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

F. A. Evans1,2, A. Rasskazov3, A. Remmelzwaal4, T. Marchetti5, A. Castro-
Ginard4, E. M. Rossi4, J. Bovy1,2
1David A. Dunlap Department of Astronomy and Astrophysics, University of
Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada
2Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON, M5S 3H4, Canada
2DAMTP, University of Cambridge, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
3Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The
Netherlands
4European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching
bei München, Germany
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We consider a scenario in which Sgr A* is in a massive black hole binary
(MBHB) with an as-of-yet undetected supermassive or intermediate-mass black
hole companion. Dynamical encounters between this MBHB and single stars in its
immediate vicinity would eject hypervelocity stars (HVSs) with velocities
beyond the Galactic escape velocity of the Galaxy. In this work, we use
existing HVS observations to constrain for the first time the existence of a
companion to Sgr A*. We simulate the ejection of HVSs via the ‘MBHB slingshot’
scenario and show that the population of HVSs detectable today depends
strongly on the companion mass and the separation of the MBHB. We demonstrate
that the lack of uncontroversial HVS candidates in Gaia Data Release 3 places
a firm upper limit on the mass of a possible Sgr A* companion. Within one
milliparsec of Sgr A*, our results exclude a companion more massive than
$1000\,\mathrm{M_{\odot}}$. If Sgr A* recently merged with a companion black
hole, our findings indicate that unless this companion was less massive than
$500\,\mathrm{M_{\odot}}$, this merger must have occurred at least $10$ Myr
ago. These results complement and improve upon existing independent
constraints on a companion to Sgr A* and show that large regions of its
parameter space can now be ruled out.

###### keywords:

Galaxy: centre, nucleus – stars: kinematics and dynamics

††pubyear: 2023††pagerange: Constraints on the Galactic Centre environment
from Gaia hypervelocity stars III: Insights on a possible companion to Sgr
A*–References

##  1 Introduction

In the centre of our Galaxy lurks Sagittarius A* (Sgr A*), a supermassive
black hole (SMBH) with a mass of $\sim$4$\times 10^{6}\,\mathrm{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
$\sim 10^{2}-10^{4}\,\mathrm{M_{\odot}}$ intermediate mass black holes (IMBHs)
formed originally in stellar clusters (Miller & Hamilton, 2002; Portegies
Zwart & McMillan, 2002) or directly in a SMBH accretion disc (Goodman & Tan,
2004; McKernan et al., 2012) can thereby form a bound 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
(Levin et al., 2005; Portegies Zwart et al., 2006; 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 of 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 $\mathrm{M_{\odot}}$ 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 $853\pm 12\,\mathrm{km\ s^{-1}}$. 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; Brown et al., 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 (Bromley et al., 2018; Shen et al., 2018; Hattori
et al., 2018; Du et al., 2019; Li et al., 2018; Luna et al., 2019; Huang et
al., 2021; Li et al., 2021, 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 atalogue are relatively well-modelled (Everall & Boubert, 2022;
Sartoretti et al., 2022; Cantat-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
mehanism. 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
Galactocentric velocity in excess of $700\ \mathrm{km\ s^{-1}}$ 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 Sec. 2 we outline on a step-by-step
basis our model for generating mock populations of HVSs ejected via the MBHB
slingshot mechanism. In Sec. 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 Sec. 4 before offering a
summary and conclusions in Sec. 5.

##  2 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
speedystar111https://github.com/fraserevans/speedystar.

###  2.1 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):

$\frac{da}{dt}\bigg{|}_{\rm HVS}=-\frac{G\rho H}{\sigma}a^{2}\,\text{,}$ (1)

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 $\rho=7\times 10^{4}\,\mathrm{M_{\odot}\
pc^{-3}}$ (Schödel et al., 2007; Feldmeier et al., 2014; Schödel et al., 2014)
and $\sigma=100\,\mathrm{km\ s^{-1}}$ (Figer et al., 2003; Schödel et al.,
2009; Feldmeier et al., 2014; Do et al., 2020) and $H$ is a dimensionless
hardening rate:

$H(\sigma,\rho,a)=\frac{\sigma}{G\rho}\frac{d}{dt}\left(\frac{1}{a}\right)\,\text{.}$
(2)

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

$a_{h}\equiv\frac{GM_{\rm c}}{4\sigma^{2}}\;\text{,}$ (3)

where $M_{\rm c}$ 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):

