# The last three years: multiband gravitational-wave observations of stellar-
mass binary black holes

Antoine Klein<EMAIL_ADDRESS>Institute for Gravitational Wave
Astronomy & School of Physics and Astronomy, University of Birmingham,
Birmingham, B15 2TT, UK  Geraint Pratten  Institute for Gravitational Wave
Astronomy & School of Physics and Astronomy, University of Birmingham,
Birmingham, B15 2TT, UK  Riccardo Buscicchio  Institute for Gravitational Wave
Astronomy & School of Physics and Astronomy, University of Birmingham,
Birmingham, B15 2TT, UK  Dipartimento di Fisica “G. Occhialini”, Universitá
degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy
Patricia Schmidt  Institute for Gravitational Wave Astronomy & School of
Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK
Christopher J. Moore  Institute for Gravitational Wave Astronomy & School of
Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK
Eliot Finch  Institute for Gravitational Wave Astronomy & School of Physics
and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK  Alice Bonino
Institute for Gravitational Wave Astronomy & School of Physics and Astronomy,
University of Birmingham, Birmingham, B15 2TT, UK  Lucy M. Thomas  Institute
for Gravitational Wave Astronomy & School of Physics and Astronomy, University
of Birmingham, Birmingham, B15 2TT, UK  Natalie Williams  Institute for
Gravitational Wave Astronomy & School of Physics and Astronomy, University of
Birmingham, Birmingham, B15 2TT, UK  Davide Gerosa  Institute for
Gravitational Wave Astronomy & School of Physics and Astronomy, University of
Birmingham, Birmingham, B15 2TT, UK  Dipartimento di Fisica “G. Occhialini”,
Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, 20126
Milano, Italy  INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, 20126
Milano, Italy  Sean McGee  Institute for Gravitational Wave Astronomy & School
of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK
Matt Nicholl  Institute for Gravitational Wave Astronomy & School of Physics
and Astronomy, University of Birmingham, Birmingham, B15 2TT, UK  Alberto
Vecchio  Institute for Gravitational Wave Astronomy & School of Physics and
Astronomy, University of Birmingham, Birmingham, B15 2TT, UK

###### Abstract

Understanding the formation and evolution of the stellar-mass binary black
holes discovered by LIGO and Virgo is a challenge that spans many areas of
astrophysics, from stellar evolution, dynamics and accretion disks, to
possible exotic early universe processes. Over the final years of their lives,
stellar-mass binaries radiate gravitational waves that are first observable by
space-based detectors (such as LISA) and then ground-based instruments (such
as LIGO, Virgo and the next generation observatories Cosmic Explorer and the
Einstein Telescope). Using state-of-the-art waveform models and parameter-
estimation pipelines for both ground- and space-based observations, we show
that (the expected handful of) these multiband observations will allow at
least percent-level measurements of all 17 parameters that describe the
binary, the possible identification of a likely host galaxy, and the
forewarning of the merger days in advance allowing telescopes at multiple
wavelengths to search for any electromagnetic signature associated to it.
Multiband sources will therefore be a gold mine for astrophysics, but we also
show that they could be less useful as laboratories for fundamental tests of
general relativity than has been previously suggested.

Introduction –  The first detection of gravitational waves (GWs), GW150914
[1], also revealed merging stellar-mass binary black holes (SmBBHs) with
masses well above ${\sim 10\,M_{\odot}}$. Subsequently, tens of BBHs with
component masses $\gtrsim 10\,M_{\odot}$ have been observed by the LIGO-Virgo
network [2]. The observed mass distribution of these black holes extends
beyond the maximum mass predicted by the onset of (pulsational) pair-
instability during the explosion of the progenitor star [3], as firmly
established by the detection of GW190521 [4]. No host galaxy of any BBH has so
far been identified, although a possible counterpart of GW190521 has been
suggested [5]. This means we have no direct information about the environment
in which these systems form.

Many potential formation and evolutionary scenarios for these BBHs have been
suggested. These range from isolated formation to dynamical assembly in dense
clusters or AGN disks, or less conventional routes such as primordial BH
formation (see Refs. [6, 7, 8] for reviews). Clearly identifying an
evolutionary path is a holy grail of the field, with profound consequences for
a range of open problems in stellar physics, binary (and higher multiplicity)
stars, dynamics in dense environments, and relativistic astrophysics.
Observationally, providing definitive evidence of a particular path is
challenging, as it would ideally include a host galaxy identification and
precise measurements of the orbital eccentricity and the masses and spins of
the component BHs.

The progenitors to SmBBH mergers are potential sources [9] for the Laser
Interferometer Space Antenna (LISA) [10]. LISA will detect SmBBHs radiating
GWs at $\sim 10\,\mathrm{mHz}$ several years 111The title of this letter
intentionally mirrors “The last three minutes” by Cutler  _et al._ [11] — a
seminal study that set the stage for GW studies of compact binaries with
ground-based detectors. prior to merger. A handful of these systems are
expected to be detected at low signal-to-noise ratios (SNRs) [12, 13, 14].
These sources will become the first multiband GW signals when they merge $\sim
3\,$years later at $\sim 100\,\mathrm{Hz}$ in the sensitivity band of the
ground-based instruments operating at the time (e.g. an improved version of
the current LIGO-Virgo-KAGRA network or next generation (3G) observatories,
such as Cosmic Explorer (CE) [15] and/or the Einstein gravitational-wave
Telescope (ET) [16]).

In this letter, we assess the scientific potential of these multiband sources
using state-of-the-art waveforms and fully realistic Bayesian parameter
estimation in both frequency regimes for the first time. We consider two
prototype sources with detector-frame (i.e. redshifted) BH masses $m_{1(2)}$
and dimensionless spins $\chi_{1(2)}$ consistent with those of GW150914
(“light”) and GW190521 (“heavy”), respectively. The minimum signal-to-noise
ratio threshold needed for the detection of a SmBBH in LISA is uncertain; we
simulate these systems with SNRs of $11.9$ and $15.3$ respectively which are
close to threshold. For both systems, we simulate them from the start of LISA
observations, which we take to be 3 years before merger, all the way to
coalescence, when the GW signal is observed by ground-based observatories at
SNR $\sim 10^{3}$. We consider two possibilities for the orbital eccentricity
(“low” and “high”) in the LISA band and assume that (due to GW emission) they
have negligible eccentricity by the time they enter the frequency band of
ground-based instruments [17]. The waveforms for these systems are shown in
Fig. 1 and the system parameters are summarised in Table 1 (full details are
given in the supplement). We simulate these multiband sources consistently
across all frequencies using waveforms that include eccentricity and spin-
induced precession and higher multipoles where relevant.

Table 1:  Summary of the properties of the injected sources. The orbital eccentricity at $10\,$mHz is denoted ${e_{0.01}}$.  Name | $m_{1}$ | $m_{2}$ | $\chi_{1}$ | $\chi_{2}$ | ${e_{0.01}}$ | $z$
---|---|---|---|---|---|---
Light | $36\,M_{\odot}$ | $29\,M_{\odot}$ | $0.132\,949\,735\,971\,405\,05$ | $0.138\,395\,242\,455\,940\,84$ | ${}^{$0.0001$}_{$0.0077$}$ | $0.032\,833\,741\,287\,371\,31$
Heavy | $85\,M_{\odot}$ | $65\,M_{\odot}$ | $0.761\,599\,575\,852\,663\,9$ | $0.854\,529\,220\,271\,256\,3$ | ${}^{$0.0001$}_{$0.0316$}$ | $0.108\,877\,671\,938\,197\,38$
Figure 1:  The multiband waveforms starting 3 years before merger chirp
through the upper end of the LISA band at $\sim 10\,\mathrm{mHz}$ into the
ground-based band (starting at $10\,\mathrm{Hz}$) where they merge. Blue (red)
solid lines show the characteristic strain for the light (heavy), high
eccentricity sources. The sources take several days to cross from the LISA to
the ground-based band. Black solid/dashed/dotted lines show the design
sensitivities for CE/ET/LISA whilst the dash-dotted lines show the target O5
sensitivities of LIGO/Virgo. Higher modes and spin precession characterize the
signal modulation for ground based detectors. In the LISA band, signal
modulation occurs due to spacecraft motion around the Sun and, for these high
eccentricity sources, contributions from subdominant harmonics visible at the
lowest frequencies.

