#  On Using Agent-based Modeling and Simulation
for Studying Blockchain Systems
Sur l’Utilisation de la Modélisation et de la Simulation basées Agents pour
Etudier les Systèmes de Chaînes de Blocs

Önder Gürcan111ORCID: 0000-0001-6982-5658

Université Paris-Saclay, CEA, List
F-91120, Palaiseau, France
<EMAIL_ADDRESS>
Bitcoin [1] is the core of decentralized crypto-currency systems. The
underlying data structure of Bitcoin is called the blockchain in which
transactions of digital coins between accounts are batched in so-called
blocks, where each block is appended to the last one in a cryptographic way to
make the malicious/accidental change of blocks content very hard. Participants
following this protocol can create together a distributed, economical, social
and technical system where anyone can join/leave and perform transactions in-
between without neither needing to trust each other nor having a trusted third
party. It is a very attractive technology since it maintains a public,
immutable and ordered log of transactions which guarantees an auditable ledger
accessible by anyone.

Technically speaking, a blockchain system is an open and distributed
transactional system composed of participants called users and block creators.
All participants of a blockchain system store unconfirmed transactions in
their memory pools and confirmed transactions in their blockchains. However,
it should be noted that, a participant may enter into several blockchain
systems at the same time and can have different objectives and expectations in
each.

Moreover, blockchain systems have the following three characteristics.

  * •

Blockchain systems are distributed systems. As Leslie Lamport says222Leslie
Lamport, Distribution, https://www.microsoft.com/en-
us/research/publication/distribution/, E-mail message sent to a DEC SRC
bulletin board at 12:23:29 PDT on 28 May 1987., a distributed system is one in
which the failure of a computer you did not even know existed can render your
own computer unusable. Most distributed systems (including blockchain systems)
are designed with fault tolerance as one of the main objectives in order to
achieve high availability and storing or processing distributed data.

  * •

Blockchain systems are social organizations. A social organization can be
defined as formal or informal groups of interrelated individuals (agents) who
pursue a collective goal and who are embedded into an environment [2].
Moreover, the blockchain (data structure) is a physical manifestation of the
interactions of users. Blockchain systems facilitates cooperation by getting
self-interested, distrustful people to work together, even when narrow self-
interest would seem to dictate that no individual should take part. Blockchain
systems have highly volatile dynamics, conflict of individual/collective goals
(e.g., users want lower fees while block creators want higher fees) and
continuous enter/exit dynamics [3].

  * •

Blockchain systems are obviously economical systems. An economical system, as
any other complex system, reflects a dynamic interaction of a large number of
different agents, not just a few key agents. The resulting systemic behavior,
observable on the aggregate level, often shows consequences that are hard to
predict (e.g., the transaction fees) which cannot be simply explained by the
behaviors of a few major agents.

Furthermore, it is a very active and dynamic ecosystem where new blockchain
platforms and algorithms are developed continuously, due to the interest of
industries to the technology.

Thus, there is a need for a simulation framework, which is develop as a
software using modern engineering approaches (e.g., modularity - i.e. model
reuse-, testing, continuous development and continuous integration, automated
management of builds, dependencies and documentation) and agile principles,
(1) to make rapid prototyping of industrial cases and (2) to carry out their
feasibility analysis in a realistic manner (i.e., to test hypothesis by
simulating complex experiments involving large numbers of participants of
different types acting in one or several blockchain systems).

Besides, we anticipate a challenging and interdisciplinary research agenda in
blockchain systems, built upon a methodology that strives to capture the rich
process resulting from the interplay between the behavior of agents and the
dynamic interactions among them. To be effective, however, simulation studies
providing insights into blockchain systems from massive data analysis, theory
encompassing the appropriate description of agents and their interactions, and
a systemic perspective bestowing a new understanding of global effects as
coming from varying network interactions are needed. We predict that such
studies will create a more unified field of blockchain systems that advances
our understanding and leads to further insight. However, such studies are not
trivial and very hard to conduct without dedicated tool support.

We claim that such a simulation framework needs to be a deployed agent-based
modeling and simulation platform which allows to recreate the dynamics of
systems at an agent level, so that the impact of actions executed by an
algorithm under different scenarios can be tested and analyzed in granular
detail. We also claim that to better capture and model the requirements of
blockchain systems, it should follow an organization centered approach [4]
rather than an agent centered one. To the best of our knowledge, there is no
such platform yet. We are currently active working on this and our first
results [5] are promising.

## References

  * [1] S. Nakamoto, “Bitcoin: A peer-to-peer electronic cash system,” 2008.  https://bitcoin.org/bitcoin.pdf.
  * [2] E. Ostrom, “A general framework for analyzing sustainability of social-ecological systems,” Science, vol. 325, no. 5939, pp. 419–422, 2009\.
  * [3] Ö. Gürcan, A. Del Pozzo, and S. Tucci-Piergiovanni, “On the bitcoin limitations to deliver fairness to users,” in On the Move to Meaningful Internet Systems. OTM 2017 Conferences (H. Panetto, C. Debruyne, W. Gaaloul, M. Papazoglou, A. Paschke, C. A. Ardagna, and R. Meersman, eds.), (Cham), pp. 589–606, Springer Int. Publishing, 2017.
  * [4] J. Ferber, O. Gutknecht, and F. Michel, “Agent/Group/Roles: Simulating with Organizations,” in ABS’03: Agent Based Simulation (M. J.P., ed.), (Montpellier (France)), p. 12, Apr. 2003.  April 28-30.
  * [5] N. Lagaillardie, M. A. Djari, and O. Gürcan, “A computational study on fairness ofthe tendermint blockchain protocol,” Information, vol. 10, no. 12, 2019.


