# Searching for rapid pulsations in solar flare X-ray data

Andrew R. Inglis Solar Physics Laboratory, Heliophysics Science Division, NASA
Goddard Space Flight Center, Greenbelt, MD, USA  Physics Department, The
Catholic University of America, Washington, DC, USA  Laura A. Hayes European
Space Agency (ESA), European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

###### Abstract

Most studies of quasi-periodic pulsations in solar flares have identified
characteristic periods in the 5 – 300s range. Due to observational limitations
there have been few attempts to probe the ¡ 5s period regime and understand
the prevalence of such short-period quasi-periodic pulsations. However, the
Fermi Gamma-ray Burst Monitor (GBM) has observed approximately 1500 solar
flares to date in high cadence 16 Hz burst mode, providing us with an
opportunity to study short-period quasi-periodic pulsations at X-ray energies.
We systematically analyse every solar flare observed by Fermi/GBM in burst
mode, estimating the prevalence of quasi-periodic pulsations in multiple X-ray
energy bands. To better understand these results, we complement this with
analysis of synthetic solar flare lightcurves, both with and without
oscillatory signals present. Using these synthetic lightcurves, we can
understand the likely false alarm and true positive rates in the real solar
GBM data. We do not find strong evidence for widespread short-period quasi-
periodic pulsations, indicating either a low base occurrence rate of such
signatures or that their typical signal-to-noise ratios must be low – less
than 1 – in Fermi/GBM data. Finally, we present a selection of the most
interesting potential quasi-periodic pulsation events that were identified in
the GBM solar X-ray data.

##  1 Introduction

Solar flares are rapidly evolving transient phenomena, capable of releasing up
to 1032 ergs of energy in a matter of minutes from the solar corona. Flares
are efficient accelerators of energetic particles, and also heat the local
solar atmosphere to millions of degrees. Thus, understanding flares is key for
our knowledge of how energy is released in the solar atmosphere, and in highly
magnetised plasma throughout the universe.

The time-variability of flare emission – and periodic behaviour in particular
– is a subject of much interest. Quasi-periodic pulsations (QPPs) are a
regularly observed phenomenon during both the impulsive and decay phases of
flare emission (Nakariakov & Melnikov, 2009; Inglis et al., 2016; Van
Doorsselaere et al., 2016; Hayes et al., 2020; Zimovets et al., 2021).
Although lacking a strict definition, the term QPPs typically refers to
evidence of periodic behaviour in flare timeseries, though this behaviour may
not be strictly oscillatory. Signals with evolving periods and anharmonic
shapes are often included under the QPP umbrella. QPPs have been noted for
over sixty years (e.g. Parks & Winckler, 1969; Chiu, 1970; Lipa, 1978) and can
be observed throughout the electromagnetic spectrum, from radio waves and
microwaves (e.g. Grechnev et al., 2003; Melnikov et al., 2005; Inglis et al.,
2008; Kupriyanova et al., 2013; Carley et al., 2019), to ultraviolet (UV)
(e.g. Brosius & Inglis, 2018), extreme-ultraviolet (EUV) (e.g. Dominique et
al., 2018), X-ray (e.g. Asai et al., 2001; Dolla et al., 2012; Simões et al.,
2015; Hayes et al., 2020; Collier et al., 2024), and gamma-ray wavelengths
(Nakariakov et al., 2010). In many cases, QPPs are a multi-wavelength feature
of flare emission, seen similarly across many pass-bands (e.g. Clarke et al.,
2021). Similar quasi-periodic signatures have also been observed on other
stars, primarily in the white light regime and occasionally in X-rays, for
example Mathioudakis et al. (2006); Pugh et al. (2015); Balona et al. (2015);
Pugh et al. (2016); Broomhall et al. (2019a); Kolotkov et al. (2021); Doyle et
al. (2022). QPPs are crucial to understand because they are signatures of
fundamental physical processes that occur during solar flare energy release
(McLaughlin et al., 2018; Zimovets et al., 2021), including magnetic
reconnection, particle acceleration, and magnetohydrodynamic (MHD) wave
generation. A comprehensive understanding of QPPs therefore allows them to be
used as potential diagnostic tools for understanding how energy is released in
solar and stellar atmospheres.

Although QPPs have been detected with a wide range of periods, from sub-second
(e.g. Knuth & Glesener, 2020) to tens of minutes (e.g. Foullon et al., 2005),
due to observational constraints most detections of QPPs have historically
been in the 5 - 300s range (e.g. Simões et al., 2015; Hayes et al., 2020).
While it is well-established that solar X-ray emission can show structure at
sub-second scales (e.g. Kiplinger et al., 1983, 1984; Dennis, 1985), and case
studies of rapid pulsations do exist (Knuth & Glesener, 2020), there have been
few attempts to systematically search solar flare data for evidence of rapid
quasi-periodic behaviour in the ¡ 5s regime. Thus, the prevalence of rapid
cadence QPPs in solar flares is currently unknown. This is crucial, since
pulsations on such timescales imply the presence of periodic drivers of
particle acceleration. In this work, we conduct one of the first such searches
using high cadence data from the Fermi Gamma-ray Burst Monitor (GBM). Fermi is
an astrophysics mission designed to observe distant transient events, but also
observes solar flares. During transient events, the GBM often triggers a high
cadence mode for certain binned data products, temporarily switching to 0.064s
(16 Hz) temporal binning. We analyse all solar flares currently listed in the
Fermi/GBM trigger catalog, a total of 1460 events. For each event, we search
for evidence of QPPs in multiple energy ranges by fitting different models to
the flare Fourier power spectra, following the methodology presented in Inglis
et al. (2015, 2016); Broomhall et al. (2019b); Hayes et al. (2020); Murphy et
al. (2020). To better verify the results and understand the potential for
false alarms, we complement this observational search with tests of the
methodology on sets of simulated lightcurves.

In Section 2 we describe the data sources and the methodology used in this
research. In Section 3 we use sets of simulated lightcurves to evaluate the
performance of our methodology and understand the potential false alarm rates.
In Section 4 we apply our methodology to the Fermi/GBM burst mode data from
1460 solar flares. We discuss the results of the statistical search for rapid
pulsations in Section 4.1, and present a selection of the most interesting QPP
events in Section 4.2. The entire work is summarized in Section 5.

##  2 Data and methodology

###  2.1 Fermi/GBM data

For this work, we use data from the Fermi/GBM instrument (Meegan et al.,
2009). GBM consists of 12 NaI detectors and 2 BGO detectors, with their
pointing axes arranged at different angles such that together they can observe
the entire visible sky and localize transient astrophysical phenomena such as
gamma-ray bursts (GRBs). The NaI detectors cover the energy range $\sim$ 4 keV
– 1 MeV, while the BGO detectors complement this with a higher energy range of
$\sim$ 200 keV – 40 MeV.

GBM produces three main science data products; the CSPEC, CTIME, and TTE
(time-tagged event) data products. TTE data provides a list of the individual
detected photons each with timestamp information. CSPEC data provides binned
count rates over time using 128 energy bins with a default temporal cadence of
$\sim 4s$. The CTIME data product sacrifices energy resolution in favour of
higher temporal resolution; it provides binned count rates in 8 energy bins
with a default time resolution of 0.256s, and 0.064s in trigger mode.

