A distance read from how grainy a galaxy looks

The surface brightness of an object does not depend on how far away it is, a fact that has been a nuisance to observers and a tool for cosmologists. Move a galaxy twice as far and its total light falls by a factor of four, but its light is concentrated in a patch four times smaller, and so each patch of its image looks equally bright. In a picture of an elliptical galaxy, the brightness of a given pixel contains no information about the distance. Measuring a distance has usually needed either a star that can be resolved or a known kind of explosion, and for most ellipticals there are neither to hand.

In 1988, John Tonry and Donald Schneider published a method that gets around this by paying attention to what has been averaged away. A pixel’s mean brightness has no memory of the distance, but its brightness is made of a finite number of stars. That number is larger for a galaxy that is further off, since the pixel covers a larger physical area. Stars are discrete, so the number in a pixel fluctuates, and the relative size of that fluctuation falls as the number rises. It should be possible to see this directly: a nearby elliptical looks mottled, and a distant one looks smooth.

This is among the simplest pieces of statistical reasoning ever put to use in cosmology. It requires no resolved star, no standard explosion and no model of the galaxy’s gas, and it reaches well beyond the distances where Cepheids can be seen. What it does need is a careful answer to the question of what mean star the grain belongs to, as it turns out to be neither the average star nor a star anyone has seen.

Counting stars without resolving one #

Take a pixel covering a patch of an elliptical at distance $d$. The number of stars in it is proportional to the area, so $\bar{N} \propto d^2$, and each star of luminosity $L$ delivers a flux $L/4\pi d^2$. The mean flux per pixel is the number times the mean flux per star, and $d$ cancels. That is the invariance of surface brightness. The number of stars actually in a given pixel is a Poisson variable, however, with a variance equal to its mean.

The variance of a sum of a Poisson number of random values is the expected number times the mean square of each value. Dividing the variance of the pixel flux by its mean, then, gives

$$\frac{\sigma_f^2}{\bar f} = \frac{\bar{N}\langle f^2\rangle}{\bar{N}\langle f\rangle} = \frac{1}{4\pi d^2}\frac{\langle L^2\rangle}{\langle L\rangle} = \frac{\bar{L}}{4\pi d^2}.$$

The right side looks exactly like the flux of a single star at the distance of the galaxy. In the argot of the subject, that is the fluctuation star, and its apparent magnitude $\bar{m}$ is what the method measures. Nothing about the galaxy’s total mass, its size, or how many stars it has remains in the formula. Only the distance and a ratio of two moments of the luminosity function remain.

This has the same logical form as a counting measurement of noise, though what is normally the error bar has become the signal. A photon-noise measurement gives the flux of an individual photon from a variance and a mean, without ever identifying a photon. This one gets the flux of a typical star without ever identifying a star. The method of using a variance as a measurement of a unit, sometimes called a shot-noise measurement, is how the charge on an electron can be read from a current’s fluctuation. The stars in a galaxy are the electrons here.

Which star the grain belongs to #

The mean in that formula is not the mean anyone expects. The luminosity weighting gives enormous importance to the brightest stars. Averaged by number, most stars in an old population are faint dwarfs far less luminous than the Sun. Averaged by light, the typical contributor is a red giant. Averaged by light squared, the contribution comes from the brightest giants of all.

This is fortunate. The giant branch is a very reproducible thing in an old population, as its tip behaves almost like a standard candle. So the variance belongs mainly to a kind of star that is similar across galaxies and whose luminosity is set by core physics more than by the mass of the star. A method based on the bright-giant population is stable in ways that one based on faint dwarfs would not be, and it means that near-infrared bands, where the giants are particularly bright, give strong signals.

The weighting also explains an initial surprise in the numbers. A fluctuation magnitude near $-1.7$ in the $I$ band is the absolute brightness of a star several hundred times the Sun’s output, and that star is a virtual one that exists as a moment of a distribution. It is an interesting kind of standard candle, with nothing at all at the position of the candle.

Separating the grain from the noise #

A measurement that treats a variance as signal has to worry about all other sources of variance. The image of a galaxy contains photon noise from sky and galaxy, readout noise from the detector, the smooth shape of the galaxy itself, globular clusters, background galaxies and dust. The technique is to subtract a model of the galaxy, divide by the square root of that model so that the fluctuations have a constant amplitude, and take a power spectrum.

