Constraining Ultralight Scalars with Black Hole Binary Mergers in Galactic Nuclei

Author(s)

Khalaf, Majed, Kuflik, Eric, Lenoci, Alessandro, Stone, Nicholas Chamberlain, Telem, Ofri

Abstract

Ultralight scalars can form long-lived, macroscopic bound states around spinning black holes, known as superradiance clouds. These clouds provide an additional channel for energy dissipation during close encounters, enhancing the black hole binary formation and merger rates in dense environments such as galactic nuclei. We show that the rate of mergers with mass ratio below $3/10$ in the LIGO-Virgo-KAGRA GWTC-5 catalog can probe ultralight scalars in the mass range $[4\times 10^{-14},10^{-13}]$ eV, complementing existing strategies based on black hole spin measurements.

Figures

Caption

Exclusion plot in ultralight boson mass $\mu$ and self-interaction strength (expressed as $1/f = \sqrt{\lambda}/\mu$). The red contours represent the excluded region requiring the predicted rate of detections with mass ratio less than 3/10 to not exceed the counts from the GWTC-5 catalog~\cite{LIGOScientific:2026wfs} at 90\% confidence level. We assume a BH mass function in GNs $\propto M^{-\beta}$ with different values of $\beta=\{0.5,1,1.5\}$, increasing from the most aggressive to the most conservative, shown as different contours. We also present the constrains obtained from spin-down arguments, from a GW event~\cite{Caputo:2025oap,Aswathi:2025nxa} in blue and inferred from X-ray measurements~\cite{Witte:2024drg} in gray.

Caption

Detection rates obtained from the LLO detections in the LVK catalog (horizontal lines) compared to prediction of merger rates in LLO from SC-enhanced captures in GNs (scatter points) as a function of the ultralight scalar mass $\mu$. For each LLO rate the thin lines show 90\% upper bounds on this rates. For illustration, we fix $f=10^{19}$ GeV, $M_{\rm max}=100M_\odot$ and $\beta=1$. We show different partial rates with different colors. In all cases except in the $e(10\ {\rm Hz})>0.03$ channel for a given range of ultralight scalar masses $\mu$, the SC enhances the rates with respect to the standard GW-capture prediction. By comparing the points to the upper bounds, we set constraints on a certain $\mu$ interval.

Caption

Solutions of the system of ODEs~\eqref{app:ceq} for a BH with initial mass $M_{\rm i}=20M_\odot$, initial spin $\chi_{\rm i}=0.998$ and ultralight boson mass $\mu=6.7\times 10^{-13}$ eV for different values of the self-interaction strength. The three panels show the cloud mass $q_{\rm c} = \alpha_{\rm i}^2\epsilon /\alpha$, normalized fine structure constant $\alpha/\alpha_{\rm i}$ and dimensionless spin $\chi$. We see that for weak self-interactions $f=10^{19}$ GeV, the 211 and 322 states arise sequentially while for stronger self-interactions there is a partial coexistence of the two states, with one dominating over the other. Note how the superradiant production of the $\ket{211}$ and $\ket{322}$ states is associated with substantial (and almost instantaneous) spin-down and mass loss of the BH, unless the self-interaction is strong.

Caption

Representation of the cloud mass $q_{\rm c} \equiv \max[q_{\rm c}^{211},q_{\rm c}^{322}]$ obtained via solving the system of ODEs~\eqref{app:ceq} for different values of initial spin, fine structure constant and self-interaction strength for $\mu=10^{-13}$ eV. For illustration we randomly choose an array of $(\chi_{\rm i}, \alpha_{\rm i})$ randomly extracted between $[0,0.998]\times[0.005,0.2]$. For visibility the linewidth increases with $f$. Solid lines represent when the cloud is predominantly in the $\ket{211}$ state while dashed lines show when it is in the $\ket{322}$ state. The gray band shows the observationally relevant range of BH ages with the lighter shade showing a younger BH population.

