---
title: LIGO and the First Detections
module: Gravitational Waves
moduleNumber: 9
lessonNumber: 3
order: 903
summary: >
  A gravitational wave is measured as a differential length change of the two
  arms of a kilometre-scale Michelson interferometer, a strain of order ten to
  the minus twenty-one that moves the mirrors by a fraction of a proton radius.
  GW150914 recorded the inspiral, merger, and ringdown of two black holes, fixing
  their masses and the energy radiated, and GW170817 with its coincident gamma-ray
  burst and kilonova opened multimessenger astronomy.
topics: [Gravitational Waves]
draft: false
sources:
  - book: LIGO
    ref: "Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016)"
  - book: Hartle
    ref: "Gravity, Ch. 16 — Gravitational Waves (detection)"
---

The quadrupole formula predicts a strain of order $10^{-21}$ reaching Earth from
a compact-binary merger. Measuring it requires sensing a length change of a
fraction of a proton radius over a baseline of kilometres. A laser
interferometer does exactly this, and on 14 September 2015 the two LIGO detectors
recorded the signal GW150914, the inspiral and merger of two black holes. This
lesson describes the interferometric measurement, the noise that limits it, the
information carried in the inspiral–merger–ringdown waveform, and the
multimessenger event GW170817.

## Interferometric detection

A gravitational wave with plus polarization arriving perpendicular to the plane of
a right-angle interferometer stretches one arm while shrinking the other. A
Michelson interferometer reads this differential change: laser light is split at a
beamsplitter, travels down two perpendicular arms of length $L$, reflects off
end mirrors that hang as freely falling test masses, and recombines. The phase
difference between the returning beams depends on the arm-length difference, and
the wave produces a differential strain

$$
\frac{\Delta L}{L} = \frac{\Delta L_x - \Delta L_y}{L} = h(t),
$$

where the two arms respond with opposite sign, doubling the signal relative to a
single arm. The output port brightness swings as the two beams move between
constructive and destructive interference, converting the strain into a measured
photocurrent.

$$
% caption: A Michelson interferometer splits laser light down two perpendicular
% arms to freely hanging mirrors; a passing wave lengthens one arm and shortens
% the other, and the recombined beams at the photodiode register the differential
% strain.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % laser
  \draw[black, thick] (-2.4,0) rectangle (-1.6,0.5);
  \node[black, anchor=center] at (-2.0,0.25) {laser};
  \draw[thick] (-1.55,0.25) -- (0,0.25);
  % beamsplitter
  \draw[black, thick] (-0.25,0.55) -- (0.25,-0.05);
  \node[black, anchor=north] at (0,-0.15) {splitter};
  % arm x (horizontal)
  \draw[acc, thick] (0.25,0.25) -- (3.4,0.25);
  \draw[black, thick] (3.4,-0.15) rectangle (3.55,0.65);
  \node[black, anchor=west] at (3.6,0.25) {mirror};
  \node[acc, anchor=south] at (1.8,0.3) {arm x};
  % arm y (vertical)
  \draw[acc, thick] (0,0.5) -- (0,3.0);
  \draw[black, thick] (-0.4,3.0) rectangle (0.4,3.15);
  \node[black, anchor=south] at (0,3.2) {mirror};
  \node[acc, anchor=west] at (0.1,1.8) {arm y};
  % photodiode below splitter
  \draw[thick] (0,-0.05) -- (0,-1.0);
  \draw[black, thick] (-0.45,-1.5) rectangle (0.45,-1.0);
  \node[black, anchor=center] at (0,-1.25) {photodiode};
\end{tikzpicture}
$$

## The sensitivity challenge

For LIGO's arm length $L=4\ \text{km}$ and a strain $h\sim10^{-21}$, the mirror
displacement is

$$
\Delta L \sim h\,L \sim 10^{-21}\times 4\times10^3\ \text{m} \sim 4\times10^{-18}\ \text{m},
$$

about one two-hundredth of a proton radius. Reaching this requires suppressing
every competing displacement of the mirrors. Three noise sources set the limit
across the band, each dominant in a different frequency range.

- **Seismic noise** dominates below about $10\ \text{Hz}$: ground motion shakes
  the apparatus, suppressed by multi-stage pendulum suspensions that isolate the
  mirrors above the pendulum resonance.
- **Thermal noise** dominates the mid-band near $100\ \text{Hz}$: random thermal
  motion of the mirror substrates and suspension fibres, reduced by high-quality
  materials and careful coating design.
- **Shot noise** dominates above a few hundred hertz: the quantum counting
  statistics of photons at the output, reduced by increasing the laser power and
  by resonant optical cavities that build up the circulating power.

