ASTR 3740 Problem Set 4

ASTR 3740 Relativity & Cosmology Spring 1999. Problem Set 4. Due Tue 13 Apr

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1. Anti-gravity

Both high redshift supernova teams reported over the past year 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 L), and that it is geometrically flat

W_M + W_L = 1

(1.1)

where W_M º r_M/r_c and W_L º r_L/r_c are the contributions to Omega in matter and vacuum. How big must W_L be for the Universe to be accelerating? [Hint: Friedmann's equation for the acceleration d²a / d t² of the cosmic scale factor a(t) is

..
a

= -

p G ( r + 3 p )

(1.2)

which shows that the Universe is accelerating if r + 3 p < 0. Ordinary matter has mass-energy density r_M but essentially no pressure, p_M = 0, while vacuum has negative pressure equal to its mass-energy density, p_L = - r_L.]

(b) Draw your own conclusion

Have a look at the websites of the two supernova teams: the Supernova Cosmology Project, and the High-Z Supernova Search, both of which I found linked at Supernova and Supernova Remnant Pages on the WWW. Check out also Ned Wright's News of the Universe for a summary of the observations. 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.

If you want to dig deeper, you can search the Los Alamos National Laboratory astrophysics preprint server. The leaders of the two supernova teams are Saul Perlmutter and Brian P. Schmidt.

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

.
a

a²

p G r .

(2.1)

Suppose further that the Universe is dominated by stuff whose mass-energy density r varies with cosmic scale factor a as

r µ 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 ¹ 0

Solve Friedmann's equation to show that, for n ¹ 0,

a µ t^2/n .

(2.3)

[Hint: You should find that Friedmann's equation can be recast in the form t = ò 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 (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

a µ e^{H t}

(2.4)

where H º (da/dt)/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 r_L?

(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 thermdynamics) that

n = 3

æ
ç
è

1 +

ö
÷
ø

(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 r_c of the critical density r_c, and W º r/r_c of Omega, to show that Friedmann's equation (including the curvature term)

H² =

p G r -

k c²

a²

(3.1)

can be rewritten in the form

W- 1

3 k c²

8 p G r a²

(3.2)

(b) Evolution of W with a

Suppose once again that r µ a^-n. Show that (a simple consequence of [3.2])

W - 1

µ a^?

(3.3)

where ? is an exponent which you should derive (in terms of n).

(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.]

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