Description
We have found some analytical solutions of the first Friedmann equation, but in most cases this is not possible. However, it is possible to explore the qualitative properties of the solutions in a simple, graphical way. With matter, spatial curvature, and a component with equation of state parameter w (the case with a cosmological constant corresponds to w = −1, Ωw0 = ΩΛ0) the first Friedmann equation is
.
By introducing new variables x = a/a0 and τ = H0t, this equation can be written in dimensionless form:
= constant. (1)
Note that the first term on the left-hand side is always ≥ 0, just like a kinetic energy. So this equation has the same form as the equation for conservation of energy of a particle moving in one dimension in a potential,
T + U = E = constant,
This gives us a quick way of checking the qualitative nature of the solution of the Friedmann equations for given values of the density parameters. Plot the ‘potential energy’ U as a function of x = a/a0, and in the same plot draw the line E(x) = 1 − Ωm0 − Ωw0 = constant. The allowed values of x (that is, a) are those for which U lies below or touches E. The scale factor cannot have values fow which U lies above E, because that would mean that the ‘kinetic energy’ would be negative.
Along with this text you will receive a table of measured luminosity distances with associated errors. The table is in the format (redshift, luminosity distance, error). The distances and the errors are given in units of Gpc (1 Gpc = 109 pc). You will now use these measurements to find empirical constraints on cosmological parameters. Let us, as an example, look at the case when we want to constrain Ωm0 and ΩΛ0. Let us call the expression for the luminosity distance based on these two parameters for our model. Given values for Ωm0 and ΩΛ0, we want to know the probability of the model, given the data, P(model|data). There is no ready recipe for calculating this probability, but a result known as Bayes’ theorem says that
P(data|model)P(model)
P(model|data) = (2)
P(data)
The second factor in the numerator is the probability we would assign to the model before obtaining the data, and it is called the prior. The factor in the denominator is known as the evidence. We will, as is quite common, consider both of these factors to be constants, and we then have the result
P(model|data) ∝ P(data|model). (3)
The probability on the right-hand side is known as the likelihood, and the point is that it is possible to work out how to calculate it. For example, we will assume that the observations are drawn from a Gaussian distribution. This means that we assume that if we measure the luminosity distance to the ith redshift zi to be with measurement error σi, then the probability
3
distribution for the true luminosity distance dL(zi) is
.
If we also assume that the measurements are uncorrelated, it can be shown that the likelihood is given by
P(data|model) = ,
where dL(zi,p~) is the model prediction for the luminosity distance to redshift zi for given parameter vector p~, and N is the number of observations. In our case, p~ = (Ωm0,ΩΛ0). To find the most probable values of Ωm0 and ΩΛ0, we want to maximize the likelihood as a function of these two parameters, and this is equivalent to minimizing the quantity
.
We cannot be sure that the most probable value is the true value. All the data allow us to find is the most probable value, and the range in which the true value probably lies. In the case you consider here, it can be shown that there is a 95 % probability that the true values of Ωm0 and Ωλ0 if found in the region which satisfies
,
where is the minimum value of χ2.
