One profile for a galaxy that has two

Massive galaxies that had stopped forming stars by a redshift of two were small: their light was concentrated inside a kiloparsec or so, three to five times more compactly than in galaxies of the same mass today. That result rests on half-light radii from fitted profiles, which escape the isophotal threshold that shrinks every naive size at high redshift. It leaves one worry that fits do not obviously escape. If those galaxies grew mostly by accreting small satellites into an extended envelope — the process that builds the faint stellar haloes seen round nearby giants at thirty magnitudes per square arcsecond — then at high redshift the envelope would sit precisely where the dimming of the expanding universe and the noise of the sky make light hardest to see, and the fits might report only the core.

That worry can be put to a model. Take a compact core with an envelope round it, dim it as the expansion dims it, cut it off where a survey’s noise would, fit it as the surveys fit galaxies — with a single Sérsic profile, whose one shape parameter sets how quickly the light falls — and compare the size the fit reports with the size the galaxy has. The answer turns out not to be the one the worry anticipated.

A galaxy with two parts #

A de Vaucouleurs profile — the law that the light of elliptical galaxies falls with the fourth root of radius — is what violent relaxation of a collapsing system tends to produce, the equilibrium reached without encounters, and the compact cores of the early quiescent galaxies follow it well. An envelope built later, from the debris of many small satellites stripped at large radii, has no reason to follow the same law. It is better described as a separate, more extended and much fainter component with a profile of its own. The galaxy is then two things superposed, and its total half-light radius depends on both: on the core’s size, the envelope’s size, and the share of the light each holds.

In the model, the envelope is an exponential with a half-light radius of ten kiloparsecs holding three-tenths of the light. That shifts the galaxy’s true half-light radius from one kiloparsec to 2.1, a doubling, while barely changing the surface brightness inside the inner few kiloparsecs. Placed at a redshift of two, every point is dimmed by 4.77 magnitudes — the fourth power of one plus the redshift, the dimming no distance can undo. The envelope then spends most of its light between 26 and 31 magnitudes per square arcsecond.

A survey reaching 27 magnitudes per square arcsecond sees the envelope only where it overlaps the core; one reaching 30, which is about as deep as the deepest fields go on scales this large, sees it out to sixteen kiloparsecs. Neither single-profile fit reports the true size. The shallow fit gives 1.3 kiloparsecs, the deep one 1.5. Three magnitudes of extra depth — sixteen times less noise per unit area — move the answer from about sixty to about seventy per cent of the truth.

Why small mergers build large envelopes #

The envelope picture is not an arbitrary choice; it is what the arithmetic of mergers predicts for galaxies that have stopped forming stars. Such a galaxy can grow only by accreting other galaxies, and the virial theorem says how its size responds. If a galaxy absorbs another of a fraction η of its mass, whose stars move with a fraction of its own velocity dispersion, the energy brought in is small compared with the mass, and the merged system has to be larger to hold its new mass at the old binding energy. In the limit of small, cold satellites the radius grows as the square of the mass: an increase in mass by half makes the galaxy more than twice as large. A merger of two equal galaxies, by contrast, grows the radius only in proportion to the mass.

So a compact galaxy that doubles its mass through many small mergers can quadruple its size, which is close to the growth observed, while doubling it through one equal merger would double its size and leave it too compact. The small satellites do not reach the centre intact. They sink by dynamical friction on orbits that shrink slowly, and the tidal field of the host strips their stars on the way, spreading them in tails and shells that phase-mix into a smooth envelope at tens of kiloparsecs. The core is left nearly untouched; the envelope is where the mass arrives.

That is why the worry was specific to envelopes, and why it mattered. The process expected to dominate the growth puts the new stars exactly where the old measurement is weakest.

Depth is not the cure #

The opening figure is one galaxy. The bias’s dependence on the envelope is the interesting part, because it does not go in one direction.

An envelope only a few times larger than the core makes the fit overestimate the size, by as much as two-thirds. A single profile asked to follow a bright core and its slightly broader skirt does it by adopting a higher index, with wings that fall more slowly than either component, and that profile’s half-light radius is larger than the pair’s. An envelope ten or more times larger than the core makes the fit underestimate, by up to half: its light is spread so thinly that, pixel for pixel, it carries little weight against the core, and the fit settles on the core’s own shape with a slight broadening. Between the two the fit crosses the truth by accident.

