Exponential expansion, de Sitter space and the inflationary scenario

To see what happens once inflation begins, treat a small region of the patch as a homogeneous Robertson–Walker universe, leave out the curvature term (it quickly becomes negligible, as we'll see) and let the false-vacuum density dominate:

The right side is a constant, so $\dot a/a$ is constant, and

With the full equations there would be other solutions for other initial conditions, but this exponential growth always dominates at late times. That is what inflation is: a period of exponential expansion.

The cosmological no-hair conjecture

Why is inflation so robust — why does it lead to the same result however it starts? One reason is the cosmological no-hair statement: if the matter has $p = -\rho c^2$ with $\rho > 0$, essentially any starting metric evolves to look, locally, like this flat, exponentially expanding spacetime. It can be proven for small initial deviations (perturbatively); beyond that it is believed but unproven, a conjecture. "Locally" matters: a closed universe filled with false vacuum stays closed forever, never literally flat, but as it grows exponentially every piece of it looks flatter and flatter.

de Sitter space

This flat, exponentially expanding spacetime is de Sitter space, after the Dutch astronomer who described it early in the history of general relativity (Guth thinks in 1917, which is right). It has properties de Sitter himself did not notice: by choosing different surfaces of equal time, the very same spacetime can be described as an open or a closed Robertson–Walker universe, homogeneous in each case. For most purposes, though, think of it as flat and exponentially expanding.

The horizons met earlier in the course are, technically, particle horizons: they concern the past — how far light can have travelled since the beginning. de Sitter space also has an event horizon, which concerns the future: two events at the same time that are far enough apart can never influence each other's future, because the expansion outruns the light.

Light's coordinate speed is $c/a(t)$, so the coordinate distance it covers between $t_1$ and $t_2$ is

As $t_2 \to \infty$ this approaches a finite limit: anything starting farther away in coordinates will never be reached by the light, however long you wait. Multiplying the limit by $a(t_1)$ gives the physical distance at time $t_1$:

— the Hubble length. Light emitted now from anything more than one Hubble length away will never reach us.

This matters because nothing travels faster than light. If a de Sitter region is surrounded by "junk" we can't predict, whatever is outside can never penetrate more than about one Hubble length into it. The interior is protected — a rigorous theorem of general relativity, Guth says — so once a sizeable region of de Sitter space exists, it never disappears, whatever happens outside.

We don't know at what energy inflation happened, and so far every observable consequence is much the same whatever the scale. Inflation was invented in the context of grand unified theories, still a plausible setting, so Guth gives grand-unified sample numbers.

The GUT scale comes from measuring the strengths of the three Standard Model interactions — one for each of SU(3), SU(2) and U(1) — at accessible energies (around 100 GeV) and extrapolating them theoretically to high energies. They meet, to good accuracy, at one point: that is the basis of grand unification, one interaction that looks like three at low energy through spontaneous symmetry breaking. The meeting is decent in the Standard Model as it is, and works much better in its minimal supersymmetric extension (where every known particle gets an unseen partner) — a piece of evidence for supersymmetry. Either way the lines meet at about $10^{16}$ GeV.

We can't calculate vacuum energies, but dimensional analysis fixes the false-vacuum mass density up to factors: the only way to make a mass density from $E_\text{GUT}$ and the constants of physics is

Putting this into $\chi$:

[Guth first labels the time as $\chi$; a student catches it, and he corrects it to $\chi^{-1}$.] All off scale by human standards — as grand unified theories are.

Start with a patch of roughly that Hubble length, $\sim 10^{-27}$ cm — some 14 or 15 orders of magnitude smaller than a proton ($\sim 10^{-13}$ cm). Inflation must last long enough for the patch to reach roughly 1 to 10 cm or more by its end. More is no problem: it just means more inflation than the minimum.

How does inflation end? On the plateau, fluctuations destabilise the field; it rolls down the hill and oscillates about the bottom. Because it interacts with other fields, it hands its energy to them — ultimately the Standard Model fields — heating them into the hot soup of particles that is the starting point of the conventional Big Bang. This reheating produces a temperature comparable to the theory's own scale: about $10^{16}$ GeV for grand unification.

After that the universe coasts, with $aT \approx$ constant, so the scale factor grows by the ratio of temperatures:

[Guth says the temperature ratio is $10^{16}$ GeV "times" $k \times 2.7$ K; it is the quotient, as written above.] Guth estimates that we need only about $40 \times 10^9$ light-years to cover the observed universe (a rough figure: its present diameter is about $9 \times 10^{10}$ light-years), so this is ten times too big — which is fine: 1 cm at the end of inflation would do, and 10 cm is generous. In short: a tiny patch inflates to a centimetre or more, reheats, and coasts into something larger than the whole region we observe.

• With the false vacuum dominating, $(\dot a/a)^2 = \frac{8\pi}{3}G\rho_f$ gives $a \propto e^{\chi t}$, $\chi = \sqrt{8\pi G\rho_f/3}$.
• By the cosmological no-hair conjecture, almost any starting geometry evolves to look locally like this flat exponential expansion: de Sitter space.
• de Sitter space has an event horizon at one Hubble length, $c\chi^{-1}$, which shields an inflating region from whatever lies outside.
• At the GUT scale: $\rho_f \sim E^4/\hbar^3c^5 \approx 2.3\times10^{81}$ g/cm³, $\chi^{-1} \approx 2.8\times10^{-38}$ s, $c\chi^{-1} \approx 8\times10^{-28}$ cm. About 65 e-folds stretch this to ~10 cm; reheating to ~$10^{16}$ GeV and coasting then give a region larger than the observable universe.