Black-body radiation

Take a box with a cavity inside and hold the box at a uniform temperature $T$. The cavity fills with electromagnetic radiation whose properties depend only on $T$ — not on what the box is made of. This is claimed, and verified experimentally. Saying the radiation "has temperature $T$" describes it completely. It goes by several names: black-body radiation, cavity radiation, or thermal (equilibrium) radiation.

A black body, in the literal sense, absorbs all radiation that hits it: it reflects nothing. It still glows, like hot iron — but that is thermal emission, a slow process in which excited atoms de-excite, not instantaneous reflection.

Guth's argument (transcript pp. 17–19):

1. Put a black body into the cavity. Whatever its starting temperature, it reaches the temperature $T$ of everything else and stays there.
2. In equilibrium it absorbs all the radiation hitting it, so to keep its temperature it must emit at exactly the same rate.
3. Emission is driven by processes inside the body. Take the body out of the cavity before it has time to cool, and it keeps emitting at the same rate — exactly the intensity of the cavity radiation it was bathed in.
4. Surround it with filters that pass only certain frequencies. Equilibrium must hold for each frequency separately, so the emitted spectrum must match too.

So a perfect absorber at temperature $T$ emits exactly cavity radiation — hence the name. A body that reflected some radiation would emit something different. (Once removed, the body does cool, and strictly its surface cools faster than its centre; Guth agrees with a student that this spoils exact equilibrium, but if the cooling is slow it can be accounted for.)

In the 1800s physicists tried to understand cavity radiation with classical physics. Put the field in a box (any big imaginary box, later taken to infinity). By Fourier's theorem any field in the box is a sum of standing waves — normal modes with a whole or half-whole number of wavelengths across the box. Each mode is a harmonic oscillator.

Classical statistical mechanics gives $\frac12kT$ per degree of freedom, and a harmonic oscillator has two (kinetic and potential energy), so each mode gets $kT$. But there is no limit to how short a wavelength can be: there are infinitely many modes, more and more at shorter wavelengths, each with $kT$. The energy in the box would be infinite. Guth calls this Jeans's paradox (also known as the ultraviolet catastrophe). A box at fixed temperature would soak up energy without limit, siphoning it into ever shorter wavelengths — "we'd all freeze to death".

Quantum mechanics saves the day. A harmonic oscillator's energy can no longer take any value:

(up to a zero-point $\frac12$, which can be dropped by moving the zero of energy — though Guth warns it will matter later). For a short-wavelength mode, $h\nu$ is large: the minimum ante to excite it is far above the classical share $kT$. Faced with putting in nothing or far more than $kT$, statistical mechanics says the likeliest choice is nothing. The short-wavelength modes are frozen out, and the energy density becomes finite.

What about very long wavelengths? A student worries about a divergence there. In a box, a wavelength can be at most twice the box size. In an infinite universe the total energy is indeed infinite — but that is fine for a finite energy density. The box cuts off long wavelengths, quantum mechanics short ones.

Treating the gas in a box as many harmonic oscillators, with quantum statistical mechanics, gives the energy density

with $\hbar = h/2\pi$ and $k$ Boltzmann's constant. The same analysis gives $p = \frac13u$, agreeing with the piston argument of Lecture 15. The number density of photons is

where $\zeta$ is the Riemann zeta function. For photons $g = g^* = 2$, but for other particles they will differ — hence the two names.

Entropy is, loosely, a measure of how many quantum microstates make up a given macroscopic state. For us its key property is that it is conserved as long as things stay at or near thermal equilibrium — as they do in the expanding early universe. Its density is

with the same $g$ as in $u$ — 2 for photons. [Here the transcript has Guth saying "protons" where he means photons.] (The powers of $k$ and $T$ don't match only because of how entropy's units are defined.) The factor 2 for photons counts their two polarisations, for example right- and left-handed; writing it out keeps the formulas "per polarisation", ready for other particles.

How can entropy be conserved when the universe cools? Conserved total entropy means the entropy density thins out like the particles, as $1/a^3$. With $s \propto T^3$ that just requires

which is consistent with $u \propto T^4 \propto 1/a^4$ from Lecture 15.

Neutrinos contribute significantly to the radiation. Until around 2000, Guth says here, they were thought to be massless (in Lecture 17 he says 1990); now we know they have a very small mass. For cosmology, at least for its effect on the expansion rate, they still act as if massless (their masses would matter for structure formation).

There are three types, or flavours: $\nu_e$, $\nu_\mu$ and $\nu_\tau$, named after the electron, the muon and the tau (heavier cousins of the electron), with which they are typically produced.

No neutrino mass has been measured directly. What is seen is one flavour turning into another in flight. A massless particle moves at the speed of light, where any clock it carried would stand still: it has no time in which to change. So neutrinos that change in transit must have mass. (Asked about photons making electron–positron pairs, Guth says a free photon never decays; a collision can create pairs — in his picture, the collision "breaks the clock".)

The particle-data tables Guth quoted in 2013 give the differences of squared masses:

These are fractions of an electron volt, against 938 MeV for a proton [Guth says "three million electron volts"; 938 MeV is 938 million]. The labels 1, 2, 3 are not $e$, $\mu$, $\tau$: the states of definite mass, $\nu_1, \nu_2, \nu_3$, are quantum superpositions of the flavour states. That a $\nu_e$ converts into other flavours is precisely the statement that it has no definite mass. (Why no $\Delta m_{31}^2$ is quoted, Guth puts down to his own lack of knowledge. [Our note: its size follows from the other two, up to the still-unknown ordering of the masses.])

For the rest of the course neutrinos are treated as massless, which is extremely accurate for their effect on the expansion. Their contribution to black-body radiation is the first new topic of Lecture 17.

• Radiation in equilibrium at temperature $T$ depends only on $T$: black-body, cavity or thermal radiation. A perfect absorber at $T$ emits exactly this.
• Classically each of infinitely many modes gets $kT$ — an infinite energy (Jeans's paradox). Quantisation, $E = nh\nu$, freezes out modes with $h\nu \gg kT$.
• $u = g\frac{\pi^2}{30}\frac{(kT)^4}{(\hbar c)^3}$, $p = u/3$, $n = g^*\frac{\zeta(3)}{\pi^2}\left(\frac{kT}{\hbar c}\right)^3$, $s = g\frac{2\pi^2}{45}\frac{k^4T^3}{(\hbar c)^3}$, with $g = g^* = 2$ for photons (two polarisations).
• Conserved entropy in an expanding universe means $T \propto 1/a$.
• Neutrinos (three flavours) change flavour in flight, so they have tiny masses; cosmologically they act as massless radiation.