††thanks: The first two authors contributed equally

# A study of the Inspiral-Merger-Ringdown Consistency Test with gravitational-
wave
signals from compact binaries in eccentric orbits

Md Arif Shaikh<EMAIL_ADDRESS>Department of Physics, Vivekananda
Satavarshiki Mahavidyalaya (affiliated to Vidyasagar University), Manikpara
721513, West Bengal, India  Department of Physics and Astronomy, Seoul
National University, Seoul 08826, Korea  International Centre for Theoretical
Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
Sajad A. Bhat<EMAIL_ADDRESS>The Inter-University Centre for Astronomy
and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India  Chennai
Mathematical Institute, Plot H1 SIPCOT IT Park, Siruseri 603103, India.
Shasvath J. Kapadia<EMAIL_ADDRESS>The Inter-University Centre for
Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India

(August 28, 2024)

###### Abstract

The Inspiral Merger Ringdown Consistency Test (IMRCT) is one among a battery
of tests of general relativity (GR) employed by the LIGO-Virgo-KAGRA (LVK)
collaboration. It is used to search for deviations from GR in detected
gravitational waves (GWs) from compact binary coalescences (CBCs) in a model-
agnostic way. The test compares source parameter estimates extracted
independently from the inspiral and post-inspiral portions of the CBC signals
and, therefore, crucially relies on the accurate modeling of the waveform.
Current implementations of the IMRCT routinely use quasicircular waveforms,
under the assumption that the residual eccentricity of the binary when the
emitted GWs enter the frequency band of the LVK detector network will be
negligible. In this work, we perform a detailed study to investigate the
typical magnitudes of this residual eccentricity that could potentially lead
to spurious violations of the IMRCT. To that end, we conduct injection
campaigns for a range of eccentricities and recover with both quasicircular
and eccentric waveforms. We find that an eccentric GW signal from a
GW150914-like system with $e_{\text{gw}}\gtrsim$ 0.04 at an orbit averaged
frequency $\langle f_{\text{ref}}\rangle=$ 25 Hz breaks the IMRCT if recovered
with quasicircular waveforms at $\gtrsim 68\%$ confidence. The violation
becomes more severe ($\gtrsim 90\%$ confidence) for $e_{\text{gw}}=$ 0.055 at
$\langle f_{\text{ref}}\rangle=$ 25 Hz. On the other hand, when eccentric
waveforms are used, the IMRCT remains intact for all eccentricities
considered. We also briefly investigate the effect of the magnitude and
orientation (aligned/antialigned) of the component spins of the binary on the
extent of the spurious violations of the IMRCT. Our work, therefore,
demonstrates the need for accurate eccentric waveform models in the context of
tests of GR.

##  I Introduction

The LIGO-Virgo Aasi  _et al._ (2015); Acernese  _et al._ (2015) network of
ground-based interferometric gravitational-wave (GW) detectors has detected
$\sim 90$ compact binary coalescence (CBC) events in its first three observing
runs (O1, O2, O3) Abbott  _et al._ (2023). The majority of these are binary
black hole (BBH) mergers, although binary neutron star (BNS) Abbott  _et al._
(2017, 2020a) and neutron star-black hole (NSBH) coalescences Abbott  _et al._
(2021a) have also been observed.

The detected GW signals, especially those pertaining to BBH mergers, come from
the last few orbits of the inspiral, as well as the merger and ringdown phases
(see, e.g., Sathyaprakash and Schutz (2009)). These GWs are well-suited to
probe general relativity (GR) in the strong-field regime, unlike most tests of
GR that use electromagnetic waves, which typically probe the weak field
regime111A notable exception to this are the results of the Event Horizon
Telescope (EHT) Akiyama  _et al._ (2019), which imaged the shadow of the
supermassive black hole at the center of the M31 galaxy and enabled EM-based
strong-field tests of GR Psaltis (2019)..

The LIGO-Virgo-KAGRA (LVK) collaboration Aasi  _et al._ (2015); Acernese  _et
al._ (2015); Akutsu  _et al._ (2021) has conducted a suite of tests of GR
across O1, O2, and O3 Abbott  _et al._ (2021b). This includes222but is not
limited to a model-agnostic residuals test, which subtracts out the best-
matched GR-modeled CBC waveform from the data containing the GW signal, and
checks if the statistical properties of the residual are consistent with noise
Abbott  _et al._ (2016, 2019a, 2020b); a test that probes the inspiral
evolution of the CBC by searching for deviations in Post-Newtonian (PN)
parameters of the GW signal Blanchet and Sathyaprakash (1995, 1994); Arun  _et
al._ (2006a, b); Yunes and Pretorius (2009); Mishra  _et al._ (2010); Li  _et
al._ (2012a, b); an inspiral-merger-ringdown consistency test (IMRCT) that
compares and assesses the consistency between the low and high-frequency
portions of the CBC signals Ghosh  _et al._ (2016, 2018); and propagation
tests of GWs that compare their speed with respect to the speed of light
Abbott  _et al._ (2019b) as well as any modulations in the waveform due to
finite-graviton-mass-driven velocity dispersion Will (1998).

In this work, we restrict our attention to the IMRCT. The IMRCT compares the
two-dimensional joint posterior on the final mass and spin of the merged
binary estimated independently from the inspiral (low-frequency) and post-
inspiral (high-frequency) portions of the waveform Ghosh  _et al._ (2016,
2018). A sufficiently large deviation from GR should result in the difference
distribution evaluated from these 2D posteriors to deviate from the origin
(zero). However, it is conceivable that waveforms that do not accurately
capture the physics of the CBC could lead to biased posteriors. We propose to
study the resulting systematics in the IMRCT. In particular, we investigate
how the neglect of eccentricity in the waveform models could cause spurious
violations of this test.

It is well-known that, as GWs carry away energy and angular momentum of a CBC,
it reduces the binary orbit’s eccentricity Peters and Mathews (1963); Peters
(1964). The prevalent expectation is that by the time GWs from the CBC enter
the frequency band of the LVK network, any initial eccentricity that the
binary may have had at the time of formation would have been reduced to a
negligibly small value. Indeed, current constraints on the rate of eccentric
mergers, from searches for such signals, suggest that observations of these in
O4 are likely to be relatively rare Abac  _et al._ (2023); Abbott  _et al._
(2019c); Nitz  _et al._ (2019). Thus, tests of GR conducted so far by the LVK
routinely use quasicircular waveforms, including the IMRCT.

On the other hand, certain formation channels, such as those pertaining to
dense stellar environments, could produce binaries whose eccentricities are
non-negligible when they enter the LVK frequency band Mapelli (2020).
Dynamical encounters in such environments could harden the binary so that GW
emission cannot completely carry away the eccentricity by the time the GWs
enter the LVK’s frequency band. Moreover, in AGN disks, eccentricity can be
amplified by binary-single encounters Samsing  _et al._ (2022); Tagawa  _et
al._ (2021), while the presence of a third object in the vicinity of the
binary could similarly boost the eccentricity via the Kozai-Lidov mechanism
Kozai (1962); Lidov (1962); Naoz (2016); Antonini  _et al._ (2017); Randall
and Xianyu (2018); Yu  _et al._ (2020); Bartos  _et al._ (2023). Such
eccentricity-enhancing processes could also leave a residual in-band
eccentricity detectable by the LVK network. Neglecting eccentricity of such
signals in (PN) parametrized tests has already been shown to lead to spurious
violations of GR both at the individual event level Saini  _et al._ (2022) and
population level inference Saini  _et al._ (2023). We here focus only on the
effect of neglect of eccentricity on the IMRCT.

Performing the IMRCT using quasicircular waveform models, could potentially
cause systematics in the IMRCT. Intuitively, the eccentricity at the time of
band-entry of the GW signal will be larger than the eccentricity in the post-
inspiral phase, where it is usually vanishingly small. Thus, one would expect
source parameters recovered with quasicircular waveform models to be biased
when using the inspiral portion of the waveform Favata (2014); Favata  _et
al._ (2022); O’Shea and Kumar (2023), while those recovered using the post-
inspiral phase to be relatively less biased, if at all. The resulting 2D
difference-distribution constructed from the (inspiral/post-inspiral) final
mass and spin posteriors should be deviated from zero.

To assess the impact of neglecting eccentricity in waveform models on the
IMRCT, we consider a set of synthetic eccentric GW signals. We inject these in
the most probable realization of (zero-mean) Gaussian noise, the so-called
“zero-noise” realization, assuming an O4-like noise power-spectral density
(PSD). We then perform the IMRCT on each of these signals, using quasicircular
templates. We find that the IMRCT is indeed violated with quasicircular
templates, as we increase the injected eccentricity, for a set of masses and
spins where the IMRCT is known not to suffer from other systematics. This
violation progressively increases, from moderate to severe, with increasing
eccentricity. To ascertain that no intrinsic waveform systematics are
contributing to this bias, we repeat the IMRCT using eccentric waveform model
for recovery. We indeed find that the IMRCT is no longer violated.

