Two coincidences with one mechanism

Two facts about the universe are individually unremarkable and jointly hard to arrange.

The first: the microwave background has the same temperature in every direction to one part in a hundred thousand, and the particle horizon at recombination subtends about a degree on the sky. Two patches two degrees apart had never exchanged so much as a photon and are nevertheless at the same temperature to five decimal places.

The second: the total density equals the critical density to a fifth of a per cent. That would be a coincidence at any time; the difficulty is that the equality is an unstable equilibrium, so matching it now requires matching it far more precisely earlier.

The horizon problem, stated properly #

The comoving Hubble radius is the distance over which physics can act now, in comoving units, and in a decelerating universe it grows monotonically. That means the comoving distance over which anything could ever have acted — the integral of it over the whole history — is dominated by the present. Regions that are in contact today were not in contact yesterday, and the further back one looks the less of the sky was ever connected.

Run that to recombination and the number is stark. The comoving Hubble radius at $z = 1090$ is 209 megaparsecs; the comoving distance to the surface of last scattering is 13,866. The ratio is 66, so the sky is divided into some thousands of patches that had no common past. There is no process operating within any of them that could have set them to a common temperature, because a process that acts at all acts inside the horizon.

It is worth being clear about which of the temperature’s properties is the problem, because the two halves of what the microwave background shows sit on opposite sides of it. That the spectrum is thermal is not mysterious: each patch was individually opaque and individually in equilibrium, and a thermal spectrum is what an opaque region produces. What is mysterious is that every patch arrived at the same temperature. Equilibrium within a patch is cheap; equality between patches that never met is not.

The flatness problem, which is about instability #

The Friedmann equation can be rearranged as

$$|\Omega(a) - 1| = \frac{|\Omega_k|}{a^2 H^2 / H_0^2},$$

so the departure from flatness grows in proportion to the square of the comoving Hubble radius. In a decelerating universe that radius grows, so flatness is repelled rather than attracted. Any small departure at early times is amplified.

The amplification is enormous. To have $|\Omega - 1| < 0.002$ today requires $|\Omega - 1| < 10^{-16}$ at nucleosynthesis and $|\Omega - 1| < 10^{-60}$ at the Planck time. That is not a coincidence of the kind that can be shrugged off; it is a demand that a dimensionless number be tuned to sixty decimal places by initial conditions.

One mechanism, because both problems are the same problem #

Both statements are about the comoving Hubble radius growing. The horizon problem says it was small in the past, so nothing large was ever connected. The flatness problem says it was small in the past, so any curvature must have been smaller still.

Make it shrink for a while, and both go away at once. A period during which $c/aH$ falls means that regions now separated were once inside a single Hubble radius and could have equilibrated; and it means $|\Omega - 1|$ was driven towards zero rather than away from it, by however many decades the radius fell.

The condition for the comoving Hubble radius to shrink is exactly $\ddot a > 0$: accelerated expansion. That is the whole definition of inflation, and everything else — scalar fields, potentials, slow roll — is machinery for producing it.

How much is needed is set by the horizon problem. The comoving Hubble radius has grown by about $e^{60}$ since the end of inflation, so it must have fallen by at least that much during it, which is 60 e-folds of expansion. Sixty e-folds is $10^{26}$: a patch smaller than a proton becomes larger than the observable universe.

There is a third fine-tuning that the same mechanism disposes of, and historically it came first. Grand unified theories generically produce magnetic monopoles at the symmetry-breaking transition, at roughly one per horizon volume, and monopoles are massive and dilute only as $a^{-3}$ — so a universe that made them would be dominated by them long before now, by many orders of magnitude. Inflation removes the problem in the crudest possible way: sixty e-folds of expansion dilutes any pre-existing relic by $e^{180}$, leaving less than one monopole in the observable universe. That was Alan Guth’s original motivation in 1980, and the horizon and flatness problems were noticed to fall out of the same mechanism afterwards. The monopole argument is the weakest of the three today, because it depends on a grand unified theory nobody has confirmed, and it is worth keeping distinct from the other two for that reason.

The prediction that makes it a theory #

An explanation that only accounts for what it was built to account for is not worth much, and if inflation did nothing but fix two fine-tunings it would be an assumption with extra steps. What makes it a scientific claim is that the same mechanism produces the fluctuations.

Quantum fields have zero-point fluctuations at every scale. During inflation, a mode’s wavelength is stretched past the Hubble radius, and once outside it there is nothing that can act across it — the fluctuation stops evolving and is frozen in as a classical density perturbation. Modes leave in order of size, and because $H$ is nearly but not exactly constant, they leave under slightly different conditions.

