# Carroll-Schrödinger Equation

Mojtaba Najafizadeh  School of Physics, Institute for Research in Fundamental
Sciences (IPM)
P.O.Box 19395-5531, Tehran, Iran  Department of Physics, Faculty of Science,
Ferdowsi University of Mashhad
P.O.Box 1436, Mashhad, Iran
<EMAIL_ADDRESS>
###### Abstract

The Poincaré symmetry can be contracted in two ways to yield the Galilei
symmetry and the Carroll symmetry. The Schrödinger equation exhibits the
Galilei symmetry and is a fundamental equation in Galilean quantum mechanics.
However, the question remains: what is the quantum equation that corresponds
to the Carroll symmetry? In this paper, we derive a novel equation, called the
Carroll-Schrödinger equation, which describes the quantum dynamics in the
Carrollian framework. We also introduce the Carroll-Schrödinger algebra, which
is a conformal extension of the centrally extended Carroll algebra, and show
that it is the symmetry algebra of the Carroll-Schrödinger equation in two
dimensions. We generalize our results to arbitrary dimensions and discuss some
possible applications and extensions of our work.

##  I Introduction

In light of the inadequacy of the Galilei symmetry to describe
electrodynamics, it was necessary to extend it to the Lorentz symmetry, or
more generally, the Poincaré symmetry. Through a process of contraction, it
was anticipated that the Poincaré symmetry would reproduce the Galilei
symmetry. This contraction is characterized by small characteristic velocities
compared to the speed of light, referred to as the $c\to\infty$ limit or the
Galilei/nonrelativistic limit.

Alternatively, the Poincaré symmetry can be contracted by considering
characteristic velocities that are much larger than the speed of light, known
as the $c\to 0$ limit or the Carroll/ultrarelativistic limit. This distinct
type of contraction was discovered by Lévy-Leblond in 1965 and is known as the
Carroll symmetry [1].

Both the Galilei and the Carroll symmetries are two distinct contractions of
the Poincaré symmetry, depending on the limit of the speed of light. In terms
of the spacetime coordinates, the limit can be expressed as
$x_{i}\to\epsilon\,x_{i}$, $t\to t$, $\epsilon\to 0$ for the Galilei case, and
$x_{i}\to x_{i}$, $t\to\epsilon\,t$, $\epsilon\to 0$ for the Carroll case.

While these two contractions are taken from the same thing, it is worth noting
that the Galilei symmetry aligns with empirical evidence observed in the
natural world, whereas the Carroll symmetry represents a theoretical framework
that has the potential to uncover novel aspects of physics.

The study of Carrollian structures and their application to diverse physical
systems is driven by several motivations. Notably, a recent motivation that
has stimulated this field of research is the fact that the Carrollian
conformal algebra is isomorphic to the BMS algebra in one dimension higher [2,
3, 4] and therefore is relevant for the celestial approach to flat space
holography [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

Another aspects of Carrollian physics has been studied by different
motivations, see e.g. [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 66, 59, 60, 61, 62, 63, 64, 65] and references
therein. In particular, it could potentially have implications for de Sitter
cosmology and the phenomenon of inflation [43, 59], as well as the Hall effect
[53], hydrodynamics [27, 28, 7], scalar fields [44, 43, 54, 66], fermionic
fields [55, 56, 61, 65], and supersymmetry [22, 57, 61].

The Carrollian physics could offer intriguing possibilities, particularly in
the context of quantum mechanics. Usually, the Schrödinger equation can be
obtained from the Klein-Gordon equation by applying a field redefinition and
taking the Galilei/nonrelativistic limit $c\to\infty$ (see e.g. [67] or a
brief review in appendix A). However, in the Carrollian sector, we find a
similar approach proves unsuccessful when starting from the Klein-Gordon
equation. Instead, one has to begin with the tachyon Klein-Gordon equation.
Consequently, the main objective of this paper is to identify a quantum
mechanic equation analogous to the Schrödinger equation in any dimension
within the Carrollian framework.

The Schrödinger algebra is a conformal extension of the Galilei algebra with a
central charge111The field theories with the Schrödinger symmetries have been
studied in [68, 69, 70]. This algebra associated with the nonrelativistic or
Galilei mass captures the symmetry of the Schrödinger equation. As another
goal, similarly, we introduce the Carroll-Schrödinger algebra in two
dimensions, which is a conformal extension of the Carroll algebra with a
central charge. In this case, the central charge, identified as the
ultrarelativistic or Carroll mass, captures the symmetry of the Carroll-
Schrödinger equation. It finds that there exists an infinite dimensional
extension of this algebra.

The shorthand notations $\partial_{t}:=\partial/\partial t$,
$\partial_{x}:=\partial/\partial x$, and $\partial_{i}:=\partial/\partial
x^{i}$ (with $i=1,\ldots,d$) are used.

##  II Carroll-Schrödinger equation

Let us take into account the Klein–Gordon equation for a complex tachyon field
$\phi$ of mass $m$ in a two-dimensional Minkowski spacetime, with the mostly
plus signature for the metric, including the speed of light $c$ and the
reduced Planck constant $\hbar$

$\displaystyle\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}\,+\,\partial_{x}^{2}\,+\,\mu^{2}\right)\phi=0\,,$
(1)

where $\mu:={mc}/{\hbar}$. By applying a field redefinition

$\displaystyle\phi=\frac{1}{\sqrt{\mu}}~{}e^{-i\mu x}\,\psi\,,$ (2)

the tachyon Klein-Gordon equation (1) reduces to

$\displaystyle\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}\,-\,2i\mu\,\partial_{x}\,+\,\partial_{x}^{2}\right)\psi=0\,.$
(3)

Using a dimensionless parameter $\epsilon$, we can first rescale
$\mu\to\mu/\epsilon^{2}$ in (3) and then take the Carroll limit

$x\to x\,,\qquad t\to\epsilon\,t\,,\qquad\epsilon\to 0\,.$ (4)

Through this process, we obtain the following equation

$\displaystyle{\boxed{\left(i\hbar\,c\,\partial_{x}+\frac{\hbar^{2}}{2mc^{2}}\,\partial_{t}^{2}\right)\psi=0\,.}}$
(5)

We will refer to this equation as the two-dimensional “Carroll-Schrödinger
equation”, because it has a Carroll-Schrödinger symmetry, as will be discussed
later. The operator acting on the field $\psi$ in (5) is Hermitian with
respect to the Hermitian conjugation
$(\partial_{t})^{\dagger}\equiv-\,\partial_{t}$, and
$(\partial_{x})^{\dagger}\equiv-\,\partial_{x}$. This suggests writing the
free Carroll-Schrödinger action as

$\displaystyle{\boxed{S=\int
dt\,dx\,\psi^{\star}\left(i\hbar\,c\,\partial_{x}+\frac{\hbar^{2}}{2mc^{2}}\,\partial_{t}^{2}\right)\psi\,.}}$
(6)

We note that the introduced $\mu$ in (1) has the length dimension
$[\mu]_{{}_{L}}=-1$. In two spacetime dimensions, the length dimension of a
scalar field is $[\phi]_{{}_{L}}=0$. Hence, the coefficient in the field
redefinition (2) is chosen to ensure that the length dimension of the Carroll-
Schrödinger field $\psi$ becomes $[\psi]_{{}_{L}}=-1/2$. This choice ensures
that the action (6) possesses the expected dimension of $\hbar$.

It is worth noting that when the above procedure is followed and a real scalar
field $\phi$ is employed, with $\phi\propto e^{-\mu x}\,\psi$, one arrives at
an equation that the operator acting on the field $\psi$ is not Hermitian, and
thus cannot be derived from an action.

Moreover, if we begin with a complex scalar field equation with a real mass,
rather than a complex tachyon field equation (1), it would result in an
equation that becomes ill-defined in the Carroll limit (4). This highlights
the necessity of initiating the analysis with a tachyon field equation (1).