LISA observations –  The early inspiral is well described by post-Newtonian
theory and is governed primarily by the chirp mass ${\cal
M}_{c}=(m_{1}m_{2})^{3/5}/(m_{1}+m_{2})^{1/5}$. The inspiralling, precessing,
and eccentric signal is modelled using an approach that leverages analytic
solutions of the conservative problem to efficiently integrate the equations
of motion over radiation-reaction timescales [18]. Following [19], we perform
Bayesian inference on the full LISA time delay interferometry outputs using
the Balrog code (for details, see the supplement).

In all cases, the chirp mass is extremely well measured, with fractional
errors of $\sim 10^{-4}$. Crucially, LISA also makes accurate measurements of
the eccentricity, ${e_{0.01}}$ (at a reference frequency of
$10\,\mathrm{mHz}$), with absolute errors of $\sim 10^{-4}-10^{-3}$. The other
intrinsic parameters (including the mass ratio and component spins) are not
well measured, see figures and tables in the supplement.

The LISA measurements of the chirp mass and eccentricity are partially
degenerate with the time of merger, $t_{m}$, which is determined to within a
few hours. When the system eventually merges in the ground-based band, $t_{m}$
is determined to within $\sim 1\,\mathrm{ms}$, breaking this degeneracy.
Therefore, LISA measurements combined with just the time of the 3G detection
(and no other information) allows a more precise determination of ${\cal
M}_{c}$ and ${e_{0.01}}$ (see Fig. 2). The improved eccentricity measurements
mean that binaries with residual eccentricities $\sim 10^{-3}$ at orbital
periods of a few minutes can be distinguished from circular.

Figure 2:  LISA posteriors (dotted blue) on the chirp mass $\mathcal{M}_{c}$,
time to merger $t_{m}$, and eccentricity $e_{0.01}$ for the “light”, high-
eccentricity source. Solid/dashed blue lines show posteriors from third
generation ET/CE ground-based instruments. The LISA measurement of
$\mathcal{M}_{c}$ (and, to a lesser extent, $e_{0.01}$) is degenerate with
$t_{m}$ which is broken when combining with the merger time measured from any
ground-based instrument leading to significant improvements in the combined,
multiband measurements (shown in black). Similar results for the “heavy”
source are shown in Fig. S1.

Ground-based observations –  It is currently uncertain what ground-based
instruments will be operating when the first multiband sources merge. We
consider 3 possibilities: an upgrade (target O5 sensitivity) of the existing
LIGO-Virgo network [20] (hereafter HLV+ ), and the 3G instruments ET or CE.
Adopting a conservative low-frequency cutoff of $10\,\mathrm{Hz}$, the
observations span $\sim 1-5\,\mathrm{s}$, ending with the extremely
relativistic merger. These signals will be extremely loud, with ground-based
SNRs in the range from several hundreds to thousands (see Tables S1 and S2 in
the supplement).

We model the signals above $10\,\mathrm{Hz}$, including the effects of spin
precession and higher order modes (the eccentricity is $\lesssim 10^{-5}$ and
is set to zero), using IMRPhenomXPHM [21, 22, 23]. We further assume the noise
and calibration in each instrument are known at a level much better than the
statistical uncertainty affecting the observations, and perform Bayesian
analysis using Bilby [24] (for details, see the supplement).

The large SNRs in the ground-based band allow for exquisite measurements of BH
masses and spins. In the 3G detectors, the individual masses are measured with
fractional errors of $\sim 0.1\%$ and the spin magnitudes and orientations at
the $1\%$ level; see Fig. 3 and the supplement. (In the HLV+ network, errors
are approximately an order of magnitude larger due to the lower SNR.)

Figure 3:  One-dimensional marginalised posteriors on the intrinsic parameters
for the “light”, high-eccentricity source: $10\,\mathrm{mHz}$ eccentricity
${e_{0.01}}$, detector frame component masses $m_{1(2)}$, dimensionless spin
magnitudes $\chi_{1(2)}$, spin tilt angles $\theta_{1(2)}$, and the angle
between the in-plane spins $\phi_{12}$. Dot/dash/solid/dot-dash lines indicate
LISA/CE/ET/HLV+ posteriors. LISA measures eccentricity extremely well from the
early inspiral (the black curve shows the improvement with the multiband
measurement of $t_{m}$) and the 3G instruments measure the other intrinsic
parameters with exquisite accuracy. The injected eccentricity is indicated;
for all other parameters, $\Delta$ denotes a shift from the injected value.
Similar results for the “heavy” source are shown in Fig. S1.

Localisation –  Multiband GW observations are expected to be restricted to a
small number of high-mass, low-redshift ($z\lesssim 0.1$) sources. The
advanced warning provided by LISA makes these highly promising targets for
multimessenger follow-up across the electromagnetic (EM) spectrum, and hence
localisation is crucial.

The LISA observations, ending several days before merger, predict the time of
the merger to within a few hours and the sky position to within a 90% area of
$\approx$4\text{\,}{\mathrm{deg}}^{2}$$ ($10\text{\,}{\mathrm{deg}}^{2}$) for
the light (heavy) source; see Fig. 4. LISA also measures the distance to these
sources with fractional errors of $\sim 30\%$. Assuming the LISA data
processing can be performed sufficiently quickly, there is time for telescopes
to be positioned before the merger.

With ground-based observations, the localisation improves further. This
improvement depends on the number and locations of the ground-based
instruments; we show results for a single triangular-shaped ET detector and
the HLV+ network (we do not consider CE, as a single L-shaped instrument
operating alone is not expected to determine the sky position accurately).

ET alone localises these extremely loud sources to a small area; however, the
posteriors contain several widely separated modes (see Fig. 4) correlated with
the GW polarisation. Only one of these modes is consistent with the LISA
measurements, and so the combined multiband (LISA-ET) network can localise the
sources to within $4\,\mathrm{deg}^{2}$ ($0.7\,\mathrm{deg}^{2}$). ET can also
measure the luminosity distances to fractional errors of 0.6% (2%) thereby
localising the sources to a 3D 50% volumes of $3\,\mathrm{Mpc}^{3}$
($50\,\mathrm{Mpc}^{3}$). For comparison, the HLV+ network can potentially
make better measurements of the sky position because of the multiple,
separated instruments (see Fig. 4 insets) but less accurate distance
measurements due to the lower SNRs; the 3D volumes are $1\,\mathrm{Mpc}^{3}$
($700\,\mathrm{Mpc}^{3}$).

These small localisation volumes mean that it may be possible to identify a
likely host galaxy from GW observations alone, even without a counterpart.
Integrating the galaxy luminosity function [25] over the range $0.001-10L_{*}$
gives a galaxy number density of $\approx 1\,\mathrm{Mpc}^{-3}$. Restrict
further to likely hosts with $L\gtrsim L_{*}$, this density reduces to $\sim
0.2\,\mathrm{Mpc}^{-3}$, giving in $\sim 1-100$ likely host galaxies in these
localisation volumes.

Multimessenger astronomy –  The form or existence of any electromagnetic
counterpart to BBH mergers is currently unknown. If a host is identified even
in the absence of a counterpart, this will yield the redshift allowing for a
measurement of the Hubble constant with a statistical error $\lesssim
1\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ for LISA-3G multiband
observations. In addition, identifying the host will also enable detailed
measurements of the environmental metallicity and star-formation history,
providing additional information on the formation channel of the binary. If
the host contains an AGN, this would suggest accretion disks may play a role
in the formation, since the number density of AGN is low at $\sim
10^{-5}\,\mathrm{Mpc}^{-3}$ [26].

Unique to multiband events is the few days advance warning of a merger with a
localization of $\sim 1\,\mathrm{deg}^{2}$. This is well-matched to several
wide-field instruments across the EM spectrum. On the shortest timescales, the
Square Kilometre Array (SKA; field-of-view (FOV) $\sim 1\,\mathrm{deg}^{2}$)
would place sensitive constraints on any associated fast radio burst (FRB),
e.g. could detect a faint FRB – like the 2020 outburst of SGR 1935+2154 [27,
28] – at $z=0.03$. Similar gamma-ray burst (GRB) tests could be done with
missions including SVOM [29], THESEUS [30], Gamow [31] and the Einstein Probe
[32], characterised by $\sim$ steradians FOV. Pointed X-ray instruments will
play an important role in constraining any shock breakout transient, e.g. from
a dense circumstellar medium. ATHENA’s FOV of $\sim 0.5\,\mathrm{deg}^{2}$
will cover most of the localization area [33, 34], and the planned ULTRASAT
[35] would provide sensitive, wide-field UV coverage.

