The epoch nobody saw moves the tilt

Every prediction of inflation that has been compared with the sky is evaluated at one moment: when the pivot scale, a comoving wavelength of about 125 megaparsecs, was stretched past the Hubble radius. How far before the end of inflation that moment came is a number, N*, and the tilt reads as a count of it once a potential is granted — 56.5 for a mass term, 54.9 for the Starobinsky plateau.

The count is not a property of the potential. It depends on how much the universe expanded after inflation ended, and part of that expansion happened during an interval nobody has seen: the time it took the energy stored in the inflaton to become the hot plasma of ordinary particles. The essay on the mechanism described that step, reheating, as the least constrained part of the whole account, with a final temperature free across more than fifteen orders of magnitude. That is true. What this essay is about is that the freedom is not invisible. It leaves a mark on the one number the sky measures best.

Why a count depends on what came after #

The logic is geometric and short. A comoving scale leaves the Hubble radius when its wavenumber equals $aH$. The pivot scale’s size today is known — it is 222 times smaller than the present Hubble radius — and so is the expansion rate during inflation, to within the factor the tensor ratio leaves open. What connects the two is the total factor by which the universe has expanded since that scale left, and that factor is a sum of pieces laid end to end: N* e-folds of inflation after the exit, whatever expansion happened during reheating, and the expansion of the radiation and matter eras since.

The last of those is not free. Once the plasma is in thermal equilibrium its entropy is conserved, so its temperature falls in inverse proportion to the scale factor, and the expansion from reheating to today is fixed by how hot reheating left the universe compared with the 2.7 kelvin of the microwave background that is its cooled remnant. The expansion during reheating is fixed by how the energy density fell while it lasted, which depends on the equation of state.

The numbers are worth having in hand. With instant reheating, the universe has expanded by about $10^{27.6}$ — 63.5 e-folds — since inflation ended: the ratio of the reheating temperature to today’s, corrected for the particle species that have since annihilated and warmed the photons. Add the 55.6 e-folds of inflation that followed the pivot scale’s exit and the scale has grown by about 119 e-folds, a factor of $10^{52}$, from a size far smaller than a proton to 125 megaparsecs. The split between the two halves of that total is what the rest of this essay is about, because the total is fixed and the halves are not.

Anything that makes the universe expand more after inflation leaves less for inflation to do, and N* falls.

An oscillating field dilutes like matter, its energy density falling as $a^{-3}$, where radiation falls as $a^{-4}$. For the same fall in density from the end of inflation to the reheating temperature, matter-like dilution requires a larger expansion than radiation would. A long, matter-like reheating therefore stretches the universe further than an instant one, today’s scales grew more after inflation, and fewer e-folds of inflation were needed to put them where they are. A stiff fluid, $w = 1$, dilutes as $a^{-6}$, stretches less, and needs more. A radiation-like epoch is indistinguishable from none, which is why its line in the opening figure is flat.

The ledger #

Written out, N* is a sum of five terms, and it is worth seeing their sizes, because the familiar “fifty to sixty” hides how it is made.

The largest term is the logarithm of a ratio of expansion rates: $H$ during inflation was about $10^{55}$ times its present value. The second largest, with the opposite sign, is the logarithm of a ratio of temperatures — from the $2.6\times10^{15}$ GeV at which instant reheating would leave the plasma to the $2.3\times10^{-13}$ GeV of the microwave background today. The number of e-folds is what remains when one is subtracted from the other. It is a small difference of two large logarithms, and that is why an unobserved term of only a few units in the ledger moves the answer by an amount that matters.

The ledger also shows which uncertainties do not matter. The expansion rate during inflation enters through a logarithm, so the whole range of energy scales left open by the tensor bound — a factor of ten in $r$, a factor of three in $H$ — shifts N* by little more than one. The disagreement over the present expansion rate, 67 against 73 kilometres a second per megaparsec, shifts it by a tenth. The species term assumes the particle content of the Standard Model when reheating ended; doubling the number of particle species, as a supersymmetric spectrum roughly would, shifts it by a quarter. The reheating term alone is unconstrained, and it can be tens.