We note that it is possible to begin with a field redefinition (2) by
considering the opposite sign of $\mu$. This would lead to the replacement of
$m\to-\,m$ in the resulting equation and action, presenting another
possibility.

It is also important to emphasize that taking the limit is a technique to
obtain a new physics. Accordingly, after taking the limit (4), the nature of
initial field has been changed. Instead of being a tachyon, it now represents
a massive Carroll field.

##  III Carroll-Schrödinger algebra

The Carroll algebra in $1+1$ dimensions, involving time translation $H$, space
translation $P$, and Carroll boost $B$, is given by $[\,P\,,\,B\,]=H$. It is
convenient to see that this algebra admits two non-trivial central extensions
as $[\,H\,,\,B\,]=c_{1}$, and $[\,H\,,\,P\,]=c_{2}$. By ignoring $c_{2}$, and
identifying $c_{1}$ with mass parameter, $c_{1}:=M$, the double centrally
extended Carroll algebra reduces to

$\displaystyle[\,P\,,\,B\,]=H\,,$ $\displaystyle[\,H\,,\,B\,]=M\,.$ (7)

We name this the “Carroll-Bargmann algebra” by the analogy with the Bargmann
algebra which is a central extension of the Galilei algebra. Next, we extend
this algebra by including the conformal generators, namely the dilatation $D$
and the special conformal transformation $K$. We find such an extended
algebra, that is a $2$-dimensional conformal Carroll algebra with the central
charge $M$, has the following nonzero brackets

${\boxed{~{}~{}~{}\begin{aligned}
&[\,P\,,\,B\,]=H\,,\quad&&[\,H\,,\,B\,]={M}\,,\\\\[0.0pt]
&[\,P\,,\,D\,]=2\,P\,,\quad&&[\,D\,,\,K\,]=2\,K\,,\\\\[0.0pt]
&[\,H\,,\,D\,]=H\,,\quad&&[\,P\,,\,K\,]=D\,,\\\\[0.0pt]
&[\,D\,,\,B\,]=B\,,\quad&&[\,H\,,\,K\,]=B\,.\end{aligned}~{}~{}~{}}}$ (8)

We will refer to this algebra as the 2-dimensional “Carroll-Schrödinger
algebra”, because it is the symmetry group of the Carroll-Schrödinger action
(6), as we will see below. The generators of the Carroll-Schrödinger algebra
(8) can be represented by

$\displaystyle H=\partial_{t}\,,\qquad$ $\displaystyle
B=x\,\partial_{t}-im\,t\,,$ (9) $\displaystyle P=\partial_{x}\,,\qquad$
$\displaystyle D=t\,\partial_{t}+2\,x\,\partial_{x}+\omega\,,$
$\displaystyle{M}=-im\,,\qquad$ $\displaystyle
K=x\,t\,\partial_{t}+x^{2}\,\partial_{x}-\tfrac{1}{2}\,im\,t^{2}+\omega x\,,$

where $\omega$ is the dilatation weight.

We can now demonstrate that the Carroll-Schrödinger action (6) is invariant
under (assuming $c=1=\hbar$)

$\delta\psi=\Big{(}\lambda_{{}_{H}}H+\lambda_{{}_{P}}P+\lambda_{{}_{B}}B+\lambda_{{}_{M}}M+\lambda_{{}_{D}}D+\lambda_{{}_{K}}K\Big{)}\psi\,,$
(10)

upon the dilatation weight $\omega=-\,1/2$. Here, $H$, $P$, $B$, $M$, $D$, $K$
are the generators of the algebra introduced in (9), and $\lambda_{{}_{H}}$,
$\lambda_{{}_{P}}$, $\lambda_{{}_{B}}$, $\lambda_{{}_{M}}$,
$\lambda_{{}_{D}}$, $\lambda_{{}_{K}}$ are the corresponding parameters
associated with each transformation. This implies that the Carroll-Schrödinger
action (6), and consequently the equation (5), have the Carroll-Schrödinger
algebra (8) as their symmetry algebra.

We note that, as demonstrated in (9), the dilatation generator $D$ scales
space and time differently $x\to\lambda^{2}x$, $t\to\lambda\,t$, i.e. $z=1/2$.
However, the corresponding generator in the conformal Carroll algebra (CCA)
scales space and time in the same way $x\to\lambda\,x$, $t\to\lambda\,t$ [4].
Moreover, the $z=1/2$ Carroll-Schrödinger algebra (8) can be made isomorphic
to the $z=2$ Schrödinger algebra by replacing the generators $P$ and $H$ with
each other. This allows us to simply write an infinite dimensional extension
of the algebra (8).

For this purpose, let us ignore the central charge $M$ and the dilatation
weight $\omega$ for simplicity. Consequently, the infinite dimensional
Carroll-Schrödinger algebra can be given by

${\boxed{~{}~{}~{}\begin{aligned}
&[\,L_{n}\,,\,L_{m}\,]=(n-m)\,L_{n+m}\,,\\\\[5.0pt]
&[\,L_{n}\,,\,Y_{\ell}\,]=\left(\,\tfrac{n}{2}-\ell\,\right)Y_{n+\ell}\,,\\\\[5.0pt]
&[\,Y_{\ell}\,,\,Y_{k}\,]=0\,,\end{aligned}~{}~{}~{}}}$ (11)

where $n,m\in\mathbb{Z}$, and $\ell,k\in\mathbb{Z}+\tfrac{1}{2}$. The
generators that fulfill the algebra (11) are

$\displaystyle L_{n}$
$\displaystyle=-\,x^{n+1}\,\partial_{x}-\,\tfrac{1}{2}\,(n+1)\,x^{n}\,t\,\partial_{t}\,,$
(12) $\displaystyle Y_{\ell}$
$\displaystyle=x^{\,(\ell+\frac{1}{2})}\,\partial_{t}\,.$ (13)

Here it is understood that the finite generators are those with $n=0,\pm\,1$
and $\ell=\pm\,\frac{1}{2}$. In other words, we can write the generators (9)
in a suggestive form

$\displaystyle P=\partial_{x}=-\,L_{{}_{-1}}\,,\qquad$ $\displaystyle
H=\partial_{t}=Y_{-\frac{1}{2}}\,,$ $\displaystyle
D=t\,\partial_{t}+2\,x\,\partial_{x}=-\,2L_{{}_{0}}\,,\qquad$ $\displaystyle
B=x\,\partial_{t}=Y_{\frac{1}{2}}\,,$ $\displaystyle
K=x\,t\,\partial_{t}+x^{2}\,\partial_{x}=-\,L_{{}_{1}}\,.\qquad$ (14)

As a result, the finite dimensional Carroll-Schrödinger algebra (8) would be a
sub-algebra of (11). Hence, the Schrödinger algebra and the Carroll-
Schrödinger algebra do indeed possess the same infinite-dimensional structure,
analogous to the Conformal Carroll Algebra (CCA) and the Galilean Conformal
Algebra (GCA). In addition, the triplet $P$, $D$, $K$ (or $L_{-1}$, $L_{0}$,
$L_{1}$), forms an $sl(2;R)$ subalgebra.

