Astronomy & Celestial Mechanics
the sky, computed
Concept

Selection function — where it appears

The probability that an object with given properties makes it into a catalogue, as a function of those properties. It is a property of the survey rather than of the sky, and any statement about a population is a statement about the data divided by it.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A sample that gets brighter with distance because the faint ones drop out. The mean absolute magnitude of a magnitude-limited sample, relative to the population it is drawn from, against distance. The population has a spread of 0.5 magnitudes about a mean of -4, and the survey stops at apparent magnitude 20. Nearby, everything is detected and the sample is unbiased. Beyond about 316228 parsecs the faint end of the distribution starts falling below the limit and the survivors are brighter than average; further out the bias deepens without limit, because eventually only the extreme tail is detectable. The horizontal line is the classical Malmquist value of 1.382 times the square of the spread, which is what the bias averages to over a magnitude-limited sample as a whole — it is a property of the sample rather than of any one object, and using it as a correction for an individual star is a common and specific mistake.

A sample brighter than the population it came from

Every survey stops at some apparent brightness. At any distance it therefore contains only the objects luminous enough to make the cut, so the average object in it is brighter than the average object in the universe — by an amount that grows with distance and that has been shortening every distance in astronomy since 1920.

starlight · Distance ladder A high end that is 2.5 in slope steeper than the one stars were born with. The mass function a star count measures against the one stars were born with, for a population that has been forming stars steadily for 10 billion years. Below about a solar mass nothing has had time to die, so the two coincide exactly. Above it the fraction still alive is the main-sequence lifetime divided by the age, and since the lifetime falls as the two-and-a-half power of the mass, the present-day function is steeper than the initial one by exactly that exponent. A count of massive stars in an old population therefore under-represents them by orders of magnitude, and reading it as an initial mass function gives a slope far too steep. The correction is large, it is calculable, and it depends on the star-formation history — which is usually the thing the mass function was going to be used to constrain.

A mass function corrected by an age

Counting stars by mass gives the stars that are alive. What every argument needs is the stars that were born, and the two differ by the fraction of each mass still on the main sequence — which is a lifetime divided by an age, and which for massive stars in an old population is a very small number.

stars · Initial mass function 1020 clusters or 372, and the survey cannot say which parameter moved. The number of clusters per unit redshift a survey finds above an integrated Compton signal of 8·10⁻⁵ arcmin² over 6 per cent of the sky, computed from the Sheth–Tormen halo count grown by the linear growth factor, the comoving volume in each redshift slice, and the calibrated relation between signal and mass. Each curve is one pair of assumptions: the amplitude of structure, σ₈, and the hydrostatic mass bias, 1 − b, which says how far the X-ray masses the relation was calibrated on fall below the true masses. With σ₈ = 0.811 and 1 − b = 0.8 the survey finds 1020; σ₈ = 0.811 with 1 − b = 0.6 gives 372; σ₈ = 0.75 with 1 − b = 0.8 gives 548. The counts fall at low redshift because there is little volume, and at high redshift because massive halos have not yet formed, and the peak sits near z = 0.24. Lowering 1 − b pushes the threshold onto more massive and rarer halos; lowering σ₈ makes every halo rarer. The two lower curves differ in total by 47 per cent and in normalised shape by at most 2 per cent of the peak. A survey that assumed 1 − b = 0.8 would read the 372 clusters of the curve with 1 − b = 0.6 as σ₈ = 0.716, with a redshift distribution that differs from it by at most 12 per cent of the peak — which is the only handle the survey has on the difference, and it is smaller than the counting noise in any redshift bin holding fewer than about 67 clusters. A total count cannot distinguish a universe with less structure from a survey that has misjudged its masses, and it is that degeneracy, not the counting, that has been argued about since the first large catalogue.

Too few clusters, or a scale that reads light

A catalogue selected on the microwave shadow is, past redshift one half, very nearly a catalogue of everything above a fixed mass — so its count by redshift is the growth of structure read almost directly. Almost, because the mass behind the threshold comes from a calibration, and a scale that reads twenty per cent light is indistinguishable from a universe with less in it.

cosmology · Sunyaev zeldovich A calibration sample, and the stars a precision cut keeps. Measured parallax against apparent magnitude for one star in four of a simulated sample of 8000 standard candles of absolute magnitude 0.5 and intrinsic scatter 0.15 mag, spread through a disc 15 kpc in radius and 300 pc thick and catalogued to magnitude 16. The parallax axis is stretched near zero so the negative measurements show. The solid curve is the parallax a star of exactly the calibrated luminosity would have at each magnitude, and the dashed curve is the cut ϖ/σ = 5: stars above it are the "good parallaxes" a calibration usually keeps. It keeps 3242 of the 8000, 41 per cent, and at faint magnitudes the curve it cuts through runs through the middle of the stars rather than below them, so the ones kept there are the ones whose errors pushed their parallaxes up. 45 stars have negative parallaxes; no inversion can use them, and a fit in parallax space uses them like any other.

A calibration chosen by the measurement it calibrates

A standard candle is calibrated on stars with good parallaxes, and "good" is decided by the parallax itself — so the calibrators are the stars whose errors happened to push them nearer. Correcting each star's distance afterwards does not repair it, because the correction needs a picture of where the sample's stars really lie, and that picture depends on the luminosity being calibrated. What works is to choose nothing, keep every star with the negative parallaxes included, and fit the candle by predicting the parallaxes rather than inverting them.

starlight · Parallax

Named alongside it

The objects these essays reach for when they reach for this one.

CompletenessMalmquist biasStandard candleSystematic errorAngular-diameter distanceCluster mass functionDegeneracyDistance ladderEddington biasFlux-limited sampleHalo mass functionHydrostatic bias

All concepts