Optical or infrared transients, if present, are expected to arise on longer
timescales (from days to weeks/months), either as a result of an off-axis GRB
afterglow or a thermal/diffusive transient, such as a merger within an AGN
disk [36, 5]. The Rubin Observatory would be an ideal instrument for repeat
monitoring to search for optical emission on a wide range of timescales, with
a FOV that fully covers the localisation area and benefiting from pre-existing
templates at full LSST 10-year depth to perform image subtraction for faint
transient detection. A high-confidence host galaxy association enables
monitoring with sensitive and high-resolution (but narrow-field) instruments,
such as JWST (which may last until the 2040s after a successful launch) or
ELT. Repeat visits could search for very faint EM emission in the optical or
infrared in the weeks following merger – a 1-hour exposure with JWST could
probe 100 times deeper than a single visit with Rubin.

Figure 4:  _Main panel_ : Mollweide projection of the sky in galactic
coordinates (ecliptic plane and poles are indicated by black line and crosses,
respectively). Blue/red plots shows results for the light/heavy sources. The
ET sky maps are multimodal with a small number of small, well-separated
regions. _Left insets_ : Zoom-in on the true location, showing the 90%
contours of LISA/ET/HLV+ measurements in dotted/solid/dot-dashed curves.
_Right insets_ : Posteriors on the luminosity distance $d_{L}$.

Astrophysical implications –  Only a few SmBBHs are expected to be observed
first by LISA and then ground-based instruments, but the rate is uncertain and
depends on the binary merger rate, the mass spectrum and the SNR detection
threshold required in LISA [12, 13, 14]. Searches of LISA archival data
targeted on systems detected with a ground-based network will further increase
this number [37, 12].

While the BH masses are highly degenerate among the many proposed formation
pathways of SmBBHs, spins and eccentricities are believed to be better
tracers. SmBBH eccentricities can be excited by dynamical encounters and are
thus unique observables to probe black-hole multi-body dynamics.
Circularization highly suppresses the eccentricity in the ground-based band,
but observing SmBBHs in the LISA band provides a new, powerful leverage [38,
39]. At $\sim 10\,\mathrm{mHz}$, binaries formed in galactic fields (dense
clusters) are expected to have eccentricities of $\sim 10^{-4}\\!-\\!10^{-3}$
($10^{-3}\\!-\\!10^{-1}$) [39]. Kozai-Lidov oscillations induced by a nearby
supermassive black-hole can increase the SmBBH’s eccentricity to
${e_{0.01}}\lesssim 1$ [40]. SmBBH spin directions also encode information on
the evolutionary mechanism [41, 42, 43, 44]. Dynamical interactions should
randomize the spin orientations. On the other hand, binaries formed in
isolation will be subject to small misalignment $\lesssim 20^{\circ}$ [44] at
merger, induced by e.g. supernova kicks [45] and tidal interactions [46]. Spin
misalignments in the ground-based and LISA bands are strongly correlated [47].

Our results show that LISA can measure ${e_{0.01}}$ with absolute errors of
$\sim 10^{-4}-10^{-3}$ while 3G detector will provide estimates of the spin
tilts down to $\sim 1\mathrm{deg}$. The resulting astrophysical inference from
either of the measurements alone will already be incredibly powerful, but a
consistent determination of the formation channel from two observables (spins
and eccentricity) and two bands (ground and space) will open a new era of
precision black-hole binary astrophysics. Identification of the galactic host
would allow for precise determination of the metallicity of the environment
where the source evolved. This will place a further constraint in pinning down
the details of the formation mechanism (e.g. [48]), possibly breaking the mass
degeneracy that affects the proposed pathways.

Tests of Fundamental Physics –  The prospect of multiband GW observations
spanning several orders of magnitude in frequency would seem to offer a unique
opportunity for testing general relativity (GR) across a wide range of scales
with a single source [49]. Indeed, several papers have suggested that these
multiband sources will be ideal for testing aspects of fundamental physics
such as parametrized post-Einsteinian deviations [50], inspiral-merger-
ringdown consistency tests [51, 52, 53], measurements of post-Newtonian
coefficients [54, 55], and the existence of dipole GW radiation [56, 57, 58].

However, our results suggest these sources are not as good for testing GR as
had been initially hoped, for several related reasons. Firstly, most previous
studies have relied on Fisher matrices, which are not well-suited to these
sources, particularly in the LISA band [59]. Secondly, the low SNR of the LISA
observations, compared to the ground-based instruments, means some multiband
tests are limited by LISA and are no more constraining than what is achievable
with 3G ground-based detectors alone, or with high-SNR LISA observations of
supermassive BBHs. This was also suggested in Ref. [59].

Consider the possibility of dipole emission (which can occur in modified
gravity theories, or if the BHs are electrically charged). It dominates at low
frequencies, accelerating the inspiral and reducing the time to merger. It has
been suggested that multiband sources, where the merger time can be predicted
from the low-frequency signal, should be ideal for constraining dipole
emission. Ref. [57], using a Fisher matrix analysis, found that LISA can
predict the time of merger for a GW150914-like source to within $\pm
10\,\mathrm{s}$ and place a constraint on the size of a dipole contribution to
the flux that is an order of magnitude better than is possible with other GW
sources (although still worse than the double pulsar constraint [60] for
relevant theories). However, as has been shown here with a full Bayesian
analysis, the LISA determination of the merger time is significantly worse at
around $\pm 3\,\mathrm{hours}$ (see Fig. 2). As the dipole constraint degrades
linearly with the time of merger uncertainty, the constraints on the dipole
could be less constraining than has been previously suggested. This should be
verified with a Bayesian analysis incorporating modifications of GR. We find
that for our light source, the dimensionless dipole coupling $B$ [56] can be
constrained to $\lesssim 10^{-7}$; this corresponds to a constraint on the BH
charges of $Q\lesssim 2\times 10^{-4}Q_{\rm max}$, where $Q_{\rm max}$ is the
extremal BH charge.

Challenges to be met –  These multi-band observations pose challenges on data
processing, waveform modelling and instrument calibration that require work to
start immediately.

The early warning that enables telescope pointing ahead of the merger requires
the LISA data processing to be done in near real time and completed within
$\sim$ a day from the time a SmBBH candidate signal exits the LISA band. We
currently do not have a search strategy for SmBBHs, although solutions are
being developed as part of the LISA Data Challenges [61]. Accurate
measurements of the coalescence time and sky position with the fully Bayesian
techniques used here are highly computationally expensive (see also [19]).
This combination of factors drives demanding requirements on LISA data
processing and data transfer. Search strategies also need to be developed in
the context of LISA archival searches for signals detected in ground-based
data sets.

The exquisitely precise characterisation of the properties of these sources
requires waveforms for eccentric and precessing systems that are phase
coherent over $\sim 10^{6}$ binary orbits and $\sim 10^{3}$ precession and
periapsis precession cycles [62]. Moreover, the extreme SNRs of $\sim
10^{2}-10^{3}$ in the ground-based observations, where one should also account
for the possible presence of small residual eccentricities, places
requirements on waveform accuracy that go well beyond what is currently
available.

Equally, one needs to ensure that other modelling components of the GW signal
as measured at the detector output introduces systematic errors below the
statistical ones. Specifically, in our analysis, we have not considered
uncertainties coming from instrument calibration errors or unknown noise
levels. We have de-facto assumed calibration is exact. Calibration errors will
need to be smaller than $0.1\%$ in amplitude and $\lesssim 0.1\,\mathrm{deg}$
in phase.

Finally, we want to point out that there is a blind (in frequency) spot in
observing these sources at $\sim 0.1-1\,\mathrm{Hz}$. Several space-based
mission concepts that can cover this frequency band are currently under study
[63, 64, 65]. Although the impact of the operation of these instruments
concurrently with LISA and ground-based instruments is beyond the scope of
this paper, unquestionably they would provide additional information.

