A spectrum that counts what no single gap can

A globular cluster torn apart by the Milky Way’s tide leaves a thin trail of stars along its orbit, a few tens of parsecs wide and tens of kiloparsecs long, and a dark subhalo passing close to the trail kicks the nearby stars along it and opens a gap. The gap is the strongest evidence that could exist for dark matter clumped on scales too small to hold any stars of its own. It is also, as its own arithmetic shows, a measurement of a product: the depth depends on the subhalo’s mass and on the time since the encounter only through their product, and the time is not observed.

That essay ended by pointing at the way out. Cold dark matter does not predict one encounter per stream; it predicts dozens, of every size, spread through the stream’s history. The density of a real stream is therefore not a smooth trail with a gap in it. It is a field in which gaps of every width overlap, and the question the stream can answer is not how heavy one subhalo was but how many subhaloes of each mass there are.

A stream crossed by a population #

The figure is a calculation, and what goes into it is the same impulse that opens a single gap. A subhalo of mass $m$ passing at impact parameter $b$ at relative speed $v$ gives the star at distance $s$ from the point of closest approach a kick along the stream — the impulse approximation, valid because the encounter is over in a few million years — of

$$\delta v(s) = \frac{2Gm\,s}{v\,(s^2 + b^2 + r_s^2)},$$

where $r_s$ is the subhalo’s own size — a subhalo is not a point, and for cold dark matter its scale radius grows roughly as the square root of its mass, about a kiloparsec at $10^8$ solar masses. In a potential with a flat rotation curve the kicked star drifts along the stream by $T\,\delta v$ after a time $T$, so stars on either side of the impact move apart and leave an underdensity between them.

A population of subhaloes is then a list of such encounters. The masses are drawn from a mass function falling as the $-1.9$ power of the mass, which is what cold-dark-matter simulations give for the subhaloes of a Milky Way-sized halo, between $10^5$ and $10^8$ solar masses. The impact parameters are uniform out to five scale radii, because a line presents a cross-section proportional to the distance from it rather than to its square. The epochs are uniform over the last four billion years. And the displacements from every encounter are added, star by star, before the stars are counted along the stream.

The one number that is not drawn is the most important: how many encounters there are. The figure takes about twenty in the stretch drawn, a few of them above a million solar masses, and calls that abundance one. It is an assumption, stated as one, because the abundance is exactly what the method exists to measure; published calculations for streams like the ones observed give numbers of this order, with an uncertainty that is the point of the exercise.

What the top trace shows is that the gaps do not stay separate. A heavy subhalo long ago opened a wide, shallow depression; a lighter one more recently cut a narrow notch inside it; the edges of each gap pile up into small overdensities where the displaced stars have gathered; and a dozen smaller encounters have roughened everything in between. Pointing at any one dip and inferring a subhalo from it would require knowing which encounters it is the sum of, and nothing in the density says. The three traces differ, clearly, but not in any feature — in how much of the stream is disturbed and on what scales.

The statistic that survives superposition #

A field made of many independent disturbances is described by its power spectrum: how much of the density’s variance lies in fluctuations of each wavelength. The density contrast along the stream, $\delta(s) = \rho(s)/\bar\rho - 1$, is Fourier-transformed, and the squared amplitude at each wavenumber is averaged.

The reason the spectrum is the right statistic is that it adds. Two encounters in different places produce density disturbances whose phases are unrelated, so their contributions to the power at each wavelength add rather than interfering; a stream with twice as many encounters has, as long as they do not overlap, twice as much power. The spectrum is blind to where along the stream each encounter happened and to when, which are the things no single gap can supply anyway, and sensitive to how many there were and how strong.

The undisturbed stream in the figure is not flat because it is perfect. It is flat because the only fluctuation left is the counting noise of a finite number of stars — a sample of $N$ stars over a length $L$ has a density contrast with power $L/N$ at every wavelength, the flat floor of an error bar that comes from counting. The model uses sixty thousand stars, so its floor is low. A real stream does not. The best-measured streams have a few thousand member stars identified from the space astrometry that separates them from the foreground by their motions, and the dashed line is that floor for three thousand. Any fluctuation below the dashed line cannot be measured in a real stream, however well the model predicts it. The encounters’ power rises above the noise only at wavelengths of about a kiloparsec and longer, and that is where the measurement is made.