Past work, using an approximate Fisher Matrix Analysis (FMA) Cutler and
Flanagan (1994) and the Cutler-Valisneri framework Cutler and Vallisneri
(2007), pointed out that the IMRCT could be violated due to neglect of
eccentricity Bhat  _et al._ (2023). Another work also found spurious
violations of the IMRCT using available numerical-relativity waveforms as
injections, while performing the IMRCT with quasicircular phenomenological
waveform models Narayan  _et al._ (2023). Our work complements and expands on
both of these studies in the following way. We perform Bayesian parameter
inference runs using large-scale nested sampler and, therefore, have more
realistic estimate of the bias compared to those in provided in Bhat  _et al._
(2023). Moreover, unlike in Bhat  _et al._ (2023), we do not neglect the
eccentricity in the post-inspiral part of the signal which may lead to over-
estimation or under-estimation of the bias. Also, because we employ an
eccentric waveform model TEOBResumS-Dalí Nagar  _et al._ (2018, 2021), we have
two advantages over the results in Narayan  _et al._ (2023). We use the same
waveform model to create the injection and then to recover it. Therefore, we
rule out any bias originating due to the waveform systematics which may be
present if the recovery template is not as accurate as the injected NR
waveforms. Also, we have the freedom to create injections at arbitrary values
of eccentricities allowing us to do an injection study on a finer grid of
eccentricity values. This, in turn, allows us to identify more accurately at
what eccentricity the IMRCT bias becomes severe. Similarly, we examine the
sensitivity of the bias for a given eccentricity on other parameters like the
magnitude of the component spins and their orientation (aligned/antialigned)
with respect to the orbital angular momentum of the binary. Moreover, we
ascertain that the IMRCT is indeed recovered as expected, when eccentricity is
included, for all eccentricities considered. We make contact with both these
works to compare the eccentricity at which the IMRCT is violated, using a
generalised definition of eccentricity that is independent of waveform models
Shaikh  _et al._ (2023).

The rest of this paper is organized as follows. Section II briefly describes
the IMRCT, and argues why one should expect spurious violations due to neglect
of eccentricity. Section III shares the results of the injection campaign
described above. Section IV summarizes the work and discusses other potential
sources of spurious IMRCT violations.

##  II Methods

###  II.1 Gravitational-wave parameter estimation

In GR, GWs have two polarizations, $h_{+}$ and $h_{\times}$. The phase
evolution of these polarizations depends on the intrinsic parameters of the
binary, viz. the masses $m_{1},m_{2}$ of the individual components
333Strictly, the masses that determine the shape of the GW signal measured at
the detector are the so-called “detector-frame” masses, which are simply the
masses multiplied by $(1+z)$, where $z$ is the cosmological redshift., and the
spin vectors of these components $\mathbf{S}_{1},\mathbf{S}_{2}$. The
amplitude of the polarizations depend on the inclination of the binary with
respect to the line of sight, $\iota$, luminosity distance $d_{L}$, the phase
$\phi_{c}$ and time $t_{c}$ at coalescence. In addition to the above mentioned
intrinsic parameters, we need two more parameters – eccentricity $e$ and mean
anomaly $l$ 444The angle variable that describes the position of the compact
objects on the orbit can be also described by other equivalent variables like
eccentric anomaly or true anomaly. Clarke  _et al._ (2022) – at a given
reference frequency to accurately describe GWs from a binary in eccentric
orbit. Also, the GW waveform shape is more sensitive to the ‘chirpmass’
$\mathcal{M}_{c}=m_{1}^{3/5}m_{2}^{3/5}/(m_{1}+m_{2})^{1/5}$ compared to
individual masses and often it is more practical to use $\mathcal{M}_{c}$ and
the mass ratio $q=m_{1}/m_{2}$ instead of $m_{1}$ and $m_{2}$ for parameter
estimation.

The response of the GW detector to each of the GW polarizations is given by
the antenna pattern functions, $F_{+},F_{\times}$. These depend on the
location of the source in the sky, viz. the right ascension $\alpha$ and
declination $\delta$, as well as the polarization angle $\psi$. The GW strain
measured by the detector is a linear combination of the two polarizations,
weighted by the antenna pattern functions:

$h(t;\vec{\theta},\vec{\lambda})=F_{+}(\vec{\lambda}_{F})h_{+}(t;\vec{\theta},\vec{\lambda}_{h})+F_{\times}(\vec{\lambda}_{F})h_{\times}(t;\vec{\theta},\vec{\lambda}_{h})$
(1)

with $t$ as the detector-frame time,
$\vec{\theta}=\\{{\mathcal{M}_{c},q,\mathbf{S}_{1},\mathbf{S}_{2},e,l\\}}$,
$\vec{\lambda}_{F}=\\{{\alpha,\delta,\psi\\}}$,
$\vec{\lambda}_{h}=\\{{d_{L},\iota,t_{c},\phi_{c}\\}}$, and
$\vec{\lambda}=\vec{\lambda}_{F}\cup\vec{\lambda}_{h}$.

Given two waveforms $h_{1,2}(t;\vec{\theta},\vec{\lambda})$, their noise-
weighted inner product is defined as:

$\langle
h_{1},h_{2}\rangle=4\Re\int_{f_{\text{min}}}^{f_{\text{max}}}\frac{\tilde{h}^{\star}_{1}(f;\vec{\theta},\vec{\lambda})\tilde{h}_{2}(f;\vec{\theta},\vec{\lambda})}{S_{n}(f)}df$
(2)

where $f$ is the frequency, and $\tilde{h}_{1,2}$ are Fourier transforms of
the corresponding time domain waveforms, and $\star$ represents complex
conjugation and $\Re$ denotes the real part. $S_{n}(f)$ is the noise power
spectral density (PSD) of the detector. The lower and the upper cutoff
frequencies, $f_{\text{min}}$ and $f_{\text{max}}$ are determined by the
source properties and the sensitivity of the detector.

Let us now consider a time-domain stretch of detector strain data:

$s(t)=n(t)+h(t;\vec{\theta},\vec{\lambda})$ (3)

known to consist of noise $n(t)$, and GW signal
$h(t;\vec{\theta},\vec{\lambda})$ whose parameters
$\vec{\theta},\vec{\lambda}$ are to be estimated. This can be achieved by
sampling the seventeen-dimensional GW posterior
$p(\vec{\theta},\vec{\lambda}|s)\propto
p(\vec{\theta},\vec{\lambda})p(s|\vec{\theta},\vec{\lambda})$, where
$p(\vec{\theta},\vec{\lambda})$ is the prior distribution on the parameters,
and the GW likelihood is given by Cutler and Flanagan (1994):

$p(s|\vec{\theta},\vec{\lambda})\propto\exp\left[-\frac{\langle
s-h(\vec{\theta},\vec{\lambda}),s-h(\vec{\theta},\vec{\lambda})\rangle}{2}\right]$
(4)

The sampling of this high-dimensional posterior is typically achieved using
large-scale Markov-chain Monte-Carlo methods, or nested-sampling methods.

###  II.2 The inspiral-merger-ringdown consistency test

The IMRCT can be thought of as a coarse-grained test of the frequency
evolution of a BBH waveform. It compares the low-frequency and high-frequency
portions of the waveform by comparing the final mass and spin of the merged
binary estimated from the inspiral and merger-ringdown regimes. Let
$M_{f}^{\text{I}}$ and
$\chi_{f}^{\text{I}}\equiv|S_{f}^{\text{I}}|/(M_{f}^{\text{I}})^{2}$ be the
estimates of the detector-frame final mass and dimensionless spin of the
merged binary, respectively, from the inspiral and $M_{f}^{\text{MR}}$,
$\chi_{f}^{\text{MR}}$ be the same from the post-inspiral part. The IMRCT
constructs a joint distribution on the following difference parameters Ghosh
_et al._ (2016, 2018):

$\frac{\Delta M_{f}}{\bar{M}_{f}}\equiv
2\frac{M_{f}^{\text{I}}-M_{f}^{\text{MR}}}{M_{f}^{\text{I}}+M_{f}^{\text{MR}}},\leavevmode\nobreak\
\frac{\Delta\chi_{f}}{\bar{\chi}_{f}}\equiv
2\frac{\chi_{f}^{\text{I}}-\chi_{f}^{\text{MR}}}{\chi_{f}^{\text{I}}+\chi_{f}^{\text{MR}}}$
(5)

The joint-distribution on these parameters should be consistent with zero if
the waveform model used to extract the signal in the data captures all the
physics associated with the production of the signal. On the other hand, a
sufficiently large deviation from GR is expected to shift this distribution to
an extent that it does not contain zero with high confidence ($\gtrsim 68\%$
or $\gtrsim 90\%$). In this work, we demonstrate that even if both the
waveform model and the signal in the data are consistent with GR, an eccentric
signal recovered with a quasicircular model will also lead to a (spurious)
violation of the IMRCT.

