Undergrad Qs re Cosmological Models Using Bayesian Probability Methods

[Buzz Bloom]

TL;DR

I have questions about how Bayesian probabilities are used when calculating the model parameters of a universe model based on the Friedmann equation at the beginning of the post body.

I am familiar with non-Bayesian methods for calculating best fit values of various parametric models, but I have not had any experience with cosmological models calculations. My understanding is that these models have five parameters:

H₀, Ω_r, Ω_m, Ω_k, Ω_Λ,​

and the last four satisfy the constraint that their sum exactly equals 1.

(If anyone is interested about how I think I would go about calculating a non-Bayesian best fit of the five parameters, I will post a description about this.)

I also have a limited understanding of calculating probabilities using Bayesian methods. I understand that to calculate a probability value (or distribution) for a variable one assumes an a priori value (or distribution) for the variable along with various contingent probabilities that depend on other variables.

Q1. What are some examples of these contingent probabilities used for cosmological modeling?

Q2. What priors are used for the variables?
I searched the paper

https://arxiv.org/pdf/1502.01589.pdf​

for use of the word “prior”, and found only one usage. (pg 4)

Thus, the CIB model used in this paper is specified by only one amplitude, A^CIB_217×217, which is assigned a uniform prior in the range 0–200 μK².​

I would much appreciate it if someone would explain what this means.

 

[kimbyd]

Aside: $\Omega_r$ is most of the time not used as a separate free parameter, because it is measurable extremely precisely by the average temperature of the CMB, which is known to a fraction of a percent (secondary aside: to be precise, $\Omega_r H_0^2$ is the value that is known with extremely high precision).

Typically, flat priors are used for most parameters in a cosmological model. There is some choice in terms of which parameters are used which does influence the outcome, however. For instance, a flat prior in $\Omega_m$ is not necessarily the same thing as a flat prior in $\rho_m$. Shifting the prior can influence the outcome by roughly one standard deviation in most reasonable cases.

Mathematically, a flat prior is the same as just ignoring the prior probability altogether.