like they do.

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.
 

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?)

• Plot up x vs y for all galaxies to give a feel for what the data look like.  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 2 and 20 Mpc, then fit a power law and see what slope you get. Describe qualitatively what this 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 'em up, fit a slope, and describe the differences. ie 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?
• Compare your values for the clustering properties of different galaxy sets (all, early, late) to those published in the literature.
  

Calculating the 2pt correlation function:

Remember the 2ptcf describes the probability of finding two galaxies seperated by a distance r over and beyond that expected from random distribution.

The simplest estimator for this is given by

1+xi(r) = DD(r)/RR(r)