Astronomy & Celestial Mechanics
the sky, computed
Galaxies

A measurement of dark matter that needs a model of galaxies

Weak lensing weighs all the matter, dark and ordinary, by how it bends light. But the ordinary matter does not sit where gravity alone would put it — galaxies' supernovae and black holes blow a large part of it out of their haloes — and the scales where that happens are the scales that carry most of a lensing survey's statistical power. Ignored, it reads as a less clumpy universe, in the direction of the disagreement lensing has with the microwave background.

Assumes Weak lensing, Large-scale structure and Density parameters.

15 min read 6 figures Measuring the unreachableLight is the only messenger

Weak gravitational lensing has one property that no other measurement of the universe’s structure shares. It responds to mass, all of it, without asking what the mass is made of. A galaxy survey counts galaxies and has to model how galaxies trace matter; an X-ray survey sees hot gas and has to model the gas; the distortion of background galaxies’ shapes by the matter in front of them is a direct integral of that matter’s density. That is why a lensing survey is described as weighing the dark matter, and why the one number it measures well, a combination of the matter density and its clumpiness called S8S_8, is taken as a clean test of the standard cosmological model.

The property has a condition hidden in it. Lensing weighs the matter as it actually is. The prediction it is compared with is computed for matter that moves under gravity alone — simulations of dark matter, with the ordinary matter treated as though it followed the dark. On large scales that is right, because on large scales nothing but gravity has had time to act. On the scales of individual haloes it is wrong, because ordinary matter is heated, pushed and expelled by the galaxies forming inside those haloes, and it takes some of the gravitating mass with it.

The matter power spectrum with galaxies' feedback taken out. The ratio of the matter power spectrum with baryonic feedback to the spectrum without it, against wavenumber in h per megaparsec on a logarithmic scale, for three strengths of feedback that span the range the large hydrodynamic simulations disagree over. Gas blown out of haloes by supernovae and active nuclei removes mass from scales below a few megaparsecs, so the ratio falls below one from about 0.5 h per megaparsec and is deepest near 4 h per megaparsec, at 9, 17 and 26 per cent; at still smaller scales gas that has cooled into galaxy centres adds mass back. The marks show the wavenumber that dominates a lensing measurement at multipoles 100, 1000, 5000, taken where the lensing kernel for sources at redshift 1 peaks: 0.09, 0.87, 4.35 h per megaparsec. The shape is a parametric stand-in placed where simulations put the effect, not a simulation.
Fig. 1 The matter power spectrum with feedback from galaxies, divided by the spectrum gravity alone would make, for three strengths of feedback that span the range the large simulations disagree over. The suppression starts near half an inverse megaparsec, is deepest at a few, where it removes 9 to 26 per cent of the power, and recovers at small scales where gas that has cooled into galaxy centres adds mass back. The marks show the wavenumber a lensing survey mainly sees at three angular scales.

The shape is not a detail of any one simulation. Every hydrodynamic simulation that forms realistic galaxies needs feedback — energy from supernovae and from accreting black holes returned to the gas — to stop the galaxies from turning all their gas into stars. That energy is large enough to push gas out of the haloes of galaxy groups entirely, and the matter that leaves takes its gravity with it, so the density of the halo falls and the power spectrum falls on the halo’s scale. Where the simulations disagree is how much. The strength parameter ε\varepsilon in the figure is a stand-in for that disagreement: the shape is placed where all of them put the effect, and the depth is the thing that is not known.

Why the dip sits where it does

The location of the suppression follows from where the expelled gas comes from. The universe’s matter is about one-sixth ordinary, and a halo formed by gravity alone would hold ordinary matter in that proportion. Galaxy clusters, the most massive haloes, are deep enough to hold on to nearly all of theirs: the hot gas measured in X-rays brings them close to the cosmic fraction, with the baryons that are not in the galaxies making up most of it. Galaxy groups, ten to a hundred times less massive, are not. Their measured gas fractions are well below the cosmic value, which means that feedback has pushed a large part of their ordinary matter out, and with it several per cent of their total mass.

Groups are also where most of the power on scales of one to ten inverse megaparsecs resides. In the halo picture of large-scale structure, the power at a given wavenumber is contributed mostly by haloes whose size is comparable to its wavelength, and haloes of 101310^{13} to 101410^{14} solar masses dominate the range the lensing survey measures most precisely. A halo that has lost a tenth of its gas has lost about one and a half per cent of its mass, and a redistribution of the rest outward lowers its central density further; power, as a square of the density contrast, loses twice the fractional change. That is how a modest mass loss per halo becomes a suppression of ten to twenty per cent in the spectrum.

