# Cosmological redshift without approximations

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?

• You should rephrase your question as it effectively states that all textbooks written by amateurs. Commented Feb 28 at 10:44
• 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. Commented Feb 28 at 10:46
• 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"."
– KDP
Commented Feb 28 at 10:57
• 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. Commented Feb 28 at 11:40

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)