# Looking for the parents of LIGO’s black holes

Vishal Baibhav<EMAIL_ADDRESS>Department of Physics and Astronomy, Johns
Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218, USA
Emanuele Berti  Department of Physics and Astronomy, Johns Hopkins University,
3400 N. Charles Street, Baltimore, Maryland 21218, USA  Davide Gerosa  School
of Physics and Astronomy & Institute for Gravitational Wave Astronomy,
University of Birmingham, Birmingham, B15 2TT, UK  Matthew Mould  School of
Physics and Astronomy & Institute for Gravitational Wave Astronomy,
University of Birmingham, Birmingham, B15 2TT, UK  Kaze W. K. Wong  Department
of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street,
Baltimore, Maryland 21218, USA

###### Abstract

Solutions to the two-body problem in general relativity allow us to predict
the mass, spin, and recoil velocity of a black-hole merger remnant given the
masses and spins of its binary progenitors. We address the inverse problem:
given a binary black-hole merger, can we use the parameters measured by
gravitational-wave interferometers to determine whether the binary components
are of hierarchical origin, i.e. whether they are themselves remnants of
previous mergers? If so, can we determine at least some of the properties of
their parents? This inverse problem is in general overdetermined. We show that
hierarchical mergers occupy a characteristic region in the plane composed of
the effective-spin parameters $\chi_{\rm eff}$ and $\chi_{\rm p}$, and
therefore a measurement of these parameters can add weight to the
hierarchical-merger interpretation of some gravitational-wave events,
including GW190521. If one of the binary components has hierarchical origin
and its spin magnitude is well measured, we derive exclusion regions on the
properties of its parents: for example we infer that the parents of GW190412
(if hierarchical) must have had unequal masses and low spins. Our formalism is
quite general, and it can be used to infer constraints on the astrophysical
environment producing hierarchical mergers.

Figure 1: Schematic representation of the inverse problem addressed in this
paper. In Sec. II we ask: if we detect GWs from a binary BH merger with mass
ratio $q$, effective spin $\chi_{\rm eff}$ and precessional spin $\chi_{\rm
p}$, can we establish whether one of the binary components (in this sketch,
the one with remnant spin $\chi_{\mathrm{r}}=\chi_{1}$) originated from a
previous merger? In Sec. III we instead ask: if indeed one of the binary
components comes from a previous merger and we measure $\chi_{\mathrm{r}}$,
can we determine the mass ratio $Q$, spin magnitudes $\xi_{1,2}$ and recoil
velocity of the parent binary? The merger remnant does not necessarily have to
be the primary component of the observed binary, as shown in this sketch
(i.e., the arguments in Sec. III also apply to the case where
$\chi_{\mathrm{r}}=\chi_{2}$).

##  I Introduction

Gravitational-wave (GW) astronomy is rapidly entering a data-driven regime
where astrophysical modeling uncertainties are becoming the key limiting
factor. The most popular astrophysical formation channels for the merging
black-hole (BH) binaries detected by LIGO and Virgo include isolated binary
evolution in the galactic field and dynamical assembly in either dense stellar
clusters or accretion disks [1, 2]. Astrophysical predictions depend on
several poorly understood phenomena (nuclear-reaction rates, supernova kicks,
common-envelope efficiency, metallicity, formation and evolution of stellar
clusters), which make the various channels largely or partially degenerate.
While there is some evidence that multiple formation channels provide
comparable contributions to the overall rate of merger events detectable in
GWs [3, 4], pinpointing the origin of specific systems observed by LIGO and
Virgo remains challenging.

The difficulty of the task is greatly alleviated by the so-called gaps in the
BH binary parameter space: regions that can be populated only by a subset of
the proposed formation channels. The mass spectrum of merging BHs is predicted
to present two such gaps. The typical timescales involved in the explosion
mechanism may prevent the formation of compact objects between $\sim
3M_{\odot}$ and $\sim 5M_{\odot}$ [5], while unstable pair production during
the advanced evolutionary stages of massive stars can impede collapse to BHs
with masses $\gtrsim 50M_{\odot}$ and $\lesssim 120M_{\odot}$ [6, 7]. The
existence of both gaps is partially supported by current observations, which
show that the BH binary merger rate is indeed suppressed in those regions [8].
However, observations also tell us that the gaps are somehow polluted, with
GW190814 [9] and GW190521 [10] presenting component masses that sit squarely
in the lower and upper mass gap, respectively.

Spin orientations are often invoked as a promising tool to distinguish
formation channels, with binaries born in the field (assembled dynamically)
being more likely to enter the LIGO band with small (large) spin-orbit
misalignments [11, 12, 13, 14]. However, the discriminating power of the spin
directions trivially fades if the spin magnitudes turn out to be vanishingly
small, as predicted by some [15, 16] (but not all [17, 18]) models of stellar
evolution. In analogy with the mass gaps highlighted above, we previously
referred to the putative absence of high-spinning BHs as the “spin gap” [19].
While the bulk of the observed population appears to be compatible with small
but nonzero spins [8], both GW190521 [10] and GW190412 [20] present strong
evidence for spin dynamics.

Hierarchical BH mergers are a natural strategy to populate both the upper mass
gap and the spin gap [21, 22] (see [23] for a review). By avoiding stellar
collapse altogether, assembling objects from older generations of BH mergers
bypasses the constraint imposed by both pair production instabilities and
core-envelope interactions. The masses of merger remnants are roughly equal to
the sum of the masses of the merging BHs, while their spins follow a
characteristic distribution that is highly peaked at $\sim 0.7$. The key
prediction is that of a positive correlation between masses and spins, with a
depleted region in the high-mass/low-spin corner of the parameter space [24,
25]. Hierarchical mergers could explain the mass/spin gap properties of both
GW190412 [26, 27] and GW190521 [28]. Some population analyses are tentatively
reporting that a subpopulation of hierarchical mergers might be present in the
data [29, 30, 31], although current evidence is inconclusive [8]. In order to
host hierarchical mergers, astrophysical environments need to possess a
sufficiently large escape speed to retain merger remnants following
relativistic recoils [32]. Nuclear star clusters, accretion disks surrounding
active galactic nuclei (AGN), and possibly globular clusters are among the
most plausible hosts [33, 34, 35, 19, 36, 37, 38].

This line of reasoning leads to two deeply connected questions, which are
illustrated schematically in Fig. 1. First, do the measured parameters of a GW
event allow us to tell whether it originated hierarchically from previous
mergers? If so, what are the properties of the parent BHs? While the main goal
of LIGO/Virgo data analysis is to infer the properties of the merging BHs from
the observed GWs, here we take one step backward in the BH family tree and try
to constrain the properties of the parent BHs that generated the observed
binary. This problem is in general overdetermined, but we show that it is
possible to infer many properties of the parent binary.

