Searching for structure in the binary black hole spin distribution Jacob Golomb LIGO Laboratory California Institute of Technology Pasadena CA 91125 USA and

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Searching for structure in the binary black hole spin distribution

Jacob Golomb

LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA and

Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA

Colm Talbot

LIGO Laboratory, Massachusetts Institute of Technology,

185 Albany St, Cambridge, MA 02139, USA and

Department of Physics and Kavli Institute for Astrophysics and Space Research,

Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA

(Dated: March 15, 2023)

The spins of black holes in merging binaries can reveal information related to the formation

and evolution of these systems. Combining events to infer the astrophysical distribution of black

hole spins allows us to determine the relative contribution from diﬀerent formation scenarios to the

population. Many previous works have modelled spin population distributions using low-dimensional

models with statistical or astrophysical motivations. While these are valuable approaches when the

observed population is small, they make strong assumptions about the shape of the underlying

distribution and are highly susceptible to biases due to mismodeling. The results obtained with

such parametric models are only valid if the allowed shape of the distribution is well-motivated (i.e.

for astrophysical reasons). Unless the allowed shape of the distribution is well-motivated (i.e., for

astrophysical reasons), results obtained with such models thus may exhibit systematic biases with

respect to the true underlying astrophysical distribution, along with resulting uncertainties not

being reﬂective of our true uncertainty in the astrophysical distribution In this work, we relax these

prior assumptions and model the spin distributions using a more data-driven approach, modelling

these distributions with ﬂexible cubic spline interpolants in order to allow for capturing structures

that the parametric models cannot. We ﬁnd that adding this ﬂexibility to the model substantially

increases the uncertainty in the inferred distributions, but ﬁnd a general trend for lower support at

high spin magnitude and a spin tilt distribution consistent with isotropic orientations. We infer that

62 - 87% of black holes have spin magnitudes less than a= 0.5, and 27-50% of black holes exhibit

negative χeﬀ . Using the inferred χeﬀ distribution, we place a conservative upper limit of 37% for

the contribution of hierarchical mergers to the astrophysical BBH population. Additionally, we ﬁnd

that artifacts from unconverged Monte Carlo integrals in the likelihood can manifest as spurious

peaks and structures in inferred distributions, mandating the use of a suﬃcient number of samples

when using Monte Carlo integration for population inference.

I. INTRODUCTION

Gravitational waves oﬀer a unique probe into the prop-

erties of merging black holes (BHs) and neutron stars.

Since the ﬁrst such detection in 2015, the LIGO-Virgo

network [1–3] has reported the detection of ∼90 bi-

nary black hole (BBH) mergers, with each gravitational-

wave (GW) signal encoding physical information about

the BHs involved, such as their masses and angular mo-

menta (spins) [4,5]. Extracting this information has has

enabled the study of properties of BBH systems on both

an individual and population-level basis. From an astro-

physical perspective, combining GW detections to infer

the mass, spin, and redshift distributions of BBH systems

can help answer questions ranging from binary formation

and stellar evolution [6,7] to the expansion rate of the

Universe and possible deviations from General Relativity

[8,9].

The spin of the BHs in a BBH system oﬀer insight into

the history of the binary. For example, BH spins can help

reveal whether the BHs in a BBH system formed directly

from core collapse of a heavy star or from the previous

merger of two lighter BHs [10–13]. Although the pro-

cesses governing the angular momentum transport out of

a stellar core during collapse are not well-constrained, re-

cent modeling work indicates that BHs resulting directly

from core collapse supernovae should have negligible spin

magnitudes [14–16]. While processes such as tidal inter-

actions and mass transfer can induce higher spins on BHs

in binary systems, it is uncertain how appreciable the re-

sulting spin-ups can be [17–20]. On the other hand, BHs

formed from the merger of two non-spinning BHs are ex-

pected to form a ﬁnal BH with a relatively high spin

magnitude [13,21,22], motivating the possibility to use

spin magnitude as a tracer of a BHs formation history.

The direction of the BH spin vectors also encode in-

formation related to the formation history of a BBH sys-

tem. Models suggest that BBH systems formed from

common evolution, in which the component BHs evolve

together from a stellar binary in an isolated environment

free from signiﬁcant dynamical interactions, should have

component spin vectors nearly aligned with the orbital

angular momentum axis, with any tilt being eﬃciently

brought into alignment by tidal interactions [23,24]. On

the other hand, BBH systems formed from dynamical en-

counters are not expected to have any correlated spins,

arXiv:2210.12287v2 [astro-ph.HE] 14 Mar 2023

such that the BH spin vectors are isotropic with respect

to the orbital angular momentum [11,25,26].

