# Increasing the detectability of long-period and nulling pulsars in next-
generation pulsar surveys

G. Grover  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia  [ N. D. R. Bhat  International Centre
for Radio Astronomy Research, Curtin University, Bentley, WA 6102, Australia
S. J. McSweeney  International Centre for Radio Astronomy Research, Curtin
University, Bentley, WA 6102, Australia

(dd Mmm YYYY; dd Mmm YYYY; dd Mmm YYYY)

###### Abstract

Recent discoveries of multiple long-period pulsars (periods ${\sim}10\,$s or
larger) are starting to challenge the conventional notion that coherent radio
emission cannot be produced by objects that are below the many theorised death
lines. Many of the past pulsar surveys and software have been prone to
selection effects that restricted their sensitivities towards long-period and
sporadically-emitting objects. Pulsar surveys using new-generation low-
frequency facilities are starting to employ longer dwell times, which makes
them significantly more sensitive to detecting long-period or nulling pulsars.
There have also been software advancements to aid more sensitive searches
towards long-period objects. Furthermore, recent discoveries suggest that
nulling may be a key aspect of the long-period pulsar population. We simulate
both long-period and nulling pulsar signals, using the Southern-sky MWA Rapid
Two-meter (SMART) survey data as reference, and explore the detection efficacy
of popular search methods such as the fast Fourier transform (FFT), fast-
folding algorithm (FFA) and single pulse search (SPS). For FFT-based search
and SPS, we make use of the PRESTO implementation, and for FFA we use RIPTIDE.
We find RIPTIDE’s FFA to be more sensitive; however, it is also the slowest
algorithm. PRESTO’s FFT, although faster than others, also shows some
unexpected inaccuracies in detection properties. SPS is highly sensitive to
long-period and nulling signals, but only for pulses with high intrinsic
signal-to-noise ratios. We use these findings to inform current and future
pulsar surveys that aim to uncover a large population of long-period or
nulling objects and comment on how to make optimal use of these methods in
unison.

###### doi:

ZZZ

###### keywords:

surveys: sky surveys – pulsars: general – techniques: algorithms

G<EMAIL_ADDRESS>dd Mmm YYYY

##  1 INTRODUCTION

Coherent radio emission from pulsars is thought to rely critically on pair
production within their magnetospheres (Ruderman & Sutherland, 1975). At
longer rotation periods ($P$), the potential difference in the magnetosphere
is suspected to be insufficient for pair production to occur, and hence, no
emission can be produced (Ruderman & Sutherland, 1975; Arons & Scharlemann,
1979). This naturally places a constraint on the rotation period $P$ and the
characteristic magnetic field $B\propto\sqrt{P\dot{P}}$ of the pulsar, where
$\dot{P}$ is the period derivative. The related constraints can be identified
as ‘death lines’ on a $P$-$\dot{P}$ diagram. Many death lines have been
proposed, each with different assumptions such as the magnetic field
orientation, assumptions about the pair production, etc (eg Ruderman &
Sutherland, 1975; Chen & Ruderman, 1993; Zhang et al., 2000; Rea et al.,
2023). However, the one that most completely encompassed the pulsar population
was by Zhang et al. (2000).

In recent years, there has been a flurry of discoveries of pulsars with very
long periods that challenge some of the most conservative death line models,
as shown in Figure 1. These include a 12.1-second pulsar J2251-3711 discovered
by Morello et al. (2020b), a 23.5-second pulsar J0250+5854 by Tan et al.
(2018) and PSR J0901-4046 with a rotation period of 76 seconds (Caleb et al.,
2022). These are currently among the slowest known radio pulsars. Along with
their relatively small $\dot{P}$, their positions on the $P$-$\dot{P}$ diagram
are also near the edge of the furthest (least stringent) death line. This
death line is shown as Zhang et al. (2000) III’ in Figure 1. The discoveries
of such pulsars also suggest the possible existence of a larger population of
long-period pulsars that could be uncovered using more optimal search
strategies.

Remarkably, all three objects show some degree of “nulling,” which is the
cessation of detectable radio emission for a certain duration or number of
rotations. For example, PSRs J2251-3711 and J0250+5854 have nulling fractions
of 67% and 27% respectively (Tan et al., 2018; Morello et al., 2020b), and PSR
J0901-4046 shows nulling occurring on time scales shorter than a single pulse
(Caleb et al., 2022). Assuming these objects are representative of the
population, nulling may be a common property among these objects. It has been
suggested that nulling is indicative of the radio emission becoming
unsustainable for a pulsar and the nulling fraction increases progressively
until there is no detectable emission from the pulsar (Ritchings, 1976; Zhang
et al., 2007). However, no convincing correlation has been found between
nulling and physical parameters such as $P$ or $\dot{P}$ (Anumarlapudi et al.,
2023; Sheikh & Macdonald, 2021), though this may be also due to observational
selection effects.

In addition to these three pulsars, a couple of pulsar-like sources with
extremely long periods have also been detected in recent years. A noteworthy
source is GLEAMX J162759.5-523504.3, with a rotation period of $\sim 18.18$
min, discovered by Hurley-Walker et al. (2022). Another similar source, with
an even longer period of $\sim$21 min, has just been discovered (Hurley-Walker
et al., 2023). The origins of this new class of sources remain currently
unknown. The first source does seem to go into a quiescent state, which could
be interpreted as an extended nulling state, but interestingly the second
source has been active for over three decades. These discoveries have
motivated Rea et al. (2023) to extend the original models for death lines.

Yet another recent addition is a $\sim 10$ s pulsar discovered by Su et al.
(2023) using the Five-hundred-metre Aperture Spherical Telescope (FAST).
Intriguingly, over a 2.29-year observational time span, there is no evidence
of nulling in this object. Regardless, it sits well beyond the current death
line models on the $P$-$\dot{P}$ diagram, thereby further strengthening the
evidence of a possible long-period pulsar population. Uncovering a larger
population of sources (both pulsars or pulsar-like objects) can help
significantly advance our understanding of coherent radio emission in pulsars
and current models for the death lines.

Death line models are typically modified to constrain new long-period pulsars
as they are discovered; however, long-period pulsar discoveries are made
difficult due to observational selection effects. Most large pulsar surveys
that scan large swaths of the skies typically have dwell times between 2 min
and 5 min and therefore are less sensitive to long-period pulsar signals
(Stovall et al., 2014; Han et al., 2021; Keith et al., 2010). Longer dwell
times naturally increase the sensitivity to these objects, particularly those
with long durations of nulling or rotation periods larger than tens of
seconds. Specifically, longer observing duration increases the chance of
detecting a larger number of pulses, especially when $P\sim 10-100$ s or
longer. The emergence of wide-field instruments at low frequencies such as the
Murchison Widefield Array (MWA; Tingay et al. 2013) and Low Frequency Array
(LOFAR; van Haarlem et al. 2013) and their adaptation to large pulsar surveys
(e.g., Bhat et al., 2023a; Sanidas et al., 2019), means longer dwell times are
now affordable, at least at low frequencies. This increases the achievable
sensitivity in general, particularly for those objects with rotation periods
$\gtrsim$ 10 s, or with long null durations ($\sim$tens of minutes). The LOFAR
Tied-Array All-Sky Survey (LOTAAS; Sanidas et al. (2019)) has a 60 min dwell
time, and the Southern-sky MWA Rapid Two-meter (SMART) survey uses an 80 min
dwell time. The SMART survey uses the MWA to observe the whole southern sky
between 140 MHz - 170 MHz. The data has 100 $\mu$s time resolution and 10 kHz
frequency resolution, and are stored as raw voltages, and can be beamformed
offline to tessellate the full field of view using thousands of sensitive
tied-array (phased array) beams. For more details on the SMART survey see Bhat
et al. (2023a). In detecting a long-period or highly intermittent population
of pulsars, these surveys will have a competitive advantage.

Figure 1: A $P\dot{P}$ diagram showing all pulsars (grey), nulling pulsars
(orange) and RRATs (blue). Also highlighted are the three recent long-period
pulsar discoveries (Tan et al., 2018; Morello et al., 2020b; Caleb et al.,
2022). The various death lines, taken from Zhang et al. (2000), are also
shown, as per the legend. These death lines are improvements based on the
works of Ruderman & Sutherland (1975) and (Chen & Ruderman, 1993).

###  1.1 Overview of Search Methods

In addition to the use of more optimal survey strategies, it is also important
to adopt search algorithms that are sensitive to sporadic, intermittent and or
long-period objects. In principle, long-duration data from surveys like the
SMART can also be exploited to search for pulsars in binary systems; however,
our focus will primarily be on isolated or single pulsars. There are three
popular pulsar searching methods, including the fast Fourier transform (FFT)
based search, the fast folding algorithm (FFA; first described by Staelin
1969) and the single pulse search (SPS). Fourier methods have been the primary
ones used in pulsar searching and have been very successful, having discovered
the vast majority of the pulsars currently known. However, in searching for
long-period pulsars, Fourier-based methods have been shown to be inherently
less sensitive towards long-period and narrow duty-cycle signals, which is
generally the case for most long-period pulsars (van Heerden et al., 2017;
Cameron et al., 2017; Singh et al., 2022; Kondratiev et al., 2009; Morello et
al., 2020a; Parent et al., 2018). A major obstacle is the presence of low-
frequency ‘red’ noise that arises from slowly varying power levels in the
system (e.g. due to gain variations), or radio frequency interference (RFI)
that is persistent in nature. This tends to mask long-period signals in
Fourier space. Moreover, this method relies on the signal to be periodic to
make a significant detection and is therefore inherently less sensitive
towards the sporadic emission from pulsars that tend to null for long
durations. PRESTO (Ransom, 2001), includes algorithms for suppressing red
noise, and has not been robustly tested for its efficacy in detecting such
objects that emit sporadic or intermittent signals.

FFA is a time-domain periodicity search. It searches for pulsars by
efficiently folding signals at multiple trial periods. There have been a few
different implementations of this algorithm: ffaGo (Parent et al., 2018),
ffancy (Cameron et al., 2017) and RIPTIDE (Morello et al., 2020a), all
highlighting the advantages of using FFA over FFT-based searches, especially
for long-period pulsars. The algorithm has been demonstrated in its ability to
find new pulsars, having recently detected 6 new pulsars in the GMRT High-
Resolution Southern-sky (GHRSS) survey (Singh et al., 2023). FFA has also been
shown to be effective at finding long-period pulsars, and in fact, has been
able to successfully make independent detections of PSRs J0250+5854 and
J2251-3711, at a higher detection significance than FFT methods (Morello et
al., 2020b; Tan et al., 2018).

Comparisons between FFA and FFT have also been explored in Singh et al. 2022
(hereafter S22). They made use of both simulations and examples of real data
and investigated the efficacies of the FFT and FFA implementations in order to
compare the quality of detections. They varied the period, duty cycle and
morphology of pulses and reported FFA to be able to detect pulsar signals at a
consistently higher S/N than FFT.

Similar to FFT, FFA is less effective at detecting sporadic signals. However,
a robust analysis comparing the efficacies of FFA’s implementations in
detecting nulling signals has not been carried out yet. Some limited studies
have been published on this front; e.g. the recent work by Singh et al. (2023)
who compare FFA and FFT in detecting nulling signals. However, this was done
for only one particular source.

