# Search for gravitational wave signals from known pulsars in LIGO-Virgo O3
data using the 5n-vector ensemble method

Luca D’Onofrio<EMAIL_ADDRESS>INFN, Sezione di Roma, I-00185 Roma,
Italy  Rosario De Rosa  Università di Napoli “Federico II”, I-80126 Napoli,
Italy  INFN, Sezione di Napoli, I-80126 Napoli, Italy  Cristiano Palomba
INFN, Sezione di Roma, I-00185 Roma, Italy  Paola Leaci  INFN, Sezione di
Roma, I-00185 Roma, Italy  Università di Roma ”Sapienza”, I-00185 Roma, Italy
Ornella J. Piccinni  Institut de Fisica d’Altes Energies (IFAE), The Barcelona
Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona)
Spain  Valeria Sequino  Università di Napoli “Federico II”, I-80126 Napoli,
Italy  INFN, Sezione di Napoli, I-80126 Napoli, Italy  Luciano Errico
Università di Napoli “Federico II”, I-80126 Napoli, Italy  INFN, Sezione di
Napoli, I-80126 Napoli, Italy  Lucia Trozzo  INFN, Sezione di Napoli, I-80126
Napoli, Italy  Jim Palfreyman  University of Tasmania  James W. McKee  E.A.
Milne Centre for Astrophysics, University of Hull, Cottingham Road, Kingston-
upon-Hull, HU6 7RX, UK  Centre of Excellence for Data Science, Artificial
Intelligence and Modelling (DAIM), University of Hull, Cottingham Road,
Kingston-upon-Hull, HU6 7RX, UK  Bradley W. Meyers  International Centre for
Radio Astronomy Research (ICRAR), Curtin University, Bentley WA 6102 Australia
Ingrid Stairs  Dept. of Physics and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, BC V6J 2B1 Canada  Lucas Guillemot
Laboratoire de Physique et Chimie de l’Environnement et de l’Espace,
Université d’Orléans/CNRS, F-45071 Orléans Cedex 02, France  Observatoire
Radioastronomique de Nançay, Observatoire de Paris, Université PSL, Université
d’Orléans, CNRS, 18330 Nançay, France  Ismaël Cognard  Laboratoire de Physique
et Chimie de l’Environnement et de l’Espace, Université d’Orléans/CNRS,
F-45071 Orléans Cedex 02, France  Observatoire Radioastronomique de Nançay,
Observatoire de Paris, Université PSL, Université d’Orléans, CNRS, 18330
Nançay, France  Gilles Theureau  Laboratoire de Physique et Chimie de
l’Environnement et de l’Espace, Université d’Orléans/CNRS, F-45071 Orléans
Cedex 02, France  Observatoire Radioastronomique de Nançay, Observatoire de
Paris, Université PSL, Université d’Orléans, CNRS, 18330 Nançay, France  LUTH,
Observatoire de Paris, PSL Research University, Meudon, France  Michael J.
Keith  Jodrell Bank Centre for Astrophysics, Department of Physics and
Astronomy, The University of Manchester, Manchester M13 9PL, UK  Andrew Lyne
Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The
University of Manchester, Manchester M13 9PL, UK  Chris Flynn  OzGrav-
Swinburne University of Technology, VIC 3166, Australia  Ben Stappers  Jodrell
Bank Centre for Astrophysics, Department of Physics and Astronomy, The
University of Manchester, Manchester M13 9PL, UK

###### Abstract

The 5n-vector ensemble method is a multiple test for the targeted search of
continuous gravitational waves from an ensemble of known pulsars. This method
can improve the detection probability combining the results from individually
undetectable pulsars if few signals are near the detection threshold. In this
paper, we apply the 5n-vector ensemble method to the O3 data set from the LIGO
and Virgo detectors considering an ensemble of 201 known pulsars. We find no
evidence for a signal from the ensemble and set a 95% credible upper limit on
the mean ellipticity assuming a common exponential distribution for the
pulsars’ ellipticities. Using two independent hierarchical Bayesian
procedures, we find upper limits of $1.2\times 10^{-9}$ and $2.5\times
10^{-9}$ on the mean ellipticity for the 201 analyzed pulsars.

††preprint: APS/123-QED

##  I Introduction

Continuous gravitational waves (CWs) signals are promising targets for the
LIGO [1] and Virgo [2] detectors defined by persistence and near-
monochromaticity over long time scales. Among the most plausible sources of
this type of signal there are spinning neutron stars with a non-axisymmetric
mass distribution [3] due to deformations from shear strains in the crust, or
from magnetic stresses [4].

The signal strain amplitude depends on the ellipticity, the physical parameter
that quantifies the mass distribution asymmetry with respect to the rotation
axis [5]. Different neutron star equations of state can allow different sizes
of deformations, i.e. of ellipticities, to be sustained in the star. The
detection of a CW signal could allow us to make physical inference about the
equation of state, and/or about the internal magnetic field strength [6].

According to the knowledge of the source parameters (sky position, rotation
parameters), different strategies can be adopted to search for a CW signal
[7]. The simplest, but also the most sensitive strategy, is the targeted
search [8]: using information about the spin evolution, astrometry, and binary
orbit gained from continued pulsar monitoring, it is possible to infer with
high accuracy the sky position and the rotation parameters including the
source spin frequency and the spin-down. Indeed, most pulsars are observed to
steadily spin-down due to the emission of electromagnetic waves (and,
possibly, also CWs). From the pulsar spin-down, it is possible to set a
theoretical upper limit to the strain amplitude, the so-called spin-down
limit, assuming that all of the rotation energy lost by the pulsar is
converted to gravitational wave (GW) energy [9].

At the source reference frame, the CW signal is almost monochromatic with
frequency proportional to the rotation frequency of the source, through a
factor which depends on the considered emission model [4]. At the detector,
the CW signal has a frequency and phase modulation mainly due to the Doppler
effect for the relative motion between the source and the Earth.

De-modulation techniques are useful for removing the phase modulation caused
by the spin-down/Doppler shift and hence to precisely unwind the apparent
phase evolution of the source. Once these corrections have been applied, a CW
signal would become monochromatic except for the sidereal modulation due to
the antenna patterns. This modulation depends on the detector position and
orientation on the Earth and it is used by the 5n-vector method to define a
detection statistic as well as to estimate the signal parameters [10, 11].

So far, no evidence for a CW signal has been found in LIGO and Virgo data. The
last targeted search from the LIGO-Virgo-Kagra (LVK) Collaboration [9], using
the data from the observing run O3 and analyzing 236 known pulsars, set 95%
credible upper limits on the amplitude and the ellipticity of each pulsar.

To improve the detection probability for the targeted search of CWs, different
strategies can be used considering an ensemble of individually undetectable
pulsars as in [12, 13, 14]. In this paper, we use the 5n-vector ensemble
method, described in [15]. This method defines an ensemble statistic, $T(k)$,
as the partial sum of the statistics of single pulsar, ranked for increasing
$p$-values.

The ensemble procedure is a rank truncation method that selects the top
ranking sources according to the significance of the corresponding individual
test. Using the statistics $T(k)$, we define a $p$-value as a function of $k$,
that is a $p$-value for the overall hypothesis of the presence of CWs from the
ensemble.

In case of no detection, we can set 95% credible upper limit on the global
parameter $\Lambda$ (defined in [15]) that describes the ensemble, and on the
mean ellipticity that returns information about the population properties
using two independent hierarchical procedures.

The paper is organized as follows. In Sec. II, we review the 5n-vector
ensemble method and describe the procedures to set upper limits in case of no
detection from the ensemble. In Sec. III, we describe the data set used for
the analysis while in Sec. IV, we show the results of the targeted search for
single pulsar using the 5n-vector method with the first application to pulsars
in binary system. In Sec. V, we report the results of the ensemble procedure
on three different sets. Since there is no evidence of a signal from the
ensemble, we set the upper limits, that are compared with the results of
previous similar searches. Conclusions are presented in Sec. VI.

##  II The 5n-vector ensemble method

The 5n-vector ensemble method [15] is a multiple test that combines the
effects of individually undetectable pulsars using the results of the single
pulsar analysis through the 5n-vector method.

The 5n-vector method [11] is a frequentist pipeline used in the LVK
Collaboration for CW searches. It is based on the splitting at the detector of
the expected GW frequency $f_{\text{gw}}$ in five components due to the
sidereal modulation of the Earth. The expected signal $h(t)$ can be written
using the polarization ellipse [10]:

$h(t)=H_{0}(H_{+}A_{+}(t)+H_{\times}A_{\times}(t))e^{i\Phi(t)}$ (1)

where $H_{+/\times}$ are the polarization functions for the plus and cross
components:

$H_{+}=\frac{\cos(2\psi)-j\eta\sin(2\psi)}{\sqrt{1+\eta^{2}}}$ (2)
$H_{\times}=\frac{\sin(2\psi)+j\eta\cos(2\psi)}{\sqrt{1+\eta^{2}}}\,.$ (3)

$\eta$ and $\psi$ are the polarization parameters [10], $A_{+/\times}$ are
linked to the antenna patterns while $\Phi(t)$ is the GW phase. The parameter
$\eta$ is defined as:

$\eta=-\frac{2\cos(\iota)}{1+\cos^{2}(\iota)}$ (4)

where $\iota$ is the angle between the star rotation axis and the line of
sight. The amplitude $H_{0}$ is linked to the classic amplitude $h_{0}$,
defined for example in [9], by:

$H_{0}=h_{0}\sqrt{\frac{1+6\cos^{2}(\iota)+\cos^{4}(\iota)}{4}}\,.$ (5)

The phase $\Phi(t)$ in Eq. 1 shows a time dependence due to different
phenomena that modulate in time the received signal frequency. Heterodine
corrections [16] are applied to account for the pulsar spin-down and Doppler
effect due to the Earth motion. The residual modulation is in the functions
$A_{+/\times}$, i.e. in the detector response, that depend on the 5
frequencies $f_{\text{gw}},f_{\text{gw}}\pm\Omega_{\oplus},f_{\text{gw}}\pm
2\Omega_{\oplus}$ where $\Omega_{\oplus}$ is the Earth’s sidereal angular
frequency.

The data 5-vector X and the signal template 5-vectors $\textbf{A}^{+/\times}$
are defined as the Fourier transforms of the data and of the template
functions $A_{+/\times}$ at the 5 frequencies where the signal power is split.
The data/template 5-vectors from each of the $n$ considered detectors are then
combined as described in Appendix A for a multidetector analysis.

The 5n-vector defines two matched filters between the data X and the signal
templates $\textbf{A}^{+/\times}$ vectors, used in order to maximize the
signal-to-noise ratio:

$\hat{H}_{+}=\frac{\textbf{X}\cdot\textbf{A}^{+}}{|\textbf{A}^{+}|^{2}}\qquad\text{and}\qquad\hat{H}_{\times}=\frac{\textbf{X}\cdot\textbf{A}^{\times}}{|\textbf{A}^{\times}|^{2}}$
(6)

The two matched filters are estimators [10] of the signal plus and cross
amplitudes. They are used to construct the detection statistic $S$:

$\small
S=\frac{|\textbf{A}^{+}|^{4}}{\sum_{j=1}^{n}\sigma_{j}^{2}\,T_{j}\,|\textbf{A}^{+}_{j}|^{2}}|\hat{H}_{+}|^{2}+\frac{|\textbf{A}^{\times}|^{4}}{\sum_{k=1}^{n}\sigma_{k}^{2}\,T_{k}\,|\textbf{A}^{\times}_{k}|^{2}}|\hat{H}_{\times}|^{2}\,.$
(7)

where $\sigma_{j}^{2}$ and $T_{j}$ are the variance of the data distribution
in a frequency band around $f_{gw}$ (usually few tenths of Hz wide) and the
observation time111Observation time refers to the amount of data; that is, the
time span not considering the periods where the detector was in NO-science
mode in the $j^{\text{th}}$ detector, respectively. In the case of Gaussian
noise, the distribution of the statistic $S$ in Eq. (7) is a Gamma
distribution $\Gamma(x;2,1)$.

Fixing the number $N$ of analyzed pulsars, the 5n-vector ensemble method ranks
these pulsars according to the $p$-values $P_{i}$ (with $i=1,...,N$),

$P_{i}=P(S_{i}\geq\overline{S}_{i}|\text{noise})$ (8)

computed from the $S_{i}$ noise distribution and the measured value
$\overline{S}_{i}$ for the $i^{\text{th}}$ pulsar. Then, the ensemble
statistic $T(k)$ is defined as the partial sum of the first $k$ largest order
statistics (statistics with the $k$ smallest p-values):

$T(k)=\sum_{j=N-k+1}^{N}S_{(j)}\,.$ (9)

From the statistics $T(k)$, we can compute the $p$-value of ensemble $P(k)$ as
a function of $k$:

$P(k)=P(T(k)\geq\overline{T}(k)|\text{noise})$ (10)

with

$\overline{T}(k)=\sum_{j=N-k+1}^{N}\overline{S}_{(j)}\,,$ (11)

and $\overline{S}_{(j)}$ is the $j^{\text{th}}$ largest measured value for the
detection statistic for a certain pulsar in the ensemble. For example, for an
ensemble of $N$ pulsars with $P_{j}\leq P_{i}\,\,\forall i$ and $P_{k}\geq
P_{i}\,\,\forall i$, the smallest and the largest order statistics values are
$\overline{S}_{(1)}\equiv\overline{S}_{k}$ and
$\overline{S}_{(N)}\equiv\overline{S}_{j}$, respectively.

