A ratio that is an energy and a distance

The spectral index measures a potential’s shape: a combination of its slope and its curvature, with the height divided out. Two very different potentials — a parabola and a plateau — reproduce it to a thousandth, one by spending its tilt on slope and the other on curvature. Separating them needs a measurement that sees the slope on its own, and there is exactly one.

During inflation the quantum fluctuations were not confined to the inflaton. Space-time itself fluctuated, and the fluctuations that crossed out of the Hubble radius were frozen in as gravitational waves, just as the field’s were frozen in as curvature. The ratio of the two amplitudes is the tensor-to-scalar ratio, $r$. It is the most sought-after number in cosmology, it has not been detected, and the upper limit already set on it has removed the simplest potential anybody ever wrote down.

Gravitational waves with nothing to source them #

The tensor fluctuations are the part of the story that needs no inflaton at all. Any quantum field in an expanding space has zero-point fluctuations, and the metric of space-time is such a field. A mode whose wavelength is stretched past the Hubble radius stops oscillating and keeps its amplitude, and the amplitude it keeps is set by the only scale around at the moment of crossing, the expansion rate:

$$\mathcal{P}_t = \frac{2H^2}{\pi^2 M^2},$$

with $M$ the reduced Planck mass. Nothing about the potential’s shape enters, only how fast space was expanding. The curvature fluctuations, by contrast, carry the factor $1/\varepsilon$ that makes a flat potential produce large fluctuations from a low height. Dividing one by the other, and using the slow-roll relation between $H$, $V$ and ε,

$$r = \frac{\mathcal{P}_t}{\mathcal{P}_\zeta} = 16\,\varepsilon.$$

That is the slope parameter alone, which is what the tilt could not supply. It is also sixteen times a number that the plateau puts at two parts in ten thousand and the mass term at nine parts in a thousand, so the two potentials that agreed on the tilt disagree on $r$ by a factor of forty.

The waves themselves are the same kind of object as the chirp from a merging pair of black holes, stretched beyond recognition. The modes that matter for the microwave background have wavelengths comparable to the observable universe and frequencies of order $10^{-18}$ hertz. The same spectrum extends, nearly flat, to every frequency, and it passes through the nanohertz band where pulsar timing arrays have found a background — but at an energy density of order $10^{-17}$ of the critical density for $r = 0.01$, many orders of magnitude below that signal, whose most likely source is pairs of supermassive black holes. The primordial waves are invisible everywhere except in the one place they have had fourteen billion years to leave a mark: the polarisation of the microwave background.

What would be seen, if they were, is worth stating plainly. A primordial tensor mode is a quantum fluctuation of the gravitational field, amplified and frozen. No experiment has ever observed a fluctuation of that kind, and a detection of $r$ would be the first direct evidence that gravity is quantised at all — obtained not from a laboratory but from a pattern in the polarisation of light emitted when the universe was four hundred thousand years old.

A fourth root #

Because the tensor amplitude depends only on $H$, and $H^2 = V/3M^2$, the ratio is a measurement of the potential’s height. With the measured scalar amplitude, $2.1\times10^{-9}$,

$$V^{1/4} = \left(\frac{3\pi^2}{2}\,\mathcal{P}_\zeta\, r\right)^{1/4} M \approx 1.0\times10^{16}\ {\rm GeV}\,\left(\frac{r}{0.01}\right)^{1/4}.$$

The fourth root is what makes the line so flat, and it cuts both ways. A detection at any ratio the planned experiments can reach would place inflation within a factor of three of $10^{16}$ GeV, which is a trillion times the collision energy of the Large Hadron Collider. A non-detection moves the upper bound on the energy down only by the fourth root of the improvement: a hundredfold better limit on $r$ lowers the ceiling by a factor of three. The expansion rate at the bound is about $5\times10^{13}$ GeV, which as a time is a Hubble time of $10^{-38}$ seconds.

The coincidence with the scale at which the forces’ strengths converge is striking and should be held at arm’s length. It says that inflation, if the tensors are there to be found, happened at energies where a grand unified theory would be relevant; it does not say that one caused the other, and neither theory predicts the other’s number.

It does have an independent check, and the check has nothing to do with the sky. A grand unified theory lets quarks turn into leptons through particles with masses near the unification scale, so the proton decays, with a lifetime that rises as the fourth power of that mass. The largest underground water detector has watched some $10^{34}$ protons for two decades without seeing one decay into a positron and a pion, which puts the lifetime for that channel above $2\times10^{34}$ years and rules out the simplest unified theories outright. A tank of water under a mountain in Japan and a polarimeter at the South Pole are constraining physics near the same energy, one by waiting for a proton to disappear and the other by looking for a curl in the oldest light there is.

