During the interval from t = 380,000 yr (CMB last surface) to the present, if the Universe were exclusively matter at critical density, the scale factor [1+z] would be proportional to t to the 2/3 power. Note that [13.7 Gyr/380,000 yr] raised to the power 2/3 corresponds to [1+z] = 1090. This of course is the correct redshift to take the CMB last surface from 3000K to the present 2.7K.(adsbygoogle = window.adsbygoogle || []).push({});

However, at t = 380,000 yr, matter probably dominated (matter was about 75% of the total then), while now, matter is only about 25-30% of the total. If the CMB photons have stretched by an extra amount due to recent acceleration/dark energy, wouldn't that cause the predicted value of [1+z] to be larger than 1090 and thus give an incorrect prediction for temperature?

To put it in numbers, in addition to time dependence, I think the scale factor [1+z] is also proportional to the matter fraction to the negative 1/3 power. Thus the starting (t = 380,000 yr) matter fraction of 0.75 would reduce [1+z] to 0.91 [0.75 to the power 1/3] of the matter-exclusive value of 1090, while the ending matter fraction of about 0.28 would multiply [1+z] by 1/[0.28 to the power 1/3] =1.53. The total correction factor is thus 0.91x1.53 = 1.39, taking the predicted value of [1+z] up to 1515. This would predict a present temperature of about 1.98K--clearly wrong.

Are photons perhaps not stretched by the recent acceleration? I think such an assumption would cause logical inconsistencies. Please comment on where I went wrong.

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# Does recent expansion change the predicted CMB temperature?

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