Vol.0 (20xx) No.0, 000–000

11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100101, China<EMAIL_ADDRESS>
22institutetext: University of Chinese Academy of Sciences, Beijing, China.
33institutetext: Institute for Frontier in Astronomy and Astrophysics, Beijing
Normal University, Beijing 102206, China.
44institutetext: Department of Physics and Astronomy, University of Nevada,
Las Vegas, NV 89154, USA<EMAIL_ADDRESS>
55institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China
66institutetext: Department of Astronomy, Peking University, Beijing 100871,
China
77institutetext: Research Center for Intelligent Computing Platforms, Zhejiang
Laboratory, Hangzhou 311100, China
88institutetext: Cahill Center for Astronomy and Astrophysics, MC 249-17
California Institute of Technology, Pasadena CA 91125, USA
99institutetext: CSIRO Space and Astronomy, PO Box 76, Epping, NSW 1710,
Australia
1010institutetext: School of Astronomy and Space Science, Nanjing University,
Nanjing 210093, China
1111institutetext: Key Laboratory of Modern Astronomy and Astrophysics
(Nanjing University), Ministry of Education, China
1212institutetext: Department of Astronomy, University of Science and
Technology of China, Hefei 230026, China
Received 20xx month day; accepted 20xx month day

# FAST observations of an extremely active episode of FRB 20201124A:
IV. Spin Period Search

Jia-Rui Niu 1122 Wei-Wei Zhu 1133 Bing Zhang 44 Mao Yuan 1122 De-Jiang Zhou
1122 Yong-Kun Zhang 1122 Jin-Chen Jiang 1166 J. L. Han 1122 Di Li 11 Ke-Jia
Lee 115566 Pei Wang 11 Yi Feng 77 Dong-Zi Li 88 Rui Luo 99 Fa-Yin Wang
10101111 Zi-Gao Dai 10101212 Chen-Chen Miao 1122 Chen-Hui Niu 11 Heng Xu 5566
Chun-Feng Zhang 5566 Wei-Yang Wang 5566 Bo-Jun Wang 5566 Jiang-Wei Xu 5566

###### Abstract

We report the properties of more than 800 bursts detected from the repeating
fast radio burst (FRB) source FRB 20201124A with the Five-hundred-meter
Aperture Spherical radio telescope (FAST) during an extremely active episode
on UTC September 25th-28th, 2021 in a series of four papers. In this fourth
paper of the series, we present a systematic search of the spin period and
linear acceleration of the source object from both 996 individual pulse peaks
and the dedispersed time series. No credible spin period was found from this
data set. We rule out the presence of significant periodicity in the range
between 1 ms to 100 s with a pulse duty cycle $\textless$ 0.49$\pm 0.08$ (when
the profile is defined by a von-Mises function, not a boxcar function) and
linear acceleration up to $300$ m s-2 in each of the four one-hour observing
sessions, and up to $0.6$ m s-2 in all 4 days. These searches contest
theoretical scenarios involving a 1 ms to 100 s isolated magnetar/pulsar with
surface magnetic field $<10^{15}$ G and a small duty cycle (such as in a
polar-cap emission mode) or a pulsar with a companion star or black hole up to
100 M⊙ and $P_{b}>10$ hours. We also perform a periodicity search of the fine
structures and identify 53 unrelated millisecond-timescale ”periods” in multi-
components with the highest significance of 3.9 $\sigma$. The ”periods”
recovered from the fine structures are neither consistent nor harmonically
related. Thus they are not likely to come from a spin period. We caution
against claiming spin periodicity with significance below $\sim$ 4 $\sigma$
with multi-components from one-off FRBs. We discuss the implications of our
results and the possible connections between FRB multi-components and pulsar
micro-structures.

###### keywords:

methods: observational - radio continuum: transients – transients: fast radio
bursts

##  1 Introduction

Fast radio bursts (FRBs) are bright radio transients with milliseconds
duration discovered since 2007 (Lorimer et al. 2007). Most of the known FRBs
are one-off events, but a small group of them repeat and are called repeating
FRBs (Spitler et al. 2016). Their origin is not yet fully understood. The
detection of FRB 200428 associated with the galactic magnetar SGR J1935+2154
suggests that magnetars could produce at least some FRBs (Bochenek et al.
2020; CHIME/FRB Collaboration et al. 2020; Lin et al. 2020; Li et al. 2021;
Tavani et al. 2021).

Active repeating FRBs are observed to emit apparently sporadic pulses
separated by seconds. Finding a spin period ($P$) from a repeating FRB source
could be the smoking gun evidence for a neutron star origin. However, no
apparent periods have been detected in the millisecond to second range for
either FRB 20121102A or FRB 20201124A, two of the most well-studied repeaters,
despite thousands of pulses have been detected (Li et al. 2021; Caleb et al.
2020; Cruces et al. 2021; Gourdji et al. 2019; Xu et al. 2021). The same
applies to FRB 20180916B (Chime/Frb Collaboration et al. 2020; Chawla et al.
2020; Marthi et al. 2020), which has a 16-day longer period. Caleb et al.
(2020) used the PRESTO111https://github.com/scottransom/presto(Ransom 2011) to
perform a Fast Fourier Transform (FFT) periodic search of FRB 20121102A using
MeerKAT on the 6th and 8th of October 2019. After folding and visual checking,
no credible period was found. They also performed a Fast Folding Algorithm
(FFA) (Staelin 1969; Morello et al. 2020) to search for periods ranging from
500 ms to 10 min, but did not detect any significant periodic pulsations above
the S/N threshold of 8. Cruces et al. (2021) performed an extensive multi-
wavelength observation in radio, optical, X-ray and $\gamma$-ray of FRB
20121102A. They searched with DM trials for the acceleration search and the
FFA search (Staelin 1969; Morello et al. 2020). No source was found among the
candidates down to an S/N of 6. Li et al. (2021) observed FRB 20121102A using
the FAST telescope and performed a period search of $P$ in 1 ms – 1000 s and
$\dot{P}$ in $10^{-12}$ – $10^{-2}$ $s\,s^{-1}$ with the Phase-Folding and the
Lomb-Scargle Periodogram (LSP) methods. They did not detect any periodicity or
quasi-periodicity and proposed that this challenges the theory related to
single rotating compact object. Jahns et al. (2022) studied the period and
waiting time characteristics of FRB 20121102A with the Arecibo Telescope. They
used the Pearson $\chi^{2}$ test and the LSP to search for periods between 10
ms and 100 s. They found no statistically significant periods from two
observations, one with 218 bursts detected in 1 hr and one with 227 in 1.38
hr. Nimmo et al. (2022) observed FRB 20200120E using the Effelsberg telescope.
The highest event rate reached 53 bursts in 40 minutes. They used PRESTO’s
accelsearch, brute force search and fold search with $\chi^{2}$ statistic.
They found no strict periodicity in the burst arrival times during the storm.

While a clear second-level spin-related periodicity has not yet been observed
from any FRB, some repeating FRBs show an active-dormant cycle on the order of
tens to hundreds of days, e.g. FRB 20180916B – 16 days (Chime/Frb
Collaboration et al. 2020) and FRB 20121102A – 160 days (Rajwade et al. 2020;
Cruces et al. 2021; Li et al. 2021). The theoretical interpretations for these
long cycles for FRBs include binary systems (Ioka & Zhang 2020; Zhang & Gao
2020; Sridhar et al. 2021), magnetar precession (Yang & Zou 2020; Levin et al.
2020; Zanazzi & Lai 2020; Sob’yanin 2020; Tong et al. 2020), asteroid
interactions Dai & Zhong (2020); Voisin et al. (2021); Du et al. (2021), and
slow rotating magnetars (Beniamini et al. 2020).

In addition to these long-term cycles, some sub-second periodic fine pulse
structures, also called multi-components, have been observed from single FRB
events. It was first reported in FRB 20121102A (Hessels et al. 2019; Gourdji
et al. 2019) and later seen in FRB 180814B (Amiri et al. 2019) and many other
bursts. Andersen et al. (2021) discovered three FRB signals with multiple
components that show a tentative periodicity. The most significant signal
comes from FRB 20191221A, with a period of 216.8(1) ms and a significance of
$6.5\,\sigma$. The authors speculated that the intriguing periodic multi-
components could be related to the spin of the source object if the same-
period multi-component bursts can be reliably observed to repeat themselves in
the future. Alternatively, (Chen et al. 2021) proposed that the micro-lensing
effect could cause such a phenomenon. The superposition of pulses from
individual micro-images produces a light curve that appears as multi-peak
FRBs. It should be noted that similar fine structures were also observed from
some normal and millisecond pulsars (Cordes et al. 1990; De et al. 2016; Liu
et al. 2022).