The GBM was designed to trigger different science operational modes when
transient events – such as GRBs or solar flares – are detected. Since GBM is
able to localize events, a solar flare can be identified when the source
location is consistent with the position of the Sun. For the CTIME product,
the time resolution is increased from 0.256s to 0.064s for a period of time
after the trigger occurs, typically 600s, though it can be shorter. It is this
high time resolution burst mode CTIME data that is the focus of this work, as
it provides us with the opportunity to search for quasi-periodic signals in
the 0.1s - 5s period regime.

We begin with the Fermi/GBM trigger
catalog111https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigtrig.html, which
at the time of analysis contained 1460 events marked as solar flares between
2008 to present. Fermi/GBM data has been previously analysed by Inglis et al.
(2016) at 1s cadence in search of quasi-periodic pulsations. In that work, the
distribution of detected periods was centered at $\sim$ 10s, with periods of
$<$ 5s largely inaccessible due to the chosen temporal cadence. The GBM burst
mode data provides us with an opportunity to explore the $<$ 5s regime in
search of pulsations. In this work, we limit our search for pulsations to this
short period regime, while acknowledging that GBM is effective at detecting
longer period pulsations in flares.

###  2.2 Time series analysis with the AFINO tool

To search for flare quasi-periodic pulsations with characteristic periods in
the ¡5s regime, we use the AFINO (Inglis et al., 2015, 2016, Automated Flare
Inference of Oscillations) analysis tool. We apply AFINO to GBM flare X-ray
data in three binned energy ranges provided by the CTIME product, 4 - 15 keV,
15 - 25 keV, and 25 - 50 keV. AFINO has been previously described in detail by
Inglis et al. (2015, 2016); Murphy et al. (2018, 2020); Hayes et al. (2020),
therefore we discuss only the key aspects of the method here.

For a time series of interest, AFINO works by fitting a selection of models,
one of which includes an oscillatory signal, to the Fourier power spectrum of
the normalized, apodized original series. The original signal is not de-
trended during this analysis, to avoid introducing artifacts into the results
(see Dominique et al., 2018, for a discussion of this issue). Normalization is
done for convenience only, however the apodization – that is, the
multiplication of the original signal by a window function – is important to
mitigate the effects of having a non-infinite time series. In this work we use
the Hanning window function. Following these steps, the Fourier power spectrum
is fit by each model using a maximum likelihood method. Then, these models are
compared using the Bayesian Information Criterion (BIC) to determine which
model is most probable given the data. The BIC is defined as

$BIC=-2\ln(L)+k\ln(n)$ (1)

where $L$ is the maximum likelihood, $k$ is the number of free parameters and
$n$ is the number of data points in the power spectrum.

In this paper, we test three models of the Fourier power spectrum, one of
which is representative of an oscillation in the original time series. These
models may be written

$S_{0}(f)=A_{0}f^{-\alpha_{0}}+C_{0}$ (2)

$S_{1}(f)=A_{1}f^{-\alpha_{1}}+B\ \exp\left(\frac{-(\ln f-\ln
f_{p})^{2}}{2\sigma^{2}}\right)+C_{1}$ (3)

$S_{2}(f)=\begin{cases}A_{2}f^{-\alpha_{b}}+C_{2},&\text{if $f<f_{break}$}.\\\
A_{2}f^{-\alpha_{b}-\alpha_{a}}f^{-\alpha_{a}}+C_{2},&\text{if
$f>f_{break}$}.\end{cases}$ (4)

for frequencies $f$, where $A_{0}$, $A_{1}$, $A_{2}$ and $B$ are amplitudes,
$\alpha_{0}$, $\alpha_{1}$, $\alpha_{a}$ and $\alpha_{b}$ are power law
indices, $C_{0}$, $C_{1}$, and $C_{2}$ are constants, and $f_{p}$ and $\sigma$
represent the location in frequency of a Gaussian peak and its width
respectively.

In descriptive terms, model $S_{0}$ is the simplest, a single power law plus a
constant term. This model is motivated by the fact that power law power
spectra are ubiquitous in astrophysical phenomena (Vaughan, 2005, 2010; Cenko
et al., 2010; Gruber et al., 2011; Huppenkothen et al., 2013). Model $S_{1}$
represents an oscillation in the data, including a localized Gaussian
frequency enhancement at $f_{p}$ in addition to a power law and a constant
term. In this work, we limit the maximum allowed period to 10s in order to
focus on short-period events. The minimum period is determined by the Nyquist
frequency. The width $\sigma$ of the Gaussian enhancement is also constrained
to lie in the range 0.03 $<\log\sigma<$ 0.15, preventing the frequency
enhancement from being unrealistically narrow or broad. Model $S_{2}$ is a
broken power law, an alternative to model $S_{0}$ that can capture more
complex features in the Fourier spectrum but does not contain an oscillation.

For each fitted model, we calculate BIC and compare the QPP-like model with
all other models, i.e. we calculate $\Delta$BIC = $BIC_{j}$ \- $BIC_{QPP}$ for
all the non-QPP models $j$. Since the fitting procedure minimizes $L$ and
therefore $BIC$, we are interested in events where $\Delta$BIC is positive for
all $j$. A typical threshold used to identify strong evidence in favour of one
model over another is $\Delta$BIC ¿ 10, as discussed in Kass & Raftery (1995);
Burnham & Anderson (2004); Inglis et al. (2015, 2016); Broomhall et al.
(2019b); Hayes et al. (2020). When analysing a timeseries, when the
oscillation model $S_{1}$ is strongly preferred to models $S_{0}$ and $S_{2}$
according to the $BIC$ comparison, we consider there to be strong evidence of
a quasi-periodic signal in that timeseries with characteristic frequency
$f_{p}$. The premise of this approach therefore is that a quasi-periodic
signal can be adequately represented by a local frequency enhancement
superimposed on a power-law background in the Fourier domain. For this to be
the case, the signal must be at least quasi-stationary. If a signal of
interest is sufficiently anharmonic or nonstationary (e.g. with period
changing substantially over time), it may not be captured by this approach.
Additionally, this method of maximum-likelihood based fitting assumes that the
data points are independent and identically distributed, which is not strictly
the case for Fourier power spectra (see e.g. Baluev, 2008; Vaughan, 2010;
Süveges et al., 2015). Therefore, absolute statements about the significance
or probability of a detection must be treated with caution.