Acknowledgments –  We thank all the developers of the Balrog codesuite,
including those who are not authors here: Janna Goldstein, Elinore Roebber,
Sebastian Gaebel, Siyuan Chen, Christopher Berry. Computational resources used
for this work were provided by the University of Birmingham’s BlueBEAR High
Performance Computing facility and by Supercomputing Wales, funded by STFC
grants ST/I006285/1 and ST/V001167/1 supporting UK involvement in the
operation of Advanced LIGO. A.K., C.J.M. and A.V. acknowledge the support of
the UK Space Agency, Grant No. ST/V002813/1. A.V. acknowledges the support of
the Royal Society and Wolfson Foundation. R.B. is supported by the Italian
Space Agency Grant “Phase A LISA mission activities”, Agreement No.
2017-29-H.0, CUP F62F17000290005. D.G. is supported by European Union’s H2020
ERC Starting Grant No. 945155–GWmining, Cariplo Foundation Grant No.
2021-0555, and Leverhulme Trust Grant No. RPG-2019-350. P. S., G.P. and A.V.
acknowledge support from STFC grant ST/V005677/1. M.N. is supported by the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 948381) and by a
Fellowship from the Alan Turing Institute. Part of this research was performed
while G.P., P.S. and L.T. were visiting the Institute for Pure and Applied
Mathematics (IPAM), supported by the NSF, Grant No. DMS-1925919.

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Supplemental material: “The last three years: multiband gravitational-wave
observations of stellar-mass binary black holes”

##  I System parameters

Here, we provide detailed values of the source parameters selected for our
inference. They are summarised in Tables S1 and S2.

In the LISA band, we perform four distinct injections of gravitational-wave
signals from binary systems. Their parameters are consistent with those of
GW150914 (“ _light_ ”) and GW190521 (“ _heavy_ ”), with the exception of the
luminosity distance that we chose such that the binaries are near the
(currently understood) detection threshold SNR in LISA. In the detector frame,
the _light_ (_heavy_) binary has a primary component mass
$m_{1}=$36\text{\,}\mathrm{M_{\odot}}$($85\text{\,}\mathrm{M_{\odot}}$)$ and a
secondary component of
$m_{2}=$29\text{\,}\mathrm{M_{\odot}}$($65\text{\,}\mathrm{M_{\odot}}$)$.
Correspondingly, their chirp mass is
$\mathcal{M}_{c}=$28.1\text{\,}\mathrm{M_{\odot}}$($64.6\text{\,}\mathrm{M_{\odot}}$)$.
The luminosity distance is set to
$d_{L}=$149.1\text{\,}\mathrm{Mpc}$~{}($521.10\text{\,}\mathrm{Mpc}$)$, in
order to give an ${\rm SNR=11.9~{}(15.3)}$ for LISA over a mission duration of
$4\text{\,}\mathrm{yr}$.

Ecliptic longitude and latitude are set to $l=$5.01\text{\,}\mathrm{rad}$$ and
$b=$-1.49\text{\,}\mathrm{rad}$$ ($l=$5.55\text{\,}\mathrm{rad}$$ and
$b=$-0.83\text{\,}\mathrm{rad}$$), respectively, and the inclination to
$2.28\text{\,}\mathrm{rad}$ ($0.68\text{\,}\mathrm{rad}$). The binary is set
to an initial orbital frequency such that it reaches merger
$1\text{\times}{10}^{8}\text{\,}\mathrm{s}$ (approximately 3 years) after the
start of the LISA observation. For convenience, times are stated shifted such
that the merger occurs at $t_{m}=0$.

The signals are injected with a small but non-zero eccentricity. We perform
injections with two different eccentricities (“high” and “low”) for each
binary: $7.7\text{\times}{10}^{-3}$ and $1\text{\times}{10}^{-4}$ for the
light binary and $3.16\text{\times}{10}^{-2}$ and $1\text{\times}{10}^{-4}$
for the heavy binary. This spread of values was chosen to illustrate our
ability to confidently distinguish small values from zero –or not– with LISA
observations.

Conserved quantities are used consistently to generate and inject signals for
the ground-based detector networks. Non-conserved parameters (spin and orbital
angular momentum magnitudes and angles, and eccentricities) are defined at
$0.01\text{\,}\mathrm{Hz}$ for LISA signals. Dimensionless spin magnitudes
amount to $\chi_{1}=c|\vec{S_{1}}|/Gm_{1}^{2}=0.1329$ and
$\chi_{2}=c|\vec{S_{2}}|/Gm_{2}^{2}=0.1384$ ($\chi_{1}=0.7616$ and
$\chi_{2}=0.8545$). Spin tilts relative to the respective binary orbital
angular momentum are
$\theta_{1}=\arccos(\hat{S}_{1}\cdot\hat{L})=$1.45\text{\,}\mathrm{rad}$$ and
$\theta_{2}=\arccos(\hat{S}_{2}\cdot\hat{L})=$1.60\text{\,}\mathrm{rad}$$
($\theta_{1}=$0.92\text{\,}\mathrm{rad}$$ and
$\theta_{2}=$1.77\text{\,}\mathrm{rad}$$), respectively. The resulting angle
between the line-of-sight unit vector and the total angular momentum is set to
$\theta_{\text{JN}}=$2.03\text{\,}\mathrm{rad}$$
($\theta_{\text{JN}}=$0.700\text{\,}\mathrm{rad}$$) and
$\theta_{\text{JN}}=$2.09\text{\,}\mathrm{rad}$$
($\theta_{\text{JN}}=$0.682\text{\,}\mathrm{rad}$$) for the low-eccentricity
and high-eccentricity binaries, respectively. Analogously, the azimuthal angle
of $\hat{L}$ with respect to $\hat{J}$ is set to
$\phi_{\text{JL}}=$0.445\text{\,}\mathrm{rad}$$
($\phi_{\text{JL}}=$0.216\text{\,}\mathrm{rad}$$) and
$\phi_{\text{JL}}=$0.365\text{\,}\mathrm{rad}$$
($\phi_{\text{JL}}=$0.192\text{\,}\mathrm{rad}$$). These parameters are then
evolved to $10\text{\,}\mathrm{Hz}$ for injection in the ground-based detector
signals [66, 18]. Dimensionless spin magnitudes for ground-based detector
signals are set to $\chi_{1}=0.1329$ and $\chi_{2}=0.1383$ ($\chi_{1}=0.7615$
and $\chi_{2}=0.8544$). Spin tilts with respect to the binary orbital angular
momentum are $\theta_{1}=$1.53\text{\,}\mathrm{rad}$$ and
$\theta_{2}=$1.50\text{\,}\mathrm{rad}$$
($\theta_{1}=$0.83\text{\,}\mathrm{rad}$$ and
$\theta_{2}=$1.85\text{\,}\mathrm{rad}$$). The resulting angle between the
line-of-sight unit vector and the total angular momentum is set to
$\theta_{\text{JN}}=$2.31\text{\,}\mathrm{rad}$$
($\theta_{\text{JN}}=$0.88\text{\,}\mathrm{rad}$$), while the azimuthal angle
of $\hat{L}$ with respect to $\hat{J}$ is set to
$\phi_{\text{JL}}=$0.77\text{\,}\mathrm{rad}$$
($\phi_{\text{JL}}=$2.03\text{\,}\mathrm{rad}$$). The eccentricity is set to
zero in the ground-based band.

Except for the luminosity distance, these parameter choices are consistent
with the posteriors of GW150914 [67] and GW190521 [68], respectively.