$\frac{da}{dt}\bigg{|}_{\rm GW}=-\frac{64}{5}G^{3}c^{5}\frac{M_{\rm
SgrA^{*}}M_{\rm c}M_{\rm total}}{a^{3}}\,\text{,}$ (4)

where $c$ is the speed of light, $M_{\rm SgrA^{*}}$ is the mass of Sgr A*,
taken here as $4\times 10^{6}\,\mathrm{M_{\odot}}$ (Eisenhauer et al., 2005;
Ghez et al., 2008), and $M_{\rm total}\equiv M_{\rm SgrA^{*}}+M_{\rm c}$ is
the total mass of the MBHB. The evolution of the MBHB semi-major axis with
time is then

$a(t)=a_{0}+\int_{t_{0}}^{t}\left(\frac{da}{dt}\bigg{|}_{\rm
HVS}+\frac{da}{dt}\bigg{|}_{\rm GW}\right)dt\,\text{,}$ (5)

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

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

$\Delta M_{\rm ej}=JM_{\rm total}\Delta\mathrm{ln}(1/a)\,\text{.}$ (6)

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:

$\Delta M_{\rm ej}=\frac{\Delta t_{\rm HVS}^{-1}}{\Delta t_{\rm
total}^{-1}}JM_{\rm total}\Delta\text{ln}(1/a)\;\text{,}$ (7)

where $\Delta t_{\rm total}$ is the total time required for the MBHB to shrink
from separation $a$ to $a-\Delta a$ and $\Delta t_{\rm HVS}>\Delta t_{\rm
total}$ 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.

Figure 1: The dependence of the dimensionless hardening rate $H$ (left) and
mass ejection rate $J$ (right) on the MBHB hardness, assuming a circular MBHB
orbit. MBHB separations are scaled to the hardening separation (Eq. 3)
Relationships for MBHBs with differing mass ratios ($q$) are shown with
different line colours.

Over a grid of MBHB mass ratios $q$ in the range $10^{-4}\leq q\leq 10^{0}$,
we determine the dimensionless hardening rate $H$ and the mass ejection rate
$J$ by perform 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 [$3\times
10^{3}v_{0}\sqrt{q/(1+q)},30v_{0}\sqrt{q/(1+q)}$] (Sesana et al., 2006), where

$v_{0}\equiv\sqrt{\frac{G(M_{\rm c}+M_{\rm MBH})}{a}}$ (8)

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 Sec. 4).
Each star is scattered off the binary and the simulation ends when i) the star
reaches a distance of 50$a$ 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 $1.6\times 10^{4}$ binary orbital periods within a
distance of 50$a$ from the MBHB. For each MBHB mass ratio $q$ we run 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 Fig. 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 $5.5\sigma=550\;\mathrm{km\
s^{-1}}$, which is approximately the escape velocity from the bulge if it is
modelled as a single isothermal sphere profile. In general $H$ at fixed
$a/a_{\rm h}$ increases with decreasing $q$ though the relationship is not
strictly monotonic. Note as well that more massive binaries still merge more
quickly since $a_{\rm h}$ decreases linearly with $q$. In agreement with
Sesana et al. (2006), we find that each can be approximated as

$\displaystyle H(a)$ $\displaystyle=A_{H}(1+a/a_{0,H})^{\gamma_{H}}\;\text{,}$
(9) $\displaystyle J(a)$
$\displaystyle=A_{J}(a/a_{0,J})^{\alpha_{J}}(1+(a/a_{0,J})^{\beta_{J}})^{\gamma_{J}}\;\text{,}$
(10)

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

Table 1: Best fit parameters for the hardening rate $H$ and stellar mass ejection rate $J$ (see Eq. 9). 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 ($v_{\rm ej}\geq 400\,\mathrm{km\ s^{-1}}$) is well-fit by a
broken power-law distribution of the form

$\frac{\mathrm{dN}}{\mathrm{dlog}v_{\rm ej}}=\begin{cases}C_{\rm
1},&\text{if}\ v_{\rm ej}<v_{\rm break}\\\ C_{\rm 2}v_{\rm
ej}^{-3.1},&\text{if}\ v_{\rm ej}\geq v_{\rm break}\end{cases}$ (11)

where $C_{1}$ and $C_{2}$ are constants, $v_{\rm
break}=1.2v_{0}\sqrt{2q}/(1+q)$ 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$\sigma$,
5$v_{0}$].