We apply AFINO to each event in the GBM solar flare burst catalog. It is
unlikely for a short-period (P ¡ 5s) quasi-periodic signal to be present
throughout the entire or even a substantial portion of the flare evolution.
Instead, we expect any QPP signature to be a transient event that may last for
only a few oscillation cycles. Therefore, to maximise the chances of detecting
such a signal, we apply AFINO to the solar flare timeseries using an
overlapping interval approach, rather than analysing the entire flare time
series at once. This is similar to a windowed Fourier transform or a dynamic
power spectrum approach (e.g. Huppenkothen et al., 2019). We first identify
the beginning of the 0.064s burst mode data for each event. Then, beginning 10
s after the burst mode trigger, we analyse the flare timeseries in a sequence
of 60 s intervals, stepping forward by 50s each time such that there is slight
overlap between each analysed timeseries interval (see Figure 1). The choice
of a 60s window balances the intention to search sub-regions of the flare with
the need for a sufficient number of data points in each temporal window. While
other window size choices are possible, a 60 s window results in a Fourier
power spectrum covering 2.5 decades in frequency space. This ensures that the
model comparison method is not biased towards a null result due to a lack of
data. The 50 s step interval is used to ensure that a potential QPP signal is
not missed due to edge effects. In principle, a smaller step size could be
used to attempt to capture frequency evolution of an oscillation. However,
this would only be effective if the oscillation duration were comparable to or
longer than the chosen temporal window size, which is unlikely to be the case
here. Based on these temporal window choices, the first timeseries interval
runs from 10s after the burst mode trigger to 70s after the trigger, while the
second interval runs from 60s after the trigger to 120s after the trigger.
This process is repeated until the cadence reverts to the nominal value of
0.256s. For an individual flare the maximum number of intervals analysed is
eleven, assuming 600s of burst mode data.

Figure 1: An example of AFINO applied in an overlapping interval fashion to 15
- 25 keV GBM X-ray data from a solar flare that occurred on 2013 October 28 at
11:35 UT. Here, a QPP signal is detected in interval 1 (blue region), but not
in interval 2 (green region).

##  3 Testing on simulated X-ray lightcurve data

Before we search the Fermi/GBM dataset in search of short-period signals, it
is important to understand whether our analysis approach is effective at
detecting such signals. To test the ability of AFINO to detect a transient
oscillatory signal with $P$ ¡ 5s in a longer timeseries, we construct sets of
simulated lightcurves containing an oscillation localized in time. These
lightcurves are generated with a sampling time of $dt$ = 0.064s, and a total
length of $t$ = 300s, mimicking the Fermi/GBM lightcurve data.

The long-term flare lightcurve evolution is represented by a two-sided
Gaussian, where the width of the rising phase side of the Gaussian is narrower
than the decay phase side. This approach has been previously suggested as an
appropriate approximation of a flare shape (e.g. Jackman et al., 2019;
Broomhall et al., 2019b). To this we add Poisson noise of varying strengths,
and a localized oscillation. The oscillation is defined by a sine wave
multiplied by a Gaussian envelope, which is added to the baseline signal at a
randomized time $t_{p}$. The width of the envelope is defined as 3 times the
oscillation period, such that the same number of oscillation cycles are always
present in the data. This synthesized signal may be written as,

$F(t)=F(t)_{osc}+F(t)_{bkg}+F(t)_{noise}$ (5)

$F_{bkg}$ is defined as,

$F(t)_{bkg}=\begin{cases}A_{0}\exp\left(-{\frac{(t-\mu)^{2}}{2\sigma_{1}^{2}}}\right),&\text{if
$t$ $<$ $\mu$}\\\
A_{0}\exp\left(-{\frac{(t-\mu)^{2}}{2\sigma_{2}^{2}}}\right),&\text{if $t$ $>$
$\mu$}\\\ \end{cases}$ (6)

where $A_{0}$ is the amplitude of the long-term flare background, $\mu$ is the
time of the peak of this background component, $\sigma_{1}$ is the left-side
width of this component, and $\sigma_{2}$ is the right-side width of this
component.

The oscillation component of the signal, $F_{osc}$, may be written as,

$F(t)_{osc}=A_{osc}\sin(2\pi
ft)\times\exp\left(-{\frac{(t-\mu_{osc})^{2}}{2\sigma_{osc}^{2}}}\right)$ (7)

where $A_{osc}$ is the amplitude of the oscillation, $f$ is the frequency of
oscillation, $\mu$ is the centroid location of that oscillation in the
timeseries, and $\sigma_{osc}$ is the width of the Gaussian envelope for the
signal, defined as three times the oscillation period, or $\sigma_{osc}$ =
3$\times$ $1/f$.

The final signal component, $F(t)_{noise}$, is drawn at each $t$ from a
Poission distribution with an expectation value $\lambda$. In this work, we
define the signal-to-noise ratio as the ratio between the oscillation
amplitude $A_{osc}$ and the standard deviation of the Poisson noise, i.e. SNR
= $A_{osc}$ / $\sqrt{(}\lambda)$.

Figure 2 shows two examples of AFINO applied to the simulated lightcurves. In
both cases, the relevant temporal window is shown in the inset, where the
transient oscillation is present. In this example, $\sigma_{1}$ = 15 is the
width of the rise phase Gaussian background, and $\sigma_{2}$ = 60s is the
width of the decay phase Gaussian background. The oscillation in this case is
centered at $\mu_{osc}$ = 90s, which is at the center of one of the AFINO time
series segments. The left panel shows the case where the signal amplitude is
five times the noise amplitude. The right panel shows the case where the
signal is only two times the background noise level. In both cases, the
transient oscillation is successfully detected.

Figure 2: Examples of simulated X-ray lightcurve data, normalized by their
maximum value. The data is simulated with a cadence of 0.064s (16 Hz). A two-
sided Gaussian is used to represent the base flare lightcurve, to which noise
is added. A transient oscillation is defined by a sine wave multiplied by a
Gaussian envelope and is inserted at $\mu_{osc}$ = 90s (see inset), with a
Gaussian envelope width of $\sigma$ = 3 times the oscillation period. the left
panel shows a simulated lightcurve with SNR = 5. The right panel shows thee
same simulated flare but with SNR = 2. In both cases, AFINO correctly
identifies the oscillation with $P$ = 1 s.

For our first test, we set $\mu_{osc}$ to a fixed value of 90s, so that we
examine the impacts of changing the noise level and oscillation frequency in
isolation. We apply the windowed AFINO method exactly as described above, and
examine the results. We test noise-to-signal (1 /SNR) ratios of 0.05, 0.1,
0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.25, 1.5, 1.75, and 2.0. We test
each of these SNRs for oscillation frequencies of 0.25 Hz, 0.5 Hz, 1.0 Hz, 1.5
Hz, 2.0 Hz, 2.5 Hz, and 3.0 Hz (i.e. 0.3 – 4s period).

The top left panel of Figure 3 illustrates the results of this test using a
standard detection threshold of $\Delta$BIC ¿ 10 (Inglis et al., 2016; Hayes
et al., 2020). The grey circles indicate each combination of oscillation
frequency and background noise level tested. The blue circles indicate
instances where the oscillation was successfully detected by AFINO using the
running window technique, and the actual frequency detected.

Figure 3: Detections of oscillations in simulated X-ray lightcurves. left
panels: detected frequency (blue circles) vs original simulated oscillation
frequency (grey circles) as a function of the signal to noise ratio. In these
panels, the transient oscillation is inserted at a fixed time in the flare
timeseries. Right panels: same as left panels, except that the timing of the
transient oscillation is randomized. The top row shows the results when the
detection threshold is set to $\Delta$BIC ¿ 10. The bottom row shows the
results of the same experiment when the detection threshold is set to
$\Delta$BIC ¿ 20.

