The tilt knows the slope and not the height

The case for inflation rests on two coincidences removed by one mechanism, and on a single number that the mechanism predicted before anyone could measure it: the primordial fluctuations are nearly, but not exactly, equally strong on every scale. Their spectral index is 0.9649 ± 0.0042, which is 3.5 per cent short of exact scale invariance and more than eight standard deviations away from it.

That number is usually described as a measurement of inflation. It is narrower than that, and more precise. It measures two dimensionless properties of whatever drove the expansion — how steeply its energy fell as the field rolled, and how quickly that steepness was changing — over the short stretch of the roll during which the scales now on the sky were leaving the Hubble radius. It does not measure how much energy there was. A potential a hundred million times higher, with the same shape, predicts exactly the same tilt.

A field falling at its terminal speed #

The mechanism is a scalar field, the inflaton, sitting on a potential $V(\phi)$. Its equation of motion in an expanding universe is that of a ball on a slope with friction,

$$\ddot\phi + 3H\dot\phi + V'(\phi) = 0,$$

and the friction coefficient $3H$ is set by the expansion rate, which is in turn set by the energy the field holds: $H^2 = V/3M^2$ while the kinetic energy is small, with $M$ the reduced Planck mass, $2.435\times10^{18}$ GeV. The field supplies the expansion, and the expansion supplies the friction that slows the field.

Slow roll is the overdamped limit of that equation. When friction dominates, the acceleration term is negligible and the field moves at the speed where friction balances the slope, $3H\dot\phi \simeq -V'$. That is a terminal velocity, and it has the property every terminal velocity has: it forgets how it started. A dust grain settling through a gas disc at the speed its drag allows carries no memory of how it was released, and neither does an inflaton on an overdamped slope. That forgetting is why the predictions below depend on the shape of the potential and not on whatever state the field began in.

Two conditions keep the approximation honest, and they are the two numbers the rest of this essay is about:

$$\varepsilon = \frac{M^2}{2}\left(\frac{V'}{V}\right)^2, \qquad \eta = M^2\,\frac{V''}{V}.$$

The first says the slope is gentle compared with the height, which keeps the kinetic energy small and the expansion accelerating. The expansion accelerates exactly while ε is below one, so ε = 1 is where inflation ends, and it is the point marked on every potential in this essay. The second says the slope is not changing quickly. A slope that steepened appreciably within one expansion time would leave the field lagging behind its own terminal velocity, and the acceleration term that was dropped would come back.

Both are ratios of the potential to its own derivatives. Multiply $V$ by any constant and neither of them moves.

There is a connection here that is worth making at once, because the same parameter turns up in a measurement made at the other end of cosmic history. A field rolling slowly has pressure nearly equal and opposite to its energy density, and its equation of state is $w = -1 + 2\varepsilon/3$. During inflation, with ε a few thousandths, that is $w \approx -0.997$. The number measured for dark energy today, $w = -1.03 \pm 0.03$, is the same quantity for whatever is accelerating the expansion now, and a slowly rolling field is one of the two readings of it. The early acceleration and the late one obey the same condition on the same kind of number, although their energy densities are more than a hundred decades apart.

The value of $w$ also decides the causal structure each epoch leaves behind. Any expansion with $w$ below $-1/3$ accelerates, shrinks the comoving Hubble radius, and surrounds every observer with an event horizon — so each inflating patch was, while it lasted, enclosed by a horizon a single Hubble length across, just as the present universe is beginning to be. The fluctuations were generated at that horizon, and that is the reason their amplitude depends on the expansion rate at all.

Where the height goes #

The fluctuations are generated as the field rolls. Each comoving scale is stretched past the Hubble radius at some moment — the crossing drawn in the diagram of the horizon problem — and the quantum jitter in the field at that moment is frozen in as a perturbation to the curvature of space. Its amplitude is

$$\mathcal{P}_\zeta = \frac{V}{24\pi^2\,\varepsilon\,M^4},$$

evaluated when that scale left. The height of the potential is there, in the numerator. So is ε, in the denominator.

The measured amplitude, $2.1\times10^{-9}$ at a pivot wavelength of about 125 megaparsecs, therefore fixes one combination only: $V/\varepsilon$. A steep potential must be high and a flat one can be low, and the same amplitude — the pattern of one part in a hundred thousand in the microwave background’s frozen standing wave — comes out of either. The smallness of the fluctuations is therefore not evidence that inflation happened at low energy. It is evidence that $V$ and ε stood in a particular ratio, and it says nothing about either of them separately.

