Gravitational waves from bodies orbiting the Galactic Center black hole and their detectability by LISA

We present the first fully relativistic study of gravitational radiation from bodies in circular equatorial orbits around the massive black hole at the Galactic Center, Sgr A* and we assess the detectability of various kinds of objects by the gravitational wave detector LISA. Our computations are based on the theory of perturbations of the Kerr spacetime and take into account the Roche limit induced by tidal forces in the Kerr metric. The signal-to-noise ratio in the LISA detector, as well as the time spent in LISA band, are evaluated. We have implemented all the computational tools in an open-source SageMath package, within the Black Hole Perturbation Toolkit framework. We find that white dwarfs, neutrons stars, stellar black holes, primordial black holes of mass larger than $10^{-4} M_\odot$, main-sequence stars of mass lower than $\sim 2.5\, M_\odot$ and brown dwarfs orbiting Sgr A* are all detectable in one year of LISA data with a signal-to-noise ratio above 10 during at least $10^5$ years in the slow inspiral towards either the innermost stable circular orbit (compact objects) or the Roche limit (main-sequence stars and brown dwarfs). The longest times in-band, of the order of $10^6$ years, are achieved for primordial black holes of mass $\sim 10^{-3} M_\odot$ down to $10^{-5} M_\odot$, depending on the spin of Sgr A*, as well as for brown dwarfs, just followed by white dwarfs and low mass main-sequence stars. The long time in-band of these objects makes Sgr A* a valuable target for LISA. We also consider bodies on close circular orbits around the massive black hole in the nucleus of the nearby galaxy M32 and find that, among them, compact objects and brown dwarfs stay for $10^3$ to $10^4$ years in LISA band with a 1-year signal-to-noise ratio above 10.


Introduction
The future space-based Laser Interferometer Space Antenna (LISA) (Amaro-Seoane et al. 2017), selected as the L3 mission of ESA, shall detect gravitational radiation from various phenomena involving massive black holes (MBHs), the masses of which range from 10 5 to 10 7 M (see e.g. Amaro-Seoane 2018; Babak et al. 2017, and references therein). It turns out that the mass of the MBH Sgr A* at the center of our galaxy lies within this range (GRAVITY Collaboration et al. 2018a,b): M Sgr A * = 4.10 ± 0.03 × 10 6 M . (1) More precisely, the angular velocity ω 0 on a circular, equatorial orbit at the Boyer-Lindquist radial coordinate r 0 around a Kerr black hole (BH) is given by the formula (Bardeen et al. 1972) where G is the gravitational constant, c the speed of light, M the BH mass and a = J/(cM) its reduced spin, J being the magnitude of the BH angular momentum (a has the dimension of a length). The motion of a particle of mass µ M on a circular orbit generates some gravitational radiation with a periodic pattern, the dominant mode of which is m = 2 and has the frequency f m=2 = 2 f 0 , where f 0 ≡ ω 0 /(2π) is the orbital frequency (details are given in Sect. 2). Combining with Eq. (2), we get f m=2 = 1 π (GM) 1/2 r 3/2 0 + a(GM) 1/2 /c . This frequency is maximal at the (prograde) innermost stable circular orbit (ISCO), which is located at r 0 = 6GM/c 2 for a = 0 (Schwarzschild BH) and at r 0 = GM/c 2 for a = a max ≡ GM/c 2 (extreme Kerr BH). Equation (3) By some nice coincidence, f ISCO,a max m=2 Kennefick 2002) and tidal effects are evaluated via the theory of Roche potential in the Kerr metric developed by . Moreover, from the obtained waveforms, we carefully evaluate the signal-to-noise ratio in the LISA detector, taking into account the latest LISA sensitivity curve (Robson et al. 2018). There is another MBH with a mass within the LISA range in the Local Group of galaxies: the 2.5 × 10 6 M MBH in the center of the galaxy M32 (Nguyen et al. 2018). By applying the same techniques, we study the detectability by LISA of bodies in close circular orbit around it. The plan of the article is as follows. The method employed to compute the gravitational radiation from a point mass in circular orbit around a Kerr BH is presented in Sect. 2, the open-source code implementing it being described in Appendix A. The computation of the signal-to-noise ratio of the obtained waveforms in the LISA detector is performed in Sect. 3, from which we can estimate the minimal detectable mass of the orbiting source in terms of the orbital radius. Section 4 investigates the secular evolution of a circular orbit under the reaction to gravitational radiation and provides the frequency change per year and the inspiral time between two orbits. The potential astrophysical sources are discussed in Sect. 5, taking into account Roche limits for non-compact objects and estimating the total time spent in LISA band. The case of M32 is treated in Appendix C. Finally, the main conclusions are drawn in Sect. 6.

Gravitational waves from an orbiting point mass
In this section and the remainder of this article, we use geometrized units, for which G = 1 and c = 1. In addition we systematically use Boyer-Lindquist coordinates (t, r, θ, ϕ) to describe the Kerr geometry of a rotating BH of mass M and spin parameter a, with 0 a < M. We consider a particle of mass µ M on a (stable) prograde circular equatorial orbit of constant coordinate r = r 0 . Hereafter, we call r 0 the orbital radius. The orbital angular velocity ω 0 is given by formula (2). In practice this "particle" can be any object whose extension is negligible with respect to the orbital radius. In particular, for Sgr A*, it can be an object as large as a solar-type star. Indeed, Sgr A* mass (1) corresponds to a length scale M = 6.05 × 10 6 km ∼ 9R , where R is the Sun's radius. Moreover, main-sequence stars are centrally condensed objects, so that their "effective" size as gravitational wave generator is smaller that their actual radius. In addition, as we shall see in Sect. 5.1, their orbital radius must obey r 0 > 34M to avoid tidal disruption (Roche limit), so that R /r 0 < 3 × 10 −3 . Hence, regarding Sgr A*, we may safely describe orbiting stars as point particles.

Gravitational waveform
The gravitational waves generated by the orbital motion of the particle are conveniently encoded in the linear combination h + − ih × of the two polarization states h + and h × . A standard result from the theory of linear perturbations of the Kerr BH (Teukolsky 1973;Detweiler 1978;Shibata 1994;Kennefick 1998;Hughes 2000;Finn & Thorne 2000;Glampedakis & Kennefick 2002) yields the asymptotic waveform as Z ∞ m (r 0 ) (mω 0 ) 2 −2 S amω 0 m (θ, ϕ) e −im(ω 0 (t−r * )+ϕ 0 ) , where (h + , h × ) are evaluated at the spacetime event of Boyer-Lindquist coordinates (t, r, θ, ϕ) and r * is the so-called "tortoise coordinate", defined as where r ± ≡ M ± √ M 2 − a 2 denote the coordinate locations of the outer (+) and inner (−) event horizons. The phase ϕ 0 in Eq. (6) can always be absorbed into a shift of the origin of t. The spin-weighted spheroidal harmonics −2 S amω 0 m (θ, ϕ) encode the dependency of the waveform with respect to the polar angles (θ, ϕ) of the observer. For each harmonic ( , m), they depend on the (dimensionless) product aω 0 of the Kerr spin parameter and the orbital angular velocity, and they reduce to the more familiar spin-weighted spherical harmonics −2 Y m (θ, ϕ) when a = 0. The coefficients Z ∞ m (r 0 ) encode the amplitude and phase of each mode. They depend on M and a and are computed by solving the radial component of the Teukolsky equation (Teukolsky 1973); they satisfy Z ∞ ,−m = (−1) Z ∞ * m , where the star denotes the complex conjugation.
Given the distance r = 8.12 ± 0.03 kpc to Sgr A* (GRAVITY Collaboration et al. 2018a), the prefactor µ/r in formula (6) takes the following numerical value:

Mode amplitudes
The factor |Z ∞ m (r 0 )|/(mω 0 ) 2 sets the amplitude of the mode ( , m) of h + and h × according to Eq. (6). The complex amplitudes, Z ∞ m , are computed by solving the Teukolsky equation (Teukolsky 1973), where, typically, the secondary is modeled as a structureless point mass. Generally, the Teukolsky equation is solved in either the time or frequency domain. Time domain calculations are computationally expensive but well suited to modeling a source moving along an arbitrary trajectory. Frequency domain calculations have the advantage that the Teukolsky equation is completely separable in this domain and this reduces the problem from solving partial to ordinary differential equations. This leads to very efficient calculations so long as the Fourier spectrum of the source is Article number, page 3 of 29 sufficiently narrow. Over short timescales 1 the trajectory of a small body with µ M orbiting a MBH is well approximated by a bound geodesic of the background spacetime. Motion along a bound geodesic is periodic (or bi-periodic (Schmidt 2002)) and so the spectrum of the source is discrete. This allows the Teukolsky equation to be solved efficiently in the frequency domain, at least for orbits with up to a moderate eccentricity (for large eccentricities the Fourier spectrum broadens to a point where time domain calculations can be more efficient (Barton et al. 2008)). Frequency domain calculations have been carried out for circular (Detweiler 1978), spherical (Hughes 2000), eccentric equatorial (Glampedakis & Kennefick 2002) and generic orbits (Drasco & Hughes 2006;Fujita et al. 2009;van de Meent 2018) and we follow this approach in this work.
In the frequency domain the Teukolsky equation separates into spin-weighted spheroidal harmonics and frequency modes. The former can be computed via eigenvalue (Hughes 2000) or continuous fraction methods (Leaver 1985). The main task is then finding solutions to the Teukolsky radial equation. Typically, this is a two step process whereby one first finds the homogeneous solutions and then computes the inhomogeneous solutions via the method of variation of parameters. Finding the homogeneous solutions is usually done by either numerical integration or via an expansion of the solution in a series of special functions (Sasaki & Tagoshi 2003). In this work we make use of both methods as a cross check. Direct numerical integration of the Teukolsky equation is numerically unstable but this can be overcome by transforming the equation to a different form (Sasaki & Nakamura 1982a,b). Our implementation is based off the code developed for Gralla et al. (2015). For the series method our code is based off of codes used in Kavanagh et al. (2016); Buss & Casals (2018). Both of these codes, as well as code to compute spin-weighted spheroidal harmonics, are now publicly available as part of the Black Hole Perturbation Toolkit 2 .
The final step is to compute the inhomogeneous radial solutions. In this work we consider circular, equatorial orbits. With a point particle source, this reduces the application of variation of parameters to junction conditions at the particle's radius (Detweiler 1978). The asymptotic complex amplitudes, Z ∞ m , can then be computed by evaluating the radial solution in the limit r → ∞. The mode amplitudes are plotted in Fig. 2 as functions of the orbital radius r 0 for 2 5, 1 m and some selected values of the MBH spin parameter a. Each curve starts at the value of r 0 corresponding to the prograde ISCO for the considered a.

Waveform for distant orbits (r 0 M)
When the orbital radius obeys r 0 M, we see from Fig. 2 that the modes ( , m) = (2, ±2) dominate the waveform (cf. the solid red curves in the four panels of Fig. 2). Moreover, for r 0 M, the effects of the MBH spin become negligible. This is also apparent on Fig. 2: the value of |Z ∞ m (r 0 )|/(mω 0 ) 2 for ( , m) = (2, 2) and r 0 = 50M appears to be independent of a, being equal to roughly 7 × 10 −2 in all the four panels. The value of Z ∞ 2,±2 (r 0 ) at the lowest order in M/r 0 is given by e.g. Eq. (5.6) of Poisson (1993a), and reads 3 The dependency with respect to a would appear only at the relative order (M/r 0 ) 3/2 (see Eq. (24) of Poisson (1993b)) and can safely be ignored, as already guessed from Fig. 2. Besides, for r 0 M, Eq.
(2) reduces to the standard Newtonian expression: Combining with Eq. (9), we see that the amplitude factor in the waveform (6) is Besides, when r 0 M, Eq.
As expected for r 0 M, we recognize the waveform obtained from the standard quadrupole formula applied to a point mass µ on a Newtonian circular orbit around a mass M µ (compare with e.g. Eqs. (3.13)-(3.14) of Blanchet (2001)).

Fourier series expansion
Observed at a fixed location (r, θ, ϕ), the waveform (h + , h × ) as given by Eq. (6) is a periodic function of t, or equivalently of the retarded time u ≡ t − r * , the period being nothing but the orbital period of the particle: T 0 = 2π/ω 0 . It can therefore be expanded in Fourier series. Noticing that the ϕ-dependency of the spheroidal harmonic −2 S amω 0 m (θ, ϕ) is simply e imϕ , we may rewrite Eq. (6) as . Waveform (left column) and Fourier spectrum (right column) of gravitational radiation from a point mass orbiting on the ISCO of a Schwarzschild BH (a = 0). All amplitudes are rescaled by r/µ, where r is the Boyer-Lindquist radial coordinate of the observer and µ the mass of the orbiting point. Three values of the colatitude θ of the observer are considered: θ = 0 (first row), θ = π/4 (second row) and θ = π/2 (third row).
an explicit Fourier series expansion 4 : where ψ is given by Eq. (14) and A + m (θ), A × m (θ), B + m (θ) and B × m (θ) are real-valued functions of θ, involving M, a and r 0 : We then define the spectrum of the gravitational wave at a fixed value of θ as the two series (one per polarization mode): We have developed an open-source SageMath package, kerrgeodesic_gw (cf. Appendix A), implementing the above formulas, and more generally all the computations presented in this article, like the signal-to-noise ratio and Roche limit ones to be discussed below. The spectrum, as well as the corresponding waveform, computed via kerrgeodesic_gw, are depicted in Figs. 3, 4 and 5 for a = 0, a = 0.90M and a = 0.98M respectively. In each figure, ϕ = ϕ 0 and three values of θ are selected: θ = 0 (orbit seen face-on), π/4 and π/2 (orbit seen edge-on).
We notice that for θ = 0, only the Fourier mode m = 2 is present and that h + and h × have identical amplitudes and are in quadrature. This behavior is identical to that given by the large radius (quadrupole-formula) approximation (15). For θ > 0, all modes with m 1 are populated, whereas the approximation (15) contains only m = 2. For θ = π/2, h × vanishes identically and the relative amplitude of the modes m 2 with respect to the mode m = 2 is the largest one, reaching ∼ 75% for m = 3 and ∼ 50% for m = 4 when a = 0.98M.
As a test of our computations, we notice the similarity of the h + waveform at θ = π/2 for a = 0.90M (lower left panel of Fig. 4) with Fig. 9 of Detweiler (1978)'s seminal work, taking into account that in Detweiler (1978) (i) the x-axis is (r * − t)/M, i.e. the opposite of ours, and (ii) the orbit is located at r 0 = 2.4M, i.e. slightly outside the ISCO, while we are considering an orbit exactly at the ISCO (r 0 = 2.321M). Other tests have been conducted by comparing with the 1.5PN waveforms obtained by Poisson (1993a) for a = 0, see Appendix A.