At still smaller scales the effect reverses. The gas that stays behind cools and condenses into stars at the halo’s centre, more concentrated than the dark matter would be, and at wavenumbers of tens of inverse megaparsecs this adds power back. That recovery sits beyond the scales current surveys use, which is why the dip, not the recovery, is what matters to them.

Which scales a survey is looking at

An angular multipole ℓ\ell on the sky corresponds, for matter at comoving distance χ\chi, to a wavenumber k=ℓ/χk = \ell/\chi. A lensing measurement at one multipole sums over all the matter between the observer and the source galaxies, weighted by how efficiently each distance lenses, so each multipole sees a spread of wavenumbers.

Which scales of matter a lensing multipole measures. How the convergence power at three angular multipoles, 200, 1000, 3000, is built up from the matter power at different wavenumbers, for sources at redshift 1, with each curve normalised to unit total and plotted per shell of lensing matter. A multipole ℓ sees wavenumber ℓ divided by the distance to the lensing matter, so each curve is spread over a factor of several in k by the spread of lens distances. Half of the signal comes from wavenumbers above 0.19, 1.06, 2.86 h per megaparsec respectively. The two higher multipoles, where a survey's statistical power is concentrated, draw most of their signal from the scales where feedback removes matter.
Fig. 2 How the lensing signal at three multipoles is built up from the matter at different wavenumbers, for sources at redshift one. The multipole of 200 draws on scales around a tenth to half an inverse megaparsec, safely linear. The multipoles of 1000 and 3000 draw half their signal from above 1 and 3 inverse megaparsecs, inside the shaded region where feedback acts.

The figure explains why the problem cannot be avoided by looking elsewhere. The statistical power of a lensing survey grows with the number of independent modes it measures, and the number of modes on a sphere up to a multipole ℓ\ell grows as ℓ2\ell^2. A survey measuring to ℓ=3000\ell = 3000 has a hundred times as many modes as one stopping at 300. Nearly all of that extra information lives at wavenumbers of one to ten inverse megaparsecs — which is where gas physics moves the matter.

A tilt that looks like a smaller amplitude

Projected onto the shear power spectrum, the suppression is smeared. Each multipole averages the matter power over a range of scales, so the sharp-edged dip in the matter spectrum becomes a gentle decline across the lensing one.

Cosmic shear power with feedback, over power without. The ratio of the convergence power spectrum with baryonic feedback to the spectrum without it, against angular multipole from 20 to 5000, for the three feedback strengths, with sources at redshift 1. Below a multipole of about a hundred the ratio is within a per cent of one. At ℓ = 1000 it is 0.948, 0.895, 0.843; at ℓ = 3000, 0.920, 0.839, 0.759. The suppression is smeared over a wide range of multipoles, because each multipole sums the matter power over a range of lens distances, so there is no sharp feature in the shear spectrum to mark where feedback starts. It enters as a gentle tilt, which is what makes it easy to mistake for a smaller amplitude.
Fig. 3 The shear power with feedback over the power without it, against multipole, for the same three strengths. Below a multipole of a hundred the difference is under a per cent. At 1000 the power is lower by 5 to 16 per cent; at 3000, by 8 to 24 per cent. There is no feature marking where the suppression begins, only a progressive loss of power toward small scales.

The absence of a feature is the problem. A lower amplitude of matter fluctuations — a smaller σ8\sigma_8 — also lowers the shear power, and lowers it more on small scales than large, because nonlinear growth amplifies small-scale power more steeply than linearly with the amplitude. So a model without feedback, fitted to a sky with feedback, finds a good fit at a lower S8S_8. The data do not complain; the parameter moves.

The clumpiness a survey infers when feedback is ignored. The fractional shift in the inferred amplitude of matter fluctuations, S₈, in per cent, when shear power with baryonic feedback is fitted by a model without it, using all multipoles from 20 up to a maximum ℓ, against that maximum; the fit is weighted by the statistical errors of a survey covering a tenth of the sky with ten galaxies per square arcminute. Every shift is downward, because suppressed power looks like less clumpy matter. With all multipoles to 3000 the three feedback strengths shift S₈ by −1.7, −3.5, −5.2 per cent; with multipoles only to 500, by −0.6, −1.3, −1.9. The disagreement between lensing surveys and the microwave background is about eight per cent in S₈, in the same direction.
Fig. 4 The shift in the inferred S8S_8 when the shear power with feedback is fitted by a model without it, using all multipoles up to a maximum, weighted by the errors of a survey over a tenth of the sky. Every shift is downward. Using multipoles to 3000 the three strengths lower S8S_8 by 1.7, 3.5 and 5.2 per cent; stopping at 500 they lower it by 0.6, 1.3 and 1.9. The lensing surveys’ disagreement with the microwave background is about eight per cent, in the same direction.

