A ratio in which the galaxies cancel

The acceleration of the expansion can be explained in two ways that have nothing in common. Either the universe contains a substance with negative pressure — a cosmological constant, or a field slowly changing — and gravity is the gravity of relativity; or there is no such substance, and gravity itself behaves differently on the scales of hundreds of megaparsecs. Both can reproduce every distance measurement there is, because a distance measures only the expansion history and both are free to produce any expansion history they like. The supernova distances that first showed the acceleration are equally well fitted by a constant, by a field thawing along a single line, or by a gravity that weakens on large scales.

What separates them is the growth of structure. Relativity leaves no freedom in how fast structure grows inside a given expansion, and the redshift-space distortions measured by galaxy surveys find growth at the rate relativity predicts, to about ten per cent in the growth index. That test has a blind side. It measures how matter falls, and matter feels only one of the two potentials a perturbed universe has. A gravity that changed the other one would leave every growth measurement untouched.

The other potential is felt by light. Combining a measurement of how galaxies fall with a measurement of how light bends around the same galaxies gives a ratio called $E_G$, and it has two properties that make it unusual among cosmological statistics. The quantities nobody knows well — how strongly the galaxies trace the matter, and how large the fluctuations are — cancel out of it exactly. And what remains is a prediction of relativity that depends on a single, well-measured number.

Two potentials, and which one each messenger feels #

A small disturbance in an expanding universe is described by two potentials. One, Ψ, is a perturbation of the rate at which clocks tick; the other, Φ, is a perturbation of the lengths of rulers. A slow-moving body — a galaxy, a cloud of dark matter — responds to Ψ alone, because for something moving much slower than light the distortion of time dominates its motion. That is the Newtonian potential, and it is what galaxies fall in.

Light responds to both equally, because for something moving at the speed of light space and time enter on the same footing. A ray passing a mass is bent by Φ + Ψ.

In relativity, for ordinary matter with no anisotropic stress, Φ and Ψ are equal. So light is bent by twice the Newtonian potential, and it is deflected twice as much as a Newtonian calculation of a particle moving at the speed of light would give. That factor of two is the one measured during the 1919 eclipse — 1.75 arcseconds at the Sun’s limb against the Newtonian 0.87 — and measured since, by radio timing past the Sun, to agree with relativity to a few parts in a hundred thousand.

The cosmological question is whether the same factor holds on scales a billion times larger, around the matter that forms galaxies and clusters, while the expansion is accelerating. Many of the alternatives to a cosmological constant answer no: they add a field that couples to matter differently from the way it couples to light, so that Φ and Ψ come apart. Their ratio, Φ/Ψ, is called the gravitational slip, and $E_G$ is built to measure it.

Three measurements with the galaxies in them #

A galaxy survey sees galaxies, not matter, and galaxies trace the matter with a bias b — more strongly clustered than the matter, by a factor of one to three depending on the galaxies chosen. Every measurement of the matter made through galaxies carries some power of b, and b is not known well.

The clustering of galaxies with each other measures $b^2\sigma_8^2$, the bias squared times the amplitude of the matter fluctuations squared. The lensing of background galaxies by the matter around the foreground ones — galaxy–galaxy lensing — measures the correlation between the foreground galaxies and the mass that bends light, $b\sigma_8^2$, and through the light it measures Φ + Ψ. The redshift-space distortions, the squashing of the clustering pattern by the galaxies’ infall, measure the ratio β = f/b, where f is the logarithmic growth rate.

Divide the lensing by β times the clustering. The b’s cancel: one in the numerator, two in the denominator, one more in the numerator from β. The $\sigma_8$’s cancel: two in each. What is left is

$$E_G = \frac{\Omega_{m0}\,\Sigma}{f(z)},$$

where Σ measures how strongly the matter bends light relative to relativity’s prediction, and f measures how fast it falls. Zhang, Liguori, Bean and Dodelson proposed the combination in 2007. It is the bending of light divided by the speed of infall, with the two unknowns that limit every other measurement removed by the way it is put together.

Why the amplitude was the obstacle #

The cancellation of $\sigma_8$ matters as much as the cancellation of bias, and for a reason specific to the present state of cosmology. Each of the ingredients on its own measures a product. The redshift-space distortions do not measure f but $f\sigma_8$, because the squashing of the clustering pattern is a velocity relative to a density contrast whose normalisation is the amplitude. Cosmic shear measures the amplitude and the density as one combination, $S_8$, and that combination is where the lensing surveys and the microwave background disagree by a few per cent.

A test of gravity built from either alone has to be compared with a prediction that assumes an amplitude, and the amplitude is precisely the number in dispute. If the lensing surveys are right that the fluctuations are a little weaker than the microwave background predicts, a growth measurement will come out low and look like a weaker pull on matter, and a lensing measurement will come out low and look like weaker bending — whether or not gravity has changed. $E_G$ is immune. A universe with smaller fluctuations has proportionally less lensing and proportionally less clustering, and the ratio is unchanged. It separates the question of how gravity works from the question of how lumpy the universe is, which every other probe of growth mixes.

