Constraints on the Galactic Centre environment from Gaia hypervelocity stars III: Insights on a possible companion to Sgr A*

Introduction

In the centre of our Galaxy lurks Sagittarius A (Sgr A*), a supermassive black hole (SMBH) with a mass of $\sim4 \times10^6 M_\odot$ (Ghez et al. 2008; Genzel et al. 2010; Akiyama et al. 2022). Such SMBHs seem to be omnipresent in the nuclei of external galaxies as well (Magorrian et al. 1998; Kormendy & Ho 2013), at least among galaxies above a particular mass (see Merritt 2013). Dynamical friction Chandrasekhar 1943 in dense stellar systems conspires to bring massive bodies towards the centres of gravitational potentials. Other SMBHs delivered via major galaxy mergers or $\sim10^2 - 10^4 M_\odot$ intermediate mass black holes (IMBHs) formed originally in stellar clusters [141] (Miller & Hamilton 2002) or directly in a SMBH accretion disc (Goodman & Tan 2004; McKernan et al. 2012) can thereby form a bound

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massive black hole binary (MBHB) consisting of either an SMBH-SMBH pair (e.g. (Begelman et al. 1980; Volonteri et al. 2003; Di Matteo et al. 2005)) or an SMBH-IMBH pair [143] (Levin et al. 2005; Arca-Sedda & Gualandris 2018). MBHBs have received considerable attention in recent years owing to the fact that colliding MBHBs are loud gravitational wave sources (Peters 1964) in a frequency range accessible to the upcoming Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2017).

There exists the possibility that Sgr A* has a SMBH or IMBH companion which, owing to the fact that the Galactic Centre (GC) makes for a challenging observational environment (see Schödel et al. 2014, for a review), has not yet been detected. The possible parameter space for such a companion has been steadily reduced over the past two decades from astrometric observations of Sgr A* itself (Hansen 2003; Reid & Brunthaler 2004, 2020), orbital modelling of the S-star cluster (Gualandris & Merritt 2009) and the S-star

cluster star S2 in particular (Gualandris et al. 2010; Naoz et al. 2020; GRAVITY Collaboration et al. 2020). Taken together, current constraints allow for a companion of at most a few hundred $M_{\odot}$ just within or outside the orbit of S2 (c.f. GRAVITY Collaboration et al. 2020, Appendix D).

An independent method of constraining a possible companion to Sgr A* is by considering its direct interactions with stars that approach it. Dynamical interactions between single stars and MBHBs tend to ‘fling’ the stars out at roughly the orbital velocity of the less massive member of the MBHB, draining energy from the system and hardening the MBHB (Quinlan 1996; Zier & Biermann 2001). Ejection velocities therefore increase as the MBHB hardens, to the point where the hard MBHB is able to eject so-called ‘hypervelocity stars’ (HVSs) with velocities in excess of the Galactic escape speed (Yu & Tremaine 2003). It should be noted that a companion to Sgr A* is not required to produce HVSs in the first place – the term HVS was coined by Hills (1988), who first theorized that the disruption of a stellar binary by a single SMBH could eject one member of the binary with an extreme velocity (see also Yu & Tremaine 2003; Kenyon et al. 2008; Sari et al. 2010; Kobayashi et al. 2012; Rossi et al. 2014; Zhang et al. 2013; Generozov & Madigan 2020; Generozov 2021). While this so-called Hills mechanism remains the most promising scenario for the ejection of fast stars from the centre of the Galaxy, the ‘MBHB slingshot’ mechanism described above is a well-researched alternative (Yu & Tremaine 2003; Levin & Beloborodov 2003; Baumgardt et al. 2006; Sesana et al. 2006, 2007a; Löckmann et al. 2008; Marchetti et al. 2018; Rasskazov et al. 2019; Darbha et al. 2019; Zheng et al. 2021; Mastrobuono-Battisti et al. 2023).

After the predictions of Hills (1988), nearly two decades would elapse until the first detection of a promising HVS candidate by Brown et al. (2005), who discovered a B-dwarf in the outer Galactic stellar halo with a heliocentric radial velocity of 853$\pm$12 km s$^{-1}$. Following this and other serendipitous HVS discoveries (Edelmann et al. 2005; Hirsch et al. 2005), other candidate stars potentially unbound to the Galaxy have trickled in courtesy of targeted surveys (Brown et al. 2006, 2009, 2012, 2014), follow-up observations of previously identified stars (e.g Heber et al. 2008; Tillich et al. 2009; Irrgang et al. 2010, 2019) and queries of large Galactic surveys (e.g. Palladino et al. 2014; Zhong et al. 2014; Huang et al. 2017; Koposov et al. 2020). See Brown (2015) for a review of these objects. While the current roster of proposed HVS candidates stands at $\sim$two dozen, only S5-HVS1 (Koposov et al. 2020) can be uncontroversially associated with an origin in the Galactic Centre.

Our understanding of the structure and kinematics of the Milky Way as a whole has been revolutionized by the European Space Agency satellite $Gaia$ (Gaia Collaboration et al. 2016). In its third and most recent data release (DR3; Gaia Collaboration et al. 2022), $Gaia$ provides five-parameter astrometry (parallax, position, proper motion) for $\sim$1.5 billion Galactic sources (Gaia Collaboration et al. 2021a) and validated heliocentric radial velocities for $\sim$34 million sources (Katz et al. 2022; Sartoretti et al. 2022). While $Gaia$ observations have been invaluable in the identification of new potential HVS candidates [105, 107] (Bromley et al. 2018; Shen et al. 2018; Hattori et al. 2018; Du et al. 2019; Luna et al. 2019; Huang et al. 2021; Li et al. 2022; Igoshev et al. 2023; Prudil et al. 2022), currently missing from all data releases of $Gaia$ radial velocity catalogues are high-confidence HVS candidates, that is, candidates with i) precise astrometric measurements (relative parallax error $<20\%$), ii) a velocity which suggests the star is unbound to the Galaxy, and iii) a trajectory which is consistent with an origin in the Galactic Centre (Marchetti et al. 2019; Marchetti 2021; Marchetti et al. 2022).

This presents a tantalizing opportunity. If Sgr A* has a hidden companion, HVSs ejected via the MBHB slingshot mechanism may be detectable by $Gaia$. Given that the selection functions of both the $Gaia$ source catalogue and radial velocity catalogue are relatively well-modelled (Everall & Boubert 2022; Sartoretti et al. 2022; Cantiat-Gaudin et al. 2022, Castro-Ginard et al. submitted), it is possible to determine with reasonable confidence which stars should appear in the catalogue and which stars should not. An absence of confident HVS candidates in $Gaia$ DR3 can therefore rule out areas of the MBHB parameter space (namely the hidden companion’s mass and separation from Sgr A*) which predict an abundance of MBHB slingshot-ejected HVSs.

This work is a companion to Evans et al. (2022a,b) and Marchetti et al. (2022), in which we exploited a similar possibility, focusing on the Hills mechanism. In Evans et al. (2022a) we used the lack of main sequence HVSs in $Gaia$ Early Data Release 3 (Gaia Collaboration et al. 2021a) to impose an upper limit on the HVS ejection rate. We expanded upon this in Evans et al. (2022b), showing that considering the lack of evolved HVS candidates improves constraints. By considering as well the existence of S5-HVS1 (Koposov et al. 2020), strengthened constraints further and additionally constrained the shape of the initial mass function (IMF) among HVS progenitor binaries in the GC. In Marchetti et al. (2022) we mined the radial velocity catalogue of the then-newly released $Gaia$ DR3 to search for new high-confidence HVS candidates. While no new HVS candidates were unearthed with precise astrometry, with a Galacticentric velocity in excess of 700 km s$^{-1}$ and with a trajectory pointing away from the GC, we showed how the lack of HVS candidates in this data release improved constraints even further. In those works we considered the Hills mechanism as the only mechanism which ejects HVSs. In this work we follow a similar philosophy considering the MBHB slingshot mechanism – by exploring how different present-day MBHB binary configurations would result in HVS populations of different size, we show that some configurations are incompatible with an absence of detected HVSs in the $Gaia$ DR3 radial velocity sample.