In summary, repeating FRBs have shown temporal behaviors on three different
time scales: 1\. Pulses were emitted with seconds separations when the source
is most active and the separation time distribution appears sporadic; 2\.
Active and dormant cycles of tens to hundreds of days have been observed from
some FRBs such as FRB 20121102A (Rajwade et al. 2020) and FRB 20180916B
(Chime/Frb Collaboration et al. 2020); 3\. Millisecond-timescale micro-
component fine structures have been observed in some bursts within a duration
of hundreds of milliseconds.

FRB 20201124A was discovered by CHIME (Chime/FRB Collabortion 2021) and
located by giant meter-wave radio telescope (uGMRT) (Wharton et al. 2021) and
European VLBI Networ (EVN) (Nimmo et al. 2021). Subsequently, Lanman et al.
(2021) published the morphology, fluences, and arrival times from the follow-
up observations by CHIME. They did not find any evidence of cyclical behavior
on timescales between 1 and 178.5 days. Hilmarsson et al. (2021) published
polarization observations by using the Effelsberg 100-m radio telescope at
1.36 GHz. Marthi et al. (2021) observed FRB 20201124A in the incoherent array
mode using the upgraded giant uGMRT. They found persistent radio emissions
associated with the host galaxy and detected 48 bursts at 550-750 MHz. They
also searched for spin periodicity by using different period search methods,
including PRESTO (Ransom 2011) accelsearch and jerk search (Andersen & Ransom
2018), Fast Folding Algorithm (FFA) analysis (Staelin 1969; Morello et al.
2020), but no significant periodic candidates were found. FAST observed bursts
of FRB 20201124A from April 1st, 2021 to June 11th, 2021. Xu et al. (2021)
reported the data and discovered significant and irregular variations of the
Faraday Rotation Measurement (RM) and the evidence of polarization Faraday
conversion. They searched for periodicity using LSP between 30 ms and 10 days
and found no obvious periodic signals.

Figure 1: FFA periodicity search in September 27th (a) and 28th (b) from 20 s
to 600 s. Comparing the results of these two days, there are no significant
matching periods.

In the September 2021 episode, FRB 20201124A exhibited an outburst and
reported by CHIME222https://www.chime-frb.ca/repeaters/FRB20201124A and
Effelsberg telescope (Main et al. 2021). We observed the source during its
outburst using FAST. We detected 25 pulses on September 25th, 59 on September
26th, 274 pulses on September 27th and 638 pulses on September 28th. In this
paper, we refer to the two rounds of FAST observations of April as the active
episode 1 and September as the active episode 2.

The number of pulses reported in this paper differs from those reported in our
companion papers (Zhou et al. 2022, Zhang et al. 2022 and Jiang et al. 2022)
because we use different criteria for isolating the pulses which suit our
purpose of searching for a period. This extremely active episode enables us to
search for the millisecond to second-level spin period from FRB 20201124A with
an unprecedentedly large and high-hourly-rate event sample. We also perform a
period acceleration search both in the raw data and from the pulse arrival
times to detect or rule out putative binary systems.

In this paper, we employ the following criteria to determine whether the
signals have a credible spin period: 1) It has a high statistical significance
despite the number of trials searched; 2) It appears in multiple epochs of
observations either as identical, close or harmonically related signals.
Abiding by these two criteria, we do not detect any credible periodic signal
from the extremely active episode of this active repeating FRB.

The paper is organized as follows: Section 2 describes the observations. In
Section 3, we present the results of the periodicity search using different
methods, including searches using dedispersed time series, using TOAs, using
the TOAs of special pulses, and using the multi-components fine structures.
Finally, we briefly discuss and summarize our results in Section 4.

Table 1: Observation table. | Date
---
(Y-M-D)
| Start Time
---
UTC MJD
| Duration
---
(s)
Pulses
2021-09-25 | 22:37:01 | 59482.942361 | 3600 | 25
2021-09-26 | 20:41:00 | 59483.861805 | 3600 | 59
2021-09-27 | 19:31:00 | 59484.813194 | 3600 | 274
2021-09-28 | 18:46:00 | 59485.781944 | 3600 | 638
2021-09-29 | 18:48:00 | 59486.783333 | 3600 | 0
2021-09-30 | 19:58:00 | 59487.831944 | 3600 | 0
2021-10-01 | 21:08:00 | 59488.880555 | 3600 | 0
2021-10-02 | 20:29:00 | 59489.853472 | 2357 | 0
2021-10-02 | 21:18:55 | 59489.888136 | 7814 | 0
2021-10-07 | 22:20:00 | 59494.930555 | 3600 | 0
2021-10-08 | 21:50:00 | 59495.909722 | 3600 | 0
2021-10-09 | 21:11:00 | 59496.924305 | 3600 | 0
2021-10-10 | 21:42:00 | 59497.904166 | 3600 | 0
2021-10-11 | 21:52:00 | 59498.911111 | 3600 | 0
2021-10-12 | 17:44:00 | 59499.738888 | 3600 | 0
2021-10-13 | 18:23:00 | 59500.765972 | 3600 | 0
2021-10-14 | 21:29:00 | 59501.895138 | 3600 | 0
2021-10-17 | 21:07:00 | 59504.879861 | 3600 | 0

##  2 Observations

Observation data used in this paper are taken using the FAST telescope with
its 19 beam receivers in 1.05-1.45 GHz (Nan et al. 2011; Li et al. 2018; Jiang
et al. 2019). Data is recorded with the re-configurable open architecture
computing hardware version 2 (ROACH2) System (Hickish et al. 2016). The
recorded data is in PSRFITS format with four polarizations, 49.152 $\mu$s
sampling interval and 4096 channels in search mode.

We utilize the psrfits_utils333https://github.com/demorest/psrfits_utils
software package to merge the four polarization raw data into one single
channel 8-bit intensity data. The program also adjusts the levels of the data
according to its running statistics. We then dedisperse and search the data
using the PRESTO444https://github.com/scottransom/presto(Ransom 2011) software
suite using a single fixed DM value of 413.5 pc cm-3 and (see Zhou et al. 2022
for the detailed analysisof the DM distribution of the pulses) an S/N
threshold of 6.5. We further search into the fine structures of the detected
pulses by using other methods. We used the
DSPSR555http://dspsr.sourceforge.net/ (van Straten & Bailes 2011) program and
the PSRCHIVE666http://psrchive.sourceforge.net/ (van Straten et al. 2012)
software package for polarization analysis. The details of the single pulse
analysis will be presented in Section 3.2. Observation information is listed
in the Table 1.

In our period search based on TOAs, we select special pulses by their
polarization properties for part of our period search. The FAST observations
was taken with four polarizations and calibrated by using a winking 10 k noise
signal injected into the feed with an on-off period of 0.2 seconds in the
first minute of our observation. We extracted the polarizations and correct
for Faraday rotation of the pulses using the software
DSPSR777http://dspsr.sourceforge.net/ (van Straten & Bailes 2011) and
PSRCHIVE888http://psrchive.sourceforge.net/ (van Straten et al. 2012). The RM
of FRB 20201124A has changed dramatically over time, as observed in the
preceding study (Xu et al. 2021). The first active episode’s RM change ranges
from -887 rad m-2 to -363 rad m-2(Xu et al. 2021), while the second time’s RM
change ranges from -650 rad m-2 to -545 rad m-2 (Jiang et al. 2022). The
majority of the pulses are strongly linearly polarized, with a narrow range of
PA variation. For periodicity search using special pulses, we extracted the
polarization signals from the detected pulses by using their best-fit RM
value.