Figure 2: The MBHB orbital decay and rate of mass in ejected stars as a
function of binary semi-major axis, for three values of the mass ratio $q$.
The coloured vertical lines show the hardening separation (Eq. 3) for the
corresponding mass ratio. The primary MBH mass in each scenario is assumed to
be $4\times 10^{6}\,\mathrm{M_{\odot}}$ and each starts at $t=0$ at a
separation $a_{0}=100\,\mathrm{pc}$. The top panel shows how the separation
shrinks with time, with the dotted horizontal lines showing when each binary
merges. The middle panel shows how the stellar mass ejection rate evolves
throughout the inspiral. The bottom panel shows how the decay rate changes
with decreasing separation, with the contributions from HVS ejections and from
GW emissions plotted separately.

In Fig. 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 $a_{\rm h}$
for the corresponding MBHB. The MBHB reaches $a=a_{\rm h}$ 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
$a=a_{\rm h}$. For small $q$ this peak occurs after $da/dt|_{\rm
HVS}=da/dt|_{\rm GW}$ (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 $10^{-3}$.

###  2.2 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 $M_{\rm c}$
started initially at separation $a_{0}=100\,\mathrm{pc}$222This is a very
large separation, chosen mostly to ensure we use an initial separation which
captures all HVS ejections. In practice, we find that only stars ejected when
the MBHB is at a separation of $\lesssim$ 0.1 pc are detectable in any Gaia
data release. at time $t_{0}$ and is now at separation $a_{\rm now}$. We
create a grid of 1000 semi-major axes from $a_{\rm now}$ 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, …, 1000. The corresponding number of ejected HVSs is

$\Delta N_{i}=\frac{\Delta M_{\mathrm{ej},i}}{\int_{M_{\rm min}}^{M_{\rm
max}}Mf(M)dM}\;\text{,}$ (12)

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

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

$t_{\rm flight,\textit{i}}=\int_{a_{i}}^{a_{\rm
now}}\left(\frac{da}{dt}\bigg{|}_{\rm HVS}+\frac{da}{dt}\bigg{|}_{\rm
GW}\right)^{-1}da\,\text{,}$ (13)

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 $t_{\rm age,ej}$ is a random fraction of its
maximum lifetime $t_{\rm life}$;

$t_{\rm age,ej}=\epsilon\cdot t_{\rm life}\;,$ (14)

where $\epsilon$ is a random number uniformly distributed in [0,1]. $t_{\rm
life}$ for each ejected star is determined using the single stellar evolution
(SSE; Hurley et al., 2000) algorithms within the
AMUSE333https://amuse.readthedocs.io/en/latest/index.htmlenvironment
(Portegies Zwart et al., 2009; Portegies Zwart et al., 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 $t_{\rm life}$ to a
maximum of 13.8 Gyr to ensure the star is not older than the Universe.

Today, the age of each star is

$t_{\rm age}=t_{\rm age,ej}+t_{\rm flight}\;.$ (15)

We remove stars for whom $t_{\rm age}>t_{\rm life}$, 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 $3\,\mathrm{pc}$ 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 Sec. 4.

###  2.3 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
GALPY444https://github.com/jobovy/galpy (Bovy, 2015) using a fifth-order
Dormand-Prince integrator (Dormand & Prince, 1980) and a timestep of
$0.1\,\mathrm{Myr}$.

We estimate the visual dust extinction at each star’s distance and sky
position using the combined15 dust map of (Bovy et al.,
2016)555https://github.com/jobovy/mwdust, 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) models666https://waps.cfa.harvard.edu/MIST/.
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 $I_{\rm c}$ bands (Bessell, 1990) and Gaia $G$ and $G_{\rm RP}$,
bands777seehttps://www.cosmos.esa.int/web/gaia/edr3-passbands (Riello et al.,
2021). We estimate each star’s magnitude in the Gaia $G_{\rm RVS}$ band from
its $V$, $I_{\rm c}$ and $G$-band magnitudes using fits from Jordi et al.
(2010).