Current spin inference is largely limited to estimates of some effective
combinations of the two spins. In Sec. II, we show that hierarchical mergers
occupy a distinct region in the plane composed of the two commonly used
parameters $\chi_{\rm eff}$ and $\chi_{\rm p}$, which can thus be used to
infer whether one or both of the merging BHs originated from a previous
merger. We apply this argument to GW190521 and show that, together with its
mass-gap properties, its spin values add considerable weight to a
hierarchical-merger interpretation. Next, in Sec. III we show that events with
a well-constrained component spin magnitude allow for a unique reconstruction
of the previous generation of BHs. Our argument relies on inverting a
numerical-relativity fit for the remnant spin. We apply this inversion to
GW190412 and find that its parent binary must have had a moderate mass ratio
and moderately low spins. Finally, in Sec. IV we present conclusions and
directions for future work. We use geometrical units $G=c=1$ and GW posterior
samples from Ref. [39].

##  II Hierarchical black-hole mergers in the $(\boldsymbol{\chi}_{\bf
eff},\,\boldsymbol{\chi}_{\bf p})$ plane

A GW signal depends in principle on all 6 degrees of freedom corresponding to
the two spin vectors $\boldsymbol{\chi}_{1}$ and $\boldsymbol{\chi}_{2}$, but
most of the discriminating power is contained in a limited number of effective
parameters. In this section we illustrate how, even with this limited set of
information, one can draw powerful constraints on the likelihood that a given
GW event originated from previous mergers.

###  II.1 Effective spins

The two spin components parallel to the binary’s orbital angular momentum are
often combined into the mass-weighted expression [40]

$\displaystyle\chi_{\rm
eff}\equiv\frac{\chi_{1}\cos\theta_{1}+q\chi_{2}\cos\theta_{2}}{1+q}\,,$ (1)

where $q=m_{2}/m_{1}\leq 1$ is the mass ratio,
$\chi_{i}=|\boldsymbol{S}_{i}|/m_{i}$ are the dimensionless Kerr parameters of
the individual BHs, and
$\theta_{i}=\arccos\small(\hat{\boldsymbol{S_{i}}}\cdot\hat{\boldsymbol{L}}\small)$
are the spin-orbit angles. The effective spin $\chi_{\rm eff}$ is a constant
of motion at second post-Newtonian (2PN) order [40, 41].

The precession of the orbital plane is encoded in the variation of the
direction of the orbital angular momentum $|{d\hat{\boldsymbol{L}}}/{dt}|$. It
is usually described in terms of a spin parameter $\chi_{\rm p}$ first defined
in a heuristic fashion as [42]

$\displaystyle\chi_{\rm
p}^{\rm(heu)}=\max\left(\chi_{1\perp},\chi_{2\perp}\right)\,,$ (2)

where

$\displaystyle\chi_{1\perp}=\chi_{1}\sin\theta_{1}\,,\quad\chi_{2\perp}=q\frac{4q+3}{4+3q}\chi_{2}\sin\theta_{2}\,,$
(3)

This parameter is routinely used in GW data analysis [8].

The quantity $\chi_{\rm p}^{\rm(heu)}\in[0,1]$ retains some, but not all,
precession-timescale variations. This inconsistency was rectified in Ref. [43]
using a complete precession average of $|{d\hat{\boldsymbol{L}}}/{dt}|$ at
2PN. Their augmented definition $\chi_{\rm p}^{\rm(av)}\in[0,2]$ has the
desirable properties of (i) being conserved on the short spin-precession
timescale of the problem, (ii) consistently including two-spin effects, (iii)
agreeing with the heuristic expression in the single-spin limit. In the large-
separation limit $r\to\infty$ (corresponding to GW emission frequencies
$f_{\rm GW}\to 0$), the quantity $\chi_{\rm p}^{\rm(av)}$ can be written in
closed form as [43]

$\displaystyle\lim_{r\to\infty}\chi_{\rm p}^{\rm(av)}$
$\displaystyle=\frac{|\chi_{1\perp}-\chi_{2\perp}|}{\pi}\,{\rm
E}\\!\left[-\frac{4\chi_{1\perp}\chi_{2\perp}}{(\chi_{1\perp}-\chi_{1\perp})^{2}}\right]$
$\displaystyle+\frac{\chi_{1\perp}+\chi_{2\perp}}{\pi}\,{\rm
E}\\!\left[\frac{4\chi_{1\perp}\chi_{2\perp}}{(\chi_{1\perp}+\chi_{2\perp})^{2}}\right]\,,$
(4)

where ${\rm E}(m)$ is the complete elliptic integral of the second kind. In
practice, this expression is accurate down to $\lesssim 0.1\%\ (1\%)$ for
$r\gtrsim 10^{6}M\ (10^{4}M)$ (where $M=m_{1}+m_{2}$ is the total mass of the
binary), and thus well describes BH binaries at the large separations where
they form. For configurations in the sensitivity window of LIGO/Virgo ($f_{\rm
GW}\gtrsim 10$ Hz), one instead needs to use $\chi_{\rm p}^{\rm(av)}$ as given
in Eq. (16) of Ref. [43].

Figure 2: Distribution of different BH generations in the $(\chi_{\rm
eff},\,\chi_{\rm p})$ plane. Green, orange, and blue regions indicate 1g+1g,
1g+2g, and 2g+2g mergers, respectively, with contours marking the 68% and 90%
levels. We consider a simple population model where $\chi_{\rm max}$ indicates
the largest BH spins at formation, and $\beta$ indicates the binary pairing
properties. In particular, we consider $(\chi_{\rm max},\beta)=(0.1,5)$ (small
spins and selective pairing, left panels), $(\chi_{\rm max},\beta)=(0.5,5)$
(large spins and selective pairing, middle panels), and $(\chi_{\rm
max},\beta)=(0.1,0)$ (small spins and random pairing, right panels). We
contrast the aligned effective spin $\chi_{\rm eff}$ with the heuristic value
of the precession parameter $\chi_{\rm p}^{(\rm heu)}$ (bottom panels) and the
asymptotic limit of the consistently averaged quantity
$\lim_{r\to\infty}\chi_{\rm p}^{\rm(av)}$. Gray contours indicate the
posterior distribution of GW190521 (68% and 90% levels). Samples have been
backpropagated to $f_{\rm GW}=0$ Hz ($r\to\infty$) to model how the source
formed.

###  II.2 Elliptical arcs

Let us assume for simplicity that first-generation (1g) BH binaries have
uniformly distributed spin magnitudes $\chi_{i}\in[0,\chi_{\rm max}]$, while
their mass ratio $q$ has a power-law distribution $p(q)\propto q^{\beta}$.
Large, positive values of $\beta$ indicate a scenario where BHs pair
selectively with companions of similar masses, as expected in mass-segregated
clusters. The case $\beta=0$ models a random pairing process, and it is
tentatively supported by current GW observations [8]. We compute the effective
spins for 1g+1g mergers by assuming the spin directions to be isotropically
distributed, as expected in dynamical formation channels.

