A count with a knee in it

The simplest question that can be asked about a population is how many of each size there are. For galaxies the answer has a shape that was fitted in 1976 and has been refined ever since, and the shape says two things at once that sound contradictory until the arithmetic is done.

The function #

Paul Schechter’s form, written in luminosity, is

$$\phi(L)\,dL = \phi^*\left(\frac{L}{L^*}\right)^{\alpha}e^{-L/L^*}\frac{dL}{L^*},$$

with three parameters: a normalisation $\phi^*$, a characteristic luminosity $L^*$, and a faint-end slope $\alpha$.

Its two limits are what make it useful, and both are read straight off the logarithmic axes the figure uses. For $L \ll L^*$ the exponential is one and the function is a pure power law, $\phi \propto L^\alpha$. For $L \gg L^*$ the exponential takes over and the count falls faster than any power. So $L^*$ marks the transition, and it is not an arbitrary fitting parameter — it is a physical scale, corresponding to an absolute magnitude near $-21$ and to a galaxy of roughly the Milky Way’s brightness.

The function is usually plotted and fitted in magnitudes, where the conversion introduces a factor of $0.4\ln 10$ and turns the power law into a straight line on a logarithmic count axis. That is the form the figure above draws.

Two integrals that behave differently #

Now the arithmetic that makes the shape interesting.

The number of galaxies fainter than some limit is $\int \phi\,dL$, whose integrand goes as $L^{\alpha}$ near zero. For $\alpha \le -1$ that integral diverges as the limit goes to zero: formally, there is no finite number of galaxies. In practice the count keeps rising as fainter systems are found, which is exactly what surveys of the Local Group have been doing for two decades.

The light those galaxies contribute is $\int L\phi\,dL$, whose integrand goes as $L^{\alpha+1}$. For $\alpha > -2$ that integral converges. So the total luminosity is finite and is dominated by galaxies near $L^*$.

Both statements are true at once with the measured $\alpha \approx -1.25$: almost every galaxy is a dwarf, and almost none of the light is in one.

That is not a paradox, it is a property of power laws, and it is worth carrying because it settles a whole class of questions in advance. Any quantity dominated by numbers — how many satellites a halo has, how many objects a survey will detect — is set by the faint end. Any quantity dominated by light or by stellar mass — the luminosity density of the universe, the total metal content — is set by the knee.

Why there is a knee at all #

Left to itself, gravity has no preferred scale. The distribution of dark-matter halo masses predicted by structure formation is a power law over many decades with a slope near $-1.9$, and no feature at $L^*$ or anywhere near it.

So the knee is not gravitational. It is the signature of the processes that stop gas from turning into stars, and the shape of the galaxy luminosity function against the halo mass function is one of the strongest constraints on those processes.

At the faint end, star formation is inefficient because supernova-driven winds can expel gas from a shallow potential well. A dwarf galaxy loses much of its gas before it can be turned into stars, so it is far fainter than its halo mass would suggest.

At the bright end, the suppression is thought to be energy from the central black hole, which can heat or expel gas from even a deep potential well. Without something of the sort, the most massive haloes would form galaxies far brighter than any that exist.

The numbers, worked #

The parameters are worth turning into quantities a reader can hold, because the function is otherwise three symbols.

The characteristic luminosity. $M^* = -20.9$ corresponds to about $1.2\times10^{10}$ solar luminosities, which is a galaxy slightly fainter than the Milky Way. So the knee sits almost exactly at the kind of galaxy this one is, which is a coincidence of the same sort as the Sun being a typical star: it is not a coincidence at all, since a large fraction of the light in the universe comes from galaxies near $L^*$ and a randomly chosen bright galaxy is likely to be one.

The normalisation. $\phi^* \approx 0.009$ per cubic megaparsec means about one $L^*$ galaxy per hundred cubic megaparsecs — a mean separation of a few megaparsecs, which is the right order for the spacing of bright galaxies in the Local Volume.

The luminosity density. Integrating $L\phi(L)$ gives about $1.5\times10^8$ solar luminosities per cubic megaparsec. Divided by the critical density of the universe, that corresponds to a mass-to-light ratio of several hundred if the universe is at critical density — which is the cosmological version of the same discrepancy the rotation curves found, arrived at by counting rather than by weighing.

Those three numbers, and the shape connecting them, are most of what a census of galaxies amounts to.

Turning a survey into a count per volume #

The hardest part of measuring this function is not the fitting. It is that a survey does not observe a volume; it observes a solid angle down to a flux limit, and a flux limit is a different luminosity limit at every distance.

