The Big Bang and its prequel — an overview of inflation

The conventional Big Bang theory says that the universe as we know it began 13 to 14 billion years ago — now pinned down, from Planck combined with other data, to

Guth stresses the qualifier "as we know it": whether anything came before the Big Bang is completely open, and the multiverse ideas at the end of the course suggest the Big Bang may be only the beginning of our local pocket universe.

According to the theory, our region of the universe was then an extremely hot, dense, uniform soup of particles, filling space and already expanding. That is not the popular cartoon of a small egg exploding into empty space. If it were, the sky would look different towards the egg than away from it. Instead it looks the same, on average, in every direction: the Big Bang happened everywhere at once.

From the simple assumptions of a hot gas in thermal equilibrium, expanding and pulled back by gravity, the theory calculates:

• how the early universe expanded and cooled — its expansion rate, temperature and density at any instant;
• how the light elements formed (the subject of Steven Weinberg's The First Three Minutes): about five isotopes of hydrogen, helium and lithium, whose predicted abundances agree well with observation. Most of the kinds of element were made much later in stars and spread by supernovae, but most of the matter — hydrogen and helium — comes from the Big Bang;
• in outline, how matter congealed into stars, galaxies and clusters — still a work in progress, which the course will not follow far.

The standard theory says nothing about what caused the expansion: everything starts out already expanding. It also assumes that every particle had some precursor from the very beginning, without saying where the matter came from. In Guth's phrase, it says nothing about what banged, why it banged, or what happened before it banged. It is really a theory of the aftermath of a bang — a "bangless" theory, despite its name.

Inflation is a small modification of the standard theory — a prequel, in Hollywood's word: a brief account of what happened just before the Big Bang, explaining the bang itself, the propulsion that drove the expansion. It rests on what Guth calls a miracle of physics — "miracle" only because it is so surprising, and because, when he was an undergraduate, nobody talked about it:

Miracle 1: gravity can be repulsive

This has been known since Einstein's general relativity. Einstein introduced the cosmological constant [Guth says 1916; Einstein's cosmological paper was 1917] to balance ordinary gravity in a static universe — which turned out to be "dead wrong" as a model, but showed that general relativity allows gravitational repulsion. In general relativity both energy density and pressure create gravity (in Newtonian physics only mass density does). Positive pressure gives normal, attractive gravity; negative pressure gives repulsive gravity. Combining general relativity with modern particle physics gives a clear indication — not quite a prediction — that at very high energy densities there are states of matter with such negative pressure.

Inflation proposes that a small patch of this repulsive-gravity material existed in the early universe. We don't know when — that is, at what energy — but a plausible choice is the energy of grand unified theories, which unify the weak, strong and electromagnetic interactions at about $10^{16}$ GeV (1 GeV being roughly the rest energy of a proton). Then:

• The patch need only be about $10^{-28}$ cm across. It needn't be the whole universe, and such patches could be very rare: everything we see would come from one.
• Its repulsion drives exponential expansion, doubling in size every $10^{-37}$ second or so.
• About 100 doublings expand it by a factor of at least $10^{28}$ — it could be much more; more inflation just means our uniform region is bigger than what we see — in about $10^{-35}$ s. At the end, the region destined to become our observed universe is about the size of a marble.
• The material is unstable and decays, like a radioactive substance, into ordinary matter. The released energy becomes the hot soup of particles: the prequel ends and the main feature, the standard Big Bang, begins.

There is a caveat. Like radioactive decay, this decay is exponential, with a half-life: however long you wait, a little remains. That tiny remnant is the seed of eternal inflation (next page).

Miracle 2: negative gravitational energy

During inflation the density of the repulsive-gravity material does not decrease. Double the radius of an ordinary gas and its density falls eightfold; this material keeps a constant density, so the energy inside the expanding region grows exponentially. Does that violate conservation of energy — a sacred principle of physics?

No: the energy of a gravitational field is negative, both in Newtonian physics and in general relativity. Newton's law of gravity has the same form as Coulomb's law, but with the opposite sign — like masses attract. The same argument that gives the positive energy density of an electric field gives a negative energy density for a gravitational field. As inflation proceeds, the positive energy of the material grows, and the negative energy of the gravitational field filling the region grows to match. Schematically, the universe has a huge positive energy in matter and radiation and a huge negative energy in gravity, and everything we know is consistent with their sum being zero.