The two bars that change in that figure change in opposite directions, and the net effect is the difference between how a matter-like fluid and radiation dilute. That is why the shift is so much smaller than either bar: fifteen of the twenty e-folds of extra expansion during reheating are simply expansion that radiation would have done anyway, over the same fall in temperature. Only the excess counts.

Which way the tilt moves #

For a given potential, fewer e-folds means the observed scales left closer to the end of inflation, where the slow-roll parameters are larger. Every potential considered in these essays has a spectral index that falls as the end approaches, so a smaller N* makes the prediction redder. For the plateau the index is close to $1 - 2/N$, so a fall of N* from 55.6 to 42 takes it from about 0.964 to about 0.952 — a shift of three times the present uncertainty, from a history that emitted nothing that survives.

That is the sense in which the tilt is a thermometer, and also the sense in which it is a poor one. The temperature enters through a logarithm, so the prediction changes by a thousandth for every factor of about thirty in temperature, and the present measurement says only that a matter-like reheating in this model probably finished above a few GeV — which is not a demanding requirement, since the light elements need it finished above a few MeV anyway. The forecast lines show why the argument is worth making now. At an uncertainty of 0.002, the same curve would cross the edge of the band many decades higher in temperature, and a statement about an epoch that emitted nothing that survives would become a limit with teeth.

A stiff reheating runs the other way, and it shows that the direction is not a convention of the calculation.

The minimum decides the direction #

Whether reheating was matter-like, radiation-like or stiff sounds like one more free parameter, and for most of its duration it is not free at all. It is fixed by the shape of the potential at its minimum, by an argument that is older than cosmology.

After inflation ends the field oscillates about the bottom of its potential, and it oscillates quickly: once the slow-roll conditions fail, the curvature of the minimum exceeds the expansion rate, so the field completes many swings in a single expansion time. The equation of state that governs the expansion is then the average over a swing, set by how the kinetic and potential energies share out over a cycle. For a potential that rises as a power of the field near its minimum, $V \propto \phi^n$, the virial theorem fixes that share: the time-averaged kinetic energy is $n/2$ times the time-averaged potential energy. The pressure is the kinetic energy minus the potential, the density is their sum, and so

$$\langle w \rangle = \frac{n-2}{n+2}.$$

A quadratic minimum, $n = 2$, gives $w = 0$: the oscillating field has no average pressure and dilutes like matter, exactly as a pendulum’s kinetic and potential energies average to equal shares. A quartic minimum, $n = 4$, gives $w = 1/3$, the equation of state of radiation, with no particles involved at all. A minimum rising as the sixth power gives $w = 1/2$, stiffer than radiation, and its delayed reheating would push the tilt blue. The whole family is one line of the same theorem that weighs a cluster of galaxies from the speeds of its members.

The Starobinsky potential, whatever it looks like on its plateau, is quadratic at its minimum. So once its inflation ends and before its energy has been transferred, it dilutes like matter, and the direction in which its prediction moves is not a choice: a delayed reheating can only redden it. A potential with a quartic minimum would behave like radiation from the start, and its prediction would hardly depend on how long reheating took. Dark energy’s equation of state is the same number measured for a field today; here it is fixed by the curvature of the bottom of a potential nobody has measured.

What the oscillation does not fix is when it stops. The field decays into particles at a rate set by its couplings to them, and the reheating temperature is set by that rate. A feeble coupling, of gravitational strength, can prolong the matter-like phase until the universe has cooled to a few GeV or less. A strong one can end it within a few swings, and not by ordinary decay. A field oscillating in time makes the effective mass of anything coupled to it oscillate too, and a mode whose mass is modulated at twice its own natural frequency grows exponentially, the way a swing is pumped by a child rising and crouching twice per cycle. That parametric resonance, worked out for inflation by Lev Kofman, Andrei Linde and Alexei Starobinsky in 1994, can transfer most of the field’s energy into particles long before any perturbative decay would have started — at which point the expansion stops being matter-like, but the resulting plasma is far from thermal and takes longer still to become the equilibrium the ledger assumes.

