I have often referred to the Cosmo-Calculator in the past. Here is a Blog post dedicated to this ''workhorse''. The picture below is a cropped screenshot, so rather look at the real-thing under the link.

You may wonder: why an update? Is there something new?

Not really, but there are a few extra bits and pieces of information in the updated calculator that may be of interest to some readers. The change was motivated by a comment (on another Blog) that enquired about using the standard Special Relativity (SR) Doppler shift to explain cosmological redshift. My immediate reaction was that it is invalid because SR only applies in flat (Minkowski) spacetime. The cosmos is in all probability 'flat' in term of space, but not in terms of spacetime. The simple reason is that there are gravity and vacuum energy (possibly) operating on the cosmic scales.

Then I remembered that in my eBook Relativity-4-Engineers, I used exactly that: the SR formula for Doppler shift as related to cosmic redshift. In chapter 16,^[1] I noted

The question is: why is this a good approximation of a cosmological recession speed? My best answer is that it is partly due to a coordinate system choice and partly due to the quasi-flatness of the present values of the ΛCDM cosmic model.

The solid curve represents ΛCDM and the dotted line is for a simple linear expansion law, where a = t/T_Hubble, with T_Hubble the inverse of the Hubble constant H₀ (in suitable units). From the recession speed v/c, one gets distance D, due to Hubble's Law: v/c = H₀D.

H₀ is normally expressed in units of km/s per Megaparsec (Mpc),^[2] i.e. recession speed per unit distance. Distances in Mpc normally means co-moving or 'proper' cosmological distances, telling us how far the source is from us today, assuming that one could hypothetically stop expansion completely and then see how long light would take to traverse the distance (or measure it with our standard meter sticks).

However, in the SR-Doppler-redshift interpretation, that distance is look-back distance, derived directly from how long it took light to travel the distance in an expanding space. It usually has the units light-year, for obvious reasons. It is simply a different coordinate system than the usual cosmological choice, but there is nothing wrong with it.

In a way, this SR interpretation is easier to comprehend than the usual co-moving interpretation. The SR value is usually qualified with the term apparent (i.e. apparent recession speed/velocity). The latter is just referred to as recession speed or sometimes proper recession speed, due to its connection with proper distance. At the risk of confusion, I will measure both distance types in million light-years (Mly), but will name them appropriately.

Now back to the cosmo-calculator, as pictured above, or under the hotlink provided. The relationship between proper recession speed and apparent (SR) recession speed is immediately visible. For z=1088 (CMB redshift, the farthest we can optically observe), the present proper distance is 45,918 - i.e. the region of early space were the radiation originated, is presently that far from us. Its proper recession speed 'now' is 3.3c. The proper recession speed 'then' was about 65c. The present look-back distance is 13,757 Mly, i.e. less that 0.4 Mly from the theoretical look-back distance of the BB. The apparent recession speed of the region is 0.999998 c.

As I said above, the latter is perhaps the more comprehensible value. The more distant the source (higher z), the closer to c the apparent recession speed becomes, yet it never reaches c. The proper recession speed 'now' levels off at around 3.4c, as we reach the horizon of the observable universe. However, the proper recession speed 'then' increases without limit against z,^[3] according to the standard ΛCDM model.

To wrap up the output, I have added the average temperatures of the CMB 'now' and 'then', which are simply related by a factor 1/a = (1+z). You can check by inputting a redshift of 9 and note that the temperature 'then' is 10 times larger than 'now'.

Any feedback would be appreciated.

-J

[1] You can read chapter 16 here; you will find the complete Relativity-4-Engineers here.

[2] One Mpc is about 3.26 light-year. See Wiki for more definition on cosmic distances.

[3]A successful quantum gravity theory may avoid these infinities - and it may imply a 'bounce' rather than a 'bang'.