##  IV Carroll-Schrödinger in $d>2$

To derive an equation in arbitrary dimensions, we begin with the Klein-Gordon
equation applied to a complex scalar tachyon field $\phi$ in any space
dimension

$\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}+\partial^{i}\partial_{i}+\mu^{2}\right)\phi=0\,.$
(15)

By applying the field redefinition

$\phi=\frac{1}{\sqrt{\mu}}~{}e^{-i\mu\sqrt{x^{2}}}\,\psi\,,$ (16)

with $x^{2}=x^{i}x_{i}$, we can arrive at the equation

$\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}-2i\mu\,\frac{1}{\sqrt{x^{2}}}\,x^{j}\partial_{j}-i\mu\,\frac{d-1}{\sqrt{x^{2}}}+\partial^{j}\partial_{j}\right)\psi=0\,.$
(17)

After rescaling $\mu\to\mu/\epsilon^{2}$ in the latter and then applying the
Carroll limit (4), we obtain the Carroll-Schrödinger equation in any dimension

${\boxed{\begin{aligned} &\left(i\hbar
c\,\nabla_{x}+\frac{\hbar^{2}}{2mc^{2}}\,\partial_{t}^{2}\right)\psi=0\,,\\\\[8.0pt]
&\text{where}~{}~{}~{}~{}~{}~{}~{}\nabla_{x}=\frac{1}{\sqrt{x^{2}}}\left(x^{i}\partial_{i}+\frac{d-1}{2}\right)\,.\end{aligned}}}$
(18)

We notice that the operator $\nabla_{x}$ possesses a length dimension
$[\nabla_{x}]_{{}_{L}}=L^{-1}$ and in the case of $d=1$ it reduces to the
standard derivative $\nabla_{x}\to\partial_{x}$, such that (18) reduces to
(5). Furthermore, it is anti-Hermitian, i.e.
$\nabla_{x}^{\dagger}=-\nabla_{x}$, with respect to the Hermitian conjugation
already introduced. This guarantees that in (18), the operator acting on the
field $\psi$ is Hermitian. Consequently, the equation (18) can be derived from
the following action:

${\boxed{S=\int dt\,d^{d}x\,\psi^{\star}\left(i\hbar
c\,\nabla_{x}+\frac{\hbar^{2}}{2mc^{2}}\,\partial_{t}^{2}\right)\psi\,.}}$
(19)

Recalling once again that in $d$-dimensional space, the length dimension of a
scalar field is $[\phi]_{{}_{L}}=(1-d)/2$, we can see from (16) that the
length dimension of the Carroll-Schrödinger field is $[\psi]_{{}_{L}}=-\,d/2$.
This satisfies the expected dimension of $\hbar$ for the action (19) and is
the same as that of the Schrödinger field (see appendix A).

The Schrödinger equation (23) does not hold invariance under the
transformation $t\to-\,t$. However, in the case of the Carroll-Schrödinger
equation (18), one finds that it is invariant under both the transformations
$t\to-\,t$ and $x_{i}\to-\,x_{i}$ for $d>1$, while it is not invariant for
$d=1$.

In our approach, we first derived the equation of motion and subsequently
obtained the action. We notice that by beginning with the action of a complex
tachyon field, applying the field redefinition (16), and taking the Carroll
limit, we could directly arrive at the action (19).

##  V Discussion

We have formulated a novel equation applicable in any dimension, which we have
named the Carroll-Schrödinger equation (18). Our derivation initially stemmed
from a relativistic tachyon equation; however, it is crucial to highlight that
the resulting equation does not exhibit tachyonic behavior. Instead, it
embodies the nature of Carrollian dynamics.

In the specific case of two dimensions, we have achieved a successful
conformal extension of the centrally extended Carroll algebra. This extension,
called the Carroll-Schrödinger algebra (8), serves as the symmetry algebra for
the derived equation. The infinite dimensional algebra was presented in (11)
demonstrating a same structure as that of the Schrödinger algebra. It becomes
an intriguing question to explore the possibilities of extending the Carroll-
Schrödinger algebra beyond two dimensions.

The action (19) demonstrates a global phase symmetry under the transformation
$\psi\to e^{i\lambda}\,\psi$. Exploring a local phase symmetry could present
an interesting research problem.

Moreover, as we know, the Schrödinger equation (23) can also be derived from
the Dirac equation [67]. Consequently, it becomes interesting to explore the
possibility of deriving the Carroll-Schrödinger equation (18) from a tachyon
Dirac equation.

In both Galilean and Carrollian field theories, it is known that there are two
distinct sectors: the electric sector and the magnetic sector. Typically, the
electric sector corresponds to theories where time derivatives play a dominant
role, whereas the magnetic sector pertains to theories where spatial
derivatives are dominant 222A comprehensive review of electric and magnetic
scalar fields, as well as electric and magnetic spinor fields, can be found in
[61].. It is noteworthy that both the electric and magnetic versions share the
same underlying symmetry, whether it be Galilean or Carrollian.

With this in mind, if we refer to the Schrödinger equation (23) as the
magnetic version, it raises the question of what the electric version of the
Schrödinger equation would be. Similarly, if we consider the obtained Carroll-
Schrödinger equation (18) as the electric version, we may inquire about its
magnetic counterpart. One possible approach to deriving such equations is
through the utilization of the seed Lagrangian method [54] (for further
details, refer to [61]).

The Carroll-Schrödinger equation can provide a promising avenue for the
development of Carrollian quantum mechanics, with potential applications in
condensed matter systems. This could open up new possibilities for addressing
various problems within the Carrollian framework, such as: the infinite
potential well333As we were finalizing this work, we came across this
reference [71] that derived the two-dimensional equation (5) by a different
method. It also solved the problem of the infinite potential well., the
Aharanov Böhm Effect, the Hydrogen atom, the harmonic oscillator and so on.

By solving the equation for different potential energy scenarios, we can
compare the results with established outcomes and examine any differences.
Notably, it seems the presence of the speed of light $c$ in the Carroll-
Schrödinger equation (18) has posed challenges in finding the effects of
Carrollian quantum mechanics so far. However, it is anticipated that a careful
analysis may reveal potential shifts in energy levels associated with this
novel framework.

Acknowledgments: We are grateful to Hamid Afshar and Ahmad Ghodsi for useful
discussions and comments, and to Shahin Sheikh-Jabbari for his support. The
author is partially supported by IPM funds, and the Iran National Science
Foundation (INSF) (Grant Number: 4000132).

##  Appendix A Schrödinger equation

Let us consider the Klein–Gordon equation for a complex scalar field $\upphi$
of mass $m$ in a $(d+1)$-dimensional Minkowski spacetime, with the mostly plus
signature for the metric, including the speed of light $c$ and the reduced
Planck constant $\hbar$

$\displaystyle\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}\,+\,\partial^{i}\partial_{i}\,-\,\mu^{2}\right){\upphi}=0\,,$
(20)

where $\mu:={mc}/{\hbar}$. Using a field redefinition

$\displaystyle\upphi=\frac{1}{\sqrt{\mu}}~{}e^{-i\mu ct}~{}\uppsi\,,$ (21)

the Klein-Gordon equation (20) reduces to

$\displaystyle\left(-\,\frac{1}{c^{2}}\,\partial_{t}^{2}\,+\,\frac{2im}{\hbar}\,\partial_{t}\,+\,\partial^{i}\partial_{i}\right)\uppsi=0\,.$
(22)

By applying the Galilei or nonrelativistic limit ($c\to\infty$) to the latter,
and multiplying by $\hbar^{2}/2m$, we will arrive at the Schrödinger equation

$\displaystyle\left(i\hbar\,\partial_{t}\,+\,\frac{\hbar^{2}}{2m}\,\partial^{i}\partial_{i}\right)\uppsi=0\,.$
(23)

This equation can be obtain from the Schrödinger action

$\displaystyle S=\int
dt\,d^{d}x~{}\uppsi^{\star}\left(i\hbar\,\partial_{t}\,+\,\frac{\hbar^{2}}{2m}\,\partial^{i}\partial_{i}\right)\uppsi\,.$
(24)

We note that the length dimension of a scalar field is
$[\upphi]_{{}_{L}}=(1-d)/2$. Therefore, following (21), the length dimension
for the Schrödinger field $\uppsi$ reads $[\uppsi]_{{}_{L}}=-\,d/2$, and as a
result the action (24) possesses the expected dimension of $\hbar$.

