The growth that tells a substance from a law

Every measurement of the dark energy discussed so far has been a measurement of distance. Supernovae give luminosity distances, the acoustic scale gives angular and radial distances, the microwave background gives one distance to the surface of last scattering, and from all of them together comes an expansion history: the scale factor as a function of time. The dark energy’s density, its equation of state, whether it has changed, whether it could be a field that has begun to roll — each is a statement about that one function.

There is a second function. While the universe expands, the small excesses of density that the microwave background shows at one part in a hundred thousand grow, by their own gravity, into the galaxies and clusters seen today. How fast they grow depends on how fast the universe is expanding, because expansion pulls matter apart while gravity pulls it together, and on how strongly matter attracts matter — on the law of gravity. If the law is known, the growth follows from the expansion. If it is not, the two can disagree, and the disagreement measures the law.

The essay on redshift-space distortions made this point in words: two models can share every distance ever measured and grow structure at rates ten per cent apart, and only a growth measurement can tell them apart. What follows computes it. Each figure solves for the growth inside a stated expansion history and a stated law of gravity, and asks a narrower question than “is it a test?” — how large is each difference, and which of the open questions about the dark energy can growth actually answer?

What relativity leaves no room to choose #

Write a region’s density as the average times $1 + \delta$. In the linear regime, where $\delta$ is much less than one, its growth with the scale factor $a$ obeys a single equation. With $N = \ln a$ and $E(a) = H(a)/H_0$ the expansion rate in units of today’s,

$$\frac{\mathrm{d}^2 D}{\mathrm{d}N^2} + \left(2 + \frac{\mathrm{d}\ln E}{\mathrm{d}N}\right)\frac{\mathrm{d}D}{\mathrm{d}N} = \frac32\,\Omega_m(a)\,D,$$

where $\delta \propto D$ and $\Omega_m(a)$ is the fraction of the density in matter at that epoch. The left side is the perturbation’s own motion with a drag term set by the expansion; the right side is the pull of its excess mass, proportional to how much of the universe is matter. A dark energy that does not itself clump enters only through $E(a)$ and $\Omega_m(a)$, which are the expansion history. Given the history, the equation has no free parameter. Every expansion history that a survey fits therefore predicts a growth history, and in relativity it predicts it exactly.

The quantity galaxy surveys measure is not $D$ itself but $f\sigma_8$: the growth rate $f = \mathrm{d}\ln D/\mathrm{d}\ln a$, which sets how fast galaxies are falling into overdense regions, multiplied by $\sigma_8$, the root-mean-square amplitude of the density fluctuations in spheres of eight megaparsecs over the Hubble constant in units of a hundred. The infall shows up as a flattening of clustering along the line of sight in redshift space, and its amplitude is proportional to the product.

The product is not a convenience; it is the only combination the data can reach. Galaxies are not the matter. They trace it with a bias $b$ — a galaxy population that lives in the densest regions is more clustered than the matter by a factor of two or three — and $b$ is not known in advance. Nick Kaiser showed in 1987 that, on large scales, the galaxies’ clustering measured in redshift space is the matter’s clustering multiplied by $(b + f\mu^2)^2$, where $\mu$ is the cosine of the angle between the pair and the line of sight. The part of the signal independent of direction measures $b\sigma_8$; the part that depends on direction measures $f\sigma_8$. Neither $f$ nor $\sigma_8$ appears alone anywhere in the data, so every survey reports their product, and every comparison between a theory and a survey is a comparison of $f\sigma_8$.

All three curves start from the same amplitude at a scale factor of 0.02, deep in the era when matter dominated and every history grows perturbations in proportion to $a$; the reference curve is normalised so that $\sigma_8 = 0.811$ today, the value the microwave background implies for a constant. From there each grows according to its own expansion and its own law.

The two relativistic predictions — a cosmological constant, and the dark energy whose equation of state crosses −1 in the fit with DES-SN5YR — lie almost on top of each other. The seven measurements give $\chi^2 = 6.4$ for the first and $6.0$ for the second: no preference at all. The third curve, with the same distances as the constant but a different law of gravity, sits nine per cent below at redshift 0.5 and gives $\chi^2 = 11.3$, a difference of about two standard deviations for a model with no extra parameters. The data neither confirm nor exclude it, but they lean against it, and they lean against it far more than they distinguish the two dark energies that the distance measurements separate at three or four standard deviations.

Why the expansion change hardly shows #

The near-coincidence of the two relativistic curves is the first thing that needs explaining, because the expansion histories are different enough to be a discovery.

The figure shows growth relative to the scale factor. In a universe containing only matter, perturbations grow in exact proportion to $a$ and the curves would stay at one for ever. They fall away because the dark energy, once it is a substantial fraction of the density, accelerates the expansion without adding anything that attracts, and the drag of the expansion on a growing perturbation outruns the pull of its own mass. By today a perturbation inside a constant’s expansion has grown to 0.784 of what it would have reached with matter alone.

