Comoving distance constant with increasing redshift?

[FritzK]

TL;DR Summary

Why remains the comoving distance the same over all times if the redshift to a galaxy will increase?

In the literature and on the Internet it is said that this remains the same for all times, since CM = coordinate distance in the "Hubble flow". On the other hand, many diagrams show that the CM increases with greater redshift.
But if I determine a CM = x to redshift = y for a galaxy today, then in the more distant future the redshift will probably have increased to it. Then how can the comoving distance stay the same?
(Surely in many millions of years I would measure a greater redshift to the galaxy than today?)

 

[Orodruin]

By definition, the comoving distance is the distance between two comoving objects today. The actual distance between those objects at any other time will be the comoving distance multiplied by the scale factor.

(This assumes a normalization such that the scale factor today is $a_0 = a(t_0)=1$)

 

[Orodruin]

Addition: Comoving distance to objects with higher redshift today is larger because they need to be further away to have a larger redshift (as the universe is expanding).

 

[FritzK]

Thank you for your quick answer.
But all popular explanations tell that comoving distance does not change with time, the Hubble flow is not taken into account, its the coordinate distance which remains fixed and so on. So, if its senseful to ask what an observer in 1 billion years would get as the numerical value for the comoving distance to my galaxy above, he will get the same numerical value as I now or not. If not, then all those popular statements make no sense in my opinion.
If the value remains the same, then I would argue that the redshift in 1 billion years should be larger than today and then with your argument above the comoving distance could not be the same.
Is there a common agreement which numerical value for the comoving distance for the same galaxy we would get in one billion years (the same as today or not)? Or is the question senseless because only with reference to a fixed time you can define a comoving distance? But even then this looks like a contradiction to the statement "no change in time".

 

[Orodruin]

But all popular explanations tell that comoving distance does not change with time

It doesn’t. It is the distance now, not in 1 billion years. Now.

 

[PeterDonis]

all popular explanations tell that comoving distance does not change with time

That's because it's defined not to. The term "comoving distance" is actually a misnomer because it does not tell you the actual distance. It is just a coordinate marker that identifies a particular comoving worldline; it does not change with time because it marks the same worldline throughout the history of the universe. The marker is chosen to be the actual distance now for convenience.

the Hubble flow is not taken into account

Yes, it is. The whole point of comoving distance as a marker (see above) is to mark out the same comoving worldline throughout the Hubble flow, i.e., as the actual distances change due to the expansion of the universe.

 

[PeterDonis]

If the value remains the same, then I would argue that the redshift in 1 billion years should be larger than today

You are mistaken about what the redshift means. The redshift tells us the factor by which the universe has expanded between the emission of the light and our seeing it. Whether or not the redshift we see in light from a particular galaxy 1 billion years from now will be larger than now depends on the factor by which the universe will have expanded in both cases. That will be different for different galaxies.

 

[PeterDonis]

many diagrams show that the CM increases with greater redshift.

Please give a specific reference. I strongly suspect that you are misinterpreting something.

 

[PAllen]

Please give a specific reference. I strongly suspect that you are misinterpreting something.

maybe just means that galaxy's seen with higher redshift today will have higher CM distance.

 

[Bandersnatch]

You can think of a distance to any galaxy as some function $r(t)$, where the distance $r$ increases with $t$ as the universe expands. It's a completely kosher way of thinking about distances, but not always the easiest.

Comoving distance is just a different way of expressing $r(t)$, where you say $r(t)=a(t)D$. You make $D$ - the comoving distance - have some arbitrarily chosen value with units of distance, and you relegate the change in time to a completely separate, dimensionless factor $a(t)$ - called the scale factor. So instead of saying a galaxy 1 billion ly away will after time $\Delta t$ have receded to 2 billion ly, you can say that the scale factor after time $\Delta t$ will have grown to twice the initial value.
This is handy, because in an expanding universe the scale factor applies to all distances. You don't have to keep in mind that this galaxy receded from 1 to 2 Gly, while that one receded from 5 to 10, and yet another one from 2.745 to 5.49. Whatever their individual values of $D$, they are all affected by the same scale factor. All you're saying with each of those example distances anyway is that they've grown by a factor of 2.
All that is left is to assign some particular distance to serve as $D$ for each galaxy. By convention the current distance $r(t_0)$ is used, so that $r(t_0)=D$. $D$ will stay constant forever. Then the scale factor expresses changes in distances w/r to the current state of the universe. More than one means larger, less than one is smaller.

BTW, comoving distance increases with greater redshifts only in the sense that you're looking at farther galaxies. That's what it's supposed to represent, after all - the relative positions of different galaxies. If you were to track the $D$ for each particular galaxy as the universe evolves in time, it'd stay the same - because the relative positions of galaxies don't change with the expansion, only the scale of distances between them.

 

[FritzK]

Hello,
here is one of many similar examples for the increasing comoving distance D with the redshift z.
I have now an idea what you are meaning and in which sense D has to be assigned to z.
I thought if I can measure today the redshift z0 of a specific galaxy, than in one billion years an observer could do the same with this galaxy. The redshift z1 then should be larger. But if I understand you right, I cannot use those D/z-Diagrams to determine D1 and to compare it with D0 at the same time.
After all, when I say r(t0) = D0 for my galaxy today, than in one billion years another observer would claim r(t1) = D1. In that sense the comoving "distance" to the same galaxy has changed from D0 to D1 (numercally different)?! It seems that it's only senseful to say r(t0) has changed to r(t1) and r(t0) is constant for all times like r(t1)...

D0.

 

[PeterDonis]

here is one of many similar examples for the increasing comoving distance D with the redshift z

The comoving distance to a given galaxy never changes, even though the redshift we observe in light from that galaxy might change. (Which does not mean it will necessarily increase in the future.)

The comoving distance to different galaxies with different redshifts is of course different. The relationship between comoving distance and redshift can change with time.

I think you are failing to distinguish these different things and it is confusing you.

 

[PeterDonis]

here is one of many similar examples

Please give a link to the source. The graph by itself is not enough.