# Demonstration of comoving volume between 2 redshifts

• I

## 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 :

##
##

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]

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

Is it right to write :

##\dfrac{1}{\sqrt{1-\kappa r^{2}(z)}} \dfrac{c \mathrm{d} z}{H(z)}=\mathrm{d} r## ??

I can't infer this relation, so if someone could help me, this would be fine. The only thing I have found is :

##\dfrac{c\text{d}z}{H(z)}\dfrac{a(t)}{\sqrt{1-\kappa r^{2}(z)}} =-c\,\mathrm{d} t##

Regards

PeterDonis
Mentor
2019 Award
The comoving volume of a region covering a solid angle ##\Omega## between two redshifts ##z_{\mathrm{i}}## and ##z_{\mathrm{f}}##
There are several possible issues here that need to be resolved before we can even do a calculation.

First, the "volume" you are talking about is a 4-dimensional spacetime volume, not a 3-dimensional spatial volume. Do you realize that?

Second, what does "a region covering a solid angle" mean? A solid angle alone isn't enough to bound a finite region; pick a solid angle in the sky and you can look in any direction within that solid angle out to infinity. (Unless you are in a closed universe, but you mention a spatially flat universe, which is not closed, so whatever calculation you are going to make has to cover that case.)

Third, what do you mean by "between two redshifts"? Do you mean "between two spacelike hypersurfaces of constant cosmological time at these two redshifts"? That would seem to be the most natural interpretation, but you should be explicit.

PAllen
2019 Award
Do you, @fab13 , mean reshift of galaxies as observed by us, or redshift of the CMB from when the universe became transparent? Two completely different things. I am guessing you meant the first.

(1) Sorry, I have a naive point of view of the comoving volume notion, i.e I thought that it was the difference between the volume of universe at ##z=z_f## and the volume of universe at ##z=z_i## with ##z_i<z_f##, wasn't it ?

By 4-dimensional spacetime volume, you mean that we have to mix the time with 3D spatial coordinates, like I said above ?. Redshift has the special property which is duality : it represents both time and distance, so this is where my confusions appear.

So from my point of view, I think I have to do calculations taking into account this duality and the difference of physical volume (classical volume of a sphere) that I have cited into (1) sentence above. Concerning the solid angle, I don't know exactly its expression in this case (I only know the classical ##\text{d}\Omega=\text{d}\theta\,\text{sin(}\theta\text{)}\text{d}\phi##).

What do you think about it ?

Regards

PeterDonis
Mentor
2019 Award
I have a naive point of view of the comoving volume notion, i.e I thought that it was the difference between the volume of universe at ##z=z_f## and the volume of universe at ##z=z_i## with ##z_i<z_f##
The spatial volume of the universe is infinite, so this makes no sense.

By 4-dimensional spacetime volume, you mean that we have to mix the time with 3D spatial coordinates, like I said above ?
It's not a matter of "mixing". Spacetime is 4-dimensional. So a subset of it will in general be 4-dimensional as well. It is possible to pick out subsets that have fewer dimensions, but it's not clear to me whether that's what you're trying to do.

Redshift has the special property which is duality : it represents both time and distance
No, by itself it represents neither. It can be correlated with either one, if you have other data as well as the redshift.

What do you think about it ?
I still can't tell what you're trying to calculate. I suspect you don't fully understand what you're trying to calculate. So I think we need to take a step back: where are you getting this "comoving volume between 2 redshifts" thing from in the first place? Why do you care about it? What higher level question are you trying to answer?

@PeterDonis

here's the source and the context of my initial post where equation ##(1)## appears :

2.2. Distance measurements
The comoving distance to an object at redshift ##z## can be computed as
##r(z)=\frac{c}{H_{0}} \int_{0}^{z} \frac{\mathrm{d} z}{E(z)}##
Although this quantity is not a direct observable, it is closely related to other distance definitions that are directly linked with cosmological observations. A distance that is relevant for our forecasts is the angular diameter distance, whose definition is based on the relation between the apparent angular size of an object and its true physical size in Euclidean space, and is related to the comoving distance by
##
D_{A}(z)=\left\{\begin{array}{ll}
(1+z)^{-1} \frac{c}{H_{0}} \frac{1}{\sqrt{\left|\Omega_{k 0}\right|}} \sin \left[\sqrt{\left|\Omega_{k, 0}\right|} \frac{H_{0}}{c} r(z)\right] & \text { if } \Omega_{x, 0}<0 \\
(1+z)^{-1} r(z) & \text { if } \Omega_{K, 0}=0 \\
(1+z)^{-1} \frac{c}{H_{0}} \frac{1}{\sqrt{\Omega_{K}, 0}} \sinh \left[\sqrt{\Omega_{K}, D_{0}} \frac{H_{0}}{c} r(z)\right], & \text { if } \Omega_{K, 0}>0
\end{array}\right.
##
Also relevant for our forecasts is the comoving volume of a region covering a solid angle ##\Omega## between two redshifts ##z_{i}## and ##z_{f},## which is given by ##V\left(z_{i}, z_{i}\right)=\Omega \int_{a_{i}}^{a} \frac{r^{2}(z)}{\sqrt{1-K r^{\prime}(z)}} \frac{c d z}{H(z)}##
(14)
for a spatially flat universe ##(K=0),## the latter becomes ##V\left(z_{i, z_{i}}\right)=\Omega \int_{r_{(i)}}^{\left(i_{i}\right)} r^{2} \mathrm{d} r=\frac{\Omega}{3}\left[r^{3}\left(z_{i}\right)-r^{3}\left(z_{i}\right)\right]##
(15)
These expressions allow us to compute the volume probed by Euclid within a given redshift interval.
I hope that you will understand better the context of my initial question and so help me.