# GW190521: tracing imprints of spin-precession on the most massive black hole
binary

Simona J. Miller  Department of Physics, California Institute of Technology,
Pasadena, California 91125, USA  LIGO Laboratory, California Institute of
Technology, Pasadena, California 91125, USA  Maximiliano Isi  Center for
Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Katerina Chatziioannou  Department of Physics, California Institute of
Technology, Pasadena, California 91125, USA  LIGO Laboratory, California
Institute of Technology, Pasadena, California 91125, USA  Vijay Varma
Department of Mathematics, Center for Scientific Computing and Data Science
Research, University of Massachusetts, Dartmouth, MA 02747, USA  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), D-14476
Potsdam, Germany  Ilya Mandel  School of Physics and Astronomy Monash
University, Clayton, VIC 3800, Australia  The ARC Centre of Excellence for
Gravitational Wave Discovery – OzGrav, Australia

###### Abstract

GW190521 is a remarkable gravitational-wave signal on multiple fronts: its
source is the most massive black hole binary identified to date and could have
spins misaligned with its orbit, leading to spin-induced precession—an
astrophysically consequential property linked to the binary’s origin. However,
due to its large mass, GW190521 was only observed during its final 3-4 cycles,
making precession constraints puzzling and giving rise to alternative
interpretations, such as eccentricity. Motivated by these complications, we
trace the observational imprints of precession on GW190521 by dissecting the
data with a novel time domain technique, allowing us to explore the morphology
and interplay of the few observed cycles. We find that precession inference
hinges on a quiet portion of the pre-merger data that is suppressed relative
to the merger-ringdown. Neither pre-merger nor post-merger data alone are the
sole driver of inference, but rather their combination: in the quasi-circular
scenario, precession emerges as a mechanism to accommodate the lack of a
stronger pre-merger signal in light of the observed post-merger. In terms of
source dynamics, the pre-merger suppression arises from a tilting of the
binary with respect to the observer. Establishing such a consistent picture
between the source dynamics and the observed data is crucial for
characterizing the growing number of massive binary observations and
bolstering the robustness of ensuing astrophysical claims.

_Introduction —_ At a total mass of ${\sim}150\,M_{\odot}$, GW190521 [3, 4] is
the current record-holder among massive black hole binaries confidently
detected through gravitational waves by LIGO [1] and Virgo [9]. Such high-mass
systems are essential probes of the role of hierarchical mergers [43, 45, 63,
28, 35] and pair-instability physics [101, 100, 13, 94, 92, 81] in binary
formation and evolution. Observationally, massive binaries merge toward the
low edge of the detectors’ bandwidth and are only detectable for a short time.
One third of the binaries in the the latest gravitational-wave catalog have a
median detector-frame total mass $>100\,M_{\odot}$ [7], corresponding to
${\sim}5$ signal cycles at more than half a standard deviation above the
noise. The short duration makes characterizing these signals and inferring
astrophysical properties such as spin challenging.

Spin is a key signature of the physics behind angular momentum transport in
stellar interiors, black-hole formation, black-hole retention in dense
environments, and more [65, 83, 82, 42, 31, 37]. Gravitational-waves provide
one of the few ways to measure spins for stellar-mass black holes directly.
Spin components parallel to the binary’s orbital angular momentum affect the
signal duration [21] and are approximately conserved during the inspiral as
the “effective spin” [79, 10]. Spin components perpendicular to the orbital
angular momentum, i.e., in the orbital plane, cause the binary to precess,
leading to signal modulations as the emission pattern varies relative to the
line of sight [11, 61]. Although typically weak [98, 97, 90], this effect is
highly sought-after: spin-induced precession and the associated in-plane spins
could differentiate between dynamical and field binary formation, e.g., [65,
104, 106].

The elusiveness of precession is exacerbated for heavy systems. The precession
timescale can be longer than the observed inspiral for large masses [46],
making modulations difficult to identify. The exact imprint of precession on
the ensuing merger and ringdown remains poorly understood and analytically
intractable, although numerical-relativity and data-analysis studies suggest
that imprints do exist [75, 15, 54, 32, 60, 51]; for example, Ref. [15]
suggests, based on simulations, that high-frequency data typically associated
with the merger-ringdown can constrain precession. This uncertainty makes it
difficult to distinguish precession from eccentricity, another highly-valuable
binary property [85, 84, 20]. Interpretation is further complicated by high
sensitivity to the system’s true parameters [15, 30, 50, 75] and the priors
[76].

In light of this, the massive GW190521 system stands out for its informative
precession constraints. Precession is quantified by the effective precessing
spin $\chi_{\mathrm{p}}$ [87, 88, 89] that is motivated by inspiral dynamics.
A value of zero (one) indicates no (maximal) precession. Under the assumption
of a quasicircular orbit, GW190521 has
$\chi_{\mathrm{p}}=0.68^{+0.25}_{-0.37}$ at 90% credibility [3, 4], the
largest inferred $\chi_{\mathrm{p}}$ and the one whose posterior is the most
informative to date [5, 6, 7];111GW200129, another massive binary [7], has a
potentially comparable constraint. When averaged over waveform models, its
$\chi_{\mathrm{p}}$ is unconstrained [7]; when restricting to the NRSur7dq4
[95] model, GW200129 is inferred to be precessing [52]. This interpretation is
however complicated by data quality issues [77]. similar conclusions are
reached under alternative parametrizations for precession [47, 40].222These
and further ways to quantify precession are elaborated upon in the
Supplementary Material. The combined high mass and large in-plane spin make
GW190521 an essential probe of hierarchical black hole mergers [69, 63, 62],
dense stellar environments such as nuclear star clusters [82, 44, 36], active
galactic nuclei disks [66, 68, 67], and more [4, 14, 81, 49, 27, 33, 74].