We note that the p-value refers only to the null hypothesis and does not make
reference to or allow conclusions about any other hypotheses, such as the
alternative hypothesis. A small p-value (for example, below the 1%) can be
used to recognize interesting ensemble of pulsars as starting point for the
detection.

The noise and the signal distribution of the statistics $T(k)$ can be inferred
using a Monte Carlo procedure starting from the $N$ experimental noise and
signal distributions for the statistics of single pulsar, as described in
[15].

The $T(N)$ statistic coincides with the simple sum of the statistics $S_{i}$;
in case of Gaussian noise, $T(N)$ follows a $\Gamma(x;2N,1)$ distribution. The
$T(N)$ signal distribution can be also expressed in closed-form and it is
proportional to a non-central $\chi^{2}$ distribution with $4N$ degrees of
freedom and non-centrality parameter $\Lambda$ [15]:

$\Lambda=\sum_{i=1}^{N}H_{0,i}^{2}\,f_{i}(\eta_{i},\psi_{i})$ (12)

where the coefficient $f_{i}$ is:

$\small{f_{i}(\psi_{i},\eta_{i})=2\left(\frac{|\textbf{A}^{+}_{i}|^{4}\,|H_{+,i}|^{2}}{\sum_{j=1}^{n}\sigma_{j}^{2}T_{j}|\textbf{A}^{+}_{j,i}|^{2}}+\frac{|\textbf{A}^{\times}_{i}|^{4}\,|H_{\times,i}|^{2}}{\sum_{k=1}^{n}\sigma_{k}^{2}T_{k}|\textbf{A}^{\times}_{k,i}|^{2}}\right)}\,.$
(13)

Even though we have defined a $p$-value of ensemble as a function of $k$, the
statistical inference concerns the entire set of considered pulsars. The claim
of a rank truncation method is the same of the Fisher’s combination method
where all the individual tests are combined together: i.e. there are some
signals among all the $N$ tests, regardless that a certain $k<N$ led to a
rejection.

###  II.1 Upper limit procedures

For the single pulsar analysis, we can set the 95% credible upper limits on
the amplitude assuming uninformative uniform priors on $\psi$ (with
$-\pi/2\leq\psi\leq\pi/2$) and $\cos(\iota)$ (with $-1\leq\cos(\iota)\leq 1$)
that defines $\eta$ (see Eq. 4). Knowing the distance and assuming a fiducial
moment of inertia, we set the 95% credible upper limits on the ellipticity.

The upper limit for the ensemble procedure must be set on population
parameters since using a multiple test, we can not infer statistical
information about the single pulsar parameters.
Considering the statistic $T(N)$, i.e. the entire set of considered pulsars,
two different approaches can be used.

First, we can constrain the parameter $\Lambda$, defined in Eq. (12), that
fixes the $T(N)$ distribution considering a mixed Bayesian-frequentist
approach as in the case of single pulsar [18].
For the Bayes’ theorem, the posterior on $\Lambda$ is:

$P(\Lambda|\overline{T}(N))\varpropto L(\overline{T}(N)|\Lambda)\Pi(\Lambda)$
(14)

where $\overline{T}(N)$ is the measured value of the ensemble statistic,
$\Pi(\Lambda)$ is the prior while $L(\overline{T}(N)|\Lambda)$ is the
likelihood, that is estimated from the theoretical $T(N)$ signal distribution
evaluated at the value $\overline{T}(N)$ for different $\Lambda$ values.

The upper limit $\Lambda^{95\%}$ on the parameter $\Lambda$ is:

$\Lambda^{95\%}\,:\,\int_{0}^{\Lambda^{95\%}}P(\Lambda|\overline{T}(N))d\Lambda=0.95\,.$
(15)

$\Lambda$ is a global parameter for the ensemble and $\Lambda^{95\%}$ can be
used to show the improvement in the method sensitivity analyzing future runs.

Secondly, assuming a common distribution for the ellipticities, we can use a
hierarchical Bayesian framework to constrain the value of the hyperparameter
that fixes the assumed common distribution. Indeed, hierarchical Bayesian
inference allows to study population properties for the analyzed ensemble of
pulsars [19].

Let us consider an ensemble of $N$ pulsars and a common distribution fixed by
the hyperparameter $\mu_{\epsilon}$ for the set of ellipticities
${\epsilon_{i}}$ (with $i=1,..,N$).

We can constrain the value of the hyperparameter $\mu_{\epsilon}$ considering
the posterior probability density function inferred from a Bayesian procedure:

$P(\mu_{\epsilon}|\overline{T}(N))\varpropto
L(\overline{T}(N)|\mu_{\epsilon})\Pi(\mu_{\epsilon})$ (16)

where $\Pi(\mu_{\epsilon})$ is the prior distribution on the hyperparameter,
while the likelihood $L(\overline{T}(N)|\mu_{\epsilon})$ can be defined as the
value of the $T(N)$ signal distribution evaluated at $\overline{T}(N)$, if the
common distribution of the ellipticities is fixed by the hyperparameter
$\mu_{\epsilon}$. The $T(N)$ signal distribution and hence, the likelihood, is
fixed by $\Lambda$ in case of Gaussian noise. It follows that:

$L(\overline{T}(N)|\mu_{\epsilon})=\int
L(\overline{T}(N)|\Lambda)\Pi(\Lambda|\mu_{\epsilon})d\Lambda\,.$ (17)

From the definition in Eq. (12), deducing the analytical prior on $\Lambda$
for an assumed distribution of the ellipticities with hyperparameter
$\mu_{\epsilon}$ is not straightforward. For a large ensemble, $N\gg 1$,
$\Pi(\Lambda|\mu_{\epsilon})$ can be approximated with a Gaussian distribution
according to the Central Limit Theorem. The mean and variance of the Gaussian
distribution depend on the assumed distribution since it is the weighted sum
of the mean and variance for the squared $\epsilon_{i}$ random variables.

The upper limit $\mu^{(\Lambda)}$ is:

$\mu^{(\Lambda)}\,:\,\int_{0}^{\mu^{(\Lambda)}}P(\mu_{\epsilon}|\overline{T}(N))d\mu_{\epsilon}=0.95\,.$
(18)

Differently, we can also consider a classic hierarchical procedure combining
the results of the single pulsar analysis as in [12]. In this case, the
posterior probability density function on the hyperparameter $\mu_{\epsilon}$
is:

$P(\mu_{\epsilon}|\overline{S}_{1},..,\overline{S}_{N})\varpropto\left(\prod_{i=1}^{N}\int
L(\overline{S}_{i}|\epsilon_{i})\Pi(\epsilon_{i}|\mu_{\epsilon})d\epsilon_{i}\right)\Pi(\mu_{\epsilon})\,.$
(19)

This posterior is independent of the ensemble procedure (i.e. of the statistic
$T(N)$) and depends on the results of the single pulsar analysis through the
likelihoods $L(\overline{S}_{i}|\epsilon_{i})$. In addition, it does not
depend on the Gaussian noise hypothesis and it can be used to constrain the
hyperparameter $\mu_{\epsilon}$ using the likelihood
$L(\overline{S}_{i}|\epsilon_{i})$ computed for the $i^{\text{th}}$ pulsar
analysis.
The upper limit $\mu^{(S)}$ is

$\mu^{(S)}\,:\,\int_{0}^{\mu^{(S)}}P(\mu_{\epsilon}|\overline{S}_{1},..,\overline{S}_{N})d\mu_{\epsilon}=0.95\,.$
(20)

##  III Data set

In this section, we describe the data set used for the analysis. It consists
of the GW data set from the LIGO and Virgo detectors, and the electromagnetic
data set from the different observatories used to determine the pulsar
ephemerides.

###  III.1 Gravitational wave data

The O3 observing run started on 2019 April 1 (MJD: 58574.625) and ended on
2020 March 27 (MJD: 58935.708) for both the LIGO and Virgo detectors. O3
consisted of two parts, separated by a one month-long commissioning break. O3a
ran from 2019 April 1, 15:00 UTC until 2019 October 1, 15:00 UTC. O3b ran from
2019 November 1, 15:00 UTC, to 2020 March 27, 17:00 UTC.

The duty cycles for this run were 76%, 71%, and 76% for LIGO Livingston (LLO),
LIGO Hanford (LHO), and Virgo, respectively. The O3 data set is publicly
available via the Gravitational Wave Open Science Center [20].

In this paper, we use the Band Sampled Data (BSD) framework [16] that consists
of band-limited, down-sampled time series of the detector calibrated data,
called BSD files. This framework can be described as a database of sub-
databases where the BSD file is a complex time series that covers 10 Hz and
spans 1 month of the original data. Using the BSD libraries, it is possible to
extract frequency sub-bands or time sub-periods in a flexible way, reducing
the computational cost of the analysis.

###  III.2 Electromagnetic data

The set of analyzed pulsars is in Table LABEL:tab:long. We consider the
ephemerides used in [9].

The observatories which have contributed to the data set 222The ephemerides
and the rotational/orbital parameters may be obtainable on request at the
discretion of the individual EM pulsar groups from the above-mentioned
observatories. are: the Canadian Hydrogen Intensity Mapping Experiment (CHIME)
[22], the Mount Pleasant Observatory 26-m telescope, the 42-ft telescope and
Lovell telescope at Jodrell Bank, MeerKAT [23], the Nançay Decimetric Radio
Telescope, the Neutron Star Interior Composition Explorer (NICER) and the
Molonglo Observatory Synthesis Telescope (as part of the UTMOST pulsar timing
program) [24, 25]. Pulsar timing solutions were determined from this data
using Tempo [26] or Tempo2 [27, 28, 29] to fit the model parameters. The
ephemerides have been created using pulse time-of-arrival observations that
mainly overlapped with O3 observing period.

To estimate the theoretical spin-down limit or the upper limit for the
fiducial ellipticity for a certain source, the pulsars’ distances are needed
[9]. For many pulsars, the distance can be found in the ATNF Pulsar Catalog
[30]. These are distances mostly based on the observed dispersion measure and
estimated using the Galactic electron density distribution model YMW16 [31] or
the NE2001 model [32], although others are based on parallax measurements, or
inferred from associations with other objects or flux measurements.

The distances used for each pulsar are given in Table LABEL:tab:long. The
distances based on the dispersion measure are estimates with uncertainties
that can amount up to a factor of two [31]. Nearby pulsars, for which parallax
measurements are possible, have smaller distance uncertainties.

We select pulsars whose rotation frequency is greater than 10 Hz to match the
sensitivity band of the GW detectors. This leads to primarily targeting
millisecond pulsars and fast spinning young pulsars. In the analyzed set,
there are 165 millisecond pulsars with expected CW frequencies above 100 Hz.

Pulsars can experience glitches during the observation time (as for the Crab
pulsar during O3 [9]), i.e. sudden, impulsive increases in their spin [33].
The general procedure for glitching pulsars is to analyze each inter-glitch
period independently and then to sum the resulting statistics [9]. In this
analysis, we have not considered glitching pulsars; a dedicated study is
needed to optimize the detection probability when glitching pulsars are
present in the ensemble.

Figure 1: Strain amplitude as a function of the GW frequency for O3 data of
the LIGO and Virgo detectors. Blue stars are the upper limits inferred in [9]
using the Bayesian method for the 223 pulsars in Table LABEL:tab:long while
the red circles are the upper limits for the 5n-vector method. The continuous
pink line is the minimum detectable amplitude for the O3 run considering a
multidetector analysis.

##  IV Single pulsar analysis

In this section, we describe the results of the analysis of single pulsars
using the 5n-vector method. Differently from the analysis in [9], we use the
normalized single pulsar statistic $S$ defined in Eq. (7) using the weighted
multidetector extension described in Appendix A. For the first time using the
5n-vectors, we also consider pulsars in binary systems.

The results, reported in Table LABEL:tab:long, are obtained considering a
single harmonic emission, i.e. the expected GW frequency is assumed exactly
twice the rotation frequency of the source, and analyzing O3 data from the two
LIGO and Virgo detectors.

For each pulsar, we consider a small frequency band (at least 0.2 Hz and at
most 0.5 Hz wide) around the expected GW frequency. The frequency band
extracted from the BSD files is chosen considering the spin-down of the source
and in case of binary systems, considering also the Doppler effect due to the
orbital motion.

Using the heterodyne correction [16], we remove the spin-down and the Doppler
frequency modulation for the expected GW signal. The Doppler correction for
pulsars in binary systems follows the model described in [34]. We do not take
into account the uncertainty estimates in the binary parameters; the possible
effects of these uncertainties on the heterodyne correction is described in
the analysis in Appendix C.

The experimental noise distribution is inferred considering a range of off-
source frequencies as described in [11]. The measured value of the statistics
is compared with the noise distribution computing the $p$-value. Using the BSD
framework, the computational cost of the analysis is reduced to a few CPU-
minutes per source per detector.

In agreement with [9], there is no evidence of a CW signal from any pulsar in
O3 data. We set 95% credible upper limit on the amplitude $h_{0}$ for each
pulsar.

For the 5n-vector method since we are considering a different statistic, the
results do not exactly coincide with the results in [9] but there is a very
good agreement. For example, the upper limit on the amplitude for the pulsar
J0711-6830 in [9] is $5.0\times 10^{-27}$, while $4.8\times 10^{-27}$ in this
work.

As shown in Fig. 1, the upper limits from the 5n-vector method (red circles)
are in good agreement with the results of the Bayesian method (blue stars as
in [9]). The differences in the results can be due to the data frameworks that
are used in the two pipelines that entail different analyzed frequency bands
for each pulsar and different pre-processing of the data.