Relativity’s prediction from one density #

In relativity Σ is one and f is close to $\Omega_m(z)^{0.55}$, so $E_G = \Omega_{m0}/f(z)$ follows from the matter density today and the growth rate at the redshift measured — and the growth rate follows from the same density.

The curve falls with redshift, from 0.60 today to 0.36 at a redshift of one, because the growth rate rises towards one as matter comes to dominate the expansion: in a universe of matter alone, f would be exactly one and $E_G$ would be the matter density itself. The fall is the dark energy’s influence on growth, and the curve is fixed once the density is.

The dashed curves show the leverage of the other coupling. A gravity that pulled matter ten per cent harder today, growing that extra pull with the dark energy’s share of the density, would make galaxies fall faster and lower $E_G$ by a few per cent at moderate redshift. A gravity that pulled ten per cent more weakly would raise it. The first measurement, made in 2010 by Reyes and collaborators from the luminous red galaxies of the Sloan survey, found 0.39 ± 0.06 at a redshift of 0.32 — consistent with relativity to within its errors, and inconsistent with some alternatives that had been proposed to explain galaxy rotation without dark matter.

Two couplings, and which measurement fixes which #

A gravity modified on large scales can change two things, and the language that has settled is to call them μ, the strength of the pull on matter relative to relativity, and Σ, the strength of the bending of light. Both are one in relativity. They are related to the slip: Σ = μ(1 + η)/2, where η = Φ/Ψ.

The growth measurements fix μ. With the amplitude of the fluctuations set at early times by the microwave background, how much structure has grown since — and how fast it is growing now — depends on how hard matter is pulled, and the band in the figure is what a five per cent measurement of $f\sigma_8$ at a redshift of one half allows. It is vertical. It constrains the pull on matter and is perfectly blind to the bending of light.

The $E_G$ lines run almost across it. A ten per cent measurement of $E_G$ picks out a horizontal strip, and the two strips together close a box around a single region of the plane. Relativity is the point (1, 1). The geometry is what makes the combination worth the effort: a growth measurement alone could find relativity’s value of μ while Σ was anything, and $E_G$ alone mixes the two, but they constrain directions nearly at right angles, and between them they test both of the things a modified gravity can change.

A slip that growth cannot see #

The case growth measurements are blind to is the one where matter feels exactly relativity’s gravity and light does not.

In each of these three universes, every redshift-space distortion measurement gives the same answer, every infall velocity is the same, every cluster count is the same. What differs is the bending of light, and therefore the lensing, by (η − 1)/2 relative to relativity. $E_G$ moves by that fraction: ten per cent today for a slip of twenty per cent, and a third of that at a redshift of one, where the dark energy’s share is smaller and the slip, in these models, smaller with it.

That is the sense in which $E_G$ is the cosmological analogue of the Eddington factor. The eclipse expedition measured the ratio of light’s bending to the Newtonian pull on the scale of the Sun; $E_G$ measures the same ratio on the scale of the galaxy distribution, normalised so that the galaxies themselves drop out. A universe in which light bent around galaxies by 1.9 times the Newtonian potential, rather than 2, would show it here and nowhere in the growth data at all.

The one number the ratio still needs #

The cancellation removes the bias and the amplitude. It does not remove the matter density, which multiplies everything.

The dependence is softer than proportional, because a lower density also lowers the growth rate, which partly compensates; but it is still strong, and stronger at higher redshift, where the growth rate is close to one for any reasonable density and the cancellation weakens. At a redshift of one half, the difference between a matter density of 0.29 and 0.3153 is a six per cent change in $E_G$.

That is a real hazard rather than a formal one, because the matter density is not as settled as its quoted precision suggests. The microwave background gives 0.315 with an uncertainty under a hundredth; some combinations of supernova distances and the baryon acoustic scale prefer values a little higher, and some lensing analyses, which measure a combination of the density and the amplitude, have pulled lower. A test of gravity through $E_G$ is a comparison of a measured ratio against a predicted one, and the prediction is only as good as the density fed into it. Within the standard model the density is constrained well enough that the prediction is good to two per cent; a test that wants to be independent of the standard model has to measure the density independently too.

A force with a range #

Some modifications of gravity act only within a range. A scalar field with a mass adds a force that falls off exponentially beyond its Compton wavelength, and theories of the chameleon type arrange for that range to be short near dense matter — hiding the force in the Solar System, where relativity is tested to five decimal places — and long in the empty spaces between galaxies. Such a force adds up to a third to the pull on matter inside its range and leaves the bending of light alone.