The high mass of GW190521 and its few observable cycles open the door to
competing astrophysical interpretations. Romero-Shaw et al. [85] and Gayathri
et al. [41] find that the data are consistent with eccentricity, though this
interpretation is not supported by Iglesias and others [56] and Ramos-Buades
et al. [80]. Gamba et al. [39] propose a hyperbolic capture scenario. Nitz and
Capano [74] suggest a highly asymmetric binary interpretation. More exotic
explanations include boson stars [19] and cosmic strings [12]. Any of these
alternatives would have important implications if confirmed [105, 86].
Additionally, random detector noise can have an outsized impact on the
inference of poorly-constrained effects, although Biscoveanu et al. [15] and
Xu and Hamilton [102] show that the inference of $\chi_{\mathrm{p}}$ away from
zero in GW190521-like systems cannot be due to Gaussian noise alone. The fact
that full-scale parameter estimation allows for competing interpretations
suggests that different physical effects can result in similar observational
imprints over GW190521’s few cycles. Similarly to precession and eccentricity,
these imprints are often not analytically tractable.

Toward bolstering the interpretation of massive binaries, it is essential to
gain intuition about the observable imprint of physical effects of interest
and how their measurability is affected by mismodeling. Lacking analytical
equations for precession in the merger phase, we introduce a novel approach
that traces its imprint along the signal and identifies the role of each cycle
on the $\chi_{\mathrm{p}}$ constraint. We dissect the data in the time-domain
and compare inference between different data subsets. We provide a cycle-by-
cycle physical picture of source dynamics and explore the interplay of
different data regions.

_Methods —_ Gravitational-wave parameter estimation is typically conducted in
the frequency-domain to leverage the stationarity of detector noise for
computational efficiency [2, 93]. However, frequency-domain methods are non-
local in time; thus isolating temporal features of source dynamics and their
imprint on the data requires nontrivial likelihood modifications [103,
23].333Frequency truncation enables consistency checks [18, 48],
investigations of data-quality issues [26, 77], or alternative studies of the
measurability of precession in simulated data [15], but this is not equivalent
to cuts in time. We instead adopt direct time-domain inference to isolate
different signal cycles, an approach originally conceived for black hole
ringdowns [58, 59, 57, 24]. We truncate data from LIGO Livingston, LIGO
Hanford, and Virgo at different times ranging from $t=-50\,M_{\mathrm{ref}}$
to $50\,M_{\mathrm{ref}}$ with respect to coalescence.444We define $t$ with
respect to geocenter GPS time $1242442967.405764\,\mathrm{s}$. Under geometric
units we adopt the median detector-frame remnant mass scale
$M_{\mathrm{ref}}=1.27$ ms [4]; in standard units
$M_{\mathrm{ref}}=258.3\,M_{\odot}$. The choice of remnant rather than total
mass, was inspired by ringdown analyses [91]. We independently infer the
signal properties solely from data before and after each cutoff as well as the
full span of data.

We model the signal with the numerical relativity surrogate model NRSur7dq4
[95], which assumes quasi-circular orbits and includes precession and higher-
order modes. Within its region of validity, NRSur7dq4 displays the lowest
mismatches against numerical relativity among existing models [95]. We adapt
the time-domain inference code from Isi et al. [57] and sample the
multidimensional posterior for the binary masses, spin magnitudes and tilt
angles, azimuthal inter-spin angle, azimuthal precession cone angle,
inclination, luminosity distance, and phase of coalescence. The time of
coalescence, right ascension, declination, and polarization angle are fixed
for computational efficiency.555We have verified that these choices do not
affect our conclusions. All parameter estimation settings, priors, and
consistency checks are given in the Supplementary Material.

We report precession constraints using the canonical effective precessing
spin, $\chi_{\mathrm{p}}$ [87, 88, 89]:

$\chi_{\mathrm{p}}=\mathrm{max}\left[\chi_{1}\sin\theta_{1},\left(\frac{3+4q}{4+3q}\right)q\,\chi_{2}\sin\theta_{2}\right]\in[0,1)\,.$
(1)

Here $\chi_{i}\in[0,1)$ are the dimensionless spin magnitudes and $\theta_{i}$
are the tilt angles between the spin and orbital angular momentum vectors.
Subscripts $i\in\\{1,2\\}$ denote each black hole with mass $m_{i}$ and
$q\equiv m_{2}/m_{1}\leq 1$.

Figure 1:  Evolution of GW190521 inference for representative cutoff times
$t\in\\{-40,-20,-10,20,30\\}\,M_{\mathrm{ref}}$ from the time of coalescence
(top to bottom; vertical black dashed lines where applicable). Left: Posterior
for $\chi_{\mathrm{p}}$ from the pre- (blue), post-cutoff (orange), and full
(black solid) analysis and prior (gray dotted). The inset shows the whitened
maximum-posterior waveform (with $\chi_{\mathrm{p}}=0.62$, detector-frame
total mass $M=267\,M_{\odot}$, and $q=0.89$) from the full analysis (black)
along with the whitened LIGO Livingston data (gray). The blue/orange shaded
regions highlight the data informing the same-color $\chi_{\mathrm{p}}$
posterior. Center and Right: Waveform reconstruction draws for LIGO Livingston
from the pre- (center, blue) and post-cutoff (right, orange) analyses and
maximum-posterior waveform for the full analysis (black). Median and 90%
credible intervals for the matched-filter network SNRs are given in-figure.
Gray shading denotes data excluded from each analysis. See Ref. [71] for an
animation of this figure including more cutoff times.

_Results —_ Figure 1 shows inferred GW190521 properties from data before
(blue) and after (orange) five representative cutoff times (vertical lines) as
well as the full signal (black) for comparison. Results for further cutoff
times are included in the Supplementary Material. Insets in the left panels
visualize the truncation in LIGO Livingston, selected as the detector in which
GW190521 is the loudest. The left column shows the posterior for
$\chi_{\mathrm{p}}$ at a reference frequency of 11 Hz [4, 3]. For the earliest
cutoff time (top row), the post-$t=-40\,M_{\mathrm{ref}}$ $\chi_{\mathrm{p}}$
posterior is almost identical to that of the full analysis, while the pre-
cutoff one is identical to the prior. This is due to the fact that the post-
cutoff analysis includes the full available signal, while none of it is
contained in the pre-cutoff data, see inset. As the cutoff moves to later
times (top to bottom), the pre- and post-cutoff posteriors gradually exchange
places as the data preceding each cutoff become more informative and the data
following become less so.