This paper is organized as follows. In § 2 we outline on a step-by-step basis our model for generating mock populations of HVSs ejected via the MBHB slingshot mechanism. In § 3 we present our analyses of these populations – we show how the population of HVSs in $Gaia$ depends on the characteristics of the MBHB binary and how the absence of high-confidence HVS candidates in $Gaia$ DR3 constrains the mass of an as-of-yet unseen companion to Sgr A* and its separation from it. We discuss these results in § 4 before offering a summary and conclusions in § 5.

Massive black hole binary Slingshot Ejection Model

In this section we describe how we model the MBHB orbital evolution, how we generate ejected HVSs, how we propagate these HVSs through the Galaxy and how we obtain mock observations of them to determine which would be detectable by Gaia. Overall our procedure resembles the approach of Marchetti et al. (2018), who also explore Gaia-detectable HVSs ejected from an MBHB in the GC. We expand upon their modelling by investigating a larger set of MBHB configurations and by using updated prescriptions for propagating HVSs through the Galaxy, performing mock photometry, and modelling the Gaia selection function. The code we use to implement the model we describe in this section is included as the MBHB module in the publicly available PYTHON package speedystar.¹

Orbital decay of the MBHB

An MBHB embedded in a collisionless, fixed stellar background loses orbital energy via i) the ejection of stars via the MBHB slingshot mechanism, and ii) the emission of gravitational waves. We model the hardening of the MBHB during phase i) following Quinlan (1996):

where $G$ is the gravitational constant, $\rho$ and $\sigma$ are the mass density and one-dimensional velocity dispersion of the stellar background, respectively, assumed here to be $\rho=7\times10^{4}\,\mathrm{M}_{\odot}\,\mathrm{pc}^{-3}$ (Schödel et al. 2007; Feldmeier et al. 2014; Schödel et al. 2014) and $\sigma=100\,\mathrm{km}\,\mathrm{s}^{-1}$ (Figer et al. 2003; Schödel et al. 2009; Feldmeier et al. 2014; Do et al. 2020) and $H$ is a dimensionless hardening rate:

As noted by Quinlan (1996), $H$ is approximately constant when the binary separation is below the ‘hardening separation’ $a_{\mathrm{h}}$:

where $M_{\mathrm{c}}$ is the mass of the less-massive companion in the MBHB. We approximate the orbital decay of the MBHB due to gravitational wave emission following Peters (1964):

where $c$ is the speed of light, $M_{\mathrm{SgrA}^{*}}$ is the mass of Sgr A*, taken here as $4\times10^{6}\,\mathrm{M}_{\odot}$ (Eisenhauer et al. 2005; Ghez et al. 2008), and $M_{\mathrm{total}}\equiv M_{\mathrm{SgrA}^{*}}+M_{\mathrm{c}}$ is the total mass of the MBHB. The evolution of the MBHB semi-major axis with time is then

where at an initial time $t_{0}$ the binary starts at separation $a=a_{0}$.

Following as well from Quinlan (1996), the total mass $\Delta M_{\mathrm{ej}}$ in stars ejected by the MBHB as it shrinks from separation $a$ to $a-\Delta a$ is modelled using the dimensionless mass ejection rate $J$:

Note that this ejection rate above assumes the MBHB orbital decay is driven entirely by HVS ejections. To account for the contribution of GW emission to the binary hardening, we must include a correction to Eq. (6):

where $\Delta t_{\mathrm{total}}$ is the total time required for the MBHB to shrink from separation $a$ to $a-\Delta a$ and $\Delta t_{\mathrm{HVS}}>\Delta t_{\mathrm{total}}$ is the time required to shrink from $a$ to $a-\Delta a$ if the decay is entirely by HVS ejections, calculated by integrating and inverting Eq. (1).

Over a grid of MBHB mass ratios $q$ in the range $10^{-4}\leq q\leq10^{0}$, we determine the dimensionless hardening rate $H$ and the mass ejection rate $J$ by performing scattering experiments. The methodology of these experiments closely follows Rasskazov et al. (2019) and we refer the reader to that work for more details. Stars, assumed to be massless, approach a MBHB of separation $a$ from infinity with a velocity $v$, impact parameter $b$, on directions randomized with respect to the binary phase and orbital plane. $v$ is drawn from a Maxwellian distribution in the range $[3\times10^{3}v_{0}\sqrt{q/(1+q)},30v_{0}\sqrt{q/(1+q)}]$ (Sesana et al. (2006)), where

is the orbital velocity of the MBHB. $b^{2}$ is drawn uniformly such that pericenter distances are constrained to the range $[0,5a]$. We assume the scattering interactions never increase the eccentricity of the binary – its orbit remains circular throughout (see discussion on this point in § 4). Each star is scattered off the binary and the simulation ends when i) the star reaches a distance of $50a$ from the MBHB with a positive total energy, or ii) the timescale for the scattering interaction exceeds 10 Gyr, or iii) when the star spends longer than $1.6\times10^{4}$ binary orbital periods within a distance of $50a$ from the MBHB. For each MBHB mass ratio $q$ we ran a total of four million such simulations using the ARCHAIN (Mikkola & Merritt 2008) algorithm, specifically developed to simulate small-$N$ systems.

The results of these simulations are shown in Figure 1, where we plot how $H$ and $J$ depend on the MBHB separation for various mass ratios. In our calculation of $J$, a star counts as ‘ejected’ if and only if its velocity at a large distance from the MBHB is larger than $5.5\sigma=550\,\mathrm{km}\,\mathrm{s}^{-1}$, which is approximately the escape velocity from the bulge if it is modelled as a single isothermal sphere profile. In general $H$ at fixed $a/a_{\mathrm{h}}$ increases with decreasing $q$ though the relationship is not strictly monotonic. Note as well that more massive binaries still merge more quickly since $a_{\mathrm{h}}$ decreases linearly with $q$. In agreement with Sesana et al. (2006), we find that each can be approximated as

where $A$, $\alpha$, $\beta$, $\gamma$ are fitting parameters. In Table 1 we share best-fit parameters for a selection of mass ratios.

mass ratio	$A_H$	$a_{0,H}/a_h$	$\gamma_H$	$A_J$	$a_{0,J}/a_h$	$\alpha_J$	$\beta_J$	$\gamma_J$
1.0	14.58	1.98	-0.62	0.21	0.14	-0.36	11.17	-0.20
0.3	15.98	3.42	-0.77	0.15	0.48	-0.35	2.77	-1.08
0.1	17.33	3.43	-0.77	0.26	0.41	-0.18	1.32	-1.57
0.03	18.34	3.86	-0.84	0.22	0.24	-0.27	2.37	-0.54
0.01	19.14	3.24	-0.77	0.20	0.21	-0.29	3.75	-0.29
0.003	18.27	4.00	-0.81	0.21	0.20	-0.30	4.31	-0.26
0.001	17.29	3.29	-0.75	0.22	0.21	-0.30	3.93	-0.30
0.0003	17.36	2.92	-0.73	0.22	0.22	-0.29	3.46	-0.36

Another result of our scattering experiments, echoing the results of Rasskazov et al. (2019), is that the high-velocity tail of the ejection velocity distribution ($v_{\mathrm{ej}} \ge400\ \mathrm{km\ s^{-1}}$) is well-fit by a broken power-law distribution of the form

where $C_1$ and $C_2$ are constants, $v_{\mathrm{break}} = 1.2v_0\sqrt{2q}/(1 + q)$ and $v_0$ is the circular velocity of the MBHB (Eq. (8)). When assigning initial velocities to ejected stars, we draw velocities at random from this distribution in the range $[5.5\sigma, 5v_0]$.

