# Eccentricity estimation for five binary black hole mergers with higher-order
gravitational wave modes

H. L. Iglesias  Center of Gravitational Physics, University of Texas at
Austin, Austin, TX 78712, USA  J. Lange  Center of Gravitational Physics,
University of Texas at Austin, Austin, TX 78712, USA  I. Bartos  Department of
Physics, University of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA
S. Bhaumik  Department of Physics, University of Florida, PO Box 118440,
Gainesville, FL 32611-8440, USA  R. Gamba  Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universit ät Jena, 07743, Jena, Germany  V.
Gayathri  Department of Physics, University of Florida, PO Box 118440,
Gainesville, FL 32611-8440, USA  Leonard E. Parker Center for Gravitation,
Cosmology, and Astrophysics, University of Wisconsin–Milwaukee, Milwaukee, WI
53201, USA  A. Jan  Center of Gravitational Physics, University of Texas at
Austin, Austin, TX 78712, USA  R. Nowicki  Center of Gravitational Physics,
University of Texas at Austin, Austin, TX 78712, USA  R. O’Shaughnessy
Rochester Institute of Technology, Rochester, NY 14623, USA  D. M. Shoemaker
Center of Gravitational Physics, University of Texas at Austin, Austin, TX
78712, USA  R. Venkataramanan  Center of Gravitational Physics, University of
Texas at Austin, Austin, TX 78712, USA  K. Wagner  Rochester Institute of
Technology, Rochester, NY 14623, USA

###### Abstract

The detection of orbital eccentricity for a binary black hole system via
gravitational waves is a key signature to distinguish between the possible
binary origins. The identification of eccentricity has been difficult so far
due to the limited availability of eccentric gravitational waveforms over the
full range of black hole masses and eccentricities. Here we evaluate the
eccentricity of five black hole mergers detected by the LIGO and Virgo
observatories for the first time using the TEOBResumSGeneral model. This model
accounts for the full eccentricity range possible and incorporates higher-
order gravitational wave modes critical to model emission from highly
eccentric orbits. The binaries have been selected due to previous hints of
eccentricity or due to their unusual mass and spin. While other studies found
marginal evidence for eccentricity for some of these events, our analyses do
not favor the incorporation of eccentricity compared to the quasi-circular
case. While lacking the eccentric evidence of other analyses, we find our
analyses marginally shifts the posterior in multiple parameters for several
events when allowing eccentricity to be non-zero.

eccentric black hole mergers

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##  1 Introduction

The LIGO (Aasi et al., 2015) and Virgo (Acernese et al., 2015) observatories
have discovered about 90 binary mergers via gravitational waves (GWs) signals
over three observing periods (Abbott et al., 2019, 2021, 2021). The properties
of these mergers have been estimated using gravitational wave templates that
assume quasi-circular binary orbits prior to merger (Ossokine et al., 2020;
Khan et al., 2019; Pratten et al., 2021). This approach has been the baseline
due to both prior expectations and technical challenges. Gravitational wave
emission is expected to circularize binaries over time, making eccentricity
unexpected for the majority of binaries. At the same time the construction of
accurate eccentric gravitational waveforms is difficult due to the required
large dynamic range, while accounting for eccentricity also requires probing
an expended parameter space. Nonetheless, searches carried out by LIGO and
Virgo are sensitive to eccentric binaries (Abbott et al., 2019), leaving the
possibility that some of the recovered mergers are secretly eccentric.

While long-lived, isolated binaries lose their eccentricity before entering
the LIGO/Virgo frequency band, measurable eccentricity is expected through
several astrophysical mechanisms. First, binary black hole systems that form
with small initial separations may not have sufficient time prior to merger to
lose their eccentricity. This scenario can occur, e.g. in chance encounters in
galactic nuclei or globular clusters (Portegies Zwart & McMillan, 2000;
Banerjee et al., 2010; Ziosi et al., 2014; Morscher et al., 2015; Mapelli,
2016; Rodriguez et al., 2016; Askar et al., 2017). Eccentricity can also be
enhanced for long-lived binaries by a nearby third object via the Kozai-Lidov
mechanism (Naoz, 2016; Antonini et al., 2017; Randall & Xianyu, 2018). In
active galactic nuclei (AGN), binaries formed via gas capture in AGN disks can
have enhanced eccentricity due to regular encounters with stars and black
holes in the galactic nucleus (Samsing et al. 2022; Tagawa et al. 2021; Bartos
et al. 2017).

Recently, several techniques have been developed and applied to probe orbital
eccentricity in binary black hole mergers detected by LIGO and Virgo. Gayathri
et al. (2022) used a bank of about 600 numerical relativity simulations
throughout the full eccentricity range and estimated their match with the
observed data for the binary merger GW190521 (Abbott et al., 2020) using the
RIFT parameter estimation suite (Lange et al., 2018, 2017; Lange, 2020). They
found that the observed data best matches a highly eccentric waveform, and
reconstructed the eccentricity of GW190521 to be $e=0.69^{+0.17}_{-0.22}$.
While applicable to GW190521, their technique was limited to binaries with
high masses due to the duration of the simulated waveforms, while the
reconstruction’s precision was limited by the limited number of discrete
eccentricity, spin and mass ratio values covered in the study.

Romero-Shaw et al. (2020) used gravitational wave template banks that covered
the full mass parameter space, while limited to eccentricities $e<0.2$ and
black hole spins aligned with the binary orbit. They performed their inference
by a novel postprocessing technique: they first analyze with the quasi-
circular model IMRPhenomD (Ajith et al., 2011) and, based on those results,
reweight their conclusions with the aligned-spin eccentric model SEOBNRE
(Payne et al., 2019; Romero-Shaw et al., 2019). A further possible limitation
of this approach was the iterative reweighting which may have been limited by
systematic errors between the two models used in the reweighted analysis
(Romero-Shaw et al., 2021; Jan et al., 2020).

O’Shea & Kumar (2021) and Gamba et al. (2021) performed parameter inference
including the continuously-explored effects of eccentricity and assuming black
hole spins aligned with the orbital axis (non-precessing spins). O’Shea &
Kumar (2021) analyzed the low-mass events GW151226 and GW170608 including
eccentricity and non-precessing spins; Gamba et al. (2021) analyzed GW190521
with non-spinning systems and sampled in initial energy and angular momentum
instead of eccentricity which could impact the prior volume and therefore the
evidence. These analytic models assumed spin-orbit alignment, limiting
attempts to disentangle the effects of orbital eccentricity and precession
(Bustillo et al., 2021).

Currently, there are two easily-accessible waveform models which predict the
strong-field merger signal for merging binary black holes which incorporate
both binary spin and orbital eccentricity: SEOBNRE and TEOBResumSGeneral. The
SEOBNRE model is an effective-one-body (EOB) (Buonanno & Damour, 1999, 2000;
Damour et al., 2000; Damour, 2001; Damour et al., 2015) model that is based on
the early quasi-circular nonprecessing model SEOBNRv1. While this model can
describe eccentric binary black hole mergers, it is limited to eccentricities
$e<0.2$ and spin magnitudes $a<0.6$ due to being based on the older v1 SEOB
model (Cao & Han (2017)). The TEOBResumSGeneral model is an EOB approximant
which can generate waveforms from non-precessing binaries coalescing along
generic orbits (Chiaramello & Nagar, 2020; Nagar et al., 2021b, 2021), or
generic-spins binaries coalescing along quasi-circular orbits (Damour & Nagar,
2014; Nagar et al., 2016, 2018, 2020a, 2020b; Riemenschneider et al., 2021;
Akcay et al., 2021; Gamba et al., 2021). In both scenarios, the model includes
tidal interactions (Damour & Nagar, 2010; Bernuzzi et al., 2015; Akcay et al.,
2019) as well as subdominant modes up to $\ell=|m|=5$ but not $m=0$ modes.
While the model can generate stable waveforms up to an eccentricity of $e=0.9$
with maximal spin magnitudes, comparisons to numerical relativity (NR)
waveforms have only been carried out up to eccentricities of $e=0.2$ and spin
magnitudes up to $a=0.7$ (Nagar et al., 2021). Another EOB model in the
literature that incorporates orbital eccentricity is SEOBNRv4EHM, which also
describes non-precessing spins and includes higher order modes (Khalil et al.,
2021; Ramos-Buades et al., 2022).

