Detector Noise and Sensitivity Curve

The sensitivity curve is the daily working object of every gravitational-wave observatory. Advanced LIGO and Advanced Virgo now inject frequency-dependent squeezed vacuum through a $300\,\mathrm{m}$ filter cavity, lowering the quantum-noise envelope below the standard quantum limit and extending the detection range by tens of percent without a single extra watt — the escape route sketched in Derivation 3, now hardware. Coating thermal noise, the mid-band floor, is the target of a global materials effort chasing lower-loss dielectric mirrors, and proposed cryogenic detectors such as Japan's KAGRA and the planned Einstein Telescope and Cosmic Explorer would cool the optics to attack the $\sqrt{k_BT\phi}$ scaling directly. Meanwhile the space mission LISA, with $2.5$-million-kilometre arms and free-falling proof masses, moves the whole curve to the millihertz band that Earth's seismic wall forever forbids, opening supermassive-black-hole mergers and galactic binaries. Every one of these frontiers is a targeted assault on one region of one curve — the plot Weiss first sketched by hand.

Think of the detector as a ruler made of light. A laser beam is split at a half-silvered mirror, sent down two perpendicular kilometre-long arms, bounced off freely-hanging test masses, and recombined. When the arms are equal the recombined light interferes destructively at the readout port — the port is dark. A passing wave lengthens one arm and shortens the other, unbalancing the ruler by $\Delta L=\tfrac12 h L$ and leaking a trickle of light onto the photodiode. The first figure fixes the scale of the problem: the length change to be measured is smaller than an atomic nucleus.

Now the second figure is what a detector builder actually stares at. Every impostor process has its own frequency signature. The ground never stops shaking, and although stacked pendulums filter that motion enormously, below about $10\,\mathrm{Hz}$ the residual seismic drive and the direct Newtonian tug of passing density fluctuations climb into an insurmountable wall. In the middle band the mirrors and their glass fibres are warm, and the fluctuation-dissipation theorem guarantees that any material with mechanical loss jitters thermally — the coatings on the mirror faces set the most sensitive floor. At high frequency the graininess of light takes over: photons arrive at random, and counting them to infer a phase has an irreducible Poisson uncertainty. The incoherent sum of all three carves the characteristic bucket.

Two of these three families are consequences of deep physics, not sloppy engineering. Thermal noise follows from the fluctuation-dissipation theorem: the same molecular friction that would damp a mirror's ringing also drives it with random thermal kicks, and you cannot have one without the other. Quantum noise follows from Heisenberg: reading a mirror's position with light disturbs its momentum, and the harder you look the harder you kick. You cannot wish these floors away — you can only cool the optics, choose low-loss materials, shape the light with squeezing, or leave the planet. The last lever, laser power, sits at the heart of the quantum trade-off. More power means more photons per second, sharper phase readout, and less shot noise — it falls as $P^{-1/2}$. But more power also means a stronger, noisier stream of photon momentum battering the mirrors, so radiation-pressure noise rises as $P^{1/2}$. Reading the ruler more clearly and disturbing it more are two faces of the same photons, and their optimal balance defines the standard quantum limit: the best a plain position measurement can do.

The chain from wave to signal begins with geodesic deviation. In the transverse-traceless gauge two freely-falling mirrors a distance $L$ apart have their proper separation modulated by the strain projected onto that arm, and a Michelson reads the difference between two perpendicular arms, doubling the effect.

where: $\Delta L$ — differential change of an interferometer arm length; $h(t)$ — TT-gauge gravitational-wave strain projected onto the arm; $L$ — arm length ($4\,\mathrm{km}$ for LIGO). Reading: the detector is a strain meter, not a length meter — its output scales with $L$, which is why arms are built as long as engineering allows. Because noise is a stationary random process, it has no waveform, only a spectrum. The Wiener-Khinchin theorem turns the autocorrelation into a power spectral density; the detector community folds it onto positive frequencies (the "one-sided" convention) and uses the equivalent statement about the covariance of Fourier bins.

where: $S_n(f)$ — one-sided noise PSD (strain$^2$/Hz), defined for $f\ge 0$; $C_n(\tau)=\langle n(t)n(t+\tau)\rangle$ — autocorrelation of the stationary noise; $\tilde n(f)$ — Fourier transform of the noise time series. Reading: the plotted sensitivity curve is $\sqrt{S_n(f)}$, with units $\mathrm{Hz}^{-1/2}$; it is a density, not a strain, and that distinction underlies the central misconception below. Detectability of a known waveform in this noise is governed by the matched filter. Weighting each frequency by inverse noise power gives the optimal statistic; the resulting signal-to-noise ratio is a single integral.

where: $\rho$ — optimal (matched-filter) SNR; $\tilde h(f)$ — Fourier transform of the expected signal; $S_n(f)$ — one-sided noise PSD. Reading: bands where $S_n$ is small dominate the reach, so the shape of the whole bucket matters, not just its minimum — the origin of the derivation below. The mid-band floor is thermal. The fluctuation-dissipation theorem relates the displacement spectrum of a suspended mirror to the dissipative part of its mechanical response, modelled as an oscillator with a small structural loss angle $\phi$.