For pulsations of 0.25 Hz and 0.5 Hz (P = 4s, P = 2s), we see that the
transient oscillation is detected by AFINO even when the noise level exceeds
the signal level (SNR ¡ 1). For pulsation frequencies of 1 Hz, the transient
oscillation is also mostly detected while SNR ¿ 1, with a few detections
missed. As frequency increases, the signal becomes harder to detect due to
it’s shorter total duration in the lightcurve (since the Gaussian envelope is
a fixed function of the oscillation frequency). At 3 Hz, the signal can only
be reliably detected once the 1/SNR value reaches 0.6. There are also two
noticeably inaccurate detections during this test, where the detected period
is different from the true period.

Having established the effect of the signal-to-noise ratio on detecting a
period at a fixed $\mu_{osc}$, we next attempt to detect a localized
oscillation in cases where $\mu_{osc}$ is randomized. This represents a more
realistic scenario that will be encountered in real solar X-ray data. We test
the same set of frequencies and noise levels as before. The results are shown
in the right hand panels of Figure 3.

We see that, as before, signals at lower frequencies are easier to detect when
the signal to noise ratio is low. At $f$ = 0.25 Hz and $f$ = 0.5 Hz, the
signal is still detectable at times when SNR ¡ 1. Generally speaking, signals
with $f$ ¿ 1 Hz can only be detected when SNR ¿ 1, and only reliably detected
when SNR ¿ 2 (or 1/SNR ¡ 0.5).

The bottom row of Figure 3 repeats both experiments with a stricter detection
threshold of $\Delta$BIC ¿ 20. In this scenario, we see that inaccurate
detections of the input signal – where the reported frequency is incorrect –
are almost eliminated. Meanwhile the correct signals are still detected at
similar SNR ratios as before.

These simulations show that AFINO – when applied using overlapping analysis
windows – is able to detect short-period ¡5s pulsations in flare-like
lightcurves for a range of background noise levels and oscillation
frequencies, even in the case where the oscillation is transient and only
present for a short time. This was demonstrated both for the case of an
oscillation at a fixed point in the flare decay phase, and also for
oscillations where the temporal location was randomized in time.

###  3.1 Testing the false alarm rate and the true positive rate

In real Fermi/GBM solar X-ray data, the true number of quasi-periodic signals
is unknown. Since we are searching a large number of timeseries, and applying
a stepping interval approach to each one, the total number of signals analysed
by AFINO is large, over 15,000 in each energy band. It also follows that, even
if a QPP is present in a flare, many of the individual flare time series
segments analysed will not contain an oscillatory signal. Therefore, it is
crucial to understand to the extent possible the rate of false alarms in the
detection method, in other words the rate of type 1 errors. When analysing so
many individual time series, if the true occurrence rate is low even a modest
false alarm rate may dominate the results, a phenomenon known as the base rate
fallacy.

We can test the false alarm rate of our approach by applying AFINO to a large
number of simulated lightcurves that by design do not contain an oscillatory
signature (see Broomhall et al. (2019a) for a similar concept). We construct
1000 flare-like lightcurves of the form described in Section 3 above, with
some key differences. First, no transient oscillation is inserted into the
signal. Second, the width parameters of the two-sided Gaussian that determines
the background flare shape are allowed to vary, creating variation in the
simulated signals. Finally, the background noise level is randomized for each
simulated signal.

Complementary to this, we can also explore the impact of changing the
detection threshold on the true positive detection rate. We achieve this in
the same manner as the false alarm testing. We construct 200 flare-like
signals with randomized two-sided Gaussian background profiles, but this time
containing a randomized, local pulsation as described in Section 3 above and
added noise at a fixed signal-to-noise ratio. We repeat this for a number of
SNR values, i.e. each SNR value is associated with 200 flare-like signals. We
then apply AFINO to these synthetic signals and evaluate the number of times
the correct oscillatory signal was identified.

We apply AFINO to each simulated timeseries – both the oscillation-free
signals and those containing a QPP – using the stepping window technique as
before. Thus, each individual timeseries is broken up into five segments. In
the case of the oscillation-free signals, AFINO is applied a total of 5,000
times, allowing us to investigate low false-alarm rates. By testing different
$\Delta$BIC thresholds for identifying a significant QPP event, we can
evaluate the false alarm rate and the true positive rate (see Figure 4).

Firstly, regarding the false positive rate we see that, when a threshold of
$\Delta$BIC $>$ 0 is used, we obtain a large false alarm rate of $>$ 5.5%.
When a more typical detection threshold of $\Delta$BIC ¿ 10 is used, this
resulted in 30 false alarms, a rate of 0.6%. Although low, in the analysis of
real Fermi/GBM data where we expect the number of true positives to be low we
must be concerned about false alarms being a significant or dominant
contribution to the total number of detections. We are analysing 1460
individual solar flares, each of which is broken up into a maximum of eleven
individual time series segments. AFINO is being applied a total of 15,358
times in each energy band studied. With a false alarm rate of 0.6%, we could
therefore expect $\sim$ 90 false alarms in each energy band if the GBM data
behaves similarly to the simulated data. If the detection threshold is raised
to a conservative value of $\Delta$BIC $>$ 20, we see that the false alarm
rate is much reduced, to $\sim$ 0.04%.

Figure 4: Top: the false alarm rate of AFINO as a function of the detection
threshold parameter $\Delta$BIC. This is estimated by applying AFINO to a
suite of simulated X-ray lightcurves that do not contain an oscillation.
Bottom: The true positive detection rate of AFINO as a function of
$\Delta$BIC, for three different signal-to-noise (SNR) ratios. This is
estimated by applying AFINO to a suite of simulated X-ray lightcurves
containing oscillations, and measuring the number of correct identifications.

We can evaluate the true positive rate in a similar fashion (see Figure 4,
bottom panel). For a true positive detection, we require that the period
identified by AFINO must match the true underlying period to within a
tolerance of 25%. We see that the overall true positive detection rate is
strongly dependent on the SNR as expected. However, for a given SNR, the
reduction in true positive detections as $\Delta$BIC increases is modest
compared to the behaviour of the false alarm rate. For signals with SNR = 2
for example, at $\Delta$BIC ¿ 0 the true positive rate is 80.5%, while at
$\Delta$BIC ¿ 10 it drops to 69.5%, and at $\Delta$BIC ¿ 20 it is reduced to
65.0%.

How this affects the statistical results of a QPP search strongly depends on
the true base occurrence rate of QPP signals in solar flares, which is unknown
and has widely varying estimates (Zimovets et al., 2021). However, these
simulations show that adopting a more conservative $\Delta$BIC value can
strongly reduce the false alarm rate with only a modest cost to the true
positive detection rate. Therefore, in the analysis of the real GBM X-ray
data, we focus on detection of QPPs using strict thresholds of $\Delta$BIC ¿
15 or $\Delta$BIC ¿ 20.

##  4 Analysis of Fermi/GBM X-ray flare trigger data

###  4.1 Statistical results

As described in Section 2, we apply AFINO to 1460 flares observed by Fermi/GBM
in trigger mode. For each flare, we split the timeseries up into a number of
overlapping intervals of 60s length, beginning 10s after the trigger time, and
analyse each interval independently. This is done in three energy channels; 4
– 15 keV, 15 – 25 keV, and 25 – 50 keV. For a flare with 600s of burst mode
data, this results in 11 time series analysis segments. In each energy range,
the total number of independent time series intervals studied with AFINO is
15358.

