Gravitational-Wave Fringes at LIGO: Detecting Compact Dark Matter by Gravitational Lensing

Utilizing gravitational-wave (GW) lensing opens a new way to understand the small-scale structure of the Universe. We show that, in spite of its coarse angular resolution and short duration of observation, LIGO can detect the GW lensing induced by small structures, in particular by compact dark matter (DM) or the primordial black hole of $10–{10}^5{M}_{\odot }$, which remains an interesting DM candidate. The lensing is detected through GW frequency chirping, creating the natural and rapid change of lensing patterns: frequency-dependent amplification and modulation of GW waveforms. As a highest-frequency GW detector, LIGO is a unique GW lab to probe such light compact DM. With the design sensitivity of Advanced LIGO, one-year observation by three detectors can optimistically constrain the compact DM density fraction $f_{\mathrm{DM}}$ to the level of a few percent.

Figure 1

Illustration of GW lensing fringe at aLIGO: Lensed (red curves) vs unlensed (black curve) waveforms. The lensing compact DM mass $M_{\mathrm{DM}}=100M_{\odot }$ and redshifted GW binary masses $M_1=M_2=20M_{\odot }$ merging at $f\simeq 320\mathrm{Hz}$. At small impact angle $y={\theta }_s/{\theta }_E$ (red-solid line) Eq. (1), frequency-dependent amplification cannot be fit by a constant rescaling $A_0$ [Eq. (4)] of unlensed waveforms (black curve); whereas, at large $y$ (dashed line), frequency-dependent modulation cannot be matched by a constant phase shift ${\varphi }_0$ [Eq. (3)]. In general (dot dashed curve), both lensing effects coexist.

Figure 2

The lensing-detection efficiency $\epsilon$ in Eq. (8) of a single aLIGO detector with design sensitivity. The DM masses are $M_{\mathrm{DM}}={10}^4$ (green), $10^3$ (orange), 100 (blue), 30 (red) $M_{\odot }$ and redshifted binary masses $M_1=M_2=1.5$ (solid line), 20 (dashed line), 40 (dot-dashed line), 80 (long-dashed line) $M_{\odot }$.

Figure 3

The volume-averaged optical depth $\overline{\tau }$ as defined in Eq. (11). It is the probability for detectable lensing, including the probability for ${\mathrm{SNR}}_{\text{test}}>3$ (dashed line) or 5 (solid line) and $\mathrm{SNR}>8$ from three aLIGOs. The upper (lower) set of curves is for $M_{\mathrm{DM}}=300(30)M_{\odot }$. $f_{\mathrm{DM}}=1$.

Figure 4

The number of GW lensing detections per year by three aLIGOs. The darker (lighter) region satisfies ${\mathrm{SNR}}_{\text{test}}>5$ (3) and $\mathrm{SNR}>8$, and vertical ranges span between the optimistic merger-rate $M1$ and the pessimistic $M3$ [40]. The blue (orange) region is for the lensing $M_{\mathrm{DM}}=300(30)M_{\odot }$. The horizontal dashed line denotes one detection per year. The total detection rate is the sum over mass bins. $f_{\mathrm{DM}}=1$.

Figure 5

Expected 95% C.L. constraints on $f_{\mathrm{DM}}$ from one-year observation by three aLIGOs. The Poisson distribution of fringe detection probability is assumed. ${\mathrm{SNR}}_{\text{test}}>3$ (dashed line) or 5 (solid line) is used with optimistic (blue) or pessimistic (red) merger-rate models. Shaded regions are existing constraints: microlensing (left blue) [21], caustic crossing (green) [41], cluster survival (orange) [42], and millilensing of quasars (right blue) [23].