Regards

PAllen
2019 Award
So this says my guess in post #4 is correct. Your recent post (showing why we emphasize a reference for discussion so much) immediately points up a misunderstanding. You say you want to see how to compute a given formula for comoving volume for a solid angle and red shift range from us, in terms of comoving distance, but the formulas you then quote are for angular size distance. Why on earth would you want to compute a comoving volume from angular size distance formulas?

I don't want to make confusions : the only thing that I wanted is just to get the demonstration which allows to find :

##V\left(z_{i}, z_{i}\right)=\Omega \int_{a_{i}}^{a} \frac{r^{2}(z)}{\sqrt{1-K r^{\prime}(z)}} \frac{c d z}{H(z)}##

nothing else.

PeterDonis
Mentor
2019 Award
here's the source
What book is this from? Just cutting and pasting doesn't tell us the source.

I am doing bibliographic researches on this paper , you will see the formulas that I have cited.

PAllen
2019 Award
Actually, looking at your post #7, I immediately noticed some inconsistencies in your volume formulas, and guessed what they ought to be. Please compare yours to the paper carefully. There are important errors in your rendition of them.

[edit: actually, in your OP you have the volume formulas correct, the error is only in post #7. So you just want to know how these are derived. You are not insisting they be derived from the irrelevant angular distance formulas. I believe the derivation is pretty straightforward, but I will not have the time to write it up any time soon.]

@PAllen . Thanks, I realized that I have written bad bounds for integral :

it is not : ##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)##

but rather :

##V\left(z_{i}, z_{i}\right)=\Omega \int_{a_{i}}^{a} \frac{r^{2}(z)}{\sqrt{1-K r^{\prime}(z)}} \frac{c d z}{H(z)}##

with bounds which refer to the scale factor, doesn't it ?

Do you agree ? However, I can't still get to demonstrate this relation. A little help wouldn't be too much.

Regards

PAllen
2019 Award
@PAllen . Thanks, I realized that I have written bad bounds for integral :

it is not : ##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)##

but rather :

##V\left(z_{i}, z_{i}\right)=\Omega \int_{a_{i}}^{a} \frac{r^{2}(z)}{\sqrt{1-K r^{\prime}(z)}} \frac{c d z}{H(z)}##

with bounds which refer to the scale factor, doesn't it ?

Do you agree ? However, I can't still get to demonstrate this relation. A little help wouldn't be too much.

Regards
NO. The first is correct, the second makes no sense. Look at the paper you referenced. As to the general method, in GR, a volume element is given by the square root of the metric determinant times a raw coordinate volume element. I would guess all they have done is to integrate the volume of a comoving constant time slice over a solid angle between two red shifts using the determinant of the induced 3 metric. All of this starting from the FLRW metric.

NO. The first is correct, the second makes no sense. Look at the paper you referenced. As to the general method, in GR, a volume element is given by the square root of the metric determinant times a raw coordinate volume element. I would guess all they have done is to integrate the volume of a comoving constant time slice over a solid angle between two red shifts using the determinant of the induced 3 metric. All of this starting from the FLRW metric.
1) I don't understand how can I demonstrate the relation (1) from the metric determinant times a raw coordinate volume element. If you could tell me more about this.

2) In general, I believed that big papers like this one were examined perfectly before published and I realize that's not the case : it might be many errors and however, the paper would be accepted : strange, isn't it ? Surely, I idealize too much the research domain.

3) @PAllen : if you had any starting point for my demonstration (equation 1), this would be nice to show it.

Regards

PAllen
2019 Award
They are doing what I described, with a few tricks.

Note, from definitions earlier in the paper, that the collection of terms near dz makes it effectively dr. Then, note that restricted to a constant cosmological time, you have one of the 3 homogeneous, isotropic 3 metrics (this is the induced 3-metric I was referring to). See, for example, the reduced circumference coordinate section of:
https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

Then, instead of working from a complete volume element, they note that holding r constant, this 3-metric gives area as just solid angle times radius squared. Solid angle is constant for all r, by specification, so it comes out as a constant. Then the relation of dr to comoving distance to complete an infinitesimal shell volume is given by the square root of the line element above holding angle constant. With these steps you arrive exactly at the formula they give. There is no mistake, and further, all of this is expected to be straightforward to the intended audience of the paper.

Last edited:
Hello,

I have just taken over this issue and I tried to make progress. Following PAllen's advises, I did the following small calculation :

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} \theta\end{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}^2\theta##

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}r\,\text{sin}\theta\,\text{d}\theta\, \text{d}\phi##

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

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

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

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 :

Where is my error ?

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

@PAllen

a volume element is given by the square root of the metric determinant times a raw coordinate volume element. I would guess all they have done is to integrate the volume of a comoving constant time slice over a solid angle between two red shifts using the determinant of the induced 3 metric. All of this starting from the FLRW metric.
What do you mean please by "a raw coordinate volume element" ?

Isn't really anyone that could help me to prove the relation ##(1)## above from my attempt in post #17 ?

PeterDonis
Mentor
2019 Award
If I consider only slice times constant
You don't seem to be, since you start out with an integral from ##0## to ##t_0##, which is an integral over multiple time slices.

Also, the expression you appear to be trying to find appears to be integrating between multiple time slices, since it is treating the Hubble value ##H(z)## as a function of redshift. But the Hubble value is the same everywhere on a slice of constant time; it's not a function of anything unless you are considering multiple time slices.

I still don't think you fully understand what you are trying to calculate.