From this population of 1g+1g mergers, we estimate the spin of the second-
generation (2g) merger remnants using fits to numerical relativity simulations
[44]. This yields the probability distribution $p(\chi_{r})$ of 2g BHs. To
construct the population of 1g+2g and 2g+2g mergers, the 2g BHs [with spin
distribution $p(\chi_{r})$] are paired with 1g and 2g BHs [with spin
distributions $U(0,\chi_{\rm max})$ and $p(\chi_{r})$] assuming the pairing
probability $p(q)\propto q^{\beta}$. In Fig. 2 we show the three populations,
1g+1g, 1g+2g, and 2g+2g, where we contrast $\chi_{\rm eff}$ against both the
heuristic expression $\chi_{\rm p}^{(\rm heu)}$ [Eq. (2)] and the asymptotic
limit of the fully averaged expression $\chi_{\rm p}^{(\rm av)}$ [Eq. (4)].

We find that 1g+1g BH binaries are confined to a region near the origin of the
$(\chi_{\rm eff},\,\chi_{\rm p})$ plane. Interestingly, the populations
involving 2g BHs cluster in arc-shaped structures extending to regions of the
plane where $\chi_{\rm p}\gtrsim 0.5$ and $|\chi_{\rm eff}|\gtrsim 0.5$.

The leftmost panels of Fig. 2 show a case where BH spins at birth are low
($\chi_{\rm max}=0.1$) and pairing is highly selective ($\beta=5$). In this
case, the 1g+2g and 2g+2g spin distributions are well separated from their
1g+1g progenitors, indicating that future GW measurements with accuracies of
$\sim 10\%$ on $\chi_{\rm eff}$ and $\chi_{\rm p}$ will allow us to
confidently pinpoint their hierarchical generation.

To better understand this analytically, let us analyze the limit where
$\chi_{\rm max}\ll 1$ and $\beta\gg 1$. Because pairing is selective, most
sources will have $q\sim 1$.

For the 1g+2g population, one has $\chi_{2}\sim 0$ (because 1g spins are
assumed to be low) and $\chi_{1}\sim 0.7$ (because the 2g BH is the remnant of
a previous merger). The condition $\chi_{2}\ll\chi_{1}$ implies that
$\chi_{\rm p}^{\rm(heu)}\simeq\lim_{r\to\infty}\chi_{\rm p}^{\rm(av)}$ [43],
thus explaining why the two orange arcs in the left panels of Fig. 2 are very
similar to each other. From the definitions (1) and (2) it follows that
$\chi_{\rm eff}\simeq\chi_{1}\cos\theta_{1}/2$ and $\chi_{\rm
p}\simeq\chi_{1}\sin\theta_{1}$, and therefore

$\left(\frac{2}{0.7}\chi_{\rm eff}\right)^{2}+\left(\frac{\chi_{\rm
p}}{0.7}\right)^{2}\simeq 1\,,$ (5)

which is the equation of an ellipse. The effective spins are thus limited by
$-0.35\lesssim\chi_{\rm eff}\lesssim 0.35$ and $0\lesssim\chi_{\rm p}\lesssim
0.7$.

For the 2g+2g case, one has $q\sim 1$ and $\chi_{1}\sim\chi_{2}\sim 0.7$. This
yields the relation

$\left(\frac{2}{0.7}\chi_{\rm
eff}+\cos\vartheta\right)^{2}+\left(\frac{\chi_{\rm
p}^{\rm(heu)}}{0.7}\right)^{2}\simeq 1\,,$ (6)

where $\cos\vartheta$ is a random number distributed uniformly in $[-1,1]$.
The additional term compared to Eq. (5) has the effect of splitting the
ellipse and shifting it horizontally in both directions, resulting in the
double-arc blue structures observed in Fig. 2. Equation (6) also implies that
$\chi_{\rm p}^{\rm(heu)}\lesssim 0.7$. However this is an artificial limit
introduced by the heuristic definition of $\chi_{\rm p}$, as evidenced by the
sharp truncation of the blue arcs in the top panels of Fig. 2. We can estimate
the upper bound of the fully averaged precession parameter $\chi_{\rm
p}^{\rm(av)}$ by imposing $\chi_{\rm eff}\sim 0$, which in this case
corresponds to setting $\cos\theta_{1}\sim-\cos\theta_{2}$ and thus
$\chi_{1\perp}\sim\chi_{2\perp}$ in Eq. (4), with the result

$\lim_{r\to\infty}\chi_{\rm p}^{\rm(av)}\lesssim\frac{4\times
0.7}{\pi}\sqrt{1-\cos^{2}\vartheta}\lesssim 0.9\,.$ (7)

This is consistent with the extended blue arcs in the bottom-left panel of
Fig. 2.

The trends that we just described remain valid, although in a more approximate
fashion, for generic values of $\beta$ and $\chi_{\rm max}$. For uniform
pairing ($\beta=0$; rightmost panels in Fig. 2), the 2g spins follow a broader
distribution extending from $\mathchar 21016\relax\,0$ to $\mathchar
21016\relax\,0.7$. This increases the area covered by the arcs in Fig. 2.
Small-spin 2g mergers end up populating the region where $\chi_{\rm eff}$ and
$\chi_{\rm p}$ are both small, leading to some overlap with the 1g+1g
distribution. If instead the 1g spins are larger ($\chi_{\rm max}=0.5$; middle
panels in Fig. 2), the 1g+1g spins end up spanning a larger region in the
$(\chi_{\rm eff},\,\chi_{\rm p})$ plane, but the 2g spins remain peaked at
large values.

###  II.3 Backpropagating GW190521

At present, the most promising candidate of hierarchical origin is GW190521
[10]. This conclusion is largely driven by its masses, at least one of which
lies in the pair-instability gap. The “effective-spin arcs” that we just
explored allow us to cross-check whether this interpretation is also
compatible with its spin properties. The measured values of $\chi_{\rm eff}$
and $\chi_{\rm p}$ for GW190521 are indicated in Fig. 2 with gray contours. In
the posterior distribution samples provided in Ref. [39], the spin directions
are provided at the arbitrary reference frequency $f_{\rm GW}=11$ Hz. We
backpropagate the spin distributions to $f_{\rm GW}=0$ Hz using the formalism
and the code of Refs. [45, 41, 46]. This is because, much like any other
astrophysical model, our populations describe BH binaries at formation, not at
detection, so it would be wrong to use the LIGO posterior samples at face
value. To the best of our knowledge, this is the first time that samples from
a real GW event have been backpropagated from detection to formation.

