Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

Alexis Boudon; Philippe Brax; Patrick Valageas; Leong Khim Wong

doi:10.48550/arxiv.2305.18540

Boudon, Alexis;Brax, Philippe;Valageas, Patrick;Wong, Leong Khim

2023-05-29 00:00:00

1 1, 2 1 1 Alexis Boudon, Philippe Brax, Patrick Valageas, and Leong Khim Wong Université Paris-Saclay, CNRS, CEA, Institut de physique théorique, 91191, Gif-sur-Yvette, France CERN, Theoretical Physics Department, Geneva, Switzerland We investigate the imprints of accretion and dynamical friction on the gravitational-wave signals emitted by binary black holes embedded in a scalar dark matter cloud. As a key feature in this work, we focus on scalar fields with a repulsive self-interaction that balances against the self-gravity of the cloud. To a first approximation, the phase of the gravitational-wave signal receives extra correction terms at−4PN and−5.5PN orders, relative to the prediction of vacuum general relativity, due to accretion and dynamical friction, respectively. Future observations by LISA and B-DECIGO have the potential to detect these effects for a large range of scalar masses 𝑚 and DM self-interaction couplings 𝜆 ; observations by ET and Advanced LIGO could also detect these effects, albeit in a more limited region of parameter space. Crucially, we find that even if a dark matter cloud has a bulk density ρ that is too dilute to be detected via the effects of dynamical friction, the imprints of accretion could still be 4 3 3 observable because it is controlled by the independent scale ρ = 4𝑚 𝑐 /(3𝜆 ℏ ). In the models we consider, DM the infalling dark matter increases in density up to this characteristic scale ρ near the Schwarzschild radius, which sets the accretion rate and its associated impact on the gravitational waveform. I. INTRODUCTION the scalar self-interacts. For negligible self-interactions, solitons are supported against gravitational collapse by the wavelike nature of the scalar field, which gives rise to a so- Perturbations to the orbits of compact objects, like black called “quantum pressure”—this is commonly referred to as holes (BHs), can serve as a dynamical probe of their local the fuzzy dark matter (FDM) scenario [48]. Allowing for environment. One important effect is dynamical friction, a repulsive, quartic interaction term introduces additional first calculated in a seminal paper by Chandrasekhar [1] for pressure effects [8,49–52], however, which can even dominate collisionless particles, and later extended to gaseous media over the quantum pressure in certain cases. This occurs in, e.g., Refs. [2–5]. In all of these cases, the compact object when the soliton size is greater than the scalar’s de Broglie decelerates as it exchanges momentum with distant particles wavelength, and this will be the regime of interest in this paper. - or “streamlines” - that are deflected by its gravitational Solitons with radii on the order of a kiloparsec may alleviate field. Equivalently, one can think of dynamical friction as some of the small-scale problems in galaxies encountered by the gravitational pull on the compact object exerted by the the standard CDM scenario, such as the core/cusp problem, resulting fluid overdensity that forms in its wake. A second the too-big-to fail problem, or even the missing satellites effect is the accretion of matter onto the compact object. problem [53–56]. We note, however, that other scenarios Naturally, the amount of influence these effects can have on suggest that solitons could also form at higher redshifts and be the compact object’s trajectory depends on the specific nature of a much smaller size (see, e.g., Ref. [57]). In this paper, we of the environment. We are interested here in the case of dark make no a priori assumptions about the size of the soliton, and matter clouds, within which most binary systems are expected will instead explore what information can be extracted from to reside. Motivated by the lack of experimental evidence GW signals for all possible values of soliton radii. for weakly interacting massive particles (see, e.g., the reviews We consider the effects of both accretion and dynamical in Refs. [6,7]), we focus on scalar-field dark matter models −20 friction on the waveform. A BH moving inside a (much larger) with a particle mass between 10 eV and 1 eV. Within soliton disturbs the distribution of dark matter both locally and this range, very large occupation numbers are needed to form further out into the bulk. Near the BH, the density of infalling a galactic halo; hence, the scalar field behaves essentially dark matter grows as ρ ∝ 1/𝑟 until it reaches a nonlinear and classically and is described by a Schrödinger wave function in relativistic regime close to the horizon [58–60]. This inner- the nonrelativistic regime. Static equilibrium solutions, also radius boundary condition sets the accretion rate onto the called “solitons,” form at the centers of these halos [8–35]. BH. At larger distances, dynamical friction arises due to the In this article, we investigate the impact on the gravitational- deflection of streamlines over the bulk of the scalar cloud. As wave (GW) signal emitted by a binary BH that is embedded in for gaseous media [2–5], neglecting the backreaction of the one of these solitons. scalar field causes the dynamical friction force to vanish in the In the wider cosmological context, the energy density of dark subsonic regime [59,60]. Both effects decrease the relative matter in these scenarios is determined by the misalignment velocity between the BH and the scalar cloud. For BHs in a mechanism [36–39], wherein the field is initially frozen but binary system, the consequence is a higher rate of orbital decay then oscillates rapidly once its mass exceeds the Hubble rate. than if the binary were to evolve solely due to the emission of For scalar-field potentials that are dominated by their mass −3 GWs. In standard post-Newtonian (PN) terminology, we find term, the energy density decays as 𝑎(𝑡) , as it does for cold that accretion first contributes to the GW phase at the −4PN dark matter (CDM), with 𝑎(𝑡) the cosmic scale factor. One thus level, while dynamical friction is a−5.5PN order effect. recovers the main predictions of the standard CDM paradigm The remainder of this paper is organized as follows. In on cosmological scales [40–47]. Meanwhile, the details of Sec. II, we begin by reviewing the self-interacting model of what transpires on smaller scales depends on how strongly arXiv:2305.18540v1 [astro-ph.CO] 29 May 2023 2 scalar-field dark matter that we consider. In Sec. III, we then in the Thomas-Fermi limit of negligible quantum pressure. Ob- solve for the motion of a binary BH in the presence of a scalar serve that such solitons are described by just three parameters: cloud. The perturbations to the phase of the emitted GWs the fundamental constants 𝑚 and 𝜆 , and the average bulk DM 4 arising from accretion and dynamical friction are derived in density ρ . The value of this last quantity—or, equivalently, Sec. IV, and finally, in Sec. V we forecast the prospects of the value of the soliton mass 𝑀 = (4/𝜋)ρ 𝑅 —depends on sol 0 sol detecting such a dark matter environment in current and future the formation history of the dark matter halo. GW experiments. We conclude in Sec. VI. If the characteristic scale 𝑟 in Eq. (2.3) is on the order of a kiloparsec or more, then these solitons form at the centers of galaxies, as in the FDM case [64], while the outer regions of II. EQUATIONS OF MOTION the dark matter halo follow an NFW density profile [65]. A A. Scalar field dark matter numerical study of such soliton-halo systems for the potential in Eq. (2.2) is presented in Ref. [35]. On scales greater than In this paper, we study the signatures imprinted on the 𝑅 and the de Broglie wavelength 𝜆 ≡ 2𝜋ℏ/(𝑚 𝑣), both gravitational waveform of a binary system of BHs by dark sol dB DM the self-interaction and quantum pressure are negligible, and matter environments associated with a self-interacting scalar so scalar-field dark matter behaves as collisionless cold dark field. The dynamics of the scalar are governed by the action matter would. Moreover, even though 𝑟 is fixed, increasingly 𝑑 𝑥√ 1 large and massive halos can form hierarchically in this model, µν 𝑆 = −g − g ∂ 𝜙∂ 𝜙− 𝑉(𝜙) , (2.1) 𝜙 µ ν as in the standard CDM paradigm [66]. ℏ𝑐 2 At the other end of the spectrum, if 𝑟 is much smaller than where we take the scalar-field potential to be the typical size of galaxies, then solitons may have formed at early times before the formation of galaxies. In a manner 2 2 𝑚 𝑐 DM 2 4 similar to the formation of primordial BHs, this could lead 𝑉(𝜙) = 𝜙 + 𝜙 , (2.2) 2 2 2 2ℏ 4ℏ 𝑐 to macroscopic dark matter objects with radii ranging from that of an asteroid to giant molecular clouds [57]. Indeed, if with coupling constant 𝜆 > 0. This gives rise to a repulsive the hierarchy of scales is sufficiently large, then many small self-interaction between dark matter particles in the nonrel- solitons may be present within galactic halos. In this scenario, ativistic limit, wherein the global behavior of dark matter is stellar-mass binary BH systems could happen to be embedded akin to that of a compressible fluid. The effective outward within such solitons. We shall investigate the impact of both pressure of this repulsive interaction can counterbalance the types of solitons—galactic sized or smaller—on the motion of attractive force of gravity, and therefore leads to the formation binary BHs. of stable, equilibrium dark matter configurations on small Several assumptions have been made to render the calcula- scales, called solitons. tions in this paper feasible. First, note that the sound speed of A detailed cosmological analysis of this dark matter model is the dark matter fluid is given by [58,59] presented in Ref. [61]. We here briefly review the main points. On cosmological scales, the oscillations of the scalar field due 2 2 𝑐 = 𝑐 , (2.5) to the quadratic mass term in 𝑉(𝜙) are dominant since at least the time of matter-radiation equality. This ensures that the scalar field behaves as dark matter with a background density ρ ¯ as would be expected for a polytropic gas with index 𝛾 = 2. −3 that decays with the scale factor 𝑎(𝑡) as ρ ¯ ∝ 𝑎(𝑡) . However, We restrict ourselves to the nonrelativistic regime wherein the pressure associated with the self-interaction term prevents 𝑐 ≪ 𝑐, and thus ρ ≪ ρ . We further limit our attention to 𝑠 0 𝑎 the growth of density perturbations below the Jeans scale the large-scalar-mass limit, 4 3 4𝑚 𝑐 −1 DM ℏ 𝑚 BH 𝑟 = √ , ρ = . (2.3) −11 𝑎 𝑎 3 𝑚 > = 7× 10 eV, (2.6) DM 4𝜋𝐺ρ 3𝜆 ℏ 𝑟 𝑐 1 𝑀 𝑠 ⊙ The characteristic scale 𝑟 actually sets both the cosmological where𝑟 ≡ 2𝐺𝑚 /𝑐 is the Schwarzschild radius of the larger 𝑠 BH Jeans length, which leads to a small-scale cutoff for cosmolog- of the two BHs embedded in the soliton. Taking this limit ical structure formation, and the radius of the soliton [8,62]. amounts to assuming that the scalar’s de Broglie and Compton In the nonrelativistic regime, the nonlinear Klein-Gordon wavelengths are smaller than the BH’s horizon, and much equation derived from the action in Eq. (2.1) reduces to the non- smaller than the size of the soliton. The analytic formulas linear Schrödinger-Poisson system. In simple configurations for the accretion rate and dynamical friction force that we use (wherein the density does not vanish), a Madelung transfor- below were derived in Refs. [58–60] and are valid only when mation [63] can be used to map this onto a hydrodynamical this holds. Conveniently, a by-product of this assumption is system, in which case the solitons correspond to hydrostatic that the only dark matter parameters affecting the binary’s equilibria. The quartic self-interaction in Eq. (2.2) gives rise to 2 motion are the two characteristic densities, ρ and ρ . 