ASTR 3740 Spring 2004 Homepage

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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Search" at http://cfa-www.harvard.edu/cfa/oir/Research/supernova/ (sadly, the website of the High-Z SN team appears out of date).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
W_{M}
and
W_{L}?
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* (r*a*^{3} ) + *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
*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.]