1. Large-scale uniformity

The CMB has the same temperature in all directions to about one part in 100,000. (Measured raw, there is a larger asymmetry, about one part in 1,000, hotter in one direction than the opposite: that is our own motion through the radiation. It has a definite angular pattern — the "dipole" — which is subtracted, and the remaining variations are one part in 100,000.) For the first ~400,000 years the universe was a plasma in which photons kept scattering off free electrons; then it became a neutral, transparent gas, so the CMB is a picture of the universe at 400,000 years — and it is bland.

You can simply assume the universe started that uniform. But to explain it within the conventional Big Bang, matter and energy would have to be moved around at about 100 times the speed of light. Inflation solves this: before inflation the observed region was so tiny that it easily became uniform, as the air in a lecture hall spreads to fill it evenly; inflation then stretched it to include everything we see.

2. Flatness

"Flat" does not mean two-dimensional: it means Euclidean, as opposed to the curved geometries general relativity allows. In two-dimensional analogues, the angles of a triangle add up to more than 180° on a sphere (closed), less on a saddle (open), and exactly 180° only in the flat case. (Asked whether three-dimensional curvature needs a fourth dimension: mathematically no — the extra dimension is a crutch for visualising, though a useful one, used later for the closed universe.) General relativity ties the geometry to the density through

where the critical density depends on the expansion rate. $\Omega = 1$ is flat, $\Omega > 1$ closed, $\Omega < 1$ open.

$\Omega = 1$ is like a pencil balanced on its tip: slightly above 1 and Ω soon rises to infinity and the universe recollapses; slightly below and Ω dribbles off to zero — either way no galaxies. To be as close to 1 as today, Ω at one second must have been 1 to about 15 decimal places. [Lecture 21 explains the numbers: with Planck's bound on Ω today it gets 18 decimal places, and a bound about a thousand times looser, as known around 1980, gives 15.]

During inflation, gravity is repulsive and Ω is driven towards 1, exponentially fast, so before inflation it could have been 2, 10, 1/10 or 100. The effect is so strong that inflation predicts Ω should be 1 today. [Guth says each e-fold brings Ω "another factor of 10 closer"; the factor is $e^2 \approx 7.4$ — roughly ten.] Before 1998 astronomers measured only $\Omega \approx 0.2$–0.3 — which made for uncomfortable dinners, Guth recalls. The difference was dark energy, discovered in 1998 because the expansion has been accelerating for about the last 5 billion years. The earlier measurements were right about what they measured — visible matter (only about 0.04) plus dark matter, inferred for example from how fast galaxies rotate — but missed dark energy. Now

— Ω equals 1 to about half a percent, or maybe 1%, Guth says, just as inflation predicts.

3. Small-scale non-uniformity

On small scales the universe is very lumpy: the Earth is about $10^{30}$ times denser than the average. These lumps grew by gravitational instability from the faint ripples, one part in 100,000, seen in the CMB. In the conventional Big Bang they had to be put in by hand. Inflation attributes them to quantum fluctuations: the uncertainty principle forbids perfect uniformity. Inflation cannot predict their amplitude without knowing more particle physics, but it predicts the spectrum — how the intensity varies with wavelength. Guth shows the comparison of theory with the WMAP seven-year data, and Planck's 2013 data — a gorgeous fit — and alternatives such as cosmic strings, which are excluded. (The graph is © Science and not reproduced here; it is described on the Lecture 23 page.) Asked whether dark energy changed inflation, Guth says inflation itself hardly changed: the observed curve combines what inflation produced with the evolution since, and dark energy changes the latter — and gets the mass density right.

• The standard Big Bang describes the aftermath of a bang — a hot, dense, uniform, expanding soup 13.82 billion years ago — but not what caused the expansion or where the matter came from.
• Inflation, the prequel, uses repulsive gravity from negative pressure to blow a ~$10^{-28}$ cm patch up by ≳ $10^{28}$ in ~$10^{-35}$ s; the material then decays into the hot soup.
• Energy is conserved because gravitational field energy is negative; the total energy may be zero.
• Evidence: inflation explains the CMB's uniformity, predicts Ω = 1 (confirmed once dark energy was found), and explains the ripples as quantum fluctuations.