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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

| (1.1) |

| (1.2) |

**(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

| (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* (r*a*^{3} ) + *p* *d* (*a* ^{3}) = 0
(which you may recognize as the equation
*dE* + *p* *dV* = 0
of thermdynamics)
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
*H*^{2} = (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)

| (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
~ 10^{32} K.
The radiation temperature of the Universe today is of course
that of the CMB, about 3 K.
If W_{0} (subscript 0 means the present time)
is of order, but not equal to, one at the present time
(W_{0} ~ 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.]

File translated from T

On 30 Mar 1999, 09:40.