Signal-to-noise ratio in the LISA detector
The results in Sect. 2 are valid for any BH. We now specialize them to Sgr A* and evaluate the signal-to-noise ratio in the LISA detector, as a function of the mass µ of the orbiting object, the orbital radius r 0 and the spin parameter a of Sgr A*.

Computation
Assuming that its noise is stationary and Gaussian, a given detector is characterized by its one-sided noise power spectral density (PSD) S n ( f ). For a gravitational wave search based on the matched filtering technique, the signal-to-noise ratio (SNR) ρ is given by the following formula (see e.g. Jaranowski & Królak 2012;Moore et al. 2015): whereh( f ) is the Fourier transform of the imprint h(t) of the gravitational wave on the detector, h(t) being a linear combination of the two polarization modes h + and h × at the detector location: In the above expression, (t, r, θ, ϕ) are the Boyer-Lindquist coordinates of the detector ("Sgr A* frame"), while F + and F × are the detector beam-pattern coefficients (or response functions), which depend on the direction (Θ, Φ) of the source with respect to the detector's frame and on the polarization angle Ψ, the latter being the angle between the direction of constant azimuth Φ and the principal direction "+" in the wavefront plane (i.e. the axis of the h + mode or equivalently the direction of the semi-major axis of the orbit viewed as an ellipse in the detector's sky) (Apostolatos et al. 1994). For a detector like LISA, where, for high enough frequencies, the gravitational wavelength can be comparable or smaller than the arm length (2.5 Gm), the response functions F + and F × depend a priori on the gravitational wave frequency f , in addition to (Θ, Φ, Ψ) (Robson et al. 2018). However for the gravitational waves considered here, a reasonable upper bound of the frequency is that of the harmonic m = 4 (say) of waves from the prograde ISCO of an extreme Kerr BH (see Fig. 5). From the value given by Eq. (5), this is f max = 2 × 7.9 15.8 mHz, the multiplication by 2 taking into account the transition from m = 2 to m = 4. This value being lower than LISA's transfer frequency f * = 19.1 mHz (Robson et al. 2018), we may consider that F + and F × do not depend on f (see Fig. 2 in Robson et al. (2018)). They are given in terms of (Θ, Φ, Ψ) by Eq. (3.12) of Cutler (1998) (with the prefactor √ 3/2 appearing in Eq. (3.11) included in them). Generally, the function S n ( f ) considered in the LISA literature, and in particular to present the LISA sensitivity curve, is not the true noise PSD of the instrument, P n ( f ) say, but rather P n ( f )/R( f ), where R( f ) is the average over the sky (angles (Θ, Φ)) and over the polarization (angle Ψ) of the square of the response functions F + and F × , so that Eq. (19) yields directly the sky and polarization average SNR by substituting |h Robson et al. (2018) for details). With Sgr A* as a target, the direction angles (Θ, Φ) are of course known and, for a short observation time (1 day say), they are approximately constant. However, on longer observation times, theses angles varies due to the motion of LISA spacecrafts on their orbits around the Sun. Moreover, the polarization angle Ψ is not known at all, since it depends on the orientation of the orbital plane around the MBH, which is assumed to be the equatorial plane, the latter being currently unknown. For these reasons, we consider the standard sky and polarization average sensitivity of LISA, S n ( f ) = P n ( f )/R( f ), as given e.g. by Eq. (13) of Robson et al. (2018), and define the effective signal-to-noise ratio ρ by whereh + ( f ) andh × ( f ) are the Fourier transforms of the two gravitational wave signals h + (t) and h × (t), as given by Eq. (6) or Eq. (16), over some observation time T :  Fig. 6, except for r 0 ranging up to 50M. Note that for r 0 > 15M, only the a = 0 curves are plotted, since the MBH spin plays a negligible role at large distance.
The effective SNR resulting from Eq. (24) is shown in Figs. 6-7. We use the value (8) for µ/r and the analytic model of Robson et al. (2018) (their Eq. (13)) for LISA sky and polarization average sensitivity S n ( f ). We notice that for a given value of the orbital radius r 0 and a given MBH spin a, the SNR is maximum for the inclination angle θ = 0 and minimal for θ = π/2, the ratio between the two values varying from ∼ 2 for a = 0 to ∼ 3 for a = 0.98M. This behavior was already present in the waveform amplitudes displayed in Figs. 3-5.
Roche limit for 1 M -star/Jupiter Roche limit for rocky body Roche limit for 0.2 M -star Roche limit for brown dwarf Another feature apparent from Figs. 6 and 7 is that at fixed orbital radius r 0 , the SNR is a decaying function of a. This results from the fact that the orbital frequency f 0 is a decaying function of a [cf. Eq. (2)], which both reduces the gravitational wave amplitude and displaces the wave frequency to less favorable parts of LISA's sensitivity curve.
At the ISCO, the SNR for θ = 0 is with the coefficient α given in Table 1. Note that if the observation time is 1 year, then the factor (T/1d) 1/2 is √ 365.25 19.1.

Minimal detectable mass
As clear from Eq. (24), the SNR ρ is proportional to the mass µ of the orbiting body and to the square root of the observing time T . It is then easy to evaluate the minimal mass µ min that can be detected by analyzing 1 year of LISA data, setting the detection threshold to where SNR 1 yr stands for the value of ρ for T = 1 yr. The result is shown in Fig. 8. If one does not take into account any Roche limit, it is worth noticing that the minimal detectable mass is quite small: µ min 3 × 10 −5 M at the ISCO of a Schwarzschild BH (a = 0), down to µ min 2 × 10 −6 M (the Earth mass) at the ISCO of a rapidly rotating Kerr BH (a 0.90M).

Radiated energy and orbital decay
In the above sections, we have assumed that the orbits are exactly circular, i.e. we have neglected the reaction to gravitational radiation. We now take it into account and discuss the resulting secular evolution of the orbits.