Figure 2: Summary plot of the results for the ensemble over 201 pulsars
considering O3 data of LLO, LHO and Virgo detectors. Top-left plot Fitted
shape (red triangles) and scale parameters (blue squares) using a Gamma
distribution to the experimental noise distribution for each pulsar as a
function of the emission frequency of the pulsar. Top-right plots Fitted shape
and scale parameters (dashed lines) with $95\%$ confidence level (coloured
area) for the $T(k)$ noise distributions for each $k$ inferred from the Monte
Carlo procedure. Bottom plot $p$-value of ensemble (blue line) compared with
the single pulsar $p$-values (red dots), ranked for increasing values.

##  V Ensemble analysis

In this section, we show the results of the 5n-vector ensemble method applied
to O3 data. From the set of pulsars in Table LABEL:tab:long, we select 201
pulsars excluding 22 binary pulsars according to the selection criteria in Eq.
48 described in Appendix C based on the uncertainties of the binary
parameters.

As described in [15], the improvement in the detection probability for the
5n-vector ensemble method depends on the number of analyzed pulsars $N$ and on
the power of the combined individual tests. In a real search, we can not
optimize $N$ since we do not know the strength of the signals a priori.

We fixed three different ensembles from the set in Table LABEL:tab:long: the
entire set for the three detectors, the set of millisecond pulsars
($f_{\text{gw}}>100$ Hz) for the LIGO detectors and the high-value pulsars set
for the LIGO detectors. High-value pulsars are the analyzed pulsars whose
ratio between the experimental upper limit on the amplitude $h_{0}^{95\%}$ and
the theoretical spin-down limit $h_{\text{sd}}$ (the so-called spin-down
ratio) is:

$h_{0}^{95\%}/h_{\text{sd}}\lesssim 1\,.$ (21)

We chose the three ensembles according to physical observations. Indeed,
millisecond pulsars are most likely to have undergone an accretion phase,
which could alter the structure of their crust and magnetic field compared to
non-recycled young pulsars [35], while high-value pulsars are promising
targets that have surpassed their spin-down limits.

The three ensembles correspond to fix $N=201$ for the first set, $N=146$ for
the millisecond pulsars set and $N=20$ for the high-value pulsars set.

After analyzing each pulsar singularly, we reconstruct the statistics $T(k)$
and compute a $p$-value of ensemble as a function of $k$. The results are
shown in the form of summary plot; the top-left plot shows the fitted shape
and scale parameters of a Gamma distribution for the experimental noise
distribution for the statistic $S$ as a function of the GW frequency, the top-
right plots show the fitted shape and scale parameters to the $T(k)$ noise
distributions as a function of $k$ while the bottom-plot compares the ordered
single pulsar $p$-values (red dots) with the ensemble $p$-values from the
$T(k)$ statistics (blue line).

Since there is no evidence of signal from the ensemble, we set 95% credible
upper limit on the global parameter $\Lambda$ and on the hyperparameter
$\mu_{\epsilon}$, that is the mean value of the assumed exponential
distribution for the ellipticities.

Figure 3: Summary plot of the ensemble composed by the millisecond pulsars in
Table LABEL:tab:long, considering O3 data and the two LIGO detectors. Top-left
plot Fitted shape (red triangles) and scale parameters (blue squares) using a
Gamma distribution to the experimental noise distribution for each pulsar as a
function of the emission frequency for the considered pulsar. Top-right plots
Fitted shape and scale parameters (dashed lines) with $95\%$ confidence level
(coloured area) for the $T(k)$ noise distributions for each $k$ inferred from
the Monte Carlo procedure. Bottom plot $p$-value of ensemble (blue line)
compared with the single pulsar $p$-value (red dots), ranked for increasing
values.

###  V.1 All pulsars, all detectors

The summary plot for the entire set of analyzed pulsars is shown in Fig. 2.
For this analysis, we consider the two LIGO and Virgo detectors.

A criterion based on noise is used to select the detectors for the multi-
detector analysis of single pulsar. Since the normalized statistic $S$, in the
hypothesis of Gaussian noise, is distributed according to a $\Gamma(x;2,1)$,
from the fitted parameters to the experimental noise $S$ distribution (top-
left plot in Fig. 2) we can easily check the Gaussianity in the selected
frequency band for each pulsar. For the multi-detector analysis, we consider
only the detectors that have fitted shape and scale parameters respectively in
the range $[1.6,2.4]$ and $[0.8,1.2]$ (that is a maximum difference of $20\%$
from the expected values for Gaussian noise). For each pulsar, this procedure
allows to exclude the detectors with large noise disturbances in the analyzed
frequency band. Indeed, experimental noise $S$ distributions that are very
different from the $\Gamma(x;2,1)$ can influence the Monte Carlo procedure for
the $T(k)$ noise distributions.

From the bottom-plot of Fig. 2, there is no evidence of a signal form the
ensemble. One pulsar has single pulsar $p$-value (red dot) below $1\%$ as
expected for a set of 201 pulsars. The $p$-value computed for the statistics
$T(k)$ are consistent with the noise hypothesis.

###  V.2 Millisecond pulsars, LIGO detectors

The summary plot for the millisecond pulsars is shown in Fig. 3. For this
analysis, we consider only the two LIGO detectors with the same noise criteria
introduced in the previous subsection, not considering the Virgo detector that
has a higher noise level. As shown in Appendix A, a multidetector analysis
does not necessarily entail a better sensitivity compared to the most
sensitive detector’s analysis.

As shown in the bottom plot in Fig. 3, there are three pulsars - J0824+0028,
J1544+4937, J1551$-$0658 - below the $1\%$ threshold. There is no evidence of
a signal for any of these pulsars (as for the Bayesian results in [9]); the
combination of the two LIGO detectors produces lower $p$-values with respect
to the three detectors case.

For example considering the pulsar J0824+0028, the $p$-values for the single-
detector single-pulsar analysis are 0.01, 0.04, 0.37 for LLO, LHO and Virgo
respectively. The multidetector analysis considering all the three detectors
entails a $p$-value of 0.01 while considering only the two LIGO detectors, the
$p$-value is 0.002.

The ensemble $p$-value has a minimum of almost $3\%$ for $k=4$ with no
evidence of a signal from the ensemble. The strong decrease for the first
values of $k$ is due to the presence of the three pulsars with $p$-values
below 1%.

Figure 4: Summary plot of the ensemble composed by the high-value pulsars with
spin-down ratio $\lesssim 1$ considering O3 data and the two LIGO detectors.
Top-left plot Fitted shape (red triangles) and scale parameters (blue squares)
using a Gamma distribution to the experimental noise distribution for each
pulsar as a function of the frequency for the considered pulsar. Top-right
plots Fitted shape and scale parameters (dashed lines) with $95\%$ confidence
level (coloured area) for the $T(k)$ noise distributions for each $k$ inferred
from the Monte Carlo procedure. Bottom plot $p$-value of ensemble (blue line)
compared with the single pulsar $p$-value (red dots), ranked for increasing
values.

###  V.3 High-value pulsars, LIGO detectors

There are 20 high-value pulsars in our dataset, that is there are 20 pulsars
with spin-down ratio $\lesssim 1$ (as in [9], excluding the glitching
pulsars).

The summary plot for the high-value pulsars is shown in Fig. 4. For this
analysis, we consider only the two LIGO detectors with the same noise criteria
introduced in the previous subsection.

As shown by the the blue line in the bottom plot of Fig. 4, the $p$-value of
the ensemble is fully consistent with the noise hypothesis.

###  V.4 Upper limits

For the upper limit computation, we consider the set of known pulsars analyzed
in Fig. 2 and the O3 dataset for the LIGO and Virgo detectors.

Using the statistic $T(N)$, that is the simple sum of the statistics for the
pulsars in the analyzed ensemble, we set 95% credible upper limit on the
parameter $\Lambda$ defined in Eq 12. The prior on $\Lambda$ is chosen uniform
while the likelihood is the value of the $T(N)$ signal distribution evaluated
at the measured value $\overline{T}(N)$. The posterior
$P(\Lambda|\overline{T}(N))$, shown in the left plot of Fig. 5, is compatible
with zero as expected for the noise-only case. The 95% credible upper limit
is:

$\Lambda^{95\%}=84.1\,.$ (22)

It is not possible to infer any information for the single pulsars from the
upper limit $\Lambda^{95\%}$ since $\Lambda$ is a global parameter for the
ensemble. $\Lambda^{95\%}$ can be used to constrain different global
parameters based on some assumptions for the single pulsar amplitudes. For
example, by assuming that each pulsar in the ensemble emits a CW signal with
the same amplitude $\overline{h}$, from the upper limit $\Lambda^{95\%}$ it
follows that $\overline{h}=1.9\times 10^{-27}$.

For hierarchical procedures, following the analysis in [12] and the stochastic
searches in [36, 37], we assume a common exponential distribution for the
ellipticities with the hyperprior $\Pi(\mu_{\epsilon})$ that is log-uniform
between $10^{-10}$ and $10^{-7}$. The posteriors
$P(\mu_{\epsilon}|\overline{T}(N))$ and
$P(\mu_{\epsilon}|\overline{S}_{1},..,\overline{S}_{N})$ are shown in the
right plot of Fig. 5.

Figure 5: Upper limits inferred from the ensemble procedure considering the
set of 201 pulsars in Figure 2 and considering the O3 data for the LIGO and
Virgo detectors. Left plot: posterior distribution for the global parameter
$\Lambda$ (defined in Eq. 12). The red point indicates the 95% credible upper
limit for $\Lambda$. Right plot: posterior distribution for the two
independent hierarchical procedures to set the upper limits on the mean
$\mu_{\epsilon}$ of the assumed exponential distribution. The continuous line
is the log-uniform hyperprior on $\mu_{\epsilon}$. The blue and yellow points
indicate the 95% credible upper limit for $\mu_{\epsilon}$.

The upper limit $\mu_{\epsilon}^{(\Lambda)}$ for the hierarchical procedure
using the $\Lambda$ parameter is

$\mu_{\epsilon}^{(\Lambda)}=2.5\times 10^{-9}\,.$ (23)

This upper limit procedure considers the theoretical signal distribution to
evaluate the likelihood. The theoretical distribution can be used in the
hypothesis of Gaussian noise in the analyzed frequency band for each pulsar.
This can be checked looking at the fitted Gamma parameters for the $T(N)$
statistic in the top-right plot of Fig. 2.

The upper limit $\mu_{\epsilon}^{(S)}$ for the hierarchical procedure using
the single pulsar results is

$\mu_{\epsilon}^{(S)}=1.2\times 10^{-9}\,.$ (24)

The results and also the posteriors in Fig. 5 can be compared with Figure 8 in
[12], where the authors analyzed 92 pulsars using data from the LIGO S6
science run showing the 90% upper limit on the mean of the assumed exponential
distribution of $3.8\times 10^{-8}$.

###  V.5 Discussion

As shown in Fig. 2-4 there is no evidence for a CW signal from the three
different considered ensemble. In the following, we report some comments on
the results found in this work.

  1. (1)

The upper limits set in this paper on the mean of the exponential distribution
for the set of analyzed pulsars are more than one order of magnitude below the
value in [12]. The combined O3 data for the LIGO and Virgo detectors are
largely more sensitive than the LIGO S6 science run data and the analyzed
ensemble is larger. A more reliable comparison between the two pipelines
should consider the same ensemble, i.e. the same pulsars, and is outside the
scope of this paper.

  2. (2)

The lowest limit on the ellipticity from the targeted search in [9] is
$3.2\times 10^{-9}$ for J0636+5129 while from the all-sky search in [38] is
$1.4\times 10^{-9}$ for a neutron star at $10\,$pc and $2047.5\,$Hz. The upper
limits on the mean ellipticity set in this work are consistent with these
values. We also note that these results approach the minimum limit on the
ellipticity for millisecond pulsars of $\approx 10^{-9}$ supposed in [39].

  3. (3)

The upper limits in [36, 37] on average ellipticity for the neutron star
population are $\mathcal{O}(10^{-8})$ from cross-correlation-based searches of
a stochastic gravitational wave background. These limits can not be directly
compared with the limits in this work since we focused on known pulsars that
represent only a small subset of the entire set of neutron stars in our
Galaxy.

  4. (4)

The upper limits on the hyperparameter for the exponential distribution do not
consider the uncertainties on the distance. The distance should be included as
a variable assuming, for example following [12], a Gaussian prior with a mean
given by each pulsar’s best fit distance, and a standard deviation of 20% of
that value with a hard cutoff at zero.

  5. (5)

The hierarchical procedures assume that all the analyzed pulsar ellipticities
are drawn from a common distribution that can be too simplistic to describe
the true $\epsilon$ distribution. For example, the population of young pulsars
and old recycled millisecond pulsars have undergone different evolution and so
have a different distribution. These populations should be treated
independently, or a more complex distribution that allows separation of the
two distributions should be used [12]. Therefore, a bimodal distribution, or
two independent exponential distributions with different mean values, could be
more appropriate and worth of study.

  6. (6)

Using the multidetector procedure, the results of single pulsar analysis in
Fig. 1 are in agreement with [9]. However, it is still not clear when a
multidetector analysis outperforms the most sensitive detector analysis. To
choose which detectors should be used in the multidetector case, criteria
based on the source sky position, on the detectors’ sensitivities and
observation times will be studied in the next future.

##  VI Conclusion

In this paper, we have described the application of the 5n-vector ensemble
method to O3 data for the LIGO and Virgo detectors considering a set of 201
known pulsars.

