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I am trying to prove that for a single component flat universe

$$\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{\frac{3 + 3w}{2}}$$

For a single component flat universe,

##q = \frac{2}{3 + 3w}##

##a(t) = (t/t_0)^q##

##t_0 = qH_0^{-1}##

##1 + z = (t_0/t_e)^q##

Now here is my approach,

$$\frac{dz}{dt_0} = qt_0^{q-1}t_e^{-q} = q (t_0/t_e)^qt_0^{-1}$$

$$\frac{dz}{dt_0} = q ( 1 + z) q^{-1}H_0 = H_0 (1 + z)$$

Can someone point it out what am I missing. Like I am trying over 3 days now. Either I suck at math or there's something that I am missing

Edit: This identity can be derived from above identities but I wanted to write

$$t_e = \frac{t_0}{(1 + z)^{3(1+ w)/2}} = \frac{2}{3(1 + w)H_0} \frac{1}{(1+z)^{3(1+ w)/2}}$$

$$\frac{dz}{dt_0} = H_0(1 + z) - H_0(1 + z)^{\frac{3 + 3w}{2}}$$

For a single component flat universe,

##q = \frac{2}{3 + 3w}##

##a(t) = (t/t_0)^q##

##t_0 = qH_0^{-1}##

##1 + z = (t_0/t_e)^q##

Now here is my approach,

$$\frac{dz}{dt_0} = qt_0^{q-1}t_e^{-q} = q (t_0/t_e)^qt_0^{-1}$$

$$\frac{dz}{dt_0} = q ( 1 + z) q^{-1}H_0 = H_0 (1 + z)$$

Can someone point it out what am I missing. Like I am trying over 3 days now. Either I suck at math or there's something that I am missing

Edit: This identity can be derived from above identities but I wanted to write

$$t_e = \frac{t_0}{(1 + z)^{3(1+ w)/2}} = \frac{2}{3(1 + w)H_0} \frac{1}{(1+z)^{3(1+ w)/2}}$$

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