The ${\sim}40\,$ms between $t=-40\,M_{\mathrm{ref}}$ and
$-10\,M_{\mathrm{ref}}$ are crucial to constrain precession for GW190521. This
region roughly corresponds to the final cycle before the onset of merger. The
$\chi_{\mathrm{p}}$ posterior obtained from data after
$t=-40\,M_{\mathrm{ref}}$ (first row, orange) is consistent with that from the
full analysis, i.e., precession is constrained. On the other hand, data after
$t=-10\,M_{\mathrm{ref}}$ (third row, orange), result in a $\chi_{\mathrm{p}}$
posterior that is nearly identical to the prior, i.e., uninformative. Between
these times, the posterior shifts smoothly between the full measurement and
the prior; e.g., the post-$t=-20\,M_{\mathrm{ref}}$ analysis (second row,
orange). The reduction in the signal-to-noise ratio (SNR) from excluding data
is negligible between $-40\,M_{\mathrm{ref}}$ and $-10\,M_{\mathrm{ref}}$
suggesting this qualitative change in precession inference is not due to an
SNR drop, see Fig. 8 in the Supplementary Material.

Neither the inspiral nor the merger/ringdown data alone are fully responsible
for precession constraints in GW190521. The data both pre- and
post-$t=-10\,M_{\mathrm{ref}}$ alone are uninformative about precession (third
row, orange and blue). Moreover, the pre-$t=30\,M_{\mathrm{ref}}$ analysis
that excludes the final ringdown cycle (fifth row, blue) is consistent with
the full analysis. It is therefore not solely the final pre-merger cycle that
informs precession, but rather its combination with the subsequent $2$ merger
and early-ringdown cycles. This does not rule out ringdown imprints of
precession that are too weak to discern at this SNR or with this waveform.

Figure 2: Results from the full (black) and from the
post-$t=-10\,M_{\mathrm{ref}}$ analysis (orange) that are informative and
uninformative about precession respectively. Top: 50% credible intervals for
the whitened waveforms in LIGO Livingston in units of standard deviations of
the noise. Data are plotted in gray. Gray shading denotes data excluded from
the post-$t=-10\,M_{\mathrm{ref}}$ analysis. The inset zooms in around
$t=-15\,M_{\mathrm{ref}}$ (blue dashed line), the minimum of the final pre-
merger cycle. Bottom: Posteriors for $\chi_{\mathrm{p}}$, the absolute value
of the whitened strain $|\hat{h}|$, and the inclination angle relative to
edge-on configurations $|\iota-\pi/2|$. Quantities labeled in blue are plotted
at $t=-15\,M_{\mathrm{ref}}$. Contours denote 50% and 90% credible regions.
The whitened strain is anticorrelated with $\chi_{\mathrm{p}}$ and correlated
with $|\iota-\pi/2|$. Large $\chi_{\mathrm{p}}$ is paired with smaller pre-
merger signal and more edge-on configurations.

The center and right columns of Fig. 1 investigate features of the waveforms.
The blue and orange waveforms are informed only by data in the unshaded
regions and extended coherently into the shaded regions. As progressively less
data are analyzed (center bottom to top, right top to bottom), the waveform
reconstructions agree less with the full-analysis waveform, eventually
becoming incoherent. The right column reveals the morphological imprint of
precession on the signal during the transition from an informative
$\chi_{\mathrm{p}}$ posterior (first row) to the prior (third row). When the
$\chi_{\mathrm{p}}$ inference returns (close to) the prior, the final pre-
merger cycle is extrapolated to be larger than when $\chi_{\mathrm{p}}$ is
constrained to take higher values, c.f., the waveform peak at
$t\sim-30\,M_{\mathrm{ref}}$ and trough at $t\sim-15\,M_{\mathrm{ref}}$. Again
there is a progression: the post-$t=-40\,M_{\mathrm{ref}}$ inferred waveforms
(orange) are consistent with the full analysis (black), while the final pre-
merger cycle subtly increases in strength toward
post-$t=-10\,M_{\mathrm{ref}}$ (top row to third row).

Figure 3: Posteriors for $\chi_{\mathrm{p}}$, $M$, and $q$ for the same cutoff
times as Fig. 1. Posteriors from the full analysis and priors are plotted in
black solid and dotted respectively. Contours denote 50% credible regions.
Information about spin-precession is lost post-$t=-10\,M_{\mathrm{ref}}$
(shaded light orange), while the total mass and mass ratio posteriors are
informative at post-$t=20\,M_{\mathrm{ref}}$ (green). See Ref. [70] for an
animation showing corner plots for mass and spin parameters for the pre- and
post-cutoff analyses at more cutoff times.

To further explore the pre-merger waveform suppression, we compare the full
analysis (black) in which precession is constrained and the
post-$t=-10\,M_{\mathrm{ref}}$ analysis where the data are uninformative about
precession in Fig. 2. In order to focus on waveform features that are
informative compared to the noise, we plot the whitened waveform.666Whitened
waveforms are obtained by dividing the Fourier-domain waveform by the noise
amplitude spectral density and then inverse Fourier-transforming. Its value
depends on the sampling rate. Though we sample the data at $2048\,\mathrm{Hz}$
for inference, we use $1024\,\mathrm{Hz}$ when plotting for comparison to Fig.
1 of Ref. [3].  In the top panel the grayed-out region denotes data available
to the full analysis but not the post-$t=-10\,M_{\mathrm{ref}}$ one. The inset
focuses around $t=-15\,M_{\mathrm{ref}}$, an extremum of the final pre-merger
cycle. The reconstructions are inconsistent at the 50% credible level, with
the post-$t=-10\,M_{\mathrm{ref}}$ analysis resulting in a larger amplitude
(in absolute value). This inconsistency only occurs at the extrema of the
final pre-merger cycle, i.e. the peak around $t\sim-30\,M_{\mathrm{ref}}$ and
trough at $t\sim-15\,M_{\mathrm{ref}}$, see Fig. 9 in the Supplementary
Material.