𝑎 0 an effective pressure 𝑃 ∝ ρ , not unlike a polytropic gas with As a BH moves inside such dark matter solitons, it slows index 𝛾 = 2. The soliton density profile then takes the form down because of two effects, the accretion of dark matter and sin(𝜋𝑟/𝑅 ) the dynamical friction with the dark matter environment. We sol ρ (𝑟) = ρ , 𝑅 = 𝜋𝑟 , (2.4) sol 0 sol 𝑎 describe these effects in the next two sections. 𝜋𝑟/𝑅 sol 3 B. Accretion drag force Because Eq. (2.9) applies only when 𝑣 > 𝑐 and 𝑟 > 𝑟 , BH 𝑠 𝑎 UV we can define a critical velocity For the particular model in Eqs. (2.1) and (2.2), it was shown   in Ref. [58] that the accretion rate of scalar dark matter onto a ! √︂ 2/3 2 𝐺𝑚 BH is given by BH   𝑣 = 𝑐 max 1, 6 (2.11) 𝑐 𝑠 . 𝑟 𝑐 2 𝑠 𝑚 = 3𝜋𝐹 ρ 𝑟 𝑐 BH ★ 𝑎 2 2 3 = 12𝜋𝐹 ρ 𝐺 𝑚 /𝑐 ★ 𝑎 below which the dynamical friction force must vanish. This BH 2 2 2 threshold is only an approximation, however, as a perturbative = 12𝜋𝐹 ρ 𝐺 𝑚 /(𝑐 𝑐), (2.7) ★ 0 BH 𝑠 treatment to higher orders, which takes the scalar field’s backreaction onto the BH into account, should smooth out where an overdot denotes differentiation with respect to time and 𝐹 ≃ 0.66 is obtained from a numerical computation the transition at 𝑐 and give a small but nonzero force in the subsonic regime [69]. Nevertheless, we expect our use of a of the critical flux, which is associated with the unique transonic solution that matches the supersonic infall at the sharp transition at 𝑣 to provide a conservative estimate for the Schwarzschild radius to the static equilibrium soliton at large impact of the dynamical friction on the motion of a BH. distances. This critical behavior is similar to that found for hydrodynamical flows in the classic studies of Refs. [67,68], III. BINARY MOTION and is closely related to the case of a polytropic gas with index 𝛾 = 2 [58,59]. However, close to the BH, the dynamics deviates We focus on a binary system of two BHs and study their dynamics in their inspiralling phase in the Newtonian regime. from that of a polytropic gas as one enters the relativistic regime. Near the Schwarzschild radius, the scalar field must Then, the Keplerian orbital motion is perturbed by the dark matter accretion and dynamical friction and by the emission be described by the nonlinear Klein-Gordon equation instead of GWs. Both effects lead to a shrinking of the BH separation, of hydrodynamics [58]. This implies that the critical flux and until their merging. In the large-distance inspiralling phase, the accretion rate in Eq. (2.7) differ from the usual Bondi result 2 2 3 we obtain the perturbations of the Keplerian motion at first 𝑚 ∼ ρ 𝐺 𝑚 /𝑐 . Indeed, this is manifest in the way the Bondi 0 BH order. This allows us to consider separately the impact of the last line of Eq. (2.7) depends on the speed of light 𝑐, which is scalar cloud and of the GWs. absent from the usual Bondi result. Now consider a BH moving with velocity v through this BH scalar cloud. In the nonrelativistic limit 𝑣 ≡ |v | ≪ 𝑐 BH BH A. Keplerian motion and in the reference frame of the cloud, the accretion of zero- To compute the perturbation of the orbits at first order, we use momentum dark matter does not change the BH momentum the standard method of osculating orbital elements [70], where but slows down its velocity as we derive the drift of the orbital elements that determine the . . shape of the orbits. To define our notations, we first recall the 𝑚 v | = −𝑚 v . (2.8) BH BH acc BH BH properties of the Keplerian orbits. At zeroth order, the binary system of the two BHs of masses {𝑚 , 𝑚 }, positions {x , x } 1 2 1 2 C. Dynamical friction and velocities {v , v }, is reduced to a one-body problem by 1 2 Dynamical friction also acts to reduce the BH’s velocity. introducing the relative distance r, As in the hydrodynamical case [2,4,5], the dynamical friction force (in the steady-state limit) vanishes for subsonic speeds r = x − x , v = v − v , (3.1) 1 2 1 2 𝑣 < 𝑐 [59] but is nonzero at supersonic speeds. The BH 𝑠 the total and reduced masses additional force on the BH in the latter regime reads [60] 2 2 𝑚 = 𝑚 + 𝑚 , µ = 𝑚 𝑚 /𝑚. (3.2) 8𝜋𝐺 𝑚 ρ . 0 𝑟 1 2 1 2 BH 𝑚 v | = − ln v , (2.9) BH BH df BH 3𝑣 UV BH This gives the equation of motion where the small-radius cutoff of the logarithmic Coulomb 𝐺𝑚 factor is given by r = − r (3.3) √︂ 3/2 2 𝐺𝑚 𝑐 BH 𝑠 for the relative separation, whereas the center of mass remains 𝑟 = 6 (2.10) UV 𝑒 𝑣 at rest if its initial velocity vanishes. Then, we also have BH 𝑚 𝑚 2 1 and 𝑒 is Euler’s number (not to be confused with the orbital x = r, x = − r, (3.4) 1 2 eccentricity 𝔢 in Sec. III). Equation (2.9) takes the same form 𝑚 𝑚 as the collisionless result by Chandrasekhar [1], except that the choosing for the origin of the coordinates the barycenter of cutoff scale 𝑟 is here determined by the physics of the scalar UV the binary system. The solution for bound orbits is the ellipse field and its effective pressure, instead of the minimum impact given by parameter 𝑏 ∼ 𝐺𝑚 /𝑣 . Meanwhile, the large-radius min BH BH cutoff 𝑟 is given by the size of the dark matter soliton, which 𝑟 = , 𝑝 = (1− 𝔢 )𝑎, (3.5) recall depends explicitly on 𝑚 and 𝜆 via Eq. (2.3). DM 4 1+ 𝔢 cos(𝜙− 𝜔) 4 . . where 𝑝 is the orbit semi-lactus rectum, 𝑎 the semi-major axis, to express x in terms of r in the last three terms, as we work 𝔢 the eccentricity and 𝜔 the longitude of the pericenter. The at first order in the perturbations 𝑚 and 𝑓 . Thus, we obtain 𝑖 𝑖 orbit takes place in the plane (e , e ) orthogonal to the axis e . an equation of motion of the form 𝑥 𝑦 𝑧 In spherical coordinates, the polar angle 𝜃 = 𝜋/2 is constant 𝐺𝑚(𝑡) while the azimuthal angle 𝜙 runs. The total angular momentum r¥ = − r− 𝐹(𝑡)r. (3.16) L is constant, Here and in the following, we assumed that at zeroth-order the L = 𝑚 x × v + 𝑚 x × v = µ h, (3.6) 1 1 1 2 2 2 center of mass of the binary is at rest in the scalar cloud, or with more generally that its velocity is small as compared with the binary orbital velocity v. Following the method of the osculating orbital elements [70], h = r× v = ℎ e , ℎ = 𝑟 𝜙, 𝑝 = . (3.7) 𝐺𝑚 we obtain the impact of the accretion and of the dynamical friction by computing the perturbations to the orbital elements. The constancy of 𝜔 is related to the conservation of the Runge- It is clear from Eq.(3.16) that the orbital plane remains constant. Lenz vector, In particular, the specific angular momentum h remains parallel v× h to e and evolves as A = − e = 𝔢(cos𝜔 e + sin𝜔 e ). (3.8) 𝑟 𝑥 𝑦 𝐺𝑚 h = −𝐹(𝑡)h, (3.17) In the following, we will also use the true anomaly defined by whereas the Runge-Lenz vector evolves as 𝜑 = 𝜙− 𝜔, (3.9) which measures the azimuthal angle from the direction of A = − + 2𝐹(𝑡) (A+ e ). (3.18) pericenter and grows with time as 𝑚 √︄ This gives next the evolution of the eccentricity and of the . 𝐺𝑚 𝜑 = (1+ 𝔢 cos 𝜑) . (3.10) semi-major axis, The period 𝑃 of the orbital motion reads 𝔢 = − + 2𝐹(𝑡) (𝔢+ cos 𝜑), √︂ 𝑚 𝑎 𝑎 . 𝑎 = − + 2𝐹(𝑡) (1+ 𝔢 + 2𝔢 cos 𝜑). (3.19) 𝑃 = 2𝜋 , (3.11) 𝑚 1− 𝔢 𝐺𝑚 which is known as Kepler’s third law. Using Eq.(3.10), the derivatives with respect to the true anomaly 𝜑 read at first order B. Drag force from the dark matter √︂ 𝑑𝔢 𝑝 𝑚 𝔢+ cos 𝜑 As seen in Sec. II, the equations of motion of the two BHs = − + 2𝐹(𝑡) , 𝑑𝜑 𝐺𝑚 𝑚 (1+ 𝔢 cos 𝜑) read √︂ x − x . . . 𝑑𝑎 𝑝 𝑚 𝑎 1+ 𝔢 + 2𝔢 cos 𝜑 2 1 𝑚 x = 𝐺𝑚 𝑚 − 𝑚 x − 𝑓 x , = − + 2𝐹(𝑡) . 1 1 1 2 1 1 1 1 3 2 2 |x − x | 𝑑𝜑 𝐺𝑚 𝑚 1− 𝔢 (1+ 𝔢 cos 𝜑) 2 1 x − x . . 1 2 (3.20) 𝑚 x¥ = 𝐺𝑚 𝑚 − 𝑚 x − 𝑓 x , (3.12) 2 2 1 2 2 2 2 2 |x − x | 1 2 The perturbations generated by the dark matter lead to where we take into account the Newtonian gravity, the accretion oscillations and secular changes of the orbital elements. The of dark matter and the dynamical friction, with cumulative drift associated with the secular effects is obtained by averaging over one orbital period, as 2 2 8𝜋𝐺 𝑚 ρ 𝑓 (𝑡) = Θ(𝑣 > 𝑣 ) ln , (3.13) 𝑖 𝑖 𝑐𝑖 Z Z 𝑃 2𝜋 3𝑣 UV𝑖 . 1 . 1 𝑑𝑎 ⟨𝑎⟩ = 𝑑𝑡 𝑎 = 𝑑𝜑 . (3.21) 𝑃 𝑃 𝑑𝜑 0 0 where Θ is a Heaviside factor with obvious notations. This gives for the separation r the equation of motion C. Effect of the accretion 𝐺𝑚 µ . 𝑚 𝑓 . 𝑚 𝑓 . 2 1 1 2 We first consider the impact of the accretion of dark matter r¥ = − r− r− r− r. (3.14) 𝑟 µ 𝑚 𝑚 𝑚 𝑚 1 2 on the orbital motion. This corresponds to both the term 𝑚/𝑚 and the contribution Here we used Eq.(3.4), which gives at zeroth-order 𝑚 𝑚 2 1 𝐹 = . (3.22) v = v, v = − v, (3.15) acc 1 2 𝑚 𝑚 5 We focus on the regime where these accretion rates vary slowly E. Effect of GWs emission as compared with the orbital motion and we take them constant As is well known, the emission of GWs makes the orbits over one period. Then, we obtain from Eqs.(3.20) and (3.21) become more circular and tighter, until the BHs merge. At . . lowest order in a post-Newtonian expansion and using the . 𝑚 µ ⟨𝔢⟩ = 0, ⟨𝑎⟩ = − + 2 𝑎. (3.23) acc acc quadrupole formula, the drifts of the eccentricity and of the 𝑚 µ semi-major axis are given by the standard results [70] Thus, the accretion of dark matter does not change the eccentricity while it reduces the size of the orbit, as we have 304ν 𝑐 𝐺𝑚 121 2 −5/2 2 ⟨𝔢⟩ = − 𝔢 (1− 𝔢 ) 1+ 𝔢 . . gw 2 2 2 15𝑎 𝑐 𝑎 304 𝑚 𝑚 + 𝑚 𝑚 . . . . 1 2 2 1 𝑚 = 𝑚 + 𝑚 > 0, µ = > 0. (3.24) (3.32) 1 2 and The result (3.23) for the semi-major axis can be recovered at 3 73 2 37 4 1+ 𝔢 + 𝔢 once for circular orbits from the constancy of the total angular . 64ν 𝑐 𝐺𝑚 24 96 √︁ ⟨𝑎⟩ = − . (3.33) gw 2 2 7/2 momentum 𝐿 = µ 𝐺𝑚𝑝 and 𝑎 = 𝑝 for 𝔢 = 0. 𝑐 𝑎 (1− 𝔢 ) where D. Effect of the dynamical friction The dynamical friction corresponds to the contribution ν = µ /𝑚 = 𝑚 𝑚 /𝑚 (3.34) 1 2 𝑚 𝑓 𝑚 𝑓 2 1 1 2 𝐹 = + . (3.25) is the symmetric mass ratio. Throughout this paper, we work df 𝑚 𝑚 𝑚 𝑚 1 2 at the lowest post-Newtonian order (3.33). This is sufficient Using the zeroth-order expressions (3.15), we can write for our purpose, which is to estimate the dark matter density thresholds associated with a significant impact on the GW 𝐴 𝐵 𝑣 df df 𝐹 (𝑡) = + ln , (3.26) signal. As discussed in Sec. IV below, the dark matter df 3 3 𝑣 𝑣 corrections are most important in the early inspiral and behave as negative post-Newtonian orders. As such, they are not with degenerate with higher post-Newtonian orders. 