##  3 Result

###  3.1 Periodicity search with raw data

We utilize PRESTO (Ransom 2011) to search the dedispersed time series from
data taken on the 27th and 28th with high burst rates and employ Fourier
transform and accelsearch on the time series. The specific steps and
parameters are as follows. Firstly, we mask the interference on the original
data using rfifind with a time block size of 1 s. Secondly, we dedisperse and
retrieve the time series with DM 413.5 pc cm-3 using prepdata. Thirdly, we run
accelsearch on the time series. We search for periodicity between 0.1 ms and
20 s using a sigma cutoff of 5, zmax parameter of 1200, wmax to 4000 to
account for the potential Doppler acceleration with the maximum number of
harmonics set to 32. Both zmax and wmax are set to the maximum allowed value
of accelsearch. Based on such setting, we can search Doppler accelerations up
to $a/c={\rm zmax}$ $P/T_{\rm obs}^{2}=1\times$10-4 s-1 and acceleration
derivatives up to $\dot{a}/c={\rm wmax}$ $P/T_{\rm obs}^{3}=8.6\times 10^{-8}$
$\rm s^{-2}$ for a sinusoidal signal of period 1 s and an $a/c=9\times$10-8
s-1, $\dot{a}=8.6\times 10^{-11}\rm s^{-2}$ for a period of 1 ms and an
observing time of $T_{\rm obs}=$3600 s (Andersen & Ransom 2018). For putative
pulsars that rotate slower than 1 ms, our PRESTO acceleration search should
detect their periodicity (assuming PRESTO $\sigma>5$) if the pulsar is in a
binary with a companion lighter than 1 $M_{\rm\odot}$ and an orbit longer than
10 hr. However, we find no significant period candidate after removing
candidates caused by radio frequency interference (RFI). Candidates caused by
RFI are identified with the best DM of 0 pc cm-3 or signal coming from a
narrow band. We list the remaining marginal candidates in Table 2. Meanwhile,
we also use the FFA algorithm
riptide999https://github.com/v-morello/riptide(Morello et al. 2020) to search
the time series for periods in the range 20 s to 600 s. In the FFA search, the
signal to noise ratio (SNR) is equivalent to the best estimator of
Z-statistic. When dealing with ideal data containing only Gaussian noise and
periodic signals, the critical threshold $\eta$ can be expressed as
$\eta_{z}(\alpha)=\bar{\Phi}^{-1}(\alpha)$ (Morello et al. 2020), where
$\bar{\Phi}^{-1}$ is the inverse survival function of the standard normal
distribution. $\eta_{z}$ represents the number of Gaussian sigmas associated
with the probability $\alpha$. However, real data often contains more than
simple Gaussian noise and the signal-to-noise ratio of periodograms often
increase with the period and may be significantly higher than the theoretical
threshold. In this paper, we use the
find_peaks101010https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.find_peaks.html
method to pick out signals that stand out from the surrounding baseline. The
find_peaks program has one parameter – prominence, which is defined as the
vertical distance between the peak and its lowest profile line. We use
observation taken on September 29, 2021 in which we detected no pulses. We
inject a mock periodic signal into this empty data set, and obtained an
artificial periodic signal 10 $\sigma$ higher than the baseline. We apply the
FFA search on the mock data and get a SNR of 41 for the artificial signal, so
we set the find_peaks prominence parameter to 41 in our search for the
September 25th-28th data. We find few prominent peaks that appear in
individual days but none simultaneously appears in the data from September
27th and September 28th, as shown in Figure 1. We list these candidates in
Table 3.

Figure 2: Lomb-Scargle periodogram search result from all TOAs from September
25th-28th. Different sub-plots correspond to different period ranges. Each
subplot shows the stack of search results from 25th-28th from top to bottom.
The vertical lines in the plot represent a significant peak in LSP identified
by FAP$=0.1$. We stack the detections from different days together to check
for recurrence of the same signal on different days.

###  3.2 Lomb-Scargle Periodicity search with TOAs

The DM of FRB 20201124A is known to fluctuate, ranging from 409 pc cm-3 to 420
pc cm-3 in FAST observation of active episode 1 (Xu et al. 2021), and few
pulses varied considerably. When obtaining TOAs from the data, we fix the DM
to 413.5 pc cm-3, a DM value that is found to optimally dedisperse many of the
pulses (Zhou et al. 2022), to avoid shifting our TOAs randomly due to their
varying pulse morphology. We treat the multi-components (pulse fine structure)
as multiple individual pulses to obtain a more finely resolved list of TOAs.
After the initial search, we obtain the time-frequency waterfall plot of the
pulses. We use the find_peaks algorithm to find significant peaks of the pulse
profile. The strength of different components varies greatly in complex multi-
components. We adjust the threshold of find_peaks to obtain the peaks of all
components. Then, we perform a multi-Gaussian fitting whose initial parameter
is the position and strength of peak values from find_peaks to determine the
best fit arrival time.

(a)

(b)

Figure 3: (a): An example of a periodic signal detected from the September
28th observation at 2543.13151 Hz that is likely a harmonic of the sampling
frequency. This one is 8 times the time resolution instead of the real period.
(b): An example of a periodic signal detected from four-day search at
293.883889 Hz ($P=3.4027$ ms). See Table 5 for specific information. The four-
day time span is large, so the number of pulses is used on the y-axis.

We search for periodicity by the LSP of the TOAs collected on September
25th-28th, shown in Figure 2 and select significant signals by using
find_peaks. The significance of the LSP detections depend on the number of
trial periods and the power of the signal. We use the False Alarm Probability
(FAP) to quantify the significance of the peaks. For normalized periodogram,
VanderPlas (2018) shows that

${\rm FAP}(z)\approx 1-\left[P_{\text{single }}(z)\right]^{N_{eff}}$

where,$P_{\text{single }}(Z)$ represents the cumulative probability of seeing
a peak less than $Z$ at a single specific frequency assuming Gaussian noise.
Taking into account the folding of $N$ number of data samples (TOAs), the
degree of freedom for null hypothesis is ${\rm NH}=N-1$, and $P_{\text{single
}}(Z)=(1-Z)^{({\rm NH}/2)}$, where $Z$ is the normalized periodogram power
value, $N_{eff}$ is the number of independent peaks and can be estimated as
$N_{eff}=f_{max}T$. $T$ is the range of the time series and $f_{max}$ is the
maximum frequency of the LSP search. We adopt the find_peaks method again and
set the prominence parameter to the $Z$ value that leads to a certain FAP
threshold, thereby reducing the influence of large differences in baselines in
different period ranges. When using an FAP threshold of 0.01, we find no
period recurs on multiple days. Considering a 10-day orbital period and a
companion star with a mass of 100 M⊙, which leads to a $a/c$ of $1\times
10^{-8}$ s-1, we estimate the error for matching $F_{0}$ should be about
1/1000 of $F_{0}$. With a FAP threshold of 0.1, we do not find a period that
appears in all four days, but find two period candidates ($P=0.7924(3)$ and
$P=0.3216(2)$) that appears in the search of two different days in Figure 2,
we zoom in on the LSP results and show in the Figure 4. In addition, we
perform simulations to generate the same four sets of random numbers as the
four-day TOAs, perform an LSP search with the same threshold of ${\rm
FAP}=0.1$, and found that in 1000 simulations, the probability of peak
matching is 1.4%. Considering that the two periods do not appear in both the
27th and 28th data, and simulation shows a small chance of such match
occurring in random signals. Thus, we do not consider them credible spin
periods, but we list them in candidates Table 4.

(a) In the LSP, we find same periods showing significant peaks on the 25th and
28th in $P=0.7924(3)$ s with ${\rm FAP}=0.1$.

(b) In the LSP, we find same periods showing significant peaks on the 26th and
28th in $P=0.3216(2)$ s with ${\rm FAP}=0.1$.

Figure 4: Two periods showing significant peaks in LSP.

###  3.3 Period and acceleration search with TOAs

A linear acceleration, described by a constant $\dot{P}$ is the first-order
approximation to a binary orbit. A real orbit would normally require at least
five Keplerian parameters to describe. But the simple linear approximation
could apply if the duration of the observation $T_{\rm obs}$ is much shorter
than ($<$1/10) the orbital period $P_{\rm b}$ (Ransom et al. 2003). Due to the
limitation of this approach, our search is only sensitive to binaries with a
period much longer than our observing time, i.e., binaries with $P_{b}>10$
hours if we consider only the TOAs within one session and $P_{b}>40$ days when
we combine the TOAs from all four days. As an example, Wang et al. (2022a)
proposed a model involving a wide orbit magnetar-Be star binary to explain FRB
20201124A’s RM variation (Xu et al. 2021). In their model, the magnetar-Be
binary could have an orbital period of 80 days and companion mass up to 30 M⊙.

