# Redshift time relation

I was reading that the relation

$$dt=\frac{dz}{H(z)(1+z)}$$

between time and redshift (H is the Hubble constant) holds.

I don't understand this. I thought the relation between time and redshift is

$$z=H(t-t_0)\Rightarrow dz=H dt$$

What am I doing wrong?

• Is the universe that you're modeling expanding? Nov 22, 2020 at 23:17
• Yes definitively
– user255856
Nov 22, 2020 at 23:19
• Then the second equation you provide is incorrect, since the expansion of the universe must be accounted for, ie. H is a function of redshift z. Otherwise you're treating H as a constant, i.e the universe is not expanding. Nov 22, 2020 at 23:26
• Maybe you don't realize that the definition of H depends on the cosmology being considered. So, for example the FLRW metric defines the scale factor $a = (1 - z)^{-1}$ and $H = \dot{a}/a$ Nov 22, 2020 at 23:32

$$dt = \frac{dt}{da}da \frac{a}{a}$$
and we know that $$1/a = 1+z$$ and $$da = -(1+z)^{-2}dz$$ and $$\frac{da/dt}{a} = H_0E(z)$$
$$dt = \frac{1}{H_0E(z)} \times \frac{-dz}{(1+z)^2}\times(1+z)$$
$$dt = -\frac{dz}{(1+z)H_0E(z)} \equiv -\frac{dz}{(1+z)H(z)}$$