Galactic and extragalactic probe of dark matter with LISA’s binary black holes near their galactic center

For a Schwarzschild black hole with mass $m$ that is moving at a supersonic velocity $v_{\infty}$ w.r.t. the asymptotic/unperturbed background medium with density $\rho_{\infty}$, matter within the impact parameter $r_{a}={2Gm}/{v_{\infty}^{2}}$ will be captured by the black hole. The resulting accretion cross-section can be estimated as $\sigma_{\rm HL}=\pi r_{a}^{2}$ which results in the Hoyle-Lyttleton mass accretion rate [42]

Meanwhile, following Bondi [43], for a black hole that has $v_{\infty}=0$, the mass accretion rate reads

where $c_{s}=c\sqrt{\gamma\Theta_{\infty}}$ is the local sound speed and

is the dark matter dimensionless temperature. Above $T,m_{\chi},k_{B},c$ are the dark matter (effective) temperature, particle mass, Boltzmann constant, and the speed of light in vacuum, respectively.

In deriving Eq. (2), it is assumed that the background medium can be treated as a polytrope with the equation of state $P\propto\rho^{\gamma}$, where $\gamma\leq 5/3$ for a non-relativistic fluid, and that dark matter can be treated as a non-relativistic medium. A relativistic extension of the Bondi formalism was later conducted by Michel in Ref. [44] (for a review, see Appendix of Ref. [45]). However, for a medium with $\Theta_{\infty}\lesssim 10^{-5}$, the mass accretion rate can be well modelled by Eq. (2) (cf., [46]). Since the current upper bound on the dark matter’s dimensionless temperature is expected to be $\Theta_{\infty}\approx\mathcal{O}(10^{-8})$ [47], thus one can consider Eq. (2) to hold for dark matter accretion irrespective of it being cold or warm.

Since $\dot{m}_{\rm HL}\propto v_{\infty}^{-3}$, to generate a larger accretion rate, for the current work, we are interested in the scenario where the black hole either moves at a subsonic or marginally supersonic velocity. To this end, using Eq. (1) and (2), a simple interpolation gives us the Bondi-Hoyle [48] mass accretion rate,

which holds for both the subsonic and supersonic scenarios. The above expression for the accretion rate would be weakly sensitive to the spin of the black holes since the accretion cross-section is set at a large distance from the black hole. Thus, the performed calculations would not be significantly impacted by the assumption of Schwarzschild geometry.

To determine $\rho_{\infty}$, we use an NFW profile for the dark matter halo; see Eq. (36). In the inner region, the halo can experience adiabatic compression due to the presence of the SMBH, resulting in a spike in dark matter density [49, 50]. To give conservative estimates of $\dot{m}$, we disregard the presence of such a density spike in this work.

If dark matter is instead modeled as a scale field, far away from the black hole, the incoming wave can be treated as a plane wave. The treatment of such long wavelength ($\lambda_{\rm DM}\gg m$) scalar field absorption by a Schwarzschild black hole is provided in [51]. The resulting mass accretion rate reads

where (with $\hbar=G=c=1$)

The capture rate of such long-wavelength scalar fields is expected to be relatively weak compared to point-particle dark matter and likely to be important when the black hole is present in dense dark matter environments, such as near the center of our galaxy. A generic prediction for such ultralight fields is that they will produce a solitonic core near the galactic center (e.g., [39, 52]). In the absence of self-interaction, the soliton is purely supported via quantum pressure, and its density can be approximated as [39]

where $\tilde{\lambda}=2^{1/8}-1$. Additionally, the core’s central density can be written as [52]

where $M_{\rm sol}$ is the mass of the soliton and $\rho_{0}=0.00440$ [52]. The core radius $r_{c}$ is defined as $\rho(r_{c})=\rho_{c}/2$ and takes the form [39]

The virial velocity of the solitonic core takes the form

with $w_{0}=0.10851$ [52]. Our target black holes will be going in a circular orbit around their galactic center with velocity $v_{c}$, with $v_{c}\ll v_{\mathrm{vir}}$ ($v_{c}$ calculated in Appendix A). Thus, we make the assumption that $v_{\infty}\equiv v_{\mathrm{vir}}$, which would otherwise be determined by $v_{c}$.