The estimate of the final mass and final spin of the merged binary determined
with the intrinsic parameters $\vec{\theta}$ extracted from the inspiral
portion of the waveform is achieved via numerical-relativity-based fits
Hofmann  _et al._ (2016); Healy and Lousto (2017); Jiménez-Forteza  _et al._
(2017), as done in Ghosh  _et al._ (2016, 2018). On the other hand, the shape
of the ringdown contained in the post-inspiral portion of the waveform depends
on the final mass and spin, which can therefore be extracted directly.

In this work, we restrict ourselves to non-precessing waveforms, i.e,
waveforms produced by binaries whose spin-vectors are aligned or antialigned
with the orbital angular momentum vector $\mathbf{L}$ ($z$-axis). Moreover, we
only consider binaries with mass-ratios $q$ that are sufficiently small
($q=m_{1}/m_{2}\sim 1$). We therefore only need to use the leading order
$(2,\pm 2)$ multipole, since higher multipoles for the types of systems we
consider will be suppressed.

The choice of the transition frequency that separates the inspiral and post-
inspiral portions of the waveform is not unique. However, all implementations
of the IMRCT use some approximation of the frequency corresponding to the
innermost stable circular orbit (ISCO) of the binary. The LVK estimates the
ISCO using the medians of intrinsic parameters $\theta$ extracted from the
full IMR waveform. Narayan et al Narayan  _et al._ (2023) use the medians of
the final mass and spin of the merged binary estimated from the full IMR
waveform, similar to what was originally done in Ghosh  _et al._ (2016, 2018).
However, as pointed out in Ghosh  _et al._ (2016, 2018); Narayan  _et al._
(2023), the IMRCT is largely insensitive to moderate variations in the choice
of the transition frequency. In this work, we use the exact values of
$\vec{\theta}$ (i.e, the injected values), as well as the numerical relativity
fits Hofmann  _et al._ (2016); Healy and Lousto (2017); Jiménez-Forteza  _et
al._ (2017), to determine a point-estimate of the final mass and spin, from
which the ISCO frequency is evaluated.

###  II.3 A standardized definition of eccentricity

In GR, there is no universally agreed-upon definition of eccentricity.
Consequently, different waveform models adopt custom internal definitions of
eccentricity, leading to model-dependent inferences about eccentricity. To
mitigate ambiguity in the inferred value of eccentricity from gravitational
wave data analysis, there is a growing effort to standardize the definition of
eccentricity. This standardization involves measuring eccentricity from a
gauge-independent quantity, such as the gravitational waveform modes Ramos-
Buades  _et al._ (2020); Islam  _et al._ (2021); Bonino  _et al._ (2023);
Shaikh  _et al._ (2023).

The waveform modes are obtained from the polarizations by decomposing the
complex combination $h_{+}-ih_{\\!\times\\!}$ on a sphere into a sum of spin-
weighted spherical harmonic modes $\mathpzc{h}_{\ell m}$, so that the waveform
along any direction $(\iota,\varphi_{0})$ in the binary’s source frame is
given by

$\mathpzc{h}(t,\iota,\varphi_{0})=\sum_{\ell=2}^{\ell=\infty}\sum_{m=-\ell}^{m=\ell}\mathpzc{h}_{\ell
m}(t)\leavevmode\nobreak\ {}_{-2}Y_{\ell m}(\iota,\,\varphi_{0}),$ (6)

where $\iota$ and $\varphi_{0}$ are the polar and azimuthal angles on the sky
in the source frame, and ${}_{-2}Y_{\ell m}$ are the spin $=-2$ weighted
spherical harmonics.

In this work, we employ the Python implementation gw_eccentricity of the
standardized definition of eccentricity from Shaikh  _et al._ (2023), which
defines the eccentricity $e_{\text{gw}}$ using the frequency of the (2, 2)
mode $\mathpzc{h}_{22}$. Initially, it computes $e_{\omega_{22}}$ as described
in Ramos-Buades  _et al._ (2020); Islam  _et al._ (2021); Bonino  _et al._
(2023); Shaikh  _et al._ (2023).

$e_{\omega_{22}}(t)=\frac{\sqrt{\omega^{\text{p}}_{22}(t)}-\sqrt{\omega^{\text{a}}_{22}(t)}}{\sqrt{\omega^{\text{p}}_{22}(t)}+\sqrt{\omega^{\text{a}}_{22}(t)}},$
(7)

Here, $\omega^{\text{p}}_{22}(t)$ and $\omega^{\text{a}}_{22}(t)$ denote the
interpolants through the frequency of the (2, 2) mode at the pericenter (point
of closest approach) and the apocenter (point of farthest approach),
respectively. The frequency $\omega_{22}$ of the (2, 2) mode is derived from
$\mathpzc{h}_{22}$ as

$\displaystyle\mathpzc{h}_{22}(t)=A_{22}(t)\,e^{-i\phi_{22}(t)},$ (8)
$\displaystyle\omega_{22}(t)=\frac{\mathrm{d}\phi_{22}(t)}{\mathrm{d}t}\,.$
(9)

After computing $e_{\omega_{22}}$ from $\mathpzc{h}_{22}$, $e_{\text{gw}}$ is
determined through the following transformation to ensure that $e_{\text{gw}}$
correctly exhibits the Newtonian limit Ramos-Buades  _et al._ (2022a).

$e_{\text{gw}}=\cos(\Psi/3)-\sqrt{3}\,\sin(\Psi/3),$ (10)

where

$\Psi=\arctan\left(\frac{1-e_{\omega_{22}}^{2}}{2\,e_{\omega_{22}}}\right).$
(11)

Since eccentricity decays via gravitational wave radiation, we must specify a
reference frequency $\langle f_{\text{ref}}\rangle$ where the value of
$e_{\text{gw}}$ is measured. Moreover, eccentricity also induces modulations
in the instantaneous frequency $\omega_{22}$, we use
$\langle\omega_{22}\rangle$ (by taking the orbital average of the
instantaneous $\omega_{22}$, see Section II.E.2 of Shaikh  _et al._ (2023) for
more details) instead of $\omega_{22}$ for defining the reference frequency
$\langle f_{\text{ref}}\rangle$. Presenting our results with a standardized
definition facilitates one-to-one comparisons with similar studies in the past
and enables future studies to do the same with the results in this work.

###  II.4 Parameter estimation setup

We obtain our results by performing Bayesian parameter inference runs using
state-of-the-art GW parameter inference framework built in Bilby Ashton  _et
al._ (2019) with a dynamical nested sampler dynesty Speagle (2020). For the
injection and recovery, we use the publicly available eccentric waveform model
TEOBResumS-Dalí Nagar  _et al._ (2018, 2021) 555We use the commit 0f19532 of
the eccentric branch of TEOBResumS available at
https://bitbucket.org/eob_ihes/teobresums/. In the following, we will denote
the eccentricity that the waveform model takes as an input by $e_{\text{eob}}$
and the frequency where $e_{\text{eob}}$ is defined by $f_{\text{ref}}$.
Additionally, we provide {$\langle f_{\text{ref}}\rangle,e_{\text{gw}}$} for
each injection to facilitate better comparisons across waveform models and
systematics studies. 666Note that we sample over the model eccentricity
$e_{\text{eob}}$ and not $e_{\text{gw}}$.

Ideally, in order to accurately describe an eccentric orbit, the waveform
model should also include mean anomaly as a free parameter Islam  _et al._
(2021); Clarke  _et al._ (2022); Shaikh  _et al._ (2023). Most of the publicly
available eccentric waveform models like TEOBResumS-Dalí, SEOBNRE Cao and Han
(2017); Liu  _et al._ (2020), EccentricTD Tanay  _et al._ (2016) do not
include mean anomaly as a free parameter but instead start at the pericenter
or apocenter. 777Ecc1dSur Islam  _et al._ (2021) and SEOBNRv4EHM Ramos-Buades
_et al._ (2022b, 2023) include mean anomaly as a free parameter but not yet
publicly available. This, however, does not affect our results since we use
the same waveform model to inject and recover.

To obtain the 2D posteriors on $\Delta M_{f}/\bar{M}_{f}$ and
$\Delta\chi_{f}/\bar{\chi}_{f}$ for the IMRCT, we use the summarytgr command
line tool from PESummary Hoy and Raymond (2021). The uninformative priors used
for the mass and spin parameters lead to non-trivial priors on $\Delta
M_{f}/\bar{M}_{f}$ and $\Delta\chi_{f}/\bar{\chi}_{f}$. To impose agnostic
priors on these inference parameters used to probe GR-violation, we reweight
the 2D posterior to a flat prior on the deviation parameters, as is routinely
done by the LVK Abbott  _et al._ (2021c, b).