None of this is cured by depth. Four magnitudes — forty times less noise — move the answer by less than a fifth of the error in each case. The reason is that the fault is in the model, not the data. One Sérsic profile cannot describe two components with different shapes and scales, and the least-squares fit, weighted as an image fit weights it, settles on whatever compromise best matches the brightest pixels. Deeper data add faint pixels with low weight; they nudge the compromise, and leave it a compromise.

What the index records #

The fit does leave a trace of the envelope, in its shape parameter.

An envelope comparable in size to the core raises the fitted index well above four, the signature of light falling more slowly at large radius than a de Vaucouleurs law allows. An extended envelope barely moves it, because the fit barely notices it. The index is therefore a reasonable diagnostic for envelopes that are already causing the fit to overestimate, and a poor one for those causing it to underestimate — the ones the original worry was about.

Measured indices do change with time in the right direction for some envelope to be appearing. Compact quiescent galaxies at a redshift of two have lower indices, on average, than their present-day counterparts of the same mass, which have higher indices and more extended wings. That trend is what the growth of an envelope from one or two kiloparsecs outward would produce in a single-profile fit: invisible at first, then raising the index as it becomes comparable to the core. It is not by itself evidence for envelopes that the fits miss, but it is the trace those envelopes would leave once they were no longer missed.

The dimming, measured on its own #

The original worry was specifically about depth: that the dimming at high redshift pushes envelopes below the sky. The model can measure that effect in isolation by holding the galaxy fixed and moving it.

The dimming alone makes this galaxy appear to grow by about a fifth between redshift two and the present at a shallow survey’s depth, and by about a twentieth at a deep one. That is a real effect, and it goes in the direction of the worry. It is also small beside the measured threefold to fivefold growth, and small beside the fit’s own offset, which keeps both fitted radii well below the true one at every redshift. Depth matters for whether the envelope is seen at all; it matters much less for the size the fit reports, because the fit was not following the envelope even where it was seen.

Growth that a fit understates #

The real question is not about an unchanging galaxy but about one that grows, and the growth the worry was about is growth by an envelope.

The result inverts the worry. While the envelope holds a minority of the light, the fit at the later epoch still follows the core, and it reports less growth than happened — 2.1-fold instead of 2.8 when the envelope holds two-fifths of the light. Once the envelope holds most of the light, the fit switches to following the envelope, and the measured growth catches up with the true one. Growth by envelope is never manufactured by the fits in this model; it is understated in the middle and recovered at the end. If the compact galaxies grew by building envelopes, the measured growth is if anything a lower limit.

That is consistent with the most direct test made so far. Deep images of compact galaxies at a redshift of two, stacked and examined for faint envelopes, have found some galaxies with extended light the standard fits missed and many without; the sizes measured from those deep images, with the envelopes included where found, have not removed the growth. The present-day comparison galaxies, meanwhile, are measured from surveys at low redshift where their envelopes are bright, and many of the nearby giants are now known to be dominated by them.

A size that depends on the weights #

The single-profile size is a property of the fit as well as of the galaxy, and the clearest way to see it is to keep the galaxy and the data fixed and change only what the fit cares about.

The three answers for the same galaxy differ by more than a factor of two. Weighting by signal-to-noise, as every fit to an image with sky noise does, favours the bright centre. Weighting every pixel equally favours the outskirts, where most of the pixels are, and lets an extended envelope pull the profile’s wings far out — overestimating in its turn. Weighting by radius sits between. None of them is wrong; each is the right answer to a different question about which part of the galaxy the single profile should reproduce. The single half-light radius of a two-component galaxy is not determined by the data. It is determined by the data and the choice.

What the half-light radius was supposed to be #

The half-light radius was adopted because it seemed to depend on nothing but the galaxy: no threshold, unlike an isophotal radius, and no assumption about shape, unlike a Petrosian radius, whose aperture already catches only four-fifths of a spheroid’s light. It needs the total light, and the total is where the trouble lies. Measured non-parametrically, from the curve of growth, the total stops at the noise and misses whatever is fainter — an aperture chosen for its signal-to-noise discards part of the light by design, and a curve of growth stopped at the sky does the same without choosing to. Measured parametrically, it comes from extrapolating a model, and the model decides what lies beyond the data.

For a one-component galaxy whose profile is well described by the model, the extrapolation is harmless and the fitted half-light radius is the true one. That is the case the method was designed for, and for most disc galaxies and many ellipticals it holds. For a galaxy with two components of very different scale the extrapolation is a guess, and the guess is made by the bright centre. The cure is to fit two components, which needs data deep enough to constrain the second, and that is where depth matters after all: not to rescue the single fit, which it cannot, but to make a two-component fit possible.