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# Accretion mediated spin-eccentricity correlations in LISA massive black hole
binaries

Authors et al  Mudit Garg,1 Christopher Tiede,2 & Daniel J. D’Orazio2
1Department of Astrophysics, University of Zurich, Winterthurerstrasse 190,
CH-8057 Zürich, Switzerland
2Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100
Copenhagen, Denmark E-mail<EMAIL_ADDRESS>
(Accepted ???. Received ???; in original form ???)

###### Abstract

We examine expected effective spin ($\chi_{{\rm eff},1{\rm yr}}$) and orbital
eccentricity ($e_{1\rm yr}$) correlations for a population of observable
equal-mass massive black hole binaries (MBHBs) with total redshifted mass
$M_{z}\sim[10^{4.5},10^{7.5}]~{}{{\rm M}_{\odot}}$ embedded in a circumbinary
disc (CBD), one-year before merging in the LISA band. We find a strong
correlation between measurable eccentricity and negative effective spin for
MBHBs that are carried to merger by retrograde accretion. This is due to the
well-established eccentricity pumping of retrograde accretion and the
formation of retrograde CBD-aligned mini-discs, as observed in hydrodynamical
simulations. Conversely, prograde accretion channels result in positive
$\chi_{{\rm eff},1\rm yr}$ and non-measurable $e_{1\rm yr}$. This clear
contrast between the two CBD orientations—and particularly the unique
signature of retrograde configurations—provides a promising way to unlock the
mysteries of MBHB formation channels in the LISA era.

###### keywords:

accretion, accretion discs – black hole mergers – gravitational waves –
quasars:general – methods: numerical

††pubyear: 2024††pagerange: Accretion mediated spin-eccentricity correlations
in LISA massive black hole binaries–References

##  1 Introduction

Galaxy mergers in the Universe lead to the formation of massive black hole
binaries (MBHBs; Begelman et al. 1980). Dynamical friction from stars and gas
can bring such MBHBs to parsec scales, but beyond this it becomes inefficient.
At these separations, gravitational waves (GWs) alone can not merge the binary
within the lifetime of the Universe. This is canonically referred to as the
final-parsec problem (Milosavljevic & Merritt, 2003). However, there are
several possibilities suggested to overcome this problem such as improvements
in modeling stellar potentials, subsequent galactic mergers, and gas discs
around binaries (see, e.g., Amaro-Seoane et al., 2023), but how this is
overcome in nature remains uncertain. These dynamical effects can shrink MBHBs
through the sub-parsec regime until GWs become strong enough to drive the
binary to merger within a Hubble time. Recently, pulsar timing arrays (EPTA
Collaboration et al., 2023; Agazie et al., 2023; Reardon et al., 2023) have
provided evidence for a stochastic GW background that is likely sourced by a
population of $\gtrsim 10^{8}{{\rm M}_{\odot}}$ inspiraling MBHBs, which have
made it through the final parsec. If this also occurs for
$10^{4}-10^{8}~{}{{\rm M}_{\odot}}$ near-equal-mass merging MBHBs, then they
will be observable up to redshift $z\sim 20$ by the Laser Interferometer Space
Antenna (LISA; Amaro-Seoane et al. 2017; Colpi et al. 2024), recently adopted
by ESA.

In the LISA milli-Hz-frequency band, GWs from MBHBs will have high signal-to-
noise ratios (SNRs), which will help to constrain source parameters and also
put constraints on merging binaries’ environments. The effects of gas are
particularly compelling in contrast with those from stellar cluster hardening
(Gualandris et al., 2022) because they can leave detectable imprints on the GW
waveform (Barausse et al., 2014; Zwick et al., 2022; Garg et al., 2022; Tiede
et al., 2024; Garg et al., 2024a). Prior to merger, gas has also been
demonstrated to excite binary eccentricities that may remain measurable in the
LISA band (Zrake et al., 2021; Garg et al., 2024b) and gas can also affect the
component BH’s spin (Bourne et al., 2023).

For near-equal-mass MBHBs, radiatively efficient gas will settle into a co-
planar circumbinary disc (CBD; D’Orazio et al. 2016) with prograde
(retrograde) CBDs leading to CBD-aligned mini-discs around each component BH
(Farris et al., 2014; Tiede & D’Orazio, 2024). While a recent study (Garg et
al., 2024a) put constraints on the CBD properties using GWs alone, it also
suggests degeneracies between disc parameters that can be broken by
electromagnetic (EM) counterparts (Haiman, 2017; Mangiagli et al., 2022;
D’Orazio & Charisi, 2023; Franchini et al., 2024; Cocchiararo et al., 2024).
Another possibility is to study the population of observed events. Realization
of MBHB merger rates at the high end of what is expected for LISA (100’s per
year; Amaro-Seoane et al. 2023) would allow us to not only analyze events
individually, but also the population and learn underlying properties of
source distributions and their environments. This strategy is already employed
by ground-based GW detectors to disentangle formation channels for the
observed stellar-mass binary BH population (Nishizawa et al., 2016; Breivik et
al., 2016; Samsing & D’Orazio, 2018; D’Orazio & Samsing, 2018; Romero-Shaw et
al., 2019, 2020; Zevin et al., 2021; Romero-Shaw et al., 2022).

Recent suites of high-resolution hydrodynamical simulations of CBDs suggest
pumping or damping of binary orbital eccentricity depending on binary and disc
parameters. In particular, whether the disc settles into a prograde or
retrograde configuration can have an important impact on the binary
eccentricity evolution, and the rate at which the binary inspirals (if it at
all; see, e.g. Muñoz et al. 2019) which could lead to distinguishable
parameters in the LISA band for the same starting binary. For prograde
accretion scenarios (Zrake et al., 2021; D’Orazio & Duffell, 2021; Siwek et
al., 2023), gas brings the binary to an equilibrium eccentricity $\sim 0.4$,
where it shrinks the binary’s semi-major axis until GWs are strong enough to
drive the binary to coalescence, re-circularizing the orbit (Peters & Mathews,
1963; Peters, 1964). In retrograde configurations (Bankert et al., 2015;
Schnittman & Krolik, 2015; Tiede et al., 2024), there is an eccentricity
pumping throughout the evolution until the GWs takeover (but, see also Nixon
et al. 2011; Roedig & Sesana 2014 for low-eccentricity behavior). In
retrograde discs, the binary also shrinks its semi-major axis at least two-
times quicker than in prograde scenarios leading to the possibility of
different final masses, spins of the component BHs, and orbital eccentricity
near-coalescence for the same starting binary. Here we conduct a quantitative
study of the differences between merging MBHBs in the LISA band when driven
towards merger via retrograde vs prograde accretion. We show that retrograde
configurations yield negative effective spin parameters ($\chi_{\rm eff}$;
Damour 2001) correlated with residual eccentricity near-merger, where both can
be readily measured via GW observations. This contrasts with prograde systems
and may be unique among all formation channels, offering a possibly distinct
indicator of the environmental effects that abetted a binary merger.

##  2 Binary Evolution Model

Consider the motion of two equal-mass spinning BHs of total redshifted mass
$M_{z}$ at redshift $z$ in a tight eccentric orbit with a detector-frame semi-
major axis111Detector-frame SMA is $(1+z)$ times the source-frame SMA, since
redshifted orbital frequency $\propto\sqrt{M_{z}/a^{3}}$. (SMA) $a$ and
eccentricity $e$ embedded in a circumbinary disc (CBD). A prograde
(retrograde) CBD implies that the disc’s angular momentum ($\vec{L}_{\rm
CBD}$) is in the same (opposite) direction as the binary’s angular momentum
($\vec{L}_{\rm Binary}$). Moreover, a prograde (retrograde) CBD leads to CBD-
aligned mini-discs around each component BH (D’Orazio & Duffell, 2021; Tiede &
D’Orazio, 2024). We also define a dimensionless spin parameter222Here $G$ is
the gravitational constant and $c$ is the speed of light. $s\equiv
cJ/Gm^{2}\in(-1,1)$ for a BH of mass $m$ and a spin angular momentum $J$ with
positive (negative) values corresponding to a prograde (retrograde) mini-disc
orientation with respect to the BH. Fig. 1 shows a schematic for our system
for a retrograde CBD configuration.