Finally, single pulse search (SPS) is quite effective in detecting pulsed,
transient-like emission, and is routinely used in transient, pulsar and fast
radio burst (FRB) surveys. It searches for emission by convolving the time
series with box car filters with various trial widths (e.g. in logarithmic
steps of $2^{n}$, where $n=0,1,2,...,n$). This method is demonstrably
effective at finding long-period pulsars, having successfully made the initial
detections of both PSRs J2251-3711 and (Tan et al., 2018) J0901-4046 (Caleb et
al., 2022). SPS has also been effective in detecting sporadically emitting
pulsars, such as Rotating Radio Transients (RRATs), which are, by definition,
better detected by SPS (Keane & McLaughlin, 2011). Although the efficacy of
SPS has been demonstrated in detecting both sporadic and long-period sources,
the efficacy of SPS towards sporadic and long-period objects has not been
robustly tested and compared with other algorithms.

Ongoing large surveys, such as the SMART survey, intend to incorporate FFA and
SPS into their search pipelines, alongside FFT-based methods, in order to
increase sensitivity towards long-period objects (Bhat et al., 2023b). Given
that recent long-period pulsar discoveries tend to show a significant amount
of nulling, it is valuable to test the efficacies of various methods in
detecting such long-period nulling pulsars. In order to test these methods, we
chose PRESTO’s implementations of FFT and SPS, and RIPTIDE’s implementation of
FFA (see justifications in Section 2.1). We perform a simulation analysis to
explore the large search parameter space, covering periods, nulling parameters
and S/Ns for a robust analysis. Section 2 describes the process and the
justifications for our analysis. For reference, we try to duplicate typical
observations from the SMART survey. The results are presented in Section 3. As
a sanity check, we also explore the effectiveness of the search methods in
detecting pulsars in real data (Section 4). Finally, we discuss our results
and make some recommendations to consider in future search processing to
increase the detectability of long-period pulsars (Section 5). Our conclusions
are summarised in Section 6.

##  2 Analysis

In order to test how well FFT, FFA and SPS would perform in detecting a
variety of long-period and nulling pulsars, we generated a large number of
simulated pulsar signals. This allows us to sample a large parameter space,
uninfluenced by selection biases. A high-level overview of our simulation and
searching procedure is shown in Figure 2.

S22 performed a similar study where they tested the detection significance of
RIPTIDE’s FFA and PRESTO’s FFT implementations using simulated data. We chose
to keep parts of our analysis similar to S22 to allow for meaningful
comparison. S22 varied periodicity, subpulse morphology and duty cycles. As we
are interested in nulling properties of long-period pulsars, we chose to vary
the pulse period, nulling fraction, nulling duration and signal strength,
searching with RIPTIDE’s FFA, PRESTO’s FFT and PRESTO’s SPS implementations.

###  2.1 Simulating pulsar signals

The simulated signals consisted of trains of Gaussian pulses (mimicking pulsar
signals) injected into Gaussian noise (receiver noise). We based our
simulations on the SMART survey, as it has the longest dwell time of any
current survey (4800 s), and therefore, in principle, offers better detection
prospects (within the sensitivity limits) for detecting long-period signals.

Our main goal is to explore the long-period and nulling parameter spaces. We,
therefore, simulate signals with varying periods ($P$), nulling properties
(nulling fraction, $n_{\rm f}$; nulling duration, $N_{\rm d}$) and average
signal-to-noise ratio per pulse (S/NPP). Within each simulated signal, we
allowed individual pulses to vary in brightness according to a modulation
index that was kept constant for all simulations. We also kept the pulse duty
cycle constant at $4\%$.

The target parameters were varied logarithmically with 6 - 7 increments in
order to encapsulate a large range of values. The periods ranged from 0.5 s -
50 s. The choice of a large value for the maximum period was driven by our
intent to simulate very long-period pulsars ($P\gtrsim 10$ s).

To emulate nulling, we varied $n_{\rm f}$ from 1% - 90% and $N_{\rm d}$ from 5
pulses - 1000 pulses. This accounts for the majority of the nulling pulsar
population as well as extremely nulling pulsars such as RRATs. For each null
sequence within a simulation, its duration was drawn from a Poissonian
distribution with an expected value $N_{\rm d}$. Burst sequence durations were
drawn from a Poissonian distribution with expected value:

$B_{\rm d}=\frac{N_{\rm d}}{n_{\rm f}}-N_{\rm d}.$ (1)

These choices guarantee that the long-term behaviour of the simulated pulse
train matches the expected behaviour for a pulsar with the selected average
$N_{\rm d}$ and $n_{\rm f}$, whilst still reflecting the randomness of nulling
behaviours observed in real pulsars.

Finally, as pulsars are known to vary in brightness, we chose S/N per pulse
(S/NPP) to range from 0.1 - 10. A large upper limit was chosen primarily
because at low frequencies ($\lesssim 300$ MHz), pulsars tend to be generally
brighter and some RRATs tend to have bright pulses (Burke-Spolaor & Bailes,
2010). Given that the number of pulses within a simulated signal was not
predetermined due to the randomness of the nulls, the S/N was chosen per pulse
rather than the signal as a whole. Varying S/NPP also allows us to more
accurately test the threshold of SPS capabilities than we would have with an
integrated S/N. The distribution of S/NPP was assumed to be log-normal, in
agreement with what is commonly observed (e.g. Burke-Spolaor et al., 2012):

$f(x)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(\frac{(\ln
x-\mu)^{2}}{2\sigma^{2}}\right).$ (2)

Here $\mu$ is the mean S/NPP (where S/NPP ranges from 0.1 - 10) and
$\sigma^{2}$ is the variance, which is dependent on the modulation index $m$
of the signal via $\sigma=m\mu$. The modulation index was chosen arbitrarily
as 0.1 as the value can range from 0 - 2 for typical and sub-pulse drifting
pulsars (Weltevrede et al., 2006).

Due to computational constraints, all signals were generated with a single
dispersion measure (DM) of 150 cm-3 pc. As we chose not include any scattering
effects, the choice of DM was arbitrary as it has little to no effect on the
search algorithms. If scattering was implemented to the simulated data, a
large DM such 150 cm-3 pc would imply considerable scatter broadening, which
would lower the peak amplitude of the pulse. The effects of scattering appear
differently in the time and frequency domain and hence would affect FFT, FFA
and SPS differently. As our primary focus is to assess the effects of nulling
on various search methods, we have not considered effects such as pulse
broadening (scattering) that alter the pulse shape (and thus impact the
detectability). As there is no scattering and the data are dedispersed before
being searched, the DM is only relevant as metadata. As long as the algorithm
is allowed to search signals of the given DM, it has no effect on the search
process. In practice, however, the exact DM of the pulsar may not be searched,
and there would be some loss of S/N, as described by Cordes & McLaughlin
(2003), in the case of searching sporadic emission. This would directly affect
the detection significance of the search methods but since we are using
simulated data, we take the ‘ideal case’ approach.

Figure 2: A flow diagram for the simulation of the synthetic pulsar signal and
their processing through FFT, FFA and SPS methods.

###  2.2 Searching Methods on Simulated Data

Once generated, these signals were processed and searched with RIPTIDE’s FFA,
PRESTO’s FFT and PRESTO’s SPS. We chose RIPTIDE as it is the latest,
progressive implementation of FFA which has parallelised computation and
improved detectability with matched filtering (see Morello et al. (2020a) for
details). This is also the implementation used in S22.

For the FFT implementation, we chose PRESTO’s accelsearch as it is the most
commonly used version and is already in use in the first pass (i.e. shallow
survey) of the SMART survey (Bhat et al., 2023a). Although accelsearch was
designed to do an additional acceleration search in order to target binary
pulsars, we do not use this feature as our simulated signals do not include
any Doppler modulation of the period due to binary orbital motion. By using
the PRESTO implementation for comparison with other searches in this work, we
could directly relate our analysis with S22.

The SPS method was originally described in Cordes & McLaughlin (2003). More
recent implementations (e.g. Michilli et al., 2018) primarily improve
additional aspects such as the machine learning classification of candidates.
We chose PRESTO’s single_pulse_search as it is essentially an implementation
of the original algorithm described in Cordes & McLaughlin (2003).

It is important to acknowledge that FFA, FFT and SPS are inherently different
searching methods that analyse different aspects of the signal and report
their results using different metrics. For example, PRESTO’s accelsearch
searches in Fourier space and reports a $\sigma_{\rm FFT}$ (the detection
significance) and, through additional code, a spectral S/N (S/NFFT). RIPTIDE’s
FFA also reports a detection period but it is accompanied by a S/NFFA measured
in the time domain. PRESTO’s single_pulse_search on the other hand, is a
transient search method, that reports a $\sigma_{\rm SPS}$ for its transient
candidate. The $\sigma$ and S/N for these methods are thus not equivalent
though they all signify detection significance. Hence, a more meaningful
comparison of these different methods involves testing whether or not the
method can detect a signal based on their relevant criteria.

####  2.2.1 FFA & FFT

For both FFA and FFT searches, detections would be successful if a candidate
was produced with the correct period, within some tolerance, and with a
detection significance $\geq$ 6$\sigma$. As FFA does not report a $\sigma$
value, we used the reported S/NFFA. A S/NFFA cut-off of $\sim$8 was recently
chosen by (Wongphechauxsorn et al., 2024) and the $\sigma_{\rm FFT}$ cut-off
can be $2\sigma_{\rm FFT}-6\sigma_{\rm FFT}$ (Sanidas et al., 2019; Stovall et
al., 2014) for surveys, depending on the instrument. It was also acceptable if
the detection was a fractional harmonic of the fundamental period (up to the
32nd harmonic) while still being within the tolerance. The 32nd harmonic limit
was chosen as the FFT algorithm was allowed to sum up to 32 harmonics to
search for periodic signals. We also lowered the default low-frequency cutoff
from one to 0.5 Hz to accommodate long-period signals. The period tolerance
was defined as

$\text{Tol}=\frac{P^{2}\times D}{T}\times 1.2$ (3)

where $D$ is the duty cycle of the signal (which was set to $\sim$ 4%) and $T$
is the total signal time (4800 s). This was formulated such that the
difference in the detected period and original period should not be larger
than the full-width half max of the pulse. This ensured that signals folded at
the detected period retained a high S/N and resembled the original signal. The
additional factor of 1.2 was enforced as the tolerance was too strict and did
not reflect a realistic tolerance for surveys, which are more flexible for the
detection period.

RIPTIDE was our chosen implementation for FFA. It comes with a search
pipeline, rffa, which uses all the tools within the package to search data for
periodic signals. This pipeline must be accompanied by a configuration file
that specifies the search parameters, such as DM, S/NFFA cut-off, running
median width, period range, etc.

####  2.2.2 SPS

SPS is not a periodicity search; for a given candidate, single_pulse_search
(PRESTO) primarily returns a DM and detection significance, $\sigma_{\rm SPS}$
and S/NSPS. The $\sigma_{\rm SPS}$ is calculated as $\rm S/N_{\rm
raw}/\rm\sqrt{bin\\_width}$ and is reported in a text file, and S/NSPS is
reported in a diagnostic plot; however, the two values are identical, implying
that the plot may have intended to display $\sigma_{\rm SPS}$ but it was
incorrectly labelled, or vice versa. Hence in this paper, we will only report
$\sigma_{\rm SPS}$. Note that the bins mentioned here refer to time bins of
the convolved time series generated by single_pulse_search. As mentioned
earlier, all our signals were generated with an arbitrary DM of 150 cm-3 pc.
If SPS produced a candidate above a significance of $6\sigma_{\rm SPS}$, that
would be considered a detection. This cutoff is similar to the cut-off of
5$\sigma_{\rm SPS}$ used by (Sanidas et al., 2019). For FFA and FFT, the S/N
cut-off of 6 was based on other surveys, however, for SPS, the cut-off was
chosen based on trials with false candidates. We intentionally de-dispersed
the signals at various incorrect DMs and found all false candidates at
$\sigma_{\rm SPS}<6$, motivating the choice of $\sigma_{\rm SPS}=6$ as our
threshold for real detections.