The standard treatment is the $1/V_{\max}$ method, and it is a good example of an idea that is obvious once seen. For each galaxy in the sample, compute the maximum distance at which it would still have passed the flux limit, and hence the volume $V_{\max}$ over which an object like it could have been found. Then count each galaxy with weight $1/V_{\max}$. A bright galaxy, detectable throughout the survey, counts once per large volume; a faint one, detectable only nearby, counts once per small volume and therefore weighs much more. The residual worry is completeness in the second variable, and it is the reason the measured faint-end slope has crept from about $-1.0$ in early work to $-1.25$ or steeper today.

The count that was made before anyone knew what was being counted #

The history is worth a paragraph, because the first version of this measurement was made when the objects being counted were of unknown nature.

Counts of “nebulae” by magnitude were being compiled in the nineteenth century, and Hubble himself used them in the 1920s and 1930s — before, during and after the argument about whether the spirals were inside the Milky Way. A count by apparent magnitude does not need to know what the objects are, and it constrains their spatial distribution: for a uniform population in Euclidean space, the number brighter than a given flux goes as $F^{-3/2}$, and departures from that slope say something about geometry or evolution or both.

That style of argument — counting sources by brightness and comparing the slope against the uniform prediction — was later the principal evidence against the steady-state cosmology, from radio-source counts in the 1950s and 1960s. The counts came out steeper than $-3/2$, meaning there were more faint sources than a uniform, unchanging population allows, and the universe had therefore been different in the past.

What the modern luminosity function adds is the third dimension: with redshifts, the count becomes a count per unit volume per unit luminosity, and the geometry drops out. It is a better measurement in every respect, and it took sixty years and a hundred thousand spectra to get.

The observation behind the number #

The modern determination comes from large redshift surveys, because a luminosity requires a distance and a distance requires a redshift.

Three systematics dominate.

The photometry of the outer parts. A galaxy’s total magnitude depends on how far out the light is integrated, and the light does not stop so much as fall below the sky, and different definitions — isophotal, Petrosian, model-fitted — differ by tenths of a magnitude, more for the most extended systems. Since the bright end is exponential, a systematic error of 0.2 magnitudes there changes the fitted $L^*$ appreciably.

The corrections for redshift and dust, both of which depend on the galaxy’s own spectrum.

And the local density. A survey of a few hundred cubic megaparsecs may sit in an overdense or underdense region, and $\phi^*$ scales directly with that. Cosmic variance of ten per cent in a modest volume is normal, and it is why the normalisation is the least well-determined of the three parameters.

The bias that lives at the other end #

The faint end has the completeness problem. The bright end has one of its own, and it is a different mechanism entirely, because the count there is falling exponentially rather than rising as a power law.

Consider a galaxy whose measured magnitude has an uncertainty of a tenth of a magnitude. It could have scattered up from a fainter true magnitude or down from a brighter one. In a population where the two are equally common those errors cancel; in a population where the fainter objects vastly outnumber the brighter ones, far more scatter up than scatter down, and the observed count at any bright magnitude is inflated.

That is Eddington bias, and its size depends on the curvature of the count in log space multiplied by the square of the photometric error. Where the function is a straight line in the log — the faint end — a symmetric error does nothing at all. Where it is bending steeply downward — the bright end — the correction can be substantial: at two magnitudes above the knee, a 0.15-magnitude error inflates the count by tens of per cent.

The practical consequence is that the fitted characteristic luminosity depends on the photometric precision of the survey that measured it, in a direction that makes shallower surveys report a brighter knee. Since $L^*$ is the physical scale the whole function exists to locate, that is not a nuisance to be footnoted.

It is also a general result rather than a fact about galaxies, and it is worth carrying: a symmetric measurement error applied to a steeply falling distribution produces an asymmetric bias in the count, and the bias is upward wherever the distribution is convex in the log. The same arithmetic inflates the bright end of a cluster mass function, the high-mass end of a stellar mass function, and the count of the most energetic cosmic rays — everywhere a population thins out faster than a power law.

What the counts are used to test #

A luminosity function is not an end in itself. It is the observable that theories of structure formation are asked to reproduce, and the comparison is made in a particular way that is worth understanding.

Structure formation predicts a halo mass function — how many dark-matter haloes of each mass there are per unit volume — and it does so robustly, because it involves only gravity. What it does not predict without further assumptions is how much light each halo contains.