Figure 1:  Schematic for a retrograde CBD configuration, where the binary with
angular momentum $\vec{L}_{\rm Binary}$ is anti-aligned with that of the
accretion disc $\vec{L}_{\rm CBD}$. The two component BHs have spin vectors
$\vec{s}_{1,2}$, which are misaligned by an angle $\theta_{1,2}\in[0,\pi/2)$
with respect to their own CBD-aligned mini-discs of angular momenta
$\vec{L}_{{\rm MD}1,2}$.

$s$ sets the innermost stable circular orbit (ISCO; $a_{\rm ISCO}$) and
radiative efficiency ($\epsilon$) of the accretion onto the BH (Fiacconi et
al., 2018):

$\displaystyle a_{\rm ISCO}$
$\displaystyle=F(s){Gm}/{c^{2}},\quad\epsilon=1-\sqrt{1-{2}/{3F}},$ (1)
$\displaystyle F(s)$
$\displaystyle=3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})},$ $\displaystyle
Z_{1}$
$\displaystyle=1+(1-s^{2})^{\frac{1}{3}}[(1+s)^{\frac{1}{3}}+(1-s)^{\frac{1}{3}}],\quad
Z_{2}=\sqrt{3s^{2}+Z_{1}^{2}},$

where upper (lower) sign represents a prograde (retrograde) mini-disc
configuration.

The evolution of a spin ($s_{i}$) that is fully (anti-) aligned with its co-
planar mini-disc (i.e., $\theta_{i}=0$) is (Fiacconi et al., 2018)

$\displaystyle\dot{s}_{i}$
$\displaystyle=\frac{\dot{M}_{z,i}}{M_{z,i}}\left(\frac{{F}^{2}\mp
2s_{i}\sqrt{F}+s_{i}^{2}}{{F}\sqrt{{F}-3\pm
2s_{i}/\sqrt{F}}}\frac{1}{1-\epsilon_{i}}-2s_{i}\right)=\frac{\dot{M}_{z,i}}{M_{z,i}}{\tilde{F}}(s_{i}),$
(2)

where again upper (lower) sign represents a prograde (retrograde) mini-disc
orientation and $M_{z,i}=M_{z}/2$. Here $\dot{M}_{z,i}$ is the accretion rate
onto the component BH and determined by the steady-state accretion rate at
infinity $\dot{M}_{0}\equiv{\rm f}_{\rm Edd}M_{z}/45~{}{\rm Myr}^{-1}$ (Jiang
et al., 2014) via $\dot{M}_{z,i}=(1-\epsilon_{i})\dot{M}_{0}/2$. Assuming
$\dot{M}_{0}$ is split evenly between both component BHs, motivated by
measured preferential accretion for both prograde and retrograde CBDs
(D’Orazio et al., 2024), we can write down the accretion rate onto the binary

$\displaystyle\dot{M}_{z}=\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{2}\right)\dot{M}_{0}=\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{2}\right)\frac{{\rm
f}_{\rm Edd}}{45~{}{\rm Myr}}M_{z},$ (3)

where ${\rm f}_{\rm Edd}$ is the Eddington ratio and $\epsilon_{i}$ is the
efficiency of the $i$-th component BH. We assume $\dot{M}_{0}$ or equivalently
${\rm f}_{\rm Edd}$ are constant throughout the binary evolution, i.e., the
feeding rate to the disc at infinity is constant.

For an equal-mass MBHB embedded in a co-planar circumbinary disc (CBD), the
details of the accretion flow and binary-disc parameters dictate if the gas
torques the binary to inspiral or outspiral. For a retrograde disc, the
evolution of SMA and eccentricity due to gas are (Tiede & D’Orazio, 2024)

$\displaystyle\dot{a}^{\rm ret}_{\rm gas}$
$\displaystyle\approx-10\frac{\dot{M}_{z}}{M_{z}}a,\quad\dot{e}^{\rm ret}_{\rm
gas}\approx\frac{\dot{M}_{z}}{M_{z}}\begin{cases}20e,&e\leq 0.15,\\\
3,&e>0.15.\end{cases}$ (4)

While for the prograde binary, we take $\dot{e}^{\rm pro}_{\rm gas}$ from the
Zrake et al. (2021) fit and the most negative value of $\dot{a}^{\rm pro}_{\rm
gas}$ from D’Orazio & Duffell (2021) to consider the fastest merger timescale:

$\displaystyle\dot{a}^{\rm pro}_{\rm gas}$
$\displaystyle=-5\frac{\dot{M}_{z}}{M_{z}}a\ .$ (5)

We also always initialize prograde binaries from their equilibrium
eccentricity $e_{\rm eq}^{\rm pro}=0.45$ (Zrake et al., 2021; D’Orazio &
Duffell, 2021; Siwek et al., 2023), where binaries only circularize when GW
emission becomes dominant over gas effects. This will result in the fastest
inspiral times for the prograde case to compare with retrograde results
computed later. Slower inspiral times will only exacerbate the differences we
later find between the two disc configurations.

GWs remove energy and angular momentum from a compact MBHB. At Newtonian-
order, orbit-averaged evolution rates for $a$ and $e$ due to emission of GWs
are (Peters & Mathews, 1963; Peters, 1964)

$\displaystyle\dot{a}_{\rm GW}$
$\displaystyle=-\frac{16}{5}\frac{G^{3}}{c^{5}}\frac{M_{z}^{3}}{a^{3}}\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right)(1-e^{2})^{-\frac{7}{2}},$
(6) $\displaystyle\dot{e}_{\rm GW}$
$\displaystyle=-e\frac{76}{15}\frac{G^{3}}{c^{5}}\frac{M_{z}^{3}}{a^{4}}\left(1+\frac{121}{304}e^{2}\right)(1-e^{2})^{-\frac{5}{2}}.$

Even though retrograde configurations in § 3 will induce high eccentricities
during binary evolution, we have checked that we never reach the relativistic
regime to require higher post-Newtonian terms.

We choose to begin our binary evolution from the self-gravitating radius
($R_{\rm sg}$) of a thin CBD. Beyond this, the disc can fragment, or
calculations including disc self-gravity are required (see, e.g. Franchini et
al., 2021). Furthermore, we argue that this is the valid limit for the
$M_{z}\lesssim 10^{7}$ MBHBs in the LISA band. For a turbulo-viscous parameter
$\alpha=0.1$ (Shakura & Sunyaev, 1973), the self-gravitating radius can be
expressed as (Bortolas et al., 2021)

$\displaystyle R_{\rm sg}[r_{s}]=$ $\displaystyle 5.32\times 10^{6}\left({{\rm
f}_{\rm Edd}}/{0.1}\right)^{-\frac{22}{45}}\left({M_{z}}/{10^{5}{{\rm
M}_{\odot}}}\right)^{-\frac{52}{45}},$ (7)

where $r_{s}\equiv 2GM_{z}/c^{2}$ is the Schwarzschild radius of the total
binary mass. For MBHBs in the LISA band, $R_{\rm sg}$ is much larger than the
separation scale where GW inspiral will outpace the CBD and gas effects are no
longer important ($a\lesssim 10^{2}R_{\rm sg}$; see, e.g. Dittmann et al.
2023). Hence, for the lower-mass LISA MBHBs, we are justified in using Eqs (4)
and (5), as a gravitationally stable CBD will surround such MBHBs during a
long duration of inspiral where gas effects can dominate over GW-driven
evolution. Therefore, we consider earlier evolution of the MBHB via stars or
gravitationally unstable gas as setting our initial conditions, which will
primarily dictate final values of binary parameters in § 3.