The plane that has been emptied #

The opening figure is the most compressed summary of what the data have done to inflationary models, and it repays being read slowly. Each potential is a short track rather than a point because the number of e-folds between the pivot scale’s exit and the end of inflation is uncertain by a few, and each potential’s prediction moves along its own track as that number changes.

The mass term is the case that settles an argument. Its tilt fits: 0.9604 at 50 e-folds, 0.9669 at 60, bracketing the measurement. Its tensor ratio does not: 0.158 at 50 e-folds and 0.132 at 60, four times the bound at the nearer end. There is no number of e-folds in the plausible range that brings it inside, so the simplest potential there is has been excluded, and not by the tilt it predicted correctly — by the gravitational waves it predicted and that are not there. Every power law with an exponent of one or more shares the fate, and so does natural inflation at the width drawn.

The exclusion took about a decade of tightening, and the sequence is worth recording because at each stage the mass term was the model the limit was compared with. The bound stood above 0.2 after WMAP’s five-year data, at 0.11 after Planck’s first release in 2013, at 0.07 in 2016 once the BICEP2 and Keck Array maps were analysed jointly with Planck’s, and at 0.036 in 2021. The mass term went from comfortable to marginal to excluded without its prediction changing at all.

On a logarithmic axis the region below the bound turns out to be large, and the models that live in it have something in common. They are flat where the observed scales left: either plateaus, whose slope falls exponentially with the field, or hilltops, whose slope vanishes at the summit. The Starobinsky potential, which arises from adding a term quadratic in curvature to the gravitational action, is the best known of the first kind and makes a prediction, $r \approx 12/N^2$, that is small but not hopelessly so: three to four thousandths, within reach of the experiments now being designed.

That small prediction is also unusually sensitive to the one thing the plateau leaves open. Because $r$ falls as the inverse square of the e-fold count, the plateau’s own track runs from 0.0042 at 50 e-folds to 0.0030 at 60 — a forty per cent spread from a range of e-folds that is itself a proxy for how the energy of inflation was turned into heat. A detection at that level would therefore measure two things at once: that the potential is flat, and roughly how long the universe took to reheat afterwards.

The tensor ratio’s relation to the slope has a direct reading on the chart from the essay on the tilt. The slope’s share of the tilt is $6\varepsilon$, and $r = 16\varepsilon$, so every slope bar in that chart is the tensor ratio multiplied by three eighths.

How a ratio becomes a distance #

The ratio fixes a speed as well as a height. In slow roll the field moves through $\Delta\phi$ per e-fold of expansion at

$$\frac{d\phi}{dN} = M\sqrt{2\varepsilon} = M\sqrt{r/8},$$

so a tensor ratio sets how many Planck masses the field crosses in each e-fold. Over the seven or eight e-folds during which the observed scales left, the field must have moved at least $\Delta N\sqrt{r/8}$ Planck masses, and if ε only grows towards the end — as it does in every potential drawn here — the whole remaining roll to the end covers at least $N\sqrt{r/8}$. David Lyth pointed this out in 1997, and the inequality carries his name.

The arithmetic is short. At the present bound, $r = 0.036$, the observed window alone requires $8\sqrt{0.0045} = 0.54$ Planck masses; the roll to the end, at the same minimum speed, requires 3.7. At $r = 0.01$ the window requires 0.28 and the whole roll 1.9. Only below $r \approx 0.003$ does the whole roll fit inside a single Planck mass. Every potential that has ever been taken seriously as a source of detectable tensors travels further than that, and even the plateau, with a ratio ten times below the present bound, crosses four and a half Planck masses on its way to the end.

The bound runs one way only, and the figure shows it. A small tensor ratio does not imply a short journey: the hilltop has $r = 0.015$ and crosses 7.3 Planck masses, and the plateau crosses 4.4 at $r = 0.0035$, because both do most of their travelling in the last few e-folds, where the slope has steepened far beyond its value in the observed window. What the inequality forbids is the other combination — a large ratio from a field that stayed nearly still — and that is exactly the combination a detection near the present bound would demand of any potential.

The reason that matters is not the size of the number but what the number is compared with. A potential is an effective description, written as a polynomial in the field valid below some scale at which unknown physics — quantum gravity, at the latest — would add terms of its own. Each such term is suppressed by a power of $\phi/\Lambda$, with $\Lambda$ at or below the Planck mass, and over a field range larger than $\Lambda$ that suppression becomes an enhancement. A potential that stays smooth across ten Planck masses is a potential in which an infinite series of corrections has been assumed to be absent, and that assumption is exactly what the effective description cannot justify. Symmetries can protect such a potential, which is why the natural and monodromy models were built around approximate shift symmetries of axion-like fields; conjectures from string theory that forbid large excursions outright exist, are actively argued about, and are unproved.