GW190521 is more easily accommodated by the 1g+2g and 2g+2g populations for
all values of $\chi_{\rm max}$ and $\beta$ explored in Fig. 2. To quantify
this statement, we compute the likelihood that GW190521 belongs to generation
$g$ as follows:

$\displaystyle P_{g}(\chi_{\rm max},\beta,|d)=$ $\displaystyle\int d\chi_{\rm
eff}\,d\chi_{\rm p}\frac{p(\chi_{\rm eff},\chi_{\rm p}|d)}{\pi(\chi_{\rm
eff},\chi_{\rm p})}p_{g}(\chi_{\rm eff},\chi_{\rm p}|\chi_{\rm max},\beta)$
(8)

where $g$={1g+1g, 1g+2g, 2g+2g}, $p(\chi_{\rm eff},\chi_{\rm p}|d)$ is the
LIGO posterior for GW190521, $\pi(\chi_{\rm eff},\chi_{\rm p})$ is the LIGO
prior, and $p_{g}(\chi_{\rm eff},\chi_{\rm p}|\chi_{\rm max},\beta)$ is the
probability distribution of $(\chi_{\rm eff},\,\chi_{p})$ for given
$(\chi_{\rm max},\,\beta)$. Note that in the expression above we are
implicitly neglecting selection effects [47, 48], which are largely irrelevant
in this case because we are integrating only over the spin parameters [49].
Since the event priors and posteriors are provided as discrete samples, the
integral in Eq. (8) is computed as a Monte Carlo summation with $p_{g}$ and
$\pi$ evaluated on the posterior samples. The prior values are estimated with
a bounded kernel density fit to the LIGO prior samples.

The likelihood ratios

$\displaystyle\mathcal{L}_{\rm 1g+2g}=\frac{P_{\rm 1g+2g}(\chi_{\rm
max},\beta,|d)}{P_{\rm 1g+1g}(\chi_{\rm max},\beta,|d)}$ (9)

and

$\displaystyle\mathcal{L}_{\rm 2g+2g}=\frac{P_{\rm 2g+2g}(\chi_{\rm
max},\beta,|d)}{P_{\rm 1g+1g}(\chi_{\rm max},\beta,|d)}$ (10)

are given in Table 1 for all values of $\chi_{\rm max}$ and $\beta$ explored
in Fig. 2. For small 1g spins and selective pairing ($\chi_{\rm max}=0.1$ and
$\beta=5$), large precessing spins of GW190521 can only be explained by 2g
populations. On the other hand, if $\chi_{\rm max}=0.5$, 1g populations have
extended support to large $(\chi_{\rm eff},\,\chi_{p})$, leading to a
reduction in $\mathcal{L}_{\rm 1g+2g}$ and $\mathcal{L}_{\rm 2g+2g}$. In the
limit $\chi_{\rm max}\to 1$, which allows for the possibility that the large-
spin BHs observed in high mass X-ray binaries [50] may contribute to multiple-
generation mergers, one has $\mathcal{L}_{\rm 1g+2g}\to 1.1$
($\mathcal{L}_{\rm 2g+2g}\to 1.1$) if $\chi_{p}=\chi_{p}^{\rm(heu)}$ or
$\mathcal{L}_{\rm 1g+2g}\to 1.3$ ($\mathcal{L}_{\rm 2g+2g}\to 1.4$) if
$\chi_{p}=\chi_{p}^{\rm(avg)}$ , blurring the lines between the spin
distribution of 1g+1g mergers and mergers involving 2g BHs. Similarly, random
pairing at $\beta=0$ also leads to smaller 2g spins, and hence relatively less
support at large spins, again reducing $\mathcal{L}_{\rm 1g+2g}$ and
$\mathcal{L}_{\rm 2g+2g}$. While large values of $\chi_{\rm max}$ and small
values of $\beta$ reduce the likelihood ratio, they might also diminish the
population of 2g BHs. This is because the larger recoils experienced by
remnants of 1g+1g mergers will make it harder for clusters to retain them.

In Table 1 we did not include a prior ratio, which is model dependent.

| $\chi_{p}=\chi_{p}^{\rm(av)}$ | $\chi_{p}=\chi_{p}^{\rm(heu)}$
---|---|---
$\\!\\!(\chi_{\rm max},\,\beta)\\!\\!$ | $\\!\\!{\mathcal{L}}_{\rm 1g+2g}\\!\\!\\!\\!$ | ${\mathcal{L}}_{\rm 2g+2g}\\!\\!$ | $\\!\\!{\mathcal{L}}_{\rm 1g+2g}\\!\\!\\!\\!$ | $\\!\\!{\mathcal{L}}_{\rm 2g+2g}\\!\\!\\!\\!$
$(0.1,\,5)$ | $5.3$ | $8.9$ | $6.1$ | $6.3$
$(0.5,\,5)$ | $3.5$ | $5.1$ | $3.4$ | $3.6$
$(0.1,\,0)$ | $3.8$ | $4.7$ | $4.5$ | $4.8$
Table 1: Likelihood ratios for GW190521 when comparing 1g+2g and 2g+2g origins
against 1g+1g origin.

##  III The parents of hierarchical black-hole mergers

Constraints on hierarchical mergers would become more informative if GW data
were to provide accurate measurements of the individual spins of the merging
BHs—not only of some effective combination thereof. In this case, one can
reconstruct not only the generation of the observed events but also the
properties of its parents.

###  III.1 Constraints on the remnant black-hole spin

Suppose that we have measured the dimensionless spin $\chi_{\mathrm{r}}$ of
one binary component. This is likely to be the spin of the heavier BH in the
observed binary (i.e. $\chi_{\mathrm{r}}=\chi_{1}$), although this is not
required for the argument presented here to be valid. Assuming that the
measured BH with spin $\chi_{\mathrm{r}}$ is the remnant of a previous merger,
we now wish to infer the properties of its parents.

