A crossing time that is the filter's

Two colours, and almost nothing between converted an empty region of a histogram into a timescale, using an argument that is exactly right: in a steady state the number of objects in an interval is proportional to the time each spends there — the same argument a gap in the exoplanet radius distribution rests on, so a deficit is a short crossing. It then flags its own weakness in one line — the valley’s width depends on the bands — and stops, having already used the same reasoning to separate the mass and environment terms that set how many galaxies make the crossing at all.

That line is the whole of this essay, and the size of the effect is larger than a caveat.

A colour is a clock, and its rate is a stellar lifetime #

A galaxy’s luminosity in a band is the sum of its stars’ contributions, weighted by how much light each contributes in that band. Since the most massive stars are both the brightest and the shortest-lived, that weighting is a function of age, and the function is very different from band to band.

In the near ultraviolet the light comes almost entirely from stars of a few solar masses and above, which live a few hundred million years. Stop forming stars and the ultraviolet collapses within that time.

In the red the light comes from stars near a solar mass and from red giants, which live and shine for ten billion years. Stop forming stars and the red band barely notices for a billion.

So each band has a memory, and a colour index formed from two bands has a response time set by the shorter of the two. The reddening after a shutdown is therefore not one curve; it is a different curve in every index, and the difference is a factor of fifteen in the rate at the start.

What the valley’s emptiness actually measures #

That earlier argument says the number of galaxies at a colour is proportional to the time spent at it. That is correct. What it does not say is which time, and the answer is the time spent at that colour in that index — which is a property of the index.

Put the two together. The observed fraction of galaxies in the valley is a few per cent in optical colours and rather larger in ultraviolet ones, and both are correct measurements of the same population. The optical valley is emptier because the optical colour crosses it faster, relative to the total range it spans.

So a crossing time inferred from a single colour is a crossing time in that colour. It is a real quantity and it is not the shutdown’s timescale, and converting one to the other requires knowing how the band’s memory relates to the history.

That sounds like an obstacle and it is a measurement.

Two filters measure a timescale one cannot #

The structure of that last figure is worth extracting because it converts a systematic into an observable.

If the shutdown is abrupt, the indices agree. If it is gradual, they disagree, and the amount they disagree by rises with how gradual it was. So a survey with both an ultraviolet and an optical colour for the same galaxies can ask a question a single-colour survey cannot: not how long does the transition take but how sharply does the star formation stop.

The answer, from the surveys that have both, is that the population contains at least two kinds of transition.

A fast one, in which a galaxy crosses in under half a billion years. These galaxies are red in every index almost simultaneously, they are predominantly early-type, and the timescale is consistent with a merger-driven burst followed by an abrupt shutdown — which is what two gas-rich galaxies colliding produces.

And a slow one, in which the crossing takes billions of years. These are predominantly disc galaxies, they linger in the ultraviolet valley while looking red in the optical, and the timescale is consistent with the gradual exhaustion that the environmental clock a satellite’s own gas supply sets implies.

Those two populations are not distinguishable in a single optical colour at all. They overlap in it, and the measured “crossing time” from an optical valley is an average over both weighted by their numbers — which is a number that describes neither.

Why the two indices disagree by more than a factor #

It helps to see where the factor comes from, because it is not the ratio of the band memories.

Consider the limit in which star formation stops instantly. Every band’s luminosity then decays according to the ageing of a fixed population, and the ratio of two bands — the colour — reddens at a rate set by the difference between the two decay rates. Nothing about the history enters, because there is no history. Every index measures the same clock, differing only in how far it runs.

Now let star formation decline gradually instead. A band whose memory is short compared with the decline tracks the declining rate directly: its luminosity is proportional to the current star-formation rate, and it falls exponentially with the shutdown’s own timescale. A band whose memory is long compared with the decline is still integrating over an era when the rate was high, and it barely moves.

So the colour between a short-memory and a long-memory band falls at the shutdown rate, while the colour between two long-memory bands falls at the difference of two much slower ageing rates. The disagreement between the indices is the difference between tracking a history and remembering one, and it exists only in the regime where the history is still changing.