Realistically, pulsar data are trialled at multiple DMs as a real pulsar
signal retains its signal strength over multiple DMs, as described by Cordes &
McLaughlin (2003). Detections of a transient at multiple DMs help confirm the
validity of a candidate. However, the aforementioned approach would only
assist the detection quality, if the quality was assessed by a machine
learning algorithm or by visual inspection. In this work, we are testing the
algorithm’s ability to find simulated signals, hence the quality of detection
is irrelevant.

##  3 RESULTS FROM SIMULATED DATA

We simulated pulsar signals with varying periods, nulling fractions, nulling
durations and S/N per pulse. Each combination of parameters had 1000 signals
generated. These were searched using PRESTO’s accelsearch (FFT), RIPTIDE’s
rffa (FFA) and PRESTO’s single_pulse_search (SPS).

The results from our simulations are summarised in Figures 3 & 4. They are
shown as heatmaps with the colour representing the percentage of simulated
signals detected out of 1000, as indicated by the colour bar. Each figure has
7 heatmap panels with a progressively increasing S/NPP. Each of the 7 plots
shows the period of the signal vs. either the nulling fraction (Figure 3) or
nulling duration (Figure 4). In either case, the nulling parameter that is not
plotted has been averaged in order to make the data presentable. For example,
Figure 3 plots the nulling fraction and rotation period, hence the information
relating to the nulling duration has been averaged.

In general, all search methods share some common characteristics. The number
of detections increases for all methods as the S/NPP increases. There are also
more detections made for shorter periods. This is reconcilable as signals with
shorter periods and brighter pulses tend to be present more frequently, even
at high values of nulling parameters (i.e. $n_{\rm f}$ and $N_{\rm d}$).

When comparing nulling duration and period, a slight decrease in the number of
detections is seen between FFA and FFT. This may be due to our approach of
drawing the nulling duration values from a Poisson distribution. At low mean
values ($\sim 5$), the Poisson distribution has a low chance of outputting 0.
Based on equation 1, if the null duration $N_{\rm d}$ is 0 pulses, the burst
duration $B_{\rm d}$ would also be 0 pulses. Effectively, this would result in
an additional null sequence, deviating the mean null duration to a larger
value. This is a possible reason why the number of detections is lower for
$N_{\rm d}=5$ pulses than for consecutive higher nulling durations.

###  3.1 Benchmarking

PRESTO’s accelsearch was the fastest process and took typically $\lesssim 0.5$
s to search a signal summing up to 8 harmonics; however, when summing up to 32
harmonics searching took up to 2-3 s on some occasions. The accelsearch
routine was used with mostly default settings, searching for signals with 0
acceleration, summing up to 32 harmonics and a lower frequency minimum of 0.5
Hz to accommodate for the long-period signals. PRESTO’s single_pulse_search
was the second fastest, computing for $\lesssim 1$ s for low S/NPP ($<0.5$)
signals and 1 s - 2 s for high S/NPP (0.5 - 1) signals. This was also used
with the default option, meaning the width of boxcar filters used were $\leq
30$ bins; if the whole range of widths were used (1 - 300 bins), the
processing time would likely be longer. Finally, RIPTIDE’s rffa was the
slowest implementation, which could range 3 s - 20 s depending on the period
ranges searched and the definition of other parameter ranges in the
configuration file. Note these process times were from a local computer and
not a supercomputer, hence computing times may change with devices and
different algorithm implementations. 111Local machine has an AMD Ryzen 6
core/12 thread CPU and 2x32GB RAM

###  3.2 FFT

Overall accelsearch performed well in detecting long-period and nulling
signals, contrary to common perception. The performance was comparable to that
of FFA’s; however, FFT had noticeably fewer detections at low S/NPP (0.1 -
0.5) and at high nulling fractions ($n_{\rm f}$ = 0.5 - 0.9). FFT’s better
performance is likely due to the very long observation length (4800 s), which
can greatly increase the sensitivity to long-period and nulling signals. We
did not consider effects such as RFI and red noise, which can heavily limit
the detection of long-period signals in the Fourier space (Parent et al.,
2018; Cameron et al., 2017; van Heerden et al., 2017). The duty-cycle was also
kept constant at 4%, rather than decreasing with increasing period; a narrow
duty-cycle has been shown to be difficult to detect by accelsearch (Singh et
al., 2022; Morello et al., 2020a).

Another subtlety we noted is more detections for weaker signals (S/NPP $<0.3$)
at higher periods at $n_{\rm f}=0.9$ (top four panels of Figure 3(a)). At
higher S/NPP, this reverses to more detections at lower periods, which is the
expected trend.

Initially, we processed the signals summing up to 8 harmonics (by default) for
all period ranges; however, this introduced some inaccuracies and artefacts in
the simulated results. As the candidate period is dependent on the highest
harmonic summed, the candidate periods were not precise (e.g. the number of
decimal places), which led to a decrement in detections (for $P=1.5$ s and $3$
s), and some periods could not be detected ($P=10$ and $50$ s) at low S/NPP.
When reprocessing the FFT search, summing up to 32 harmonics, we found a
significant increase in the number of long-period detections and an increase
in the precision of the detection period.

###  3.3 FFA

RIPTIDE’s rffa was very sensitive to all periods, signal strengths and nulling
properties. It had the same if not more detections than accelsearch for each
parameter set and was more consistent than single_pulse_search in detecting
signals at multiple periods and of multiple strengths. Aside from the expected
decrease in the number of detections for very high nulling properties at high
periods, FFA was efficacious for a large range of parameters in comparison to
FFT and SPS.

###  3.4 SPS

PRESTO’s single_pulse_search was primarily only effective at high S/NPP, as
one would expect. Periodicity searches fold individual pulses to gain an
integrated S/N, however, single_pulse_search does not benefit from that. At
best, the algorithm can downsample the time series in order to increase the
combined $\sigma_{\rm SPS}$. This is why single_pulse_search has very few
detections until S/NPP= 0.5.

The algorithm shows a clear preference for signals with periods between 1.5 s
- 10 s as shown in the top right plots of Figures 3(c) & 4(c). The
single_pulse_search routine evaluates $\sigma_{\rm SPS}$ of the pulse based on
the matched filters of different widths and their convolution with the pulse.
All pulses have the same duty cycle, meaning the pulse width scales with the
period. Hence a matched filter would favour the larger period, which would
have wider pulses. This is likely the reason why there are fewer detections
for smaller periods at small S/NPP for single_pulse_search.

However, there are a discrete number of matched filters with different widths,
and as mentioned previously, the maximum widths of the filters by default
range up to 30 bins. As the increment step ($dt$) of the time series is
$3.2\times 10^{-3}$ s, the width of the pulses for the periods under
consideration go as 6.25, 12.5, 18.75, 37.5, 62.5, 125 and 625 bins. For $P=5$
s where the pulse is $\sim$60 bins wide, which is double the maximum width
allowed for the box car filter, the filtering returns large $\sigma_{\rm SPS}$
which can allow for better detections; however, for larger periods, the pulse
is too large and the maximum filter width of 30 is not able to encapsulate a
significant enough portion of the pulse. This limits the $\sigma_{\rm SPS}$
and hence the detectability. Hence the feature we see is due to the choices of
the filters and the shape of the pulses.

At large S/NPP, single_pulse_search performs effectively for both nulling
parameters, and at times even performs better than rffa. Although it may be
less meaningful to compare SPS with FFA or FFT as their methods have different
criteria for detections, it is important to compare the efficacy of these
methods to have the best likelihood of finding long-period and nulling
pulsars.

(a) FFT

(b) FFA

(c) SPS

Figure 3: This figure shows the result of searching the simulated data with
FFT (a), FFA (B) and SPS (c). The x-axis shows the period (s) and the y-axis
shows the nulling fraction. The 7 plots inside the subfigures show heatmaps
with the colour of each pixel indicating the percentage of signals, with the
indicated period and nulling fraction, detected with the corresponding search
method. This is accompanied by the colour bar on the right-hand side. The 7
plots represent different S/NPP and are labelled as such.

##  4 RESULTS FROM REAL DATA

###  4.1 Observational data

We have compared the performance of various search methods on real survey
data; we made use of example data from the ongoing SMART survey and measured
the detection significance to assess the search methods. Specifically, we made
use of data from the discovery and follow-up observations of new pulsars from
the SMART survey. In all cases, the original detections were made using
PRESTO’s FFT implementation; the main purpose is to assess whether the
inclusion of other search methods would have led to quicker or more
significant detections of these pulsars. Observations of real pulsars used in
this study are listed in Table 1.

The observation lengths varied from 600 s to 4800 s, depending on the data
that were readily available for analysis. Although all five pulsars have
rotation periods of $\sim 1$ s - 2 s, two pulsars (J0026-1955 & J0452-3418 )
also show significant amounts of nulling ($\sim$ 50% and 33% respectively for
PSRs J0026-1955 & J0452-3418 ; cf. McSweeney et al. 2022; Grover2024, making
them good targets for testing the efficacy of the methods explored in this
work.

Table 1: The details of the observations and pulsars therein were used for the sanity check analysis. For more on these pulsars (excluding J0452-3418 ) see Bhat et al. (2023b). Analysis on J0452-3418 (Grover2024). Pulsar | Period (s) | DM cm-3 pc | S/NFFA | S/NFFT | $\sigma_{\rm FFT}$ | $\sigma_{\rm SPS}$ | Observation length (s) | Observation MJD
---|---|---|---|---|---|---|---|---
J0036-1033 | 0.90 | 23.1(2) | 15.0 | 22.2 | 9.7 | 5.3 | 600 | 58774
J0036-1033 | 0.90 | 23.1(2) | 17.5 | 23.7 | 10.6 | 6.0 | 2000 | 58792
J0026-1955 | 1.30 | 23.81(1) | 14.5 | 21.5 | 9.4 | 9.2 | 4800 | 58792
J0026-1955 | 1.30 | 23.81(1) | 62.0 | 54.6 | 26.4 | 14.0 | 4800 | 58799
J1002-2036 | 1.67 | 42.28 | 14.6 | 18.9 | 8.0 | 5.7 | 600 | 58890
J1357-2530 | 0.91 | 16.04 | 11.5 | 19.0 | 8.2 | 5.3 | 900 | 59309
J0452-3418 | 1.66 | 19.77(2) | 100.9 | 118.2 | 58.1 | 10.1 | 4200 | 58757

(a) FFT

(b) FFA

(c) SPS

Figure 4: This figure is very similar to Figure 3. It shows the result of
searching the simulated data with FFT (a), FFA (B) and SPS (c). The x-axis
shows the period (s) and the y-axis shows the nulling duration. The 7 plots
inside the subfigures show heatmaps with the colour of each pixel indicating
the percentage of signals, with the indicated period and nulling duration,
detected with the corresponding search method. This is accompanied by the
colour bar on the right-hand side. The 7 plots represent different S/NPP and
are labelled as such.