In this study, we analyzed a group of GW events to produce full posteriors
using the eccentric model TEOBResumSGeneral in tandem with parameter
estimation carried out using RIFT. Events were selected based on signs of
possible eccentricity in previous studies, or due to their unusual mass/spin
properties. For each event, we run three different analyses (two show in the
figures): a non-precessing, non-eccentric analysis (from now on called
TEOBResumS-GIOTTO); a precessing, non-eccentric analysis (from now on called
TEOBResumSP-GIOTTO); and a non-precessing, eccentric analysis (from now on
called TEOBResumS-DALI). To quantify any evidence in these analyses, we
calculate Bayes’ factors for eccentricity as well as precession. We include
the latter to make sure any evidence of eccentricity does not also yield
evidence of precession. This could potentially happen if an event is in a
region of parameter space where precession and eccentricity can mimic each
other in the analysis (Romero-Shaw et al., 2023).

This paper is organized as follows. In Section 2, we introduce the
TEOBResumSGeneral model and review the use of RIFT in this study. In Section
3, we present the results of parameter inference as well as Bayes’ factors for
our five events. In Section 4, we summarize our results and conclude with some
brief remarks about our future work.

##  2 Methods

A coalescing binary black hole system in a quasi-circular orbit can be
completely characterized by its intrinsic ($\lambda$) and extrinsic ($\theta$)
parameters. By intrinsic parameters we refer to the binary’s masses $m_{i}$
and spins. By extrinsic parameters we refer to the seven numbers needed to
characterize its space-time location and orientation. We will express masses
in solar mass units and dimensionless spins in terms of cartesian components
$\chi_{i,x},\chi_{i,y},\chi_{i,z}$, expressed relative to a frame with
$\hat{\mathbf{z}}=\hat{\mathbf{L}}$ and (for simplicity) at the orbital
frequency corresponding to the earliest time of computational interest (e.g.,
an orbital frequency of $\simeq 10\,{\rm Hz}$). We will use $\lambda,\theta$
to refer to intrinsic and extrinsic parameters, respectively.

###  2.1 Waveform model

In order to extract the eccentricity feature from detected GW events, we are
using the TEOBResumSGeneral waveform model (Nagar et al., 2021a, 2018;
Albanesi et al., 2022), which includes eccentricity features, aligned spin,
and higher order modes. This waveform family is based on the EOB formalism and
it can be used to simulate quasi-circular, eccentric or hyperbolic compact
binaries systems. This model includes higher multipoles up to 5 throughout
binary phases (inspiral, plunge, merger and ringdown) and eccentricity
parameter space (i.e., up to $e\simeq 1$; see figure 19 in Nagar et al.
(2021a)).

TEOBResumSGeneral waveforms have been validated with Numerical Relativity
waveforms from the Simulating eXtreme Spacetime collaboration with $e\leq
0.2$, and mass ratio $q\leq 3$. For our study we used TEOBResumSGeneral
waveforms with three different configurations, corresponding to $e=0$ with
non-precessing spins (TEOBResumS-GIOTTO), $e=0$ with precessing spins
(TEOBResumSP-GIOTTO), and $e>0$ with non-precessing spins (TEOBResumS-DALI).

To generate orbital dynamics from eccentric initial conditions – an initial
eccentricity $e_{0}$ at initial orbital frequency $f_{0}$ – the TEOBResumS-
DALI code evolves an orbit starting at apastron, with $r_{0}=p_{0}/(1-e_{0})$,
$p^{0}_{\phi}=j_{0}$ the adiabatic angular momentum implied by angular
momentum conservation, and $p_{r_{*}}=0$. In these expressions, $p_{0}$ is
evaluated numerically from the Hamiltonian such that
$\partial_{p_{\phi}}H(r_{0}(p_{0}),j_{0}(p_{0}),p_{r_{*}}=0)=f_{0}$. The
orbital dynamics produce an associated asymptotic gravitational wave strain
$h(t,\hat{n})=\sum_{lm}{}^{-2}Y_{lm}(\hat{n})h_{lm}(t)$, where the $h_{lm}(t)$
depend on the intrinsic parameters. In practice, we characterize the emission
direction relative to the orbital angular momentum direction with two polar
angles ($\iota,\phi_{c}$), so the binary’s radiation relative to its fiducial
inertial frame is fully specified with an initial frequency (to define
conventions), the initial eccentricity at that frequency, two masses, both
spins, and these two polar angles. While this current parameterization fixes
the angle of periapsis (one degree of freedom available for eccentric orbits
associated with the direction of the Runge-Lenz vector in the orbital plane),
this angle is expected to be observationally inaccessible in the near future
(see, e.g., Clarke et al., 2022).

###  2.2 RIFT

RIFT consists of a two-stage iterative process to estimate the source
parameters $\bm{\theta},\lambda$ responsible for gravitational wave
observations $d_{k}$ via comparison to predicted gravitational wave signals
$h_{k}(\bm{\lambda},\bm{\theta})$ where $k=1\ldots N$ indexes the observing
instruments.

Assuming a Gaussian, stationary noise model, the log-likelihood can be
expressed as

$\displaystyle\ln$ $\displaystyle{\cal L}(\lambda,\theta)=$
$\displaystyle-\frac{1}{2}\sum_{k}\left<h_{k}(\lambda,\theta)-d_{k}|h_{k}(\lambda,\theta)-d_{k}\right>_{k}-\left<d_{k}|d_{k}\right>_{k}$
(1)

where we have omitted normalization constants. (RIFT assumes the input
detector noise power spectrum is known, and does not currently marginalize
over the accuracy of that estimate, nor over calibration uncertainty.)

In these expressions, the inner products represent the noise-weighted inner
product derived from the $k$th detector’s noise power spectrum

$\displaystyle\left<a|b\right>_{k}\equiv\int_{-\infty}^{\infty}2df\frac{\tilde{a}(f)^{*}\tilde{b}(f)}{S_{n,k}(|f|)}\,,$

is a $S_{n,k}(f)$, where $\tilde{a}(f)$ is the Fourier transform of $a(t)$,
$\tilde{a}(f)^{*}$ denotes complex conjugation of $\tilde{a}(f)$, and $f$ is
frequency; see, e.g., Pankow et al. (2015) for more details. We adopt a low-
frequency cutoff $f_{\rm low}$ such that all inner products are modified to

$\displaystyle\left<a|b\right>_{k}\equiv 2\int_{|f|>f_{\rm
low}}df\frac{\tilde{a}(f)^{*}\tilde{b}(f)}{S_{n,k}(|f|)}\,.$ (2)

The two iterative stages of RIFT construct the necessary ingredients for an
iterative evaluation of Bayes’ theorem for this likelihood. In one stage of
RIFT, many worker codes evaluate the _marginal_ likelihood $\ln{\cal
L}_{\alpha}\equiv\ln{\cal L}_{\rm marg}(\lambda_{\alpha})$ for
$\lambda_{\alpha}$ in some grid of evaluation points, via

$\displaystyle{\cal L}_{\rm marg}(\lambda)\equiv\int d\theta p(\theta){\cal
L}(\lambda,\theta)$ (3)

In the other iterative stage of RIFT, an interpolation algorithm provides a
current-best-estimate $\hat{\cal L}(\lambda)$ based on current training data
$\\{(\lambda_{\alpha},{\cal L}_{\alpha}\\}$, and employs this estimate in
Bayes’ theorem to construct an approximate posterior distribution over the
intrinsic parameters $\lambda$:

$\displaystyle\hat{p}_{\rm post}(\lambda)\simeq\frac{\hat{{\cal
L}}(\lambda)p(\lambda)}{\int d\lambda\hat{{\cal L}}(\lambda)p(\lambda)}$ (4)

Again using Monte Carlo integration, RIFT produces independent fair draws from
this estimated posterior distribution, thus providing a new grid for the
evaluation stage. After several iterations, $\ln\hat{\cal L}$ will converge to
the the true log-likelihood in the neighborhood where the posterior has its
support, and the intrinsic samples will be the true posterior distribution.

##  3 Results

In this section, we present the results from our eccentric reanalysis of the
following events: GW150914, GW190521, GW190620, GW190706, GW190929. GW190521
and GW190620 were analyzed due to past evidence or hints of eccentricity
(Gayathri et al. (2022); Romero-Shaw et al. (2021)); GW150914, GW190706, and
GW190929 were also picked due to specific characteristics (HOMs, unequal
masses, high mass, positive spins, etc.). All the data and power spectral
density (PSDs) are the same files used in the LVK’s GWTC-2.1 paper (The LIGO
Scientific Collaboration et al. (2021)) and are available on GWOSC (Abbott et
al., 2021a). Due to this, we chose 4 seconds of data around each event except
for GW190521 where we chose 8 seconds of data111We chose to use 8 seconds of
data instead of following GWTC-2.1 settings since the original analysis of
this event was done with 8 seconds including Gayathri et al. (2022)..
GW190521, GW190706, GW190929 uses data from all three HLV detectors with the
HL data using the C01 calibration with noise subtracted out around 60 Hz (The
LIGO Scientific Collaboration et al. (2021); Littenberg & Cornish (2015);
Chatziioannou et al. (2019)). For these events, the V data uses the online
calibration data except for GW190929 that uses 1A calibration. Finally
GW150914 uses HL data using the C02 calibration, and GW190620 uses LV data
using the C01 calibration with noise subtracted out around 60 Hz for L and
using the online calibration for V. The PSDs were generated using BayesWave
(Littenberg & Cornish (2015); Chatziioannou et al. (2019)) on the same segment
of data used in each analysis.