The bottom panel shows marginal posteriors for select quantities: the
effective precessing spin $\chi_{\mathrm{p}}$, the absolute value of the
whitened strain $|\hat{h}|$ (units of standard deviations $\sigma$ of the
noise), and the difference between $\iota$—the angle between the direction of
maximum signal emission and the line of sight [17]—and $\pi/2$ at
$t=-15\,M_{\mathrm{ref}}$. As expected from Fig. 1, $\chi_{p}$ and $|\hat{h}|$
are anticorrelated (albeit weakly): when the full data are analyzed (black),
the final cycle is constrained to be weaker, resulting in a larger
$\chi_{\mathrm{p}}$; when this cycle is excluded from the analysis (orange),
the extrapolated waveform is not required to have such a small value,
obviating the need for a higher $\chi_{\mathrm{p}}$. In summary, precession in
GW190521 is informed by a suppression of the gravitational-wave in the
observed waveform’s last cycle before merger. This is also the region in which
the waveform is overall quietest: the whitened signal is less than
$1$-$\sigma$ above the noise, compared to the subsequent merger cycles that
are over $2$-$\sigma$.

The origin of the signal suppression can be attributed to the evolution of the
emission direction. As a binary precesses, the angle between the dominant
emission direction and the line-of-sight evolves, changing the amplitude of
the observed signal. Systems with $\iota\sim\pi/2$ are quieter than those with
$\iota\sim 0$ or $\sim\pi$. To explore these dynamics, we plot the absolute
difference between the inclination angle and $\pi/2$ at the time
$t=-15\,M_{\mathrm{ref}}$ in Fig. 2. As expected, $\chi_{\mathrm{p}}$ and
$|\iota-\pi/2|$ are anti-correlated, while $|\hat{h}|$ and $|\iota-\pi/2|$ are
correlated. When precession is constrained, $\iota$ is found to be closer to
edge-on at the last pre-merger cycle than when precession is unconstrained,
leading to a suppressed cycle and smaller $|\hat{h}|$.

We investigate the impact of data truncation on other source parameters in
Fig. 3. Information about $\chi_{\mathrm{p}}$, $M$ and $q$ is not lost or
gained in lockstep as a function of cutoff time. At
post-$t=-20\,M_{\mathrm{ref}}$ (dark orange), the $\chi_{\mathrm{p}}$
posterior shifts away from the full analysis posterior; however, for $M$ and
$q$, this does not happen until multiple cycles later.
Post-$t=-10\,M_{\mathrm{ref}}$ (shaded light orange), i.e., at the end of the
final pre-merger cycle, the $\chi_{\mathrm{p}}$ posterior is close to the
prior, while $M$ and $q$ both resemble the posteriors from the full analysis.
Thus, the lack of an informative $\chi_{\mathrm{p}}$ posterior
post-$t=-10\,M_{\mathrm{ref}}$ does not simply arise from poor parameter
constraints over-all due to lower SNR; rather, the suppression of the final
pre-merger cycle is informative specifically about precession. We also confirm
that the $\chi_{\mathrm{p}}$ inference is not driven by a conditional
measurement based on the typically better-measured aligned spins in the
Supplementary Material.

_Conclusions —_ Precession inference for the massive, distant binary black
hole signal GW190521 is subtle. It originates from contrasting a
${\sim}40~{}\mathrm{ms}$ slice of data from the final pre-merger cycle between
$t=-40\,M_{\mathrm{ref}}$ and $-10\,M_{\mathrm{ref}}$ with the loud merger
cycles following it. The merger of GW190521 is $2$ loud cycles that reach
$2.5\sigma$ above the noise and is informative about the source’s masses.
However, precession is only constrained away from the prior when the merger is
observed in tandem with the final pre-merger cycle, which does not rise more
than $1\sigma$ above the noise. The measurement is linked to a relative
suppression of the aforementioned final cycle, caused by the binary tilting
toward an edge-on configuration due to precession. This picture qualitatively
agrees with the interpretation posited in Ref. [3] by comparing precessing and
spin-aligned waveform reconstructions for the full signal, and is supported by
simulations [20].

Siegel et al. [91] carried out a complementary study seeking a description of
the the GW190521 ringdown consistent with the NRSur7dq4 full
analysis.777Capano et al. [23, 22] provided an alternative interpretation
based on waveform models of the Phenom family [78, 29]. Ringdown mode content
encodes information about the preceeding binary dynamics [60, 16, 8], meaning
it is (in theory) possible to identify signatures of precession in the
ringdown. Siegel et al. [91] found support for the presence of at least two
modes; consistency with NRSur7dq4 suggests a configuration including the 220
and 210 fundamental modes. A large 210 mode amplitude could be expected under
strong precession [91]. The fact that past GW190521 ringdown-only analyses
cannot unequivocally infer precession is consistent with our finding that a
post-peak analysis is not sufficient to constrain precession.

Our study highlights the delicate nature of precessional imprints on observed
signals, providing a new view of spins in massive systems beyond the frequency
domain [15, 102, 3]. For GW190521, the difference between the most informative
precession inference to date and the prior boils down to a single, quiet pre-
merger cycle that needs to be measured to better than half a standard
deviation, cf., the difference between the black and orange waveforms in Fig.
2. This is quantitatively in agreement with the conclusions of Payne et al.
[77] who explored the impact of data quality on our ability to obtain an
unbiased measurement at that level.

Our novel time-domain approach of tracing the observational imprint of
interesting physical effects cycle-by-cycle can provide physical intuition
about how key source properties are inferred in relation to observed data
features. In anticipation of further massive observed signals, such a
correspondence between source dynamics and observed data can help pinpoint the
most informative data in order to assess data quality and waveform
systematics, and enable us to morphologically study competing physical
interpretations that are likely to keep arising.

