Experiment: Gravitational Waves
The prediction of Gravitational-Wave Theory — that accelerated masses radiate energy in ripples of spacetime — was confirmed twice, by two utterly different techniques separated by four decades. First indirectly: a binary pulsar, timed for years with radio telescopes, was observed to lose orbital energy at exactly the rate demanded by the quadrupole formula of The Quadrupole Formula for Gravitational Radiation. Then directly: kilometre-scale laser interferometers recorded the strain of a passing wave itself. This chapter presents both records in full phenomenological detail; no derivations are performed here.
The Hulse–Taylor Binary Pulsar (PSR B1913+16) (1974–2016)
Tests Phenomenon 46.1 and Equation (46.6).
A pulsar is a rotating neutron star whose radio beam sweeps past the Earth once per rotation, providing a clock of extraordinary stability. A pulsar bound in a tight orbit around a second neutron star is therefore a moving clock in a strong gravitational field — a ready-made laboratory for relativistic orbital dynamics, including the slow decay of the orbit by gravitational radiation.
Apparatus
The \(305\,\mathrm{m}\) radio telescope at Arecibo, Puerto Rico, equipped for precision pulsar timing: dispersion-corrected radio receivers and time-tagging of pulse arrival times against maser clocks, later linked to international atomic time. The pulsar PSR B1913+16 was found in a systematic survey for new pulsars conducted by Hulse and Taylor in the summer of 1974 [Hulse:1975].
Procedure
The pulsar's spin period of about \(59\,\mathrm{ms}\) was observed to shift periodically by up to \(80\,\mu\mathrm{s}\), revealing orbital Doppler motion: the pulsar orbits an unseen companion, with orbital period \(7.75\,\mathrm{h}\) and orbital eccentricity \(e \approx 0.617\) [Hulse:1975]. Over the following four decades, pulse arrival times were accumulated and fitted to a fully relativistic timing model, whose post-Keplerian parameters — periastron advance, combined time dilation and gravitational redshift, and the secular decay of the orbital period — are measured without assuming general relativity and only then compared with its predictions [Taylor:1982] [Weisberg:2016].
Observations and data
The key fitted parameters are collected below [Hulse:1975] [Weisberg:2016].
| Quantity | Value |
|---|---|
| Pulsar spin period | \(59\,\mathrm{ms}\) |
| Orbital period $P_b$ | \(7.75\,\mathrm{h}\) |
| Orbital eccentricity $e$ | \(0.617\) |
| Periastron advance $\langle\dot\omega\rangle$ | \(4.226^\circ\) per year |
| Orbital period decay $\dot P_b$ | $\approx -2.4\times 10^{-12}\,\mathrm{s}/\mathrm{s}$ |
| Observed/predicted decay ratio | $0.9983 \pm 0.0016$ |
The periastron of the orbit advances by \(4.226^\circ\) per year — in a single day, roughly what Mercury's perihelion accumulates in a century. The orbital period shrinks by about \(-2.4\times 10^{-12}\,\mathrm{s}/\mathrm{s}\), i.e. some \(76.5\,\mu\mathrm{s}\) per year. The most vivid presentation of the timing data is the cumulative shift of periastron passage times relative to a constant-period orbit: the data points trace a downward-opening parabola, reaching a cumulative shift of about \(-40\,\mathrm{s}\) after roughly thirty years of observation, with the general-relativistic prediction drawn through them as a solid curve from which no point visibly departs [Weisberg:2016]. After correcting for the small relative acceleration of the solar system and the pulsar in the Galaxy, the ratio of the observed period decay to that predicted by the quadrupole formula is \(0.9983 \pm 0.0016\) [Weisberg:2016].
Interpretation
The orbit decays because the system radiates energy in gravitational waves, at precisely the rate given by the quadrupole formula (The Quadrupole Formula for Gravitational Radiation): agreement at the level of \(0.16\%\), with no adjustable parameters. This was the first evidence — indirect but quantitative — for the existence of gravitational radiation, and it earned Hulse and Taylor the 1993 Nobel Prize in Physics. The steadily improving timing residuals of [Weisberg:2016] remain among the most stringent tests of the radiative sector of general relativity.
Every post-Keplerian parameter entering these fits is now derived in this book. The orbital decay \(\dot{P}_b\) — the parameter carrying the gravitational-wave result quoted above — follows from the quadrupole formula proved in The Quadrupole Formula for Gravitational Radiation, including the eccentricity enhancement factor \(f(e)\) evaluated there at the measured \(e=0.617\). The remaining three rest on the classical-test machinery of Experiment: The Classical Tests of General Relativity and are short enough to derive here.