###  4.2 Quantitative performance

S22 compared the efficacies of FFA and FFT using S/N as their detection
significance. They report RIPTIDE’s S/NFFA to be much larger than PRESTO’s
S/NFFT for both simulated and real data. PRESTO’s accelsearch reports a
detection probability in the form of a different parameter, $\sigma_{\rm
FFT}$, which may mistakenly be compared with rffa’s S/NFFA. Here, we have
presented both values for a clear comparison (see Table 1).

We find S/NFFA to be consistently larger than $\sigma_{\rm FFT}$, which is not
meaningful since $\sigma_{\rm FFT}$ is essentially a probability statistic.
However, we also find that S/NFFT is larger in most cases than the S/NFFA,
which is opposite to the findings of S22.

SPS, on the other hand, provides a $\sigma_{\rm SPS}$, which is a multiple of
the S/Nraw. This value is consistently lower than $\sigma_{\rm FFT}$ value;
however, the $\sigma_{\rm FFT}$ is based on many other parameters as well as
the integrated profile, whereas the $\sigma$SPS is based on individual pulses
and the widths of the filters. Interestingly, we note that SPS returns a very
high $\sigma$SPS value for both nulling pulsars, with $\sigma$SPS $\sim$ 9 and
10 for J0026-1955 and J0452-3418 , respectively (as shown in Table 1).

##  5 DISCUSSION

On the basis of the analysis, we now comment on the efficacies of various
search methods in detecting long-period and/or nulling pulsars. This is
especially relevant for searches using wide-field instruments that also employ
longer dwell times (e.g., the SMART and LOTAAS surveys).

###  5.1 Nulling duration vs Nulling fraction

As described earlier (Section 2), nulling was characterised using two basic
parameters: nulling fraction and nulling duration. Our results are shown in
Figures 3 & 4, where the detections (for signals with varying $n_{\rm f}$ and
$N_{\rm d}$) are a function of the rotation period. At first glance, the
number of detections for both parameters is similar for all search methods,
particularly as they approach larger values of $n_{\rm f}$ and $N_{\rm d}$ at
large S/NPP. It is difficult to compare $n_{\rm f}$ and $N_{\rm d}$ directly
as the definition of ’large’ is different for the two parameters. $n_{\rm f}$
has a clearly defined minimum and maximum value, whereas the range of $N_{\rm
d}$ values are based entirely on observations.

Based on our analysis, the detections tend to vary more with an increase in
$n_{\rm f}$ than that in $N_{\rm d}$, for the periodicity searches. At low
S/NPP, the number of detections is roughly constant for all values of $N_{\rm
d}$. For larger S/NPP, the detections tend to plateau near two to three
values, where the number of detections remains consistent for some consecutive
values of $N_{\rm d}$. This is only visible for FFT at higher S/NPP, whereas
for FFA, this is also observed at low S/NPP. The plateauing is likely due to
coarse increment sizes for $N_{\rm d}$.

The increments in $N_{\rm d}$ could be increased to better sample the range
covered, allowing for a clear gradual change in the number of detections. In
any case, the range we cover likely encapsulates the majority of nulling
pulsars, as only a small percentage are known to null for longer durations,
and as such, for larger values, we may expect to see only a small number of
pulses from them. This is discussed further in Section 5.3.

###  5.2 Comparison of the search methods on simulated data

To analyse the efficacy of FFA, FFT and SPS implementations in detecting long-
period and nulling pulsars, we used analysis of simulated data. As there were
$1.764\times 10^{6}$ individual time series streams to process, the
computation time for the three search methods was essential to understand.
This is a key factor that will influence the consideration of these methods
for future surveys. Of all the three methods, the FFT implementation is the
fastest and performs up to an order of magnitude faster compared to the other
two. FFA implementation is the slowest. Although the results in Figures 3(b) &
4(b) show FFA to be more sensitive than FFT, FFT is computationally efficient.
For long-duration surveys (e.g., the SMART survey, which can produce tens of
millions of candidates), the processing efficiency may outweigh the benefits
of extra sensitivity.

The results presented in Section 3 suggest that FFA and FFT are surprisingly
effective at finding pulsars with high nulling properties at low to moderate
rotation periods. At larger periods and high nulling properties, both methods
lose effectiveness as the signals become less periodic and less frequent.

To further elaborate, rffa accurately and consistently detects pulsars
throughout all S/NPP. PRESTO’s accelsearch is also very effective, however, it
has fewer detections at low signal strengths (S/NPP=0.1 - 0.5) and high
nulling fractions ($n_{\rm f}$ = 0.5 - 0.9). For longer periods ($P\geq 10$
s), it heavily relies on the harmonics sums to be $\geq 16$ for the signal to
be detected accurately at the fundamental frequency.

Our analysis also suggests that SPS is most effective in detecting long-period
signals at high nulling fractions and durations, albeit this applies primarily
at high S/NPP. Although SPS may miss a large majority of generic and low-
luminosity pulsars that FFT and FFA may detect, it is effective in detecting
bright pulses, especially when emission is sporadic. This is vividly
demonstrated by the way the extremely long-period pulsar, J0901-4046, was
discovered, where the initial detection was made by searching for individual
dispersed pulses (Caleb et al., 2022).

###  5.3 Critique of the Simulation

Our choice of a detection significance above 6 was based on generic cut-offs
for candidate selection typically employed in search pipelines. For SPS, a
threshold of 6$\sigma$ was based on experiments with incorrectly de-dispersed
data as described in Section 2.2.2. If the threshold was to be increased from
6 to 7 for example, it would primarily affect SPS. As FFA and FFT integrate
the signal and base their significance values on the sum, their values are
generally higher than 6, especially for simulated data in ideal noise. SPS’s
detection significance is based on the strength of individual pulses. Hence
increasing the threshold from 6 to 7 would likely further decrease the
detections made by SPS at low S/NPP. However, by definition, this cut-off
would also lead to more probable detections.

The periods analysed in this study were spaced logarithmically to cover a
large range of values. However, this led to large jumps in periods,
specifically between 10 s - 50 s, which reflect as drastic changes in the
detectability of the search methods in Figures 3 & 4. Conducting a simulation
for these intermediate periods would reveal more precisely how sensitive the
methods are to detecting pulsars like J2251-3711 (Tan et al., 2018) and
J0901-4046 (Caleb et al., 2022).

The limitation in terms of time and computational resources prompted us to
consider simpler characteristics of pulsar signals; for example, we considered
pulse shapes that are composed only of repeating Gaussian-shaped pulses with
an additive Gaussian noise component to mimic receiver noise. This was
sufficient for our purpose. Furthermore, as mentioned earlier in Section 2,
the data were de-dispersed at a single DM of 150 cm-3 pc. Exploring the impact
of incorrect dedispersion (i.e. true DM different from an assumed DM) will
involve more computation and resources, but can help to assess loss in
detection significance resulting from an error in DM.

Our analysis did not consider the effect of _red noise_ that may arise from
variations in receiver noise or gain or varying levels of RFI. As such, such
effects tend to be specific to the characteristics of the instrument and
therefore a generalisation may not be meaningful. In any case, we note that
the inclusion of red noise would have affected the periodicity searches,
making detections more difficult (van Heerden et al., 2017). We also note that
S22 performed some useful comparison in this regard, between RIPTIDE and
PRESTO, highlighting a significant reduction in the detection S/N in PRESTO
searches than in RIPTIDE searches.

Furthermore, in our simulations, we allowed nulling to occur randomly (see
Section 2 for details). The choice of our considerations also meant that for
some data sets, no pulses were present, due to either a high nulling fraction
or a large nulling duration. In some ways, this reflects the realistic
expectations in searching for objects with high degrees of intermittency. As a
result, detections for S/NPP = 5 and 10 look very similar for all three
searches, and highlights the difficulty in finding such signals, regardless of
their brightness.

The modulation was fixed to a value of 0.1, and we did not test how changing
it might affect the different search methods. We can anticipate, however, that
it would not have a strong effect on periodicity searches, which primarily
rely on the integrated signal power and only to a lesser extent on the
“equivalent” red noise introduced by the modulation. However, a change in
modulation would affect the SPS; e.g., a larger modulation index will allow
certain pulses in the signal to be more readily detectable. I.e.,
intrinsically low S/NPP signals would still have a higher probability of
detection by SPS, especially when certain pulses are temporally boosted in
brightness as a result of modulation. This can be relevant for objects like
giant pulse emitters and RRATs that tend to modulate heavily and emit bright
pulses.

The duty cycle of the pulse was also kept constant in our simulations. We note
that previous studies (e.g. Morello et al. 2020a, S22) have demonstrated the
loss of sensitivity in FFT searches for narrow duty cycles. We therefore chose
to keep this constant. Long-period pulsars tend to have very small duty
cycles, $\lesssim 1\%$; however, a recently discovered pulsar-like source with
a period of $\sim 20$ min showed a duty cycle of up to 25%. For such small
duty cycles, FFT’s detection ability will be somewhat hindered, specifically
for lower harmonics (Morello et al., 2020a).

###  5.4 Real data

In order to verify the performance of various search methods, we tested rffa,
accelsearch and single_pulse_search on examples of real data, chosen from
SMART data sets that are currently being processed for a first-pass shallow
survey (cf. Bhat et al. 2023b).

As summarised in Table 1, both FFT and FFA comfortably found all five pulsars
in all observations that we used. Based on the criteria employed for the
simulated analysis, all sources would be classified as successful detections.
We note that SPS finds only two pulsars above a significance of 6; both
nulling and bright enough to conduct single pulse analysis. This demonstrates
that SPS is primarily sensitive to detecting bright pulsars.

In general, we find S/NFFT to be higher than S/NFFA. The difference in S/N may
arise due to the differences in the measurement of this value. For instance,
FFA integrates the pulses in the time domain and measures the S/N from the
combined profile, therefore the S/NFFA is a function of the trial period and
the trial width of the matched filters and is based on a defined Z-statistic
(Morello et al., 2020a). FFT instead sums the square root of the amplitude of
the harmonics in the Fourier space to calculate S/NFFT (Ransom, 2001).
Theoretically, for an S/N measured in the Fourier domain to equal one in the
time domain, the power in all harmonic bins (up to infinity) should be summed.
This implies either RIPTIDE is underestimating the S/N or PRESTO is
overestimating the S/N.

However, S22 report that S/NFFA is up to orders of magnitude larger than
S/NFFT for a majority of their sources. A similar difference is seen in our
case only when we compare S/NFFA to $\sigma$FFT, the latter being the commonly
adopted metric in accelsearch candidates.

###  5.5 Boolean vs Quantitative comparison

As mentioned earlier, comparing the different search methods by merely using
commonly adopted detection significance may be less meaningful than using a
standard applicable to all methods. Unravelling how the different values are
connected is a complex task as there are many parameters and calculations
involved. This can lead to unfair comparisons between search methods as many
generally assume a larger detection significance means a stronger detection;
however, if the detection significance parameters are defined differently by
two methods, then the larger value may not always mean a better detection.

Further, this can also lead to confusion about which numbers to compare when
conducting a thorough analysis. SPS lists $\sigma$SPS as a detection
significance of individual pulses; however, FFT or FFA perform periodicity
searches where they consider the sum of the pulses. Hence it is not meaningful
to directly compare their S/N’s and $\sigma$ values. Similarly, the
$\sigma$FFT should not be compared to S/NFFA as they also measure different
properties. This is made further difficult as the PRESTO implementation of FFT
does not initially report a S/NFFT; it is obtained once candidates are pruned
with quick_prune_cands or ACCEL_sift.