Total radiated power
The total power (luminosity) emitted via gravitational radiation is given by (Detweiler 1978): where S r is the sphere of constant value of r and an overdot stands for the partial derivative with respect to the time coordinate t, i.e.ḣ +,× ≡ ∂h +,× /∂t. Substituting the waveform (6) into this expression leads to Thanks to the orthonormality property of the spin-weighted spheroidal harmonics, the above expression simplifies to Note thatL is a dimensionless function of x ≡ r 0 /M, the dimension of Z ∞ m being an inverse squared length (see e.g. Eq. (9)) and ω 0 being the function of r 0 /M given by Eq. (2). Moreover, the functionL(x) depends only on the parameter a/M of the MBH.
As a check of Eq. (30), let us consider the limit of large orbital radius: r 0 M. As discussed in Sect. 2.3, only the terms ( , m) = (2, ±2) are pertinent in this case, with Z ∞ 2,±2 (r 0 ) given by Eq. (9) and ω 0 related to r 0 by Eq. (10). Equation (30) reduces then to We recognize the standard result from the quadrupole formula at Newtonian order (Landau & Lifshitz 1971) (see also the lowest order of formula (314) in the review by Blanchet (2014)). The total emitted power L (actually the functionL(r 0 /M)) is depicted in Figs. 9-10. Note that for large values of r 0 all curves converge towards the curve of the quadrupole formula (31) (dotted curve), as they should. Another test of our computations, at low radii this time, is provided by the comparison with Figs. 6 and 7 of Detweiler (1978)'s study. At the naked eye, the agreement is quite good, in particular for the values of L at the ISCO's.

Secular evolution of the orbit
For a particle moving along any geodesic in Kerr spacetime, in particular along a circular orbit, the conserved energy is E ≡ −p a ξ a , where p a is the particle's 4-momentum 1-form and ξ a the Killing vector associated with the pseudo-stationarity of Kerr spacetime (ξ = ∂/∂t in Boyer-Lindquist coordinates). Far from the MBH, E coincides with the particle's energy as an inertial observer at rest with respect to the MBH would measure. For a circular orbit of radius r 0 in the equatorial plane of a Kerr BH of mass M and spin parameter a, the expression of E is (Bardeen et al. 1972) where µ ≡ (−p a p a ) 1/2 is the particle's rest mass. Due to the reaction to gravitational radiation, the particle's worldline is actually not a true timelike geodesic of Kerr spacetime, but is slowly inspiralling towards the central MBH. In particular, E is not truly constant. Its secular evolution is governed by the balance law ( whereĖ ≡ dE/dt, L is the gravitational wave luminosity evaluated in Sect. 4.1 and L H is the power radiated down to the event horizon of the MBH. It turns out that in practice, L H is quite small compared to L. From Table VII of Finn & Thorne (2000), we notice that for a = 0, one has always |L H /Ė| < 4 × 10 −3 and for a = 0.99M, one has |L H /Ė| < 9.5 × 10 −2 , with |L H /Ė| < 10 −2 as soon as r 0 > 7.3M. In the following, we will neglect the term L H in our numerical evaluations ofĖ. From Eq. (32), we havė E = µM 2r 2 0 1 − 6M/r 0 + 8aM 1/2 /r 3/2 0 − 3a 2 /r 2 0 1 − 3M/r 0 + 2aM 1/2 /r 3/2 0 3/2ṙ 0 .
In view of Eq.
(2), the secular evolution of the orbital frequency By combining successively Eqs. (35), (34), (33) and (30), we geṫ where we have introduced the rescaled horizon flux functionL H , such that This relative change in orbital frequency is depicted in Figs. 11-12, with a y-axis scaled to the mass (1) of Sgr A* for M and to µ = 1 M . Note thatḟ 0 diverges at the ISCO. This is due to the fact that E is minimal at the ISCO, so that dE/dr 0 = 0 there. At this point, a loss of energy cannot be compensated by a slight decrease of the orbit.
Another representation of the orbital frequency evolution, via the adiabaticity parameter ε ≡ḟ 0 / f 2 0 , is shown in Fig. 13. The adiabaticity parameter ε is a dimensionless quantity, the smallness of which guarantees the validity of approximating the inspiral trajectory by a succession of circular orbits of slowly shrinking radii. As we can see on Fig. 13, ε < 10 −4 except very near the ISCO, whereḟ 0 diverges.

Inspiral time
By combining Eqs. (34), (33), (37) and (30), we get an expression forṙ −1 0 = dt/dr 0 as a function of r 0 . Once integrated, this leads to the time required for the orbit to shrink from an initial radius r 0 to a given radius r 1 < r 0 : whereā ≡ a/M = J/M 2 is the dimensionless Kerr parameter. We shall call T ins (r 0 , r 1 ) the inspiral time from r 0 to r 1 . For an object whose evolution is only driven by the reaction to gravitation radiation (e.g. a compact object, cf. Sect. 5.3), we define then the life time from the orbit r 0 as T life (r 0 ) ≡ T ins (r 0 , r ISCO ).
Indeed, once the ISCO is reached, the plunge into the MBH is very fast, so that T life (r 0 ) is very close to the actual life time outside the MBH, starting from the orbit of radius r 0 . The life time is depicted in Figs. 14-15, which are drawn for M = M Sgr A * . The dotted curve corresponds to the value obtained for Newtonian orbits and the quadrupole formula (31)

Potential sources
Having established the signal properties and detectability by LISA, let us now discuss astrophysical candidates for the orbiting object. A preliminary required for the discussion is the evaluation of the tidal effects exerted by Sgr A* on the orbiting body, since this can make the innermost orbit to be significantly larger than the ISCO. We thus start by investigating the tidal limits in Sect. 5.1. Then, in Sect. 5.2, we review the scenarios which might lead to the presence of stellar objects in circular orbits close to Sgr A*. The various categories of sources are then discussed in the remaining subsections: compact objects (Sect. 5.3), main-sequence stars (5.4), brown dwarfs (Sect. 5.5), accretion flow (5.6), dark matter (Sect. 5.7) and artificial sources (Sect. 5.8).

