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Modern cosmology relies heavily on the redshift scale-factor relation $$a=\frac{1}{1+z}$$

But what is the experimental evidence for it?

It's derived in textbooks from General Relativity, but for all other predictions of GR there have been stringent tests. The bending of light by stars, precession of the perihelion of Mercury, the equivalence principle, time dilation, etc... have all been subject to high precision tests. Even if GR is correct, it's possible that the derivations misinterpret something...

Have any tests been done, or are any planned, that that look for direct experimental evidence of the relation?

The question is different to this one What experiment would disprove Friedmann model of cosmology? as it asks specifically about attempts to experimentally verify the redshift scale-factor relation.

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First of all, the scaling law is as follows. For a photon detected emitted at $t_0$ and detected at $t_1$ we have the redshift $z$ of the photon given by the scale factors $a$ at times $t_0, t_1$ as $$\frac{a(t_0)}{a(t_1)} = \frac{1}{1 + z}$$ That is, the redshift is derived from the ratio of scale factors. The issue with confirming this relation is that $a(t)$ is a metric component, and metric components are typically not directly observable. One has to be very careful - what exactly is meant by "testing" the $a-z$ relation? What are the observations that we are comparing?

Let's take a look, the scale factor $a(t)$ is special, since it chosen to correspond to the rescaling of distances in the cosmological comoving frame. The $a-z$ relation could then be taken as a definitory statement, since it really corresponds to the stretching of distances between the wavecrests of the photon. In that case, one would just need to verify the isotropy and other assumptions of the FLRW cosmology to test whether such a definition of the scale factor is self-consistent. This is actually how cosmology is built in practice, astronomers essentially do not refer to $a$, they refer to $z$ and define the cosmological eras using the redshift as a "coordinate".

Another operational definition of $a(t)$ (up to a rescaling) could be also from the average distance between ordinary matter such as galaxies and clusters of galaxies. If one is able to track all the matter in one era and measure its distances, compare this with another era, one essentially finds the ratio of $a(t)$s between these two eras. Does this ratio agree with the redshift observed between them?

Unfortunately, this simple idea is extremely difficult to realise. At short distances you can cover things well but you do not have the statistics (you need Gigaparsec volume averages for FLRW cosmology to start making sense), at larger distances you start having selection effects (for instance, you notice less bright things less). And when you pass between eras with very different redshifts, you have different selection effects for every era: you use different instruments with different capabilities because the wavelengths become different (and you need different sensitivities), and even the objects themselves change, since they may be glowing more or less dependent on the era you are in. In short, there is no good experimental test of a definition of $a(t)$ based on average distances, you have to take the redshift relation as more of a definition.

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  • $\begingroup$ It's true that an experiment might be difficult...About this part "The a−z relation could then be taken as a definitory statement, since it really corresponds to the stretching of distances between the wavecrests of the photon." it seems to miss the experimental check - perhaps quantum effects change things, i.e. perhaps photons are needed in the derivation instead of classical waves. $\endgroup$
    – John Hunter
    Commented Jun 2, 2021 at 10:42
  • $\begingroup$ @JohnHunter One of the tenets of QFT is that free quantum fields have exactly the same kinematics as the classical fields. It is true, though, that the a-z formula applies only to light propagating in vacuum and wavelengths shorter than cosmological scales. However, virtually all light has such wavelengths, and the propagation in media such as dust and dilute gases never leads to uniform changes across all frequencies, so one can spot it by looking at the entire spectrum. One thing which is done is testing the luminosity distance as a function of redshift, but that also assumes Friedmann eqs. $\endgroup$
    – Void
    Commented Jun 2, 2021 at 12:04
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The point here is that redshift and scale factor are the same thing. If you assume that light speed is constant everywhere, which is a reasonable well tested assumption, then the correlation between scale factor and redshift follows. Imagine two galaxies A and B. Imagine to send a a laser beam with a certain wavelength. Now imagine to divide the distance between A and B by 𝛌 and call this number n (number of wavelength contained between A and B). If n is fixed, stretching the space between A and B will give you a different value of 𝛌. As you may see if space is stretching the wavelength varies.

