The flatness problem

Guth quotes the best value from the Planck team (combined, as for all his Planck numbers, with some other data; the satellite alone gives slightly looser limits):

The uncertainty is a little more than half a percent. [Planck published this as a value of $\Omega_k = 1 - \Omega$, so strictly the central value of $\Omega_0 - 1$ has the opposite sign; the size of the bound, which is all that matters here, is the same.]

Guth stresses that the argument does not need this precision. When inflation was invented around 1980, Ω was known far less precisely — only to within roughly an order of magnitude, with a lower end around 0.1 — and the problem was nearly as strong. [The transcript gives the 1980 range as "0.1 and 0.2"; since Guth calls it still close to 1 and later says the bound was three orders of magnitude weaker, a range from about 0.1 to about 2 is probably meant.]

What makes this a problem is the dynamics: $\Omega = 1$ is an unstable equilibrium. An exactly flat universe stays exactly flat, so if $\Omega = 1$ exactly, it stays 1 forever. But if Ω in the early universe were a tiny bit above 1, it would rise and quickly reach infinity — which is what happens when a closed universe reaches its maximum size — and the universe would recollapse. A tiny bit below 1, and Ω would rapidly trail off towards 0. Finding Ω near 1 today is like finding a pencil still almost upright after standing on its tip for a billion years: it must have started incredibly close to vertical. The question is how close.

Guth evaluates Ω at $t = 1$ second, following Dicke and Peebles, whose 1979 discussion he credits with introducing the problem. One second is when the processes of nucleosynthesis begin, and the successful predictions of the light-element abundances are real empirical evidence that the conventional model works back to one second after the Big Bang.

Nothing new is needed: only the Friedmann equation and how temperature and density behave in the matter- and radiation-dominated eras. Start from

together with entropy conservation:

($aT$ changes by a factor of order one when electron–positron pairs disappear — the $(4/11)^{1/3}$ of the neutrino temperature — which does not matter here.)

Reshuffling the Friedmann equation, with no approximation, gives a form whose time dependence is easy to read off:

The temperature cancels between $A$ and $T^2$, so this is exactly the Friedmann equation; it is written this way because $A$ is (nearly) constant, so $(\Omega - 1)/\Omega$ follows $T^2/\rho$. (Divide $H^2 = \frac{8\pi}{3}G\rho - kc^2/a^2$ by $\frac{8\pi}{3}G\rho$ to check.) Since Ω is close to 1, the Ω in the denominator hardly matters.

Today. The Planck limit is largest on the negative side, $|\Omega_0 - 1| \le 0.0075$, which gives $|\Omega_0 - 1|/\Omega_0 \approx 0.0076$. Guth rounds up:

Matter era, from about 50,000 years to now. For a flat matter-dominated universe $a \propto t^{2/3}$, with $T \propto 1/a$ and $\rho \propto 1/a^3$. So $T^2/\rho \propto a \propto t^{2/3}$: $\Omega - 1$ grows with time. (Matter–radiation equality was about 50,000 years; the class's own estimate was about 47,000.)

Radiation era, from one second to 50,000 years. Now $a \propto t^{1/2}$ and $\rho \propto 1/a^4$, so $T^2/\rho \propto a^2 \propto t$ — growing even faster than in the matter era.

Putting the pieces together:

To be in the allowed range today, Ω at one second must have equalled 1 to 18 decimal places — and no known physics forces Ω to be 1. With the 1980 knowledge, the present bound was about three orders of magnitude weaker, giving $10^{-15}$ instead: a thousand times larger, but still an incredibly small number, and the argument sounds exactly the same.

As with the horizon problem, ignorance can be blamed on the singularity: assume the universe started with Ω exactly 1, or 1 to some extraordinary accuracy. But that is a hope for an explanation, not an explanation. Inflation, Guth says, provides a mechanism that drives the early universe towards $\Omega = 1$.

A student suggests that $\Omega = 1$ is natural because a flat universe has zero total energy. Guth disagrees: a closed universe also has zero total energy, so $\Omega = 1$ is not special that way. And $k$ can take any value on the real line — positive (closed) or negative (open) — while exactly flat is the single point $k = 0$, hardly what you would expect from a random draw. Ultimately we can't know without understanding the singularity; you are free to speculate, but inflation removes the need to.

• $\Omega = 1$ is an unstable equilibrium: any departure grows with time.
• Exactly rewritten, the Friedmann equation reads $(\Omega - 1)/\Omega = AT^2/\rho$ with $A = 3kc^2/(8\pi Ga^2T^2)$ nearly constant, so $(\Omega - 1)/\Omega \propto t^{2/3}$ in the matter era and $\propto t$ in the radiation era.
• With $|\Omega_0 - 1| < 0.01$, at $t = 1$ s (the start of nucleosynthesis) $|\Omega - 1| \lesssim 10^{-18}$; even 1980's rough knowledge gives $10^{-15}$.
• The conventional Big Bang can only assume this; inflation explains it.