Tidal radius and Roche radius
In Sects. 2-4, we have considered an idealized point mass. When the orbiting object has some extension, a natural question is whether the object integrity can be maintained in presence of the tidal forces exerted by the central MBH. This leads to the concept  of tidal radius r T , defined as the minimal orbital radius for which the tidal forces cannot disrupt the orbiting body. In other words, the considered object cannot move on an orbit with r 0 < r T . The tidal radius is given by the formula where M is the mass of the MBH, ρ the mean density of the orbiting object and α is a coefficient of order 1, the value of which depends on the object internal structure and rotational state. From the naive argument of equating the self-gravity and the tidal force at the surface of a spherical Newtonian body, one gets α = (3/(2π)) 1/3 = 0.78. If one further assumes that the object is corotating, i.e. is in synchronous rotation with respect to the orbital motion, then one gets α = (9/(4π)) 1/3 = 0.89. Hills (1975) uses α = (6/π) 1/3 = 1.24, while Rees (1988) uses α = (3/(4π)) 1/3 = 0.62. For a Newtonian incompressible fluid ellipsoid in synchronous rotation, α = 1.51 (Chandrasekhar 1969). This result has been generalized by Fishbone (1973) to incompressible fluid ellipsoids in the Kerr metric: α increases then from 1.51 for r M to 1.60 (resp. 1.56) for r = 10M and a = 0 (resp. a = 0.99M) (cf. Fig. 5 of Table 2. Roche radius r R for different types of objects orbiting Sgr A*. The first three lines give the mass µ, the mean radius R and the mean mass density ρ, all in solar units. χ = 0 stands for an irrotational body and χ = 1 for a corotating one. See the text for the chosen characteristics of the red dwarf and the brown dwarf.  Fishbone (1973), which displays 1/(πα 3 )). Taking into account the compressibility decreases α: α = 1.34 for a polytrope of index n = 1.5 (Ishii et al. 2005). For a stellar type object on a circular orbit, a more relevant quantity is the Roche radius, which marks the onset of tidal stripping near the surface of the star, leading to some steady accretion to the MBH (Roche lobe overflow) without the total disruption of the star . For centrally condensed bodies, like main-sequence stars, the Roche radius is given by the condition that the stellar material fills the Roche lobe. In the Kerr metric, the volume V R of the Roche lobe generated by a mass µ on a circular orbit of radius r 0 has been evaluated by , yielding to the approximate formula 5 V R µM 2 V R , with where r ISCO is the radius of the prograde ISCO, χ ≡ Ω/ω 0 is the ratio between the angular velocity Ω of the star (assumed to be a rigid rotator) with respect to some inertial frame to the orbital angular velocity ω 0 and F(a, χ) is the function defined by Note that χ = 1 for a corotating star. The Roche limit is reached when the actual volume of the star equals the volume of the Roche lobe. If ρ stands for the mean mass density of the star, this corresponds to the condition µ = ρV R , or equivalently Solving this equation for r 0 leads to the orbital radius r R at the Roche limit, i.e. the Roche radius. Note that the mass µ has disappeared from Eq. (44), so that r R depends only on the mean density ρ and the rotational parameter χ. For r 0 r ISCO , we can neglect the second term in the square brackets in Eq. (42) and obtain an explicit expression: r R 1.14 1 + χ 2.78 Note that Eq. (45) has the same shape as the tidal radius formula (41). Using Sgr A* value (1) for M, we may rewrite the above formula as where ρ ≡ 1.41 × 10 3 kg · m −3 is the mean density of the Sun. The numerical resolution of Eq. (44) for r R has been implemented in the kerrgeodesic_gw package (cf. Appendix A) and the results are shown in Fig. 16 and Table 2. The straight line behavior in the left part of Fig. 16 corresponds to the power law r R ∝ ρ −1/3 in the asymptotic formula (46). In Table 2, the characteristics of the red dwarf star are taken from Fig. 1 of Chabrier et al. (2007) -it corresponds to a main-sequence star of spectral type M4V. The brown dwarf model of Table 2 is the model of minimal radius along the 5 Gyr isochrone in Fig. 1 of Chabrier et al. (2009). This brown dwarf is close to the hydrogen burning limit and to the maximum mean mass density ρ among brown dwarfs and main-sequence stars. We note from Table 2 that it has a Roche radius very close to the Schwarzschild ISCO. We note as well that r R < M for a white dwarf. This means that such a star is never tidally disrupted above Sgr A*'s event horizon. A fortiori, neutron stars share the same property.

Presence of stellar objects in the vicinity of Sgr A*
The Galactic Center is undoubtably a very crowded region. For instance, it is estimated that there are ∼ 2 × 10 4 stellar BHs in the central parsec, a tenth of which are located within 0.1 pc of Sgr A* (Freitag et al. 2006). The recent detection of a dozen of X-ray binaries in the central parsec (Hailey et al. 2018) supports these theoretical predictions. The two-body relaxation in the central cluster causes some mass segregation: massive stars lose energy to lighter ones and drift to the center (Hopman & Alexander 2005;Freitag et al. 2006). Accordingly BHs are expected to dominate the mass density within 0.2 pc. However, they do not dominate the number density, main-sequence stars being more numerous than BHs (Freitag et al. 2006; Amaro-Seoane 2018). The number of stars or stellar BHs very close to Sgr A* (i.e. located at r < 100M) is expected to be quite small though. Indeed the central parsec region is very extended in terms of Sgr A*'s length scale: 1 pc = 5.1 × 10 6 M, where M is Sgr A*'s mass. At the moment, the closest known stellar object orbiting Sgr A* is the star S2, the periastron of which is located at r p = 120 au 3 × 10 3 M (GRAVITY Collaboration et al. 2018a). The most discussed process for populating the vicinity of the central MBH is the extreme mass ratio inspiral (EMRI) of a (compact) star or stellar BH (Amaro-Seoane et al. 2007;Amaro-Seoane 2018). In the standard scenario (see e.g. Amaro-Seoane (2018) for a review), the inspiralling object originates from the two-body scattering by other stars in the Galactic Center cluster. It keeps a very high eccentricity until the final plunge in the MBH, despite the circularization effect of gravitational radiation (Hopman & Alexander 2005). Such an EMRI is thus not an eligible source for the process considered in the present article, which is limited to circular orbits.
Another kind of EMRI results from the tidal separation of a binary by the MBH (Miller et al. 2005). In such a process, a member of the binary is ejected at high speed while the other one is captured by the MBH and inspirals towards it, on an initially low eccentricity orbit. Gravitational radiation is then efficient in circularizing the orbit, making it almost circular when it enters LISA band. Such an EMRI is thus fully relevant to the study presented here. The rate of formation of these zero-eccentricity EMRIs is very low, being comparable to those of high-eccentricities EMRIs (Miller et al. 2005), which is probably below 10 −6 yr −1 (Amaro-Seoane 2018; Hopman & Alexander 2005). However, as discussed in Sect. 5.3, due to their long life time (> 10 5 yr) in the LISA band, the probability of detection of these EMRIs is not negligibly small.
Another process discussed in the literature and leading to objects on almost circular orbits is the formation of stars in an accretion disk surrounding the MBH (see e.g. Collin & Zahn 1999;Nayakshin et al. 2007;Collin & Zahn 2008, and references therein). Actually, it was particularly surprising to find in the inner parsec of the Galaxy a population of massive (few 10 M ) young stars, that were formed ≈ 6 Myr ago (Genzel et al. 2010). Indeed, forming stars in the extreme environment of a MBH is not obvious because of the strong tidal forces that would break typical molecular clouds. A few scenarios were proposed to account for this young stellar population; see Mapelli & Gualandris (2016) for a recent dedicated review. Among these, in situ formation might take place in a geometrically thin Keplerian (circularly orbiting) accretion disk surrounding the MBH (Collin & Zahn 1999;Nayakshin et al. 2007;Collin & Zahn 2008). Such an accretion disk is not presently detected, and would have existed in past periods of AGN activity at the Galactic Center (Ponti et al. 2013(Ponti et al. , 2014. Stellar formation in a disk is supported by the fact that the massive young stellar population proper motion was found to be consistent with rotational motion in a disk (Paumard et al. 2006). It is interesting to note that the on-sky orientation of this stellar disk is similar to the orientation of the orbital plane of a recently detected flare of Sgr A* (GRAVITY Collaboration et al. 2018b).