Caption

The energy transfer kernels $\cal E$ as function of pericenter distance in Bohr radii units $r_{\rm p}/r_{\rm B}$ computed for two different values of $\alpha$ and for the $\ket{211}$ (\textbf{left} panels) and $\ket{322}$ (\textbf{right} panels) SC states. We consider $\omega=0$, vary $\iota$, fix $q_{\rm c}=0.01$ and show $q=\{0.1,0.3,1\}$ from top to bottom. The colored lines show the kernel \textit{predicted} from the one computed at the reference value. The percent-level agreement is shown by overlapping the actual calculation with markers: $\epsilon$ is the median error. Transparent dashed lines and empty circles markers indicate the region where $P_{\rm tot}^s>0.1$ and the perturbative formalism is not valid anymore.

Caption

The angular momentum transfer in the interaction between the SC and the perturber in units of $GM^2$ as function of pericenter distance in Bohr radii units $r_{\rm p}/r_{\rm B}$ computed for two different values of $\alpha$ and for the $\ket{211}$ (\textbf{left} panels) and $\ket{322}$ (\textbf{right} panels) SC states. We consider $\omega=0$, vary $\iota$, fix $q_{\rm c}=0.01$ and show $q=\{0.1,0.3,1\}$ from top to bottom. As in Fig.~\ref{fig:dE_transfer}, transparent dashed lines indicate the region where $P_{\rm tot}^s>0.1$ and the perturbative formalism is not valid anymore. The gray-filled area shows where the angular momentum loss by GW emission is larger than the SC-mediated one while the black lines (dashed and dotted) show $|L_{\rm k}|/GM^2$. The SC angular momentum exchange is always within those two lines.

Caption

The eccentricity of the BH-perturber system after the parabolic encounter as a function of the normalized pericenter distance. \textbf{Left} panels show the encounter when the SC of the primary BH is in the $\ket{211}$ state, \textbf{right} show the case of $\ket{322}$ state. We show examples for $q=0.1$ (\textbf{top} panels) and $q=1$ (\textbf{bottom} panels). The angles of the encounter are fixed to $(\omega,\iota,\Omega)=(0,0,0)$. The relative velocity is varied according to the color map. Solid lines show the eccentricity when the SC-enhanced case dominates the energy transfer with dashed lines showing when this violates the perturbativity requirements. Dotted lines show the classic GW-capture case. Inset panels show the behavior of the curves very close to $e_0=1$.

Caption

\textbf{Left}: the maximum relative velocity that allows capture, $w_{\rm max}$, as a function of normalized pericenter distance $r_{\rm p}/r_{\rm B}$ for a SC in $\ket{211}$ state (top left) or $\ket{322}$ state (bottom left). The gray shaded area shows the distances where the GW emission dominates the capture dynamics. \textbf{Right}: the $w_{\rm max}$ enhancement $\cal W$ over pure GW capture, for a SC in the $\ket{211}$ state (top right) or $\ket{322}$ state (bottom right) as a function of $r_{\rm p}/r_{\rm B}$. In all the plots we show the results varying the inclination of the orbit $\iota$ as visible in the color scale. Solid lines show where $P_{\rm tot}<0.1$,while dashed show $P_{\rm tot}>0.1$.

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Histogram of $\tau_{\rm 3rd}/\tau_{\rm GW}$ of two simulations with 200000 binaries showing that about than 1 binary in a thousand can get ionized by an encounter with a third body in the case of SC-enhanced capture, and less than one per million in the GW-capture case. Even at the high densities of a GN, the timescale for binary ionization is several orders of magnitude larger than the time to merger.

Caption

Detection rates obtained from the LLO detections in the GWTC-5 catalog (horizontal lines) compared to prediction of merger rates in LLO from SC-enhanced captures in GNs (scatter points) as a function of the ultralight scalar mass $\mu$. For each LLO rate the thin lines show 90\% upper bounds on this rates. We fix $f=10^{19}$ GeV, $M_{\rm max}=100M_\odot$ and vary $\beta=\{0.5,1,1.5,2\}$. We show different partial rates with different colors. We can exclude intervals in the ultralight scalar mass $\mu$ by comparing the points to the horizontal lines that represent upper bounds on the observed detection rates.

