MATHEMATICAL TRIPOS Part II PAPER 4 Before you begin read ...

5
10E Cosmology
The number density of a species ⋆ of non-relativistic particles of mass m, in
equilibrium at temperature T and chemical potential µ, is
n ⋆ = g ⋆
( 2πmkT
h 2 ) 3/2
e (µ−mc2 )/kT ,
where g ⋆ is the spin degeneracy. During primordial nucleosynthesis, deuterium, D, forms
through the nuclear reaction
p + n ↔ D ,
where p and n are non-relativistic protons and neutrons.
between the chemical potentials in equilibrium.
Write down the relationship
Using the fact that g D = 4, and explaining the approximations you make, show that
(
n D h
2
) 3/2 ( )
BD
≈
exp ,
n n n p πm p kT
kT
where B D is the deuterium binding energy, i.e. B D = (m n + m p − m D )c 2 .
Let X ⋆ = n ⋆ /n B where n B is the baryon number density of the universe. Using the
fact that n γ ∝ T 3 , show that
( )
X D
∝ T 3/2 BD
η exp ,
X n X p kT
where η is the baryon asymmetry parameter
η = n B
n γ
.
Briefly explain why primordial deuterium does not form until temperatures well below
kT ∼ B D .
Part II, Paper 4
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