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10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 However, such a scenario suffers from the fact that the young stars observed have a median eccentricity of 0.36 ± 0.06 (Bartko et al. 2009), while formation in a Keplerian disk leads to circular orbits. On the other side, the recently detected X-ray binaries (Hailey et al. 2018) mentioned above are most probably quiescent BH binaries. These BHs are likely to have formed in situ in a disk (Generozov et al. 2018), giving more support to the scenario discussed here. A population of stellar-mass BHs will form after the death of the most massive stars born in the accretion disk. These would be good candidates for the scenario discussed here, provided the initially circular orbit is maintained after supernova explosion. The recent study of Bortolas et al. (2017) shows that BHs formed from the supernova explosion of one of the members of a massive binary keep their initial orbit without noticeable kink from the supernova explosion. Given that a large fraction (tens of percent) of the Galactic Center massive young stars are likely to be binaries (Sana & Evans 2011), this shows that circular-orbiting BHs are likely to exist within the framework of the Keplerian in-situ star formation model. This scenario was already advocated by Levin (2007), which considers the fragmentation of a self-gravitating thin accretion disk that forms massive stars, leading to the formation of BHs that inspiral in towards Sgr A*, following quasi-circular orbits, in a typical time of ≈ 10 Myr.

Compact objects
As discussed in Sect. 5.1, compact objects -BHs, neutron stars and white dwarfs -do not suffer any tidal disruption above the event horizon of Sgr A*. Their evolution around Sgr A* is thus entirely given by the reaction to gravitational radiation with the timescale shown by Figs. 14-15.
Let us define the entry in LISA band as the moment in the slow inspiral when SNR 1 yr reaches 10, which is the threshold we adopt for a positive detection [Eq. (26)]. The orbital radius at the entry in LISA band in plotted in Fig. 17 as a function of the mass µ of the inspiralling object. It is denoted by r 0,max since it is the maximum radius at which the detection is possible. Some selected values are displayed in Table 3. The mass of the primordial BH has been chosen arbitrarily to be the mass of Jupiter (10 −3 M ), as a representative of a low mass compact object.
For a compact object, the time T in-band spent in LISA band is nothing but the inspiral time from r 0,max to the ISCO: where T ins is given by Eq. (38) and T life by Eq. (39). The time in LISA band is depicted in Fig. 18 and some selected values are given in Table 3. The trends in Fig. 18 can be understood by noticing that, at fixed initial radius, the inspiral time is a decreasing function of µ [as µ −1 , cf. Eq. (38)], while it is an increasing function of the initial radius [as r 4 0 at large distance, cf. Eq. (40)], the latter being larger for larger values of µ, since r 0 marks the point where SNR 1 yr = 10, the SNR being an increasing function of µ [cf. Eq. (24)]. The behavior of the T in-band curves in Fig. 18 results from the balance between these two competing effects. The maximum is reached for masses around 10 −3 M for a = 0 (max T in-band ∼ 9 × 10 5 yr) and around 10 −5 M for a = 0.98M (max T in-band ∼ 2 × 10 6 yr) , which correspond to hypothetical primordial BHs.
The key feature of Fig. 18 and Table 3 is that the values of T in-band are very large, of the order of 10 5 yr, except for very small values of µ (below 10 −4 M ). This contrasts with the time in LISA band for extragalactic EMRIs, which is only 1 to 10 2 yr. This is of course due to the much larger SNR resulting from the proximity of the Galactic Center. This large time scale counter-balances the low event rate for the capture of a compact object by Sgr A* via the processes discussed in Sect. 5.2: even if a compact object is Table 3. Orbital radius r 0,max at the entry in LISA band (SNR 1 yr 10), the corresponding gravitational wave frequency f m=2 (r 0,max ) and the time spent in LISA band until the ISCO, T in-band , for various compact objects orbiting Sgr A*. The numbers outside (resp. inside) parentheses are for Sgr A* spin parameter a = 0 (resp. a = 0.98M). driven to the close vicinity of Sgr A* every 10 6 yr, the fact that it remains there in the LISA band for ∼ 10 5 yr makes the probability of detection of order 0.1. Given the large uncertainty on the capture event rate, one can be reasonably optimistic. One may stress as well that white dwarfs, which are generally not considered as extragalactic EMRI sources for LISA because of their low mass, have a larger value of T in-band than BHs (cf. Table 3). Given that they are probably more numerous than BHs in the Galactic Center, despite mass segregation (cf. the discussion in Sect. 5.2 and Freitag (2003b)), they appear to be good candidates for a detection by LISA.

Main-sequence stars
As discussed in Sect. 5.1 (see Table 2), main-sequence stars orbiting Sgr A* have a Roche limit above the ISCO. Away from the Roche limit, the evolution of a star on a quasi-circular orbit is driven by the loss of energy and angular angular momentum via gravitational radiation, as for the compact objects discussed above. The orbit thus shrinks until the Roche limit is reached. At this point, the star starts to loose mass through the Lagrange point L 1 (Hameury et al. 1994;) (standard accretion onto the MBH by Roche lobe overflow) and possibly through the outer Lagrange point L 2 as well for stars of mass µ 1M (Linial & Sari 2017). In any case, the mass loss is stable and proceeds on a secular time scale (with respect to the orbital period). The net effect on the orbit is an increase of its radius (Hameury et al. 1994;Linial & Sari 2017), at least for masses µ < 5.6 M (Linial & Sari 2017). Accordingly, instead of an EMRI, one may speak about an extreme mass ratio outspiral (EMRO) , or a reverse chirp gravitational wave signal (Linial & Sari 2017;Jaranowski & Krolak 1992) when describing the evolution of such systems after they have reached the Roche limit.
For stars, let us denote by T ins in-band the inspiral time from the entry in LISA band (r 0 = r 0,max , cf. Fig. 17) to the Roche limit (r 0 = r R , cf. Table 2). T ins in-band is a lower bound for the total time spent in LISA band, the latter being T ins in-band augmented by the massloss time at the Roche limit, which can be quite large, of the order of 10 5 yr . The values of T ins in-band are given in Table 4 for three typical main-sequence stars: a Sun-like one, a red dwarf (µ = 0.2 M , same as in Table 2) and a main-sequence Table 4. Inspiral time to the Roche limit in LISA band for a µ = 0.062 M brown dwarf and different types of main-sequence stars. The numbers outside (resp. inside) parentheses are for Sgr A* spin parameter a = 0 (resp. a = 0.98M). T ins in-band appears to be very large, of the order of 10 5 yr, except for the 2.4 M -star for the inclination angle θ = π/2, which has r R > r 0,max , i.e. it is not detectable by LISA. As already argued for compact objects, this large value of the time spent in LISA enhances the detection probability.
Regarding main-sequence stars, we note that the recently claimed detection of a 149 min periodicity in the X-ray flares from Sgr A* (Leibowitz 2018) has been interpreted as being caused by a µ = 0.18 M star orbiting at r 0 = 13.2M, where it is filling its Roche lobe (Leibowitz 2018). If such a star exists, we can read on Fig. 7 that LISA can detect it with a SNR equal to 76 (resp. 35) for θ = 0 (resp. θ = π/2) in a single day of data!

