# Help with some Numbers

1. Oct 21, 2008

### mysearch

Hi,
Would appreciate any clarification of the following numbers and the logic?

1. If a CMB photon left the surface of last scattering at +380,000 years and arrives on Earth today, is the total travel time 13.7-0.38 billion years?

2. If the photon always travels at [c], there is an initial assumption that the photon must have travelled [ct], where [t] equals 13.3 billion years, i.e. 13.3 billion lightyears?

3. However, the estimated temperature of the CMB at decoupling was ~3000K, while today this temperature has fallen to 2.725K due to the expansion of the universe. This expansion is also reflected in the photon wavelength and corresponds to a redshift of 3000/2.725=1090?

4. The redshift factor therefore reflects the expansion of the path travelled by the photon, i.e. 13.3 billion/1090 = 12.5 million lightyears, i.e. the source of CMB photon was originally only 12.5 million lightyears away from Earth?

5. However, this source will also have receded away due to the expansion of the universe, after the photon was emitted, with a recession velocity [v] that is linked to the Hubble constant H=v/r. As such, the recessional velocity is a function of radius and time. Therefore, the current distance to the source is related to [ct] and [vt], where [t] equals 13.3 billion years?

6. Based on a couple of on-line cosmology calculator, the age of a redshift of 1090 is stated more accurately as 377,000 years. Does anybody know how this event is positioned so accurately without an equally accurate understanding of the how thermodynamics of the universe drove the expansion as a function of time, i.e. volume and radius?

7. The same calculators also suggest that H=1,329,466km/s/Mpc at 377,000 years in comparison with today’s value of 71. Today, the Hubble radius R=c/H is ~13.7 billion lightyears based on H=71km/s/Mpc, but using the number 1,329,466, this would reduce to 733,000 lightyears. As such, the source of the CMB photon was original receding away much faster than [c], i.e. v/c~17, and therefore this source was outside the definition of the visible universe?

8. While we can readily calculate [ct], the value of [vt] is a complex issue requiring the aggregation of the expansion rate over the time [t], but in line with the description of the particle horizon, the net effect of [v] would result in [vt] being some 3 times greater than [ct] suggesting that the source must now be ~40-50 billion lightyears from Earth?

2. Oct 21, 2008

### marcus

That's very much the sort of question you should be asking! Going thru the reasoning step by step. In this case the answer is no!

380,000 is not equal to 0.38 billion.
It is equal to 0.00038 billion.
13.7 - 0.00038 is approximately 13.7.
Would you go thru your chain of reasoning using the corrected figure and re-post? I would like to see your questions reformulated with the more accurate figure for the travel time. Thanks

BTW your step #4 does not make sense. You need to reconsider that one. Each segment of the path has a different age and is subject to a different amount of expansion. Before considering the effect of expansion, the path is 13.7 billion LY long. But different segments are subject to different ratios. Early segments have increased roughly 1000 fold, but very recent segments have increased almost not at all. The integrated effect will bring it up to just about 46 billion LY.

But let's not move ahead too fast. First please rewrite your sequence of steps making the indicated corrections and then let's have a look.

Last edited: Oct 21, 2008
3. Oct 22, 2008

### mysearch

That silly mistake has misled me, as I had interpreted your earlier response to imply that the CMB photons were associated with the age of the universe and not decoupling. Whereas you were simply pointing out the near equivalence of (13.7-0.00038) to 13.7 billion. Thanks
1) I think I understand your position from yet another earlier response. Decoupling must be pretty close to what is referred to as the particle horizon, estimated at ~46 billion lightyears. Therefore, 46 billion/1090 gives 42 million lightyears not 12.5 million?

The reason I was pursuing this issue was based on a perception that in some sense the photon had always travelled at [c] for 13.7 billion years, i.e. 13.7 billion lightyears. Given that the redshift is associated with the wavelength of the photon, I simply wondered whether it was this distance that had expanded by the factor 1090, hence 12.5 million lightyears.

2) In essence, I was concerned that using the figure of 46 billion included a rate of expansion, i.e. linked to H=v/r, that neither the photon nor the space it was travelling through had ever experienced?

