Detection Methods

1. Introduction — The Challenge of Detecting h ~ 10¹²¹

A gravitational wave with dimensionless strain amplitude $h \sim 10^{-21}$ stretches a 4-km arm by a mere $\Delta L = \tfrac{1}{2}hL \sim 2\times 10^{-18}\,\text{m}$, roughly one-thousandth the diameter of a proton. Detecting such absurdly small displacements is perhaps the most demanding measurement ever attempted in experimental physics. The key insight, due to Rainer Weiss (1972), is that a Michelson interferometer converts differential arm-length changes into intensity variations of a laser beam's interference pattern — an approach that naturally rejects common-mode disturbances (e.g., laser frequency noise) while amplifying the differential signal.

Modern ground-based detectors (LIGO, Virgo, KAGRA) operate as power-recycled Fabry–Pérot Michelson interferometers with arm lengths $L = 4\,\text{km}$ (LIGO) or $L = 3\,\text{km}$ (Virgo). They are sensitive in a band $f \sim 10\,\text{Hz} - 10\,\text{kHz}$, set at low frequency by seismic and gravity-gradient noise and at high frequency by photon shot noise.

Space-based detectors such as LISA (Laser Interferometer Space Antenna) will use million-kilometre arm lengths to access the millihertz band ($10^{-4} - 10^{-1}\,\text{Hz}$), targeting massive black-hole mergers and galactic binaries. Pulsar timing arrays (PTAs) exploit millisecond pulsars as naturally stable clocks to probe the nanohertz regime ($10^{-9} - 10^{-7}\,\text{Hz}$), where the stochastic gravitational-wave background from supermassive binary black holes is expected to dominate.

In what follows we derive the response of a laser interferometer to a gravitational wave, characterize the dominant noise sources, develop the theory of matched filtering and signal-to-noise ratios, and finally compute the Hellings–Downs curve for pulsar timing arrays.

2. Derivation: Michelson Interferometer Response

2.1 Phase Shift in a Single Arm

Consider an arm of length $L$ oriented along the $x$-axis. A plus-polarized gravitational wave propagating along the $z$-axis produces the TT-gauge metric perturbation:

A photon traveling along the $x$-arm satisfies $ds^2 = 0$ with $dy = dz = 0$, giving:

The round-trip travel time for a photon in the $x$-arm (emitted at $t_0$, returning at $t_0 + \tau_x$) is:

where $t_{\rm out}(x) = t_0 + x/c$ and $t_{\rm ret}(x) = t_0 + (2L - x)/c$. In the long-wavelength limit ($L \ll \lambda_{\rm GW}$), the strain is approximately constant during the light travel time: $h_+(t) \approx h_+(t_0)$, so:

Similarly for the $y$-arm: $c\,\tau_y \approx 2L - h_+(t_0)\,L$. The differential round-trip time is:

2.2 Phase Difference at the Photodetector

The electric field of the laser has angular frequency $\omega_0 = 2\pi c/\lambda$. The phase accumulated by the photon in one arm is $\Phi = \omega_0 \tau$. The differential phase between the two arms is:

For LIGO ($L = 4\,\text{km}$, $\lambda = 1064\,\text{nm}$) and a strain $h = 10^{-21}$:

This tiny phase shift is amplified by a factor of $\mathcal{F}/\pi \approx 150$ when Fabry–Pérot cavities are placed in each arm (where $\mathcal{F} \approx 450$ is the finesse), effectively increasing the arm length to $L_{\rm eff} = \mathcal{F}L/\pi \approx 600\,\text{km}$.

2.3 Transfer Function for Finite Arm Length

For a monochromatic gravitational wave $h(t) = h_0\,e^{2\pi i f t}$, we must account for the variation of the strain during the photon's round trip. Integrating the accumulated phase perturbation over the outgoing and return legs:

Evaluating the integrals yields the single-arm transfer function:

The magnitude of the transfer function is the well-known sinc function:

The first zero of $|T(f)|$ occurs at $f = c/L$. For LIGO ($L = 4\,\text{km}$), this is $f \approx 75\,\text{kHz}$, safely above the detection band.

2.4 Optimal Arm Length

The detector's response is maximized when the round-trip light travel time equals half the gravitational-wave period: $2L/c = 1/(2f)$, giving:

For a $100\,\text{Hz}$ signal, $L_{\rm opt} \approx 750\,\text{km}$ — far longer than any terrestrial arm. This motivates the use of Fabry–Pérot cavities, which fold the light path to achieve an effective length $L_{\rm eff} \gg L$ without extending the physical baseline. For LISA targeting $f \sim 10^{-2}\,\text{Hz}$, one needs$L_{\rm opt} \sim 7.5\times 10^9\,\text{m}$, comparable to LISA's planned $2.5\times 10^9\,\text{m}$ arms.

