Problem sheet 5 - University of Nottingham

F34 COS Introduction to Cosmology – 2010
Problems 5
1. In the lectures on Nucleosynthesis we have made use of the thermal distribution of
non-relativistic particles at temperature T . In particular for particles of mass m satisfying
m T , we wrote the number density n as
m
n ∝ (mT )3/2 e− T .
(1)
Lets prove it – the relativistic case is described in Summary 9 of the lectures.
Start with the number density for bosons:
g
(exp[E/T ] − 1)−1 k 2 dk
2π 2
Z ∞
(E 2 − m2 )1/2
g
EdE
2π 2 m (exp[E/T ] − 1)
Z
n =
=
(2)
(3)
where g is the spin degeneracy factor determining the number of relativistic particles
present at any given temperature T , E is the energy of the particle and in going from
(2) to (3) I have used the fact that E 2 = k 2 + m2 . Recall we are interested in the nonrelativistic regime, m T .
(1a). In (3) change variable to x = E/T and show that
2
Z ∞
(x2 − m
)1/2
T2
gT 3
n=
2π 2
m/T
(exp[x] − 1)
xdx
(4)
(1b). Now show that (4) can be very well approximated by
n =
=
gT 3
2π 2
Z ∞
gT 3
6π 2
Z ∞
e
m2
x − 2
T
!1/2
−x
e
m2
x − 2
T
!3/2
−x
m/T
m/T
2
2
xdx
(5)
dx
(6)
where (6) emerges after an integration by parts.
(1c). Given the integral
Z ∞
e
u
−µx
2
2 ν−1
(x − u )
1
dx = √
π
2u
µ
ν− 1
2
Γ(ν)Kν− 1 (uµ)
(7)
2
where Γ(u) is the Gamma function and Kν (µu) is a Bessel function of imaginary order
show that (6) becomes
gT 3 1
2m 2 5
m
√
n=
Γ( )K2 ( )
(8)
6π 2 π T
2
T
√
(1d). Now using the fact that Γ(z + 1) = zΓ(z) and Γ( 12 ) = π show that (8) reduces to
n=
g
m
m2 T K2 ( )
2
2π
T
(9)
(1e) Finally we need to think about the Bessel function. Given that we are considering the
regime m T we use the asymptotic expansion of the Bessel function for large argument,
namely
r
π −z
lim Kν (z) '
e
2z
|z|1
Use this in (9) to obtain the desired result:
mT
n=g
2π
3/2
m
e− T .
(10)
2. This question introduces you to the role of inflation in the early universe. Consider
a simplified model of the history of a flat universe involving a period of inflation. The
history is split into 4 periods: (a) 0 < t < t3 radiation only; (b) t3 < t < t2 vacuum energy
dominates, with an effective cosmological constant Λ = 4t32 ; (c) t2 < t < t1 a period of
3
radiation domination; (d) t1 < t < t0 matter domination.
(i) Show that in (c) ρ(t) = ρr (t) = 3/32πGt2 and in (d) ρ(t) = ρm (t) = 1/6πGt2 .
(ii) Give simple analytical formulae for a(t) which are approximately true in these four
phases.
(iii) Show that, during the inflationary phase (b) the universe expands by a factor
t2 − t3
a(t2 )
= exp
a(t3 )
2t3
(iv) Derive an expression for Λ in terms of t2 , t3 and ρ(t2 ).
(v) Show that
ρr (t0 )
9
=
ρm (t0 )
16
t1
t0
2/3
.
(vi) If t3 = 10−35 seconds, t2 = 10−32 seconds, t1 = 104 years and t0 = 1010 years, give a
sketch of log a against log t marking any important epochs.
(vii) Define what is meant by the particle horizon and calculate how it behaves for this
model. Indicate this behaviour on the sketch you made in (vi). How does the period of
inflation ‘solve’ the horizon problem?
3a. Discuss the properties of the three types of weakly interacting relic particles that
could make up the dark matter.
3b. What is a coherence length in the context of dark matter particles?
3c. Describe how a relic dark-matter particle can impose a coherence scale on structure
in the Universe. What does the scale depend on?
3d. Using the temperature time relationship for the radiation era, where the time in
seconds is related to the temperature in Kelvin, calculate the coherence scale in comoving
units as a function of the mass of the dark matter particle, giving your result in comoving
Mpc and with mass units of eV.
3e. Using your answer to part (c) explain why it is we don’t believe that the universe is
dominated by hot dark matter.

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