Given the posterior samples obtained in the parameter estimation from the
inspiral and post-inspiral part, summarytgr uses an NR informed fit to obtain
the final mass and spin posteriors. This fit is based on quasicircular NR
simulations only and does not take into account any eccentric corrections
which has been shown to be negligible at the quasi-Newtonian level with small
eccentricity approximation Bhat  _et al._ (2023). Currently, we lack a
sufficient number of NR-simulations of merging eccentric binaries required for
obtaining the eccentricity corrected NR-fit for final mass and final spin. We
measure {$\langle f_{\text{ref}}\rangle,e_{\text{gw}}$} for a given
{$f_{\text{ref}},e_{\text{eob}}$} using gw_eccentricity Shaikh  _et al._
(2023).

##  III Results

The main result of this work can be divided into two parts. The first part,
Sec. III.1, discusses the effect of neglecting eccentricity in the waveform
template on the IMRCT. The second part, Sec. III.2, demonstrates the
robustness of the IMRCT performed on GW data containing an eccentric signal
using a waveform template that includes physics required to describe GWs
coming from binaries on an eccentric orbit. We summarize the results of the
study of bias in IMRCT due to neglecting eccentricity in Table 2.

###  III.1 Effects of neglecting eccentricity in templates on IMRCT

Figure 1: Bias in IMRCT due to neglecting eccentricity in recovery template.
The top and the bottom row represent the IMRCT result for the aligned spin
configurations with dimensionless component spins
$(\chi_{1},\chi_{2})=(0.4,0.3)$ and $(\chi_{1},\chi_{2})=(0.2,0.1)$,
respectively. The contours represent the joint probability distribution of the
final mass deviation $\Delta M_{f}/\bar{M}_{f}$ and final spin deviation
$\Delta\chi_{f}/\bar{\chi}_{f}$ at 90% and 68% confidence on the left and the
right panels, respectively. For $(\chi_{1},\chi_{2})=(0.4,0.3)$, GR value is
excluded at 68% confidence at $e_{\text{eob}}=0.09$. For
$(\chi_{1},\chi_{2})=(0.2,0.1)$, GR value is excluded at 68% and 90%
confidence for $e_{\text{eob}}=0.05$ and $0.07$, respectively.

####  III.1.1 Aligned spin case

First, we focus on studying the cases where the spins of the binary components
are aligned with the orbital angular momentum of the binary. It is convenient
to define the tilts $\theta^{\mathbf{S}\mathbf{L}}_{i}$ as the angle between
of the component spin $\mathbf{S}_{i}$ and the orbital angular momentum
$\mathbf{L}$

$\theta^{\mathbf{S}\mathbf{L}}_{i}=\cos^{-1}\left(\frac{\mathbf{L}.\mathbf{S}_{i}}{|\mathbf{L}||\mathbf{S}_{i}|}\right)\qquad
i=\\{1,2\\}$ (12)

Since TEOBResumS-Dalí can describe only non-precessing systems,
$\theta^{\mathbf{S}\mathbf{L}}_{i}$ can take only two values $0$ (aligned) or
$\pi$ (antialigned). In this section, we consider binary systems with
$\theta^{\mathbf{S}\mathbf{L}}_{1,2}=0$.

We choose the component masses to be similar to that of the gravitational wave
event GW150914. For this study we choose a total mass of system that lies
within the ‘golden mass’ range $M\in[50-100]M_{\odot}$ which ensures high
enough SNRs both in the inspiral and post-inspiral parts of the signal Ghosh
_et al._ (2016). The injection parameters and the priors are noted in Table 1.
The injection eccentricities $e_{\text{eob}}\in$ {0.05, 0.06, 0.07, 0.08,
0.09, 0.1} are defined at a reference frequency of $f_{\text{ref}}=$ 20 Hz.
These injected waveforms have $e_{\text{gw}}$ values {0.040, 0.048, 0.055,
0.063, 0.071, 0.079} 888We used the ResidualAmplitude method within
gw_eccentricity to measure these eccentricities. measured at an orbit averaged
frequency of $\langle f_{\text{ref}}\rangle=$ 25 Hz. To understand the effect
of spin magnitudes on IMRCT systematics, we consider two different choices of
dimensionless component spins, one low $(\chi_{1},\chi_{2})=(0.2,0.1)$ and one
moderate $(\chi_{1},\chi_{2})=(0.4,0.3)$, for the set of injected
eccentricities. All the injections considered in this study have HLV network
(at O4 design sensitivity) signal to noise ratio (SNR) in the range $27-52$.

While recovering the injection, we use a Dirac delta prior on eccentricity
peaked at $e_{\text{eob}}=10^{-4}$ making the parameter estimation procedure
equivalent to one using a quasicircular template. 999For $e_{\text{eob}}\leq
10^{-4}$, the corresponding $e_{\text{gw}}$ is $10^{-6}$ implying the
waveforms to be essentially quasicircular. See fig. 10 in Shaikh  _et al._
(2023)

Parameter | Injected value | Prior
---|---|---
$\mathcal{M}_{c}(M_{\odot})$ | 26.36 | $\mathcal{U}(12,45)$
$q$ | 0.8 | $\mathcal{U}(0.25,1)$
$\chi_{1}$ | {0.2, 0.4} | $\mathcal{U}(0,0.99)$
$\chi_{2}$ | {0.1, 0.3} | $\mathcal{U}(0,0.99)$
$\theta^{\mathbf{S}\mathbf{L}}_{1}$ (rad) | 0 | 0
$\theta^{\mathbf{S}\mathbf{L}}_{2}$ (rad) | 0 | 0
$d_{L}$(MPc) | 410 | $\mathcal{U}(50,2000)$
$\iota$ | $0.78$ | $\mathcal{\sin}(0,\pi)$
$\alpha$(rad) | 4.89 | $\mathcal{U}(0,2\pi)$
$\delta$ | -0.218 | $\mathcal{\cos}(-\pi/2,\pi/2)$
$\psi$(rad) | 0.54 | $\mathcal{U}(0,\pi)$
$t_{c}$(s) |  $1126259462$ ($\equiv t_{0}$) | $\mathcal{U}(t_{0}-2,t_{0}+2)$
$\phi_{c}$(rad) | 4.205 | $\mathcal{U}(0,2\pi)$
$e_{\text{eob}}$ | {0.05, 0.06, 0.07, 0.08, 0.09, 0.10} | $10^{-4}$
$f_{\text{ref}}$(Hz) | 20 | 20
Table 1:  Injection values of the binary parameters and corresponding priors
used for parameter estimation with Bilby Ashton  _et al._ (2019).
$\mathcal{U}(a,b)$ stands for uniform prior in the range $(a,b)$.
$\mathcal{\sin}(0,\pi)$ and $\mathcal{\cos}(-\pi/2,\pi/2)$ stand for the Sine
and Cosine priors, respectively on the ranges mentioned. Single values
represent Dirac delta priors peaked at those values. Different values in the
the curly braces represent different injections. The table, therefore,
represents 12 injections – 6 different eccentricities each with 2 different
spin configurations.

The results of these IMRCTs are shown in Fig. 1. The top row shows the results
for $(\chi_{1},\chi_{2})=(0.4,0.3)$ and the bottom row shows the same for
$(\chi_{1},\chi_{2})=(0.2,0.1)$. In each of these rows, the left panel shows
the iso-probability contours of the 2D probability distribution on the final
mass deviation $\Delta M_{f}/\bar{M}_{f}$ (along $x$-axis) and the final spin
deviation $\Delta\chi_{f}/\bar{\chi}_{f}$ (along $y$-axis) plane at 90%
confidence. The right panel shows the same at 68% confidence. Different
contours in each panel correspond to the injected eccentricities
$e_{\text{eob}}\in$ {0.05, 0.06, 0.07, 0.08, 0.09, 0.1} as noted on the top of
the right panel. Fig. 1 also shows the marginalised 1D posterior of $\Delta
M_{f}/\bar{M}_{f}$ on top of the left panel and that of
$\Delta\chi_{f}/\bar{\chi}_{f}$ on the right of the right panel.

