Distance to surface of last scattering in +, -, and flat uni's

[thecourtholio]

Homework Statement

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

i) Open universe, Ω_Λ= 0.65, Ω_m = 0.30
ii) Closed universe, Ω_Λ = 0.75, Ω_m = 0.30
ii) Flat universe, Ω_Λ = 0.75, Ω_m = 0.25

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

Homework Equations

d_A=d_hor(t₀)/z_ls
d_hor(t₀)= c ∫_te^t₀ dt\a(t)
H²/H₀² = Ω_m/a²+(1-Ω_m-Ω_Λ)/a²+Ω_Λ
H₀t=∫₀^a da[Ω_m/a + Ω_Λa²+(1-Ω_m-Ω_Λ)]^-1/2
z_ls=1100

The Attempt at a Solution

Honestly, I'm not even sure where to start. My main problem is with part 1 and 2 for the open and closed universes and trying to calculate the horizon distances in those universes. I think I'm just having problems understanding how to relate the friedmann equation (the third one listed above) to the horizon distance (the second equation). In my textbook it says that for the benchmark model d_hor(t₀) = 3.24c/H₀ but where did they get the 3.24?
If someone could just kind of point me to the right direction that would be super appreciated. Also, my integral skills are a little rusty so any help with those would be awesome.

 

[mark726]

Thank you for your question. Calculating the angular diameter distance to the last scattering surface in different cosmological models is a complex task that requires knowledge of various equations and concepts. I will try to provide some guidance on how to approach this problem.

Firstly, it is important to understand the concept of the angular diameter distance. This is the distance at which an object of known physical size appears to have a certain angular size. In this case, we are interested in the angular diameter distance to the last scattering surface of the cosmic microwave background (CMB) radiation. This is the surface at which the universe became transparent to photons, about 380,000 years after the Big Bang.

To calculate the angular diameter distance, we need to use the Friedmann equations, which describe the evolution of the universe. These equations relate the expansion rate of the universe (Hubble parameter) to the energy content of the universe (represented by Ωm and ΩΛ). The equations also involve the scale factor (a), which describes the size of the universe at a given time.

For the open universe (i), the Friedmann equation can be written as:

H2/H02 = Ωm/a2 + (1-Ωm-ΩΛ)/a2 + ΩΛ

where H0 is the current value of the Hubble parameter. To calculate the horizon distance (dhor) for this model, we need to integrate this equation from the present time (t0) to the time of last scattering (tls). This will give us the comoving distance (dcom) to the last scattering surface, which is related to the angular diameter distance (dA) by the following equation:

dA = dcom/(1+zls)

where zls is the redshift of the last scattering surface. Since we know that zls = 1100, we can use this equation to calculate the angular diameter distance.

For the closed universe (ii), the Friedmann equation takes a slightly different form:

H2/H02 = Ωm/a2 + (1-Ωm-ΩΛ)/a2 + ΩΛ - k/a2

where k is the curvature of the universe. For a closed universe, k is a positive value. The rest of the process is the same as for the open universe.

For the flat universe (iii), the Friedmann equation simplifies to:

H