Velocities that see past the edge of the map

Every galaxy has two velocities along the line of sight, and a spectrum measures their sum. One is the recession the expansion of the universe gives it, which is the Hubble constant times its distance. The other is its own motion through the expanding space around it — the peculiar velocity — which it has because gravity has been pulling it towards the denser regions nearby for thirteen billion years. The peculiar velocity is a few hundred kilometres a second. It is also the most direct record there is of what gravity has done to the matter, because it responds to all of the mass, dark and luminous, and to nothing else.

A redshift survey cannot see peculiar velocities one at a time, because it uses the redshift as the distance. What it sees instead is their statistical imprint on its map — the squashing of structure on large scales and the fingers pointing at the observer on small ones. The distortion measures the growth rate of structure, but it measures it divided by how strongly the galaxies cluster relative to the matter, and it needs a large volume at moderate redshift to do so.

There is another route, and it is older. If a galaxy’s distance can be found without its redshift, the peculiar velocity is simply the difference: the redshift velocity minus the Hubble constant times the distance. Do that for thousands of galaxies and the velocity field of the local universe is measured directly, galaxy by galaxy. The trouble is in the word distance, and the whole character of the method follows from it.

An error that grows with the thing it is measured against #

A redshift is measured to a few kilometres a second. A distance is not measured at all, in the sense a redshift is; it is inferred from a relation that ties some observable property of a galaxy to its luminosity or size. For spirals it is the rotation speed read off the width of the hydrogen line, whose relation to luminosity — the Tully–Fisher relation — has a scatter of about 20 per cent in distance. For ellipticals it is the fundamental plane, which ties size, surface brightness and velocity dispersion together and gives distances to about 25 per cent. The best single indicator in routine use, the peak brightness of a type Ia supernova once its light curve has standardised it, gives five to seven per cent, and supernovae are rare.

A fractional distance error becomes a velocity error by multiplication. A galaxy at 100 megaparsecs (in units where the Hubble constant is 100 h kilometres a second per megaparsec, the natural ones here because they remove the constant’s own disputed value from the problem) recedes at 10,000 kilometres a second, and a 20 per cent distance error is 2,000 kilometres a second of velocity error. The velocity being sought is typically 300. The figure makes the comparison at every distance. With the Tully–Fisher relation a single galaxy’s peculiar velocity is bigger than its own uncertainty only within about 15 megaparsecs — the Local Group, the Virgo cluster’s outskirts and not much else. Beyond it every individual velocity is noise with a signal buried in it, and the noise keeps growing.

That settles the shape of every peculiar-velocity survey. Individual velocities are useless beyond the immediate neighbourhood; what can be measured is an average over many galaxies — the mean motion of everything in a sphere, the correlation of velocities at a given separation, the slope of measured against predicted velocities — and the averaging has to beat an error that grows linearly with depth. The number of galaxies grows as the cube of the depth, so in principle the average wins, but only if the galaxies are there to be measured and only if the error really is random. Neither is quite true, and the rest of the method is about the ways it is not.

Waves longer than the survey #

The obvious objection is that a survey reaching a hundred megaparsecs, with errors like these, can say little that a redshift survey of a billion light-years cannot say better. It would be decisive if velocities and densities responded to structure in the same way. They do not.

In linear theory a wave of density contrast δ with wavenumber $k$ drives a flow along its gradient of amplitude

$$v_k = \frac{f H\, a\, \delta_k}{k},$$

where $f$ is the growth rate. A long wave of a given contrast moves matter faster than a short one of the same contrast, because the matter has further to fall. So the variance of the velocity at a point is an integral over the power spectrum weighted by $k^{-2}$ relative to the density’s, and the two respond to entirely different scales. The density at a point is dominated by the smallest scales that exist — its contribution per decade is still climbing at a wavenumber of one, where galaxies and groups live. The velocity is dominated by wavelengths of around a hundred megaparsecs, and half of it comes from waves longer than that.