While only a couple of events in the third gravitational-

wave transient catalog individually feature conﬁdently

high spin magnitudes or anti-alignment (i.e. a spin vec-

tor pointing opposite the angular momentum), hierarchi-

cally combining observations of GW events while folding

in selection eﬀects can reveal the degree to which these

parts of spin parameter space contribute to the astro-

physical distribution of BH spins. Previous work has

used these inferred contributions to estimate the fraction

of BBH systems in the local Universe which may have

been formed hierarchically, dynamically, and by isolated

evolution [6,10,11,27]. However, recent publications

have disagreeing estimates for the contributions of anti-

aligned and non-spinning BBHs to the astrophysical pop-

ulation.

In [6,7], the authors conclude that the BBH distribu-

tion must feature anti-aligned spins at >90% credibility,

in contrast to the conclusion drawn in [28] that such anti-

alignment is not evident in the population. In addition,

[29] ﬁnds evidence for a non-spinning subpopulation of

BHs, a conclusion which was challenged by [30]. While

technical diﬀerences exist between works, a major possi-

ble contribution to some of these diﬀering conclusions is

model misspeciﬁcation (see, e.g. [31,32]); that is, diﬀer-

ent assumptions being imposed on the functional form of

the spin distribution.

The Default model in [6,7] models the distribution of

the magnitude of the BH spin vector and the tilt angle be-

tween the spin vector and the orbital angular momentum.

They adopt a Beta distribution for the spin magnitude

model [6,33],

π(a1,2|αχ, βχ) = Beta(a1,2|αχ, βχ),(1)

where a1(a2) is the magnitude of the spin vector of the

primary (secondary) BH, and αχand βχare population

hyperparameters determining the structure of the Beta

distribution. The model for the distribution of tilt angles,

θ, is motivated by two subpopulations: one preferentially

aligned (cos(θ)≈1) and one isotropic [6,7,34]. The

model is parameterized as:

π(cos θ1,2|ξ, σt) = ξGt(cos θ1|σt)Gt(cos θ2|σt) + 1−ξ

(2)

where Gtis a truncated Gaussian centered at cos θ= 1

with standard deviation σtand bounded in [−1,1], and

ξis the relative mixing fraction between the subpopula-

tions. The second term corresponds to the contribution

from the uniform (isotropic) distribution.

This population model has been extended in other

work to allow for other astrophysically-motivated fea-

tures to help draw conclusions related to the diﬀerent

formation scenarios present in the astrophysical distribu-

tion [28–30,35]. Adopting an astrophysically-motivated,

strongly parametric model necessarily limits the possible

features resolvable in the inferred distribution to what

the chosen function can model. Accordingly, in this work,

we consider a strongly parametric model to be one that

has a speciﬁc, prior-determined shape as provided by the

parameterization (e.g. a normal distribution), which is

then constrained by the data. When using such a dis-

tribution to draw astrophysical conclusions from the in-

ferred population, this is a reasonable and intended con-

sequence, as the model is chosen to encode prior beliefs

on the parameters that should govern the astrophysical

distribution; however, if additional features exist in the

true astrophysical distribution and a strongly paramet-

ric model cannot account for them, such features can be

missed and a biased result may be obtained.

Previous work has shown that substructures in the BH

mass distribution can be captured by cubic splines acting

as a perturbation on top of a simpler parameteric model

[6,36]. In [36], the authors consider an exponentiated

spline perturbation modulating an underlying power law

in the mass distribution. In this work, we model the spin

magnitude and tilt distributions using exponentiated cu-

bic splines modulating a ﬂat distribution to obtain a more

data-driven result for the inferred population of BH spins.

In doing so, we limit the potential bias caused by mis-

modeling the spin distribution and allow for the possi-

bility of capturing features not accessible with a strongly

parametric model.

The remainder of the paper is organized as follows. In

Section II, we detail the functional form and implemen-

tation of the cubic spline model. We provide the back-

ground of hierarchical Bayesian inference in Section III,

as it applies to population inference with GW sources. In

Section IV we present the resulting spin distributions we

obtain for various spline models adopted in this work. Fi-

nally, we use these results to draw conclusions related to

the astrophysical distribution of BBH spins and provide

a relevant discussion in Section V. We additionally sup-

ply three appendices; the ﬁrst provides additional details

about an eﬃcient caching technique for the cubic spline

model, the second explores the eﬀect of uncertainty in

our estimation of the selection function, and the third

describes robustness of our results to diﬀerent choices of

prior distribution.

II. MODELS

Following the model for the black hole mass distribu-

tion considered in [36], we ﬁt the distribution of spin

magnitudes and cosine tilts using exponentiated cubic

splines

p(x)∝ef(x).(3)

A spline is a piecewise polynomial function deﬁned by a

set of node positions, the value of the function at those

nodes, and boundary conditions at the end nodes. We

use a cubic spline as it is the lowest order spline that

enforces continuity of the function and its ﬁrst derivative

everywhere.

A. Node positions and amplitudes

In this work we consider models with 4, 6, 8, and 10

nodes spaced linearly in the domain of the parameter

space. For the two distributions being modelled with

splines, this gives 16 unique spin models (4 amplitude

node placement models ×4 tilt node placement models).

