Section 5

Question 1

Jean’s Criterion

Answer 1

GM/L^2 ρL > ρkT/m

GρL ≥ kT/m =cs^2

cs = sound speed

Jeans length

L ~ √ cs^2/ρG

Jeans mass

M ~ cs^3/(ρ^1/2 G^3/2)

Question 2

Solution for small perturbations of non-expanding universe

Answer 2

assume ρ1 has the form of a plane wave

ρ¨1 = (−iω)^2 ρ1 = −ω^2 ρ1

∇^2 ρ1 = (ik)^2 ρ1 = −k^2 ρ1

ω^2 ρ1 − ρ0 4πG ρ1 + kT/µmH k^2 ρ1 = 0

criterion for instability occurs when ^ <0

k^2 < 4πG ρ0 / cs^2

giving critical-scale length

λJ = √(πcs^2/Gρ0) = √(πkT/Gρ0µmH)

Question 3

If the characteristic length, L, of the perturbation is larger than Jeans Length then

Answer 3

the Jeans instability Criterion reveals that gravitational collapse will occur

Question 4

Jeans Mass

Answer 4

MJ = ρ0λ^3 = π^(3/2)cs^3 ρ0^(−1/2) G^(−3/2)

Question 5

sound speed

Answer 5

cs = dP/dρ = γρ^( γ-1)

Question 6

Evolution of density pertubation

Answer 6

flat universe

ρ = 3H^2/8πG

4πGρ =3H^2/2 = 2/3t^2

cs^2 = dP/dρ and P = Poργ

cs^2 ~ ρ^( γ-1) ~ t^(-2(γ-1))

gives

s(double dot) +4/3t s(dot) + (η^2/t^(2(γ-1/3)) - 2/3t^2) s = 0

Question 7

cs at decoupling

Answer 7

= c/√3 for radiated-dominated

= √(kT/mH) for matter dominated

Question 8

∇P1 =

Answer 8

cs^2 ∇p1

Question 9

MJ definition

Answer 9

defines a maximum stable mass concentration in which stable density perturbations can be supported

Any mass M > MJ should gravitationally collapse

Question 10

P ∝ p^γ

Answer 10

MJ ∝ (∂P/∂p)^(3/2) p0^(-1/2)

MJ ∝ (p0^(γ-1))^(3/2) p0^(-1/2)

Question 11

γ < 4/3

Answer 11

new MJ decreases as p increases

the fragments lose heat in order for gravitational collapse to occur

Question 12

γ > 4/3

Answer 12

new MJ increases as p increases

gravitational collapse halts

Question 13

If Hubble flow is taken into account

Answer 13

produce time-dependent coefficients in the density evolution eq

experimental law is replaced by a power law

jeans mass persists but allowing for an expanding universe means that fragmentation is less plausiable as an explanation for the observed mass distribution in the universe

Question 14

Newtonian cosmology

Answer 14

the study of the evolution of the Universe using only Newton’s laws to describe gravity

Question 15

matter-dominated

Answer 15

the epoch where the evolution of the universe is dominated by non-relativistic particles

Question 16

radiated-dominated

Answer 16

the epoch where the evolution of the universe is dominated by relativistic particles

Question 17

pm p,Ω

Answer 17

pm = pm,0/a^3 and pΩ = pΩ,0/a^2

pΩ,0/a^2 = pm,0/a^3

hence Ω(Ω,0)/a^4 = Ω(m,0)/a^3

a = Ω(R,0)/Ω(m,0)