# A Stochastic Approach To Reconstruct Gamma Ray Burst Lightcurves

Maria G. Dainotti  National Astronomical Observatory of Japan,2 Chome-21-1
Osawa, Mitaka, Tokyo 181-8588, Japan.  Ritwik Sharma  Department of Physics,
Deshbandhu College, University of Delhi, New Delhi, India.  Aditya Narendra
Astronomical Observatory of Jagiellonian University, Krakow.  Delina Levine
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Enrico Rinaldi  Interdisciplinary Theoretical & Mathematical Science Program,
RIKEN (iTHEMS),
2-1 Hirosawa, Wako, Saitama, Japan 351-0198.  Agnieszka Pollo  Astronomical
Observatory of Jagiellonian University, Krakow.  National Centre for Nuclear
Research, Warsaw.  Gopal Bhatta  Institute of Nuclear Physics Polish Academy
of Sciences, PL-31342 Krakow, Poland.

###### Abstract

Gamma-Ray Bursts (GRBs), being observed at high redshift ($z=9.4$), vital to
cosmological studies and investigating Population III stars. To tackle these
studies, we need correlations among relevant GRB variables with the
requirement of small uncertainties on their variables. Thus, we must have good
coverage of GRB light curves (LCs). However, gaps in the LC hinder the precise
determination of GRB properties and are often unavoidable. Therefore,
extensive categorization of GRB LCs remains a hurdle. We address LC gaps using
a “stochastic reconstruction,” wherein we fit two pre-existing models
(Willingale 2007; W07 and Broken Power Law; BPL) to the observed LC, then use
the distribution of flux residuals from the original data to generate data to
fill in the temporal gaps. We also demonstrate a model-independent LC
reconstruction via Gaussian Processes. At 10% noise, the uncertainty of the
end time of the plateau, its correspondent flux, and the temporal decay index
after the plateau decreases, on average, by 33.3% 35.03%, and 43.32%,
respectively for the W07, and by 33.3%, 30.78%, 43.9% for the BPL. The slope
of the plateau decreases by 14.76% in the BPL. After using the Gaussian
Process technique, we see similar trends of a decrease in uncertainty for all
model parameters for both the W07 and BPL models. These improvements are
essential for the application of GRBs as standard candles in cosmology, for
the investigation of theoretical models and for inferring the redshift of GRBs
with future machine learning analysis.

$\gamma$-ray bursts— statistical methods— relativistic processes—cosmology:
cosmological parameters— light curve reconstruction

##  1 Introduction

Gamma-Ray Bursts (GRBs) are transient astrophysical events that can be
observed up to redshift $z=8.2$ (Tanvir et al., 2009; Salvaterra et al., 2009)
and $z=9.4$ (Cucchiara et al., 2011). Thus, they are excellent candidates for
cosmological tools that can be utilized to probe the early universe. Moreover,
a comprehensive characterization of GRBs can provide insight into Population
III stars, the most ancient stars observed at the epoch of re-ionization.
Having a reliable taxonomy of GRB classes and good data coverage will favor
future population studies and thus will enhance the determination of the
cosmological evolution of GRB properties and the investigation of their
emission mechanism and/or progenitors. All these are foundational topics for
astrophysics.

The lack of a robust GRB classification scheme, the incompleteness of redshift
information in the existing sample of observed GRBs, gaps in the light curves
(LCs), and the need for a unique database repository are many of the difficult
challenges that astrophysicists need to overcome. In this paper, we attempt to
solve the issue related to the temporal gaps in the LCs.

Observationally, GRB emission can be divided into two phases: the prompt and
the afterglow. The prompt is the primary phase observed from high-energy
$\gamma$-rays to X-rays and sometimes in optical bands (Vestrand et al., 2005;
Blake et al., 2005; Beskin et al., 2010; Gorbovskoy et al., 2012; Vestrand et
al., 2014). The prompt is followed by an afterglow phase (Costa et al., 1997;
van Paradijs et al., 1997; Piro et al., 1998; Gehrels et al., 2009), observed
in multiple wavelengths such as X-rays, optical, and sometimes radio bands.