In total, using a strict detection threshold of $\Delta$BIC ¿ 20, we find
strong evidence of rapid pulsations for 13 flares in the 4 – 15 keV energy
band, 8 flares in the 15 – 25 keV band, and 5 flares in the 25 – 50 keV band.

Considering our estimated false alarm rate of 0.04% based on simulated
lightcurves with a threshold of $\Delta$BIC = 20, the expected number of false
alarms in each energy band is $\sim$ 6\. In the 4 – 15 keV energy band, we do
identify more QPP-like signals than expected from the false alarm rate, but
the difference is only a factor of 2. For the other energy bands, the
detection rate is inline with that expected from false alarms. Thus, while we
may have identified some real QPP signals, we do not find strong evidence that
such signals are widespread with periods in the $<$ 5s regime using this
approach.

If we adopt a less stringent threshold of $\Delta$BIC = 15, we find instead 39
events in the 4 – 15 keV band, 17 events in the 15 – 25 keV band, and 11
events in the 25 – 50 keV band. Comparing again with the estimated false alarm
rate – this time of 0.2% – the expected number of false alarms is $\sim$ 30 in
each energy band. Thus, again our results are inline with false alarm rate
expectations, and we have not found strong evidence of a large underlying
short-period QPP population.

Figure 5: The expected detection rate of short-period QPP in flares as a
function of the QPP base rate and the QPP signal-to-noise ratio, based on
synthetic lightcurve analysis (see Section 3.1). Top left: The expected number
of flares, out of 1460, featuring a QPP detection, as a function of SNR and
base rate, using a detection threshold of $\Delta$BIC ¿ 20. Top right: The
expected percentage of flares featuring a QPP detection as a function of SNR
and base rate. Bottom row: Same as top row, but for a less stringent detection
threshold of $\Delta$BIC ¿ 15.

As our application of AFINO to simulated flare lightcurves showed (see Figure
4), the expected true positive rate of QPP detections is strongly dependent on
the mean signal-to-noise ratio of the QPP signals. It was also shown in
Broomhall et al. (2019b) that, while AFINO had a low false alarm rate, it was
also more conservative regarding real detections than other methods.
Therefore, it is likely that there are a number of real QPP signals present in
this flare dataset that we were unable to positively identify.

We can explore this further in order to put constraints on the possible QPP
base rates in the GBM data as well as the likely signal-to-noise ratios of
those signals in the GBM burst mode dataset. We achieve this by using the true
positive detection rates estimated from simulations of QPP-like signals in
synthetic flares described in Section 3.1 and shown in Figure 4. By assuming
different base occurrence rates of QPPs in GBM flare data, and using our
estimates of true positive detection rates as a function of signal-to-noise
ratio, we can estimate the percentage of detections we would expect from the
GBM flare dataset, as well as the absolute number of detections. This is shown
in Figure 5.

The top row of Figure 5 shows the expected number (left panel) of true
positive detections – out of 1460 solar flares – as a function of the assumed
QPP base rate and the SNR of those QPP signals. The right panel expresses the
same result as a percentage. These values are calculated assuming that an
individual QPP signal is present in only a single flare time series segment.
For the top row, this is estimated using a $\Delta$BIC ¿ 20 threshold. The
bottom row provides the same estimates for a less stringent threshold of
$\Delta$BIC ¿ 15.

Comparing the number of QPP events we have detected in the GBM data with
Figure 5 clearly indicates that observationally we have some combination of a
low SNR and low base rate of QPPs. Either the true number of short-period QPP
signals present in the GBM dataset is low, or the typical SNR of signals is
sufficiently low (i.e. SNR ¡ 1) that detection is very rare. These results
show that short-period QPPs with strong signal-to-noise ratios would have been
detected in large numbers, even for modest base rates of 10–20%. For a base
rate of 20% percent and an SNR of 2.0 for example, 189 true positive
detections would be expected using a $\Delta$BIC ¿ 20 threshold, and 194 using
a $\Delta$BIC ¿ 15 threshold.

In summary, our analysis of 1460 solar flares observed with GBM burst mode
data does not reveal strong statistical evidence for a large number of short-
period QPPs. We surmise that either the true occurrence rate of such signals
is low, or that for the burst mode data they are usually present with signal-
to-noise ratios of less than 1 and are therefore challenging to detect and
distinguish from false alarms.

###  4.2 Case studies of individual events

Although our statistical study of Fermi/GBM flare data did not find evidence
for a large underlying population of QPPs with periods in the $<$ 5s regime,
we have nevertheless identified several interesting QPP events that we are
confident are real solar signals, and not false alarms. Here, we explore a
selection of these events in more detail, although we do not assign a strict
statistical significance to these events. Each flare presented below showed
strong evidence of QPPs in at least one GBM energy channel (meeting at least a
threshold of $\Delta$BIC ¿ 15), and additionally shows visually compelling
evidence of QPPs. We identify each event using its GBM catalog burst number,
which is based on the date and time (expressed as fraction of day) of the
observation. For example, BN110616429 corresponds to 2011-06-16 at 10:17:45
UT.

####  4.2.1 BN110616429

This burst was associated with a C7.1-class solar flare that occurred on
2011-06-16, beginning at 10:13 UT according to the GOES flare catalog and
continuing until 10:29 UT. The Fermi/GBM trigger was at 10:17:46 UT. This was
one of a series of flares occurring from AR 11237, as noted by Panesar et al.
(2015).

Figure 6 shows the Fermi/GBM lightcurves of this event in the 4-15 keV, 15-25
keV, and 25-50 keV energy ranges. In the top panel, the red shaded area
highlights the portion of the flare timeseries where the QPP was detected. The
center panel shows a zoomed in timeseries of the red shaded area from the top
panel. The bottom panel of Figure 6 shows the result of the AFINO power
spectrum model fitting procedure. For this flare, a period of $P$ = 2.1s was
detected in the 25 – 50 keV energy band.

Figure 6: Summary of the BN110616429 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). Here, a period of $P$ = 2.1s was detected in the 25 – 50
keV energy band. In this plot and all subsequent summary plots, the absolute
y-values of the GBM lightcurves have been adjusted by adding or subtracting
constant factors as needed for visual clarity.

####  4.2.2 BN120510849

This burst trigger was associated with a solar flare from 2012-05-10. The
flare was of GOES class M1.7 and began at 20:20 UT, continuing until 20:30 UT.
The flare was sourced from AR 11476.

Figure 7 shows the Fermi/GBM lightcurves of this event and the best-fit model
to the Fourier power spectrum in the bottom panel. The periodic structure was
detected in the first timeseries segment of this flare, as shown by the red
highlighted area. A periodic structure with $P$ = 2.6s was detected by AFINO
in the 25-50 keV band. Visually, a corresponding set of pulsations can be
clearly observed between $t$ = 40s and $t$ = 50s.

