A map that is not of positions

The rung below this one asked whether the galaxy distribution is homogeneous and measured the scale at which it becomes so. This rung is about the coordinate system that measurement was made in.

A redshift survey has three coordinates and only two of them are angles. The third is a redshift, converted to a distance by assuming the redshift is entirely due to the expansion — and it is not. Every galaxy also moves under the gravity of everything around it, at a few hundred kilometres a second, and that motion adds to the observed redshift and is indistinguishable from being further away.

At $H_0 = 70$ km s⁻¹ Mpc⁻¹, a peculiar velocity of 700 kilometres a second is ten megaparsecs of spurious displacement. That is not a small correction to a map whose interesting structures are tens of megaparsecs across.

Two regimes, opposite in sign #

The two effects are the same physics at different amplitudes, and the sign difference is worth deriving rather than asserting.

On large scales the motions are coherent. A region denser than average pulls its surroundings inwards. A galaxy on the far side of an overdensity is moving towards the observer, so its redshift is reduced and it appears nearer; one on the near side is moving away, so it appears further. Both are displaced towards the overdensity’s centre, and the structure is compressed along the line of sight.

The amplitude follows from continuity: the infall velocity is proportional to the growth rate of the density perturbation, so the compression measures how fast structure is currently growing. In linear theory the observed overdensity in redshift space is

$$\delta_s \;=\; \left(b + f\mu^2\right)\delta_m,$$

with $b$ the galaxy bias, $f$ the growth rate, and $\mu$ the cosine of the angle to the line of sight. Everything else in this essay follows from that one line.

On small scales the motions are virialised and have lost all memory of the infall that produced them. A cluster’s galaxies move at a dispersion of several hundred to a thousand kilometres a second in random directions, so the cluster is stretched by an amount that has nothing to do with its size — several megaparsecs, when the cluster is one or two across.

Why it is a test of gravity #

The growth rate $f = d\ln D/d\ln a$ measures how fast perturbations are growing now, and it is not a free parameter. In general relativity it is very nearly

$$f \;\simeq\; \Omega_m(z)^{\gamma}, \qquad \gamma \approx 0.55,$$

with the exponent following from the equations of general relativity and essentially nothing else. It is insensitive to the dark-energy equation of state, to the curvature, and to the details of the expansion history — those enter through $\Omega_m(z)$ and are measured independently.

A theory of gravity that differs from general relativity on cosmological scales while matching every solar-system test changes $\gamma$ and nothing else observable. That is what makes the measurement a test rather than a parameter estimate: the expansion history and the growth history are separately predicted by a given theory, and a mismatch between them is a failure that no adjustment of the energy content can repair.

This matters because the alternative explanation of the accelerating expansion is not a substance at all. Either the universe contains something with negative pressure — in which case the geometry and the growth are related as general relativity says — or gravity itself behaves differently at large separations, in which case they are not.

What is actually measured, and the product that gets in the way #

What a survey measures is not $f$. It is $f\sigma_8$.

Two degeneracies force that. The observed clustering amplitude is $b\sigma_8$ — the galaxies’ bias times the matter clustering amplitude — and neither factor is separately observable, because nobody has an independent measurement of how galaxies trace matter. The redshift-space quadrupole is proportional to $f\sigma_8 \times b\sigma_8$, and dividing by the monopole’s $b\sigma_8$ leaves $f\sigma_8$ with the bias cancelled and $\sigma_8$ still attached.

So the reported quantity is a product of the growth rate and the amplitude of clustering, and interpreting it as a constraint on $\gamma$ requires $\sigma_8$ from somewhere else.

What was actually found #

Measurements of $f\sigma_8$ now exist from about redshift 0.02 to 1.5, from galaxy surveys, from quasars and from the Lyman-α forest, at precisions of five to ten per cent each.

They are consistent with general relativity. Fitting $\gamma$ freely gives values near 0.55 with an uncertainty of order 0.05 to 0.1, depending on which data are combined and what is assumed about the expansion history. There is no detection of a departure.

There is, however, a persistent low pull. The measured $f\sigma_8$ values sit slightly below the prediction from the microwave background’s best-fit cosmology, by something like five to ten per cent — the same direction and roughly the same size as the discrepancy weak lensing reports in $\sigma_8$. Whether that is one effect or two, and whether it is a systematic in the tracers or something physical, is unresolved.

