Fuzzy dark matter soliton as gravitational lens

The Schrödinger-Poisson (SP) equations predict fuzzy dark matter (FDM) solitons. Given the FDM mass $\sim10^{-20}\rm~{eV}/c^2$, the FDM soliton in the Milky Way is massive $\sim 10^7M_{\odot}$ but diffuse $\sim 10{\rmpc}$. Therefore, such FDM soliton can serve as a gravitational lens for gravitational waves (GWs) with frequency $\sim10^{-8}{\rmHz}$. In this paper, we investigate its gravitational lensing effects by numerical simulation of the propagation of GWs through it. We find that the maximum magnification factor of GWs is very small $\sim10^{-4}$, but the corresponding magnification zone is huge $\sim6{\rmpc}$ for FDM with mass equal to $8\times10^{-21}\rm~{eV}/c^2$. Consequently, this small magnification factor compounding over such large magnification zone results in a small antisotropy of $\sim10^{-4}$ over a large solid angle in the GW background. That level of antisotropy is out of the sensitivity, $<20%$, of the pulsar timing arrays today.

💡 Research Summary

This paper investigates the gravitational lensing effect of fuzzy dark matter (FDM) solitons on ultra‑low‑frequency gravitational waves (GWs) with frequencies around 10⁻⁸ Hz, a band targeted by pulsar timing arrays (PTAs). FDM is modeled as an ultralight scalar field (mass ≈10⁻²⁰ eV/c²) whose dynamics obey the coupled Schrödinger–Poisson (SP) equations. Solutions of the SP system predict a ground‑state soliton at the center of a Milky Way‑size halo with a mass of order 10⁷ M☉ and a characteristic radius of ∼10 pc. Because the GW wavelength at 10⁻⁸ Hz is ≈1 pc, the soliton’s size is comparable, suggesting a possible weak lensing effect.

The authors first compute soliton density profiles for three particle masses (8, 10, 12 × 10⁻²¹ eV/c²) using a shooting method on the dimensionless SP equations, and then translate these profiles into physical gravitational potentials Φ(r) and spatial‑curvature perturbations Ψ(r). Both potentials are shallow, with amplitudes ≈10⁻⁶, implying that any Shapiro time delay will be negligible.

To treat GW propagation correctly, the paper derives a modified wave equation for the GW strain tensor in the weak‑field, transverse‑traceless gauge. By applying the eikonal approximation, the tensorial problem reduces to a scalar wave equation for the amplitude u: ∇²u + ∇(Φ − Ψ)·∇u − (1 − 2Φ − 2Ψ)⁻¹c⁻²∂²_t u = 0. The authors introduce an effective wave speed a and a vector b = Φ − Ψ, recasting the equation into a form suitable for finite‑element analysis.

Numerical simulations are performed with a customized version of the GWsim code, which implements a Crank–Nicolson time integrator (θ = ½) and second‑order spatial discretization on a 3‑D cylindrical domain (radius 7.5 pc, length 15 pc). The mesh is refined to 2⁸ levels, yielding ≈1.7 × 10⁸ degrees of freedom; each run consumes ~320 CPU cores for ~14 000 CPU‑hours. Plane GWs of amplitude 1 and frequency 10⁻⁸ Hz are injected along the cylinder axis. Three soliton configurations are examined, differing in radius (3.5, 4.4, 5.8 pc) corresponding to the three particle masses.

Results show that along the propagation direction (x‑y plane) the flat potential produces no discernible Shapiro delay or amplitude change. However, in the transverse (y‑z) plane a clear magnification pattern emerges. The magnification factor, defined as F(x)=|u − u_v|/u_v (where u_v is the unlensed amplitude), reaches a maximum of ≈10⁻⁴. The “magnification zone”—the region where the amplitude is noticeably enhanced—extends to ≈6 pc for the lightest particle mass (8 × 10⁻²¹ eV/c²), ≈5 pc for 10 × 10⁻²¹ eV/c², and ≈4 pc for the heaviest case. Thus, while the individual amplification is tiny, it covers a large solid angle, leading to an overall anisotropy in the stochastic GW background of order ΔΩ/Ω ≈ 10⁻⁴.

Current PTA experiments (e.g., NANOGrav, EPTA) have sensitivity at the 20 % level, far above the predicted anisotropy, rendering detection infeasible with present data. The authors conclude that the method provides a novel, indirect probe of FDM in the mass range ∼10⁻²⁰ eV/c², but practical constraints make it unlikely to be realized soon. They also emphasize that traditional analytic lensing formulas, which ignore the differing propagation medium inside the lens, are insufficient; direct numerical integration of the full wave equation is required. Future work could explore higher‑mass solitons, multi‑lens configurations, or the use of more sensitive detectors such as space‑based interferometers (LISA) to improve prospects for observing these subtle effects.