In a spatially flat universe containing only matter, the Friedmann equation can be used to describe the expansion of the universe. The equation can be solved to obtain the scale factor in terms of the expansion rate today. Formulas for the comoving distance, luminosity distance, and angular diameter distance of a source observed today as a function of its redshift can also be derived.
In a spatially flat universe containing only matter, we can use the Friedmann equation to describe the expansion of the universe. The Friedmann equation is given by:
H^2 = (8πGρ)/3
where H is the expansion rate of the universe, G is the gravitational constant, and ρ is the density of matter. To solve this equation, we can assume that the density of matter is constant, and write it as ρ = ρ0/a^3, where ρ0 is the present density of matter and a is the scale factor. Substituting this into the Friedmann equation, we get:
H^2 = (8πGρ0)/3a^3
We can also express the expansion rate today H0 in terms of the present density of matter ρ0, using the equation H0^2 = (8πGρ0)/3. Dividing the two equations, we get:
(H/H0)^2 = 1/a^3
Taking the square root of both sides, we get:
H/H0 = 1/a^(3/2)
Therefore, we can express the scale factor as a function of the expansion rate today H0:
a(t) = (H/H0)^(-2/3)
The comoving distance de(z) is given by:
de(z) = c * ∫(0, z) (1/H(z')) dz'
where c is the speed of light, and the integral is taken from 0 to z. The luminosity distance dL(z) is given by:
dL(z) = (1+z) * de(z)
The angular diameter distance dA(z) is given by:
dA(z) = de(z)/(1+z)