Figure 7: Summary of the BN120510849 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). Here, a period of $P$ = 2.6s was detected in the 25 – 50
keV energy band.

####  4.2.3 BN131028192

The GBM trigger BN131028192 is associated with a strong solar flare of GOES
class M5.1, originating from AR 11875 on 2013-10-28. The Fermi trigger time
for this event was 04:35:51 UT, while the GOES catalog lists the flare
beginning at 04:32 UT and ending at 04:46 UT. This active region was also
associated with a number of other powerful solar flares, including several
others on this day, some of which were associated with longer period QPPs, on
the order of 10’s of seconds (e.g. Hayes et al., 2016).

Figure 8 shows the Fermi/GBM lightcurves of the event and the best-fit AFINO
result. In this case, AFINO detects a significant oscillation with $P$ = 2s in
the 15 – 25 keV energy band, during the first flare time series segment.
Inspection of this segment via the center panel of Figure 8 shows the presence
of such an oscillation in the 15 – 25 keV lightcurve (orange curve) between
approximately $t$ = 45s and $t$ = 65s after the burst trigger. Similar
features are seen in the 25 – 50 keV energy band, but are not as strongly
identified by AFINO.

Figure 8: Summary of the BN131028192 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). A period of $P$ = 3.7s was detected in the 15 – 25 keV
energy band.

This event was also observed by the Nobeyama Radioheliograph (NoRH) and
Nobeyama Radiopolarimeters (NoRP). Both of these instruments observe in the
GHz range and can provide observations with 0.1s time resolution during
certain events. Figure 9 shows the 17 GHz and 9.4 GHz time series data from
NoRP, showing clear bursty structure on a range of timescales. Applying AFINO
to this 0.1s cadence radio data in 60s overlapping intervals reveals the
presence of a period of P $\approx$ 4.5 - 5.0s betwen 04:38 and 04:39 UT. This
does not temporally overlap with the period found in the GBM data, which was
in the interval 04:36 UT to 04:37 UT. This may be due to the fact that the
NoRP 0.1s mode data did not begin until around 04:36:30 UT. Nevertheless, the
combined X-ray and radio data reveals an interesting, multi-periodic solar
flare.

Figure 9: Lightcurves of the solar flare of 2013 October 28 as observed at
radio frequencies of 9.4 GHz (red) and 17 GHz (black) by the Nobeyama
Radiopolarimeters. A significant period of 4.5 - 5.0s is detected by our
analysis method during the interval 04:38 to 04:39 UT.

####  4.2.4 BN131028483

This solar flare trigger is associated with an solar flare from 2013-10-28
with a GOES class of M1.4. In the GOES catalog, the flare began at 11:32 UT
and ended at 12:39 UT. The Fermi trigger time for this event was 11:35:26 UT.

Figure 10 shows the Fermi/GBM lightcurves of this event and the best-fit AFINO
model as before. A periodic structure is detected in the first timeseries
segment of this flare, with $P$ = 3.7s. These pulsations are most strongly
evident in the 15 - 25 keV energy band. In the center panel of Figure 10, a
clear sequence of pulsations can be seen from $t$ = 10s to $t$ = 50s,
consistent with the AFINO detection. These pulsations can also be seen
visually in the 25 – 50 keV band.

Figure 10: Summary of the BN131028483 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). A period of $P$ = 3.7s was detected in the 15 – 25 keV
energy band.

####  4.2.5 BN141020790

This burst trigger was associated with a solar flare from 2014-10-20. The
flare was of GOES class M1.4 and began at 18:55 UT, continuing until 19:04 UT.
The source of the solar flare was AR 12192. The Fermi trigger time was
18:57:08 UT.

Figure 11 shows the Fermi/GBM lightcurves of this event and the best fit AFINO
model in the bottom panel. From the lightcurves alone, very clear pulsating
structures can be seen in the 25 - 50 keV energy band, and less prominently in
the 15 - 25 keV band. This is confirmed by the best-fit AFINO model to the
Fourier power spectrum, which identifies a signal with a period of $P$ = 1.8s.

Figure 11: Summary of the BN141020790 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). Here, a period of $P$ = 1.8s was detected in the 25 – 50
keV energy band.

####  4.2.6 BN150929278

This burst trigger was associated with a solar flare from 2015-09-29. The
flare was of GOES class M1.4 and began at 06:39 UT, continuing until 06:46 UT.
The flare was sourced from AR 12422.

Figure 12 shows the Fermi/GBM lightcurves of this flare and the best-fit AFINO
model to the Fourier power spectrum. A QPP is detected in the second
timeseries segment of this flare, shown by the red highlighted area in the
lightcurve. The center panel of Figure 12 shows a pronounced sequence of
pulsations in the 25 - 50 keV band on top of a longer $\sim$ 20s pulse of hard
X-rays. The best-fit model in the AFINO analysis indicates a period of $P$ =
1.1s.

Figure 12: Summary of the BN150929278 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). Here, a period of $P$ = 1.1s was detected in the 25 – 50
keV energy band.

####  4.2.7 BN220520322

This burst trigger was associated with a solar flare from 2022-05-20. The
flare was of GOES class M3.0 and began at 07:35 UT, continuing until 07:49 UT.
AFINO indicates the presence of a periodic signal with $P$ = 1.85s in the 25 –
50 keV band. This is consistent with the lightcurve shown in the center panel
of Figure 13, which shows a sequence of periodic bursts between approximately
$t$ = 45s and $t$ = 60s.

Figure 13: Summary of the BN220520322 solar flare burst. Top: Normalized count
rates in the 4-15 keV (green), 15-25 keV (orange), and 25-50 keV (blue) energy
ranges at 0.064s cadence. The time series segment where a significant QPP was
detected is shaded in red. Center: Zoomed-in lightcurve showing only the time
series segment where the QPP was detected, i.e. the red shaded area from the
previous panel. Bottom: The Fourier power spectral density (PSD) of the flare
time series segment (blue), with the best fit of the QPP-like AFINO model
overplotted (orange). Here, a period of $P$ = 1.8s was detected in the 25 – 50
keV energy band.

Figure 13 also indicates the presence of a prominent, longer period pulsation
with a triangular profile occurring throughout the event, most strongly
observed in the 25 – 50 keV energy band. This was not found by AFINO due to
the restrictions on the maximum allowed period in the best-fit model, which
was 10s. If we remove this period requirement and apply AFINO to the entire 25
– 50 keV time series between $t$ = 40s and $t$ = 140s, we find strong evidence
for a signal with $P$ $\sim$ 9s. Therefore, BN220520322 is a multi-periodic
flare, although we note that the two periods do not appear to occur
simultaneously; the short-period 1.8s pulsation occurs during the initial hard
X-ray peak centered at $t$ = 50s, while the longer 9s pulsation occurs later,
from $t$ = 60s onwards.

This longer period structure is also observed in the soft X-ray flux observed
by GOES at 1s cadence, in both the long (1 – 8 Å) and short (0.5 – 4.0 Å)
wavelength channels. This is illustrated in Figure 14, which compares the
derivative of the GOES soft X-ray flux with the raw GBM 25 – 50 keV flux. Hard
X-ray solar flare emission is often strongly correlated with the the
derivative of thermal soft X-ray flux, a phenomenon known as the Neupert
effect (Neupert, 1968). In this flare, we see that multiple peaks in the GOES
derivative are temporally co-aligned with the peaks in the raw 25 – 50 keV
count rates, particularly from $t$ = 70s after the burst trigger onwards.