8𝜋𝐺 ρ 𝑟 0 𝑎 𝐴 = 𝑚 Θ(𝑣 > 𝑣 ) ln df 1 𝑐1 3µ 𝑅 𝑐1 F. Relative impact of dark matter and GWs +𝑚 Θ(𝑣 > 𝑣 ) ln , (3.27) 2 𝑐2 2 1. Accretion 𝑐2 From Eqs.(3.23) and (3.33), the ratio of the semi-major axis 4𝜋𝐺 ρ drifts due to the accretion of dark matter and to the emission 3 3 𝐵 = 𝑚 Θ(𝑣 > 𝑣 )+ 𝑚 Θ(𝑣 > 𝑣 ) , (3.28) df 1 𝑐1 2 𝑐2 1 2 2 of GWs reads 1/3 2 1/3 where we introduced the characteristic radii ρ 𝑐 𝐺 𝑚 acc 𝑎 √︂ ∼ , (3.35) 5/2 8/3 𝐺𝑚 𝑎 𝑚 𝑓 2 < gw 𝑅 = 6 . (3.29) 𝑐𝑖 3/2 µ 𝑐 where 𝑚 = max(𝑚 , 𝑚 ), 𝑚 = min(𝑚 , 𝑚 ), and 𝑓 = 2/𝑃 > 1 2 < 1 2 Because of the velocity dependence of the dynamical friction is the GW frequency (which is twice the orbital frequency). force (3.26), it is not possible to obtain explicit analytical Here we assumed the eccentricity to be small, 𝔢 ≲ 1. This expressions for the secular drifts (3.21). However, we can gives obtain explicit expressions for the series expansion in powers 1/3 −1 of the eccentricity 𝔢. At lowest order, this gives 𝑎 ρ 𝑚 𝑚 acc 𝑎 > < −4 ∼ 2× 10 " !# . √︂ −3 1 𝑀 1 𝑀 𝑎 1 g· cm ⊙ ⊙ gw 3/2 𝑎 𝐺𝑚 1 ⟨𝔢⟩ = 3𝔢 𝐴 + 𝐵 ln −8/3 df df df 1/3 𝐺𝑚 𝑎 𝑒 𝑐 × . (3.36) (3.30) 1 Hz and " !# Thus, we can see that the impact of the accretion of dark matter √︂ 3/2 . 𝑎 𝐺𝑚 1 on the orbital dynamics is typically much smaller than that of ⟨𝑎⟩ = −2𝑎 𝐴 + 𝐵 ln . (3.31) df df df 𝐺𝑚 𝑎 𝑐 the emission of GWs. It increases for smaller masses and low frequencies. This implies that it is most important at the early Thus, the dynamical friction increases the eccentricity, if 𝔢 > 0, stages of the inspiral phase. and reduces the size of the orbit. We assume in this paper that the impact of the dark matter cloud on the binary is smaller than 2. Dynamical friction that of the emission of GWs, which decreases the eccentricity. Therefore, in the following, we consider circular orbits with From Eqs.(3.31) and (3.33), the ratio of the semi-major axis 𝔢 = 0. drifts due to the dynamical friction and to the emission of GWs 6 reads 4. Comparison of accretion and dynamical friction . We can note that whereas the accretion effect (3.36) only 4/3 𝑎 ρ 𝑐 𝑚 df 0 depends on the characteristic density ρ defined in Eq.(2.3), ∼ 𝑎 2/3 11/3 𝑎 𝐺 𝑚 𝑓 gw the dynamical friction effect (3.37) depends on both ρ , which < 0 4/3 −3 −11/3 sets its magnitude, and ρ because of the conditions (3.41) and ρ 𝑚 𝑚 𝑓 0 > < ∼ 47 . (3.43). −3 1 g· cm 1 𝑀 1 𝑀 1 Hz ⊙ ⊙ This follows from the fact that the scalar field dark matter (3.37) model (2.2) can be mapped in the nonrelativistic regime to a hydrodynamical system with a polytropic equation of state Thus, the impact of the dynamical friction is also typically 𝑃 ∝ ρ [58]. For a radial accretion flow, this stiff equation of much smaller than that of the emission of GWs. It again state implies that the critical transonic point is actually close increases for smaller masses and low frequencies and is most to the Schwarzschild radius, in the relativistic and nonlinear important at the early stages of the inspiral phase. regime. There, the infall speed is of the order of the speed From Eqs.(3.30) and (3.32), we obtain for the eccentricity of light whereas the dark matter density is of the order of the characteristic density ρ , where the quartic self-interaction . . 4/3 𝔢 ρ 𝑐 𝑚 𝑎 part is of the same order as the mass term in the potential 𝑉(𝜙) df 0 df ∼ ∼ . (3.38) . . . 2 2/3 11/3 of Eq.(2.2). This explains at once the scaling 𝑚 ∼ ρ 𝑟 𝑐 BH 𝑎 𝐺 𝑚 𝑓 𝑎 𝑠 gw gw recalled in Eq.(2.7) for the accretion rate and why the accretion effect only depends on ρ , independently of the density ρ in 𝑎 0 As expected, this shows the same behavior as for the semi- the bulk of the scalar cloud. major axis and the impact of the dynamical friction is again In contrast, the dynamical friction force (2.9) is a large-scale typically small. effect associated with the deflection of distant streamlines, up to the radius of the dark matter cloud [58], as for the classical 3. Conditions for dynamical friction case of collisionless particles [1]. This is why its magnitude is set by the bulk density ρ of the cloud. We have ρ < ρ , From Eq.(3.15) we have 0 0 𝑎 −3 and typically ρ ≪ ρ and ρ ≪ 1 g· cm . For dark matter 0 𝑎 0 𝑚 𝑚 2 1 1/3 solitons that are of kpc size, the density ρ would be of the 𝑣 = 𝑣, 𝑣 = 𝑣, 𝑣 = (𝜋𝐺𝑚 𝑓 ) . (3.39) 1 2 𝑚 𝑚 order of the typical dark matter density in galaxies and would be very difficult to measure from the gravitational waveforms, Then, the first condition (2.11) for the mass 𝑚 to be in the as seen in Sec. V E below. However, if the scalar clouds formed supersonic regime, 𝑣 > 𝑐 , reads 1 𝑠 at high redshifts and have a much smaller size, their typical density ρ could be much higher than the current dark matter 3 3 𝑚 𝑐 1 density measured over galactic sizes or at the background level. 𝑣 > 𝑐 : 𝑓 > , (3.40) 1 𝑠 3 4 𝜋𝐺ν 𝑚 Then, it could be detected. On the other hand, even if ρ is of the order of the mean dark matter density, ρ can be much which gives greater because it is set by the fundamental parameters of the scalar field Lagrangian, see Eq.(2.3), and only reached near 3 −4 3 𝑚 𝑚 𝑐 the BH horizon as the dark matter density increases along the 1 𝑠 −6 −3 𝑓 > 2× 10 ν Hz infall onto the BH. Therefore, as we shall find in Sec. V E, the 1 𝑀 1 𝑀 100 km/s ⊙ ⊙ (3.41) dynamical friction effect is typically smaller than the accretion The second condition (2.11) for the mass 𝑚 to be above the effect and GW interferometers are more likely to measure ρ 1 𝑎 critical value (i.e. 𝑟 > 𝑟 ) reads than ρ . 𝑎 UV 0 2 5 288𝐺 𝑚 ρ IV. GW PHASE AND THE IMPACT OF DARK 𝑟 > 𝑟 : 𝑓 > , (3.42) 𝑎 UV1 3 4 2 𝑒ν 𝑚 𝑐 𝑐 MATTER A. Constant mass approximation which gives As we work at first order in all perturbations, we can sum the −1 ρ 𝑐 𝑎 𝑠 contributions from the accretion of dark matter, the dynamical −32 −3 𝑓 > 7× 10 ν −3 friction and the emission of GWs. This gives the total drift of 100 km/s 1 g· cm the orbital radius 5 −4 𝑚 𝑚 × Hz. (3.43) . . . . 1 𝑀 1 𝑀 ⊙ ⊙ ⟨𝑎⟩ = ⟨𝑎⟩ + ⟨𝑎⟩ + ⟨𝑎⟩ . (4.1) acc df gw We can see that this condition is typically much less stringent This drift depends on the masses of the two BHs and their than the supersonic condition (3.41). Therefore, dynamical accretion rates. However, for small accretion rates we can take friction usually occurs as soon as the BH reaches velocities 𝑚 and 𝑚 to be constant over the duration of the measurement. 𝑖 𝑖 above 𝑐 and satisfies the condition (3.41). Assuming this spansN orbital periods, with typicallyN ∼ 100, 𝑠 7 we require that 𝑚 N 𝑃 ≪ 𝑚 . From the accretion rate (2.7) with 𝑖 𝑖 this gives 𝔣 = . (4.7) 𝜋𝐺𝑚 𝑐 𝑓 ρ ≪ , (4.2) 24𝜋𝐹 𝐺 𝑚 N ★ > Integrating the phase Φ(𝑡) = 2𝜋 𝑑𝔣 (𝔣/𝔣) and the time 𝑡 = where 𝑓 = 2/𝑃 is the GW frequency (which is twice the orbital 𝑑𝔣 (1/𝔣) over the GW frequency [81], we obtain frequency) and 𝑚 = max(𝑚 , 𝑚 ). This gives > 1 2 6𝜋 6𝜋𝐶 3𝜋𝐶 2 3 −5/3 −13/3 −16/3 Φ(𝔣) = Φ − 𝔣 + 𝔣 + 𝔣 −1 2 2 5𝐶 13𝐶 8𝐶 𝑚 𝑓 1 10 −1 −3 1 1 ρ ≪ 6× 10 N g· cm . (4.3) 1𝑀 1 Hz 1 1 × 𝐴 + 𝐵 + ln(𝔣/𝔣 ) (4.8) df df ★ 16 3 The strongest limitation is associated with the case of Massive Binary Black Holes (MBBH) to be detected with the space and −4 interferometer LISA, at frequencies 𝑓 ≳ 10 Hz. This gives 3 3𝐶 3𝐶 2 3 3 −8/3 −16/3 −19/3 the upper bound ρ ≪ 0.01 g/cm , which is much beyond the 𝑡(𝔣) = 𝑡 − 𝔣 + 𝔣 + 𝔣 𝑎 𝑐 2 2 8𝐶 1 16𝐶 19𝐶 expected dark matter densities. For instance, the dark matter 1 1 −24 density in the Solar System is about 10 g/cm [71–79]. On 1 1 × 𝐴 + 𝐵 + ln(𝔣/𝔣 ) , (4.9) df df ★ the other hand, accretion disks around supermassive BHs can 19 3 −9 3 have baryonic densities up to 10 g/cm for thick disks and −1 3 where Φ and 𝑡 are the phase and the time at coalescence 𝑐 𝑐 10 g/cm for thin disks [80]. Therefore, the bound (4.3) is time, and we introduced well satisfied up to the baryonic densities found in accretion . . disks. At higher densities, we should explicitly take into 8/3 96𝜋 𝑚 µ 3 5/3 account the time dependence of the BH masses and accretion 𝐶 = (𝐺M) , 𝐶 = 2 +3 , 𝐶 = (4.10) 1 2 3 5𝑐 𝑚 µ 𝜋𝐺𝑚 rates. This would further enhance the deviation from the signal 3/5 associated with the binary system in vacuum and increase where M = ν 𝑚 is the chirp mass. Equations (4.8)-(4.9) the dark matter impact on the waveform. Therefore, our provide an implicit expression for the function Φ(𝑡), describing computation provides a conservative estimate of the detection the GWs phase as a function of time. Here, we considered the threshold. In fact, Eqs.(3.36) and (3.37) show that the presence regime where the dark matter contribution to the frequency of the dark matter environment will be detected much before drift is weaker than the GW contribution and we linearized the condition (4.3) is violated. In fact, we checked numerically over the coefficients 𝐶 and 𝐶 , associated with the accretion 2 3 that taking 𝑚 (𝑡) = 𝑚 (𝑡 )+𝑚 (𝑡−𝑡 ) with a constant accretion 𝑖 𝑖 0 𝑖 0 and the dynamical friction. As seen in (3.36) and (3.37), rate does not change our results, presented in Sec. V E below. this is the case in realistic configurations. Besides, this is sufficient for the purpose of estimating the dark matter density thresholds required for detection. At much higher densities, B. Phase and coalescence time our computation of the frequency drift is no longer reliable but In the limit of small eccentricity, 𝔢 ≪ 1, the drift (4.1) reads the presence of dark matter would remain clear in the data. . . We recover the fact that the dark matter contributions are 3/2 . 64ν 𝑐 𝐺𝑚 𝑚 µ 𝑎 𝑎 = − − + 2 𝑎 − 2𝑎 more important during the early stages of the inspiral, that is, 5 𝑚 µ 𝐺𝑚 𝑐 𝑎 at low frequencies. This means that relativistic corrections to " !# √︂ the orbital motion would not change our results for the dark 𝐺𝑚 1 × 𝐴 + 𝐵 ln . (4.4) df df matter detection thresholds. 𝑎 𝑐 The GW signal is of the form ℎ(𝑡) = A(𝑡) cos[Φ(𝑡)], where Φ(𝑡) is implicitly determined by Eqs.(4.8)-(4.9) and A(𝑡) ∝ From Kepler’s third law (3.11), we have 2/3 𝔣 if we neglect the dark matter corrections in the amplitude . . [70]. The Fourier-space data analysis considers the Fourier 𝔣 1 𝑚 3 𝑎 𝑖2𝜋 𝑓𝑡 = − . (4.5) transform ℎ( 𝑓 ) = 𝑑𝑡 𝑒 ℎ(𝑡). In the stationary phase 𝔣 2 𝑚 2 𝑎 𝑖Ψ( 𝑓 ) approximation [81], one obtains ℎ( 𝑓 ) = A( 𝑓 )𝑒 , with where 𝔣 = 2/𝑃 is again the frequency of the GWs. We use −7/6 A( 𝑓 ) ∝ 𝑓 , Ψ( 𝑓 ) = 2𝜋 𝑓 𝑡 − Φ(𝑡 )− 𝜋/4, (4.11) ★ ★ a gothic font in this section to distinguish 𝔣, the function of time describing the frequency sweep, from 𝑓 , the Fourier- where the saddle-point 𝑡 is defined by 𝔣(𝑡 ) = 𝑓 , as Φ = 2𝜋𝔣. ★ ★ transform variable used below in the Fourier-space analysis of Using Eqs.(4.8)-(4.9) we obtain the time-sequence data. This gives Ψ( 𝑓 ) = 2𝜋 𝑓 𝑡 − Φ − + Ψ + Ψ + Ψ , (4.12) . . 𝑐 𝑐 gw acc df 8/3 𝔣 96𝜋 ν 𝑚 µ 3 5/3 8/3 −1 = (𝐺𝑚) 𝔣 + 2 + 3 + 𝔣 𝔣 𝑚 µ 𝜋𝐺𝑚 5𝑐 where the different contributions are 𝐵 𝔣 df 3 −5/3 × 𝐴 + ln , (4.6) df Ψ = (𝜋𝐺M 𝑓/𝑐 ) , (4.13) gw 3 𝔣 128 8 3 2 75𝜋𝐹 (1+ ν ) ρ 𝐺 M 𝑎 decorrelated from the other parameters {𝜃 } [81]. Therefore, 3 −13/3 𝑖 Ψ = − (𝜋𝐺M 𝑓/𝑐 ) , (4.14) acc 3/5 6 we do not consider the amplitude any further. 