_Data Release —_ Posterior samples can be found on Zenodo at Ref. [72].
Scripts to generate the waveform reconstructions and inclination angles are on
Github at Ref. [73], as are notebooks to generate all figures.

_Acknowledgements —_ We thank Harrison Siegel for helpful discussions on
ringdown analyses of GW190521, Sylvia Biscoveanu for insights about inference
on high-mass gravitational-wave sources, Sophie Hourihane for assistance
whitening gravitational-wave signals, Davide Gerosa for insight on alternative
measures of precession, and Will Farr for insights about time-domain
inference. We also extend thanks to Jacob Lange, Christopher Berry, Carlos
Lousto, Juan Calderon Bustillo, and Salvatore Vitale for their helpful
comments on our manuscript.

This material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. S.M. and K.C.
were supported by NSF Grant PHY-2110111. K.C. acknowledges support from the
Sloan Foundation. The Flatiron Institute is funded by the Simons Foundation.
V.V. acknowledges support from NSF Grant No. PHY-2309301, and the European
Union’s Horizon 2020 research and innovation program under the Marie
Skłodowska-Curie grant agreement No. 896869. I.M. is a recipient of the
Australian Research Council Future Fellowships (FT190100574). The authors are
grateful for computational resources provided by Cardiff University and
supported by STFC grant ST/I006285/1. Software: emcee [34], LALSuite [64],
numpy [53], scipy [96], h5py [25], matplotlib [55], seaborn [99], ringdown
[59, 58], gwtools [38].

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##  I Supplement to “GW190521: tracing imprints of spin-precession on the most
massive black hole binary”

###  I.1 Data release

Our work is based on selecting cutoff times along GW190521 and independently
analyzing the data before and after these cutoffs. For the main text, we
display select results for cutoff times in intervals of
$10\,M_{\mathrm{ref}}=0.0127$ seconds – as defined in Ref. [2] – from the
coalescence GPS geocenter time $t_{0}=1242442967.405764$ s, ranging from
$t=-50\,M_{\mathrm{ref}}$ to $50\,M_{\mathrm{ref}}$. Cutoff times in intervals
of $2.5\,M_{\mathrm{ref}}$ between $t=-30\,M_{\mathrm{ref}}$ and
$20\,M_{\mathrm{ref}}$ were additionally explored to more finely resolve the
transition between informative and uninformative $\chi_{\mathrm{p}}$
posteriors. Data products for all of these results are available on Zenodo at
Ref. [26]. Code to generate waveform reconstructions and time-evolution of the
inclination angles, as well as notebooks to make all figures can be found on
Github at Ref. [27]. The repository additionally contains animations showing
results from for all pre- and post-cutoff analyses in intervals of
$t=2.5\,M_{\mathrm{ref}}$:

  * •

Ref. [25]: Animation of $\chi_{\mathrm{p}}$ posteriors, whitened
reconstructions, and colored reconstructions for all time slices; similar to
Fig. 1.

  * •

Ref. [24]: Animation of a corner plot for $\chi_{\mathrm{p}}$,
$\chi_{\textrm{eff}}$, $M$, and $q$ at all time slices; similar to Figs. 3 and
6.

###  I.2 Parameter estimation settings and priors

During parameter estimation, we sample the masses, spin magnitudes and tilt
angles, azimuthal inter-spin angle, azimuthal precession cone angle,
inclination angle, luminosity distance, and phase of coalescence. The time of
coalescence, right ascension, declination, and polarization angle are fixed
for computational efficiency; we have ensured that further sampling over these
parameters does not alter our conclusions, as is shown in Fig. 4. Priors for
all parameters are given in Table 1. We use slightly different mass and
distance priors than Ref. [1] but – as we show in Fig. 5 – obtain consistent
results when analyzing the full signal.888We use the NRSur7dq4 [33] posterior
samples released by LIGO/Virgo at Ref. [3] for this comparison.

Parameter estimation settings are listed in Table 2. All analyses make use of
the three detector network of LIGO Livingston, LIGO Hanford, and Virgo, using
strain and event power spectral densities publicly available on the
Gravitational-wave Open Science Center [9, 8].999Specifically we download the
“32 sec, 16 KHz” strain data uploaded to Ref. [7]. We use a lower and
reference frequency of 11 Hz, a maximum frequency of 1024 Hz, and a sampling
rate of 2048 Hz.101010We opt for a minimum frequency of 11 Hz which is
consistent with Refs. [4, 6], but not the more recent Ref. [5]. Each analysis
conducted before a given cutoff time begins at the GPS time at geocenter
$1242442966.9077148$ s; those conducted after a given time all end at
$1242442967.607715$ s. To translate the cutoff times from geocenter to the
times at the three detectors, we use a right ascension of $\alpha=6.075$ rad,
a declination of $\delta=-0.8$ rad, and polarization angle of $\psi=2.443$
rad.