Figure 14: Comparison of the GBM 25 – 50 keV flux for the BN220520322 event
with the derivative of the solar soft X-ray flux as measured by the GOES/XRS
instrument. The derivative of the GOES long channel (1 – 8 Å) is shown in red,
while the derivative of the short channel (0.5 – 4.0 Å) is shown in magenta.
Multiple co-temporal pulses are visible in these lightcurves from
approximately $t$ = 70s onwards.

##  5 Summary

We have analysed 1460 solar flares observed by the Fermi/GBM instrument at
X-ray energies, in search of short period (¡5s) quasi-periodic pulsations. We
analyse each flare using the AFINO analysis method, applying it to each flare
in a sequence of overlapping temporal windows. To better understand our
observational results, we perform tests of our analysis approach on sets of
synthetic flare lightcurves, in order to understand our ability to detect such
signals and the potential false alarm rates. The main conclusions of this work
can be summarized as follows:

  * •

Based on synthetic flare lightcurves, our analysis method is able to detect
short-period, short-lived pulsations in flare lightcurve data (see Figure 3).
The detection effectiveness depends on both the oscillation frequency and the
signal-to-noise ratio, with lower frequencies easier to detect at low SNR.
Oscillation frequencies of 0.25 Hz and 0.5 Hz were detectable even when SNR ¡
1, while oscillation frequencies up to 3 Hz (P = 0.33s) were also detectable
by this method provided the SNR was moderate, above $\sim$ 2.

  * •

We also estimate the false positive rate of this method as a function of
detection threshold $\Delta$BIC. This is achieved by applying the method to
1000 flare-like signals that do not contain any oscillations. In addition, we
estimate the true positive rate as a function of both $\Delta$BIC and SNR, by
applying AFINO to sets of synthetic lightcurves with a fixed SNR. We show that
the false alarm rate is more strongly dependent on $\Delta$BIC than the true
positive detection rate (see Figure 4). This leads us to choose stricter
detection thresholds of $\Delta$BIC = 15 and $\Delta$BIC = 20 when searching
for pulsations in the real Fermi/GBM data.

  * •

We do not find strong statistical evidence for a large number of short-period
QPP events in the Fermi/GBM burst mode data. Using a strict detection
threshold of $\Delta$BIC ¿ 20, we find 13 events in the 4 – 15 keV energy
range, 8 events in the 15 – 25 keV range, and 5 events in the 25 – 50 keV
range. This compares to our estimated false alarm rate of 0.04% – or 6 events
– per energy band. If we relax the detection threshold to $\Delta$BIC ¿ 15, we
find 39 events in the 4 – 15 keV energy range, 17 events in the 15 – 25 keV
range, and 11 events in the 25 – 50 keV range. Again, this is inline with the
false alarm expectation of 30 events, given a false alarm rate of 0.2%.

  * •

Using our analysis of synthetic lightcurves, we can estimate the expected
number of detections for a given signal to noise level and base QPP occurrence
rate (see Figure 5). Given the low number of QPP detections in the real GBM
X-ray data, we can surmise that either the underlying base occurrence rate of
short-period QPPs is very low, or the typical signal to noise ratios of such
signals is SNR ¡ 1. This highlights that good signal-to-noise is required in
addition to high time resolution when searching for pulsations in solar
flares.

  * •

Finally, we present a selection of 7 case study QPP events. Each of these
flares exhibits a QPP that was detected by AFINO in at least one energy
channel with a minimum threshold of $\Delta$BIC ¿ 15. Additionally, each event
shows visually compelling evidence of pulsations, indicating a true positive
detection and not a false alarm. The periods of these events range from 1 – 4
s. At least two of these flares – the M5.1 flare of 2013 October 28 at 04:32
UT, and the M3.0 flare of 2022 May 20 at 07:35 UT – appear to be multi-
periodic events. Observations of multi-periodicity have been relatively rare
to date and merit further investigation.

This work presents some probable examples of quasi-periodic pulsations in
solar flares in the P ¡ 5s range, including two events that may be multi-
periodic. However, the overall prevalence of such signatures in flare emission
remains unclear. This analysis did not uncover strong evidence for widespread
short-period QPPs, due to the difficulty in distinguishing between genuine QPP
signals and false alarms when analysing large samples of data. This is partly
due to the signal-to-noise ratio available in high cadence solar X-ray data.
Future high temporal cadence data with an improved signal to noise ratio are
needed in order to resolve this question. Current instruments such as STIX on
Solar Orbiter (Krucker et al., 2020) can provide additional high-cadence
observations of QPPs (e.g. Collier et al., 2023), with the advantage of close
approaches to the Sun improving the signal to noise ratio. Future focusing
optics X-ray telescopes (e.g. Shih et al., 2023), with intrinsically low
background, are a promising avenue for exploring this phenomenon in more
detail (Inglis et al., 2023).

Understanding short-period QPPs is key because they pose questions regarding
the fundamental physics of solar flare energy release. Of the many proposed
explanations for QPPs, only some can produce periodic behaviour on this
timescale in the X-ray regime. Compared to longer period QPPs, short-period
events could represent a different population or regime of this phenomenon,
strongly implying that multiple different mechanisms produce QPPs depending on
the timescale. Confirmed short-period QPPs can therefore provide additional
insight and constraints into the timescales of particle acceleration and
energy release in flares.

ARI acknowledges support from the Fermi Guest Investigator program, NASA grant
number 80NSSC20K1601. LAH is supported by an ESA Research Fellowship. The
authors are grateful to the anonymous referee, who’s comments improved the
final manuscript.