Our choice for the prior on the amplitude of each node

is a unit Gaussian distribution. Comparisons with other

node amplitude prior choices are detailed in Appendix C.

In order to fully characterize a cubic spline, the ﬁrst

and second derivatives must be determined at each node.

For all but the endpoints, these derivatives are speciﬁed

by requiring continuity in the spline and its derivative. At

the endpoints, there is no unique way to determine this

and a range of boundary conditions are commonly used.

For our implementation, we want the prior distribution

of the derivatives at the endpoints to match that of the

internal nodes. This requires providing two additional

free parameters at each end of the spline. In practice, we

add two additional nodes outside each boundary, with

amplitudes that are free to vary according to the prior.

Throughout this work, the number of nodes in a model

refers to the number of nodes within the domain (i.e.

not including these outside nodes). The spacing between

these nodes is the same as that between nodes within the

domain.

B. Modeling spins with splines

In this work, we use the spline model detailed above to

model the population of spin magnitudes aand tilt angles

cos θ. Consistent with [6,7], we model these parameters

as independent and identically distributed. The total

spin population model is

πspin(η|Λs) = pa(a1)pa(a2)pt(cos θ1)pt(cos θ2),(4)

where Λsis the set of population hyperparameters con-

trolling the spline node location and amplitudes. The

functions pare determined from Eq. 3. The domain of the

spin magnitude distribution extends over a∈[0,1] and

that of the spin tilt distribution covers cos θ∈[−1,1].

III. METHODS

A. Hierarchical Bayesian Inference

In order to constrain the spin magnitude and tilt dis-

tribution, we carry out hierarchical Bayesian inference

in which we calculate the likelihood of the entire ob-

served dataset given a set of population hyperparameters

Λ while marginalizing over the uncertainty in the physical

parameters of each event. After analytically marginaliz-

ing over the total merger rate Rwith a prior π(R)∝R−1,

Parameter Description Prior

αPower Law index for m1(-4, 12)

βPower Law index for q(-2, 7)

Mmax Maximum mass (60, 100)

Mmin Minimum mass (2, 7)

λFraction of sources in Gaussian peak (0, 1)

Mpp Location of Gaussian peak (20, 50)

σpp Standard deviation of Gaussian peak (1, 10)

δmMinimum mass turn-on length (0, 10)

TABLE I. Priors for mass distribution used in hierarchical

inference, consistent with those used in [6]. Priors on the spin

distribution are described in Section II. Priors are uniform

between the bounds listed in the third column.

we express the likelihood of the hyperparameters Λ pa-

rameterizing the population is expressed as (e.g. [37]):

L({d}|Λ) ∝pdet(Λ)−N

iZL(di|ηi)π(ηi|Λ)dηi.(5)

Here, L(di|ηi) is the likelihood of observing the data d

from the ith event, given physical (i.e. single-event) pa-

rameters ηi. In this work, ηiconsists of masses, spins, and

redshift of the ith event. The quantity pdet(Λ) encodes

the sensitivity of the search algorithm that identiﬁed the

signals and is described in more detail in Sec. III B.

Our population model π(η|Λ) describes the astrophysi-

cal distribution of masses, redshifts, and spins. We model

the primary mass distribution with the Powerlaw + Peak

model [38], the mass ratio (q=m2

m1) distribution with

a power law, and the redshift distribution also with a

power law, with source-frame comoving merger rate den-

sity R(z)∝(1 + z)3[6,7,39]. We choose to ﬁx the

redshift distribution because we use our own injection

set to estimate sensitivity, thresholding on SNR rather

than FAR to determine “found” injections. Since this

makes the threshold used to select real events (FAR <1

yr) slightly diﬀerent from that used to threshold sensitiv-

ity injections, and the redshift distribution is particularly

sensitive to the near-threshold events, we ﬁx the redshift

distribution in order to avoid biases (see [40] for an ex-

ample of where a similar approximation was used). See

Section III B for details on sensitivity injections. We list

the hyperparameters Λ and their corresponding priors in

Table I.

An initial choice that must be made when computing

Eq. 5is which events to include in the analysis. Typically

this is done by establishing some detection threshold on

the Signal-to-Noise Ratio (SNR) or False Alarm Rate

(FAR) and including all events that pass this threshold.

We choose to include the 59 events in the third observing

run (O3) of the LIGO-Virgo network which have a False

Alarm Rate of less than 1 year−1and are included in the

main BBH analysis of [6]. We limit ourselves to events

in O3 for self-consistency, as the injections we perform to

evaluate selection eﬀects (see Section III B) use O3a and

O3b detector sensitivities.

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SearchingforstructureinthebinaryblackholespindistributionJacobGolombLIGOLaboratory,CaliforniaInstituteofTechnology,Pasadena,CA91125,USAandDepartmentofPhysics,CaliforniaInstituteofTechnology,Pasadena,CA91125,USAColmTalbotLIGOLaboratory,MassachusettsInstituteofTechnology,185AlbanySt,Cambridge,MA02139,...

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