Gravitational Waves from Inspiraling Binaries

General relativity predicts that accelerating masses radiate gravitational waves, propagating distortions of spacetime that travel at the speed of light. The waves are extraordinarily weak, because the coupling constant of gravity to matter is $G/c^4\sim 10^{-45}{\mathrm{s}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-1}$, and only the most violent astrophysical events, the merger of two compact objects, produce a signal detectable on Earth. This lesson derives the quadrupole formula and the reason gravitational radiation is so faint, describes the two polarizations and their effect on free masses, follows the orbital decay that drives an inspiral into a rising chirp, and extracts the chirp mass from the frequency sweep. It closes with the interferometric detection principle and the first observed event, GW150914.

Linearized gravity and the quadrupole formula

Far from any source, spacetime is nearly flat and the metric is a small perturbation on the Minkowski background,

Linearizing the Einstein field equations in $h_{\mu \nu }$ and choosing the harmonic (Lorenz) gauge reduces them to a wave equation with the stress-energy tensor as source,

where ${\bar{h}}_{\mu \nu }$ is the trace-reversed perturbation. In vacuum the right side vanishes and $h_{\mu \nu }$ propagates as a wave at speed $c$. The lowest radiating multipole is not the monopole or dipole. Mass conservation forbids monopole radiation, and momentum conservation forbids mass-dipole radiation, because the mass dipole's second derivative is $\ddot{\sum m_i{\vec{r}}_i}=\dot{\vec{P}}=0$ for an isolated system. Gravitational radiation therefore begins at the quadrupole order.

The prefactor $G/c^5$ carries the units of inverse power and has the numerical value $1/L_0$ with $L_0=c^5/G\approx 3.6\times 10^{52}\mathrm{W}$, an enormous luminosity scale. Gravitational-wave luminosities are large only when a substantial fraction of a compact object's rest energy is set into quadrupolar motion at the dynamical frequency, which happens only in the final orbits of a compact-binary merger.

Two polarizations and their effect on free masses

A gravitational wave propagating along $z$ has two independent polarization states, transverse and traceless. In the transverse-traceless gauge the perturbation is

The wave does not move test masses through space; it changes the proper distances between them. Consider a ring of freely floating masses in the plane transverse to the propagation direction. The plus polarization stretches the ring along one axis while squeezing it along the perpendicular axis, then reverses each half-cycle. The cross polarization does the same along axes rotated by $45^{\circ }$. A displacement $L$ between two free masses oscillates by

so the wave amplitude $h$ is a dimensionless strain, the fractional change in length.

Orbital decay and the inspiral chirp

A binary of masses $m_1$ and $m_2$ in a circular orbit of separation $a$ has a mass quadrupole that rotates at the orbital frequency, and the quadrupole formula gives its gravitational-wave luminosity. Evaluating $⟨{\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}_{ij}⟩$ for the orbit yields

This power is drawn from the orbital energy $E=-G{m}_1{m}_2/2a$. Setting $\dot{E}=-L_{\mathrm{GW}}$ and solving for the shrinking separation gives

The orbit decays slowly at first and then catastrophically as $a$ shrinks, since the rate diverges as $a^{-3}$. Integrating from an initial separation $a_0$ to coalescence gives a finite inspiral time

As $a$ falls, the orbital frequency rises (Kepler's law, ${\omega }^2=GM/a^3$), and so does the gravitational-wave frequency, which is twice the orbital frequency because the quadrupole returns to its configuration twice per orbit. The amplitude grows in concert. The signal is a chirp: an oscillation that sweeps upward in both frequency and amplitude until merger.

The chirp mass

The rate at which the frequency sweeps depends on a single combination of the two masses. Rewriting the decay in terms of the gravitational-wave frequency $f={\omega }_{\mathrm{GW}}/2\pi$ gives

where the mass dependence collapses entirely into the chirp mass.

The chirp mass is the best-measured parameter of any inspiral, because it controls the phase evolution that a matched-filter analysis tracks over hundreds or thousands of cycles. The individual masses, the spins, and the distance enter at higher order and are extracted from the full waveform including the merger and ringdown. Measuring $\dot{f}$ and $f$ gives $\mathcal{M}$; combined with the observed amplitude $h$, which scales as ${\mathcal{M}}^{5/3}{f}^{2/3}/r$, the analysis also yields the luminosity distance $r$, making a compact binary a self-calibrating standard siren, the tool developed in the next lesson.

Interferometric detection

A strain of $h\sim 10^{-21}$ changes a $4\mathrm{km}$ interferometer arm by $\delta L\sim 10^{-18}\mathrm{m}$, a thousandth of a proton radius. A Michelson interferometer measures exactly this differential arm-length change. Laser light is split and sent down two perpendicular arms, reflected, and recombined; a passing wave lengthens one arm and shortens the other, shifting the interference fringe at the output. LIGO, Virgo, and KAGRA fold the light through Fabry-Pérot cavities in each arm, so the light traverses the arm hundreds of times and multiplies the accumulated phase shift, and add power recycling to build up the circulating laser power.

The overwhelming challenge is noise. Seismic vibration dominates at low frequency, thermal motion of the mirror suspensions in the middle band, and photon shot noise at high frequency. The mirrors hang as pendulums to isolate them from ground motion, and the detectors are sensitive between roughly $10$ and $1000\mathrm{Hz}$, the band in which stellar-mass compact binaries chirp through merger. A real signal is confirmed by coincident detection in widely separated instruments, which also triangulates the sky position from the arrival-time differences.

GW150914

On 14 September 2015 the two LIGO detectors recorded a chirp sweeping from $35$ to $250\mathrm{Hz}$ over about $0.2\mathrm{s}$, matching the template for two inspiraling black holes. The measured chirp mass and full waveform fit gave component masses of about $36M_{\odot }$ and $29M_{\odot }$ merging into a final black hole of about $62M_{\odot }$.

GW150914 confirmed three predictions at once: gravitational waves exist and travel as the quadrupole formula requires, stellar-mass black holes exist and form binaries that merge within a Hubble time, and the full inspiral-merger-ringdown waveform matches the numerical solution of the Einstein equations. It opened gravitational-wave astronomy as an observational field.

Summary

Gravitational radiation begins at quadrupole order, with strain $h=(2G/c^{4}r){\ddot{I}}_{ij}$ and luminosity $L_{\mathrm{GW}}=(G/5c^5)⟨{\stackrel{...}{I}}_{ij}{\stackrel{...}{I}}_{ij}⟩$; the tiny factor $G/c^5$ makes the waves weak. The plus and cross polarizations stretch and squeeze a ring of free masses along axes $45^{\circ }$ apart, producing a fractional strain $\delta L/L=h/2$. A binary radiates orbital energy, so the separation shrinks as $\dot{a}\propto -a^{-3}$ and the gravitational-wave frequency sweeps upward in a chirp whose rate $\dot{f}\propto {\mathcal{M}}^{5/3}{f}^{11/3}$ fixes the chirp mass $\mathcal{M}=(m_1{m}_2{)}^{3/5}/(m_1+m_2{)}^{1/5}$. Laser interferometers with Fabry-Pérot arms measure the $h\sim 10^{-21}$ strain, and GW150914 recorded two merging black holes of $\sim 36$ and $29M_{\odot }$ radiating $\sim 3M_{\odot }{c}^2$, launching gravitational-wave astronomy.