Distances in cosmology I want to make sure that I understand the different distance measures is cosmology.
To do that I consider the FLRW metric:
$$ ds^2=dt^2-R(t)^2\left(\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2\right)$$
My first question is how can I see that the ($r,\theta,\phi$) in the FLRW metric are comoving coordinates? Furthermore in my book (Kolb and Turner) it is said that $R(t)$ has dimension length and $r$ is dimensionless and goes from $0$ to $1$?
Next I consider a light source $S$ and a light detector $D$. At the beginning of our considerations $S$ and $D$ do not move with respect to each other in the comoving coordinates. The light propagates along a trajectory with $d\theta=0=d\phi$.
The comoving distance ($d_c$) between $S$ and $D$ is then
$$d_c=\int_{t_S}^{t_D}\frac{dt}{R(t)}=\int_{r_S}^{r_D}\frac{dr}{\sqrt{1-k r^2}}\overset{k=0}{=}r_D-r_S$$
where the light is emitted at $t_S$ and received at $t_D$. We have shown explicitly that for $k=0$ the comoving distance does not change.
Short aside: The times $t_D$ and $t_S$ can be measured as follows: $D$ and $S$ have a clock attached to each other. They will be synchronized at the very beginning and then $t_S$ will just be measured by the clock which sits at $S$ and $t_D$ will be measured by the clock which sits at $D$. is ts correct?
The physical distance (also called proper distance, correct?) between $S$ and $D$ at the time of emission $t_S$ is
$$d_p=R(t_S)\int_{r_S}^{r_D}\frac{dr}{\sqrt{1-k r^2}}=R(t_S)\int_{t_S}^{t_D}\frac{dt}{R(t)}=R(t_S)d_c$$
This would be the distance which I would measure if I would stop the time and measure the distance with a normal ruler, correct? But to calculate this distance I have to know the comoving distances?
The luminosity distance $d_L$ is defined as
$$d_L^2=\frac{\mathcal{L}}{4\pi\mathcal{F}}$$
where $\mathcal{L}$ is the total emergy emitted by a source per time and $\mathcal{F}$ is the energy that a detector receives on a specific detector area per time. If the universe would not expand this distance would be the physical distance. In my book (Kolb and Turner) they give an equation for $\mathcal{F}$:
$$\mathcal{F}=\frac{\mathcal{L}}{4\pi R^2(t_D)r_S^2(1+z)^2}$$
how is this derived? They say it follows from energy conservation but I cannot see this.
If I accept the previous equation it follows:
$$d_L^2=R^2(t_D)r_S^2(1+z)^2$$
They set $r_D=0$ and therefore $r_S=d_c$. This would mean that this is a relation between luminosity distance and comoving distance. Is this correct?
One last question: In Kolb and Turner they derive the Hubble law with $d_L$.
$$H_0 d_L=z+\frac{1}{2}(1-q_0)z^2$$
Does this mean that the distance in the Hubble law is actually not a physical distance that one would measure with a ruler? The physical distance would be obtained by using the relations above which I have worked out!?
 A: 
My first question is how can I see that the (,,) in the FLRW metric are comoving coordinates?

Comoving distances remain constant with time. The metric in the form you wrote it places all the time dependence into the function $R(t)$, and the coordinates $r,\theta,\phi$ are independent of time. So any distance expressed in terms of $r,\theta,\phi$ is a comoving distance.

Furthermore in my book (Kolb and Turner) it is said that () has dimension length and  is dimensionless and goes from 0 to 1?

There are several conventions for where to put the units. It doesn't actually matter too much for spatially flat universes ($k=0$); this is the most relevant case because we now know the Universe is spatially flat to very good precision. But one choice is certainly to define $R(t)$ to have units of length, in which case $r, \theta, \phi, k$ are all dimensionless. However, $r$ certainly must range from $0$ to $\infty$ when $k=0$, otherwise there will be a boundary of the manifold at $r=1$ which a geodesic observer could reach in finite proper time. Perhaps the book was saying that $k$ can take on discrete values of $-1, 0, +1$? (There is another convention where $R(t)$ is dimensionless and $r$ and $k^{-1/2}$ have units of length). Or, the book may have been focused on the case when $k=+1$, in which case $r=1$ is at an infinite spatial distance.

Short aside: The times  and  can be measured as follows:  and  have a clock attached to each other. They will be synchronized at the very beginning and then  will just be measured by the clock which sits at  and  will be measured by the clock which sits at . is ts correct?

Sounds reasonable, but there isn't a need to be super careful about this since $D$ and $S$ are both at rest in the cosmic rest frame so have the same time coordinate.

The physical distance (also called proper distance, correct?)

Yes.

This would be the distance which I would measure if I would stop the time and measure the distance with a normal ruler, correct?

Yes.

But to calculate this distance I have to know the comoving distances?

If you know one distance measure, and the scale factor, you can calculate all the others. But it's not the case that you will always be given the comoving distances and need to calculate some other distance measure. Observationally we never know the comoving distances directly.

how is this derived? They say it follows from energy conservation but I cannot see this.

To be honest I think invoking "energy conservation" is confusing here, since energy is not conserved in the FLRW Universe. Here's how I would try to explain it in a few words...
Imagine photons emitted from the surface of a star. If the emitted power integrated over the entire surface is $P$, then the power that propagates through a surface at proper distance $r$ must also be $P$. This is true if we work in comoving coordinates.
However in physical coordinates, the photons lose energy as they propagate because the expansion of the Universe causes their wavelength to expand by a factor of $1+z$. Additionally, power is an energy flux per unit time. The time interval between the emission of two photons as seen by the observer is longer than the time interval between the emission of two photons at the source by a factor of $1+z$ Therefore the luminosity (power flux) through a surface is really smaller than we would expect in comoving coordinates, by a factor of $(1+z)^2$.
Since the luminosity distance $d_L$ is proportional to the inverse square root of the luminosity, the luminosity distance is larger than the comoving distance $d_C$ by a factor of $1+z$. The relationship in equations is $d_L = (1+z) d_C$.
Here is a source I found on google which describes the derivation of different distance metrics in more detail: https://wwwmpa.mpa-garching.mpg.de/~gamk/TUM_Lectures/Lecture3.pdf

Does this mean that the distance in the Hubble law is actually not a physical distance that one would measure with a ruler? The physical distance would be obtained by using the relations above which I have worked out!?

Hubble's law, in the sense $v = H d$, is only valid when the distances are small enough that $z \ll 1$, in which case all the distance metrics are approximately the same. However you are right that to measure the acceleration of the Universe (which goes beyond Hubble's law), you have to be more precise The supernova measurements which were the first evidence of the acceleration of the Universe used the luminosity distance.

Finally, as a friendly tip for the future: in general it's a good idea to ask focused questions with 1 or at most 2 questions per post. For little niggling doubts about definitions, often doing a few problems is a good way to build confidence and clean up confusion, and filter for larger conceptual issues that are harder to answer on your own.