So the tensor ratio is a statement about the ultraviolet completion of gravity as well as about inflation. A detection near the present bound would say the field travelled super-Planckian distances on a smooth potential, which some theory of quantum gravity must permit. A ratio below a thousandth would be comfortable for every theory and would leave the energy scale of inflation within a factor of a few of where it already is.

What is actually measured #

Nothing in the plane measures a gravitational wave directly. It measures a pattern in the polarisation of the microwave background, and the pattern is separable because of geometry.

Thomson scattering of radiation with a quadrupole anisotropy polarises it, and a polarisation field on the sky can be split, like any field of headless vectors, into a part with no handedness and a part with it — E-modes and B-modes. Density perturbations produce only E-modes at linear order, because a compression has no handedness. Gravitational waves stretch space in one direction and squeeze it in another, and they produce both. A primordial B-mode on degree scales is therefore a signature that no density perturbation can mimic, and that is why the whole effort is organised around it. The tensor signal peaks near multipole 80, on scales a little larger than the patches of the first acoustic peak, where the modes that were entering the Hubble radius at last scattering left their mark, with a second bump at the largest scales from the ionised gas at the end of reionisation.

The measurement is the one an optical polarimeter makes of starlight — recovering the direction a photon count throws away — carried out on a signal of a few tenths of a microkelvin at most, where the telescope’s own polarisation is far larger than the sky’s. An instrument more polarised than the sky is the whole engineering problem of a B-mode experiment, and the answer is the same as for a stellar polarimeter: modulate the sky’s signal and not the instrument’s, with a spinning half-wave plate in front of the detectors or with detector pairs rotated through a set of angles, so that only a component tied to the sky survives the demodulation.

Two things produce B-modes that are not primordial. Gravitational lensing by intervening structure shears the E-mode pattern — the same shear that a million galaxies are averaged to see — and converts part of it into B, and at a tensor ratio below about 0.01 the lensing signal is larger than the primordial one at every scale; removing it requires reconstructing the lensing from the same maps. The second is dust in the Milky Way, whose grains are aligned by the Galaxy’s magnetic field — the alignment that polarises starlight in the first place — and emit polarised light with a different spectrum, which the retracted 2014 detection mistook for the signal. The present limit, $r < 0.036$ at 95 per cent from the BICEP and Keck experiments with Planck and WMAP data for the foregrounds, is a limit set by that separation as much as by sensitivity; adding the baryon acoustic scale and Planck’s own lensing tightens it to 0.032.

What the plane leaves out #

The limits are drawn as a box. The published constraint is a two-dimensional likelihood in which the index and the ratio are correlated, and it is somewhat tighter than a vertical band crossed with a horizontal ceiling in the corner where the tilt is high and the ratio is near its bound.

The tracks assume a range of e-folds. The span from 50 to 60 is a convention standing in for an unobserved thermal history after inflation, and a history outside it would stretch each track further.

The ratio is quoted at a scale. Both the index and the ratio are defined at a pivot wavenumber, and with a tilted tensor spectrum the numerical value of $r$ depends slightly on which pivot is chosen. The bound here is at 0.05 Mpc⁻¹.

The planned sensitivity is set by lensing, not by detectors. Below a ratio of about 0.01 the lensing B-modes behave as an irreducible noise at every scale, and adding detectors only measures that noise better. What pushes the limit further is subtracting the lensing itself, reconstructed from the small-scale distortion of the E-mode pattern in high-resolution maps; how completely that subtraction can be done is what puts the designed uncertainty near a thousandth rather than at any lower number.

And the relation r = 16ε is itself a prediction. Single-field slow roll also fixes the tilt of the tensor spectrum, $n_t = -r/8$, a second consistency condition that would test the whole framework and that cannot be measured until $r$ has been detected well above the planned sensitivity.

Still open: whether a plateau is a prediction or the place left over #

The potentials that survive are the flat ones, and there are two ways to read that. One is that the data have selected a class of models that were motivated independently, from corrections to gravity or from the geometry of field space, and have confirmed that class. The other is that flat potentials are simply what remains after everything with a large tensor signal has been removed, and would have been declared the favourite whatever they were. The plateau’s own prediction, three to four thousandths, is the measurement that decides between the readings, and it sits just above the sensitivity the next generation of polarisation experiments has been designed for — close enough that a non-detection there would be the first result to put pressure on the survivors rather than on the models already gone.