Every thawing field falls on one line

When a survey reports that the dark energy may be changing, what it reports is two numbers: $w_0$, the equation of state today, and $w_a$, how much it differed in the past, in the form $w(a) = w_0 + w_a(1 - a)$. That form is a description, chosen because it is simple and because it behaves sensibly at both ends of cosmic history, and it was adopted when the only established fact was that the expansion had stopped decelerating. It is not a theory, and a straight line in the scale factor fitted to a curved history can show features — a crossing of $-1$, a density that peaked and fell — that the history it was fitted to never had.

The alternative is to start from a theory and see where it lands. The simplest dynamical dark energy is a single scalar field $\phi$ filling space, with a potential energy $V(\phi)$ and a kinetic energy $\tfrac12\dot\phi^2$. It is the same kind of object the inflaton was supposed to be, somewhere between eighty and a hundred and ten orders of magnitude lower in energy density, and it has the same equation of motion:

$$\ddot\phi + 3H\dot\phi + \frac{\mathrm{d}V}{\mathrm{d}\phi} = 0.$$

The field rolls downhill like a ball in a bowl of treacle. The slope of the potential pushes it; the term $3H\dot\phi$ — the expansion rate times the field’s velocity — resists it, and plays the part of friction. Its equation of state is the ratio of its pressure to its density,

$$w = \frac{\tfrac12\dot\phi^2 - V}{\tfrac12\dot\phi^2 + V},$$

which is exactly $-1$ while the field is at rest and rises towards zero, or beyond, as the kinetic energy grows.

What follows takes this equation seriously. Each history below is a solution: the field starts at rest in the early universe, the potential’s height is adjusted until the dark energy makes up sixty-nine per cent of the present density — the larger part of a budget whose familiar part is five per cent — and one parameter of its shape is adjusted until $w$ today has a chosen value. Nothing else is tuned. Then each solution is put through the same mock survey used to examine the published fits — sixteen hundred supernovae, acoustic-scale distances at five redshifts, the microwave background’s shift parameter — and described by the two numbers the surveys report.

A field that cannot move while the universe is young #

The friction term decides almost everything, and it is easiest to see in the simplest potential: a mass term, $V = \tfrac12 m^2\phi^2$, a parabola with its minimum at zero and the field placed somewhere up its side.

When the expansion rate is much larger than the mass, the friction term is much larger than the push. A ball in treacle that is thick enough does not roll at all; it sits where it was put. The field’s velocity is damped to almost nothing, its kinetic energy is a vanishing fraction of its potential energy, and its equation of state is $-1$ to high precision. In the figure, at redshift six, where the expansion is about ten times faster than today, the field is still within a part in a thousand of a cosmological constant. A universe containing it would be indistinguishable from one with a constant at every epoch in which the dark energy was too small to matter.

The expansion slows as the universe dilutes, and the friction slows with it. When $H$ falls to within a factor of a few of $m$, the push and the friction become comparable and the field starts to move. In the figure that happens near redshift 2.4, and by today the field is rolling fast enough that $w = -0.79$. Nothing about the field changed in between. The universe slowed down until it could no longer hold the field still. This is the reason such models are called thawing: the field was frozen by the expansion and is thawing as the expansion lets go of it.

That picture carries a number with it. For the field to thaw in the last few billion years rather than long ago or far in the future, its mass must be of order the present expansion rate. In the figure $m = 1.2\,H_0$, and $\hbar H_0$ is $1.4 \times 10^{-33}$ electronvolts, so the field’s mass is about $1.7 \times 10^{-33}$ eV. That is some thirty orders of magnitude lighter than the lightest neutrino, and its Compton wavelength is the size of the observable universe. A particle that light would mediate a force with the reach of the universe, and it must couple to ordinary matter extremely weakly or that force would already have been seen in tests of gravity in the solar system. The field that solves the dark-energy problem dynamically has to be the lightest particle there is, by a margin no other particle approaches, and the fact that it thaws now rather than at any other time is another way of stating the coincidence the constant already had.

Four potentials that give one history #

The quadratic potential was a choice, and the obvious worry is that everything just said depends on it. The test is to solve other potentials and compare. Four are drawn here: the mass term; a linear slope, $V \propto 1 - s\phi$; an exponential, $V \propto e^{-s\phi}$; and the top of a cosine, $V \propto 1 + \cos\phi$, the potential of a field that started near the crest of a hill. They are not variations on one function. One has a minimum and the others do not; one is curved downward, two upward, one not at all.

Shot to the same present value, they are hard to tell apart. Every one was frozen, every one began moving at about the same time, and every one rose to $-0.8$ along nearly the same curve. The largest disagreement between any two, anywhere below redshift three, is 0.027 in $w$ — a quantity no survey yet planned could separate.

