Derivations

Question 1

Derive luminosity distance

Answer 1

d^2(lum) = L/S

S = L/(a^2(0)r^2(0))

photons lose energy ∝ (1+z) so have less energy on arrival, and

photons arrive less frequently, also ∝ (1+z)

S = L/[a^2(0)r^2(0)(1+z)^2]

giving formula on crib sheet

Question 2

Derive the angular diameter distance

Answer 2

d(diam) = l/sinθ ~ l/θ

l = r(0)a(t(e))θ

θ = l/r(0)a(t(e)) = l(1+z)/a(0)r(0)

giving formula on crib sheet

Question 3

derive the galaxy number counts

Answer 3

N = n(0) 4/3 πr^3 Ω/4π

F = L/4πr^2

N(>F) = const n(0) L^(3/2) F^(-3/2) Ω

N(>F) = const n(0)F^(-3/2) Ω(∞ ∫ 0) f(L) L^(3/2) dL

giving formula on crib sheet

Question 4

Derive the Friedmann equation

Answer 4

K = 1/2mr^2(dot) = 1/2 ma^2(dot) s^2

U = -GMm/r = -4/3 π a^2s^2Gpm

total energy = constant

giving the Friedmann equation COS2 equation sheet

Question 5

Solution of Friedmann’s equation for k = 0

Answer 5

(da/dt)^2 = 8πGpa^2/3

Mass is proportional to pa^3

(da/dt)^2 = A/a

Question 6

Redshift in flat Universe

Answer 6

a = 1/1+z

1+z = λ(obs)/λ(em)

p(z) = p(0)(1+z)^3

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

1 + z = (t/t(0))^(-2/3)

Question 7

the critical density of the universe

Answer 7

when k = 0

(a(dot)/a)^2 = 8πGp/3

H = (a(dot)/a)

p = 3H^2/8πG

p(c,0) = 3H(0)^2/8πG

Question 8

Derivation of Hubble-lamaitre law

Answer 8

start from the fluid equation

p = p(0) + U/c^2

p(dot) + (p+P/c^2)∇ . v(v) = 0

∇ . v(v) = ∂xvx+∂yvy+∂zvz

∂xvx = ∂yvy = ∂zvz = H(t)

δv = H(t) δr

Question 9

Derive the Friedmann equation starting from the momentum equation

Answer 9

starting from the momentum/acceleration equation equation

from fluid equation rearrange for P and substitute in

multiply by a(dot)

perfect differential

leads to fried equation

Question 10

Derive the Einstein De-Sitter solutions

Answer 10

k = 0 in the friedmann equation

p = p(0) (a(0)/a)^η

sqrt to find a(dot)

get a on LHS and rest on RHS

derivative in terms of t

Question 11

Derive q(t) = 1/2Ω(t)

Answer 11

substitute a(d dot) and a(dot) into q(t)

identify Ω

Question 12

Derive Ω = Ω(0)/[Ω(0) + (1-Ω(0))(1+z)^(2-η)]

Answer 12

equate present Friedmann equations

replace a/a(0) for 1/1+z

replace H^2/H(0)^2

replacing p/p(0) = (a(0)/a)^η

Question 13

Derive the Jeans Length

Answer 13

PV = NkT

P = Gp^2L^2

Gp^2L^2 ≥ pkT/m

gives Jeans length

and thus jeans mass as M = pL^3 or M(J) = p(0)λ^3(J)

Question 14

Derive a single equation for the evolution of the density perturbation.

Answer 14

start from the continuity equation

p(dot) + p(0) ∇.(v) = 0

take derivative

substitute the momentum equation

substitute Poisson’s equation

substitute for sound speed

∇ P(1) = c^2(s)∇ p(1)

Question 15

Derive Ω - 1 = [Ω(0) - 1]/[1+Ω(0)z]

Answer 15

Friedman equation = Friedmann equation present time

substitute in the density parameter

substitute redshift (1+z) factor

(H(0)/H)^2 = p(0)/p Ω/Ω(0) = Ω/Ω(0) (1+z)^(-3)

Question 16

from the single equation for the evolution of the density perturbation derive the critical mass M(J)

Answer 16

assume plane wave with p(1) = exp{i(kr(v).ωt)}

substitute into the single equation for the evolution of the density perturbation

get ω on one side

for stability RHS > 0

gives λ(J)

M(J) = λ^3(J)p(0)

Question 17

Demonstrate that M(J) ∝ p(0)^(3γ-4)/2

Answer 17

M(J) = π^3/2 c^3(s) p(0)^(-1/2) G^(-3/2)

P ∝ p^γ

M(J) ~ p^(-1/2) p^(3(γ-1)/2)

~ p^1/2(3γ-4)

M(J) ∝ p(0)^(3γ-4)/2

Question 18

starting from the Friedmann equation demonstrate that the expansion of a flat universe under the influence of gravity aloe proceeds more rapidly if the universe is matter dominated than if radiated-dominated.

Answer 18

Friedmann equation for flat k = 0

matter-dominated p ~ 1/a^3

radiated-dominated p ~ 1/a^4

a on LHS and constant on RHS

integrate

a(t) ~ t^2/3

and a(t) ~ t^1/2

Question 19

linearisation of continuity equation

Answer 19

continuity equation

∇.(Hr) = 3H

p0(dot) + 3Hp(0) = 0

s(dot) + Hr.∇s + ∇.u = 0

Question 20

a(d dot) = -4πGp(0)/3a^2

gives

a^2(dot) - 8πGp(0)/3a = -H^2(0)(Ω-1)

Answer 20

take integral

rearrange and substitute

Question 21

derive

M = 3σ(vr)^2Re/G

Answer 21

<K> = 3/2Mσ(vr)^2

<u> = -GM^2/Re

0 = <K>+1/2<u>
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