The temperature of the early universe and decoupling

In a flat radiation-dominated universe, $\rho = 3/(32\pi Gt^2)$; for black-body radiation, $\rho c^2 = u = g\frac{\pi^2}{30}\frac{(kT)^4}{(\hbar c)^3}$. Equating the two and solving for $kT$:

Here $G$ is Newton's constant and $g$ the effective number of degrees of freedom: spin states and particle–antiparticle pairs of every species present, with fermions counted at 7/8. Strictly $g$ changes as the universe cools through thresholds where particle types drop out, so near a threshold the formula is only approximate. But well inside any one era it is very accurate.

About one second after the Big Bang, photons, neutrinos and electron–positron pairs are all present, still at a common temperature, so $g = 10\frac34$ and

[Computed exactly, the formula above gives $0.859$ MeV and $9.97 \times 10^{9}$ K; the lecture rounds them to 0.860 MeV and $9.98 \times 10^{9}$ K, the values used on this page.]

Since $0.86 \approx 1$: at one second, $kT$ was about 1 MeV, above the electron's rest energy of half an MeV. The pairs were still largely present then, and began to disappear right afterwards.

The universe's ordinary matter is, in Guth's round numbers, about 20% helium and the rest hydrogen. [By mass the helium fraction is closer to 25%.] Until the temperature fell to about 4,000 K, the hydrogen was ionised. That figure is not a property of hydrogen alone — the ionisation temperature depends on the density — but it is the value for the densities of the early universe. When the hydrogen becomes neutral is called recombination, though, as Guth points out, it had never been combined before; the "re" is pure convention.

Recombination is gradual, and is usually dated to the point where about half the hydrogen has recombined. More interesting is decoupling: the point after which a typical photon never scatters again. Because half the hydrogen is still ionised at 4,000 K, plenty of free electrons remain to scatter photons, so decoupling comes later, at a lower temperature — about 3,000 K.

To date decoupling, Guth first adds a useful relation. Entropy is conserved, so as the universe expands the entropy density thins out as $1/a^3$, like the mass density of matter — this is always true. If in addition $g$ is fixed, $s \propto T^3$, and then

As long as the number of species contributing is fixed, the temperature simply falls as $1/a$.

To estimate when, treat the universe as matter-dominated from decoupling until today. Decoupling comes long after matter–radiation equality, and dark energy (not yet introduced in the course) is ignored. Then $a \propto t^{2/3}$, so $T \propto t^{-2/3}$:

[The transcript has "13.8 trillion years"; the age of the universe is 13.8 billion.] Guth gives the result as about 380,000 years (with $T_0 = 2.725$ K it is 378,000), and notes that this crude estimate lands almost exactly on the number people quote from careful calculations. He then rounds: decoupling was about 400,000 years after the Big Bang.

From decoupling on, the photons have travelled essentially in straight lines. So when we look at the cosmic microwave background, we see an image of the universe at decoupling — in exactly the way you see someone's face by photons travelling straight from it to your eyes. That is why the time of decoupling is such an important number.

• Equating $\rho = 3/(32\pi Gt^2)$ with $u = g\frac{\pi^2}{30}\frac{(kT)^4}{(\hbar c)^3}$ gives $kT = (45\hbar^3c^5/16\pi^3gG)^{1/4}/\sqrt t$, accurate well inside any era of fixed $g$.
• With $g = 10\frac34$: $kT = 0.860$ MeV$/\sqrt{t\,[\text{s}]}$, $T = 9.98 \times 10^9$ K$/\sqrt{t\,[\text{s}]}$ — about 1 MeV at one second.
• Conserved entropy and fixed $g$ give $T \propto 1/a$.
• Recombination (~4,000 K, half the hydrogen neutral) is followed by decoupling (~3,000 K), when photons stop scattering — about 380,000 years after the Big Bang by a matter-dominated estimate.
• The CMB is an image of the universe at decoupling.