Demonstration of comoving volume probed between 2 redshifts

1) I can’t manage to find/justify the relation (1) below, from the common relation (2) of a volume.

2) It seems the variable r is actually the comoving distance and not comoving coordinates (with scale factor R(t) between both).

The comoving volume of a region covering a solid angle $Omega$ between two redshifts $z_{mathrm{i}}$ and $z_{mathrm{f}},$is given by

$$
Vleft(z_{mathrm{i}}, z_{mathrm{f}}right)=Omega int_{z_{mathrm{i}}}^{z_{mathrm{f}}} frac{r^{2}(z)}{sqrt{1-kappa r^{2}(z)}} frac{c mathrm{d} z}{H(z)}quadquad(1)
$$

for a spatially flat universe $kappa=0)$ the latter becomes

$$Vleft(z_{mathrm{i}}, z_{mathrm{f}}right)=Omega int_{rleft(z_{mathrm{i}}right)}^{rleft(z_{mathrm{f}}right)} r^{2} mathrm{d} r=frac{Omega}{3}left[r^{3}left(z_{mathrm{f}}right)-r^{3}left(z_{mathrm{i}}right)right]
quadquad(2)$$

I would like to demonstrate it from the comoving distance with :

$$D_{mathrm{A}}(z)=left{begin{array}{ll}
{(1+z)^{-1} frac{c}{H_{0}} frac{1}{sqrt{left|Omega_{mathrm{K}, 0}right|}} sin left[sqrt{left|Omega_{mathrm{K}, 0}right|} frac{H_{0}}{c} r(z)right],} & {text { if } Omega_{mathrm{K}, 0}<0}
{(1+z)^{-1} r(z),} & {text { if } Omega_{mathrm{K}, 0}=0}
{(1+z)^{-1} frac{c}{H_{0}} frac{1}{sqrt{Omega_{mathrm{K}, 0}}} sinh left[sqrt{Omega_{mathrm{K}, 0}} frac{H_{0}}{c} r(z)right]} & {text { if } Omega_{mathrm{K}, 0}>0}
end{array}right.
$$

UPDATE 1: It’s been a long time that I posted this question.

I recently took over this isssue and I have done little progress, at least I think.

The FLRW metric can be expressed under following (0,2) tensor form :

$left[begin{array}{cccc}1 & 0 & 0 & 0 0 & -frac{R^{2}(t)}{1-k r^{2}} & 0 & 0 0 & 0 & -R^{2}(t) r^{2} & 0 0 & 0 & 0 & -R^{2}(t) r^{2} sin ^{2} thetaend{array}right]$

If I consider only slice times constant, my goal is to compute the volume probeb by a satellite between 2 redshifts.

1. We can easily find that :

$$int_{0}^{z_0}frac{cdz}{H(z)} = int_{0}^{t_0}frac{cdt}{R(t)}$$

2. Then, If I consider a volume with $text{d}r$, $text{d}theta$ and $text{d}phi$ coordinates, I have the following expression for determinant :

$$g=text{det}[g_{ij}] = -dfrac{R(t)^6}{1-kr^2},r^4,text{sin}^2theta$$

Which means that I have :

$$text{d}V=sqrt{-g}text{d}^3x = dfrac{R(t)^3}{sqrt{1-kr^2}},r^2,text{d}rtext{sin}theta,text{d}theta, text{d}phi$$

$$V=inttext{d}V= int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2,text{d}rtext{sin}theta,text{d}theta, text{d}phi$$

$$V = int text{d}Omega int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2,text{d}r$$

$$rightarrowquad V = Omega int dfrac{R(t)^3}{sqrt{1-kr^2}},r^2,text{d}r$$

with $Omega$ the solid angle considered.

But as you can see, I am far away from the expression $(1)$ that I would like to find, i.e :

$$Vleft(z_{mathrm{i}}, z_{mathrm{f}}right)=Omega int_{z_{mathrm{i}}}^{z_{mathrm{f}}} frac{r^{2}(z)}{sqrt{1-kappa r^{2}(z)}} frac{c mathrm{d} z}{H(z)}quadquad(1)$$

Any help would be fine, I am stucked for the moment.