Carried along by the Hubble flow.

marcus

Convenient language is so important! I've been keeping track of Hongbao Zhang (and a frequent co-author named Song He) for several years---Perimeter Institute connections, Potsdam Max Planck Institute connections, and they somehow stand out. I like how Zhang uses the phrase measured by isotropic observers here right around equation (2.7)

==quote Zhang==
...the magnitude of momentum of massive particle measured by the isotropic observers. Whence we know that for a freely traveling massive particle in an expanding FLRW universe, it is its momentum rather than energy that goes like¹
p ~ 1/a (2.7)
It is noteworthy that this result is also obtained in [Weinberg], where, however, the method employed seems somewhat complicated, and some approximations are also made.

3. Thermal Evolution
Let us assume that during the evolution of our universe, there exists a last scattering surface at the time t_L when some kinds of massive particles such as neutrinos and antineutrinos suddenly went from being in thermal equilibrium to a decoupled expansion.

Footnote 1: Of course, the momentum of a massless particle shares the same behavior...
==endquote==
Isotropic observers would be ones for whom the CMB looks the same temperature in all directions (no Doppler hotspot), and the recession rate pattern of galaxies looks the same in all directions. IOW the isotropic observers are the ones who are not moving relative to background. They are at universal rest.

So it is momentum measured by isotropic observers which very gradually tapers off as the spatial geometry expands. That is a good way to put it, that is not so bewildering I think as saying momentum as measured by observers "at rest with respect to the Hubble flow". The "Hubble flow" is not motion in the ordinary sense (nobody gets anywhere) so calling it a "flow" is a linguistic disaster which makes listeners imagine motion. The "Hubble flow" means everybody stays at their own latitude and longitude on the balloon, everybody stays PUT. It is an oldfashioned expression used mainly before 1970. after that people knew about the CMB so they could say as measured by observers "at CMB rest".

But I think saying "as measured by isotropic observers" may even beat that, convenience-wise.

Naty1

Marcus:
yes, So that explains the Tamara Davis one liner :

or something very close...

very nice!

Marcus:
I will try and read, er, that is, 'understand', the Zhang paper but before I do, can you confirm that this result is applicable for all cosmological time...in other words, in earlier matter dominated expansion as well as our current energy dominated expansion....a[t] varies over time so the redshift pattern of momentum decline [redshift] also varies over time, right.....that also provides a nice insight about the cumulative effects of expansion on redshift that I did not really appreciate previously.

edit: sure. it is ok for varying cosmological time periods and the Hubble parameter is related since H[t] = a'[t]/a[t]...

[This almost makes sense!!.]

PS: Wasn't that you who previously mentioned 'isotropic observers' in another thread??....no "hotspots'....anyway, somebody did and that perspective made it into my personal notes!!!! [ If you try and deny it I will be forced to look it up in my notes and see if I have an attribution!]

marcus

I wouldn't urge anyone to read Hongbao Zhang's paper, too technical. But on page 2 it has equation (2.7) which says that, as measured by isotropic observers
p ~ 1/a

After that, he goes into how the distribution of energies remains THERMAL which is kind of interesting. But not essential.
You know that lop-sided bell curve mound that Planck discovered describes the distribution of energies of photons in thermal equilibrium (say in a hot box).
It is a kind of beautiful fact that the CMB still has that thermal shape after all these years.
The shape was established when the photons actually were in equilibrium with hot gas. But they have kept that same distribution for 13.7 billion years during which they have NOT been in contact and have not been thermalized and made to be in equilibrium with anything. Just flying free.

Well, he goes thru some math to show that neutrinos, even though they lose energy and momentum differently, would ALSO retain a thermal distribution. If we could ever see the cosmic neutrino background, we would find that it too (like the CMB) had a nice lopsided bellcurve distribution.

But that, tho nice to know, is not essential. I would just read (2.7) and glance at some verbal context, and be lazy about the rest. Life is short.

In answer to your questions: I confirm as well as I can (as non-expert retired guy who loves cosmology) that equation (2.7) would work for all the time that cosmology normally covers.

(Up to near the start of expansion where the classical GR geometry fails and you need a quantum cosmology extension.)

And I don't recall having used the phrase isotropic observers before, but I could have and forgotten. You might find the phrase in your notes if you took the time to look. It's great you keep notes.

Naty1

I skimmed the paper....if you like advanced math, go for it; otherwise, there are few additional insights described beyond what Marcus posted....