Using a linear acceleration search would enable us to test models such as Wang
et al. (2022a) up to a certain range of the companion mass $M_{c}$. If the
magnetar is subject to gravitational pull of its companion, then it would
experience a gravitational acceleration of
$a=(2\pi/P_{b})^{2}\,R\,\eta=GM_{c}/R^{2}$, where R is the distance between
the two objects and $\eta=M_{c}/(M_{p}+M_{c})$ with $M_{p}$ being the pulsar
mass (usually $\simeq 1.4$ M⊙). These equations give us
$a=(2\pi/P_{b})^{2}\,\eta\,(GMP_{b}^{2}/(2\pi)^{2}/\eta)^{1/3}$.

Such acceleration could lead to a $\dot{P}_{b}/P_{b}=a/c$, where $c$ is the
speed of light. We choose an upper limit of $M_{c}=100\,$M⊙, which is a large
mass even for a Be star or a stellar black hole.

We find that the upper limit of $a/c$ is $7\times 10^{-7}$ s-1, assuming
$P_{b}=$ 10 hours, $a/c=2\times 10^{-9}$ s-1 for $P_{b}=$ 4 days.

In the meantime, a negative linear acceleration, or a positive $\dot{P}$ could
also come from a spinning down pulsar or magnetar. The spin down rate $F_{1}$
is determined by the surface magnetic field strength of the pulsar:

$B_{s}=3.2\times 10^{19}{\rm G}\sqrt{P\dot{P}}$

. Most pulsars have magnetic field strength around $10^{12}$ G and most known
magnetars have $B_{s}$ around $10^{13}$ G to $10^{15}$ G. Assuming that we are
searching for a magnetar with $B_{s}=10^{15}$ G, we could derive that the
pulsar would experience an $F_{1}$ of $F_{1}=-F_{0}^{3}(B_{s}/3.2\times
10^{19}{\rm G})^{2}$.

Pulses experiencing a linear acceleration are affected by the parameters $P$
and $\dot{P}$. We write the phase $\phi_{i}$ as a function of $F_{0}=1/P$ and
spin frequency derivative $F_{1}=-\dot{P}/P^{2}$ in the following way:
$\phi_{i}=F_{0}\,t+0.5\,F_{1}\,t_{i}^{2}$. To complete our acceleration
search, we need to search $F_{0}$ and $F_{1}$ up to the intended accelerations
($F_{1}/F_{0}=-\dot{P}/P=a/c$). We fold the TOAs into a pulse density profile
using $\phi_{i}$ and use the $H$-test to determine the significance of the
resulting pulse profile. The $H$-test provides a better sensitivity to low-
significance pulsar candidates. So it is extensively used for periodicity
search in X-ray and $\gamma$-ray pulsar research (De Jager & Büsching 2010;
Bachetti et al. 2021). The $H$-test statistic is defined as:

$H=\max\left(\mathrm{Z}_{m}^{2}-4\mathrm{~{}m}+4\right),1\leq m\leq 20,$

where $Z_{n}^{2}$ is defined as:

$Z_{n}^{2}=\frac{2}{N}\sum_{k=1}^{n}\left[\left(\sum_{j=1}^{N}\cos
k\phi_{j}\right)^{2}+\left(\sum_{j=1}^{N}\sin k\phi_{j}\right)^{2}\right],$

where N is the number of detected photons, and $\phi_{i}$ (i $=1,...,N$) is
the pulse phase. De Jager & Büsching (2010) obtained the cumulative
probability of $H$ statistic by simulations as

$\operatorname{Prob}(>H)=\exp(-0.4H).$

To simplify the phase computation, we utilize the frequency $F_{0}$ and its
derivative $F_{1}$.

When conducting acceleration search for the one hour sessions, we search
$F_{0}$ in the range between 0.01 Hz and 1000 Hz. $F_{1}$ is selected to cover
the range of $a/c$ in [ -1$\times$10-6, 1$\times$10-6 ] s-1. Such an
acceleration exists if the neutron star is being pulled by a 200
$M_{\rm\odot}$ companion in a 10-hr orbit. We also estimate the resolution
needed for the search by calculating how much of a change in $F_{0}$ and
$F_{1}$ would lead to a small change in phase $\delta\phi_{\rm last}$ for a
$t=3600$ s. It is clear that the smallest change in $F_{0}$ and $F_{1}$ that
we need to search for $\delta\phi_{\rm last}$ not to exceed 0.2 is $\delta
F_{0}=5.6\times 10^{-5}$ s-1 and $\delta F_{1}=3\times 10^{-8}$ s-2. The
number of trials of the $F_{0}$-$F_{1}$ test reaches 7.2$\times$1011. This is
a substantial computing challenge. We split the computation into smaller
blocks of the size $\Delta F_{0}=1$ Hz by $\Delta F_{1}=1\times 10^{-5}$ s-2,
and use GPUs to conduct the phase calculation and compute the $H$ value for
the folded profiles. For each of the smaller blocks, we evaluate the folded
profile from a fine grid of 20480$\times$320 of $F_{0}$ and $F_{1}$ to achieve
the aforementioned resolution.

Figure 5:  Panel a is the $H$-test result of $P$-$\dot{P}$ search with 638 raw
TOAs including all the micro-components. The gray line represents the
threshold $H$ statistic that leads to a unity null-hypothesis probability when
taking the number of trials into account. Panel b shows the $H$ map of
searching with 638 TOAs augmented with ms-level random noises. Panel c shows
the $H$ map of searching with 424 grouped and augmented TOAs that do not
include all the micro-components. Panel d right plot shows the $H$ map results
of September 28th using the augmented grouped TOAs. We search the data of all
four days, respectively, and present here the result of September 28th as an
example. The search is conducted for $F_{0}$ in the range of 0.01$\sim$1000 Hz
and $\rm a/c$ in the range of [-1, 1] $\rm\times 10^{-6}~{}s^{-1}$. Here we
display $H$ greater than 74 as red circles. This threshold is determined by
the probability of the $H$-test based on the number of trials. The dash curve
represent the $F_{0}$–$F_{1}$ value corresponds to a surface dipole magnetic
field strength of $10^{13}$ G, $10^{14}$ G and $10^{15}$ G. The left plot of
panel d shows an example of the grid search block, in which the highest $H$
value pixel is marked with star. Panel e shows the search results for the
augmented and grouped TOAs combining all four days. The $H$-test result is on
the folded profile for assumed $F_{0}$ in the range of 0.01$\sim$333 Hz from
the TOAs of September 25th-28th. The $\rm a/c$ is in the range of [-2, 2]
$\rm\times 10^{-9}~{}s^{-1}$. The left plot is a zoomed-in display of the grid
search for each pixel of the right plot. Panel e right plot shows the
$P$-$\dot{P}$ search results. Here we display $H$ greater than 79 as red
circles. The dashed curve represent the $F_{0}$–$F_{1}$ values correspond to a
surface dipole magnetic field strength of $10^{13}$ G and $10^{14}$ G.

When conducting an acceleration search for TOAs, we conduct three initial test
searches, each with a different setting of TOAs from September 28th, the
highest event rate. For the first test, we search all TOAs, including
individual TOAs from the multi-components. The number of trials reaches
$10^{12}$ with the search range and resolution described above. The
theoretical false alarm threshold, i.e. null-hypothesis probability times
number of trials ($10^{12}$), reaches unity when $H=68$. However, we find many
signals with $H$ values larger than 68 (as shown in Figure 5 panel a). The
folding results look like multiple frequencies of smaller periods. So we
increase the search range to 1 – 5500 Hz with the number of trials of
$10^{13}$. In particular, we find a strong signal that seems to be the
harmonics of the fundamental frequency $F_{0}=5086.263$ Hz. This signal
happens to correspond to a period of 196.6 us – 4 times of our sample time.
Therefore, we suspect that this is an artifact of our data’s finite sampling
frequency. Figure 3 (a) shows a candidate with a period that is four times the
sample time. We check the folded profile by eye for the candidates with $H>$
200 and find most high $H$-test statistics are harmonically related to the
sampling artifact. Therefore, we conclude that we have to remove this
significant artifact signal to find any real period.

In the second test, we augment the TOAs by adding a random, normally
distributed noise to the TOAs with a Gaussian $\sigma$ of 1 ms. We call these
TOAs augmented TOAs hereafter. Using these augmented TOAs, we run our search
again, and this time the $5086.263$ Hz signal, and its harmonics are all gone.
Surprisingly, there are still hundreds of periodic signals with $H>$ 68 (as
shown in Figure 5 panel b), some of which appeared both in the September 27th
and 28th data. We inspect these profiles and find that they are likely to be
caused by the presence of a few clusters of pulses, i.e., the multi-components
of the pulses. These small clusters of TOAs tend to significantly boost the
counts in a single profile bin, leading to many high $H$ signals in the low
spin frequency end. Such artificial signals also prevent us from correctly
detecting any real spin period.