Inside the evolving fit’s expansion it has grown to 0.786 — a difference of a quarter of a per cent. The fit’s dark energy is not the same as a constant at any single epoch: its density rose to a maximum near redshift 0.4 and has been falling since, so it was denser than a constant in the recent past and thinner earlier. Those two departures act on growth in opposite directions — denser dark energy suppresses growth sooner, thinner dark energy lets it continue longer — and over the interval in which most of the suppression happens they nearly cancel. The fit also has a slightly higher matter density, 0.319 against 0.315, which helps growth a little. The net is almost nothing.

That cancellation is not a coincidence of these particular numbers. An evolving dark energy that fits the same distances must reproduce, to within the errors, the same integrated expansion between the present and the microwave background, and growth is another integral over nearly the same interval. Two histories that agree on one integral tend to agree on the other. It is the same reason the acoustic scale and the supernovae are needed together to see the dark energy change at all: what changes is a redistribution of the expansion in time, not its total, and growth mostly sees the total.

The size of each difference #

The quantity that decides whether growth can answer a question is how large the difference is compared with how well growth can be measured.

The evolving dark energy, kept within relativity, changes $f\sigma_8$ by between minus two and plus two per cent at every redshift below two. It changes sign twice, because the redistribution of the expansion moves growth one way while the dark energy is denser than a constant and the other way while it is thinner. A measurement of $f\sigma_8$ that could distinguish it from a constant at three standard deviations, at its largest, would need an error of about 0.7 per cent — ten times better than the best single measurement in the figure, and several times better than any survey now running will reach in one redshift bin.

The modified-gravity growth, with the expansion held fixed, differs by 8.6 per cent at redshift 0.5 and by nearly twenty per cent at the present day. That is comparable to a single good measurement, and a survey with ten bins each measured to two or three per cent would separate it from relativity at many standard deviations.

The arithmetic of combining bins does not rescue it. Ten independent bins, each measured to two per cent, combine to about 0.6 per cent on an overall amplitude — but the evolving fit’s difference is not an overall amplitude. It is $-2$ per cent at low redshift and $+2$ per cent above redshift one, and a survey whose bins straddle the change of sign averages part of the signal away. What survives is the difference in shape, and the shape is the part of the growth that the distance measurements constrain better already.

So growth answers the two open questions about the dark energy very unequally. Whether the dark energy is changing is a question about the expansion, and growth inherits the expansion’s answer at about a fifth of the strength. Whether there is a dark energy at all — whether the acceleration is a substance with its own density, or a property of gravity on the largest scales — is a question growth answers directly and distances cannot answer at all. Growth is a weak test of the first and a strong test of the second, which is the opposite of the test the distance measurements make.

A number that belongs to the gravity #

The growth rate in relativity is very well approximated by

$$f(a) = \Omega_m(a)^{\gamma},$$

with $\gamma$ a constant called the growth index. The approximation is not exact, but it is accurate to a fraction of a per cent across a wide range of expansion histories, and the reason it is accurate is the useful part. The growth rate is driven by the matter fraction, which the expansion history sets; the exponent is what is left over, and it depends on how strongly matter responds to its own density — on the law.

The value 0.55 is not arbitrary. In the limit where matter dominates and the dark energy is a small correction, the growth equation can be solved as a series in the dark energy’s fraction, and the leading term gives $\gamma = 3(1 - w)/(5 - 6w)$ — Limin Wang and Paul Steinhardt’s result of 1998. For a cosmological constant, $w = -1$, that is $6/11 = 0.545$. As the universe becomes dominated by the dark energy the series gains higher terms, and $\gamma$ drifts up by about a hundredth.

Computed rather than assumed, $\gamma$ in relativity stays between 0.539 and 0.558 inside every one of the four histories at every redshift below two. The three evolving fits, which differ from a constant by three to four standard deviations in their distances, differ from it in $\gamma$ by less than a hundredth. The figure shows both effects. The constant’s curve approaches 0.545 in the past, where the matter fraction approaches one, and rises to 0.554 today. The evolving fits’ curves fall below it at high redshift, to 0.539, and the Wang–Steinhardt formula says why: at redshift two the fit with DES-SN5YR has $w = -1.33$, and $3(1 + 1.33)/(5 + 6 \times 1.33) = 0.539$. The growth index carries the dark energy’s equation of state as a correction in the third decimal place, which is exactly why it is useless for measuring that equation of state and well suited to measuring something else.

Eric Linder showed in 2005 that, for a dark energy with constant equation of state, $\gamma \approx 0.55 + 0.05(1 + w)$ — so a dark energy with $w = -0.8$ shifts $\gamma$ by one hundredth — and the figure shows the same insensitivity holding for histories that change.

In the modified theory marked by the dashed line, $\gamma$ is 0.68. That theory, proposed by Gia Dvali, Gregory Gabadadze and Massimo Porrati in 2000, puts ordinary matter on a four-dimensional surface in a five-dimensional space and lets gravity leak into the fifth dimension on scales larger than a crossover distance. On one branch of its solutions the leakage accelerates the expansion with no dark energy at all. Its own expansion history has since been ruled out by distance measurements, but its growth index remains the standard example of what a change of law does: a shift of 0.13 in a number that a change of substance moves by 0.01.