What happened to the unknown time #

The single gap failed because its depth fixes $mT$ and not $m$: a light subhalo long ago and a heavy one recently draw the same hole. The spectrum does not escape that degeneracy for any one encounter. It sidesteps it by averaging over them. If encounters happen at a steady rate through the stream’s life, their epochs are spread evenly between the stream’s formation and the present, and every encounter’s $T$ is drawn from the same known distribution. The spectrum then depends on the mass function and on that distribution of epochs together, and the second is assumed rather than measured.

So the time that could not be measured for one gap has become an assumption about the whole population: that the rate of encounters has been roughly constant. It has not been exactly. Subhaloes orbiting the Galaxy are stripped by its tide and destroyed near its disc, more of them in the past than now, and a stream that formed early met a younger, denser subhalo population than one that formed late. The dependence is gentler than the single-gap degeneracy, because it enters as a weighting over billions of years rather than as a single unknown date, but it is the same missing clock, moved from the individual to the ensemble. A stream’s own age sets the length of that clock, and for streams from globular clusters it is known only to a factor of about two.

A proportion, then a ceiling #

If each encounter adds its own power, the power should scale with the abundance, and the calculation can be run to see whether it does.

It does, and then it stops. Between 0.3 and 1 times the assumed abundance the mean power rises almost exactly in proportion. Beyond three times, it barely rises at all: once the gaps overlap, a stretch already emptied cannot be emptied again, and the density contrast is bounded below by minus one. A stream crossed by far more subhaloes than cold dark matter predicts would not look proportionately more disturbed — it would look thoroughly disturbed, and the measurement would lose its leverage exactly where an excess would be most interesting.

The more important feature is the spread of the points. At the assumed abundance, twelve realisations of the same subhalo population — same mass function, same abundance, different random draws of which subhaloes happened to pass — give band powers scattered by 82 per cent about their mean. A few encounters with the heaviest subhaloes dominate, and whether a stream happened to meet one of them is luck. One stream is one realisation of a random process, and it measures the abundance only to within that scatter, however many of its stars are counted. It is the same limit as the cosmic variance of a sky map with only one sky: the error is not in the measurement but in the sample, and the only remedy is more samples. Several streams on different orbits are several independent realisations, and together they narrow the scatter as the square root of their number.

Which masses put power where #

The spectrum is more than an amplitude. Its shape encodes which subhalo masses did the work, because each mass disturbs the stream over its own characteristic length.

A subhalo of $10^8$ solar masses has a scale radius of about a kiloparsec, passes at a kiloparsec or more, and displaces stars over several kiloparsecs of stream: its power lies at long wavelengths. One of $10^5$ is a thirtieth of that size and makes a dent a few hundred parsecs across. There are many more light subhaloes than heavy ones — the mass function guarantees it — and their many small dents add to a spectrum that is broad and peaks at shorter wavelengths, while the rarer heavy encounters put a steep rise into the longest scales.

So the spectrum’s shape is a map from subhalo mass onto wavelength, and reading the shape is reading the mass function. That is the reason the method matters beyond confirming that the mass that is not the light is clumped at all. The alternatives to cold dark matter differ precisely in how many small subhaloes they allow. Dark matter made of lighter, faster particles — warm dark matter — erases structure below a mass set by the particle’s free-streaming, and for particles of a few kiloelectronvolts that suppression begins around $10^8$ solar masses and removes nearly everything below $10^7$. In the figure that would delete the two lower curves and most of the power at wavelengths shorter than a few kiloparsecs. An analysis of the two best-measured streams in exactly these terms found power consistent with cold dark matter’s abundance and placed a lower bound of a few kiloelectronvolts on a warm particle’s mass — comparable to bounds from the Lyman-α forest and from counting faint satellite galaxies, and resting on entirely different physics.

The difficulty is also on the figure. The masses below a million solar masses, the ones that host no stars and are the purest test of what dark matter is, put their power nearest the counting noise. Seeing them needs streams with many more stars than any yet mapped, which is the case for the deep imaging surveys now beginning, and for the spectroscopy that adds velocities to positions and so doubles the information per star.

Clumps the stream made itself #

Not every fluctuation in a stream’s density was put there by something passing. The progenitor makes some of its own.

