Once again, this problem set may take you some time to complete, so please do not wait until the last day to start it.
Once again you will need access to the WWW to complete this problem set. If this is a problem for you, please talk to me in good time and I will provide you with a printout of some of the relevant pages (but I prefer you to do your own investigation on the web).
While you are visiting Ned Wright's pages on the web, I encourage you to go over his excellent cosmology tutorial. You may like to print out a hardcopy of this tutorial.
Both high redshift supernova teams have recently reported that their observations of the Hubble diagram of supernovae at high redshift suggest that the Universe may be accelerating.
(a) Condition for an accelerating Universe
Suppose that the Universe contains only matter energy (M)
and vacuum energy
(a cosmological constant
and that it is geometrically flat
(b) Draw your own conclusion
Check out Ned Wright's News of the Universe for a summary of the observations (note that there are two relevant reports on Ned Wright's page, the ``ANTI-GRAVITY? - Report on DM98'' piece and the ``More on Distant Supernovae'' piece below it). Describe in your own words what you think the supernova people have found, and, in the light of the first part of this question, draw your own conclusion as to whether the Universe may be accelerating.
2. Solutions to Friedmann's equations in a Flat Universe
Suppose that the Universe is flat,
so that Friedmann's energy equation reduces to
Solve Friedmann's equation to show that, for
(b) Deceleration or acceleration?
For what range of n is the Universe decelerating () or accelerating ()? 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
(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
(which you may recognize as the equation
dE + p dV = 0 of thermdynamics) that
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
of the critical density
to show that Friedmann's equation (including the curvature term)
(b) Evolution of with a
Suppose once again that
Show that (a simple consequence of [3.2])
(c) Here's the flatness problem
Suppose that the currently observable Universe, ~ 1026 meters in radius, started from something about a Planck length ~ 10-35 meters in size at the Big Bang. If (subscript 0 means the present time) is of order, but not equal to, one at the present time ( ~ 0.3, say), roughly how close to one was at the Big Bang? [Hint: The two sizes tell you how much the cosmic scale factor a has changed from the Big Bang to now. Define the small quantity , and use (3.3) to estimate at the Big Bang. Note that for tiny , you can approximate . For simplicity, you may suppose that the Universe has been mainly radiation-dominated, , during its expansion from the Big Bang.]