That is a strange and useful fact. A galaxy’s motion is set mostly by structure bigger than any survey that measures it. The Local Group moves at 627 kilometres a second relative to the microwave background, and the Virgo cluster, the nearest large mass, accounts for only a fraction of it; the rest comes from the Great Attractor region at 50 megaparsecs, the Shapley concentration at 150, and whatever lies beyond. A velocity survey of a hundred-megaparsec volume is a detector for mass outside the volume — which is exactly the regime where a redshift survey has the fewest independent modes and where any systematic error in its selection is largest.

It is also the reason the velocity field keeps structure that the homogeneity scale of the galaxy distribution seems to erase. Counts of galaxies in spheres of a hundred megaparsecs agree with one another to a few per cent, and the universe looks uniform at that scale; the same spheres still move at a hundred and fifty kilometres a second relative to one another, because a velocity is the density divided by the wavenumber and the division amplifies whatever small contrast the longest waves still carry.

It also means the velocity field is smooth. Dominated by long waves, it varies slowly from place to place, so velocities averaged over a few tens of megaparsecs are nearly as informative as the individual velocities would be if they could be measured. That smoothness is what lets the averaging work at all, and it is why the velocity field can be reconstructed on a grid from data too noisy to use one galaxy at a time.

One sphere, and the universe’s own scatter #

The simplest average is the bulk flow: the mean peculiar velocity of everything inside a sphere of radius $R$ round the observer. In linear theory it is a Gaussian random vector with an rms speed fixed by the power spectrum filtered through the sphere,

$$\sigma_B^2(R) = \frac{(fH_0)^2}{2\pi^2}\int P(k)\, W^2(kR)\, dk,$$

where $W$ is the Fourier transform of a sphere. A larger sphere averages over more of the long waves that move it, so the expectation falls with radius.

The expectation is 290 kilometres a second at 50 megaparsecs and 156 at 150. The band around it is what matters more. The bulk velocity is a three-component Gaussian vector, so its magnitude follows a Maxwell distribution, and one sphere can have a bulk speed anywhere between a third of the rms and 1.6 times it without being unusual. There is only one sphere round the observer. No survey, however complete, narrows that band: it is the scatter of the universe, not of the measurement.

With a catalogue of the density of the largest modern compilations — tens of thousands of galaxies within a few hundred megaparsecs — the noise from distance errors falls below the expected signal, so a bulk flow is measurable, and the measured value can be compared with the band. The comparison has been made many times, with results that range from comfortably inside it to uncomfortably above. One analysis of the largest catalogue found about 400 kilometres a second at 150 megaparsecs, well outside the 90 per cent band drawn; another analysis of the same data with a different estimator found a value consistent with it. The difference between the two lay not in the data but in how the far galaxies, with their enormous errors, were weighted, and in how the sky left unsampled behind the Milky Way’s disc was filled in. That the answer depends on the estimator is not an embarrassment peculiar to one result. It is the general condition of the method, and the next section says why.

A galaxy at rest, reported as falling in #

A distance indicator is built on magnitudes — a luminosity predicted from a line width, compared with an apparent brightness — and magnitudes are logarithms. Its scatter is therefore symmetric in the logarithm of the distance: an indicator that is right on average in $\log d$ is as likely to put a galaxy 20 per cent too far as 20 per cent too near in the logarithm, which means further in absolute terms when it overestimates than when it underestimates.

The consequence is drawn for a galaxy that has no peculiar velocity at all. Its redshift is exactly the Hubble velocity, and the direct estimate, redshift minus $H_0$ times the inferred distance, inherits the asymmetry of the distance error with its sign flipped. The overestimates of distance, which are the larger ones, become large negative velocities; the distribution is skewed, and its mean is 202 kilometres a second towards the observer. Average a thousand resting galaxies at a hundred megaparsecs and the survey reports that the shell is falling in at 200 kilometres a second — a signal comparable to the one being looked for, and growing as the square of the distance.

