What the instrument actually hears

The wave that stretches one way and squeezes the other draws the quadrupole pattern: a ring of test masses distorted into an ellipse, one axis longer while the other is shorter. The natural next thought is that an interferometer measures that directly — put a mirror on each axis, and the beams take different times to return.

That thought is right in its conclusion and wrong in its reasoning, and the difference matters for what the instrument can and cannot do.

The objection that has to be answered first #

The standard objection to the naive picture is worth meeting head on, because anybody who has thought about it for five minutes has raised it.

If space is stretched along one arm, then the wavelength of the light in that arm is stretched by the same factor at the same moment. A ruler made of light would stretch with what it is measuring and report nothing. Where does the signal come from?

The answer is that the light is not a ruler. It is a clock: what is compared is the arrival time of two beams that left together, and a time is not stretched by a metric perturbation in space. Solving the null geodesic equation in the wave’s own gauge gives a round-trip time that depends on the wave, and that difference is what the interferometer converts into a phase.

A local measurement of a length has no meaning here; a comparison of two round-trip times does. That is why the calculation is done in terms of light travel time rather than in terms of mirror positions, and it is also why the mirrors’ coordinate positions do not move at all in the natural gauge — a fact that sounds like a paradox and is merely a choice of labels.

Why the response has zeros #

Once the measurement is understood as an average over the light’s stay, the shape of the frequency response follows without any further physics.

A wave slow enough to be effectively steady while the light is in the arm is measured in full. A wave that completes a whole cycle during the stay contributes as much positively as negatively, and the response is exactly zero. In between it is a sinc, and the figure checks both the flat part and the first null rather than drawing a curve of the right general shape.

For a single pass down a four-kilometre arm the first null is at thirty-seven kilohertz, which is far above anything astrophysical, so the response is flat across the whole band of interest. That is not an accident of design; it is why a single-pass interferometer would work at all.

The real instruments store the light for far longer, which is where the trade appears.

The trade the storage time makes #

Storing the light for $N$ round trips multiplies the signal by $N$, because the phase accumulated is the sum over trips. It also divides the null frequency by $N$, because the averaging time is longer.

That is a genuine trade rather than a free gain, and it is the central design decision in a detector. A long storage time buys sensitivity where the sources are loud and slow — a pair of black holes in their last orbits, at tens to hundreds of hertz — and loses it where they are fast, which is the last milliseconds of a neutron-star merger where the most interesting physics is.

Modern instruments do not use a fixed number of bounces; they use a Fabry–Perot cavity, whose effective storage time and hence whose response shape can be tuned by changing the mirror transmissions, and a signal-recycling mirror that shapes the response further. The essential arithmetic is unchanged: the light’s stay in the arm sets a corner frequency, and everything above it is bought back at a cost.

The shorter arm in the figure is the same statement about scale. A six-hundred-metre instrument has a corner frequency nearly seven times higher for the same number of bounces, which is one reason the smaller detectors in a network are useful — they are less sensitive overall and better at high frequency.

Where the instrument is deaf #

The antenna pattern is the other half of what a single instrument cannot do.

An L-shaped interferometer compares two arms, so it is blind to any wave that changes both equally. There are four such lines in the sky for the plus polarisation, and the pattern vanishes on them exactly. The cross polarisation is blind on different lines, and the two sets do not overlap — so the instrument always hears something, at some strength, from every direction.

What it cannot do is know which. A single detector’s output is one number at each instant, and that number mixes the source’s direction, its distance, its inclination and its two polarisations. Even the loudest signal from one instrument leaves the sky position essentially unconstrained.

The sky-averaged value of the squared pattern is the number that turns a strain sensitivity into a range, and the figure sums it over a hundred and sixty thousand directions rather than quoting it: the two polarisations’ averages add to two fifths, and averaged over polarisation angle as well each contributes a fifth. A detector’s quoted reach is roughly half its reach for a source directly overhead, and the factor is entirely geometrical.

How small the signal is #

The numbers are worth putting side by side because they explain why the measurement took a century to make.