It can be noted that in the case of moderate spin
$(\chi_{1},\chi_{2})=(0.4,0.3)$, for smaller eccentricities
($e_{\text{eob}}<0.08$), the contours at 90% and 68% contain the point (0, 0)
denoted by $+$ which stands for zero-deviation from GR. We will denote this
point of zero-deviation from GR as the ‘GR value’. However, for larger
eccentricities ($e_{\text{eob}}\gtrsim 0.08$), the contours are shifted
towards positive deviation in $\Delta M_{f}/\bar{M}_{f}$ and the GR value is
excluded at 68% confidence for $e_{\text{eob}}\geq 0.09$. For smaller spins
$(\chi_{1},\chi_{2})=(0.2,1)$, the bias is more severe. For the low spin case
(bottom row in Fig. 1), the GR value is excluded at 68% confidence for
$e_{\text{eob}}=0.05$ and at 90% confidence for $e_{\text{eob}}=0.07$.

Figure 2: GR Quantile $Q_{\texttt{GR}}$ vs injected eccentricity for different
component spins for the aligned spin systems. Small $Q_{\texttt{GR}}$
indicates better agreement with GR. Circles denote the quasicircular recovery
and the ellipses denote the eccentric recovery. The low spin
$(\chi_{1},\chi_{2})=(0.2,0.1)$ and moderate spin
$(\chi_{1},\chi_{2})=(0.4,0.3)$ case is denoted by violet and orange,
respectively. When recovered with quasicircular templates, low spin cases show
more bias compared to moderate spin case.

One can also construct a statistical measure of the deviation from GR using
the GR quantile $Q_{\texttt{GR}}$ which is the fraction of the posterior
samples contained within the iso-probability contour that passes through the
GR value (no-deviation point $(0,0)$). A smaller $Q_{\texttt{GR}}$ implies a
better agreement with GR since the GR value is contained within a smaller iso-
probability contour. In this work, we compute the GR quantile in a slightly
different way. Instead of using the samples directly, we use an interpolated
histogram of the 2D probability distribution. In particular, we use the
implementation in summarytgr within PESummary Hoy and Raymond (2021).

In Fig. 2, we plot the $Q_{\texttt{GR}}$ for different component spin
configurations as a function of the injection eccentricities. It shows the
overall trend of $Q_{\texttt{GR}}$ increasing with $e_{\text{eob}}$. While for
moderate spins $(\chi_{1},\chi_{2})=(0.4,0.3)$, $Q_{\texttt{GR}}$ becomes
$\approx 68\%$ at $e_{\text{eob}}=0.08$, for low spins
$(\chi_{1},\chi_{2})=(0.2,0.1)$, $Q_{\texttt{GR}}$ becomes $>68\%$ at
$e_{\text{eob}}=0.05$ and $\gtrsim 90\%$ at $e_{\text{eob}}=0.07$.

####  III.1.2 Antialigned spin case

Figure 3: IMRCT with quasicircular recovery template for injections with
antialigned spin configurations.The panels from the top to bottom correspond
to primary antialigned, secondary antialigned and both antialigned spin
configurations, respectively. For all the three cases, the dimensionless
component spin magnitudes are $(\chi_{1},\chi_{2})=(0.4,0.3)$. Due to
neglecting eccentricity in the recovery template, the GR value is excluded at
68% confidence at $e_{\text{eob}}=0.07,0.09$ and $0.08$ for primary
antialigned, secondary antialigned and both antialigned cases, respectively.
The GR value is also excluded at 90% confidence for secondary antialigned at
$e_{\text{eob}}=0.1$. Figure 4: GR Quantile $Q_{\texttt{GR}}$ vs injected
eccentricity for different component spin alignments for
$(\chi_{1},\chi_{2})=(0.4,0.3)$. Small $Q_{\texttt{GR}}$ indicates better
agreement with GR. Circles denote the quasicircular recovery and the ellipses
denote the eccentric recovery. When recovered with quasicircular template, the
$Q_{\texttt{GR}}$ tend to increase with increasing eccentricity implying
increasing bias in IMRCT due to neglecting eccentricity in the waveform model.
On the other hand, for eccentric recovery, the $Q_{\texttt{GR}}$ remains
within $68\%$ for all eccentricities. For all cases of eccentric recovery, the
$Q_{\texttt{GR}}$ is $\lesssim 10\%$. In the secondary antialigned and both
antialigned cases, the $Q_{\texttt{GR}}$ becomes even smaller ($<5\%$)
implying excellent agreement with GR.

In this section, we study how the bias in the IMRCT due to neglecting
eccentricity in the recovery waveform model depends on the orientation of the
spin with respect to the angular momentum of the system. Let the pair
$(\theta^{\mathbf{S}\mathbf{L}}_{1},\theta^{\mathbf{S}\mathbf{L}}_{2})$ denote
the tilts of the spin $\mathbf{S}_{1}$ of the primary and the spin
$\mathbf{S}_{2}$ of the secondary, respectively, with respect to the orbital
angular momentum $\mathbf{L}$ of the system. We consider the following three
cases

  1. 1.

$(\pi,0)$ – $\mathbf{S}_{1}$ is antialigned and $\mathbf{S}_{2}$ is aligned.

  2. 2.

$(0,\pi)$ – $\mathbf{S}_{1}$ is aligned and $\mathbf{S}_{2}$ is antialigned.

  3. 3.

$(\pi,\pi)$ – Both $\mathbf{S}_{1}$ and $\mathbf{S}_{2}$ are antialigned.

We consider only the moderate spin case where the dimensionless spins are
(0.4, 0.3). The other parameters remain the same as in Table 1. We use a Dirac
delta prior peaked at the injected values for the tilts. The number of
injections for each of these tilts is 6 for six different eccentricities
making a total of 18 injections for these three tilt configurations.

The rows from the top to bottom in Fig. 3 show the IMRCT results for the cases
of primary antialigned, secondary antialigned and both antialigned,
respectively. In each of these figures, the left panel shows the iso-
probability contours at 90% confidence and the right panel shows the same at
68% confidence in the $\Delta M_{f}/\bar{M}_{f}-\Delta\chi_{f}/\bar{\chi}_{f}$
plane. The general trend is similar to that of aligned spin cases in Sec.
III.1.1.

Dimensionless component spins | Tilts | Threshold eccentricity for GR violation at $90\%$ | Threshold eccentricity for GR violation $68\%$
---|---|---|---
$(\chi_{1},\chi_{2})$ | $(\theta^{\mathbf{S}\mathbf{L}}_{1},\theta^{\mathbf{S}\mathbf{L}}_{2})$ |  $e_{\text{eob}}$ at $f_{\text{ref}}=20$Hz |  $e_{\text{gw}}$ at $\langle f_{\text{ref}}\rangle=25$Hz |  $e_{\text{eob}}$ at $f_{\text{ref}}=20$Hz |  $e_{\text{gw}}$ at $\langle f_{\text{ref}}\rangle=25$Hz
(0.4, 0.3) | (0, 0) |  |  | 0.09 | 0.071
(0.2, 0.1) | (0, 0) | 0.07 | 0.055 | 0.05 | 0.040
(0.4, 0.3) | $(\pi,0)$ |  |  | 0.07 | 0.056
(0.4, 0.3) | $(0,\pi)$ | 0.10 | 0.079 | 0.09 | 0.071
(0.4, 0.3) | $(\pi,\pi)$ |  |  | 0.08 | 0.064
Table 2: Summary of the study of bias in IMRCT due to neglecting eccentricity.
It provides the threshold injection eccentricity ($e_{\text{eob}}$ at 20 Hz)
at or above which the IMRCT result is biased at 68% and 90% confidence. It
also provides the eccentricity $e_{\text{gw}}$ in terms of standardized
definition Shaikh  _et al._ (2023) at an orbital averaged frequency $\langle
f_{\text{ref}}\rangle=25$ Hz corresponding to each of these threshold
injections. As an example, for an aligned low spin
$(\chi_{1},\chi_{2})=(0.2,1.0)$ system, GR is excluded at 68% confidence for
$e_{\text{eob}}=0.05$ ($e_{\text{gw}}=0.04$) and GR is excluded at 90%
confidence for $e_{\text{eob}}=0.07$ ($e_{\text{gw}}=0.055$).

###  III.2 IMRCT with eccentric signal

In the previous section, we showed that IMRCT shows spurious deviation from GR
when the parameters from an eccentric gravitational wave signal is inferred
using a quasicircular waveform model (using a delta function prior on
eccentricity fixed at $e_{\text{eob}}=10^{-4}$). In this section, we show that
such spurious deviations go away when eccentric waveform model is used instead
of quasicircular model by allowing the sampler to sample from a uniform prior
on eccentricity as described in Tab. 3.

Parameter | Injected value | Prior
---|---|---
$e_{\text{eob}}$ | {0.05, 0.06, 0.07, 0.08, 0.09, 0.10} | $\mathcal{U}(8\times 10^{-3},0.15)$
$f_{\text{ref}}$(Hz) | 20 | 20
Table 3: Injection values of the eccentricities and priors for IMRCTs with
eccentric recovery.