We assume that the binary is always fully (anti-) aligned with its CBD at
least by the time the system is initialized at the self-gravitating radius.
For mild initial misalignment, it takes around a few Myr for complete (anti-)
alignment (Martin & Lubow, 2019). Moreover, we assume that the magnitude and
misalignment angle of the component BH’s spin vector evolve independently from
each other. We also always assume that the BH spins (anti-) align with their
minidiscs when the binary reaches the LISA band. The spin-alignment timescale
is at maximum $\sim 60$ Myr for an already maximally spinning BH with almost
$\pi/2$ radians misalignment for ${\rm f}_{\rm Edd}=0.1$ and azimuthal Mach
number $10$. Therefore, for almost all parameters considered, the BH spins
will fully align with their mini-discs, and only for a narrow set of
parameters will some residual misalignment remain near merger. This can be
seen later in Fig. 2, where only for initial eccentricities $e_{0}\gtrsim
0.6$, inspiral timescales are less than $60$ Myr. Our spin-alignment
assumption is further supported by Bourne et al. (2023), where they almost
always find spin alignment irrespective of the disc orientation and initial
eccentricity.

To study the evolution of binary parameters under the influence of both GWs
and CBD, we evolve the following coupled equations from $a=R_{\rm sg}$ to $1$
year before merger333SMA, one year before merging, is calculated by
integrating $\dot{a}_{\rm GW}$ with final separation $a_{\rm ISCO}$. in the
LISA band:

$\displaystyle\dot{a}$ $\displaystyle=\dot{a}_{\rm GW}+\dot{a}_{\rm
gas},~{}\dot{e}=\dot{e}_{\rm GW}+\dot{e}_{\rm gas},$ (8)
$\displaystyle~{}\frac{\dot{M}_{z}}{M_{z}}$
$\displaystyle=\left(1-\frac{\epsilon_{1}+\epsilon_{2}}{2}\right)\frac{{\rm
f}_{\rm Edd}}{45~{}{\rm
Myr}},~{}\dot{s}_{1}=\frac{\dot{M}_{z,1}}{M_{z,1}}{\tilde{F}}(s_{1}),~{}\&~{}\dot{s}_{2}=\frac{\dot{M}_{z,2}}{M_{z,2}}{\tilde{F}}(s_{2}),$

where $\dot{a}_{\rm gas}$ is $\dot{a}^{\rm pro}_{\rm gas}$ ($\dot{a}^{\rm
ret}_{\rm gas}$) for the prograde (retrograde) CBD.

In the next section, we evolve our systems of interest under the influence of
both gas and GWs to find expected binary parameters in the LISA band.

##  3 Results

###  3.1 Relevant timescales

We solve the coupled equations in Eq. (8) for the retrograde system for ${\rm
f}_{\rm Edd}=0.1$, $z=1$, an initial total mass $M_{z,0}=10^{5}~{}{{\rm
M}_{\odot}}$, initial spins $s_{1,2}=0$, and for various initial
eccentricities $e_{0}$. In Table 1, we show the corresponding inspiral
timescale $t_{\rm ins}$, eccentricity one-year before merger ($e_{1\rm yr}$),
the final-to-initial total-mass ratio, and final spins. Here, $M_{z}/M_{z,0}$
can be well approximated by $\exp({\rm f}_{\rm Edd}t_{\rm ins}/50~{}{\rm
Myr})$. We can infer from $s_{1,2}>0$ that retrograde accretion leads to
component BH spins aligned with their mini-discs and anti-aligned with the
binary. Increasing the total mass to $M_{z,0}=10^{6}~{}{{\rm M}_{\odot}}$ does
not meaningfully alter $t_{\rm ins}$ for retrograde binaries,444Except for
$e_{0}=0$ case, where $t_{\rm ins}\sim 250$ Myr. but $e_{1\rm yr}$ decreases
by a factor of 5 from the $M=10^{5}~{}{{\rm M}_{\odot}}$ case for a given
$e_{0}$. While for a prograde binary, $e_{1\rm yr}\sim 10^{-3}$, irrespective
of the initial total mass. Increasing the accretion rate to ${\rm f}_{\rm
Edd}=1$ results in a nearly ten times faster inspiral time as well as slightly
higher eccentricities for retrograde CBDs and $e_{1\rm yr}\sim 10^{-2.5}$ for
prograde scenarios. While the final spins are almost independent of ${\rm
f}_{\rm Edd}$, we further discuss spin evolution in § 3.2.

$e_{0}$ | $t_{\rm ins}{\rm[Myr]}$ | $e_{1\rm yr}$ | $M_{z}/M_{z,0}$ | $s_{1,2}$
---|---|---|---|---
0 | 370 | 0 | 2.03 | 0.99
0.1 | 149 | 0.03 | 1.35 | 0.73
0.2 | 130 | 0.05 | 1.31 | 0.67
0.3 | 113 | 0.06 | 1.26 | 0.62
0.4 | 97 | 0.09 | 1.22 | 0.55
0.5 | 81 | 0.12 | 1.18 | 0.48
0.6 | 64 | 0.16 | 1.14 | 0.41
0.7 | 48 | 0.21 | 1.11 | 0.32
0.8 | 32 | 0.27 | 1.07 | 0.22
Table 1: The initial eccentricity ($e_{0}$), the inspiral timescale ($t_{\rm
ins}$), the eccentricity one year before merger ($e_{1\rm yr}$), the final-to-
initial total mass ratio, and final spins assuming ${\rm f}_{\rm Edd}=0.1$,
initial total mass $M_{z,0}=10^{5}~{}{{\rm M}_{\odot}}$, and zero initial
spins, for retrograde evolution.

In Fig. 2, as a function of $e_{0}$, we compare spin-up timescales from
different initial spin magnitudes, binary lifetimes for different
configurations, and a typical CBD lifetime based on observations of AGN. We
see that the prograde systems live longer than their spin-up timescales for
any $s_{0}$, while retrograde systems have lifespans shorter than the spin-up
timescale. This implies that prograde systems will always maximally spin up
their component BHs to align with the orbital angular momentum while in
retrograde scenarios this happens only for initially highly spinning component
BHs aligned with their minidiscs (and so anti-aligned with the binary, i.e.,
$s_{0}\sim 1$). We further explore retrograde spin-up in the next sub-section.

The observationally motivated CBD lifetime for ${\rm f}_{\rm Edd}\gtrsim 0.1$
is around $100$ Myr (Hopkins & Hernquist, 2009), for masses relevant to LISA.
This implies that for ${\rm f}_{\rm Edd}=0.1$, MBHBs initialized from the
self-gravitating radius with either prograde CBDs or retrograde CBDs with low
initial binary eccentricity ($e_{0}\lesssim 0.3$) will not reach the LISA
band. To consider this, § 3.3 shows population results for three scenarios: no
restriction on inspiral timescale, $t_{\rm ins}\lesssim 100$ Myr restriction
by starting longer inspiral systems closer than $R_{\rm sg}$, and only
considering $R_{\rm sg}$ initialized MBHBs that merge within the CBD lifetime,
i.e., $t^{\rm sg}_{\rm ins}<100$ Myr.