###  2.4 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 $3500\,\mathrm{K}\leq T_{\rm eff}\leq 6900\,\mathrm{K}$,
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 GaiaUnlimited888https://github.com/gaia-unlimited/gaiaunlimited 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 $p_{\rm source}$ 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 $G\lesssim 3$ (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 $p_{\rm vrad}$ that a star in the
DR3 source catalogue at a given sky position, $G$-band magnitude and $G-G_{\rm
RP}$ colour is in the radial velocity catalogue as well. We manually set
$p_{\rm vrad}$ to zero for mock HVSs in sky position/magnitude/colour bins
entirely unpopulated in the DR3 radial velocity catalogue, since in the
GaiaUnlimited statistical model $p_{\rm vrad}$ 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 $p_{\rm source}$
and $p_{\rm vrad}$. 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 5x5 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)999see
https://github.com/gaiaverse/scanninglaw, 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 PyGaia101010https://github.com/agabrown/PyGaia.

We also determine which stars will be detectable in the fourthGaia data
release, DR4. Radial velocity measurements in this survey would be available
for all mock HVSs cooler than $6900\,\mathrm{K}$ down to the $G_{\rm
RVS}=16.2$ 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 $G_{\rm RVS}=14$.
We estimate DR4 astrometric errors using Everall et al. (2021) DR3 astrometric
spread function, reducing the errors according to the predicted Gaia
performance111111https://www.cosmos.esa.int/web/gaia/science-performance, see
also Brown (2019)..

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, 2015).

  * •

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
$\textbf{v}_{\odot}\equiv[U_{\odot},V_{\odot},W_{\odot}]=[12.9,245.6,7.78]\,\mathrm{km\
s^{-1}}$ (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, however, 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 the 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.

##  3 Results

###  3.1 An existing companion to Sgr A*

Figure 3: The population of high-confidence HVSs (see Sec. 2.4) expected to
appear in Gaia DR3 and DR4. Panels show how NHVS depends on the mass of the
companion (left), the current separation $a_{\rm current}$ between Sgr A* and
its companion (middle) and the slope $\kappa$ of and initial mass function
among HVS progenitors (right). Parameters are fixed to their fiducial values
when not being varied. Shaded regions span the 16th to 84th quantiles over 40
iterations. The vertical dashed line in the middle panel shows the hardening
separation for this binary. Figure 4: Contour lines show how the population of
HVSs detectable in Gaia DR3 depends on the Sgr A* companion mass $M_{\rm c}$
and the current separation $a_{\rm current}$ between Sgr A* and its companion,
averaged over 40 realizations and smoothed over the grid. The colourbar shows
how the 1$\sigma$ scatter of NHVS. The white dashed line shows where the
1$\sigma$ lower bound of $N_{\rm HVS}$ reaches one – the parameter space above
this line is consistent with zero HVS detections in Gaia DR3.

Having outlined our model for generating mock HVS populations from the MBHB
slingshot mechanism, we start in Fig. 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 $1\,\mathrm{mpc}$ and the IMF of HVS
progenitors has a power-law slope of -1.7 (Lu et al., 2013). Below a companion
mass of $\simeq 800\,\mathrm{M_{\odot}}$, we expect $\leq 1$ 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 Mc is fairly
intuitive – as the companion mass increases, the total ejected stellar mass
increases and the ejection velocity distribution shifts towards higher
velocities (Fig. 2, Eq. 11). A strong dependence on the current separation
$a_{\rm current}$ 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. $N_{\rm HVS}$ plateaus for separations smaller than the binary
hardening separation (vertical line) because the binary merges shortly
thereafter (see Fig. 2). For an HVS IMF slope of -1.7 and a companion mass of
$4000\,\mathrm{M_{\odot}}$, >1 HVS is expected in Gaia DR3 as long as the MBHB
separation is less than a few milliparsecs. This number should reach $\sim$60
in DR3 or several hundred in DR4 if the MBHB is just about to merge within the
next decade121212A small point of possible confusion – for simplicity, we
assume the MBHB separation does not shrink throughout the Gaia mission
lifetime. Given typical typical MBHB coalescence times (see Fig. 2), this
assumption is valid. The analysis presented in this work assumes a scenario in
which Gaia DR3 and DR4 are ‘snapshots’ of the Milky Way at a time when the
MBHB separation is $a_{\rm current}$, rather than observations collected over
the span of 34 or 66 months, respectively..