GRBs have, historically, been divided into two classes depending on their
prompt duration, $T_{90}$, the time interval during which a burst releases
$90\%$ of its total background-subtracted counts, beginning after $5\%$ of the
total counts have been measured (Mazets et al., 1981; Kouveliotou et al.,
1993). Short GRBs (SGRBs), with $T_{90}\leq 2$s, are produced by the merging
of compact objects (Duncan & Thompson, 1992; Narayan et al., 1992; Usov, 1992;
Thompson, 1994; Levan et al., 2008; Metzger et al., 2011; Bucciantini et al.,
2012; Perna et al., 2016), and long GRBs (LGRBs), with $T_{90}\geq 2$s, result
from collapsing massive stars (Woosley, 1993; Paczyński, 1998; MacFadyen &
Woosley, 1999; Bloom et al., 2002; Hjorth et al., 2003; Woosley & Bloom, 2006;
Woosley & Heger, 2006; Bucciantini et al., 2008; Kumar et al., 2008; Hjorth &
Bloom, 2012; Cano et al., 2017; Lyman et al., 2017; Perna et al., 2018; Aloy &
Obergaulinger, 2021; Ahumada et al., 2021). However, this historical
classification in LGRBs and SGRBs simplifies a more complex and realistic
picture where many more classes have been discovered in the literature since
then (see Section 1.3 for more details). In the analysis proposed here, the
reconstruction of GRB LCs will improve GRB characterization to cast further
light between different classes.

The Neil Gehrels Swift Observatory (Swift, Gehrels et al. (2004)) is crucial
for observing GRB temporal properties. The Swift Burst Alert Telescope (BAT,
15-150 keV, (Barthelmy et al., 2005)) enables rapid detection of the prompt
emission and a fast follow-up of the afterglow by the X-ray (XRT, 0.3-10 keV,
(Burrows et al., 2005)) and Ultra-Violet telescopes (UVOT 170 - 600 nm,
(Roming et al., 2005)). Furthermore, due to rapid afterglow follow-up in
several wavelengths, Swift data has shown new features in the GRB LCs
(Tagliaferri et al., 2005; Nousek et al., 2006; Troja et al., 2007).

Many X-ray LCs have a steep flux decay after the prompt emission ends,
sometimes followed by flares and a plateau (Zhang et al., 2006; O’Brien et
al., 2006; Nousek et al., 2006; Sakamoto et al., 2007; Liang et al., 2007;
Willingale et al., 2007a; Dainotti et al., 2008, 2010a, 2016, 2017).
Specifically, XRT detections are available for 81% of Swift GRBs: 42% with
X-ray plateaus (Evans et al., 2009; Li et al., 2018b). The X-ray plateau
generally lasts $10^{2}-10^{5}$ s and has a subsequent phase characterized by
a power-law (PL) decay. In addition, about 30% of optical LCs observed by UVOT
and ground-based facilities also exhibit a shallow decay phase (Vestrand et
al., 2005; Kann et al., 2006; Zeh et al., 2006; Panaitescu & Vestrand, 2008;
Kann et al., 2010; Panaitescu & Vestrand, 2011; Kann et al., 2011; Li et al.,
2012; Oates et al., 2012; Zaninoni et al., 2013; Margutti et al., 2013;
Melandri et al., 2014; Li et al., 2015, 2018a; Si et al., 2018; Dainotti et
al., 2020b). The plateau can be fitted with a broken power-law (BPL) (Zhang et
al., 2006; Racusin et al., 2009), a smoothly broken power-law (SBPL), or the
Willingale et al. (2007) phenomenological model (W07, Willingale et al.
(2007a)). The W07 model determines the time at the end of the plateau
($T_{a}$), its corresponding flux ($F_{a}$), and the temporal index after the
plateau ($\alpha_{a}$). The BPL determines $T_{a}$, $F_{a}$, and the slope of
the LC during the plateau ($\alpha_{1}$) and after the plateau ($\alpha_{2}$).
These models are briefly described in Section 2.1.

The plateau is usually explained via the magnetar model, based on the dipole
radiation emitted by the rotational energy of a newly-born neutron star (NS)
In this model, the plateau ends when the NS reaches its critical spin-down
timescale; the uncertainties on $T_{a}$ can be ascribed to the uncertainties
on the magnetar spin period and magnetic field.

However, the characterization of the plateau emission can be hindered by
temporal gaps which can occur in the beginning, during, and at the end of the
plateau. These may arise from the orbital period of satellites, lack of fast
follow-up studies, atmospheric turbulence, and instrumental errors or
failures. Thus, extensive characterization of GRB LCs remains a bottleneck.

