Fuzzy dark matter soliton as gravitational lens

TL;DR

This work investigates how a fuzzy dark matter (FDM) soliton in the Milky Way can gravitationally lens nanohertz gravitational waves (GWs) detected by pulsar timing arrays. By solving the Schrödinger–Poisson equations to obtain soliton density profiles and then numerically propagating GWs through the soliton with a finite-element formulation of the GW propagation equation, the authors quantify lensing effects. They find a maximum GW magnification of about $\sim10^{-4}$, occurring over a large magnification zone ($\sim4$–$6$ pc) for $m=8$–$12\times10^{-21}\rm\ eV/c^2$, while Shapiro time delays are negligible due to the soliton’s shallow potential. This lensing induces a small antisotropy, $\sim10^{-4}$, in the low-frequency GW background over a large solid angle, which currently lies below PTA sensitivity but could provide an independent constraint on the FDM mass with future improvements.

Abstract

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 $\sim 10^7~M_{\odot}$ but diffuse $\sim 10{\rm~pc}$. Therefore, such FDM soliton can serve as a gravitational lens for gravitational waves (GWs) with frequency $\sim10^{-8}{\rm~Hz}$. 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 $\sim6{\rm~pc}$ 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.

Figures (5)

• Figure 1: FDM soliton profiles for a set of FDM masses $m$, where the gravitational potential $\Phi({\bf{r}})$ decays approximately $\propto-\frac{GM(\bf{r})}{c^2|\bf{r}|}$ and the perturbation of the spatial curvature $\Psi({\bf{r}})=-\frac{GM(\bf{r})}{c^2|\bf{r}|}$Guzman:2004wj.
• Figure 2: Last snapshots of $u$ in the $x$-$y$ plane at $z=0$ for simulations with FDM soliton with $m=12\times10^{-21}{\rm~eV/c^2}$. This figure, compared with Fig. \ref{['fig:xyl']}, illustrates that due to the gravitational potential flatness of the FDM soliton, any gravitational lensing effects cannot be distinguished in the $x$-$y$ plane, see text.
• Figure 3: Last snapshots of $u$ in the $y$-$z$ plane with $r=7.5{\rm~pc}$ at $x=7.25{\rm~pc}$ for simulations without FDM soliton (upper left) and with FDM soliton with $m=8\times10^{-21}{\rm~eV/c^2}$ (upper right), $m=10\times10^{-21}{\rm~eV/c^2}$ (lower left) and $m=12\times10^{-21}{\rm~eV/c^2}$ (lower right) respectively.
• Figure 4: Formation of the nHz GW background anisotropy induced by the lensing effect of the Galactic Center soliton. The anisotropy proceeds from the maximum angle of magnified GWs sweeping the PTA.
• Figure 5: Last snapshots of $u$ in the $x$-$y$ plane at $z=0$ for simulations in vacuum.