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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?

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    $\begingroup$ Is the universe that you're modeling expanding? $\endgroup$
    – Daddy Kropotkin
    Nov 22, 2020 at 23:17
  • $\begingroup$ Yes definitively $\endgroup$
    – user255856
    Nov 22, 2020 at 23:19
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    $\begingroup$ 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. $\endgroup$
    – Daddy Kropotkin
    Nov 22, 2020 at 23:26
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    $\begingroup$ 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 $H = \dot{a}/a$ $\endgroup$
    – Daddy Kropotkin
    Nov 22, 2020 at 23:32

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$$dt = \frac{dt}{da}da \frac{a}{a}$$

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

so we obtain

$$dt = \frac{1}{H_0E(z)} \times \frac{-dz}{(1+z)^2}\times(1+z)$$

$$dt = -\frac{dz}{(1+z)H_0E(z)} \equiv -\frac{dz}{(1+z)H(z)} $$

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