Instrumental Noise Sources

Detector noise is best organized by a coupling path, not merely by a name on a sensitivity curve. A fluctuating physical quantity must reach the differential arm readout through mechanics, optics, electronics, control, or gravity before it appears as equivalent strain. This viewpoint separates a disturbance source from its transfer function, exposes correlated terms, and tells an experimentalist which measurement can confirm or reject a proposed explanation.

Required background. Probability and Inference for Gravity supplies random processes, covariance, and conditional reasoning.

Helpful background. Gravitational Radiation and Waveforms fixes strain normalization. Computational Relativity supplies numerical filtering, convergence tests, and reproducible data workflows.

From a disturbance to equivalent strain

Let $q_a(t)$ be a physical disturbance: ground displacement, laser-frequency fluctuation, coating motion, magnetic field, or an electronic control signal. In a locally linear regime its contribution to calibrated strain is

$$\widetilde n_a^{\,h}(f) =T_{h q_a}(f;t)\widetilde q_a(f),$$

where $T_{h q_a}$ is a complex, possibly time-dependent coupling function. If the sources are mutually uncorrelated, their one-sided power spectral densities add:

$$S_h(f)=\sum_a \left\lvert T_{h q_a}(f)\right\rvert^2S_{q_a}(f).$$

In general the cross terms matter:

$$S_h(f)=\sum_{a,b} T_{h q_a}(f)T_{h q_b}^*(f)S_{q_aq_b}(f).$$

This is why adding amplitude spectral densities is wrong and why a budget can exceed or undershoot the measured curve when correlations or omitted couplings are important. The units provide an immediate check: if $q$ is measured in metres and strain is dimensionless, $T_{hq}$ has units $\mathrm{m}^{-1}$, and $\lvert T_{hq}\rvert^2S_q$ has units $\mathrm{Hz}^{-1}$.

A measured strain spectrum says what fluctuation is present at the output. A noise budget makes the stronger claim that identified sources, projected through measured or modeled couplings, explain that output. The latter needs uncertainty bands, epoch information, and closure tests.

Quantum measurement noise

Vacuum fluctuations entering unused optical ports produce fluctuations in detected phase and in radiation-pressure force. In a simple tuned interferometer, the high-frequency sensing or shot-noise amplitude scales schematically as

$$\sqrt{S_h^{\rm shot}}\propto P^{-1/2},$$

while radiation-pressure displacement noise scales as

$$\sqrt{S_x^{\rm rad}}\propto \frac{P^{1/2}}{m f^2},$$

for circulating power $P$ and test-mass mass $m$. Increasing power improves photon counting but increases fluctuating force. These scalings motivate the standard quantum limit for an uncorrelated position measurement; they are not a universal bound on engineered interferometers. Signal recycling, squeezed states, frequency-dependent readout, optical loss, and correlations between sensing and back-action alter the full quantum-noise curve.

The Advanced LIGO optical configuration and its design noise budget are documented in LIGO Scientific Collaboration 2015, §§2.1–2.3 and §3, PDF. That source describes a reference instrument; measured dominance at a later epoch must be established from dated spectra and configuration records.

Thermal motion

Mechanical dissipation implies fluctuating motion through the fluctuation–dissipation theorem. The relevant quantity is not temperature alone but the dissipative part of the response sampled by the optical beam. Major families include:

• coating and substrate Brownian noise in the test masses;
• thermoelastic and thermorefractive fluctuations;
• suspension thermal noise from fibers, joints, and other lossy elements;
• thermal motion of auxiliary optics that couples through imperfect common-mode rejection.

For a generalized displacement $x$ with mechanical admittance $Y(2\pi f)$ in thermal equilibrium and in the classical limit $2\pi\hbar f\ll k_{\rm B}T$, the one-sided spectrum per hertz is

$$S_x(f)=\frac{k_{\rm B}T}{\pi^2 f^2}\operatorname{Re}Y(2\pi f),$$

where $Y$ is velocity divided by generalized force. Equivalent forms use the imaginary part of a displacement susceptibility; spectral density per hertz versus per radian per second and one- versus two-sided normalization must be checked before comparing formulas. The decisive experimental inputs include mechanical loss angle, beam size, material properties, temperature, mode shape, and readout projection. Saulson’s fluctuation–dissipation treatment gives the foundational suspension-noise calculation in Saulson 1990, Eqs. (7)–(12).

For a concrete check, consider a viscously damped oscillator,

$$m\ddot x+m\gamma\dot x+m\omega_0^2x=F, \qquad Y(\omega)=\frac{-i\omega}{m(\omega_0^2-\omega^2-i\gamma\omega)}.$$

Here $\gamma>0$ is a damping rate and $\omega=2\pi f$. Substitution gives

$$S_x(f)=\frac{4k_{\rm B}T\gamma/m} {[\omega_0^2-(2\pi f)^2]^2+\gamma^2(2\pi f)^2}.$$

The thermal motion is enhanced near resonance, and integrating over positive frequencies gives $\langle x^2\rangle=k_{\rm B}T/(m\omega_0^2)$, the equipartition result. A real suspension can have a different loss law, so this viscous example checks normalization without replacing a measured mechanical-loss model.

Ground motion and gravity fluctuations

Seismic motion reaches the test masses through finite mechanical isolation, suspension resonances, scattered-light paths, and control couplings. Passive and active isolation can strongly suppress direct displacement above their control bands, but large low-frequency motion can also spoil lock or up-convert nonlinearly into the observing band.

