The spectrum of the cosmic microwave background

Guth's review of Lecture 17 (slides 2–8) sharpens one definition. In $g$ and $g^*$, the factor out front is 1 for bosons and $\frac78$ (for $g$) or $\frac34$ (for $g^*$) for fermions, and it multiplies the number of particle types — where a "type" is a complete specification: species, particle or antiparticle (if they differ), and spin state.

Two questions from the review:

• Why do electron–positron pairs act like radiation at high temperature? Radiation means effectively massless particles. The typical thermal energy is $kT$, so when $m_ec^2 \ll kT$ the electrons and positrons behave as if massless. At high enough temperatures, all particles do.
• If neutrinos are Majorana particles, does anything change? No: the factor 2 then counts two spin states instead of particle and antiparticle, with the same product. Their (tiny) masses would only matter once $kT$ fell below $mc^2$ — at most around electron volts.

When Arno Penzias and Robert Wilson discovered the CMB in 1965 — its 50th anniversary was coming up, Guth notes — they measured it at one frequency only. Convincing themselves that the hiss in their detector was not electrical noise was a tour de force. They compared it with a cold load cooled by liquid helium and found the comparison worked as expected, and the signal was the same in every direction, pointing to a cosmological rather than local source. (Weinberg's book recounts that they even suspected pigeon droppings in the antenna.)

They were not yet convinced it came from the Big Bang. Their paper, describing the experiment, appeared back to back with a theory paper by Dicke, Peebles, Roll and Wilkinson, which proposed that the radiation was left over from the Big Bang [the transcript spells the first name "Dickie"]. Within the same year, Guth says, Roll and Wilkinson measured it at a second frequency.

For a long time the evidence that the radiation really had a black-body spectrum was thin. Guth's own tabulation of all data known in 1975 (slide 9) shows the expected black-body curve for 2.726 K with all the historical measurements crowded in its low-frequency corner. Besides the Penzias–Wilson and Roll–Wilkinson points it includes the cyanogen measurements described by Weinberg, a 1971 balloon flight by MIT's Rainer (Ray) Weiss — the first evidence that the spectrum turns over and falls as a black body should — and a 1974 balloon flight with very broad error bars. In 1975, Guth sums up, you could believe it was black-body radiation if you wished, but there was not a lot of evidence.

Then, in 1987, a rocket flight by a Berkeley–Nagoya collaboration measured three points (slide 10), all well above the black-body curve, with small error bars. As effective temperatures, two of them were

against the 2.7 K that fit lower frequencies — one point roughly 16 standard deviations off. The team checked their analysis exhaustively and found nothing wrong. Some theorists built models to explain the excess; others, Guth among them, waited for it to go away. They were right.

The Cosmic Background Explorer (COBE) — the first satellite dedicated to the CMB, as Guth describes it — was launched in 1989 and released its first data in January 1990. Before arXiv, results circulated as preprints: photocopies mailed to perhaps a hundred institutions. COBE's preprint (slide 11) showed a spectrum (slide 12) that fitted a black body fantastically well. It changed cosmology, Guth says, from a field of approximate ideas into a precise science. John Mather presented it in January 1990 to a standing ovation [Guth says the American Physical Society meeting; it was the American Astronomical Society's]; he shared the 2006 Nobel Prize with George Smoot, who led another COBE experiment. A couple of years later the full data set cut the error bars by a factor of 10 — still a perfect fit.

What went wrong in 1987? The group never really published an explanation, Guth says; the widespread rumour was that the rocket had seen its own exhaust.

The shape of the spectrum is fixed by thermal equilibrium: a black body in equilibrium must emit not just the right total amount of radiation but the right amount at every frequency (the filter argument of Lecture 16). The energy density between frequencies $\nu$ and $\nu + d\nu$ is

(Guth wrote it as $\rho_\nu$; a student confirmed the units are energy density, so $u_\nu$ is the clearer name.) For the energy density per wavelength interval, an extra factor from $d\lambda/d\nu$ is needed; Weinberg's book plots that version, which looks different.

So precise a black-body spectrum proves that the early universe really was in thermal equilibrium to high accuracy — which can only happen if it was very dense, as the model, going back to infinite density, predicts. More quantitatively: all important energy-releasing processes must have happened before about one year after the Big Bang. Anything later would have left a visible distortion in the spectrum. So the measured spectrum confirms the Big Bang model back to about one year.

As of 2013, Guth notes, COBE's remained the best measurement of the spectrum. Later missions, WMAP and Planck, were dedicated to the CMB's tiny non-uniformities (anisotropies), which COBE had also first measured; Guth promises to return to them later in the course.

• $g$ and $g^*$ count particle types (species, particle/antiparticle, spin state), with fermion factors $\frac78$ and $\frac34$.
• Penzias and Wilson (1965) measured one frequency; until 1987 the data were sparse, and a 1987 rocket experiment disagreed with a black body by up to 16 standard deviations.
• COBE (data January 1990) found a near-perfect black body; Mather and Smoot won the 2006 Nobel Prize.
• $u_\nu\,d\nu = \frac{16\pi^2\hbar\nu^3}{c^3}\frac{d\nu}{e^{2\pi\hbar\nu/kT}-1}$; today's spectrum peaks at 160 GHz.
• The perfect fit shows the early universe was dense and in thermal equilibrium, with no major energy release after about one year.