A crossing that may belong to the fitting form

For twenty-five years every measurement of the dark energy has returned a number close to minus one. The number is $w$, the ratio of the dark energy’s pressure to its energy density, and minus one is the value a cosmological constant has — a density that does not dilute as space expands, the same in every cubic metre at every epoch. Measured as a single constant, $w$ came out at $-1.03 \pm 0.03$, and the interest was in the width rather than the centre.

A constant $w$ was always an assumption, and the obvious relaxation is to let it change. The form every survey now fits is

$$w(a) = w_0 + w_a\,(1 - a),$$

with $a$ the scale factor — one today, a half at redshift one. It has two parameters: $w_0$, the value now, and $w_a$, how much it differed in the early universe, where $a \to 0$ and $w \to w_0 + w_a$. A cosmological constant is the single point $w_0 = -1$, $w_a = 0$. In 2025 the second year of acoustic-scale measurements from the largest spectroscopic galaxy survey yet made was combined with the microwave background and with each of three supernova compilations, and all three combinations put the best fit well away from that point.

The preference over a constant is quoted at 2.8, 3.8 and 4.2 standard deviations. All three curves have the same shape. The dark energy is less negative than a constant today — $w_0$ between $-0.84$ and $-0.67$ — and more negative in the past, falling through $-1$ near redshift 0.4 and heading for $-1.3$ to $-1.5$ by redshift two. If the fits are describing the universe, the dark energy is not a constant, and it did something along the way that is harder to explain than a constant ever was.

What follows is an account of what the fits actually hold, taken one piece at a time — because the same three curves can be read as a discovery, as the shape of a fitting function, or as a calibration difference between nearby supernovae and distant ones, and the figures are what separates the readings.

A number measured well at one redshift and extrapolated to the others #

The first thing to know about a fit of $w_0$ and $w_a$ is that neither parameter is what the data measure. Distances respond to the dark energy through an integral over its whole history, and a survey samples that history over a finite range of redshift — supernovae mostly below one, acoustic-scale distances out to about two and a third, the microwave background once, at 1090. The equation of state is constrained where those measurements have leverage, and the two-parameter form then carries that constraint to places where there is none.

The survey in the figure is a mock, and it says so: sixteen hundred supernovae spread evenly between redshift 0.02 and 1.1 with a scatter of 0.14 magnitudes each; transverse and radial acoustic-scale distances at five redshifts to about a per cent; and the microwave background compressed to its shift parameter, the one distance to last scattering it pins to a quarter of a per cent. The supernovae carry a free absolute magnitude and the acoustic distances a free normalisation, because neither is known independently of the fit. With those choices the mock gives $\pm0.063$ on $w_0$ and $\pm0.24$ on $w_a$, close to the published errors, which is the reason to trust its shape.

Its shape is a waist. The uncertainty on $w$ at redshift $z$ is

$$\sigma_w^2(z) = \sigma_{w_0}^2 + 2\,(1-a)\,C_{0a} + (1-a)^2\,\sigma_{w_a}^2,$$

and because the covariance $C_{0a}$ between the two parameters is large and negative, there is one value of $1 - a$ at which the terms nearly cancel. That redshift is the pivot. In the mock it is 0.33, and there $w$ is known to $\pm0.021$ — three times better than its value today, which is an extrapolation from the pivot back to $a = 1$, and ten times better than the early-universe value $w_0 + w_a$, which is an extrapolation in the other direction to where there are almost no data at all.

The pivot is not a feature of the fitting form. It is where the data are: the redshift at which the combination of probes has the most leverage on the dark energy’s density, weighted by how much of the universe the dark energy was then. Below it the volume is small and the dark energy is still being measured only through the nearest supernovae; above it the dark energy is a shrinking fraction of the total, and its density at redshift three is a few per cent of the budget. The three published curves converge near redshift a half — within a few hundredths of one another, and of $-1$ — for exactly this reason. What they disagree about, and what the fits report as $w_0$ and $w_a$, is how the curve continues away from the one place it is held.

Two numbers that are really one and a lever arm #

Plotted in the plane of its own two parameters, the constraint is not a circle.

A correlation of $-0.94$ makes the ellipse a needle. Along its short axis the data constrain $w$ at the pivot; along its long axis they constrain almost nothing, because a larger $w_0$ and a more negative $w_a$ can be traded against each other with the value at the pivot held fixed. The long axis runs from upper left to lower right, and the published fits lie along its continuation: each is a history with the same $w$ near redshift a third as a constant has, tilted about that point.

The inverse area of this ellipse is the figure of merit that a task force on dark energy defined in 2006, and every survey designed since has been specified by how much it shrinks it. The quantity has a practical virtue and a hidden cost. It rewards a survey for narrowing the needle in both directions, which is a sensible target. It also invites the plane to be read as though its two axes were two measurements, when along one of them the answer is almost entirely set by the fitting form’s assumption that $w$ changes linearly in the scale factor.

