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The classical proof of cosmological redshift that leads to the relation:

$\frac{\lambda_0}{\lambda_e}=1+z=\frac{a\left(t_0\right)}{a\left(t_e\right)}$

is quite well known as for example (among many other textbooks) available here https://people.ast.cam.ac.uk/~pettini/Intro%20Cosmology/Lecture05.pdf.

However all textbooks (including that PDF) make the assumption that the change in $a(t)$ during the time intervals between two successive crests can be "safely neglected".

However this is not the case in an inflationary universe (at the beginning of "time"). So I was looking for a more rigorous expression for the cosmological redshift used by professional cosmologists or astrophysicists, because I cannot believe they use the relation above (that is an approximation) to publish scientific papers.

Does anyone know what is the real relation for the cosmological redshift used by professional scientists?

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  • $\begingroup$ You should rephrase your question as it effectively states that all textbooks written by amateurs. $\endgroup$
    – my2cts
    Commented Feb 28 at 10:44
  • $\begingroup$ No for sure! But i know that sometimes they oversimplify on purpose because they assume the reader doesn't have the knowledge to deal with the real advanced maths. $\endgroup$
    – Vincent ISOZ
    Commented Feb 28 at 10:46
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    $\begingroup$ A less emotive title might be something like "Cosmological redshift without neglecting change in a(T). " or "Cosmological redshift without treating change in a(T) as negligible"." $\endgroup$
    – KDP
    Commented Feb 28 at 10:57
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    $\begingroup$ It's also worth mentioning that approximations made in physics are often not hiding "real advanced maths", but rather the result of having performed that "advanced math" and having realized that the deviations from the simplied result are effectively insignificant. You can go on and use full-fledged GR to calculate (approximatively) the trajectory of an apple falling off a tree, but you'll be wasting your time. $\endgroup$
    – Albert
    Commented Feb 28 at 11:40

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You can not access $z$ at the beginning of time as any observation of light beyond the CMB is impossible. Unless in very specific cases, and for all practical purpose, "professional scientists" always use

$z=\frac{1}{a}-1$

(and $a(t_0)=1$ by construction)

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    $\begingroup$ Ok that makes sense if we assume indeed that inflation stopped before the CMB. Thanks a lot! $\endgroup$
    – Vincent ISOZ
    Commented Feb 28 at 10:38
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    $\begingroup$ Indeed, and inflation has to stop (way) before CMB to be in agreement with observational data. $\endgroup$
    – Léo Vacher
    Commented Feb 28 at 11:01
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    $\begingroup$ This is not an answer to (what I thought was) the main part of the question. Gravitational waves emerge from the inflationary epoch, are redshifted and can potentially be observed $\endgroup$
    – ProfRob
    Commented Feb 28 at 13:03
  • $\begingroup$ And how on earth could this mean that inflation was not over at CMB? I did not say that inflation had no observable consequences. I said that it HAS to happen way before CMB emission. $\endgroup$
    – Léo Vacher
    Commented Feb 28 at 13:06
  • $\begingroup$ "You can not access z at the beginning of time as any observation of light beyond the CMB is impossible." Correct, but that doesn't stop you defining or measuring things that are redshifted from before that epoch. $\endgroup$
    – ProfRob
    Commented Feb 28 at 13:09

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