That produces a spectrum of primordial amplitudes that is nearly but not exactly scale-invariant, and the deviation is predicted to be small and negative. The mechanism is the same one that freezes a fluctuation outside the Hubble radius and returns it later, and the tilt records how much $H$ had fallen between one mode leaving and the next. The measured value is

$$n_{\rm s} = 0.9649 \pm 0.0042,$$

which is 3.5 per cent below 1 and excludes exact scale invariance at more than eight standard deviations. That number is the strongest quantitative support inflation has, and it was a genuine prediction — the tilt was expected before it was measurable, and its sign follows from $H$ decreasing as the field rolls.

What is measured, and what is assumed #

The honest accounting is short. Inflation as a mechanism has not been observed. What has been observed is a set of properties that it predicts and that are otherwise unexplained: spatial flatness, a nearly scale-invariant spectrum with a measured tilt, adiabatic and Gaussian fluctuations, and phase coherence on super-horizon scales at recombination.

What has not been observed is the one prediction that would identify the energy scale. Inflation also generates tensor fluctuations — gravitational waves — with an amplitude set by the expansion rate during inflation, and those would imprint a curl-like pattern in the polarisation of the microwave background. The current bound is $r < 0.032$, which excludes the simplest models and detects nothing.

The history of that search is a cautionary one worth recording. In 2014 the BICEP2 experiment announced a detection at $r = 0.20$, which would have fixed the energy scale of inflation at $2\times10^{16}$ GeV and been among the most important measurements ever made. It was polarised emission from aligned dust grains in the Galactic foreground, and the reason it was mistaken for a signal is that the collaboration had only one observing frequency and therefore no way to separate a foreground with a different spectrum from a cosmological one. The result was retracted within a year, jointly with the Planck team whose multi-frequency dust maps settled it. The foreground is not a nuisance around the signal here; it is larger than the signal at every frequency, and every subsequent experiment has been designed around that fact rather than around sensitivity. The comoving picture drawn over fourteen billion years shows the past. Drawn over sixty it shows what the same cosmology does next, and the two problems look different from there.

That is the same accelerated expansion the flatness problem is about, running forward rather than backward, and it makes the parallel with inflation explicit. A de Sitter phase drives the comoving Hubble radius down, which is what inflation does in the first fraction of a second and what dark energy is doing now. The difference is entirely one of scale: sixty e-folds then, a handful over the remaining life of the universe, and the same equation.

The comoving event horizon converging on a fixed distance also settles a question the earlier figures leave open. If the universe is spatially infinite, the observable part is a finite and permanently bounded sample of it, and no observation made at any future time will enlarge that sample by more than a few per cent. Cosmology therefore has a hard limit on how much of its subject it can ever see, and the limit is not technological. That is an unusual position for a physical science and it is worth stating plainly rather than leaving as a footnote to a diagram.

The third problem, which came first #

The two coincidences in the title are the ones the mechanism is usually introduced by, and neither was what prompted it.

Grand unified theories of the early 1980s predicted that as the universe cooled through the unification temperature, the symmetry would break in different directions in regions that were not in causal contact — and wherever the mismatched regions met, a topological defect would be left behind. The simplest such defect is a magnetic monopole, and the number produced follows from the horizon size at that epoch.

The number is catastrophic. Monopoles of the predicted mass, at the predicted density, would dominate the energy density of the universe by an enormous factor — they would have closed it long ago and there would be no universe of the kind observed to argue about.

Inflation disposes of them by dilution. Any relic produced before the expansion is diluted by the same factor everything else is, which for the standard parameters is more than $10^{78}$ — so a density that would have overclosed the universe becomes a density of less than one monopole in the entire observable volume.

That was the original argument, and the horizon and flatness problems were noticed as further consequences of the same dilution. It is a good illustration of how the mechanism works: it does not explain why the monopoles are absent, it makes the region they were produced in vastly larger than what is now observable, so there is no reason to expect one here.

Ending it #

A mechanism that expands the universe by sixty e-folds has to stop, and stopping it is the part that is least constrained and hardest to check.

The expansion continues while the field driving it is far from the minimum of its potential and rolling slowly. As it approaches the minimum the potential steepens, the slow-roll conditions fail, and the field begins oscillating about the minimum instead of drifting towards it.

Those oscillations are what ends the inflationary phase, and their energy has to be converted into ordinary particles — reheating — because the universe at the end of inflation is empty, cold and containing nothing but the field. The conversion happens through the field’s couplings to everything else, and the result is a hot plasma with a temperature set by how strong those couplings are.

Almost nothing about that step is determined by observation. The reheating temperature could be anywhere from the unification scale down to a few MeV — the lowest value consistent with nucleosynthesis happening afterwards — which is more than fifteen orders of magnitude of freedom.

What the observations constrain is the era before the end and the era after it, and not the transition between them. The primordial fluctuations were laid down during the slow roll and are measured precisely; the abundances were laid down after reheating and are measured precisely; and the process linking the two is a gap in the account that every model fills differently.

