The closed universe as a cycloid

Start from the Friedmann equation with pressureless matter, for which $\rho a^3$ is constant:

For a closed universe $k > 0$. Strip away the arbitrary notch units by choosing new variables that are both lengths:

Since $\rho a^3$ and $k$ are constant, so is $\alpha$; it has units of length and is the only parameter left.

Separate the variables and integrate from the Big Bang ($\tilde t = 0$, $\tilde a = 0$) to an arbitrary time $\tilde t_f$:

(Near $\tilde a = 0$ the integrand behaves like $\sqrt{\tilde a}$, so the integral converges — Guth's reminder to always check.) Two substitutions crack it:

1. Complete the square: $2\alpha\tilde a - \tilde a^2 = \alpha^2 - (\tilde a - \alpha)^2$, and set $x = \tilde a - \alpha$.
2. Trigonometric substitution: $x = -\alpha\cos\theta$, so $\sqrt{\alpha^2 - x^2} = \alpha\sin\theta$ and the integrand becomes simply $\alpha(1 - \cos\theta)\,d\theta$.

Integrating, and matching the upper limits ($\tilde a_f - \alpha = -\alpha\cos\theta_f$), gives the parametric solution:

The angle $\theta$ is a parameter that drives both time and size. You cannot solve $\theta - \sin\theta = ct/\alpha$ for $\theta$ with ordinary functions (a computer can do it numerically), so this pair is "as good as it gets". (In the lecture the integral was first written down wrongly and students caught it — a useful reminder to check every step.)

The pair of equations describes a cycloid: the path traced by a point on the rim of a wheel of radius $\alpha$ rolling without slipping along a line.

• Horizontally: after the wheel has turned through angle $\theta$, its contact point has moved $\alpha\theta$; the marked point sits $\alpha\sin\theta$ behind the centre, so $x = \alpha(\theta - \sin\theta) = ct$.
• Vertically: up $\alpha$ to the centre, then down $\alpha\cos\theta$, so $y = \alpha(1 - \cos\theta) = a/\sqrt k$.

Guth calls $\theta$ the development angle. For every closed universe it runs from 0 at the Big Bang, through $\pi$ at maximum size, to $2\pi$ at the Big Crunch.

The solution seems to have two parameters, $\alpha$ and $k$, but $k$ can be set to 1 by choosing the notch. Closed matter-only universes form a one-parameter family, labelled by $\alpha$:

The geometric meaning of $a/\sqrt k$ — it will turn out to be the radius of curvature of space — comes in later lectures.

• With $\tilde a = a/\sqrt k$ and $\tilde t = ct$, the closed-universe Friedmann equation becomes $(d\tilde a/d\tilde t)^2 = 2\alpha/\tilde a - 1$ with one length parameter $\alpha$.
• Completing the square and a trigonometric substitution give $ct = \alpha(\theta - \sin\theta)$, $a/\sqrt k = \alpha(1 - \cos\theta)$: a cycloid.
• The development angle runs from 0 (Big Bang) through $\pi$ (maximum size $2\alpha$) to $2\pi$ (Big Crunch, at $t = 2\pi\alpha/c$).
• At early times the solution matches the flat-universe law $a \propto t^{2/3}$.