Newtonian noise is different: changing mass density in the ground, atmosphere, or nearby structures changes the local gravitational force on a test mass. Mechanical isolation cannot shield gravity. Arrays of seismometers or microphones can estimate relevant fields, and subtraction is possible only to the extent that the array samples the field and the propagation model is accurate. The physical mechanisms and subtraction geometry are reviewed in Harms 2015, §§2–4.

The distinction is experimentally useful. If a disturbance disappears when a mechanical path is interrupted, it was not irreducible Newtonian coupling; if it persists coherently with a modeled density field, direct gravitational coupling becomes plausible.

Technical and environmental couplings

Technical noise is not one residual category. It includes laser frequency and intensity noise, oscillator and timing noise, electronics, digitization, actuator noise, control-loop reinjection, beam jitter, angular-to-length coupling, residual gas effects, and scattered light. Environmental fields—acoustic, magnetic, electric, seismic, and anthropogenic—matter only through a demonstrated or bounded path to strain.

Many paths are bilinear. For example, beam jitter $\theta(t)$ coupling through a slowly drifting beam offset $a(t)$ gives

$$n_h(t)\propto a(t)\theta(t).$$

The effective transfer function then changes as alignment drifts, so a stationary linear projection measured at one time may fail later. Scattered light is similarly nonlinear: phase noise can depend on both a moving surface and the amount of stray light reaching it. Sidebands, arches in time–frequency maps, and state dependence can be stronger evidence than a broadband spectrum alone.

The observed taxonomy of stationary lines, transient glitches, and persistent broadband contributions is documented in LIGO Scientific Collaboration and Virgo Collaboration 2020, §§3 and 6, PDF. Its examples refer to the epochs covered there and should not be promoted to present-tense claims about an instrument.

Establishing a noise coupling

A defensible attribution usually combines four tests:

1. Witness correlation: a physical or control channel is coherent with strain at the feature of interest.
2. Injection: a controlled excitation produces the predicted strain response without saturating or changing the operating state.
3. Scaling: amplitude, phase, frequency, or environmental dependence follows the proposed mechanism.
4. Intervention: mitigating or disabling the path reduces the feature while other conditions remain comparable.

For a transfer measurement, estimate

$$\widehat T_{hq}(f)=\frac{\widehat S_{hq}(f)}{\widehat S_{qq}(f)}$$

only where the witness has adequate signal-to-noise ratio and the system is approximately linear and stationary. Then compare the projected spectrum $\lvert\widehat T_{hq}\rvert^2\widehat S_{qq}$ with the measured strain spectrum. Report averaging, coherence, statistical uncertainty, injection amplitude, detector state, and the interval over which the transfer is assumed valid.

Instrumental examples and public data products in this chapter are bounded by sources available on 2026-08-18. A current noise budget requires a dated official measurement; neither a design curve nor an older characterization paper establishes present dominance.

Common pitfalls

Calling every low-frequency feature seismic. Direct seismic motion, Newtonian gravity fluctuations, control saturation, and scattered-light up-conversion have different paths and mitigation strategies.

Treating coherence as causation. Two channels can share a third driver, and digital cross-talk can mimic a physical path. Injection and intervention tests are needed for attribution.

Forcing budget closure by tuning uncertain projections. A budget should expose unexplained residuals and covariance, not hide them inside optimistic or duplicated terms.

Exercises

1. Add correlated sources. Two equivalent-strain noises have equal one-sided PSD $S_0$ and complex coherence $\gamma_{12}=S_{12}/\sqrt{S_{11}S_{22}}$. Find their total PSD and evaluate it for $\gamma_{12}=0.5$, $0$, and $-0.5$.

Solution

For $n=n_1+n_2$,

$$S_n=S_{11}+S_{22}+2\operatorname{Re}S_{12} =2S_0\left(1+\operatorname{Re}\gamma_{12}\right).$$

The three values are $3S_0$, $2S_0$, and $S_0$. Independent powers add only in the middle case. An amplitude spectral density budget that simply sums $\sqrt{S_0}$ would give $4S_0$ after squaring and would be wrong in all three cases.

2. Change laser power. In the simple scalings above, circulating power is increased by a factor of four. How do the shot-noise and radiation-pressure amplitude spectra change?

Solution

Shot-noise amplitude scales as $P^{-1/2}$ and therefore falls by a factor of two. Radiation-pressure displacement amplitude scales as $P^{1/2}$ and therefore rises by a factor of two at fixed mass and frequency. The exercise isolates the elementary tradeoff; a real interferometer requires the full optical response, losses, quadrature, and correlations.

References

• Harms, Jan. “Terrestrial Gravity Fluctuations.” Living Reviews in Relativity 18 (2015): 3. DOI. Open PDF.
• LIGO Scientific Collaboration. “Advanced LIGO.” Classical and Quantum Gravity 32 (2015): 074001. DOI. Open PDF.
• LIGO Scientific Collaboration and Virgo Collaboration. “A Guide to LIGO–Virgo Detector Noise and Extraction of Transient Gravitational-Wave Signals.” Classical and Quantum Gravity 37 (2020): 055002. DOI. Open PDF.
• Saulson, Peter R. “Thermal Noise in Mechanical Experiments.” Physical Review D 42 (1990): 2437–2445. DOI.