The reason is the friction again. A field that was held still until recently has not yet travelled far. It has felt the slope of the potential where it started and a little of the curvature, and nothing else: the shape of the potential a long way from the starting point, which is where the four differ, has not yet come into the field’s history at all. To the extent that the field has only just begun to move, its history is set by how fast it has started moving — that is, by $w_0$ — and very little by which potential it is moving in. Robert Scherrer and Anjan Sen made this precise in 2008: for a thawing field whose equation of state is still close to $-1$, the whole history $w(a)$ is a single function multiplied by $1 + w_0$. Sourish Dutta and Scherrer extended it the same year to fields that start near the top of a hill, where the only addition is a weak dependence on the potential’s curvature at the starting point.

That is the useful half of the result, and the uncomfortable half follows from it. Measuring $w(a)$ for a thawing field tells very little about the potential. The dark energy, if it is a field, could be any of these and the history would barely say which.

The same limitation has been met once before, by the same kind of field at the other end of cosmic history. The fluctuations left by inflation record the slope of the inflaton’s potential at the moment the observable scales left the horizon, and not its height, and a decade of increasingly precise measurements of the microwave background has narrowed the slope without fixing the potential. A slowly rolling field reports its local gradient and keeps the rest of its potential to itself, whether it rolled in the first $10^{-32}$ seconds or is rolling now. In both cases the observations measure how fast the field is moving, and in both cases a family of very different potentials agrees on that.

Leaving −1 is a different shape from approaching it #

The single function can be seen directly by dividing each history by its own present distance from $-1$. The quantity $(1 + w)/(1 + w_0)$ is zero for a frozen field and one today, whatever $w_0$ is.

Twelve thawing solutions, spanning four potentials and a factor of five in $1 + w_0$, collapse into a narrow bundle. At half the present scale factor — about eight billion years ago — every one of them was between a fifth and three tenths as far from $-1$ as it is now. Their spread is mostly the exponential potential again, and mostly at the fastest-rolling end, where the approximation that the field has not travelled far is weakest.

Above the bundle sits a different family. Freezing fields roll down potentials like $V \propto \phi^{-n}$, which have no minimum and grow shallower as the field rolls. Such a field is not frozen in the early universe; it rolls from the start, settling onto a tracker — a solution in which its density follows the matter’s at a fixed ratio, with $w$ well above $-1$ — and then, as the potential flattens, slowing and approaching $-1$ from above. Its history is the thawing history reversed in direction: $1 + w$ was larger in the past than it is now. The three tracker solutions in the figure, with exponents a half, one and two, all lie between 1.19 and 1.28 at half the present scale factor.

These are the two ways a single field with ordinary kinetic energy can behave near $-1$. It can leave it or approach it. The equation of motion does not allow it to pass through, because a field at rest has $w = -1$ exactly and moving in either direction raises it. Linder and Caldwell, who drew this division in 2005, pointed out that it splits the possible histories into two regions that a sufficiently good measurement could tell apart without knowing anything about the potential at all.

Physics draws a line where a fit draws an ellipse #

Each solution can now be put through the mock survey and described by the two numbers the surveys report. A fit of $w_0$ and $w_a$ has no knowledge of fields or potentials; it only knows distances. What it returns for a field is the straight line in the scale factor that best reproduces that field’s supernova, acoustic-scale and microwave-background distances.

Twenty-eight solutions, four potentials, and they land on a line. That follows from the last two figures: if every thawing history is one function times $1 + w_0$, then its best straight-line description is one pair of numbers times $1 + w_0$, and $w_a$ is proportional to $1 + w_0$. In this survey the constant of proportionality is 1.49, with no solution more than 0.08 from it; a published calibration of the same relation by Roland de Putter and Eric Linder, made with a different mock survey, found 1.58. The freezing solutions fall on the other side of the cross, with $w_a$ positive, as their reversed shape requires.

So a thawing field does not predict a region of this plane. It predicts a line through the point of a cosmological constant, and the only thing a thawing field leaves free is how far along the line it sits. That changes what the published ellipses mean. A fit of $w_0$ and $w_a$ spends one of its two parameters on a direction no thawing field can move in; a thawing field has one parameter, and the data’s answer to “how far along” is a measurement of a single number.

The figure also shows something about crossings. The thawing line is steeper than the diagonal $w_0 + w_a = -1$, which is where the fitted early-universe value is exactly $-1$, so every point on the line is fitted with $w_0 + w_a < -1$ — a fit that crosses $-1$ in the past. That is the effect the straight line produces when it follows a curved history, measured now across every solution rather than one. But because the line runs through the constant, the crossing always happens early: the fits of these solutions cross $-1$ between redshift 1.7 and 3.3, where a line of slope 1.49 through the constant must, near $z = 1/(1.49 - 1) \approx 2$. The published fits cross between redshift 0.35 and 0.44. A thawing field, put through the same fitting form, produces a crossing in roughly the wrong place by a factor of five in redshift.

And the published points are not on the line. The fit with DES-SN5YR, at $w_0 = -0.75$, has $w_a = -0.86$; a thawing field fitted the same way with that $w_0$ has $w_a \approx -0.37$. All three published fits lie below the line by between 0.38 and 0.59 in $w_a$, in the direction of a steeper fall into the past than any rolling field can produce.

