Astronomy & Celestial Mechanics
the sky, computed
Concept

Inflation — where it appears

A conjectured epoch of accelerated expansion in the very early universe. It is invoked to explain why the observable universe is flat and homogeneous on scales that were never in causal contact. It also supplies a spectrum of primordial perturbations that the microwave background then measures.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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 Thirty e-foldings erase the memory of a seed. Field strength against time for three seed fields 8 orders of magnitude apart, amplified at one e-folding every 3·10⁸ years — a galactic dynamo's measured turnover rate — and stopped at the 3·10⁻⁶ gauss the disc actually has. In 10 billion years the budget is 33 e-foldings, which is a factor of 3·10¹⁴. That is the finding: the three tracks reach the same ceiling within 5.5 billion years of one another, so the field a galaxy has today carries essentially no information about the field it started with. Any seed above about 10⁻²⁰ gauss will do, and mechanisms that produce far less than that are the only ones ruled out. The measurement that does constrain a seed has to be made where no dynamo ever ran, which means the voids between clusters — and the limit there comes from gamma rays that never arrived.

The seed that cannot be remembered

A galactic dynamo affords something like thirty e-foldings over the age of a galaxy, which multiplies any seed field above a ten-thousandth of a billionth of a microgauss up to the microgauss actually observed. The field a galaxy has today therefore says nothing about the field it started with, and the only place a seed survives unamplified is the emptiness between clusters.

cosmology · Primordial fields The same history, on the clock that straightens light. Conformal time upward against comoving distance sideways, for the Planck 2018 cosmology. Conformal time is ∫dt/a, which is exactly the comoving distance light covers, so on these axes every photon moves at forty-five degrees — at every epoch, whatever the expansion is doing. That single property turns every curved thing in the ordinary space-time diagram into a straight one. The universe began at η = 0 and is now at η = 46.1 Gly of conformal time; it will ever accumulate only 62.8, because the integral ∫dt/a converges once Λ dominates, and that finite ceiling is the whole reason an event horizon exists. The particle horizon is the 45° line from the origin and the event horizon is the 45° line back from the ceiling, so the two horizons that were curves are now the two edges of one light cone drawn twice. The shaded wedges are the past light cones of two points on the last scattering surface, at conformal time 0.914 Gly and comoving distance 45.2 Gly from here. They do not overlap. Two points on that surface separated by more than 2η_rec were never in causal contact, which subtends 2.31° on the sky, and the microwave sky therefore contains about 9,805 patches that have no common past and the same temperature to one part in a hundred thousand. That is the horizon problem, and in these coordinates it is a statement about whether two triangles intersect.

The clock on which light travels in straight lines

Cosmic time makes light cones bulge and horizons curve. There is another time coordinate on which a photon's worldline is a forty-five degree line at every epoch, and on it the horizon problem stops being a piece of arithmetic and becomes a question about whether two triangles overlap.

cosmology · Horizons 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 A family that runs the full height of the plane. The spectral tilt and the tensor-to-scalar ratio (logarithmic) predicted by inflationary potentials, against the measured tilt (0.9649 ± 0.0042, shaded, two standard deviations) and the upper bound on the ratio (0.036, solid line). The two curves are the T-model α-attractor, V ∝ tanh²(φ/√6α), for α from ten thousandths to a thousand, at 50 and 60 e-folds before the end of inflation: at large α the model is a quadratic potential and sits above the bound; as α falls the ratio drops without limit while the tilt settles at 1 − 2/N. The dots on each curve are α = 1, close to Starobinsky's model, at a ratio of 0.0046 and 0.0032. The short segments are power-law potentials, all excluded or nearly so. The dashed line at 0.001 is a target of the next generation of polarisation experiments: a null result there would exclude α = 1 and leave every smaller α standing.

A null that removes a model and not its family

The plateau potentials that survive every measurement of inflation predict a tensor-to-scalar ratio a few thousandths high, and the next polarisation experiments are designed to reach it. A null result there would exclude Starobinsky's model. It would not exclude the family it belongs to: the α-attractors stretch one shape by one parameter, their tilt stays where the measurement put it, and their ratio slides down as α does, without a floor.

cosmology · Inflation A running the measurement cannot yet tell from zero. The running of the spectral tilt, dnₛ/d ln k, against the tilt itself. The ellipses are the microwave-background measurement, nₛ = 0.9649 ± 0.0042 and a running of −0.0045 ± 0.0067, at one and two standard deviations. The three curves are what slow-roll inflation predicts once the tilt is fixed: a running of −(1 − nₛ)²/β, with β = 2 for the plateaus and every α-attractor (and for φ²), 4/3 for the steepest power law still drawn and 4 for a cubic hilltop. At the measured tilt they give −0.0009 to −0.0003, all within a band thinner than a line at this scale, 11 times smaller than the measurement's error.

A drift in the tilt that is the tilt squared

The primordial spectrum's tilt has been measured to half a per cent. Whether the tilt itself changes with scale has not, and slow-roll inflation says why it will be hard: the running is fixed by the tilt, at about six ten-thousandths, ten times below the present error. The one measurement that reaches far enough along the spectrum to feel it — the μ-distortion of the microwave background's colour — would need a precision no planned instrument has.

cosmology · Inflation

Named alongside it

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

E-foldsScalar spectral indexSlow-rollReheatingTensor to scalar ratioHorizon problemMicrowave backgroundPrimordial fluctuationsB-modesBiermann batteryBlazarCausal contact

All concepts