The same census, taken in two places

Two counts that are not the same shape compared galaxies against halos and found one efficiency curve. The assumption underneath that comparison is that a halo’s mass is the only thing that decides what galaxy it holds.

It is not, and the second variable is not a property of the galaxy at all. It is what else is nearby.

What a cluster does to the galaxies it is given #

A cluster is not a place where galaxies formed. It is a place they fell into, and several things happen to a galaxy when it does.

Its gas is stripped. The intracluster medium exerts a ram pressure on a galaxy moving through it, and where that pressure exceeds the gravitational restoring force on the disc’s gas, the gas leaves. The stars do not, because stars are not a fluid and feel no pressure — so the galaxy emerges with its stellar disc intact and its star formation stopped from the outside in.

Its supply is cut. Even where the disc gas survives, the reservoir of hot gas in the galaxy’s own halo is removed on infall, so nothing replenishes what is consumed. That is a slower process and it produces a slower fade.

Its outskirts are pulled off. Repeated fast encounters with other cluster members transfer energy to the stars in the outer parts of a galaxy, which is called harassment and matters most for the smallest members, whose binding energies are lowest.

All three suppress star formation and none of them removes a galaxy from a catalogue. What they do is make it redder and fainter, which moves it down the luminosity function rather than out of it.

Why that produces a shallower faint end #

A galaxy that stops forming stars fades. The massive stars that dominated its light die within a few tens of millions of years, and the luminosity falls by a factor of several over the following gigayear before settling onto the slow decline of an old population.

For a bright galaxy that is a shift of half a magnitude or so, and it moves the knee. For a dwarf it is worse, because dwarfs have a larger fraction of their light in young stars — so the same quenching costs a dwarf more magnitudes than it costs a giant.

The consequence is that quenching tilts the luminosity function. Faint galaxies move further than bright ones, the faint end is depopulated relative to the bright end, and the fitted slope comes out shallower without a single galaxy having been destroyed.

The arithmetic is worth doing once, because the size of the tilt is not obvious. A galaxy forming stars steadily for ten billion years has roughly a third of its optical light from stars younger than a gigayear. Stop the star formation and that third goes within a gigayear, then the remainder declines as roughly $t^{-0.8}$ as the main-sequence turn-off moves down the mass function. After three gigayears the galaxy is about a magnitude fainter than it was; after eight it is nearer two.

Two magnitudes is a large displacement on a luminosity function, and it is larger for a galaxy whose recent star formation was a larger fraction of its light. The differential fading between a giant and a dwarf is perhaps half a magnitude, and half a magnitude applied over the four magnitudes between the knee and the faint end drawn is enough to change a fitted slope by more than the observational uncertainty on it.

Which means the tilt is a prediction rather than a hypothesis. Given a quenching history and a stellar population model, the change in the faint-end slope follows, and comparing it against the measured change is a test. The comparison has been made, and fading alone accounts for most but not all of the observed difference — the residual is what the destruction argument is for.

The alternative that predicts the same thing #

Destruction predicts a shallower faint end too, and by a mechanism that has nothing to do with fading.

A dwarf galaxy passing close to the cluster centre, or close to a massive member, can be tidally disrupted outright. Its stars are not destroyed — they join the cluster’s diffuse intracluster light, which is a real and measured component carrying perhaps a tenth of a cluster’s total stellar mass — but the galaxy ceases to exist as a catalogued object.

So the two explanations differ in where the light went, and not in what the luminosity function looks like. Fading keeps the galaxy and dims it; disruption keeps the light and disperses it.

Distinguishing them requires an accounting of the total stellar mass rather than of the galaxies, which is why the intracluster light is measured at all. If the dwarfs were destroyed, their stars are in it; if they faded, they are not.

A third reading nobody can exclude #

Fading and destruction both assume the cluster’s dwarfs began as the field’s dwarfs. There is a third possibility, and it is the hardest to test because it concerns a time before any cluster existed.

The regions that became clusters were overdense from the beginning. Structure grows from the same density field everywhere, and a region destined to become a cluster crossed every collapse threshold earlier than an average region did — so its galaxies formed earlier, in denser surroundings, from gas that was reionised earlier and was hotter for longer.

A cluster’s dwarfs may therefore be scarce because fewer were ever made, not because any process removed them. That is a statement about the initial conditions, and no observation of a present-day cluster can test it directly.

