The supernova evidence for acceleration

Guth recalls that the 1998 teams assumed every type Ia supernova has the same intrinsic power output. That assumption rests on observation and some guesswork; there is no good theory behind it. With it, the flux formula of Lecture 19 predicts how bright a supernova should look at each redshift, for any choice of the Ω's. He again treats only the closed universe; the open and flat cases are similar, and he assigns them as homework.

Substituting $\tilde a(t_0) = cH_0^{-1}/\sqrt{-\Omega_{k,0}}$ into the flux formula puts it entirely in measurable terms:

with $\psi(z_S)$ the integral over redshift from Lecture 19 (slide 5).

Guth then shows data from the High-Z Supernova Search Team, one of the two 1998 teams (slide 7). Its vertical axis is brightness, but expressed the way astronomers like it. First, the luminosity distance $d_L$ is defined by what the flux would be in a static, flat, Euclidean universe:

So $d_L$ is the distance the source would have to be at, in a static Euclidean universe, to look as bright as it does. The distance modulus $m - M$, measured in magnitudes, is a logarithm of it, with a conventional offset of 25:

Because of the minus sign, a larger $m - M$ means a dimmer source — one that looks farther away. Astronomers, Guth says, see brightness and think distance.

The plot also carries the label MLCS, for multicolor light-curve shape. The High-Z team found, from nearby type Ia supernovae, that a supernova's absolute brightness is correlated with the shape of its light curve (its brightness against time). By measuring the light curve of each supernova in the study, they could apply a small correction to its intrinsic brightness instead of assuming every one is identical.

The team compared the data with three models, described in the Ω language of this course (radiation negligible):

• a flat vacuum-dominated model, $\Omega_\Lambda = 0.76$ (what we call $\Omega_\text{vac}$) and $\Omega_m = 0.24$;
• an open model with no vacuum energy and $\Omega_m = 0.2$ — essentially the favoured model of the time, an open universe with a fifth to a quarter of the critical density;
• a flat matter-only model, $\Omega_m = 1$.

On the raw plot the three curves are hard to tell apart, so the team re-plotted everything relative to the open model, which becomes a straight line, and showed deviations from it. Magnified that way, the vacuum-dominated model — called ΛCDM, for Λ plus cold dark matter (cold dark matter being what this course calls non-relativistic matter) — fits best. By eye the advantage is not overwhelming, but statistically the fit is much better, and the astronomers were convinced. Asked when the data were from, Guth says 1998; they have improved a great deal since, and the case is now much more clear-cut.

In the data, the distant supernovae are dimmer than either model without vacuum energy predicts, by a few tenths of a magnitude. Each tenth of a magnitude is roughly 10% in brightness, so the supernovae are some 20–30% dimmer than expected.

That dimmer is the right sign for acceleration is not obvious, and Guth admits that thinking at fixed redshift confuses him too. It is clearer to fix the distance and ask about the redshift. Redshift is strongly tied to the velocity of the source. In an accelerating universe the expansion was slower in the past than it would otherwise have been — it has speeded up to reach today's rate. So a galaxy at a given distance was receding more slowly when it emitted its light, and has a lower redshift. Moving a data point to lower $z$ at fixed height puts it above the curve, which is the same as shifting it upward: dimmer than expected.

So dramatic a conclusion demands that other causes of dimming be ruled out. Two were discussed seriously at the time.

Dust. Looking through dust makes things dimmer. Two arguments weighed against it:

1. No reddening. Dust usually absorbs blue light more than red, so it makes objects look redder. The spectra of the distant supernovae, studied carefully, looked just like those of nearby ones — redshifted, but otherwise undistorted. (A student asks how reddening differs from redshift: redshift moves the whole spectrum down by the same factor, while reddening suppresses the blue end relative to the red, changing the spectrum's shape.) "Gray" dust, which absorbs all colours equally, would need unusually large grains, and nobody has found a source of such grains.
2. No extra scatter. Most relevant dust would be in the supernova's own host galaxy, since intergalactic space holds little dust. Then the dimming would depend on where in its galaxy each supernova happened to lie, producing a large spread in brightness. The spread actually seen matched the known uncertainties in supernova brightness and in the instruments, with nothing left over for a dusty host.

Evolution. Galaxies gradually enrich themselves with elements heavier than helium, made in stars and spread by supernova explosions; the early universe was almost entirely hydrogen and helium. That could in principle change how supernovae behave. But (a) every distance-independent property astronomers could measure — spectra, light-curve patterns — looked the same for the distant supernovae of billions of years ago as for nearby ones; and (b) nearby galaxies have a wide range of heavy-element abundances, and among nearby type Ia supernovae those abundances had no perceptible effect.

Opinion gradually shifted, and Guth says it is now almost unanimous that the acceleration is real. It is confirmed independently by the CMB anisotropies measured by ground-based experiments and by the WMAP and Planck satellites. There is a detailed theory of those ripples, and fitting it requires essentially the same amount of vacuum energy as the supernovae.

Two causes of the acceleration are discussed. The simplest is vacuum energy. The other is quintessence (a made-up word): a nearly uniform scalar field, changing so slowly that it looks constant, whose energy density mimics vacuum energy — in principle the same mechanism as inflation in the early universe. In 2013 Guth called it the minority view, adding parameters that seem unnecessary, but "up for grabs".

• With $\tilde a(t_0)$ substituted, $J = PH_0^2|\Omega_{k,0}|/[4\pi(1+z_S)^2c^2\sin^2\psi(z_S)]$, assuming all type Ia supernovae have the same power output (corrected for light-curve shape by MLCS).
• Astronomers quote brightness as luminosity distance, $J = P/4\pi d_L^2$, and distance modulus, $m - M = 5\log_{10}(d_L/1\text{ Mpc}) + 25$; larger means dimmer.
• The 1998 High-Z data fit the flat model with $\Omega_\Lambda = 0.76$, $\Omega_m = 0.24$ much better than an open $\Omega_m = 0.2$ model or a flat matter-only model: distant supernovae are 20–30% dimmer than models without vacuum energy predict.
• Dimmer means accelerating: slower past expansion gives a lower redshift at a given distance.
• Dust (no reddening, no extra scatter) and evolution (distant and nearby supernovae alike; heavy elements have no effect) were ruled out, and the CMB confirms the result. Vacuum energy is the simplest explanation; quintessence is the main alternative.