The single pulsar analysis can be seen as an hypothesis test where the null
hypothesis of pure noise is tested against the alternative hypothesis of the
presence of a CW signal in the data. To improve the detection probability, we
apply the 5n-vector ensemble method, a multiple test that combines multi-
detector single pulsar statistics defined through the 5n-vector method.

For the analysis of single pulsar, the single harmonic search (i.e.
gravitational wave frequency exactly twice the rotation frequency of the
source) on the 223 known pulsars shows good agreement with the Bayesian
results in [9]. For the first time, the 5n-vector method is applied to binary
systems for the targeted search and compared with the Bayesian results. The
dual harmonic search (GW frequency at both once and twice the rotation
frequency) is not implemented for the 5n-vector method. In Appendix C, we
describe a preliminary analysis to quantify the effect of the binary
parameters uncertainties on the detection sensitivity. We exclude from the
ensemble analysis 22 binary pulsars whose combined uncertainties on $t_{p}$
and $\omega$ do not respect the selection criteria in Eq. 48.

The ensemble statistics $T(k)$ for the rank truncation method is defined as
the partial sum of the $k$ largest order statistics to control the look-
elsewhere effect. Reconstructing the $T(k)$ noise distributions for each $k$,
using a Monte Carlo procedure, we compute the $p$-value of ensemble as a
function of $k$, that is a $p$-value for the overall hypothesis of the
presence of CWs from the ensemble.

In case of no detection, we propose different procedures to set upper limits
using the ensemble procedure and the statistic $T(N)$, i.e. the statistic for
the entire ensemble. Using a mixed frequentist-Bayesian procedure, we can
constrain the value of the global parameter $\Lambda$ that fixes the $T(N)$
signal distribution. Assuming a common exponential distribution for the
ellipticities, we propose two independent hierarchical procedures to set upper
limit on the mean $\mu_{\epsilon}$ of the assumed distribution.

Section V shows the application of the ensemble procedure to the set of 223
pulsars in Table LABEL:tab:long considering O3 data. Glitching pulsars, as the
Crab, are not considered in the single pulsar analysis and therefore, also in
the ensemble analysis. In future searches, glitching pulsars can be included
in the ensemble procedure considering the resulting statistic for the single
pulsar analysis, or considering each inter-glitch period as an independent
pulsar.

Three different ensembles are analyzed: the full set for the LIGO and Virgo
detectors, the millisecond pulsar set for the two LIGO detectors and the high-
value pulsars set for the two LIGO detectors. The results are shown in the
summary plots in Fig. 2, Fig. 3 and Fig. 4.

There is no evidence of a CW signal from the ensemble; the $p$-values as a
function of $k$ are well above the assumed 1% threshold.

The upper limits procedures are applied to the entire set of pulsars
considering the LIGO and Virgo detectors and O3 data. The posterior on the
$\Lambda$ parameter and on $\mu_{\epsilon}$ are shown in Fig. 5. The upper
limit set on $\Lambda$ is $\Lambda^{95\%}=84.1$ while the upper limit on
$\mu_{\epsilon}$ is $2.5\times 10^{-9}$ for the hierarchical procedure using
the $\Lambda$ parameter, and $1.2\times 10^{-9}$ for the hierarchical
procedure using the single pulsar results. These results are more than one
order of magnitude below the upper limit in [12], where the authors considered
a classic hierarchical Bayesian procedure on an ensemble of 92 pulsars and
data from the LIGO S6 science run.

The 5n-vector ensemble method improves the detection probability for the
targeted search of CWs from known pulsars. Application of this procedure on
the next observing runs will improve the possibility to detect a CW signal
from known pulsars for the first time.

## Acknowledgement

This research has made use of data obtained from the Gravitational Wave Open
Science Center (https://www.gw-openscience.org/ ), a service of LIGO
Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration.
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.
JWM gratefully acknowledges support by the Natural Sciences and Engineering
Research Council of Canada (NSERC), [funding reference #CITA 490888-16].
Pulsar research at UBC is supported by an NSERC Discovery Grant and by the
Canadian Institute for Advanced Research.
Pulsar research at UoM is supported by a Consolidated Grant from STFC.
The Nançay Radio Observatory is operated by the Paris Observatory, associated
with the French Centre National de la Recherche Scientifique (CNRS). We
acknowledge financial support from “Programme National de Cosmologie et
Galaxies” (PNCG) and “Programme National Hautes Energies” (PNHE) of CNRS/INSU,
France.
We acknowledge that CHIME is located on the traditional, ancestral, and
unceded territory of the Syilx/Okanagan people. We are grateful to the staff
of the Dominion Radio Astrophysical Observatory, which is operated by the
National Research Council of Canada. CHIME is funded by a grant from the
Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund (Project 31170)
and by contributions from the provinces of British Columbia, Québec and
Ontario. The CHIME/FRB Project, which enabled development in common with the
CHIME/Pulsar instrument, is funded by a grant from the CFI 2015 Innovation
Fund (Project 33213) and by contributions from the provinces of British
Columbia and Québec, and by the Dunlap Institute for Astronomy and
Astrophysics at the University of Toronto. Additional support was provided by
the Canadian Institute for Advanced Research (CIFAR), McGill University and
the McGill Space Institute thanks to the Trottier Family Foundation, and the
University of British Columbia. The CHIME/Pulsar instrument hardware was
funded by NSERC RTI-1 grant EQPEQ 458893-2014. This research was enabled in
part by support provided by WestGrid (www.westgrid.ca) and Compute Canada
(www.computecanada.ca).

##  Appendix A Weighted multidetector extension

Considering $n$ detectors, the classic 5n-vector definition, described in
[40], combines together the 5-vectors $\mathbf{X}_{i}$ from the
$i^{\text{th}}$ detector:

$\mathbf{X}=[\mathbf{X}_{1},...,\mathbf{X}_{n}]\,.$ (25)

Since the detectors can have different sensitivities, the statistic based on
the 5n-vector can reduce the signal to noise ratio compared to the statistic
based only on the 5-vector of the most sensitive detector.

In this paper, we define the weighted data 5n-vector as:

$\mathbf{X}=[c_{1}\mathbf{X}_{1},...,c_{n}\mathbf{X}_{n}]$ (26)

where the weights $c_{j}$ are defined as:

$c_{j}=\frac{\sqrt{n}}{\sqrt{\sum\limits_{i=1}^{n}\left(\frac{T_{i}}{S_{i}}\right)}}\sqrt{\frac{T_{j}}{S_{j}}}=\sqrt{\mathcal{H}}\,\sqrt{\frac{T_{j}}{S_{j}}}\,.$
(27)

$S_{j}$ and $T_{j}$ are the variance of the data in a small frequency band
around the expected GW frequency and the observation time in the
$j^{\text{th}}$ detector, respectively. $\mathcal{H}$ is the harmonic mean of
the time-weighted variances.

The matched filter is (similarly for $\hat{H}_{\times}$):

$\begin{split}\hat{H}_{+}&=\frac{\textbf{X}\cdot\textbf{A}^{+}}{|\textbf{A}^{+}|^{2}}=\frac{\sum\limits_{j=1}^{n}c_{j}\textbf{X}_{j}\,(\textbf{A}_{j}^{+})^{*}}{\sum\limits_{k=1}^{n}\textbf{A}_{k}^{+}\,(\textbf{A}_{k}^{+})^{*}}=\\\
&=\frac{1}{|\textbf{A}^{+}|^{2}}\left(c_{1}|\textbf{A}_{1}^{+}|^{2}\hat{H}_{+,1}+...+c_{n}|\textbf{A}_{n}^{+}|^{2}\hat{H}_{+,n}\right)\,.\end{split}$
(28)

In the hypothesis of Gaussian noise for the $j^{\text{th}}$ detector with
variance $\sigma_{j}^{2}$, the two complex estimators $\hat{H}_{+/\times}$
have also Gaussian distributions,

$\hat{H}_{+/\times}\sim Gauss\left(x;\,0,\sigma^{2}_{+/\times}\right)$ (29)

with

$\sigma^{2}_{+/\times}=\sum\limits_{j=1}^{n}\frac{(c_{j}\sigma_{j})^{2}\,T_{j}\,|\textbf{A}_{j}^{+/\times}|^{2}}{|\textbf{A}^{+/\times}|^{4}}\,.$
(30)

Using the $c_{j}$, we re-define the noise variance in each detector:

$(c_{j}\sigma_{j})^{2}\varpropto
c_{j}^{2}\,S_{j}=\frac{n\,T_{j}}{\sum\limits_{i=1}^{n}\left(\frac{T_{i}}{S_{i}}\right)}\,.$
(31)

If the observation time was the same ($\forall\,j,\,T_{j}=t$), this
corresponds to ”equalize” the noise in each detector. In this case, it follows
that:

$\begin{split}\sigma^{2}_{+/\times}&=\sum\limits_{j=1}^{n}\frac{(c_{j}\sigma_{j})^{2}\,T_{j}\,|\textbf{A}_{j}^{+/\times}|^{2}}{|\textbf{A}^{+/\times}|^{4}}=\sum\limits_{j=1}^{n}\frac{n\,t\,|\textbf{A}_{j}^{+/\times}|^{2}}{\sum\limits_{i=1}^{n}\left(\frac{1}{\sigma_{i}^{2}}\right)|\textbf{A}^{+/\times}|^{4}}=\\\
&=\frac{n\,t}{\sum\limits_{i=1}^{n}\left(\frac{1}{\sigma_{i}^{2}}\right)|\textbf{A}^{+/\times}|^{2}}=\frac{n\,t}{\sum\limits_{i=1}^{n}\left(\frac{1}{\sigma_{i}^{2}}\right)\sum\limits_{k=1}^{n}|\textbf{A}_{k}^{+/\times}|^{2}}\,.\end{split}$
(32)

Let us consider the toy case of $n$ co-located detectors where
$\,|\textbf{A}_{k}^{+/\times}|^{2}=|\textbf{A}^{+/\times}_{1}|^{2}\,,\forall\,k$;
the variances are

$\sigma^{2}_{+/\times}=\frac{t}{\sum\limits_{i=1}^{n}\left(\frac{1}{\sigma_{i}^{2}}\right)|\textbf{A}^{+/\times}_{1}|^{2}}\,.$
(33)

This corresponds to the case of one detector with observation time $t$ and
variance $V^{2}$:

$V^{2}=\frac{1}{\sum\limits_{i=1}^{n}\left(\frac{1}{\sigma_{i}^{2}}\right)}=\frac{\mathcal{H}}{n}$
(34)

where $\mathcal{H}$ is the harmonic mean of the variances. Since there is the
condition:

$min\\{\sigma_{1}^{2},...,\sigma^{2}_{n}\\}\leq\mathcal{H}\leq
n\,min\\{\sigma_{1}^{2},...,\sigma^{2}_{n}\\}$ (35)

this means that:

$\frac{min\\{\sigma_{1}^{2},...,\sigma^{2}_{n}\\}}{n}\leq\frac{\mathcal{H}}{n}\leq
min\\{\sigma_{1}^{2},...,\sigma^{2}_{n}\\}\,.$ (36)

It follows that for $n$ co-located detectors with the same observation time,
we always have an improvement in the detection sensitivity using the
coefficients $c_{j}$.

For example, let us consider the case of two co-located detectors $n=2$ (equal
to consider different datasets of the same detector), with
$\sigma_{2}^{2}=C\,\sigma_{1}^{2}$ and $C>1$:

$\sigma^{2}_{+/\times}=\frac{C\,\sigma_{1}^{2}\,t}{(C+1)\,|\textbf{A}^{+/\times}_{1}|^{2}}\,.$
(37)

The minimum detectable signal is $h_{min}$:

$h_{min}\varpropto\sqrt{\frac{C}{C+1}\frac{\sigma_{1}^{2}}{t}}\,.$ (38)

It is clear that in the general case (different detectors’ locations and
observation times), the multi-detector analysis not necessarily outperforms
the most sensitive detectors.

It is important to briefly describe the signal distribution for
$|\hat{H}_{+/\times}|^{2}$, see also [11]. If a CW signal is present into the
analyzed data, the distributions of the two complex estimators
$\hat{H}_{+/\times}$ are:

$\hat{H}_{+/\times}\sim
Gauss\left(x\,;\,H_{0}\,e^{j\gamma}\,H_{+/\times}\,M_{+/\times}\,,\,\sigma^{2}_{+/\times}\right)$
(39)

where $H_{+/\times}$ are the polarization functions, $H_{0}$ is the amplitude,
$\gamma$ the phase and

$M_{+/\times}=\sum\limits_{j=1}^{n}\frac{c_{j}\,|\textbf{A}_{j}^{+/\times}|^{2}}{|\textbf{A}^{+/\times}|^{2}}$
(40)

is a known factor for the targeted search.
$|\hat{H}_{+/\times}|^{2}$ have a non central-$\chi^{2}$ distribution (apart
from the factor $k_{+/\times}$):

$|\hat{H}_{+/\times}|^{2}\sim\frac{k_{+/\times}}{2}e^{-\frac{k_{+/\times}x+\lambda_{+/\times}}{2}}I_{0}(\sqrt{k_{+/\times}\lambda_{+/\times}x})$
(41)

where $I_{0}$ is the modified Bessel function of the first kind, and

$k_{+/\times}=\frac{2}{\sigma^{2}_{+/\times}}\qquad\lambda_{+/\times}=k_{+/\times}\,|H_{+/\times}|^{2}\,H_{0}^{2}\,M_{+/\times}\,.$
(42)

With respect to the classic definition of the 5n-vector, the distributions do
not change. There is just a re-definition of the variances for the detectors
noise. This means that using the coefficients $c_{j}$, it is changed how
$\lambda_{+/\times}$ depends on the detectors’ sensitivity.