Therefore, we propose a method to reconstruct LCs starting from the plateau
emission, which due to its theoretical interpretation within the magnetar
model, is grounded in fundamental physics. Morphologically, the plateau has
more standard features among diverse GRBs (e.g., its length and flatness) than
the prompt properties. The plateau features have attracted attention due to
their use in building relevant correlations with the plateau parameters and
their application as cosmological tools. Specifically, Dainotti et al. (2008,
2010b, 2011, 2013, 2015, 2017) and Li et al. (2018b) explored the luminosity
at the end of the plateau, $L_{X,a}$ vs. its rest-frame time $T^{*}_{X,a}$
(known as the Dainotti relation or 2D L-T relation) 111the rest-frame time is
denoted with an asterisk. The 2D relation has also been discovered in optical
plateau emissions (Dainotti et al., 2020b, 2022f). Within the theoretical
magnetar scenario, Rowlinson et al. (2014) showed that the X-ray Dainotti
relation is reproduced with a slope for $L_{a,X}$-$T^{*}_{a,X}$ of $-1$. This
correlation has already been applied in the cosmological framework to
construct a GRB Hubble diagram out to $z>8$ (Cardone et al., 2009, 2010;
Postnikov et al., 2014; Dainotti et al., 2013).

An extension of the 2D L-T relation, obtained by adding the peak prompt
luminosity, $L_{X,peak}$, has led to the Dainotti 3D relation (Dainotti et
al., 2016, 2017, 2020a, 2022f). This 3D relation has also been successfully
applied to constrain cosmological parameters (Dainotti et al., 2022b, d, e;
Cao et al., 2022a, b; Dainotti et al., 2023b). Importantly, Dainotti et al.
(2022d) have shown that if we reduce the uncertainties on the parameters of
the plateau emission by 47.5%, we will reach the same precision on the
cosmological value of $\Omega_{M}$ quoted in Conley et al. (2011) even now
compared to the same accuracy which we would reach in 2037 based on current
observation rates and parameter uncertainties. This shows how appealing a more
robust LC reconstruction (LCR) can be, given that it would save us 15 years of
observations to reduce the uncertainties on cosmological parameters to reach
the same precision achieved by SNe Ia (for details see Dainotti et al., 2020).

In addition, the plateau emission has been the object of investigation
concerning closure relations (relations between the temporal and spectral
index during or after the plateau region), which allows us to test the
viability of the standard fireball model for the observations carrying plateau
emission in high-energy $\gamma$-rays, X-rays, or optical wavelengths (Tak et
al., 2019; Srinivasaragavan et al., 2020a; Dainotti et al., 2021c, a, 2022c;
Willingale et al., 2007a; Evans et al., 2009; Racusin et al., 2009; Kumar &
Barniol Duran, 2010; Oates et al., 2012; Gao et al., 2013; Wang et al., 2015;
Misra et al., 2021; Fraija et al., 2020; Ryan et al., 2020; Levine et al.,
2022; Dainotti et al., 2023a). Although closure relations are a quick way to
test the standard fireball model, a more precise characterization of the decay
index after the plateau emission would allow for better precision in
determining these relations.

Lastly, another relevant application is to use the reconstructed LCs to train
machine-learning (ML) models for redshift estimation and classification of all
redshifts, especially of high-${z}$ GRBs.

Because of the many applications, both from a theoretical and a cosmological
perspective, the plateau parameters must be well-constrained. In this regard
reconstructing LCs with plateaus in the plateau region is extremely important.
Indeed, LCs with gaps are, in many circumstances, not usable for cosmological
applications. In addition, LCs with temporal gaps cannot be reliably used to
test theoretical models that attempt to explain the GRB emission (e.g., the
standard fireball model, Cavallo & Rees (1978); Meszaros & Rees (1993);
Meszaros (1997); Piran (1999); Panaitescu & Kumar (2000); Mészáros (2001);
Panaitescu & Kumar (2001); Zhang & Meszaros (2002); Piran (2004); Zhang &
Mészáros (2004); Zhang et al. (2006); Scargle (2020)). Thus, LCs with better
coverage will facilitate modeling and determining correlations among critical
physical properties.

Our reconstruction method gives us a glimpse into what the data likely would
have been in these gaps and increases the overall density distribution of the
LCs over time. As a result, we can improve how well LCs can be used as
standard candles and for theoretical modeling.

In this paper, we take the first step toward full LCR: We develop a
reconstruction technique and test its performance on our dataset. This paper
is divided into the following sections: in §1.1 we provide the definition of
the LCR, in §1.2 scientific motivation for the LCR, and in §1.3 we outline the
GRB classification. Then, in §2, we outline the data sample, Willingale and
BPL models (§2.1), and in detail, the algorithms used to obtain the
reconstructed LCs, including the functional form toy model (§2.2) and the
Gaussian Processes (§2.3). The results are summarized in §3, of the functional
form toy model approach (§3.1) and the Gaussian Processes (§3.2). We draw our
conclusions in §4.

