Question

In an accelerating or decelerating Universe, the redshift z of a particular source will slowly change over time to, as measured by an observer.

a. Show that the rate of change is
dz/dt₀ H₀(1+z) - H(z)
where H(z) = Rₑ/Rₑ is the Hubble parameter at the time of emission.
Hint: Differentiate the definition of redshift, 1 Z R₀/Rₑ, with respect to to. Use the chain rule to deal with expressions such as dRₑ/dt₀.

Answer 1

We have shown that the rate of change of redshift z with respect to t₀ is given by:

dz/dt₀ = H₀(1 + z) - H(z

Here's the derivation of the rate of change of redshift:

Start with the definition of redshift:

z = 1 + R₀/Rₑ

Differentiate both sides with respect to t₀:

dz/dt₀ = d/dt₀ (1 + R₀/Rₑ)

Apply the chain rule to dRₑ/dt₀:

dz/dt₀ = 0 + (1/Rₑ) * dR₀/dt₀ - (R₀/Rₑ²) * dRₑ/dt₀

Express dR₀/dt₀ in terms of H₀:

dR₀/dt₀ = H₀ * R₀

Express dRₑ/dt₀ in terms of H(z):

dRₑ/dt₀ = H(z) * Rₑ

Substitute these expressions into the equation:

dz/dt₀ = (1/Rₑ) * H₀ * R₀ - (R₀/Rₑ²) * H(z) * Rₑ

Simplify:

dz/dt₀ = H₀ - H(z) * (R₀/Rₑ)

Replace R₀/Rₑ with 1 + z from the definition of redshift:

dz/dt₀ = H₀ - H(z) * (1 + z)

Rearrange to match the desired form:

dz/dt₀ = H₀(1 + z) - H(z)

Therefore, we have shown that the rate of change of redshift z with respect to t₀ is given by:

dz/dt₀ = H₀(1 + z) - H(z)