The two lines drawn across the plane divide it by what the history does rather than by where the parameters are. Everything to the right of $w_0 = -1$ has $w > -1$ today. Everything above the diagonal $w_0 + w_a = -1$ had $w > -1$ in the early universe. The quadrant that satisfies both is the one a single, ordinary scalar field can occupy. The three published points are to the right of the vertical and below the diagonal, in the quadrant of histories that start phantom and end not phantom — and it is the diagonal, not the distance from the cross, that makes the result strange.

Measured in the mock’s own metric the three fits are 2.6, 5.5 and 4.0 standard deviations from a constant. Those are not the published significances, which come from full likelihoods with correlated errors this mock does not have, and the order of the three is different. What the mock does reproduce is that all three are several standard deviations away along the needle, and that the needle is long enough for a factor of two in $w_a$ to cost only a fraction of a standard deviation.

A density that grew #

The equation of state is a derived quantity. What the expansion actually responds to is the dark energy’s density, and for this form of $w$ the density has an exact expression:

$$\frac{\rho(a)}{\rho_0} = a^{-3(1 + w_0 + w_a)}\,\exp\!\big[-3\,w_a\,(1 - a)\big].$$

Read from right to left, which is forward in time, the fitted density rises from about half its present value at redshift three to a maximum four to five billion years ago, then declines. A density that grows while space expands is what $w < -1$ means: the continuity equation for any component is

$$\dot\rho = -3H\,(1 + w)\,\rho,$$

so with $H > 0$ the density falls if $w > -1$, stays fixed if $w = -1$, and rises if $w < -1$. The fitted dark energy spent the first nine billion years of cosmic history doing the third, turned over, and is now doing the first.

For the ordinary candidates that is not merely unusual. The simplest dynamical dark energy is a single scalar field rolling in a potential — quintessence — and its equation of state is

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

a ratio of kinetic minus potential energy to kinetic plus potential. With a positive kinetic term the ratio is bounded below by $-1$ and reaches it only when the field is momentarily at rest. A field can approach $-1$ from above and leave again, but it cannot go through: the only route below $-1$ is a negative kinetic energy, the ghost that makes a field theory’s vacuum unstable, which is the objection to phantom dark energy set out where $w = -1.03$ was first measured. A crossing requires two fields, a field coupled to the matter, or a modification of gravity itself. None is excluded. Each is a larger step than the one the fits appear to ask for.

That is the reason the three curves matter more than their significance alone suggests, and also the reason to look hard at whether the crossing is in the data at all.

Three compilations, one difference, and where it lives #

The three combined fits share their galaxy data and their microwave-background data. What differs is the supernova sample: Pantheon+, a compilation of about sixteen hundred objects assembled from many surveys; Union3, a re-analysis of about two thousand with a different treatment of their systematics; and the five-year sample of the Dark Energy Survey, about sixteen hundred photometrically classified supernovae at intermediate redshift, anchored at low redshift by a separate external sample. Remove the supernovae entirely and the galaxy and background data alone prefer an evolving dark energy at about three standard deviations. With supernovae the preference ranges from 2.8 to 4.2 depending on which, which means the supernovae are moving the answer by about as much as they are adding to it.

A supernova distance enters the fit only as a difference. The absolute magnitude of a standardised type Ia is not known independently of the fit — it is calibrated through a chain of nearer distance measurements, and in a dark-energy analysis it is left free — so a constant offset in every distance modulus is absorbed and invisible. What a supernova sample constrains is the shape of the distance modulus against redshift, and a change in shape is what an evolving dark energy has to produce.

The shape the evolving fit asks for is concentrated at the near end. Relative to a constant, and after the free offset is removed, the nearest supernovae must be about forty millimagnitudes fainter than the distant ones — four per cent in flux, two per cent in distance. Beyond redshift 0.3 the residual is nearly flat. The dashed line is not a cosmology at all: it is what a forty-millimagnitude difference between a low-redshift sample and a high-redshift one looks like once the same offset has been taken out. Over the range where most of the objects are, the two are hard to tell apart.

That is precisely the size of calibration question the field has argued about for two decades. The nearby supernovae come from different telescopes, different filter systems and different years from the distant ones, and are tied to them through the zero points of the photometric system, which are uncertain at the ten-millimagnitude level. They sit where a galaxy’s own motion is a real fraction of its redshift, so their distances carry a correction for peculiar velocities that shares neighbours and does not average away. Their host galaxies are, on average, of different mass and age from those of distant supernovae, and the standardisation depends on the host. One re-analysis of the objects common to two of the compilations found their distance moduli offset by about four hundredths of a magnitude between the low- and high-redshift ends, which is the size of the signal.