In Figure 2 we summarize our modelling of the MBHB orbit, showing the orbital decay and ejected stellar mass for MBHBs with $q = 10^{-1}$, $q = 10^{-2}$ and $q = 10^{-3}$. Each binary has an initial separation of $a_0 = 100\ \mathrm{pc}$ at $t = 0$, and for the purposes of this study it is sufficient to say the MBHB has ‘merged’ when its separation reaches $10^{-6}\ \mathrm{pc}$, as after this the separation will drop to zero within a year. The top panel shows the elapsed time required for the MBHB to shrink to a separation $a$. The total time to merge is quite sensitive to the companion mass – the $q = 10^{-1}$, $q = 10^{-2}$ and $q = 10^{-3}$ binaries merge within 34, 60 and 120 Myr, respectively. The vertical lines show the hardening separations $a_h$ for the corresponding MBHB. The MBHB reaches $a = a_h$ at earlier stages of the inspiral for progressively more massive companions of Sgr A*. Middle panel shows that the HVS mass ejection rate profile peaks shortly after $a = a_h$. For small $q$ this peak occurs after $da/dt|_{\mathrm{HVS}} = da/dt|_{\mathrm{GW}}$ (see bottom panel) and as $q$ increases this peak occurs while the slingshot mechanism is still the dominant hardening mechanism. The total ejected HVS mass also depends strongly and nonlinearly on the MBHB companion mass – a binary with a mass ratio of $10^{-2}$ ejects 330 times more HVS mass in total than a binary with a mass ratio of $10^{-4}$, but only eight times more mass than a binary with a mass ratio of $10^{-3}$.

Generating the HVS sample

The previous subsection described how the mass ejected in HVSs by a MBHB is related to the characteristics and evolution of the binary. Here we describe how we apply this to generate individual mock HVSs. Our approach is as follows.

We assume a MBHB composed of Sgr A* and a companion of mass $M_c$ started initially at separation $a_0=100$ pc² at time $t_0$ and is now at separation $a_{\mathrm{now}}$. We create a grid of 1000 semi-major axes from $a_{\mathrm{now}}$ to $a_0$ uniformly spaced in log-space. We use Eq. (7) to determine the mass $\Delta M_{\mathrm{ej},i}$ ejected while the binary was in the $i$’th separation bin, $i=1,2,\ldots,1000$. The corresponding number of ejected HVSs is

where $f(M)$ is the assumed IMF, defined between minimum and maximum stellar masses $M_{\min}$ and $M_{\max}$. We adopt a single power-law IMF with slope $\kappa$, i.e. $f(m) \propto m^{-\kappa}$ and minimum and maximum masses of $0.1\,M_{\odot}$ and $100\,M_{\odot}$, respectively.

If the MBHB is currently at separation $a_{\mathrm{now}}$, the flight time of all stars ejected while the binary was in the $i$’th separation bin have the same flight time $t_{\mathrm{flight},i}$:

where $a_i$ is the midpoint of the $i$’th separation bin. We assume a star is equally likely to be ejected at any point in its lifetime, therefore we say the age of each star at ejection $t_{\mathrm{age,ej}}$ is a random fraction of its maximum lifetime $t_{\mathrm{life}}$:

where $\epsilon$ is a random number uniformly distributed in $[0,1]$. $t_{\mathrm{life}}$ for each ejected star is determined using the single stellar evolution (SSE; Hurley et al. 2000) algorithms within the AMUSE³ environment (Portegies Zwart et al. 2009, 2013; Pelupessy et al. 2013; Portegies Zwart & McMillan 2018), taking the start of the asymptotic giant branch phase as the ‘end’ of a star’s life. We cap $t_{\mathrm{life}}$ to a maximum of 13.8 Gyr to ensure the star is not older than the Universe.

Today, the age of each star is

We remove stars for whom $t_{\mathrm{age}}>t_{\mathrm{life}}$, i.e. stars which were ejected as main sequence or evolved stars but are stellar remnants in the present day.

After drawing an ejection velocity for each surviving star following Eq. (11), stars are initialized on a sphere 3 pc in radius centred on Sgr A* with a velocity pointing radially away from the GC. We assume stars are ejected isotropically – we comment further on this assumption in § 4.

Orbital integration and mock photometry

After initializing our mock HVSs, our scheme for propagating them forward in time through the Galactic potential and obtaining synthetic photometry of them remains essentially unchanged from the method described in Evans et al. (2022b). We refer the reader to that work for more detailed explanations and briefly summarize the approaches here.

We propagate each star forward in time for its flight time assuming the Galactic potentials of McMillan (2017), who use a Monte Carlo Markov Chain (MCMC) method to fit a many-component potential to various kinematic data. For each realization, we draw a potential at random from the McMillan (2017) MC chain (P. McMillan, private communication). All ejected stars are integrated through this potential with the PYTHON package GALPY (Bovy 2015) using a fifth-order Dormand-Prince integrator (Dormand & Prince 1980) and a timestep of 0.1 Myr.

We estimate the visual dust extinction at each star’s distance and sky position using the COMBINED15 dust map of (Bovy et al. 2016)⁴, itself a combination of the Galactic dust maps of Drimmel et al. (2003); Marshall et al. (2006) and Green et al. (2015). We then determine mock apparent magnitudes for each mock HVS in the photometric bands using the MESA Isochrone and Stellar Tracks, or MIST (Dotter 2016; Choi et al. 2016) models⁵. With each star’s luminosity, surface gravity and effective temperature (determined with AMUSE depending on each star’s mass, age and metallicity), as well as its visual extinction, we interpolate the MIST bolometric correction tables to determine each star’s apparent magnitude in the Johnsons-Cousins $V$ and $I_{\mathrm{c}}$ bands (Bessell 1990) and Gaia $G$ and $G_{\mathrm{RP}}$ bands⁶ (Riello et al. 2021). We estimate each star’s magnitude in the Gaia $G_{\mathrm{RVS}}$ band from its $V$, $I_{\mathrm{c}}$ and $G$-band magnitudes using fits from Jordi et al. (2010).

HVSs identifiable by Gaia

With mock magnitudes and stellar parameters for our synthetic HVS populations, we now determine which HVS candidates should be detectable in the different Gaia data releases with measured radial velocities. While we use an updated selection function, this approach is similar in philosophy to Evans et al. (2022b).

