The radiation-dominated universe

Asked why neutrinos, which have mass, count as radiation, Guth points to $|\vec p|^2 - E^2/c^2 = -(m_0c)^2$ (slide 2). When a particle's energy is large compared with its rest energy, the right side is negligible and the particle behaves as if it were massless: $E \approx c|\vec p|$. That is the case for neutrinos in the early universe — and, at early enough times, even for electron–positron pairs. Any particle acts like radiation when its energy is large compared with its rest energy (transcript p. 5).

The review ends with the corrected Friedmann equation from Lecture 15 (slide 8):

As long as the universe is homogeneous and isotropic, Guth says, this is exact as far as we know — it is what general relativity gives. A student asks why the derivation used $dU = -p\,dV$ and not the full first law $dU = T\,dS - p\,dV$. Because the expanding gas stays very close to thermal equilibrium with no heat flowing in or out — it expands adiabatically — its entropy does not change, and the $T\,dS$ term is negligible.

The equation contains a startling fact: in general relativity pressure, not only mass density, produces gravity. $\ddot a$ measures how fast gravity slows the expansion, and pressure adds to the slowing.

• In the radiation era, $p = \frac13\rho c^2$, so $3p/c^2 = \rho$: the pressure is exactly as important as the mass density in slowing the expansion. Today the pressure of ordinary matter is negligible.
• Energy densities are, for all the laws of physics we know, positive. But pressures can be negative. A material with negative pressure would, by the same equation, produce gravitational repulsion and make the expansion accelerate.

This is not the mechanical effect of pressure. Mechanical forces come only from pressure differences: the air pressure in a room is large, but we don't feel it because it pushes equally in all directions. In a homogeneous universe the mechanical effect is zero. What remains is gravitational — the term carries Newton's $G$.

And it matters: for about the last five billion years, our universe has been accelerating. The only explanation we have, Guth says, is the repulsive gravity of some negative-pressure material, called dark energy — very likely vacuum energy, a topic for later in the course.

In the piston, the gas's lost energy went into pushing the walls. The universe has no walls. For a small region, the answer is that it pushes on the regions around it. But where does the energy ultimately go? Guth's answer: into gravitational potential energy. Picture a gas cloud held together by its own gravity and made to expand: pulling the particles apart against their mutual attraction raises the gravitational potential energy. So, all in all, energy is still conserved (transcript pp. 12–13).

Now solve the Friedmann equations for a flat ($k = 0$) universe dominated by radiation. With $k = 0$ the first equation is just $H^2 = \frac{8\pi}{3}G\rho$, and since $\rho \propto 1/a^4$,

Integrating, $\frac12a^2 = \text{const}\cdot t + \text{const}'$. Choosing $t = 0$ at the moment $a = 0$ — the standard way of setting the clocks — makes $\text{const}' = 0$, so

replacing $t^{2/3}$ for flat matter domination. For a flat universe the constant of proportionality means nothing: it just fixes the size of a notch. It drops out of every physical quantity, starting with

The horizon distance — how far light has travelled since $t = 0$, in physical units — is

compared with $ct$ in a static universe and $3ct$ in a flat matter-dominated one.

Put $H = 1/(2t)$ back into $H^2 = \frac{8\pi}{3}G\rho$:

This is remarkable: we don't need to know what the radiation is — photons, neutrinos, whatever — only that it falls as $1/a^4$ and that the universe is flat. The flatness is crucial; without it the density could be anything. (Asked about an almost flat universe, Guth says that a quantitative "almost" would give a correspondingly approximate formula.)

• Particles act like radiation when their energy is far above their rest energy; early on that includes neutrinos and even electron–positron pairs.
• In $\ddot a = -\frac{4\pi}{3}G(\rho + 3p/c^2)a$, pressure gravitates. In the radiation era it doubles the deceleration compared with pressureless matter of the same density; negative pressure would accelerate the expansion — the dark-energy explanation of today's acceleration.
• Uniform pressure has no mechanical effect; the energy lost by an expanding gas ends up, Guth suggests, as gravitational potential energy.
• Flat and radiation-dominated: $a \propto t^{1/2}$, $H = 1/(2t)$, horizon $2ct$, and $\rho = 3/(32\pi G t^2)$ regardless of what the radiation is.