where: $S_x^{\rm th}$ — displacement PSD from thermal motion; $k_B T$ — thermal energy ($T\approx290\,\mathrm{K}$ at room temperature); $Z(\omega)$ — mechanical impedance of the mode; $\omega_0,\phi$ — resonant angular frequency and loss angle; $m$ — effective modal mass. Reading: off resonance ($\omega\gg\omega_0$) this falls steeply, so low loss $\phi$ and heavy masses push thermal energy out of the measurement band and into a narrow resonance spike. At high frequency the floor is photon shot noise. Counting a finite number of photons to infer a phase carries Poisson uncertainty, which translates into an equivalent strain noise for a simple Michelson.

where: $\hbar$ — reduced Planck constant; $\lambda$ — laser wavelength ($1064\,\mathrm{nm}$); $P$ — laser power at the beam splitter; $L$ — arm length; $c$ — speed of light. Reading: shot noise falls as $P^{-1/2}$, which is why detectors run kilowatts of circulating power and add optical cavities to build it up further. Push the power up and radiation-pressure back-action climbs to meet the falling shot noise. Optimizing power frequency-by-frequency yields a power-independent envelope, the standard quantum limit for a free test mass.

where: $m$ — test-mass mass; $f$ — gravitational-wave frequency; $L$ — arm length. Reading: the SQL depends only on $m$, $f$, $L$ and $\hbar$ — no laser parameter survives — and it is a direct fingerprint of the position-momentum uncertainty relation on the mirror. Finally, the science payoff. Since a source's amplitude scales as $\tilde h\propto1/D$, holding the SNR at a detection threshold defines a horizon distance set by the inverse-PSD integral over the source's band.

where: $D_{\rm hor}$ — maximum distance at which a fiducial source reaches threshold SNR; $\tilde h_{1\,\rm Mpc}$ — the waveform scaled to $1\,\mathrm{Mpc}$. Reading: improve $\sqrt{S_n}$ by a factor across the band and the horizon grows by the same factor, while the surveyed volume — and the event rate — grows by its cube.

Each band of the sensitivity curve is ruled by a different mechanism with its own frequency scaling; reading the curve means knowing which term wins where.

1 · How small is the signal, and how much light does it take to see it? (fully worked)

GW150914 reached a peak strain $h\approx1.0\times10^{-21}$. Take LIGO's $L=4\,\mathrm{km}$ arms and $\lambda=1064\,\mathrm{nm}$ light. Find the differential arm-length change, the optical phase it produces, and an order-of-magnitude estimate of the light power needed to resolve that phase against shot noise in a $\sim0.01\,\mathrm{s}$ merger.

Step 1. Length change from equation (1): $\Delta L=\tfrac12 hL=\tfrac12(1.0\times10^{-21})(4000\,\mathrm{m})=\mathbf{2.0\times10^{-18}\,m}$. Compared with a proton radius $\approx0.84\times10^{-15}\,\mathrm{m}$, this is about $1/420$ of a proton — the headline number that makes the whole enterprise sound impossible.

Step 2. Optical phase. A differential arm change produces $\Delta\phi=4\pi\Delta L/\lambda$ (a factor 2 for the round trip, a factor 2 for the differential readout): $\Delta\phi=4\pi(2.0\times10^{-18})/(1.064\times10^{-6})=\mathbf{2.4\times10^{-11}\ rad}$. The interferometer must resolve a phase eleven orders of magnitude below a full fringe.

Step 3. Photons required. Poisson counting gives phase precision $\delta\phi\sim1/\sqrt{N}$, so resolving $\Delta\phi$ needs $N\gtrsim1/\Delta\phi^2=1/(2.4\times10^{-11})^2\approx\mathbf{1.8\times10^{21}\ photons}$ accumulated during the burst.

Step 4. Convert to power. Each $1064\,\mathrm{nm}$ photon carries $E=hc/\lambda=1.87\times10^{-19}\,\mathrm{J}$, so $N$ photons carry $NE\approx335\,\mathrm{J}$, and delivering that in $\tau\approx0.01\,\mathrm{s}$ demands $P\approx NE/\tau\approx\mathbf{3.4\times10^{4}\ W}\approx34\,\mathrm{kW}$. The signal is a $2\times10^{-18}\,\mathrm{m}$, $2.4\times10^{-11}\,\mathrm{rad}$ wobble, and beating shot noise on it needs tens of kilowatts of light — which is exactly why Advanced LIGO stores tens of kilowatts of circulating power using arm cavities and power recycling rather than a bare Michelson.