Parameter |  Symbol |  Value
---|---|---
Detector-frame total mass |  $M(1+z)$ |  $U(200,500)\,M_{\odot}$
Mass ratio |  $q$ |  $U(0.17,1)$
Primary spin magnitude |  $\chi_{1}$ |  $U(0,1)$
Secondary spin magnitude |  $\chi_{2}$ |  $U(0,1)$
Primary spin tilt |  $\theta_{1}$ |  isotropic
Secondary spin tilt |  $\theta_{2}$ |  isotropic
Azimuthal inter-spin angle |  $\phi_{12}$ |  $U(0,2\pi)$
Azimuthal cone precession angle |  $\phi_{JL}$ |  $U(0,2\pi)$
Inclination angle |  $\iota$ |  isotropic
Luminosity distance |  $d_{L}$ |  $U(10^{3},10^{4})$ Mpc
Phase of coalescence |  $\varphi$ |  $U(-\pi,\pi)$
Table 1: Priors used in parameter estimation. $U(a,b)$ means uniform between $a$ and $b$. An isotropic prior for an angle $x$ means that the prior on $\cos x$ is uniform between $0$ and $\pi$. Right ascension, declination, polarization, and time of coalescence are fixed to the values given in Table 2. Parameter |  Symbol |  Value
---|---|---
Start GPS time of “before” segments |  $t_{\mathrm{start}}$ |  1242442966.907715 s
End GPS time of “after” segments |  $t_{\mathrm{end}}$ |  1242442967.607715 s
Coalescence GPS time |  $t_{0}$ |  1242442967.405764 s
Mass-time scaling relation |  $1\,M_{\mathrm{ref}}$ |  0.00127 s
Right ascension |  $\alpha$ |  6.075 rad
Declination |  $\delta$ |  -0.800 rad
Polarization angle |  $\psi$ |  2.443 rad
Minimum frequency |  $f_{\mathrm{min}}$ |  11 Hz
Maximum frequency |  $f_{\mathrm{max}}$ |  1024 Hz
Sampling rate |  $f_{\mathrm{samp}}$ |  2048 Hz
Reference frequency |  $f_{\mathrm{ref}}$ |  11 Hz
Table 2: Settings for the time-domain parameter estimation for GW190521. The
pre-cutoff analyses all begin at $t_{\mathrm{start}}$, while the post-cutoff
ones end at $t_{\mathrm{end}}$. These times are calculated with respect to the
geocenter time $t_{0}$ and then shifted between detectors using the extrinsic
parameters $\alpha$, $\delta$, and $\psi$. Figure 4:  Posteriors for the
detector-frame total mass $M$, mass ratio $q$, effective spin
$\chi_{\textrm{eff}}$, and effective precessing spin $\chi_{\mathrm{p}}$ for
parameter estimation with (lighter color) and without (darker color) sampling
over time of coalescence $t_{0}$ and sky position – referring to the right
ascension, declination, and polarization angle – for the full signal (black;
top row), the post-$t=-10\,M_{\mathrm{ref}}$ (orange; middle row), and the
post-$t=-40\,M_{\mathrm{ref}}$ (pink; bottom row). All results are consistent,
the only difference being a slightly wider total mass posterior. Figure 5:
Posteriors for the detector-frame total mass $M$, mass ratio $q$, luminosity
distance $D_{L}$, and effective precessing spin $\chi_{\mathrm{p}}$ from this
work (black), Ref. [1] (LIGO/Virgo data release; light blue), and this work
reweighted to the priors of Ref. [1] (dark blue). Our inference is consistent
with Ref. [1] under the same priors.  Figure 6: Posteriors for
$\chi_{\mathrm{p}}$ and $\chi_{\textrm{eff}}$ for GW190521 informed by data
after five representative cutoff times:
$t=\\{-40,-20,-10,20,30\\}\,M_{\mathrm{ref}}$ in pink, red, orange, green,
blue respectively, compared to the full signal analysis (solid) and the prior
(dotted) in black. Contours of the 50% credible regions are shown for the two-
dimensional posteriors. The two spin parameters are not correlated, meaning
that inferences on $\chi_{\mathrm{p}}$ at the various cutoff times do not
simply arise from conditional priors from $\chi_{\textrm{eff}}$.

Figure 7: From top to bottom: posteriors for $\chi_{\mathrm{p}}$, generalized
$\chi_{\mathrm{p}}$, and $||\vec{\chi}_{\perp}||$ for the full analysis
(black), and the post-$t=-40\,M_{\mathrm{ref}}$ (pink) and
post-$t=-10\,M_{\mathrm{ref}}$ (orange) analyses. All parametrizations of
precession lead to qualitatively similar conclusions about spin inference and
GW190521.

###  I.3 Quantifying precession

Spins aligned with the orbit are characterized at leading order by the
effective spin $\chi_{\textrm{eff}}$,

$\chi_{\mathrm{eff}}=\frac{\chi_{1}\cos\theta_{1}+q\,\chi_{2}\cos\theta_{2}}{1+q}\in(-1,1)\,,$
(2)

which is conserved to at least the second post-Newtonian order in the inspiral
[28, 10]. Precession – stemming from mis-aligned spins – is typically
characterized via the “effective precessing spin” parameter [31], as defined
in Eq. (1) in the main text. Posteriors for $\chi_{\textrm{eff}}$ versus
$\chi_{\mathrm{p}}$ for five representative cutoff times (compared to the full
signal posterior and the prior) are shown in Fig. 6. Though typically
$\chi_{\textrm{eff}}$ is better constrained than $\chi_{\mathrm{p}}$, this is
not the case for GW190521: $\chi_{\textrm{eff}}$ is never inferred to be far
from its prior with only slight deviations from the prior that occur at
different times from when we best constrain $\chi_{\mathrm{p}}$. Furthermore,
because the $\chi_{\mathrm{p}}$ and $\chi_{\textrm{eff}}$ are not correlated
at any time slice, inferences of $\chi_{\mathrm{p}}$ – and thus our main
conclusions – do not arise from a conditional measurement driven by
$\chi_{\textrm{eff}}$ [17]. In other words, we are not just identifying
differences in the spins from $\chi_{\textrm{eff}}$ that then propagate into
differences in $\chi_{\mathrm{p}}$, but rather we are directly identifying
differences in the in-plane spin components’ inference.

However, $\chi_{\mathrm{p}}$ is a parameter motivated by inspiral precession
dynamics [29, 30, 31]. The angles $\theta_{i}$ associated with each black
hole’s spin vector – used to calculate $\chi_{\mathrm{p}}$ – are defined with
respect to the binary’s orbital angular momentum, a quantity that becomes
meaningless during the merger as the two black holes cease orbiting each
other. Indeed, the canonical equations describing precession dynamics (see Eq.
(11) of Ref. [11]) are only accurate through the second post-Newtonian order,
an expansion which is only valid when the black holes are moving substantially
slower than the speed of light, i.e. during the inspiral [11, 23, 13, 12, 14].
Once entering the merger, precession becomes no longer analytically tractable.