For Gaia DR3, we start by discarding all HVSs with effective temperatures outside the range $3500\ \mathrm{K} \le T_{\mathrm{eff}} \le6900\ \mathrm{K}$, since the Gaia spectroscopic pipeline does not assign validated radial velocities to stars outside this temperature range (Katz et al. 2022; Sartoretti et al. 2022). Next, we use the selection functions made available by the GaiaUnlimited⁸ project (Cantat-Gaudin et al. 2022; Castro-Ginard et al. 2023) to identify stars which would be detectable in the Gaia DR3 radial velocity catalogue. In short, the DR3 empirical selection function is calibrated against the Dark Energy Camera Plane Survey Data Release 1 (Schlafly et al. 2018), and querying it yields the probability $p_{\mathrm{source}}$ that a star at a given sky position with a given Gaia $G$-band magnitude would appear in the DR3 source catalogue, i.e. it would have at least a measured position, magnitude and colour. Since it compares to a deeper survey, We note that this approach only characterizes the faint-end selection function of Gaia. At the bright end, saturation effects lead to incompleteness for sources brighter than $G \lesssim3$ (see Fabricius et al. 2021; Gaia Collaboration et al. 2021b). We remove all mock HVSs brighter than $G = 3$, however we note that it is extraordinarily rare for our model to produce HVSs this bright. Likewise, querying the spectroscopic selection function yields the probability $p_{\mathrm{vrad}}$ that a star in the DR3 source catalogue at a given sky position, $G$-band magnitude and $G-G_{\mathrm{RP}}$ colour is in the radial velocity catalogue as well. We manually set $p_{\mathrm{vrad}}$ to zero for mock HVSs in sky position/magnitude/colour bins entirely unpopulated in the DR3 radial velocity catalogue, since the GaiaUnlimited statistical model $p_{\mathrm{vrad}}$ will be non-zero but small and highly uncertain. The total probability $p$ that a mock HVS would appear in the Gaia DR3 radial velocity subsample is then the product of $p_{\mathrm{source}}$ and $p_{\mathrm{vrad}}$. For each mock HVS in our sample we draw a random number $0 < \epsilon< 1$ and designate the HVS as DR3-detectable if $\epsilon< p$.

After deciding which mock HVSs would have radial velocities in DR3, we determine each one’s $5{\times}5$ astrometric covariance matrix (position, parallax and proper motion uncertainties and correlations amongst them) by querying the Gaia DR3 astrometric spread function of Everall et al. (2021)⁹, which computes uncertainties based on the sky position and $G$-band magnitude of the source. Finally, we determine the DR3 radial velocity uncertainty for each star using the PYTHON package PyGaia¹⁰.

We also determine which stars will be detectable in the fourth Gaia data release, DR4. Radial velocity measurements in this survey would be available for all mock HVSs cooler than 6900 K down to the $G_{\mathrm{RVS}} = 16.2$ mag faint-end magnitude limit of the Gaia radial velocity spectrometer (Cropper et al. 2018; Katz et al. 2019). For hotter HVSs, radial velocities would be available for stars brighter than $G_{\mathrm{RVS}} = 14$. We estimate DR4 astrometric errors using Everall et al. (2021) DR3 astrometric spread function, reducing the errors according to the predicted Gaia performance¹¹.

To be labelled as a ‘detectable’ high-confidence HVS in a particular Gaia DR3 or DR4, an ejected star must:

• be brighter than the faint-end apparent magnitude limit of the radial velocity catalogue of the data release.

• have an effective temperature range within the bounds imposed by each data release (see above).

• have a relative parallax uncertainty below 20%. For larger uncertainties, estimating distances (and therefore total velocities) becomes problematic (see Bailer-Jones [6]).

• have a total velocity in the Galactocentric rest frame in excess of $700\ \mathrm{km\ s^{-1}}$. We sample over the astrometric and radial velocity uncertainties of each mock star and compute its total velocity assuming a distance from the Sun to the GC of 8.122 kpc (GRAVITY Collaboration et al. 2018), a height of the Sun above the Galactic disc of 20.8 pc (Bennett & Bovy 2019) and a rest-frame velocity of the Sun of $\mathbf{v}_{\odot} \equiv[U_{\odot},V_{\odot},W_{\odot}] = [12.9,245.6,7.78]\ \mathrm{km\ s^{-1}}$ (Reid & Brunthaler 2004; Drimmel & Poggio 2018). Upon sampling, in at least 80% of realizations the total velocity of the star must be above $700\ \mathrm{km\ s^{-1}}$.

These above cuts resemble those used to search for HVS candidates in Gaia Data Release 2 (Marchetti et al. 2019), Early Data Release 3 (Marchetti 2021) and DR3 (Marchetti et al. 2022). We remark that Marchetti et al. (2019), Marchetti (2021) and Evans et al. (2022a,b) selected HVSs by searching for stars likely to be moving faster than the Galactic escape speed at their position. In Marchetti et al. (2022) and in this work, we focus instead on stars moving faster than $700\ \mathrm{km\ s^{-1}}$. Determining the boundedness of a star requires assuming a Galactic potential, whereas a flat velocity cut instead allows agnosticism towards the potential. This cut at $700\ \mathrm{km\ s^{-1}}$ is conservative – in reasonable models of the Milky Way potential the escape velocity is $<700\ \mathrm{km\ s^{-1}}$ everywhere except perhaps within the innermost kpc of the Galaxy in heavier models (e.g. McMillan 2017).

For brevity, in the remainder of this work we use the terms

‘Gaia DR3/DR4’ to refer exclusively to the radial velocity subsamples, and by the term ‘HVS’ we refer only to those stars which satisfy the criteria listed above.

Results

An existing companion to Sgr A*

Having outlined our model for generating mock HVS populations from the MBHB slingshot mechanism, we start in Figure 3 by illustrating how the HVS population detectable in Gaia DR3 and DR4 depends on our model assumptions. In the left panel we show how the Gaia HVS population depends on the IMBH companion mass when the MBHB separation is fixed at 1 mpc and the IMF of HVS progenitors has a power-law slope of $-1.7$ (Lu et al. 2013). Below a companion mass of $\sim800\,M_{\odot}$, we expect $\leq1$ HVS in both DR3 and DR4. This expected DR3 population can rise to several tens of thousands if Sgr A* is in a near-equal mass MBHB. The steep dependence on $M_{\mathrm{c}}$ is fairly intuitive – as the companion mass increases, the total ejected stellar mass increases and the ejection velocity distribution shifts towards higher velocities (Figure 2, Eq. (11)). A strong dependence on the current separation $a_{\mathrm{current}}$ of the MBHB (middle panel) is digestible as well – a smaller current separation means the MBHB has ejected more HVSs in the recent past, and HVS ejection velocities are larger since the MBHB circular velocity is larger. $N_{\mathrm{HVS}}$ plateaus for separations smaller than the binary hardening separation (vertical line) because the binary merges shortly thereafter (see Figure 2). For an HVS IMF slope of $-1.7$ and a companion mass of $4000\,M_{\odot}$, $>1$ HVS is expected in Gaia DR3 as long as the MBHB separation is less than a few milliparsecs. This number should reach $\sim60$ in DR3 or several hundred in DR4 if the MBHB is just about to merge within the next decade¹².

In the right panel of Figure 3 we show the dependence of $N_{\mathrm{HVS}}$ on the IMF power law slope $\kappa$. The turnover at $\kappa\simeq-1.7$ can be explained by two competing factors. Firstly, as $\kappa$ increases, ejected stars are on average more massive, and therefore more luminous and more likely to be brighter than Gaia’s faint-end magnitude limit of $\sim14$ for the DR3 radial velocity sample. Secondly, however, the total number of ejected HVSs decreases as $\kappa$ increases, since the ejected stellar mass is locked up in fewer, more massive stars (see Eq. (7)). The turnover represents the point where the latter effect overcomes the former.