How much of the preference a rolling field can take #

Lying below the line is not, by itself, a verdict, because the errors are large and correlated. The question that matters is quantitative: if the dark energy were a thawing field, the best one available, how much worse would it describe the data than the two-parameter fit does, and how much better than a constant?

The figure measures the three published fits against the line in the mock survey’s own metric. That metric reproduces the published uncertainties, not the full published likelihoods with their correlated systematics, and the numbers should be read as the shape of the answer rather than the answer. The shape is clear. For each compilation there is a nearest thawing field, and it is close to a constant: $w_0$ between $-0.97$ and $-0.87$, a field that has only just begun to roll. It is preferred over a constant — the square root of the difference in $\chi^2$, for the one extra parameter, is 0.9, 3.9 and 2.4 standard deviations for Pantheon+, Union3 and DES-SN5YR. And it is still 2.5 to 3.9 standard deviations from the published two-parameter fit.

That divides the published preference for an evolving dark energy into two parts. One part can be taken by a field that has begun to thaw: most of it for Union3, much of it for DES-SN5YR, very little for Pantheon+. The other part is the steep early fall below $-1$ that the two-parameter form also contains, and no thawing field can take any of it. Analyses that fit physical thawing models directly to the real likelihoods report a similar division: a thawing field is preferred over a constant, by somewhat less than the two-parameter form is, with $w_0$ a little above $-1$.

Whether that second part is real is the question the calibration of the nearest supernovae bears on, because a few hundredths of a magnitude between low and high redshift is enough to produce it. If it goes, what is left is exactly what a thawing field predicts: a dark energy that sat at $-1$ for nine billion years and has recently begun to move, measured by one number, its distance along the line.

Three futures that agree today #

A cosmological constant has one future: the expansion accelerates for ever, distances between unbound galaxies grow exponentially, and the galaxies already out of reach stay out of reach. A field that has begun to roll has whatever future its potential gives it, and the potential is exactly what the history does not reveal.

The three universes share their matter density, their dark-energy density and, for the two fields, their present equation of state, $w = -0.8$. The figure follows each into the future. The constant does what constants do. The field in the quadratic potential keeps rolling, reaches the bottom of its well and oscillates about it; an oscillating field’s pressure averages to zero, so it behaves like a gas of very light, very cold particles, and the acceleration ends. The expansion continues, decelerating, as though the universe contained only matter. The field on the linear slope rolls past zero potential energy into negative values; the total energy density falls towards zero, the expansion slows, stops at 2.26 times its present size about twenty-three billion years from now, and the universe begins to contract.

These are not exotic choices. A linear potential is the first term of any potential’s expansion about the field’s starting point, and the four potentials in the earlier figure gave the same history up to today. Renata Kallosh, Andrei Linde and their colleagues pointed out in 2003 that a universe with such a field could be as close to its end as it is to its beginning, and that no measurement of the present expansion could rule it out. The thawing figures sharpen that: the history is set by $w_0$ and the future is set by the shape of the potential, and the shape is the thing the history hardly depends on. A measurement of the equation of state today constrains how fast the field is rolling. It does not say where the field is going.

What the solutions leave out #

The fields here are the simplest dynamical dark energy and nothing more. Each is a single scalar with an ordinary kinetic term, minimally coupled to gravity and not coupled to matter; each starts at rest; and each lives in a flat universe with the matter density held at 0.31. Fields that couple to dark matter, fields with non-standard kinetic terms and theories in which gravity itself is modified can all do things these cannot, including crossing $-1$, and none of them is examined here.

The mock survey is also the same simplified one used before: uniform redshift distributions, independent errors and no selection function. It reproduces the published uncertainties reasonably and the published significances only roughly. The slope of the thawing line depends weakly on the survey through which it is fitted, and the published calibration differs from 1.49 by about a tenth for that reason. The distances in standard deviations are therefore approximate, and the qualitative statements — that the solutions lie on a line, that the published points lie below it, that a nearly frozen field takes part of the preference and not all of it — are the parts the figures establish.

And none of the figures says that the dark energy is a field. They say that if it is one of these fields, the next measurements should find the published points moving onto a line through the constant, and should find a single number, $1 + w_0$, rather than two.

Still open: what the field would have to weigh #

The thawing line turns the question of whether the dark energy is changing into a measurement of one number, how far along the line the universe sits. A survey that measures $1 + w_0$ to a few hundredths, at the pivot where the acoustic ruler measured along and across the line of sight is most sensitive, would either find it consistent with zero or find a field that has begun to roll and measure the mass of the lightest particle there could be — about $10^{-33}$ electronvolts, a value with no explanation in any theory of particles and a smallness that needs protecting as badly as the constant’s own value does. Two further measurements would then separate thawing from what lies off the line. A redshift watched as it drifts over a decade or two measures the expansion rate directly, with no calibration, at the redshifts where the published fits and the thawing line disagree most. And the growth of structure inside the same expansion — the subject that follows — would say whether whatever is accelerating the expansion is a substance at all, or a change in gravity that happens to produce the same distances. If the published points settle onto the line, the dark energy has a mass and a history but no predictable future, because the future belongs to the part of the potential the field has not yet reached.