LIGO and the First Detections

The quadrupole formula predicts a strain of order $1 0^{- 21}$ 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{Δ L}{L} = \frac{Δ L _{x} - Δ 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.

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.

The sensitivity challenge

For LIGO's arm length $L = 4 km$ and a strain $h \sim 1 0^{- 21}$ , the mirror displacement is

Δ L \sim h L \sim 1 0^{- 21} \times 4 \times 1 0^{3} m \sim 4 \times 1 0^{- 18} 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 $10 Hz$ : 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 $100 Hz$ : 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 $1 0^{- 23}$ level per root hertz in the most sensitive band.

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.

GW150914

The signal recorded on 14 September 2015 appeared in both the Hanford and Livingston detectors, separated by the $7 ms$ 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 $250 Hz$ over $0.2 s$ , the unmistakable chirp of an inspiral, then peaked and decayed. Matched-filtering against general-relativity waveform templates fixed the source parameters.

Component masses: $3 6_{- 4}^{+ 5} M_{⊙}$ and $2 9_{- 4}^{+ 4} M_{⊙}$ , read from the chirp mass and the higher-order inspiral evolution.
Final black hole: $6 2_{- 4}^{+ 4} M_{⊙}$ , spinning, its mass smaller than the sum because $3. 0_{- 0.5}^{+ 0.5} M_{⊙} c^{2}$ of energy was radiated as gravitational waves.
Peak luminosity: $3. 6_{- 0.4}^{+ 0.5} \times 1 0^{49} W$ , briefly exceeding the combined electromagnetic luminosity of all the stars in the observable universe.
Distance: a luminosity distance of $41 0_{- 180}^{+ 160} Mpc$ , redshift $z \approx 0.09$ , 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.

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.

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 $M$ , 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.

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.

Multimessenger astronomy

On 17 August 2017 the event GW170817 recorded the inspiral of two neutron stars, with a chirp mass identifying components near $1.4 M_{⊙}$ 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 $1.7 s$ 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 $1 0^{15}$ .
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.

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.

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.