Since the impact of $\kappa$ on $N_{\mathrm{HVS}}$ in DR3 is comparatively weak (particularly in the vicinity of $\kappa\approx-1.7$), for the remainder of this work we explore only constraints on $a_{\mathrm{current}}$ and $M_{\mathrm{c}}$. We run simulations over a $M_{\mathrm{c}}/a_{\mathrm{current}}$ grid, marginalizing over $\kappa$ by sampling it at random in each iteration from a Gaussian distribution centred on $\kappa= -1.7$ with a standard deviation of 0.2 (Lu et al. 2013). In Figure 4 we show the results of this simulation suite. The contours show lines of constant $N_{\mathrm{HVS}}$ throughout $M_{\mathrm{c}} - a_{\mathrm{current}}$ space and the colourbar indicates the $1\sigma$ scatter. For our fiducial choices of $M_{\mathrm{c}} = 4000\,M_{\odot}$ and $a_{\mathrm{current}} = 1\,\mathrm{mpc}$, we determine that $8 \pm4$ high-confidence HVSs should have been uncovered in the radial velocity catalogue of Gaia DR3. Following from Figure 3, the expected DR3 HVS population ranges from $<1$ (for low-mass companions at large separations) to $\sim$thousands (for a near-equal-mass, hardened MBHB). The relative scatter is $\sigma(N_{\mathrm{HVS}})/N_{\mathrm{HVS}} \simeq0.2$ for $N_{\mathrm{HVS}} \gtrsim100$, rising to $\simeq0.5$ and $\simeq1$ if $\sim$ tens or only a few HVS were uncovered, respectively. For $M_{\mathrm{c}} \lesssim10^{5}\,M_{\odot}$ $N_{\mathrm{HVS}}$ increases monotonically with decreasing $a_{\mathrm{current}}$. For larger companion masses, however, $N_{\mathrm{HVS}}$ decreases for $a_{\mathrm{current}}$ less than $\sim$ a few $\times10^{-1}$ mpc since the MBHB slingshot mass ejection rate peaks and begins to decline before GW emission takes over and the binary quickly merges (see Figure 2 middle panel). The dashed white line indicates where the $1\sigma$ lower limit of $N_{\mathrm{HVS}}$ is equal to one, i.e. where one HVS at least should be in the survey. Since no such HVSs have been detected in Gaia DR3, MBHB binaries below this line can be excluded. A companion to Sgr A* cannot exist within 1 mpc (2000 AU) of the GC unless it is quite low-mass ($M_{\mathrm{c}} \lesssim2000\,M_{\odot}$). A near-equal mass MBHB remains possible only for separations $\gtrsim0.05$pc.

Prior works have constrained a companion to Sgr A*. In Figure 5 we place our constraints in context with these. In all, large regions of the parameter space we exclude in this work have previously been excluded, particularly cases where Sgr A* has a fairly massive companion just inside or outside the orbit of S2. These configurations, if true would result in an astrometric ‘wobble’ of Sgr A* (Hansen & Milosavljević 2003; Reid & Brunthaler 2004, 2020), would impact the orbit of S2 (Gillessen et al. 2009), its stability against perturbation (Gualandris et al. 2010; Naoz et al. 2020; GRAVITY Collaboration et al. 2020) and the stability of the S-star cluster in general (Gualandris & Merritt 2009). HVS observations are an independent tool to measure the Galactic Centre and the constraints they impose on a possible MBHB in the GC reinforce and somewhat sharpen these prior constraints.

Of the parameter space that has not already been excluded by previous works, we mainly rule out companions separated from Sgr A* by less than a milliparsec. Such configurations, depending on the mass of the companion, could only persist for a short amount of time between gravitational wave emission drives the MBHB to coalescence. We annotate Figure 5 with lines of constant gravitational wave coalescence time. Some prior works have disregarded configurations below these lines outright, as it would be fortuitous if, in the present day, Sgr A* were about to merge with a companion in the very near future. Regardless, a lack of HVS candidates in Gaia DR3 observations exclude such fortunate scenarios and demonstrates the advantage of using HVS observations alongside traditional probes. Combining prior constraints with constraints determined in this work, the allowed configurations of a MBHB in the GC include i) a $\lesssim5 \times10^{4}\,M_{\odot}$ companion at a separation greater than a few tens of milliparsecs from Sgr A*, ii) a $M = 100 - 200\,M_{\odot}$ companion just within or outside the orbit of S2 (970 AU, or 4.7 mpc), or iii) a $M \lesssim500\,M_{\odot}$ companion at a separation of less than a few hundred microparsecs from Sgr A*.

A former companion to Sgr A*

As the MBHB spirals in, the HVS mass ejection rate peaks at a small separation (Figure 2, see also Gualandris et al. 2005; Baumgardt et al. 2006; Levin 2006). Post-merger, this final ‘gasp’ of HVSs propagates outwards through the Galaxy. Over the next $\sim$ tens of Myr, these HVSs can still be detected from Earth. This means HVS observations can probe not only an existing companion to Sgr A*, but a former companion as well. We illustrate this in Figure 6. In the left panel we show how $N_{\mathrm{HVS}}$ depends on the time $t_{\mathrm{since}}$ since the MBHB in the GC merged, holding $M_{\mathrm{c}}$ and the IMF index $\kappa$ fixed. A negative $t_{\mathrm{since}}$ indicates the binary has not yet merged. $N_{\mathrm{HVS}}$ is maximized just before the MBHB coalesces. Otherwise, from the HVS population size alone one cannot determine whether the MBHB exists in the present day or whether it already merged in the recent past. Given a substantial population of HVS candidates, the distribution of flight times could discriminate between these two scenarios (see § 4). In the right panel of Figure 6 we show how $N_{\mathrm{HVS}}$ depends on both $M_c$ and $t_{\mathrm{since}}$ when we sample over $\kappa$. The dashed line indicates where the $1\sigma$ lower limit of $N_{\mathrm{HVS}}$ reaches one. Any configuration below this line can be ruled out, since at least one HVS ejected before the MBHB merged should still be detectable in Gaia DR3. If Sgr A* ever had a companion more massive than 1000 $M_\odot$, it must have merged with Sgr A* more than 12 Myr ago. This lower limit on $t_{\mathrm{since}}$ increases with increasing companion mass. To our knowledge this is the first direct observational constraint on the specific merger history of Sgr A* within the last $\sim$ tens of Myr. If Sgr A* merged with a companion more than 30 Myr ago, we cannot offer constraints on how massive that companion could have been. Similarly, if Sgr A* recently accreted a companion less massive than $\sim200 M_\odot$, current HVS observations cannot constrain how recently this merger occurred. The coloured bands show the parameter space allowed if 1, 10 $\pm\sqrt{10}$, 100 $\pm10$ or 1000 $\pm\sqrt{1000}$ HVSs are found in DR4. Given the lack of HVSs in Gaia DR3, our simulations suggest we should expect no more than 9, 17, or 40 HVSs in DR4 if the companion mass is $10^3$, $10^4$, or $10^5 M_\odot$, respectively. If zero high-confidence HVSs are discovered in Gaia DR4, regions of parameter space below the lower edge of the ‘1’ band will be excluded. Otherwise, since $N_{\mathrm{HVS}}$ depends so intimately on $M_c$ and $a_{\mathrm{current}}/t_{\mathrm{since}}$, strict (albeit degenerate) constraints can be placed on these parameters if a non-zero number of HVSs are uncovered in DR4.

Prospects for DR4

Having established constraints on a possible existing or former Sgr A* companion from Gaia DR3 HVS observations (or lack thereof), we can project forward and explore how these constraints may update with the improved astrometric precision and deeper faint-end magnitude limit of Gaia DR4, expected $\sim2026$. In Figure 7 we show how constraints on the presumed (existing or former) Sgr A* companion mass and either its current separation from Sgr A* (if the MBHB still exists) or the time since coalescence (if the MBHB has already merged) will change depending on the number of high-confidence HVS uncovered in Gaia DR4.