Brown dwarfs
Brown dwarfs are less massive than main-sequence stars, their mass range being ∼ 10 −2 to ∼ 0.08 M (Chabrier & Baraffe 2000;Chabrier et al. 2009). Accordingly, they enter later (i.e. at smaller orbital radii) in the LISA band. However, they are more dense than main-sequence stars, so that their Roche limit is closer to the MBH, as already noticed in Sect. 5.1: the µ = 0.062 M brown dwarf of Table 2 has a Roche radius of order 7M, i.e. quite close to the Schwarzschild ISCO. In this region the SNR is quite high, despite the low value of µ: for µ = 0.062 M and θ = 0, SNR 1 yr = 7.4 × 10 3 (resp. SNR 1 yr = 5.4 × 10 3 ) at the Roche limit with χ = 0 (resp. χ = 1). For θ = π/2, these numbers become SNR 1 yr = 3.7 × 10 3 (χ = 0) and SNR 1 yr = 2.6 × 10 3 (χ = 1). Moreover, brown dwarfs stay longer in this region than compact objets since the inspiral time is inversely proportional to the mass µ of the orbiting object (cf. Eq. (38)). As we can see from the values in Table 4, the inspiral time in LISA band of brown dwarfs is even larger than that of main-sequence stars: T ins in-band ∼ 5 × 10 5 yr for θ = 0 and T ins in-band ∼ 3 × 10 5 yr for θ = π/2. Even if we do not know the capture rate of brown dwarfs by Sgr A*, these large values make us conclude that brown dwarfs on close circular orbits could be good candidates for a detection by LISA.

Inner accretion flow
Sgr A*'s accretion flow is known for generating particularly low-luminosity radiation, orders of magnitude below the Eddington limit, and orders of magnitude below what could be available from the gas supply at a Bondi radius (Falcke & Markoff 2013). This means that accretion models should be very inefficient in converting viscously dissipated energy into radiation. This energy will rather be stored in the disk as heat, so that Sgr A* accretion flow must be part of the hot accretion flow family (Yuan & Narayan 2014). Such systems are made of a geometrically thick, optically thin, hot (i.e. close to the virial temperature) accretion flow, probably accompanied by outflows. A plethora of studies have been devoted to modeling the hot flow of Sgr A*, see Falcke & Markoff (2000); Vincent et al. (2015); Broderick et al. (2016); Ressler et al. (2017); Davelaar et al. (2018), among many others, and references therein.
There is reasonable agreement between these different authors regarding the typical number density and geometry of the geometrically thick hot flow in the close vicinity of Sgr A*. The electron maximum number density is of order 10 8 cm −3 (to within one order of magnitude), and the density maximum is located at a Boyer-Lindquist radius of around 10 M (to within a factor of a few). It is thus straightforward to give a very rough estimate of the mass of the flow, which is of the order of ≈ 5 × 10 −11 M (where we consider a constant-density torus with circular cross section of radius 4 M, such that its inner radius is at the Schwarzschild ISCO). This extremely small total mass of Sgr A*'s accretion flow makes it impossible to detect gravitational waves from orbiting inhomogeneities. Figure 7 shows that the LISA SNR would be vanishingly small, assuming for instance an inhomogeneity of 10% of the total mass.

Dark matter
The dark matter (DM) density profile in the inner regions of galaxies is subject to debate. There is a controversy between observations and cold-dark-matter simulations regarding the value of the DM density power-law slope in the inner kpc, observations advocating a cored profile ρ(r) ∝ r 0 , while simulations predict ρ(r) ∝ r −1 (de Blok 2010). The parsec-scale profile is even less well known. Gondolo & Silk (1999) have proposed a model of the interaction of the central MBH with the surrounding DM distribution for the Milky Way. According to these authors, the presence of the MBH should lead to an even more spiky inner profile, with a scaling of ρ(r) ∝ r −2.3 . Such a dark matter spike can be constrained by high-angular resolution observation at the Galactic Center (Lacroix 2018). Figure 1 of Lacroix (2018) shows the enclosed DM mass at the Galactic Center as a function of radius, for various DM models: either non-annihilating DM, or self-annihilating DM (with particle mass equal to 1 TeV) for various cross sections. Weaklyinteracting DM ( σv < 10 −30 cm 3 s −1 ) leads to an enclosed mass higher than 10 −4 M in the inner 10 M. Figure 7 shows that this leads to SNR 1 yr > 0.2, assuming that 10% inhomogeneities would appear in the DM distribution and orbit circularly around the MBH around 10 M. For non-annihilating DM, the SNR values can be as high as SNR 1 yr ∼ 10 4 . This makes a DM spike an interesting candidate for a potential gravitational wave source at the Galactic Center, to be studied in details in a forthcoming article (Le Tiec & et al. 2019).

Artificial sources
The MBH Sgr A* is indubitably a unique object in our Galaxy. If 6 an advanced civilization exists, or has existed, in the Galaxy, it would seem unlikely that it has not shown any interest in Sgr A*. On the contrary, it would seem natural that such a civilization has put some material in close orbit around Sgr A*, for instance to extract energy from it via the Penrose process. Whatever the reason for which the advanced civilization acted so (it could be for purposes that we humans simply cannot imagine), the orbital motion of this material necessarily emits gravitational waves and if the mass is large enough, these waves could be detected by LISA. Given the SNR values obtained in Sect. 3 and assuming that Sgr A* is a fast rotator, an object of mass as low 7 as the Earth mass orbiting close to the ISCO is detectable by LISA. This scenario is discussed further by Abramowicz et al. (2019), which consider a long lasting Jupiter-mass orbiter, left as a "messenger" by an advanced civilization, which possibly disappeared billions of years ago.