The tilt is the logarithmic slope of that amplitude against scale. Scales leave one after another, about one e-fold of expansion for each factor of $e$ in wavelength, so a derivative with respect to $\ln k$ is a derivative with respect to the number of e-folds. To first order in the small parameters,

$$n_s - 1 = \frac{d\ln\mathcal{P}_\zeta}{d\ln k} = 2\eta - 6\varepsilon.$$

A constant factor in $V$ is a constant term in $\ln\mathcal{P}_\zeta$, and a derivative removes it. The tilt is the one number the sky gives about inflation from which the height has been taken out by construction.

The two potentials share almost nothing. One is a parabola that climbs without limit, the other a plateau that approaches a ceiling it never reaches. They predict tilts a thousandth apart, far inside the measurement’s uncertainty of four thousandths, and their energy densities differ by a factor of about forty. The fourth power hides how different that is: 2.0 and 0.79 in units of 10¹⁶ GeV look like neighbours, and $(2.0/0.79)^4 = 41$.

The history of the prediction runs in an instructive order. Harrison, Zel’dovich and Peebles argued around 1970 that the primordial spectrum should be exactly scale-invariant, on the grounds that nothing in the early universe picked out a length, and that proposal was written down before inflation existed. Mukhanov and Chibisov computed the fluctuations of the Starobinsky model in 1981 and found a small logarithmic departure from it — a tilt, towards larger amplitude on larger scales. It took until the WMAP satellite’s later data releases for a departure to show at a few standard deviations, and until Planck for the present figure. The inflationary prediction was never “scale invariance”. It was a small departure of a particular sign, and the departure is the measurement.

Scale invariance is easy to mistake for a claim that the universe is lumpy on every scale, and it is the opposite. What is equally strong on every scale is the fluctuation in the gravitational potential. The density contrast that potential produces on a given scale grows with the square of the scale’s wavenumber, so the largest scales carry the smallest density contrasts and a scale-invariant universe becomes smooth above a hundred megaparsecs without anything having to smooth it. A tilt towards the red strengthens the largest scales very slightly and leaves that conclusion untouched.

Two numbers along the last sixty e-folds #

The slow-roll parameters are not constants. They grow as the field approaches the end, since inflation ends precisely when ε reaches one, and what the sky samples is their value over a short stretch well before that point.

For a power law, $V\propto\phi^p$, the whole calculation closes. Integrating the number of e-folds from the end of inflation gives the field value $N$ e-folds before the end as $\phi^2 = 2p\,(N + p/4)$ in Planck units, and then

$$\varepsilon = \frac{p}{4N+p}, \qquad \eta = \frac{2(p-1)}{4N+p}, \qquad n_s = 1 - \frac{2p+4}{4N+p}.$$

The mass term, $p = 2$, has η exactly equal to ε, which is why its two curves coincide. A quartic has η one and a half times ε. A linear potential has no curvature at all, and a power below one has a curvature of the opposite sign. Every member of the family gets its tilt mostly from the slope, and the slope’s share, $6\varepsilon$, falls as $1/N$.

That pair of figures is the reason a single measured number cannot identify a potential. The mass term makes its tilt from the slope. The plateau makes the same tilt from the curvature. The observable is the combination $2\eta - 6\varepsilon$, and there are as many ways to reach 0.035 as there are ways to split one number into two.

One tilt, shared out #

Setting several potentials side by side makes the degeneracy concrete, and it also shows which of them the tilt alone already excludes.

Read row by row, the chart is an argument about what a tilt excludes and what it does not. The quartic is too red by five standard deviations and needs no further measurement to be set aside. The power of two-thirds is too blue by a comparable margin. Natural inflation, a cosine potential of the kind an axion would have, sits just outside the band at this value of its width. The mass term and the plateau both fit, and they have almost nothing else in common: the first spends a tilt of 0.054 on its slope and gives a third of it back through its curvature, the second spends essentially nothing on its slope and takes the whole tilt from curvature.

The hilltop row is worth separating from the rest, because it fails for the opposite reason from the quartic.

On a hill the curvature is negative from the start and nearly constant, so the tilt is fixed almost entirely by the width of the hill, $\eta \approx -2M^2/\mu^2$, and hardly at all by how many e-folds remain. A hill ten Planck masses wide is too narrow. That is a statement about a length in field space, and it foreshadows something that the tensor amplitude makes explicit: whether the field travelled more or less than a Planck mass is not an aesthetic question about a model, because the effective description of the potential is only trustworthy over distances shorter than the scale at which unknown physics would enter.

The tilt read as a count #

Every figure so far has put the observed scales between 50 and 60 e-folds before the end, and that range was an assumption rather than a measurement. Turning the relation round shows what it was hiding.