Such inherent differences prompted us to consider a Boolean approach in our
simulations where we ascertain the efficacies of the search methods based on
their ability to make a detection. Although there may still be some bias due
to the definition of the detection criteria, it is a more meaningful
comparison for survey needs.

###  5.6 Recommendations for future surveys

Based on the performance of FFT, FFA and SPS on our simulated data, we outline
some considerations to guide their effective use in long-duration surveys such
as the SMART survey (see Table 2).

Being a computationally efficient and inexpensive algorithm, FFT has been very
effective for pulsar searching. Our analysis suggests that it is still the
most effective method for searching pulsars with $P\lesssim 10$ s. Summing up
to 32 harmonics, we found many long-period and nulling candidates, however,
there was no red noise injected and the duty cycle of signals was kept
constant, which can affect the effectiveness of FFT as noted by Singh et al.
(2022); Morello et al. (2020a). FFT also detected fewer signals than FFA at
low S/NPP (0.1 - 0.5) and at high nulling fractions ($n_{\rm f}=0.5-0.9$).
Overall, the maximum period range that could be searched with FFT should be
$P\lesssim 10$ s. This accounts for the vast majority of the pulsar
population. This upper limit also means that the number of harmonics summed
can be smaller (16 instead of 32), hence the processing can be faster.

FFA, in general, shows sensitivity throughout the parameter space; however, it
is the slowest of the three search methods (by 1-2 order of magnitudes on
local machine). As FFT had fewer detections when summing weaker and highly
nulling signals than FFA has been shown to be more sensitive to faint pulsars,
which was also shown by Morello et al. (2020a), FFA should have some overlap
with the FFT searching periods e.g. $P\gtrsim 1$ s, to ensure a more thorough
search. This is also computationally a less expensive option as rffa can be
more than 6 times faster for $P\gtrsim 1$ s (Morello et al., 2020a).

SPS is also a computationally inexpensive algorithm that is independent of the
exact period and can be quite effective in detecting sporadically emitting
sources such as RRATs and FRBs. Most ongoing pulsar surveys routinely use this
as part of their processing pipelines (eg Sanidas et al., 2019; Bhattacharyya
et al., 2016), and as such it is among the future processing plans envisioned
for SMART Bhat et al. (2023a). However, we note it is primarily effective at
detecting bright single pulses. This can be useful in detecting giant pulse
emitters or bright RRATs. Considering the sporadicity aspect, SPS will offer
increased sensitivity for detecting new objects, for longer observations. It
appears that a higher detection threshold ($>$6 $\sigma_{\rm SPS}$) can be a
more efficient way of finding bright sporadically emitting objects.

Incorporation of the above considerations will help long-duration survey
processing attain an increased detection sensitivity to a wider spectrum of
pulsars and search a large part of the parameter space. This can be beneficial
for future next-generation surveys with MeerKAT or SKA-Low, which have the
potential to uncover the comprehensive population of radio-emitting neutron
stars in our Galaxy. These can also be applicable to non-traditional search
strategies such as those considered by CHIME where they take daily
observations of the same position in the sky and sum their power spectra
(similar strategies to Cadelano et al., 2018). This allows for an essentially
longer time series which can be useful in detecting long-period and nulling
objects. Consideration of such extended search strategies will also enable
ongoing and future searches to explore possible populations of pulsars that
have been missed earlier due to selection effects.

Search method | Software | Period range |  $\sigma$ or S/N threshold
---|---|---|---
FFT | PRESTO |  $\lesssim$ 10 s |  $\sim$ 6
FFA | RIPTIDE |  $\gtrsim$ 1 s |  $\sim$ 6
SPS | PRESTO | - |  $>$ 6
Table 2: A summary of the recommendations for the three search algorithms
based on this work.

##  6 CONCLUSION

To develop a more comprehensive picture of pulsar emission properties and
various aspects relating to their populations, it is important to extend
current search approaches to include algorithms and techniques that will allow
a more extensive exploration of the long-period parts of the parameter space.
This can potentially lead to an improved understanding of pulsar death lines
and the related physical modes, alongside yielding new insights into linking
nulling to long periods, as hinted by some of the recent discoveries. With the
advent of wide-field radio-astronomy facilities like the MWA and LOFAR, and
their adoption for large (untargeted) pulsar surveys, longer dwell times per
pointing become more affordable, and this can be well exploited for increasing
the detection sensitivity to such long-period and nulling pulsars. Popular
search algorithms and software tend to vary both in their efficacy and
sensitivity to the related classes of objects. Motivated by this, we explored
the efficacies of different search methods in detecting long-period and
nulling pulsars, using both simulations and examples of real data. The
software used includes PRESTO’s FFT implementation (accelsearch), RIPTIDE’s
FFA implementation (rffa) and PRESTO’s SPS implementation
(single_pulse_search). The simulation analysis was approached from the
perspective of exploring a wider range in parameter space.

Our analysis finds FFA to be most sensitive to pulsar signals over a much
wider range in period ($\sim$ tens of seconds) even when the signal strengths
are low; however, it is also slower than FFT. While FFT is computationally
more efficient and generally sensitive towards long periods, it loses
sensitivity for very weak and highly nulling signals . SPS on the other hand
is primarily sensitive to sporadic or intermittent emission, but largely
sensitive to high signal strengths. Therefore, a more effective (if not ideal)
search strategy could include the use of FFT searches for $P\lesssim 10$ s,
FFA searches for $P\gtrsim 1$ s, alongside SPS with a detection threshold of
$>6\sigma$, and preferably with long observations ($>10$ minutes), for
increased sensitivity to intermittent or sporadic emitters. While such a
strategy was derived from simulations and examples of real data selected from
SMART survey data sets, they are also applicable for many of the current and
planned large pulsar surveys, including those envisioned with SKA-Low. In
general, more extensive explorations of the long-period parts of the search
parameter space can help overcome some of the inherent observational biases in
detecting long-period or nulling pulsars, and thus uncover a larger population
(and variety) of objects that will help us to develop a better understanding
of pulsar population and emission physics.

The authors would like to acknowledge Scott Ransom and Vincent Morello for
discussions and advice on optimal ways to run the search software. Bradley
Meyers and Chia Min Tan for their help in the analysis and review process, and
Edwin Dickinson for their useful comments.

This scientific work uses data obtained from Inyarrimanha Ilgari Bundara / the
Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamaji
People as the Traditional Owners and native title holders of the Observatory
site. Establishment of CSIRO’s Murchison Radio-astronomy Observatory is an
initiative of the Australian Government, with support from the Government of
Western Australia and the Science and Industry Endowment Fund. Support for the
operation of the MWA is provided by the Australian Government (NCRIS), under a
contract to Curtin University administered by Astronomy Australia Limited.
This work was supported by resources provided by the Pawsey Supercomputing
Research Centre with funding from the Australian Government and the Government
of Western Australia.

This work was supported by resources awarded under Astronomy Australia Ltd’s
ASTAC merit allocation scheme on the OzSTAR national facility at Swinburne
University of Technology. The OzSTAR program receives funding in part from the
Astronomy National Collaborative Research Infrastructure Strategy (NCRIS)
allocation provided by the Australian Government, and from the Victorian
Higher Education State Investment Fund (VHESIF) provided by the Victorian
Government.

## References

  * Anumarlapudi et al. (2023) Anumarlapudi, A., Swiggum, J. K., Kaplan, D. L., & Fichtenbauer, T. D. J. 2023, ApJ, 948, 32
  * Arons & Scharlemann (1979) Arons, J., & Scharlemann, E. T. 1979, ApJ, 231, 854
  * Bhat et al. (2023a) Bhat, N. D. R., Swainston, N. A., McSweeney, S. J., et al. 2023a, PASA, 1
  * Bhat et al. (2023b) Bhat, N. D. R., Swainston, N. A., Mcsweeney, S. J., et al. 2023b, Publications of the Astronomical Society of Australia, 1
  * Bhattacharyya et al. (2016) Bhattacharyya, B., Cooper, S., Malenta, M., et al. 2016, ApJ, 817, 130
  * Burke-Spolaor & Bailes (2010) Burke-Spolaor, S., & Bailes, M. 2010, MNRAS, 402, 855
  * Burke-Spolaor et al. (2012) Burke-Spolaor, S., Johnston, S., Bailes, M., et al. 2012, MNRAS, 423, 1351
  * Cadelano et al. (2018) Cadelano, M., Ransom, S. M., Freire, P. C. C., et al. 2018, ApJ, 855, 125
  * Caleb et al. (2022) Caleb, M., Heywood, I., Rajwade, K., et al. 2022, Nature Astronomy, doi:10.1038/s41550-022-01688-x
  * Cameron et al. (2017) Cameron, A. D., Barr, E. D., Barr, E. D., et al. 2017, Monthly Notices of the Royal Astronomical Society, 468, 1994
  * Chen & Ruderman (1993) Chen, K., & Ruderman, M. 1993, The Astrophysical Journal, 402, 264
  * Cordes & McLaughlin (2003) Cordes, J. M., & McLaughlin, M. A. 2003, The Astrophysical Journal, 596, 1142
  * Han et al. (2021) Han, J. L., Wang, C., Wang, P. F., et al. 2021, Research in Astronomy and Astrophysics, 21, 107
  * Hurley-Walker et al. (2023) Hurley-Walker, N., Rea, N., & McSweeney, S. 2023, Nature, 619, doi:10.1038/s41586-023-06202-5
  * Hurley-Walker et al. (2022) Hurley-Walker, N., Zhang, X., Bahramian, A., et al. 2022, Nature, 601, 526
  * Keane & McLaughlin (2011) Keane, E. F., & McLaughlin, M. A. 2011, Bulletin of the Astronomical Society of India, 39, 333
  * Keith et al. (2010) Keith, M. J., Jameson, A., van Straten, W., et al. 2010, MNRAS, 409, 619
  * Kondratiev et al. (2009) Kondratiev, V. I., McLaughlin, M. A., Lorimer, D. R., et al. 2009, ApJ, 702, 692
  * McSweeney et al. (2022) McSweeney, S. J., Bhat, N. D. R., Swainston, N. A., et al. 2022, ApJ, 933, 210
  * Michilli et al. (2018) Michilli, D., Hessels, J. W. T., Lyon, R. J., et al. 2018, MNRAS, 480, 3457
  * Morello et al. (2020a) Morello, V., Barr, E. D., Stappers, B. W., Keane, E. F., & Lyne, A. G. 2020a, Monthly Notices of the Royal Astronomical Society, 497, 4654
  * Morello et al. (2020b) Morello, V., Keane, E. F., Enoto, T., et al. 2020b, Monthly Notices of the Royal Astronomical Society, 493, 1165
  * Parent et al. (2018) Parent, E., Kaspi, V. M., Ransom, S. M., et al. 2018, The Astrophysical Journal, 861, 44
  * Ransom (2001) Ransom, S. M. 2001, PhD thesis, Harvard University, Massachusetts
  * Rea et al. (2023) Rea, N., Hurley-Walker, N., Pardo-Araujo, C., et al. 2023, arXiv e-prints, arXiv:2307.10351
  * Ritchings (1976) Ritchings, R. T. 1976, MNRAS, 176, 249
  * Ruderman & Sutherland (1975) Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51
  * Sanidas et al. (2019) Sanidas, S., Cooper, S., Bassa, C. G., et al. 2019, A&A, 626, A104
  * Sheikh & Macdonald (2021) Sheikh, S. Z., & Macdonald, M. G. 2021, Monthly Notices of the Royal Astronomical Society, 502, 4669
  * Singh et al. (2023) Singh, S., Roy, J., Bhattacharyya, B., et al. 2023, ApJ, 944, 54
  * Singh et al. (2022) Singh, S., Roy, J., Panda, U., et al. 2022, ApJ, 934, 138
  * Singh et al. (2023) Singh, S., Roy, J., Sunder, S., Bhattacharyya, B., & Kudale, S. 2023, The Astrophysical Journal, 947, 88
  * Staelin (1969) Staelin, D. 1969, Proceedings of the IEEE, 57, 724
  * Stovall et al. (2014) Stovall, K., Lynch, R. S., Ransom, S. M., et al. 2014, ApJ, 791, 67
  * Su et al. (2023) Su, W. Q., Han, J. L., Wang, P. F., et al. 2023, arXiv e-prints, arXiv:2305.16754
  * Tan et al. (2018) Tan, C. M., Bassa, C. G., Cooper, S., et al. 2018, The Astrophysical Journal, 866, 54
  * Tingay et al. (2013) Tingay, S. J., Goeke, R., Bowman, J. D., et al. 2013, PASA, 30, e007
  * van Haarlem et al. (2013) van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2
  * van Heerden et al. (2017) van Heerden, E., Karastergiou, A., & Roberts, S. J. 2017, MNRAS, 467, 1661
  * Weltevrede et al. (2006) Weltevrede, P., Edwards, R. T., & Stappers, B. W. 2006, A&A, 445, 243
  * Wongphechauxsorn et al. (2024) Wongphechauxsorn, J., Champion, D. J., Bailes, M., et al. 2024, MNRAS, 527, 3208
  * Zhang et al. (2007) Zhang, B., Gil, J., & Dyks, J. 2007, MNRAS, 374, 1103
  * Zhang et al. (2000) Zhang, B., Harding, A. K., & Muslimov, A. G. 2000, The Astrophysical Journal, 531, L135