13312ν 𝑐 As compared with the study presented in [83], we neglect and the effective spin 𝜒 ≡ (𝑚 𝜒 + 𝑚 𝜒 )/𝑚, which is only eff 1 1 2 2 considered to calculate the last stable orbit using the analytical 3 2 25𝜋[C ( 𝑓 )+C ( 𝑓 )] ρ 𝐺 M 1 2 0 3 −16/3 Ψ = − (𝜋𝐺M 𝑓/𝑐 ) , PhenomB templates [84]. This is because our results for df 11829248ν 𝑐 the accretion rate and the dynamical friction have only been (4.15) derived for Schwarzschild BHs. However, we expect the order with of magnitude that we obtain for the dark matter densities to " !# 3 remain valid for moderate spins. A second difference from [83] 𝑚 𝑓 𝑟 C ( 𝑓 ) = 𝜃(𝑣 > 𝑣 ) 105+ 304 ln . 𝑖 𝑖 𝑐𝑖 is that in addition to the dark-matter density ρ , which describes 2 0 𝑚 𝔣 𝑐𝑖 the bulk of the cloud, we also have a second characteristic (4.16) density ρ . It describes the dark matter density close to the The coefficients C ( 𝑓 ) depend on the frequency 𝑓 through the Schwarzschild radius and it is directly related to the strength of logarithmic term and the Heaviside factor, as from Eq.(3.39) the dark-matter self-interaction. The dynamical friction (2.9) the BH velocities 𝑣 grow with the frequency 𝑓 and can go depends on the bulk density ρ , as in the standard cases, but also from the subsonic to the supersonic regime during the inspiral. on ρ through the dependence of the Coulomb logarithm and √︁ In terms of post-Newtonian (PN) contributions, the accretion of the supersonic condition on the sound speed 𝑐 = 𝑐 ρ /ρ . 𝑠 0 𝑎 acts as a -4PN contribution and the dynamical friction as a The accretion rate (2.7) only depends on ρ . Therefore, in the -5.5PN contribution. This expresses the fact that higher-order subsonic regime the bulk density ρ is unconstrained. From post-Newtonian contributions are increasingly important at −1 the Fisher matrix we obtain the covariance Σ = Γ , 𝑖𝑗 high frequencies, in the late stage of the inspiral, whereas the 𝑖𝑗 which gives the standard deviation on the various parameters dark matter contributions are increasingly important at low 2 1/2 as σ = ⟨(Δ𝜃 ) ⟩ = Σ . 𝑖 𝑖 𝑖𝑖 frequencies, in the early stage of the inspiral. This means that these two types of corrections are not degenerate and one should be able to discriminate efficiently between both B. Gravitational-wave detectors effects. In practice, in this paper we do not include higher-order The gravitational-wave detectors that we consider are Adv- relativistic corrections as we focus on the early stages of the LIGO [85], ET [86], LISA [87] and B-DECIGO [88]. We inspiral, where the environmental effects are most important. use the noise spectral densities presented in [89–92]. The frequency ranges are given in Table I, where the PhenomB V. FISHER INFORMATION MATRIX inspiral-merger transition value 𝑓 is defined in [84] and 𝑓 = 1 obs 5 3 − − 8 8 −5 M obs A. Fisher analysis 4.149× 10 is the frequency at a given 1 yr 10 𝑀 observational time before the merger, as defined in [93]. We We use a Fisher matrix analysis to estimate the dark take 𝑇 = 4 yr in our computations. matter densities ρ and ρ that could be detected through obs 𝑎 0 the measurement of GWs emitted by binary BHs in the inspiral phase. The Fisher matrix is given by [81,82] Frequency 𝑓 (Hz) 𝑓 (Hz) min max Detector max ˜ ˜ 𝑑𝑓 ∂ℎ ∂ℎ Adv-LIGO 10 𝑓 Γ = 4 Re , (5.1) 𝑖𝑗 𝑆 ( 𝑓 ) ∂𝜃 ∂𝜃 𝑛 𝑖 𝑗 min ET 3 𝑓 −5 2 LISA max 2× 10 , 𝑓 min 10 , 𝑓 obs 1 where {𝜃 } is the set of parameters that we wish to measure −2 B-DECIGO 10 min (1, 𝑓 ) and 𝑆 ( 𝑓 ) is the noise spectral density, which depends on the GW interferometer. The signal-to-noise ratio is TABLE I: Gravitational waves frequency band considered for the Adv-LIGO, ET, LISA and B-DECIGO interferometers, max 𝑑𝑓 2 2 where 𝑓 is the frequency of the binary 4 years before the obs (SNR) = 4 |ℎ( 𝑓 )| . (5.2) 𝑆 ( 𝑓 ) 𝑛 merger [93] and 𝑓 is the PhenomB inspiral-merger transition 𝑓 1 min value [84]. −7/6 𝑖Ψ( 𝑓 ) Writing the gravitational waveform as ℎ( 𝑓 ) = A 𝑓 𝑒 , as in Eqs.(4.11)-(4.12), we obtain 2 𝑓 max (SNR) 𝑑𝑓 ∂Ψ ∂Ψ C. Events −7/3 Γ = 𝑓 (5.3) 𝑖𝑗 R max 𝑑 𝑓 −7/3 𝑆 ( 𝑓 ) ∂𝜃 ∂𝜃 𝑛 𝑖 𝑗 𝑓 min We focus on the description of 6 events, 2 ground based and 𝑓 𝑆 ( 𝑓 ) min 𝑛 4 space based, the last ones being for LISA since its detection where the parameters that we consider in our analysis are{𝜃 } = range differs from the others. All the events are BH binaries. {ln(M), ln(ν ), 𝑡 , Φ , ρ , ρ }. The amplitude A would be an The virtual events correspond to different types of binaries: 𝑐 𝑐 0 𝑎 0 additional parameter. However, the Fisher matrix is block- Massive Binary Black Holes (MBBH), Intermediate Binary diagonal as Γ = 0 and the amplitude A is completely Black Holes (IBBH), an Intermediate Mass Ratio Inspiral A ,𝜃 0 0 𝑖 9 (IMRI) and an Extreme Mass Ratio Inspiral (EMRI). All of E. Detection prospects these events are of the same type as the ones considered by The dark matter parameters ρ and ρ must be positive 0 𝑎 [83]. We focus on BH binaries and do not consider neutron and satisfy the hierarchy ρ ≪ ρ , so that 𝑐 ≪ 𝑐 and the 0 𝑎 𝑠 star binaries. The details of the events are given in Table II. For bulk of the scalar cloud can be described in the nonrelativistic completeness, we included the spins and 𝜒 , which sets the eff regime. These constraints, the large range of possible dark upper frequency cutoff of the data analysis. The SNR values matter densities and the complex dependence on {ρ , ρ } of 0 𝑎 for each of these events are taken from [83] and summarized the gravitational waveform contributions (4.14)-(4.15) make in Table III. a standard forecast analysis, where elliptic contours around a fiducial choice in the plane (ρ , ρ ) are drawn from the 0 𝑎 Properties Fisher matrix, not very convenient. Instead, we show in Figs. 1 𝑚 (M ) 𝑚 (M ) 𝜒 𝜒 𝜒 ⊙ ⊙ 1 2 1 2 eff Event and 2 the regions in the (ρ , ρ ) plane where either (ρ < 0 𝑎 0 6 5 σ , ρ < σ ), (ρ < σ , ρ > σ ), or (ρ > σ , ρ > σ ) 0 𝑎 𝑎 0 0 𝑎 𝑎 0 0 𝑎 𝑎 MBBH 10 5× 10 0.9 0.8 0.87 where σ and σ are respectively the standard deviations of 0 𝑎 4 3 IBBH 10 5× 10 0.3 0.4 0.33 ρ and ρ . Indeed, we find that there are no cases where 0 𝑎 IMRIs 10 10 0.8 0.5 0.80 (ρ > σ , ρ < σ ), that is, it is always easier to constrain ρ 0 0 𝑎 𝑎 𝑎 EMRIs 10 10 0.8 0.5 0.80 than ρ (which by definition has a smaller value). Thus, these GW150914 35.6 30.6 0.13 0.05 0.09 maps show the regions in the plane (ρ , ρ ) of the two dark 0 𝑎 matter parameters where dark matter can be detected or not, GW170608 11 7.6 0.13 0.50 0.28 and when this is possible whether both (ρ , ρ ) or only ρ 0 𝑎 𝑎 TABLE II: Details on masses and spins of the considered can be measured. In other words, for each point in the plane events. The information on GW150914 and GW170608 are (ρ , ρ ) taken as a fiducial value, we examine whether these 0 𝑎 taken from [94]. densities can be measured within a factor of order unity. We first consider the space mission LISA in Fig. 1 and next the ground-based interferometers B-DECIGO, ET, and Adv-LIGO in Fig. 2. In Fig. 1 for LISA, we present a map of the detection Detector LISA B-DECIDO ET Adv-LIGO prospects for the dark matter density parameters ρ and ρ , Event 0 a for the MBBH, IBBH, IMRIs and EMRIs events. Let us first MBBH 3× 10 × × × describe the MBBH case, which also illustrates the general IBBH 708 × × × behavior. As explained above, the white area below the IMRIs 22 × × × diagonal ρ = ρ is excluded because it has no physical 𝑎 0 EMRIs 64 × × × meaning (ρ > ρ would correspond to a sound speed greater 0 𝑎 than the speed of light). Then, in the upper left half-plane we GW150914 × 2815 615 40 have three areas distinguished by increasingly darker shades GW170608 × 2124 502 35 of blue, associated with the three cases (ρ < σ , ρ < σ ), 0 0 𝑎 𝑎 TABLE III: Value of the signal-to-noise ratio (SNR) of the (ρ < σ , ρ > σ ), and (ρ > σ , ρ > σ ). Thus, the lower 0 0 𝑎 𝑎 0 0 𝑎 𝑎 considered events for each detector, taken from [83]. left brighter area at low densities corresponds to no detection of the dark matter environment, the upper left intermediate shade of blue corresponds to a detection of dark matter with a measurement of the density ρ only, and the darker triangle corresponds to a measurement of both ρ and ρ . 𝑎 0 D. Relativistic corrections Let us now describe these domains and the critical lines displayed in the plot. For the sake of the argument, we can The dynamical friction formulae used here are valid in the assume that the gravitational wave data correspond to a given nonrelativist limit 𝑣 ≪ 𝑐. Relativistic corrections typically 2 2 2 frequency (and BH masses), hence to given velocities 𝑣 of give a corrective prefactor 𝛾 (1+𝑣 ) in the dynamical friction 𝑖 the two BHs. As the accretion effect (4.14) on the waveform [95–97]. This can be obtained in the collisionless case from only depends on ρ , in a first analysis the detection threshold the relativistic formula for the scattering deflection angle 𝑎 for ρ due to the accretion corresponds to an horizontal line and the relativistic Lorentz boost between the fluid and BH 𝑎 in the plane (ρ , ρ ). This agrees with the horizontal lower frames [95]. This should remain a good approximation in 0 𝑎 boundaries of the region (ρ < σ , ρ > σ ) found in Fig. 1, the highly supersonic case, where the streamlines at large 0 0 𝑎 𝑎 although the threshold ρ appears to take two values. We will radii follow collisionless trajectories as pressure effects are 𝑎★ 2 2 come back to the explanation of these two levels below. small. For velocities as high as 𝑣 ∼ 0.137 𝑐 this only Let us now move horizontally, at a fixed ρ , from the right gives a multiplicative factor of about 1.5. As the dark matter 𝑎 boundary ρ = ρ to the left towards low bulk densities ρ → 0. contributions are most important in the early inspiral, we can 0 𝑎 0 As ρ decreases, the sound speed of the dark matter fluid (in see that relativistic corrections can be neglected and will not 0 the bulk of the soliton) decreases from 𝑐 to zero, see Eq.(2.5). change the order of magnitude of our results. In practice, we 2 2 Therefore, close to the right diagonal boundary, 𝑐 ≃ 𝑐 and both cut the analysis below the frequency 𝑓 where 𝑣 = 0.137 𝑐 , 𝑠 BH velocities are subsonic. Within our approximation (2.11), to ensure relativistic corrections remain modest. 10 FIG. 1: Maps of the detection prospects with LISA for different events, focusing on the dark matter parameters ρ and ρ . The 0 𝑎 lower right white area is not physical. The upper left area shows three shades of blue, associated with i) no detection of the dark matter environment (bright blue), ii) measurement of ρ (intermediate shade), iii) measurement of both ρ and ρ (dark 𝑎 𝑎 0 blue). Detection is deemed successful when the standard deviation is smaller than the corresponding parameter value. this means that both BHs violate the supersonic condition frequency 𝑓 ). The parallel dashed line corresponds to the max (3.40) and their dynamical friction vanishes. Then, the only bulk density ρ such that the sound speed 𝑐 is equal to the 0 𝑠 impact of the dark matter environment on the waveform is smaller BH velocity at the beginning of the data sequence due to the accretion, which only depends on ρ , and the bulk (when 𝑣 is minimum, at the lowest frequency 𝑓 ). Therefore, 𝑎 min density ρ is irrelevant and cannot be constrained. This is why in-between these two lines the smaller BH is supersonic only at in the left neighborhoud of the diagonal boundary we cannot late times in the data sequence, whereas to the left of the dashed measure ρ . We either measure ρ at high density or do not line it is supersonic throughout all the data sequence. Thus, 0 𝑎 detect dark matter at low density. the dynamical friction experienced by the smaller BH becomes Since the smaller BH has the higher velocity from (3.39), increasingly present between these two lines and fully active to as we decrease 𝑐 while moving towards the left, we first the left of the dashed line. Next, as we move further to the left encounter the threshold 𝑐 = 𝑣(𝑚 ). Thus, we first enter the and further decrease 𝑐 , the larger BH also becomes supersonic. 