Gravitational-Wave Lens Fringes

If one gravitational-wave chirp travels two paths around a mass, what does the detector hear? The detector hears the two copies added together. When the delay is longer than a wave period, they reinforce at some frequencies and cancel at others, so the loudness swings up and down across the chirp. When the wave is longer than the lens can bend, the copies blur into one and the signal barely changes.

Drag a point-mass lens across the line of sight. Watch two delayed copies of one chirp add up, and see fringes appear in the amplification across frequency.

Delay between the two paths $19.9\,\mathrm{ms}$

Loudness of the second copy $0.61$

Match with the unlensed chirp $0.84$

Lens signature strength $230$

Model assumptions

An isolated point mass lenses a synthetic equal-mass inspiral. The amplification factor is the exact point-lens wave-optics result, switched to the geometric-optics limit above a declared argument, where the measured difference stays below 0.2 percent. The chirp is the leading-order stationary-phase inspiral, cut at the innermost stable orbit, with no merger, ringdown, spin or detector antenna pattern. The noise curve is a published fit to the Advanced LIGO design sensitivity, not recorded detector noise. The match is maximized only over arrival time, phase and amplitude, so the distinguishability shown is an optimistic upper value. Distances, angles and packet sizes in the scene are schematic.

Gain at each frequencyCalculated · wave optics and two-image reference

The phase the lens addsCalculated · wave optics and two-image reference

The chirp with and without the lensSynthetic signal · noise-free

Signal against detector noiseSynthetic signal · design noise curve

Cyan: full wave calculation. Dashed gold: two-image limit, an ideal reference without diffraction. Gray: no lens.

What the match and lens signature mean

The match is the noise-weighted overlap between the lensed chirp and the unlensed chirp, maximized over arrival time, phase and amplitude. The lens signature strength is the squared lensed signal-to-noise ratio times one minus the squared match: the drop in fit quality an unlensed template would suffer on noise-free data.

Here the unlensed template keeps the true black-hole masses. A real search also varies masses and spins, which can absorb part of the lens effect, so these values are an optimistic upper bound. In the offline tutorial, pure noise beat the lens threshold of the tutorial in about one trial in nine.

Focus question: If you make the lens ten times lighter, do the fringes simply become ten times wider?
One-sentence intuition: The amplification depends on frequency only through the product of lens mass and frequency; when that product is small the wave flows around the lens instead of splitting.

Core mathematical model

Amplification factor

F(w,\vec y)=\frac{w}{2\pi i}\int d^2x\,\exp\!\bigl[i w\,T(\vec x,\vec y)\bigr],\qquad T=\tfrac12|\vec x-\vec y|^2-\ln|\vec x|-\phi_m(y)

Every point of the lens plane contributes a wavelet with its own arrival time; the lensed spectrum is the unlensed one multiplied by this complex sum.

Dimensionless frequency

w=\frac{8\pi G M_{Lz} f}{c^3}\approx 1.24\times10^{-4}\,\frac{M_{Lz}}{M_\odot}\,\frac{f}{1\,\mathrm{Hz}}

When w is well below one the wave is longer than the lens can bend and F stays close to one. When w is large the integral splits into two images.

Two-image limit

F\simeq|\mu_+|^{1/2}-i\,|\mu_-|^{1/2}\,e^{2\pi i f\,\Delta t_d}

The later saddle image carries an extra quarter-cycle phase. Its interference with the first image makes the magnitude of F oscillate with period one over the delay.

Common difficulties

Fringes are not proof of a lens

Typical misconceptionA wiggle in the measured amplitude across frequency proves the signal passed a lens.

Better mental modelNoise fluctuations, an imperfect waveform model or detector calibration can produce similar wiggles. The search compares the best lensed and the best unlensed template on the same data and calibrates the result with injections into noise.

Run the experiment

01
Drag the lens toward the axis
Start from Two paths, clear fringes and drag the lens slowly toward the line of sight. Predict what happens to the delay and to the second copy.
What to observe: The delay shrinks, the second copy grows nearly as loud as the first, and the fringes in the amplification widen and deepen.
02
Make the lens light
Choose Waves too long to split and compare the solid wave-optics curve with the dashed two-image curve.
What to observe: The dashed two-image curve predicts a gain of about one and a half, but the true amplification stays close to one at low frequency and rises only slowly: diffraction stopped the wave from splitting.
03
Ask whether it is detectable
Choose A faint second copy, open Source and loudness, and lower the loudness without a lens toward 10.
What to observe: The match stays near one and the lens signature drops to a few units, about what noise alone can produce.

Gravitational-Wave Lens Fringes

Two copies of one chirp add up at the detector

Core mathematical model

Amplification factor

Dimensionless frequency

Two-image limit

Common difficulties

Fringes are not proof of a lens

Run the experiment

Drag the lens toward the axis

Make the lens light

Ask whether it is detectable

Related experiments and articles

Articles

From two paths to a lens search