- #1

- 204

- 4

## Summary:

- I would like to get help about an expression of comoving volume between 2 redshifts

## Main Question or Discussion Point

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}},## to find is :

##

V\left(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)}\quad\quad(1)

##

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

##V\left(z_{\mathrm{i}}, z_{\mathrm{f}}\right)=\Omega \int_{r\left(z_{\mathrm{i}}\right)}^{r\left(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]

\quad\quad(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.

##

Anyone could give me some clues/tracks/suggestions to get it ?

Regards

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}},## to find is :

##

V\left(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)}\quad\quad(1)

##

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

##V\left(z_{\mathrm{i}}, z_{\mathrm{f}}\right)=\Omega \int_{r\left(z_{\mathrm{i}}\right)}^{r\left(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]

\quad\quad(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.

##

Anyone could give me some clues/tracks/suggestions to get it ?

Regards