A null that removes a model and not its family

Inflation is tested by two numbers measured from the microwave sky. The spectral tilt, 0.965, says how the amplitude of the primordial fluctuations changes with scale, and it knows the slope of the inflaton’s potential and not its height. The tensor-to-scalar ratio says how strong the primordial gravitational waves were relative to the density fluctuations, and it is an energy and a distance: the height of the potential, and how far the field rolled. The ratio has not been measured. It is bounded, below 0.036, and that bound has removed the simple power-law potentials that were once the textbook examples.

What survives are the flat potentials, the plateaus — and in particular the model Alexei Starobinsky proposed in 1980, before inflation had its name, from a correction to Einstein’s equations quadratic in the curvature. It predicts a tilt of about 0.965 and a ratio of three to four thousandths. The essay on the ratio ended on the question that prediction raises: whether the plateau is a prediction the data confirm or merely the region left over after everything else was excluded. The next generation of polarisation experiments is designed to reach a ratio of about one thousandth, which would detect Starobinsky’s gravitational waves or exclude them.

This essay is about what an exclusion would mean. It would remove the model. It would not remove the idea, because the model is one member of a family whose ratio can be anything below the present bound while its tilt stays exactly where the measurement put it.

Why the survivors are plateaus #

Inflation was proposed around 1980 to explain two coincidences with one mechanism: the uniformity of regions of the sky that should never have been in contact, and the flatness of a universe whose curvature should have grown. A brief epoch of accelerated expansion, driven by the potential energy of a scalar field, solves both, and it predicts, as a by-product, a nearly scale-invariant spectrum of density fluctuations and a background of gravitational waves.

The first popular versions used the simplest potentials — a mass term, $m^2\phi^2$, or a quartic, $\lambda\phi^4$ — because they need no special structure: put a field high up a smooth hill and let it roll. Those potentials predict ratios of a tenth or more, and they were the models the polarisation experiments first set out to test. They are now excluded. The quartic went first, on its tilt as much as its ratio; the quadratic fell when the ratio bound reached about 0.07; and the gentler power laws are at or beyond the edge of the present bound.

What is left is the class of potentials that become flat at large field values, and the reading of that fact is what the preceding essays left open. The α-attractors make the reading precise. If the plateau is a coincidence — merely what remains — then nothing in particular should be expected about how flat it is. If it is the signature of the geometry that defines the attractors, then its flatness is parameterised by one number, and that number has a natural range.

One shape, stretched #

The family is the α-attractors, introduced in 2013. Their potentials have a plateau that flattens exponentially towards large field values and a minimum where inflation ends, and the simplest of them is

$$V(\phi) \propto \tanh^2\!\left(\frac{\phi}{\sqrt{6\alpha}}\right),$$

in units of the reduced Planck mass. The single parameter $\alpha$ sets how wide the edge of the plateau is: the potential rises from its minimum to its plateau over a distance of about $\sqrt{6\alpha}$ Planck masses.

The model is solvable exactly in the slow-roll approximation, which is part of its appeal. With $s = \sqrt{6\alpha}$ and $x = \phi/s$, the number of e-folds remaining at a field value is $N = (s^2/8)(\cosh 2x - \cosh 2x_{\rm end})$, inflation ends where the first slow-roll parameter reaches one, and the observables follow in closed form. For small $\alpha$ they reduce to

$$n_s \approx 1 - \frac{2}{N}, \qquad r \approx \frac{12\alpha}{N^2}.$$

The tilt does not contain $\alpha$. The ratio is proportional to it.

A family that runs the full height of the plane #

Drawn on the plane the measurements are plotted in, the family is a near-vertical line.

That picture is the answer to the essay’s question. A measurement of the ratio at the level of a thousandth will either detect gravitational waves — in which case it measures $\alpha$, through $r = 12\alpha/N^2$ — or it will not, in which case it bounds $\alpha$ below about a quarter. Neither outcome excludes the family, because the family has a member at every ratio below the present bound, all at the same tilt.

This is not a fine-tuning invented to escape a test. The α-attractors were introduced because the same mathematical structure appears in supergravity theories, where $\alpha$ is related to the curvature of the space of fields, and they were noticed precisely because their predictions are insensitive to the details of the potential: many different functions, stretched by the same geometry, land at the same point in the plane. The name means what it says. Predictions converge on $n_s = 1 - 2/N$ as on an attractor.