For all events except GW190521, we used a low-frequency cutoff of 20 Hz,
motivated by the lack of observationally-accessible information at lower
frequencies for typical sources. For GW190521, we use a low-frequency cutoff
of 11 Hz. The high-frequency cutoff was 896, 224, 448, 896, and 896 Hz for
GW150914, GW190521, GW190620, GW190706, and GW190929 respectively. We analyzed
each event with the non-precessing, eccentric TEOBResumS-DALI include all the
$\ell_{\rm max}\leq 4$ except the $m=0$ modes. For comparison, we also
analyzed each event with the non-precessing, non-eccentic TEOBResumS-GIOTTO
including the same amount of modes as the eccentric analysis. An additional
analysis with the precessing, non-eccentric TEOBResumSP-GIOTTO was conducted
to ascertain evidence of precession in these events.

We adopted conventional masses and distance priors (uniform in detector-frame
mass and in the cube of the luminosity distance; see, e.g., Veitch et al.,
2015). For the precessing spins, we assume the spin vectors to be isotropic on
the sphere and uniform in spin magnitude
$\chi_{i,\\{x,y,z\\}}\in[-0.99,0.99]$. For the non-precessing spins, we employ
an aligned-spin prior called “zprior,” described in Appendix A of Lange et al.
(2018); we use this alternative prior to enable comparison to precessing spins
as it is equivalent to the uniform spin magnitude prior after marginalizing
out non-aligned spins. For eccentricity, we adopted a uniform prior over the
range $e\in[0.0,0.6]$ for each event. Although the TEOBResumS-DALI model can
generate stable waveforms with eccentricities up to $e\sim 0.9$, issues with
the validity of these highly eccentric waveforms led to our choice to
constrain the prior range; this is further explained in Appendix A.

###  3.1 Masses

Since we carried out analyses with both TEOBReumS-GIOTTO and TEOBResumS-DALI,
we were able to make a direct comparison of the effect of the inclusion of
eccentricity on each binary’s component masses. For mosts events, the inferred
mass parameters do not differ significantly depending on whether an analysis
allowed for or excluded eccentricity. The only two examples of the mass
parameters changing is GW150914 and GW190521. The right panels of Figures 1
and 2 shows the inferred 90% credible intervals for the joint $M_{\rm
source}-q$ distribution, derived from inference with both TEOBResumS-GIOTTO
and TEOBResumS-DALI. In the GW150914 case, the $M_{\rm source}$ distributions
are noticeably different with TEOBResumS-DALI slightly shifted toward larger
masses while the $q$ distributions remain largely consistent. In GW190521
case, the $M_{\rm source}$ distributions are quite consistent in shape and
support, while the $q$ distributions show modest differences in their shape
and credible interval with TEOBResumS-GIOTTO shifted towards more unequal
masses.

To quantify the relative change in parameter $x$ between the two analyses, we
define $\epsilon_{x}=|\mu_{1}-\mu_{2}|/\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}$,
where $\mu_{k},\sigma_{k}$ are the posterior mean and standard deviation of
$x$ for models $1$ (here, eccentric) and $2$ (here, non-eccentric)
respectively. Between the events included in this study, we found the the
largest relative shift for the total mass and mass ratio to be
$\epsilon_{M}=0.472$ from GW150914 (See the right panel of Figure 1) and
$\epsilon_{q}=0.432$ from GW190521 (See the right panel of Figure 2)
respectively.

(a)  (b)

Figure 1:  GW150914: Corner plots of 2D and 1D marginal posteriors of $M_{\rm
source},q,e,S_{\rm hu}$ using TEOBResumS-GIOTTO in black and TEOBResumS-DALI
in blue. The left panel shows only the analysis using TEOBResumS-DALI and
includes eccentricity in the corner plot along with masses and spin
parameters. The right panel includes both the TEOBResumS-DALI and TEOBResumS-
GIOTTO analyses including only the masses and spin parameters. The contours
represent the 90% confidence intervals for each joint distribution. When
eccentricity is included, it shifts the posterior of the masses and spins.

(a)  (b)

Figure 2:  GW190521: Corner plots of 2D and 1D marginal posteriors of $M_{\rm
source},q,e,S_{\rm hu}$ using TEOBResumS-GIOTTO in black and TEOBResumS-DALI
in blue. The left panel shows only the analysis using TEOBResumS-DALI and
includes eccentricity in the corner plot along with masses and spin
parameters. The right panel includes both the TEOBResumS-DALI and TEOBResumS-
GIOTTO analyses including only the masses and spin parameters. The contours
represent the 90% confidence intervals for each joint distribution. When
eccentricity is included, it shifts the posterior of the mass ratio.

(a)  (b)

Figure 3:  GW190620: Corner plots of 2D and 1D marginal posteriors of $M_{\rm
source},q,e,S_{\rm hu}$ using TEOBResumS-GIOTTO in black and TEOBResumS-DALI
in blue. The left panel shows only the analysis using TEOBResumS-DALI and
includes eccentricity in the corner plot along with masses and spin
parameters. The right panel includes both the TEOBResumS-DALI and TEOBResumS-
GIOTTO analyses including only the masses and spin parameters. The contours
represent the 90% confidence intervals for each joint distribution. When
eccentricity is included, it slightly shifts the posterior of the masses and
spins.

(a)  (b)

Figure 4:  GW190706: Corner plots of 2D and 1D marginal posteriors of $M_{\rm
source},q,e,S_{\rm hu}$ using TEOBResumS-GIOTTO in black and TEOBResumS-DALI
in blue. The left panel shows only the analysis using TEOBResumS-DALI and
includes eccentricity in the corner plot along with masses and spin
parameters. The right panel includes both the TEOBResumS-DALI and TEOBResumS-
GIOTTO analyses including only the masses and spin parameters. The contours
represent the 90% confidence intervals for each joint distribution. When
eccentricity is included, it slightly shifts the posterior of the masses and
spins.

(a)  (b)

Figure 5:  GW190929: Corner plots of 2D and 1D marginal posteriors of $M_{\rm
source},q,e,S_{\rm hu}$ using TEOBResumS-GIOTTO in black and TEOBResumS-DALI
in blue. The left panel shows only the analysis using TEOBResumS-DALI and
includes eccentricity in the corner plot along with masses and spin
parameters. The right panel includes both the TEOBResumS-DALI and TEOBResumS-
GIOTTO analyses including only the masses and spin parameters. The contours
represent the 90% confidence intervals for each joint distribution. When
eccentricity is included, it shifts the posterior of the mass ratio.

###  3.2 Spins

Similar to Section 3.1, we can also directly compare the analyses to examine
how the inclusion of eccentricity affects the recovery of the spin parameters.
For our Figures, we use the spin parameter $S_{\rm hu}$ (Healy & Lousto
(2018)) and is defined as:

$M^{2}S_{\rm
hu}=\left((1+\frac{1}{2q})\vec{S}_{1}+(1+\frac{1}{2}q)\vec{S}_{2}\right)\cdot\hat{L}.$
(5)

The $S_{\rm hu}$ parameter is similar to the widely used $\chi_{\rm eff}$;
however, $S_{\rm hu}$ describes the leading order effect of hangup on the full
NR waveforms.

The inferred spin and mass-spin distributions are frequently quite consistent
in shape and support when using a model including eccentricity with only
GW150914 being the outlier having a $\epsilon_{S_{\rm hu}}=0.834$.
Interestingly, the inclusion of eccentricity shifts the spin parameter
posterior to more positive values in the case of GW150914. Unsurprisingly few
finely-tuned accidents occur, so binaries with notable shifts in inferred
masses usually also have notable shifts in inferred spin, and vice-versa.