3. Derivation: Noise Sources and Sensitivity

The detector output is $s(t) = h(t) + n(t)$, where $n(t)$ is the noise. We characterize the noise by its one-sided power spectral density (PSD) $S_n(f)$, defined via:

The noise PSD has units of $\text{Hz}^{-1}$ (strain-squared per Hz). The amplitude spectral density is $\sqrt{S_n(f)}$ with units $\text{Hz}^{-1/2}$.

3.1 Photon Shot Noise

The fundamental quantum limit at high frequencies is photon shot noise. Each photon carries energy $E = h\nu = hc/\lambda$ (here $h$ is Planck's constant, not the strain!). With laser power $P_{\rm laser}$ and photodetector quantum efficiency $\eta$, the photon arrival rate is $\dot{N} = \eta P_{\rm laser}/(h\nu)$. The shot noise in the photon count over time $T$ is $\delta N = \sqrt{\dot{N}\,T}$, producing a phase uncertainty:

Converting to equivalent strain noise using $\Delta\Phi = (4\pi L/\lambda)\,h$:

Key features: $S_{\rm shot}$ is frequency-independent (white noise), decreases with increasing laser power $P_{\rm laser}$ and arm length $L$. For Advanced LIGO ($P_{\rm laser} \approx 200\,\text{W}$, $L = 4\,\text{km}$, $\lambda = 1064\,\text{nm}$), one obtains $\sqrt{S_{\rm shot}} \sim 10^{-23}\,\text{Hz}^{-1/2}$ at $f \gtrsim 200\,\text{Hz}$.

3.2 Radiation Pressure Noise

At low frequencies, quantum fluctuations of the laser field exert a fluctuating radiation pressure force on the mirrors. The momentum kick per photon reflection is $2h\nu/c$. Fluctuations in photon number $\delta N$ produce a force noise:

This force displaces a mirror of mass $m$ by $\delta x = F_{\rm rad}/(m\omega^2)$ at frequency $f = \omega/(2\pi)$. Converting to strain:

This noise increases with laser power, scales as $f^{-4}$, and dominates at low frequencies. The standard quantum limit (SQL) arises from the trade-off between shot noise (decreasing with power) and radiation pressure noise (increasing with power). The optimal power gives $S_{\rm SQL}(f) \propto 1/(m\,f^2\,L^2)$.

3.3 Seismic Noise

Ground vibrations from human activity, wind, ocean waves (microseismic peak at $\sim\!0.15\,\text{Hz}$), and earthquakes produce displacement noise. The ground displacement PSD above $\sim\!1\,\text{Hz}$ is approximately:

The mirrors are isolated by multi-stage pendulum suspensions. Each pendulum stage acts as a second-order low-pass filter with corner frequency $f_0$, attenuating seismic motion by a factor $(f_0/f)^2$ above resonance. With $N$ stages:

Advanced LIGO uses a 4-stage pendulum isolation system ($N = 4$) with $f_0 \sim 1\,\text{Hz}$, giving attenuation $\sim (1/f)^8$ above 1 Hz. The strain PSD from seismic noise then scales as:

For $N = 4$: $S_{\rm seismic} \propto f^{-20}$, producing a very steep “seismic wall” below $\sim\!10\,\text{Hz}$.

3.4 Thermal (Brownian) Noise

The fluctuation-dissipation theorem relates the thermal displacement noise of a mechanical system at temperature $T$ to its dissipation. For a pendulum suspension with quality factor $Q$ and resonant frequency $f_0$:

Well above resonance ($f \gg f_0$), this reduces to:

Coating thermal noise from the dielectric mirror coatings is currently the dominant noise source in Advanced LIGO's most sensitive band (50–200 Hz). Mirror substrates also contribute via thermo-elastic and thermo-refractive fluctuations. The total thermal noise budget is:

3.5 Total Noise Curve

The total noise PSD is the sum of all independent noise contributions:

The gravity gradient (Newtonian) noise arises from local mass density fluctuations (seismic waves, atmospheric density changes) that produce time-varying gravitational fields at the test masses. This noise is irreducible below $\sim\!10\,\text{Hz}$ for surface detectors, motivating underground designs such as the Einstein Telescope, where seismic surface waves are greatly attenuated.

4. Derivation: Signal-to-Noise Ratio and Matched Filtering

4.1 The Inner Product

We define a noise-weighted inner product between two time series $a(t)$ and $b(t)$:

where $\tilde{a}(f) = \int_{-\infty}^{\infty} a(t)\,e^{-2\pi i f t}\,dt$ is the Fourier transform. This inner product naturally down-weights frequency bins where the noise is large.

