The horizon distance

The horizon distance is the present physical distance of the most distant objects that could, in principle, be seen — limited only by the speed of light, not by telescopes. "Physical" (or proper) distance means distance measured now by rulers at rest with respect to the local matter.

Why the present distance, when we see these objects as they were long ago? Guth's answer: it is a useful figure of merit for picturing the universe as it is now.

The naive guess, $c$ times the age, would be right only in a static universe. In an expanding one the answer is larger.

The earliest light we could see left its source at $t = 0$. Follow it in comoving coordinates, where it moves at the coordinate speed of light:

By today, $t_0$, it has crossed a coordinate distance $x_\text{max} = \int_0^{t_0} c\,dt/a(t)$. Multiply by today's scale factor to turn notches into metres:

For the flat matter-dominated universe, $a = b\,t^{2/3}$:

The constant $b$ — with its notch units — cancels, as it must. The factor 3 arises because light covered much of its path early on, when everything was much closer together, and the source has been receding ever since.

The plot shows where the earliest light is, as a physical distance from us, while it travels. Use units with $c = 1$ and today's age $t_0 = 1$, so $a = t^{2/3}$ and the light's comoving position is $3(1 - t^{1/3})$ from us.

This answers a puzzle a student raised in the lecture: has the source moved three times its own distance? No — when the light left, the source was at essentially zero distance from us. The light first receded, then came back.

At $t = 0$ the scale factor vanishes, so all distances were zero. Could every part of the universe once have exchanged signals? Guth's answer: nobody knows what happened at the singularity itself. But start the calculation at a small time $\varepsilon$ instead of 0, and the horizon distance changes only slightly, shrinking to zero change as $\varepsilon \to 0$. Although separations go to zero at early times, the recession velocities ($H = 2/(3t)$) grow without limit. Light needs exactly the time the integral says, so the horizon stays finite. This finite horizon will become important when we meet the horizon problem, one of the motivations for inflation.

• The horizon distance is the present physical distance of the most distant objects whose light could have reached us.
• In general $\ell_{p,\text{horizon}} = a(t_0)\int_0^{t_0} c\,dt/a(t)$; for a flat matter-dominated universe it is $3ct_0 = 2c/H_0$.
• It exceeds $ct_0$ because light covered much of its path when the universe was small; the earliest light first moved away from us before turning back.
• The horizon stays finite even though all distances vanish at $t = 0$, because recession speeds then grow without limit.