####  III.2.1 Aligned spin case

Figure 5: IMRCT with eccentric recovery template for aligned spin injections.
The top row corresponds to $(\chi_{1},\chi_{2})=(0.4,0.3)$ and the bottom row
corresponds to $(\chi_{1},\chi_{2})=(0.2,0.1)$. Same as Fig. 1, but now the
injections are recovered with eccentric templates instead of quasicircular
templates. GR value is included within the iso-probability contours at both
the 68% and 90% confidence intervals for all injections for these two spin
configurations.

First, we consider the case of aligned spin where both the primary and the
secondary spin are aligned with the angular momentum of the system, i.e.,
$\theta^{\mathbf{S}\mathbf{L}}_{1,2}=0$. In Fig. 5, we show the results of
IMRCTs with a set of eccentric aligned spin signals. The top row shows the
result for injections with moderate component spins
$(\chi_{1},\chi_{2})=(0.4,0.3)$ and the bottom row shows the result for low
component spins $(\chi_{1},\chi_{2})=(0.2,0.1)$. The set of injected
eccentricities and the prior on the eccentricity for both of these spin
configurations are provided in Tab. 3. The other parameters remain the same as
in Sec. III.1.1.

In each row of Fig. 5, the left (right) panel shows the iso-probability
contours of the 2D probability distribution in the $\Delta
M_{f}/\bar{M}_{f}-\Delta\chi_{f}/\bar{\chi}_{f}$ plane at 90% (68%)
confidence. The 1D posterior on $\Delta M_{f}/\bar{M}_{f}$ is shown on top of
left panel and the 1D posterior on $\Delta\chi_{f}/\bar{\chi}_{f}$ is shown on
the right of the right plot. In both of these spin configurations, for all the
injected eccentricities, we find IMRCTs to be unbiased both at 68% and 90%
confidence.

For these two aligned spin configurations, the GR quantile $Q_{\texttt{GR}}$,
when recovered with eccentric waveform template, is shown using ellipses in
Fig. 2. $Q_{\texttt{GR}}$ is plotted as a function of the injection
eccentricities. It can be seen that for all the injected eccentricities, the
$Q_{\texttt{GR}}$ remains $<68\%$ for low spins
$(\chi_{1},\chi_{2})=(0.2,0.1)$ and $<30\%$ for moderate spins
$(\chi_{1},\chi_{2})=(0.4,0.3)$ implying an agreement with GR.

####  III.2.2 Antialigned spin case

Figure 6: IMRCT with eccentric recovery template for injections with
antialigned spin configurations. The rows from the top to bottom correspond to
primary antialigned, secondary antialigned and both antialigned spin
configurations, respectively. For all the three cases, the dimensionless
component spin magnitudes are $(\chi_{1},\chi_{2})=(0.4,0.3)$. Same as Fig. 3
but now the injections are recovered with eccentric templates instead of
quasicircular templates. Note that the GR value is included at both 68% and
90% confidence for all the injections for these three configurations.

We now turn our focus to the potential effects of the orientation of the
component spins on IMRCTs with eccentric signal when recovered with eccentric
waveform models. We consider the same injections as in Sec. III.1.2; we
consider three possible configurations where the pair of tilts may take the
values $(\pi,0),(0,\pi)$ or $(\pi,\pi)$ with the component dimensionless spins
$(\chi_{1},\chi_{2})=(0.4,0.3)$. In Sec. III.1.2, these injections were
recovered using a quasicircular template which resulted in spurious bias in
the IMRCT result. Now we recover these injections with eccentric template
using a prior on eccentricity as shown in Tab. 3.

The rows from the top to bottom in Fig. 6 shows the IMRCT contours for primary
antialigned, secondary antialigned and both antialigned configurations,
respectively. For a given row, the left (right) panel shows the iso-
probability contours of the 2D probability distribution in the $\Delta
M_{f}/\bar{M}_{f}-\Delta\chi_{f}/\bar{\chi}_{f}$ plane at 90% (68%)
confidence. The 1D posterior on $\Delta M_{f}/\bar{M}_{f}$ is shown on top of
the left panel and the 1D posterior on $\Delta\chi_{f}/\bar{\chi}_{f}$ is
shown on the right of the right panel. Different contours in each panel
correspond to different injected eccentricity as denoted on top of the right
panel of each row. Unlike the results in Fig. 3, all the contours in Fig. 6
include the GR value.

We also plot the $Q_{\texttt{GR}}$ (denoted by ellipses) values of the IMRCTs
corresponding to these injections in Fig. 4. Note that the $Q_{\texttt{GR}}$
values for these cases of recovery with eccentric templates are very small
compared to the quasicircular recovery (denoted by circles). The
$Q_{\texttt{GR}}$ for eccentric recovery is $<68\%$ for all considered
eccentricities in these three antialigned spin configurations. While for the
primary antialigned cases, the $Q_{\texttt{GR}}$ remains $\lesssim 10\%$, for
the secondary antialigned and both antialigned cases, the GR quantile is even
smaller, $Q_{\texttt{GR}}\lesssim 5\%$ implying excellent agreement with GR.

##  IV Summary and Outlook

The IMRCT Ghosh  _et al._ (2016, 2018) is one among several tests of GR with
GWs used by the LVK. While model-agnostic in the sense that it is not based on
any particular alternative theory of gravity, it does nevertheless crucially
rely on the CBC template waveform being an accurate representation of the GW
signal. Thus, intuitively, one would expect spurious violations of this test
when physical and environmental effects are not accounted for in the waveform
model.

To demonstrate this, in this work, we focus our attention on GWs from
eccentric compact binaries. We inject, in zero-noise assuming an O4-like noise
PSD, synthetic GW150914-like CBC signals but with a range of residual
eccentricities at a reference frequency of $f_{\mathrm{ref}}=20$ Hz. The spins
are always assumed to lie along the orbital angular momentum ($z-$) axis,
although we consider two different sets of spin magnitudes, and all possible
orientation(aligned/antialigned)-combinations of the binary components along
the $z-$ axis. We then perform the IMRCT with quasicircular templates (these
being routinely used by current implementations of the IMRCT), as well as
eccentric templates.

We find that, as expected, the IMRCT gets progressively biased with increasing
injected eccentricity while recovering with quasicircular templates. At $68\%$
confidence, we find that the IMRCT starts to get violated for eccentricities –
as defined by the waveform model used – between $e_{\text{eob}}\sim 0.05-0.09$
at a reference frequency $f_{\text{ref}}=20$ Hz, for our chosen CBC systems.
Converting this to a standardized definition of eccentricity Shaikh  _et al._
(2023), these values become $e_{\text{gw}}\sim 0.04-0.07$ at an orbital
averaged frequency $\langle f_{\text{ref}}\rangle=25$ Hz. Indeed, as the
eccentricity is increased, the violations occur even at $90\%$ confidence.

On the other hand, when recovering with eccentric templates, we find that, the
IMRCT is always satisfied, regardless of the spin magnitude, orientation, or
eccentricity value, at $68\%$ confidence. Our work therefore demonstrates the
need for accurate eccentric waveform models for IMRCT and possibly other tests
of GR. Moreover, our work complements and expands on previous work studying
the effect of neglect of eccentricity on the IMRCT. These either use
approximate GW PE likelihoods Bhat  _et al._ (2023), or only study the
potential bias in the IMRCT caused by recovering a few eccentric numerical
relativity waveforms with phenomenological waveforms Narayan  _et al._ (2023).

Two important limitations of our work must be pointed out. The first is that
the injections are performed in zero-noise rather than real-noise. This could
potentially introduce additional fluctuations in the IMRCT as a consequence of
a particular noise realisation. The second is that the recovery of the IMRCT
has been tested using identical injection and recovery waveform models. It
would be of interest to consider eccentric numerical-relativity injections
recovered with other eccentric waveform models that are nominally used for GW
PE. Such a study might potentially demonstrate non-trivial biases in the IMRCT
even when eccentric waveforms are used, and if not, would ascertain the
feasibility of using current eccentric waveform models for tests of GR. We
plan to address the said limitations in future work.

###### Acknowledgements.

We thank Pankaj Saini for a careful reading of the draft. We also thank
Prayush Kumar, Nathan K. Johnson-McDaniel, Aditya Vijaykumar, and members of
the ICTS-TIFR Astrophysical Relativity Group, for useful discussions. M.A.S.’s
research was supported by the Department of Atomic Energy, Government of India
under project no. RTI4001 and the National Research Foundation of Korea under
grant No. NRF-2021R1A2C2012473. S.A.B. acknowledges support from the
Department of Science and Technology and the Science and Engineering Research
Board (SERB) of India via Swarnajayanti Fellowship Grant No.
DST/SJF/PSA-01/2017-18 and support from Infosys foundation. The authors
acknowledge the public code at https://github.com/osheamonn/eobresum_bilby/, a
modified version of which was used as the source model for injection study
with Bilby Ashton  _et al._ (2019). The authors are grateful for computational
resources provided by the LIGO Laboratory and supported by National Science
Foundation Grants PHY-0757058 and PHY-0823459. The authors also acknowledge
the use of the Alice cluster at the International Centre for Theoretical
Sciences.