𝑠 < 𝑠 domain where the smaller BH is supersonic. Because the GW This corresponds to the black solid and dashed lines, again data span a range of time and frequency, hence of orbital radius associated with the finite frequency range [ 𝑓 , 𝑓 ]. When min max and velocity, this threshold is not marked by a single line but the dynamical friction contribution Ψ to the waveform comes df by the diagonal band between the red solid and dashed lines. into play, it brings a new factor that depends on the bulk density The solid line corresponds to the bulk density ρ such that ρ , see Eq.(4.15). Therefore, to the left of these supersonic 0 0 the sound speed 𝑐 is equal to the small BH velocity at the threshold diagonal lines we can measure ρ . However, the 𝑠 0 end of the data sequence (when 𝑣 is maximum, at the highest magnitude of the dynamical friction decreases linearly with ρ 0 11 FIG. 2: Maps of the detection prospects for three different interferometers (from left-to-right: B-DECIGO, ET, and Adv- LIGO), for the two events GW150914 (upper row) and GW170608 (lower row). The color convention is the same scheme as for Fig. 1. (neglecting the impact of the Coulomb logarithm). This leads of ρ is set by the dynamical friction term. to a lower limit on ρ , that is, a left vertical boundary in the The IBBH case, in the lower left panel, is very similar to plot, for the domain where ρ can be measured. the MBBH case in all of these aspects, including the shape of This dark blue domain where ρ can be measured is also the detection areas. While both of them permit detection of −25 3 bounded on the upper left by critical lines with a lower slope. ρ at dark matter densities close to ρ ∼ 6.7× 10 g/cm , 𝑎 0 They correspond to the second criterium (3.42). The effective the expected density of dark matter in our neighborhood in the pressure of the dark matter gives rise to a small-scale cutoff Milky Way [71–79], no detection of ρ at this typical value is in the Coulomb logarithm, that is, to a lower radius below possible. In the best case, an MBBH event can give information −15 3 which the streamlines do not generate significant dynamical for ρ > 10 g/cm . Of course, doing the same calculation friction. As seen from Eq.(3.43), moving vertically to larger with bigger BHs might lessen this constraint, but such events ρ we first encounter the threshold associated with the larger are very unlikely to happen. BH and second the one associated with the smaller BH. Again, Let us now examine the case of IMRIs, in the upper because of the finite frequency width [ 𝑓 , 𝑓 ] of the data, right panel. Because of the large mass ratio, the critical min max these thresholds do not correspond to a single line but to a lines associated with the two BHs are now widely separated. finite band, marked by the solid and dashed lines. In this upper The threshold for the measurement of ρ is again mostly left corner, where the dynamical friction is negligible in our independent of ρ and of the same order as for IBBH, as it is approximation, our computation is not very accurate because determined by the accretion onto the most massive BH, which we can expect the flow around each BH to be perturbed by has not changed. In contrast, the domain where ρ can be the other BH and our expression (2.9) would not be valid. measured is now split into two widely separated triangular Then, the dynamical friction may not vanish and it may be still regions, associated with the dynamical friction experienced possible to constrain ρ to some degree. by either the smaller or the larger BH. The large upper right Coming back to the two levels for the detection threshold triangle is due to the dynamical friction experienced by the ρ , we note that it is somewhat higher in the regions where smaller BH, as for smaller BH mass the supersonic condition 𝑎★ dynamical friction takes place. This is because of the partial (3.41) moves horizontally to larger 𝑐 and ρ while the radius 𝑠 0 degeneracy between the two parameters ρ and ρ , which condition (3.43) moves vertically to larger ρ . On the other 𝑎 0 𝑎 makes it more difficult to obtain an accurate measurement of hand, because the magnitude of the dynamical friction (2.9) each of them. scales linearly with the BH mass, it requires a larger bulk Overall, the detection of ρ is governed by the accretion term density ρ for the dynamical friction experienced by the smaller 𝑎 0 in both supersonic and subsonic regimes, while the detection BH to have a significant impact on the waveform. Therefore, 12 Detector LISA B-DECIDO ET Adv-LIGO Event −14 3 MBBH 3.7× 10 g/cm −9 3 IBBH 2.5× 10 g/cm −12 3 IMRIs 9.3× 10 g/cm −12 3 EMRIs 2.5× 10 g/cm −11 3 −1 3 2 3 GW150914 4.8× 10 g/cm 1.3× 10 g/cm 2.6× 10 g/cm −12 3 −3 3 1 3 GW170608 4.2× 10 g/cm 9.5× 10 g/cm 4.4× 10 g/cm −25 3 TABLE IV: Assuming a bulk dark matter density ρ = 7× 10 g/cm , value ρ of the minimum density parameter ρ that 0 𝑎★ 𝑎 can be measured and enables a detection of the presence of a dark matter cloud. We show our results for all events presented in Figs. 1 and 2. Detector LISA B-DECIDO ET Adv-LIGO Event −15 3 MBBH 2.3× 10 g/cm −10 3 IBBH 1.3× 10 g/cm −18 3 8.5× 10 g/cm IMRIs −9 3 1.9× 10 g/cm −7 3 EMRIs 1.0× 10 g/cm −14 3 −1 3 3 3 GW150914 8.3× 10 g/cm 2.7× 10 g/cm 3.1× 10 g/cm −15 3 −3 3 2 3 GW170608 1.8× 10 g/cm 9.8× 10 g/cm 1.1× 10 g/cm TABLE V: Value ρ of the minimum density parameter ρ that can be measured. We show our results for all events presented 0★ 0 in Figs. 1 and 2. There are two values for IMRIs as we have two very distinct detection regions this detection region is cut by a vertical line at the rather high matter cloud cannot be detected. −7 3 detection threshold ρ ≃ 10 g/cm . The small lower left We show in Fig. 2 the detectability maps for GW150914 0★ dark triangle is due to the dynamical friction experienced by and GW170608 for other detectors. The shape of the results is the larger BH, as its dynamical friction only operates at lower similar to what was obtained for the MBBH and IBBH cases, densities. Because of its higher mass, its dynamical friction can as the BH mass ratio is again of the order of unity, making −17 3 still make an impact at these low densities, ρ ≃ 10 g/cm . the large and small BH critical lines close to one another. −18 However, this region is almost immediately cut at ρ ≃ 10 However, the much smaller BH masses change the location of 0★ g/cm as even for this massive BH the bulk dark matter density these thresholds in the (ρ , ρ ) plane. B-DECIGO shows more 0 𝑎 becomes too low to make a significant effect. In the MBBH promising forecasts than ET and Adv-LIGO, as the latter can and IBBH cases, where the mass ratio is only a factor two, only measure ρ for very high densities. In fact B-DECIGO these two regions overlap and form a single detection domain appears to be as promising as LISA for detecting both dark for ρ . matter density parameters. The EMRIs case, shown in the lower right panel, is similar For all events, there is no domain where we can measure ρ to the IMRIs case, with an even larger mass ratio. The two but not ρ . This is because the measurement of ρ is governed 𝑎 a domains where each BH experiences dynamical friction are by the mass accretion effect onto the waveform, which is always even more widely separated. We find again in the upper right present, while ρ needs an efficient dynamical friction. This corner a region at high densities where ρ can be measured, only occurs in a limited part of the parameter space, in the thanks to the dynamical friction on the smaller BH. The region supersonic regime, and with a lower magnitude. Morever, on where the larger BH experiences dynamical friction is pushed the theoretical side, we can expect the density ρ near the BH −22 3 to even lower densities, ρ ≃ 10 g/cm . This makes the horizons to be much greater than the density ρ in the bulk of 0 0 signal too weak to enable a measurement of ρ . At these low the scalar cloud. This makes it much more likely to detect such densities, even ρ can no longer be measured and the dark a dark matter environment through the accretion contribution 𝑎 13 (4.14) to the waveform, and to obtain an estimate of the density (2.6), which also reads ρ if such a cloud is present. ℏ𝑐 For each event we present in Tables IV and V the minimum 𝑚 > . (5.6) DM 2𝐺𝑚 densities ρ and ρ that can be detected and measured. the 𝑎 0 measurement of ρ is only possible at much higher densities This ensures the validity of the accretion rate (2.7) and of the than the typical dark matter density on galaxy scales, which is dynamical friction (2.9), derived in [58–60] in the large-mass −26 −23 3 about 10 to 10 g/cm [65,98–100]. For comparison, we limit ∂ ≪ 𝑐𝑚 /ℏ. This condition excludes the green area 𝑟 DM also note that accretion disks have a baryonic matter density marked by a vertical line on the left in the figures. 