Problem #1 [Structure Formation I: Radiation Era]

(b). At late time (y ≫ 1), compare your solutions for δ m with those derived in class.
if y ≫ 1 the universe becomes matter dominated and the above expression becomes
d 2 δ m
dy 2
+ 3 dδ m
2y dy − 3
2y 2 δ m = 0
if we assume that this has a power solution of the form
δ m ∝ y α
dδ m
dy = αyα−1
d 2 δ m
dy 2
= α(α − 1)yα−2
we find that
which can be factored as
α 2 + 1 2 α − 3 2 = 0
(
(α − 1) α + 3 )
= 0
2
giving the two solutions as
α = 1 α = −3/2
and so we can see that
δ + m ∝ y and δ− m ∝ y−3/2
and we find a growing solution and a decaying solution. The two solutions from class are
δ + m ∝ a δ − m ∝ a −3/2
and so we know that when y ≫ 1 implies ρ m ≫ ρ r and we recover the solutions from class.
(c). Verify that δ m ∝ y+2/3 is a solution to the equation in part (a) in general. Can δ m grow
much in the radiation dominated era?
we can verify this solution by doing
δ m = y + 2 3
˙ δ m = 1
¨ δ m = 0
and plugging this into the differential equation yields
(
2 + 3y
2y(y + 1) − 3
y + 2 )
2y(y + 1) 3
= 0
0 = 0
thus, this is a solution to the differential equation.
We know that for radiation dominated era y ≪ 1 and since the solution for δ + m
can see it does not grow very much during the radiation dominated regime.
is linear we
2