That direction is the reason this has become one of the most argued-over corrections in cosmology. The lensing surveys of the last decade have found S8S_8 lower than the value the microwave background predicts under the standard model, by about eight per cent, at a significance of two to three standard deviations depending on the survey and the analysis — a smaller cousin of the disagreement over the expansion rate, with the same two epochs on either side of it. Feedback lowers the inferred S8S_8. The surveys do model it, and their published values carry its uncertainty. What has been argued since is whether the feedback in nature is stronger than the models allowed — and an analysis that let the suppression be as strong as the data wanted found that it could remove most of the disagreement, if the real suppression were at the strong end of the simulated range or beyond it.

The size of the effect was itself a discovery, and a recent one. Before about 2011 the working assumption was that ordinary matter followed dark matter closely enough on the relevant scales for the difference to be a per cent or two, well inside other errors. Then simulations that included feedback from active galactic nuclei — added because without it the simulated groups had too many stars and too bright an X-ray glow — showed the matter power suppressed by twenty per cent or more at a few inverse megaparsecs, far larger than anything the gravity-only picture allowed. The feedback had been put in to fix the galaxies; its side effect on the matter distribution was an order of magnitude larger than anyone had budgeted for in lensing forecasts.

What makes that history instructive is the direction the evidence came from. Nothing in the lensing data had flagged a problem. The shear measurements at the time were too noisy for a twenty per cent change at small scales to be visible as a failure of the fit, so the correction would have been absorbed silently into the amplitude. It was found only because a different part of astrophysics, trying to get galaxy groups right, needed an ingredient that turned out to move mass.

The scale cut

A survey that does not trust its model of galaxy physics can refuse to use the scales where galaxy physics acts. That is the scale cut, and every lensing analysis makes one. The figure below is the arithmetic of choosing it.

The scale at which the model of galaxies becomes the error. For feedback of strength ε = 0.2, the statistical uncertainty on S₈ from shear multipoles up to a maximum ℓ — with every other parameter held fixed, the idealised limit; marginalising over the rest multiplies it by several — and the size of the bias from ignoring the feedback, both in per cent, against that maximum. Adding smaller scales shrinks the uncertainty — from 1.2 per cent at ℓ = 200 to 0.2 per cent at 3000 — and grows the bias, from 0.41 to 3.5 per cent. The two cross at ℓ ≈ 317; beyond it, the survey's answer is more wrong than it is uncertain. A survey either stops there, discarding the multipoles that carry most of its statistical power, or it models the feedback and pays for the model's freedom in precision.
Fig. 5 For the middle feedback strength, the statistical uncertainty on S8S_8 from multipoles up to a maximum and the bias from ignoring the feedback, both in per cent. The uncertainty falls from 1.2 to 0.2 per cent as the maximum rises from 200 to 3000, and the bias rises from 0.4 to 3.5 per cent. They cross near a multipole of 320. The uncertainty drawn is the idealised one with every other parameter fixed; a real analysis, marginalising over the others, is several times less precise and can go further before the bias dominates.

The crossing is the point past which a survey’s answer is more wrong than it is uncertain. Beyond it, adding data makes the result worse. The idealised uncertainty drawn here puts the crossing uncomfortably low; the realistic uncertainty, several times larger, moves it to a multipole of about a thousand, and published analyses cut there or model the feedback with free parameters out to a few thousand. Both choices cost precision. The first discards most of the survey’s modes. The second spends them on constraining the feedback rather than the cosmology — a lensing survey that fits the suppression with free parameters becomes, in part, a measurement of how much gas galaxies expel.

So the premise of weak lensing as a clean weighing of dark matter holds on large scales and fails at exactly the scales that would make it most precise. The survey’s precision is purchased with a model of galaxies, and the price is paid either in scales not used or in parameters not known.

A deeper survey is more exposed, not less

A natural hope is that better data will settle the question by themselves, and for this problem better data make it harder. The bias from an unmodelled suppression does not shrink with survey size; it is a property of the sky. The statistical uncertainty does shrink, and the surveys now running will measure shapes over a third of the sky with two to three times the source density of their predecessors, reducing the uncertainty on S8S_8 by a factor of several. The crossing in the figure above therefore moves to lower multipoles, not higher.