The remedy most analyses now use is to change what is estimated. The logarithm of the ratio of the redshift distance to the indicator distance is Gaussian, because the error is Gaussian in the logarithm, and multiplying it by the redshift velocity gives an estimator that is unbiased and symmetric. The cost is that it is only linear in the true peculiar velocity while that velocity is small compared with the recession velocity, which at the depths where the bias matters it always is. The two estimators give different bulk flows from the same catalogue, and the difference is a large part of why published bulk flows disagree.

This is one member of a family the literature on distance indicators knows by the name of Malmquist, whose best-known member is the brightness bias of a flux-limited sample. The other member is worse, because it depends on the universe rather than on the indicator. A galaxy scattered to a larger inferred distance is more likely to have come from a denser region nearby than from an empty one, because there are more galaxies in the dense regions to be scattered. So the inferred distances are pulled towards the peaks of the density field, and the inferred velocities acquire a spurious flow into every concentration. That bias can be corrected only with a model of the density field — and the density field is what the velocities are being used to test. The modern analyses break the circle by fitting the density and the velocities jointly, sampling over the true distance of each galaxy given everything else, which turns a correction into a model assumption stated openly.

A growth rate from motions rather than from a map #

The most powerful use of a velocity catalogue is not the bulk flow but a comparison. A redshift survey of the same volume maps the galaxy density, and from a density map linear theory predicts the velocity at every point: the flow is the gravitational acceleration from the mapped density times $f$, divided by how much more strongly galaxies cluster than matter. The ratio $\beta = f/b$ is then measured by asking how much the predicted velocities have to be scaled to match the measured ones.

The figure is a simulation built to show where the information comes from, not a reconstruction of any real catalogue. Galaxies within about forty megaparsecs, with velocity errors of a few hundred kilometres a second, lie close to a line; those beyond it scatter over thousands and contribute almost nothing, however many of them there are. A fit that weights each galaxy by its own error recovers the slope that was put in; one that weights them equally is pulled about by the far galaxies’ noise. Multiplying β by the galaxies’ own clustering amplitude, measured from the same redshift survey, cancels the bias and gives $f\sigma_8$ — the growth rate times the amplitude of matter fluctuations — at redshift zero.

The published values of that product from such comparisons lie between about 0.35 and 0.45, with uncertainties near ten per cent, and they are the only measurements of the growth rate at the present epoch. Redshift-space distortions cannot reach redshift zero, because the volume inside a small redshift is too small to hold enough independent structures; the velocity comparison can, because it uses the individual galaxies rather than their statistics. And because the density and the velocity trace the same structures in the same volume, much of the cosmic variance that limits either alone cancels in their ratio. That cancellation is the method’s quiet advantage, and it is why combining the two in one volume beats either one in a much larger volume.

Reconstructing a flow nobody can see whole #

The individual velocities are radial. A spectrum records motion along the line of sight and nothing across it, so a catalogue of peculiar velocities is a catalogue of one component of a three-dimensional field, sampled irregularly and with errors larger than the signal. Recovering the whole field from that looks hopeless, and it would be if the field could be anything. It cannot. A flow generated by gravity from small initial fluctuations has no curl — it is the gradient of a potential — and a field with no curl is fixed by its radial component alone. Integrating the radial velocities outward along each line of sight gives the potential; its gradient gives the full velocity; and the divergence of the full velocity gives the density of all the matter, dark and luminous, with no assumption about how the galaxies trace it.

That chain of reasoning produced the first maps of the mass in the local universe drawn from motions alone, and its structures — the Great Attractor, the Perseus–Pisces ridge, the voids between them — matched the galaxy maps well enough to confirm that galaxies do trace the mass on those scales, and poorly enough in the details to show that the bias between them is not a single number. It also showed how fragile the method is. Every step differentiates or integrates noisy data, and the smoothing needed to control the noise erases structure smaller than about ten megaparsecs. Modern reconstructions work the other way round, forward rather than backward: they sample over initial density fields, evolve each one, and keep those whose predicted galaxies and velocities agree with both catalogues at once. The irrotational flow is still the principle underneath. It is the reason one component is enough.