That is why the curves in the figure above converge at the left and diverge at the right, and it is why the divergence is a measurement rather than a nuisance. A survey that finds two indices agreeing has found an abrupt shutdown; one that finds them disagreeing by a factor of two has measured how gradual it was.

The population that gives the game away #

There is a class of object that makes the argument concrete, because it sits in a part of the diagram that only a fast shutdown can reach.

A post-starburst galaxy has strong hydrogen absorption lines and no emission lines. The absorption says its light is dominated by A-type stars, which live about a billion years; the absence of emission says nothing is ionising gas, which means no stars above about ten solar masses, which means nothing has formed for at least a few hundred million years.

That combination is only possible if star formation was substantial and then stopped abruptly — over less than a few hundred million years — leaving a population of intermediate-age stars with nothing younger behind them. A gradual decline never produces it, because a gradual decline always leaves some ionising stars.

These galaxies sit in the middle of the optical green valley and near the blue edge of the ultraviolet one, which is exactly what the figures above predict for a fast shutdown observed a few hundred million years in. They are a small fraction of the population — under one per cent — and they are the cleanest evidence that at least some transitions are abrupt.

What was actually measured #

Broadband photometry in several filters, for the same galaxies, with redshifts.

Three difficulties dominate and all three are worse in the ultraviolet than in the optical.

The ultraviolet has to be observed from space, and the surveys that did it are shallower and cover less area than the optical ones. So the sample with both colours is a subset, selected differently, and the selection has to be propagated.

Dust attenuates the ultraviolet enormously. An extinction that costs a tenth of a magnitude in the red costs most of a magnitude in the near ultraviolet, so an ultraviolet colour is far more sensitive to dust than an optical one — and the correction, which is fitted from the same photometry it is correcting, is the leading systematic in every ultraviolet-based measurement — dust being moved light rather than lost light does not help, because what is moved goes to a wavelength the survey does not have.

And the models supplying the band memories are the same models being tested. The conversion from a colour to a time since quenching runs through a stellar population synthesis code, which needs the stellar lifetimes, the mass distribution at birth, the metallicity and the treatment of the thermally pulsing giants that dominate the near infrared. The uncertainty on the absolute timescale is a factor of order 1.5, and the uncertainty on the ratio between two indices’ timescales is smaller, because much of the model cancels.

That last point is the reason the two-filter measurement is more robust than either filter alone. A ratio of two timescales derived from one model is less model-dependent than either, which is the usual consolation and it is real here — the same consolation a colour offers over a magnitude.

The valley is not one place #

Once the crossing time is understood to be a property of the coordinate, a second question follows: whether the objects found in the valley are one population caught in transit or several with different reasons for being there.

Four kinds of galaxy occupy the optical green valley, and only one of them is what the argument assumes.

Galaxies genuinely in transit, whose star formation has recently declined. These are the population the timescale argument is about, and they are a minority.

Dusty star-forming galaxies, reddened into the valley by extinction rather than by age. They are still forming stars vigorously, and they are numerous — an inclined spiral full of dust is one of the commonest objects in any red-selected sample.

Composite galaxies, whose light is the sum of a red bulge and a blue disc. Neither component is in the valley and the average is, which is an artefact of measuring a whole galaxy with one number.

And rejuvenated galaxies, which were red and have acquired gas and started again. They cross the valley in the wrong direction, and a steady-state argument that assumes a one-way flow counts them with the wrong sign.

Separating the four needs more than a colour: a resolved image separates the third, an infrared measurement separates the second, and a spectrum separates the fourth. Each of the discriminants exists and each is available for a much smaller sample than the colour is, which is the trade this subject keeps running into — the measurement that removes the systematic exists, and it exists for a hundredth as many objects.

The practical consequence is that the fraction of valley galaxies which are genuinely in transit has to be estimated rather than counted, and it is estimated at somewhere between a third and a half. Any crossing time inferred from the valley’s total occupancy is therefore too long by a factor of two or three before any of this essay’s argument is applied.

