Matter–radiation equality and the pressure of light

Guth quotes the present radiation energy density (how it follows from the temperature of the cosmic microwave background comes later in the course):

This includes neutrinos as well as photons: in the early universe neutrinos behaved just like radiation. The photon part is measured; the neutrino part rests on theory, discussed later. From here on, the subscript $\gamma$ means photons and $r$ means all radiation.

Compared with water, at 1 g/cm³, that is, in Guth's words, $10^{-34}$ "or maybe $10^{-33}$" times as dense. It is tiny even compared with the critical density,

Why write $H_0$ through a dimensionless $h_0$? Partly so astronomers needn't keep saying "kilometres per second per megaparsec", and partly so formulas stay valid as the measured value moves — it changed, Guth notes, every time he taught the course. Dividing gives the radiation share of the critical density:

using the 2013 Planck value $h_0 = 0.67$ (the earlier accepted value was about 0.70). About one part in $10^4$ of today's energy density is radiation.

With $\Omega_m \approx 0.30$ for (non-relativistic) matter, today's ratio is $\Omega_r/\Omega_m \approx 3.1 \times 10^{-4}$. The ratio scales as $1/a$, so at any time

Equality is when this ratio is 1. Since $1 + z = a(t_0)/a(t)$,

[The transcript first has Guth saying "about 32,000"; one sentence later he says 3,200, which is right.] To turn this into a time, Guth treats the whole history since then as matter-dominated, $a \propto t^{2/3}$, with $t_0 = 13.8 \times 10^9$ years (a number he admits he had forgotten to give):

This is only an estimate, with two known errors: it ignores the recent acceleration due to dark energy, and close to equality the radiation itself already affects the expansion. Ryden's textbook, including everything, gets 47,000 years — mostly different, Guth says, because she used $h_0 = 0.72$. Either way: the radiation era lasted tens of thousands of years, long compared with a human life but a tiny fraction of cosmic history.

To follow a radiation-dominated universe, the Friedmann equations must be adapted. (The details are an 8.286 homework problem; the lecture gives the logic.) For non-relativistic matter we have

together with $\rho \propto 1/a^3$. Writing $\rho = b/a^3$ with a constant $b$ and differentiating:

These three are not independent. Differentiate (1) in time, use (3) for $\dot\rho$, and out comes (2); any two give the third. For radiation, $\rho \propto 1/a^4$ turns (3) into

Now the set is inconsistent: if (3) changes, (1) or (2) must change too. Which one?

Shouldn't $\dot\rho$ be fixed by conservation of energy? It is — but energy conservation includes work done by pressure. A gas in a cylinder pushes on a piston with force $pA$; if the piston moves out by $dx$, the gas does work $pA\,dx = p\,dV$ and loses that energy:

The formula holds for any container, however irregularly it grows. Dust at rest exerts no pressure; photons racing around do. Apply it to a box fixed in comoving coordinates, whose physical volume is $V = a^3(t)\,V_\text{coord}$ and whose energy is $U = V\rho c^2$:

With $p = 0$ this is (3). To get the 4 of (3′), the pressure must be one third of the energy density:

which agrees with any other way of calculating the pressure of light.

A rigorous answer needs general relativity, but Guth motivates it. Equation (1), first order in time, is an energy equation, like $\frac12mv^2 + V(r) =$ constant; (2), second order, is a force equation.

Now imagine sticks of TNT spread through all of space, set off at the same cosmic time. The pressure jumps instantly; the energy density does not. If pressure appeared in (1), nothing else in it could jump to match, so (1) would be violated. If it appears in (2), the jump just changes $\ddot a$ suddenly — exactly as a sudden force changes a particle's acceleration. So the pressure belongs in (2). Requiring (1), (2′) and the new $\dot\rho$ equation to be consistent fixes the form:

These are the Friedmann equations general relativity gives directly. [Guth momentarily says equation 1 must change before correcting himself to equation 2.] Read off the equation: a positive pressure adds to $\rho$ as a source of deceleration — it makes gravity stronger (a point Guth draws out in Lecture 16).

• Today radiation (photons + neutrinos) has $u_r = 7.01 \times 10^{-14}$ J/m³, $\rho_r = 7.80 \times 10^{-31}$ kg/m³, $\Omega_r = 4.15 \times 10^{-5}h_0^{-2} \approx 9.2 \times 10^{-5}$.
• $\rho_r/\rho_m \propto 1/a$; with $\Omega_m \approx 0.30$, equality was at $z_\text{eq} \approx 3{,}200$, roughly $t_\text{eq} \approx 75{,}000$ years (Ryden: 47,000) — an order-of-magnitude estimate.
• With pressure, energy conservation reads $\dot\rho = -3(\dot a/a)(\rho + p/c^2)$; radiation's $1/a^4$ needs $p = \frac13\rho c^2$.
• Pressure enters the second-order equation: $\ddot a = -\frac{4\pi}{3}G(\rho + 3p/c^2)a$, while $(\dot a/a)^2 = \frac{8\pi}{3}G\rho - kc^2/a^2$ is unchanged.