Where the compact galaxies went #

There is a second, independent line of evidence that the compact galaxies grew, and it does not depend on any fit. Compact quiescent galaxies were common at a redshift of two — a large share of all massive quiescent galaxies then — and galaxies that compact are rare today. Counted per unit volume in the nearby universe, the dense, old, massive galaxies that would be the unchanged descendants of the early population are found at a small fraction of the density the early population had, and the few that exist are known as relics precisely because they are exceptional. Since stars do not disappear and quiescent galaxies do not easily restart their star formation, the early compact galaxies must still be there, inside larger galaxies, which is to say they grew.

The census argument constrains how many grew, not how much, and it is free of the profile-fitting problem entirely. It agrees with the fits that most of the population changed, which is reassuring. What the fits add is the amount, and the model says their amount is likely an understatement when the growth was by envelope.

Two components, and what fitting them costs #

Fitting two components instead of one replaces a biased answer with an uncertain one, and the uncertainty is where depth finally earns its keep. A two-component model has two more free numbers — a second size and the share of light between the components — and the data have to constrain both. They can do so only if the envelope’s own profile is traced over a range of radii where it dominates, which for the model galaxy means seeing it clearly beyond about five kiloparsecs, at 27 to 29 magnitudes per square arcsecond after the dimming. Above that range the envelope is degenerate with the core’s wings; below it the envelope is noise.

The degeneracy has a specific form. A slightly larger core with a slightly fainter envelope fits the same pixels as a slightly smaller core with a brighter one, and the total half-light radius moves along that trade. With a survey reaching 30 the trade is narrow and the total size is measured to a modest error; at 27 it is broad, and the two-component size is honest about not knowing. That is still an improvement on a single fit, which reports a precise number that is wrong.

The sky itself sets the last limit. At thirty magnitudes per square arcsecond the envelope is a thousandth of the sky’s brightness, and an error of a part in ten thousand in the sky subtracted from an image creates or removes an envelope of that size. Deep imaging programmes designed for faint outskirts spend as much effort on the sky as on the galaxy, and the envelopes of compact galaxies at high redshift are an extreme case of that problem: small on the sky, faint, and seen through the light of everything else in the field.

The same problem in the nearby giants #

The problem is not confined to high redshift. The most massive galaxies today, at the centres of groups and clusters, have envelopes that merge smoothly into the diffuse light between the cluster’s galaxies, and their half-light radii depend dramatically on where the galaxy is deemed to end. A single profile fitted to such a galaxy from a survey of ordinary depth gives a size that is a fraction of what a deep image with a two-component fit gives, and different analyses of the same galaxies disagree by factors of two for exactly the reasons drawn here. The intracluster light, counted as part of the galaxy or not, can hold a fifth to a third of a cluster’s stars.

This matters for the comparison. If the present-day giants used as the endpoint of size growth are measured with single profiles, their envelopes are partly missed too, and the measured growth is the growth of the cores plus whatever fraction of the envelopes the fits happen to catch. The ratio of two compromises is not the ratio of two sizes. Comparisons made with the same method at both ends, on galaxies selected the same way, cancel part of this; comparisons that mix methods do not.

What the model leaves out #

The model galaxy is spherically symmetric, unblurred and noise-free inside its limit; real images are blurred by a point-spread function comparable to the core’s size at high redshift, which moves the fitted index and radius on its own. The core is a pure de Vaucouleurs profile and the envelope a pure exponential, with no colour gradient; real envelopes are bluer or redder than their cores, so the same galaxy has different sizes in different bands. No correction is made for the change of the observed band with redshift, or for the fading of stellar populations, both of which change the relative brightness of core and envelope over ten billion years. The fit is a grid over two parameters with the amplitude solved exactly, weighted as described; real fitting codes use the image’s pixels, its noise map and a model of the blurring. None of these restores the single-profile size to the true one, and several make the bias larger.

Still open: how many of the compact galaxies already had envelopes #

The model says that a compact galaxy at a redshift of two with an extended envelope would be measured as compact, by any single-profile fit at any depth, and that the growth measured since would understate the true growth. What it does not say is how many of them had such envelopes. Deep images in the near-infrared from space, now reaching the rest-frame optical light of these galaxies at depths and resolutions that resolve a kiloparsec core and trace light to twenty kiloparsecs, make two-component fits possible for individual galaxies rather than only for stacks. Whether the envelopes are common, and whether their light is old — accreted from satellites that formed their stars elsewhere — or young, is the next measurement, and it decides whether the growth of massive galaxies since was mostly the building of envelopes, as the minor-merger picture predicts, or something that changed their cores.