In the right panel of Fig. 3 we show the dependence of NHVS 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 $\sim 14$ 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 NHVS 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 $a_{\rm current}$ and $M_{\rm c}$. We
run simulations over a $M_{\rm c}/a_{\rm current}$ 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 Fig. 4 we show the results of this simulation suite. The
contours show lines of constant $N_{\rm HVS}$ throughout $M_{\rm c}-a_{\rm
current}$ space and the colourbar indicates the 1$\sigma$ scatter. For our
fiducial choices of $M_{\rm c}=4000\,\mathrm{M_{\odot}}$ and $a_{\rm
current}=1$ mpc, we determine that $8\pm 4$ high-confidence HVSs should have
been uncovered in the radial velocity catalogue of Gaia DR3. Following from
Fig. 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 $\sigma(N_{\rm HVS})/N_{\rm HVS}\simeq
0.2$ for $N_{\rm HVS}\gtrsim 100$, rising to $\simeq 0.5$ and $\simeq 1$ if
$\sim$tens or only a few HVS were uncovered, respectively. For $M_{\rm
c}\lesssim 10^{5}\,\mathrm{M_{\odot}}$ $N_{\rm HVS}$ increases monotonically
with decreasing $a_{\rm current}$. For larger companion masses, however,
$N_{\rm HVS}$ decreases for $a_{\rm current}$ less than $\sim$ a few $\times
10^{-1}$ mpc since the MBHB slingshot mass ejection rate peaks and begins to
decline before GW emission takes over and the binary quickly merges (see Fig.
2 middle panel). The dashed white line indicates where the 1$\sigma$ lower
limit of $N_{\rm HVS}$ 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_{\rm
c}\lesssim$2000 $\mathrm{M_{\odot}}$). A near-equal mass MBHB remains possible
only for separations $\gtrsim 0.05\mathrm{pc}$.

Figure 5: Constraints on the mass of a black hole companion to Sgr A* and its
separation from Sgr A* in angle (left axis) and physical space (right axis).
Adapted from GRAVITY Collaboration et al. (2020), adapted in turn from
Gualandris & Merritt (2009). The lack of confident HVS candidates in Gaia DR3
excludes the region below the black line. The dashed horizontal line
highlights the semimajor axis of S2 at 125 mas ($\approx 4.7\,\mathrm{mpc}$
GRAVITY Collaboration et al., 2020). Yu & Tremaine (2003) previously excluded
the orange region by remarking that the barycentre of a presumed MBHB in the
GC cannot be significantly displaced from Sgr A*, otherwise the cusp of
stellar density in the GC would not be coincident with Sgr A*. The cyan, blue
and dark blue regions are excluded from astrometric observations of Sgr A*
(Hansen & Milosavljević, 2003; Reid & Brunthaler, 2004, 2020). The violet
region is excluded by the orbit of S2 (Gillessen et al., 2009). The green
region is excluded by Naoz et al. (2020) by requiring that the orbit of S2 is
stable against perturbations by a Sgr A* companion (a fading colour denotes
weakening constraints). The solid gold region is excluded by Gualandris &
Merritt (2009) using the orbital eccentricity distribution of the S-star
cluster. Companion masses to the right of the dotted red line are excluded by
Gualandris et al. (2010) from the orbit of S2. Following the 2018 pericentric
passage of S2, these constraints were improved by GRAVITY Collaboration et al.
(2020). The diagonal dotted lines show lines of constant gravitational wave
inspiral time for the MBHB.

Prior works have constrained a companion to Sgr A*. In Fig. 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 Fig. 5 with lines of
constant gravitational wave coalescence times. 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 $\lesssim 5\times
10^{4}\,\mathrm{M\odot}$ companion at a separation greater than a few tens of
milliparsecs from Sgr A*, ii) a $M=100-200\,\mathrm{M_{\odot}}$ companion just
within or outside the orbit of S2 (970 AU, or 4.7 mpc), or iii) a $M\lesssim
500\,\mathrm{M_{\odot}}$ companion at a separation of less than a few hundred
microparsecs from Sgr A*.

###  3.2 A former companion to Sgr A*

Figure 6: Left: The dependence of $N_{\rm HVS}$ on the time $t_{\rm since}$
since the MBHB has merged with a companion. A negative $t_{\rm since}$
indicates that the MBHB has not yet merged – when this is the case, the top
horizontal axis indicates the MBHB separation. Shaded regions span the 16th to
84th quantiles over 50 iterations. Right: contour lines show how the
population HVSs detectable in Gaia DR3 depends on the former Sgr A* companion
mass $M_{\rm c}$ and time elapsed since the merger $t_{\rm since}$, averaged
over 20 realizations and smoothed over the grid. The colourbar shows how the
1$\sigma$ scatter of NHVS. The dashed line shows where the 1$\sigma$ lower
bound of $N_{\rm HVS}$ equals one.

