Lecture 5

Question 1

The age of the universe: (derivation)

Answer 1

t0 = (t0 ∫ 0) dt

dt = - dz/(HoE(z)(1+z))

=> t0 = (t0 ∫ 0) dt = 1/Ho (∞ ∫ 0) dz/((1+z)E(z))

Question 2

Is a single statement of E(z) valid for all z -> ∞

Answer 2

if Ωr0 is important we should use

H(z) = Ho [ Ωm0(1+z)^3 + Ωr0(1+z)^4 + Ωk0(1+z)^2 + ΩΛ0]^(1/2)

but this is only true at very high redshift so can be used for z -> ∞

if Ωr0 is not important the term can be dropped

in the end we come to

t0 = 9.78h^-1 Ƒ(Ωm0,ΩΛ0,Ωk0) Gyr

Question 3

Ƒ(Ωm0,ΩΛ0,Ωk0) =

Answer 3

(∞ ∫ 0)dz/((1+z)E(z))

for a de-sitter universe Ƒ = 2/3

t0 = 6.52h^-1 Gyr

Question 4

why is h > 0.8 a big problem

Answer 4

we already know the Milky Way is older

we assume the universe is empty and thus Ƒ = 1 and the predicted age of the universe grows

Question 5

For the concordance model Ƒ =

Answer 5

0.97

giving t0 = 13.5 Gyr

Question 6

The Einstein Static Universe (derivation)

Answer 6

p = Λ/4πG ; a = c/√(4πGp) = c/√Λ

if we perturb this solution we can show that

ε(d dot) = Aε(t) where A = 2/3 Λ

the outcome is the perturbations grow exponentially with

ε(t) ∝ √At

Question 7

The de sitter universe (derivation)

Answer 7

if we set p = 0 and k = 0 then the FM equation reduces to

H^2 = Λ/3

consequently ΩΛ = 1

taking the positive square root and noting that Λ is constant we find

a(t) = a0 exp[ √Λ/3 (t-t0) ]

if dark energy really is described by a constant Λ then the universe is tending towards a de Sitter model in the future as the matter density becomes less and less important.

Question 8

cosmological constant problem.

Answer 8

if the universe expands and mass density remains constant new mass is being produced where is all this mass coming from

comparing p(vacuum) and pΛ they differ by 120 orders of magnitude

why is the observed equivalent mass density of the vacuum energy so much smaller

if its so small can we set it to zero

it cant be set to zero as we have observed it hence it can’t not exist.

Question 9

the equation of state of dark energy

Answer 9

pΛ = -pΛc^2

this makes things worse - now we have something called a dark fluid which acts as if it has negative pressure

we visualise pressure as the result of an outward force

but if the vacuum pressure is negative, wouldn’t that have the effect of ‘pulling’ the piston inwards instead out pushing it out

but if that were the case, then why does the expansion accelerate

Question 10

more general equation of state

Answer 10

P = ωpc^2

where ω = ω(z) is the model of quintessence.

determining if ω = ω(z) is a current topic of research