###  3.3 Eccentricity

Despite the impact of eccentricity on other parameters, our inferred
eccentricity distributions are frequently quite modest. For example, the
inferred distribution of eccentricity for GW150914 strongly supports a non-
eccentric origin. Even for GW190521, where the inferred posterior eccentricity
peaks near $e\simeq\,0.2$, the posterior distribution contains significant
support for $e\simeq 0$. A Bayes factor analysis performed both by direct
integration and by the Savage-Dickey ratio suggests these two events are most
likely non-eccentric, with Bayes’ factors of 0.153 and 0.174, respectively. In
fact, as Table 1 shows, none of the events strongly prefer the eccentric
hypothesis over the non-eccentric hypothesis. Our conclusions about these
events are reasonably consistent with prior studies using _non-precessing_
binaries, which generally find at best modest evidence for eccentricity when
precession is not allowed. While on the surface our numerical Bayes factors
for these events are in notable tension with the results of Romero-Shaw et al,
we highlight the substantial systematic and methodological differences
associated with our approach using a different eccentric model with a wider
prior range and a different starting frequency. While a direct apples-to-
apples comparison between these results would need a conversion between the
two models’ definition of eccentricity, a closer comparison would require
similar run settings. Such a conversion has been investigated in Knee et al.
(2022), who developed a framework to translate between the different
definitions of eccentricity in the eccentric waveform models SEOBNRE and
TEOBResumSGeneral.

While not fully understood, it is expected that eccentricity could mimic
precession. To ensure the evidences presented here are not mistaken for
evidences of misaligned spins, we conduct analyses of GW150914, GW190521,
GW190620, GW190706, and GW190929 using the precessing waveform model
TEOBResumSP-GIOTTO; this allows us to perform a comparison between the
eccentric and precession hypotheses. The posterior distributions obtained with
all three models, along with a discussion of the results from the precession
analyses, can be found in Appendix B.

In addition to the eccentric vs. non-eccentric Bayes’ factors $\text{BF}_{E}$,
Table 1 contains the Bayes’ factors $\text{BF}_{P}$ of the quasi-circular,
spin-precessing hypothesis, calculated with the waveform model TEOBResumSP-
GIOTTO, against the spin-aligned hypothesis, calculated with the waveform
model TEOBResumS-GIOTTO, for GW150914; GW190521; GW190620; GW190706; and
GW190929. We have also included the relative Bayes’ factors $\text{BF}_{E/P}$
of the eccentric hypothesis, calculated with the waveform model TEOBResumS-
DALI, against the precessing hypothesis. Our results for $\text{BF}_{P}$
indicate a marginal preference for the spin-precession hypothesis for
GW150914, GW190706, and GW190929; furthermore, our results for
$\text{BF}_{E/P}$ indicate a marginal preference for the spin-precession
hypothesis for all of the events. However, none of the events exhibit a strong
preference for either hypothesis, so we cannot conclusively state whether any
of these events are eccentric, precessing, or manifest both eccentricity and
spin-precession.

(a)  (b)

Figure 6:  1D marginal distribution of eccentricity: Each distribution is the 1D marginal distribution of eccentricity for each event: GW150914, GW190521, GW190620, GW190706, GW190929 (this is the same as one of the histograms in the left panels of Figures 1-5). The left panel is only of GW150914 due to the significantly different scaling between this and the other distributions. See Table 1 for the Bayes’ Factors comparing eccentric vs non-eccentric.  Table 1: Bayes’ factors for each event comparing evidences of eccentric vs. non-eccentric ($\text{BF}_{E}$), precessing vs. non-eccentric ($\text{BF}_{P}$), and eccentric vs. precessing ($\text{BF}_{E/P}$).  Event |  $\text{BF}_{E}$ |  $\text{BF}_{P}$ |  $\text{BF}_{E/P}$
---|---|---|---
GW150914 |  0.153 |  1.626 |  0.094
GW190521 |  0.174 |  0.492 |  0.357
GW190620 |  0.134 |  0.539 |  0.248
GW190706 |  0.326 |  1.088 |  0.299
GW190929 |  0.616 |  1.235 |  0.498
|  |  |

##  4 Conclusions

In this work, we presented a state-of-the-art model-based assessment of the
presence and impact of eccentricity in several promising BBH events, based on
the TEOBResumSGeneral model, including higher-order modes. We compare two
analyses with the same model that include and exclude eccentricity. While we
do see some shifts in the posterior when including eccentricity, we do not
find evidence in favor of eccentric dynamics for any of the events presented,
instead favoring a quasi-circular result.Our results are in tension
numerically with previously-presented results for several of these events.
Specifically we do not find evidence of eccentricity for either GW190521 or
GW190620; however, numerical differences are most likely due to the
differences in prior range and starting frequencies, and it could also reflect
systematics, as we adopted different waveforms and algorithms.

For example, Gayathri et al. (2022) found higher likelihoods for eccentric,
precessing systems compared to quasi-circular, precessing systems for
GW190521, using direct comparison to precessing numerical relativity
simulations; by contrast, our analysis assumes spin-orbit alignment and finds
the full parameter posterior. Likewise, Gamba et al. (2021) argued that
GW190521 is highly eccentric based on an analysis with TEOBResumSGeneral,
excluding higher-order modes, starting from an initially unbound configuration
at extremely low initial frequencies; by contrast, our analysis exclusively
uses only the bound, later-time evolution (and thus higher starting frequency
and corresponding lower eccentricity) and the higher-order-mode form of that
model. Similarly, Romero-Shaw et al. (2021) found evidence of eccentricity for
GW190620 and GW190521 based off analyses that used a re-weighting technique
with the eccentric model SEOBNRE, a model that excludes higher-order modes,
using a different configuration with different priors and initial frequencies;
by contrast, our analysis again uses a starting frequency of 5 Hz and higher-
order-mode form of a different model as well as a larger uniform-in-e range.
To do a fair comparison, one would need to use the same settings of one of the
above analyses.

In future work, we plan to analyze the rest of the public events available
from the LVK (Abbott et al., 2023, 2021b). We plan to include the same three
configurations as in this paper: the non-eccentric, non-precessing case; the
eccentric, non-precessing case; and the non-eccentric, precessing case to make
sure any evidence of eccentricity is not mixing up evidence of precession.

The authors thank Juan Caldron Bustillo and Aaron Zimmerman for useful
feedback and conversations. We are thankful to the waveform developers of
TEOBResumSGeneral, specifically Alessandro Nagar and Sebastiano Bernuzzi for
their help with implementing their model. The git hash of TEOBResumSGeneral
used was bc4fcd0f0d97f9f0351e11c2e880ab0a6422ac20 for non-eccentric and
39e6d7723dacb23220ff5372e29756e5f94cb004 for eccentric I.B., V.G., and S.B.
acknowledges the support of the National Science Foundation under grants
PHY-1911796 and PHY-2110060, and the Alfred P. Sloan Foundation. HLI, JL, AJ,
RN, DS, and RV thanks NSF PHY-2114581, PHY-2207780 and XSEDE TG-PHY120016. G.V
acknowledges the support of the National Science Foundation under grant
PHY-2207728. ROS is supported by NSF PHY-2012057, PHY-1912632, and
AST-1909534. This material is based upon work supported by NSF’s LIGO
Laboratory which is a major facility fully funded by the National Science
Foundation. This research has made use of data or software obtained from the
Gravitational Wave Open Science Center (gw-openscience.org), a service of LIGO
Laboratory, the LIGO Scientific Collaboration, the Virgo Collaboration, and
KAGRA. LIGO Laboratory and Advanced LIGO are funded by the United States
National Science Foundation (NSF) as well as the Science and Technology
Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS),
and the State of Niedersachsen/Germany for support of the construction of
Advanced LIGO and construction and operation of the GEO600 detector.
Additional support for Advanced LIGO was provided by the Australian Research
Council. Virgo is funded, through the European Gravitational Observatory
(EGO), by the French Centre National de Recherche Scientifique (CNRS), the
Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef,
with contributions by institutions from Belgium, Germany, Greece, Hungary,
Ireland, Japan, Monaco, Poland, Portugal, Spain. The construction and
operation of KAGRA are funded by Ministry of Education, Culture, Sports,
Science and Technology (MEXT), and Japan Society for the Promotion of Science
(JSPS), National Research Foundation (NRF) and Ministry of Science and ICT
(MSIT) in Korea, Academia Sinica (AS) and the Ministry of Science and
Technology (MoST) in Taiwan. We are grateful for computational resources
provided by the Leonard E Parker Center for Gravitation, Cosmology and
Astrophysics at the University of Wisconsin-Milwaukee. We acknowledge the use
of IUCAA LDG cluster Sarathi for the computational/numerical work. We are also
grateful for the computational resources provided by California Institute of
Technology at Pasadena, California as well as the LIGO Livingston Observatory.