3 −9 3 below ∼ 0.1 g/cm for thin disks, and below 10 g/cm for Observations of cluster mergers, such as the bullet cluster, −16 3 thick disks [80], with a lower bound around 10 g/cm . Thus, provide an upper bound on the dark matter cross-section, gravitational waveforms can only probe ρ for densities that 0 2 σ /𝑚 ≲ 1 cm /g [102]. This gives the upper bound [61] DM are much above the mean dark matter densities but below those of baryonic accretion disks. However, because of the −12 𝜆 < 10 , (5.7) lack of dissipative and radiative processes, the mechanisms 4 1 eV enabling dark matter to reach such high densities must be different from that of the baryonic disks. Such dark matter shown by the dashed red line in the upper left corner of the figures. clouds could instead form in the early universe, as discussed for instance in [57,101] for several scenarios. In contrast with Another observational limit, shown by the upper left red solid line, is the maximum size of the dark matter solitons. the standard CDM case, the dark matter density field would be extremely clumpy, in the form of a distribution of small As we wish such solitons to fit inside galaxies, we require 𝑅 < 10 kpc. This gives the upper bound and dense clouds (in a manner somewhat similar to primordial sol BHs or macroscopic dark matter scenarios, but with larger-size 𝑅 𝑚 objects). sol DM 𝜆 < 0.03 . (5.8) 10 kpc 1 eV F. Detection threshold for ρ and parameter space This condition is actually parallel to the detection threshold (5.5) and somewhat above it in the Figs. 1 and 2. Therefore, In this section, we compare the detection threshold ρ 𝑎★ the largest solitons would not be detected by GW. This will be obtained in Table IV with the allowed parameter space of our more clearly seen in Sec. V G below. dark matter model, in the (𝑚 ,𝜆 ) plane. This allows us DM 4 Our derivation of the accretion rate (2.7) and of the to check wether this scenario can be efficiently probed by the dynamical friction (2.9) assumes that the self-interaction measurement of the gravitational waves emitted by BH binary dominates over the quantum pressure [58–60], in contrast systems embedded in such dark matter clouds. Our results are displayed in Figs. 3 and 4, representing the outcomes for with FDM scenarios where the latter dominates and the self- interactions are neglected. The self-interaction potential LISA and the other experiments respectively. Various colored regions on the figures correspond to distinct limits based on reads Φ = 𝑐 ρ /ρ , whereas the quantum pressure reads 𝐼 𝑎 √ √ 2 2 2 Φ = −ℏ ∇ ρ /(2𝑚 ρ ). This gives the condition either observational constraints or the regime considered in 𝑄 DM 2 2 2 2 our calculations. As seen in Figs. 1 and 2, the detection 𝑐 ρ /ρ > ℏ /(𝑟 𝑚 ), where ρ and 𝑟 are the density and DM threshold for ρ is mostly independent of ρ , although it can length scale of interest. This condition near the BH horizon, 𝑎 0 move somewhat between domains where dynamical friction with ρ ∼ ρ and 𝑟 ∼ 𝑟 , coincides with the condition (5.6) and 𝑎 𝑠 is important or not. For definiteness, we adopt the threshold is thus already enforced. Requiring that this also holds over −25 3 for ρ associated with the value ρ = 7× 10 g/cm , which the bulk of the soliton, at density ρ and radius 𝑟 ∼ 𝑅 , gives 𝑎 0 0 sol is the typical local dark matter density in the Milky Way the additional constraint [71–79]. From Eq.(2.3), a constant detection floor ρ for ρ 𝑎★ 𝑎 8𝑚 𝐺 corresponds to an upper ceiling for 𝜆 that scales as 𝑚 , 4 DM DM 𝜆 > , (5.9) √ √ 3 𝜋ℏ ρ 4 3 4𝑚 𝑐 DM ρ is measured if 𝜆 < , (5.4) which reads 𝑎 4 3ρ ℏ 𝑎★ ρ 𝑚 0 DM −26 𝜆 > 7× 10 . (5.10) which reads −25 3 1 eV 7× 10 g/cm −1 ρ 𝑚 𝑎★ DM −8 This excludes the blue region in the bottom right corner of the 𝜆 < 3× 10 . (5.5) −11 3 1 eV figures, below the blue solid line. This limit is the only one 10 g/cm that depends on the density parameter ρ , and it becomes less This ceiling is shown by the black solid line labeled ρ = σ stringent as ρ increases. Therefore, it moves down if ρ is 𝑎 𝑎 0 0 in Figs. 3 and 4. much larger than the mean dark matter density of the Milky Way. We now describes the constraints that determine the param- Below this threshold the model itself is not excluded, but our eter space of the model, with the exclusion domains shown by derivation of the dynamical friction must be revised as the bulk the colored regions in the plots. First, we require the condition of the soliton is now governed by the quantum pressure instead 14 FIG. 3: Domain over the parameter space (𝑚 , 𝜆 ) where our derivations are applicable, in the case of the LISA interferom- DM 4 −25 3 eter and assuming a bulk dark matter density ρ = 7 × 10 g/cm . The white area represents the allowed parameter space. The upper left red region is excluded by observational constraints. In the lower right blue region the scalar dark matter model is allowed but the assumptions used in our computations must be revised. The black line corresponds to the detection limit obtained in Fig. 1. Parameter values above this line are beyond the detectability range of the interferometer. of the self-interactions. Nevertheless, this should not change must thus consider that much our results, because the dynamical friction form (2.9) 4 2 16𝐺𝑐𝑚 𝑟 is actually very general and common to most models in the DM orbit 𝜆 > , (5.11) supersonic regime, where it is similar to the classical result by 3𝜋ℏ Chandrasekhar for collisionless particles [1]. This is because in which reads the supersonic regime the details of the self-interactions and of pressure terms are not important. Only the Coulomb logarithm 2 𝑚 𝑟 DM orbit −10 can change and depends on the details of the physics. Therefore, 𝜆 > 3× 10 . (5.12) 1 eV 1 pc this line does not really exclude the model nor changes the fact that the region below it in the (𝑚 ,𝜆 ) plane leads to a DM 4 For 𝑟 we take the maximum orbital radius, computed with orbit measurement of ρ by the gravitational waves interferometer, Kepler’s third law at the earliest measurement time, associated through the accretion effects for which our assumptions still with the frequency 𝑓 (4 yr). This constraint is parallel to the obs apply. soliton-size condition (5.8) and to the detection threshold ρ 𝑎★ Lastly, the area below the dashed blue line represents the in Eq.(5.5). parameter space where the soliton size is smaller than the initial Hence, the white area in the parameter space indicates where orbit of the binary system during the measurement. To ensure the dark matter model is realistic and all our calculations apply the applicability of our calculation across all frequencies, we successfully. More precisely, the upper bounds, associated with 15 FIG. 4: Domain over the parameter space (𝑚 , 𝜆 ) where our derivations are applicable and detection threshold, as in Fig. 3 DM 4 but for the interferometers B-DECIGO, ET, and Adv-LIGO. the red exclusion regions, correspond to unphysical regions Way. Again, LISA and B-DECIGO probe a large fraction of the parameter space, whereas the lower bounds, associated of the parameter space, while ET and Adv-LIGO typically with blue exclusion regions, only correspond to regions where probe smaller soliton sizes. Whereas LISA probes models −17 some of our computations should be revised. However, where with a scalar mass 10 ≲ 𝑚 ≲ 1 eV, B-DECIGO is DM −13 they fall within the detection domain, below the black solid restricted to 10 ≲ 𝑚 ≲ 1 eV, and ET and Adv-LIGO to DM −8 −4 line, it should remain possible to measure ρ . 10 ≲ 𝑚 ≲ 1 eV and 10 ≲ 𝑚 ≲ 1 eV, respectively. 𝑎 DM DM We can see in Figs. 3 and Fig.4 that in all cases the detection threshold ρ runs through the white area. In particular, it 𝑎★ H. Comparison with other results is parallel but below the upper bound associated with the soliton size limit and above the lower bound associated with Our results for the minimal value ρ of the bulk density ρ 0★ 0 the orbital radius limit. Thus, whereas the largest solitons that can be measured (i.e., its detection threshold) is close to cannot be detected, a large part of the available parameter the results for σ obtained in [83] from collisionless dynamical space could lead to detection by interferometers such as LISA friction, for the B-DECIGO, ET and ADv-LIGO events and and B-DECIGO. The detection prospect is less favorable for for the LISA interferometer in the MBBH and IBBH cases. ET and Adv-LIGO. Indeed, as noticed above, the expression (2.9) for the dynamical friction drag force is quite general and applies to most media, from collisionless particles to gaseous media and scalar-field G. Constraints on the soliton radius dark matter scenarios, up to some multiplicative factor. This is The two parameters 𝑚 and 𝜆 also determine the soliton not surprising, since in the supersonic regime pressure forces DM 4 size 𝑅 , as seen in Eqs.(2.3) and (2.4). As 𝑅 is more relevant and self-interactions are negligible. However, the Coulomb sol sol for observational purposes than the coupling 𝜆 , we show in logarithm and validity criteria depend on the medium and its Figs. 5 and 6 the application domain of our computations and detailed properties. For instance, for collisionless particles the detection threshold ρ in the parameter space (𝑚 , 𝑅 ), with a monochromatic velocity distribution, that is, a Dirac 𝑎★ DM sol instead of the plane (𝑚 ,𝜆 ) shown in Figs. 3 and 4 above. peak at velocity 𝑣 (which plays the role of 𝑐 ) the classical DM 4 𝑐 𝑠 We can see that no experiment can probe galactic-size result for the dynamical friction [1] vanishes if the compact soltons, 𝑅 ≳ 1 kpc, that could be invoked to alleviate object moves at a velocity 𝑣 < 𝑣 . In our case, the dynamical sol 𝑐 the small-scale problems encountered by the standard CDM friction vanishes if 𝑣 < 𝑐 [59]. Because for the B-DECIGO, scenario. At best, LISA with MBBH events could probe ET and ADv-LIGO events and for the LISA interferometer in models up to 𝑅 = 0.1 kpc. Typically, LISA and B-DECIGO the MBBH and IBBH cases, the threshold ρ that we obtain in sol 0★ −6 can probe models associated with 10 ≲ 𝑅 ≲ 10 pc. These Figs. 1 and 2 is within the region where the dynamical friction sol astrophysical scales range from a tenth of astronomical units is efficient for both BHs, we thus recover results similar to [83] to ten times the typical distance between stars in the Milky (where the Coulomb logarithm is taken to be of order unity). 16 FIG. 5: Domain over the parameter space (𝑚 , 𝑅 ) where our derivations are applicable and detection threshold, in the case DM sol −25 3 of the LISA interferometer and assuming a bulk dark matter density ρ = 7× 10 g/cm , as in Fig. 3 For the IMRIs case with the LISA interferometer, we also (ρ , ρ ) plane is given by a single upper right triangle, with 0 𝑎 −17 3 recover the result of [83] with ρ ∼ 10 g/cm , which a clear threshold (ρ , ρ ). This shows that the conditions 0★ 0★ 𝑎★ corresponds to the second small detection area where both for efficient dynamical friction, such as the criteria (3.40) and BHs experience dynamical friction. However, as seen in (3.42), can have a significant impact on the detection prospects. the plot, this detection domain does not extend to arbitrarily larger densities. Instead, it only appears for a narrow range of VI. CONCLUSION densities because at higher bulk density ρ the larger BH is subsonic and has a greatly reduced dynamical friction The detection of GWs has already given important results (zero within our approximation). The main detection domain for fundamental physics, e.g. the near equality between the −9 3 appears at much higher densities, ρ ≳ 10 g/cm , when the speed of GWs and the speed of light [103–105]. In this paper, dynamical friction experienced by the smaller BH becomes we suggest that future experiments could reveal some key large enough to make an impact on the gravitational waveform. properties of dark matter. As an example, we focus on scalar −7 3 For the EMRIs case, we obtain ρ ≃ 10 g/cm whereas 0★ dark matter with quartic self-interactions and assume that the −20 3 [83] find σ ≃ 10 g/cm . This is because in our case the dark matter density of the Universe is due to the misalignment domain where the larger BH experiences dynamical friction mechanism for the scalar field. Locally inside galaxies, these −23 3 is pushed to very low densities, below 10 g/cm , where it models can give rise to dark matter solitons of