## References

  * Asai et al. (2001) Asai, A., Shimojo, M., Isobe, H., et al. 2001, ApJ, 562, L103
  * Balona et al. (2015) Balona, L. A., Broomhall, A.-M., Kosovichev, A., et al. 2015, MNRAS, 450, 956
  * Baluev (2008) Baluev, R. V. 2008, MNRAS, 385, 1279
  * Broomhall et al. (2019a) Broomhall, A. M., Thomas, A. E. L., Pugh, C. E., Pye, J. P., & Rosen, S. R. 2019a, A&A, 629, A147
  * Broomhall et al. (2019b) Broomhall, A.-M., Davenport, J. R. A., Hayes, L. A., et al. 2019b, ApJS, 244, 44
  * Brosius & Inglis (2018) Brosius, J. W., & Inglis, A. R. 2018, ApJ, 867, 85
  * Burnham & Anderson (2004) Burnham, K. P., & Anderson, D. R. 2004, Sociological Methods & Research, 33, 261
  * Carley et al. (2019) Carley, E. P., Hayes, L. A., Murray, S. A., et al. 2019, Nature Communications, 10, 2276
  * Cenko et al. (2010) Cenko, S. B., Butler, N. R., Ofek, E. O., et al. 2010, AJ, 140, 224
  * Chiu (1970) Chiu, Y. T. 1970, Sol. Phys., 13, 420
  * Clarke et al. (2021) Clarke, B. P., Hayes, L. A., Gallagher, P. T., Maloney, S. A., & Carley, E. P. 2021, in American Astronomical Society Meeting Abstracts, Vol. 53, American Astronomical Society Meeting Abstracts, 303.03
  * Collier et al. (2023) Collier, H., Hayes, L. A., Battaglia, A. F., Harra, L. K., & Krucker, S. 2023, A&A, 671, A79
  * Collier et al. (2024) Collier, H., Hayes, L. A., Yu, S., et al. 2024, arXiv e-prints, arXiv:2402.10546
  * Dennis (1985) Dennis, B. R. 1985, Sol. Phys., 100, 465
  * Dolla et al. (2012) Dolla, L., Marqué, C., Seaton, D. B., et al. 2012, ApJ, 749, L16
  * Dominique et al. (2018) Dominique, M., Zhukov, A. N., Dolla, L., Inglis, A., & Lapenta, G. 2018, Sol. Phys., 293, 61
  * Doyle et al. (2022) Doyle, J. G., Irawati, P., Kolotkov, D. Y., et al. 2022, MNRAS, 514, 5178
  * Foullon et al. (2005) Foullon, C., Verwichte, E., Nakariakov, V. M., & Fletcher, L. 2005, A&A, 440, L59
  * Grechnev et al. (2003) Grechnev, V. V., White, S. M., & Kundu, M. R. 2003, ApJ, 588, 1163
  * Gruber et al. (2011) Gruber, D., Lachowicz, P., Bissaldi, E., et al. 2011, A&A, 533, A61
  * Hayes et al. (2016) Hayes, L. A., Gallagher, P. T., Dennis, B. R., et al. 2016, ApJ, 827, L30
  * Hayes et al. (2020) Hayes, L. A., Inglis, A. R., Christe, S., Dennis, B., & Gallagher, P. T. 2020, ApJ, 895, 50
  * Huppenkothen et al. (2013) Huppenkothen, D., Watts, A. L., Uttley, P., et al. 2013, ApJ, 768, 87
  * Huppenkothen et al. (2019) Huppenkothen, D., Bachetti, M., Stevens, A. L., et al. 2019, ApJ, 881, 39
  * Inglis et al. (2023) Inglis, A., Hayes, L., Guidoni, S., et al. 2023, in Bulletin of the American Astronomical Society, Vol. 55, 181
  * Inglis et al. (2016) Inglis, A. R., Ireland, J., Dennis, B. R., Hayes, L., & Gallagher, P. 2016, ApJ, 833, 284
  * Inglis et al. (2015) Inglis, A. R., Ireland, J., & Dominique, M. 2015, ApJ, 798, 108
  * Inglis et al. (2008) Inglis, A. R., Nakariakov, V. M., & Melnikov, V. F. 2008, A&A, 487, 1147
  * Jackman et al. (2019) Jackman, J. A. G., Wheatley, P. J., Pugh, C. E., et al. 2019, MNRAS, 482, 5553
  * Kass & Raftery (1995) Kass, R. E., & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773
  * Kiplinger et al. (1984) Kiplinger, A. L., Dennis, B. R., Frost, K. J., & Orwig, L. E. 1984, ApJ, 287, L105
  * Kiplinger et al. (1983) Kiplinger, A. L., Dennis, B. R., Frost, K. J., Orwig, L. E., & Emslie, A. G. 1983, ApJ, 265, L99
  * Knuth & Glesener (2020) Knuth, T., & Glesener, L. 2020, ApJ, 903, 63
  * Kolotkov et al. (2021) Kolotkov, D. Y., Nakariakov, V. M., Holt, R., & Kuznetsov, A. A. 2021, ApJ, 923, L33
  * Krucker et al. (2020) Krucker, S., Hurford, G. J., Grimm, O., et al. 2020, A&A, 642, A15
  * Kupriyanova et al. (2013) Kupriyanova, E. G., Melnikov, V. F., & Shibasaki, K. 2013, Sol. Phys., 284, 559
  * Lipa (1978) Lipa, B. 1978, Sol. Phys., 57, 191
  * Mathioudakis et al. (2006) Mathioudakis, M., Bloomfield, D. S., Jess, D. B., Dhillon, V. S., & Marsh, T. R. 2006, A&A, 456, 323
  * McLaughlin et al. (2018) McLaughlin, J. A., Nakariakov, V. M., Dominique, M., Jelínek, P., & Takasao, S. 2018, Space Sci. Rev., 214, 45
  * Meegan et al. (2009) Meegan, C., Lichti, G., Bhat, P. N., et al. 2009, ApJ, 702, 791
  * Melnikov et al. (2005) Melnikov, V. F., Reznikova, V. E., Shibasaki, K., & Nakariakov, V. M. 2005, A&A, 439, 727
  * Murphy et al. (2020) Murphy, K. R., Inglis, A. R., Sibeck, D. G., Watt, C. E. J., & Rae, I. J. 2020, Journal of Geophysical Research (Space Physics), 125, e27887
  * Murphy et al. (2018) Murphy, K. R., Inglis, A. R., Sibeck, D. G., et al. 2018, Journal of Geophysical Research (Space Physics), 123, 6457
  * Nakariakov et al. (2010) Nakariakov, V. M., Foullon, C., Myagkova, I. N., & Inglis, A. R. 2010, ApJ, 708, L47
  * Nakariakov & Melnikov (2009) Nakariakov, V. M., & Melnikov, V. F. 2009, Space Sci. Rev., 149, 119
  * Neupert (1968) Neupert, W. M. 1968, ApJ, 153, L59
  * Panesar et al. (2015) Panesar, N. K., Sterling, A. C., Innes, D. E., & Moore, R. L. 2015, The Astrophysical Journal, 811, 5
  * Parks & Winckler (1969) Parks, G. K., & Winckler, J. R. 1969, ApJ, 155, L117
  * Pugh et al. (2016) Pugh, C. E., Armstrong, D. J., Nakariakov, V. M., & Broomhall, A.-M. 2016, MNRAS
  * Pugh et al. (2015) Pugh, C. E., Nakariakov, V. M., & Broomhall, A.-M. 2015, ApJ, 813, L5
  * Shih et al. (2023) Shih, A. Y., Glesener, L., Krucker, S., et al. 2023, in Bulletin of the American Astronomical Society, Vol. 55, 364
  * Simões et al. (2015) Simões, P. J. A., Hudson, H. S., & Fletcher, L. 2015, Sol. Phys., 290, 3625
  * Süveges et al. (2015) Süveges, M., Guy, L. P., Eyer, L., et al. 2015, MNRAS, 450, 2052
  * Van Doorsselaere et al. (2016) Van Doorsselaere, T., Kupriyanova, E. G., & Yuan, D. 2016, Sol. Phys., 291, 3143
  * Vaughan (2005) Vaughan, S. 2005, A&A, 431, 391
  * Vaughan (2010) —. 2010, MNRAS, 402, 307
  * Zimovets et al. (2021) Zimovets, I. V., McLaughlin, J. A., Srivastava, A. K., et al. 2021, Space Sci. Rev., 217, 66