# Killing tensors of generalized Lense-Thirring spacetime

Saeedeh Sadeghian<EMAIL_ADDRESS>Department of Theoretical Physics,
Faculty of Science, University of Mazandaran, P.O. Box 47416-95447, Babolsar,
Iran  ICRANet-Mazandaran, University of Mazandaran, P. O. Box 47415-416,
Babolsar, Iran.

###### Abstract

We investigate the Hamilton–Jacobi equation of a probe particle moving on
$d$-dimensional generalized Lense-Thirring metric. This spacetime is different
from the slowly rotating Myers-Perry black hole at second order in rotation
parameters. We show that the dynamics of the probe particle along the timelike
geodesic of the generalized Lense-Thirring spacetime is superintegrable and
has more constants of motion with respect to the same dynamics on Myers-Perry
black hole. We also discuss the second rank Killing tensors associated with
the generalized Lense-Thirring metric.

##  I Introduction

The spacetime outside a slowly rotating massive object is described by the
Lense-Thirring metric [1]. This metric is also given by the slow rotation
limit of a Kerr black hole to first order in rotation parameter. Hence, this
metric is an approximate solution to the field equation of the pure Einstein
theory. Another description for this spacetime is that it appears as the
asymptotic (large $r$) behavior of the Kerr metric. Since stationary, this
spacetime does not follow the four-dimensional uniqueness theorem. Therefore,
different rotating objects can be described by Lense-Thirring metric
asymptotically. So, we can appropriately add a coordinate-dependent term of
second order in the rotation parameter to the metric which makes the analysis
of the new metric simpler[2, 3]. Although this change gives a different
spacetime, it still satisfies Einstein’s equation to the first order in
rotation parameter.

This spacetime inherits the rich symmetry structure from the Kerr black hole.
The study of geodesic equations on Lense-Thirring spacetime shows that the
dynamics of the probe particle along the timelike geodesic is integrable[3]
and one constant of motion is associated with the nontrivial Killing tensor of
the second rank. This is similar to the Carter constant [4] for the probe
particle dynamics on a Kerr black hole.

The Killing tensors of rank-$n$ which satisfy the following equation,

$\displaystyle\nabla^{\\{\mu}K^{\nu_{1}\cdots\nu_{n}\\}}=0\,,$ (1)

are the generators of some symmetry actions on the phase space of the probe
particle of the curved spacetime, as the generalization of the Killing
vectors. However, the projection of their action on the configuration space is
null, unlike the action of Killing vectors. In this sense, they are associated
with _hidden_ symmetries. There are some cases in which the second rank
Killing tensor of the $d$-dimensional spacetime, $K^{ab}$, is written as the
“square” of a Killing-Yano tensor, $Y$, of rank $(d-2)$, i.e.

$\displaystyle K=Y\cdot
Y\,;\quad\nabla_{(\mu}Y_{\nu)\nu_{1}\cdots\nu_{d-3}}=0\,.$ (2)

Here ($\cdot$) means the contraction of the $(d-3)$ number of the indices. In
such cases, the integrability of the Klein-Gordon equation is guaranteed by
the integrability of the geodesic equation. Then, the Killing-Yano tensor,
$Y$, can be written as the Hodge dual of a second rank closed conformal
Killing tensor $h$,

$\displaystyle Y=\star h\,,$ (3)

the so-called principal tensor[5]. This can be locally given by a 1-form
potential $b$ through

$\displaystyle h=db\,.$ (4)

The existence of principal tensor $h$, plays a crucial role in the
integrability of other probes like Maxwell field on the curved spacetime which
entails hidden symmetries [6, 7, 8, 9, 10, 11] (see [12] for recent review).

The generalization of the four-dimensional Lense-Thirring metric has been
studied by adding Maxwell field, cosmological constant, or by going to the
higher dimensions[13]. This study shows that in higher dimensions some new
hidden symmetries emerge as a result of the slow rotation limit.

In the present work, we start with a brief review of the derivation of
$d$-dimensional generalized Lense-Thirring spacetime from the slowly rotating
Myers-Perry black hole. Then, we develop the hidden symmetry study in [13] by
investigating the Hamilton–Jacobi equation of the probe particle. We show that
this equation is separable in an appropriate coordinate system. Using the
separation constants, we read the associated Killing tensors of the
generalized Lense-Thirring spacetime in $d=2m+1+\epsilon$ dimensions. The
remarkable property of this dynamics is that a U$(m)$ symmetry emerges on the
$t,r=$constant subspace. Therefore, in addition to the mentioned separation
constant, there are more independent constants associated with the second rank
Casimirs of this U$(m)$ symmetry which renders the dynamics superintegrable.
We present the explicit form of the associated second rank Killing tensors and
discuss how their number is restricted by the phase space dimension of the
probe particle moving along the timelike geodesic.

##  II Generalized Lense-Thirring metric

###  II.1  In four dimensions

Expanding the Kerr metric in rotation parameter ($a$), while keeping mass
parameter finite, we have an approximate solution of the Einstein theory up to
order $O(a^{2})$:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+2a\sin^{2}\\!\theta(f(r)-1)dtd\phi$
(5) $\displaystyle+r^{2}\sin^{2}\\!\theta
d\phi^{2}+r^{2}d\theta^{2}+O(a^{2})\,,$

where the metric function $f(r)$ is given by

$\displaystyle f(r)=1-\frac{2M}{r}\,.$ (6)

The Killing tensor of the approximate solution is given by

$\displaystyle K=2\ a\ \partial_{t}\
\partial_{\phi}+\frac{1}{\sin^{2}\\!\theta}(\partial_{\phi})^{2}+(\partial_{\theta})^{2}+O(a^{2})\,.$
(7)

The first term is trivial in the sense that it is the tensor product of two
Killing vectors, $\partial_{t}$ and $\partial_{\phi}$. The spherical symmetry
can be seen in the slow rotation limit ($a\to 0$) as the second and third term
of the above Killing tensor is the metric on a 2-sphere. Based on (4), the
associated principal tensor is generated by

$\displaystyle 2\ b={r^{2}}dt-{ar^{2}\sin^{2}\\!\theta}d\phi+O(a^{2})\,.$ (8)

Adding a term of order $a^{2}$ to the metric to complete the square of the
$t-\phi$ part of the line element, we get the following metric:

$\displaystyle
ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\sin^{2}\\!\theta\Bigl{(}d\phi\\!+\\!\frac{a(f(r)-1)}{r^{2}}dt\Bigr{)}^{2}\\!\\!$
$\displaystyle\hskip 34.1433pt+\ r^{2}d\theta^{2}\,,$ (9)

where the function $f(r)$ is given by (6). This is also a solution to the
Einstein’s equations to the linear order in $a$. The advantage of adding the
$O(a^{2})$ term to the metric is that its analysis is easier than the one in
(5) [2, 3, 13]. Also, the truncation of the Killing tensor in (7) satisfies
the exact Killing tensor equation.

###  II.2 In higher dimensions

We start from the rotating Myers-Perry black hole [14] in $d=2m+1$ dimensions,

$\displaystyle
ds^{2}=-dt^{2}+\sum_{i=1}^{m}(r^{2}+a_{i}^{2})\left(d\mu_{i}^{2}+\mu_{i}^{2}d\phi_{i}^{2}\right)+\frac{\Pi
F}{\Pi-2Mr^{2}}dr^{2}$ $\displaystyle\hskip 28.45274pt+\frac{2Mr^{2}}{\Pi
F}\left(dt+\sum_{i=1}^{m}a_{i}\mu_{i}^{2}d\phi_{i}\right)^{2}\,,$ (10)

where $\mu_{i}$ are azimuthal coordinates, restricted by
$\sum_{i=1}^{m}\mu_{i}^{2}=1$. Here, $M$ and $a_{i}$ are the mass and rotation
parameter, respectively and the metric functions $F$ and $\Pi$ are defined
through

$\displaystyle\Pi=\prod_{i=1}^{m}(r^{2}+a_{i}^{2}),\hskip
11.38092ptF=1-\sum_{i=1}^{m}\frac{a_{i}^{2}\mu_{i}^{2}}{r^{2}+a_{i}^{2}}\,.$
(11)

The isometries of this spacetime are generated by $\partial_{t}$ and
$\partial_{\phi_{i}}$. This metric also admits the principal tensor generated
by the following potential [5]:

$\displaystyle
2b=\left(r^{2}+\sum_{i=1}^{m}a_{i}^{2}\mu_{i}^{2}\right)dt+\sum_{i=1}^{m}a_{i}\mu_{i}^{2}\left(r^{2}+a_{i}^{2}\right)d\phi_{i}\,.\
$ (12)

Taking the slow rotation limit to linear order, we have

$\displaystyle
ds^{2}=-\left(1-\frac{2M}{r^{2m-2}}\right)dt^{2}+r^{2}\sum_{i=1}^{m}\left(d\mu_{i}^{2}+\mu_{i}^{2}d\phi_{i}^{2}\right)$
$\displaystyle\hskip
28.45274pt+\frac{dr^{2}}{1-2M/r^{2m-2}}+\frac{4M}{r^{2m-2}}\sum_{i=1}^{m}a_{i}\mu_{i}^{2}d\phi_{i}dt+O(a_{i}^{2})\,.$

The same procedure as in the four-dimensional case leads to the generalized
Lense-Thirring metric in $d=2m+1+\epsilon$ dimensions[13],

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\sum_{i=1}^{m}\mu_{i}^{2}\Bigl{(}d\phi_{i}+\frac{2Ma_{i}}{r^{2m+\epsilon}}dt\Bigr{)}^{2}$
$\displaystyle\hskip
28.45274pt+r^{2}\,\sum_{i=1}^{m+\epsilon}\\!d\mu_{i}^{2}\\!\,,$ (14)

where $\epsilon=0,1$ for odd/even spacetime dimensions, respectively. The
metric function $f(r)$ is given by

$\displaystyle f(r)\\!=\\!1-\frac{2M}{r^{2m-2+\epsilon}}\,.\ $ (15)

The restriction on $\mu_{i}$ coordinates is

$\displaystyle\sum_{i=1}^{m+\epsilon}\mu_{i}^{2}=1\,,$ (16)

which hints at the spherical symmetry on a submanifold spanned by $\mu_{i}$.
The Killing vectors of this spacetime are $\partial_{t}$ and
$\partial_{\phi_{i}}$. Later, we discuss the hidden symmetries associated with
the Killing tensors of the above metric.