That is the precise form of the claim that growth separates a substance from a law. A substance, however it evolves, enters growth only through the expansion it produces, and in $\gamma$ the expansion is very nearly divided out. A law enters directly, and shows up in $\gamma$ undiluted. Measuring $\gamma$ is measuring the law of gravity on scales of tens of megaparsecs, with the expansion history almost cancelled from the answer.

What seven measurements say about the exponent #

The measurements in the first figure can be fitted for $\gamma$ directly. The amplitude is fixed at early times by the microwave background, the expansion is fixed by the distance measurements, and $\gamma$ is the one free parameter.

The seven measurements prefer $\gamma = 0.55$, the value relativity predicts, with a one-standard-deviation range from 0.49 to 0.61. They place 0.68 at about $\Delta\chi^2 = 5$, a little over two standard deviations away. And the choice of expansion history — a constant or the evolving fit — moves the preferred value by one hundredth, less than a sixth of the width of either curve. That is the previous figure measured rather than computed: whatever the dark energy is doing to the expansion, the growth data return the same exponent.

The width is the present limitation. Seven numbers with errors between eight and thirty per cent locate $\gamma$ to about ±0.06. The surveys now running measure $f\sigma_8$ in more redshift bins, with errors of a few per cent each, and published forecasts put their combined precision on $\gamma$ at a few hundredths — enough to exclude the self-accelerating theory at many standard deviations if relativity holds, or to detect a departure from 0.55 of the size modified theories typically produce.

The fit hides one important choice. The amplitude was fixed from the microwave background, so a lower measured $f\sigma_8$ is read as slower growth and pushes $\gamma$ upwards. A survey that measured clustering alone, without the microwave background’s amplitude, would find $\gamma$ and $\sigma_8$ degenerate, because the data measure their combination. The same degeneracy is behind the tension between weak-lensing surveys and the microwave background in $S_8$: if structure today is a few per cent less clumpy than the background predicts, that could be slower growth, a smaller early amplitude, or astrophysical feedback that redistributes matter on small scales. A modified law of gravity with $\gamma$ above 0.55 would lower $S_8$ in the direction of the tension, which is why the tension is examined as a possible sign of new gravity — and why it cannot be read as one until the feedback is measured.

Growth that is not about gravity #

Two other effects lower growth without any change of law, and either could be mistaken for one.

Massive neutrinos do not fall into structures smaller than the distance they travel freely, so a sum of neutrino masses of a tenth of an electronvolt suppresses growth on small scales by a few per cent. The suppression depends on scale, which a change of $\gamma$ does not, and that is how the two are separated in principle; in practice a measurement of $f\sigma_8$ at a single effective scale sees both as one number.

And a dark energy that clumps — a field with a sound speed much less than the speed of light, rather than one that stays smooth — enters the growth equation directly and changes $\gamma$. A thawing field of the kind that fits on the thawing line has a sound speed equal to the speed of light and does not clump on the scales measured here, so for it the relativistic calculation above holds. The statement “growth separates a substance from a law” is exact only for substances that do not cluster. A clustering dark energy is a substance that behaves partly like a law, and growth alone would not tell the two apart.

What the calculation leaves out #

The growth here is linear. Real measurements of $f\sigma_8$ come from scales where the density contrast is not small, and the models fitted to them include non-linear corrections and a bias relating galaxies to matter; the published values carry those models with them. The three predictions share one early amplitude and ignore the small difference in the transfer function that a different matter density produces, which would shift the evolving fit’s curve by a fraction of a per cent. The modified-gravity growth uses the $\gamma$ parameterisation with a constant exponent rather than solving any theory’s equations, and real modified theories usually have scale-dependent growth and an expansion of their own. The seven measurements are drawn with symmetric errors and treated as independent, which the overlapping BOSS bins are not quite.

None of these changes the sizes that matter: a change of expansion within relativity moves growth by one or two per cent, and a change of law moves it by several times that.

Still open: whether the law holds where the acceleration is #

The present growth measurements agree with relativity at the level of about ten per cent in $\gamma$, and the expansion history they are interpreted inside barely matters to that statement. The next generation of spectroscopic surveys will measure $f\sigma_8$ to a few per cent in a dozen bins; weak-lensing surveys will measure growth through the lensing of shapes rather than through velocities; and the Doppler shift that moving clusters imprint on the microwave background measures the same infall through a third channel, proportional to $f\sigma_8^2$. Together they can test the law of gravity on scales of hundreds of megaparsecs to a few per cent — on exactly the scales where the acceleration happens, and where gravity has never been tested independently of the cosmology that assumes it. If $\gamma$ stays at 0.55, the acceleration is a substance with a density and perhaps a history, and the question returns to the distances. If $\gamma$ moves, the substance was never there, and every figure about its equation of state was a measurement of gravity misread as a measurement of matter.