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##  Appendix A Waveform model

The TEOBResumS-DALI model employed in the main text can generate waveforms
over a large portion of the parameter space, including systems with large
eccentricities ($e\sim 0.9$). However, extreme care should be applied when
such regions are explored for systems with large total detector frame mass.
Indeed, if $M$ is large and the initial (reference) frequency is high, the
initial separation $r_{0}$ at which the EOB model will begin to evolve the
dynamics of the system may be close to the last stable orbit (LSO) of the
binary. In this scenario, bodies evolving along eccentric orbits may plunge
almost immediately, completing very few orbital cycles and – as a consequence
– emitting very similar waveforms irrespectively of the starting input
eccentricity. A clear example of this can be seen in Fig. 7, where we display
the waveforms and EOB dynamics obtained with the parameters corresponding to
the two posterior peaks of the GW190706 analysis. To avoid a multi-modal
eccentricity estimation that is a direct consequence of evolving the system
with separation close to its LSO, we lowered the initial reference frequency
in our analysis from 18 Hz to 5 Hz.

(a)

(b)

Figure 7:  GW190706 waveform and dynamics: Waveforms (left) and EOB dynamics
(right) corresponding to the two eccentricity peaks of the GW190706 analysis
($e_{0}=0.1$, $e_{0}=0.4$). Because of the low radial separation at which the
evolution of the system is started, the waveforms emitted by the two systems
look remarkably similar, in spite of the rather different values of input
eccentricity.

##  Appendix B Precession analysis

We perform quasi-circular, spin-precessing analyses of the events GW150914,
GW190521, GW190620, GW190706, and GW190929, using the precessing waveform
model TEOBResumSP-GIOTTO. For the precessing spins, we assume the spin vectors
to be isotropic on the sphere and uniform in spin magnitude
$\chi_{i,\\{x,y,z\\}}\in[-0.99,0.99]$. Figures 8, 9, 10, 11, and 12 show
posterior distributions for mass and spin parameters calculated from all three
analyses for our five selected events. Across all five events, the posteriors
for the mass and spin parameters using the spin-precessing model are fairly
consistent in shape and support when compared to the posterior distributions
generated using the aligned spin model.

Figure 8: GW150914 (all analyses): A corner plot of 2D and 1D marginal
posteriors of $M_{\rm source},q,S_{\rm hu}$ using TEOBResumS-GIOTTO in black,
TEOBResumSP-GIOTTO in red, and TEOBResumS-DALI in blue. The contours represent
the 90% confidence intervals for each joint distribution. When eccentricity is
included, it shifts the posterior of the masses and spins. Figure 9: GW190521
(all analyses): A corner plot of 2D and 1D marginal posteriors of $M_{\rm
source},q,S_{\rm hu}$ using TEOBResumS-GIOTTO in black, TEOBResumSP-GIOTTO in
red, and TEOBResumS-DALI in blue. The contours represent the 90% confidence
intervals for each joint distribution. When eccentricity is included, it
shifts the posterior of the mass ratio. Figure 10: GW190620 (all analyses): A
corner plot of 2D and 1D marginal posteriors of $M_{\rm source},q,S_{\rm hu}$
using TEOBResumS-GIOTTO in black, TEOBResumSP-GIOTTO in red, and TEOBResumS-
DALI in blue. The contours represent the 90% confidence intervals for each
joint distribution. When eccentricity is included, it slightly shifts the
posterior of the masses and spins. Figure 11: GW190706 (all analyses): A
corner plot of 2D and 1D marginal posteriors of $M_{\rm source},q,S_{\rm hu}$
using TEOBResumS-GIOTTO in black, TEOBResumSP-GIOTTO in red, and TEOBResumS-
DALI in blue. The contours represent the 90% confidence intervals for each
joint distribution. When eccentricity is included, it slightly shifts the
posterior of the masses and spins. Figure 12: GW190929 (all analyses): A
corner plot of 2D and 1D marginal posteriors of $M_{\rm source},q,S_{\rm hu}$
using TEOBResumS-GIOTTO in black, TEOBResumSP-GIOTTO in red, and TEOBResumS-
DALI in blue. The contours represent the 90% confidence intervals for each
joint distribution. When eccentricity is included, it shifts the posterior of
the mass ratio.


# Quantum Mechanics in Pseudotime

A. K. Kapoor
School of Basic Sciences
Indian Institute of Technology, Bhubneswar
Bhubaneswar 751007, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Based on some results on reparmetrisation of time in Hamiltonian path integral
formalism, a pseudo time formulation of operator formalism of quantum
mechanics is presented. Relation of reparametrisation of time in quantum with
super symmetric quantum mechanics is established. We show how some important
concepts such as shape invariance and tools like isospcetral deformation
appear in pseudo time quantum mechanics.

##  1 Introduction

The classical trajectory of a point particle can be given in terms a
parameter, other than time, to be called pseudotime in this paper. A
Lagrangian formulation, called homogeneous formalism [1] is available and
leads to a singular Lagrangian. A Hamiltonian formulation can be written down
using Dirac’s treatment of such systems. Quantisation of such a system can
then proceed by setting the fundamental commutator brackets equal to $i\hbar$
times the Dirac brackets. A relativistic particle, string theory, general
relativity constitute important examples which are manifestly invariant under
a reparametrisation of time[2]. Many investigations of reparametrisation in
quantum mechanics start with the homogeneous formalism of classical mechanics
and use Dirac quantisation.

Non relativistic point particle is not mainfestly invariant under a
reparametrisation of time. However it found important applications in the path
integral formulation of quantum mechanics[3]. Duru and Kleinert [4] formulated
reparamterisation of time within the Hamiltonian path integral approach and
used it as a tool, along with K-S transformation[5], to obtain a path integral
solution for H atom problem. Following this idea several authors used
Lagrangian as well as Hamiltonian path integral to write path integral
solution for several exactly solvable potential models.

In [6] a scheme of setting up Hamiltonian path integral (HPI) was proposed.
Later quantisation scheme in arbitrary coordinates using reparametrised
Hamiltonian path integral (RHPI) and it was found that correct quantisation
could be carried out without a need to add $O(\hbar^{2})$ terms [7]. Further
results on RHPI and applications have been given in [8].

In this paper we wish to formulate reparametrisation of time in the operator
formalism of quantum mechanics closely following the ideas of
reparametrisation in the path integral approach. The central results, that we
need from earlier works, are Duru Kleinert formula and operator ordering.
These are summarised in Sec.3.

Path integral representation derives its importance from the fact that is it
can be used as scheme of quantisation too. It is this fact that will be of
crucial importance to us in this paper.

In this article we are concerned with reparametrisation in operator formalism
of nonrelativistic quantum mechanics. We will connect work on Hamiltonian path
integral with some well known results in quantum mechanics of one dimensional
systems. It should then be possible to extend those results to higher
dimensional systems.

The aim of this paper is to propose a pseuto-time formalism of quantum
mechanics and to link available results of Hamiltonian path integrals to
corresponding results in the operator formalism and is organised as follows.
The homogeneous formulation of classical mechanics of a point particle is
briefly described in the next section. Section 3 summarises results about
using reparametrisation of time in the Hamiltonian path integral approach. In
next section several know results is established with pseudotime quantum
mechanics. The reparamterisation of time appears to be intimately related to
supersymmetric qunatum mechanics (SUSYQM)[9]. We to establish a connection of
shape invariance, isospcetral deformation [10] with pseudo-time formalism. The
quantum Hamilton Jacobi equation(QHJ)[11], [12] is related to pseudo time
quantum mechanics in a simple and straight forward manner. A connection with
exceptional polynomials [13] is briefly mentioned. In Sec. 5 the mathematical
structure of quantum theory in pseudo-time is presented. Several possible
different interpretations of the mathematical formulation of pseudo-time
quantum mechanics are outlined in Sec.6. Directions for possible further study
and concluding remarks are given the last section of this paper.

##  2 Reparametrisation in classical mechanics

In this section we briefly describe the homogeneous formalism[1]. In classical
mechanics the trajectory of a point particle is found by solving the
Hamilton’s equations for generalised coordinates and momenta as functions of
time.