###  1.1 LC Reconstruction

LCR offers an innovative solution to the problem of temporal gaps in LCs. In
several domains of astronomy, statistical LCR methods have been used to
account for missing data. Such methods have also been applied to Cepheids
using simulated annealing and Fourier decomposition techniques (Ngeow et al.,
2003). A similar application of reconstruction methods on LCs can be seen in
Huber et al. (2010) for planetary eclipse mapping and Geiger & Schneider
(1996) for measurement of time delay in gravitational lens systems. However,
an application of reconstruction techniques on GRB LCs has yet to be
undertaken because GRB LCs have very variable features. They present a wide
range of shapes, including smooth single pulses and multiple peaks with
different widths occurring at different times, while others exhibit more
complex behavior.

As a first step to address this problem and evaluate the performance of LCR,
we have used a relatively simple methodology that chooses a model to fit the
LCs. Many groups have classified LC morphology with a simple PL, BPL, smoothly
BPL, or W07 model; some with plateaus, and some with double breaks (Giblin et
al., 2002; Nousek et al., 2006; De Pasquale et al., 2008; Evans et al., 2009;
Racusin et al., 2009; Liang et al., 2009; Cenko et al., 2010). Although our
results depend on the underlying model chosen for the reconstruction, we can
give a first estimate of how the LCR can enhance the determination of the
plateau parameters. Thus, we show how to enable more accurate morphological LC
classification and determination of GRB parameters. Our LCR analysis reduced
the uncertainties on the parameters of the plateau by 37.22% for 10% noise
level on average among all plateau parameters for the reconstruction with the
W07 functional form toy model and 30.69% for 10% noise level for the BPL
functional form toy model. Furthermore, using the Gaussian Processes, the
uncertainties on all plateau parameters are decreased, on average, by 31.43%
for the W07 and 21.99% for the BPL (see §3).

###  1.2 The advantage of the LCR in theoretical models

We can anticipate that with the new light curves reconstructed, we can use
them directly to test theoretical models. We here limit ourselves to provide
two examples because this topic is far beyond the current focus of the paper,
and it will be investigated in a forthcoming paper. One of the most immediate
advantages is the use of the parameters of the BPL obtained with the
reconstructed lightcurves with Gaussian processes or the other functional toy
model LCR modeling to test the standard fireball model via the closure
relationships, which are the relationships among the temporal, $\alpha$, and
spectral indices, $\beta$, of the lightcurves. For a precise new estimate of
the modeling one could check the differences between the results obtained in
X-rays from previous papers (Srinivasaragavan et al., 2020b; Dainotti et al.,
2021b) and the ones from the LCR. We envision that the prediction on the
closure relationships will be around 40% more accurate compared to the current
estimate in the literature. Indeed, 43.32% is the percentage reduction on the
uncertainty on the temporal index, $\alpha$, after the plateau phase if we
consider, for example, the W07 functional toy model for the reconstruction,
and for the BPL, the uncertainty in the slope of the LC after the plateau
phase, $\alpha_{2}$, decreases by 43.9%. For the Gaussian Processes, the
uncertainty in W07 $\alpha$ parameter decreases by 41.5%, and the uncertainty
in the $\alpha_{2}$ parameter decreases by 35.92%. There are cases for which
it is not possible to uniquely determine the outcome of the closure
relationships, namely whether the closure relationship favors a scenario with
a fast cooling or slow cooling or if a constant medium or a wind medium is
preferred. We can anticipate that a fraction of cases with the LCR will be
able to discern better the interstellar medium and the regime and will be able
to remove part of the degeneracy in the theoretical model investigation.
Srinivasaragavan et al. (2020b) found that $>50\%$ follow 11 out of the 16
closure relationships tested. It is interesting to check how the exact
percentage of fulfillment could change when LCR is adopted in this scenario.
Another model that can benefit from the LCR is the magnetar model. Indeed, it
was shown in Stratta et al. (2018) that due to the temporal gaps in some of
the LC, the magnetic field, the spin period parameters, or the electron energy
fraction could not be determined very precisely; namely, their uncertainties
are so large that the estimate cannot be considered reliable. In this
scenario, it is important to have full coverage of the lightcurve so that a
combination of the magnetic field, the spin period and the electron energy
fraction, could solve cases in which, due to these uncertainties on the
parameters, it is even hard to distinguish between models (magnetars vs. black
hole). The LCR can alleviate the degeneracy of the parameters of the model by
reducing the uncertainties of the parameters. However, some degree of
degeneracy inherent to the model will still remain unavoidable. Here, we
remind that this LCR will be approached as a first step, but it can also be
extended in high-energy $\gamma$-rays, optical and radio wavelengths for the
determination of the closure relationship in these respective energy bands
(Dainotti et al., 2021d, 2023a, 2022c; Levine et al., 2023).