The trick that makes this possible lies in the telescope’s own blurring. Every star’s light has been spread into the point-spread function, and so the stellar fluctuations are correlated from pixel to pixel in a known way, whereas the photon noise is independent in each pixel. In Fourier space, one is a constant and the other has the known shape of the telescope’s response. So the separation does not rest on knowing the amplitude of either in advance. The fit returns both, and it can also be checked, as the shape must match a star’s image in the same frame.

It is here that a space telescope has its largest advantage, as the blur is much smaller, and the stellar signal is spread over fewer pixels. The largest ground telescopes were limited to perhaps twenty or forty megaparsecs in good conditions. Hubble’s sharp infrared imaging could reach to about a hundred megaparsecs, and the James Webb Space Telescope, working where the giants are at their brightest, should extend this well beyond. The method’s reach is then a matter of how sharp the picture is as well as how deep, a mix that is unusual among distance methods.

A candle corrected by its colour #

Since the fluctuation star is a moment of a population, it depends on that population’s age and metal content. A galaxy’s stars can differ from another’s, and this has to be corrected for. The saving grace, recognised early on, is that a galaxy’s colour moves in step with the fluctuation magnitude, at least in the $I$ band, where age and metallicity changes act in nearly the same direction.

So the method’s accuracy depends on a one-parameter calibration of the population, and that accounts for its appeal in ellipticals, where the population is relatively simple. Spiral galaxies with young stars and dust are much less suited to it, and a young population makes the relation break down. The relation’s zero point has to come from elsewhere: Cepheid distances to a few galaxies in common, or the tip of the red-giant branch, which form links to the chain of calibrated distances. That is one reason the method is not strictly independent of Cepheid-based measurements, although it is not tied to any one of them.

What is left in the grain #

Anything else that varies from pixel to pixel adds to the variance and makes the fluctuation star look brighter, so the galaxy seems nearer. Globular clusters are the chief such contaminant, along with faint background galaxies. They are bright point-like or small objects, and each contributes a large share of variance to the pixel it falls in. The practical step is to find and mask them, down to as faint a level as the image allows, and to estimate what remains from their known luminosity functions.

The square root softens that effect but does not remove it. The residual has to be computed, and it is one of the largest terms in the error budget for all but the deepest images. Dust has a different signature, since a patch of dust lowers the local brightness and adds variance. The usual remedy is to measure in the near-infrared, or to mask regions where a dust lane can be seen in a colour map.

A small background galaxy is a subtler matter. Such galaxies are numerous at faint levels, and the number that lie behind the galaxy being measured rises with the depth of the exposure. Going deeper removes more contaminants but it also reveals more of them, so the improvement depends on how the galaxy’s own signal compares with the count of faint objects. In this respect the measurement resembles the confusion a crowded sky imposes, and it benefits from sharper images for much the same reason. Different images of one galaxy can be compared, and if a variance measurement depends on depth in the way the correction predicts, the correction is working.

A method that needs a dull galaxy #

The requirements amount to a particular kind of target. An elliptical or a lenticular has a nearly constant population over its whole face, almost no dust, no star formation, and a smooth light distribution that can be modelled and subtracted. Each of these helps. A uniform population makes the fluctuation star the same across the image, so a single number describes every pixel. A lack of dust means that nothing absorbs light in a patchy way. Smoothness makes the model subtraction clean, so that what remains can be analysed.

A spiral fails on all counts. Young star clusters are bright discrete objects with huge variance, dust lanes imprint structure of their own at all scales, and the arm pattern is a large fluctuation with nothing to do with counting statistics. Attempts have been made using spiral bulges and the smoother parts of dwarfs, and recent work on dwarf galaxies in nearby groups is useful for establishing group membership. In dwarfs the stars are few enough that the number of giants in each pixel is low, and care is needed since the fluctuation itself departs from Gaussian form.

