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Do we know the expansion rate of the Universe at each moment of time $t$ of its history?

We only measured the expansion rate for a few decades (let's say), which isn't much compared to the age of the Universe.

Is the expansion rate extrapolated from the Hubble's Law?

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We can fit the expansion using a model with various empirical parameters. Having measured all the parameters, one then has the whole expansion history according to that model. However then one has to think about the fact that the model itself was somewhat of an idealization, or maybe just wrong.

A model called $\Lambda$CDM (lambda cold dark matter) fits observations to within experimental uncertainty (few percent level in Hubble parameter, larger uncertainties in other parameters), and this model, with the aid of numerical calculations, is suitable back to early universe but not necessarily very early universe. Here by 'very early' I mean where the energy scales begin to go beyond where we have confident knowledge of fundamental physics.

In the above sense we know, therefore, subject to the uncertainty associated with the model itself, the entire expansion history since that very early time, to within some experimental precision.

The uncertainty associated with the $\Lambda$CDM model itself is mostly in the dark energy contribution. The evidence suggests it behaves like a cosmological constant, but there is also the fact that measurements can be thrown off somewhat by other factors, such as whether our local part of the universe happens to be a bit over-dense or under-dense, and the effect of dust on measurements of light received from distant supernovas, and things like that.

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  • $\begingroup$ "A model called $\Lambda$CDM (lambda cold dark matter) fits observations to within experimental uncertainty", it's an inaccurate description of the current status! The local measurement of the Hubble parameter is 3.4 sigma higher than predicted by $\Lambda$CDM. See here: arxiv.org/abs/1604.01424. To put it bluntly, $\Lambda$CDM is in dire crisis. $\endgroup$ – MadMax Jun 20 at 14:11
  • $\begingroup$ @MadMax This is a useful comment and thanks for it. I wouldn't quite say "dire crisis" but I agree there is a very interesting tension here. I'll update my answer after waiting to see if any more comments come in. $\endgroup$ – Andrew Steane Jun 20 at 15:05
  • $\begingroup$ Cosmologists are coming up with a whole zoo of ad hoc models to resolve the "tension". As far as I am concerned, these sort of tinkering with $\Lambda$CDM involves way too many fudge factors. $\endgroup$ – MadMax Jun 20 at 15:32
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The expansion rate can in principle be measured at all times for which there are objects visible with which to measure it. Distant objects are viewed as they were in the past. There is an implicit assumption here of homogeneity, that is that on large scales every bit of the universe behaves like every other bit at the same epoch.

The measured redshift of an object depends on the history of the expansion rate. The details of this are spelled out in According to Hubble's Law, how can the expansion of the Universe be accelerating?

The inference of dark energy and an accelerating expansion is an example of this. Observations of type Ia supernovae tell us about the history of the expansion rate and reveal that the expansion rate first decreased and then increased at later epochs.

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    $\begingroup$ This doesn't seem right to me. Cosmological redshifts aren't like kinematic redshifts in SR. They don't give you a snapshot in time. Their interpretation depends on the entire history of the spacetime between emission and absorption. $\endgroup$ – Ben Crowell Jun 20 at 12:10
  • $\begingroup$ @BenCrowell correct - they depend on $a(t)$ which is exactly what the question is asking. If you are arguing that actually they tell you what the integral of that is, well fair enough. $\endgroup$ – Rob Jeffries Jun 20 at 15:39

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