The present epoch in FRW metric

[victorvmotti]

In an expanding universe that is modeled by the FRW metric we assume that scale factor of the "present epoch" is unity which is equivalent to a zero redshift. Therefore, most observed galaxies with nonzero redshifts are in our past light cone.

But it is unclear to me how much back in time or far away in physical distance can we go to assume that scale factor is unity.

Alternatively, I need to know when back in time we can no longer assume that proper distance is equal to co-moving distance. What is the physical distance limit of "today universe"?

 

[PeterDonis]

we assume that scale factor of the "present epoch" is unity which is equivalent to a zero redshift.

Why do you think this? Scale factor at a single epoch is not what determines the observed redshift of an object.

The scale factor is just the "current size" of the universe; more precisely, it is the factor by which coordinate distances are multiplied to get proper distances within a given spacelike hypersurface of constant "comoving" time (i.e., FRW coordinate time).

The observed redshift of an object depends on the ratio of *two* scale factors: the scale factor at the time of emission of the light from the object, and the scale factor at the time of observation of the light.

Therefore, most observed galaxies with nonzero redshifts are in our past light cone.

Um, *all* observed galaxies (like all other observed objects) are in our past light cone (more precisely, the points on their worldlines at which the light we see was emitted are in our past light cone).

But it is unclear to me how much back in time or far away in physical distance can we go to assume that scale factor is unity.

The scale factor changes with time; if you go back in time *at all*, it will be different.

The scale factor does not change at all in space; it is the same everywhere in a given spacelike hypersurface of constant time.

Alternatively, I need to know when back in time we can no longer assume that proper distance is equal to co-moving distance.

It always is, within a given spacelike hypersurface of constant time. "Comoving distance" *is* proper distance for comoving objects.

("Time" means FRW coordinate time in all of the above.)

What is the physical distance limit of "today universe"?

It depends on whether the universe is spatially closed. Our current best fit model says it is not, so there is no "physical distance limit"; each spacelike hypersurface of constant time is infinite in size.

 

[victorvmotti]

The scale factor changes with time; if you go back in time *at all*, it will be different.

Thanks a lot.Yes, I know this.

But let me rephrase what I was meaning.

Our solar system could be, with a good approximation, our today.

My question and ambiguity was about approximation, given the current value of Hubble constant, how many years should pass on Earth, so that we can infer that the proper distance between Sun and say Alpha Centauri has increased significantly? Does "significantly" make sense at all here?

Also, given the effect of a changing scale factor, if we measure interstellar physical distances, for example between Sun and Centauri or intergalactic physical distances, say, between Milky Way and Andromeda,

How can we compare the rates of increasing proper distances between interstellar physical distances and intergalactic physical distances based on the FRW metric? Are these rates the same?

Back to solar system, the proper distance between Earth and Sun is not changing with time, but apparenlty FRW metric concerns the whole universe and is not applicable to solar system evolution of spacetime.

 

[PeterDonis]

Our solar system could be, with a good approximation, be our today.

I'm not sure this rephrasing makes things any clearer. I think what you mean to say is that our solar system, to a good approximation, is following a "comoving" worldline; that is, it is, to a good approximation, at rest in an FRW coordinate system describing the universe. This is true for many purposes, yes; however, note that we can easily detect the error in this approximation, because we can easily measure a dipole anisotropy in the CMBR, which indicates that the velocity of the solar system, relative to a "comoving" worldline, is a few hundred kilometers per second.

The real issue, though, is not how good an approximation the solar system is to a "comoving" object, but how large a distance scale you have to look at before you are looking at dynamics that are primarily governed by the Hubble flow, as opposed to looking at local dynamics of gravitationally bound systems. You appear to recognize that the solar system is a gravitationally bound system, so motions within it are not due to the Hubble flow; but the same is also true for much larger systems. See below.

My question and ambiguity was about approximation, given the current value of Hubble constant, how many years should pass on Earth, so that we can infer that the proper distance between Sun and say Alpha Centauri has increased significantly?

This is the wrong proper distance to look at, because the Sun and Alpha Centauri are both part of a gravitationally bound system, our galaxy. The Hubble constant describes the relative expansion of objects which are *not* part of gravitationally bound systems; that basically means the "objects" are galaxy clusters. So the question should be, how many years would pass on Earth before we can infer that the proper distance between our local galaxy cluster and neighboring galaxy clusters, not gravitationally bound to ours, has increased significantly?

Also, given the effect of a changing scale factor, if we measure interstellar physical distances, for example between Sun and Centauri or intergalactic physical distances, say, between Milky Way and Andromeda

The Milky Way and the Andromeda galaxy are also gravitationally bound; they are both part of the "Local Group" of galaxies. The Andromeda galaxy doesn't even show a redshift at all; it is moving *towards* the Milky Way (because of the gravity of the two galaxies), not away from it.

How can we compare the rates of increasing proper distances between interstellar physical distances and intergalactic physical distances based on the FRW metric? Are these rates the same?

Neither of these proper distances can be related to the change in the FRW metric scale factor; you have to look on much larger distance scales. See above.

Back to solar system, the proper distance between Earth and Sun is not changing with time, but apparenlty FRW metric concerns the whole universe and is not applicable to solar system evolution of spacetime.

Yes, that's true, and the same is true for our galaxy, our "Local Group" of galaxies, and even, according to our best current models, a larger galactic cluster of which our Local Group is a part. Which means most of the questions you are asking are the wrong questions; again, you need to look at much larger distance scales. See above.

 

[victorvmotti]

Thank you very much.

It is all clear now, the FRW metric deals with expansion of objects which are *not* part of gravitationally bound systems.