That gap is also where most of the model-building freedom lives, which is why the predictions that are quoted as tests of inflation are all statements about the fluctuations rather than about the transition.

It is also why a measurement of the primordial gravitational waves would be worth so much more than another measurement of the density fluctuations: the amplitude of the waves is set by the energy scale at which the expansion happened, which is the one number that would pin the mechanism to a particular era of physics rather than to a shape of potential.

Where the objections are #

Inflation’s difficulties are not observational and it is worth being clear about that, because the observations have gone its way.

It is a framework, not a model. There are hundreds of specific potentials, they predict a range of $n_{\rm s}$ and $r$ wide enough to accommodate most plausible measurements, and the data have excluded a great many of them without selecting one.

The initial conditions for inflation are themselves a question. Inflation requires a patch of the right size dominated by the right kind of energy, and whether that is easier to arrange than the flatness it explains is genuinely argued about.

And generically it does not stop. In most models the field continues inflating somewhere for ever, producing an unbounded number of causally disconnected regions with different properties. That is eternal inflation, and it undermines the predictive claim: if everything happens somewhere, a measurement here constrains nothing without a measure over the possibilities, and no accepted measure exists.

None of that makes the two problems in the first half of this essay go away, which is the argument for taking it seriously in spite of the objections: it is the only proposal that addresses both with one mechanism and makes a numerical prediction that has been confirmed.

What the pictures cannot show #

The left half of the hero figure is a schematic and the caption says so. The height of the inflationary plateau is the energy scale of inflation, which is unmeasured, and the number of e-folds drawn is chosen to be the minimum needed rather than derived. Only the slope — $-1$ in the log — is a consequence of accelerated expansion.

No figure here shows a fluctuation being generated, because the generation is quantum and the drawn quantities are classical. The transition between the two is the least well understood step in the whole account, and it is glossed in every presentation including this one.

And the diagram of two disconnected patches proves less than it appears to. It shows that no causal process connected them under the post-inflationary expansion history. It cannot show that no process connected them at all, because the pre-inflationary history is unknown, which is precisely the freedom inflation exploits.

The generalisation #

The structure of the argument is worth extracting because it is a common and often misused move: two unexplained coincidences that share a mechanism count for far more than twice one coincidence.

The reason is that a mechanism invented to fix one problem has free parameters, which can usually be chosen to fix it; a mechanism that fixes two problems of different kinds with the same parameter choice has spent its freedom. Here the parameter is the number of e-folds, the horizon problem sets a lower bound on it, and the flatness problem is then fixed for free by the same number.

This collection has met the pattern before. The rotation curves and the cluster masses are two discrepancies of quite different kinds, at scales differing by a hundred, and one missing component resolves both. A resonant chain of planets explains both the period ratios and the fact that the planets are where they cannot have formed. In each case the strength of the inference is in the joint fit, not in either half.

The corresponding warning is that the move only works if the two problems are genuinely independent. Here they are not fully independent — both are statements about the comoving Hubble radius growing — which is why the honest version of the argument says “one mechanism removes both” rather than “two independent confirmations”. The tilt of the spectrum is the part that is independent, and it is the part that does the real work.

It is worth seeing what an insufficient amount of inflation looks like, because the number of e-folds is the one quantity the mechanism does not predict.

Sixty e-folds is therefore a requirement rather than a prediction, and it is a requirement with no upper bound: inflation that ran for six hundred e-folds would satisfy every observation equally well and would put the boundary of the smooth region unimaginably far outside the horizon. So the theory is constrained from below by the observations and not at all from above, which is one of the more common objections to it and one of the harder ones to answer.

What the mechanism does predict, and what the e-folds do not enter, is the shape of the perturbation spectrum — nearly scale-invariant, slightly tilted, adiabatic, Gaussian. That is why the tilt is the measurement that matters and the e-fold count is not, and it is why a theory with a free parameter in its duration is nonetheless testable. The tilt is measured to be 0.965 rather than 1, a four-per-cent departure from exact scale invariance that is now established at high significance, and it is the closest thing the subject has to a direct measurement of the inflaton potential’s slope. Everything else about inflation is a consistency condition; that one number is data.

Where the ladder goes next #

The fluctuations inflation lays down become galaxies, and the pattern they make has a scale above which it stops being lumpy. The next essay measures where.

Later rungs on this anchor: slow roll as a condition on a potential, and the two parameters that summarise it; the tensor-to-scalar ratio and what a detection would fix; non-Gaussianity as a discriminant between models; the monopole problem, which is the third fine-tuning inflation was originally invented for and which depends on a grand unified theory nobody has confirmed; and eternal inflation and the measure problem.