###  1.3 The connection of the LCR and the GRB classification and Properties

Often the classification of GRB properties is challenging because of the
general lack of data points in the LC or, more often, for the lack of data
points in a crucial part of the LC for which a feature is expected, e.g., the
beginning or end of the plateau emission, etc. This section aims to clarify
how many GRBs classified in the literature as, for example, LGRBs, SGRBs, or
the other classes defined below correspond to the morphological classes
defined here. Indeed, this analysis is relevant because, with this study, we
can check what percentage of the Good GRBs fall in a given class.

Besides the Long and Short classification, we point out that other sub-classes
have been discovered. Intrinsically SGRBs (IS) have $T_{90}/(1+z)<2s$
(Levesque et al., 2010; Zhang et al., 2021; Rossi et al., 2021). SGRBs with
extended emission (SEE) (Norris & Bonnell, 2006; Norris et al., 2010, 2000;
Levan et al., 2007; Dichiara et al., 2021) are SGRBs with mixed properties
between LGRBs since they have $T_{90}>2$s, but they are harder in their
spectral properties than the LGRBs, and in this regard, they are similar to
SGRBs. From the observational point of view, they have a spike in the prompt
followed by a long tail. The spike of the initial emission is followed by a
brief pause in emission, lasting for $\simeq$10 seconds, after which the
emission increases again on a timescale of 30-50 seconds (Norris & Bonnell,
2006). Ultra-long GRBs (ULGRBs) have $T_{90}\geq 1000$s (Stratta et al., 2013;
Gendre et al., 2013; Nakauchi et al., 2013; Piro et al., 2014; Levan et al.,
2014; Greiner et al., 2015; Kann et al., 2018; Gendre et al., 2019; Zhang et
al., 2014). These GRBs may originate from engine-driven explosions of stars
much larger than LGRB progenitors (Levan et al., 2014). X-ray flashes (XRFs)
(Soderberg et al., 2004; Chincarini et al., 2010) are LGRBs with greater
fluence in X-rays (2–30 keV) than in $\gamma$-rays (30-400 keV), while X-ray
rich (XRR) GRBs are an intermediate class between XRFs and LGRBs that display
very strong X-ray emission. SNe-GRBs are LGRBs associated with Supernovae of
type Ic (SNe Ic) for which the SNe has been clearly observed (Dainotti et al.,
2007, 2022a). These sub-classes may imply diverse progenitors or the same
progenitor in different circumburst media.

Zhang et al. (2007a, b, 2009) unified the classification in terms of
progenitors, with Type I and Type II GRBs classes. Type I include SGRBs, SEE,
and IS; Type II includes LGRBs, XRFs, XRRs, GRB-SNe, and ULGRBs. For schematic
pictures of the classes, see Fig. 1.

Figure 1: Schematic of GRB classes divided in Type I and Type II and their
related sub-categories.

##  2 Methodology

We aim to fill temporal gaps in GRB LCs by compiling all observations from
Swift XRT taken from the BAT+XRT repository. To account for the missing data
points, we adopt a stochastic approach where the reconstructed points are
built from a chosen LC model. To account for the realistic variation of the
data, a noise function that considers the residuals from the initially
observed flux data points is used to inject additive noise into the
reconstructed points.

###  2.1 Data Sampling, the Willingale Model and the Broken-Power law model

We take a sample of 455 GRBs (Table. 1 from Srinivasaragavan et al. (2020a))
with X-ray plateaus (222 with known redshift and 233 without known redshift),
originally obtained from the Swift BAT-XRT repository (Evans et al., 2009,
2007). These GRBs are fitted with the W07 function, described in Eq. 1 as
presented in Willingale et al. (2007b):

$f(t)=\left\\{\begin{array}[]{ll}\displaystyle{F_{i}\exp{\left(\alpha_{i}\left(1-\frac{t}{T_{i}}\right)\right)}\exp{\left(-\frac{t_{i}}{t}\right)}}{\hskip
28.45274pt\rm for}\ \ t<T_{i}\\\ \\\
\displaystyle{F_{i}\left(\frac{t}{T_{i}}\right)^{-\alpha_{i}}\exp{\left(-\frac{t_{i}}{t}\right)}}{\hskip
28.45274pt\rm for}\ \ t\geq T_{i},\\\ \end{array}\right.$ (1)

Where $T_{i}$ and $F_{i}$ are the times and fluxes, respectively, either at
the end of the prompt (denoted with $p$) or at the end of the plateau emission
(denoted with $a$). The parameter $\alpha_{i}$ is the temporal index after
$T_{i}$. The initial rise time is marked by the time $t_{i}$, which reads as
$t_{a}$ for the afterglow emission. The maximum flux occurs at
$t_{i}=\sqrt{tT_{c}/a_{i}}$.