The length change is four times $10^{-18}$ metres. A proton is two hundred times wider. The mirrors are made of atoms whose thermal motion is enormously larger, the coating on their faces is thicker than the signal by fifteen orders of magnitude, and the ground the whole apparatus stands on moves by micrometres.

None of that is an obstacle in the way it sounds, because none of those motions is the measurement. What is measured is the difference between two arms, at a specific frequency, averaged over many cycles, with the laser supplying enough photons that the phase of the average is known far better than the phase of any one. The thermal motion of a mirror is a real noise source and enters at a level set by its dissipation rather than by its amplitude — which is the fluctuation–dissipation theorem doing the essential work.

The signal is not small compared with the noise; it is small compared with things that are not noise, and separating those two categories is most of what the instrument’s design is about.

The limit the light itself sets #

The phrase “enough photons” in the last section hides the whole of the instrument’s ultimate limit, and it is worth unpacking because it is a quantum limit reached by a kilometre-scale machine.

Light arrives in countable lumps — light arrives in lumps — so the number reaching a photodiode in a given time fluctuates, and the phase of the interference is uncertain by roughly one over the square root of that number. More laser power means more photons and a better-known phase, so shot noise falls as the square root of the power. That is why these instruments run hundreds of kilowatts of circulating light in the arms.

More power has a cost that appears at the other end of the band. The photons carry momentum, they arrive in fluctuating numbers, and the fluctuation pushes the mirrors about. That radiation-pressure noise rises with power and dominates at low frequency, so the total is minimised at one particular power and no lower.

The trade between them is the uncertainty relation applied to a measurement: knowing the phase better means disturbing the mirror more, and sharpness has to be paid for is the same statement in its original setting. The minimum of the sum is the standard quantum limit, and current instruments sit within a factor of a few of it across part of their band.

Beating it requires squeezed light, in which the phase is measured better than the shot-noise limit at the cost of the amplitude being measured worse. That is now routine: the observatories inject squeezed vacuum into the dark port and gain tens of per cent in range, which is one of the few places where a laboratory quantum-optics technique is doing production astronomy.

What a second detector buys #

Two instruments do three things a single one cannot, and only one of them is confirmation.

They discriminate against local disturbances, including the seismic motion that a layer a parcel cannot leave would describe in a fluid, since a genuine wave arrives at both with the right delay and a truck driving past one does not. That is the obvious benefit and it was the reason two were built.

They separate the polarisations, because two differently oriented instruments have different antenna patterns and their two outputs constrain the two polarisation amplitudes.

And they localise, badly. An arrival-time difference fixes the angle between the source and the baseline, which is a ring on the sky — thousands of square degrees, a few degrees wide. The figure computes the ring for a given delay, and shows that the curve is flattest at the ends, so a source nearly along the baseline is the hardest of all to place.

Three detectors give two rings and hence two patches. A fourth chooses between them and shrinks the remaining patch to tens of square degrees, which is the point at which a telescope can be pointed. The multi-messenger observations that followed a neutron-star merger were possible because three instruments were running, and would not have been with two.

What comes out of a detection #

It is worth listing what the instrument actually delivers, because the list is shorter and stranger than an image from a telescope.

The chirp mass, very precisely. The rate at which the frequency sweeps upward is set by a particular combination of the two masses, and the orbit that has to shrink computes it. That combination is measured to a per cent or better even for a modest signal, because it is read from a frequency evolution rather than from an amplitude.

The individual masses, badly. Separating them requires the later part of the waveform where the two masses enter differently, and the uncertainty is often tens of per cent — which is why the mass ratio of most detected binaries is poorly known while their chirp mass is not.

The distance, degenerate with the inclination. The amplitude depends on both, and a single detector cannot separate them; a network partially can, through the polarisation, and the residual degeneracy is the largest uncertainty in using these events to measure the expansion of the universe.

The spins, partially, through their effect on the phase evolution and on the precession of the orbital plane.

And the equation of state, when neutron stars are involved, through the tidal deformation in the last few orbits — a direct constraint on what the mass no cold matter can hold up is asking about, obtained from the phase of a waveform rather than from any observation of a star’s surface.

