# Approximate conversion of redshift 'z' to a time and/or distance, when reading papers?

I'm looking for some form of "rough and ready" formula to convert between redshift z value, years since BB, and distance, so that when I read an astronomy paper and it discusses an event that occurred at z=10+/-0.5, or a quasar at z=7, I can get a rough idea when the event occurred and how far away they are saying the quasar is, as context.

I'm aware that this is an inherently vague question, since the times and distances depend on the model and convention chosen, and also that I'm not specifying proper distances/comoving coordinates, etc.

My assumption is that most current models and their parameters, that are mainstream accepted and used, will be based on very similar parameters of very similar models (if not the same model), such as latest Planck parameters, or similar. They might differ more due to assumptions about very early times, but redshift is inherently >~380 k years, and if they did vary a lot for times beyond that timeframe, we'd have real issues. In other words, I suspect/hope that any differences in answers due to model variation, won't significantly change the answer to this question. So the only question will (hopefully) be, which convention or type of metric/distance will be appropriate.

If an assumption needs making (such as the meaning of 'distance' to be applied, or the zero point used for time: BB or end of inflation etc) please make an assumption most likely to help me, and I'll edit the question to clarify if needed, once I see what is too vague in the question as initially worded.

Thank you !

Use one of the "cosmology calculators". The conversions depend on what you assume for the cosmological parameters.

Here's one that will do what you want. http://home.fnal.gov/~gnedin/cc/

e.g. For the default parameters, $z=7$ corresponds to a look back time of 13.01 billion years, whilst $z=10.5$ corresponds to a look back time of 13.33 Gyr, with the current age of the universe being 13.78 Gyr. Choosing the "luminosity distance" option, we that the object has a luminosity distance of 112 Gpc.

If you want the proper distance you could use http://www.astro.ucla.edu/~wright/CosmoCalc.html According to this calculator the $z=10.5$ object is currently at a proper distance of 9.76 Gpc (31.8 billion light years).

I suspect that converting red-shift to distance, for any particular object, is going to have a lot of error. Especially for z > 1. Even for 0.05 < z < 1, the errors are often over 10%.

There are two reasons that can make a red shift: (1) a moving object (Doppler effect) and (2) space expansion, they are equivalent.

If an object (at this moment) is at a distance of $$d$$ from us, it's getting farther from us with the speed of,

$$v \ =\ d*Hubble\ Constant$$,

and it's light will arrive to us after $$t$$ time. During $$t$$, space will expand enough to convert the wavelength of $$w_0$$ to $$w_1$$, so that,

$$\frac{(w_1-w_0)}{w_0}$$ = $$\sqrt{\frac{(1+v/c)}{(1-v/c)}}-1$$

• This answers the question only partially. The OP was asking about a relation between redshift and age of the universe, can you address this point? Moreover, this relation is only valid approximately at low redshift (z<0.5), surely not for quasars at z=7, as mentioned by OP Jul 18, 2022 at 13:03