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  • $\begingroup$ I think the OP is asking, what is the experimental evidence that "redshift and scale factor are the same thing", as you put it. $\endgroup$
    – Guy Inchbald
    Commented May 31, 2021 at 10:51
  • $\begingroup$ Mainstream cosmology would say that the received wavelength is proportional to the scale factor at time of emission and reception, not sure if the above is the same. Derivations are risky so, @ Guy Inchbald Yes the 'experimental evidence' is important. $\endgroup$
    – John Hunter
    Commented May 31, 2021 at 12:36
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In the context of FRW cosmology, the equation of you wrote is just the definition of the variable $z$. It is straightforward to show that in the FRW metric, as light travels, its wavelength is stretched by the scale factor $a$, which gives a redshift.

But I see that you are looking specifically for experimental evidence. That's tricky, because the scale factor $a(t)$, or $a(z)$ is a property of the FRW metric. The FRW is an meant to be an approximate model for our universe, where the universe is homogeneous and isotropic. Obviously, that is not the case on scales less than a Mpc, so the "scale factor" doesn't strictly exist.

On the other hand, you can define an approximate scale factor on large scales. We usually measure it in terms of redshift, i.e. we assume FRW because this is the model we are testing.

But I suppose that if there were an object whose size were known, we could compare its apparent angular diameter to its intrinsic angular diameter, whose ratio should be $a(z)$. We could then compare the apparent wavelength of its light to the intrinsic wavelength of its light, whose ratio should be $1+z$. I guess that would be a test of $a=1+z$. But this sounds tricky. I'm not aware of such a test being done, but maybe somebody knows one?

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  • $\begingroup$ They might have tried it using BAO - a 'standard ruler', but the results lead to possible problems in modern cosmology (problems already there), such as dark energy or the Hubble tension - that's one reason the question is asked, to see how reliable the relation is. $\endgroup$
    – John Hunter
    Commented May 31, 2021 at 12:39
  • $\begingroup$ BAO is a standard ruler only if CMB input is used, correct? Then I agree, in the case of BAO the relation fails (to 6%). $\endgroup$
    – Eric David Kramer
    Commented May 31, 2021 at 13:23
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Edwin Hubble discovered the law which bears his name in the 1930s.

By determining the apparent brightness of a star and its distance away, its intrinsic brightness or luminosity can be calculated. At the time of Hubble, the distances to many nearby stars had been calculated, for example through measurement of geometrical parallax from different observation points.

Stars of a type called Cepheid variables pulse steadily in their luminosity. Using the above method, they had been found to have a direct correlation between their pulse rate and luminosity, so by measuring the pulse rate their luminosity could be determined.

This allows the distance of a far more distant Cepheid variable to be determined, simply by measuring its pulse rate and apparent brightness. Such stars can thus be used as standard candles to determine how far away they are.

Hubble correlated these calculated distances with their measured spectral redshifts, to discover their direct relation.

His final step was to interpret redshift as indicating a receding velocity (a phenomenon already known), leading to his discovery that the Universe is expanding. But that goes beyond the redshift-scale factor relation asked about here.

Subsequent measurements, using these and other standard candles, have confirmed and refined his results.

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  • $\begingroup$ This is the so called 'distance ladder', most of these measurements are quite nearby (in cosmological terms) and not sure whether these could distinguish experimentally between the relation in the question or alternative relations that give a similar result at low redshift. Subsequent measurements have extended things to higher redshift, but then there are unexplained phenomenon discovered by these higher redshift measurements such as the Hubble tension... $\endgroup$
    – John Hunter
    Commented May 31, 2021 at 12:43
  • $\begingroup$ @JohnHunter Yes, the distance ladder gets periodically recalibrated. But that is precisely because the quality and strength of the empirical evidence improve with time. See the answer's last paragraph. $\endgroup$
    – Guy Inchbald
    Commented May 31, 2021 at 14:06

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