The masses of black holes considered here, along with the expression for $v_{\infty}$ in Eq. (10), suggests $\xi\ll 1$, implying that Eq. (5) can be reduced to

Substituting the density of the soliton in Eq. (11) thus results in the accretion rate

Based on numerical simulations, $M_{\rm sol}$ can be calculated using the soliton - halo mass relation as [53]

where $M_{\rm halo}$ is the mass of the dark matter halo. This implies that $M_{\rm sol}\propto m_{\chi}^{-1}$ and, as we later discuss, for Milky Way like galaxy with $M_{\rm halo}\approx 10^{12}M_{\odot}$ results in a negligible impact on $\mathcal{M}$ of the binary. To this end, for the sake of demonstration of the formalism discussed here, we also consider an alternative scenario where we fix the Milky Way’s soliton mass to $M_{\rm sol}=10^{9}M_{\odot}$ (the latter value is based on Ref. [54])).

Since LISA’s binaries will have an orbital period of $10-10^{4}$ s [35], this implies $r_{a}\gg a$, $a$ being the binary’s semi-major axis. Thus far away from the binary, the infalling point-particle dark matter will perceive the binary as a single object with mass $M=m_{1}+m_{2}$ [55, 56, 57]. This gives the total mass accretion rate on the BBH as

where $0\leq\delta\leq 1$.

As the matter falls into the total potential of mass $M$, far away from the BBH, $\delta=1$. Then, to maintain a steady-state mass accretion rate, this captured mass would eventually have to be accreted. The mass contained in the last term in Eq. (14) would be distributed among the component masses. As the current treatment only serves as a test of LISA’s capability to detect dark matter, in the following, for simplicity, we present our results assuming $\delta=0$, which provides a floor for the expected evolution of the BBH mass.

A similar effect would manifest during the descent of wave-like dark matter on the binary’s collective gravitational potential, cf. Eq. (11). Nevertheless, as previously, we treat the infall and absorption of dark matter on each black hole separately for the latter case as well.

For BBHs orbiting in a Keplerian orbit, the orbital frequency $f_{\rm orb}$ satisfies

Thus, if the binary accretes dark matter, the evolution of $f_{\rm orb}$ can be calculated as

where in the scenario when the BBH lives in a vacuum, we can set $\dot{M}=0$. From Eq. (21), one can then calculate the gravitational wave frequency evolution, noting that the gravitational wave frequency in the $n$th harmonic should be given by $f=nf_{\rm orb}$.

To restate, we are interested in determining the changes in the evolution of the BBH properties, owing to the capture of surrounding dark matter. The leading order orbital evolution of a gravitational wave emitting binary is determined by its chirp mass $\mathcal{M}=\mu^{3/5}M^{2/5}$ and in the case where $\dot{M}=0$, $\mathcal{M}$ remains invariant over time. Observationally, for certain binaries, LISA may be able to measure their gravitational wave frequency $f$ as well as the frequency drift $\dot{f}$ yielding

which LISA can measure with the resolution [60]

Moreover, for circular binaries, $\dot{f}$ can be measured with the resolution

where $\bar{\rho}_{2}$ is the SNR of the gravitational wave of the binary in the second orbital harmonic and is defined later in Eq. (29). This yields

implying that a longer observation time will lead to a smaller error in the measured value of $\mathcal{M}$. Assuming that we measure $\mathcal{M}$ for two different observation times $t_{1},t_{2}$ (with $t_{2}>t_{1}$), we can then calculate the change in the binary’s chirp mass over the observation time as $\Delta\mathcal{M}=\mathcal{M}(t_{2})-\mathcal{M}(t_{1})$, where the change in $\mathcal{M}$ is driven by the capture of dark matter. For a successful detection of this change, we require that the net measurement uncertainty remain smaller than the measured value, i.e.,

where $\sigma_{\mathcal{M}}(t_{1})>\sigma_{\mathcal{M}}(t_{2})$. For calculational simplicity, in the current work, we adopt a more stringent criterion, i.e.,

for a successful detection.

