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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?

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# Help with some Numbers

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