Chapter 9

Case study: LIGO and noise budgets

Everything from chapters 4–8 — PSDs, noise colors, sensitivity in $\mathrm{Hz}^{-1/2}$, in-loop caveats — assembles into the way a flagship experiment actually presents its noise: one sensitivity curve, and behind it a noise budget. We build a schematic Advanced-LIGO-like budget you can turn the knobs on, and end with the single integral that unifies gravitational-wave detectors, locked lasers, atomic clocks and qubits.

The most famous ASD in physics

LIGO measures strain: the fractional stretch $h(t) = \Delta L / L$ of its $L = 4\,$km arms. Strain is a dimensionless time series like any voltage record in chapter 4, so its noise is quoted the same way: as an amplitude spectral density $\sqrt{S_h(f)}$ with units $1/\sqrt{\mathrm{Hz}}$. For Advanced LIGO the floor of that curve sits at a few$\,\times 10^{-24}\,/\sqrt{\mathrm{Hz}}$ in the "bucket" around 100–300 Hz.

You read this curve exactly like every ASD since chapter 4: the RMS noise in a band is the square root of the area under the PSD,

$$ h_{\mathrm{rms}}[f_1, f_2] \;=\; \sqrt{\int_{f_1}^{f_2} S_h(f)\, df\,}. $$

On a roughly flat floor this is just (floor level)$\,\times\sqrt{\text{bandwidth}}$. GW150914 — the first detected black-hole merger — reached a strain amplitude of about $10^{-21}$ near 100–250 Hz, comfortably above a floor whose in-band RMS is around $10^{-22}$. And note how the comparison is drawn: the signal too is plotted per root hertz (for a sweeping chirp one plots a "characteristic strain", essentially $|\tilde h(f)|\sqrt{f}$, which accounts for how long the chirp spends near each frequency). Only then are signal and noise apples to apples on the same axes.

The readout above is worth staring at: an RMS strain of $\sim 10^{-22}$ over 4 km is a length noise of a few $\times 10^{-19}\,$m — a few ten-thousandths of a proton radius. Nothing about the mathematics changes at that scale; the PSD bookkeeping of chapter 4 simply keeps working.

Noise budgets: variances add

Where does the curve come from? A LIGO-class instrument has dozens of noise sources: the ground shakes, the mirror coatings jiggle thermally, the laser delivers photons at random times. The sources are (to a good approximation) statistically independent, and independent noises add in power — variances add, so PSDs add:

$$ S_{\mathrm{total}}(f) = \sum_i S_i(f), \qquad \sqrt{S_{\mathrm{total}}} = \sqrt{\textstyle\sum_i \big(\sqrt{S_i}\big)^2}. $$

ASDs add in quadrature, never linearly. This single fact is why every big experiment publishes a "noise budget": a plot of many labeled component curves whose quadrature sum — the thick "total" — should hug the measured sensitivity. Wherever it doesn't, there is unexplained noise and a graduate student is dispatched. The whole idea fits in a few lines of code:

// the entire "noise budget" idea: PSDs add, so ASDs add in quadrature
const total = fGrid.map(f => {
  const components = [seismic(f), suspension(f), coating(f),
                      shot(f), radiationPressure(f)];   // ASDs, 1/√Hz
  let sumOfPSDs = 0;
  for (const a of components) sumOfPSDs += a * a;       // variances add
  return Math.sqrt(sumOfPSDs);
});

Here is a schematic aLIGO-like budget with the classic cast of characters, each a power law with a representative slope and magnitude:

• Seismic wall: ground motion, filtered by stacked isolation stages, rises extremely steeply below $\sim$15 Hz (modeled $\propto f^{-10}$). Below the wall there is simply no data.
• Suspension thermal: Brownian motion of the fused-silica fibers the mirrors hang from ($\propto f^{-2.5}$ here), important in the 10–40 Hz decade. Thermal ASDs scale as $\sqrt{T}$.
• Coating Brownian thermal: thermal jitter of the mirror coatings ($\propto f^{-1/2}$), the floor of the bucket.
• Quantum shot noise: photon counting noise — flat in phase, so in strain it is flat below the arm-cavity pole ($\sim$400 Hz) and rises $\propto f$ above it: $A_{\mathrm{shot}}\sqrt{1 + (f/400\,\mathrm{Hz})^2}$ with $A_{\mathrm{shot}} \propto 1/\sqrt{P}$. More laser power, more photons averaged, less shot noise.
• Quantum radiation pressure: the same photon-number fluctuations physically push the mirrors, $\propto \sqrt{P}/f^2$. More power, more push.

Signal-to-noise of a known waveform

LIGO does not just stare at the noise floor waiting for a bump — for binary inspirals the waveform shape is known from general relativity, and the optimal strategy is matched filtering: correlate the data with the expected template, weighting each frequency by how quiet the detector is there. The achievable signal-to-noise ratio is

$$ \mathrm{SNR}^2 \;=\; 4 \int_0^{\infty} \frac{|\tilde h(f)|^2}{S_h(f)}\, df . $$

Read it as a competition held frequency by frequency: the signal only scores where it has power, and each frequency's contribution is divided by the noise PSD there. The sensitivity curve is a weighting function. An inspiral chirp has $|\tilde h(f)| \propto f^{-7/6}$ up to roughly the last stable orbit, $f_{\mathrm{isco}} \approx 4.4\,\mathrm{kHz}/(M/M_\odot)$ for total mass $M$ — so heavy binaries live at low frequency, exactly where the seismic wall looms.

Control loops shape noise: why there are bumps

One more ingredient before you can read a real noise budget: almost every curve on it has passed through a feedback loop. The mirrors must be held on the interferometer's operating point, the laser on the cavity resonance — so error signals are measured and fed back. Describe the loop by its open-loop gain $G(f)$: a complex number per frequency saying by how much, and with what phase delay, a wiggle at $f$ comes back around the loop (chapter 2's phasors, now applied to a feedback path — magnitude is gain, angle is accumulated delay). With open-loop gain $G(f)$, a disturbance $d$ entering the loop appears at the output suppressed to

$$ \frac{d}{1 + G(f)}, $$

while the sensor's own noise $n$ is injected as $-\tfrac{G}{1+G}\, n$: inside the bandwidth, where $|G| \gg 1$, the loop faithfully writes the sensor noise onto the mirror. And $|1/(1+G)|$ is less than one inside the bandwidth but typically exceeds one near the unity-gain frequency, where $G \approx -1$ is nearly reached: that is the notorious servo bump. How close $G$ gets to exactly $-1$ is measured by the phase margin — how many degrees the loop's phase at unity gain stands clear of $180°$ of inversion. The smaller the margin (extra delay eats it), the bigger the bump.

Filter functions beyond LIGO

Step back: this chapter has computed the same object three times, and chapter 7 makes a fourth. In every case the answer was

$$ \text{(variance your protocol accumulates)} \;=\; \int_0^\infty S(f)\, |W(f)|^2 \, df, $$

where $S(f)$ is set by what the noise is and the filter function $|W(f)|^2$ entirely by how you measure. Every measurement protocol is a spectral weighting applied to the noise PSD:

The last two rows are the qubit/clock workhorses, and comparing them is the payoff: the echo's $\pi$-pulse flips the accumulated phase halfway through, which moves the weight off $f = 0$. That is exactly why echoes extend coherence in a $1/f$ environment — the weight is removed from where that noise lives. Watch the two weights work on the same noise:

One thread is left hanging. This chapter treated feedback loops as something noise passes through on its way to a noise budget. But a loop can also be the whole instrument: take two oscillators, a discriminator and an integrator, close the ring, and you have built a synthesizer, a laser lock, or an atomic clock. That construction — the phase-locked loop, and the block diagrams every lab drawing of one is made of — is the final chapter.

Exercises