We also fit the GRBs with the simple broken power law model (BPL):

$f(t)=\left\\{\begin{array}[]{ll}\displaystyle{F_{i}\left(\frac{t}{T_{i}}\right)^{-\alpha_{1}}}&{\rm
for}\ \ t<T_{i}\\\
\displaystyle{F_{i}\left(\frac{t}{T_{i}}\right)^{-\alpha_{2}}}&{\rm for}\ \
t\geq T_{i},\\\ \end{array}\right.$ (2)

Where $T_{i}$ and $F_{i}$ are the times and fluxes, respectively, at the break
time (coincident with the end time of the plateau), $\alpha_{1}$ is the slope
of the LC before the break, and $\alpha_{2}$ is the slope of the LC after the
break. In both our Willingale and BPL fit, as in W07, we take the logarithm in
base 10 of these functions.

GRB NAME | $z$ | $T_{90}$ | $\frac{T_{90}}{(1+z)}$ | LONG/SHORT | Type | Reference
---|---|---|---|---|---|---
GRB050128 | - | 28.00 | - | L | SNe-GRB | [1]
GRB050315 | 1.95 | 95.40 | 32.35 | L | XRR | [2]
GRB050318 | 1.44 | 40.00 | 16.37 | L | XRR | [1]
GRB050319 | 3.24 | 151.74 | 35.79 | L | XRF | [2]
GRB050401 | 2.90 | 33.30 | 8.54 | L | L | [1]
GRB050416A | 0.65 | 6.62 | 4.01 | L | XRF/SNe-GRB, D | [1], [3]
GRB050502B | 5.20 | 17.72 | 2.86 | L | XRR | [2]
GRB050505 | 4.27 | 58.90 | 11.18 | L | L | [1]
GRB050607 | - | 48.00 | - | L | XRR | [2]
Table 1: Full sample of 455 GRBs, including redshift $z$, observer-frame burst
duration $T_{90}$ and rest-frame duration $\frac{T_{90}}{(1+z)}$ long vs.
short classification, GRB type, and reference. Full table is available online.
References: [1] Sakamoto et al. (2008); [2] Bi et al. (2018); [3] Dainotti et
al. (2022a)

We focus our reconstruction efforts on the plateau region of the GRB LCs,
where our methodology works effectively. Before the plateau, the prompt
emission has significant variability compared to the afterglow, and thus the
prompt is difficult to model accurately. The stochastic approach used in this
paper will also not account for flares and bumps in the afterglow since such
variations are randomized to a very high degree. This is a first step towards
understanding the feasibility of LCR. More complex methods that can generalize
this first attempt to the prompt emission or flares in the afterglow phase
will be explored in future works.

The 455 LCs were then separated into categories based on the afterglow
features. For subsequent analysis, we categorize the GRB afterglow LCs into
those having i) good approximation with the W07 model (hereafter called Good
GRBs), ii) flares or bumps throughout the afterglow region, iii) a double
break at the end of the LC, iv) flares/bumps along with a double break. Fig. 2
shows LCs belonging to each category with the W07 model fit superimposed. We
take the Willingale function parameter values (for each GRB LC) from
Srinivasaragavan et al. (2020a).

Figure 2: LCs divided into four categories depending on the afterglow feature.
i) Good GRBs (top left); ii) A break in the end of the LC (top right); iii)
Flares/Bumps in the afterglow (bottom left); iv) Flares/Bumps with a Double
Break at the end of the LC (bottom right).

We segregate the GRBs into classes to better understand the characterization
of the Good LCs, which make up 48% of our total sample of 455 GRBs. We show a
breakdown of our sample into classes in Fig. 3 \- the top left chart shows
classification by duration into LGRBs, SGRBs, SEEs and IS GRBs. The top right
chart shows classification into morphological classes. The bottom two charts
show classification by type (XRRs, XRFs, ULGRBs, SNe-GRBs) for all LGRBs
(left) and all Good GRBs (right). For the Good GRBs, we see that the largest
fraction of the GRBs are XRR GRBs (39%), with the second-most frequent class
being simply LGRBs (38%). The third-most frequent class is the XRF (5%). This
would indicate that many LCs in our Good sample have very strong emissions in
X-rays. There is no strong preference for any other class, with the other
classes comprising $<5\%$ of the Good sample.

Figure 3: Top left: breakdown of all 455 GRBs by duration: long (L), short
(S), short with extended emission (SEE) and intrinsically short (IS); top
right: breakdown of all 455 GRBs by morphological class; bottom left:
breakdown of all Long GRBs into types: X-ray rich (XRR), X-ray flash (XRF),
supernova-associated (SNe-GRB), ultra-long (UL), SEE, IS, and combinations
thereof; bottom right: breakdown of only Good GRBs used for reconstruction
into types.