To minimize this micro-component effect, we conduct a third test. This time,
we group the TOAs from a cluster of multi-components into one. The one TOAs is
derived from the weighted center of the multi-components group. In this third
test, only a few period candidates slightly exceed $H$ of 68 (as shown in
Figure 5 panel c). Similarly, only a few detections come from the other three
days and no matching signals from different days. We show the $H$ distribution
from the third test in Figure 5. So again, no credible spin period was
detected after considering a large range of $P$ and $\dot{P}$. In Figure 5 d,
we show the $H$-test result from September 28, 2021 for $P$ between 1 ms and
100 s and the range of $\dot{P}$ limited by $|a/c|<1\times 10^{-6}$ s-1. In
the September 28th results presented here, we also expandedthe search to a
significant negative acceleration ($F_{1}$) that translate to surface magnetic
fields $B_{s}$ up to $10^{15}$ G for an isolated magnetar with $P$ between 3
ms and 100 s.

Finally, we conduct an $P$-$\dot{P}$ acceleration search for the grouped and
augmented TOAs from all four days. We search $F_{0}$ between 0.01 Hz and 333
Hz. $F_{1}$ is selected to cover the range of $a/c$ in [ -2$\times$10-9,
2$\times$10-9 ] s-1, which corresponds to companion mass up to 100 M⊙ and
orbital period longer than 40 days. Due to the longer observing time when
combining all 4 days, the resolution in $F_{0}$ and $F_{1}$ needed to be much
higher: $\delta F_{0}=6\times 10^{-7}$ s-1 and $\delta F_{1}=3\times 10^{-12}$
s-2. To complete this search, we split the computation into smaller blocks of
the size $\Delta F_{0}=1$ Hz by $\Delta F_{1}=1\times 10^{-5}$ s-2, each
containing 65536$\times$256 individual pairs of $F_{0}$ and $F_{1}$ that can
be evaluated in a single GPU run. This leads to a substantially higher number
of trials and changes the threshold for $H$ to 79. We show the search result
in Figure 5 panel e.

###  3.4 Periodicity search with special pulses

This section summarizes the periodicity search in pulses selected based on
frequency and polarization. We observed that some of the FRB pulses show up
only in part of our observable band, as shown in Figure 19, e.g. burst #(29).
The four pulses burst in #(29) appear in the low frequency part with periods
of 54.2 ms. Pulses with different spectral shapes may originate from different
rotational phases of the source object. In light of this possibility, we split
pulses based on their frequency range: high frequency and low frequency.
Pulses appearing in the full bandwidth are not considered here. We perform the
Lomb-Scargle periodicity search in different frequency bands and list the top
candidates. The periodicity search results of pulse categorized by frequency
on September 28th are shown in Figures 10. No obvious period was found after
folding.

For pulsars, the phases where their circular polarization changes sign usually
correspond to their polar cap’s phase center. Xu et al. (2021) found
considerable circular polarisation and quick position angle swing in FRB
20201124A. Wang et al. (2022b, 2021) and Tong & Wang (2022) discussed the
geometric and non-geometric origin of FRB circular polarization. If the FRB is
caused by a pulsar emitting in all rotational phases, we might not be able to
find its period through our methods. However, its pulses with unique
polarisation features could still show periodicity. We identify the pulses
with sign-changing circular polarization from the September episode. We then
use the LSP search and perform $P$-$\dot{P}$ search based on its barycentered
pulse arriving TOAs. The results are shown in Figure 9. Only 11 pulses are
selected, and the period candidates are not statistically significant. We show
the folding results of two possible top period candidates (top plot: 0.10852 s
and bottom plot: 4.11649 s) in Figure 9. To verify the significance of our
detection, we randomly pick samples of 11 pulses from all TOAs and test the
significance of the periodicity search. We compare the highest $H$ value from
the randomly selected samples and the pulses that we found using sign-changing
circular polarization. The highest $H$ of the random sample is larger than the
$H$ of circular polarization search for 1455 times in 4000 trails. Therefore,
we conclude that the number of special circular polarization pulses is too
small to obtain a significant detection. Still, this method may be worth
trying after more such pulses are collected in the future.

###  3.5 Periodicity analysis of multi-components

Hessels et al. (2019); Gourdji et al. (2019); Amiri et al. (2019) discovered
FRB multi-component phenomena from FRB 20121102A, FRB 180814\. Recently,
Pastor-Marazuela et al. (2022) reported the FRB 20201020A with five components
and quasi-periodicity of 0.415 ms. But multi-components have not been
systematically studied since bright repeated bursts are uncommon in FRBs.

The FRB multi-component phenomena are similar to the micro-structure already
observed in pulsars (Craft et al. 1968). Micro-structures have been noticed in
the observations of pulsars and millisecond pulsars (MSPs) (Cordes et al.
1990; De et al. 2016; Liu et al. 2022). Angular radiation and temporal
modulation are some possible theoretical explanations (Benford 1977; Rickett &
Cordes 1981; Boriakoff 1976; Cheng & Ruderman 1980; Arons 1979).

In this paper, we analyze the periodicity of multi-component bursts if they
are composed of at least 4 components with the longest waiting time less than
1 second. Many fine structures are composed of three consecutive peaks, but we
do not include them. This is because we can always find a common denominator
period from three pulses. Thus the resulting period is not a periodic signal
with statistical significance.

We find a total of 53 multi-components in the second active episode (Figure
19). The bottom panels show the pulse profiles; the solid black line in the
middle indicates the pulse profile, the solid red line represents linear
polarization, and the solid blue line represents circular polarization. The
top panel represents Position Angle (PA). As indicated in the Table 6, we
count the quasi-period information of all multi-components. We define the
residual $Res=\sqrt{\sum r_{i}^{2}}$ and Dimensionless residual $\overline{R}$
to characterize the quality of the quasi-period.

$\overline{R}=\frac{t_{\rm bin}\times\sqrt{\sum r_{i}^{2}}}{(N-1)\,P}$

Figure 6: An example of a typical pulse containing multi-components, the total
pulse width reaches 150 ms, there are 11 apparent structures, that can well
meet the interval of 3.060 ms. The red line in the figure represents linear
polarization, the blue line represents circular polarization, and the sub-
figure on the top is the position angle.

(a) Detailed analysis of multi-components in Figure 6.

(b) Detailed analysis of multi-components (29).

(c) Detailed analysis of multi-components (25).

(d) Detailed analysis of multi-components (10).

Figure 7: In each subplot, the top left plot is the pulse profile. The lower
left is the Fourier transform of the profile, and the scatter plot in the
middle is the residual of TOAs and the straight line fitting of cycle and
TOAs. The upper right is the auto-correlation analysis of the profile, and the
lower right is the result of the quasi-periodic hypothesis testing.

Andersen et al. (2021) presented discoveries of multi-components in three
CHIME FRBs. One of these FRB 20191221A shows periodicity with a confidence
level of $\rm 6.5\,\sigma$. The other two sources have lower confidences of
2.4 $\sigma$, and 1.3 $\sigma$, respectively. They pointed out that the sub-
bursts of multi-components are often downward-drifting in the 400–800 MHz. We
adopt the same analysis method to check for the significance of periodicity
found in FRB20201124A’s multi-components. We obtain the TOAs of the profile
peaks and found an intriguing quasi-periodic pattern. Figure 6 is one of the
most complicated structures in our observations. Both the downward-drifting
and upward-drifting components exist at this multi-component, and its period
is 3.060 ms. Detailed periodicity analysis of this multi-component is shown in
Figure 7(a). The picture in the upper left corner is the pulse profile. We
separated the components to limit the impact of the baseline on the
periodicity search. The Fourier transform of the separated profile is shown in
the lower-left corner. The red dotted line is the frequency where the quasi-
period is located. The picture in the upper-middle part is the residual
analysis of the arrival time. Influenced by the sub-pulse strength, not all
FFT results are obvious, so we determine the best quasi-period where the
multi-component residuals are smallest. The black dots in the lower middle of
the figure are pulses in the 11-component burst. X-axis is the integer-valued
vector of the burst. $n_{i}$ $=$ (0, 3, 7, 9, 14, 16, 20, 26, 29, 40, 41),
Y-axis is the TOA $t_{i}$. The period is given by the minimum TOA residual. In
order to compare with Andersen et al. (2021), we use the same statistical
analysis method, and fit the gap $n_{i}$ and TOA $t_{i}$ with a straight line,
$t_{i}=f^{-1}n_{i}+T_{0}+r_{i}$. Define a statistic value $L$ that measures
the difference to the straight line