Heavy binary
---
| Injected value | Posterior
|  0010mHz00.00010Hz0 | LISA | ET | CE | HLV+
| Eccentricity |  | High ecc. | Low ecc. |  |  |
| High $\mid$ Low |  |  |  |  |  |
$\rho_{\text{opt}}$ |  |  | $14^{+2}_{-2}$ | $14^{+2}_{-2}$ | $968^{+2}_{-2}$ | $4372^{+2}_{-2}$ | $248^{+2}_{-2}$
$\mathcal{M}_{\text{c}}$ $\left[M_{\odot}\right]$ | $64.59$ | $64.594^{+0.001}_{-0.002}$ | $64.592^{+0.002}_{-0.001}$ | $64.59^{+0.05}_{-0.05}$ | $64.59^{+0.01}_{-0.01}$ | $64.7^{+0.4}_{-0.4}$
$M_{\texttt{tot}}$ $\left[M_{\odot}\right]$ | $150.00$ | $151^{+14}_{-3}$ | $154^{+18}_{-6}$ | $150.00^{+0.11}_{-0.12}$ | $150.00^{+0.03}_{-0.03}$ | $150.2^{+1.0}_{-0.9}$
$q$ | $0.76$ | $>0.46$ | $>0.38$ | $0.76^{+0.01}_{-0.01}$ | $0.765^{+0.001}_{-0.001}$ | $0.77^{+0.02}_{-0.03}$
$m_{1}$ $\left[M_{\odot}\right]$ | $85.00$ | $90^{+30}_{-10}$ | $100^{+30}_{-20}$ | $84.99^{+0.31}_{-0.32}$ | $85.00^{+0.08}_{-0.07}$ | $85.0^{+1.5}_{-1.4}$
$m_{2}$ $\left[M_{\odot}\right]$ | $65.00$ | $60^{+10}_{-10}$ | $60^{+10}_{-10}$ | $65.01^{+0.26}_{-0.26}$ | $65.00^{+0.06}_{-0.06}$ | $65.2^{+1.1}_{-1.2}$
$\chi_{1}$ | $0.76$ | $0.3^{+0.3}_{-0.2}$ | $0.3^{+0.3}_{-0.2}$ | $0.76^{+0.02}_{-0.02}$ | $0.762^{+0.005}_{-0.004}$ | $0.7^{+0.2}_{-0.1}$
$\chi_{2}$ | $0.85$ | $0.3^{+0.5}_{-0.3}$ | $0.4^{+0.4}_{-0.4}$ | $0.85^{+0.02}_{-0.02}$ | $0.854^{+0.004}_{-0.004}$ | $0.7^{+0.2}_{-0.5}$
$\iota$ $\left[\text{rad}\right]$ | $0.542$ | $0.68$ | $0.9^{+0.2}_{-0.5}$ | $0.9^{+0.3}_{-0.5}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $0.71^{+0.11}_{-0.08}$
$\theta_{1}$ $\left[\text{rad}\right]$ | $0.92$ | $0.83$ | $0.8^{+1.3}_{-0.6}$ | $0.8^{+1.4}_{-0.5}$ | $0.83^{+0.02}_{-0.02}$ | $0.83^{+0.01}_{-0.01}$ | $0.8^{+0.3}_{-0.2}$
$\theta_{2}$ $\left[\text{rad}\right]$ | $1.77$ | $1.85$ | $1.0^{+1.3}_{-0.7}$ | $2.0^{+0.8}_{-1.2}$ | $1.85^{+0.03}_{-0.03}$ | $1.85^{+0.01}_{-0.01}$ | $2.0^{+0.8}_{-0.2}$
$\phi_{12}$ $\left[\text{rad}\right]$ | $2.95$ | $1.48$ | $2.0^{+1.0}_{-2.0}$ | $2.0^{+1.0}_{-2.0}$ | $1.48^{+0.03}_{-0.03}$ | $1.48^{+0.01}_{-0.01}$ | $1.3^{+4.5}_{-1.0}$
$\phi_{\text{JL}}$ $\left[\text{rad}\right]$ | $0.192\mid 0.216$ | $2.03$ | $0.2^{+0.3}_{-0.2}$ | $0.2^{+0.3}_{-0.2}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $1.9^{+0.4}_{-0.2}$
$\theta_{\text{JN}}$ $\left[\text{rad}\right]$ | $0.682\mid 0.700$ | $0.88$ | $1.0^{+0.4}_{-0.5}$ | $0.9^{+0.4}_{-0.5}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $0.91^{+0.06}_{-0.06}$
$\chi_{\text{p}}$ | $0.617$ | $0.60$ | $0.2^{+0.3}_{-0.1}$ | $0.2^{+0.3}_{-0.1}$ | $0.60^{+0.01}_{-0.01}$ | $0.604^{+0.003}_{-0.003}$ | $0.6^{+0.2}_{-0.1}$
$\chi_{\text{eff}}$ | $0.188$ | $0.19$ | $0.16^{+0.1}_{-0.07}$ | $0.22^{+0.05}_{-0.04}$ | $0.188^{+0.003}_{-0.003}$ | $0.1885^{+0.0008}_{-0.0008}$ | $0.19^{+0.02}_{-0.02}$
$e_{0.01}$ | $0.0316\mid 0.0001$ | N/A | $0.0316^{+0.0002}_{-0.0001}$ | $<0.0023$ | N/A | N/A | N/A
$d_{L}$ $\left[\text{Mpc}\right]$ | $521.1$ | $400^{+200}_{-100}$ | $500.0^{+200.0}_{-100.0}$ | $521^{+4}_{-4}$ | $\mathrm{MM}$ | $520^{+20}_{-20}$
$z$ | $0.11$ | $0.09^{+0.04}_{-0.02}$ | $0.10^{+0.03}_{-0.03}$ | $0.1090^{+0.0008}_{-0.0008}$ | $\mathrm{MM}$ | $0.109^{+0.004}_{-0.005}$
$\alpha$ (RA)$\left[\text{rad}\right]$ | $6.26$ | $6.27^{+0.01}_{-0.02}$ | $6.26^{+0.02}_{-0.02}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $6.26^{+0.01}_{-0.01}$
$\delta$ (DEC)$\left[\text{rad}\right]$ | $-1.15$ | $-1.147^{+0.008}_{-0.006}$ | $-1.144^{+0.007}_{-0.006}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $-1.146^{+0.002}_{-0.002}$
$\Delta t_{m}$ $\left[\text{s}\right]$ | 0 | $-2000^{+4000}_{-3000}$ | $0^{+2000}_{-4000}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $-0.0001(3)$
$\Delta\Omega$ $\left[\text{sq. deg.}\right]$ | N/A | $10.7$ | $10.5$ | $6.5$ | N/A | $1.7$
Table S1:  Injected and recovered source parameters for the heavy binary. For
most parameters, we choose to quote the median posterior values with the 90%
confidence intervals in either of the equivalent notations
$0.1234^{+5}_{-5}=0.1234(5)$, as appropriate. For some parameters that can’t
be confidently measured to be away from a prior boundary, we instead choose to
quote 90% limits. No values are given for parameters where posteriors are
extremely multi-modal (MM) and where it doesn’t make sense to quote a single
range; we refer the interested reader to the plots instead. All spin-related
quantities are quoted at a two reference frequencies, $10\,\mathrm{Hz}$ and
$10\,\mathrm{mHz}$, corresponding to the LISA and ground-based bands.
Eccentricities are quoted at a reference frequency of $10\,\mathrm{mHz}$.
Light binary
---
| Injected value | Posterior
|  0010mHz00.00010Hz0 | LISA | ET | CE | HLV+
| Eccentricity |  | High ecc. | Low ecc. |  |  |
| High $\mid$ Low |  |  |  |  |  |
$\rho_{\text{opt}}$ |  |  | $10^{+2}_{-2}$ | $10^{+2}_{-2}$ | $2701^{+2}_{-2}$ | $6837^{+3}_{-3}$ | $401^{+2}_{-2}$
$\mathcal{M}_{\text{c}}$ $\left[M_{\odot}\right]$ | $28.10$ | $28.096^{+0.001}_{-0.001}$ | $28.0954^{+0.0009}_{-0.0008}$ | $28.096^{+0.001}_{-0.001}$ | $28.0955^{+0.0006}_{-0.0006}$ | $28.10^{+0.04}_{-0.05}$
$M_{\texttt{tot}}$ $\left[M_{\odot}\right]$ | $65.00$ | $66^{+11}_{-2}$ | $68^{+13}_{-3}$ | $65.00^{+0.01}_{-0.01}$ | $65.00^{+0.01}_{-0.01}$ | $65.01^{+0.12}_{-0.12}$
$q$ | $0.81$ | $>0.36$ | $>0.33$ | $0.805^{+0.003}_{-0.003}$ | $0.806^{+0.001}_{-0.001}$ | $0.81^{+0.01}_{-0.01}$
$m_{1}$ $\left[M_{\odot}\right]$ | $36.00$ | $40^{+19}_{-8}$ | $40^{+20}_{-10}$ | $36.01^{+0.06}_{-0.06}$ | $36.00^{+0.03}_{-0.03}$ | $36.0^{+0.3}_{-0.3}$
$m_{2}$ $\left[M_{\odot}\right]$ | $29.00$ | $26^{+6}_{-7}$ | $24^{+7}_{-6}$ | $28.99^{+0.05}_{-0.05}$ | $29.00^{+0.02}_{-0.03}$ | $29.0^{+0.3}_{-0.3}$
$\chi_{1}$ | $0.13$ | $0.2^{+0.5}_{-0.2}$ | $0.3^{+0.5}_{-0.3}$ | $0.133^{+0.003}_{-0.003}$ | $0.133^{+0.002}_{-0.002}$ | $0.12^{+0.04}_{-0.04}$
$\chi_{2}$ | $0.14$ | $0.4^{+0.5}_{-0.3}$ | $0.4^{+0.5}_{-0.3}$ | $0.139^{+0.005}_{-0.005}$ | $0.138^{+0.002}_{-0.002}$ | $0.14^{+0.06}_{-0.06}$
$\iota$ $\left[\text{rad}\right]$ | $2.32$ | $2.28$ | $2.3^{+0.5}_{-0.4}$ | $2.3^{+0.5}_{-0.3}$ | $2.28^{+0.01}_{-0.01}$ | $2.275^{+0.005}_{-0.005}$ | $2.28^{+0.07}_{-0.03}$
$\theta_{1}$ $\left[\text{rad}\right]$ | $1.45$ | $1.53$ | $1.0^{+1.0}_{-1.0}$ | $1.1^{+1.1}_{-0.7}$ | $1.54^{+0.05}_{-0.05}$ | $1.53^{+0.03}_{-0.03}$ | $1.5^{+0.3}_{-0.3}$
$\theta_{2}$ $\left[\text{rad}\right]$ | $1.60$ | $1.50$ | $1.0^{+1.0}_{-1.0}$ | $1.3^{+1.2}_{-0.9}$ | $1.50^{+0.07}_{-0.07}$ | $1.50^{+0.04}_{-0.04}$ | $1.5^{+0.4}_{-0.3}$
$\phi_{12}$ $\left[\text{rad}\right]$ | $0.88$ | $5.52$ | $2.0^{+1.0}_{-2.0}$ | $2.0^{+1.0}_{-1.0}$ | $5.52^{+0.09}_{-0.09}$ | $5.52^{+0.05}_{-0.05}$ | $5.5^{+0.5}_{-0.6}$
$\phi_{\text{JL}}$ $\left[\text{rad}\right]$ | $0.365\mid 0.445$ | $0.77$ | $0.3^{+0.6}_{-0.3}$ | $0.3^{+0.5}_{-0.3}$ | $0.77^{+0.02}_{-0.02}$ | $0.78^{+0.01}_{-0.01}$ | $\mathrm{MM}$
$\theta_{\text{JN}}$ $\left[\text{rad}\right]$ | $2.09\mid 2.03$ | $2.31$ | $2.2^{+0.6}_{-0.6}$ | $2.2^{+0.6}_{-0.6}$ | $2.31^{+0.01}_{-0.01}$ | $2.312^{+0.005}_{-0.005}$ | $2.31^{+0.03}_{-0.02}$
$\chi_{\text{p}}$ | $0.132$ | $0.13$ | $0.3^{+0.5}_{-0.2}$ | $0.3^{+0.4}_{-0.2}$ | $0.133^{+0.003}_{-0.003}$ | $0.133^{+0.002}_{-0.002}$ | $0.13^{+0.04}_{-0.04}$
$\chi_{\text{eff}}$ | $0.0070$ | $0.01$ | $0.0^{+0.2}_{-0.2}$ | $0.1^{+0.1}_{-0.1}$ | $0.0071^{+0.0005}_{-0.0005}$ | $0.0070^{+0.0003}_{-0.0003}$ | $0.01^{+0.01}_{-0.01}$
$e_{0.01}$ | $0.0077\mid 0.0001$ | N/A | $0.007^{+0.002}_{-0.002}$ | $<0.0047$ | N/A | N/A | N/A
$d_{L}$ $\left[\text{Mpc}\right]$ | $149.1$ | $170^{+70}_{-60}$ | $170^{+70}_{-50}$ | $148.6^{+0.5}_{-0.4}$ | $52.03^{+0.03}_{-0.04}$ | $149^{+3}_{-3}$
$z$ | $0.03$ | $0.04^{+0.01}_{-0.01}$ | $0.04^{+0.02}_{-0.01}$ | $0.0328(1)$ | $0.011654(7)$ | $0.0329(7)$
$\alpha$ (RA)$\left[\text{rad}\right]$ | $1.50$ | $1.50^{+0.01}_{-0.013}$ | $1.50^{+0.01}_{-0.01}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $1.495^{+0.002}_{-0.002}$
$\delta$ (DEC)$\left[\text{rad}\right]$ | $-1.24$ | $-1.239^{+0.004}_{-0.004}$ | $-1.24^{+0.004}_{-0.004}$ | $\mathrm{MM}$ | $\mathrm{MM}$ | $-1.239^{+0.001}_{-0.001}$
$\Delta t_{m}$ $\left[\text{s}\right]$ | 0 | $-1000^{+6000}_{-7000}$ | $1000^{+4000}_{-5000}$ | $0.00005^{+0.00015}_{-0.00015}$ | $\mathrm{MM}$ | $0.00000(6)$
$\Delta\Omega$ $\left[\text{sq. deg.}\right]$ | N/A | $4.3$ | $4.1$ | $58.9$ | N/A | $0.2$
Table S2:  Injected and recovered source parameters for the light binary. For
most parameters, we choose to quote the median posterior values with the 90%
confidence intervals in either of the equivalent notations
$0.1234^{+5}_{-5}=0.1234(5)$, as appropriate. For some parameters that can’t
be confidently measured to be away from a prior boundary, we instead choose to
quote 90% limits. No values are given for parameters where posteriors are
extremely multi-modal (MM) and where it doesn’t make sense to quote a single
range; we refer the interested reader to the plots instead. All spin-related
quantities are quoted at a two reference frequencies, $10\,\mathrm{Hz}$ and
$10\,\mathrm{mHz}$, corresponding to the LISA and ground-based bands.
Eccentricities are quoted at a reference frequency of $10\,\mathrm{mHz}$.

