Astr/Phys 328/428 Homework #1

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 to get an analytic expression 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, remembering that R(t)~t2/3 in such a universe. While this cosmology is 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 and the expression for t0 to turn it into r=f(z,H0).

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

2. The Flatness of the Universe

Starting with the Friedmann equantion for a Lambda=0 universe, show that the Hubble parameter can be written as:

Helpful hints:

Then use that to show that

Helpful Hints:

So we see that Omega varies with time. Now do the following:

3. Apparent Magnitudes and sizes of distant objects

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

The giant elliptical galaxy M87 has a V band luminosity of 1011 Lsun and an effective radius of about 10 kiloparsecs. Imagine moving M87 to different redshifts, and plot its apparent V magnitude as a function of redshift from z=0.1 to z=2.0 for universes with q0=0.1, 0.5, and 1.5. Ignore K-corrections for this calculation, but describe qualitatively what the K-correction will do to your plot. Also, explain qualitatively how these universes differ in terms of their expansion history and spatial curvature.

Make a second plot showing the angular size of M87 (in arcseconds) as a function of redshift over that same redshift range.

For each of these different universes, answer the following:

4. Distant objects

Do this calculation twice. Once analytically for a OmegaM=1, OmegaL=0 universe (show your work!), and then use the astropy cosmology calculator or the Cosmology Calculator to get the values for a OmegaM=0.3, OmegaL=0.7 universe. Compare how the values differ in the different universes.

One of the most distant radio galaxies is 8C 1435+63, at a redshift of z=4.25. Answer the following:

ASTR 428 additional question:

5. Lookback Times and Age Constraints

Use either the astropy cosmology package or the Cosmology Calculator for this exercise.