That selection shapes what the method can say, and it marks a difference from the Cepheid route, which works in spirals only. The two methods have almost no overlap in their target types, so comparison needs a group of galaxies in which an elliptical and a spiral lie at the same distance. Group membership then carries a small additional uncertainty, of the order of a group’s depth. It is also one reason the comparison has been valuable, since the sets of possible systematic errors barely touch. If a problem in the expansion rate arose from something about young stars or star-forming hosts, this method would not see it. If it arose from something about old populations, Cepheids would not. Only a comparison, with the group’s depth allowed for, says which of them is the odd one out.

What it has been used for #

The method’s greatest early success was mapping the local neighbourhood. Tonry’s survey of about 300 galaxies, published by 2001, supplied relative distances out to about 40 megaparsecs and gave a detailed picture of the flows around the Virgo Cluster and the Great Attractor. It also resolved the depth of the Virgo Cluster itself, with its subclumps at somewhat different distances. Since then a survey of Virgo and Fornax with Hubble has put those clusters at 16.5 and 20 megaparsecs, with galaxy-to-galaxy distance errors of a few per cent. That is good enough for the three-dimensional shape of a cluster to be read directly, which no other method could then do.

Its most striking recent use involved a single galaxy. When the merger of two neutron stars produced a gravitational-wave signal in 2017, the host galaxy NGC 4993 needed a distance, in order to compare with the distance that the wave itself had provided. A surface brightness fluctuation measurement from Hubble images gave 40.7 megaparsecs, with an uncertainty of about six per cent, and that was consistent with the gravitational-wave estimate. It also fed one of the first measurements of the Hubble constant by that route, and it has since been among the more direct calibration routes for the far universe. In 2021, John Blakeslee and collaborators used infrared fluctuation measurements of massive ellipticals, calibrated through the tip of the red-giant branch and Cepheids, to reach out to about 100 megaparsecs, and they obtained a Hubble constant of about 73 kilometres per second per megaparsec with about a three per cent uncertainty. That lies on one side of the argument over the expansion rate, and it did so using ellipticals, which have nothing to do with the spiral galaxies that host Cepheids and most calibrating supernovae. For that reason it makes a useful test of whether the disagreement comes from some property particular to those hosts.

The surprise in a blurred image #

The conceptual pleasure of the method is in what the original observers would have thought was merely noise. The grainy appearance of a nearby galaxy in a good photograph was not, for a century of observers, something to measure. A smooth model of the galaxy was the object, and anything left over was to be subtracted. It took the argument about counting to see that the residual holds a quantity that the smooth model cannot, the size of the unit of which the galaxy is composed. The discovery that a galaxy’s fluctuation is measurable also connects it with the dispersion and scale of an elliptical on the fundamental plane. Both rely on the fact that a pressure-supported galaxy has a simple population and a smooth shape, the same property making it boring to a visual observer and useful to a statistician.

What the picture cannot show #

These simulations use a power-law luminosity function with no giant branch, no horizontal branch and no asymptotic giants, and it is chosen to give a mean star at about the right level. Actual fluctuation magnitudes come from stellar population models and empirical calibration, and the real relation depends strongly on the observing band. The scans are one-dimensional and contain no blur, no photon noise and no galaxy gradient, and each pixel is independent, so their structure is not what a real image looks like. The power-spectrum figure is schematic, with a Gaussian in place of a true point-spread function.

The colour relation is taken from published fits, and it applies in a limited range of colour, for early-type galaxies measured in that band. Younger populations follow different relations, and the scatter drawn here is simulated. The residual-variance figure gives a simple correction, assuming a stable fluctuation amplitude. A real measurement’s errors also include uncertainty in the zero point, which dominates the absolute distances and does not shrink with more images.

Still open: what the fluctuation star does in the far infrared #

The newest space telescope offers fluctuation measurements in near-infrared bands with a resolution and depth never previously available for ellipticals at more than 100 megaparsecs. There the signal is brighter by several magnitudes, and it is dominated by asymptotic-giant-branch stars, which are less well understood than ordinary red giants, and whose numbers depend sensitively on a population’s age. That could make the relation noisier than in $I$, or it could reveal an intermediate-age population that an optical colour conceals. Whether the calibration can be carried over from nearby galaxies, and whether it holds without any change in the zero point across ages and masses, will determine whether the method reaches the Hubble flow on its own, with an independent calibration through the tip of the giant branch. If so, it would give a measurement of the expansion rate in which neither a Cepheid nor a supernova has played any part at all.