Consider a parent binary with masses $M_{1,2}$, spins $\boldsymbol{\xi}_{1,2}$
and mass ratio $Q=M_{2}/M_{1}\leq 1$ (recall that $m_{1,2}$,
$\boldsymbol{\chi}_{1,2}$ and $q=m_{2}/m_{1}$ denote the corresponding
quantities for the observed binary; cf. Fig. 1). Using the numerical-
relativity fits of Ref. [51], $\chi_{\mathrm{r}}$ is given by

$\displaystyle\chi_{\mathrm{r}}=\min\Bigg{\\{}1,\Bigg{|}\frac{\boldsymbol{\xi}_{1}+Q^{2}\boldsymbol{\xi}_{2}}{(1+Q)^{2}}+\eta\boldsymbol{\hat{L}}\Bigg{[}2\sqrt{3}+t_{2}\eta+t_{3}\eta^{2}$
$\displaystyle+\\!s_{4}\left(\\!\frac{\boldsymbol{\xi}_{1}+Q^{2}\boldsymbol{\xi}_{2}}{1+Q^{2}}\\!\right)^{2}\\!\\!\\!+\\!(s_{5}\eta+t_{0}+2)\\!\left(\\!\frac{\boldsymbol{\xi}_{1}+Q^{2}\boldsymbol{\xi}_{2}}{1+Q^{2}}\\!\right)\\!\cdot\\!\boldsymbol{\hat{L}}\Bigg{]}\Bigg{|}\Bigg{\\}}\,,$
(11)

where $\eta=Q/(1+Q)^{2}$ and $\boldsymbol{L}$ denotes the angular momentum of
the parent binary. The fitting coefficients are $t_{0}=-2.8904$,
$t_{2}=-3.51712$, $t_{3}=2.5763$, $s_{4}=-0.1229$, and $s_{5}=0.4537$ [51]. We
use the fits of Ref. [51] as opposed to newer fits like Refs. [44, 52] to keep
the analytical treatment of the problem tractable. This has a negligible
effect on our analysis, because the systematic error in the fits is smaller
than typical statistical errors in the measurement of $\chi_{r}$.

When both spins of the parent binary are either aligned
($\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{1}=\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{2}=+1$,
henceforth denoted by $\uparrow\uparrow$) or antialigned
($\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{1}=\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{2}=-1$
denoted by $\downarrow\downarrow$) relative to the orbital angular momentum,
$\chi_{\mathrm{r}}$ is given by

$\displaystyle\chi_{\mathrm{r}}^{\uparrow\uparrow/\downarrow\downarrow}$
$\displaystyle=\Bigg{|}2\sqrt{3}\eta+t_{2}\eta^{2}+t_{3}\eta^{3}+\frac{s_{4}\eta}{(1-2\eta)^{2}}\tilde{\xi}^{2}$
$\displaystyle\pm\left[1+\eta\frac{(s_{5}\eta+t_{0}+2)}{1-2\eta}\right]\tilde{\xi}\Bigg{|}\,,$
(12)

where

$\tilde{\xi}=\frac{\xi_{1}+Q^{2}\xi_{2}}{(1+Q)^{2}}\,.$ (13)

The plus and minus signs correspond to aligned ($\uparrow\uparrow$) and
antialigned ($\downarrow\downarrow$) spins, respectively. We dropped the
minimum operation that was present in Eq. (11) and, consequently, our
inversion is valid only for $\chi_{\mathrm{r}}<1$.

In most of the three-dimensional parameter space $(Q,\,\xi_{1},\,\xi_{2})$,
the remnant spin $\chi_{\mathrm{r}}$ is maximal when the binary spins are
aligned ($\chi_{\mathrm{r}}=\chi_{\mathrm{r}}^{\uparrow\uparrow}$) and it is
minimal when they are antialigned
($\chi_{\mathrm{r}}=\chi_{\mathrm{r}}^{\downarrow\downarrow}$). In the corner
of the parameter space with $Q/(1+Q)^{2}\lesssim 0.28\tilde{\xi}$ , the
minimum $\chi_{\mathrm{r}}$ is instead obtained when the primary BH is
antialigned and the secondary BH is aligned, i.e.
$\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{1}=-\boldsymbol{\hat{L}}\cdot\boldsymbol{\hat{\xi}}_{2}=-1$
(denoted by $\downarrow\uparrow$). The value of
$\chi_{\mathrm{r}}^{\downarrow\uparrow}$ can be obtained by replacing
$\tilde{\xi}\to({\xi_{1}-Q^{2}\xi_{2}})/({1+Q)^{2}}$ in the expression for
$\chi_{\mathrm{r}}^{\downarrow\downarrow}$. Because

$\frac{\xi_{1}-Q^{2}\xi_{2}}{(1+Q)^{2}}=\tilde{\xi}+\mathcal{O}(Q^{2})$ (14)

and the condition
$\chi_{\mathrm{r}}^{\downarrow\uparrow}<\chi_{\mathrm{r}}^{\downarrow\downarrow}$
requires unequal mass binaries, in the following we approximate the minimum of
$\chi_{\mathrm{r}}$ with $\chi_{\mathrm{r}}^{\downarrow\downarrow}$ for all
values of $Q$, $\xi_{1}$, and $\xi_{2}$. The error introduced by this
approximation is $\Delta\chi_{\mathrm{r}}\lesssim 0.07$, and it is largest for
the extremal spins $\xi_{1}\sim 1$ and $Q\sim 0.26$.

Figure 3: Bounds on the mass ratio $Q$ and spin combination $\tilde{\xi}$ of
the parent binary producing a remnant with spins $\chi_{\mathrm{r}}=0.5$
(solid black line), $\chi_{\mathrm{r}}=0.69$ (thin dashed line), and
$\chi_{\mathrm{r}}=0.9$ (thin dash-dotted line). The lower and upper bounds
are given by the functions $\mathcal{F}_{\pm}$ defined in Eq. (16). For
clarity, in this figure we explicitly label these functions only when
$\chi_{\mathrm{r}}=0.5$. Blue lines represent further upper limits that can be
imposed if the parent BH spins are bounded by $\xi_{1,2}<\chi_{\rm max}$ with
$\chi_{\rm max}=0.2$, $0.5$ and $1$ (in different shades of blue): cf. Eq.
(18). For given values of $\chi_{\mathrm{r}}$ and $\chi_{\rm max}$, the
allowed region for the progenitors lies within the wedges above the black
curves corresponding to the given $\chi_{\mathrm{r}}$, and below the blue
curves corresponding to the given $\chi_{\rm max}$.  Figure 4: Allowed range
of the parent binary mass ratio $Q$ for a given value of the observed spin
$\chi_{\mathrm{r}}$. A single value $Q=Q_{0}$ (solid black curve) is allowed
if both parents are nonspinning, i.e. $\chi_{\rm max}=0$. As $\chi_{\rm max}$
increases, the allowed range of $Q$ for a given $\chi_{\mathrm{r}}$ (shown in
different shades of blue) widens.