Nothing on that list is an image, and nothing is a position better than a patch of sky. The instrument is a spectrometer for a single number’s evolution in time, and everything astronomical is inferred from the shape of that evolution.

Why the arms are perpendicular #

The L shape looks like a convenience of construction and is not: it is the shape that maximises the signal, and the reason is in the quadrupole pattern itself.

A wave stretches one direction while squeezing the perpendicular one. Two arms at right angles therefore see changes of opposite sign, and the difference between them is twice what either arm gives alone. Two arms at any other angle see a smaller difference, and two arms parallel see none at all — they are one arm measured twice.

The factor of two is worth having and it is not the main point. The main point is that the common motion of the two arms cancels. Anything that changes both arms equally — the laser’s own frequency drifting, the whole apparatus expanding with temperature, the ground moving both end stations together — is subtracted out, and only the differential part survives. The perpendicularity is a noise-rejection scheme as much as a signal-enhancement one, and it is the reason the instrument can work at all against disturbances fifteen orders of magnitude larger than the signal.

That is also why the blind directions exist. They are exactly the directions from which a wave changes both arms equally, and a common-mode-rejecting instrument must be deaf to them. The nulls in the antenna pattern are the price of the rejection, and any instrument that removed them would have given up the thing that makes the measurement possible.

Where the model stops #

The antenna pattern is for a wavelength much longer than the arm. Above the corner frequency the pattern itself becomes frequency-dependent and the simple angular factors stop applying, which matters for a space-based detector whose arms are millions of kilometres long and whose sources are correspondingly slow.

The response is drawn for a fixed number of bounces. A Fabry–Perot cavity’s response is a pole rather than a sinc, with the same corner frequency and a different shape, and modern instruments add a signal-recycling mirror that can move the peak sensitivity around. The trade between storage time and bandwidth is unchanged.

The detector is treated as rigid. The mirrors are suspended and the suspensions have resonances, the mirrors themselves have internal modes at tens of kilohertz, and the real noise curve is a landscape of peaks that none of this describes.

And the network figure assumes timing is the only information. In practice the relative amplitudes and phases at the detectors constrain the direction as well, and a full localisation uses all of it — which improves the ring into an arc rather than replacing the geometry.

What the pictures cannot show #

The sky maps are drawn with the response’s magnitude and discard its sign, which carries the phase. Two directions with the same shading give signals of opposite sign, and it is the sign that lets a network separate a source from its antipode.

The frequency-response figure draws the instrument’s answer to a wave and not the noise it competes with. A detector’s usable band is set by the ratio of the two, and the noise rises steeply at low frequency for reasons that have nothing to do with the arm — seismic motion, and the fluctuating gravitational pull of moving air and ground. The response is flat there and the instrument is deaf anyway.

A third omission is time. Every figure treats the detector as fixed and the sky as fixed, and neither is: the Earth turns, so the antenna pattern sweeps across the sky over a day, and a source that is in a blind direction at one hour is not at the next. For the signals seen so far — a fraction of a second long — that rotation is irrelevant and the pattern is a snapshot. For a continuous source, a spinning neutron star emitting for months, the rotation is the whole of the localisation: the daily modulation of the amplitude and the annual Doppler shift from the Earth’s orbit are what pin the source down, and one detector suffices. The same instrument localises badly in a millisecond and superbly over a year, and no figure drawn here can carry the difference.

Where the ladder goes next #

The gravitational-waves ladder began with the wave that stretches one way and squeezes the other, which is the quadrupole pattern and the two polarisations, and continued with the orbit that has to shrink, where the energy carried away is computed and checked against a binary pulsar. This rung asks what an instrument can extract from a passing wave and finds the answer bounded by geometry and by the light’s own stay in the arm. The rungs after it: the noise budget, where the quantum limit and the thermal limit meet; the matched filter, which is how a signal a tenth the size of the noise is found at all; and the space-based detectors, whose arms are long enough that everything here is evaluated in the opposite limit.

The habit worth carrying away is that an instrument’s response is a physical object with its own structure. The zeros in the frequency response and the blind directions in the sky are not imperfections, they are what a comparison of two paths must have — and knowing where they are is what turns a detection into a measurement.