Lecture 4

Question 1

angular diameter distance (formula)

Answer 1

dA(z) = ds/dθ(z)

Question 2

we want f(r) in the RW metric as a function of redshift z: (derivation)

Answer 2

(t0 ∫ t) cdt/a(t) = (r ∫ 0) dr’/√(1-kr’^2) = f(r)

we need to change variables from r to z:

a0/a(t) = 1+z => a(t) = a0/(1+z) => da/dz = - a0/(1+z)^2

now re-write the Hubble parameter as a function of z:

ȧ/a = H(z) - 1/a da/dz dz/dt

so

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

now that z = 0 and t = t0 and using a0 = a(t)(1+z) and we find that

k = 0: dL(z) = cHo^-1 (1+z) (z ∫ 0) dz’/E(z’)