###  III.2 Inferring the parents’ mass ratio from the remnant spin

Our next goal is to find the spin combinations that yield a given
$\chi_{\mathrm{r}}$. We can invert the relation

$\chi_{\mathrm{r}}^{\downarrow\downarrow}(Q,\tilde{\xi})\leq\chi_{\mathrm{r}}\leq\chi_{\mathrm{r}}^{\uparrow\uparrow}(Q,\tilde{\xi})$
(15)

to obtain the $\tilde{\xi}$ needed to form a remnant with spin
$\chi_{\mathrm{r}}$:

${\mathcal{F}}_{-}(\chi_{\mathrm{r}},Q)\leq\tilde{\xi}\leq{\mathcal{F}}_{+}(\chi_{\mathrm{r}},Q)\,,$
(16)

where the functions ${\mathcal{F}}_{-}(\chi_{\mathrm{r}},Q)$ and
${\mathcal{F}}_{+}(\chi_{\mathrm{r}},Q)$ are obtained by inverting Eq. (III.1)
and are shown in Fig. 3. The detection of a BH with spins $\chi_{\mathrm{r}}$
restricts the progenitor’s properties to a specific wedge of the
($Q,\tilde{\xi}$) plane.

There is a special value of the mass ratio $Q_{0}$ for which
${\mathcal{F}}_{-}(\chi_{\mathrm{r}},Q_{0})=0$. This case corresponds to the
merger of nonspinning BHs ($\xi_{1}=\xi_{2}=0$). From Eq. (11) one gets

$\chi_{\mathrm{r}}=\eta_{0}\left(2\sqrt{3}+t_{2}\eta_{0}+t_{3}\eta_{0}^{2}\right)\,,$
(17)

where $\eta_{0}=Q_{0}/(1+Q_{0})^{2}$. For instance, for
$\chi_{\mathrm{r}}=0.5$ one has $Q_{0}=0.28$. If the parent binary had a mass
ratio $Q_{0}$, there would be no restriction on the component spin magnitudes
of the progenitors that could have formed the observed remnant. For all other
values of $Q\neq Q_{0}$, the parents must have had nonzero spin magnitudes.

When the remnant spin reaches a value $\chi_{\mathrm{r}}\simeq 0.69$, we find
that $Q_{0}=1$. Equal-mass parents can form this remnant irrespective of their
spin magnitudes, but larger BH spins are required if $Q$ is smaller. If
$\chi_{\mathrm{r}}\gtrsim 0.69$, the equation
${\mathcal{F}}_{-}(\chi_{\mathrm{r}},Q_{0})=0$ does not admit solutions with
$Q_{0}\in[0,1]$. Nonspinning BHs cannot possibly form such a remnant. If
interpreted as a hierarchical merger, a putative GW observation of a BH with
$\chi_{\mathrm{r}}\gtrsim 0.69$ would indicate that the BHs of the previous
generation were also spinning.

One might want to further impose an upper bound on the spin magnitudes of the
previous-generation BHs, say $\xi_{1,2}\leq\chi_{\rm max}$. If the parent’s
binary consists of 1g BHs, the parameter $\chi_{\rm max}$ has the clear
physical interpretation of the largest BH spin resulting from stellar
collapse. This additional condition translates to

$\tilde{\xi}\leq\frac{1+Q^{2}}{(1+Q)^{2}}\,\chi_{\rm max}\,,$ (18)

and further restricts the allowed region in the ($Q,\tilde{\xi}$) plane. The
upper bounds corresponding to selected values of $\chi_{\rm max}$ are shown in
Fig. 3 as solid colored lines. For instance, if $\chi_{\mathrm{r}}=0.5$ and
$\chi_{\rm max}=0.1$, the parent’s mass ratio must satisfy $0.22\leq Q\leq
0.34$.

This point is further explored in Fig. 4. For a given value of
$\chi_{\mathrm{r}}$, there is a finite range of allowed mass ratios $Q$. The
width of this range depends on the largest spin magnitude of the progenitor
$\chi_{\rm max}$. In particular, it shrinks to zero for $\chi_{\rm max}=0$
(corresponding to the condition $Q=Q_{0}$) and widens progressively if larger
progenitor spins are allowed.

Figure 5: Probability density function of the primary spin in a putative 1g+2g
GW event as a function of the mass ratio $Q$ and spin magnitude $\xi_{1,2}$ of
the parent binary. Blue (green) curves refer to $Q=0.2$ ($Q=0.8$), and solid
(dashed) lines refer to $\xi_{1}=\xi_{2}=0.2$ ($\xi_{1}=\xi_{2}=0.5$).
Vertical lines show the remnant spin values for the simpler case where both
parents have zero spin, $\xi_{1}=\xi_{2}=0$.  Figure 6: Left panel: robability
$p(\chi_{\mathrm{r}}=0.5|Q,\xi_{1}=\xi_{2})$ (in arbitrary units) as a
function of the parent’s mass ratio $Q$ and spins $\xi_{1}=\xi_{2}$ (here
taken to be equal to each other for simplicity), assuming isotropic spin
directions. Right panel: kick velocity $v_{\rm kick}$ imparted to BHs with
spin $\chi_{\mathrm{r}}=0.5$ as a function of the mass ratio $Q$ and spins
$\xi_{1}=\xi_{2}$ of the merging binary. Areas of high probability (dark
regions in the left panel) correlate with areas of low kick (light regions in
the right panel). The white areas beyond the solid black lines represent
forbidden regions where a remnant with $\chi_{\mathrm{r}}=0.5$ cannot be
formed.  Figure 7: Assuming that the primary BH in GW190412 is of hierarchical
nature, we show (in blue) the posterior distribution of the mass ratio $Q$ and
spin magnitudes $\xi_{1,2}$ of its parents. Contours indicate the $50$% and
$90$% confidence intervals. The medians and 90% credible intervals of the
marginalized distributions are $Q=0.21_{-0.16}^{+0.32}$,
$\xi_{1}=0.37^{+0.45}_{-0.30}$, and $\xi_{2}=0.50_{-0.40}^{+0.40}$. The green
line shows the posterior of $Q$ under the assumption that $\xi_{1,2}=0$, with
median and $90\%$ confidence intervals $Q=0.22_{-0.11}^{+0.14}$.

###  III.3 Not all parents are equally likely

The exclusion regions on the parents’ properties presented so far were
obtained by requiring that there exists some orientation of the progenitor
spins $\boldsymbol{\hat{\xi}}_{1,2}$ that can produce a given remnant. The
distribution $p(\chi_{\mathrm{r}}|Q,\,\xi_{1},\,\xi_{2})$ is shown in Fig. 5,
where isotropic spin directions are assumed, as expected for dynamical
formation channels that may produce hierarchical mergers.

As $Q\to 1$, the probability is narrowly peaked at around $0.7$. The location
of the peak value of $\chi_{\mathrm{r}}$ decreases at smaller mass ratios.
Larger progenitor spins also lead to broader distributions of remnant spins
(see e.g. [53, 24]).