Alternative parameterizations of precession have been proposed, though they
are all still inspired by intuition gained from the inspiral precession
equations. Gerosa et al. [19] note that in the definition of
$\chi_{\mathrm{p}}$, some – but not all – of the precession-timescale
oscillations are averaged over. To rectify this and retain all variation over
the precessional timescale [18], they define a “generalized
$\chi_{\mathrm{p}}$” in their Eq. (15). Defined on the range $[0,2)$, the
generalized $\chi_{\mathrm{p}}$ has the advantage that it distinguishes
between binaries in which one (gen. $\chi_{\mathrm{p}}<1$) versus both (gen.
$\chi_{\mathrm{p}}>1$) spins are precessing. Thomas et al. [32] amend the fact
that $\chi_{\mathrm{p}}$ does not accurately account for higher order multi-
polar modes by defining the “effective precession spin vector”
$\vec{\chi}_{\perp}$ – their Eq. (9). This parameter accounts for more degrees
of freedom and could facilitate a better representation of precession in the
strong-field regime. The magnitude of the effective precession spin vector
$||\vec{\chi}_{\perp}||$ is a scalar measure of precession, analogous to the
traditional $\chi_{\mathrm{p}}$. A third way to quantify the observability of
spin-precession is through the precessional SNR $\rho_{\mathrm{p}}$, as
defined in Fairhurst et al. [15, 16]. Based on the idea that precession is
only inferred when two gravitational-wave harmonics are observed,
$\rho_{\mathrm{p}}$ is related to the SNR in the second most significant
waveform harmonic, defined in Eq. (39) of Ref. [16].

Posteriors for generalized $\chi_{\mathrm{p}}$ and $||\vec{\chi}_{\perp}||$
are plotted in Fig. 7 for the post-$t=-40\,M_{\mathrm{ref}}$ (pink) and
$-10\,M_{\mathrm{ref}}$ (orange) analyses, compared to the full analysis
posteriors (black). We do not plot $\rho_{\mathrm{p}}$, as the current code to
compute it [21] necessitates a frequency domain waveform, which NRSur7dq4 is
not [33]; Hoy et al. [20] presented $\rho_{\mathrm{p}}$ constraints for
GW190521 using a different set of waveform models. Both gen.
$\chi_{\mathrm{p}}$ and $||\vec{\chi}_{\perp}||$ follow the same trend as the
traditional $\chi_{\mathrm{p}}$, shown in the top panel of Fig. 7 for
comparison. The inference of precession is closer to that from the full signal
in the post-$t=-40\,M_{\mathrm{ref}}$ analysis than the
$t=-10\,M_{\mathrm{ref}}$.

The above quantities are all defined based on the inspiral dynamics;
precession representations motivated by merger dynamics are not available to
the best of our knowledge.

Figure 8: (Lower subplot) Network matched-filter SNR for the pre-cutoff (blue)
and post-cutoff (orange) analyses as a function of the cutoff time $t$. Points
indicate the median SNR and error bars represent the 90% credible region. The
horizontal black line represents the median SNR recovered by the full
analysis, with the 90% credible region shaded in gray. (Upper subplot)
Whitened strain data in LIGO Livingston (gray), and whitened waveform
reconstruction for the maximum posterior (abbr. “max $P$”) draw from the full
analysis (black), plotted to help visualize the signal alongside the SNR
evolution. In both subplots, the red shaded region – between
$t=-40\,M_{\mathrm{ref}}$ and $-10\,M_{\mathrm{ref}}$ – is that which is
identified as crucial for constraining $\chi_{\mathrm{p}}$. Figure 9:
Identification of regions in time (pink shaded) during which the 50% credible
intervals (CI) of the reconstructed waveform strain $\hat{h}$ are inconsistent
between the full (black; labeled $\hat{h}_{\mathrm{full}}$) and
post-$t=-10\,M_{\mathrm{ref}}$ (orange; labeled $\hat{h}_{\mathrm{-10}}$)
analyses. This occurs around the two extrema of the final pre-merger cycle
($t\sim-30\,M_{\mathrm{ref}}$ and $t\sim-15\,M_{\mathrm{ref}}$; indicated with
vertical black dashed lines), when $\delta\hat{h}$—the difference between the
75% CI of one analysis and the 25% CI of the other (green and blue)—is
negative (enlarged in the inset).

###  I.4 Signal strength over time

We track the time-bounded matched-filter signal-to-noise ratio (SNR) in the
LIGO Livingston, LIGO Hanford, and Virgo detector network recovered by our
analyses, as defined in Eqs. (52) and (53) of Isi and Farr [22]. Figure 8
shows the SNR evolution for the pre- (blue) and post-cutoff (orange) analyses.
The change in SNR is negligible between $t=-40\,M_{\mathrm{ref}}$ and
$-10\,M_{\mathrm{ref}}$ (red shaded region) for both the pre- and post-cutoff
analyses, suggesting that the change in inferred $\chi_{\mathrm{p}}$ between
these times is not simply due to a drop in SNR.

To investigate correlations between signal strength and $\chi_{\mathrm{p}}$
(see Fig. 2 in the main text), we compare the 50% credible intervals (CI) of
the reconstructed waveform strain for two analyses, one in which precession is
inferred (full analysis) and one in which it is not
(post-$t=-10\,M_{\mathrm{ref}}$ analysis) . The top panel in Fig. 9 shows the
50% CIs for the strain $\hat{h}$ for these two analyses in black and orange
respectively. In the bottom panel, we plot the differences $\delta\hat{h}$
between the 75% CI of one analysis and the 25% CI of the other in green and
blue. When either difference is negative, the two sets of waveform
reconstructions are inconsistent at the 50% CI. This inconsistency only occurs
at the extrema of the final pre-merger cycle, i.e. the peak around
$t\sim-30\,M_{\mathrm{ref}}$ and trough at $t\sim-15\,M_{\mathrm{ref}}$, as
indicated by the time-slices shaded in pink. These regions are enlarged in the
inset.

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