$\hat{L}[n]=\frac{1}{2}\log\left(\frac{\sum_{i}\left(t_{i}-\bar{t}_{i}\right)^{2}}{r_{i}^{2}}\right),$

and define the periodicity-sensitive statistic $\hat{S}$:

$\hat{S}=\max_{n}(\hat{L}[n]).$

We sample the time series from the uniform distribution, and obtain the null-
hypothesis probability by comparing the $\hat{S}$ of the simulated events with
the observed data. We get a confidence of $\rm 3.3\,\sigma$ for the pulses in
the bottom right corner Figure 7(a). The figure in the upper right corner
shows the results of auto-correlation analysis of the pulse profile. The
definition of the auto-correlation function (ACF) is $R$ = $\sum
I(i)~{}I({i+\tau})$. Other bursts and the same analysis are shown in Figures
7(b),7(c),7(d). We show in Figure 8 the distribution of multi-component
periods and find a peak at 5 ms with a wide distribution.

The highest confidence quasi-periodic structure that we found is $\rm
3.9\,\sigma$, less than the $\rm 6.5\,\sigma$ level found from FRB 20191221A
by CHIME. However, multiple multi-components of FRB 20201124A are found to
show periods with the confidence higher than 3 $\sigma$. As shown in Figure 8,
the multi-component periods do not seem to be related to each other. Since
these multi-components come from the same source that lacks a global period,
these period signals are false positives. Our result shows that we should be
cautious when linking low-significance quasi-periodicity in FRB micro-
components to the rotation of the source object at least for signals with
significance below $\sim 4\,\sigma$.

Figure 8: Multi-components quasi-period statistical histogram. Figure 9: The
left image shows the observed pulse with the sign changing of the circular
polarization, where both the linearly and circular polarization are shown. The
middle plot only shows the circular polarization when zoomed in. The right
side shows the folded events based on two top period candidate the
$F_{0}$-$F_{1}$ search graph. Figure 10: Lomb-Scargle periodogram for the high
frequency and low frequency pulses detected on September 28th. Red line
represents high frequencies, blue line represents low frequencies, and circles
represent significant peaks. Figure 11: The figure is divided into upper,
middle and lower panels. The upper panel is the waiting time of the first
active episode, the middle is the second active episode, and the lower is the
waiting time of September 28th with the highest burst rate. We show the
results for different numbers of bins. For the 28th with the highest burst
rate, we perform a Gaussian fit and plot the mean values of the double peaks.
Figure 12: The K-S test results of the simulated time series and the observed
time series. The figure shows K-S test null-hypothesis probability of the
waiting time distribution of the second active episodes. We simulate the two
peaks independently since they may relate to different physics. The left panel
corresponds to (c) in Figure 11, while the right panel corresponds to (d).

###  3.6 Test putative periods using waiting times

Waiting times are the time lapses between consecutive pulses. They often are
strongly affected by the FRB repeating rate. Cruces et al. (2021); Li et al.
(2021) studied the waiting time distribution of FRB 20121102A, focusing on the
fitting of the distribution. They get the shape factor of the Weibull
distribution to study their aggregation characteristics. In this section, we
simulate pulses from a rotating object of different spin periods and compare
their waiting times to the observed waiting times to gain insight into the
range of the possible spin period for FRB 20201124A.

This work focuses on episode 2 (September 25-28th, 2021) of the two active
episodes from FRB 20201124A, which reaches an order of magnitude higher peak
event rate than episode 1 in April 2021 (Xu et al. 2021). The drastically
different burst rates lead to significantly different waiting time
distributions. We plot the waiting time distribution in Figure 11. The first
and second panels show the overall waiting times from the active episode 1 and
2, respectively, and the bottom panel shows those from the one hour
observation on September 28th, 2021, from which the event rate is at its peak.
The waiting time property has the following features: 1. two significant
bumps, one peaking in milliseconds and the other peaking in seconds. 2. the
left millisecond peak is almost identical in the two episodes, indicating that
these small waiting times may be caused by intrinsic physics that is not
affected by the burst rate. 3. the right, seconds-level peak become smaller in
episode 2, driven by the change in the event rate. The millisecond waiting
time peak reported in this work appears to be less asymmetric than the same
waiting time distribution reported for episode 1 Xu et al. (2021). This
apparent change is likely because we split the multi-components into
individual pulses when doing this paper’s analysis, so we tend to get shorter
waiting times than Xu et al. (2021).

Figure 13: We generate periodic time series with different duty cycles and
then use the previous methods $H$-test and LSP to search for periods,
respectively. The figure shows the $H$ statistic and the power of the LSP,
which becomes smaller when the duty cycle increases.

The waiting time distribution itself does not clearly show a spin periodicity.
If a time series has a clear period, sharp peaks shall appear in the waiting
time distribution. Considering that for some pulsars with a significant duty
cycle and nulling behavior the pulses could appear in a range of phases and
may not appear in every period, the peak of waiting time distribution may be
broadened into the observed shape.

The double-peaked waiting time distribution is likely the result of physics of
two different time scales. There are two possible scenarios: 1\. the
millisecond-scale waiting time peak is caused by the spin of the source, while
the second-scale waiting time peak is caused by the source turning on and off
in timescales of many seconds, similar to the nulling phenomenon in pulsars.
2\. the second-scale waiting time peak is caused by the rotation of the
source, while the millisecond-scale peak comes from pulse micro-structures
similar to those observed in some pulsars (Liu et al. 2022).

For scenario 1, we only simulate a millisecond pulsar. This will only generate
waiting time distribution analogous to the left waiting time peaks. In the
scenario 2, we simulate an intermittent pulsar with period of seconds, and
ignore the millisecond left peak, which is presumably related to the intrinsic
radiation process. We simulate pulses of various periods, burst rate and duty
cycles. We omit $\dot{P}$ because it is a negligible effect in a timescale of
seconds and should not affect the distribution of waiting times. We generate
samples of pulse arrival times following the von-Mises distribution for a
given duty cycle, burst rate and 1 hour integration time. Whether a pulse
appears in the current period depends on a Poisson chance based on the burst
rate. Then we employ a Kolmogorov–Smirnov (K-S) test to compare the simulated
pulses’ waiting time distributions with those observed from the active episode
2. The result is shown in Figure 12. The three plots in the left panel of the
Figure correspond to the K-S test results from fitting the left waiting time
peak for episode 2 marked with (c) in Figure 11, and the right panel
corresponds to the result of fitting the right waiting time peak of episode 2
marked with (d) in Figure 11.

The larger the null-hypothesis probability, the higher the degree of agreement
between the observation and the simulation. One can see that only a narrow
range of periods are compatible to the observed sample. We find that possible
compatible period range of the left and right peaks in [0.29 ms, 10.36 ms] and
[0.31s, 3.04 s ], assuming a minimum null-hypothesis probability of 0.05.

###  3.7 Large duty cycle and period detectability

To investigate the potential reasons for a non-detection of the spin period,
we simulate periodic time series of different duty cycles and then perform the
aforementioned searches on these time series using the $H$-test and LSP. We
keep the simulated observing time at one hour and the total number of pulses
close to the September 28th burst number (424 after grouping together the
micro-components). We fold and compute the $H$ and LSP power of the mock time
series. The resulting statistics of different periods and duty cycles are
displayed in Figure 13. We employ the $H$ statics threshold of 72 in 27th and
74 in 28th, and LSP power of 0.17 in 27th and 0.1 in 29th and mark the region
where the $H$ and LSP power of the time-series start to drop below thresholds
with the transparent gray band in Figure 13. The selection of the $H$
statistical threshold is based on the cumulative probability and the number of
trials. The LSP threshold is chosen to be the highest spectral power from the
LSP search of that day. These simulations show that the spin would not have
been detectable if the pulse duty cycle is larger than $0.49\pm 0.08$ if we
use the $H$ statics and $0.58\pm 0.05$ if using the LSP.

Like the PRESTO S/N, the von-Mises duty cycle is an indication of the signal
significance. A smaller duty cycle corresponds to a sharper and more
significant pulse shape, while a larger duty cycle corresponds to a wider and
less significant pulse shape. Since there are only 424 independent TOAs (from
September 28th), Figure 13 shows that we can only detect the period signal if
its pulse distribution is sharp enough.