A cluster on an eccentric orbit loses stars fastest at pericentre, where the tide is strongest, and the stars released together drift away as a group whose members oscillate about the mean track on epicycles. The result is a string of overdensities along each arm, spaced by a characteristic length set by the cluster’s mass and orbit. These epicyclic overdensities are real, have been identified in simulations of every well-studied stream, and look like the edges of gaps.

In a power spectrum they are a spike at their own wavelength, and a spike is distinguishable in principle from the broad spectrum of a subhalo population — provided it is tall enough not to be one of the peaks that noise alone produces when many wavenumbers are searched. In practice the spacing is not exactly constant, because the orbit’s period and the cluster’s mass both change as the cluster dissolves, so the spike broadens; and the wavelength at which it falls, around a kiloparsec for a typical globular cluster, is just where the power from subhaloes of a few million solar masses lies. Removing it requires a model of the progenitor — its orbit, its mass and its history of mass loss — which is information the stream’s own track is also used to measure. The subtraction and the measurement are not independent.

The progenitor is not the only impostor. Streams that pass through the inner Galaxy are perturbed by the rotating bar, which can open a gap as convincing as a subhalo’s; streams that pass through the disc meet giant molecular clouds, which are as massive as small subhaloes and far more numerous there; and the Large Magellanic Cloud, massive enough to drag the Milky Way’s own halo — a tidal interaction of the kind that draws bridges and tails, seen from inside — bends and stretches every stream in the outer Galaxy. A clean measurement uses streams on orbits that avoid the bar and the disc, which is why the analyses so far have used a very small number of streams.

A second spectrum in the velocities #

Density is not the only thing an encounter changes. The kick that opens a gap also changes the stars’ velocities, both along the stream and across it, and the stars kicked sideways leave the stream’s track altogether for a while, forming a short spur beside it. One of the best-measured streams shows exactly that: a gap with a spur of stars beside it, a few degrees long, whose shape was fitted by an encounter with a compact object of between a million and a hundred million solar masses a few hundred million years ago — too dense, if the fit is right, to be anything but a dark subhalo, since no known cluster or satellite passed there.

For a population of encounters the velocities carry a spectrum of their own. The along-stream velocity perturbation is the derivative of the displacement rather than the displacement itself, so it weights recent encounters more and old ones less, in a different proportion from the density. Measuring both, for the same stream, therefore constrains the distribution of epochs rather than assuming it — the missing clock read statistically. It requires radial velocities to a few kilometres a second for thousands of stars along each stream, which spectroscopic surveys are beginning to supply, and it is the most direct way the method has of paying back the assumption it borrowed.

What the figures do not include #

The model is the simplest one that superposes encounters: displacements along the stream only, added linearly, in a potential where the drift after a kick is exactly the kick times the elapsed time. Real encounters also kick stars perpendicular to the stream and in velocity, producing the spurs and kinks that have been seen beside some gaps; real gaps first compress and then expand, and a gap’s growth slows once the stream’s own orbit carries the kicked stars through several radial oscillations. The subhalo population is drawn with a fixed mass function, uniform epochs and uniform impact parameters, whereas real subhaloes are destroyed by the Galaxy’s tides over time, concentrated towards its outer parts, and moving on orbits whose relative speeds with the stream vary. The normalisation is an assumption rather than a prediction. And the counting noise drawn is the ideal case of pure star counts; a real stream’s membership is probabilistic, contaminated by foreground stars and incomplete in ways that vary along its length, and those variations have spectra of their own.

What survives those simplifications is the structure of the argument: that the power adds while the gaps are separate and saturates when they overlap, that each mass decade occupies its own range of wavelengths, that the light subhaloes sit nearest the noise, and that one stream is one realisation.

Still open: whether the smallest clumps are there #

Two streams measured this way have given a subhalo abundance consistent with cold dark matter and a lower bound on how warm dark matter can be. The measurement they could not make is the one the method was invented for: the abundance below a million solar masses, where subhaloes host no stars and nothing but gravity can find them. Strong lenses whose images are distorted by small clumps along the path probe the same masses in other galaxies, and their answer so far is also consistent with cold dark matter, with uncertainties just as large. The streams will reach that mass range only with more of them and more stars per stream, and with a model of each progenitor good enough to subtract its own clumps at the one wavelength where the smallest subhaloes and the stream’s own history write in the same ink. Until then, the power that separates cold dark matter from a warmer alternative lies below a noise line that is drawn by nothing more than how many stars have been counted.