Astr 328 Homework #1

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1. Volume of the Universe

Step 1: Find the volume of the Universe out to a comoving distance r. Using the Robertson-Walker metric, we can define the differential volume element as

which means the volume out to a comoving distance r is given by

Solve for V(r) for k=-1,0,1.

Step 2: Find the relationship between comoving distance r (which you've calculated the volume for, but is unobservable) to redshift z (which is observable).

Start by using the R-W metric and integrating along the path of a light ray.  Do this exercise using a flat, matter dominated OmegaM=1 universe. While this cosmology is almost certainly not the correct one, it is a useful "benchmark" to compare to (and it is analytically tractable!). This should give you a relationship like this: r=f(R,t0). Then use the Lemaitre equation to turn it into r=f(z,t0).

Step 3: Combine the relationship in steps 1 and 2 to show that

• V(z) is different in different universes. For example, using the same techniques, but with messier math, we can show that for an empty (non-Lambda) universe, the volume is

While for positively curved, OmegaM=2  universe we have

• Plot the volume (in Mpc³) out to a redshift z in each universe. A log plot might help here...
  
• If the distribution of galaxies in the universe is completely homogeneous, now plot the relative number of galaxies you see as a function of redshift in each case.
• In the absence of magnitude selection effects (yeah, right!), if you could see all the bright galaxies in the universe out to a redshift of z=2, what would the median redshift of the galaxies be in these three universes? Explain qualitatively why this makes sense.
  
• Why is this test (called the count-redshift test) a difficult test to perform?

2. Lookback Times and Age Constraints

Use the "Cosmo" applet for this exercise.

• It's 1990. You are a good, party-line cosmologist and "know" that Omega=1, and that there is no such thing as the cosmological constant. You also "know" that globular clusters are 14-16 Gyr old. What can you say about H0?
• A few years later, you decide that the measurements of H0 are getting better, so you need to believe them. So take H0=55, and tell me what constraint you can place on Omega. What about if you believe H0=65? H0=75?
• And then Hipparcos comes along (around 1997) and tell us that GCs are further away -- this means the ages of the globular clusters get revised downwards to 11-13 Gyr. Why does the further distance mean a younger age? At the same time, you decide, for better or worse, that you like H0=65. Now what constraints can you place on Omega?
• Later, in 1998 the high redshift galaxy LBD53W091 was discovered. It lives at a redshift of z=1.55, and its was determined to have a minimum age of 3.5 Gyr. How does this change your constraints on Omega?
• Six months later, the age of LBD53W091 was revised downwards, and became 1.7 +/- 0.3 Gyr. NOW how do your constraints on Omega change?
• Finally, this wacky Lambda idea starts to take off, so you have to consider OmegaL as well. Given your GC ages and H0, pick values of OmegaL and, for those values, place limits on OmegaM. Make a plot of your results on an OmegaL-OmegaM plane, showing regions of "allowed cosmology" given your GC age constraints.

In Lambda=0 universes, the "luminosity distance" (ie the distance measure you use when calculating brightnesses of objects) can be expressed analytically, and is given by

Use this to derive an expression for the magnitude-distance relationship m-M=fn(z,q0,H0).

Now plot the apparent magnitude of M87 as a function of redshift (out to a redshift of, say 1.5) for a few different reasonable values of q0. Ignore bandshifting for this exercise.

Grad students: Factor in luminosity evolution. Stellar population synthesis models can describe how the luminosity of a galaxy changes with time. Here, for example, is the Bruzual-Charlot model for how a single burst, solar metallicity population evolves with time. Assume M87 was made at very high redshift (so that its age is simply the age of the Universe). The use the B-C model to correct your plot of M87's magnitude as a function of redshift for the q=0.5 case. Compare the change in that curve due to evolution to the differences between different cosmologies w/out evolution.