$q=q(t),\quad p=p(t),\qquad t_{1}\leq t\leq t_{2}.$ (1)

These are parametrised curves in phase space. The trajectories can also be
specified by choosing another parameter $\sigma$, to be called pseudo-time. In
this description the coordinates and momenta are to be solved as functions of
$\sigma$ and the answer is to be written as

$q=q(\sigma),\quad p=p(\sigma),\qquad\sigma_{1}\leq\sigma\leq\sigma_{2}.$ (2)

To connect with (1), we require another equation of the form

$t=t(\sigma),\qquad t(\sigma_{1})=t_{1},t(\sigma_{2})=t_{2}.$ (3)

treating $t$ like a coordinate. For this purpose, one adds an equation for
evolution of $t$ in pseudotime

$\frac{dt}{d\sigma}=f(q(t))$ (4)

where the function $f$ is a positive function of coordinates $q$ and will be
called local time scaling function (LTSF). A Hamiltonian formalism can be set
up by using the pseudo Hamiltonian

${\mathscr{H}}_{E}=f(q)(H-E)$ (5)

The pseudo energy is a constant of motion for dynamics in pseudo time
$\sigma$. The Hamilton’s equations in pseudo time, using pseudo
Hamiltonian(5), give the dynamics correctly when the pseudo energy is set
equal to zero. It may be noted that setting pseudo energy equal to zero means
using the energy conservation equation $H(q,p)-E=0$. Thus classical equations
of motion can be set up equally well in pseudotime without changing physical
content.

##  3 Reparametrisation in path integral formalism

In 1979 Duru and Kleinert used K-S transformation and Hamiltonian path
integral representation in pseudo time to arrive at an exact path integral
solution for H-atom problem [4]. Crucial to their derivation was a formula
which we call Duru Kleinert formula, see (6). This formula related the path
integral in time $t$ to Hamiltonian path integral in pseudo time, and was
derived by Duru and Kleinert by means of formal manipulations. It was
therefore a fortunate circumstance that it worked for H atom.

Later the same approach was employed and relation between path integrals in
time $t$ and pseudo time $\sigma$ was carefully derived using the accepted
rules for path integrals within the time slicing approach[14]. In general,
terms of order $\hbar^{2}$ were required to be added to the potential when
writing setting up Duru-Kleinert formula. A number of authors used
Hamiltonian, as well as Lagrangian, path integral formulations to obtain exact
solution for the propagator for many problems [15]. It may be recalled that
previous to the work by Duru and Kleinert, only a small class of quantum
mechanical problems could be solved using the path integral representation
[16]. Use of reparametrisation opened the way to obtain path integral solution
for large class of problems known to be exactly solvable.

It may be mentioned that reparametrisation, articles on the path integral
approach used a few other techniques, notably addition of new degrees of
freedom, for arriving at the solutions of potential problems in quantum
mechanics.

It has been known that a careful point transformation, and also use of
Hamiltonian path integral to quantise a system in non-Cartesian coordinates
require addition of $O(\hbar^{2})$ terms to the classical Hamiltonian. It is
to be noted that the same problem also appears when we attempt to use
canonical quantisation directly in the non-Cartesian coordinates [6]. In a
series of papers on Hamiltonian path integral quantisation, it was shown that
$O(\hbar^{2})$ terms were not required if one suitably that combined
reparametrisation with the Hamiltonian path integral. Thus reparamterisation
of time seems to be intimately tied to point transformations in Hamiltonian
path integrals.

We recall a few central points and quote a few results from a previous study
of Hamiltonian path integral quantisation within time slicing approach.

### Hamiltonian path integral

Given a classical Hamiltonian $H(q,p)$, a scheme of constructing a particular
path integral has been suggested and investigated in [6]. Using a particular
form, $(qt\\!\parallel\\!q_{0}t_{0})$, for short time propagator, Hamiltonian
path integral (HPI) $K_{H}(qt,q_{0}t_{0})$ was defined as summation over all
paths from $q_{0}$ at time $t_{0}$ to $q$ at time $t$. However this did not
lead to correct Schrodinger equation in non Cartesian coordinates.

In [7] another path integral representation was introduced by means for Duru-
Kleinert formula

${\mathscr{K}(qt,q_{0}0)}\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{-\infty}^{+\infty}\left(\frac{dE}{2\pi\hbar}\right)\exp(-iEt/\hbar)\int_{0}^{\infty}[\sqrt{f(q)f(q_{0})}]\times
K_{{\mathscr{H}}_{E}}(q\sigma,q_{0}0)d\sigma.$ (6)

Here the expression$K_{{\mathscr{H}}_{E}}(q\sigma,q_{0}\sigma_{0})$, appearing
in the right hand side is the HPI constructed using the pseudo Hamiltoinian
${\mathscr{H}}_{E}=f(q)(H-E)$ corresponding to the LTSF function $f(q)$ and
will be called reparametrised Hamiltonian path integral (RHPI). It may be
noted that RHPI reduces to HPI for the special case of LTSF $f(q)=1$.

The ${\mathscr{K}(qt,q_{0}0)}$ defined by (6) will be called DK propagator
and, to simplify the notation, its dependence on Hamiltonian $H(q,p)$ and LTSF
$f(q)$ will not be shown explicitly .

We will now give a brief summary of main results obtained earlier.

### Operator ordering

Here we list the operator ordering implicit in the above discussion for the
special case of potential problems in $n$\- dimension. Corresponding
statements for more general case can be written down but will not be required
here. The Hamiltonian function $H({\bf x,p})$ 111Here x denotes Cartesian
coordinates will be assumed to be of the form.

$H({\bf x,p})=\frac{{\bf p}^{2}}{2m}+V({\bf x})$ (7)

As already remarked the HPI obeys the correct Schrödinger equation and the
corresponding operator is

$\widehat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V({\bf x}).$ (8)

The RHPI $K_{{\mathscr{H}}_{E}}$, satisfies the Schrödinger equation with
pseduo Hamiltonian operator $\widehat{\mathscr{H}}_{E}$ given by

$\widehat{\mathscr{H}}_{E}=\frac{1}{2m}{\bf\nabla}f({\bf x}){\bf\nabla}+f({\bf
x})\Big{(}V({\bf x})-E)\Big{)}.$ (9)

### Normalisation of RHPI $K_{{\mathscr{H}}_{E}}$:

In [8] is was shown that the path integral RHPI $K_{{\mathscr{H}}_{E}}$
appearing in the right hand side of (6) satisfies the Schrödinger equation

$i\hbar\frac{d}{dt}K_{{\mathscr{H}}_{E}}({\bf x}t,{\bf
x}_{0}t_{0})=\widehat{H}_{E}K_{{\mathscr{H}}_{E}}({\bf x}t,{\bf x}_{0}t_{0})$
(10)

and has the normalisation

$\lim_{t\to t_{0}}K_{{\mathscr{H}}_{E}}({\bf x}t,{\bf x}_{0}t_{0})=(f({\bf
x}))^{-1}\delta^{(n)}({\bf x}-{\bf x}_{0}).$ (11)

Thus it is seen that out that the path integral RHPI constructed with pseudo
Hamiltonian is has a normalisation different from what is required. This
explains appearance factor $\sqrt{f({\bf x})f({\bf x}_{0})}$ in the right hand
side of Duru-Kleinert formula.

### DK Propagator:

$\widetilde{H}_{E}=\frac{1}{\sqrt{f}}\left(\frac{1}{2m}{\bf\nabla}f({\bf
x}){\bf\nabla}+f({\bf x})\Big{(}V({\bf x})-E\Big{)}\right)\frac{1}{\sqrt{f}}$
(12)

Writing this last expression as $\widetilde{H}_{E}\equiv\widetilde{H}-E$, the
expression for $\widetilde{H}-E$ can be rearranged in alternate forms

$\displaystyle\widetilde{H}$ $\displaystyle=$
$\displaystyle\frac{1}{f}\frac{1}{2m}(\nabla-{\mathbf{w}})f({\bf
x})(\nabla-{\mathbf{w}})+V({\bf x})$ (13) $\displaystyle=$
$\displaystyle\frac{1}{2m}(\nabla+{\bf w})(\bf\nabla-{\bf w})+V({\bf x})$ (14)
$\displaystyle=$ $\displaystyle\frac{1}{2m}\nabla^{2}+V(x)+\Delta V$ (15)

where $w_{k}=\partial_{k}\Omega({\bf x}),\quad\Omega(x)=\frac{1}{2}\ln f({\bf
x})$ and

$\displaystyle\Delta
V=\frac{\hbar^{2}}{2m}\Big{\\{}(\nabla\Omega(x))^{2}-\nabla^{2}\Omega(x)\Big{\\}}$
(16)

The DK propagator ${\mathscr{K}({\bf x}t,{\bf x}_{0}t_{0}})$ of (6) obeys the
Schrödinger equation with $\widetilde{H}$ as the Hamiltonian

$i\hbar\frac{d\mathscr{K}({\bf x}t,{\bf
x}_{0}t_{0})}{dt}=\widetilde{H}{\mathscr{K}({\bf x}t,{\bf x}_{0}t_{0})}$ (17)

and is normalised as

${\mathscr{K}({\bf x}t,{\bf x}_{0}t_{0})}|_{t=t_{0}}=\delta({\bf x}-{\bf
x}_{0}).$ (18)

### Qunatisation in arbitrary coordinates:

Identifying HPI $K_{H}$ as propagator for the corresponding quantum problem
leads to correct quantisation scheme in Cartesian coordinates. However the
same scheme, when used for quantisation in non-Cartesian coordinates, did not
give correct quantisation scheme. It was became necessary to add
$O(\hbar^{2})$ terms to the Hamiltonian[6].