The scale at which the model of galaxies becomes the error. For feedback of strength ε = 0.3, the statistical uncertainty on S₈ from shear multipoles up to a maximum ℓ — with every other parameter held fixed, the idealised limit; marginalising over the rest multiplies it by several — and the size of the bias from ignoring the feedback, both in per cent, against that maximum. Adding smaller scales shrinks the uncertainty — from 1.2 per cent at ℓ = 200 to 0.2 per cent at 3000 — and grows the bias, from 0.61 to 5.2 per cent. The two cross at ℓ ≈ 276; beyond it, the survey's answer is more wrong than it is uncertain. A survey either stops there, discarding the multipoles that carry most of its statistical power, or it models the feedback and pays for the model's freedom in precision.
Fig. 6 The same trade for the strongest feedback of the three. The bias reaches the idealised statistical uncertainty at a lower multipole than for the middle strength and grows to more than five per cent by a multipole of 3000. A survey that did not model feedback at this strength would have to stop before the scales where most of its information is.

The implication is that the next surveys are designed around the feedback model. Their analyses will either fit the suppression with flexible parameters calibrated on many simulations, or use external data on the gas to fix it, or both. A lensing measurement of cosmology at a precision of one per cent is, in practice, a lensing measurement of galaxy formation at a precision of ten per cent, whether that is its intention or not.

Measuring the missing gas

The suppression is a statement about where the ordinary matter went, and that can be measured another way. Gas pushed out of haloes does not vanish; it spreads into a diffuse, hot, ionised medium around and between them. Half of the universe’s ordinary matter was long unaccounted for, and fast radio bursts, whose dispersion measures count every free electron along their paths, have found it in exactly that diffuse form.

The hot gas also scatters the microwave background. The thermal Sunyaev–Zel’dovich effect measures its pressure, and the kinematic effect, which depends on the gas’s bulk motion, measures its density profile around galaxies directly. Stacking the kinematic signal around hundreds of thousands of galaxies has shown the gas extended well beyond the haloes’ virial radii — more extended than several of the simulations that set the middle of the range drawn here, and in the direction that implies stronger suppression. X-ray measurements of the gas fractions of galaxy groups point the same way.

So three measurements that know nothing about lensing — radio-burst dispersion, the scattered background and X-ray gas fractions — bear on a correction that enters a lensing survey’s central number. The practical route now is to fit them together, constraining the feedback with the gas and the cosmology with the shear. What began as a nuisance parameter for lensing has become a joint measurement of galaxy formation and structure growth.

What the figures simplify

The matter power spectrum is computed from a fitting formula mapped from a linear spectrum, not from a simulation, and it carries the formula’s errors of five to ten per cent on the smallest scales. Those errors matter for absolute predictions and hardly at all for the ratios drawn here, which compare the same spectrum with and without a multiplicative suppression.

The suppression itself is a single parametric shape, with its location fixed and only its depth varied. Real feedback moves the location as well, and it depends on redshift, since the gas is expelled progressively as galaxies form. A survey whose sources span a range of redshifts sees a mixture. The single source plane at redshift one is the simplest geometry that shows the effect.

The bias is computed for a fit of the amplitude alone. In a real analysis the matter density, the spectral index, the intrinsic alignment of galaxies — the shapes galaxies were given before they were lensed — and the calibration of photometric redshifts are fitted at the same time, and some of the suppression is absorbed by them rather than by S8S_8. The size of the shift is therefore an illustration of its sign and its scale dependence, not a prediction for any particular survey.

What the figures cannot show at all is the thing the whole argument turns on: how strong feedback is in the real universe. The three curves are the range that simulations with different recipes produce, and nature is not obliged to fall inside it. The observation that would settle it is the gas itself, measured around haloes of every mass, and it is being made now.

What this does to the word clean

Lensing was supposed to escape the uncertainty of galaxy physics by not relying on galaxies, and on large scales it does. The lesson is that no measurement of matter on small scales can escape it, because on small scales the ordinary matter is not a passive tracer. It is pushed, and it pushes the dark matter’s gravitational potential a little in return, and the separation of the universe into a dark component governed by gravity and a luminous one governed by astrophysics does not hold where the two share a halo.

That is a more general statement than it looks. A cluster weighed three ways — by its galaxies’ motions, its gas’s temperature and its lensing — gives three answers whose disagreements are the physics of the gas. A galaxy survey’s small scales encode how galaxies occupy haloes. The lensing survey’s small scales encode where the gas went. In every case the most precise part of the data is the part most entangled with the physics a cosmologist would rather not model.