###  2.2 The Reconstruction method with the functional forms

We segregate the 455 GRBs into the previously-mentioned classes to understand
how many Good GRBs belong to a given class. Then, for simplicity, we limit the
application of the reconstruction technique to only the Good GRB LCs, which
still constitute a large fraction (48%; 218 GRBs) of our total sample. Indeed,
we aim for a procedure that works for well-behaved morphological LCs and can
be extended to more complex cases in the future.

We compute the flux residual for each LC. The flux residual is defined as the
difference between the logarithm of the flux value of the original LC and the
corresponding logarithmic value of the flux given by the W07 fit for a given
instant of time $t$ (in $\log_{10}$ scale). This can be represented by:

$\log_{10}F_{\rm{res}}=\log_{10}F^{obs}_{t}-\log_{10}f(t)$ (3)

where $\log_{10}F_{\rm{res}}$ is the logarithm of the flux residual,
$\log_{10}f(t)$ is the W07 model or the BPL model flux at time $t$ and
$\log_{10}F^{\rm{obs}}_{t}$ is the observed log flux at time $t$.

In the left panels of Fig. 4, we show as an example the fitting of a Good LC,
GRB 121217A with the W07 model (top; shown in red) beginning from the plateau
emission and the BPL model (bottom; shown in black). In the right panel of
Fig. 4, we show the corresponding $\log_{10}$ flux residual histograms, or
Normalized histogram of $F_{(res)}$ detailed in Eq. 3, with the best-fit
Gaussian overlaid for this GRBs. The purpose of generating these histograms is
to check the deviation of original flux values from the fitted W07 or BPL
model. Based on these histograms, we performed the fitting using the best-fit
distribution for each GRB, which resulted in the Gaussian/Normal distribution:

$P(x)=\frac{1}{{\sigma\sqrt{2\pi}}}e^{{{-\left({x-\mu}\right)^{2}}\mathord{\left/{\vphantom{{-\left({x-\mu}\right)^{2}}{2\sigma^{2}}}}\right.\kern-1.2pt}{2\sigma^{2}}}}$
(4)

where $\mu$ is the mean of the distribution, $\sigma$ is the standard
deviation and $\frac{1}{\sigma\sqrt{2\pi}}$ is the normalization constant.

W07 Fit and residuals

BPL Fit and residuals

Figure 4: The left panels show the LCs of GRB GRB 121217A starting from the
plateau emission, and the best-fit W07 model is shown in red on the upper
panels and the BPL fit shown in black on the lower panels. The right panels
show the log(flux) residual histogram and the best fit Gaussian distribution
shown in black.

We draw random variate samples from the fitted Gaussian to generate our
reconstructed data points based on our previous assumption that the flux has
Gaussian-distributed errors. We checked that the Gaussian is indeed the best-
fit distribution with the Mathematica 222using Wolfram Mathematica 12.3
function FindDistribution and find that the values for normality for
GRB121217A are $\mu=-0.0016$, and $\sigma=0.12$.

For a given time, $t$, the reconstructed flux at that time is defined in Eq. 5
as the sum of the fitted flux value and the noise based on the random variate
sampled from the flux residual Gaussian distribution:

$\log_{10}F^{\rm{recon}}_{t}=\log_{10}f(t)+(1+n)\times RV_{\mathcal{N}}$ (5)

Here, $F^{\rm{recon}}_{t}$ is the reconstructed value of flux at time $t$,
$f(t)$ is the W07 or BPL flux value at time $t$, $n$ is the noise level, and
$RV_{\mathcal{N}}$ is the random variate sampled from the Normal (Gaussian)
distribution.

We add the noise parameter to mimic the realistic variations seen in observed
LCs. We assume the noise to be stationary, meaning that the noise on the flux
does not change with time and is random without temporal autocorrelation.
Usually, it is common to encounter cases in which the noise level is increased
by 10% or 20%. We acknowledge that the choice of noise level is arbitrary, but
the purpose here is to show how LCR works with realistic cases. Anyone with
the detailed procedure we have described can change the noise level for the
particular LCs they are reconstructing.

The $RV_{\mathcal{N}}$ for this noise is generated at every instant of time,
starting from the beginning of the plateau from the distribution of the
residuals fitted with a Gaussian. Thus, the reconstructed flux value is
different for each point. In this manner, we can reconstruct data points
within the temporal gaps of each LC.