4.2 Wiener Optimal Filter

Given data $s(t) = h(t) + n(t)$, we filter with a template $K(t)$ to form the statistic $\hat{\rho} = \int s(t)\,K(t)\,dt$. The SNR is:

By the Cauchy–Schwarz inequality, this is maximized when $\tilde{K}(f) \propto \tilde{h}(f)/S_n(f)$, the Wiener optimal filter (or matched filter). The optimal SNR is then:

This result is fundamental to gravitational-wave data analysis. It tells us that the detectability of a signal is determined by the integral of $|\tilde{h}(f)|^2/S_n(f)$over the detector bandwidth. A signal is detectable when $\rho$ exceeds a threshold (typically $\rho_{\rm thr} \sim 8$ for a single detector, corresponding to a false alarm rate of $\sim\!1$ per century).

4.3 Template Banks

For compact binary coalescences, the waveform depends on intrinsic parameters$\boldsymbol{\theta} = (m_1, m_2, \mathbf{S}_1, \mathbf{S}_2, \ldots)$. The match between a signal $h(\boldsymbol{\theta}_s)$ and a template $h(\boldsymbol{\theta}_t)$ is:

A template bank is constructed to cover the parameter space such that for any signal parameters $\boldsymbol{\theta}_s$, there exists at least one template with$\mathcal{M} \geq 1 - \epsilon$ (typically $\epsilon = 0.03$, i.e., no more than 3% SNR loss). The required number of templates scales as the parameter-space volume divided by the metric-determined volume element.

4.4 Fisher Information Matrix

For loud signals ($\rho \gg 1$), parameter estimation errors are governed by the Fisher information matrix:

The covariance matrix of parameter errors is bounded by the Cramér–Rao bound:

For example, the chirp mass $\mathcal{M}_c$ of a binary is the best-measured parameter (to $\sim 0.1\%$ for GW150914), because it determines the leading-order phase evolution. Sky localization from a network of detectors relies on time-of-arrival triangulation, with typical error boxes of $\sim 10\text{--}100\,\text{deg}^2$ for two detectors and $\sim 1\text{--}10\,\text{deg}^2$ for three.

5. Derivation: Pulsar Timing Array Sensitivity

5.1 Timing Residuals from Gravitational Waves

A pulsar timing array monitors an ensemble of millisecond pulsars, each serving as an extraordinarily stable clock. A passing gravitational wave induces a shift in the observed pulse times of arrival (TOAs). For a GW propagating in direction $\hat{\Omega}$, the redshift of pulses from a pulsar in direction $\hat{p}$ is:

where $\Delta h_{ij}(t) = h_{ij}(t_{\rm Earth}) - h_{ij}(t_{\rm pulsar})$ is the difference between the metric perturbation at Earth and at the pulsar (the “Earth term” and “pulsar term”). The timing residual is the integral of the redshift:

5.2 The Hellings–Downs Curve

For a stochastic, isotropic, unpolarized gravitational-wave background, the expected cross-correlation of timing residuals between two pulsars separated by angle $\theta$on the sky depends only on $\theta$. This is the Hellings–Downs correlation, derived by Hellings & Downs (1983).

The derivation proceeds by averaging the product of redshifts for two pulsars over all GW propagation directions and polarizations. Let $x = (1 - \cos\theta)/2$. The angular integral yields:

where the $\delta(\theta)$ term represents the pulsar's auto-correlation (same pulsar correlated with itself, $\theta = 0$). Expanding:

Key features of this curve:

• $C(0) = 1/2$ (plus the auto-correlation term) — pulsars in the same direction are positively correlated
• $C(90°) \approx -0.125$ — orthogonal pulsars are slightly anti-correlated
• $C(180°) \approx 0$ — anti-podal pulsars show near-zero correlation
• The curve crosses zero near $\theta \approx 50°$ and $\theta \approx 150°$

The Hellings–Downs curve is the “smoking gun” signature of a gravitational-wave background in PTA data. In 2023, NANOGrav, EPTA, PPTA, and CPTA all reported strong evidence for a common-spectrum stochastic process, with emerging evidence for the HD inter-pulsar correlations — consistent with a nanohertz GW background, most likely from a cosmic population of supermassive black-hole binaries.

6. Applications

7. Historical Context

8. Python Simulation: Noise Curves & Hellings–Downs Correlation

The following simulation plots the characteristic strain noise curves $h_c(f) = \sqrt{f\,S_n(f)}$for five current and planned GW detectors, overlays representative signal tracks, and plots the Hellings–Downs angular correlation function. All calculations use analytic approximations to the published noise budgets. Only numpy is used — no scipy.