There is a second physical effect that removes small-scale power, and it makes the entanglement worse. Neutrinos have mass, and a massive neutrino streams freely out of any structure smaller than the distance it travels while it is relativistic, so it does not cluster on small scales and slows the growth of the matter that does. The result is a suppression of the matter power spectrum on scales below about a tenth of an inverse megaparsec, by roughly eight times the fraction of the matter in neutrinos — a few per cent for the smallest masses allowed. Lensing surveys are expected to measure the sum of the neutrino masses this way. But a smooth loss of small-scale power is also what feedback does, and at the precision required, the two must be separated by their different shapes: the neutrino suppression is a step that starts at larger scales and flattens, the feedback a dip that starts smaller and recovers. The same galaxy physics that stands between lensing and S8S_8 stands between lensing and the neutrino mass.

Still open: whether the feedback is strong enough

The disagreement between lensing and the microwave background is a few per cent in S8S_8, and it has three candidate explanations that are not exclusive: a statistical fluctuation, a systematic error in the surveys’ measurements of shapes or redshifts, or new physics that slows the growth of structure. Strong feedback is a fourth, and the only one that is both known to exist and known to push in the right direction. Whether it is strong enough depends on how much gas the haloes of galaxy groups have lost, and the kinematic scattering measurements and radio-burst dispersions now accumulating are the first data that measure that on the relevant scales. If they settle on the strong end, the disagreement is largely a statement about galaxies. If they settle in the middle, it is still a statement about cosmology.

Where this goes next

Two bands 2.1 standard deviations apart, and neither is a point. The plane of the matter density against the amplitude of matter fluctuations, with the constraints from a weak-lensing survey and from the microwave background drawn as bands. Lensing measures the shear produced by structure along the line of sight, and that shear depends on how much matter there is and on how clumpy it is in a fixed combination — more matter arranged less clumpily gives the same signal. The locus is a power law of exponent one half, and the combination it fixes is written S₈. The two bands are separated by 2.1 standard deviations, and whatever that separation is, it is a statement about the growth of structure between recombination and now rather than about either measurement's precision. Neither band alone determines either quantity, which is why the disagreement is quoted in the combination rather than in the parameters. Two parameters that lensing measures as one A weak-lensing survey measures how much the shapes of distant galaxies are distorted by the matter in front of them, and that distortion depends on how much matter there is and on how clumpily it is arranged. The two enter as a product. What the survey determines is one number, and the disagreement between surveys and the microwave background is stated in that number because neither measures either quantity alone. A coherent one-per-cent distortion, invisible on every galaxy in the picture. 150 background galaxies behind a lens of Einstein radius 14″, each drawn at its own ellipticity: an intrinsic shape with a dispersion of 0.3 per component, plus the reduced shear the lens adds. The strongest shear on any galaxy here is 0.035, one part in 8 of the intrinsic scatter, so no object in this field is measurably distorted and the tangential alignment cannot be seen by eye at all. Averaged over these 150, the mean tangential ellipticity is 0.0088 ± 0.0245 against the 0.0130 the lens model predicts — consistent with the lens and equally consistent with nothing, because 150 galaxies buy a precision of 0.024 and the signal is 0.013. Detecting it at five sigma takes about 13,275 of them, which is not a picture anybody can draw. The cross component — every shape rotated by 45°, which gravitational lensing cannot produce — averages −0.0016 ± 0.0245, consistent with nothing, and that null is what separates a mass from a badly figured optic. The signal is not in any galaxy. It is in the sum, and the whole design of a lensing survey follows from that. A one-per-cent distortion, and a million galaxies to see it A galaxy's own shape is unknown and scatters with a dispersion of about 0.3, so a coherent one-per-cent shear is thirty times smaller than the noise on any single measurement. Nothing is ever measured about one object; the estimator is an average, and the whole design of a survey follows from 0.3 over the square root of N. A delay that counts the gas nothing can see. The mean dispersion measure of a radio pulse against the redshift it comes from, for several shares of the baryons residing in diffuse ionised gas between galaxies. A pulse is delayed by free electrons in proportion to the column it crosses, and that column is an integral over the expansion history of a density the baryon budget fixes — so the only unknown in the whole expression is the share itself. At z = 1 the relation gives 1085 pc cm⁻³ if every baryon is out there and 543 at 50 per cent. The measurement runs the other way: a burst with a known host redshift and a measured delay returns the fraction, and the answer came out consistent with nucleosynthesis. A pulse a millisecond long weighs the half of the ordinary matter that no survey could find, and it does so because the thing that delays it is the thing that does not shine. Half the ordinary matter was missing, and a millisecond found it Nucleosynthesis fixes how many baryons there are to better than a per cent. Every survey of where they are came up about half short for two decades — and what closed the gap was the delay a radio pulse picks up crossing gas too thin and too hot for any telescope to have seen.

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AGN feedbackBaryonic feedbackCosmic shearMatter power spectrumScale cutSigma 8Systematic errorWeak lensing