ASTR 3740 Spring 2004 Homepage
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ASTR 3740 Problem Sets
This problem set may take you some time to complete, so please do not wait until the last day to start it.
1. Anti-gravity
(a) Condition for an accelerating Universe
Suppose that the Universe contains only matter energy (M) and vacuum energy (a cosmological constant L), and that it is geometrically flat
| (1.1) |
| (1.2) |
(b) Draw your own conclusion
Papers describing latest results from the two high-redshift supernova teams are at http://arXiv.org/abs/astro-ph/0309368 and http://arXiv.org/abs/astro-ph/0402512 . The websites of the two teams are the "Supernova Cosmology Project" at http://supernova.lbl.gov and the "High-Z Supernova Search" at http://cfa-www.harvard.edu/cfa/oir/Research/supernova/ (sadly, the website of the High-Z SN team appears out of date). A good place to start searching for more information about supernovae is "Supernova and Supernova Remnant Pages on the WWW" http://rsd-www.nrl.navy.mil/7212/montes/sne.html .
What are the latest results from the two high-redshift supernova teams? Do they agree with regard to their measurements of WM and WL? What is this w thing that they both report?
2. Solutions to Friedmann's equations in a Flat Universe
Suppose that the Universe is flat, k = 0, so that Friedmann's energy equation reduces to
| (2.1) |
| (2.2) |
(a) Case n ¹ 0
Solve Friedmann's equation to show that, for n ¹ 0,
| (2.3) |
(b) Deceleration or acceleration?
For what range of n is the Universe decelerating (da/dt < 0) or accelerating (da/dt > 0)? Is the Universe decelerating or accelerating in the particular cases of a matter-dominated (n = 3) or radiation-dominated (n = 4) Universe?
(c) Case n = 0
The case n = 0 corresponds to vacuum density, which remains constant as the Universe expands. Solve Friedmann's equation for this case to show that
| (2.4) |
(d) For your information (no credit)
You may be wondering whether there is a relation between the index n in this question and the pressure p in the Anti-Gravity question. The answer is yes. It is straightforward to show (but I'm not asking you to do this) from the energy equation d (ra3 ) + p d (a 3) = 0 (which you may recognize as the equation dE + p dV = 0 of thermodynamics) that
| (2.5) |
3. Flatness Problem
An amusing statement of this cosmological problem can be found on Ned Wright's graph.
(a) Yet another version of Friedmann's equation
Use the definitions H2 = (8/3) p G rc of the critical density rc, and W º r/rc of Omega, to show that Friedmann's equation (including the curvature term)
| (3.1) |
| (3.2) |
(b) Evolution of W with a
Suppose once again that r µ a-n. Show that (a simple consequence of [3.2])
| (3.3) |
(c) Here's the flatness problem
Suppose that the temperature at the moment of the Big Bang was about the Planck temperature ~ 1032 K. The radiation temperature of the Universe today is of course that of the CMB, about 3 K. If W0 (subscript 0 means the present time) is of order, but not equal to, one at the present time (W0 ~ 0.3, say), roughly how close to one was W at the Big Bang? [Hint: Define the small quantity e º W - 1, and use (3.3) to estimate e at the Big Bang. Note that for tiny e, you can approximate 1 + e » 1. Assume that T µ a-1 during the expansion of the Universe, and assume for simplicity that the Universe has been radiation-dominated for most of that expansion, so that n » 4.]