The model that came first #

Starobinsky’s model was not built to be a plateau. It came from asking what happens if Einstein’s action for gravity, which is linear in the spacetime curvature, acquires a small term quadratic in it — the kind of correction quantum effects of matter fields are expected to generate. The extra term behaves, after a change of variables, like a new scalar field, the scalaron, with a potential that is flat at large values and has a minimum at zero. A universe that starts with the scalaron high on its plateau inflates, solving the horizon problem the way every inflationary model does, and the predictions — tilt about 0.965, ratio about 0.004 — were made decades before they could be checked.

That history is part of why the model matters. It is not a fit; its single free parameter is fixed by the amplitude of the fluctuations, and the rest is predicted. When the Planck satellite measured the tilt, the measurement landed on the prediction. Among the attractor family it is the member at α = 1, and the family’s other members share its tilt because they share its asymptotic shape, but only this one has a reason, independent of the data, to sit at that particular α.

So a null result at a thousandth would have a specific casualty: the one plateau model that was a prediction rather than an accommodation. What would survive would be members of the family whose α was chosen, in the end, by the data — which is a weaker kind of success. It would be a real constraint on the field-space geometry, and an honest account would say that the family had become harder to test.

The ratio as a measurement of one number #

If the family is right, then the ratio is a measurement of $\alpha$, and nothing else.

The discrete values are the interesting part of the target. In some constructions the field space of the inflaton is built from up to seven identical pieces, and each contributes to $\alpha$ in a fixed unit, so that $3\alpha$ is an integer from one to seven. If that geometry is right, the possible ratios form a list, the smallest about 0.0013 — just above the next generation’s design sensitivity. A detection at one of those values would be a measurement of a discrete number in a theory of quantum gravity, made with a microwave telescope. A null result below 0.001 would exclude the whole list.

So the family as a whole cannot be excluded by the ratio, and particular, well-motivated members of it can. That is a different situation from the one the textbook simple potentials were in, where one measurement excluded a whole idea; it is closer to the situation of a theory with a free parameter that the data are beginning to bound.

How far the field travelled #

A small ratio has a second consequence, discussed in the essay on the ratio as a distance: it limits how far the inflaton moved during the last e-folds. Large field excursions — more than a Planck mass — are uncomfortable for theory, because an effective description of the inflaton has to be trusted over distances larger than the scale where unknown physics is expected to change it.

The plot shows that the α-attractors are large-field models over most of their range, even at ratios far below the next generation’s reach. Their small ratios do not come from a short excursion but from a flat potential: the field travels several Planck masses across a plateau so flat that its motion barely disturbs spacetime. Whether that is acceptable depends on whether the flatness is protected by a symmetry — which in the supergravity constructions it is, by the geometry that defines $\alpha$ — or is an accident that quantum corrections would spoil.

The tilt counts e-folds #

The other number tells a different story, because the family’s tilt depends only on how many e-folds before the end the observed scales left the horizon.

The number of e-folds is not a free parameter of the potential; it depends on what happened after inflation ended — how long the inflaton took to decay into radiation, and what the universe was made of meanwhile. Standard reheating histories put it between about fifty and sixty. The measured tilt from the Planck satellite, 0.965, sits comfortably in that range. A tilt of 0.974 does not: seventy-seven e-folds requires a post-inflationary epoch in which the universe expanded more slowly than if it were radiation, an unusual history.

That is why the tilt may test the plateau before the ratio does. The ratio can always be lowered by lowering α, and the family survives. The tilt cannot be moved by α at all; it is pinned to $1 - 2/N$, and if the measured value moves away from what plausible reheating histories allow, every member of the family is under pressure at once. Analyses in 2025 that combined the Atacama Cosmology Telescope’s data with the Planck satellite’s and with large-scale structure reported a tilt near 0.974, and the question of whether that strains Starobinsky-like models — and the whole attractor family — or reflects a tension among the data sets themselves, is open.

A higher tilt #

The figure above uses the tilt measured by the Planck satellite. A tilt nearer 0.974 changes the picture without changing the family.

That is the sense in which the attractors’ prediction is sharp. They have one free parameter that moves the ratio and none that moves the tilt, so the tilt is a direct test of the reheating history they need. A family that can accommodate any ratio cannot accommodate any tilt.

The other number that does not move #

There is a third observable the attractors predict and cannot adjust: the non-Gaussianity of the primordial fluctuations. Any model with a single field that sets both the expansion and the fluctuations predicts a skewness tied to the tilt, of order a hundredth, below anything the sky has enough independent modes to measure. That test can only fail one way: a detected non-Gaussianity of order one would exclude every single-field model at once, attractors included, and a null result confirms nothing. The attractors are single-field models and live or die with the rest of the class on that test.

