The Chirp: Reading a Merger From a Rising Whistle

Why a Shrinking Orbit Sings a Rising Note

Two compact objects on a circular orbit of separation a carry a mass quadrupole moment that returns to itself twice per orbit. Einstein's 1918 quadrupole formula gives the power such a system pours into gravitational waves: L_GW = (32/5)(G/c⁵) μ² a⁴ Ω⁶, where μ = m₁m₂/M is the reduced mass and Ω the orbital angular frequency. The steep dependence on Ω and a is the whole story: this is a switch that is essentially off for wide orbits and violently on for tight ones.

Now follow the bookkeeping. The orbital energy of a bound pair is E = −Gm₁m₂/2a. Radiating energy makes E more negative, which means a shrinks. But Kepler's third law, Ω² = GM/a³, says a smaller orbit is a faster orbit. Gravitationally bound systems have an effective negative heat capacity: take energy out and they speed up. Combining the two gives the inspiral rate da/dt = −(64/5) G³ m₁m₂M / (c⁵a³). Because the shrink rate itself scales as a⁻³, the process feeds on itself — slow for aeons, then catastrophic in the last seconds.

For a circular orbit the dominant radiation comes out at twice the orbital frequency, f_GW = 2f_orb, because the quadrupole pattern is symmetric under a half-turn. Amplitude climbs alongside frequency, since the wave strain scales as the orbital velocity squared. A tone that sweeps upward in frequency is exactly what a radar engineer calls a chirp — here with rising loudness thrown in — and the name stuck.

One detail makes the signal clean rather than messy: eccentric orbits radiate hardest at pericentre, which drains angular momentum preferentially and circularises the orbit. Philip Peters showed in 1964 that this happens long before merger, so a binary formed from an isolated stellar pair typically arrives in LIGO's band with eccentricity below ~10⁻⁴. The result is a smooth, monotonic sweep rather than a train of bursts.

The Chirp Equation and the Chirp Mass

Differentiating the inspiral relation and rewriting it in terms of the wave frequency gives the equation that defines the phenomenon:

df/dt = (96/5) π^8/3 (GM_c/c³)^5/3 f^11/3

Remarkably, the two masses do not appear separately. They enter only through one combination, the chirp mass:

M_c = (m₁m₂)^3/5 / (m₁+m₂)^1/5 = η^3/5 M, with η = m₁m₂/M².

Integrating gives the time left before coalescence from any frequency: τ = (5/256)(GM_c/c³)^−5/3(πf)^−8/3. Put GW150914's detector-frame chirp mass of ~31 M☉ and f = 35 Hz into that and you get 0.17 s — which is why the most consequential signal in modern astronomy lasted about a fifth of a second. Put GW170817's 1.19 M☉ and f = 24 Hz in instead and you get ~100 s, matching the observed duration almost exactly.

The number of wave cycles above a given frequency scales as M_c^−5/3f^−5/3: roughly 16,000 cycles above 10 Hz for a 1.2 M☉ chirp mass, but only about eight for GW150914 above the 35 Hz at which it entered LIGO's band. Every one of those cycles constrains the phase, which is why M_c is by far the best-measured parameter of any event — GW170817's is 1.186 M☉ to about 0.1%, a precision that would be the envy of most of astronomy.

The individual masses are a different matter. They enter the post-Newtonian phase expansion only at higher order (η appears at 1PN, spins at 1.5PN), and the effect of increasing the mass ratio can be partly cancelled by changing the aligned spin. This mass-ratio–spin degeneracy means GW170817's component masses are known only to roughly ±10% even though their chirp-mass combination is nailed to four digits.

Amplitude Gives Distance: The Standard Siren

The other half of the waveform is its size. To leading order the strain from a circular inspiral is

h ≈ (4/D_L)(GM_c/c²)^5/3(πf/c)^2/3 × geometry factors,

where D_L is luminosity distance. Every quantity on the right except D_L is already determined by the phase of the signal. So a gravitational-wave chirp measures its own distance absolutely, with no calibration against Cepheids or Type Ia supernovae — the idea Bernard Schutz published in 1986; the name standard siren came later, with Holz and Hughes in 2005. It is one of the very few distance indicators that carries its own zero point — derived from the physics of the source rather than borrowed from a lower rung of the cosmic distance ladder.

Two caveats bite. First, the amplitude also depends on the binary's inclination: a face-on system looks louder than an edge-on one, so distance and inclination are strongly correlated. Breaking that degeneracy requires extra information — for GW170817 it came from very-long-baseline radio imaging of the superluminally moving jet with the VLBA, HSA and EVN, which pinned the viewing angle and sharpened the Hubble constant to roughly 69 ± 5 km s⁻¹ Mpc⁻¹, compared with 70 (+12/−8) from the gravitational data alone.