##  II LISA data analysis

Detector – We consider an equal-arms constellation with arm-length
$L=$$2.5\text{\times}{10}^{6}\text{\,}\mathrm{km}$, and reference noise
spectral density as defined in the ESA Science Requirements Document [69].
Unresolved galactic binaries’ confusion noise does not affect SmBBHs
considered here, and is therefore not included in the noise budget (although,
to help guide the eye, it is plotted in Fig. 1). The LISA response is modelled
following the rigid adiabatic approximation (see, for example, Ref [70]). We
set the mission duration to $1.26\text{\times}{10}^{8}\text{\,}\mathrm{s}$ and
use a sampling cadence of $0.3\text{\,}\mathrm{s}$.

Waveforms – In the LISA sensitivity window $\sim$
$1\text{\times}{10}^{-4}\text{\,}\mathrm{Hz}0.1\text{\,}\mathrm{Hz}$, SmBBHs
are in their early inspiral. Current astrophysical understanding suggests that
in the LISA band these sources could have non-zero eccentricity and the BHs
spins have generic orientation with respect to the orbital angular momentum.
We therefore employ EFPE [18], a state-of-the-art frequency-domain waveform
model that incorporates eccentricity and leading order relativistic spin-
precession effects, with phasing accurate up to 3 post-Newtownian order.

