- The paper demonstrates that realistic galaxy-scale macrolenses convert ordinary CDM subhalo perturbations into percent-level frequency-dependent amplitude changes and phase shifts of about 10⁻² radians in LISA-band gravitational waves.
- The analysis finds that subhalos of 10⁴–10⁷ solar masses dominate the signal, with contributions saturating below 10⁴ solar masses because lighter objects produce time delays too short to affect the relevant frequencies.
- The paper shows that near-critical macro amplification is essential: with magnifications of roughly 3–5 and intrinsic signal-to-noise ratios above 100, these distortions can be detected at several-sigma significance in favorable lensed massive-black-hole-binary events.
Overview
This paper by Ando (2603.04267) demonstrates that strongly lensed gravitational waves (GWs) provide a generically favorable setting for detecting wave-optics (WO) signatures of cold dark matter (CDM) subhalos. The central result is that realistic galaxy-scale macrolenses, populated with statistically generated CDM subhalo populations, produce percent-level frequency-dependent amplitude and phase distortions in the LISA band (10−4–10−1 Hz), dominated by subhalos in the mass range 104–107M⊙. The key physical claim is that these signatures arise not from exotic compact objects but from the interplay between ordinary subhalo perturbations and macro critical amplification near caustics: removing the external macro field suppresses the frequency dependence to below 10−3, even for identical subhalo realizations.
The fiducial system consists of a lens at zL=0.5 modeled as an NFW halo with M200c=1012M⊙ and concentration c200c=10/(1+zL), plus a singular isothermal sphere (SIS) galaxy with σv=250kms−1, lensing a source at zS=1.5 with dimensionless offset 10−10. Subhalo populations are drawn from the semi-analytic SASHIMI model, which self-consistently predicts the subhalo mass function and tidal evolution; each subhalo is a tidally truncated NFW profile. Massive subhalos (10−11) are folded into the macrolens potential and treated in the geometric-optics (GO) limit, while low-mass subhalos (10−12–10−13) near the macro minimum image are propagated explicitly through the diffraction integral using the GLoW framework.
The amplification factor is evaluated in local coordinates centered on the macro minimum:
10−14
where 10−15 is the macrolens Jacobian at the image and 10−16 the local subhalo potential. This external-field decomposition incorporates the macrolens exactly to second order while avoiding double counting. Subhalo sampling uses a mass-dependent selection radius combining a Fresnel-scale criterion (10−17), a magnification-perturbation radius 10−18 requiring at least a 1% GO magnification perturbation, and an internal-structure floor—guaranteeing inclusion of all dynamically relevant perturbers.
Wave-optics signatures and their origin
Across 200 independent realizations, the median amplification tracks the GO expectation, but the 68% and 95% ranges reveal percent-level relative amplitude modulations, most pronounced at 10−19 Hz, accompanied by phase shifts of order 1040 rad. A mass-threshold scan shows that lowering 1041 from 1042 to 1043 increases the modulation amplitude, which then saturates; extending to 1044 adds nothing. The interpretation follows from time-delay scaling: a perturber of mass 1045 induces delays 1046, so only 1047–1048 objects produce delays commensurate with LISA-band periods, while lighter subhalos remain effectively in GO.
The decisive control experiment removes the quadratic external term from the integral. With 1049, the same subhalo realizations yield 107M⊙0 with modulations suppressed below 107M⊙1: the Fermat surface stays nearly quadratic and the system remains in stationary-phase (GO). Near a critical curve, however, the large inverse Jacobian amplifies small deviations from quadratic structure, redistributing the local time-delay pattern into coherent WO distortions. A complementary time-domain analysis expresses 107M⊙2 as the Fourier transform of the Fermat-potential distribution 107M⊙3; subhalos distort the shape of 107M⊙4 without changing its normalization, directly producing the observed frequency-dependent structure. The implication is that macro criticality—not merely subhalo presence—is the essential mechanism converting subdominant perturbations into observable signals.
Detectability and event rates
For a lensed event with intrinsic signal-to-noise ratio 107M⊙5 and amplification 107M⊙6, fractional amplitude perturbations are detectable at significance 107M⊙7, and phase perturbations once 107M⊙8. With typical modulations of order 107M⊙9, magnifications 10−30–10−31, and 10−32—routine for massive black-hole binaries in LISA, with favorable systems reaching 10−33—the distortions are measurable at several-10−34 significance or better.
The apparent fine-tuning concern about the small source offset is addressed quantitatively. For a uniform source distribution under the SIS model, 10−35. However, magnification bias reshapes the detection-weighted distribution to 10−36, since the accessible volume scales as 10−37. Imposing a finite maximum magnification 10−38–10−39 yields zL=0.50–zL=0.51: near-critical events can constitute order 10% of detectable strongly lensed systems. Extending the offset to zL=0.52 raises the detection-weighted rate by only a factor of zL=0.53 while substantially reducing the WO amplitude, so zL=0.54 configurations dominate the science yield. Individual-realization studies confirm that although the detailed interference pattern varies stochastically, the overall modulation scale is robust across Monte Carlo draws.
Limitations and open questions
The analysis is restricted to the macro minimum image; saddle images, which are intrinsically more sensitive to perturbations via Morse-theoretic phase shifts, are deferred to future work. The fiducial configuration assumes a single host halo mass, redshift pair, and SIS-plus-NFW macrolens, so the quoted percent-level amplitudes are specific to this geometry rather than a population-averaged forecast. The saturation of the signal below zL=0.55 depends on the truncated NFW subhalo structure assumed by SASHIMI; alternative density profiles or baryonic effects on subhalo survival could shift the contributing mass range. Finally, the detection-rate argument relies on the SIS magnification scaling and an imposed zL=0.56, with finite source size and deviations from exact isothermality acknowledged as reducing the conservative estimate toward the lower end of the 0.15–0.3 range.
Conclusion
The paper establishes that strongly lensed GWs in the LISA band carry generic, percent-level WO imprints of standard CDM substructure at masses zL=0.57–zL=0.58, driven by macro critical amplification rather than exotic compact perturbers. Because the signal scales with perturber compactness, scenarios yielding denser substructure—primordial black holes or gravothermal core collapse in self-interacting dark matter—would enhance it further, making the observable a differential diagnostic between dark matter models. The work leaves open quantitative extension to saddle images, alternative dark matter scenarios, and full population-level forecasts of event rates.