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# Arithmetic Hodge-Iwasawa Moduli Stacks

Xin Tong

###### Abstract

In this article, we are going to construct arithmetic moduli stacks of
$G$-bundles after our previous construction on Hodge-Iwasawa theory. These
stacks parametrize certain Hodge-Iwasawa structures in a coherent way.

###### Contents

  1. 1 Introduction
  2. 2 Arithmetic Moduli Stacks
    1. 2.1 Arithmetic Moduli Stacks of Filtered Frobenius Bundles
    2. 2.2 Arithmetic Hecke Stacks, Arithmetic Moduli Stacks of Shtukas
  3. 3 Variants

##  1 Introduction

We discuss in this article moduli stacks of $G$-isocrystals over arithmetic
families of Fargues-Fontaine curves after the constructions in our previous
consideration in [T1] and [T2]. The initial motivation after [Iwa], [T], [F]
is a combination reasonable of the Iwasawa deformation after Burns-Flach-
Fukaya-Kato [BF1], [BF2], [FK] and some arithmetic deformation after Kedlaya-
Pottharst-Xiao and Kedlaya-Pottharst [KPX], [KP], of the structures after
[KL1] and [KL2]. When we consider the integration effect of the deformation we
will expect some sort of moduli stacks in arithmetic situation, which is the
analogy of those considered in [He1], [PR], [Dr], [SW], [Ked], [RZ], [G],
[Dr1], [Dr2], [Har], [Har1], [HV], [M], [FS], [Sch1].

##  2 Arithmetic Moduli Stacks

###  2.1 Arithmetic Moduli Stacks of Filtered Frobenius Bundles

We now consider arithmetic moduli stacks of vector bundles after [Dr], [Dr1],
[Dr2], [He1], [PR], [EG], [EGH], [HHS] in the foundation in [HP], [FF], [KL1]
and [KL2] with generalization initialized in [KP], [T1] and [T2]. Let $K$ be a
complete adic field with valuation which we will assume to be discrete. We use
the notation $k$ to denote the residue field of $K$ which we assume to be
overall perfect in our consideration in this article. We then have the tower
$*_{\mathrm{toric}}$ toric attached to this field. Recall we have the Fargues-
Fontaine curve $\mathrm{FF}_{K,\mathrm{toric}}$ attached to $K$ in [KL1] and
[KL2] which is defined to be Frobenius quotient of the space:

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Spa}(\widetilde{\Pi}^{I}_{\mathrm{toric}},\widetilde{\Pi}_{\mathrm{toric}}^{I,+}).$
(1)

Then after [W], [He1] we consider the sites carrying ffqc topology (faithfully
flat and quasicompact topology):

$\displaystyle\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}},\mathrm{RigidAn}_{\mathbb{F}_{p}((t)),\mathrm{ffqc}}$
(2)

which consist of all the rigid analytic spaces. There are also animated
versions of these categories as in [CS1], [CS2] and [CS3], while in this
derived setting we mainly consider the affinoid algebras which are defined to
be the animation of usual rigid analytic affinoids in [Ta]. In notation we use
the notations as in the following to denote the opposite
$(\infty,1)$-categories:

$\displaystyle\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}},\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{F}_{p}((t)),\mathrm{ffqc}}.$
(3)

######  Definition 1.

We now define the $(\infty,1)$-prestack
$\mathrm{Finiteproj}_{\mathrm{FF}_{K,\mathrm{toric}}}(-)$ fibered over

$\displaystyle\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$
(4)

to be the functor sending $X$ to $(\infty,1)$-groupoid of vector bundles over
$\mathrm{FF}_{K,\mathrm{toric},X}$ being regarded as Clausen-Scholze analytic
space in [CS1], [CS2] and [CS3]. One then has the definition for equal
characteristic situation. When specializing a filtration $\mathrm{Fil}$ with
respect to some cocharacter of the group scheme $\mathrm{GL}$, we can define
the prestack
$\mathrm{Finiteproj}^{\mathrm{Fil}}_{\mathrm{FF}_{K,\mathrm{toric}}}(-)$.

######  Definition 2.

We now define the $(\infty,1)$-prestack
$\mathrm{Finiteproj}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over

$\displaystyle\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$
(5)

to be the functor sending $X$ to $(\infty,1)$-groupoid of vector bundles over
$\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric},X}$. The imperfect space we
consider here is defined to be the Frobenius quotient of

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Spec}^{\mathrm{CS}}({\Pi}^{I}_{\mathrm{toric}}\otimes^{\blacksquare}\mathcal{O}_{X},{\Pi}_{\mathrm{toric}}^{I,+}\otimes^{\blacksquare}\mathcal{O}_{X}).$
(6)

One then has the definition for equal characteristic situation. When
specializing a filtration $\mathrm{Fil}$ with respect to some cocharacter of
the group scheme $\mathrm{GL}$, we can define the prestack
$\mathrm{Finiteproj}^{\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$.

######  Definition 3.

We now define the prestack
$\mathrm{Finiteproj}^{\mathrm{disc}}_{\mathrm{FF}_{K,\mathrm{toric}}}(-)$
fibered over $\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to be the
functor sending $X$ to groupoid of vector bundles over
$\mathrm{FF}_{K,\mathrm{toric},X}$. One then has the definition for equal
characteristic situation. When specializing a filtration $\mathrm{Fil}$ with
respect to some cocharacter of the group scheme $\mathrm{GL}$, we can define
the prestack
$\mathrm{Finiteproj}^{\mathrm{disc},\mathrm{Fil}}_{\mathrm{FF}_{K,\mathrm{toric}}}(-)$.

######  Definition 4.

We now define the prestack
$\mathrm{Finiteproj}^{\mathrm{disc}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over $\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to be the
functor sending $X$ to groupoid of vector bundles over
$\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric},X}$. The imperfect space we
consider here is defined to be the Frobenius quotient of

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Spec}^{\mathrm{CS}}({\Pi}^{I}_{\mathrm{toric}}\otimes^{\blacksquare}\mathcal{O}_{X},{\Pi}_{\mathrm{toric}}^{I,+}\otimes^{\blacksquare}\mathcal{O}_{X}).$
(7)

One then has the definition for equal characteristic situation. When
specializing a filtration $\mathrm{Fil}$ with respect to some cocharacter of
the group scheme $\mathrm{GL}$, we can define the prestack
$\mathrm{Finiteproj}^{\mathrm{disc},\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$.

######  Proposition 1.

The prestack
$\mathrm{Finiteproj}^{\mathrm{disc},\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over $\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ is an Artin
stack. The parallel result in equal characteristic sitaution holds true as
well.

###### Proof.

Our construction could be regarded as relative version of the construction in
[He1, Section 4.1].

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}}}\times\left(\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}},E}/\mathrm{Parab}^{\mathrm{rigidification}}_{\mathrm{Fil}}\right)$
(8)
$\displaystyle/\bigcup_{I=[s,r]}\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}},E}$
(9)

reflects the certain stackification with $E/\mathbb{Q}_{p}$ a suitable
extension. ∎

We then consider the quotient in the $(\infty,1)$-categories to derive the
corresponding Artin $(\infty,1)$-stacks generalizing the previous discussion
above.

######  Proposition 2.

The $(\infty,1)$-prestack
$\mathrm{Finiteproj}^{\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over
$\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ is an
$(\infty,1)$-stack. The parallel result in equal characteristic sitaution
holds true as well.

###### Proof.

For instance in the mixed-characteristic situation, by the results above we
now form the quotient:

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}}}\times\left(\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}},E}/\mathrm{Parab}^{\mathrm{rigidification}}_{\mathrm{Fil}}\right)$
(10)
$\displaystyle/^{\mathrm{der}}\bigcup_{I=[s,r]}\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}},E}$
(11)

in the derived sense, which gives the result. ∎

######  Remark 1.

These $(\infty,1)$-stacks actually admit presentations in some smooth sense in
the derived sense. With the same proof one can have the corresponding
representability of the prestacks in the equal characteristic situations,
which we can omit the details.