Just like its four-dimensional counterpart, the metric introduced in (II.2) is
an approximate solution to the Einstein’s equations to the linear order in
$a_{i}$.

##  III Hamilton–Jacobi equation of the probe particle

For the case of $\epsilon=0$, the restriction (16) is written as

$\displaystyle\sum_{I=1}^{m-1}\mu_{I}^{2}+\mu_{m}^{2}=1\,.$ (17)

We solve this for $\mu_{m}$ and rewrite the metric II.2 in terms of $\mu_{I}$
with $I\in\\{1,\cdots,m-1\\}$. Then, the Hamilton–Jacobi equation for the
probe particle,

$\displaystyle-m_{0}^{2}=g^{ab}p_{a}p_{b}\,,$ (18)

moving on this background metric is given by

$\displaystyle-
m_{0}^{2}=-f(r)^{-1}\left(p_{t}-\sum_{i=1}^{m}\frac{2Ma_{i}}{r^{2m}}\,p_{\phi_{i}}\right)+f(r)\,p_{r}^{2}$
$\displaystyle+r^{-2}\left(\sum_{I,J=1}^{m-1}h^{IJ}\,p_{\mu_{I}}p_{\mu_{J}}+\sum_{i=1}^{m}\mu_{i}^{-2}\,p_{\phi_{i}}^{2}\right)\,,$
(19)

where

$\displaystyle h^{IJ}=\delta_{IJ}-\mu_{I}\mu_{J}\,.$ (20)

Since the metric is stationary and axisymmetric along $\phi_{i}$, the energy
and angular momentum of the point particle on this background metric are
conserved and $p_{t}$, $p_{\phi_{i}}$’s are some constants. If we define

$\displaystyle\mathcal{U}(r)\equiv
f(r)^{-1}\left(p_{t}-\sum_{i=1}^{m}\frac{2Ma_{i}}{r^{2m}}\,p_{\phi_{i}}\right)-f(r)\,p_{r}^{2}\,,$
(21)

then the $(t,r,\phi_{i})$ part of Eq. (III) separates from the rest,

$\displaystyle r^{2}\left(\mathcal{U}(r)-m_{0}^{2}\right)=\mathcal{C}\,,$ (22)

by introducing a separation constant $\mathcal{C}$. The $\mu_{i}$-dependent
part of Eq. (III) is

$\displaystyle\left(1-\sum_{K=1}^{m-1}\mu_{K}^{2}\right)\,\sum_{I,J=1}^{m-1}\left(p_{\mu_{I}}^{2}-\mu_{I}\,\mu_{J}\,p_{\mu_{I}}\,p_{\mu_{J}}+\mu_{I}^{-2}\,p_{\phi_{I}}^{2}\right)$
$\displaystyle\hskip
28.45274pt+\sum_{I=1}^{m-1}\mathcal{C}\,\mu_{I}^{2}=\mathcal{C}-p_{\phi_{m}}^{2}\,.$
(23)

Obviously, this equation is not separable in $\mu_{I}$ coordinates. Then, we
change to $\theta_{i}$ coordinates in which the equation (III) is separable,

$\displaystyle\mu_{m}=\cos{(\theta_{m})}\,,\quad\mu_{I}=\hat{x}_{I}\sin{(\theta_{m})}\,,$
(24)

where $\hat{x}_{I}$ is constrained by

$\displaystyle\sum_{I=1}^{m-1}\hat{x}_{I}^{2}=1\,.$ (25)

To solve this, we introduce another $\theta$ coordinate, $\theta_{m-1}$ and so
on. Therefore, we need $m-2$ number of $\theta_{i}$’s (with $i\in\\{3,m\\}$)
which ranges in $[0,\pi]$ plus one azimuthal angle
$\theta_{2}=\tilde{\phi}\in[0,2\pi]$. In this coordinate, the $\theta_{i}$
part of Eq. (18) becomes separable,

$\displaystyle
p_{\theta_{i}}^{2}+\frac{p_{\phi_{i}}^{2}}{\cos{\theta_{i}}^{2}}+\frac{\mathcal{K}_{(i-1)}}{\sin{\theta_{i}}^{2}}=\mathcal{K}_{(i)}\,,\quad
i\in\\{2,m\\}\,,$ (26)

where $\mathcal{K}_{(i)}$ are some separation constants and
$\mathcal{K}_{(1)}=p_{\phi_{1}}^{2}$. We note that $\mathcal{K}_{(m)}$ is
nothing but the separation constant $\mathcal{C}$, introduced in (22) for the
radial part. Up to here, the existence of $(m-1)$ number of
$\mathcal{K}_{(i)}$’s, in addition to $m_{0},p_{t},p_{i}$, makes the dynamics
along the geodesic integrable. However, there are more constants of motion
associated with the the generators of U$(m)$ symmetry which will be discussed
in the next section (these are similar to the hidden constants of motion
introduced in [15, 16]).

##  IV Hidden symmetries and Killing tensors

Using the recursion relation between the constants $\mathcal{K}_{(i)}$’s in
Eq. (26), we read the related Killing tensors,

$\displaystyle
K_{(k)}^{ab}\partial_{a}\partial_{b}=\partial_{\theta_{k}}^{2}+\frac{\partial_{\phi_{k}}^{2}}{(\cos{\theta_{k}})^{2}}+\left(\prod_{l=0}^{k-2}\sin{\theta_{k-l}}\right)^{-2}\partial_{\phi_{1}}^{2}+$
$\displaystyle\hskip
28.45274pt+\sum_{q=0}^{k-3}\left[\prod_{l=0}^{q}\sin{\theta_{k-l}}\right]^{-2}\left[\partial_{\phi_{k-q-1}}^{2}+\frac{\partial_{\phi_{k-q-1}}^{2}}{\left(\cos{\theta_{k-q-1}}\right)^{2}}\right]\,,$

for $3\leq k\leq m$ and

$\displaystyle
K_{(2)}=\partial_{\theta_{2}}^{2}+\cos{\theta_{2}}^{-2}\partial_{\phi_{2}}^{2}+\sin{\theta_{2}}^{-2}\partial_{\phi_{1}}^{2}\,.$
(28)

Furthermore, the generalized Lense-Thirring metric (II.2) admits some
additional Killing tensors associated with the mentioned U$(m)$ symmetry. To
write them explicitly, we first introduce coordinates $x^{i},y^{i}$ in which
the mentioned U$(m)$ symmetry is more clear as in [16],

$\displaystyle x^{i}=\mu_{i}\,\cos{(\phi_{i})}\,,\qquad
y^{i}=\mu_{i}\,\sin{(\phi_{i})}\,.$ (29)

Here, $i$ runs from 1 to $(m-1)$. If one writes the reduced metric on
$(\mu_{i}-\phi_{i})$ subspace in complex coordinates $z^{j}=x^{j}+i\,y^{j}$,

$\displaystyle
ds^{2}\Big{|}_{t,r=const.}=d\mu_{i}^{2}+\mu_{i}^{2}d\phi_{i}^{2}=dz_{i}d\bar{z}_{i}\,,$
(30)

then the U$(m)$ symmetry manifests.

The second rank Casimir of the U$(m)$ symmetry, $I_{ij}$, is given by

$\displaystyle I_{ij}=-\frac{1}{4}(\xi_{ij}^{2}+\rho_{ij}^{2})\,,\qquad(i<j)$
(31)

in which the vector fields $\xi_{ij}$ and $\rho_{ij}$ are defined by

$\displaystyle\xi_{ij}\equiv
x^{i}\frac{\partial}{\partial{x^{j}}}-x^{j}\frac{\partial}{\partial{x^{i}}}+y^{i}\frac{\partial}{\partial{y^{j}}}-y^{j}\frac{\partial}{\partial{y^{i}}}\,,$
$\displaystyle\rho_{ij}\equiv
x^{i}\frac{\partial}{\partial{y^{j}}}-y^{j}\frac{\partial}{\partial{x^{i}}}+x^{j}\frac{\partial}{\partial{y^{i}}}-y^{i}\frac{\partial}{\partial{x^{j}}}\,.$
(32)

The vector $\rho_{ij}$ is symmetric under the exchange of $i,j$ and $\xi_{ij}$
is antisymmetric under this exchange. As $\xi_{ij}$ does not contribute to the
diagonal components of $I_{ij}$, it simplifies considerably. A simple algebra
shows that $I_{ii}$ is the trivial Killing tensor since
$I_{ii}=\partial_{\phi_{i}}\partial_{\phi_{i}}$ (there is no summation on the
repeated indices). Therefore, the nontrivial Killing tensors in $\mu_{i}$
basis are given by

$\displaystyle-4\
I_{ij}=(\mu_{i}\partial_{\mu_{j}}-\mu_{j}\partial_{\mu_{i}})^{2}+\left(\frac{\mu_{j}}{\mu_{i}}\partial_{\phi_{i}}+\frac{\mu_{i}}{\mu_{j}}\partial_{\phi_{j}}\right)^{2},\
$ (33)

for $i<j$. However, _all_ of the $I_{ij}$ components (when $i<j$) do not lead
to functionally independent constants of motion for the Hamilton–Jacobi
equation. In [16], it has been shown that the constants associated with
$I_{ij}$ not only includes $(m-1)$ number of $\mathcal{K}_{(i)}$ but also it
contains $(m-2)$ number of new _independent_ constants of motion constructed
out of $I_{(i-1)i}$. The explicit form of $I_{(i-1)i}$ in $\theta_{i}$
coordinates is given by

$\displaystyle
I_{(i-1)i}=\left(\sin{\theta_{i-1}}\cot{\theta_{i}}\frac{\partial}{\partial{\theta_{i-1}}}-\cos{\theta_{i-1}}\frac{\partial}{\partial{\theta_{i}}}\right)^{2}$
$\displaystyle\hskip
34.14322pt+\left(\cos{\theta_{i-1}\tan{\theta_{i}}\frac{\partial}{\partial{\phi_{i}}}+\frac{\cot{\theta_{i}}}{\cos{\theta_{i-1}}}}\frac{\partial}{\partial{\phi_{i-1}}}\right)^{2}\,,\
\ \ \ $ (34)

where $i\in\\{2,m\\}$. One can explicitly check that it satisfies the Killing
tensor equation (1).

