The age of the Universe is about 13.8 billion years, measured by light emitted from the time it emerged from opaqueness. But how was the time from the "beginning" to 380,000 years (era of recombination) calculated? By extrapolation, using $c$ as the constant? From the initial point to 380,000, there was an opaque quark-gluon soup, in which a photon (and its descendants) would have taken far more time to traverse, leaving our universe to be much older than calculated. By analogy, our sun is about 149 million km - 500 light-seconds - plus the radius (696,000 km or 2.3 light-seconds) away from us. Yet if we extrapolate that extra, using light speed, we'd have to add about a million years extra, since that's the time it takes a photon to cross the distance from the centre to the surface of the sun.
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$\begingroup$ Measuring the optical age of very distant objects is a method of confirming what we get by measuring the way everything is moving apart, curve-fitting it, and drawing the curve backwards through time until everything is in the same place. I don't know what method was used to get kiloyear precision for a segment of that estimate or what other measurements and theories improve our ability to be precise, but that's the basic idea. $\endgroup$– g sCommented Jun 11, 2022 at 23:41
2 Answers
The CMB signal dates from the time of recombination (a misnomer) which was when the temperature had dropped enough to allow stable hydrogen atoms to form and not get immediately re-ionized. This assigns a well-defined temperature (3000 K) to the redshift (~1100) that the CMB exhibits today.
Then, an equation of state can be written for the ionized plasma which relates its temperature, pressure, and density. Assuming then that at each little step in the process the universe was in thermal equilibrium with the photons that were zinging around, the universe's clock can be run backwards from that point to earlier and still earlier times and furnishes predictions of how the temperature, pressure and density of that plasma rose higher and still higher at those times.
At some point things are so hot and heavy that protons themselves get blasted apart, but as long as you can furnish approximate equations of state for what exists earlier and earlier than that, you can push the clock further backwards and by this method get an estimate of how long before recombination the big bang itself happened (at a redshift of infinity): ~370,000 years.
The age of the universe at the age of recombination is calculated to be approximately 370,000 years by using the equation:
dt = (1/H_0) (da/a) (1/SQRT(Omega_m/a^3)).
This is actually an incorrect result. The correct integration required is
dt = (1/H_0) (da/a) (1/SQRT(Omega_r/a^4 + Omega_m/a^3)).
I do not know the variable values used to get 370,000 years.
One reason why 370,000 is wrong is because there is a significant age period at which Omega_r/a^4 >> Omega_m/a^3, and omitting Omega_r gives a definitely wrong result. The variable values I used are:
Omega_m = 0.3103,
Omega_r = 8.24 x 10^-5,
1/H_0 = 14.4 x 10^9 years, and
a_rec (a at recombination) = 0.000756505.
The integration is from 0 to a_rec.
The result omitting Omega_r is 358,590 years.
The result including Omega_r is 273,341 years.
The likely definitely correct value would only include three digits followed by ",000".