Parameter estimation – We perform fully Bayesian inference on simulated data,
injecting the signal in zero noise data. We use the Balrog codesuite, which
provides a rigid adiabatic approximation response model for LISA [71],
interfaces to a number of stochastic samplers [72, 73, 74, 75], time- and
frequency- domain waveforms for double white dwarfs and stellar mass binary
black holes, infrastructure for time-delay-interferometry [76], and access to
the LDC codebase under development within the LISA Consortium [61]. Following
closely the methodology in [19], we employ a Gaussian likelihood on the time
delay interferometry (TDI) variables $A,E,T$, constructed as noise orthogonal
combination of the Michelson TDI variables $X,Y,Z$. For each binary analysed,
we generate data in frequency ranges broad enough to capture entirely the
simulated signal. For the “light” (“heavy”) binaries, we set minimum and
maximum frequencies to $f_{\rm min}=$0.0185\text{\,}\mathrm{Hz}$$ and $f_{\rm
max}=$0.1\text{\,}\mathrm{Hz}$$ ($f_{\rm min}=$0.0111\text{\,}\mathrm{Hz}$$
and $f_{\rm max}=$0.1\text{\,}\mathrm{Hz}$$), respectively. To reduce the
computational cost, we evaluate inner products through a modified Clenshaw-
Curtis integration scheme [77]. We sample parameters uniformly in their
respective ranges (see Table S3 for details) the following parameters: chirp
mass $\mathcal{M}_{c}$, dimensionless mass difference
$\delta\mu=(m_{1}-m_{2})/(m_{1}+m_{2})$, time to merger $t_{m}$, squared
eccentricity at $0.01\text{\,}\mathrm{Hz}$ $e^{2}_{0.01}$, square roots of
circularly polarized GW amplitudes $\sqrt{A_{L,R}}=\sqrt{A}(1\pm\cos\iota)$,
polarization angle $\psi$, initial orbital phase $\phi_{0}$, initial argument
of periastron $\phi_{e}$, ecliptic longitude $l$, sine of ecliptic latitude
$\sin b$, dimensionless spin magnitudes $\chi_{1},\chi_{2}$, sine of spin
latitudes $\sin b^{\chi}_{1},\sin b^{\chi}_{2}$, and spin longitudes
$l^{\chi}_{1},l^{\chi}_{2}$.

We use a nested sampling algorithm [78] as implemented in CPnest, with 16000
livepoints multithreaded on $20$ cores for each of the four runs. Each run
took approximately 3 weeks to reach its completion (with an evidence
uncertainty of $\mathrm{d}Z=\pm 0.1$), except for the heavy binary with high
eccentricy, which required approximately $3$ months. In Table S1 and S2
parameters posterior median and 90% confidence intervals are quoted, alongside
the corresponding injected values.

Table S3:  Priors for the LISA parameter estimation.  Name | Light binary | Heavy binary
---|---|---
|  High Ecc. |  Low Ecc. |  High Ecc. |  Low Ecc.
$\mathcal{M}_{c}~{}[M_{\odot}]$ | $\text{Uniform}(28.092,28.098)$ | $\text{Uniform}(64.589,64.595)$
$\delta\mu$ | $\text{Uniform}(0,0.9)$ | $\text{Uniform}(0,0.9)$
$t_{m}~{}[s]$ | $\text{Uniform}(-10^{4},10^{4})$ | $\text{Uniform}(-10^{4},10^{4})$
$e^{2}_{0.01}$ |  $\text{Uniform}(0,3\times 10^{-4})$ |  $\text{Uniform}(0,3\times 10^{-5})$ |  $\text{Uniform}(0.00098,0.00102)$ |  $\text{Uniform}(0,3\times 10^{-5})$
$\sqrt{A_{R}}$ | $\text{Uniform}(0,1.158\times 10^{-4})$ | $\text{Uniform}(0,6.1951\times 10^{-5})$
$\sqrt{A_{L}}$ | $\text{Uniform}(0,1.158\times 10^{-4})$ | $\text{Uniform}(0,6.1951\times 10^{-5})$
$\psi~{}[\text{rad}]$ | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$\phi_{0}~{}[\text{rad}]$ | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$\phi_{e}~{}[\text{rad}]$ | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$l~{}[\text{rad}]$ | $\text{Uniform}(4,6)$ | $\text{Uniform}(5.4,5.7)$
$\sin b$ | $\text{Uniform}(-0.999,-0.993)$ | $\text{Uniform}(-0.84,-0.82)$
$\chi_{1},\chi_{2}$ | $\text{Uniform}(0,1)$ | $\text{Uniform}(0,1)$
$l^{\chi}_{1},l^{\chi}_{2}~{}[\text{rad}]$ | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$\sin b^{\chi}_{1},\sin b^{\chi}_{2}$ | $\text{Uniform}(-1,1)$ | $\text{Uniform}(-1,1)$
Table S4:  Priors for the ground-based parameter estimation. Note that whilst we use uniform priors in the component masses, following [79] we sample in $\\{\mathcal{M},q\\}$ as these variables are less correlated.  Name . | Light | Heavy
---|---|---
$m_{1}\,[M_{\odot}]$ | $\text{Uniform}(20,50)$ | $\text{Uniform}(70,100)$
$m_{2}\,[M_{\odot}]$ | $\text{Uniform}(20,50)$ | $\text{Uniform}(50,80)$
$\chi_{1},\chi_{2}$ | $\text{Uniform}(0,0.99)$ | $\text{Uniform}(0,0.99)$
$\theta_{1},\theta_{2}\,[\rm rad]$ | $\text{Sine}(0,\pi)$ | $\text{Sine}(0,\pi)$
$\phi_{12},\phi_{\rm JL}\,[\rm rad]$ | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$\theta_{\rm JN}\,[\rm rad]$ | $\text{Sine}(0,\pi)$ | $\text{Sine}(0,\pi)$
$\psi$ [rad] | $\text{Uniform}(0,\pi)$ | $\text{Uniform}(0,\pi)$
$\alpha$ [rad] | $\text{Uniform}(0,2\pi)$ | $\text{Uniform}(0,2\pi)$
$\delta\,[\rm rad]$ | $\;\text{Cosine}(-\pi/2,\pi/2)\;$ | $\;\text{Cosine}(-\pi/2,\pi/2)\;$

##  III Ground-based data analysis

Detectors – For the ground-based analysis, we consider the A+ detector network
consisting of LIGO-Hanford, LIGO-Livingston and Virgo (HLV+), the triangular
ET-D configuration [80] placed at the current Virgo site, or a single L-shaped
CE detector [15] in the location of LIGO-Hanford. For Hanford and Livingston,
we adopt the A+ design target for the fifth observing run [20, 81], and for
Virgo we use its high-limit target sensitivity [20, 81]. For the 3G detectors,
we use the ET-D and Cosmic Explorer noise curves taken from [82].

Waveforms – By the time the signals considered here reach a GW frequency of
$10\,\mathrm{Hz}$, the orbit will have circularised [17] and hence we use a
state-of-the-art waveform model for the inspiral, merger and ringdown of
quasi-circular binaries, IMRPhenomXPHM [21, 22, 23]. Following the general
framework to build waveform models that include general-relativistic spin-
precession effects [83, 84], the model applies a time-dependent rotation that
encapsulates the precession of the orbital plane to a set of approximate
aligned-spin $h_{\ell m}$-modes. The approximate aligned-spin modes calibrated
against numerical relativity simulations in IMRPhenomXPHM are the
$(\ell,|m|)=\\{(2,2),(2,1),(3,3),(3,2),(4,4)\\}$ modes.

Parameter estimation – We perform Bayesian inference utilising the Bilby
inference library [24] in combination with the nested sampler dynesty [74] to
sample the BBH parameter space and infer the posterior densities. In order to
help improve convergence of the posteriors, we use
$\sim\mathcal{O}(\text{few}\times 10^{3})$ live points and adopt a stopping
criterion on the evidence $Z$ of $\Delta\ln Z=0.1$. Due to the very large SNRs
of the injected signals, the nested sampling algorithm is extremely slow to
converge and more efficient sampling methods, such as slice sampling, offer
significant computational improvement over the default random-walk proposals.
Each run took $\sim 2-3$ weeks on $32$ cores. The simulated signals are
injected into zero noise, which should be broadly equivalent to the results
obtained by averaging over a large number of injections into different
realizations of Gaussian noise. The respective PSD enters only in the
likelihood calculation. For both signals, we analyse data segments of
$8\,\mathrm{s}$ duration with a sampling rate of $1024\,\mathrm{Hz}$. We
sample in the (detector-frame) chirp mass $\mathcal{M}_{c}$ and mass ratio $q$
but adopt priors that are uniform in the component masses [79]. The complete
set of priors is given in Table S4. We assume the binaries are observed at a
Global Positioning System (GPS) reference time of 1706962960.0 at the
geocenter.

##  IV Detailed results for all analyses

In the main text, we have presented the overall results, presenting some of
the key illustrative plots. Here we show detailed results for all the analyses
that we have carried out in the context of this study.