Discussion

Contribution from the Hills mechanism

The results of the previous section assume that all detectable HVSs are ejected via the MBHB slingshot mechanism. This is not necessarily (nor is it expected to be) true, as multiple possible HVS ejection mechanisms exist and given a sizeable HVS population it may be difficult to disentangle which was ejected via which mechanism. In practice, the ejection of HVSs from the GC may to due to a myriad of mechanisms, each occurring in tandem and coupling to each other. These other possible avenues include the dynamical encounters between single stars and a ‘swarm’ of stellar-mass black holes in the GC ((O’Leary & Loeb 2007)) or dynamical interactions between a single SMBH or MBHB and a globular cluster which has sunk toward the GC ((Capuzzo-Dolcetta & Fragione 2015; Fragione & Capuzzo-Dolcetta 2016)). Chief among these alternative mechanisms, however, is the Hills mechanism ((Hills 1988; Gould & Quillen 2003; Yu & Tremaine 2003), which involves the tidal separation of a stellar binary following a

રસ close encounter with a single SMBH. Indeed, it is difficult to contrive a scenario wherein ejections via the MBHB slingshot ejection occur but ejections via the Hills mechanism do not. While there are intrinsic differences in the kinematics and properties of stars ejected via the two mechanisms (see Brown 2015; Rasskazov et al. 2019, and references therein), in practice this is difficult to discern directly, given that current observations can detect only a biased subsample of these populations, i.e. those which are bright and relatively nearby. When considering null HVS detections this is not a problem, as a complete dearth of HVSs can constrain all ejection mechanisms simultaneously. For non-null detections, however, as may be the case for Gaia DR4, this becomes worthy of discussion. Confident constraints on an HVS ejection mechanisms cannot be imposed using a particular HVS candidate without a notion of which mechanism(s) may be responsible ejecting it. The HVS candidate S⁵-HVS1 is an example of this conundrum, and the next subsection below will be devoted to it. While it is beyond the scope of this work to develop a fully self-consistent ejection model including both the Hills mechanism and the MBHB mechanism simultaneously, in this subsection we consider whether HVS populations ejected via these different mechanisms can be meaningfully disentangled.

To compare the two populations, we use a population of Hills mechanism-ejected HVSs generated by Marchetti et al. (2022). We refer the reader to that work for more details concerning the generation of this catalogue, but it assumes an IMF index among HVS progenitors of $\kappa= -1.7$ and generates HVS progenitor binaries assuming power-law distributions of the binary mass ratio and log-orbital period. In Figure 8 we compare the properties of the Gaia DR4-detectable HVS population ejected via the Hills mechanism (black) to the population ejected via the MBHB slingshot mechanism (orange/yellow/blue). Shown are the distributions of the detectable stars’ stellar masses, Gaia $G$-band magnitudes, heliocentric distances, heliocentric radial velocities and stellar ages, stacked over 40 iterations. When we vary $M_{\mathrm{c}}$ or $a_{\mathrm{current}}$ (top and middle rows), differences between the populations are subtle. Hills-ejected HVSs are on average slightly more massive and younger than MBHB slingshot-ejected ones, and if the Sgr A* companion is particularly massive and/or its separation from Sgr A* is small, slingshot-ejected HVSs will on average exhibit a narrower range of radial velocities and will span a wider range of distances.

If Sgr A* and its companion have already merged, differentiating between the mechanisms becomes easier. The bottom row of Figure 8 shows that as the time $t_{\mathrm{since}}$ since coalescence increases, the range of stellar masses among detectable MBHB slingshot-ejected HVSs narrows considerably (massive HVSs will have already left the main sequence, low-mass HVSs will be too far away to detect), the typical distance increases (the final ‘gasp’ of HVSs will be further away), the range of radial velocities narrows (they are less impacted by the Galactic potential and farther away, so their velocity is more in the radial direction) and their typical ages increase (they can’t be younger than $t_{\mathrm{since}}$). However, given that we expect $5^{+11}_{-4}$ Hills mechanism-ejected HVSs in the radial velocity catalogue of Gaia DR4 (Evans et al. 2022b) and at most a few tens of MBHB slingshot-ejected HVSs (Figure 7), uncontroversially assigning each HVS to a corresponding ejection mechanism will be difficult in any case.

The case of S⁵-HVS1

Thus far in this work, the only HVS observational data we have considered is the lack of high-quality HVS candidates in Gaia DR3. Both null and non-null HVS detections exist in other searches and surveys, however, which can in principle strengthen constraints. For example, the MMT HVS Survey (Brown et al. 2009, 2012, 2014) identified several dozen HVS candidates. We choose not to include these in our analysis since proper motion measurements for these candidates are not sufficiently precise to conclusively associate any with an ejection from the Galactic Centre.¹³

Worth mentioning as well is the HVS candidate S5-HVS1, a $\sim2.35\,M_\odot$ star first identified by Koposov et al. (2020) in the $S^5$ survey (Li et al. [106]). $S^5$-HVS1 is notable in that it is the first HVS candidate for whom an origin in the GC is uncontroversial – its trajectory points directly away from the GC and implies an ejection 4.8 Myr ago with a velocity of $v_{\mathrm{ej}} \simeq1800\ \mathrm{km\ s^{-1}}$. As an unambiguous HVS detection, we showed in Evans et al. (2022b) that penalizing models which predict zero or $\gg1$ HVSs similar to $S^5$-HVS1 in the $S^5$ survey significantly improved constraints on the IMF slope in the GC and the ejection rate of HVSs via the Hills mechanism. In this subsection we comment on how and whether the inclusion of this star can improve constraints on a current of former companion to Sgr A*.

To identify mock HVSs which would have appeared as promising HVSs in $S^5$ by the analysis of Koposov et al. (2020), we first compute each star’s apparent magnitude in the Dark Energy Camera (DECam) $g$ and $r$ bands (Abbott et al. [1]) using the MIST models as done in § 2.3. We then roughly reproduce the $S^5$ selection function, selecting HVSs within the $S^5$ sky footprint (see Li et al. [106], table 2) which have Gaia parallaxes satisfying $\varpi< 3\sigma_\varpi+ 0.2$, and DECam photometry satisfying $15 < g < 19.5$ and $-0.4 < (g-r) < +0.1$. We then take only those mock HVSs with heliocentric radial velocities larger than $800\ \mathrm{km\ s^{-1}}$, since these were the ones selected for further inspection by Koposov et al. (2020).

The results of these selections are shown in Figure 9. Assuming the MBHB slingshot mechanism is the only mechanism ejecting HVSs, the left panel shows the number of $S^5$-HVS1 analogues predicted to appear in $S^5$ depending on the Sgr A* companion mass and the current separation of the MBHB. Configurations between the white lines are consistent within $1\sigma$ with finding exactly one similar to $S^5$-HVS1 in $S^5$, and herein lies the issue: a majority of these configurations exist within the parameter space already excluded by the lack of HVSs in the radial velocity catalogue of Gaia DR3. Assuming the MBHB slingshot mechanism is the only mechanism responsible for the ejection of HVSs, explaining both the lack of HVSs in Gaia DR3 and the existence of $S^5$-HVS1 is only possible if Sgr A* has a $200\,M_\odot\lesssim M_{\mathrm{c}} \lesssim500\,M_\odot$ companion at a separation of $\lesssim0.3$ pc.