What can be tested is a consequence. If the suppression happened early, it should be visible in the cluster’s bright galaxies too, as an older mean stellar age at fixed mass — and it is. Cluster ellipticals are measured to have formed their stars earlier and over a shorter interval than field ellipticals of the same mass, by an amount consistent with the collapse-time argument.

So all three readings have supporting evidence, which is the honest state of it: some of the difference was there at formation, some was made by fading, and some by destruction. The relation between where the mass is and where the light is carries the same ambiguity one level up, and for the same reason — a present-day map records an outcome and not a sequence.

The measurement’s own difficulty #

Counting galaxies in a cluster is harder than counting them in the field, and every difficulty pushes the same way.

Membership has to be decided. A cluster occupies a small patch of sky containing many foreground and background galaxies, and assigning membership from imaging alone means using colour — which is the quantity under investigation. A red-sequence membership criterion finds red galaxies, and then the cluster’s population is measured to be red.

Crowding removes faint objects. A dwarf galaxy projected near a giant one is harder to detect, and a cluster core is where the giants are. The completeness at the faint end is therefore lower in exactly the environment whose faint end is in question, and it is lower by an amount that has to be measured by injecting artificial galaxies into the images and counting how many come back — the same experiment a transit survey runs on its light curves, for the same reason.

And the surface-brightness limit bites hardest on the objects the argument is about. A faded dwarf is not merely fainter; it is more diffuse, and a fixed surface-brightness limit removes diffuse objects preferentially whatever their total magnitude.

Each of those makes a cluster’s faint end look shallower than it is, and none of them is the astrophysics the measurement is trying to find.

What the environment is, and how much of it is mass #

There is a prior difficulty that is conceptual rather than instrumental. “Environment” is not one quantity.

A galaxy near the centre of a rich cluster, a galaxy in the outskirts of the same cluster, a galaxy in a group of five, and a satellite of an isolated giant are in four different situations, and the processes acting on them differ in kind rather than in degree. Ram pressure requires a dense intracluster medium and does essentially nothing in a small group; tidal harassment requires high encounter speeds and is weak in groups, where encounters are slow enough to merge instead.

So a comparison between “cluster” and “field” is a comparison between two ends of a continuum that has several variables along it, measured by whichever proxy the survey used — local galaxy density, distance from a cluster centre, or membership of a catalogued group.

And a good part of what looks like environment is halo mass. A galaxy in a cluster is a satellite of a very massive halo, and the efficiency curve already says that massive halos convert few of their baryons into stars. Disentangling “this galaxy is in a dense place” from “this galaxy is in a massive halo” requires samples split on both at once, and the samples are small.

The one that goes the other way #

Everything so far concerns the faint end. The knee moves too, and it moves in the direction fading cannot explain.

A cluster’s characteristic magnitude is brighter than the field’s. Fading makes galaxies dimmer, so something else is raising the bright end, and the something else is mergers.

The centre of a cluster is where the brightest galaxy in it sits, and that galaxy grew by consuming others. A brightest cluster galaxy is typically two magnitudes brighter than the knee, far too bright to be drawn from the same Schechter function as the rest, and it sits at the bottom of the potential where dynamical friction delivers its neighbours.

So a cluster’s luminosity function is two processes acting in opposite directions on opposite ends: fading and stripping at the faint end, merging at the bright one. That is the reason the ratio curve drawn above has a slope rather than an offset, and it is why a single number — “clusters have fewer dwarfs” — understates what the comparison contains.

Two fits, and the diagonal they are correlated along #

Both curves are Schechter fits, and it is worth being exact about what a fit is here.

The measurement is a histogram of galaxy counts per magnitude bin, corrected for completeness and divided by a volume. In a cluster the volume is not well defined — the cluster is an overdensity rather than a region — so cluster luminosity functions are usually quoted per cluster or per unit cluster mass rather than per cubic megaparsec, and the normalisation drawn here is a convention.

The parameters are then correlated. A Schechter function’s faint-end slope and characteristic magnitude are not independently determined by a fit; a steeper slope can be partly compensated by a brighter knee, and the confidence region in the two-parameter plane is a long diagonal ellipse. So “shallower faint end and brighter knee” is one statement rather than two, and quoting the two parameters with independent error bars overstates what has been measured.

