Lecture 3

Question 1

finding Hubble lemaitre law from the robertson walker metric (derivation)

Answer 1

D = a(t) (rE ∫ 0) dr/√(1-kr^2) = a(t0)f(r)

differentiate D and extracting velocity

v = dD/dt = ȧ0 f(r)

multiply the LHS by a0/a0 to get

v = (ȧ/a)0 a(t0) f(r) = HoD

Question 2

The redshift of a spectral line is (formula)

Answer 2

z = (λo - λe) / λe = v/c

Question 3

Scale factor - Redshift Relation (derivation)

Answer 3

set the RW metric: ds^2 = 0

c^2dt^2 = a^2(t) (dr^2/(1-kr^2))

so

(tE+ΔtE ∫ tO+ΔtO) cdt/a(t) = (rO ∫ 0) dr/√(1-kr^2)

cΔtO/a(tO) = cΔtE/a(tE)

giving

1+z = a(t0)/a(tE)

Question 4

The HST Key project

Answer 4

H0 = 100h kms^-1 Mpc^-1

with h = 0.7

Question 5

A critical density universe (derivation)

Answer 5

its a density such that k = 0 in the FM eq

for a critical density assume Λ = 0

pc(t) = 3H^2/8πG

Question 6

Matter density makes up how much of the critical density

Answer 6

30% of the critical density

Question 7

Luminous matter makes up how much of the critical density

Answer 7

< 1% of the critical density

Question 8

Dark matter is mostly described as

Answer 8

non-baryonic and cold

Question 9

The universe has what kind of geometry

Answer 9

flat geometry

Question 10

Evolution of the Density parameters (derivation)

Answer 10

8πGp/3 = 8πGp/3 p0Ho^2/p0Ho^2

= Ho^2 (8πGp0/3Ho^2) x (p/po)

= Ho^2 Ωm0 (p/p0)

assuming mass conservation: pa^3 = p0a0^3 => (1+z)^3

therefore we can relate matter density to redshift

8πGp/3 = Ho^2 Ωm0 (1+z)^3

Question 11

Evolution of the Density parameter derivation can be used similarly to relate for curvature (formula)

Answer 11

-kc^2/a^2 = Ho^2 Ωk0 (1+z)^2

Question 12

Evolution of the Density parameter derivation can be used similarly to relate for cosmological constant (formula)

Answer 12

Λ/3 = Ho^2 ΩΛ0

Question 13

We can rewrite the Friedman equation purely in terms of redshift z:

Answer 13

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

Question 14

we can relate the present day values of the omega parameters to their values at redshift z: Ωm(z) formula

Answer 14

Ωm(z) = Ωm0(1+z)^3/E(z)^2

Question 15

:we can relate the present day values of the omega parameters to their values at redshift z: Ωk(z) formula

Answer 15

Ωk0(1+z)^2/E(z)^2

Question 16

we can relate the present day values of the omega parameters to their values at redshift z: ΩΛ(z) formula

Answer 16

ΩΛ(z) = ΩΛ0/E(z)^2

Question 17

following the discovery of expansion by hubble, cosmologists expected that the universe was

Answer 17

decelerating

Question 18

if we Taylor expand the scale factor a(t) we can extract a deceleration parameter from the second term in the expansion.

Answer 18

q0 = -ä(t0)/a(t0) 1/Ho^2 =
-a(t0)ä(t0)/ȧ^2(t0)

Question 19

if we assume a matter-dominated universe with a cosmological constant we find that q0 =

Answer 19

q0 = -0.55

negative deceleration parameter hence the universe is accelerating