Using the inversion formalism described above, we can now explore which of the
possible parents are more or less likely. The left panel of Fig. 6 illustrates
how the probability of forming a BH of spin $\chi_{\mathrm{r}}=0.5$ depends on
the progenitors’ spins and mass ratio. We have fixed $\xi_{1}=\xi_{2}$ to
reduce the number of parameters for illustrative purposes, but the argument
can be made more general. The solid black lines represent the bounds derived
from Eq. (16). There are two distinct regions of parameter space where the
probability is highest:

  1. (1)

When the mass ratio is close to the critical value $Q\simeq Q_{0}$ and the
spins are small, i.e. $\xi_{1,2}\simeq 0$.

  2. (2)

When $Q\simeq 0$ and $\xi_{1}\simeq\chi_{\mathrm{r}}$, corresponding to the
trivial case of a very small body merging into a larger Kerr BH with spin
$\chi_{\mathrm{r}}$.

The spin orientations lose meaning in both the zero-spin (case 1) and test-
particle (case 2) limits, which implies a higher concentration of possible
remnants.

Crucially, these are the very same limits that minimize the kick imparted to
the BH remnant. This is illustrated in the right panel of Fig. 6, where we use
the kick fitting formula of Ref. [46], selecting the spin directions that can
produce the targeted $\chi_{\mathrm{r}}$. The kick is suppressed in both of
the “special” regions near $Q\simeq 0$ and $\xi_{1}=\xi_{2}\simeq 0$. Parents
with these properties are not only more likely to form a remnant with the
targeted properties, but also more likely to form a remnant that is retained
in its astrophysical host—a prerequisite condition in the scenario depicted
here, where we are observing the GWs emitted by the subsequent merger.

###  III.4 Application to GW190412

The arguments above require that the spin of one BH binary component can be
reliably measured from the data. As explored in Sec. II, we often have access
only to the effective spins. For $q\ll 1$, the spin of the secondary is
negligible and one can reconstruct the spin of the primary BH as
$\chi_{1}\simeq\sqrt{\chi_{\rm eff}^{2}+\chi_{\rm p}^{2}}$.

Although most of the binaries detected to date are compatible with having
comparable masses, there are two notable exceptions: GW190412 with
$q=0.28_{-0.07}^{+0.13}$, and GW190814 with $q=0.112^{+0.008}_{-0.008}$ [39].
In particular, GW190412 was shown to be compatible with a hierarchical merger
[26] and was interpreted as such in both cluster [27] and AGN [54] formation
models. Using the spin inversion formalism derived in this paper and the LIGO
measurement $\chi_{1}=0.44_{-0.22}^{+0.16}$ [39], can we infer the properties
of the parents of GW190412?

Given the event data $d$, we calculate

$p(Q,\xi_{1},\xi_{2}|d)\propto\pi(Q,\xi_{1},\xi_{2})\int
p(\chi_{\mathrm{r}}|Q,\xi_{1},\xi_{2})\,p(\chi_{\mathrm{r}}|d)\,d\chi_{\mathrm{r}}\,,$
(19)

where $p(\chi_{1}|d)$ denotes the LIGO posterior for GW190412,
$p(\chi_{\mathrm{r}}|Q,\xi_{1},\xi_{2})$ is the probability distribution
introduced in Sec. III.3 (where we assumed isotropic orientations), and
$\pi(Q,\xi_{1},\xi_{2})$ is a prior, which we assume to be flat in $Q\in[0,1]$
and $\xi_{1,2}\in[0,1]$. We sample $p(Q,\xi_{1},\xi_{2}|d)$ using Markov-chain
Monte Carlo [55].

Figure 7 shows the resulting constraints on the parents of GW190412’s primary
BH. In particular, we infer that the progenitor binary must have had a
relatively small ratio $Q=0.21_{-0.16}^{+0.32}$. This follows mainly from the
small value of $\chi_{\mathrm{r}}\approx 0.44$. As illustrated in Fig. 6, the
regions around $Q\simeq Q_{0},\,\xi_{1,2}\simeq 0$ and $Q\simeq
0,\,\xi_{1}\simeq\chi_{\mathrm{r}}$ are favored. Similar features can also be
seen in Fig. 7, although these are somewhat washed out by measurement errors
on $\chi_{\mathrm{r}}$. Small values of $\xi_{1}$ are favored, while $\xi_{2}$
cannot be inferred. If the mass ratio $Q$ is small, one has
$\tilde{\xi}\approx\xi_{1}$, and the likelihood becomes largely independent of
$\xi_{2}$.

Now that we have (samples of) the parent binary’s properties, we can estimate
the energy $E_{\rm rad}$ that was dissipated in GWs during the earlier merger
that formed the primary component of GW190412. For each sample in the three-
dimensional parameter space $(Q,\xi_{1},\xi_{2})$ of Fig. 7, we solve for the
spin directions $\hat{\boldsymbol{\xi}}_{1,2}$ that correspond to a given
remnant spin $\chi_{\mathrm{r}}$ in the LIGO posterior. We then plug the
resulting values of $(Q,\boldsymbol{\xi}_{1},\boldsymbol{\xi}_{2})$ into the
numerical-relativity fit of Ref. [56] to estimate $E_{\rm rad}$. When combined
with the samples of $m_{1}=30.1^{+4.6}_{-5.3}$ [39] provided by LIGO, this
allows us to reconstruct the masses

$\displaystyle M_{1}$ $\displaystyle=\frac{m_{1}}{1+Q}(1-E_{\rm rad})\,,\ $
(20) $\displaystyle M_{2}$ $\displaystyle=\frac{Qm_{1}}{1+Q}(1-E_{\rm rad})\ $
(21)

of the parent BHs.

The results of this procedure are shown in Fig. 9. We find that the parents of
the primary BH in GW190412 had masses $M_{1}=25.24^{+3.55}_{-4.83}\,{\rm
M}_{\odot}$ and $M_{2}=5.61^{+5.67}_{-3.98}\,{\rm M}_{\odot}$. The negative
correlation between $M_{1}$ and $M_{2}$ apparent in Fig. 9 follows from the
constraint $M_{1}+M_{2}\simeq m_{1}$, which is accurate up to small
corrections due to the radiated energy $E_{\rm rad}\lesssim 5\%$.

Using the procedure that we just described, we can also estimate how much
linear momentum was emitted during the merger of the parents, and hence the
kick imparted to the primary component of GW190412 when it formed. Using the
numerical-relativity fit assembled in Ref. [46], we find $v_{\rm
kick}=158^{+240}_{-125}\;\rm km/s$. The resulting distribution is shown in
Fig. 9. The observed bimodality is a direct consequence of the distributions
of $Q$ found in Fig. 7. The peak at $Q\sim 0.2$ is responsible for the
dominant mode at $v_{\rm kick}>100$ km/s [57], while the tail at $Q\gtrsim 0$
produces the secondary mode at $v_{\rm kick}\gtrsim 0$, as predicted in the
test-particle limit.