finite size where can no longer make a significant impact on the gravitational gravity and the repulsive self-interaction pressure balance waveform. We noticed numerically that if we discard the exactly. This regime applies when the size of the solitons cutoffs on the dynamical friction we recover instead results is much larger than the de Broglie wavelength of the scalar similar to [83]. In that case, the detection domain in the particles. In this case, these solitons could be pervasive in each 17 FIG. 6: Domain over the parameter space (𝑚 , 𝑅 ) where our derivations are applicable and detection threshold, in the case DM sol of the B-DECIGO, ET and ADv-LIGO events, as in Fig. 4. galaxy and BHs could naturally be embedded within these dark matter density on galactic scales. Nevertheless, such high scalar clouds when inspiralling towards each other in binary densities could be reached in scenarios where the dark matter systems. The scalar clouds have two effects on the orbits of the clumps are much smaller and more dense than the averaged binary systems. First, this dark matter accretes onto the BHs galactic halos. This corresponds to models where these clumps and slows them down. Second, in the supersonic regime the would form at high redshifts, giving rise to a very clumpy dark dynamical friction due to the gravitational interaction between matter distribution. The fact that we have not detected such the BHs and distant streamlines further slows them down. Both dark matter effects in the ET and LIGO events suggests that effects can lead to significant deviations of the binary orbits either these dark matter clouds are rare (or absent) or that ρ and therefore to perturbations of the GW signal emitted by the is below 0.01 g/cm , see Table IV. pair of BHs. The accretion gives a -4PN effect whereas the Perturbations to the gravitational waveforms may result dynamical friction gives a -5.5PN contribution. As such, they from diverse environments, including gaseous clouds or dark are not degenerate with the relativistic corrections that appear matter halos associated with other dark matter models. In all at higher post-Newtonian orders. cases where such environments are present, we can expect For a large part of the scalar dark matter parameter space, accretion and dynamical friction to occur and slow down the the future experiments such as LISA, B-DECIGO, ET or Adv- orbital motion. It would be interesting to study whether one LIGO should be able to observe the impact on GW of these dark can discriminate between these different environments. As matter environments, provided binary systems are embedded shown in this paper, to do so we could use the magnitude within such scalar clouds. This would give new clues about of these two effects and also the parts in the data sequence the nature of dark matter. Within the framework of the scalar where dynamical friction appears to be active or not. Indeed, field models with quartic self-interactions studied in this paper, depending on the medium dynamical friction is expected to be this would give indications on the value of the bulk dark matter negligible in some regimes, such as subsonic velocities. If one density ρ as well as the characteristic density ρ of Eq.(2.3), can extract such conditions from the data, one may gain some 0 𝑎 that is, the combination 𝑚 /𝜆 . This would also give an useful information on the environment of the binary systems. DM indirect estimate of the size 𝑅 of the solitons, from Eq.(2.4). We leave such studies to future works. sol However, whereas ρ seems within reach of planned GW experiments for a large part of the parameter space of these AKNOWLEDGMENTS dark matter scenarios (provided such clouds exist), the bulk density ρ seems less likely to be measured. Indeed, this would A.B. would like to thank Andrea Maselli for his help in the only be possible for densities much higher than the typical first stage of this project. 18 [1] S. Chandrasekhar, Dynamical Friction. I. General Considera- SPARC galaxies with fuzzy and multistate (ultralight boson) tions: the Coefficient of Dynamical Friction, Astrophys. J. 97, scalar field dark matter, Mon. Not. Roy. Astron. Soc. 475, 1447 255 (1943). (2018), arXiv:1701.00912. [2] V. P. Dokuchaev, Emission of Magnetoacoustic Waves in the [22] P. Mocz, M. Vogelsberger, V. H. Robles, J. Zavala, M. Boylan- Motion of Stars in Cosmic Space., SvA 8, 23 (1964). Kolchin, A. Fialkov, and L. Hernquist, Galaxy formation [3] M. A. Ruderman and E. A. Spiegel, Galactic Wakes, Astrophys. with BECDM – I. Turbulence and relaxation of idealized J. 165, 1 (1971). haloes, Mon. Not. Roy. Astron. Soc. 471, 4559 (2017), [4] Y. Rephaeli and E. E. Salpeter, Flow past a massive object and arXiv:1705.05845. the gravitational drag, ApJ 240, 20 (1980). [23] K. Mukaida, M. Takimoto, and M. Yamada, On Longevity of [5] E. C. Ostriker, Dynamical friction in a gaseous medium, I-ball/Oscillon, JHEP 03 (2017) 122, arXiv:1612.07750. Astrophys. J. 513, 252 (1999), arXiv:astro-ph/9810324. [24] J. Vicens, J. Salvado, and J. Miralda-Escudé, Bosonic dark [6] L. Roszkowski, E. M. Sessolo, and S. Trojanowski, WIMP matter halos: excited states and relaxation in the potential of dark matter candidates and searches—current status and the ground state, (2018), arXiv:1802.10513. future prospects, Rept. Prog. Phys. 81, 066201 (2018), [25] N. Bar, D. Blas, K. Blum, and S. Sibiryakov, Galactic rotation arXiv:1707.06277. curves versus ultralight dark matter: Implications of the [7] G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, soliton-host halo relation, Phys. Rev. D 98, 083027 (2018), M. Pierre, S. Profumo, and F. S. Queiroz, The waning of the arXiv:1805.00122. WIMP? A review of models, searches, and constraints, Eur. [26] J. Eby, K. Mukaida, M. Takimoto, L. C. R. Wĳewardhana, and Phys. J. C 78, 203 (2018), arXiv:1703.07364. M. Yamada, Classical nonrelativistic effective field theory and [8] J. Goodman, Repulsive dark matter, New Astron. 5, 103 (2000), the role of gravitational interactions, Phys. Rev. D 99, 123503 arXiv:astro-ph/0003018. (2019), arXiv:1807.09795. [9] H.-Y. Schive, T. Chiueh, and T. Broadhurst, Cosmic Structure [27] B. Bar-Or, J.-B. Fouvry, and S. Tremaine, Relaxation in as the Quantum Interference of a Coherent Dark Wave, Nature a Fuzzy Dark Matter Halo, Astrophys. J. 871, 28 (2019), Phys. 10, 496 (2014), arXiv:1406.6586. arXiv:1809.07673. [10] H.-Y. Schive, M.-H. Liao, T.-P. Woo, S.-K. Wong, T. Chiueh, [28] D. J. E. Marsh and J. C. Niemeyer, Strong Constraints on Fuzzy T. Broadhurst, and W. Y. P. Hwang, Understanding the Core- Dark Matter from Ultrafaint Dwarf Galaxy Eridanus II, Phys. Halo Relation of Quantum Wave Dark Matter from 3D Simula- Rev. Lett. 123, 051103 (2019), arXiv:1810.08543. tions, Phys. Rev. Lett. 113, 261302 (2014), arXiv:1407.7762. [29] P.-H. Chavanis, Predictive model of BEC dark matter halos [11] A. Arbey, J. Lesgourgues, and P. Salati, Quintessential haloes with a solitonic core and an isothermal atmosphere, Phys. Rev. around galaxies, Phys. Rev. D 64, 123528 (2001), arXiv:astro- D 100, 083022 (2019), arXiv:1810.08948. ph/0105564. [30] R. Emami, T. Broadhurst, G. Smoot, T. Chiueh, and H. N. Luu, [12] P.-H. Chavanis, Mass-radius relation of Newtonian self- Soliton solution for the central dark mass in 47-Tuc globular gravitating Bose-Einstein condensates with short-range inter- cluster and implications for the axiverse, Phys. Rev. D 101, actions: I. Analytical results, Phys. Rev. D 84, 043531 (2011), 063006 (2020), arXiv:1806.04518. arXiv:1103.2050. [31] D. G. Levkov, A. G. Panin, and I. I. Tkachev, Gravitational [13] P. H. Chavanis and L. Delfini, Mass-radius relation of Bose-Einstein condensation in the kinetic regime, Phys. Rev. Newtonian self-gravitating Bose-Einstein condensates with Lett. 121, 151301 (2018), arXiv:1804.05857. short-range interactions: II. Numerical results, Phys. Rev. D [32] T. Broadhurst, I. de Martino, H. N. Luu, G. F. Smoot, 84, 043532 (2011), arXiv:1103.2054. and S. H. H. Tye, Ghostly Galaxies as Solitons of Bose- [14] D. J. E. Marsh and A.-R. Pop, Axion dark matter, solitons and Einstein Dark Matter, Phys. Rev. D 101, 083012 (2020), the cusp–core problem, Mon. Not. Roy. Astron. Soc. 451, 2479 arXiv:1902.10488. (2015), arXiv:1502.03456. [33] K. Hayashi and I. Obata, Non-sphericity of ultralight axion dark [15] E. Calabrese and D. N. Spergel, Ultra-Light Dark Matter in matter haloes in the Galactic dwarf spheroidal galaxies, Mon. Ultra-Faint Dwarf Galaxies, Mon. Not. Roy. Astron. Soc. 460, Not. Roy. Astron. Soc. 491, 615 (2020), arXiv:1902.03054. 4397 (2016), arXiv:1603.07321. [34] N. Bar, K. Blum, J. Eby, and R. Sato, Ultralight dark [16] S.-R. Chen, H.-Y. Schive, and T. Chiueh, Jeans Analysis for matter in disk galaxies, Phys. Rev. D 99, 103020 (2019), Dwarf Spheroidal Galaxies in Wave Dark Matter, Mon. Not. arXiv:1903.03402. Roy. Astron. Soc. 468, 1338 (2017), arXiv:1606.09030. [35] R. G. García, P. Brax, and P. Valageas, Solitons and halos for [17] B. Schwabe, J. C. Niemeyer, and J. F. Engels, Simulations of self-interacting scalar dark matter, (2023), arXiv:2304.10221. solitonic core mergers in ultralight axion dark matter cosmolo- [36] J. Preskill, M. B. Wise, and F. Wilczek, Cosmology of the gies, Phys. Rev. D 94, 043513 (2016), arXiv:1606.05151. Invisible Axion, Phys. Lett. B 120, 127 (1983). [18] J. Veltmaat and J. C. Niemeyer, Cosmological particle-in-cell [37] L. F. Abbott and P. Sikivie, A Cosmological Bound on the simulations with ultralight axion dark matter, Phys. Rev. D 94, Invisible Axion, Phys. Lett. B 120, 133 (1983). 123523 (2016), arXiv:1608.00802. [38] M. Dine and W. Fischler, The Not So Harmless Axion, Phys. [19] A. X. González-Morales, D. J. E. Marsh, J. Peñarrubia, and Lett. B 120, 137 (1983). L. A. Ureña López, Unbiased constraints on ultralight axion [39] A. Arvanitaki, S. Dimopoulos, M. Galanis, L. Lehner, mass from dwarf spheroidal galaxies, Mon. Not. Roy. Astron. J. O. Thompson, and K. Van Tilburg, Large-misalignment Soc. 472, 1346 (2017), arXiv:1609.05856. mechanism for the formation of compact axion structures: [20] V. H. Robles and T. Matos, Flat Central Density Profile and Signatures from the QCD axion to fuzzy dark matter, Phys. Constant DM Surface Density in Galaxies from Scalar Field Rev. D 101, 083014 (2020), arXiv:1909.11665. Dark Matter, Mon. Not. Roy. Astron. Soc. 422, 282 (2012), [40] W. Hu, R. Barkana, and A. Gruzinov, Cold and fuzzy arXiv:1201.3032. dark matter, Phys. Rev. Lett. 85, 1158 (2000), arXiv:astro- [21] T. Bernal, L. M. Fernández-Hernández, T. Matos, and M. A. ph/0003365. Rodríguez-Meza, Rotation curves of high-resolution LSB and [41] M. C. Johnson and M. Kamionkowski, Dynamical and 19 Gravitational Instability of Oscillating-Field Dark Energy and [63] E. Madelung and I. Frankfurt, 322 Quantentheorie in hydrody- Dark Matter, Phys. Rev. D 78, 063010 (2008), arXiv:0805.1748. namischer Form, Ann. d. Phys 79 (1926). [42] J.