Where does the 379,000 year recombination prediction of the Big Bang theory come from?

Question

The Big Bang theory postulates that recombination happened 379,000 years after the Big Bang.

However, I have never seen a single statement as to how this number came about.

I don't imagine they genuinely have error bars of 0.002% on the universe's timeline throughout its entire history (it seems rather like the the dinosaur fossil joke...) so I imagine something else must be going on here.

But given that we can't "see" before that time period in the electromagnetic spectrum and we don't have gravitational telescopes that can see evidence from further back either, I can only imagine it's the result of some theoretical calculations (what formulas are we talking about here?) and/or computer simulations (which would be rather intriguing to hear more about).

So my question is: where does this figure come from?

Answer 1

What we can see is the cosmic microwave background, which in combination with the measured rate of expansion of the universe tells us where on the temperature scale the black body energy spectrum was centered at different times in the universe's expansion history. We also know by experiment at what temperature hydrogen and helium condense out of an optically opaque plasma or "recombine" and become optically transparent gases. (This is important because at that point in the cooldown from the big bang, the CMB first became free to propagate through space without scattering.) These facts then let you back-calculate how old the universe was at the point that recombination occurred, within the error bars set by the measurements noted above.

Stephen Weinberg's book The First Three Minutes covers these topics in far more detail and is highly recommended.

Answer 2

The first thing to calculate is the temperature at which hydrogen atoms formed. For a first approximation, you can use the Saha ionization equation for the density of atoms in the $i$th state of ionization when they are in thermal equilibrium, and approximate hydrogen as having only the $n=1$ state at -13.6 eV. As explained here, this gives

$$\frac{x_e^2}{1-x_e} = \frac{1}{n}\left(\frac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2}e^{-E_1/k_BT}$$

where $x_e$ is the fraction of hydrogen that is ionized, $n$ is the number density of hydrogen (both ionized and un-ionized), $m_e$ is the mass of an electron, $k_B$ is the Boltzmann constant, $T$ is the temperature, $\hbar$ is the reduced Planck constant, and $E_1=13.6$ eV is the ionization energy of the $n=1$ state.

On the right-hand-side, the two quantities that vary with the evolution of the universe are $n$ and $T$. To find the temperature when the ionization fraction is 50% (so that the left-hand-side is 0.5), we need to know $n$ in terms of $T$.

According to modern cosmological theories, the number density is believed to vary as the inverse cube of the Friedmann scale factor, $n \propto a^{-3}$. The temperature is believed to vary as the inverse of the scale factor, $T \propto a^{-1}$. Thus the number density varies as the cube of the temperature, $n \propto T^3$. So we can write

$$n=n_1\left(\frac{T}{T_1}\right)^3$$

The current number density of hydrogen $n_1$ is measured as $1.6$ per cubic meter, and the current temperature $T_1$ of the universe is measured as $2.7$ K, so the ionization fraction was 50% when $T$ satisifies

$$\frac{1}{2}=\frac{1}{n_1}\left(\frac{T_1}{T}\right)^3\left(\frac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2}e^{-E_1/k_BT}$$

If you put in the various values, you find that the solution is $T=3944$ K.

Here is a plot showing how, in this model, the hydrogen ionization fraction $x_e$ falls from nearly 1 to nearly 0 as the temperature drops from $5000$ K to $3000$ K in the expanding universe:

The second thing to calculate is when was the temperature of the universe $3944$ K. This temperature is $1460$ times greater than the current temperature of $2.7$ K. Since the Friedmann scale factor $a(t)$ varies inversely with the temperature, hydrogen formed when the scale factor was $1460$ times smaller than it is today.

So, when was the universe $1460$ times smaller than it is today? A decent approximation is that, since the time hydrogen formed, the universe has been matter-dominated. The Friedmann equations tell us that $a\propto t^{2/3}$ for a matter-dominated universe. This means that the time $t$ when the universe was $1460$ times smaller was when the time was a fraction $1/1460^{3/2}=1/55,800$ of what it is now.

Since the current age of the universe is $13.8$ billion years, this means that hydrogen formed around $247,000$ years after the Big Bang.

Better estimates yielding $379,000$ years take into account other ionization states of hydrogen, other elements, the fact that universe was not completely matter-dominated after recombination, and probably other details.