##  4 Summary and Discussion

In this study we report a systematic search of pulsar spin periodicity from an
extremely active episode of FRB20201124A. We attempted different periodicity
search strategies for the data of FRB 20201124A taken during its active
episode in September 2021 by the FAST telescope: 1. searching the time series
dedispersed from the original data; 2. searching for periodicity based on the
pulse TOAs; 3. searching for periodicity in the multi-component fine pulse
structures; 4. Periodicity Search on selected pulses. Despite various
attempts, we did not find a credible period with high confidence. Here we
consider a period credible if 1\. it is statistically significant; 2\. it
appears in the search result for multiple days; and 3\. it is not caused by
obvious artificial reasons such as the sampling frequency.

Based on all the pulse TOAs from four days, we have folded and searched for a
period in the range of 0.01 Hz to 333 Hz and a linear period derivative equal
to the effect of a Doppler acceleration of $a/c$ in the range of [-2,
2]$\times 10^{-9}$ s-1, with sufficient resolutions to resolve a 0.2 phase
shift for the last TOA. This is the maximum acceleration produced by the pull
of a 100 M⊙ mass star or a black hole companion in an orbit of 40 days around
the neutron star. Any lighter companion star or longer orbit would lead to a
smaller maximum acceleration. We also folded and searched the TOAs from
individual observing sessions, for a period in the range 0.01 Hz to 1000 Hz
and period derivative for Doppler acceleration $a/c$ in [-1,1]$\times$10-6s-1.
This acceleration is equal to the pull of a 200 M⊙ mass star or a black hole
companion in an orbit of 10 hours. Both searches yield no credible spin
period. We suspect that the possible reasons for a non-detection, assuming
that the FRB is from a rotating neutron star, include a potentially large
pulse duty cycle ($>0.49$) or that the pulsar has a heavier companion or a
shorter orbit than we have assumed.

Although no credible period was identified, we provide in Table 3 a list of
top period candidates that we could neither positively identify nor rule out.
We present the periods discovered by various approaches as well as the $H$
value of the fold result.

From the fine structures of pulses, we have identified 53 bursts with fine
multi-components through which tentative periods could be derived from more
than four components. The multi-components of FRBs share many similarities
with the micro-structures of pulsars. They are observed in different
frequencies (Lange et al. 1998) and both exhibit quasi-periodic properties.
However, there are also some slight differences. First, the characteristic
timescale of pulsar micro-structure ranges from a few microseconds to hundreds
of microseconds (Johnston et al. 1996). Its typical quasi-period ranges from
several microseconds to several thousand microseconds (Popov et al. 2002,
1987). The quasi-periods observed in FRB 20201124A are on the order of
milliseconds, albeit only a sample of FRBs exhibit micro-components. Perhaps
microsecond-level fine structures exist in FRBs but are not yet observable due
to DM and scatter smearing in most observations. Additionally, the
polarization properties are also different. Kramer et al. (2002) found a
strong correlation between Stokes parameters $V$ and $I$ in the micro-
structure of Vela, and they did not find the handedness of the circular
intensity changes within a micro-pulse. In FRB 20201124A, there are circular
polarization sign changes in multi-components, and the circular polarization
has no obvious correlation with $I$.

In this paper, we find that the periods of multi-components vary in a wide
range and do not seem to correlate with each other. Therefore, we conclude
that the low-significance periodicity observed in the pulse micro-structures
is likely due to radiation mechanisms, not to the spin of the source object.
Some Low-significance periods have also been found in some FRBs. We advocate
caution when linking such quasi-periodic phenomenon to the spin period of the
source.

We searched for spin period systematically between 1 ms and 100 s and linear
spin acceleration up to a $F_{1}=3.33\times 10^{-4}$ s-2 from the second
active episode of FRB 20201124A and found no convincing spin period that
appeared in two separate observations. From the 424 bursts detected in the one
hour FAST observation taken on 2020 September 28th and 186 bursts detected on
2020 September 27th, our periodicity search methods should have found a
significant pulsar signal if it had a duty cycle less than 0.49 and with $P$
and $\dot{P}$ in our search ranges. Most pulsars show a duty cycle much
smaller than this (Rankin 1990; Kramer et al. 1998), but large duty cycles do
exist for few special cases where strong bridge emission exist or the pulsar
is an aligned rotator (Kou et al. 2021; Wang et al. 2022c; Han et al. 2021).
Some millisecond pulsars also show relatively wide pulses Johnston et al.
(1993); Kramer et al. (1998); Stairs et al. (1999). Single pulses from the
magnetar PSR J1622-4950 were found to spread in a wide range of phases (Levin
et al. 2012). Our result contests the FRB models invoking a pulsar with
typical polar cap emission or a spinning down magnetar with $B_{s}<10^{15}$ G.
For magnetars with $P>1$ s, we also rule out detection with duty cycle $<0.49$
and $B_{s}<10^{16}$ G. Note that all magnetars known to date have $P>1$ s and
$B_{s}<10^{16}$ G
111111https://www.physics.mcgill.ca/~pulsar/magnetar/main.html(Olausen & Kaspi
2014). The acceleration we searched also allows us to contest binary models of
which the orbital period is longer than 10 hours and companion below 100 M⊙.
In summary, our results contest many of the contemporary models of repeating
FRBs in a wide range of parameter space.

It is worth mentioning that the above statements apply to pulsar-like emission
models of FRBs that invoke emission of FRBs from the open field line regions
of magnetars (e.g. Kumar et al. 2017; Yang & Zhang 2018; Wang et al. 2019; Lu
et al. 2020). However, since FRBs are much more energetic than radio pulsar
emission, an FRB originating from the inner magnetosphere of a magnetar would
likely greatly distort the field lines (Qu et al. 2022), making the well-
defined open field line region fuzzier and enlarging the observed duty cycles.
Alternatively, if the FRB emission region is outside of the magnetopshere as
some models suggest (e.g. Metzger et al. 2019; Beloborodov 2020), then the
emission phase could be more random. As a result, the non-detection of periods
from this source does not necessarily rule out a magnetar as the FRB source,
even though other possibilities may be also considered (Katz 2022).

Table 2: Presto search Candidate list

Index | Harm |  $P_{bary}$ (ms) |  $\dot{P}_{bary}$ (s/s) | Date | Sigma
---|---|---|---|---|---
1 | 32 | 126.06423(11) | $-5.6(2.3)\times 10^{-10}$ | 59484 | 6.2
2 | 32 | 162.078505(96) | $-4.04(21)\times 10^{-9}$ | 59484 | 17.1
3 | 16 | 50.025310(27) | $-1.13(59)\times 10^{-10}$ | 59484 | 5
4 | 32 | 71.575233(38) | $-4.74(84)\times 10^{-10}$ | 59484 | 15.9
5 | 32 | 99.047556(69) | $-5.0(1.5)\times 10^{-10}$ | 59484 | 11.2
6 | 32 | 88.044581(40) | $-2.029(90)\times 10^{-9}$ | 59484 | 14.4
7 | 32 | 76.511895(42) | $-1.807(93)\times 10^{-9}$ | 59484 | 11.5
8 | 32 | 61.905820(42) | $1.185(93)\times 10^{-9}$ | 59484 | 11.1
9 | 32 | 117.055470(68) | $-2.19(15)\times 10^{-9}$ | 59484 | 17.6
10 | 32 | 91.625691(41) | $1.299(92)\times 10^{-9}$ | 59484 | 12.2
11 | 32 | 82.147963(36) | $3.86(80)\times 10^{-10}$ | 59484 | 13.4
12 | 32 | 114.055239(60) | $-4.67(13)\times 10^{-9}$ | 59484 | 13.9
13 | 32 | 90.044511(60) | $-2.51(13)\times 10^{-9}$ | 59485 | 13.2
14 | 2 | 499.89481(70) | $4.24(16)\times 10^{-8}$ | 59485 | 60.9