In the right-hand panel we show similar results in $M_{\mathrm{c}}$–$t_{\mathrm{since}}$ space. Sgr A* having merged within the last few tens of Myr with a $\sim200-1000\,M_\odot$ companion is consistent with both the existence of $S^5$-HVS1 and the lack of HVSs in DR3. Note, however, that $S^5$-HVS1 has a well-constrained flight time of 4.8 Myr. If it were ejected via the MBHB slingshot mechanism shortly before a merger, this merger must have occurred at most 4.8 Myr ago. Within this timeframe, only a 200 $M_\odot\lesssim M_c \lesssim500\,M_\odot$ companion can reconcile the existence of an HVS in S$^5$ with the non-detection of HVSs in Gaia DR3.

In conclusion, our simulations only support a MBHB slingshot origin for S$^5$-HVS1 in specific circumstances. Additionally, our criteria to select S$^5$-HVS1 analogues above makes no consideration for its extreme Galactocentric velocity of 1750 km s$^{-1}$. A velocity this large is quite hard to achieve in the MBHB slingshot mechanism for a companion mass of $\lesssim500\,M_\odot$ – only $\sim20$ per cent of our S$^5$-HVS1 analogues in this mass range are as fast as S$^5$-HVS1. Typical ejection velocities via the Hills mechanism, on the other hand, are comparatively larger and can more easily accommodate this star. A more rigourous investigation of whether S$^5$-HVS1 could be produced by this mechanism is warranted and does not yet exist. Generozov & Madigan (2020) used S$^5$-HVS1’s flight time as an observational constraint and found that a 1000 $M_\odot$ companion separated from Sgr A* by 0.01 pc could efficiently reproduce the observed eccentricity distribution of the S-star cluster in the GC. This, however, does not necessarily imply S$^5$-HVS1 was ejected via the MBHB slingshot mechanism.

The impact of model simplifications

Our model for the ejection of HVSs from the MBHB slingshot mechanism includes a number of implicit and explicit simplifying assumptions. Works in recent years have shown that some of these assumptions i) impact the evolution of MBHBs, and ii) may not hold strictly true in the centre of our own Galaxy. In this subsection we comment on the degree to which our results could be impacted by these simplifications.

We assume that our MBHB is non-accreting and in a gas-free environment. Dynamical friction driven by a dense gaseous disc and accretion onto the MBHB impact the separation, eccentricity and inclination evolution of the MBHB (see Escala et al. 2005; Dotti et al. 2006, 2007; Muñoz et al. 2019) and would influence the stellar ejection of HVS progenitors (see Cantiello et al. 2021). Accretion onto the MBHB in the GC would not be strong enough to impact its evolution – X-ray observations of the Galactic Centre indicate an accretion rate of $10^{-6} - 10^{-5}\,M_\odot\,\mathrm{yr}^{-1}$ (Baganoff et al. 2003; Quataert 2004) at the Bondi radius (0.04 pc) and polarization measurements limit the accretion rate to $\sim10^{-8}\,M_\odot\,\mathrm{yr}^{-1}$ near the Schwarzschild radius (Quataert & Gruzinov 2000; Bower et al. 2003; Marrone et al. 2007). We can check the impact of this accretion with a back of the envelope calculation. Assuming that the inspiral of a MBHB is driven entirely by gas accretion, differentiating the angular momentum of the MBHB with respect to time yields

where $\dot{a}$ is the hardening rate, $\dot{M}$ is the rate of accretion onto the MBHB, $M$ is the total MBHB mass and $\ell=L/M$ is the specific angular momentum of the binary. $2(\dot{\ell}/\ell- 3/2)$ is a constant of order unity depending on gas accretion physics (see D’Orazio & Duffell 2021). Plugging in characteristic numbers for the accretion rate and MBHB mass, the accretion-driven hardening rate is never more than ten orders of magnitude smaller than the total slingshot + gravitational wave hardening rate. While evidence exists for an episode of nuclear activity in the GC in the last few Myr (see Bland-Hawthorn et al. 2019, and references therein), this flare was short-lived. On the scale of detectable HVS flight times, it is valid to assume a putative MBHB in the GC would be non-accreting and in a gas-poor environment. See Naoz et al. (2020) for further discussion on the impact of accretion on a presumed black hole companion to Sgr A*.

We have assumed for simplicity that the presumed MBHB in the GC is on an initially circular orbit and never deviate from a circular orbit until coalescence. This assumption is valid since while non-zero eccentricity increases the binary hardening rate at small separations, it leads to only minor variations in the total stellar mass ejected via the slingshot mechanism (Quinlan 1996; Sesana et al. 2006; Rasskazov et al. 2019). In any event, the impact of eccentricity is only relevant for MBHB mass ratios larger than $10^{-3}$ – smaller mass ratio MBHBs tend to circularize with time (Rasskazov et al. 2019; Bonetti et al. 2020).

Our model of the GC assumes the MBHB is embedded within a spherical, nonrotating nuclear star cluster (NSC). In actuality, the NSC is slightly flattened along the Galactic vertical axis (Schödel et al. 2014) and rotates more or less parallel with the Galactic disc (Trippe et al. 2008; Schödel et al. 2009; Chatzopoulos et al. 2015; Fritz et al. 2016). A nuclear cluster which co-(counter-)rotates with the MBHB enhances (suppresses) eccentricity growth of the MBHB (Sesana et al. 2011; Rasskazov & Merritt 2017; Rasskazov et al. 2019; Bonetti et al. 2020) while the impact of counter- and corotation of the NSC on the MBHB hardening rate is less clear (see contradictory results in Holley-Bockelmann & Khan 2015; Mirza et al. 2017; Rasskazov & Merritt 2017; Rasskazov et al. 2019; Varisco et al. 2021). The MBHB slingshot mechanism mass ejection rate is decreased slightly at small separations ($a \lesssim0.3a_{\mathrm{h}}$) if the NSC is maximally corotating (Rasskazov et al. 2019), in which case our assumption of a non-rotating NSC may be overestimating $N_{\mathrm{HVS}}$. If the NSC is maximally counter-rotating, the mass ejection rate at small separations is increased by a factor of up to $\sim$ two (Rasskazov et al. 2019). This is less of a concern, given that MBHBs are expected to align their rotation with the angular momentum of the NSC (see below).

The density $\rho$ and velocity dispersion $\sigma$ of the NSC impact the MBHB hardening rate (Eq. (1) - Eq. (3)); we assume single values for these, when in fact they both vary with radial distance from the GC (Schödel et al. 2009, 2014; Feldmeier et al. 2014). We have confirmed that variations of $\sigma$ and $\rho$ within observational uncertainties do not meaningfully affect the results of this work. Along with this, we neglect to consider mass segregation as well within the NSC. A mass segregated stellar environment accelerates hardening timescales among highly unequal-mass MBHBs but has the opposite impact on $\sim$equal mass MBHBs (Mukherjee et al. 2023). We assume as well that the MBHB centre of mass is always centred on the exact geometric centre of the Galaxy. In reality, the MBHB will ‘drift’ randomly in a nonrotating stellar nucleus (Merritt 2001; Chatterjee et al. 2003) or on a closed orbit if the nucleus is rotating (Holley-Bockelmann & Khan 2015; Mirza et al. 2017; Khan et al. 2020; Varisco et al. 2021). The center of mass displacement from the Galactic barycentre is typically on the order of the MBHB radius of influence or smaller, i.e. not large enough to meaningfully affect the detectable HVS population. There are very few cases where shifting the positions of a Gaia-detectable mock HVS by $\sim$ a few pc in any direction renders it undetectable.