In relativity $E_G$ is the same on every scale: the ratio of bending to infall does not care how large the structure is. A force with a range makes it depend on scale, lower inside the range and equal to relativity’s far outside it. The drop in the figure is five per cent — modest, because the extra force grows only with the dark energy and has been acting for only the last few billion years — but its signature is a step in scale rather than a shift in level, and a measurement of $E_G$ in several bins of separation looks for exactly that.

The same step is what makes the separation from other effects possible. Massive neutrinos also make growth depend on scale, suppressing it on small scales rather than enhancing it, and they do so without changing the bending of light either; a scale-dependent $E_G$ distinguishes the two by the sign of the step.

What the measurements have found #

Since the first measurement in 2010, $E_G$ has been measured perhaps a dozen times, at redshifts from 0.2 to about 1, with the lensing supplied either by the shapes of background galaxies or by the lensing of the microwave background itself. The second option uses the most distant source there is, which makes it possible to measure lensing around galaxies at redshifts where background galaxies are too few.

The measurements have mostly agreed with relativity at the ten to twenty per cent level. One, in 2016, used the lensing of the microwave background around galaxies at a redshift of about 0.57 and came out low by more than two standard deviations; later analyses of the same galaxies, treating differently the systematic effects in each of the three ingredients, did not confirm the deficit. The episode is a fair picture of the method’s real limit. Its statistical errors are set by the number of galaxies and are falling quickly; the one-per-cent distortions of background shapes that the lensing ingredient is built from are now measured for hundreds of millions of galaxies. Its systematic errors live in the details of each ingredient: the modelling of the redshift-space distortions on non-linear scales, the photometric redshifts of the background sources, the calibration of shape measurements, the magnification of the foreground sample by matter in front of it. The cancellation of bias assumes that the same galaxies, weighted the same way, enter all three, and in practice they come from different surveys with different selections.

How precise it has to become #

The size of the effect to look for sets the precision needed.

A slip of ten per cent between the two potentials changes $E_G$ by five per cent. The first measurements, at fifteen per cent, would see it at a third of a standard deviation. Three standard deviations need about 1.7 per cent in a single measurement, or six per cent in each of a dozen independent redshift bins. That is within the stated aims of the imaging and spectroscopic surveys now overlapping on the same sky — one measuring the shapes of a billion galaxies, the other the redshifts of tens of millions. It is also the level at which every systematic listed above has to be controlled to a per cent.

What the ratio does not cancel #

The bias is assumed to be linear and the same in all three measurements. On large scales that is a good approximation; on the scales where the lensing signal is strongest it is not, and the cancellation is then approximate. Measurements are usually restricted to separations of more than about ten megaparsecs for that reason.

Lensing measures a projection. Galaxy–galaxy lensing is an integral along the line of sight, and the clustering it is divided by has to be projected the same way. The redshift-space distortion parameter, by contrast, is a three-dimensional measurement. Matching them needs care in how the radial information is discarded.

The figures parameterise rather than derive. The modifications drawn turn on with the dark energy’s share of the density, as is conventional, and the scalar force has a fixed range. Real theories have their own time dependence, and the chameleon screening that hides a force near dense matter would weaken it inside massive haloes, where much of the lensing signal comes from.

Baryons move the small-scale lensing. Gas pushed out of haloes by feedback lowers the matter power on small scales and with it the lensing around galaxies, while leaving the large-scale clustering and infall nearly untouched. Restricting the measurement to large separations protects against it, at the cost of statistical power.

A slip has other sources. Free-streaming neutrinos produce a tiny anisotropic stress and a correspondingly tiny slip, about a thousandth, far below any measurement. A slip detected at the per cent level would be a new physical effect.

Still open: whether light and matter feel the same gravity between the galaxies #

Growth has tested the pull on matter at the ten per cent level across redshifts from zero to almost two, and found relativity’s value. $E_G$ has tested the bending of light relative to that pull at the fifteen per cent level, mostly at one or two redshifts, and found relativity’s value there too. Neither yet reaches the few per cent at which most proposed alternatives would differ.

A third route to the infall is already being added: the Doppler shift that moving clusters impress on the microwave background measures the same velocities as the redshift-space distortions with none of their modelling of galaxy positions. The overlap of the large imaging and spectroscopic surveys means that, within this decade, the same galaxies will be measured all three ways at many redshifts. If $E_G$ stays at $\Omega_{m0}/f$ at the level of a couple of per cent, the factor of two Eddington measured at the Sun’s limb holds on scales of a hundred megaparsecs, and whatever is accelerating the expansion does it without changing how light and matter feel each other’s gravity. If $E_G$ moves while growth does not, the acceleration is a property of gravity — and the first evidence will be that light, crossing the space between galaxies, is bent by an amount the galaxies’ own motions do not account for.