For a given value of $f,\mathcal{M},e$ and source luminosity distance $D_{L}$, the dimensionless gravitational wave strain amplitude of the binary in the $n=2$ orbital harmonic (i.e., the most dominant harmonic at low eccentricities) is [61, 36]

In calculating $h_{2}$ for a given binary, we will treat $D_{L}$ to remain constant over the course of observation. Moreover, since we are not interested in a particular target, here we consider the gravitational wave SNR to be averaged over inclination, sky location, and gravitational wave polarization over an observing time $T_{\rm obs}$. This yields [36]

where $S_{n}(f)$ is LISA’s noise power spectral density and is defined in Appendix B. For higher harmonics, $h_{n}\propto\sqrt{g(n,e)}/n$ and become increasingly important for $e\gtrsim 0.2$ (see figure 3 in Ref. [62], the expression for $g(n,e)$ can be also be found therein). For simplicity below, we present our calculation only for the case $e=0$ for which the binary radiates at $n=2$. As the intensity of the emitted radiation is directly proportional to $e$, so our results can be treated as the floor of LISA’s sensitivity.

Given the inverse dependence of $h_{2}$ on $D_{L}$, binaries within the galaxy will be the optimal target. As mentioned earlier, we assume such binaries to be in a circular orbit around the galactic center with velocity $v_{c}$, which will be the source of $v_{\infty}$ in Eq. (4). In addition to a large $M$, Eq. (4) and (11) suggests that a larger change in $\mathcal{M}$ requires $\rho_{\infty}$ to be large and $v_{c}$ to be small. Thus, the ideal location to probe dark matter using LISA would be near the galactic center.

The calculation of $v_{c}$ for black holes residing in the Milky Way’s disk is presented in Appendix A. Since the role of the galactic bulge remains uncertain in the inner region, we assume that it is the NFW halo that determines the value of $v_{c}$ near the galactic center. Nevertheless, we compare $v_{c}$ resulting from the presence of an NFW halo and the bulge independently in Fig. 5. As discussed in Section II, in contrast to particle-like dark matter, absorption of ultralight scalar fields will only be important when the BBH resides within the galactic soliton. For such a case, $v_{\rm vir}\gg v_{c}$, thus the effect on a non-zero $v_{c}$ can be ignored.

Fig. 1 shows the variation in $\mathcal{M}$ of a BBH with $m_{1},m_{2}=25M_{\odot}$ due to the presence of particle-like dark matter. The shaded area in the figure shows the region of detectability by LISA and is dependent on the emitted gravitational wave frequency. The change in $\mathcal{M}$ depends on the magnitude of $\Theta_{\infty}$ and $r_{\rm center}$ (the distance of the BBH from the galactic center) with smaller values resulting in more favorable outcomes. We note that the kink in the curve with initial frequency $f_{\rm in}=2$ mHz is an artifact and results from the interpolation carried over the LISA sensitivity data presented in Table 1 of Ref. [36].

Similar to Fig. 1, Fig. 2 shows the variation in $\mathcal{M}$ for the case where the BBH is immersed in a wave-like dark matter surrounding of $M_{\rm sol}=10^{9}M_{\odot}$ with the corresponding dark matter masses shown in the figure legend. As the mass of dark matter increases, the density of the solitonic core increases. However, this also decreases the value of $r_{c}$ in Eq. 9. Thus, we place the BBH closer to the galactic center for a larger value of $m_{\chi}$. For both Fig. 1 and 2, we find that within 4-10 yr of operation, LISA will be able to probe the surroundings of the BBHs if such systems are detected during the observational run. However, $M_{\rm sol}$ is expected to be a dynamical quantity that changes as one varies $m_{\chi}$ [53]. For the sake of demonstration, let us pick $m_{\chi}=10^{-19}$ eV as this mass results in a detectable effect in Fig. 2. Then, based on Eq. (13), the corresponding $M_{\rm sol}=1.25\times 10^{6}M_{\odot}$. This reduces the mass accretion rate in Eq. (12) by a factor of $\approx 10^{12}$ compared to the case where $M_{\rm sol}=10^{9}M_{\odot}$ and it gets worse as $m_{\chi}$ increases. As the value of $M_{\rm sol}$ has a direct impact on $\rho_{c}$ in Eq. (8), adopting a larger value of $M_{\rm sol}$ results in a larger $\rho$. This effect is akin to the impact of adiabatic compression of the soliton on dark matter density, where the compression is driven by the SMBH at the galactic center [50]. Thus, we conclude that wave-like dark matter is unlikely to cause any noticeable variation in $\mathcal{M}$ unless the soliton gets adiabatically compressed to result in larger values of $\rho$ near the galactic center.