What the B-mode experiments will actually measure #

The ratio is measured from the polarisation of the microwave background. Thomson scattering at recombination polarises the light only where the glow was lopsided, and density fluctuations produce a pattern without curl. Primordial gravitational waves twist the polarisation pattern into a curl-like component, the B-modes, that density fluctuations at linear order cannot produce. The signal at a ratio of a thousandth is a few nanokelvin, at angular scales of a few degrees.

Two foregrounds dominate it. Polarised emission from dust grains in the Milky Way, aligned by the Galaxy’s magnetic field, is larger than the signal at every frequency where the background is bright, and has to be removed by measuring at several frequencies and using the different spectra; the 2014 claim of a detection at a ratio of 0.2 was dust. And gravitational lensing by the large-scale structure between the observer and the last-scattering surface twists some of the ordinary polarisation into B-modes, a signal that has to be modelled from the lensing itself and subtracted, a process called delensing.

A null result at a thousandth therefore means a measurement whose uncertainty, after dust and lensing have been removed, is below a few ten-thousandths. The experiments planned to reach it — a Japanese-led satellite mission scheduled for the 2030s, and ground-based telescopes in Chile and at the South Pole — are designed around those two foregrounds as much as around the signal.

What a null result would still say #

Suppose the ratio is shown to be below a thousandth. Starobinsky’s model and the discrete values $3\alpha = 1$ to $7$ are excluded. What is left?

The family below $\alpha \approx 0.25$, which predicts gravitational waves too weak for any planned experiment. Other plateau models with small ratios, which the attractor’s success has made more attractive. Models in which the tensor spectrum is suppressed by some other mechanism. And the general statement that inflation, if it happened, happened at an energy below about $6 \times 10^{15}$ GeV — about two-fifths of the energy at the present bound.

That last statement is the most robust. It does not depend on the family at all; it follows from the ratio as an energy. A null result moves the upper bound on the energy of inflation down by a factor of about 1.8 for each factor of ten in the ratio — the energy goes as the fourth root of the ratio — and the bound becomes progressively less useful as a test of theories whose natural energy scale is the grand unification scale, near $10^{16}$ GeV.

Where the model stops #

One member of a class. The figures use the T-model, one of several α-attractor potentials; the E-model, which includes Starobinsky’s exactly at α = 1, gives slightly different values at the same α — a ratio a little higher, a tilt nearly identical — and the family conclusions do not change.

Slow roll to the end. The closed-form solution uses the slow-roll approximation throughout, including near the end of inflation where it is least accurate. The number of e-folds at which the observed scales left the horizon depends on that end point at the level of an e-fold, which moves the predictions by less than the measurement’s uncertainty.

N as a parameter. The figures draw fifty, fifty-five and sixty e-folds as stand-ins for a reheating history that is unknown. A full prediction ties N to the reheating temperature and the equation of state after inflation, as the essay on that epoch does; the family’s tilt inherits that dependence exactly.

The discrete values as reported. The seven values of 3α are a prediction of specific constructions, not of supergravity in general; they are drawn as a target list, not as an expectation.

The surprising connection #

The α-attractors’ insensitivity to the shape of the potential is the same mathematical fact that makes a hyperbolic plane look flat to anyone standing in it: the inflaton’s field space has a constant negative curvature set by α, and in a negatively curved space, distances near the boundary are exponentially stretched. A potential of any shape, mapped onto that space, acquires a plateau stretching towards the boundary, and the plateau’s flatness is a property of the geometry, not of the function. M. C. Escher’s woodcuts of a disc filled with ever-smaller fish, crowding towards an edge that is infinitely far away, are a picture of the field space of an α-attractor. The tilt the microwave background measures is, in that picture, a measurement of how many fish the inflaton swam past before it reached the centre.

Still open: whether the tilt is moving #

The ratio will be measured to a thousandth within about a decade, and whatever it shows will leave the α-attractor family standing. The tilt is the sharper test of the family, because the family cannot adjust it, and its measured value is no longer settled: 0.965 from the Planck satellite alone, higher in combinations that add newer ground-based polarisation data and measurements of the baryon acoustic scale. If the higher value holds, the plateau models need more e-folds than ordinary reheating supplies, and either the post-inflationary universe was stranger than assumed or the plateau is not the right shape. Whether the shift is real, or a tension between data sets that will resolve as each improves, is a question for the next round of microwave and large-scale-structure measurements, and it is one the α-attractors answer only through N.