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**1. 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

| (1.1) |

| (1.2) |

| (1.3) |

| (1.4) |

**2. 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 5, 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.]

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

Show that Friedmann's equation can be written in the form (compare Problem Set 6, Question 3a)

| (2.1) |