When generating mock HVS populations, we take no consideration of the orientation and phase of the MBHB. If stars are ejected isotropically in the MBHB slingshot mechanism, the orientation of the MBHB is irrelevant. While our model assumes isotropic ejection, theoretically the fastest-ejected stars are ejected preferentially in the plane of the binary, with a complex polar angle distribution depending on its separation, eccentricity and mass ratio (Sesana et al. 2006; Rasskazov et al. 2019; Darbha et al. 2019). Ejections are likely non-axisymmetric as well, though the azimuthal distribution of fast ejections remains unclear (see discussion in Rasskazov et al. 2019). Without independent constraints on the inclination/phase of the MBHB, we would be compelled to sample them at random, and therefore any angular dependence would get washed out in our results after averaging over many iterations. There is reason to believe, however, that a putative MBHB in the GC would be aligned with the Galactic disc. For a spherical, nonrotating NSC, the orientation of the MBHB orbit drifts on the order of $\sqrt{m_{\star}/M_{\mathrm{MBHB}}}\sim10^{-4}$ rad relative to its initial orientation, where $m_{\star}$ is the typical mass of a star in the NSC and $M_{\mathrm{MBHB}}$ is the total mass of the MBHB (Merritt 2001). In a rotating NSC, however, the angular momentum of the MBHB aligns with the angular momentum of the nucleus (Gualandris et al. 2012; Wang et al. 2014; Cui & Yu 2014; Rasskazov & Merritt 2017). Having established above that the Milky Way NSC rotates parallel to the Galactic disc, it is not unreasonable to assume a putative MBHB would rotate parallel with the disc as well. Another point in favour of this is the existence of a nuclear stellar disc embedded within the NSC itself (Bartko et al. 2009; Lu et al. 2009; Yelda et al. 2014). This stellar disc rotates in the direction of the overall Galactic disc as well (Schönrich et al. 2015; Schultheis et al. 2021; Sormani et al. 2022) and resonant relaxation processes would align any IMBH-SMBH binary with such a disc (Szölgyén et al. 2021; Magnan et al. 2022). If the MBHB orbit were indeed aligned with the Galactic mid-plane, our assumption of isotropic ejections would slightly underestimate $N_{\mathrm{HVS}}$, since in actuality a greater proportion of HVSs are ejected on angles pointed more flattened toward the Earth.

Finally, our model assumes the phase space of low-angular momentum orbits which bring GC stars on a close ($\lesssim a$) approach to the MBHB is efficiently repopulated over time. Only stars within this ‘loss cone’ experience strong dynamical interactions with the MBHB. This loss cone is refilled over time, primarily via two-body relaxation in the NSC Sgr A* (see Lightman & Shapiro 1977; Merritt 2013) or resonant relaxation processes in the young clockwise disc (Rauch & Tremaine 1996; Madigan et al. 2009, 2011, 2014). In principle, if these scattering mechanisms are not efficient, the loss cone can empty and the MBHB hardening (and therefore the ejections of HVSs) can slow down or cease entirely (Milosavljević & Merritt 2003). This problem was investigated in the context of HVS ejections by Sesana et al. (2007b), who found that without loss cone refilling the MBHB stalls after ejecting only half the MBHB reduced mass in stars, $\sim10^{2} - 10^{6}\,M_{\odot}$ in our case depending on the assumed mass ratio. This is smaller than the total mass ejected in the full loss cone regime by a factor of a few ($\sim$10) if the mass ratio is small (large) and the ejected stars would have smaller typical velocities. Concerns about loss cone depletion among MBHBs has largely been alleviated in recent years, as simulations have shown that relaxation-driven refilling of the loss cone is sufficient to avoid loss cone depletion in triaxial potentials (Berczik et al. 2006; Vasiliev et al. 2015; Gualandris et al. 2017), or even biaxial potentials (Khan et al. 2013). Regardless, our models in this work assume a full loss cone, and this can lead to an overestimation of $N_{\mathrm{HVS}}$ in cases of partial depletion. This assumption is likely the largest source of systematic uncertainty in our modelling. Given the strong dependence of $N_{\mathrm{HVS}}$ large range of HVS population sizes As noted by Vasiliev et al. (2015), it is relatively straightforward to adjust $H$ and $J$ (Eq. (9)) to account for loss cone depletion (see also Rasskazov et al. 2019).

Conclusions

Massive black hole binaries (MBHBs) are a natural consequence of galaxy evolution. Dynamical interactions between stars and an MBHB in the centre of the Milky Way could eject hypervelocity stars (HVSs) detectable by the Gaia space satellite. In this work, we use existing HVS observations for the first time as a probe of a possible supermassive or intermediate mass companion black hole to Sgr A*, the supermassive black hole located in the Galactic Centre (GC). Building upon previous work, we realistically simulate the ejection of HVSs from an MBHB assuming a variety of MBHB mass ratios and separations. We focus in particular on HVSs which would have appeared in the radial velocity catalogue of the third data release from Gaia. Considering that zero HVSs with precise astrometry were unearthed in the radial velocity catalogue of Gaia DR3 (Marchetti et al. 2022), MBHB configurations which predict too many HVSs in this data release can be excluded. Our conclusions are as follows:

• The number of HVSs detectable by Gaia depends strongly on the MBHB separation and companion mass. It is comparatively less sensitive to the shape of the assumed initial mass function (IMF) (Figure 3).

• For a fiducial Sgr A* companion mass of 4000 $M_\odot$ and MBHB separation of 0.001 pc, $8 \pm4$ HVSs should have been detected in the Gaia DR3 radial velocity catalogue with precise astrometry (Figure 4).

• The lack of such HVSs in DR3 excludes a companion within 1 mpc of Sgr A* unless it has a mass of $\lesssim1000\,M_\odot$, complementing and extending prior constraints on a Sgr A* companion (Figure 5).

• The lack of confident HVS detections in Gaia DR3 also allows us to constrain a former companion to Sgr A* for the first time. If Sgr A* merged with a companion in the recent past, either of the following must be true: i) the former companion had a mass of $\lesssim500\,M_\odot$, or ii) the merger must have happened more than $\sim10$–$30$ Myr ago (Figure 6).

• If Sgr A* has an existing companion or had a former companion, the forthcoming fourth Gaia data release will contain at most a few tens of HVSs ejected via the MBHB slingshot mechanism in its radial velocity catalogue with precise astrometry (Figure 7).

The constraints we place in this work on a possible companion to Sgr A*, especially when combined with constraints obtained over the last few years (Naoz et al. 2020; Reid & Brunthaler 2020; GRAVITY Collaboration et al. 2020), appear to be closing the door on the existence of such a companion. While a massive companion up to $\sim10^5 M_\odot$ is still allowed at separations larger than $\sim0.1$ pc, a hardened MBHB in the GC appears unlikely unless the MBHB mass ratio is extreme. With the future Gaia data releases and their synergy with both forthcoming and currently operational spectroscopic facilities and surveys (e.g. WEAVE; Dalton et al. 2012, 4MOST; [184], SDSS-V MWM; Kollmeier et al. 2017), more and more HVSs will be detected. If Sgr A* does not have a companion of significant mass and did not have one in the recent past, the MBHB slingshot mechanism can be definitively ruled out as an avenue for HVS ejections and future research can focus on more realistic mechanisms.

Acknowledgements

The authors thank J.R. Westpachter-Schneider for helpful discussion. FAE acknowledges support from the University of Toronto Arts & Science Postdoctoral Fellowship program and the Dunlap Institute. TM acknowledges an ESO fellowship. EMR acknowledges that this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101002511 – VEGA P). JB acknowledges financial support from NSERC (funding reference number RGPIN-2020-04712).

Data availability

The simulation outputs underpinning this work can be shared upon reasonable request to the corresponding author. These simulations were produced using the speedystar package, publicly available at https://github.com/fraserevans/speedystar.

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