Comparing the two functions therefore defines an efficiency: the stellar mass a halo of each mass ends up with. Doing that comparison — abundance matching, in the simplest version — ranks haloes by mass and galaxies by luminosity and pairs them off in order. The efficiency curve that comes out peaks near $L^*$ and falls steeply on both sides, by a factor of ten or more.

That curve is the census’s central result, and it is a strange one: galaxy formation is inefficient everywhere and least inefficient in one narrow range of halo mass. Something suppresses it in small haloes and something else suppresses it in large ones, and the two mechanisms have to conspire to leave a peak at a mass corresponding to a galaxy like this one.

Two of the function’s three parameters are worth moving on their own, because each of them changes a different one of the integrals the essay is about.

What is at the bottom of the count #

The faint end deserves a closer look, because the objects defining it are not measured the way the rest of the function is and the difference matters.

Below about an absolute magnitude of $-8$ a galaxy cannot be detected as a smooth patch of light at all. What is detected is a statistical overdensity of individually resolved stars sharing a position, a distance and a colour–magnitude sequence — a search run over a star catalogue rather than over an image, with a filter matched to the expected sequence of an old, metal-poor population. Systems found that way have total luminosities of a few thousand suns, less than a single bright star cluster, spread over hundreds of parsecs.

Two things follow. The first is that the method only works where individual stars can be resolved, which means the Local Group and nothing beyond it. The faint end of the universal luminosity function is measured in one place and assumed to hold everywhere.

The second is that the completeness correction is enormous and is a calculation rather than an observation. A survey covering a fraction of the sky, to a limiting stellar magnitude, detects a system of a given luminosity only within a distance that depends on both — so the count has to be corrected by a factor derived from where such systems are expected to sit, which is taken from a simulation. The corrections applied to the observed few dozen objects give inferred totals of several hundred, and the inferred total is what gets compared against theory.

A measurement whose correction factor is an order of magnitude, and whose correction comes from the model it is testing, is not a clean test. The response has been to make the comparison in the other direction — to predict what a given survey should have found, and compare that against the raw detections — which puts the model rather than the data through the incompleteness, and is the right way round.

Where the picture stops #

A single Schechter function does not fit well any more. Splitting the population by colour gives two functions with different parameters — the red one steeper at the bright end, the blue one steeper at the faint end — and the sum of two Schechter functions fits better than one. That is a statement about the two populations rather than a failure of the form.

The faint end is not measured, it is extrapolated, beyond the Local Group. The faintest galaxies known are detectable only as resolved star counts within a megaparsec or so, and whether the same slope continues elsewhere is an assumption. It is a testable one — the satellite counts of nearby galaxies are the test — and it is being tested now.

And the function drawn here is for the local universe. It evolves: $L^*$ was brighter in the past and $\phi^*$ lower, so galaxies have been converting into fewer, brighter systems. Quoting a luminosity function without a redshift is like quoting a rotation curve without a radius.

The generalisation #

The shape of the argument is transferable and appears wherever a population spans decades: a power law with an exponent between $-2$ and $-1$ divides the questions into two, and which end dominates depends only on how many powers of the variable are in the integrand.

The stellar initial mass function is the same shape and gives the same division: most stars are small, most light comes from the rare large ones. The distribution of asteroid sizes is the same shape, and most of the belt’s mass is in the four largest bodies while most of the number is in objects nobody has catalogued. The distribution of crater sizes, of cloud masses, of star-cluster masses — all the same.

The habit worth taking is to write the integral before arguing about the population. A claim that “most X are small” and a claim that “most of the Y is in large X” are frequently both true, are frequently presented as if one refuted the other, and can be settled in one line by counting powers.

And the same reasoning ran through the exoplanet census, where the occurrence rate is a count per star per unit period and radius, corrected for a detection efficiency computed in advance. The machinery is identical; only the population differs.

And the colour distribution at an intermediate red fraction, since the bimodality is the observation the single-function description leaves out.

Where the ladder goes next #

The next rung is the stellar mass function, which replaces luminosity with the quantity theories of galaxy formation actually predict — and which requires the mass-to-light ratio to be modelled for every galaxy in the sample, so it trades a measurement problem for an inference one.

Later rungs on this anchor: the halo mass function against the galaxy luminosity function, and the efficiency curve between them; the faint-end slope and the missing-satellite problem; luminosity functions split by environment and colour; the evolution of $L^*$ with redshift; the luminosity density and what fraction of the universe’s light has been emitted; and the two ends compared directly against the physics thought to shape them.