Derivation of the remaining post-Keplerian parameters. (i) Periastron advance. Equation (53.1) gives the relativistic advance per orbit, \(\Delta\varphi = 6\pi GM/[c^{2}a(1-e^{2})]\); for a binary the relative orbit obeys the same equation with \(M = m_{1}+m_{2}\) at leading order. The secular rate is \(\dot\omega = \Delta\varphi/P_b\), and eliminating the semi-major axis through Kepler's third law \((2\pi/P_b)^{2}a^{3} = GM\) gives
Evaluated at the measured \(P_b\), \(e\) and the fitted masses \(m_{1} = 1.4398\,M_{\odot}\), \(m_{2} = 1.3886\,M_{\odot}\) [Weisberg:2016], this gives \(4.23^\circ\) per year (computed here), against the fitted \(4.2266^\circ\).
(ii) Einstein delay. The pulsar's proper time runs against coordinate time as \(\dd\tau/\dd t = 1 - v_{1}^{2}/2c^{2} - Gm_{2}/(c^{2}r)\), the two corrections being special-relativistic time dilation and the companion's redshift (The Equivalence Principle and Classical Tests). With \(v_{1} = (m_{2}/M)\,v\) on the relative orbit and the vis-viva relation \(v^{2} = GM(2/r - 1/a)\), the \(1/r\) terms combine with coefficient \(Gm_{2}(m_{1}+2m_{2})/(Mc^{2})\). On the Keplerian ellipse \(r = a(1 - e\cos E)\) with \(nt = E - e\sin E\) one has \(\int \dd t/r = E/(na)\), whose periodic part is \((e/na)\sin E\) — the constant rates are unobservable, absorbed into the fitted spin period. The arrival times are therefore modulated by \(\Delta\tau = -\gamma_{\mathrm{E}}\sin E\) with
which at the fitted masses is \(\approx4.3\,\mathrm{ms}\) (computed here), as measured.
(iii) Shapiro range and shape. Equation (53.5) applied with the companion as the deflector adds a propagation delay \(2(Gm_{2}/c^{3})\ln[\,\cdot\,]\) to pulses crossing the companion's potential; near superior conjunction the logarithm's argument is fixed by the orbital geometry through \(\sin i\) alone, so the timing model parametrizes the effect by the range \(r_{\mathrm{S}} = Gm_{2}/c^{3}\) and the shape \(s = \sin i\). In PSR B1913+16 the inclination makes the effect only marginally visible; it is measured superbly in the double pulsar (Compact Stars and Relativistic Astrophysics).
∎Together with \(\dot P_b\) from The Quadrupole Formula for Gravitational Radiation, the timing model is thus derived end to end: three parameters (Equations (47.1) and (47.2), and \(r_{\mathrm{S}},s\)) overdetermine the two masses, and the gravitational-wave test is that the fourth, \(\dot P_b\), then agrees with no freedom left.
Primary references
[Hulse:1975] [Taylor:1982] [Weisberg:2016].
Direct detection: the instrument
The Laser Interferometer Gravitational-Wave Observatory (LIGO) consists of two detectors of identical design, one at Hanford, Washington, and one at Livingston, Louisiana, separated by about \(3000\,\mathrm{km}\) — a light travel time of about \(10\,\mathrm{ms}\) that both provides a coincidence veto against local disturbances and encodes the sky direction of a source in the inter-site arrival-time difference [Aasi:2015].
Each detector is a Michelson interferometer whose two \(4\,\mathrm{km}\) arms are Fabry–Perot resonant cavities, multiplying the phase accumulated by the laser light in each arm by the cavity finesse. A power-recycling mirror at the input builds up the circulating laser power, and a signal-recycling mirror at the output shapes the instrument's frequency response [Aasi:2015]. A passing gravitational wave of strain \(h\) changes the differential arm length by \(\Delta L = hL/2\) (Gravitational-Wave Theory); at the sensitivity reached in 2015 this corresponds to measurable differential displacements of order \(10^{-18}\,\mathrm{m}\) — about one thousandth of the diameter of a proton.
The noise budget divides the band into three regimes [Aasi:2015]. Below roughly \(10\,\mathrm{Hz}\) rises the seismic wall: ground motion, suppressed by active isolation platforms and by suspending each \(40\,\mathrm{kg}\) mirror from a four-stage pendulum, still dominates and sets the low-frequency edge of the observation band. In the middle of the band, from tens to a couple of hundred hertz, thermally driven motion of the fused-silica suspension fibres and of the mirror coatings is the limiting noise. At high frequencies the limit is quantum: photon shot noise in the readout, improved by raising the circulating power. The quietest region of the resulting sensitivity curve, around \(100\,\mathrm{Hz}\), is precisely where a stellar-mass compact binary sweeps through its final orbits.
GW150914 (2015)
Tests Phenomenon 46.1 and Equation (46.8).