Second, gravitational waves are redshifted like everything else, and mass and frequency scale inversely. A detector always measures the redshifted chirp mass (1+z)M_c; the intrinsic value requires an independent redshift, from an identified host galaxy, from a statistical association with a galaxy catalogue (a “dark siren”), or from a known mass scale such as the neutron-star tidal signature. GW150914's 30.9 M☉ in the detector frame becomes ~28 M☉ at the source once z ≈ 0.09 is folded in.

Digging the Whistle Out of the Noise

A peak strain of 10⁻²¹ moves the ends of LIGO's 4 km arms by about 4×10⁻¹⁸ metres, roughly two thousandths of the diameter of a proton. That is well below the instantaneous noise floor of the Advanced LIGO detectors at Hanford and Livingston (and Virgo's 3 km arms near Pisa, and KAGRA underground in Kamioka). Nobody sees a chirp by staring at a strain time series.

The recovery technique is matched filtering: cross-correlate the data with a bank of predicted waveforms, weighting each frequency by the inverse of the measured noise power spectral density. Because the noise is close to stationary and Gaussian while the template phase-tracks the signal, the signal-to-noise ratio accumulates roughly as the square root of the number of cycles — which is why a hundred-second neutron-star inspiral can be dug out even though it is far quieter than a black-hole merger. Advanced LIGO's first observing run used a bank of about 250,000 templates spanning component masses and aligned spins; modern searches (PyCBC, GstLAL, MBTA, SPIIR) run banks several times larger.

Significance comes from coincidence and from time-slid background estimation: shift one detector's data relative to the other by more than the 10 ms light-travel time, and any surviving coincidences are noise. GW150914's false-alarm rate came out at less than one per 203,000 years, with a network SNR of 24; GW170817 reached SNR 32.4. Notably, GW150914 was first flagged not by matched filtering but by the unmodelled burst pipeline coherent WaveBurst, within three minutes of the data arriving — Marco Drago's alert email is now a piece of history.

The method has failure modes. Short instrumental transients called blip glitches mimic the few-cycle signals expected from very massive binaries, which is why high-mass candidates get the heaviest vetting. GW170817 itself had a loud Livingston glitch about 1.1 s before merger that had to be modelled and subtracted before parameter estimation. And matched filtering is only as good as its bank: a strongly eccentric or strongly precessing binary can lose SNR or be biased, because the templates assume quasi-circular orbits.

Where the Chirp Ends: Merger and Ringdown

The post-Newtonian expansion that produces the chirp equation is a series in v/c, and it degrades badly once the orbital speed approaches half the speed of light. As a rough marker, the wave frequency at the innermost stable circular orbit of a Schwarzschild spacetime is f ≈ 4.4 kHz × (M☉/M_total), which is ~70 Hz for GW150914's 65 M☉ system and ~1.6 kHz for a 2.8 M☉ neutron-star pair. Beyond that the objects plunge, and only numerical relativity — the 2005 breakthrough by Frans Pretorius and the independent moving-puncture results from the Brownsville and Goddard groups a few months later — can supply the waveform. Modern analyses use hybrid families such as the effective-one-body models (EOBNR) and the phenomenological IMRPhenom series, both calibrated against numerical simulations.

After merger the remnant is a single distorted black hole that settles by radiating quasinormal modes: the ringdown. Only the mass and spin matter, per the no-hair theorem. GW150914's remnant of 62 M☉ and dimensionless spin ~0.67 rings in its dominant ℓ=m=2 mode at ~250 Hz with a damping time of ~4 ms — which is why the chirp appears to stop abruptly at its loudest.

The energy budget is worth stating plainly: 36 + 29 = 65 M☉ went in, 62 M☉ came out, and about 3 M☉c² ≈ 5×10⁵⁴ erg left as gravitational waves in roughly 0.2 s. The peak luminosity, ~3.6×10⁵⁶ erg s⁻¹ (about 200 M☉c² per second), briefly exceeded the combined light output of every star in the observable universe.

Neutron-star chirps end differently. In the last cycles each star is tidally stretched by its companion, which drains extra energy and makes the frequency rise slightly faster than a point-mass template predicts. This tidal deformability enters the phase at 5PN order and gave GW170817 its constraint of Λ-tilde ≤ 720 (90% credible, low-spin prior), implying radii near 11–12 km for a 1.4 M☉ neutron star and ruling out the stiffest equations of state.