Granted a shape, the tilt is a count. If the potential is a mass term, the measured index says the observed scales left 56.5 e-folds before the end; if it is the plateau, 54.9. The count is not a free parameter of the potential, though. It is fixed by how much the universe expanded between the end of inflation and today, which depends on how the energy of the field was turned into the hot plasma that set the abundances of the light elements — an interval nobody has observed and that no figure here draws. A spread of a few e-folds in that history moves every curve in the chart along its own track, which is why the band is quoted with the words “between fifty and sixty” attached.

That is the least obvious consequence of the whole construction. The tilt was introduced as a probe of the potential; read backwards it becomes a probe of the thermal history just after inflation, provided the potential is known. Neither is known, and the two uncertainties trade against each other.

The size of the trade is easy to state for the curves drawn. Differentiating the closed form gives $dn_s/dN = 4(2p+4)/(4N+p)^2$, which for the mass term at 55 e-folds is $6.5\times10^{-4}$ per e-fold; the plateau’s index, close to $1 - 2/N$, moves by $2/N^2$, which is $6.6\times10^{-4}$. Five e-folds — roughly the difference between an energy transfer that finished at once and one that dragged on through a long phase in which the oscillating field behaved like matter — move either prediction by about 0.003, three quarters of the present uncertainty. The measurement is already close to the precision at which the history after inflation stops being a nuisance in the fit and starts being a quantity the fit returns.

What is actually measured #

The index is not read off the sky directly. It is one of six parameters fitted jointly to the temperature and polarisation spectra of the microwave background, and it is correlated with the others in ways that decide how far it can be trusted.

The strongest correlation is with the amplitude and, through it, with reionisation. The temperature spectrum measures the primordial amplitude multiplied by $e^{-2\tau}$, where τ is the optical depth of the ionised gas the light crossed on its way, so an amplitude and a depth arrive multiplied and have to be separated by polarisation on the largest angular scales. The index is correlated with that product because tilting the spectrum also changes the ratio of large-scale to small-scale power. The second correlation is with the density of ordinary matter, which shapes the relative heights of the acoustic peaks, and with the damping of the smallest scales that the thickness of the last-scattering shell produces. A tilt and a stronger damping both remove power at high multipoles, and it is the shape of the removal that tells them apart.

Planck’s 2018 analysis of temperature, polarisation and lensing together gives 0.9649 ± 0.0042; adding the baryon acoustic scale moves the central value to 0.9665 and trims the uncertainty slightly. The departure from one is real at more than eight standard deviations in either version, and it has been stable across every analysis since the first one that detected it.

What has not been measured is whether the tilt itself changes with scale. Slow roll predicts a running, $dn_s/d\ln k$, of second order in the small parameters — for the mass term at 55 e-folds, about $-6\times10^{-4}$. Planck measures $-0.0045 \pm 0.0067$, which is consistent with zero and ten times less precise than the prediction is small.

What the potentials leave out #

The sky sees about eight e-folds of the roll. The multipoles of the microwave background and the scales of the galaxy surveys span a factor of roughly a thousand in wavelength, which is seven or eight e-folds of the roughly sixty in the figures. Everything drawn outside the shaded bands — including the end of inflation, where ε reaches one — is an extrapolation of the potential’s formula, not a constraint on it.

The formulas are first order. The expressions for the tilt and the amplitude drop terms of order $\epsilon^2$ and $\eta^2$, which for these potentials are a few parts in ten thousand. That is below the present uncertainty on the index and not far below the uncertainty the next generation of polarisation experiments is designed to reach, at which point the second-order expressions have to be used.

The field is assumed to be on its terminal velocity already. A field that started with a large kinetic energy reaches the slow-roll speed within a few e-folds, which is why the assumption is safe for scales that left fifty e-folds before the end; it is not safe for the very largest scales if inflation lasted only a little longer than the minimum, and the slight deficit of power at the lowest multipoles has been read that way, without being significant enough to settle anything.

The six potentials are examples, not a census. Each is a single field with an ordinary kinetic term. A field whose fluctuations propagate slower than light, or several fields sharing the roll, changes the relation between the potential and the observables, and some of those changes are what the non-Gaussianity of the fluctuations would reveal.

And the height is not unconstrained. It is fixed once ε is fixed, through the amplitude. What the tilt cannot supply is ε itself, so the height enters every figure here as an output that depends on the model rather than as a measurement.

Still open: whether the slope or the curvature made the tilt #

Separating the two shares needs a measurement of ε alone, and there is one. The same quantum jitter that perturbed the inflaton also perturbed space-time itself, and those gravitational waves have an amplitude that depends on the expansion rate and nothing else. Their ratio to the curvature fluctuations is $r = 16\varepsilon$ — the slope’s share of the tilt, multiplied by eight thirds. The mass term predicts 0.144 at 55 e-folds and the plateau 0.0035, a factor of forty apart, and the polarisation of the microwave background has already been measured with a precision that distinguishes them.