Angular diameter distance to surface of last scattering

[thecourtholio]

Homework Statement

1) Calculate the angular diameter distance to the last scattering surface in the following cosmological models:

i) Open universe, ΩΛ= 0.65, Ωm = 0.30
ii) Closed universe, ΩΛ = 0.75, Ωm = 0.30
ii) Flat universe, ΩΛ = 0.75, Ωm = 0.25

Describe how the CMB power spectrum changes in each of these models. Compare your results to the Benchmark model, ΩΛ = 0.7, Ωm = 0.3

Homework Equations

d_A=d_hor(t₀)/z_ls
d_hor(t₀)= c ∫ dt\a(t)
H2/H₀2 = Ω_r/a⁴+Ω_m/a³+(1-Ω_r-Ω_m-Ω_Λ)/a²+Ω_Λ
H₀t=∫₁^a_ls da[Ω_r/a^2+Ω_m/a + Ω_Λa²+(1-Ω_r-Ω_m-Ω_Λ)]^-1/2
z_ls=1100
a_ls= 1/(1+z_ls)

The Attempt at a Solution

First off, is is it safe to assume that H_ot is the horizon distance (or proper distance)? Because that's what I'm going off of so if that's not correct then everything I've done is wrong anyway.
So far I have tried doing the integration of the 4th equation listed above for the open universe but I keep getting a negative number. Does the negative just mean that its in the past time? And if my assumption that Ht is not the horizon distance, then how do I relate the answer from the integral to the equation for the horizon distance (the 2nd eq listed above)? I think that the integration I did (eq 4) gives me t(a) rather than a(t) but then do I need to get a(t) in order to do the integral for d_hor?

Sorry if my questions are confusing. I am lost in a sea of equations and integrations.

 

[phyzguy]

I strongly recommend Hogg's paper "Distance Measures in Cosmology" , which goes through these things in detail. You are much better off working in redshift space, which makes the integrals much easier. Then, you can write the comoving distance to an object at redshift z (see Hogg's equations 14, and 15) as:

DC = \frac{c}{H_0}\int_0^z \frac{dz'}{\sqrt{\Omega_M(1+z')^3 + \Omega_k(1+z')^2 + \Omega_\Lambda}}

You can then convert to angular diameter distance using Hogg's equations 16 and 18. You may still have to do the integral numerically, but this is much simpler than the approach you are taking.