This problem of appearance of $O(\hbar^{2})$ terms is not specific to the
scheme that was used in reference [6]. A need for these extra $O(\hbar^{2})$
terms has been well known in the Hamiltonian path integral literature. In fact
the problem reappears even in canonical qunatisation scheme. In a general case
this difficulty of canonical qunatisation is masked by ordering problems and
can be seen most clearly in polar coordinates in two dimensions, a model not
having any ordering problem for the Hamiltonian.[6, 3].

It was demonstrated in reference [7], that the DK-propagator with the choice,
$f(q)=\rho(q)$ as LTSF function, leads to correct quantisation in arbitrary
coordinates without need to add any $O(\hbar^{2})$ terms. This scheme worked
with the classical Hamiltonian $H(q,p)$ directly in arbitrary coordinates;
setting up a Hamiltonian path integral in Cartesian coordinates and changing
variables was not required.

##  4 Connection of RHPI with operator formalism

In this section we establish relationship of some of the important concepts in
operator formalism and DK propagator.

### Supersymmetric quantum mechanics

Supersymmetric quantum mechanics has been an active area of research for
several decades and the concept of shape invariance continues to attract a
great deal of attention. It is easy to see the place that supersymmetric
partners and shape invariance have in the pseudotime path integral framework.
Starting from a free particle Hamiltonian $V(x)=0$ to set up a DK propagator,
scaling functions $f(x)$ and $1/f(x)$ lead to quantum system whose time
evolution is governed by Hamiltonian $H_{+}$ and $H_{-}$, respectively which
are given by

$H_{\pm}=\frac{p^{2}}{2m}+\frac{\hbar^{2}}{2m}(w^{2}\mp w^{\prime}),$ (19)

where $w=\frac{d\Omega}{dx}$ and $\Omega=\ln f$.

$\displaystyle H(q,p)\stackrel{{\scriptstyle f}}{{\longrightarrow}}\text{DK
Propagator for }H_{+}$ (20) $\displaystyle H(q,p)\stackrel{{\scriptstyle
1/f}}{{\longrightarrow}}\text{DK Propagator for }H_{-}$ (21)

In the terminology of SUSYQM, the above two Hamiltonians will be recognised as
supersymmetric partners with $w(x)$ playing the role of superpotential.

### Darboux transformation

The Darboux transformation [17] gives a relation between the eigenfunctions of
SUSY partner Hamiltonians $H_{\pm}$. The propagators for $H_{\pm}$ are both
related to free particle HPI in pseudo time with scaling functions $f(x)$ and
$1/f(x)$ respectively, see (19)-(20).

The Darboux transformation and its generalisations by Crum and by Krien are
powerful results that have found large number of applications to several areas
including exactly integrable models. Darboux’s result is implicitly contained
in Duru Kleinert formula. It is of interest to establish at a direct and
explicit correspondence between Darboux transformation and DK formula.

### Isospectral deformation of a potential

The isospectral deformation of a potential $V(x)$ generates a new potential
having exactly same spectrum as the original potential $V(x)$. This process
makes use of results from SUSYQM.

Consider a model with potential $V(x)$ corresponding to superpotential $w(x)$.
We now consider a two RHPI with $H(q,p)$ as free particle with certain scaling
functions $f(x)$ and $1/f(x)$. This will result in DK-propagator for
potentials $V_{\pm}(x)$. Demanding that $V_{+}(x)$ coincide with $V(x)$, a
solution for $f(x)$ will lead to $V_{-}(x)$ which will be the isospectral
deformation of the original potential $V(x)$. The steps for arriving at the
required solution for $f(x)$ will closely follow the steps known in the
literature for the isospectral deformation and no further explanation is
required.

### Exceptional Polynomials

In an an earlier paper [18], a systematic procedure for deformation of radial
oscillator potential was given in the framework of QHJ was presented. It was
found that demanding shape invariance be preserved under the deformation, led
to the isospectral shift of the radial oscillator potential. All these steps
can in principle be translated and followed in the pseudotime formalism as
presented here. Starting directly form the differential equation for classical
orthogonal polynomials and using using pseudo time framework, to will be
interesting to find a direct route to the exceptional polynomials.

### Time dependent sypersymmetry

Time dependent supersymmetry and time dependent Darboux transformation have
been studied[19]. These studies will be connected with reparametrisation of
time with LTSF which is a function of time, $\frac{dt}{d\sigma}=f(q,t)$. The
basic equations in the Hamiltonian path integral formalism will then have
correspondence with equations of [19].

### Quantum Hamilton Jacobi equation

Quantum Hamilton Jacobi formalism provides a scheme of computing energy
eigenvalues without solving for wave functions [11] and has been studied
extensively [12] If we substitute $\psi(x)=\exp(iS(x)/\hbar)$ the Schrödinger
equation for a potential problem $V(\bf x)$

$-\frac{\hbar^{2}}{2m}\nabla^{2}\psi({\bf x})-V(x)\psi({\bf x})=E\psi({\bf
x}).$ (22)

gets transformed into an equation for $S({\bf x})$

$\frac{1}{2m}(\nabla S({\bf x})\big{)}^{2}+\frac{i\hbar}{2m}\nabla^{2}S({\bf
x})+V({\bf x})-E=0.$ (23)

This equation is known as the quantum Hamilton Jacobi (QHJ) equation for
potential $V({\bf x})$. Consider the Schrodinger equation

$-\frac{\hbar^{2}}{2m}\nabla^{2}\psi({\bf x})+V(x)\psi({\bf x})=E\psi({\bf
x}).$ (24)

If we set up DK propagator and demand that LTSF $f({\bf x})$ be such that
$\psi({\bf x})=$ constant is a solution of Schrodinger equation for
$\widetilde{H}$, we get the following equation for the scaling function
$f({\bf x})$

$\frac{\hbar^{2}}{2m}\big{[}(\nabla\Omega(x))^{2}-\nabla^{2}\Omega(x)\big{]}+V({\bf
x})-E=0.$ (25)

Identifying $\Omega({\bf x})$ with $iS({\bf x})/\hbar$, the above equation
becomes identical with the QHJ equation (23).

##  5 General structure of quantum mechanics in pseudotime

In this section we will work with non-Cartesian coordinates coordinates and
canonical momenta ${\bf q,p}$ in place of Cartesian coordinates. The symbol
$\rho({\bf q})$ will denote the volume element defined by $d{\bf x}=\rho({\bf
q})d{\bf q}$. The states of system will be elements of Hilbert space
$\mathcal{H}$ of all square integrable functions $f({\bf q})$:

$\int|\langle q|\psi\rangle|\rho(q)\,dq<\infty$

Given a system with classical dynamics governed by Hamiltonian function
$H({\bf q,p})$ and a scaling function $f({\bf q})$, introduce pseudotime
$\sigma$ and pseudo Hamiltonian ${\mathscr{H}}_{E}$

$\displaystyle\frac{dt}{d\sigma}$ $\displaystyle=$ $\displaystyle f({\bf q}),$
(26) $\displaystyle{\mathscr{H}}_{E}$ $\displaystyle=$ $\displaystyle f({\bf
q})(H({\bf q,p})-E)=f({\bf q})({\mathscr{H}}-E),$ (27) $\displaystyle\text{
where }{\mathscr{H}}$ $\displaystyle=$ $\displaystyle f({\bf q})H({\bf q,p}).$
(28)

We will work in the Heisenberg picture. Therefore the propagator will be given
by $\langle qt|q_{0}t_{0}\rangle$, where $|qt\rangle$ is eigenvector of the
position operator $\hat{q}(t)$ at time $t$:

$\hat{q}_{k}(t)|{\bf q}t\rangle=q_{k}|{\bf q}t\rangle,k=1,2,\ldots.$ (29)

A path integral representation for propagator is constructed out of short time
propagator $({\bf q}t|{\bf q}t_{0})$ in the usual fashion by using time
slicing approach and summing over all paths.