A potential and a history, tested together #

The consequence for comparing models with the sky is that a model is not a point, or even the short track in e-folds drawn in the plane of tilt and tensor ratio. It is a line parameterised by its reheating temperature.

Two things in that figure are not obvious from any single number. The first is that a long reheating raises the tensor ratio as well as reddening the tilt, because the scales left where the slope was steeper. A model that sits just below the tensor bound with instant reheating can cross it if its reheating was slow, and a model rescued in the tilt by a slow reheating pays for the rescue in tensors.

The second is that the tracks are long compared with the band. The plateau’s track spans almost three standard deviations of the tilt. Quoting a potential as “consistent with the data” without saying how it reheated is therefore a statement about a segment of that track, usually the instant end, and the consistency is a joint property of the potential and an assumption.

Natural inflation shows the ordinary case. With instant reheating its index, 0.9614, is already near the lower edge of the band, and a matter-like reheating carries it out of one sigma below about $7\times10^{13}$ GeV and out of two sigma below about $2\times10^{5}$ GeV. For that potential the tilt prefers a quick reheating, and its tensor ratio of 0.073 is above the bound whatever the reheating did, so the tilt’s preference is academic. The power law with an exponent of two-thirds shows the trade in its sharpest form.

In the tilt alone, that potential is compatible with the data only if the universe took almost the longest time it could to reheat — a reheating that ended a few MeV above the point at which the helium abundance and the count of light particle species would have gone wrong. At that end of its track, though, its tensor ratio is 0.062, well above the bound. The slow reheating that rescues its tilt condemns it in tensors, and the plane shows both at once. The model is excluded not by either number but by the fact that no single reheating history satisfies both.

What is actually measured #

Nothing about reheating itself has been observed, and nothing in this essay changes that. The inference runs entirely through arithmetic that is well tested at the other end: entropy conservation in an expanding plasma, which is the same physics that makes four light-element abundances fit one free parameter, and the relation between an equation of state and the rate at which density falls.

The two ends of the temperature axis are measured, and they are measured quite differently. The lower end, a few MeV, comes from the requirement that neutrinos had time to reach thermal equilibrium before nucleosynthesis; a later reheating would leave too few of them, which would change both the expansion rate during element formation and the damping of the microwave background’s smallest scales, and neither change is seen. The upper end is the energy density at the end of inflation, which for the plateau is fixed by the measured amplitude of the fluctuations once the potential is chosen.

Between those ends the assumption that matters most is that entropy was conserved after reheating. Any later injection of entropy — a heavy particle that decayed after the plasma formed, a phase transition that released latent heat — stretches the universe in the same way a slow reheating does and would be read, in this ledger, as one. The microwave background’s spectrum limits energy injected later than a few months after the beginning, because a blackbody that perfect could not survive it; it says nothing about injections in the first second.

What the ledger leaves out #

The equation of state is held constant. A real reheating passes through stages: coherent oscillation, a resonant burst of particle production, a period of turbulence, and thermalisation, and its average $w$ drifts from zero towards one third as it goes. The figures bracket the possibilities with constant values rather than following any one history.

Instant reheating is a limit, not a scenario. The highest temperature drawn assumes all of the energy at the end of inflation became radiation at once, which no known coupling achieves exactly; the real curve starts slightly to the left of the point where the lines meet.

The species count is the Standard Model’s. If there are particles beyond it, the entropy they carried into the plasma changes the cooling term by a third of the logarithm of the extra count — a fraction of an e-fold for any plausible spectrum.

And some theories cap the temperature from above. In supersymmetric theories a reheating temperature much above $10^9$ GeV overproduces gravitinos, whose late decays would disrupt the light elements. That bound depends on a theory with no experimental support, and it is not drawn.

Still open: what the inflaton decayed into #

The ledger needs only a temperature and an equation of state, and the tilt can in principle constrain both. What it cannot reach is the physics that set them: the couplings through which the inflaton gave its energy to the Standard Model, which decide whether the transfer was a resonant burst or a slow leak and whether it produced the particles that survive today. A second field that decayed later, like the curvaton that would leave the sky lopsided, would add its own chapter to the same ledger. How the potential energy of a field became every particle now in existence is still a question with a temperature attached and no mechanism.