# PS09 ```Physics 444: Problem Set #9
due December 1, 2021
1. In the lecture, I said that, around the time of BBN, the Universe was
10
T (t) ≈ 10
K
t
1s
1/2
.
Using our numeric solution in the Benchmark Universe, calculate the
actual temperature when the Universe was 1 second old. What was the
scale factor a at 1 second? What are the density parameters Ωr , Ωm ,
and ΩΛ at this time? What was the energy density of radiation at this
time r ?
2. Suppose that the number of inflationary e-foldings was the value typically assumed: N = 62. Assume further that inflation started at a
time ti = 10−36 s when the Hubble parameter was Hi = t−1
i . and that
prior to this time, the Universe was radiation-dominated.
(a) Calculate the physical particle horizon size at ti (just as inflation
starts) dhor (ti ), and after inflation ends dhor (tf ).
(b) Today, the distance to the CMB is dp (t0 , tCMB ) = 13.9 Gpc.
In order for all of the volume that we now see to have been in
causal contact before inflation, the horizon size at the end of inflation must translate to a physical size today at least as large
as dp (t0 , tCMB ). What must have been the maximum scale factor
after inflation? What was the minimum temperature (in K) must
the Universe have been immediately after inflation? What energy
does this correspond to (in GeV)?
(c) Suppose that, instead of 62 e-foldings, the Universe inflated for 60
e-foldings, starting at the same time as previously assumed in part
(a) and still ending with horizon size after inflation translating to
a physical size today at least as large as dp (t0 , tCMB ).
What now is the minimum temperature (in K) and energy (in
GeV) of the Universe after inflation? Is this minimum temperature
larger or smaller than the temperature you calculated assuming
the minimum number of e-foldings?
1
(d) Let us continue assuming that there were 60 e-foldings, rather than
62. However, rather than the temperature after inflation being the
minimum calculated in part (c), let us assume that, after inflation,
the Universe has the minimum temperature which you calculated
in part (b). In that case, what is the physical distance today
corresponding to the horizon distance after inflation dhor (tf )? In
this case, would we be seeing regions in the CMB which were never
in causal contact?
3. The dark energy active in the Universe today shares many similarities to inflation. Assuming that we are dark energy-dominated today,
how long would we have to wait for 62 e-foldings to occur? How far
away would the present-day CMB surface-of-last-scattering be after
this many e-foldings?
2
```