We recall (6) and the comment on normalisation of RHPI. It has been noted that
the HPI in real time $t$ and RHPI in pseudo time $\sigma$ are normalised
differently, and that one needs to multiply by factor $\sqrt{f({\bf x})f({\bf
x}_{0})}$ at the end. We introduces a second set of position eignevectors
${\mathscr{B}}=\\{{|\bf q\succ}\\}$ defined by

${|\bf q\succ}=(f({\bf q}))^{-1/2}|q\rangle$

which are normalised differently:

$\prec\bf q|\bf q_{0}\succ=(f(\bf q))^{-1}\rho^{-1}({\bf
q})\delta^{(n)}(q-q_{0}).$ (30)

The completeness relations now take the form

$\displaystyle\int\rho({\bf q})|\bf q\rangle\langle\bf
q|dq=\hat{I},\qquad\int\rho(\bf q)f(\bf q){|\bf q\succ}{\prec\bf q|}=\hat{I}.$
(31)

The new eigenvectors of position ${|\bf q\succ}$ span a different Hilbert
space ${\mathcal{H}}_{f}$ with changed scalar scalar product
$\langle\psi|\phi\rangle$ defined as:

$\int\psi({\bf q})\phi({\bf q})\rho({\bf q})\,d{\bf
q}\longrightarrow\int\psi({\bf q})\phi({\bf q})\rho({\bf q})f({\bf q})\,d{\bf
q}.\ $ (32)

Thus we interpret the change to basis ${|\bf q\succ}$ as a switch to different
Hilbert space of functions square integrable with a new measure $\rho(q)f(q)$.

The process of setting up the RHPI with LTSF $f({\bf q})$ can now be regarded
as consisting of following steps.

Noting that $\langle\bf q|\psi\rangle\in{\mathcal{H}}$ and $\prec\bf
q|\psi\succ\in{\mathcal{H}}_{f}$ representing a state vector $|\psi\rangle$ in
the two Hilbert spaces are related by

$\prec{\bf q}|\psi\succ=(f({\bf q}))^{-1/2}\langle\bf q|\psi\rangle,$ (33)

we define a mapping of the operators in the two Hilbert spaces by

${\mathscr{X}}_{f}=(f({\bf q}))^{-1/2}\,\hat{X}\,(f({\bf q}))^{1/2}.$ (34)

The vector space equations will then be preserved when a transition is made
from the original Hilbert space to another one labelled by $f({\bf q})$.

If $\hat{X}$ is written as an ordered expression $\hat{X}\big{(}{\bf
q},\frac{\partial}{\partial\bf q}\big{)}$, the above equation translates into

${\mathscr{X}}_{f}=X({\bf q},{\bf D})$ (35)

where

$D_{k}=\partial_{k}+\omega_{k}\quad\text{and}\quad\omega_{k}=\frac{1}{2}\frac{\partial\,(\ln
f)}{\partial q_{k}}.$ (36)

Setting up of the the propagator as RHPI and using DK propagator can now be
interpreted as a sequence of the following steps:

  1. 1.

Change from original Hilbert space ${\mathcal{H}}$ to a another Hilbert space
${\mathcal{H}}_{f}$ labelled by the LTSF $f({\bf q})$, and defined by a
different square integrability requirement

$\int|\psi({\bf q})|^{2}\rho({\bf q})dq<\infty\longrightarrow\int|\psi({\bf
q})|^{2}\rho({\bf q})f({\bf q})d{\bf q}<\infty.$ (37)

  2. 2.

Set up the propagator in the new Hilbert space ${\mathcal{H}}_{f}$ as a path
integral RHPI in pseudotime $\sigma$.

  3. 3.

Revert back to original Hilbert${\mathcal{H}}$ space by using DK propagator
for time evolution in time $t$.

##  6 Interpretations of pseudotime quantum mechanics

So far we have looked at the mathematical structure of quantum theory as
suggested by Hamiltonian path integral in pseudo time. Now we discuss a few
different possible interpretations of our equations in a manner which
incorporates reparametrisation of time in quantum theory.

The classical formulations in different pseudo times are all equivalent in the
sense that they give rise to the same trajectory. In quantum theory they
solutions of dynamical equations look different for different choices of
pseudo times because of appearance of $O(\hbar^{2})$ terms appearing in the
potential. For example a free particle in real time $t$ will, in pseudo time,
appear as a particle moving in a non constant potential $\Delta V$.

Simplest interpretation would of our equations will be that, for a particular
choice of Hamiltonian $H({\bf q,p})$, only DK propagator is to be regarded as
having physical significance. The RPHI is introduced solely for technical
purposes for setting up the propagator and that it appears in intermediate
steps of quantisation and evaluation of the propagator using path integrals. A
possibility along these lines is that for each choice of coordinates ${\bf
q}$, there is one ( or more ) preferred choice(s) of LTSF $f({\bf q})$
determined by the requirement $\Delta V=0$, as is indicated by quantisation in
arbitrary coordinates.

Another way to interpret the equations and $\Delta V$ term is that the quantum
mechanical equations are not invariant under the reparametrisation of time.
The DK propagator for different LTSF function $f({\bf q})$ will give the same
results if we subtract a $\Delta V$ term from the potential while writing the
pseudo Hamiltonian. This would not be a desirable feature for theories which
have reparametrisation invariance built in the classical theory and the
symmetry needs to be preserved in the quantum theory too.

In a third approach to interpretation, one can view the quantum theory as
fundamental and interactions being linked in some way to choice of physical
time, or to a particular frame of reference. In this case one would still need
to discover a manifestly covariant form of equations in quantum theory. [21]

### A gauge covariant formalism for pseudo time dynamics:

There is yet another approach to setting up equations in pseudo time. The
states and operators representing dynamical variables transform according to
the rule in (33) and (34). In coordinate representation this is like a
nonunitary, but invertible, gauge transformation

$\psi(x)\to\psi_{f}(x)=\exp(\Omega(x))\psi(x),$ (38)

where $\Omega$ is a real function of $x$.

Recall that we are free to choose the overall normalisation of the wave
function in any fashion we like, $\psi(x)$ and $N\psi(x)$ give rise to same
physics. The transformation (38) amounts to gauging this freedom in choice of
normalisation constant. This, in fact, is close to the original suggestion of
Hermann Weyl and was used by him in a different context and with a different
motivation.[22]

Thus one can think of introducing a gauge field required to maintain the new
gauge invariance. Such a ’potential field’ can always be introduced as a
mathematical construction to do book keeping and the potential having
different values in different gauges (pseudo times). Whether this ’field’ can
be assigned dynamical properties can only decided by further investigation and
confirmation of its existence by experiments.

##  7 Concluding remarks

It is obvious that SUSYQM has initmate connection with reparametrisaton of
time in quantum mechanics. The tower of SUSY partners correspond to choices
LTSF $f^{n}(x)$ for different values of $n$. It has been shown in this paper
that this connection becomes transparent when one uses the Hamiltonian path
integral formalism of quantum mechanics. Since the path integral approach is
not restricted to one dimension, it offers a possibility of formulating SUSYQM
in higher dimensions. In [20] the concept of shape invariance has been
extended to quantum mechanics in arbitrary dimensions. However, a
generalisation of a few other results such as intertwining of partner
potentials, is needed in order to carry forward work done in SUSYQM one
dimension. Darboux transformation is powerful tool and has a wide ranging
applications in different areas of mathematical physics and exactly solvable
models. It will be useful to further study connection of Darboux
transformation with RHPI.

An area where results of our work on reparamterisation will be useful is to
the systems where the action has singularities [21] and standard path integral
needs to be regularised. Working in pseudotime, with a suitable choice of
LTSF, can be used to regularize singularities. Use of path integral to
quantise and shifting to operator formalism should makes it technically easier
as compared to using the path integral formalism alone.

One of the important approaches to reparametrisation is using the homogeneous
formalism and applying Dirac quantisation. The Dirac canonical quantisation
fixes commutators, leaving questions related to ordering open. The Hamiltonian
path integral quantisation approach followed here is powerful framework and
goes beyond canoincal quantisation. It determines an operator ordering too.
Besides fixing operator ordering, the formalism proposed here gives a way of
relating results of pseudotime quantum mechanics with those in time $t$; an
explicit formula, DK formula, has been written down to relate the time
evolutions in time $t$ and pseudotime. Though the ordering scheme depends on
the way Hamiltonian path integral is set up, the scheme of [7, 8] appears to
useful for establishing connections with SUSYQM and other areas in the
literature. Further investigation of RHPI and it correspondence with operator
formalism appears to be promising for a study of several areas of exactly
solvable systems and systems with reparametrisation invariance.

Acknowledgement: I thank Pankaj Sharan for fruitful discussions.

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