Section 3

Question 1

Consider a spherical shell of radius r within a spherical region of the universe filled with pressureless matter of uniform density

Answer 1

The universe’s expansion proceeds in the same way for all shells

Question 2

Mass conservation in a matter-dominated universe

Answer 2

a^3(t)ρ(t) = a^3(t₀)ρ(t₀) = ρ₀

Question 3

A simple pressureless model of the Universe

Answer 3

Assume a homogeneous and isotropic universe filled with ‘dust’ of uniform density ρ(t)

Let universe expand:

Conservation of total energy E

E = K(t) + U(t)

where K(t) = 1/2mv^2(t) and U(t) = -G Mrm/r(t)

Mr = 4/3 π r^3(t)ρ(t)

Question 4

Friedmann’s equation derivation

Answer 4

K = 1/2 mṙ^2 = 1/2 mȧ^2 s^2

U = -GMm/r = - 4/3π a^2 s^2 Gρm

E total = 1/2 ms^2 [ ȧ^2 - (8πGρa^2)/3 ] = const

giving

ȧ^2/a^2 - 8πGρ/3 = -kc^2/a^2

where k is a constant

Question 5

What does Friedmann’s equation describe

Answer 5

it describes how gravity slows the rate of the expansion of the Universe

Question 6

the constant k determines the curvature of the universe describe the three classes of solutions

Answer 6

k > 0 closed Universe, positive curvature
k < 0 open Universe, negative curvature
k = 0 flat Universe, zero curvature

Question 7

show that a(t) = Kt^(2/3)

Answer 7

(da/dt)^2 = A/a

ȧa^(1/2) = A^(1/2)

a^(1/2) da = A^(1/2) dt

∫ a^(1/2) da = ∫ A^(1/2) dt

2/3 a^(3/2) = A^(1/2) t

a^(3/2) = 3/2 A^(1/2) t

a(t) + Kt^(2/3)

Question 8

Solution of Friedmann’s equation for k = 0

Answer 8

(da/dt)^2 = (8πGρa^2)/3

assuming the universe is matter dominated

(da/dt)^2 = A/a

Question 9

Redshift in a flat Universe

Answer 9

a = 1/1+z

ρ(z) = ρ₀(1+z)^3

at t₀ , a/a₀ = (t/t₀)^(2/3)

1 + z = (t/t₀)^(-2/3)

Question 10

Critical density of the Universe

Answer 10

(ȧ/a)^2 = 8πGρ/3

H = ȧ/a

ρ = 3H^2 / 8πG

ρ is the critical density denoted by ρc

present day value is

ρc₀ = 3H₀^2/8πG

Question 11

If ρ > ρc

Answer 11

the universe recollapses

Question 12

if ρ < ρc

Answer 12

the universe expands indefinitely

Question 13

Density Parameter

Answer 13

Ω(t) = ρ(t)/ρc(t) = 8πGρ/3H^2

Question 14

Ω > 1

Answer 14

Universe is closed

Question 15

Ω < 1

Answer 15

Universe is open

Question 16

Ω = 1

Answer 16

Universe is flat

Question 17

Although Ω(t) can change with time, it can be shown that its state of being closed, open or flat

Answer 17

cannot change

Question 18

Derivation of fluid equations

Answer 18

we have a first fluid equation governing the flow and continuity of mass

ρ(dot) + ∇.(ρ→v) = -(P/c^2)∇.→v

ρ is the denstiy and includes the internal energy

ρ = ρ₀ + U/c^2

P is the pressure

→v is the velocity field

Question 19

Derivation of Hubble-Lemaitre Law

Answer 19

∇.(ρ→v) = ∇ρ.→v + ρ∇.→v = ρ∇.→v

therefore the mass continuity equation becomes

ρ(dot) + (ρ + P/c^2) ∇.→v = 0

by isotropy

∂xvx = ∂yvy = ∂zvz = H(t)

So 𝛿→v = H(t)𝛿→r

Question 20

Continuity of mass derivation

Answer 20

∇.→v = 3H(t)

this gives us the equation for continuity of mass in its final form

ρ(dot) + 3(ρ + P/c^2) H = 0

Question 21

Equation of state

Answer 21

typical equation of state

P(ρ) = {Kρ^(5/3) adiabatic case
{ Kρ isothermal case

Question 22

The momentum equation derivation

Answer 22

The equation of motion four our shell is then

ȧ^2 = -GM/a^2 = -G/a^2 [ 4π/3 a^3 (ρ+3P/c^2)]

therefore the momentum equation can be written as

ä = -4πGa/3 (ρ+3P/c^2)

aρ(dot)/ȧ + 3(ρ + P/c^2) = 0

P/c^2 = - aρ(dot)/3ȧ - ρ

insert into the momentum equation

ä = -4π/3 Ga (-aρ(dot)/ȧ - 2ρ)

äȧ = 4π/3 G(a^2ρ(dot) + 2ρȧa)

is a perfect differential equation

d/dt(1/2ȧ^2) = d/dt (4π/3G(a^2ρ)

ȧ^2 - 8π/3 Ga^2ρ + kc^2 = 0

Question 23

The momentum equation depends on the constant k the solutions tell us that

Answer 23

k < 0 ȧ →c √-k - open universe
k > ȧ = 0 at some critical radius followed by a callapse phase - closed universe
k = 0 ȧ→0 at t →∞ - flat universe

Question 24

The momentum equation can be written as

Answer 24

H^2(t) = 8π/3 Gρ(t) - k c^2/a^2

Question 25

Einstein-De Sitter Universe

Answer 25

is a special case of a homogeneous, isotropic universe of zero curvature described by Newtonian gravity ρ = ρ₀(a₀/a)^η ȧ^2 - 8π/3 Ga^2ρ + kc^2 = 0 ȧ^2 = 8π/3 Ga^2ρ ρ₀a₀^η a^(-η) a^(η/2 -1) ȧ = (8π/3 Gρ₀a₀^η)^1/2 2/η ∂/∂t a^(η/2) = (8π/3 Gρ₀a₀^η)^1/2 a^(η/2) = η/2 (8π/3 Gρ₀a₀^η)^1/2 t + const

Question 26

The temporal behaviour f the scale factor is given by

Answer 26

a(t) ∝ t^(2/η) a(t) ∝ { t^2/3 matter-dominated { t^1/2 radiated -dominated

Question 27

The deceleration paramter

Answer 27

q(t) = -aä/ȧ^2

Question 28

for a matter dominated flat universe

Answer 28

p~a^-3 and pa^2 = p0/a substitute into fried for k =0 a(dot) = (8πGp/3)^1/2 a^(-1/2) a(dot) a^1/2= (8πGp/3)^1/2 integrate a(t) ~ t^3/2

Question 29

for a radited dominated flat universe

Answer 29

p~a^-4 and pa^2=p0/a^2 substitute into fried for k =0 a(dot) = (8πGp/3)^1/2 a^-1 a(dot) a= (8πGp/3)^1/2 integrate a(t) ~ t^3/2

Question 30

a(dot)^2 =

Answer 30

H^2 a^2

Question 31

H0^2/H^2 =

Answer 31

Ω/Ω0 (1+z)^(-3)