##  Appendix B Test of the binary correction

Hardware injections are simulated signals added to the detectors data
physically displacing the detectors’ test masses [41]. Differential
displacement of the test masses mimics the detectors’ response to a GW signal.
Continuous hardware injections can be used to test CW analysis methods.

During the O3 run, $17$ hardware injections that simulate different CW signals
from spinning neutron stars were added in the two LIGO detectors. The list of
injected pulsar parameters can be found in [42].

In this section, we consider two different hardware injections to test the
5n-vector pipeline. We report the results for the injected pulsar P03 (SNR(1
yr) $\approx$ 30) that simulates an isolated spinning neutron stars with
$f_{\text{gw}}\simeq 108.8$ Hz and for P16 (SNR(1 yr) $\approx$ 68), a neutron
star in binary system with $f_{\text{gw}}\simeq 234.5$ Hz.

Figure 6: Amplitude spectral density for P03 and LLO data after the corrections for Doppler and spin-down effect. The right plot is the zoom around the GW frequency; the red dots show the expected 5 frequencies due to the Earth sidereal motion. Central peak frequency is the expected GW frequency. Figure 7: Amplitude spectral density for P16 and LLO data after the corrections for Doppler and spin-down effect. The right plot is the zoom around the GW frequency; the red dots show the expected 5 frequencies due to the Earth sidereal motion. Central peak frequency is the expected GW frequency. HI | Detector | $\epsilon_{H_{0}}$ | $\epsilon_{\eta}$ | $\epsilon_{\psi}$
---|---|---|---|---
P03 | LLO | 0.93 | 0.54% | 1.49%
LHO | 0.95 | 1.27% | 0.39%
P16 | LLO | 0.91 | 0.66% | -
LHO | 0.93 | 0.90% | -
Table 1:  Table of the parameters mismatch for the hardware injections (HI)
P03 and P16 analyzing O3 data from the two LIGO detectors. $\epsilon_{H_{0}}$
is the ratio between the estimated and injected amplitude, while
$\epsilon_{\eta}$ and $\epsilon_{\psi}$ are the normalized relative errors for
the polarization parameters

Figures 6 and 7 show the amplitude spectral density for LLO detector in the
frequency band around the expected GW frequency for P03 and P16 after the
Doppler and spin-down corrections using the BSD heterodyne method. The red
dots are the theoretical expected 5 frequency peaks for the GW signal due to
the sidereal modulation. The different ”heights” of corresponding peaks in the
two detectors depend on the antenna pattern, i.e. on the signal 5-vector
templates $\textbf{A}^{+/\times}$. It is important to note that the formalism
used to construct the hardware injections’ signal is independent from the
5-vector method.

In Table 1, there are the results obtained for the two hardware injections
considering the O3 datasets of the two LIGO detectors. The estimation of the
intrinsic parameters (see [10]) is not influenced by the choice of the
detection statistic since it depends only on the estimators
$\hat{H}_{+/\times}$.

The small discrepancies (below 10%) in Table 1 for the amplitude fall within
the uncertainties of the actuation system used for the injections. For P16,
the $\psi$ parameter is not well defined, since $\eta\approx 1$ and the signal
is nearly circularly polarized.

Figure 8: Uncertainties for the orbital parameters provided by the T2 model.
a) Relative errors on $t_{p}$ and $\omega$ as a function of the eccentricity.
b) Relative errors on $P$ and $a_{p}$ as a function of the eccentricity. c)
1-$\sigma$ errors on the eccentricity as a function of the eccentricity
values.

##  Appendix C Orbital parameters uncertainties

In this Appendix, we describe a preliminary analysis on the possible effects
of the uncertainties on the heterodine de-modulation correction in the
5-vector context. We also describe the selection criteria used in this paper
to include pulsars in binary systems for the ensemble procedure.
Indeed, large orbital parameters uncertainties could affect the sensitivity of
targeted searches since large mismatches on the binary parameters could reduce
the signal-to-noise ratio.
The heterodine correction is applied by multiplying the data by the
exponential factor $e^{-i\Phi_{corr}(t)}$. The total signal phase correction
$\Phi_{corr}(t)$ can be written as the sum of the spin-down and the Doppler
contribution. The Doppler contribution is the sum of the terms due to the
Earth motion and due to the pulsar orbital motion $\Phi_{bin}(t)$. For the
linear phase model and the small-eccentricity limit, $e\ll 1$ (details in
[34]):

$\small{\Phi_{bin}(t)\approx 2\pi
f\left[-a_{p}\left(\sin\psi(t)+\frac{k}{2}\sin 2\psi(t)-\frac{\beta}{2}\cos
2\psi(t)\right)\right]}$ (43)

where $a_{p}$ is the semi-major axis whike $k$ and $\beta$ are the Laplace-
Lagrange parameters:

$k\equiv e\cos(\omega)\qquad\beta\equiv e\sin(\omega)$ (44)

The mean orbital phase is

$\psi(t)=\Omega(t-t_{asc})$ (45)

measured from the time of ascending nodes $t_{asc}$. For small eccentricity
and mean orbital angular velocity $\Omega=\frac{2\pi}{P}$ ($P$ is the orbital
period), $t_{asc}$ is:

$t_{asc}=t_{p}-\frac{\omega}{\Omega}$ (46)

$t_{p}$ and $\omega$ are the time and the argument of periapse, respectively.
There are two main models for the orbital parameters [29]: the T2 model that
returns the parameters ($P$, $a_{p}$, $t_{p}$, $\omega$, $e$) and the ELL1
model that returns the parameters ($P$, $a_{p}$, $t_{asc}$, $k$, $\beta$).
Indeed in the low-eccentricity limit, the T2 model returns highly correlated
values for $t_{p}$ and $\omega$, and the ELL1 model was developed for pulsars
in such orbits.
In our preliminary analysis on the effects of the uncertainties, we focus on
the T2 model since the ELL1 model provides better constrained parameters. For
the ensemble of pulsars considered in our manuscript, almost 80 pulsars
follows the T2 model. The parameters with the larger relative uncertainties
are $t_{p}$ and $\omega$ as shown in Figure 8 (particularly when the orbital
eccentricity is below 0.01 as expected). These two parameters are also
expected to be more relevant since they contribute directly to the orbital
phase $\psi(t)$. We note that for 15 pulsars from the T2 model, we do not have
an estimation of $\Delta t_{p}$ while 11 pulsars are in circular orbits with
$e=\omega=0$.
Using the P16, we have computed the coherence333 The coherence is a number
between 0 and 1 that measures the resemblance between the shape of the
expected signal and the data. Differently form the p-value, the noise
distribution of the coherence is not flat between 0 and 1 (see [10]). It is a
supplemental parameter to show the significance of the detection. for a wide
range of the uncertainties $\Delta t_{p}$ and $\Delta\omega$. The results show
minima in the coherence corresponding to certain values of the combination:

$\Delta\psi=\Omega\Delta t_{p}+\Delta\omega$ (47)

The values of $\Delta\psi$ corresponding to the minima can be analytically
computed from the phase $\Phi_{bin}$ considering a mismatch $\Delta t_{p}$ and
$\Delta\omega$.
We fix a threshold using the value of $\Delta\psi$ associated to the first
minimum:

$\Delta\psi\sim F\cdot\frac{3}{8fa_{p}}$ (48)

We can take the factor $F=5$, which seems reasonable considering that the
first minima are very sharp.
Considering the 223 pulsars analyzed in Table LABEL:tab:long, 80 pulsars have
parameters form the T2 model; 22 pulsars (J0218$+$4232, J1017$-$7156,
J1125$-$5825, J1312$+$0051, J1529$-$3828, J1545$-$4550, J1603$-$7202,
J1614$-$2230, J1630$+$3734, J1708$-$3506, J1709$+$2313, J1737$-$0811,
J1750$-$2536, J1751$-$2857, J1835$-$0114, J1840$-$0643, J1904$+$0412,
J1946$+$3417, J1949$+$3106, J2019$+$2425, J2053$+$4650, J2129$-$5721) have a
$\Delta\psi$ above the threshold in Eq. 48. We have excluded these pulsars
from the ensemble analysis.
We stress that the uncertainties provided by the binary model are 1$\sigma$
Gaussian errors while in our relation, $\Delta t_{p}$ and $\Delta\omega$ are
considered as an absolute error.
The preliminary results described in this response consider only the $t_{p}$
and $\omega$ parameters; it remains unclear how the uncertainties of the
entire set of parameters combine together in the heterodyne correction. A
precise and independent analysis is still required and falls beyond the scope
of our manuscript.