None of this shows that the evolving fit is a calibration error. The galaxy data prefer it without any supernovae, and the three compilations, which differ in their objects and in their treatment of the same ones, all point the same way. What the figure shows is why the number moves with the compilation, and why the argument about whether the dark energy is changing has become, for the moment, an argument about how forty millimagnitudes are shared between the nearest supernovae and the rest.

A straight line fitted to a curve #

The second question is whether a crossing in the fit is a crossing in the universe, and it can be asked without any data at all. Take a dark energy that certainly never crosses $-1$, put its distances through the same mock survey, and fit the same two parameters.

The field in the figure is a thawing model: frozen by the expansion’s friction at $w = -1$ for most of cosmic history, released recently as the friction fell, rolling now so that $w = -0.8$. It is the most natural shape for a quintessence field, and it has $w \geq -1$ at every epoch by construction. Its equation of state is flat in the past and bends up steeply in the last few billion years — a curve in the scale factor, where the fitting form is a straight line.

The best straight line is $w_0 = -0.83$, $w_a = -0.30$. It matches the field closely where the data are, near the pivot, and it matches the distances to within a fraction of their errors everywhere. It also crosses $-1$ at redshift 1.28 and continues downward, to $-1.13$ in the early universe, because a straight line that follows a curve’s steep recent rise must be too steep in the flat past. The crossing is a property of the line, not of the field. Any history that is flat early and bends late will be fitted by a line that crosses, and thawing is exactly that shape.

That resolves less than it seems to. The faint curve is the published fit with DES-SN5YR, and it is a different object: it crosses at redshift 0.41 rather than 1.28, and it reaches $-1.61$ in the early universe, about five times further below $-1$ than the projection of the thawing field. A thawing field can produce an apparent crossing; it does not easily produce one this steep and this recent. What is left is not a verdict but a measurement to be made — whether a physical thawing model, fitted directly rather than through the two-parameter form, describes the data about as well. The analyses that have tried it report a preference of similar but somewhat smaller size, with $w$ a little above $-1$ today, which is the reading in which nothing crossed anything and the dark energy has simply begun to move.

What the change would also change #

An evolving dark energy does not stay inside the dark-energy question. The same fits that prefer it also change what the data say about the sum of the neutrino masses. With a constant, the combination of galaxy and background data pulls the neutrino mass towards zero — to an upper bound barely above the minimum that oscillation experiments on the Earth require, with the most likely value below it, which is a tension in its own right. Freeing the dark energy relaxes it, because a dark energy that was weaker in the past slows the late expansion in a way that partly mimics what massive neutrinos do. The two anomalies may be one: a hint of physics beyond a constant, or two faces of the same misestimated systematic.

It also changes the history of the dark energy’s own coincidence. The cosmological constant’s value sits near the matter density now, and a density that rose to a maximum four to five billion years ago and is now falling would make the present epoch special in a second way — the era of the dark energy’s own peak — which is not the kind of explanation of a coincidence anyone was hoping for.

And it changes what the next measurements should be. The acoustic ruler measured along and across the line of sight already gives $H(z)$ directly at several redshifts, and adding redshifts below the pivot, where the fits disagree most, is worth more than adding precision above it. A redshift watched as it changes over a decade would measure $\dot a$ with no calibration at all. Distances from gravitational waves share nothing with a supernova’s photometry. Each of these can see a dark energy that has changed; none has to argue about the nearest forty millimagnitudes.

What the figures cannot show #

Every figure here is either a published central value or a calculation in a mock survey, and neither is the thing the argument is about. The mock has no correlated errors, no selection function and no redshift distribution beyond a uniform one; it reproduces the published uncertainties closely and the published significances only roughly, and the order of the three fits’ distances from a constant is not the published order. The published points are drawn with symmetric errors where the originals are asymmetric.

The deeper limitation is the form itself. Two parameters cannot represent an arbitrary history, and every statement above about $w_0$, $w_a$, the crossing and the density’s peak is a statement about the best two-parameter description of the data rather than about the dark energy. The pivot is the one quantity that is largely independent of that choice, and at the pivot the three fits and a cosmological constant agree to a few hundredths.

Still open: whether the dark energy has begun to move #

The fits prefer an equation of state that has changed, at between three and four standard deviations, and the preference survives removing any one supernova compilation and removing all of them. What it does not yet survive is the question of form: the crossing of $-1$ that makes it strange is the part the data hold least well, and a field that never crosses, fitted with a straight line, would show one. Three things would decide it. A supernova sample calibrated end to end on one photometric system, so that the nearby objects and the distant ones share their zero points; acoustic-scale distances below redshift a third from a survey volume large enough to beat its own cosmic variance; and a direct fit of physical models — thawing fields, coupled fields, modified gravity — in place of a line in the scale factor. If a thawing field fits as well as the line, the result is that the dark energy is not a constant and has only recently begun to roll. If the crossing survives a physical fit, the result is larger than that, and nobody yet has the theory to hold it.