The most secure statements are therefore differences in quantities that avoid the fit altogether: the ratio of dwarfs to giants within fixed magnitude limits, which needs no functional form at all.

A smooth fit to a population that is two populations #

Both functions are drawn as smooth fits to a population that is two populations. A cluster’s red sequence and its remaining blue galaxies have different luminosity functions, and the sum of two Schechter functions with different parameters is not a Schechter function. Fitting one to the sum returns parameters that belong to neither.

Nothing here is projected. A real cluster luminosity function is measured in projection, with a background subtraction, and the subtraction is the dominant uncertainty at the faint end where the background counts are large.

And no figure separates the three mechanisms. Ram pressure, starvation and harassment all produce a shallower faint end and a redder population, and the shape of the luminosity function is the same under each. What distinguishes them is a timescale, and a count taken once has no timescale in it.

What the field is, when it is the comparison #

One half of every comparison in this essay is a control sample, and controls deserve the same scrutiny as the thing being tested.

“The field” is usually defined as everything not in a catalogued cluster or group, which makes it a residual rather than a population. It contains isolated galaxies, satellites of galaxies too faint to have been catalogued as groups, and the outskirts of structures whose centres are outside the survey. A field luminosity function is therefore an average over environments, weighted by however many of each the survey happened to contain.

That matters because the environmental effect is not a step. It begins well outside a cluster’s virial radius — galaxies are observed to be redder and more gas-poor out to two or three times it, which is usually attributed to their having already passed through the centre once and come back out — so a field sample drawn from near clusters is partly a cluster sample.

The cleanest controls are therefore defined by distance from anything, and the price of that definition is that the sample becomes small and the volume becomes large, so the two functions being compared are measured over different volumes with different systematics. There is no version of this comparison in which both halves are measured the same way.

A count that measures a history it cannot date #

The recurring limitation of every count in this essay is worth stating plainly at the end.

A luminosity function is a snapshot. It says what the population is now, with great precision and over a large dynamic range. What every question asked of it wants to know is how the population got that way, and a single snapshot contains no time. The same is true of the colour bimodality, which says that galaxies change quickly without saying when any particular one did.

The escape is always the same: find a second axis that correlates with time. Colour is one, because a population reddens as it ages, which is how quenching is detected at all. Redshift is another and the strongest, because looking further away is looking earlier, so measuring luminosity functions at several redshifts turns a snapshot into a sequence.

Environment is a third, and the least direct, because a cluster is not a later version of the field — it is a different sample of the same initial population, selected by where it started. Comparing them measures what environment does only under the assumption that the two populations were the same to begin with, and that assumption is the whole of the biased-formation argument: halos that became clusters were denser regions from the start, so the galaxies in them formed earlier, and some of the difference was there before any cluster existed.

Still open: which of the three is dominant #

The mechanisms are all established in the sense that each has been observed operating on individual galaxies. Ram-pressure stripping is visible directly as one-sided gas tails; starvation is inferred from galaxies with normal stellar discs and no halo gas; harassment is seen in the distorted outskirts of cluster dwarfs.

What is not established is their relative importance to the population as a whole, and the answer probably depends on the galaxy’s mass and on the host’s, so it is a function rather than a number.

The most promising discriminator is the timescale each implies, read off the colour distribution rather than the luminosity function. Ram pressure acts on a crossing time — a few hundred million years — and produces a population caught in transition; starvation acts over several gigayears and produces almost none, because the fading is slow enough that a galaxy is rarely observed midway. So the fraction of cluster galaxies with intermediate colours is a clock, and the clock reads short.

That points at the fast mechanism, and it is consistent with the emptiness of the valley between the two colour populations in the field as well as in clusters — which is either a hint that the same fast process operates everywhere, or a reminder that two arguments reaching the same conclusion from the same statistic are not two arguments.

From here: the measurement with a time axis in it #

What would follow is the measurement with a time axis in it: the luminosity function at several redshifts, and the evolution of the knee. It turns every question above from a comparison into a history, and it introduces a difficulty none of the local versions has — at high redshift the band a survey observes in is not the band it would like, so a luminosity function at $z = 2$ measured in the optical is a luminosity function in the rest-frame ultraviolet, and comparing it with a local one requires a correction that depends on the very populations being measured.