Fabry–Perot cavities in each arm fold the light path so the effective baseline is
hundreds of kilometres, and power recycling increases the stored laser power.
Together they push the strain sensitivity to the $10^{-23}$ level per root hertz
in the most sensitive band.

$$
% caption: The strain sensitivity curve is a bucket: seismic noise walls off low
% frequencies, mirror and suspension thermal noise limits the mid-band, and
% photon shot noise rises at high frequency, leaving a sensitive band from tens to
% hundreds of hertz.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.2,0) node[anchor=north] {frequency};
  \draw[->, black] (0,0) -- (0,3.4) node[anchor=east] {noise};
  % seismic wall (steep falling at low f)
  \draw[black, thick, densely dotted] (0.4,3.2) .. controls (0.9,1.4) and (1.4,0.7) .. (2.2,0.55);
  \node[black, anchor=west, rotate=-70] at (0.55,2.1) {seismic};
  % thermal (flat mid) 
  \draw[black, thick, dashed] (1.4,0.9) .. controls (2.6,0.5) and (4.0,0.5) .. (5.2,0.75);
  \node[black, anchor=north] at (3.2,0.28) {thermal};
  % shot noise rising at high f
  \draw[black, thick] (4.2,0.6) .. controls (5.6,0.8) and (6.6,1.6) .. (7.6,3.2);
  \node[black, anchor=south, rotate=60] at (7.1,2.1) {shot};
  % envelope (sensitive band) accent
  \draw[acc, very thick] (1.6,1.2) .. controls (2.6,0.55) and (4.0,0.55) .. (5.4,1.0);
  \node[acc, anchor=south] at (3.4,1.1) {sensitive band};
\end{tikzpicture}
$$

## GW150914

The signal recorded on 14 September 2015 appeared in both the Hanford and
Livingston detectors, separated by the $7\ \text{ms}$ light-travel time between
them, with a combined signal-to-noise ratio of $24$. The waveform swept upward in
frequency and amplitude from about $35$ to $250\ \text{Hz}$ over $0.2\ \text{s}$,
the unmistakable chirp of an inspiral, then peaked and decayed. Matched-filtering
against general-relativity waveform templates fixed the source parameters.

- **Component masses**: $36^{+5}_{-4}\,M_\odot$ and $29^{+4}_{-4}\,M_\odot$,
  read from the chirp mass and the higher-order inspiral evolution.
- **Final black hole**: $62^{+4}_{-4}\,M_\odot$, spinning, its mass smaller than
  the sum because $3.0^{+0.5}_{-0.5}\,M_\odot c^2$ of energy was radiated as
  gravitational waves.
- **Peak luminosity**: $3.6^{+0.5}_{-0.4}\times10^{49}\ \text{W}$, briefly
  exceeding the combined electromagnetic luminosity of all the stars in the
  observable universe.
- **Distance**: a luminosity distance of $410^{+160}_{-180}\ \text{Mpc}$,
  redshift $z\approx0.09$, placing the merger over a billion years in the past.

The masses are too large for neutron stars and the objects merged at a separation
of a few hundred kilometres, so the source is a pair of black holes. This was the
first direct detection of gravitational waves and the first observation of a
binary black hole.

$$
% caption: The GW150914 strain has three phases: the rising inspiral chirp, the
% brief high-amplitude merger as the horizons coalesce, and the damped ringdown
% as the final black hole settles to a Kerr geometry.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.6,0) node[anchor=north] {time};
  \draw[->, black] (0,-1.5) -- (0,1.6) node[anchor=east] {strain};
  % inspiral (rising)
  \draw[acc, thick]
    (0.3,0) sin (0.9,0.25) cos (1.5,0) sin (2.1,-0.25) cos (2.7,0)
    sin (3.1,0.45) cos (3.5,0) sin (3.9,-0.45) cos (4.3,0)
    sin (4.6,0.75) cos (4.9,0) sin (5.2,-0.8) cos (5.5,0);
  % merger peak
  \draw[acc, very thick] (5.5,0) sin (5.75,1.3) cos (6.0,0) sin (6.25,-1.15) cos (6.5,0);
  % ringdown (damped)
  \draw[acc, thick] (6.5,0) sin (6.75,0.55) cos (7.0,0) sin (7.25,-0.3) cos (7.5,0) sin (7.7,0.12) cos (7.9,0);
  \node[black, anchor=north] at (2.4,-0.4) {inspiral};
  \node[acc, anchor=south] at (5.85,1.35) {merger};
  \node[black, anchor=north] at (7.3,-0.35) {ringdown};
\end{tikzpicture}
$$

## Inspiral, merger, and ringdown

The waveform separates into three regimes, each governed by different physics and
each constraining different parameters.

