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After reading several explanations for the so-called "Hubble-radius", and still being confused, (as I reckon are some of the folks who tried to answer THAT question !!), I have a related question, which I hope might help clarify this issue. It's said that the average wavelength of photons at the time when our universe became transparent to light, [i.e., at approx. 300,000 years after the Big Bang started], was about 1000 times less than what it is now, and that each photon carried about 1000 times as much energy. These are, of course, the photons which constitute the so-called "cosmic microwave backround" [CMB] radiation. If our universe has been expanding at approx. the same rate since year-300,000 it seems like the factor would be more like (13 billion) / (300 thousand), which is much larger than 1000. Am I missing something important here?

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    $\begingroup$ What you are missing is that the expansion rate has not been constant. Very soon after the BB there was a period called inflation that caused expansion at an unbelievably high rate. You should google the CMB where you will see a wine bottle shaped graph that shows how we think the expansion rate varied from the earliest moments after the BB to the present. $\endgroup$ – Lewis Miller Jul 19 '16 at 20:22
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    $\begingroup$ Here is a link to the graph: en.wikipedia.org/wiki/Chronology_of_the_universe $\endgroup$ – Lewis Miller Jul 19 '16 at 20:35
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    $\begingroup$ @LewisMiller: about your 1st comment , inflation preceded the CMB emission time. $\endgroup$ – user46925 Jul 19 '16 at 22:08
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    $\begingroup$ not (13 billion) / (300 thousand) but ( 13 billion - 300 thousand) by some factor, called Hubble constant ( product some units' factor ) $\endgroup$ – user46925 Jul 19 '16 at 22:11
  • $\begingroup$ HMMMM: my first impulse was to regard "inflation" as not applicable here, as it occurred before our universe became transparent to light ... I see that one commenter (above) + one answerer (below) seem to agree ... in fact, I'm skeptical of the whole concept of "inflation" !! $\endgroup$ – PERFESSER CREEK-WATER Jul 20 '16 at 16:45
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See the lookback time to redshift relation in https://en.m.wikipedia.org/wiki/Redshift

You can ignore inflation if you get redshifts, temperatures and universe size (radius, scale) ratios between now and times in the past after inflation. For recombination the relations of 1+z to the scale ratios and temperature ratios are linear and direct . So for T(then)/Tnow) = 3000K/3K = 1000 you get z about 1000, or more exactly closer to 1100.

Similarly, the scale ratio of 1000 would give you the size of the universe at recombination of about 13.8 million light years (again, you can use exact numbers for better accuracy).

But the lookback time is not linear. That Wikipedia article has the equation. The Dodelson book derives it exactly and dos more calculations, and it's not that hard. There is also an online calculator, but it won't teach you anything. The dependence is that lookback time is proportional to (1+z)^(-3/2).

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    $\begingroup$ Yes: 1000^(3/2) = 3,1600, which is exactly the ratio of 13 billion to 410 thousand, which is close-enough to 300 thousand for me !! $\endgroup$ – PERFESSER CREEK-WATER Jul 20 '16 at 17:31
  • $\begingroup$ Yes, and keep in mind it's closer to 1100 $\endgroup$ – Bob Bee Jul 20 '16 at 22:27
  • $\begingroup$ Yes!! that works $\endgroup$ – PERFESSER CREEK-WATER Jul 23 '16 at 15:11

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