Discussion and conclusions
We have conducted a fully relativistic study of gravitational radiation from bodies on circular orbits in the equatorial plane of the 4.1 × 10 6 M MBH at the Galactic Center, Sgr A*. We have performed detailed computations of the SNR in the LISA detector, taking into account all the harmonics in the signal, whereas previous studies (Freitag 2003b;Linial & Sari 2017;Kuhnel et al. 2018) were limited to the Newtonian quadrupole approximation, which yields only the m = 2 harmonic for circular orbits. The Roche limits have been evaluated in a relativistic framework as well, being based on the computation of the Roche volume in the Kerr metric . This is specially important for brown dwarfs, since their Roche limit occurs in the strong field region.
Setting the detection threshold to SNR 1 yr = 10, we have found that LISA has the capability to detect orbiting masses close to Sgr A*'s ISCO as small as ten Earth masses or even one Earth mass if Sgr A* is a fast rotator (a 0.9M). Given the strong tidal forces at the ISCO, these small bodies have to be compact objects, i.e. small BHs. Planets and main-sequence stars have a Roche limit quite far from the ISCO: r R ∼ 34 M for a solar-type star (or Jupiter-type planet) and r R ∼ 13 M for a 0.2 M star. However, even at these distances, main-sequence stars are still detectable by LISA, the entry in LISA band (defined by SNR 1 yr = 10) being achieved for r 0,max ∼ 47 M for a solar-type star and at r 0,max ∼ 35 M for a 0.2 M main-sequence star, assuming an inclination angle θ = 0. Because they are more dense, brown dwarfs have a Roche limit pretty close to the ISCO, the minimal Roche radius being r R ∼ 7 M, which is achieved for a 0.062 M brown dwarf. For such an object, the entry in LISA band occurs at r 0,max ∼ 28 M.
Beside the SNR at a given orbit, a key parameter is the total time spent in LISA band, i.e. the time T in-band during which the source has SNR 1 yr 10. We have found that, once they have entered LISA band from the low frequency side, all the considered objects, be they compact objects, main-sequence stars or brown dwarfs, spend more than 10 5 yr in LISA band 8 . The minimal time in-band occurs for high-mass BHs (µ ∼ 30 M ), for which T in-band ∼ 1 × 10 5 yr (assuming θ = 0) and the maximal one, of the order of one million years, is achieved for a Jupiter-mass BH (µ ∼ 10 −3 M ) if Sgr A* is a slow rotator (a/M 1): T in-band ∼ 9 × 10 5 yr, or for a µ ∼ 10 −5 M BH if Sgr A* is a rapid rotator (a/M 0.9): T in-band ∼ 2 × 10 6 yr. These small BH masses regard primordial BHs. Among stars and stellar BHs, the maximum time spent in LISA band is achieved for brown dwarfs: T in-band 5 × 10 5 yr, just followed by low-mass main-sequence stars (red dwarfs) and white dwarfs, for which T in-band 3 × 10 5 yr. These large values of T in-band contrast with those for extragalactic EMRIs, which are typically of the order of 1 to 10 2 yr. This is of course due to the much larger SNR resulting from the proximity of Sgr A*, which allows to catch compact objects at much larger orbital radii, where the orbital decay is not too fast, and to catch main-sequence stars above their Roche limit.
To predict some LISA detection rate from T in-band , one shall know the rate at which the considered objects are brought to close circular orbits around Sgr A* ("capture" rate). While we have briefly described some scenarios proposed in the literature in Sect. 5.2, it is not the purpose of this work to make precise estimates. Having those is probably very difficult, given the involved uncertainties, both on the observational ground (strong absorption in the direction of the Galactic Center) and the theoretical one (dynamics of the tens of thousands of stars and BHs in the central parsec). Some optimistic scenarios mentioned in Sect. 5.2 predict a capture rate of the order of 10 −6 yr −1 for BHs. For T in-band ∼ 10 5 yr, this would result in a detection probability of 0.1 by LISA. For white dwarfs, low mass main-sequence stars and brown dwarfs, the capture rate could possibly be higher (Freitag 2003b), leading to a significant detection probability by LISA, especially for brown dwarfs. Instead of making any concrete prediction, we prefer an "agnostic" approach, stating that Sgr A* is definitely a target worth of attention for LISA, which may reveal various bodies orbiting around it.
Let us point out that Amaro-Seoane (2019) has recently performed a study of gravitational radiation from main-sequence stars and brown dwarfs orbiting Sgr A*. He finds results similar to ours regarding the SNR in LISA. Also, he derives the event rate for the Galactic Center taking into account the relativistic loss-cone and eccentric orbits, which are more typical in an astrophysical context.
In Appendix C, we have considered bodies in close circular orbit around the 2.5 × 10 6 M MBH in the center of the nearby galaxy M32. We found that main-sequence stars with µ 0.2 M are not detectable by LISA in this case, while compact objects and brown dwarfs are detectable, with a lower probability: the time they are spending in LISA band with SNR 1 yr 10 is 10 3 to 10 4 years, i.e. two orders of magnitude lower than for Sgr A*.
A natural extension of the work presented here is towards non-circular orbits. Another one would be to study the gravitational emission from a (stochastic) ensemble of small masses, such as brown dwarfs, in the case they are numerous around Sgr A*, or from dark matter clumps as mentioned in Sect. 5.7 (Le Tiec & et al. 2019).
where sinc stands for the cardinal sine function: sinc(x) ≡ sin x/x. The square of the modulus ofh + ( f ), which appears in the SNR formula (22), is then where the functions ∆ T, f * ( f ) are defined for any pair of real parameters (T, f * ) by For each value of f * , the ∆ T, f * constitute a family of nascent delta functions, i.e. they obey 11 These two properties imply that, for any integrable function F, In other words, when T → +∞, ∆ T, f * tends to the Dirac delta distribution centered on f * . Considering successively the four terms that appear in Eq. (B.5) and gathering them two by two by means of ±, we have then −− θ = 0 − − θ = π/4 θ = π/2 a = 0 a = 0.50 M a = 0.90 M a = 0.98 M Roche limit for 1 M -star/Jupiter Roche limit for rocky body Roche limit for 0.2 M -star Roche limit for brown dwarf  10 -3 10 -2 10 -1 10 0 10 1 10 2 Article number, page 27 of 29 Table B.2. Inspiral time to the Roche limit in LISA band (SNR 1 yr 10) for the brown dwarf and red dwarf models considered in Sect. 5.1, when orbiting M32 MBH. The numbers outside (resp. inside) parentheses are for M32 MBH spin parameter a = 0 (resp. a = 0.98M).
Regarding the detection probability, the important parameter is the time T in-band spent in LISA band, i.e. the time elapsed between the orbit at which the object starts to be detectable by LISA (cf. Fig. B.3) and either the ISCO (for a compact object, cf.  Table B.1) or the Roche limit (brown dwarfs and red dwarfs, cf. Table B.2). From Fig. B.4, the largest values of T in-band are T in-band ∼ 1. × 10 4 yr (resp. T in-band ∼ 2 × 10 4 yr) for a = 0 (resp. a = 0.98 M) and are achieved for µ ∼ 0.1 M (resp. µ ∼ 10 −3 M ), which corresponds to hypothetical primordial BHs. We note that for a 0.5 M white dwarf, T in-band ∼ 1 × 10 4 yr. For stellar mass BHs, T in-band is of the order of a few 10 3 yr.
For the 0.2 M red dwarf, we conclude from Table B.2 that it can be detected by LISA only if the inclination angle θ is small and if it is not corotating (|χ| 1). One has then T in-band > T ins in-band ∼ 2 × 10 3 yr. Regarding the 0.062 M brown dwarf, we read in Table B.2 that T in-band > T ins in-band ∼ 1 × 10 4 yr for low inclinations and ∼ 3 × 10 3 yr for large inclinations.