Stochastic and Resolvable Gravitational Waves from Ultralight Bosons

Ultralight scalar fields around spinning black holes can trigger superradiant instabilities, forming a long-lived bosonic condensate outside the horizon. We use numerical solutions of the perturbed field equations and astrophysical models of massive and stellar-mass black hole populations to compute, for the first time, the stochastic gravitational-wave background from these sources. In optimistic scenarios the background is observable by Advanced LIGO and LISA for field masses $m_s$ in the range $\sim [2\times {10}^{-13},{10}^{-12}]$ and $\sim 5\times [{10}^{-19},{10}^{-16}]\mathrm{eV}$, respectively, and it can affect the detectability of resolvable sources. Our estimates suggest that an analysis of the stochastic background limits from LIGO O1 might already be used to marginally exclude axions with mass $\sim {10}^{-12.5}\mathrm{eV}$. Semicoherent searches with Advanced LIGO (LISA) should detect $\sim 15(5)$ to 200(40) resolvable sources for scalar field masses $3\times {10}^{-13}$ $({10}^{-17})\mathrm{eV}$. LISA measurements of massive BH spins could either rule out bosons in the range $\sim [{10}^{-18},2\times {10}^{-13}]\mathrm{eV}$, or measure $m_s$ with 10% accuracy in the range $\sim [{10}^{-17},{10}^{-13}]\mathrm{eV}$.

Figure 1

GW strain produced by BH-boson condensates compared to the Advanced LIGO PSD at design sensitivity [46] and to the nonsky averaged LISA PSD [12] (black thick curves), assuming a coherent observation time of $T_{\mathrm{obs}}=4\mathrm{yr}$ in both cases. Nearly vertical lines represent BHs with initial spin ${\chi }_i=0.9$. Each line corresponds to a single source at redshift $z\in (0.001,3.001)$ (from right to left, in steps of $\delta z=0.2$), and different colors correspond to different boson masses $m_s$. Thin lines show the stochastic background produced by the whole population of astrophysical BHs under optimistic assumptions (cf. main text for details). The PSD of DECIGO [47] (dashed line) is also shown for reference.

Figure 2

Left panel: Stochastic background in the LIGO and LISA bands. For LISA, the three different signals correspond to the optimistic (top), less optimistic (middle), and pessimistic (bottom) astrophysical models. For LIGO, the different spectra for each boson mass correspond to a uniform spin distribution with (from top to bottom) ${\chi }_i\in [0.8,1]$, [0.5, 1], [0, 1], and [0, 0.5]. The black lines are the power-law integrated curves of Ref. [73], computed using noise PSDs for LISA [12], LIGO’s first two observing runs (O1 and O2), and LIGO at design sensitivity (O5) [76]. By definition, ${\rho }_{\text{stoch}}>1$ (${\rho }_{\text{stoch}}=1$) when a power-law spectrum intersects (is tangent to) a power-law integrated curve. Right panel: ${\rho }_{\text{stoch}}$ for the backgrounds shown in the left panel. We assumed $T_{\mathrm{obs}}=2\mathrm{yr}$ for LIGO and $T_{\mathrm{obs}}=4\mathrm{yr}$ for LISA.

Figure 3

Resolvable events for the same astrophysical models used in Fig. 2. Shaded areas correspond to exclusion regions from four-year LISA massive BH spin measurements, using either the popIII (brown) or Q3-nod (light blue) models of Ref. [48]. For reference we also show with brackets the constraints that can be placed by spin measurements of massive or stellar-mass BHs [37, 80].