Apparatus
The two Advanced LIGO interferometers at Hanford and Livingston, described in Section 47.2 [Aasi:2015], in their first observing run and operating in coincidence [Abbott:2016].
Procedure
Two independent analyses ran over the coincident data [Abbott:2016]. A matched-filter search correlated the strain data against a bank of about \(250000\) relativistic waveform templates for compact-binary coalescences; a separate, unmodelled burst search looked for coherent excess power in both detectors without waveform assumptions. Both searches recovered the same event, and the significance was established by measuring the background of accidental coincidences with time-shifted data.
Observations and data
On 2015 September 14 at 09:50:45 UTC both detectors recorded the same transient [Abbott:2016]:
| Quantity | Value |
|---|---|
| Peak strain amplitude | \(10^{-21}\) |
| Frequency sweep | \(35\text{–}250\,\mathrm{Hz}\) in about \(0.2\,\mathrm{s}\) |
| Cycles observed | $\sim 8$ |
| Hanford–Livingston delay | $6.9^{+0.5}_{-0.4}\,\mathrm{ms}$ |
| Combined signal-to-noise ratio | 24 |
| False-alarm rate | $< 1$ per \(203000\)~yr ($> 5.1\sigma$) |
| Primary mass $m_1$ | $36^{+5}_{-4}\,M_\odot$ |
| Secondary mass $m_2$ | $29^{+4}_{-4}\,M_\odot$ |
| Final black-hole mass | $62^{+4}_{-4}\,M_\odot$ |
| Radiated energy | $3.0^{+0.5}_{-0.5}\,M_\odot c^2$ |
| Peak luminosity | $3.6^{+0.5}_{-0.4}\times 10^{49}\,\mathrm{W}$ |
| Luminosity distance | $410^{+160}_{-180}\,\mathrm{Mpc}$ ($z \approx 0.09$) |
The signal swept upward in frequency from \(35\,\mathrm{Hz}\) to \(250\,\mathrm{Hz}\) over about \(0.2\,\mathrm{s}\) — roughly eight cycles — increasing in amplitude to a peak strain of \(10^{-21}\), then rang down. The Livingston detector recorded it first; Hanford followed \(6.9^{+0.5}_{-0.4}\,\mathrm{ms}\) later, comfortably within the \(10\,\mathrm{ms}\) light travel time between the sites. The waveforms observed in the two detectors, once shifted by this delay and inverted for the relative orientation of the instruments, lie on top of one another [Abbott:2016].
Interpretation
The chirp-mass relation of Gravitational-Wave Theory applied to the observed frequency and frequency sweep gives \(\mathcal{M} \approx 30\,M_\odot\), and hence a total mass so large, in so small an orbit at merger, that only a pair of black holes fits: at \(75\,\mathrm{Hz}\) orbital frequency the objects, of \(36^{+5}_{-4}\) and \(29^{+4}_{-4}\) solar masses, were only a few hundred kilometres apart. GW150914 is thus simultaneously the first direct detection of gravitational waves and the first observation of a binary black-hole merger. The full signal — inspiral, merger, and the damped ringdown of the final \(62^{+4}_{-4}\,M_\odot\) black hole — matches the inspiral–merger–ringdown waveform computed from general relativity, with about \(3.0 \pm 0.5\,M_\odot c^2\) radiated in a fraction of a second at a peak luminosity of \(3.6^{+0.5}_{-0.4}\times 10^{49}\,\mathrm{W}\), briefly exceeding the combined light output of the observable Universe [Abbott:2016]. The detection was recognized with the 2017 Nobel Prize in Physics.
Primary references
[Abbott:2016], the discovery; [Aasi:2015], the detector.
The multimessenger outlook
Two years after the discovery, on 2017 August 17, the LIGO–Virgo network recorded GW170817, an inspiral lasting about \(100\,\mathrm{s}\) with component masses of \(1\)–\(2\,M_\odot\): a binary neutron-star merger [Abbott:2017]. A short gamma-ray burst arrived \(1.7\,\mathrm{s}\) after the merger, and the optical counterpart — a kilonova in the galaxy NGC 4993 at about \(40\,\mathrm{Mpc}\) — was followed across the electromagnetic spectrum for weeks; the near-simultaneity of the gravitational and gamma-ray signals after a journey of more than a hundred million years constrains the speed of gravitational waves to equal \(c\) to within a few parts in \(10^{15}\) [Abbott:2017]. Since then the catalogue of compact binary coalescences has grown from a single event to on the order of a hundred, with black-hole masses ranging from a few to over a hundred solar masses.
Gravitational-wave astronomy has thus passed, within a decade, from heroic first detection to routine observation: a new spectrum, opened on a sector of the Universe — coalescing black holes — that emits no light at all. Its implications for the largest scales, where the same field equations govern the expansion of the Universe itself, are taken up in Evidence-Based Cosmology.