The false vacuum and the mechanism of inflation

In present-day particle theory (not yet string theory) every particle is a quantum excitation of a field — the photon of the electromagnetic field is the familiar example; for relativistic interacting particles, it is the only consistent description we know. A scalar field describes a spinless particle and is just a number at each point of space. The only one seen in nature is the Higgs field, and inflation is modelled on it. The field that drives inflation, called the inflaton, is probably not the Standard Model Higgs, though in the few years before 2013 several papers proposed that it might be. It is at least a cousin of the Higgs, and like it, its properties are summed up by its potential energy function: energy density as a function of the field value $\phi$.

Guth draws two kinds of potential:

• New inflation: a broad, flat plateau with a peak, usually put at $\phi = 0$.
• Original inflation (Guth's first paper — "it does not work", he says, but it is worth seeing why): a local minimum at $\phi = 0$ that is not the lowest point, with lower true minima on either side.

In both cases the state of interest is the false vacuum: the field sitting at $\phi = 0$. With a local minimum it is classically completely stable — no energy is available to lift it over the barrier — but it can tunnel through quantum mechanically. That process, worked out by Sidney Coleman and collaborators, does not happen everywhere at once (the probability of that is zero; distant regions can't coordinate). It happens in a tiny region, which then grows as a bubble at nearly the speed of light. On the plateau the false vacuum is classically metastable: the smallest fluctuation starts the field rolling. The interesting case is a hill so gentle (small second derivative at the top) that rolling off takes long compared with the Hubble time set by the energy density. So in both cases the field is, for a while, simply stuck.

"Vacuum" to a particle physicist means the state of lowest energy density. A false vacuum is a temporary one: for a time long by early-universe standards, its energy density can't decrease. That forces a large negative pressure. Guth gives two arguments.

From the equations. The energy-conservation equation of the expanding universe is

As the universe expands the field stays stuck, so the density stays at the false-vacuum value: $\dot\rho = 0$. Then

where $u_f = \rho_f c^2$ is the energy density and $\rho_f$ the mass density.

A thought experiment for your roommate. Fill a piston chamber with false vacuum of energy density $u_f$, with ordinary vacuum outside. (Our vacuum's energy is not quite zero, but it is incredibly small compared with $u_f$.) Pull the piston out by $\Delta V$, with the false vacuum somehow attached so that it fills the new volume. The energy inside rises by $u_f\,\Delta V$. Energy conservation says this equals the work done on the contents, $-p\,\Delta V$:

With positive pressure the piston would push on you and you would do negative work. For the energy inside to rise, you must pull against a suction: negative pressure.

A student asks why the energy density doesn't simply drop as the volume grows. For an ordinary gas it would. But here the energy is the field's potential energy, and the only way to lower it is for the field to cross the barrier (waiting for a slow quantum process) or to roll off the plateau (which takes time). On the time scales considered it can't, so $p = -u_f$ holds to good accuracy — not exactly on the plateau, where the density can change a little, slowly.

The second Friedmann equation tells us what gravity does:

For the false vacuum the pressure term is negative and three times as large as the density term, so the bracket becomes $-2\rho_f$: instead of slowing the expansion, gravity accelerates it. This is exactly how the vacuum energy behaves today. False vacuum means gravitational repulsion — and that is the mechanism of inflation.

Asked whether, on the plateau, it takes a long time to start rolling or to reach the bottom, Guth says both, however you divide it: the field may have an infinitesimal velocity from the start, which grows. The whole process is long compared with the time for exponential expansion to set in.

Guth's original model used the potential with a local minimum, which seemed generic, created inflation and, as he realised, solved several cosmological problems. But inflation also has to end. In that model it ended only by tunnelling, in small regions that grow as bubbles — much like water boiling: tiny bubbles form, grow, collide and make a frothy mess. He hoped the bubbles would collide while small and merge into a uniform, hot region of the new phase. They don't [the transcript has him say this "turned out to be the case"; he means the opposite]: bubble formation produces terrible inhomogeneities, with no apparent cure. That was the downfall of the original model, and the difficulty became known as the graceful exit problem.

Soon afterwards Andrei Linde in the Soviet Union and, independently, Andreas Albrecht and Paul Steinhardt in the US proposed new inflation, with the flat-plateau potential — a more special assumption. [Guth says "a few years later"; the new-inflation papers appeared in 1982, about a year after his own, published in January 1981.] Inflation then ends not by tunnelling but by small fluctuations pushing the field down the hill. Two features make it work. The fluctuations are spatially correlated, so over some small coherence region they are essentially uniform. And once the field starts to roll, the hill is still nearly flat, so a great deal more inflation happens after the coherent region forms: a microscopic coherence region is stretched enormously. That was the first solution of the graceful exit problem; later solutions work in very similar ways.

The honest answer is that we don't know: there is still no real theory of initial conditions for any cosmology. But inflation allows a much broader set of initial conditions than the conventional Big Bang, which needs them very precisely specified. Ideas include:

• Chaotic initial conditions (Andrei Linde, now at Stanford): the universe began in some random, chaotic state, and inflation started somewhere where the field happened to be right. Reasonable, Guth thinks, but vague.
• Creation from nothing (Alex Vilenkin of Tufts and, independently, Linde): the universe begins by quantum tunnelling from absolutely nothing — not empty space, which in general relativity is a dynamical thing that bends and stretches like rubber, but no matter, no space and no time; perhaps a closed universe in the limit of zero size.
• The wave function of the universe (Jim Hartle of UC Santa Barbara and Stephen Hawking): since space and time were not defined "before" the universe, it makes no sense to say it was created; it simply has an earliest time. Otherwise it is close to the tunnelling idea: a quantum origin sets the initial state.

None of these is precise — tunnelling while space itself changes needs a quantum theory of gravity we don't have. For inflation, what we need, from whatever source, is a patch at least about a Hubble length ($c/H$) across, expanding fast enough not to collapse, in which the field averages (over any rapid fluctuations) about $\phi \approx 0$. Given that, inflation begins — and once it begins, how it began hardly matters.

• Inflation is driven by a scalar field (the inflaton), a cousin of the Higgs, stuck at a high-energy false vacuum — either a local minimum or the top of a very flat plateau.
• Because its energy density can't drop quickly, $\dot\rho = 0$ gives $p = -\rho_f c^2$; the piston argument gives the same result.
• In $\ddot a = -\frac{4\pi}{3}G(\rho + 3p/c^2)a$ this negative pressure makes gravity repulsive.
• Old inflation ended by bubble nucleation and produced hopeless inhomogeneities (the graceful exit problem); new inflation ends by a slow roll off a flat plateau, stretching a coherent region.
• Inflation needs only an expanding patch at least about a Hubble length across, near the false vacuum, to start.