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The standard model of Cosmology predicts that the redshift of a single source at fixed comoving distance varies with observation time according to

$$ \dot z = \left( 1+z - \frac{H(z)}{H_0} \right) H_0.$$ (see e.g. this answer or this paper). For all commonly observed redshifts (say, up to $z \sim 10$), the factor in parentheses has order of magnitude at most $\sim 10$, so that $$|\dot z| \lesssim 10 H_0 = 1000h \frac{\text{km/s}}{\text{Mpc}} = h \cdot \frac{10^{-9}}{\text{yr}},$$ which is exceedingly small. Theorists have been discussing this rate of change, its utility, and the fact that it's generally out of experimental reach (though perhaps it's recently become doable with an experiment on the timescale a decade or so?) for a while. I have not found sources discussing empirical bounds on such a quantity, however.

So, I'm seeking references for what kind of observational constraints exist-- how closely has the redshift of a particular source been investigated over time? Evidently any variations can't be too large, or else we'd clearly have noticed. Have the rates of change for some particular sources been rigorously assessed, presumably giving upper bounds on $|\dot z|$? If so, in which studies has this been done?

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  • $\begingroup$ High z redshift errors of $10^{-3}$ are thought to be possible soon for supernovae. That is the easy case. $\endgroup$
    – eshaya
    Commented Feb 18, 2023 at 19:31
  • $\begingroup$ @eshaya I assume that refers to potential error in a single absolute redshift measurement? Do you know if astronomers have systematically measured the redshift of a single supernovae (or what have you) multiple times, with a delay between measurements, to affirm it does not vary more than their error limits (thereby effectively obtaining an upper bound on its $|\dot z|$)? $\endgroup$
    – jawheele
    Commented Feb 18, 2023 at 21:22
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    $\begingroup$ Sure. Galaxies and supernovae have their z measured every time someone takes a spectrum of it. Many have been done dozens of times over a decade or two. If any changed by more than 3 sigma, we would definitely hear about it. $\endgroup$
    – eshaya
    Commented Feb 19, 2023 at 4:44
  • $\begingroup$ @eshaya Right, thank you. Do you know of any study which has specifically assessed this and collated comparisons into a bound (say, that would be appropriate for me to cite if I were doing work related to this time evolution), or is it essentially left at a passive internal consistency check whenever spectra are taken anew which would only be brought up if a discrepancy were found with high confidence? $\endgroup$
    – jawheele
    Commented Feb 19, 2023 at 5:27
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    $\begingroup$ Right. I would not think anyone would publish until a discrepancy is found for several cases. Sometimes outflow from the active galactic center change so care would be needed. $\endgroup$
    – eshaya
    Commented Feb 19, 2023 at 12:56

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