Table 2: Table of the parameters (name, GW frequency in Hz, distance in kpc, spin-down limit $h_{\text{sd}}$) and the results inferred using the 5n-vector method (upper limit on the amplitude $h_{0}^{95\%}$, upper limit on the ellipticity $\epsilon^{95\%}$, $p$-value) for the 223 analyzed pulsars. Some pulsars don’t have a value for $h_{\text{sd}}$, and $\epsilon^{95\%}$ since the corresponding distance value (or the frequency derivative value) is not available in the ATNF Catalogue [30]. More details about the references for the ephemerides and distance estimation for the analyzed pulsars are in [9]. Pulsar name | $f_{\text{gw}}$ | Distance | $h_{\text{sd}}$ | $h_{0}^{\,95\%}$ | $\epsilon^{95\%}$ | P-value
---|---|---|---|---|---|---
(J2000) | (Hz) | (kpc) |  |  |  |
|  |  |  |  |  |
J0023+0923 | $655.7$ | $1.1$ | $1.3\cdot 10^{-27}$ | 8.2$\cdot 10^{-27}$ | $2.0\cdot 10^{-8}$ | $0.68$
J0030+0451 | $411.1$ | $0.3$ | $3.6\cdot 10^{-27}$ | 6.5$\cdot 10^{-27}$ | $1.2\cdot 10^{-8}$ | $0.71$
J0034$-$0534 | $1065.4$ | $1.4$ | $8.9\cdot 10^{-28}$ | 1.0$\cdot 10^{-26}$ | $1.2\cdot 10^{-8}$ | $0.93$
J0101$-$6422 | $777.3$ | $1.0$ | $9.7\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $1.9\cdot 10^{-8}$ | $0.11$
J0102+4839 | $674.7$ | $2.3$ | $6.8\cdot 10^{-28}$ | 7.3$\cdot 10^{-27}$ | $3.5\cdot 10^{-8}$ | $0.81$
J0117+5914 | $19.7$ | $1.8$ | $1.1\cdot 10^{-25}$ | 6.6$\cdot 10^{-25}$ | $2.8\cdot 10^{-3}$ | $0.66$
J0125$-$2327 | $544.2$ | $0.9$ | $2.0\cdot 10^{-27}$ | 8.1$\cdot 10^{-27}$ | $2.4\cdot 10^{-8}$ | $0.48$
J0154+1833 | $845.8$ | $1.6$ | $3.4\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $2.5\cdot 10^{-8}$ | $0.4$
J0218+4232 | $860.9$ | $3.1$ | $1.5\cdot 10^{-27}$ | 8.0$\cdot 10^{-27}$ | $3.2\cdot 10^{-8}$ | $1$
J0340+4130 | $606.2$ | $1.6$ | $7.2\cdot 10^{-28}$ | 9.1$\cdot 10^{-27}$ | $3.7\cdot 10^{-8}$ | $0.33$
J0348+0432 | $51.1$ | $2.1$ | $9.3\cdot 10^{-28}$ | 1.4$\cdot 10^{-26}$ | $1.0\cdot 10^{-5}$ | $0.2$
J0406+3039 | $766.7$ | - | - | $8.0\cdot 10^{-27}$ | - | $0.85$
J0407+1607 | $77.8$ | $1.3$ | $1.1\cdot 10^{-27}$ | 9.3$\cdot 10^{-27}$ | $1.9\cdot 10^{-6}$ | $0.2$
J0437$-$4715 | $347.4$ | $0.2$ | $8.0\cdot 10^{-27}$ | 7.4$\cdot 10^{-27}$ | $9.3\cdot 10^{-9}$ | $0.24$
J0453+1559 | $43.7$ | $0.5$ | $3.1\cdot 10^{-27}$ | 9.7$\cdot 10^{-27}$ | $2.5\cdot 10^{-6}$ | $1$
J0509+0856 | $493.1$ | $0.8$ | $9.6\cdot 10^{-28}$ | 6.7$\cdot 10^{-27}$ | $2.1\cdot 10^{-8}$ | $0.83$
J0509+3801 | $26.1$ | $1.6$ | $5.3\cdot 10^{-27}$ | 4.0$\cdot 10^{-26}$ | $8.7\cdot 10^{-5}$ | $0.7$
J0557+1550 | $782.4$ | $1.8$ | $7.5\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $3.2\cdot 10^{-8}$ | $0.33$
J0557$-$2948 | $45.8$ | $4.3$ | $2.4\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $2.1\cdot 10^{-5}$ | $0.65$
J0609+2130 | $35.9$ | $0.6$ | $2.9\cdot 10^{-27}$ | 1.8$\cdot 10^{-26}$ | $7.3\cdot 10^{-6}$ | $0.73$
J0610$-$2100 | $518.0$ | $3.3$ | $1.3\cdot 10^{-28}$ | 3.2$\cdot 10^{-26}$ | $3.6\cdot 10^{-7}$ | $0.63$
J0613$-$0200 | $653.2$ | $0.6$ | $2.2\cdot 10^{-27}$ | 6.9$\cdot 10^{-27}$ | $9.2\cdot 10^{-9}$ | $0.96$
J0614$-$3329 | $635.2$ | $0.6$ | $3.0\cdot 10^{-27}$ | 8.2$\cdot 10^{-27}$ | $1.2\cdot 10^{-8}$ | $0.56$
J0621+1002 | $69.3$ | $0.4$ | $2.4\cdot 10^{-27}$ | 8.8$\cdot 10^{-27}$ | $7.3\cdot 10^{-7}$ | $0.34$
J0636$-$3044 | $506.9$ | $0.7$ | $2.6\cdot 10^{-27}$ | 3.1$\cdot 10^{-26}$ | $8.1\cdot 10^{-8}$ | $0.71$
J0636+5129 | $697.1$ | $0.2$ | $4.2\cdot 10^{-27}$ | 6.3$\cdot 10^{-27}$ | $2.6\cdot 10^{-9}$ | $1$
J0645+5158 | $225.9$ | $1.2$ | $3.7\cdot 10^{-28}$ | 4.7$\cdot 10^{-27}$ | $1.1\cdot 10^{-7}$ | $0.86$
J0709+0458 | $58.1$ | $1.2$ | $2.2\cdot 10^{-27}$ | 9.1$\cdot 10^{-27}$ | $3.1\cdot 10^{-6}$ | $0.78$
J0711$-$6830 | $364.2$ | $0.1$ | $1.2\cdot 10^{-26}$ | 4.8$\cdot 10^{-27}$ | $3.8\cdot 10^{-9}$ | $0.91$
J0721$-$2038 | $128.7$ | $2.7$ | $5.1\cdot 10^{-28}$ | 6.0$\cdot 10^{-27}$ | $9.2\cdot 10^{-7}$ | $0.59$
J0732+2314 | $489.0$ | $1.1$ | $8.5\cdot 10^{-28}$ | 6.8$\cdot 10^{-27}$ | $3.1\cdot 10^{-8}$ | $0.76$
J0740+6620 | $693.1$ | $1.1$ | $1.1\cdot 10^{-27}$ | 1.0$\cdot 10^{-26}$ | $2.3\cdot 10^{-8}$ | $0.21$
J0751+1807 | $574.9$ | $0.6$ | $1.9\cdot 10^{-27}$ | 8.4$\cdot 10^{-27}$ | $1.4\cdot 10^{-8}$ | $0.62$
J0824+0028 | $202.8$ | $1.7$ | $1.8\cdot 10^{-27}$ | 1.1$\cdot 10^{-26}$ | $4.2\cdot 10^{-7}$ | $0.01$
J0835$-$4510 | $22.4$ | $0.3$ | $3.4\cdot 10^{-24}$ | 1.1$\cdot 10^{-25}$ | $5.6\cdot 10^{-5}$ | $0.26$
J0921$-$5202 | $206.6$ | $0.4$ | $2.9\cdot 10^{-27}$ | 5.1$\cdot 10^{-27}$ | $4.5\cdot 10^{-8}$ | $0.73$
J0931$-$1902 | $431.2$ | $3.7$ | $1.8\cdot 10^{-28}$ | 5.9$\cdot 10^{-27}$ | $1.1\cdot 10^{-7}$ | $0.84$
J0955$-$6150 | $1000.3$ | $2.2$ | $9.9\cdot 10^{-28}$ | 1.3$\cdot 10^{-26}$ | $2.6\cdot 10^{-8}$ | $0.51$
J1012$-$4235 | $644.9$ | $0.4$ | $3.1\cdot 10^{-27}$ | 7.9$\cdot 10^{-27}$ | $6.7\cdot 10^{-9}$ | $0.61$
J1012+5307 | $380.5$ | $0.7$ | $1.6\cdot 10^{-27}$ | 7.2$\cdot 10^{-27}$ | $3.3\cdot 10^{-8}$ | $0.3$
J1017$-$7156 | $855.2$ | $3.5$ | $2.5\cdot 10^{-28}$ | 7.3$\cdot 10^{-27}$ | $3.3\cdot 10^{-8}$ | $0.97$
J1022+1001 | $121.6$ | $0.6$ | $1.8\cdot 10^{-27}$ | 6.4$\cdot 10^{-27}$ | $2.6\cdot 10^{-7}$ | $0.56$
J1024$-$0719 | $387.4$ | $1.2$ | - | $6.1\cdot 10^{-27}$ | $4.6\cdot 10^{-8}$ | $0.75$
J1035$-$6720 | $696.4$ | $1.5$ | $2.2\cdot 10^{-27}$ | 6.5$\cdot 10^{-27}$ | $1.8\cdot 10^{-8}$ | $0.96$
J1036$-$8317 | $586.7$ | $0.9$ | $2.6\cdot 10^{-27}$ | 6.6$\cdot 10^{-27}$ | $1.7\cdot 10^{-8}$ | $0.74$
J1038+0032 | $69.3$ | $6.0$ | $2.1\cdot 10^{-28}$ | 7.6$\cdot 10^{-27}$ | $8.9\cdot 10^{-6}$ | $0.59$
J1045$-$4509 | $267.6$ | $0.6$ | $2.1\cdot 10^{-27}$ | 4.8$\cdot 10^{-27}$ | $3.7\cdot 10^{-8}$ | $0.91$
J1101$-$6101 | $31.8$ | $7.0$ | $4.2\cdot 10^{-26}$ | 2.3$\cdot 10^{-26}$ | $1.5\cdot 10^{-4}$ | $0.62$
J1101$-$6424 | $391.4$ | $2.2$ | $2.2\cdot 10^{-28}$ | 6.1$\cdot 10^{-27}$ | $8.1\cdot 10^{-8}$ | $0.6$
J1103$-$5403 | $589.5$ | $1.7$ | $5.0\cdot 10^{-28}$ | 5.4$\cdot 10^{-27}$ | $2.5\cdot 10^{-8}$ | $1$
J1125$-$5825 | $644.7$ | $1.7$ | $2.0\cdot 10^{-27}$ | 7.0$\cdot 10^{-27}$ | $2.8\cdot 10^{-8}$ | $0.78$
J1125$-$6014 | $760.3$ | $1.4$ | $7.1\cdot 10^{-28}$ | 7.7$\cdot 10^{-27}$ | $1.8\cdot 10^{-8}$ | $0.78$
J1125+7819 | $476.0$ | $0.9$ | - | $7.3\cdot 10^{-27}$ | $2.7\cdot 10^{-8}$ | $0.48$
J1142+0119 | $394.1$ | $2.2$ | - | $1.0\cdot 10^{-26}$ | $1.3\cdot 10^{-7}$ | $0.067$
J1207$-$5050 | $413.0$ | $1.3$ | $6.9\cdot 10^{-28}$ | 8.4$\cdot 10^{-27}$ | $5.9\cdot 10^{-8}$ | $0.17$
J1216$-$6410 | $565.1$ | $1.1$ | $5.0\cdot 10^{-28}$ | 5.9$\cdot 10^{-27}$ | $1.9\cdot 10^{-8}$ | $0.92$
J1231$-$1411 | $542.9$ | $0.4$ | $2.9\cdot 10^{-27}$ | 9.4$\cdot 10^{-27}$ | $1.3\cdot 10^{-8}$ | $0.29$
J1300+1240 | $321.6$ | $0.6$ | - | $6.1\cdot 10^{-27}$ | $3.4\cdot 10^{-8}$ | $0.64$
J1302$-$3258 | $530.4$ | $1.4$ | $7.4\cdot 10^{-28}$ | 7.1$\cdot 10^{-27}$ | $3.4\cdot 10^{-8}$ | $0.67$
J1302$-$6350 | $41.9$ | $2.3$ | $7.7\cdot 10^{-26}$ | 9.9$\cdot 10^{-27}$ | $1.2\cdot 10^{-5}$ | $0.97$
J1312+0051 | $473.0$ | $1.5$ | $7.6\cdot 10^{-28}$ | 9.8$\cdot 10^{-27}$ | $6.1\cdot 10^{-8}$ | $0.21$
J1327$-$0755 | $746.8$ | $25.0$ | - | $7.9\cdot 10^{-27}$ | $3.3\cdot 10^{-7}$ | $0.9$
J1327+3423 | $48.2$ | - | - | $1.3\cdot 10^{-26}$ | - | $0.5$
J1337$-$6423 | $212.2$ | $5.9$ | $1.9\cdot 10^{-28}$ | 7.5$\cdot 10^{-27}$ | $9.3\cdot 10^{-7}$ | $0.13$
J1400$-$1431 | $648.5$ | $0.3$ | $5.1\cdot 10^{-28}$ | 6.7$\cdot 10^{-27}$ | $4.0\cdot 10^{-9}$ | $0.98$
J1411+2551 | $32.0$ | $1.1$ | $8.5\cdot 10^{-28}$ | 2.8$\cdot 10^{-26}$ | $2.9\cdot 10^{-5}$ | $0.3$
J1412+7922 | $33.8$ | $2.0$ | $9.5\cdot 10^{-26}$ | 2.0$\cdot 10^{-26}$ | $3.3\cdot 10^{-5}$ | $0.48$
J1420$-$5625 | $58.6$ | $1.3$ | $8.5\cdot 10^{-28}$ | 8.0$\cdot 10^{-27}$ | $2.9\cdot 10^{-6}$ | $0.88$
J1421$-$4409 | $313.2$ | $2.1$ | $3.8\cdot 10^{-28}$ | 5.9$\cdot 10^{-27}$ | $1.2\cdot 10^{-7}$ | $0.59$
J1431$-$5740 | $486.6$ | $3.5$ | $2.8\cdot 10^{-28}$ | 7.6$\cdot 10^{-27}$ | $1.1\cdot 10^{-7}$ | $0.44$
J1435$-$6100 | $214.0$ | $2.8$ | $4.6\cdot 10^{-28}$ | 9.1$\cdot 10^{-27}$ | $5.3\cdot 10^{-7}$ | $0.024$
