Astronomy & Celestial Mechanics
the sky, computed
Concept

E-folds — where it appears

A unit of expansion: one e-fold is growth of the scale factor by a factor of e, about 2.72. Inflation is counted in them because its growth was exponential, and the observed scales left the Hubble radius some fifty to sixty e-folds before it ended, a count that depends on how the universe reheated afterwards.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as slow-roll, tensor to scalar ratio — the same set of essays touches all of them, so they are one junction rather than several.

One length, leaving and returning. The comoving Hubble radius c/aH against the scale factor, both logarithmic. Everything to the right of the kink is exact and is the same cosmology as every other figure here: after inflation the comoving Hubble radius grows, as a in the radiation era and as √a in the matter era, then turns over once Λ takes hold. Everything to the left is a schematic — the energy scale of inflation is unmeasured, so neither the height of the plateau nor the 62 e-folds drawn is a number anybody has — but the shape is not negotiable: in any accelerated expansion H is nearly constant, so c/aH falls as 1/a. The consequence is the mechanism. The solid horizontal line is a fixed comoving length of 209 Mpc: it starts inside the Hubble radius, is carried outside during inflation, sits frozen while nothing can act across it, and re-enters at z = 1090, which is last scattering — it is the scale the first acoustic peak is made of. The dashed line above it is the comoving size of the whole observable universe, 14148 Mpc, and the figure's quiet second finding is that it is still outside the Hubble radius: the very largest angular scales in the microwave background have never re-entered, which is why they show the primordial spectrum with no acoustic processing on it at all. One crossing does two jobs: it makes the sky uniform, because everything now visible was once in causal contact, and it makes it not quite uniform, because a quantum fluctuation stretched beyond the horizon has nothing left that can smooth it out.

Two coincidences with one mechanism

Opposite sides of the microwave sky were never in causal contact and have the same temperature to one part in a hundred thousand. The total density sits on the knife edge of flatness, which is an unstable equilibrium. Two fine-tunings of quite different kinds, and one epoch of accelerated expansion removes both.

cosmology · Inflation m²φ²: where the observed scales left, and where inflation ends. The m²φ² potential, drawn as a shape with its height divided out, against the field in reduced Planck masses. The field rolls downhill towards zero and inflation ends where the slow-roll parameter ε reaches one, at φ = 1.41. The shaded band is the stretch of field the scales now seen on the sky left the Hubble radius from: 60 e-folds before the end at φ = 15.56 and 50 before it at φ = 14.21. Nothing about the sky depends on the rest of the curve. Across that band the two numbers the tilt is made of are ε = 9.01e-3 and η = 0.0090, which give a spectral index of 0.9640 and a tensor-to-scalar ratio of 0.1441 at 55 e-folds. The height is not in either: it is fixed separately by the amplitude of the fluctuations, which puts the potential at (2.0 × 10¹⁶ GeV)⁴ there — and multiplying the whole curve by any constant leaves the band, the tilt and the ratio exactly where they are, because every slow-roll quantity is a ratio of the potential to its own derivatives. The field travels 13.49 Planck masses from the middle of the band to the end.

The tilt knows the slope and not the height

The measured spectral index, 0.965, is quoted as the strongest evidence for inflation, and it is a statement about two dimensionless numbers — how steeply the potential fell and how sharply that slope was changing, over the few e-folds the sky can see. The height of the potential is not in it at all, which is why potentials that look nothing alike reproduce it.

cosmology · Inflation 6 potentials against the tilt and the tensor bound. Predictions in the plane of spectral index and tensor-to-scalar ratio, each drawn as a short track from 50 e-folds to 60, the range usually allowed for the pivot scale to have left before the end of inflation. The vertical band is the measured index, 0.9649 ± 0.0042 with its two-sigma extent, and the shaded region above 0.036 is excluded at 95 per cent by the B-mode polarisation limit — drawn as two independent limits, which the published constraint is not quite: the real likelihood is a correlated contour, and it is somewhat tighter than this box in the corner where the tilt is high. λφ⁴: nₛ 0.9412, r 0.3137 at 50 e-folds; 0.9508, 0.2623 at 60 — outside; m²φ²: nₛ 0.9604, r 0.1584 at 50 e-folds; 0.9669, 0.1322 at 60 — outside; linear φ: nₛ 0.9701, r 0.0796 at 50 e-folds; 0.9751, 0.0664 at 60 — outside; φ^⅔: nₛ 0.9734, r 0.0532 at 50 e-folds; 0.9778, 0.0443 at 60 — outside; natural, f = 7: nₛ 0.9569, r 0.0906 at 50 e-folds; 0.9628, 0.0670 at 60 — outside; Starobinsky: nₛ 0.9616, r 0.0042 at 50 e-folds; 0.9678, 0.0030 at 60 — inside. The simplest potential of all, a mass term, is excluded over its whole range of e-folds, and not by the tilt, which it matches — by the tensors.

A ratio that is an energy and a distance

The tensor-to-scalar ratio is the one inflationary observable that measures the height of the potential rather than its shape, and the quantity it fixes is an energy — a ratio of 0.01 means inflation happened at 10¹⁶ GeV. It fixes a second thing as well, how far the field travelled, and near the present bound that distance is several Planck masses, which is where the theory stops being able to vouch for itself.

cosmology · Inflation Starobinsky: e-folds before the end, against how reheating went. N, the number of e-folds between the pivot scale leaving the Hubble radius and the end of inflation, for the Starobinsky potential, against the temperature at which reheating finished, for 3 equations of state during it. All the lines meet on the right at instant reheating, 2.6 × 10¹⁵ GeV, where N = 55.6. The left edge is 5 MeV, below which nucleosynthesis would not have happened. w = 0, oscillating field: 42.0 at 5 MeV; w = ⅓, like radiation: 55.6 at 5 MeV; w = 1, kination: 69.0 at 5 MeV. The reason is how far the universe stretches while the energy density falls: an oscillating field dilutes like matter, as a⁻³, so for the same fall in density it expands further than radiation would, more of the growth of today's scales happens after inflation, and fewer e-folds of inflation are needed to put them where they are. A stiff epoch with w = 1 dilutes as a⁻⁶, stretches less, and needs more. A radiation-like epoch changes nothing. None of this epoch has been observed; the lines are the arithmetic of energy and entropy, and the spread between them is how much an unobserved history moves a quantity the spectral index depends on.

The epoch nobody saw moves the tilt

Between the end of inflation and the hot universe that made the light elements lies an interval nothing has observed, in which the energy of the inflaton became radiation. How long that took changes how many e-folds before the end the observed scales left — by as many as fourteen — and that moves every model's predicted spectral index by more than the measurement's uncertainty. A potential is never tested by the tilt alone; a potential and a reheating history are tested together.

cosmology · Inflation

Named alongside it

The objects these essays reach for when they reach for this one.

Scalar spectral indexSlow-rollTensor to scalar ratioInflatonReheatingEquation of stateInflationPrimordial fluctuationsB modesBig bang nucleosynthesisComoving hubble radiusEnergy scale of inflation

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