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This may be a stupid question, but it seems to me that we could calculate the value of the scale factor at any stage in the history of the universe by analysing the relationship between recessional velocity and distance in faraway galaxy clusters. Basically, use the same method Hubble used to get to his Hubble's Law: take a base point and then look at how distance to other galaxies varies with their velocities relative to that point? Why can't we do that?

Edit: I realised we can measure the scale factor directly using redshift, but it requires that we know the light travel distance.

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    $\begingroup$ Is your question then: "does Hubble law hold for other galaxies?" $\endgroup$ – caverac Mar 18 '18 at 23:37
  • $\begingroup$ @caverac Well, I assume it does, seeing as the universe is isotropic $\endgroup$ – Max Mar 18 '18 at 23:57
  • $\begingroup$ @caverac alright, if that is my question, can you give me an answer? Any information would be appreciated, because this question is bothering me a lot $\endgroup$ – Max Mar 19 '18 at 22:01
  • $\begingroup$ If you start assuming the universe is isotropic and homogeneous, then Hubble's law will hold everywhere, right? Are you looking for an experimental proof of that? $\endgroup$ – caverac Mar 19 '18 at 22:03
  • $\begingroup$ @caverac Well, if that makes it easier for you to answer, then, yes, let's say that's my question. $\endgroup$ – Max Mar 20 '18 at 19:00
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The question of recession velocity vs distance for distant galaxies that the question is posing is not very different than the continuing uncertainties in measuring the current Hubble constant, $H_0$. The former (velocity vs distance) is problematic because the distances measurements to far off galaxies depend on the distance ladder, i.e., determining cosmological distances accurately, whereas the second depends on distance measurements as well, but also on the cosmological model as to how the acceleration/deceleration changed over cosmological times, and the dependence of that on the relative mass and energy densities - i.e. they have some model dependence.

What is fairly clear though is that the dark energy started having enough of a noticeable effect about 5 or 6 billion years ago, based on distance measurements using the distance ladder (cepheid so to supernova and so on). So it is known, from measurements which are semi model-independent (somewhat because little in astrophysics and cosmology can be, you still depend on 'known physics' to get the distance ladder, and sometimes on simplifying assumptions), that the recession velocities are indeed time, or z dependent, and it is modeled by how the scale factor grows when the universe was radiation dominated, matter dominated, or more recently becoming dark energy dominated.

So the answer comes back to how non-model dependent can one measure cosmological distances. The fact is that there are still significant uncertianTies in the measurements of the Hubble constant using the astronomical ladder distances, and using distances estimated by the CMB-related/estimated acoustic density perturbations, and the time delay measurements of gravitationally focused multiple paths in CMB observations. The first measures about 72 Kms/sec/Mpsec, while the latter is around 67 Kms/sec/Mpsec.

Two interesting articles are on the Hubble constant measurement, the first more historical, the second more on the current Hubble constant uncertianTies, where both elucidate how measuring cosmological distances are at the heart of the issue, and the model dependencies. I've seen papers on the issues/uncertianTies of the distance and velocity measurements for different cosmological epochs, and don't remember them to be extremely more accurate, nor to be totally model independent. I don't see that, and neither do cosmologists/astrophysicists, as a big weakness, but more as the normal interplay of theory and measurement in physics. Still, while it is true that we still don't a totally accurate and certain picture of the expansion, any more accuracy in the measurements will refine the models, or find some new physics.

The two references are:

-http://www.pnas.org/content/101/1/8

-http://www.sciencemag.org/news/2017/03/recharged-debate-over-speed-expansion-universe-could-lead-new-physics

-A quick reference to the standard cosmology equations showing the model dependencies, at https://ned.ipac.caltech.edu/level5/Peacock/Peacock3_2.html

-A reference to the distance ladder, but not much on the most recent attempts to go further, in https://en.m.wikipedia.org/wiki/Cosmic_distance_ladder

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  • $\begingroup$ I thought we didn't have to use any cosmological models to measure the Hubble constant? For small distances, i.e z<0.5, proper distance is roughly equal to light travel distance - regardless of model chosen (this is true for all models). So where does the uncertainty come from? $\endgroup$ – Max Mar 22 '18 at 18:55
  • $\begingroup$ They at least go out to distances where supernovas are used as standard candles, so there's astrophysics assumptions on the physics providing the inherent luminosity. See some of it at nytimes.com/2017/02/20/science/…. Even at a z of 0.5 the model makes a difference, and further out it makes more of one. See the Hubble article witH the history of how much it's varied, and some of the assumptions/methods. See cfa.harvard.edu/~dfabricant/huchra/hubble $\endgroup$ – Bob Bee Mar 23 '18 at 3:02
  • $\begingroup$ alright, you convinced me that there are large uncertainties when measuring distances to faraway objects. However, they are still a fair approximation for actual values of distance. Are the errors so high that when we try to establish the light travel distance, the error bars are so large that uncertainty range includes most cosmological models? It can't be THAT large, right? I mean, the value of the Hubble Constant is more or less agreed upon to be around 72mpckm^-1s^-1 for standard calibrations and around 67mpckm^-1s^-1 for CMB calibrations, right? $\endgroup$ – Max Mar 23 '18 at 21:28
  • $\begingroup$ And, actually, the value of the Hubble Constant doesn't even depend on distance measures. We can measure it to a sufficiently good degree of accuracy using CMB. $\endgroup$ – Max Mar 23 '18 at 21:52
  • $\begingroup$ So, the fact that the CMB number is significantly different than the distance based measurements is actually a point of controversy. Nobody knows which is the right one, it can't be both, something is wrong in one or both. The CMB numbers and the distance ladder numbers not agreeing is a bad, unexplained, problem. So you see, you do need distance measures, if that can lead to a number. The wrong one may depend on which wrong assumption was made in the physics modeling of one or the other - they both depend on physics models. As I explained in my answer. $\endgroup$ – Bob Bee Mar 27 '18 at 23:58

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