ASTR 3740 Relativity & Cosmology Spring 1998. Problem Set 3.
Due Tue 14 Apr

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.

1. Anti-gravity

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 Lambda), and that it is geometrically flat
      Omega_M + Omega_Lambda = 1 (1.1)
where Omega_M == rho_M/rho_c and Omega_Lambda == rho_Lambda/rho_c are the contributions to Omega in matter and vacuum. How big must Omega_Lambda be for the Universe to be accelerating? [Hint: Friedmann's equation for the acceleration $\ddot a == d^2 a / d t^2 of the cosmic scale factor a(t) is
      \ddot a / a = - (4/3) pi G ( rho + 3 p ) (1.2)
which shows that the Universe is accelerating if rho + 3 p < 0. Ordinary matter has mass-energy density rho_M but essentially no pressure, pM = 0, while vacuum has negative pressure equal to its mass-energy density, p_Lambda = - rho_Lambda.]

(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, $kappa = 0$, so that Friedmann's energy equation reduces to
      \dot a^2 / a^2 = (8/3) pi G rho . (2.1)
Suppose further that the Universe is dominated by stuff whose mass-energy density rho varies with cosmic scale factor a as
      rho proportional to a^-n (2.2)
as the Universe expands, with n a constant. For example, n = 3 for ordinary matter, n = 4 for radiation, and n = 0 for vacuum energy.

(a) Case n != 0

Solve Friedmann's equation to show that, for n != 0,
      a proportional to t^(2/n) . (2.3)
[Hint: You should find that Friedmann's equation can be recast in the form t = integral f(a) da where f(a) is some function of cosmic scale factor a. You may set a = 0 at t = 0, which says that the Universe had zero size at zero age.]

(b) Deceleration or acceleration?

For what range of n is the Universe decelerating (\ddot a < 0) or accelerating (\ddot a > 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
      a proportional to e^(H t) (2.4)
where H == \dot a/a$, the Hubble constant, is in this case a constant in time as well as space. What is the Hubble constant H here in terms of the vacuum energy rho_Lambda?

(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 (rho a^3) + p d (a^3) = 0 (which you may recognize as the equation dE + p dV = 0 of thermdynamics) that
      n = 3 [1 + (p/rho)] . (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 H^2 = (8/3) pi G rho_c of the critical density rho_c, and Omega == rho/rho_c of Omega, to show that Friedmann's equation (including the curvature term)
      H^2 = (8/3) pi G rho - kappa c^2 / a^2 (3.1)
can be rewritten in the form
        (Omega - 1) / Omega = 3 kappa c^2 / (8 pi G rho a^2) . (3.2)

(b) Evolution of Omega with a

Suppose once again that rho proportional to a^-n. Show that (a simple consequence of [3.2])
        (Omega - 1) / Omega proportional to a^? (3.3)
where ? is an exponent which you should derive (in terms of n).

(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 Omega_0 (subscript 0 means the present time) is of order, but not equal to, one at the present time (Omega_0 ~ 0.3, say), roughly how close to one was Omega 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 epsilon == Omega - 1, and use (3.3) to estimate epsilon at the Big Bang. Note that for tiny epsilon, you can approximate 1 - epsilon approx 1. For simplicity, you may suppose that the Universe has been mainly radiation-dominated, rho approx proportional to a^-4, during its expansion from the Big Bang.]

Andrew Hamilton