Expert: James Gort - 1/10/2011

Question
QUESTION: Hello. I am reading an astronomy text and have come across what appears to be a contradiction, and I imagine that it seems this way only because I am missing some vital point. I would really appreciate it if you could elucidate this for me. The issue is the distance determined by the Hubble equation.

The author says that from the red shift value z you compute recessional velocity as v=(z+1)^2-1/(z+1)^2+1 and then plug this figure into the Hubble equation to get distance, as d=v/Ho, with d in Mpc and v in km/s. Then he says that the number of light-years the Mpc figure converts to  “tells you the lookback time of that object, that is, how far into the past you are looking when you see that object.”  But then in the next paragraph he says that the d in the Hubble equation represents NOT the distance at which we see the object (isn’t that the lookback time?) but rather the distance between us and the object now, i.e. the commoving radial distance, which factors in the expansion of space since the detected photons left the object.

Further, as an example the book posits in a chart an object with z=3 and thus  a recessional velocity of .882 c.   Plugging that figure into the Hubble equation you get d=3520Mpc. The book then says that this is the “distance at which we see the object,” and also gives 6500Mpc as the commoving radial distance. If this chart is correct, then the d computed by the Hubble equation is the lookback time, and NOT  the commoving radial distance, contrary to the second statement set out in the above paragraph.

Any light you can shed will be greatly appreciated.

David

ANSWER: Hi David,

Interesting question.

First of all, I want to remind you that the origin of the universe by a "Big Bang" is very much still a theory, and there are alternative theories.  Some observations do not support an expanding universe or the Big Bang.  There are several references concerning this, but perhaps the most authoritative person is Halton Arp, a leading astronomer and researcher on galaxies, who wrote "Seeing Red".  That book is highly recommended to get an alternative view.  Or read "A Different Approach to Cosmology" by Hoyle, Burbidge, and Narlikar.  Another great book which gives a scientific view on how the universe has always been in a steady state.

That said, I'll try and answer your question.  The author says "that the d in the Hubble equation represents NOT the distance at which we see the object (isn’t that the lookback time?) but rather the distance between us and the object now, i.e. the commoving radial distance, which factors in the expansion of space since the detected photons left the object".

I don't agree with this sentence - I agree with the first sentence in your second paragraph.  My understanding is that the Hubble equation gives the distance which the object was when the photons left, NOT the distance corrected for recession.  In other words, it gives the "look-back" time.  Your third paragraph seems to confirm my view.

I just think the author made an error.  Glad you found it by very critical reading.  If you find something to the contrary, I'd be very happy to hear about it.

Cheers,

Prof. James Gort         


---------- FOLLOW-UP ----------

QUESTION: Thanks very much for clarifying that for me, and for pointing me to those interesting-sounding books.
I do have one follow-up question. Once you determine from the Hubble equation that an object has a look back time of x Mpc, how do you convert that to commoving radial distance? Is there a simple formula or is there some kind of elaborate process?

Thanks again.
David

Answer
Hi David,

Because relativity is involved, it's not straightforward.  You probably are aware of this page, but just in case, see http://en.wikipedia.org/wiki/Comoving_distance.  As mentioned there, comoving distance is defined to be equal to the proper distance (the D in Hubble's Law) at the present time.  So if you just talk about the present time (and not how distance changes over time), just use Hubble's Law to get the comoving distance (today).   The comoving distance doesn't change, so if you want to know where the object will be in 10 years (the proper distance that Hubble's Law would give you, if you observed the object 10 years from now), then you need to use the formula for 'chi' in that wiki page.

Hope that's not too confusing.  

Prof. James Gort