A top level summary of the symmetric 90% credible intervals of the parameters
recovered in all the analyses – LISA and the different ground-based
observatories, HLV+, ET and CE – for all the sources is given in Tables S1 and
S2.

###  IV.1 Summary plots for the heavy binary

In the main text we have shown plots based on the results for the light binary
for the chirp mass, time to merger and eccentricity (Fig. 2), and intrinsic
(Fig. 3) parameters. Here we show the same plots, but for the heavy binary.
Figure S1 (right panel) shows the posterior distribution on the chirp mass,
merger time and eccentricity (right panel) for the heavy source, equivalent to
Fig. S1 in the main text. The right panel of Figure S1 shows the marginalised
posterior distributions of the physical parameters of the heavy source, the
equivalent of Fig. 3 in the main text.

Figure S1:  Selected posterior density functions for the “heavy”
($85\,M_{\odot}-65\,M_{\odot}$), high-eccentricity ($e_{0.01}=0.0316$) source,
same as Figs. 2 and 3, which refer to the “light” binary. Left: LISA
posteriors (dotted red) on the chirp mass $\mathcal{M}_{c}$, time to merger
$t_{m}$, and eccentricity $e_{0.01}$. Solid/dashed red lines show posteriors
from third generation ET/CE ground-based instruments. The LISA
$\mathcal{M}_{c}$ measurement is strongly degenerate with $t_{m}$ (and, to a
lesser extent, with $e_{0.01}$) which is broken when combining with the merger
time measured from any ground-based instrument leading to significant
improvements in the combined, multiband measurements (shown in black). Right:
One-dimensional marginalised posteriors on the intrinsic parameters:
$10\,\mathrm{mHz}$ eccentricity ${e_{0.01}}$, detector frame component masses
$m_{1(2)}$, dimensionless spin magnitudes $\chi_{1(2)}$, spin tilt angles
$\theta_{1(2)}$, and the angle between the in-plane spins $\phi_{12}$.
Dot/dash/solid/dot-dash lines indicate LISA/CE/ET/HLV+ posteriors. LISA
measures eccentricity extremely well from the early inspiral (the black curve
shows the improvement with the multiband measurement of $t_{m}$) and the 3G
instruments measure the other intrinsic parameters with exquisite accuracy.
The injected eccentricity is indicated; for all other parameters, $\Delta$
denotes a shift from the injected value.

###  IV.2 Individual mass measurements

In Fig. S2 we show the posterior density functions on the individual masses,
$m_{1}$ and $m_{2}$, for LISA and ET observations. ET is meant to represent
the quality of observations for a 3G ground-based observatory operating in the
LISA time-frame. Note that results obtained with CE are comparable, but
actually better by a factor $\approx 2$, see Tables S1, S2 and Figs. S4 and S7

Figure S2:  Marginalised posterior density functions on the component masses
of the two binary systems considered in this study obtained in observations
with different instruments. _Left_ : posteriors for the light
($36\,M_{\odot}-29\,M_{\odot}$), high-eccentricity ($e_{0.01}=0.007$) binary
recovered with LISA (blue) and ET (green) observations. _Right_ : posteriors
for the heavy ($85\,M_{\odot}-65\,M_{\odot}$) high-eccentricity
($e_{0.01}=0.0316$) binary obtained with LISA (red) and ET (green).

###  IV.3 Measuring the eccentricity

While regardless of the eccentricity of the binary at formation, we expect
that the orbit has circularised due to radiation reaction by the time the
signal enters the band of ground-based observatories, some residual
eccentricity may well be present in the LISA sensitivity window. For this
reason, for both the heavy and light binary we consider two different
eccentricity values, labelled high and low (see Tables 1 and S1). The results
concerning the measurement of the eccentricity for both LISA-only and multi-
band observations are shown in Fig. S3. They clearly show that for the “light”
(“heavy”) binary an eccentricity of $7\times 10^{-3}$ ($3\times 10^{-2}$) can
be measured and distinguished from zero at high confidence, well above $99\%$
confidence. Conversely, for an eccentricity of $1\times 10^{-4}$ one can only
provide an upper-limit to the eccentricity of the binary. These values can
provide a guide as of what is the level of eccentricity for SmBBHs that will
be measurable with LISA, and the one for which only upper-limits can be set.
We need however to caution the reader that the exact values will depend on a
source-by-source basis, as they are affected by SNR, signal duration in band,
values of masses and spins.

Figure S3:  Marginalised posterior density functions on the eccentricity for
the light ($36\,M_{\odot}-29\,M_{\odot}$) binary system (plots on the left)
and the heavy ($85\,M_{\odot}-65\,M_{\odot}$) binary system (plots on the
right) with high (top row) and low (bottom row) eccentricity (see Tables 1 and
S1 and S2 for the parameter values, also shown as black dashed lines). Dotted
lines are the posteriors recovered with LISA observations only (blue: light
binary; red: heavy binary). The multi-band posterior density function, i.e.
the result obtained combining both LISA and ground-based observations (here we
use specifically the results obtained with ET) are shown by the solid black
line. Also show for reference in the bottom left panel is the prior
distribution.

###  IV.4 (Almost) full corner plots

The binaries considered in this study are described by the full set of 17
parameters (15 in the context of ground-based observations, where we assume
the orbit has circularised below the limit of detection of eccentricity, and
we therefore consider exactly circular binaries). Full corner plots would
therefore be illegible, and we are not presenting them here. We show however
corner plots for the most of the parameters of astrophysical interest below
for the full set of runs.

Fig S4 show results for the physical parameters of the light source for the
three different configurations of ground-based observatories: ET, CE and HLV+.
The equivalent corner plots for the same source in the case of LISA
observations are Fig S5 (for the high-eccentricity case) and Fig S6 (for the
low-eccentricity case). 90% credible intervals are summarised in Table S2.

Fig S7 show results for the physical parameters of the heavy source for the
three different configurations of ground-based observatories: ET, CE and HLV+.
The equivalent corner plots for the same source in the case of LISA
observations are Fig S8 (for the high-eccentricity case) and Fig S9 (for the
low-eccentricity case). 90% credible intervals are summarised in Table S1.

Figure S4:  Marginalised posterior density functions on the mass and spin
parameters (details are provided in Table S2) for observations of the light
($36\,M_{\odot}-29\,M_{\odot}$) binary system with different ground-based
observatories: HLV+ (grey contours: 50% and 90% probability regions) ET (blue)
CE (red) and O5 (grey).

Figure S5:  Marginalised posterior density functions on the eccentricity, mass
and spin parameters (details are provided in Table S2) for observations of the
light ($36\,M_{\odot}-29\,M_{\odot}$), high eccentricity ($e_{0.01}=0.0077$)
binary system with LISA. The dark (light) blue areas in the two-dimensional
plots refer to the 50% (90%) posterior probability regions.

Figure S6:  Marginalised posterior density functions on the eccentricity, mass
and spin parameters (details are provided in Table S2) for observations of the
light ($36\,M_{\odot}-29\,M_{\odot}$), low eccentricity ($e_{0.01}=1\times
10^{-4}$) binary system with LISA. The dark (light) blue areas in the two-
dimensional plots refer to the 50% (90%) posterior probability regions.

Figure S7:  Marginalised posterior density functions on the mass and spin
parameters (details are provided in Table S1) for observations of the heavy
($85\,M_{\odot}-65\,M_{\odot}$) binary system with different ground-based
observatories: HLV+ (grey contours: 50% and 90% probability regions) ET (blue)
CE (red) and O5 (grey).

Figure S8:  Marginalised posterior density functions on the eccentricity, mass
and spin parameters (details are provided in Table S1) for observations of the
heavy ($85\,M_{\odot}-65\,M_{\odot}$), high eccentricity ($e_{0.01}=0.0316$)
binary system with LISA. The dark (light) red areas in the two-dimensional
plots refer to the 50% (90%) posterior probability regions.

Figure S9:  Marginalised posterior density functions on the eccentricity, mass
and spin parameters (details are provided in Table S1) for observations of the
heavy ($85\,M_{\odot}-65\,M_{\odot}$), low eccentricity ($e_{0.01}=1\times
10^{-4}$) binary system with LISA. The dark (light) red areas in the two-
dimensional plots refer to the 50% (90%) posterior probability regions.