# Expresion of comoving distance

I have a simple question :

How to prove the following relation :

The comoving distance to an object at redshift $$z$$ can be computed as

$$r(z)=\dfrac{c}{H_{0}} \int_{0}^{z} \dfrac{\mathrm{d} z}{E(z)}$$

from the relation :

$$r(t)=\int_{0}^{t} \dfrac{c\mathrm{d} t}{R(t)}$$

I tried to use with the definition : $$1+z= \dfrac{R_{0}}{R(t)}$$ but I can't conclude.

Any help is welcome.

UPDATE 1 : @Tosic's demonstration seems to be correct. But the factor $$R_{0}$$ is not disappearing. Indeed, If I do :

$$\dfrac{\text{d}(1+z)}{\text{d}t} = \dfrac{\text{d}z}{\text{d}t} = -\dfrac{H(t)}{R(t)}\,R_{0}$$

which implies :

$$\int_{0}^{z} \dfrac{c\text{d}z}{H(z)} = \int_{0}^{t}c\text{d}t\dfrac{R_{0}}{R(t)} = R_{0} \int_{0}^{t}\dfrac{c\text{d}t}{R(t)}$$

How to get rid of the factor $$R_{0}$$ ? Since if I multiply the comoving coordinate $$r(t)$$ by $$R_{0}$$, I get the cosmological horizon (the limit of observable universe if I integrate up to $$z=1100$$), don't I ?

According to this, the Hubble constant for redshift z is $$H_0E(z)$$. Meaning we need to prove that $$\int_{0}^{z_0}\frac{cdz}{H(z)} = \int_{0}^{t_0}\frac{cdt}{R(t)}$$ Take the first derivative of both sides of your second equation to obtain, by the chain rule, and the equation (this is the definition of the Hubble constant) $$H = \frac{R(t)'}{R(t)}$$ the following: $$\frac{dz}{dt} = -\frac{1}{R(t)^2}*R'(t) = -\frac{H(t)}{R(t)}$$ That the redshift is zero for the current time, and some value greater than zero for some time before the current time should explain the minus sign. After multiplying with $$c$$ and placing the small changes on both sides this becomes $$\frac{cdz}{H(z)}=-\frac{cdt}{R(t)}$$, which looks like it can be integrated to obtain what we need to prove (the integral from z to 0 is the one from 0 to t).
• Thanks for your quick answer. Your demonstration seems to be correct. Unfortunately, the factor $R_{0}$ puts the mess. I point this in my UPDATE 1. – youpilat13 Jul 4 at 17:23
• from your point of view, what is the dimension of $R(t)$ and $R_{0}$ ? since when we want to compute the cosmological horizon, i.e a physical distance, we multiply the comoving coordinate $r(t)$ by $R_{0}$ with $R_{0}$ the scale factor of today. Or is it the comoving coordinate $r(t)$ that has the demension of a length and $R(t)$ just a scale factor as it is called ? – youpilat13 Jul 4 at 18:12