The horizon problem

Guth's summary slides restate the case for acceleration as three arguments that together have persuaded almost everyone — he knows a few people who aren't convinced, but calls them oddballs:

1. Distant type Ia supernovae are 20–30% dimmer than expected.
2. The CMB anisotropies give a vacuum fraction close to the supernova value, and show $\Omega_\text{tot} = 1$ to about half a percent — very hard to account for without dark energy, since there isn't nearly enough of anything else.
3. $\Omega_\text{vac} \approx 0.7$ makes the age of the universe, 13.7–13.8 billion years, consistent with the oldest stars.

Vacuum energy is the simplest explanation; quintessence, a slowly evolving scalar field, would show up as an evolution of the dark energy, which nobody had seen. He recalls the three contributions to the vacuum energy, and that a Planck-scale cutoff overshoots by more than 120 orders of magnitude — his students compute this on the homework, and it comes out a little more than 120.

He then states the anthropic idea more explicitly (slide 4). Many cosmologists — including Steven Weinberg, one of the first to advocate it, and Guth himself, a recent convert to taking it seriously — consider that there may be many types of vacuum, perhaps $10^{500}$ in string theory, most with vacuum energies around 120 orders of magnitude larger than ours. A fraction of roughly $10^{-120}$ would have a near-perfect cancellation by coincidence, and life might only be possible there: a large positive vacuum energy blows the universe apart, and a large negative one makes it implode, before structures can form. A student asks whether the different vacua would also change the other forces. Yes — everything: each vacuum would have its own version of the Standard Model, since particle physics describes small perturbations about whichever vacuum you are in. These ideas are widely discussed, Guth says, but by no means agreed on.

He also recaps the Einstein–Friedmann dates (slide 5).

The universe is observed to be incredibly uniform, and this shows up most clearly in the CMB. Its temperature, measured as a function of direction, is the same to about one part in $10^5$. Asked whether that is just the limit of the measurements, Guth says no: fluctuations at that level are actually seen.

The CMB photons have travelled in straight lines since decoupling, about 370,000 years after the Big Bang, so they show a snapshot of the universe then: an almost perfectly smooth density and temperature with tiny ripples of amplitude about $10^{-5}$ on top. Those ripples matter: the universe is gravitationally unstable, so a slightly denser region pulls in extra matter, grows denser still, and the process cascades into galaxies and clusters. How the ripples arose is left for the discussion of inflation. The question here is how the universe got so uniform.

Objects left alone do reach a uniform temperature — that is why pizza cools after it leaves the oven. Could that explain the CMB? To find out, trace back two CMB photons arriving from opposite directions to the points where they were emitted at $t_d$, and ask how far apart those points were. By symmetry one calculation is enough.

At decoupling the temperature was about 3,000 K (the temperature at which hydrogen becomes neutral at the densities of the early universe). Because the universe stays close to thermal equilibrium, entropy is conserved and $aT$ is roughly constant, so the redshift follows from the temperatures:

— far larger than the redshifts of galaxies or supernovae.

For the present distance to the emission point, Guth uses a formula derived on an earlier homework set for a flat, matter-dominated universe:

This is an approximation: the universe was matter-dominated at decoupling but not all the way to the present, since dark energy took over a few billion years ago. [The transcript says "5,000 or 6,000 years ago"; billions is meant.] It is good enough to make the point. For $z \approx 1100$ the bracket is about 0.97, so $\ell_p$ is close to $2cH_0^{-1}$ — the horizon distance itself, which corresponds to $z = \infty$, and equal to $3ct_0$ in a flat matter-dominated universe.

With the Planck value $H_0 = 67.3$ km/s/Mpc, $\ell_p(t_0) \approx 28.2$ billion light-years. [The transcript reads 57.3 km/s/Mpc, but 28.2 billion light-years follows from 67.3, the Planck 2013 value.] Distances fixed in comoving coordinates scale with $a$, so at decoupling

The horizon distance is the farthest anything could have travelled, at the speed of light, since the Big Bang. For a matter-dominated universe, at decoupling,

so

At decoupling, each emission point was about 23 horizon distances from the spot that would become our location, so the two points were about 46 horizon distances apart. Nothing at one point could have known anything about the other. Yet somehow they emitted photons at the same temperature, to one part in 100,000. That is the mystery.

There is a way out: assume the singularity that started everything produced a perfectly homogeneous universe. Since we don't understand the singularity, we may attribute anything to it. But that is an assumption, not an explanation. If you want to explain the uniformity, causality — the speed-of-light limit — makes it impossible in the conventional Big Bang.

A student asks how a singularity could fail to be uniform. Think of the time reverse: does a non-uniform object collapse to a singularity? That was debated for a long time, until Hawking and Penrose proved theorems showing that non-homogeneous solutions collapse to singularities too. So a singularity need not be uniform; uniformity has to be put in by hand.

Inflation, Guth promises, gives a dynamical explanation.

• The CMB is uniform to one part in $10^5$; the observed ripples at that level are the seeds of structure.
• $1 + z_d = T_d/T_0 \approx 1100$, and in a flat matter-dominated model the emission point is now at $\ell_p(t_0) = 2cH_0^{-1}(1 - 1/\sqrt{1+z}) \approx 28$ billion light-years — about 25 million light-years at decoupling.
• The horizon at decoupling was only $3ct_d \approx 1.1$ million light-years: the two emission points on opposite sides of the sky were about 46 horizon distances apart.
• The conventional Big Bang can only assume the uniformity, by attributing it to the singularity; inflation explains it.