Caption

\textbf{Top} panels: the differential merger rate (in $M_1$, \textbf{left} and $q$, \textbf{right}) histograms from a binary BH population of $10^{6}$ samples in the SC-enhanced capture (blue) and GW-capture scenarios (orange) compared to the populations shown in the GWTC-5 paper~\cite{LIGOScientific:2026ctl}. The table at the \textbf{bottom} shows integrated merger rates in the different cases. The data extracted from the plot in~\cite{LIGOScientific:2026ctl} is represented as a 90\% confidence interval while the results from our Monte Carlo simulation are shown with the associated Monte Carlo error over 10 batches. The fields compatible with the LIGO rates are shown in {\color{Green}green}, tensions in {\color{Orange}orange}, while the ones overshooting the LIGO merger rates are shown in {\color{Red}red} and allow to constrain the ultralight scalar. Note that the shown histogram and results correspond to our fiducial choice of astrophysical parameters ($\beta=1$, $M_{\rm max}=100M_\odot$, $p_0=0.5$, radius of influence from~\cite{Hannah+24}, $\chi\in {\cal U}[0,0.998]$, $\log_{10}(t_{\rm BH}/{\rm yr})\in {\cal U}[8,10]$).

Caption

Exclusion plot in ultralight boson mass $\mu$ and self-interaction strength (expressed as $1/f = \sqrt{\lambda}/\mu$) for different values of the segregation parameter $p_0=0.3$ (\textbf{left}) and $p_0 = 0.7$ (\textbf{right}). The red contours represent the excluded region requiring the predicted rate of detections with mass ratio less than 3/10 to not exceed the LLO counts from the GWTC-5 catalog~\cite{LIGOScientific:2026wfs} at 95\% confidence level. We assume a BH mass function in GNs $\propto M^{-\beta}$ with different values of $\beta=\{0.5,1,1.5\}$, shown as different contours.

Caption

Exclusion plot in ultralight boson mass $\mu$ and self-interaction strength (expressed as $1/f = \sqrt{\lambda}/\mu$) for different values of the segregation parameter $p_0=0.3$ (\textbf{left}) and $p_0 = 0.7$ (\textbf{right}). The red contours represent the excluded region requiring the predicted rate of detections with mass ratio less than 3/10 to not exceed the LLO counts from the GWTC-5 catalog~\cite{LIGOScientific:2026wfs} at 95\% confidence level. We assume a BH mass function in GNs $\propto M^{-\beta}$ with different values of $\beta=\{0.5,1,1.5\}$, shown as different contours.

Caption

Differential merger rates (in distance from the center of the GN, total mass, mass ratio and SMBH mass) for different choices of the segregation parameter $p_0=\{0.5,0.1,0.3,0.7,0.9\}$ obtained in a MonteCarlo fashion with 200000 binary samples. We consider the vanilla GW-capture case (\textbf{top panels}) and the case of SC-enhanced capture with $\mu=10^{-13}$ eV and $f=10^{19}$ GeV (\textbf{bottom panels}). On the side of the plots we show the total merger rates (integrated over all the parameters).

Caption

Differential merger rates (in distance from the center of the GN, total mass, mass ratio and SMBH mass) for different choices of the segregation parameter $p_0=\{0.5,0.1,0.3,0.7,0.9\}$ obtained in a MonteCarlo fashion with 200000 binary samples. We consider the vanilla GW-capture case (\textbf{top panels}) and the case of SC-enhanced capture with $\mu=10^{-13}$ eV and $f=10^{19}$ GeV (\textbf{bottom panels}). On the side of the plots we show the total merger rates (integrated over all the parameters).