Look-alikes, and What the Chirp Is Not

It is not sound. Gravitational waves are oscillations of spacetime geometry, not pressure waves in a medium. The audio renderings LIGO releases work only because stellar-mass compact binaries happen to merge at tens to hundreds of hertz, coincidentally overlapping human hearing. Shift the masses up by a factor of 10⁵ and the same physics plays out at millihertz frequencies for LISA.

It is not a Doppler effect. Nothing is approaching you faster and faster. The orbit itself genuinely accelerates as it tightens; the rising pitch is the source's own clock, not a projection of its motion.

It sweeps the wrong way to be a radio dispersion sweep. Fast radio bursts and pulsar pulses also produce a swept tone, but there the delay from ionised plasma scales as ν⁻², so the signal arrives at high frequency first and sweeps downward. Magnetospheric whistlers — the original falling tones of radio science — do the same. A gravitational-wave chirp always rises.

It is not what pulsar timing arrays hear. The supermassive black-hole binaries that dominate the nanohertz band evolve so slowly that df/dt is negligible over a human career; those sources are effectively monochromatic, and NANOGrav's 2023 evidence is for a stochastic background, not a resolved chirp.

And not every merger chirps audibly. GW190521, an 85 + 66 M☉ merger at ~5 Gpc, delivered only about four cycles between 30 and 80 Hz before ringing down — a thud rather than a whistle, and correspondingly harder to distinguish from a glitch.

The chirp's own prehistory is worth keeping straight too. Russell Hulse and Joseph Taylor discovered PSR B1913+16 in 1974 and, by timing its orbital decay, showed the period shrinking by about 2.4×10⁻¹² s per second — matching general relativity's prediction to better than 0.3%. That was the same physics measured indirectly, over decades, through pulse arrival times, and it earned the 1993 Nobel Prize. Detecting the waves themselves took another 22 years and the 2017 Nobel Prize for Rainer Weiss, Barry Barish and Kip Thorne.

What Chirps Still Have Left to Tell Us

Eccentricity and precession. Binaries assembled dynamically in globular clusters or in the gas discs of active galactic nuclei can enter the detector band with residual eccentricity, and with spins misaligned from the orbital angular momentum. Both leave fingerprints — eccentricity modulates the frequency evolution, precession amplitude-modulates the waveform — and both are how astronomers hope to separate formation channels. Current template banks mostly assume circular orbits, so measuring these is an active frontier rather than a solved problem.

The mass gaps. Electromagnetic surveys of X-ray binaries suggested a scarcity of compact objects between about 2.5 and 5 M☉. Chirps are testing that: GW190814's secondary of ~2.6 M☉ sits squarely in the disputed range and remains unclassified as the heaviest known neutron star or the lightest known black hole.

Waveform systematics. As sensitivity improves, the limiting error stops being noise and becomes the accuracy of the templates themselves. A next-generation instrument such as Cosmic Explorer or the Einstein Telescope would recover neutron-star inspirals with SNR in the hundreds, at which point small errors in the modelled phase would bias the inferred masses and tidal parameters.

Multiband astronomy. ESA adopted LISA in January 2024 for launch in the mid-2030s. Its millihertz band will hear massive black-hole binaries chirp for weeks and, tantalisingly, will catch stellar-mass binaries like GW150914's progenitors years before they reach the ground-based band — predicting the exact date of a merger a ground detector will later record.

The through-line is the same in every case: because the phase of a chirp is a clock, and because that clock is governed by so few parameters, a fraction of a second of a rising tone remains one of the most information-dense signals in all of physics.

How the source masses set the shape of the chirp

Source	Component masses (M☉)	Chirp mass (M☉)	What the detector hears
GW150914 (LIGO, 2015)	36 + 29	~28	~0.2 s, 35→250 Hz, ~8 cycles
GW170817 (LIGO–Virgo, 2017)	~1.46 + ~1.27	1.186 ± 0.001	~100 s, from 24 Hz, ~3,000 cycles
GW190425 (binary neutron star)	~2.0 + ~1.4	~1.44	Tens of seconds; found in one detector
GW190521 (2019)	85 + 66	~64	~0.1 s, ~4 cycles — a bang, not a whistle
PSR B1913+16 (Hulse–Taylor)	1.44 + 1.39	~1.23	7.75 h orbit, f ~ 72 µHz — inaudible to LIGO
Massive BH binary (LISA target)	10⁵ + 10⁵	~9×10⁴	Millihertz chirp lasting weeks to months