2 · The standard quantum limit at 100 Hz

Advanced LIGO uses $m=40\,\mathrm{kg}$ test masses and $L=4\,\mathrm{km}$ arms. Evaluate the free-mass SQL at $f=100\,\mathrm{Hz}$ and compare it to the design floor of $\sqrt{S_n}\approx2\times10^{-23}\,\mathrm{Hz}^{-1/2}$.

Step 1. Angular frequency: $\omega=2\pi f=628\,\mathrm{rad/s}$, so $\omega^2=3.95\times10^{5}\,\mathrm{s^{-2}}$.

Step 2. Denominator of equation (6): $m\,\omega^2 L^2=(40)(3.95\times10^{5})(4000)^2=2.53\times10^{14}$ in SI units.

Step 3. Numerator $8\hbar=8(1.055\times10^{-34})=8.44\times10^{-34}$, so $S_h^{\rm SQL}=8.44\times10^{-34}/2.53\times10^{14}=3.3\times10^{-48}\,\mathrm{Hz^{-1}}$, and $\sqrt{S_h^{\rm SQL}}=\mathbf{1.8\times10^{-24}\ Hz^{-1/2}}$.

Step 4. Interpretation: the SQL sits about a factor of ten below the achieved floor at $100\,\mathrm{Hz}$, so at this frequency Advanced LIGO is limited by coating thermal noise, not by quantum mechanics — there is genuine headroom, and squeezed light plus better coatings are the two levers being pushed toward the SQL.

3 · What a factor-of-two upgrade buys

A detector upgrade lowers $\sqrt{S_n(f)}$ by a uniform factor of 2 across the band. By how much do the SNR, the horizon distance, and the expected event rate change?

Step 1. From equation (3), $\rho\propto1/\sqrt{S_n}=1/(\sqrt{S_n})$, so halving the amplitude spectral density everywhere doubles $\rho$.

Step 2. Because $\tilde h\propto1/D$, holding $\rho$ at the detection threshold means the reachable distance scales with $\rho$: the horizon distance doubles.

Step 3. For a roughly uniform distribution of sources the surveyed volume scales as $D^3$, so the rate rises by $2^3=\mathbf{8}$. A modest broadband factor of two multiplies the catalogue by roughly eight — the cubic leverage that justifies expensive upgrades to the whole curve.

The sensitivity curve did not arrive with the first detection; it arrived four decades earlier as a hand-written noise budget by the man who realized that measuring a strain of $10^{-21}$ was, mechanism by mechanism, not obviously impossible.

Texts. Misner, Thorne & Wheeler, Gravitation, §37 (detection of gravitational waves) · Schutz, A First Course in General Relativity, 2nd ed., ch. 9 · Maggiore, Gravitational Waves Vol. 1, ch. 7 (interferometers and their noise) · Saulson, Fundamentals of Interferometric Gravitational Wave Detectors · Carroll, Spacetime and Geometry, ch. 7 (gravitational radiation) for the source side.

Provenance. The matched-filter inner product and the factor-4 one-sided-PSD convention follow Maggiore Vol. 1, §7.3, and Allen et al., Phys. Rev. D 85, 122006 (2012). The fluctuation-dissipation displacement spectrum is Saulson, Phys. Rev. D 42, 2437 (1990), reproduced in Maggiore §7.4. The SQL form $8\hbar/(m\omega^2L^2)$ and the shot-noise scaling match Maggiore §7.5 and Saulson's textbook; squeezing below the SQL is Caves, Phys. Rev. D 23, 1693 (1981). Advanced LIGO parameters ($\sqrt{S_n}\approx2\times10^{-23}\,\mathrm{Hz}^{-1/2}$ near $100$–$300\,\mathrm{Hz}$, seismic wall $\sim10\,\mathrm{Hz}$, $\lambda=1064\,\mathrm{nm}$, $L=4\,\mathrm{km}$, $40\,\mathrm{kg}$ fused-silica test masses) are from Aasi et al. (LIGO Scientific Collaboration), Class. Quantum Grav. 32, 074001 (2015). GW150914's peak strain and frequency sweep are from Abbott et al., Phys. Rev. Lett. 116, 061102 (2016). A verifier should confirm: $\Delta L=2\times10^{-18}\,\mathrm{m}$ and $\Delta\phi=2.4\times10^{-11}\,\mathrm{rad}$ for $h=10^{-21}$; $\sqrt{S_h^{\rm SQL}}=1.8\times10^{-24}\,\mathrm{Hz}^{-1/2}$ at $100\,\mathrm{Hz}$ for $m=40\,\mathrm{kg}$; and the factor-8 volume scaling.

Builds on: Transverse-Traceless Gauge · Geodesic Deviation and Gravitational Waves · Plus and Cross Polarizations · Quadrupole Formula. Leads to: Quadrupole Formula · Isaacson Stress-Energy of Gravitational Waves.

Ask the tutor: Ask me to walk through why a factor-of-two sensitivity gain multiplies the merger catalogue by eight, or to derive the standard quantum limit from the shot-versus-radiation-pressure trade-off step by step.