J1439$-$5501 | $69.8$ | $0.7$ | $2.7\cdot 10^{-27}$ | 8.5$\cdot 10^{-27}$ | $1.1\cdot 10^{-6}$ | $0.21$
J1446$-$4701 | $911.3$ | $1.5$ | $1.1\cdot 10^{-27}$ | 9.6$\cdot 10^{-27}$ | $1.6\cdot 10^{-8}$ | $0.69$
J1453+1902 | $345.3$ | $1.3$ | $8.0\cdot 10^{-28}$ | 6.9$\cdot 10^{-27}$ | $7.0\cdot 10^{-8}$ | $0.44$
J1455$-$3330 | $250.4$ | $1.0$ | $1.3\cdot 10^{-27}$ | 6.7$\cdot 10^{-27}$ | $1.0\cdot 10^{-7}$ | $0.41$
J1502$-$6752 | $74.8$ | $7.7$ | $3.0\cdot 10^{-28}$ | 5.8$\cdot 10^{-27}$ | $7.5\cdot 10^{-6}$ | $0.84$
J1513$-$2550 | $943.8$ | $4.0$ | $6.5\cdot 10^{-28}$ | 1.4$\cdot 10^{-26}$ | $5.7\cdot 10^{-8}$ | $0.26$
J1514$-$4946 | $557.2$ | $0.9$ | $1.6\cdot 10^{-27}$ | 1.0$\cdot 10^{-26}$ | $2.9\cdot 10^{-8}$ | $0.19$
J1518+0204A | $360.1$ | $8.0$ | $2.8\cdot 10^{-28}$ | 5.6$\cdot 10^{-27}$ | $3.3\cdot 10^{-7}$ | $1$
J1518+4904 | $48.9$ | $1.0$ | $6.3\cdot 10^{-28}$ | 9.7$\cdot 10^{-27}$ | $3.7\cdot 10^{-6}$ | $0.97$
J1525$-$5545 | $176.1$ | $3.1$ | $8.7\cdot 10^{-28}$ | 4.9$\cdot 10^{-27}$ | $4.7\cdot 10^{-7}$ | $0.81$
J1528$-$3146 | $32.9$ | $0.8$ | $2.1\cdot 10^{-27}$ | 2.5$\cdot 10^{-26}$ | $1.7\cdot 10^{-5}$ | $0.41$
J1529$-$3828 | $235.7$ | $4.3$ | $3.4\cdot 10^{-28}$ | 4.5$\cdot 10^{-27}$ | $3.3\cdot 10^{-7}$ | $0.97$
J1537+1155 | $52.8$ | $1.1$ | $6.0\cdot 10^{-27}$ | 1.3$\cdot 10^{-26}$ | $4.8\cdot 10^{-6}$ | $0.17$
J1543$-$5149 | $972.3$ | $1.1$ | $1.9\cdot 10^{-27}$ | 9.2$\cdot 10^{-27}$ | $1.1\cdot 10^{-8}$ | $0.85$
J1544+4937 | $926.2$ | $3.0$ | $3.1\cdot 10^{-28}$ | 1.7$\cdot 10^{-26}$ | $5.7\cdot 10^{-8}$ | $0.019$
J1545$-$4550 | $559.4$ | $2.2$ | $1.4\cdot 10^{-27}$ | 9.5$\cdot 10^{-27}$ | $6.3\cdot 10^{-8}$ | $0.32$
J1547$-$5709 | $466.1$ | $2.7$ | $3.9\cdot 10^{-28}$ | 7.8$\cdot 10^{-27}$ | $9.2\cdot 10^{-8}$ | $0.32$
J1551$-$0658 | $281.9$ | $1.3$ | $1.0\cdot 10^{-27}$ | 8.5$\cdot 10^{-27}$ | $1.3\cdot 10^{-7}$ | $0.15$
J1600$-$3053 | $555.9$ | $3.0$ | $4.0\cdot 10^{-28}$ | 9.9$\cdot 10^{-27}$ | $9.1\cdot 10^{-8}$ | $0.25$
J1603$-$7202 | $134.8$ | $3.4$ | $2.5\cdot 10^{-28}$ | 4.5$\cdot 10^{-27}$ | $8.0\cdot 10^{-7}$ | $0.92$
J1614$-$2230 | $634.8$ | $0.7$ | $1.2\cdot 10^{-27}$ | 7.6$\cdot 10^{-27}$ | $1.2\cdot 10^{-8}$ | $0.76$
J1618$-$3921 | $166.8$ | $5.5$ | $3.1\cdot 10^{-28}$ | 9.3$\cdot 10^{-27}$ | $1.7\cdot 10^{-6}$ | $0.036$
J1618$-$4624 | $337.2$ | $3.0$ | $1.9\cdot 10^{-28}$ | 6.1$\cdot 10^{-27}$ | $1.5\cdot 10^{-7}$ | $0.56$
J1622$-$6617 | $84.7$ | $4.0$ | $2.9\cdot 10^{-28}$ | 6.0$\cdot 10^{-27}$ | $3.2\cdot 10^{-6}$ | $0.69$
J1623$-$2631 | $180.6$ | $1.8$ | $1.3\cdot 10^{-27}$ | 5.0$\cdot 10^{-27}$ | $2.6\cdot 10^{-7}$ | $0.92$
J1628$-$3205 | $622.7$ | $1.2$ | - | $7.1\cdot 10^{-27}$ | $2.1\cdot 10^{-8}$ | $0.79$
J1629$-$6902 | $333.3$ | $1.0$ | $1.1\cdot 10^{-27}$ | 6.0$\cdot 10^{-27}$ | $4.9\cdot 10^{-8}$ | $0.79$
J1630+3734 | $602.8$ | $1.2$ | $1.1\cdot 10^{-27}$ | 7.9$\cdot 10^{-27}$ | $2.5\cdot 10^{-8}$ | $0.53$
J1640+2224 | $632.2$ | $1.5$ | $3.4\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $3.9\cdot 10^{-8}$ | $0.17$
J1641+3627A | $192.7$ | $7.1$ | $3.2\cdot 10^{-28}$ | 4.3$\cdot 10^{-27}$ | $7.7\cdot 10^{-7}$ | $1$
J1643$-$1224 | $432.7$ | $1.2$ | $1.3\cdot 10^{-27}$ | 1.1$\cdot 10^{-26}$ | $6.6\cdot 10^{-8}$ | $0.045$
J1652$-$4838 | $528.4$ | - | - | $9.7\cdot 10^{-27}$ | - | $0.14$
J1653$-$2054 | $484.4$ | $2.6$ | $5.0\cdot 10^{-28}$ | 7.5$\cdot 10^{-27}$ | $8.0\cdot 10^{-8}$ | $0.6$
J1658$-$5324 | $819.9$ | $0.9$ | $1.9\cdot 10^{-27}$ | 1.1$\cdot 10^{-26}$ | $1.4\cdot 10^{-8}$ | $0.3$
J1705$-$1903 | $806.4$ | $2.4$ | $9.8\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $4.0\cdot 10^{-8}$ | $0.33$
J1708$-$3506 | $443.9$ | $3.3$ | $3.4\cdot 10^{-28}$ | 6.2$\cdot 10^{-27}$ | $9.9\cdot 10^{-8}$ | $0.74$
J1709+2313 | $431.9$ | $2.2$ | $1.9\cdot 10^{-28}$ | 6.4$\cdot 10^{-27}$ | $7.1\cdot 10^{-8}$ | $0.68$
J1713+0747 | $437.6$ | $1.0$ | $1.1\cdot 10^{-27}$ | 7.0$\cdot 10^{-27}$ | $3.5\cdot 10^{-8}$ | $0.59$
J1719$-$1438 | $345.4$ | $0.3$ | $2.6\cdot 10^{-27}$ | 8.0$\cdot 10^{-27}$ | $2.1\cdot 10^{-8}$ | $0.25$
J1721$-$2457 | $572.0$ | $1.4$ | - | $8.0\cdot 10^{-27}$ | $3.1\cdot 10^{-8}$ | $0.59$
J1727$-$2946 | $73.8$ | $1.9$ | $1.3\cdot 10^{-27}$ | 9.3$\cdot 10^{-27}$ | $3.0\cdot 10^{-6}$ | $0.2$
J1729$-$2117 | $30.2$ | $1.0$ | $1.3\cdot 10^{-27}$ | 2.6$\cdot 10^{-26}$ | $2.7\cdot 10^{-5}$ | $0.83$
J1730$-$2304 | $246.2$ | $0.5$ | $2.0\cdot 10^{-27}$ | 5.8$\cdot 10^{-27}$ | $4.2\cdot 10^{-8}$ | $0.64$
J1732$-$5049 | $376.5$ | $1.9$ | $6.2\cdot 10^{-28}$ | 4.6$\cdot 10^{-27}$ | $5.8\cdot 10^{-8}$ | $1$
J1737$-$0811 | $479.0$ | $0.2$ | $5.4\cdot 10^{-27}$ | 8.4$\cdot 10^{-27}$ | $7.3\cdot 10^{-9}$ | $0.44$
J1738+0333 | $341.9$ | $1.5$ | $1.1\cdot 10^{-27}$ | 7.1$\cdot 10^{-27}$ | $8.5\cdot 10^{-8}$ | $0.44$
J1741+1351 | $533.7$ | $1.1$ | $2.1\cdot 10^{-27}$ | 6.4$\cdot 10^{-27}$ | $2.3\cdot 10^{-8}$ | $0.9$
J1744$-$1134 | $490.9$ | $0.4$ | $2.6\cdot 10^{-27}$ | 1.1$\cdot 10^{-26}$ | $1.8\cdot 10^{-8}$ | $0.087$
J1745$-$0952 | $103.2$ | $0.2$ | $7.5\cdot 10^{-27}$ | 5.7$\cdot 10^{-27}$ | $1.2\cdot 10^{-7}$ | $0.82$
J1745+1017 | $754.1$ | $1.2$ | $6.0\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $2.2\cdot 10^{-8}$ | $0.38$
J1747$-$4036 | $1215.4$ | $7.2$ | $2.9\cdot 10^{-28}$ | 2.1$\cdot 10^{-26}$ | $9.8\cdot 10^{-8}$ | $0.036$
J1748$-$2446A | $173.0$ | $6.9$ | $3.3\cdot 10^{-28}$ | 5.3$\cdot 10^{-27}$ | $1.2\cdot 10^{-6}$ | $0.76$
J1748$-$3009 | $206.5$ | $5.0$ | - | $8.0\cdot 10^{-27}$ | $9.0\cdot 10^{-7}$ | $0.13$
J1750$-$2536 | $57.6$ | $3.2$ | $5.1\cdot 10^{-28}$ | 8.9$\cdot 10^{-27}$ | $8.2\cdot 10^{-6}$ | $0.74$
J1751$-$2857 | $510.9$ | $1.1$ | $1.2\cdot 10^{-27}$ | 6.3$\cdot 10^{-26}$ | $2.5\cdot 10^{-7}$ | $0.088$
J1753$-$1914 | $31.8$ | $2.9$ | $1.6\cdot 10^{-27}$ | 2.6$\cdot 10^{-26}$ | $7.2\cdot 10^{-5}$ | $0.56$
J1753$-$2240 | $21.0$ | $3.2$ | $8.0\cdot 10^{-28}$ | 1.7$\cdot 10^{-25}$ | $1.2\cdot 10^{-3}$ | $0.33$
J1755$-$3716 | $156.4$ | $8.2$ | $1.5\cdot 10^{-28}$ | 6.4$\cdot 10^{-27}$ | $2.0\cdot 10^{-6}$ | $0.41$
J1756$-$2251 | $70.3$ | $0.7$ | $6.6\cdot 10^{-27}$ | 8.8$\cdot 10^{-27}$ | $1.2\cdot 10^{-6}$ | $0.41$
J1757$-$1854 | $93.0$ | $19.6$ | $4.6\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $2.5\cdot 10^{-5}$ | $0.018$
J1757$-$5322 | $225.5$ | $0.9$ | $1.5\cdot 10^{-27}$ | 6.0$\cdot 10^{-27}$ | $1.0\cdot 10^{-7}$ | $0.45$
J1801$-$1417 | $551.7$ | $1.1$ | $7.4\cdot 10^{-28}$ | 8.9$\cdot 10^{-27}$ | $3.0\cdot 10^{-8}$ | $0.41$
J1801$-$3210 | $268.3$ | $6.1$ | - | $6.2\cdot 10^{-27}$ | $5.0\cdot 10^{-7}$ | $0.52$
J1802$-$2124 | $158.1$ | $0.8$ | $2.5\cdot 10^{-27}$ | 6.4$\cdot 10^{-27}$ | $1.8\cdot 10^{-7}$ | $0.49$
J1804$-$0735 | $86.6$ | $7.8$ | $2.9\cdot 10^{-28}$ | 7.8$\cdot 10^{-27}$ | $7.7\cdot 10^{-6}$ | $0.39$
J1804$-$2717 | $214.1$ | $0.8$ | $1.9\cdot 10^{-27}$ | 6.1$\cdot 10^{-27}$ | $1.0\cdot 10^{-7}$ | $0.51$
J1807$-$2459A | $653.7$ | $3.0$ | $7.6\cdot 10^{-28}$ | 9.4$\cdot 10^{-27}$ | $6.3\cdot 10^{-8}$ | $0.41$
J1809$-$1917 | $24.2$ | $3.3$ | $1.4\cdot 10^{-25}$ | 5.6$\cdot 10^{-26}$ | $3.0\cdot 10^{-4}$ | $0.85$
J1810+1744 | $1202.8$ | $2.4$ | $5.5\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $1.8\cdot 10^{-8}$ | $0.82$
J1810$-$2005 | $60.9$ | $3.5$ | $2.9\cdot 10^{-28}$ | 8.5$\cdot 10^{-27}$ | $7.6\cdot 10^{-6}$ | $0.69$
J1811$-$2405 | $751.7$ | $1.8$ | $9.9\cdot 10^{-28}$ | 7.2$\cdot 10^{-27}$ | $2.2\cdot 10^{-8}$ | $0.98$
J1813$-$1749 | $44.7$ | $6.2$ | $2.2\cdot 10^{-25}$ | 1.1$\cdot 10^{-26}$ | $3.1\cdot 10^{-5}$ | $0.89$
J1813$-$2621 | $451.5$ | $3.2$ | - | $5.4\cdot 10^{-27}$ | $8.0\cdot 10^{-8}$ | $0.98$
J1821+0155 | $59.2$ | $1.7$ | $4.3\cdot 10^{-28}$ | 9.1$\cdot 10^{-27}$ | $4.2\cdot 10^{-6}$ | $0.92$
J1823$-$3021A | $367.6$ | $7.9$ | $2.9\cdot 10^{-28}$ | 8.6$\cdot 10^{-27}$ | $4.8\cdot 10^{-7}$ | $0.15$
J1824$-$2452A | $654.8$ | $5.5$ | $4.1\cdot 10^{-28}$ | 8.7$\cdot 10^{-27}$ | $1.1\cdot 10^{-7}$ | $0.55$
J1825$-$0319 | $439.2$ | $3.9$ | $2.6\cdot 10^{-28}$ | 5.5$\cdot 10^{-27}$ | $1.0\cdot 10^{-7}$ | $0.98$