Figure 8: Masses $M_{1}$ and $M_{2}$ of the parents of the primary component
in GW190412 (blue) under the uniform-spin priors. Contours indicate the $50$%
and $90$% confidence intervals. The medians and 90% credible intervals of the
marginalized distributions are $M_{1}=25.24^{+3.55}_{-4.83}\,{\rm M}_{\odot}$
and $M_{2}=5.61^{+5.67}_{-3.98}\,{\rm M}_{\odot}$. The green lines show $90$%
confidence intervals when the progenitors are nonspinning:
$M_{1}=25.24^{+2.35}_{-3.35}\,{\rm M}_{\odot}$ and
$M_{2}=5.44^{+3.41}_{-2.80}\,{\rm M}_{\odot}$.

Figure 9: Kick imparted to the primary BH in GW190412, assuming that it
originates from a previous BH merger. The blue curve represents the uniform-
spin prior, while the green curve corresponds to the zero-spin prior. The
median and 90% credible interval are $v_{\rm kick}=158^{+240}_{-125}\;\rm
km/s$ (uniform-spin prior) and $v_{\rm kick}=141^{+33}_{-71}\,\rm km/s$ (zero-
spin prior).

This estimate implies that, if GW190412 is indeed a hierarchical merger, it
must have formed in an environment with escape speed $v_{\rm esc}\gtrsim 100$
km/s (otherwise the primary BH would have been ejected, preventing the
formation of GW190412 itself), in agreement with previous work by Gerosa  _et
al._ [26]. Astrophysical environments with such escape speeds include nuclear
star clusters and AGN disks, but not globular clusters [58], with their escape
speeds ranging from $10$ to $100$ km/s [59]. In particular, we find $p(v_{\rm
kick}>10{\rm km/s})=0.94$, $p(v_{\rm kick}>50{\rm km/s})=0.87$, and $p(v_{\rm
kick}>100{\rm km/s})=0.75$.

Under a more restrictive—but arguably astrophysically motivated—prior where
the progenitors are assumed to be nonspinning ($\xi_{1}=\xi_{2}=0$), we obtain
a similar estimate for the mass ratio: $Q=0.22_{-0.11}^{+0.14}$ (green line in
the top panel of Fig. 7). The masses and recoil velocities calculated for
nonspinning progenitors are consistent with those obtained under uniform-spin
priors, but with smaller errors: we find $M_{1}=25.24^{+2.35}_{-3.35}\,{\rm
M}_{\odot}$, $M_{2}=5.44^{+3.41}_{-2.80}\,{\rm M}_{\odot}$ and $v_{\rm
kick}=141^{+33}_{-71}\,\rm km/s$.

When interpreting our results, one should keep in mind that our error budget
takes into account only the statistical uncertainties on the parameters of the
GW events. Much like anything else in science, our procedure also suffers from
systematics (for instance, in the specific numerical-relativity fits used
here).

##  IV Conclusions

Most of the literature on compact binaries is concerned with determining the
properties of a merger remnant from the properties of its progenitors. In this
paper we addressed the inverse problem in two steps (see Fig. 1):

  1. (a)

If we observe a binary system that ends up merging in LIGO, can we tell from
the observed properties of the binary—and in particular from the effective-
spin parameters $(\chi_{\rm eff},\,\chi_{\rm p})$—-whether one of the BHs came
from a previous (hierarchical) merger?

  2. (b)

If indeed one of the BHs came from a previous merger, can we determine at
least some of the properties of its parents?

Given the parameters of a binary system, the direct problem has a well-defined
solution that can be found by solving the field equations of general
relativity. On the contrary, the inverse problem is in general overdetermined,
because the only information available to us comes from the gravitational
radiation emitted by the observed binary. We have demonstrated that this is
enough to infer at least some of the properties of the parent binary.

In particular, our key results are as follows:

  1. (a)

Hierarchical mergers occupy a distinct region in the $(\chi_{\rm
eff},\,\chi_{\rm p})$ plane. Therefore, a measurement of these spin parameters
can add considerable weight to a hierarchical-merger interpretation of an
event located in the pair-instability mass gap. Notably, this is the case for
GW190521.

  2. (b)

If one of the BH binary components (not necessarily the primary) does indeed
have hierarchical origin and its spin magnitude is well measured, one can
place significant constraints on the properties of its parents. For example,
we can infer that the parents of the heavier BH in GW190412 must have had a
moderate mass ratio $Q=0.21_{-0.16}^{+0.32}$ and moderately low spins.

We stress that, while we focused on the examples of GW190521 and GW190412, the
formalism developed in this work is quite general.

The inversion procedure presented in Sec. III requires estimates of the
individual spin components. While challenging at the current detector
sensitivity, these are bound to become routine in the future. For stellar-mass
BH binaries, third-generation ground-based detectors will measure component
spin magnitudes within $\sim 1\%$ (see, e.g., [60]). A similar accuracy is
expected for supermassive BH binaries observed with the spaceborne
interferometer LISA [61]. In that case, the BH parents problem is arguably
even more relevant than in the LIGO context, given that supermassive BHs are
firmly believed to grow hierarchically following the assembly of large-scale
structures in the Universe [62].

Measurements of the masses, spins, and kicks of the progenitor of a
hierarchical merger can be translated into constraints on their birthplace. In
the LIGO context, for example, they can be used to infer bounds on the escape
speeds and masses of the clusters that may have produced such mergers via
dynamical interactions [32]. We have illustrated this point through a
constraint of the cluster’s escape speed for GW190412, but a more rigorous
implementation of this idea requires better astrophysical modeling, which will
be the subject of future work.

The identification of the parents’ properties can be used to validate or
reject the hierarchical origin of a GW event, because the inferred parents
should presumably be part of the same population of merging BHs observed by
our detectors. A posterior predictive check using the rest of the observed GW
catalog could be used to verify this hypothesis (see e.g. [63]). If ad hoc
population outliers are required, this would rule out (or cast serious doubt
on) the assumed hierarchical origin for the observed event. Such a test is a
natural follow-up of this work.

###### Acknowledgements.

We thank Daria Gangardt for discussions. V.B., E.B. and K.W.K.W. are supported
by NSF Grants No. PHY-1912550 and AST-2006538, NASA ATP Grants No.
17-ATP17-0225 and 19-ATP19-0051, NSF-XSEDE Grant No. PHY-090003, and NSF Grant
PHY-20043. D.G. and M.M. are supported by European Union’s H2020 ERC Starting
Grant No. 945155–GWmining and Royal Society Grant No. RGS-R2-202004. D.G. is
supported by Leverhulme Trust Grant No. RPG-2019-350. We would like to
acknowledge networking support by the GWverse COST Action CA16104, “Black
holes, gravitational waves and fundamental physics.” Computational work was
performed on the University of Birmingham BlueBEAR cluster and the Maryland
Advanced Research Computing Center (MARCC).

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