Until now, we assumed the BBH mass ratio $q\equiv m_{2}/m_{1}=1$. To investigate the impact of a lower value of $q$, we reproduce Fig. 1 and 2 in Fig. 6 in Appendix C, assuming $q=0.1$. As anticipated, a lower mass ratio reduces the prospects of detectability, meaning for smaller $q$ values, a higher value of $M$ would be desirable to probe the surrounding dark matter.

LISA will be sensitive to IMBHs out to redshift two [38]. Thus, in this section, we apply the above-discussed methodology to larger values of $D_{L}$. For simplicity, we take the Milky Way’s dark matter profile as a prototype for all galaxies at extragalactic distances.

The resulting detectability prospects are shown in Fig. 3 and 7 (in Appendix C) for $q=1,0.1$ respectively, where we again set $M_{\rm sol}=10^{9}M_{\odot}$. For demonstration purposes, here we have chosen a subset of $f_{\rm in}$ values and assumed a fixed observation time of $T_{\rm obs}=4,10$ yr. At the end of $T_{\rm obs}$, $\Delta\mathcal{M}$ is calculated as $\Delta\mathcal{M}(T_{\rm obs})=\mathcal{M}(T_{\rm obs})-\mathcal{M}(T_{\rm obs}-0.1)$. Many BBHs merge within this timespan and are thus removed from the figure, as is evident from the clipped region at the top of the shaded area.

In principle, one could fill other regions of Fig. 3 and 7 by calculating $\Delta\mathcal{M}$ as one continuously varies $T_{\rm obs}$. These figures show that observing $\mathcal{M}$ variations allow to probe²²2Maximum value of $\Theta_{\infty}$ is estimated to be $\mathcal{O}(10^{-8})$ [47]. $\Theta_{\infty}\lesssim 10^{-8}$ for the case of particle dark matter and $m_{\chi}\in[10^{-20},5\times 10^{-19}]$ eV for wave-like dark matter. Such variations can be probed to a distance of $\approx 1$ Gpc. The latter depends on the value of $r_{\rm center}$, and one can imagine that the detection prospects would be more favorable for smaller values of $r_{\rm center}$.

Solving Eq. (4) yields the mass evolution of the black hole as

where $m_{i}$ is the mass of the black hole at some initial time $t_{i}$ and $\alpha=\frac{4\pi\lambda G^{2}\rho_{\infty}}{(c_{s}^{2}+v_{\infty}^{2})^{3/2}}$. The accretion of the interstellar medium (ISM) will be most efficient when the black hole comoves w.r.t. to its surrounding medium. Thus, for the case of baryonic matter accretion, we set $v_{\infty}=0$ in Eq. (30). Moreover, the value of $\rho_{\infty}$ for the baryonic environment will depend on the nature of ISM in the vicinity of the black hole. A favorable scenario occurs when the ISM is cold and hence contains neutral particles, typically hydrogen. Such a medium is referred to as a cold neutral medium (CNM) and contains an average particle number density $n_{H}\approx(20-50)$ cm^-3 and a temperature $T\approx 100$ K [63].

It is likely that most of the surrounding CNM would have already been accreted by the time the BBH enters the LISA frequency band. Nevertheless, it is worthwhile to investigate how the black hole mass evolution compares to the scenario when the mass growth is supported by dark matter and baryonic matter accretion, respectively. To this end, Fig. 4 shows the mass evolution of black holes with $m_{i}=100M_{\odot},\,1000M_{\odot}$ immersed in a baryonic and particle-like dark matter medium with variable properties.