Figure 2:  Timescales of interest for ${\rm f}_{\rm Edd}=0.1$ as a function of
initial eccentricity $e_{0}$. We show inspiral timescales (dashed lines) for
two initial total masses $M_{z,0}=10^{5}~{}{{\rm M}_{\odot}}$ (light purple)
and $M_{z,0}=10^{6}~{}{{\rm M}_{\odot}}$ (dark purple) in a retrograde CBD
starting from the self-gravitating radius of the disc. For comparison, we show
prograde results ($\times$’s) for the same masses initialized at $e_{0}=0.45$.
We also show spin-up timescales (solid lines) for a component BH to increase
its spin to $s_{i}=0.99$ from three initial spins. We draw an expected, $100$
Myr, CBD lifetime as the dot-dashed orange line.

###  3.2 Spin evolution

Despite retrograde binary lifetimes being shorter than the typical spin-up
time, a significant imprint on the effective spin in the LISA band can still
arise for retrograde systems.

The effective spin of the binary $\chi_{\rm eff}$ assuming complete (anti-)
alignment is (Damour, 2001)

$\displaystyle\chi_{\rm eff}$
$\displaystyle=\frac{s_{1}+s_{2}}{2}\times\begin{cases}+1,&{\rm
for~{}prograde~{}CBD},\\\ -1,&{\rm for~{}retrograde~{}CBD}.\end{cases}$ (9)

Fig. 3 plots the expected effective spin one-year before merger ($\chi_{{\rm
eff},1{\rm yr}}$) for retrograde evolution for a few systems of interests. Low
initial eccentricities lead to longer inspiral timescales (see Fig. 2),
allowing more time for the mini-discs to spin-up the BHs, resulting in more
negative $\chi_{{\rm eff},1{\rm yr}}$ for retrograde binaries.

Figure 3: Effective spin $\chi_{{\rm eff},1{\rm yr}}$, one year before merger
for a $M_{z,0}=10^{5}~{}{{\rm M}_{\odot}}$ (solid line) MBHB with ${\rm
f}_{\rm Edd}=0.1$ retrograde CBD, as a function of the initial eccentricity
$e_{0}$ for different initial spin configurations. We also include
$M_{z,0}=10^{6}~{}{{\rm M}_{\odot}}$ (dashed medium blue) with zero initial
spins for comparison.

###  3.3 Population

We are interested in estimating the expected distribution for the effective
spin ($\chi_{{\rm eff},1{\rm yr}}$) and the eccentricity ($e_{\rm 1yr}$), one-
year before merger, when LISA can probe it. To do so we realize a population
by drawing the parameters described in Sec. 2 as follows:

  * •

For initial total masses, we relate $M_{z,0}$ to the MBHB’s luminosity $L={\rm
f}_{\rm Edd}L_{\rm Edd}$, where the Eddington luminosity $L_{\rm Edd}\equiv
6.3\times 10^{4}M_{z,0}~{}{\rm ergs\,g^{-1}\,s^{-1}}$ and then draw $L$ from
the observed quasar luminosity function (QLF) at redshift $z=1$ and $z=2$
(Hopkins et al., 2007), where expected merger rate for gas-embedded MBHBs is
highest (see, e.g. Salcido et al., 2016). Moreover, to have events in the LISA
band, we only consider initial total masses
$M_{z,0}\sim[10^{4.5},10^{7}]~{}{{\rm M}_{\odot}}$ for retrograde MBHBs and
$M_{z,0}\sim[10^{4},10^{7}]~{}{{\rm M}_{\odot}}$ for prograde MBHBs. Different
initial mass ranges are needed to get the same distribution of total masses
$\sim[10^{4.5},10^{7.5}]~{}{{\rm M}_{\odot}}$ in the LISA band since the
prograde inspiral timescale is much larger than in the retrograde case leading
to larger increases in mass.

  * •

We draw initial spin magnitudes and eccentricities from uniform distributions:
$s_{0}\sim{\rm Uniform}(-1,1)$ and $e_{0}\sim{\rm Uniform}(0,0.8)$.

We initialized $100$ MBHBs from their CBD’s self-gravitating radius, such that
all events have SNR of at least $8$ at $z=1$ in the LISA band to be observable
(Garg et al., 2022) and to mimic expected MBHB detections by LISA (Amaro-
Seoane et al., 2023). For both retrograde and prograde MBHBs, we show
$\chi_{{\rm eff},1{\rm yr}}$ vs $e_{\rm 1yr}$ in Fig. 4 at redshift $z=1$ and
${\rm f}_{\rm Edd}=0.1$. We find that prograde CBDs lead to almost positive
unity effective spin and non-measurable $e_{\rm 1yr}$, and in the retrograde
case, we mostly have negative $\chi_{{\rm eff},1{\rm yr}}$ and measurable
eccentricities.555An equal-mass merging MBHB with $M_{z}=10^{5}~{}{{\rm
M}_{\odot}}$ has the minimum measurable eccentricity $10^{-2.75}$ at $z=1$ for
a one-year observation window (Garg et al., 2024a).

Figure 4: $\chi_{{\rm eff},1{\rm yr}}$ vs $e_{\rm 1yr}$ for 100 LISA-
observable MBHBs at $z=1$ for prograde (blue triangles) and retrograde (red
circles) CBDs with ${\rm f}_{\rm Edd}=0.1$ Eddington ratio. The marker size
represents how large $M_{z}$ is in the range $\sim[10^{4.5},10^{7.5}]~{}{{\rm
M}_{\odot}}$. Filled symbols corresponds to measurable eccentricity ($e^{\rm
det}_{\rm 1yr}$) as per Garg et al. (2024b, a). For retrograde systems, we
draw solid red bars indicating $68$ per cent credible intervals of spin-
eccentricity distribution for measurable eccentricity events with intersection
indicating median values.

Assuming all binaries are brought to merger by an accretion phase, Fig. 4
shows that one could distinguish between prograde and retrograde from their
$\chi_{{\rm eff},1{\rm yr}}$ vs $e_{\rm 1yr}$ correlations in the LISA band,
as they occupy different parts of the parameter space. Note that, even for the
fastest inspiral case in prograde, we predict almost unity effective spin such
that this should also be the case for more complex evolutions $\dot{a}^{\rm
pro}_{\rm gas}$ (see, e.g. D’Orazio & Duffell, 2021). Similarly, $e_{1\rm yr}$
for prograde scenarios should be the same for equal-mass MBHBs (Zrake et al.,
2021) as it is only a function of the equilibrium value $e_{\rm eq}^{\rm
pro}=0.45$ (see also Siwek et al., 2023; Valli et al., 2024). Thus we expect
that our prograde results are robust despite the simplifications.

In Fig. 5, we repeat this experiment for $10^{4}$ LISA-observable events for
varying scenarios to better ascertain the underlying distributions and
minimize variance from drawing fewer events. In comparison to our fiducial
retrograde case with Eddington ratio ${\rm f}_{\rm Edd}=0.1$ and redshift
$z=1$, we consider ${\rm f}_{\rm Edd}=1$, $z=2$, and CBD-lifetime restricted
inspirals: either $t_{\rm ins}<100$ Myr or only considering events which merge
with $t^{\rm sg}_{\rm ins}<100$ Myr, when initialized from the self-gravity
radius. We also consider different maximum equilibrium eccentricities (dubbed
$e^{\rm ret}_{\rm eq}$) for the retrograde CBD. That is to say, one might
imagine that retrograde discs can only pump eccentricity to some maximum value
$e^{\rm ret}_{\rm eq}$ at which shock-dissipation at pericenter (or some other
mechanism) could shut off eccentricity growth. In such a scenario, the MBHB
will stall at $e^{\rm ret}_{\rm eq}$, keep shrinking due to $\dot{a}^{\rm
ret}_{\rm gas}$, and will circularize when GWs become dominant. Lowering
$e^{\rm ret}_{\rm eq}$ results in longer inspiral timescales and more negative
$\chi_{{\rm eff},1{\rm yr}}$, but smaller $e^{\rm det}_{1\rm yr}$. We note,
however, that each of these scenarios in Fig. 5 retains an in-band
eccentricity above the $e_{1\rm yr}>10^{-2.75}$ detection threshold for an
equal-mass $10^{5}M_{\odot}$ binary.

