How inflation solves the problems of the Big Bang

The problem was that the early universe expanded so fast that its parts had no time to communicate: when the CMB was released, points on opposite sides of the sky were about 50 horizon distances apart (46 in Guth's calculation), so nothing could explain their equal temperatures.

Inflation inserts an extra phase into the history of the universe, and at its beginning the problem simply isn't there — so it never develops. By assumption, the starting patch was about a horizon length across. If there was enough inflation to make 10 cm out of it — ten times more than needed — then our whole observed universe came from a region only about a tenth of a Hubble length across, well inside the horizon. There was plenty of time for it to reach a uniform temperature by ordinary thermal processes. Inflation then stretched that tiny, uniform region until it encompassed everything we see. So all the regions we observe were once in causal contact.

In the conventional Big Bang $|\Omega - 1|$ always grows, so it had to be about $10^{-18}$ at one second. That came from the Friedmann equation,

During inflation the equation is the same in form, but the density is completely different, and things are reversed. The density is fixed at $\rho_f$, so the first term is constant; the curvature term falls as $1/a^2$, and $a$ grows exponentially. With the sample numbers — expansion from $10^{-27}$ cm to 10 cm, a factor of $10^{28}$ — the curvature term falls by

relative to the density term. By the end of inflation it is negligible and the universe is flat: Ω is driven towards 1, exponentially fast.

Kibble argued that roughly one monopole forms per horizon volume: monopoles are knots in the scalar fields, and the fields can't organise themselves over distances larger than the horizon. That crude argument gave far too many monopoles, and Guth knows of no way to avoid the formation itself. What saves us is that the inflationary model can easily be arranged so that most of the inflation happens after the monopoles form. Then the exponential expansion thins them out. The rest of the matter is not diluted: inflation happens at constant energy density, so reheating at the end still produces the normal amount of other matter.

With the sample numbers, lengths grow by $10^{28}$, so volumes grow by

and the monopoles are diluted by that fantastic factor.

A fourth problem was never stated as one: the universe is not smooth on small scales. Stars gather in galaxies, galaxies in clusters, clusters in superclusters. The CMB shows the early version of these non-uniformities: ripples of one part in 100,000, now measured very accurately. The universe is gravitationally unstable — a slightly denser region pulls in more matter, strengthening its pull — so these ripples can grow into all the structure we see. But where did the ripples come from? In conventional cosmology there was no idea; they had to exist to explain structure, and when they were finally seen, they were just right.

Inflation smooths everything out, and for a while those working on it worried that it would produce a perfectly smooth universe with no seeds for galaxies. Then it was realised that quantum theory comes to the rescue. Classically, inflation predicts a uniform density everywhere. Quantum mechanics is fundamentally probabilistic, so that prediction becomes an almost uniform density, slightly higher in some places and slightly lower in others — qualitatively just what the CMB shows. And it can be calculated.

Guth ends by showing the comparison — a graph made with Max Tegmark for an article by Guth and David Kaiser in Science (2005), updated with WMAP seven-year data. (The graph is © Science and excluded from MIT's Creative Commons licence, so it is described here, not reproduced.) [He calls it "Planck seven-year data"; it is WMAP's seven-year data, as the slide says.]

• It plots the amplitude of the CMB temperature fluctuations against their angular wavelength on the sky — really a multipole expansion — with long wavelengths at the left.
• The data, with their error bars, follow the theoretical curve from inflation, computed with the amount of dark energy also needed for the supernova data. [The transcript says "dark matter" here; the context is the dark energy measured with supernovae.] Guth was so delighted that he shows a little "Eureka" figure.
• Alternatives, added with Tegmark's help, fail badly: an open universe with Ω around 0.2–0.3, as many believed before 1998; inflation without dark energy, with Ω = 1 made of matter; and cosmic strings as the source of the fluctuations, which these data ruled out completely.
• Planck's data, released in March 2013, fit even better: the more accurately the fluctuations are measured, the better they match.

For truth in advertising, Guth notes that the curve is partly fitted. It is a six-parameter fit; four parameters are largely fixed by other features, and two are adjusted to the data: one is the opacity of space between us and the surface of last scattering, which changes the shape a little, and the other is the height of the curve. Inflation could predict the height if we knew the inflaton's potential in detail, but we don't. The positions of the peaks, however, come straight out of the theory.

His conclusion: inflation looks like a very good explanation of the very early universe. It is bizarre to talk about times like $10^{-35}$ seconds after the Big Bang, an incredible extrapolation from known physics — and yet it agrees fantastically with what astronomers measure.

• Horizon: before inflation, the region that became our observed universe was inside the horizon and could equilibrate; inflation then stretched it.
• Flatness: during inflation $\rho_f$ is constant while $kc^2/a^2$ falls as $1/a^2$, so Ω → 1 exponentially; a $10^{28}$ expansion suppresses curvature by $10^{56}$.
• Monopoles: formed before most of the inflation, they are diluted by the volume factor ~$10^{84}$; reheating still produces normal matter.
• Structure: quantum fluctuations during inflation seed the density ripples; the predicted CMB spectrum, with dark energy included, fits the data beautifully, with only the overall height and the optical depth fitted.