As the MBHB spirals in, the HVS mass ejection rate peaks at a small separation
(Fig. 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 Fig. 6. In the
left panel we show how $N_{\rm HVS}$ depends on the time $t_{\rm since}$ since
the MBHB in the GC merged, holding $M_{\rm c}$ and the IMF index $\kappa$
fixed. A negative $t_{\rm since}$ indicates the binary has not yet merged.
$N_{\rm HVS}$ 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 Sec. 4).

In the right panel of Fig. 6 we show how $N_{\rm HVS}$ depends on both $M_{\rm
c}$ and $t_{\rm since}$ when we sample over $\kappa$. The dashed line
indicates where the 1$\sigma$ lower limit of $N_{\rm HVS}$ 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 $1000\,\mathrm{M_{\odot}}$, it must have
merged with Sgr A* more than 12 Myr ago. This lower limit on $t_{\rm since}$
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 $\sim$200 $\mathrm{M_{\odot}}$, current HVS observations cannot constrain
how recently this merger occurred.

###  3.3 Prospects for DR4

Figure 7: The coloured bands show configurations of $M_{\rm c}$ and $a_{\rm
current}$ (left) or $M_{\rm c}$ and $t_{\rm since}$ (right) space consistent
within 1$\sigma$ with finding the labelled number of HVSs in Gaia DR4. The
shaded grey regions are already excluded by the lack of HVSs in Gaia DR3.

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 $\sim$2026\. In Fig. 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. The coloured bands show the parameter
space allowed if $1$, $10\pm\sqrt{10}$, $100\pm 10$ 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 $10^{5}\,\mathrm{M_{\odot}}$
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 $N_{\rm HVS}$ depends so intimately on $M_{\rm c}$ and
$a_{\rm current}$/$t_{\rm since}$, strict (albeit degenerate) constraints can
be placed on these parameters if a non-zero number of HVSs are uncovered in
DR4.

##  4 Discussion

###  4.1 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 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 Fig. 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 $M_{\rm c}$ or $a_{\rm
current}$ (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 Fig. 8 shows that as the time
$t_{\rm since}$ 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
$t_{\rm since}$). However, given that we expect $5_{-4}^{+11}$ 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 (Fig. 7),
uncontroversially assigning each HVS to a corresponding ejection mechanism
will be difficult in any case.

Figure 8: Among HVSs detectable in Gaia DR4, distributions of HVS stellar mass
$m$, Gaia $G$-band magnitude, heliocentric distance $d_{\rm hel}$, line-of-
sight velocity $v_{\rm rad}$ and HVS stellar age $t_{\rm age}$. Black curves
show distributions for HVSs ejected via the Hills mechanism and coloured
curves show distributions for HVSs ejected via the MBHB slingshot mechanism
when the companion mass (top row), MBHB separation (middle row) and time since
the MBHB merger (bottom row) are varied.

###  4.2 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 Centre131313Note, however, that unless one invokes
an interaction with a massive black hole, it becomes quite difficult to
explain the extreme velocities of young HVS candidates with well-constrained
total velocities such as HVS1 (Brown et al., 2005)..

Worth mentioning as well is the HVS candidate S5-HVS1, a $\sim
2.35\,M_{\odot}$ star first identified by Koposov et al. (2020) in the S5
survey (Li et al., 2019). S5-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 $4.8\,\mathrm{Myr}$
ago with a velocity of $v_{\rm ej}\simeq 1800\,\mathrm{km\ s^{-1}}$. As an
unambiguous HVS detection, we showed in Evans et al. (2022b) that penalizing
models which predict zero or $\gg$1 HVSs similar to S5-HVS1 in the S5 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 S5 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.,
2018) using the MIST models as done in Sec. 2.3. We then roughly reproduce the
S5 selection function, selecting HVSs within the S5 sky footprint (see Li et
al., 2019, 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 Fig. 9. Assuming the MBHB
slingshot mechanism is the only mechanism ejecting HVSs, the left panel shows
the number of S5-HVS1 analogues predicted to appear in S5 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 star similar to S5-HVS1 in S5, 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
S5-HVS1 is only possible if Sgr A* has a $200\,\mathrm{M_{\odot}}\lesssim
M_{\rm c}\lesssim 500\,\mathrm{M_{\odot}}$ companion at a separation of
$\lesssim 0.3$ pc.