Here, the ISM is considered to be cold with $T=100$ K. However, there is a possibility that baryonic matter may form an accretion disk around the binary (by borrowing angular momentum from the BBH’s orbit) and thus heat up the surrounding gas. This will cause the temperature to rise, and for $n_{H}\approx 1$, the baryonic matter will have a subdominant impact on the black hole once $T\gtrsim 200$ K.

To be quantitative, let us assume that only a fraction $\nu$ of the accreted mass is assimilated into the accretion disk (with the rest getting accreted directly). Then, one finds that the mass assimilation rate in the disk w.r.t. the Eddington rate should evolve as [64]

For illustrative purposes, assuming $\nu=0.1$, $T=100$ K, $m\in[10M_{\odot},1000M_{\odot}]$ and the accretion efficiency $\epsilon=1/16$, yields $f\in(0.00024n_{H},0.024n_{H})$. Noting that for $f\gtrsim 0.001$, one excepts the accretion disk to radiate as a Shakura & Sunyaev accretion flow [65, 66], thus, for black holes with $m\gtrsim 100M_{\odot}$, the effect of feedback could be very strong (even for $n_{H}=1$ cm^-3), likely shutting off accretion of baryonic ISM due to a rise in its temperature.

Regardless of the presence of a baryonic environment near the black holes, LISA not detecting any $\mathcal{M}$ evolution during the course of observation would imply that the underlying dark-matter model used for performing the calculation here is not the dominant form of dark matter around the BBHs. While it can still be a subdominant component, it has to be below the threshold at which $\mathcal{M}$ evolution can be observed. Depending on the value of $M,q$, and $D_{L}$, the method proposed here can be used to perform such a check for a large parameter space of dark matter.

However, LISA alone will be unable to localize most BBHs to a subgalactic scale if observed at larger distances. Following equation 15 in Ref. [60] suggests that of the BBH mass range considered here, only the massive IMBHs within $D_{L}\lesssim 1$Mpc will have sufficient angular resolution to be localized to an $\approx 10$ pc scale. Thus, a null result will only be useful for such values of $D_{L}$. This is because, as we showed, BBHs at larger distances from their galactic center would not interact strongly with their surrounding dark matter. Thus, for them, one may not expect to see any appreciable $\mathcal{M}$ variation anyway.

There remains a possibility that if the BBH is located close to its galactic center (i.e., within $\lesssim$ 100 Schwarzschild radii of the SMBH), the SMBH-BBH system might radiate gravitational waves such that the inspiral of the BBH into the SMBH also becomes resolvable by LISA. This will result in two distinct gravitational wave signals, both coming from the same location in space but from two different sources. This can be a strong clue that the BBH signal may arise from the center of the host galaxy hosting the SMBH. In such a scenario, the null result mentioned above becomes a useful tool for much larger values of $D_{L}$.

In this work, we have attempted to estimate the floor of the possible $\mathcal{M}$ variation caused by dark matter accretion onto LISA’s BBHs and whether such a variation is detectable. In particular, we set $\delta=0$ - Eq. (14), assumed circular orbits, i.e., $e=0$, and considered no adiabatic compression of the dark matter halo [49, 50], the latter leading to larger dark matter density near the galactic center (although see Section V.1).

Our analysis demonstrates that the BBH surrounding environment can be probed by measuring variation in the binary’s chirp mass over the course of LISA’s observation. As the impact of baryonic matter accretion may be subdominant for IMBHs (Section VI.1), any significant variation in their $\mathcal{M}$ value can be attributed to the presence of dark matter. These variations depend on the intrinsic properties of dark matter, allowing us to probe them up to a luminosity distance of approximately 1 Gpc, depending on the BBH’s location relative to the galactic center and the particle physics property of dark matter. For the case where the BBH is very close to the galactic SMBH, the presence of an AGN disk can contaminate the resulting gravitational wave signal. It remains to be seen if meaningful information about the surrounding dark matter can still be recovered under such a scenario.