##  V Discussion

In this work, we studied the Hamilton–Jacobi equation of the probe particle on
the generalized Lense-Thirring metric in $d=2m+1$ dimensions. This metric is
the solution to the pure Einstein theory to the linear order in the rotation
parameter $a_{i}$. It would be interesting to analyze the geodesic of the
extended Lense-Thirring as the solution to the other theories such as
Einstein-Maxwell-Lambda to the first order in rotation parameter[13, 18, 17].

We observed that the dynamics of the probe particle along the timelike
geodesic of the generalized Lense-Thirring spacetime in $d$-dimensions is
superintegrable. For a system with $(m-1)$ degrees of freedom, it is maximally
superintegrable if it has a $2(m-1)-1$ number of independent constants of
motion. This is the case for the reduced phase space related to the
independent $\mu_{i}$’s as $(m-1)$ number of $\mathcal{K}_{(i)}$’s and $(m-2)$
number of constants associated with $I_{i(i-1)}$’s renders the dynamics
maximally superintegrable.

Regarding the principal tensor ($h$), one idea is to start from the principal
tensor associated with Myers-Perry black hole(12) and take the slow rotation
limit which gives

$\displaystyle h=d\,b\,,\qquad
b=\frac{r^{2}}{2}\left(dt+\sum_{i=1}^{m}a_{i}\,\mu_{i}^{2}d\phi_{i}\right)\,,$
(35)

to linear order in $a_{i}$. A straightforward calculation shows that the
Killing tensor associated with (35) _in the slow rotation limit_ is

$\displaystyle
K^{ab}\partial_{a}\partial_{b}=2\sum_{i=1}^{m}a_{i}\partial_{t}\partial_{\phi_{i}}$
$\displaystyle\hskip
34.14322pt+\sum_{I,J=1}^{m-1}(\delta_{IJ}-\mu_{I}\mu_{J})\partial_{\mu_{I}}\partial_{\mu_{J}}+\sum_{i=1}^{m}\frac{\left(\partial_{\phi_{i}}\right)^{2}}{\mu_{i}^{2}}\,,\
\ \ $ (36)

in odd dimensions. Here, we imposed the restriction (16), so that the indices
of $\partial_{\mu}$ runs from 1 to $(m-1)$. Therefore, the second term is the
metric on an $m$-sphere. Changing the bound of summation in (35) does not
yield a different result for the Killing tensor other than (V) _to the linear
order in $a_{i}$_. An interesting question is, are the Killing tensors
$I_{ij}$ constructed of some principal tensors? We will come back to this
question in the future.

Finally, we note that the discussion in even dimensions is very similar to the
odd-dimensional case. To avoid repetition, we left the details in the
Appendix.

## Acknowledgments

I am grateful to M.M. Sheikh-Jabbari to bringing to my attention the results
of [13] which lead to the current project. The author also acknowledges the
collaboration of H. Golchin and H. Demirchian in the early stages of this
research. I would like to thank D. Kubiznak for his comments on the draft. The
support of ICTP program network scheme NT-04 is appreciated.

##  Appendix A Geodesic equation in even dimensions

In even dimensions ($\epsilon=1$) , the restriction (16) reads

$\displaystyle\sum_{i=1}^{m}\mu_{i}^{2}+\nu^{2}=1\,,$ (37)

where, for convenience, we replaced $\mu_{m+1}$ by $\nu$. This can be solved
for $\nu$, then the metric reduces to

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}\sum_{i=1}^{m}\mu_{i}^{2}\Bigl{(}d\phi_{i}+\frac{2Ma_{i}}{r^{2m+1}}dt\Bigr{)}^{2}$
$\displaystyle\hskip 34.14322pt+\
r^{2}\sum_{i,j=1}^{m}h_{ij}\,\\!d\mu^{i}\,d\mu^{j}\,,$ (38)

where the metric functions are

$\displaystyle\\!\\!\\!\
f(r)\\!=\\!1-\frac{2M}{r^{2m-1}}+\frac{r^{2}}{\ell^{2}}\,,\quad
h_{ij}=\delta_{ij}+\frac{\mu_{i}\,\mu_{j}}{\nu^{2}}\,,$ (39)

The Hamilton–Jacobi equation of the probe particle on this background metric
is given by

$\displaystyle-
m_{0}^{2}=-f(r)^{-1}\left(p_{t}-\sum_{i=1}^{m}\frac{2Ma_{i}}{r^{2m+1}}\,p_{\phi_{i}}\right)+f(r)\,p_{r}^{2}$
$\displaystyle+r^{-2}\left(\sum_{i,j=1}^{m}h^{ij}\,p_{\mu_{i}}p_{\mu_{j}}+\sum_{i=1}^{m}\mu_{i}^{-2}\,p_{\phi_{i}}^{2}\right)\,,$
(40)

where $h^{ij}=\delta_{ij}-\mu_{i}\mu_{j}\,$. Since the metric is stationary
and axisymmetric along $\phi_{i}$’s the energy and angular momentum of the
point particle on this background metric are conserved and $p_{t}$,
$p_{\phi_{i}}$’s are some constants. If we define

$\displaystyle\mathcal{U}(r)\equiv
f(r)^{-1}\left(p_{t}-\sum_{i=1}^{m}\frac{2Ma_{i}}{r^{2m+1}}\,p_{\phi_{i}}\right)-f(r)\,p_{r}^{2}\,,\,$
(41)

then the $r$-dependent part separates from the rest of Eq. (A) by introducing
the separation constant $\mathcal{C}$,

$\displaystyle r^{2}\left(\mathcal{U}(r)-m_{0}^{2}\right)=\mathcal{C}\,.$ (42)

The $\mu_{i}$ part of Eq. (A),

$\displaystyle\sum_{i,j=1}^{m}\left(p_{\mu_{i}}^{2}+\mu_{i}\,\mu_{j}\,p_{\mu_{i}}\,p_{\mu_{j}}+\mu_{i}^{-2}\,p_{\phi_{i}}^{2}\right)=\mathcal{C}\,,$
(43)

is not separable in $\mu_{i}$ coordinates.

Then, we change the coordinates to $\theta_{i}$ coordinates,

$\displaystyle\nu=\cos{(\theta_{m+1})}\,,\ \
\mu_{i}=\hat{x}_{i}\sin{(\theta_{m+1})}\,;\ \
\sum_{i=1}^{m}\hat{x}_{i}^{2}=1\,.\ \ \ \ $ (44)

In these coordinates, the $\theta_{i}$ part of the Hamilton–Jacobi equation
(18) is written as

$\displaystyle
p_{\theta_{i}}^{2}+\frac{p_{\phi_{i}}^{2}}{\cos{\theta_{i}}^{2}}+\frac{\mathcal{K}_{(i-1)}}{\sin{\theta_{i}}^{2}}=\mathcal{K}_{(i)}\,,\quad
i\in\\{2,m+1\\}\,,\ \ \ \ \ $ (45)

with

$\displaystyle p_{\phi_{m+1}}=0\,,\quad\mathcal{K}_{(1)}=p_{\phi_{1}}^{2}\,.$
(46)

The Killing tensors related to these constants are

$\displaystyle
K_{(k)}^{ab}\partial_{a}\partial_{b}=\partial_{\theta_{k}}^{2}+\frac{\partial_{\phi_{k}}^{2}}{(\cos{\theta_{k}})^{2}}+\sum_{q=0}^{k-3}\left(\prod_{l=0}^{q}\sin{\theta_{k-l}}\right)^{-2}\\!\\!\\!\\!\partial_{\theta_{k-q-1}}^{2}$
$\displaystyle\hskip
42.67912pt+\sum_{q=0}^{k-3}\frac{\left(\prod_{l=0}^{q}\sin{\theta_{k-l}}\right)^{-2}}{\left(\cos{\theta_{k-q-1}}\right)^{2}}\partial_{\phi_{k-q-1}}^{2}$
$\displaystyle\hskip
42.67912pt+\left(\prod_{l=0}^{k-2}\sin{\theta_{k-l}}\right)^{-2}\partial_{\phi_{1}}^{2}\,,$
(47)

where $(2\leq k\leq m+1)$ while $\mathcal{K}_{(m+1)}$ is equal to
$\mathcal{C}$ which appeared as the separation constant for the radial part,
in Eq. (42).

## References

  * [1] Hans Thirring and Josef Lense, Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie, Phys. Z. 19, 156 (1918), https://inspirehep.net/literature/45452.
  * [2] J. Baines, T. Berry, A. Simpson and M. Visser, Painlevé–Gullstrand form of the Lense-Thirring Spacetime, Universe 7, 105 (2021).
  * [3] J. Baines, T. Berry, A. Simpson and M. Visser, Killing Tensor and Carter Constant for Painlevé–Gullstrand Form of Lense–Thirring Spacetime, Universe 7, 473 (2021).
  * [4] B. Carter, Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10, 280 (1968).
  * [5] V. P. Frolov and D. Kubiznak, Hidden Symmetries of Higher Dimensional Rotating Black Holes, Phys. Rev. Lett. 98, 011101 (2007).
  * [6] P. Krtous, D. Kubiznak, D. N. Page and V. P. Frolov, Killing-Yano Tensors, Rank-2 Killing Tensors, and Conserved Quantities in Higher Dimensions, JHEP 02 (2007), 004.
  * [7] D. Kubiznak and P. Krtous, On conformal Killing-Yano tensors for Plebanski-Demianski family of solutions, Phys. Rev. D 76, 084036 (2007).
  * [8] M. Cariglia, P. Krtous and D. Kubiznak, Dirac Equation in Kerr-NUT-(A)dS Spacetimes: Intrinsic Characterization of Separability in All Dimensions, Phys. Rev. D 84, 024008 (2011).
  * [9] V. P. Frolov, P. Krtous and D. Kubiznak, Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes, JHEP 02 (2007), 005.
  * [10] P. Krtouš, V. P. Frolov and D. Kubizňák, Separation of Maxwell equations in Kerr–NUT–(A)dS spacetimes, Nucl. Phys. B 934, 7 (2018).
  * [11] O. Lunin, Maxwell’s equations in the Myers-Perry geometry, JHEP 12 (2017), 138.
  * [12] V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Rev. Relativity 20, 6 (2017).
  * [13] F. Gray and D. Kubiznak, Slowly rotating black holes with exact Killing tensor symmetries, Phys. Rev. D 105, 064017 (2022).
  * [14] R. C. Myers and M. J. Perry, Black Holes in Higher Dimensional Space-Times, Annals Phys. 172, 304 (1986).
  * [15] T. Hakobyan, O. Lechtenfeld, A. Nersessian, A. Saghatelian and V. Yeghikyan, Integrable generalizations of oscillator and Coulomb systems via action-angle variables, Phys. Lett. A 376, 679 (2012).
  * [16] A. Galajinsky, A. Nersessian and A. Saghatelian, Superintegrable models related to near horizon extremal Myers-Perry black hole in arbitrary dimension, JHEP 06 (2013), 002.
  * [17] F. Gray, R. A. Hennigar, D. Kubiznak, R. B. Mann and M. Srivastava, Generalized Lense-Thirring metrics: higher-curvature corrections and solutions with matter, JHEP 04 (2022), 070.
  * [18] D. Kubiznak, T. Tahamtan and O. Svitek, Slowly rotating black holes in nonlinear electrodynamics, Phys. Rev. D 105, 104064 (2022).