J1826$-$2415 | $425.9$ | $2.7$ | $5.7\cdot 10^{-28}$ | 7.0$\cdot 10^{-27}$ | $10.0\cdot 10^{-8}$ | $0.53$
J1828$-$1101 | $27.8$ | $4.8$ | $7.7\cdot 10^{-26}$ | 3.2$\cdot 10^{-26}$ | $1.9\cdot 10^{-4}$ | $0.91$
J1829+2456 | $48.8$ | $0.9$ | $10.0\cdot 10^{-28}$ | 1.4$\cdot 10^{-26}$ | $5.2\cdot 10^{-6}$ | $0.57$
J1832$-$0836 | $735.5$ | $1.6$ | - | $1.4\cdot 10^{-26}$ | $3.8\cdot 10^{-8}$ | $0.093$
J1833$-$0827 | $23.4$ | $4.5$ | $5.9\cdot 10^{-26}$ | 6.1$\cdot 10^{-26}$ | $4.7\cdot 10^{-4}$ | $0.91$
J1835$-$0114 | $390.9$ | $3.5$ | $2.7\cdot 10^{-28}$ | 6.4$\cdot 10^{-27}$ | $1.4\cdot 10^{-7}$ | $0.71$
J1838$-$0655 | $28.4$ | $6.6$ | $1.0\cdot 10^{-25}$ | 3.4$\cdot 10^{-26}$ | $2.6\cdot 10^{-4}$ | $0.72$
J1840$-$0643 | $56.2$ | $5.0$ | $2.6\cdot 10^{-28}$ | 8.5$\cdot 10^{-27}$ | $1.3\cdot 10^{-5}$ | $0.82$
J1841+0130 | $67.2$ | $4.2$ | $3.2\cdot 10^{-27}$ | 1.2$\cdot 10^{-26}$ | $1.1\cdot 10^{-5}$ | $0.037$
J1843$-$1113 | $1083.6$ | $1.3$ | $1.4\cdot 10^{-27}$ | 2.2$\cdot 10^{-26}$ | $2.2\cdot 10^{-8}$ | $0.019$
J1843$-$1448 | $365.5$ | $3.5$ | - | $8.3\cdot 10^{-27}$ | $2.0\cdot 10^{-7}$ | $0.23$
J1849$-$0001 | $51.9$ | $7.0$ | $7.0\cdot 10^{-26}$ | 1.5$\cdot 10^{-26}$ | $3.8\cdot 10^{-5}$ | $0.08$
J1853+1303 | $488.8$ | $1.3$ | $8.9\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $6.1\cdot 10^{-8}$ | $0.058$
J1856+0245 | $24.7$ | $6.3$ | $1.1\cdot 10^{-25}$ | 5.7$\cdot 10^{-26}$ | $5.6\cdot 10^{-4}$ | $0.69$
J1857+0943 | $373.0$ | $1.2$ | $1.2\cdot 10^{-27}$ | 7.2$\cdot 10^{-27}$ | $5.8\cdot 10^{-8}$ | $0.48$
J1902$-$5105 | $1147.8$ | $1.6$ | $1.1\cdot 10^{-27}$ | 1.3$\cdot 10^{-26}$ | $1.5\cdot 10^{-8}$ | $0.59$
J1903+0327 | $930.3$ | $6.1$ | $3.9\cdot 10^{-28}$ | 1.4$\cdot 10^{-26}$ | $9.3\cdot 10^{-8}$ | $0.24$
J1903$-$7051 | $555.9$ | $0.9$ | $1.3\cdot 10^{-27}$ | 6.7$\cdot 10^{-27}$ | $1.9\cdot 10^{-8}$ | $0.77$
J1904+0412 | $28.1$ | $4.6$ | $2.2\cdot 10^{-28}$ | 2.9$\cdot 10^{-26}$ | $1.6\cdot 10^{-4}$ | $0.98$
J1905+0400 | $528.5$ | $1.1$ | $8.0\cdot 10^{-28}$ | 7.3$\cdot 10^{-27}$ | $2.6\cdot 10^{-8}$ | $0.68$
J1909$-$3744 | $678.6$ | $1.1$ | $6.6\cdot 10^{-28}$ | 8.1$\cdot 10^{-27}$ | $1.9\cdot 10^{-8}$ | $0.64$
J1910+1256 | $401.3$ | $1.5$ | $7.2\cdot 10^{-28}$ | 8.1$\cdot 10^{-27}$ | $7.2\cdot 10^{-8}$ | $0.29$
J1911$-$1114 | $551.6$ | $1.1$ | $1.3\cdot 10^{-27}$ | 1.2$\cdot 10^{-26}$ | $4.0\cdot 10^{-8}$ | $0.067$
J1911+1347 | $432.3$ | $1.4$ | $1.1\cdot 10^{-27}$ | 7.1$\cdot 10^{-27}$ | $4.9\cdot 10^{-8}$ | $0.54$
J1913+1011 | $55.7$ | $4.6$ | $5.3\cdot 10^{-26}$ | 8.6$\cdot 10^{-27}$ | $1.2\cdot 10^{-5}$ | $0.82$
J1914+0659 | $108.0$ | $8.5$ | $1.2\cdot 10^{-28}$ | 8.1$\cdot 10^{-27}$ | $5.6\cdot 10^{-6}$ | $0.23$
J1915+1606 | $33.9$ | $5.2$ | $1.9\cdot 10^{-27}$ | 2.6$\cdot 10^{-26}$ | $1.1\cdot 10^{-4}$ | $0.21$
J1918$-$0642 | $261.6$ | $1.1$ | $1.3\cdot 10^{-27}$ | 6.7$\cdot 10^{-27}$ | $1.0\cdot 10^{-7}$ | $0.45$
J1923+2515 | $528.0$ | $1.2$ | $9.1\cdot 10^{-28}$ | 1.0$\cdot 10^{-26}$ | $4.1\cdot 10^{-8}$ | $0.15$
J1925+1720 | $26.4$ | $5.1$ | $5.9\cdot 10^{-26}$ | 7.0$\cdot 10^{-26}$ | $4.8\cdot 10^{-4}$ | $0.039$
J1928+1746 | $29.1$ | $4.3$ | $8.2\cdot 10^{-26}$ | 3.4$\cdot 10^{-26}$ | $1.6\cdot 10^{-4}$ | $0.54$
J1933$-$6211 | $564.4$ | $0.7$ | $1.1\cdot 10^{-27}$ | 6.9$\cdot 10^{-27}$ | $1.3\cdot 10^{-8}$ | $0.66$
J1935+2025 | $25.0$ | $4.6$ | $1.5\cdot 10^{-25}$ | 4.9$\cdot 10^{-26}$ | $3.4\cdot 10^{-4}$ | $0.66$
J1939+2134 | $1283.9$ | $4.8$ | $1.4\cdot 10^{-27}$ | 1.4$\cdot 10^{-26}$ | $4.0\cdot 10^{-8}$ | $0.59$
J1943+2210 | $393.4$ | $6.8$ | $1.6\cdot 10^{-28}$ | 6.4$\cdot 10^{-27}$ | $2.6\cdot 10^{-7}$ | $0.62$
J1944+0907 | $385.7$ | $1.2$ | $7.4\cdot 10^{-28}$ | 6.1$\cdot 10^{-27}$ | $4.8\cdot 10^{-8}$ | $0.75$
J1946+3417 | $630.9$ | $6.9$ | $6.5\cdot 10^{-29}$ | 9.3$\cdot 10^{-27}$ | $1.5\cdot 10^{-7}$ | $0.34$
J1946$-$5403 | $737.8$ | $1.1$ | $7.0\cdot 10^{-28}$ | 7.1$\cdot 10^{-27}$ | $1.4\cdot 10^{-8}$ | $0.89$
J1949+3106 | $152.2$ | $7.5$ | $2.9\cdot 10^{-28}$ | 5.8$\cdot 10^{-27}$ | $1.8\cdot 10^{-6}$ | $0.67$
J1950+2414 | $464.6$ | $7.3$ | $2.4\cdot 10^{-28}$ | 6.8$\cdot 10^{-27}$ | $2.2\cdot 10^{-7}$ | $0.67$
J1952+3252 | $50.6$ | $3.0$ | $1.0\cdot 10^{-25}$ | 1.3$\cdot 10^{-26}$ | $1.4\cdot 10^{-5}$ | $0.22$
J1955+2527 | $410.4$ | $8.2$ | $1.5\cdot 10^{-28}$ | 7.7$\cdot 10^{-27}$ | $3.5\cdot 10^{-7}$ | $0.36$
J1955+2908 | $326.1$ | $6.3$ | $2.9\cdot 10^{-28}$ | 6.6$\cdot 10^{-27}$ | $3.7\cdot 10^{-7}$ | $0.46$
J2007+2722 | $81.6$ | $7.1$ | $7.1\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $1.1\cdot 10^{-5}$ | $0.037$
J2010$-$1323 | $382.9$ | $1.2$ | $5.3\cdot 10^{-28}$ | 8.4$\cdot 10^{-27}$ | $6.3\cdot 10^{-8}$ | $0.22$
J2017+0603 | $690.6$ | $1.4$ | $9.6\cdot 10^{-28}$ | 8.9$\cdot 10^{-27}$ | $2.5\cdot 10^{-8}$ | $0.6$
J2019+2425 | $508.3$ | $1.2$ | $4.4\cdot 10^{-28}$ | 5.4$\cdot 10^{-26}$ | $2.3\cdot 10^{-7}$ | $0.046$
J2022+2534 | $755.9$ | - | - | $1.2\cdot 10^{-26}$ | - | $0.2$
J2033+1734 | $336.2$ | $1.7$ | $5.5\cdot 10^{-28}$ | 5.2$\cdot 10^{-27}$ | $7.6\cdot 10^{-8}$ | $0.92$
J2039$-$3616 | $610.7$ | $1.7$ | $7.6\cdot 10^{-28}$ | 7.6$\cdot 10^{-27}$ | $3.3\cdot 10^{-8}$ | $0.65$
J2043+2740 | $20.8$ | $1.5$ | $6.3\cdot 10^{-26}$ | 1.7$\cdot 10^{-25}$ | $5.6\cdot 10^{-4}$ | $0.32$
J2045+3633 | $63.1$ | $5.6$ | $6.2\cdot 10^{-28}$ | 9.1$\cdot 10^{-27}$ | $1.2\cdot 10^{-5}$ | $0.34$
J2047+1053 | $466.6$ | $2.8$ | $6.4\cdot 10^{-28}$ | 7.0$\cdot 10^{-27}$ | $8.5\cdot 10^{-8}$ | $0.69$
J2053+4650 | $158.9$ | $3.8$ | $7.8\cdot 10^{-28}$ | 4.6$\cdot 10^{-27}$ | $6.6\cdot 10^{-7}$ | $0.92$
J2055+3829 | $957.3$ | $4.6$ | $1.1\cdot 10^{-28}$ | 1.2$\cdot 10^{-26}$ | $5.8\cdot 10^{-8}$ | $0.36$
J2124$-$3358 | $405.6$ | $0.4$ | $2.3\cdot 10^{-27}$ | 8.0$\cdot 10^{-27}$ | $2.0\cdot 10^{-8}$ | $0.28$
J2129$-$5721 | $536.7$ | $7.0$ | $2.6\cdot 10^{-28}$ | 7.8$\cdot 10^{-27}$ | $1.8\cdot 10^{-7}$ | $0.41$
J2144$-$5237 | $396.7$ | $1.7$ | $6.5\cdot 10^{-28}$ | 6.0$\cdot 10^{-27}$ | $6.0\cdot 10^{-8}$ | $0.63$
J2145$-$0750 | $124.6$ | $0.8$ | $1.3\cdot 10^{-27}$ | 6.5$\cdot 10^{-27}$ | $3.3\cdot 10^{-7}$ | $0.52$
J2150$-$0326 | $569.7$ | - | - | $8.9\cdot 10^{-27}$ | - | $0.46$
J2205+6012 | $828.0$ | $3.5$ | $6.5\cdot 10^{-28}$ | 8.9$\cdot 10^{-27}$ | $4.3\cdot 10^{-8}$ | $0.7$
J2214+3000 | $641.2$ | $0.6$ | $2.7\cdot 10^{-27}$ | 6.8$\cdot 10^{-27}$ | $9.3\cdot 10^{-9}$ | $0.92$
J2222$-$0137 | $60.9$ | $0.3$ | $2.0\cdot 10^{-27}$ | 7.2$\cdot 10^{-27}$ | $5.0\cdot 10^{-7}$ | $0.91$
J2229+2643 | $671.6$ | $1.8$ | $3.1\cdot 10^{-28}$ | 1.1$\cdot 10^{-26}$ | $4.2\cdot 10^{-8}$ | $0.19$
J2229+6114 | $38.7$ | $3.0$ | $3.3\cdot 10^{-25}$ | 1.3$\cdot 10^{-26}$ | $2.4\cdot 10^{-5}$ | $0.8$
J2234+0611 | $559.2$ | $1.5$ | $8.2\cdot 10^{-28}$ | 1.3$\cdot 10^{-26}$ | $5.9\cdot 10^{-8}$ | $0.071$
J2234+0944 | $551.4$ | $0.8$ | $1.9\cdot 10^{-27}$ | 7.6$\cdot 10^{-27}$ | $1.9\cdot 10^{-8}$ | $0.68$
J2235+1506 | $33.5$ | $1.5$ | $6.5\cdot 10^{-28}$ | 2.3$\cdot 10^{-26}$ | $2.9\cdot 10^{-5}$ | $0.53$
J2236$-$5527 | $289.5$ | $2.0$ | $4.6\cdot 10^{-28}$ | 5.0$\cdot 10^{-27}$ | $1.2\cdot 10^{-7}$ | $0.78$
J2241$-$5236 | $914.6$ | $1.1$ | $1.2\cdot 10^{-27}$ | 1.4$\cdot 10^{-26}$ | $1.7\cdot 10^{-8}$ | $0.097$
J2256$-$1024 | $871.6$ | $1.3$ | $1.3\cdot 10^{-27}$ | 1.6$\cdot 10^{-26}$ | $2.7\cdot 10^{-8}$ | $0.057$
J2302+4442 | $385.2$ | $0.9$ | $1.5\cdot 10^{-27}$ | 7.0$\cdot 10^{-27}$ | $3.9\cdot 10^{-8}$ | $0.39$
J2317+1439 | $580.5$ | $2.2$ | $3.3\cdot 10^{-28}$ | 7.5$\cdot 10^{-27}$ | $4.6\cdot 10^{-8}$ | $0.72$
J2322+2057 | $415.9$ | $1.0$ | $6.5\cdot 10^{-28}$ | 6.2$\cdot 10^{-27}$ | $3.4\cdot 10^{-8}$ | $0.77$
J2322$-$2650 | $577.5$ | $0.2$ | $1.2\cdot 10^{-27}$ | 7.0$\cdot 10^{-27}$ | $4.6\cdot 10^{-9}$ | $0.78$

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  * Note [3] The coherence is a number between 0 and 1 that measures the resemblance between the shape of the expected signal and the data. Differently form the p-value, the noise distribution of the coherence is not flat between 0 and 1 (see [10]). It is a supplemental parameter to show the significance of the detection.