Figure 5: Spin-eccentricity correlations for MBHBs with detectable eccentricity ($e^{\rm det}_{1\rm yr}$) for different scenarios in comparison to our fiducial retrograde CBD (solid red) with ${\rm f}_{\rm Edd}=0.1$ at $z=1$. We show ${\rm f}_{\rm Edd}=1$ (dashed light purple), $z=2$ (dotted dark purple), $t_{\rm ins}<100$ Myr (dot-dashed orange), $t^{\rm sg}_{\rm ins}<100$ Myr (solid medium blue), and assuming the maximum eccentricity in retrograde ($e^{\rm ret}_{\rm eq}$) to be either $0.9$ (solid dark green), $0.99$ (solid medium green), and $0.999$ (solid light green). Bars are similar to Fig. 4 and we summarize these bounds in Table 2.  Scenario | $\log_{10}e^{\rm det}_{1\rm yr}$ | $\chi_{{\rm eff},1{\rm yr}}$
---|---|---
Fiducial | $-1.16_{-0.57}^{+0.48}$ | $-0.56_{-0.25}^{+0.32}$
${\rm f}_{\rm Edd}=1$ | $-1.03_{-0.57}^{+0.48}$ | $-0.57_{-0.24}^{+0.32}$
$z=2$ | $-1.16_{-0.56}^{+0.48}$ | $-0.56_{-0.24}^{+0.33}$
$t_{\rm ins}<100$ Myr | $-0.96_{-0.76}^{+0.40}$ | $-0.43_{-0.27}^{+0.30}$
$t^{\rm sg}_{\rm ins}<100$ Myr | $-0.90_{-0.58}^{+0.34}$ | $-0.40_{-0.27}^{+0.31}$
$e^{\rm ret}_{\rm eq}=0.9$ | $-2.38_{-0.02}^{+0.01}$ | $-0.95_{-0.04}^{+0.06}$
$e^{\rm ret}_{\rm eq}=0.99$ | $-1.79_{-0.03}^{+0.01}$ | $-0.75_{-0.15}^{+0.23}$
$e^{\rm ret}_{\rm eq}=0.999$ | $-1.21_{-0.50}^{+0.02}$ | $-0.58_{-0.23}^{+0.29}$
Table 2: Measurable eccentricity ($e^{\rm det}_{1\rm yr}$) and effective spin
distributions for different scenarios in Fig. 5.

Both redshifts produce similar spin-eccentricity correlations, implying that
our results are independent of redshift. This is because once we consider
detector-frame quantities, a fixed range for total masses
$M_{z}\sim[10^{4.5},10^{7.5}]~{}{{\rm M}_{\odot}}$, and the same number of
observable events, then only QLF, SNR, and the minimum measurable eccentricity
changes with redshift. QLF’s shape depends weakly on $z$, SNRs are still
higher than our threshold of $8$ at $z=2$, and the minimum measurable
eccentricity only increases slightly at $z=2$, effectively, not changing
results at different redshifts. While spin distributions are almost the same
for both Eddington ratios ${\rm f}_{\rm Edd}=0.1$ and ${\rm f}_{\rm Edd}=1$,
$e^{\rm det}_{1\rm yr}$ is slightly higher for the latter and this could help
to distinguish between Eddington ratios.666For prograde case, ${\rm f}_{\rm
Edd}=1$ still has unity $\chi_{{\rm eff},1{\rm yr}}$ but $e_{1\rm yr}\sim
10^{-2.5}$ instead of $e_{1\rm yr}\sim 10^{-3}$ for $0.1$ Eddington ratio.
Limiting the inspiral timescale to typical AGN lifetimes of $100$ Myr leads to
higher effective spin and higher measurable eccentricities. The former is
because smaller initial eccentricities are time limited and not able to spin
BHs to high negative effective spin as per Figs 2 and 3 and the latter because
higher initial eccentricities are selected because they merge more quickly.

##  4 Discussion and Conclusion

We find that MBHBs with a retrograde CBD typically have negative $\chi_{{\rm
eff},1{\rm yr}}$ and measurable eccentricity in the LISA band in comparison to
prograde scenarios, which results in positive effective spin and usually non-
measurable $e_{1\rm yr}$. Moreover, the spin-eccentricity distribution of
binaries evolving through a retrograde accretion channel occupy a unique part
of parameter space that appears to be robust to environmental or evolutionary
parameters such as ${\rm f}_{\rm Edd}$, redshift, or $e^{\rm ret}_{\rm eq}$.
However, different parameters could change the exact quantitative spin-
eccentricity correlations, which may update with advances in simulating CBD-
driven orbital evolution.

An observation of even a single negative effective spin event with detectable
eccentricity combined with a measurement of a GW’s phase perturbation at the
$-4$ post-Newtonian order (Garg et al., 2024a) could provide a strong hint
towards a retrograde CBD. The latter can help to distinguish over other
channels (Zwick et al., 2023), such as hardening due to a third MBH (Bonetti
et al., 2019) or chaotic accretion (King & Pringle, 2006). Additionally, if we
detect an EM counterpart for an MBHB that confirms the presence of an
accretion disc (see, e.g. Mangiagli et al., 2022), then a positive (negative)
effective spin can provide evidence towards prograde/retrograde CBD
orientations, even for a non-measurable eccentricity. The full population of
observations may also yield insights on the relative importance of disc-
mediated phases in massive binary evolution.

We have relied on a number of assumptions that might be relaxed in the future:
While the hydrodynamical simulation results in Tiede & D’Orazio (2024) are
only computed for eccentricity up to $0.8$, we extrapolate them to higher
eccentricities. Furthermore, existing hydrodynamical results that span a
sufficient range of orbital eccentricities derive from 2D, isothermal
hydrodynamical treatments of the CBD. While the key ingredients of retrograde
eccentricity pumping (for, e.g., 3D, self-gravitating disks, including
magnetic feilds, or unequal masses Nixon et al., 2011; Roedig & Sesana, 2014;
Bankert et al., 2015; Amaro-Seoane et al., 2016) and retrograde mini-discs
(Tiede & D’Orazio, 2024) appear to be robust, inclusion of more complete
physics in future simulations will be necessary to improve the accuracy of the
predictions presented here. Moreover, we add evolution rates from gas and GWs
linearly as its not possible to infer cross terms from current simulations for
MBHBs. However, studies by Derdzinski et al. (2019, 2021) suggest that for
extreme mass ratio inspirals cross terms should be weak, which may also extend
to MBHBs. Peters (1964) results for the binary evolution under GW emission may
also breakdown for extremely high eccentricities due to errors in the orbital
averaging not capturing the correct rates (Fumagalli & Gerosa, 2023). However,
this is not clear how it will affect our results and we leave this for future
work. We additionally only consider equal-mass MBHBs, due to the limitation of
retrograde results and under the assumption that before reaching the self-
gravitating radius the binary has become close to equal-mass through
preferential accretion.

In summary, we have demonstrated a possibly unique signature of the retrograde
CBD accretion channel comprising MBHB mergers with negative effective spin and
measurable eccentricities that will help to distinguish from other binary
formation scenarios such as prograde accretion, stellar hardening, and third-
body interactions.

## Data availability statement

The data underlying this article will be shared on reasonable request to the
authors.

## Acknowledgements

MG acknowledge support from the Swiss National Science Foundation (SNSF) under
the grant 200020_192092. D.J.D. and C.T. received support from the Danish
Independent Research Fund through Sapere Aude Starting grant No. 121587. The
authors also acknowledge use of the NumPy (Harris et al., 2020) and Matplotlib
(Hunter, 2007). 0.94

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