Where the model stops #

The history drawn is one exponential. Real galaxies have bursty histories, and a burst on top of a decline produces a colour that moves backwards — a galaxy can leave the valley toward the blue, which no monotonic model allows. Those galaxies exist and they are a known contaminant of any sample selected by colour.

A galaxy is not one stellar population. It has a bulge and a disc with different histories, and the colour of the whole is a luminosity-weighted average dominated by whichever is brighter in the band. A galaxy whose disc has quenched while its bulge was always red is a different object from one that quenched throughout, and a single colour cannot tell them apart.

The band memories here are a two-parameter fit to a complicated calculation. Each band is modelled as a knee and a power-law decline, which reproduces the qualitative behaviour and not the details — in particular it misses the near-infrared’s dependence on the giant branch, which is the part of population synthesis least under control.

And the steady-state assumption is still there. Reading a deficit as a rate requires that galaxies be entering the valley at a constant rate. Over the last few billion years that is roughly true; at higher redshift it is not, and the same argument applied there has to fold in an evolving supply.

Reading the two together #

The practical upshot is a procedure rather than a caveat, and it is worth stating as one because it is what a survey with both colours actually does.

Take a galaxy with a measured position in an optical colour and in an ultraviolet one. Each position corresponds to a time since quenching, under an assumed shutdown timescale — and the two times will agree only for the right timescale. So the pair of colours determines both the time and the timescale, where either alone determines neither.

That is a two-parameter fit to two numbers, which is exactly determined and therefore carries no goodness of fit: any pair of colours returns some pair of parameters. The check has to come from a third band, and the surveys that have one find that a substantial fraction of galaxies cannot be fitted by any single exponential decline at all.

Those are the galaxies with a burst on top of a decline, or with a rejuvenation, or with a bulge and a disc doing different things. The failure of the two-parameter fit is where the interesting objects are, which inverts the usual reading of a poor fit as a nuisance.

The procedure also explains why the ultraviolet surveys were worth flying despite covering a fraction of the sky the optical ones do. The information a band adds is not proportional to how many galaxies it sees; it is proportional to how different its response time is from what is already there, and the near ultraviolet is an order of magnitude away from every optical band while the optical bands are within a factor of a few of one another.

The generalisation #

The shape worth carrying is that an observable’s response time is part of what it measures, and that a measurement made with one instrument’s response time cannot separate the signal’s timescale from the instrument’s.

A passband is a filter in the signal-processing sense as well as the optical one: it integrates the star-formation history against a kernel, and the width of the kernel is the band’s memory. Measuring a rate with a single such filter gives a number that depends on both, and there is no amount of precision in one filter that separates them. Two filters with different kernels do separate them, and the separation is the difference between the two answers.

The same structure appears wherever a rate is inferred from an integrated quantity. A photometric variability amplitude depends on the cadence; a line ratio’s response to an ionising continuum depends on the recombination time; a proxy for a climate depends on the residence time of whatever carries it. In each case the useful move is not a better single measurement but a second one with a different response, and the first question about any inferred timescale is what the instrument’s own was.

The second reading is the one the colour census got right and the wider literature keeps getting wrong. A deficit in a histogram measures a rate in the coordinate the histogram is drawn in. Change the coordinate and the deficit changes, not because the galaxies did but because the map from time to that coordinate is different. The valley is not a place galaxies avoid; it is a stretch of a particular clock’s face that the hands cross quickly.

Still open: the diagram cut by shape instead #

What comes next splits the population by morphology rather than by colour, and the two classifications agree about eighty per cent of the time. The twenty per cent is the interesting part: red spirals, which have kept their structure and lost their gas, and blue ellipticals, which are usually recent mergers with a burst still fading.

Following these timescales into that split is what connects the two — whether a galaxy’s colour changes before its shape or after it, and by how much. The answer is that colour changes first and by a lot, which is the sense in which what happens to a galaxy’s gas is more decisive than what happens to its stars’ orbits, and the size of the lag is a clock on a process nobody has watched.