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but we vary the prescriptions for the influence radius $r_0$ which determines the $r_{\rm max}$ in the GN BH number density. We indicate in the legend the values of $(A,B)$ to substitute in Eq.~\eqref{eq:rinfl_prescriptions} to obtain the different prescriptions. Hannah+24~\cite{Hannah+24} (in blue) is our fiducial prescription while the one in green refers to a prescription obtained from a larger dataset that includes estimates from the $M_{\rm SMBH}-\sigma$ relation rather than direct measurements of $M_{\rm SMBH}$. Rasskazov+19 (orange) is the prescription adopted in Ref.~\cite{Rasskazov:2019gjw} and Thomas+16 is the prescription obtained from the results presented in Ref.~\cite{Thomas+16}. The rates depend sensibly on the $r_0$ prescription due to the different number density.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but we vary the prescriptions for the influence radius $r_0$ which determines the $r_{\rm max}$ in the GN BH number density. We indicate in the legend the values of $(A,B)$ to substitute in Eq.~\eqref{eq:rinfl_prescriptions} to obtain the different prescriptions. Hannah+24~\cite{Hannah+24} (in blue) is our fiducial prescription while the one in green refers to a prescription obtained from a larger dataset that includes estimates from the $M_{\rm SMBH}-\sigma$ relation rather than direct measurements of $M_{\rm SMBH}$. Rasskazov+19 (orange) is the prescription adopted in Ref.~\cite{Rasskazov:2019gjw} and Thomas+16 is the prescription obtained from the results presented in Ref.~\cite{Thomas+16}. The rates depend sensibly on the $r_0$ prescription due to the different number density.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but we now vary $M_{\rm max} = 80 M_\odot$ instead of the fiducial value of $M_{\rm max} = 100 M_\odot$. Comparing this to the summary plot in Fig.~\ref{fig:summary_mu_inv_f} in the main text, we see that the excluded parameter space is reduced.

Caption

Histograms of the three initial spin distributions considered in our work.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but we compare the exclusion plots obtained under different assumptions for the initial spin distribution. We consider the GWTC-5 empirical distribution from the LVK binaries and a simple Gaussian with mean 0.7 and standard deviation of 0.15 as the pessimistic (\textbf{left}) and optimistic case (\textbf{right}). As evident from Fig.~\ref{fig:summary_mu_inv_f} in the main text, the agnostic uniform distribution represents a conservative choice in the middle of the ones presented here. The distributions are shown in Fig.~\ref{fig:spin_distributions}.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but we compare the exclusion plots obtained under different assumptions for the initial spin distribution. We consider the GWTC-5 empirical distribution from the LVK binaries and a simple Gaussian with mean 0.7 and standard deviation of 0.15 as the pessimistic (\textbf{left}) and optimistic case (\textbf{right}). As evident from Fig.~\ref{fig:summary_mu_inv_f} in the main text, the agnostic uniform distribution represents a conservative choice in the middle of the ones presented here. The distributions are shown in Fig.~\ref{fig:spin_distributions}.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f}, but the distribution of black hole ages is extended to a log-uniform distribution between $10^6$ yr and $10^{10}$ yr. The impact of this modification on the exclusion contours is minor.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f},, but we adopt the fiducial model without enforcing the perturbativity condition on the total transition probability $P_{\rm tot}^s<0.1$ (\textbf{left} panel). As explained in the main text, this is \textit{not} a correct assumption and we show this results just for illustration of the capability of the method. In the other panels we vary the cutoff by a factor of 2 $P_{\rm tot}^s<0.2$ (\textbf{center}, relaxed cutoff ) and $P_{\rm tot}^s<0.05$ (\textbf{right}, aggressive cutoff) to show how the exclusion contours are affected by the cutoff choice.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f},, but we adopt the fiducial model without enforcing the perturbativity condition on the total transition probability $P_{\rm tot}^s<0.1$ (\textbf{left} panel). As explained in the main text, this is \textit{not} a correct assumption and we show this results just for illustration of the capability of the method. In the other panels we vary the cutoff by a factor of 2 $P_{\rm tot}^s<0.2$ (\textbf{center}, relaxed cutoff ) and $P_{\rm tot}^s<0.05$ (\textbf{right}, aggressive cutoff) to show how the exclusion contours are affected by the cutoff choice.

Caption

Same as Fig.~\ref{fig:summary_mu_inv_f},, but we adopt the fiducial model without enforcing the perturbativity condition on the total transition probability $P_{\rm tot}^s<0.1$ (\textbf{left} panel). As explained in the main text, this is \textit{not} a correct assumption and we show this results just for illustration of the capability of the method. In the other panels we vary the cutoff by a factor of 2 $P_{\rm tot}^s<0.2$ (\textbf{center}, relaxed cutoff ) and $P_{\rm tot}^s<0.05$ (\textbf{right}, aggressive cutoff) to show how the exclusion contours are affected by the cutoff choice.