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Once again, this problem set may take you some time to complete, so please do not wait until the last day to start it.

**1. Cosmological Numbers**

The purpose of this question is to allow you to figure out for yourself the values of some basic numbers in cosmology. You should find all the necessary formulae in the notes; part of the question is to figure out what formula to use.

Assume where necessary that the Hubble constant is
*H*_{0} = 65 km s^{-1}Mpc^{-1}.

Here are
links to values of physical constants.
Possibly helpful numbers:

*c* = 299,792,458 m s^{-1},

(^{h}/_{2p}) = 1.054 572 66 ×10^{-34} J s,

*G* = 6.672 59 ×10^{-11} m^{3} kg^{-1}s^{-2}.

In 1976, the IAU (International Astronomical Union) defined the
AU (Astronomical Unit), the mean Sun-Earth distance, to be
1 AU ș 149,597,870 km.

The pc (parsec) is defined to be the distance at which
1 AU subtends one second of arc, whence
1 pc ș 648,000/p AU = 3.085 677 57 ×10^{13} km.

1 yr = 31,556,930 s.

The prefix M on a unit, as in Mpc (megaparsecs) denotes mega = 10^{6}.

The prefix G on a unit, as in Gyr (gigayears) denotes giga = 10^{9}.

**(a) Age of the Universe**

What is the age of the Universe
(i) for a wide open Universe, W = 0,
for which *t*_{0} = 1/*H*_{0},
and
(ii) for a flat matter-dominated Universe,
W = W_{M} = 1,
for which
*t*_{0} = 2/(3 *H*_{0})?
Compare your answer to the age
*t*_{0} = 11.5 ± 1.3 Gyr
of globular cluster stars determined post-Hipparcos by
Chaboyer et al. (1998, Astrophysical Journal, 494, 96).

**(b) Critical density**

What is the critical density
r_{c} of the Universe?
Express your answer in kg m^{-3}.
[Just for fun (no credit):
To bring this number down to Earth,
estimate (order of magnitude) how much volume your body would
occupy if it were spread out to the critical density;
compare that volume to something familiar.]

**(c) Omega in the CMB**

Figure out W_{CMB} today.
The mass-energy density of photons in thermodynamic equilibrium
at temperature *T* is
r = *a* *T*^{4}/*c*^{2},
where
*a* = p^{2} *k*^{4}/(15 *c*^{3} (^{h}/_{2p})^{3}) = 7.56591×10^{-16} J m^{-3} K^{-4}
is the radiation density constant.

**(d) Redshift of matter-radiation equality**

At what redshift were the densities of CMB radiation and matter equal,
if Omega in matter today is W_{M} = 0.3?

**(e) Omega in matter at Recombination**

What was W in matter at Recombination?
Assume that the Universe has been matter-dominated since
Recombination, with W_{0} = 0.3
in matter today,
and that Recombination occurred at a redshift of
1 + *z*_{R} = 1300.
[Hint:
use the formula
(W - 1) / W = 3 k *c*^{2} / (8 p *G* r *a*^{2})
you derived in Problem Set 4, with
r ” *a*^{-3}.
Recall that
1 + *z* = *a*_{0}/*a*.]

**2. Horizon at Recombination**

What angle does the horizon at Recombination subtend on the CMB today?
Assume a flat, matter-dominated Universe.
Express your answer first in terms of the redshift factor
1 + *z*_{R} of Recombination,
and then translate your answer into degrees for the case
1 + *z*_{R} » 1300.
[Hint:
In a flat, matter-dominated Universe, the comoving distance to the
horizon at a time when the cosmic scale factor is *a*
and the Hubble parameter is *H* is

| (2.1) |

| (2.2) |

| (2.3) |

| (2.4) |

**3. Horizon Problem**

**(a) Expansion factor**

The temperature of the CMB today is *T*_{0} » 3 K.
By what factor has the Universe expanded (i.e. what is *a*_{0}/*a*)
since the temperature was the Planck temperature
*T* » 10^{32} K?
[Hint: you already did this in Problem Set 4, Question 3c.]

**(b) Hubble distance**

By what factor has the Hubble distance *c*/*H* increased
during the expansion of part (a)?
Assume that the Universe has been mainly radiation-dominated
during this period, and that the Universe is flat.
[Hint:
For a flat Universe
*H*^{2} ” r,
and for radiation-dominated Universe
r ” *a*^{-4}.]

**(c) Comoving Hubble distance**

Hence determine by what factor the comoving Hubble distance
*x*_{H} = *c*/(*aH*)
has increased during the expansion of part (a).

**(d) Comoving Hubble distance during inflation**

During inflation the Hubble distance *c*/*H* remained constant,
while the cosmic scale factor *a* expanded exponentially.
What is the relation between the comoving Hubble distance
*x*_{H} = *c*/(*aH*)
and cosmic scale factor *a* during inflation?
[You should obtain an answer of the form *x*_{H} ” *a*^{?}.]

**(e) e-foldings to solve the Horizon Problem**

By how many *e*-foldings must the Universe have inflated
in order to solve the Horizon Problem?
Assume again, as in part (a), that the Universe has been mainly
radiation-dominated during expansion from the
Planck temperature to the current temperature,
and that this radiation-dominated epoch
was immediately preceded by a period of inflation.
[Hint:
Inflation solves the Horizon Problem if the currently observable Universe
was within the Hubble distance at the beginning of inflation,
i.e. if the comoving *x*_{H,0}
now is less than the comoving Hubble distance
*x*_{H,i} at the beginning of inflation.
The `number of *e*-foldings' is
ln(*a*_{f} / *a*_{i}),
where ln is the natural logarithm,
and
*a*_{i} and *a*_{f}
are the cosmic scale factors at the beginning
(*i* for initial) and end (*f* for final) of inflation.]

**4. Relation between Horizon and Flatness Problems**

Show that Friedmann's equation can be written in the form (compare Problem Set 4, Question 3a)

| (4.1) |

File translated from T

On 13 Apr 1999, 09:49.