In the right-hand panel we show similar results in $M_{\rm c}-t_{\rm since}$
space. Sgr A* having merged within the last few tens of Myr with a $\sim
200-1000\,\mathrm{M_{\odot}}$ companion is consistent with both the existence
of S5-HVS1 and the lack of HVSs in DR3. Note, however, that S5-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
$200\,\mathrm{M_{\odot}}\lesssim M_{\rm c}\lesssim 500\,\mathrm{M_{\odot}}$
companion can reconcile the existence of an HVS in S5 with the non-detection
of HVSs in Gaia DR3.

In conclusion, our simulations only support a MBHB slingshot origin for
S5-HVS1 in specific circumstances. Additionally, our criteria to select
S5-HVS1 analogues above makes no consideration for its extreme Galactocentric
velocity of $1750\,\mathrm{km\ s^{-1}}$. A velocity this large is quite hard
to achieve in the MBHB slingshot mechanism for a companion mass of $\lesssim
500\,\,\mathrm{M_{\odot}}$ – only $\sim$20 per cent of our S5-HVS1 analogues
in this mass range are as fast as S5-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
S5-HVS1 could be produced by this mechanism is warranted and does not yet
exist. Generozov & Madigan (2020) used S5-HVS1’s flight time as an
observational constraint and found that a $1000\,\mathrm{M_{\odot}}$ 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 S5-HVS1 was ejected via the MBHB slingshot mechanism.

Figure 9: The colorbar shows the population of HVSs predicted to appear in the
S5 survey (see Sec. 4.2 for details) in both $M_{\rm c}$-vs-$a_{\rm current}$
space (left) and $M_{\rm c}$-vs-$t_{\rm since}$ space (right). Configurations
below the red dashed lines are excluded by the results of this work. The
white, hashed areas show regions of parameter space consistent with finding
exactly one HVS in S5. The gold horizontal line in the right plot shows the
inferred $S^{5}$-HVS1 flight time of $4.8\,\mathrm{Myr}$ (Koposov et al.,
2020).

###  4.3 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; Dotti et al., 2007; Muñoz et al.,
2019) and would influence the stellar evolution 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 $10^{-6}-10^{-5}\,\mathrm{M_{\odot}\ yr^{-1}}$
(Baganoff et al., 2003; Quataert, 2004) at the Bondi radius
($0.04\,\mathrm{pc}$) and polarization measurements limit the accretion rate
to $\sim 10^{-8}\,\mathrm{M_{\odot}\ yr^{-1}}$ 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

$\frac{\dot{a}}{a}=2\left(\frac{\dot{\ell}}{\ell}-\frac{3}{2}\right)\frac{\dot{M}}{M}\;\text{,}$
(16)

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 co-
rotation 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 ($a\lesssim 0.3a_{\rm h}$) if the NSC is maximally corotating
(Rasskazov et al., 2019), in which case our assumption of a non-rotating NSC
may be overestimating $N_{\rm HVS}$. 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 $\sqrt{m_{*}/M_{\rm MBHB}}\sim 10^{-4}$
rad relative to its initial orientation, where $m_{*}$ is the typical mass of
a star in the NSC and $M_{\rm MBHB}$ 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 the 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 midplane, our assumption
of isotropic ejections would slightly underestimate $N_{\rm HVS}$, 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 interaction 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;
Madigan et al., 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, $\sim
10^{2}-10^{6}\,\mathrm{M_{\odot}}$ 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 $N_{\rm HVS}$ in
cases of partial depletion. This assumption is likely the largest source of
systematic uncertainty in our modelling. Given the strong dependence of
$N_{\rm HVS}$ 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).

##  5 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) (Fig. 3)

  * •

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

  * •

The lack of such HVSs in DR3 excludes a companion within $1\,\mathrm{mpc}$ of
Sgr A* unless it has a mass of $\lesssim 1000\,\mathrm{M_{\odot}}$,
complementing and extending prior constraints on a Sgr A* companion (Fig. 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 $\lesssim 500\,\mathrm{M_{\odot}}$, or ii) the
merger must have happened more than $\sim$10-30 Myr ago (Fig. 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 (Fig. 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 $\sim 10^{5}\mathrm{M_{\odot}}$ is still
allowed at separations larger than $\sim 0.1\,\mathrm{pc}$, 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; de Jong et al. 2019, 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 definitely ruled out as an avenue for HVS ejections and future research can
focus on more realistic mechanisms.

## Acknowledgements

The authors thank J.R. Westernacher-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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