What the velocity field does not say #

The figures take linear theory for the velocities, and inside a few megaparsecs of a cluster it fails: galaxies falling into Virgo or orbiting inside Coma have velocities of a thousand kilometres a second that linear theory does not predict, and every analysis either excludes those regions, groups the galaxies into haloes and uses the mean, or adds a dispersion term. The simulation fixes that term at 150 kilometres a second; in the real universe it varies with environment, and a fit that assumes it constant misattributes some of the difference.

They also treat the distance indicator as a relation with a fixed, known scatter and no dependence on anything else. The Tully–Fisher relation varies with galaxy type and colour; the fundamental plane varies with the stellar population’s age; and both have to be calibrated on the same galaxies they are then used to measure, with the zero point set by a smaller sample of galaxies whose distances are known independently. An error in the zero point is an error in $H_0 d$ for every galaxy at once, and it appears in the velocity field as a spurious monopole — a uniform expansion or contraction of the whole volume — which is why the monopole is usually discarded and only the dipole and higher moments of the flow are interpreted.

And the band around the bulk flow is a statement about a Gaussian random field with a known power spectrum. If the power on scales of hundreds of megaparsecs were larger than the standard model’s, or if the observer happened to sit in an unusual place, the band would be wider. A measured bulk flow outside it says one of three things — that the estimator is biased, that the observer’s sphere is rare, or that the power on the largest scales is not what the model says — and it cannot say which.

Where the velocities come from #

The most striking thing about a velocity map of the local universe is how much of what it shows is being done from outside it. The reconstructions assign the Local Group’s motion to a sequence of structures at increasing distance, and each extension of the survey has moved the answer outwards: first Virgo, then the Great Attractor, then the Shapley concentration and the void on the opposite side of the sky, whose emptiness pushes as effectively as a concentration pulls. The velocity the microwave background’s dipole records is the total, and the budget for it is still not closed — the flows mapped out to 200 megaparsecs account for most of it but not all, which means that some of the Local Group’s motion is caused by structure further away than anything mapped with individual distances.

That is the same point made by the dipole in quasar counts, which comes out twice as large as the one the microwave background implies. Motions at the edge of every survey are driven by what lies beyond it, and each measurement of a velocity is also a measurement of the mass it does not include. The velocity field is the one quantity in cosmology whose natural scale is larger than the volume in which it is measured.

Distances that need no calibration chain #

Two developments are changing the method’s arithmetic. The first is supernovae in quantity. Wide-field surveys now find hundreds of nearby type Ia supernovae a year, each with a five to seven per cent distance, and at that precision a single supernova’s peculiar velocity is measured to better than its size out to sixty megaparsecs and usefully out to two hundred. A catalogue of a few thousand such distances would measure the growth rate at low redshift to a few per cent from a sample a thousandth the size of a Tully–Fisher catalogue. The limit then becomes the supernovae’s own systematics — how their brightness depends on the host galaxy, which matters as much here as for the expansion history the same supernovae measure.

The second is gravitational-wave sirens, whose distances come from the waveform itself with no calibration against anything nearer. Their individual errors are large, but they are errors of a completely different kind — dominated by the unknown inclination of the orbit rather than by a calibration — and a velocity field measured with them would share none of the systematics of the magnitude-based methods. They are too few for that yet.

Still open: whether the flow at the edge is too fast #

The measurements that matter most are the ones at the largest radii, where the expected bulk flow is smallest and a measured value above the band would most strongly suggest something outside the standard model — more power on the scale of hundreds of megaparsecs, or an observer in an unusually moving region. They are also the measurements where the distance errors are largest, the Malmquist corrections biggest and the unobserved region behind the Galaxy most influential, so they are where the choice of estimator most changes the answer. Whether the fast flows reported at 150 to 250 megaparsecs survive a single analysis that handles the log-normal errors, the inhomogeneous bias and the incomplete sky together — or dissolve into them — is not settled, and a supernova catalogue deep and dense enough to answer it without the Tully–Fisher relation at all is only now being assembled.