###  2.2 Arithmetic Hecke Stacks, Arithmetic Moduli Stacks of Shtukas

Now we consider the analogues of the Hecke stacks and moduli of shtukas in
[Laff], [Dr], [Dr1], [Dr2], [SW], [RZ], [G], [A], [La] in our arithmetic
deformation situation in the following. First we can drop the filtration in
the previous discussion to define arithmetic moduli of vector bundles, namely
with the notations above we have the prestacks over the spaces:

$\displaystyle\widetilde{Y}:=\bigcup_{I=[s,r]}\mathrm{Spa}(\widetilde{\Pi}^{I}_{\mathrm{toric}},\widetilde{\Pi}_{\mathrm{toric}}^{I,+}),Y:=\bigcup_{I=[s,r]}\mathrm{Spa}({\Pi}^{I}_{\mathrm{toric}},{\Pi}_{\mathrm{toric}}^{I,+}).$
(12)

######  Definition 5.

We now define the $(\infty,1)$-prestack
$\mathrm{Finiteproj}_{\widetilde{Y}}(-)$ fibered over
$\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to be
the functor sending $X$ to $(\infty,1)$-groupoid of vector bundles over
$\widetilde{Y}_{X}$ being regarded as Clausen-Scholze analytic space in [CS1],
[CS2] and [CS3]. One then has the definition for equal characteristic
situation.

######  Definition 6.

We now define the $(\infty,1)$-prestack $\mathrm{Finiteproj}_{{Y}}(-)$ fibered
over $\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to
be the functor sending $X$ to $(\infty,1)$-groupoid of vector bundles over
${Y}_{X}$. The imperfect space we consider here is defined to be the Frobenius
quotient of

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Spec}^{\mathrm{CS}}({\Pi}^{I}_{\mathrm{toric}}\otimes^{\blacksquare}\mathcal{O}_{X},{\Pi}_{\mathrm{toric}}^{I,+}\otimes^{\blacksquare}\mathcal{O}_{X}).$
(13)

One then has the definition for equal characteristic situation.

######  Definition 7.

We now define the prestack
$\mathrm{Finiteproj}^{\mathrm{disc}}_{\widetilde{Y}}(-)$ fibered over
$\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to be the functor sending
$X$ to groupoid of vector bundles over $\widetilde{Y}_{X}$. One then has the
definition for equal characteristic situation.

######  Definition 8.

We now define the prestack $\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-)$
fibered over $\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ to be the
functor sending $X$ to groupoid of vector bundles over $Y_{X}$. The imperfect
space we consider here is defined to be the Frobenius quotient of

$\displaystyle\bigcup_{I=[s,r]}\mathrm{Spec}^{\mathrm{CS}}({\Pi}^{I}_{\mathrm{toric}}\otimes^{\blacksquare}\mathcal{O}_{X},{\Pi}_{\mathrm{toric}}^{I,+}\otimes^{\blacksquare}\mathcal{O}_{X}).$
(14)

One then has the definition for equal characteristic situation.

######  Proposition 3.

$\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-)$ is an Artin stack over
$\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$. The parallel result in
equal characteristic sitaution holds true as well.

###### Proof.

One can form the quotient Artin stacks
$[*/\bigcup_{I=[s,r]}\mathrm{Res}_{\Pi^{I}_{\mathrm{toric}}/\mathbb{Q}_{p}}\mathrm{GL}^{\mathrm{rigidification}}_{\Pi^{I}_{\mathrm{toric}}}]$
which gives the result. ∎

Then one defines the following Hecke stacks. Let $I$ be a finite set, and we
consider a disjoint decomposition of $I$ into $k$-subsets:

$\displaystyle I=\bigcup_{i=1}^{k}I_{i}.$ (15)

######  Definition 9.

We now define the $(\infty,1)$-prestack
$H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{\widetilde{Y}}(-)$ fibered over

$\displaystyle\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$
(16)

to be the functor sending $X$ to $(\infty,1)$-groupoid of the objects:

$\displaystyle(M_{0},M_{1},...,M_{k},\\{p_{i}\\}_{i\in I},f_{1},...,f_{k})$
(17)

where each $M_{i},i=1,...,k$ is a vector bundle over the base space
$\widetilde{Y}_{X}$, with $I$ points $\\{p_{i}\\}_{i\in
I}\subset\widetilde{Y}_{X}$, with for each $i=1,...,k$ the isomorphism:

$\displaystyle f_{i}:M_{i}|_{\widetilde{Y}_{X}-\cup_{j\in
I_{i}}p_{j}}\overset{\sim}{\longrightarrow}M_{i-1}|_{\widetilde{Y}_{X}-\cup_{j\in
I_{i-1}}p_{j}}.$ (18)

One then has the definition for equal characteristic situation. Following this
in similar fashion directly one can define
$H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{{Y}}(-)$,
$H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{\widetilde{Y}}(-)$
with $H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-)$.

######  Definition 10.

We then define the moduli stacks of arithmetic shtukas. This is defined as the
fiber product along the following two maps:

$\displaystyle H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{\widetilde{Y}}(-)$
$\displaystyle\rightarrow\mathrm{Finiteproj}_{\widetilde{Y}}(-)\times\mathrm{Finiteproj}_{\widetilde{Y}}(-)$
(19) $\displaystyle(M_{0},M_{1},...,M_{k},\\{p_{i}\\}_{i\in
I},f_{1},...,f_{k})$ $\displaystyle\mapsto(M_{0},M_{k})$ (20)

and

$\displaystyle\mathrm{Finiteproj}_{\widetilde{Y}}(-)$
$\displaystyle\rightarrow\mathrm{Finiteproj}_{\widetilde{Y}}(-)\times\mathrm{Finiteproj}_{\widetilde{Y}}(-)$
(21) $\displaystyle M$ $\displaystyle\mapsto(M,\mathrm{Fro}^{*}M).$ (22)

We use the notation
$\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{\widetilde{Y}}(-)$ to
denote this $(\infty,1)$-prestack. One then has the definition for equal
characteristic situation. Following this in similar fashion directly one can
define $\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{{Y}}(-)$,
$\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{\widetilde{Y}}(-)$
with
$\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-)$.

######  Remark 2.

One also has the construction in direct parallel fashion the stacks in Kummer
setting as in [EG].

##  3 Variants

We have many different types of variants of the constructions above. For
instance we can replace the Robba rings above with the following:

  * A

(Extended Fargues-Fontaine Curves)
$(\Pi^{I}_{\mathrm{toric}}[(\log(1+T))^{1/2}]\widehat{\otimes}_{\mathbb{Q}_{p}}E)\otimes^{\blacksquare}\mathcal{O}_{X}$
as in [BS] where $E$ contains square roots of $\varphi(\pi)$ and $\gamma(\pi)$
as in [BS];

  * B

(Logarithmic Fargues-Fontaine Curves)
$(\Pi^{I}_{\mathrm{toric}}\left<\log(\log(1+T))\right>)\otimes^{\blacksquare}\mathcal{O}_{X}$,
the notation $\left<\right>$ means the Tate algebra over the variable
$\log(\log(1+T))$, after [Fon];

  * C

(Product Fargues-Fontaine Curves)
$(\Pi^{I}_{\mathrm{toric}}\widehat{\otimes}_{\mathbb{Q}_{p}}\Pi^{I^{\prime}}_{\mathrm{toric}})\otimes^{\blacksquare}\mathcal{O}_{X}$
as in [CKZ] and [PZ].

######  Proposition 4.

The parallel statements for these variant situations hold true.

  * $\square$

The prestack
$\mathrm{Finiteproj}^{\mathrm{disc},\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over $\mathrm{RigidAn}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ is an Artin
stack. The parallel result in equal characteristic sitaution holds true as
well.

  * $\square$

The $(\infty,1)$-prestack
$\mathrm{Finiteproj}^{\mathrm{Fil}}_{\mathrm{FF}^{\mathrm{imper}}_{K,\mathrm{toric}}}(-)$
fibered over
$\mathrm{AnimaRigidAn}^{\mathrm{aff}}_{\mathbb{Q}_{p},\mathrm{ffqc}}$ is an
$(\infty,1)$-stack. The parallel result in equal characteristic sitaution
holds true as well.

We have the corresponding spaces in the following as well:

$\displaystyle
H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{{Y}}(-),H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{\widetilde{Y}}(-),H_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-),$
(23)
$\displaystyle\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}_{{Y}}(-),\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{\widetilde{Y}}(-)\mathrm{Sht}_{I,I_{1},...,I_{k}}\mathrm{Finiteproj}^{\mathrm{disc}}_{Y}(-).$
(24)

## Acknowledgements

We thank Professor Kedlaya for conversation on the different perspectives of
Hodge-Iwasawa Theory. We mentioned the corresponding aspects discussed in this
paper in our paper [T2]. The author thanks Eugen Hellmann for the discussion
around the construction of the rigid analytic Artin moduli stacks in the
cyclotomic tower situation. The work [EGH] contains the construction of such
analytic stack when we consider the corresponding filtration as in [He1]. We
thank Professor Sorensen for the suggestions on generalizing the work of [BS].

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