Derivations

Question 1

derive the fluid equation

Answer 1

dU + pdV = TdS

dS = 0

U = 4/3 πa^3pc^2

V = 4/3πa^3

differentiate and substitute

divide by

c^2a^3/3

Question 2

show

8πGp/3 = H(0)^2 Ω(m,0) (1+z)^3

Answer 2

multiply by p(0)/p(0) H(0)^2/H(0)^2

assuming mass conservation pa^3 = p(0)a(0)^3

Question 3

starting from

ma(d dot) = -GmM/a^2

derive the Friedmann equation

Answer 3

cancel m and multiply by 2a(dot)

integrate with respect to time

constant = -kc^2

Question 4

starting from the Robertson-walker metric show that for small r(E) the proper distance is D ~ a(t)r(E)

Answer 4

binomially expand for small r
(1+x)^n ~ 1+nx

throw away cubic term

Question 5

identify a necessary and sufficient condition that must be satisfied by the equation of state if q(0) is to be negative

Answer 5

require q(0) < 0

q(0) = - a(d dot)/a 1/H(0)^2

replace a(d dot)/a with acceleration equation

solve

Question 6

prove that in a pressureless universe with a cosmological constant, the deceleration parameter is given by

q(0) = Ω(m,0)/2 - Ω(Λ,0)

Answer 6

start from the acceleration equation plus

assume P = 0

assume present time

sub into q(0)

Question 7

show that the solution to the Friedmann equation for a universe with p = 0, k = 0 but Λ ≠ 0 predict exponential expansion

a(t) = a(0) exp(t √Λ/3)

Answer 7

start from Friedmann equation

set k = 0 and p = 0

sqrt

derivative

solve

Question 8

In the radiation era, it can be shown that time and temperature are related by t ∝ 1/T^2

If the universe expands adiabatically show the expansion rate is H ∝ T^2/2 where H is the Hubble parameter

Answer 8

for an adiabatically-expanding universe

aT = constant

a(dot)T + T(dot)a = 0

t ∝ 1/T^2

looking at Hubble’s constant

Question 9

estimate the age of the universe at decoupling as a fraction of its current age for an Einstein de Sitter universe

Answer 9

a(dec)/a(0) = (t(dec)/t(0))^2/3

z = 1000

Question 10

Show the Y4 ratio is given by

Y4 = 2Nn/Nn+Np

Answer 10

He = 4(Nn/2)

H = (Np-Nn)

Y4 = He/H+He

Question 11

Starting from

|Ω(total) -1| = |k|c^2/a^2H^2

show that if the universe undergoes any period of accelerated expansion with a(d dot) > 0 then during that accelerated expansion |Ω(total) -1|is a decreasing function of time t

Answer 11

rewrite H

take the derivative

all acceleration > 0

hence

d/dt|Ω(total) -1|< 0

Question 12

show that

a(t) = a(t(i)) exp [H(t-t(i))]

Answer 12

da/a = Hdt

ln(a(t)) = Ht + const

then gives as required

Question 13

derive the acceleration equation

Answer 13

differentiate the Friedmann equation with respect to time

replace p(dot) with fluid equation

Question 14

starting from the Roberston-walker metric derive the Hubble-lemaitre law.

Answer 14

differentiate with respect to time

Question 15

derive the age of the universe

Answer 15

t(0) =(t(0) ∫ 0) dt

where dt = -dz/[H(0)E(z)(1+z)]

gives the formula on the formula sheet

Question 16

derive aT = constant

Answer 16

reversible and adiabatic process

U = a^3T^4

V = a^3

and p = T^4/3