Table 3: FFA Candidate list

Index | Period(s) | Date | SNR | Index | Period(s) | Date | SNR
---|---|---|---|---|---|---|---
1 | 26.51(2) | 59484 | 70.62 | 13 | 106.07(2) | 59484 | 67.94
2 | 31.84(1) | 59484 | 63.31 | 14 | 123.1(1) | 59484 | 67.00
3 | 34.02(1) | 59484 | 61.38 | 15 | 212.1(3) | 59484 | 65.96
4 | 37.86(1) | 59484 | 68.61 | 16 | 265.2(6) | 59484 | 77.31
5 | 38.40(1) | 59484 | 66.28 | 17 | 344.1(3) | 59484 | 66.81
6 | 52.18(1) | 59484 | 63.78 | 18 | 424.4(8) | 59484 | 67.45
7 | 53.02(9) | 59484 | 74.36 | 19 | 591.5(5) | 59484 | 71.23
8 | 67.50(2) | 59484 | 63.69 | 20 | 36.714(3) | 59485 | 62.91
9 | 72.25(3) | 59484 | 86.92 | 21 | 70.0(1) | 59485 | 71.69
10 | 75.76(4) | 59484 | 64.99 | 22 | 136.41(8) | 59485 | 66.50
11 | 88.4(1) | 59484 | 65.61 | 23 | 227.3(4) | 59485 | 67.59
12 | 98.56(3) | 59484 | 62.60 | 24 | 301.2(7) | 59485 | 66.44

Table 4: LSP Candidate list

Index | Period(s) | Date1 | Date2 | FAP
---|---|---|---|---
1 | 0.7924(3) | 59482 | 59485 | 0.1
2 | 0.3216(2) | 59483 | 59485 | 0.1

Table 5: $P-\dot{P}$ search Candidate list

Index |  $F_{0}$(Hz) |  $F_{1}$($s^{-}2$) | Date |  $H$ value
---|---|---|---|---
1 | 185.729821 |  -92.037618$\times 10^{-6}$ | 59485 | 76.894955
2 | 144.325944 |  735.642633$\times 10^{-6}$ | 59485 | 78.167674
3 | 222.950632 |  -1540.698504$\times 10^{-6}$ | 59485 | 78.905402
4 | 316.941093 |  143.882353$\times 10^{-9}$ | 4day | 79.78374
5 | 293.883889 |  478.658824$\times 10^{-9}$ | 4day | 83.259686
6 | 279.292725 |  -315.964706$\times 10^{-9}$ | 4day | 80.035248
7 | 274.766622 |  -522.439216$\times 10^{-9}$ | 4day | 83.077999
8 | 246.126386 |  352.545098$\times 10^{-9}$ | 4day | 89.229852

  * •

The errors of $F_{0}$ and $F_{1}$ are the precision of our search.

Table 6: Multi-components of FRB 20201124A

Index |  N$\ast$ |  $P$ (ms) | $\overline{R}$ | MJD
---|---|---|---|---
1 | 5 | 7.164 | 0.0240 | 59484.81606
2 | 7 | 2.638 | 0.0411 | 59484.81650
3 | 5 | 2.471 | 0.0685 | 59484.82173
4 | 4 | 11.996 | 0.0368 | 59484.82257
5 | 6 | 5.587 | 0.0415 | 59484.82357
6 | 5 | 3.801 | 0.0347 | 59484.82633
7 | 6 | 3.504 | 0.0716 | 59484.83085
8 | 4 | 5.637 | 0.0077 | 59484.83419
9 | 5 | 5.182 | 0.0337 | 59484.84094
10 | 8 | 13.521 | 0.0272 | 59484.84349
11 | 5 | 6.409 | 0.0308 | 59484.84719
12 | 4 | 8.077 | 0.0609 | 59484.84938
13 | 4 | 12.532 | 0.0532 | 59484.85148
14 | 4 | 9.736 | 0.0748 | 59485.78325
15 | 5 | 2.766 | 0.0198 | 59485.78338
16 | 5 | 4.013 | 0.0761 | 59485.78488
17 | 8 | 14.655 | 0.0238 | 59485.78561
18 | 4 | 9.308 | 0.0381 | 59485.78653
19 | 6 | 48.734 | 0.0058 | 59485.78912
20 | 5 | 4.869 | 0.0436 | 59485.78933
21 | 4 | 6.446 | 0.0931 | 59485.79068
22 | 7 | 21.804 | 0.0355 | 59485.79374
23 | 5 | 9.623 | 0.0456 | 59485.79432
24 | 5 | 9.678 | 0.0552 | 59485.79515
25 | 10 | 1.465 | 0.0471 | 59485.79612
26 | 5 | 4.255 | 0.0748 | 59485.79922
27 | 5 | 5.115 | 0.0209 | 59485.79949
28 | 4 | 20.35 | 0.0087 | 59485.79996
29 | 4 | 54.227 | 0.0292 | 59485.80002
30 | 6 | 4.915 | 0.0120 | 59485.80020
31 | 4 | 5.999 | 0.0629 | 59485.80131
32 | 6 | 2.311 | 0.0463 | 59485.80133
33 | 4 | 6.764 | 0.0444 | 59485.80161
34 | 4 | 21.211 | 0.0045 | 59485.80171
35 | 4 | 39.408 | 0.0172 | 59485.80309
36 | 5 | 12.762 | 0.0208 | 59485.80323
37 | 7 | 5.069 | 0.0078 | 59485.80379
38 | 5 | 7.867 | 0.0674 | 59485.80521
39 | 5 | 6.075 | 0.0679 | 59485.80717
40 | 6 | 21.629 | 0.0080 | 59485.80729
41 | 5 | 3.457 | 0.0429 | 59485.80764
42 | 4 | 4.967 | 0.0888 | 59485.80808
43 | 5 | 6.835 | 0.0148 | 59485.80927
44 | 4 | 8.007 | 0.0439 | 59485.81131
45 | 4 | 6.675 | 0.0213 | 59485.81193
46 | 4 | 4.045 | 0.0574 | 59485.81267
47 | 6 | 7.354 | 0.0171 | 59485.81270
48 | 4 | 3.547 | 0.0763 | 59485.81270
49 | 6 | 5.155 | 0.0141 | 59485.81702
50 | 5 | 33.551 | 0.005 | 59485.81785
51 | 4 | 4.956 | 0.0169 | 59485.81890
52 | 7 | 5.918 | 0.0061 | 59485.82070
53 | 4 | 4.677 | 0.0258 | 59485.82138

  * •

$\ast$ Number of pulses.

(a)

(b)

(c)

(d)

Figure 14: Presto candidates fold results.

Figure 19: Examples of multi-components.

###### Acknowledgements.

We thank the referee, Scott Ransom, for helpful comments. This work made use
of data from the FAST, a Chinese national mega-science facility, built and
operated by the National Astronomical Observatories, Chinese Academy of
Sciences. This is work is supported by the National SKA Program of China No.
2020SKA0120200, 2020SKA0120100 the National Nature Science Foundation grant
No. 12041303, 11873067, 12041304 the National Key R$\&$D Program of China No.
2017YFA0402600, 2021YFA0718500, 2017YFA0402602, the CAS-MPG LEGACY project,
the Max-Planck Partner Group, the Key Research Project of Zhejiang Lab no.
2021PE0AC0. J. L. Han is supported by the National Natural Science Foundation
of China (NSFC, Nos. 11988101 and 11833009) and the Key Research Program of
the Chinese Academy of Sciences (Grant No. QYZDJ-SSW-SLH021); D. J. Zhou is
supported by the Cultivation Project for the FAST scientific Payoff and
Research Achievement of CAMS-CAS.

## Authors contributions

Jia-Rui Niu did the main data processing and analysis and the article writing.
Wei-Wei Zhu led the writing of this paper and conducted the $P$-$\dot{P}$
search section of the data analysis. Bing Zhang and Wei-Wei Zhu proposed and
chaired the FAST FRB key science project. Bing Zhang, Wei-Wei Zhu, J. L. Han,
Di Li and Ke-jia Lee coordinated the teamwork, the observational campaign, co-
supervised data analyses and interpretations. Mao Yuan helped in interference
mitigation. De-Jiang Zhou analyzed burst morphology and the results are
presented in paper I of this series. Yong-Kun Zhang analyzed the energy
distribution and the details are presented in paper II of this series. Jin-
Chen Jiang analyzed the polarization and the results are presented in paper
III. Pei Wang, Yi Feng, Dong-Zi Li, Chen-Chen Miao, Chen-Hui Niu, Heng Xu, Rui
Luo, Chun-Feng Zhang, Wei-Yang Wang, Bo-Jun Wang, Jiang-Wei Xu participated in
the discussion and revision of the paper.

## Data availability

All data for the plots in this paper, including these in appendix, can be
obtained from the authors with a kind request.

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