Concept

Luminosity distance — where it appears

The distance a measured flux implies if the inverse-square law is taken to hold, larger than the comoving one by a factor of 1+z. It is what a standard candle returns, and the factor of 1+z arises because the photons arrive both redshifted and spread over a longer interval.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

$GW150914: 33 Hz to 250 Hz in 0.22 seconds. The strain of GW150914 — two black holes — through the last 0.22 seconds before merger, computed from the quadrupole sweep at the chirp mass its fit returned, 28.716 solar masses, and drawn at the luminosity distance it returned, 440 megaparsecs. Two things rise together and neither is free to rise on its own: the frequency goes from 33 Hz to 250 Hz, and the envelope — the outer curve — grows by a factor of 3.9, because the amplitude goes as f^2/3 and nothing else in it changes over so short a span. The vertical axis is in units of 10⁻²¹, so the peak here is a fractional length change of about 2.9·10⁻²¹: over the four kilometres of an interferometer arm that is 1.2·10⁻¹⁷ metres, a thousandth of the width of a proton. The chirp mass is not fitted to the amplitude at all — it comes from the spacing of these zero crossings, which is why it is the best-determined number in the whole event and why the distance, which does come from the amplitude, is the worst.$

A distance with no ladder under it

The frequency sweep of an inspiral fixes the chirp mass with no distance in it, and the amplitude then gives the luminosity distance directly, because one expression fixes both. That is a distance measured with nothing calibrated beneath it — and its error budget is one angle.

gravitation · Gravitational waves

One measured number, and every pair of masses that produces it. The plane of the two component masses of an inspiralling binary, with three curves of constant chirp mass across it. The middle one is GW150914's value of 28.7 solar masses, and the chirp mass recovered from the coordinates of the drawn curve varies along its whole length by 7.4e-14 per cent — which is the point: every binary on that line radiates the same frequency sweep at leading order, so the early inspiral cannot tell them apart. Two of them are marked. An equal pair of 33.0 and 33.0 solar masses and a lopsided pair of 63.6 and 18.4 sit on the same contour, and their total masses differ by a factor of 1.24. What separates them is the mass ratio, which enters the phasing only at the first post-Newtonian order, suppressed by the square of the orbital speed in units of the speed of light — small through the hundreds of cycles that carry most of the signal, and appreciable only in the last few, where that speed approaches a third of c. So the chirp mass is a measurement and the individual masses are an inference from the end of the signal, which is exactly the part a detector's high-frequency noise eats first.

One number where two masses were

The hundreds of orbits an inspiralling binary completes inside a detector's band depend on its two masses only through one combination of them. Every pair on that contour radiates an identical sweep, so the early signal — which carries nearly all the signal-to-noise — cannot say which pair it was.

gravitation · Gravitational waves

Four distances to the same galaxy

Inside the Local Group the word "distance" has one meaning. Past a redshift of about a tenth it has four, they disagree by factors of a hundred by the time the light is old, and one of them stops increasing and starts coming back.

cosmology · Distance ladder

Distance modulus against redshift, for three universes. The distance modulus μ = 5 log₁₀(D_L/10 pc) against redshift for three universes, all with H₀ = 67.36 km/s/Mpc, with 60 model supernovae drawn from the ΛCDM curve with 0.15 magnitudes of scatter. The point of the figure is how little difference there is: across two decades of redshift the three curves stay within a few tenths of a magnitude, and at z = 0.5 the accelerating and decelerating cases differ by 0.387 mag. A cosmology is not read off this plot. It is read off the residual, which is the next figure.

An expansion that was supposed to be slowing

Gravity is attractive, so an expanding universe full of matter must be decelerating, and the only question was by how much. Two teams set out to measure the deceleration and both found a quarter of a magnitude of extra faintness at redshift half — which is the wrong sign.

cosmology · Dark energy

One amplitude, and every point on this curve fits it. The set of distances and inclinations that produce the same measured amplitude in one detector. A binary seen face-on radiates most strongly along its spin axis, so it can be twice as far away as an edge-on binary and still arrive with the same strain — the curve is (1 + cos²ι)/2 and the factor between its ends is exactly two. Nothing in a single detector's data distinguishes the two ends. What makes this worse rather than merely awkward is that the prior pulls the other way: an isotropically oriented population has half its members beyond 60 degrees, so most binaries really are closer to edge-on, and a posterior that combines a flat likelihood along this curve with that prior returns a distance biased low with an error bar that understates the range. The whole business of standard-siren cosmology is the business of cutting across this curve.

A distance tangled with an angle

A gravitational wave carries its own distance, with no ladder underneath it and nothing to calibrate. What it also carries, inseparably, is the orientation of the orbit that made it — and one detector cannot tell a nearby binary seen edge-on from one twice as far away seen face-on.

gravitation · Gravitational waves

Four spectra, and 3.9 magnitudes between them at the same redshift. The K-correction — the magnitude that has to be added to compare a redshifted object with a nearby one through the same filter — against redshift, for four power-law spectra. At zero redshift every correction is zero by construction. Beyond that they diverge, because a filter at a fixed observed wavelength samples a different part of the source's own spectrum at every redshift, and how much flux is there depends on the spectrum. A flat spectrum needs no correction at all at any redshift, and the two extremes drawn differ by 3.9 magnitudes by z = 1.2. The circularity is the point: applying the correction requires the spectrum, and the spectrum is what a magnitude is being used to constrain. The dashed line is the way out — observe in a band chosen so that it lands on the rest-frame band of interest, and the spectral term cancels, leaving only the bandwidth stretch.

A magnitude in a band the source never had

A filter passes light at a fixed observed wavelength, and a redshifted source emitted that light at a shorter one. Comparing a distant galaxy with a nearby one through the same filter therefore compares two different parts of two spectra — and the correction between them needs the spectrum, which is what the magnitude was going to be used to find out.

starlight · Magnitudes

Named alongside it

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

Standard siren Chirp mass Degeneracy Hubble constant Inspiral K-correction Matched filtering Accelerating expansion Angular-diameter distance Antenna pattern Bandpass Bandwidth stretch

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