- **Inspiral**: the two bodies orbit at separations large compared with their
  horizons, and the post-Newtonian expansion of the quadrupole formula applies.
  The chirp rate $\dot f$ fixes the chirp mass $\mathcal M$, and higher-order
  terms separate the individual masses and spins.
- **Merger**: the horizons touch and coalesce, a strong-field, fully nonlinear
  regime accessible only to numerical relativity. The amplitude peaks here, and
  the total mass sets the frequency of the peak.
- **Ringdown**: the distorted final black hole radiates away its asymmetry through
  damped **quasi-normal modes**, oscillations whose frequency and decay time are
  fixed by the mass and spin of the final Kerr black hole. Measuring the ringdown
  tests whether the remnant is the Kerr black hole general relativity predicts.

Reading the three phases together determines the initial masses and spins, the
final mass and spin, the radiated energy, the distance, and the sky location,
the last from the arrival-time difference between detectors.

$$
% caption: The three waveform phases each fix different parameters: the inspiral
% chirp gives the chirp mass and spins, the merger peak gives the total mass, and
% the ringdown quasi-normal modes give the final black hole's mass and spin.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \x/\ph/\prm in {0/inspiral/chirp mass, 2.9/merger/total mass, 5.8/ringdown/final spin} {
    \draw[thick] (\x,1.0) rectangle (\x+2.4,1.9);
    \node[anchor=center] at (\x+1.2,1.45) {\ph};
    \draw[black] (\x+1.2,0.95) -- (\x+1.2,0.55);
    \node[black, anchor=north] at (\x+1.2,0.5) {\prm};
  }
  \draw[->, black] (2.45,1.45) -- (2.85,1.45);
  \draw[->, black] (5.35,1.45) -- (5.75,1.45);
\end{tikzpicture}
$$

## Multimessenger astronomy

On 17 August 2017 the event GW170817 recorded the inspiral of two neutron stars,
with a chirp mass identifying components near $1.4\,M_\odot$ each and a signal
lasting about a minute in band. Its significance is the coincidence: the Fermi
gamma-ray satellite recorded a short gamma-ray burst, GRB 170817A, arriving $1.7\
\text{s}$ after the merger, and telescopes across the electromagnetic spectrum
located an optical **kilonova** in the host galaxy over the following days, the
glow of radioactive heavy nuclei forged in the neutron-rich ejecta.

- The near-simultaneous arrival of the gravitational wave and the gamma-ray burst
  over $130$ million light-years constrains the speed of gravitational waves to
  equal the speed of light to within one part in $10^{15}$.
- The kilonova confirmed neutron-star mergers as a site of rapid-neutron-capture
  nucleosynthesis, the origin of a large share of the elements heavier than iron.
- The joint gravitational-wave and electromagnetic distance measurement provides
  an independent route to the Hubble constant.

A single event observed in gravitational waves, gamma rays, and visible light is
the founding example of multimessenger astronomy, in which the same source is
studied through independent channels that constrain each other.

$$
% caption: GW170817 arrived first as a gravitational-wave inspiral, then as a
% gamma-ray burst 1.7 seconds later, and over the following days as an optical
% kilonova, the same neutron-star merger observed through three channels.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[anchor=north] {time};
  % GW at t=0
  \draw[acc, very thick] (0.6,0.1) -- (0.6,1.2);
  \node[acc, anchor=south] at (0.6,1.25) {wave};
  \fill[acc] (0.6,0) circle (0.07);
  % gamma 1.7 s later
  \draw[black, thick] (1.7,0.1) -- (1.7,0.9);
  \node[black, anchor=south] at (1.7,0.95) {gamma burst};
  \fill[black] (1.7,0) circle (0.06);
  \node[black, anchor=north] at (1.15,-0.05) {1.7 s};
  % kilonova days later
  \draw[black, thick] (5.4,0.1) -- (5.4,0.8);
  \node[black, anchor=south] at (5.4,0.85) {kilonova};
  \fill[black] (5.4,0) circle (0.06);
  \node[black, anchor=north] at (3.6,-0.05) {days};
  \draw[black, densely dotted] (1.9,-0.55) -- (5.2,-0.55);
\end{tikzpicture}
$$

Direct detection turned gravitational waves from a prediction of linearized
theory into an observational field. Each merger measures masses, spins, and
distances inaccessible to electromagnetic astronomy, and the growing catalogue of
black-hole and neutron-star coalescences maps a population of compact objects
across cosmic time, the observational counterpart to the theory built through this
subject.

[^ligo-gw150914]: **LIGO/Virgo (Abbott et al.)**, _Observation of Gravitational Waves from a Binary Black Hole Merger_, Phys. Rev. Lett. **116**, 061102 (2016), <https://doi.org/10.1103/PhysRevLett.116.061102> — the GW150914 detection, source parameters, radiated energy, and detector description.

[^hartle-detect]: **Hartle**, _Gravity_, Ch. 16 — interferometric detection of gravitational waves, the strain-sensitivity requirement, and the noise sources limiting a ground-based detector.