The arithmetic of the two regimes #

Both effects follow from one number — the peculiar velocity divided by the Hubble constant — and the difference between them is entirely in whether the velocities of neighbouring galaxies point the same way.

Coherent flows. The infall velocity onto a perturbation of overdensity $\delta$ on a scale $R$ is of order $\tfrac{1}{3}f H R\,\delta$, so the fractional compression of a structure along the line of sight is $\tfrac{1}{3}f\delta$ — a few per cent for a mild overdensity and tens of per cent for a rich supercluster. Crucially it is proportional to $\delta$ itself, which is why it appears in the statistics of the density field rather than as a fixed displacement, and why it can be written as a modification of the power spectrum’s angular dependence.

Virial motions. A cluster of mass $10^{15}$ solar masses has a dispersion near 1000 km s⁻¹, which is 14 megaparsecs of smearing at the present epoch, against a virial radius of about 2. The finger is therefore an object of axis ratio seven, and its length depends on the cluster’s mass rather than on its size — so a finger’s length is a mass measurement, which is the one use anybody makes of a feature that is otherwise pure damage.

The scale at which the two swap over is where the coherent infall velocity equals the virial dispersion, which is the turnaround radius of the structure in question: a few megaparsecs for a group, ten or more for a rich cluster. Below it, everything is random; above it, everything is coherent; and there is no clean separation because real structures span the range.

What the exponent can and cannot represent #

Writing the growth rate as $\Omega_m^{\gamma}$ and fitting $\gamma$ is the standard way of reporting these measurements, and it is worth being clear about what that parameterisation assumes, because it is easy to read it as more general than it is.

The form is an excellent approximation within general relativity across a wide range of expansion histories, which is why it is useful: a single number summarises a function, and a measured departure from 0.55 is a compact statement. It is not, however, a general parameterisation of what a modified theory can do.

Two limitations matter. First, the form has no scale in it — it asserts that perturbations of every wavelength grow at the same rate, which is true in general relativity with cold dark matter and false in a large class of alternatives. A theory with a new length scale, which most modifications of gravity have, produces growth that depends on wavenumber, and a fit that assumes it does not will return some average with no clear meaning.

Second, the exponent is being fitted to data spanning a limited range of redshift, over which $\Omega_m(z)$ itself varies by a factor of a few. A theory whose deviation from general relativity turns on at a particular epoch produces a growth history that no constant exponent describes, and the fitted value will be a compromise between the epochs sampled.

The response has been to report the raw measurements — $f\sigma_8$ at each redshift, with its covariance — alongside the summary, so that a theory can be tested against the observations rather than against a compression of them. A one-parameter summary of a function is a convenience for comparing experiments and a poor instrument for excluding theories, and the distinction matters most exactly where a departure would be most interesting.

It is worth noticing what that reporting convention costs. A set of $f\sigma_8$ values with a covariance matrix is not a measurement of anything on its own — it is a measurement of a product, at redshifts where the survey happened to have galaxies, in a fiducial cosmology used to convert the redshifts into distances in the first place. Comparing it against a theory means re-running that conversion for the theory’s own cosmology, and the correction for having assumed the wrong one is applied through the same Alcock–Paczynski geometry the analysis is separately fitting.

So the published numbers are conditional in a way a plotted point does not advertise, and a compilation of them from several surveys is a compilation of quantities defined against different fiducial cosmologies. The joint fits that use them carry the conversions explicitly; a reader taking the points off a figure does not, which is why the summary exponent, for all its limitations, remains what most cross-comparisons are actually made on.

Where the model stops #

Linear theory is not enough. The expression $\delta_s = (b + f\mu^2)\delta_m$ holds only where the perturbations are small and the velocities coherent, which is scales above about thirty megaparsecs. Below that the two regimes overlap and the modelling is a fitted description rather than a derivation — and most of the information is at those scales, so the analysis is a compromise between using the data and trusting the model.

The fingers are modelled by a fudge. The standard treatment convolves the linear result with a phenomenological function representing the small-scale velocity dispersion, with a free width. It works and it is not derived, and the fitted width is not the cluster dispersion measured any other way.

Bias is not a number. Treating the galaxy overdensity as a constant times the matter overdensity is a first term; at the precision of current surveys the second-order terms matter, and they introduce parameters that partially reabsorb the signal.