-c. Hwang and H. Noh, Axion as a Cold Dark Matter candidate, [64] H. Y. Schive, T. Chiueh, and T. Broadhurst, Cosmic structure Phys. Lett. B 680, 1 (2009), arXiv:0902.4738. as the quantum interference of a coherent dark wave, Nature [43] C.-G. Park, J.-c. Hwang, and H. Noh, Axion as a cold dark Physics 2014 10:7 10, 496 (2014), arXiv:1406.6586. matter candidate: low-mass case, Phys. Rev. D 86, 083535 [65] J. F. Navarro, C. S. Frenk, and S. D. M. White, The Structure (2012), arXiv:1207.3124. of cold dark matter halos, Astrophys. J. 462, 563 (1996), [44] R. Hlozek, D. Grin, D. J. E. Marsh, and P. G. Ferreira, A search arXiv:astro-ph/9508025. for ultralight axions using precision cosmological data, Phys. [66] P. Peebles, The large-scale structure of the universe (Princeton Rev. D 91, 103512 (2015), arXiv:1410.2896. University Press, 1980). [45] J. A. R. Cembranos, A. L. Maroto, and S. J. Núñez Jareño, [67] H. Bondi, On spherically symmetrical accretion, Mon. Not. Cosmological perturbations in coherent oscillating scalar field Roy. Astron. Soc. 112, 195 (1952). models, JHEP 03 (2016) 013, arXiv:1509.08819. [68] F. C. Michel, Accretion of matter by condensed objects, [46] L. A. Ureña López and A. X. Gonzalez-Morales, Towards Astrophysics and Space Science 15, 153 (1972). accurate cosmological predictions for rapidly oscillating scalar [69] L. Berezhiani, B. Elder, and J. Khoury, Dynamical Friction in fields as dark matter, JCAP 07 (2016) 048, arXiv:1511.08195. Superfluids, JCAP 10 (2019) 074, arXiv:1905.09297. [47] L. A. Ureña López, Brief Review on Scalar Field Dark Matter [70] E. Poisson and C. M. Will, Gravity: Newtonian, Post- Models, Front. Astron. Space Sci. 6, 47 (2019). Newtonian, Relativistic (Cambridge University Press, 2014). [48] L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Ultralight [71] R. Catena and P. Ullio, A novel determination of the local dark scalars as cosmological dark matter, Phys. Rev. D 95, 043541 matter density, JCAP 08 (2010) 004, arXiv:0907.0018. (2017), arXiv:1610.08297. [72] M. Weber and W. de Boer, Determination of the Local Dark [49] B. Li, T. Rindler-Daller, and P. R. Shapiro, Cosmological Matter Density in our Galaxy, Astron. Astrophys. 509, A25 Constraints on Bose-Einstein-Condensed Scalar Field Dark (2010), arXiv:0910.4272. Matter, Phys. Rev. D 89, 083536 (2014), arXiv:1310.6061. [73] P. Salucci, F. Nesti, G. Gentile, and C. F. Martins, The dark [50] A. Suárez and P.-H. Chavanis, Hydrodynamic representation matter density at the Sun’s location, Astron. Astrophys. 523, of the Klein-Gordon-Einstein equations in the weak field limit: A83 (2010), arXiv:1003.3101. General formalism and perturbations analysis, Phys. Rev. D 92, [74] J. Bovy and S. Tremaine, On the local dark matter density, 023510 (2015), arXiv:1503.07437. Astrophys. J. 756, 89 (2012), arXiv:1205.4033. [51] A. Suárez and P.-H. Chavanis, Cosmological evolution of a [75] M. Pato, F. Iocco, and G. Bertone, Dynamical constraints on complex scalar field with repulsive or attractive self-interaction, the dark matter distribution in the Milky Way, JCAP 12 (2015) Phys. Rev. D 95, 063515 (2017), arXiv:1608.08624. 001, arXiv:1504.06324. [52] A. Suárez and P.-H. Chavanis, Jeans type instability of a [76] P. F. de Salas, K. Malhan, K. Freese, K. Hattori, and M. Valluri, complex self-interacting scalar field in general relativity, Phys. On the estimation of the Local Dark Matter Density using Rev. D 98, 083529 (2018), arXiv:1710.10486. the rotation curve of the Milky Way, JCAP 10 (2019) 037, [53] L. Hui, Unitarity bounds and the cuspy halo problem, Phys. arXiv:1906.06133. Rev. Lett. 86, 3467 (2001), arXiv:astro-ph/0102349. [77] H.-N. Lin and X. Li, The Dark Matter Profiles in the [54] W. J. G. de Blok, The Core-Cusp Problem, Adv. Astron. 2010, Milky Way, Mon. Not. Roy. Astron. Soc. 487, 5679 (2019), 789293 (2010), arXiv:0910.3538. arXiv:1906.08419. [55] D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de [78] M. Cautun, A. Benitez-Llambay, A. J. Deason, C. S. Frenk, Naray, and A. H. G. Peter, Cold dark matter: controversies A. Fattahi, F. A. Gómez, R. J. J. Grand, K. A. Oman, J. F. on small scales, Proc. Nat. Acad. Sci. 112, 12249 (2015), Navarro, and C. M. Simpson, The Milky Way total mass profile arXiv:1306.0913. as inferred from Gaia DR2, Mon. Not. Roy. Astron. Soc. 494, [56] A. Del Popolo and M. Le Delliou, Small scale problems of 4291 (2020), arXiv:1911.04557. the ΛCDM model: a short review, Galaxies 5, 17 (2017), [79] Y. Sofue, Rotation Curve of the Milky Way and the Dark Matter arXiv:1606.07790. Density, Galaxies 8, 37 (2020), arXiv:2004.11688. [57] P. Brax, J. A. R. Cembranos, and P. Valageas, Nonrelativistic [80] E. Barausse, V. Cardoso, and P. Pani, Can environmental effects formation of scalar clumps as a candidate for dark matter, Phys. spoil precision gravitational-wave astrophysics?, Phys. Rev. D Rev. D 102, 083012 (2020), arXiv:2007.04638. 89, 104059 (2014), arXiv:1404.7149. [58] P. Brax, J. A. R. Cembranos, and P. Valageas, Fate of scalar [81] E. Poisson and C. M. Will, Gravitational waves from dark matter solitons around supermassive galactic black holes, inspiraling compact binaries: Parameter estimation using Phys. Rev. D 101, 023521 (2020), arXiv:1909.02614. second postNewtonian wave forms, Phys. Rev. D 52, 848 (1995), [59] A. Boudon, P. Brax, and P. Valageas, Subsonic accretion and arXiv:gr-qc/9502040. dynamical friction for a black hole moving through a self- [82] M. Vallisneri, Use and abuse of the Fisher information interacting scalar dark matter cloud, Phys. Rev. D 106, 043507 matrix in the assessment of gravitational-wave parameter- (2022), arXiv:2204.09401. estimation prospects, Phys. Rev. D 77, 042001 (2008), arXiv:gr- [60] A. Boudon, P. Brax, and P. Valageas, Supersonic accretion qc/0703086. and dynamical friction for a black hole moving through a [83] V. Cardoso and A. Maselli, Constraints on the astrophysical self-interacting scalar dark matter cloud, (to be submitted). environment of binaries with gravitational-wave observations, [61] P. Brax, J. A. R. Cembranos, and P. Valageas, Impact of kinetic Astron. Astrophys. 644, A147 (2020), arXiv:1909.05870. and potential self-interactions on scalar dark matter, Phys. Rev. [84] P. Ajith, M. Hannam, S. Husa, Y. Chen, B. Brügmann, D 100, 023526 (2019), arXiv:1906.00730. N. Dorband, D. Müller, F. Ohme, D. Pollney, C. Reisswig, [62] P.-H. Chavanis, Growth of perturbations in an expanding L. Santamaría, and J. Seiler, Inspiral-merger-ringdown wave- universe with Bose-Einstein condensate dark matter, Astron. forms for black-hole binaries with nonprecessing spins, Phys. Astrophys. 537, A127 (2012), arXiv:1103.2698. Rev. Lett. 106, 241101 (2011). 20 [85] J. Aasi et al. (LIGO Scientific), Advanced LIGO, Class. Quant. [97] D. Traykova, K. Clough, T. Helfer, E. Berti, P. G. Ferreira, Grav. 32, 074001 (2015), arXiv:1411.4547. and L. Hui, Dynamical friction from scalar dark matter in [86] M. Punturo et al., The Einstein Telescope: A third-generation the relativistic regime, Phys. Rev. D 104, 103014 (2021), gravitational wave observatory, Class. Quant. Grav. 27, 194002 arXiv:2106.08280. (2010). [98] J. F. Navarro, C. S. Frenk, and S. D. M. White, A Universal [87] P. Amaro-Seoane et al. (LISA), Laser Interferometer Space density profile from hierarchical clustering, Astrophys. J. 490, Antenna, (2017), arXiv:1702.00786. 493 (1997), arXiv:astro-ph/9611107. [88] S. Kawamura et al., Current status of space gravitational [99] T. P. K. Martinsson, M. A. W. Verheĳen, K. B. Westfall, wave antenna DECIGO and B-DECIGO, PTEP 2021, 05A105 M. A. Bershady, D. R. Andersen, and R. A. Swaters, The (2021), arXiv:2006.13545. DiskMass Survey. VII. The distribution of luminous and dark [89] L. Barsotti, S. Gras, M. Evans, and P. Fritschel, The updated matter in spiral galaxies, Astron. Astrophys. 557, A131 (2013), Advanced LIGO design curve, (2018). arXiv:1308.0336. [90] S. Hild et al., Sensitivity Studies for Third-Generation [100] P. Salucci, The distribution of dark matter in galaxies, Astron. Gravitational Wave Observatories, Class. Quant. Grav. 28, Astrophys. Rev. 27, 2 (2019), arXiv:1811.08843. 094013 (2011), arXiv:1012.0908. [101] V. S. Berezinsky, V. I. Dokuchaev, and Y. N. Eroshenko, [91] K. G. Arun et al. (LISA), New horizons for fundamental physics Small-scale clumps of dark matter, Phys. Usp. 57, 1 (2014), with LISA, Living Rev. Rel. 25, 4 (2022), arXiv:2205.01597. arXiv:1405.2204. [92] S. Isoyama, H. Nakano, and T. Nakamura, Multiband [102] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and Gravitational-Wave Astronomy: Observing binary inspirals M. Bradac, Constraints on the Self-Interaction Cross-Section with a decihertz detector, B-DECIGO, PTEP 2018, 073E01 of Dark Matter from Numerical Simulations of the Merging (2018), arXiv:1802.06977. Galaxy Cluster 1E 0657-56, Astrophys. J. 679, 1173 (2008), [93] E. Berti, A. Buonanno, and C. M. Will, Estimating spinning arXiv:0704.0261. binary parameters and testing alternative theories of gravity [103] B. P. Abbott et al. (LIGO Scientific, VIRGO), GW170104: Ob- with lisa, Phys. Rev. D 71, 084025 (2005). servation of a 50-Solar-Mass Binary Black Hole Coalescence [94] B. P. Abbott et al. (LIGO Scientific, Virgo), GWTC-1: A at Redshift 0.2, Phys. Rev. Lett. 118, 221101 (2017), [Erratum: Gravitational-Wave Transient Catalog of Compact Binary Phys.Rev.Lett. 121, 129901 (2018)], arXiv:1706.01812. Mergers Observed by LIGO and Virgo during the First and [104] B. P. Abbott et al. (LIGO Scientific, Virgo), Tests of General Second Observing Runs, Phys. Rev. X 9, 031040 (2019), Relativity with the Binary Black Hole Signals from the LIGO- arXiv:1811.12907. Virgo Catalog GWTC-1, Phys. Rev. D 100, 104036 (2019), [95] D. Syer, Relativistic dynamical friction in the weak scattering arXiv:1903.04467. limit, Mon. Not. Roy. Astron. Soc. 270, 205 (1994), arXiv:astro- [105] X. Liu, V. F. He, T. M. Mikulski, D. Palenova, C. E. Williams, ph/9404063. J. Creighton, and J. D. Tasson, Measuring the speed of [96] E. Barausse, Relativistic dynamical friction in a collisional fluid, gravitational waves from the first and second observing run Mon. Not. Roy. Astron. Soc. 382, 826 (2007), arXiv:0709.0211. of Advanced LIGO and Advanced Virgo, Phys. Rev. D 102, 024028 (2020), arXiv:2005.03121.

http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

High Energy Physics - Theory arXiv (Cornell University)

http://www.deepdyve.com/lp/arxiv-cornell-university/gravitational-waves-from-binary-black-holes-in-a-self-interacting-kmK1HX0aM3

Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

Loading next page...

References (0)

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

eISSN: ARCH-3336
DOI: 10.48550/arxiv.2305.18540
Publisher site: See Article on Publisher Site

Abstract

Journal

High Energy Physics - Theory – arXiv (Cornell University)

Published: May 29, 2023

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

Gravitational waves from binary black holes in a self-interacting scalar dark matter cloud

References (0)

Abstract

Journal

Recommended Articles

There are no references for this article.

Our policy towards the use of cookies