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I am working through the sticky “Effort to get us all on the same page” and endeavouring to do so. I will hold off making any direct comments until I have finished but would like to paraphrase a couple of key issues that are the centre of my concerns as listed in: https://www.physicsforums.com/showpost.php?p=1921707&postcount=16

3) Purely on the basis of expansion, if the universe was the size of a singularity some 13.7 billion years ago, but is now much much larger, doesn’t this imply a rate of expansion in the size of the universe with time, even if nobody can put a figure on it?

4) The previous question is not automatically implying an edge to the universe as I am assuming that the issue of the universe being infinite, boundless and without any edge is a separate issue associated with the geometry of the universe?

4. Oct 22, 2008

### marcus

You might get something out of Ned Wright's FAQ. Your reasoning here is having difficulty because of using mathematically undefined concepts. Wright's website would help increase your level of sophistication. Also a great new cosmology outreach site is at the Albert Einstein Institute. Explains what cosmologists commonly mean when they use the word singularity! Do check it out. URL in my sig.

One problem here is that if you are talking with professionals then "the size of a singularity" can in effect be an infinitely large spatial volume. What those words imply has no fixed meaning, it could be various things depending on context.
Commercial popularizations have given the general public an extremely distorted notion, many people think the size of a singularity is a mathematical point. This was never intended, it just happened by a process of commercialization and dumbing down.

The issue of spatial finite versus infinite remains unresolved, but I am very hopeful that we will know more in the fairly near future (assuming the Planck satellite is launched April 2009 as planned.) For the time being we must keep our minds running on two parallel tracks, taking into account both possibilities.

In fact most cosmologists show a slight preference for assuming the universe is spatial infinite, if they show any preference at all. The language of research papers tends to be carefully worded, numbers are calculated using different assumptions. Tables have multiple columns. But current sentiment probably favors infinite. In that case, of course, the initial singularity is infinite volume, and has no definite size.

And of course there is no such thing, in that case, as the size of the universe. Space has an infinite volume.

The way professionals have always gotten around this problem is to use the scalefactor. There is no reliable well-defined notion of the size of the universe, yet they need to be able to describe the expansion of distances, so they establish a convenient number which is normalized to equal ONE at the present time and is calculated according to the Friedmann eqns. as you go back in time.

The scalefactor is so simple and convenient to use, that even if we were eventually able to determine that the universe had a definite finite size, and even if we could pinpoint that size, I suspect astronomers would not switch over to using that size as their measure. They would probably continue to use the scalefactor!

After all the Friedmann eqns. are equations ABOUT the scalefactor, and they are the basic equations of cosmology---the whole field revolves around those two equations. And those core equations describe the time-evolution of the scalefactor. They do not describe the time-evolution of the size of the universe (which is currently not a well-defined concept.)

5. Oct 22, 2008

### mysearch

Thank you. I feel I may now have some insight to this issue, although I won’t pretend that I necessarily have, as yet, any intuitive feel for some of the concepts you have outlined. However, having even some inkling of this perspective may now allow me to spend some worthwhile time following up on the links recommended.

As mentioned, I have started to work through some of postings in the following sticky:
https://www.physicsforums.com/showthread.php?t=261161.

However, from my perspective, the issues you have outlined in the previous post are worthy of development within its own sticky thread positioned at the front of this forum, possibly entitled “New to Cosmology – Then Read Me First!”

I believe a paper within a post from George Jones is very useful for getting some wider insight into the issues associated with expanding space: https://www.physicsforums.com/showpost.php?p=1905863&postcount=21
The actual paper is http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf
While the entire paper explains some the confusion and difficulties associated with this issue, I believe the following extract from the conclusion underlines the importance of this issue to people like myself, who come to the PF cosmology forum without the benefit of any formal education in this subject:
While this thread is probably not the place to debate any of the points you have raised any further, I was interested in the implications of 2 further extracts taken from the paper above, which might become a topic of debate within any subsequent sticky thread?
Again, many thanks for your help and patience.

Last edited: Oct 23, 2008
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