And the survey geometry enters everything. The distortion is defined with respect to the line of sight, so any error in the selection function that varies with position on the sky, or with redshift, imitates an anisotropy. The measurement is of an anisotropy in a quantity whose isotropy is assumed, and every observational systematic is a candidate.

What kind of redshift is required #

The whole construction assumes the radial coordinate is a redshift measured well enough that the peculiar velocity is not swamped by the measurement error, and that condition excludes most of the galaxies anybody has catalogued.

A spectroscopic redshift is obtained by identifying features in a spectrum and is good to some tens of kilometres a second — a fraction of the peculiar velocities being measured. A photometric redshift is obtained by comparing brightnesses in several broad bands against template spectra, and it is good to a few per cent in $1+z$ — which at redshift 0.5 is tens of thousands of kilometres a second, three orders of magnitude worse.

That difference is not a matter of degree. A photometric survey’s radial coordinate is smeared by far more than the distortion, so the anisotropy is destroyed rather than degraded, and no amount of statistical power recovers it. A survey with a hundred million photometric redshifts and one with a million spectroscopic ones are not two versions of the same instrument: the first can measure clustering projected on the sky and lensing, and it cannot measure a growth rate this way at all.

The consequence shapes the instruments. A spectroscopic survey has to place a fibre or a slit on each target individually, so its cost scales with the number of galaxies rather than with the area, and the surveys that do this work are defined by how many spectra they can take at once — which is why the field’s progress is measured in thousands of simultaneous fibres rather than in aperture.

It also introduces a systematic with no analogue elsewhere. Fibres cannot be placed arbitrarily close together, because each occupies a physical space in the focal plane, so the closest pairs of galaxies are systematically under-sampled — and the closest pairs are exactly the ones inside clusters, which carry the small-scale distortion. The instrument that measures the anisotropy is blind in precisely the regime where the anisotropy is largest, and correcting for it requires re-observing a subset of fields to establish what was missed.

Why the effect was noticed before it was wanted #

The distortion has been visible in every redshift survey ever made, and for its first two decades it was purely an embarrassment.

The fingers were the first thing anyone saw. The earliest redshift surveys of clusters, in the 1970s, produced maps with elongated features all pointing at the origin, and the natural first reaction — that they were real filaments radial to the observer — was quickly abandoned because the observer is not a special place. By the 1980s the standard practice was to collapse them: identify a cluster, replace all its members with the cluster’s mean redshift, and proceed.

The coherent squashing was harder to see and more consequential. It was derived in 1987, in a paper whose result was that the amplitude of clustering measured in a redshift survey is systematically higher than the true one by a factor depending on $\Omega_m$ — which meant that every published clustering amplitude from a redshift survey was biased, in a way nobody had corrected.

The reinterpretation from nuisance to measurement took another decade and one change of attitude: the correction contains a parameter that is not otherwise measurable. Once that was recognised the practice reversed. Collapsing the fingers destroys information; the modern analysis models them, and the elongation of a cluster along the line of sight is used as one more constraint rather than removed as one more artefact.

The generalisation #

The situation here is a general one: a measurement whose worst systematic is also its signal.

For someone mapping the universe, peculiar velocities are pure contamination — they misplace every galaxy along the one coordinate that is hardest to check, and no amount of care in the observation reduces them. For someone measuring the growth of structure, they are the only handle available, and the map is the nuisance.

Which of those it is depends entirely on the question, and the same reversal appears elsewhere. A star’s finite size ruins an occultation light curve and measures its diameter. Atmospheric turbulence destroys an image and, through its speckles, restores the resolution. A star’s own activity imitates a planet and measures its rotation.

The pattern is worth naming because it suggests where to look: a systematic that is coherent, has a predicted amplitude, and is proportional to something interesting is not a systematic. It is a measurement that has not yet been recognised, and the transition usually happens when someone computes what its amplitude ought to be and finds that the prediction is sharper than the nuisance is annoying.

Where this ladder goes next #

Later rungs on this anchor: the void–galaxy cross-correlation, where the distortion is larger and the modelling easier because void interiors are still linear; direct peculiar-velocity surveys using distance indicators, which measure the velocity field itself rather than its statistical signature; the $S_8$ tension in full, and the several ways non-linear modelling could produce it; scale-dependent growth, which is what a theory with a new length scale predicts and which a single number for $f\sigma_8$ cannot detect; and the relativistic corrections to the distortion, which are tiny, are predicted exactly, and become measurable in surveys covering a substantial fraction of the horizon.