Applying this method will result in all the reconstructed data points in the
afterglow LC being, statistically, the same distance from the best-fit of the
W07 or BPL model as the initially-observed data points. We choose the time of
the reconstructed points along the fitting line with a time range distributed
equally in the log scale. To this end, for practicality, we use the function
in Python called geomspace, with the minimum and maximum time set at the
beginning of the plateau and the last data point of the observed LC,
respectively. Thus, each reconstructed LC is customized to each observed LC,
which ensures the realistic nature of these reconstructed LCs for future use
in cosmological studies.

Due to the random noise added to the reconstructed data points, we perform the
reconstruction steps mentioned above for each GRB 100 times. This helps in de-
randomizing the results and stabilizing them. Then, the chosen function is fit
on each iteration of the reconstructed LC, providing us with 100 new fit
parameters and error estimates for each GRB. It should be noted that the new
fit is performed on the LC, which contains the reconstructed flux and the
original observations. So the combined LC is used here. The flux uncertainties
for each reconstructed data point are generated randomly from a Gaussian
distribution which is the best-fit distribution of the uncertainties of the
observed fluxes.

Thus, with the LCR procedure, we combine the reconstructed and original data
points to create a new enhanced LC, which we refit with the chosen model with
a least-squares regression. For this, we use the minimize() method from the
lmfit library (Newville et al. (2016)), which is an interface for fitting
curves in Python. It is a further extension of the optimization techniques
available in the methods of scipy.optimize. The minimize() function is used as
an optimization function that minimizes the residuals to the fit line while
doing the refitting. We provide this function with the model we want to fit,
the initial guess for the parameters, the data to fit (the reconstructed LCs),
and the method to be used for fitting. This function returns the updated
values of the chosen model parameters and their associated uncertainties.

###  2.3 The Reconstruction method with the Gaussian processes

We here briefly describe Gaussian Processes and how we use them. Gaussian
process (hereafter called GP) is a generic supervised learning method that can
generate precise predictions, and is designed to solve regression problems. It
is based on the mathematical properties of the Gaussian/Normal distribution.

Due to its probabilistic nature, a GP model does not produce merely a singular
prediction but rather calculates the likelihood for each possible prediction.

A GP works on the postulates of Bayesian inference. It begins by inspecting
the prior, which contains the leverage of knowledge of existing trends in the
initial data, and using Bayesian inference to compute a posterior, a
probabilistic description of the outcomes that agree with both the data and
the prior. Each prediction has an associated confidence interval, which limits
the region of the likelihood of the prediction.

The capacity to find confidence intervals/regions is the true utility of a GP
model. The set of possible predictions might be a Normal distribution with the
mean as the model’s prediction and a variance that characterizes the deviation
of the predictions from the mean. A lower uncertainty corresponds to a thinner
confidence region, and the model is said to be confident with its prediction.

Covariance functions, also known as Kernels, are a fundamental requirement to
use a GP model. A Kernel can be defined as a measure of the degree of
similarity between two given input space data points. This contains the
assumption that two similar data points should yield two similar output
values. Kernels can be of different types: those depending directly on the
difference between the two data points, that is, $x-x^{\prime}$, and those
depending on the specific values of the data points $x$ and $x^{\prime}$
themselves. Kernels depending upon just their difference, are translation-
invariant in the input space. Furthermore, if the kernel in question depends
merely on $|x-x^{\prime}|$, then the kernel is said to be isotropic in the
input space. An example of such a kernel would be a Radial Basis Function
(RBF) kernel.

We here use the RBF plus the contribution of a White noise Kernel because we
assume that the noises of the flux are independent and identically normally
distributed. We use the built-in function GaussianProcessRegressor, which
enables prediction without prior fitting but only with prior knowledge of the
GP. The prior in our analysis is not normalized, meaning that the mean is
centered at zero. Here normalizing means that we would normalize the fluxes by
removing the mean and scaling to unit-variance. We choose the default option
as false. Then we fit the GP regression model with the built-in fit function
to our data. Next, we have used the in-built predict function to obtain the
reconstructed data points.

To reconstruct the LC with GP, we have chosen as the interval for the
reconstruction 95% confidence interval. We have populated the data points by
using the Eq. 5 again, but here the function $f(t)$, instead of being the W07
or the BPL function, is the function found via the GP. To avoid the
distribution of the data points in the Gaussian processes depending heavily on
the shape of the function obtained with the Gaussian Regressor function, a
built-in function in Python, for each different LC, we perform 100 MCMC
simulations of the reconstructed LC. Then, we randomly pick one value for the
data point and its associated uncertainty.

##  3 Results

###  3.1 Results from functional form reconstruction

Applying our LCR methodology to the 218 Good GRBs in our sample, we expect to
see a reduction in the uncertainties on the chosen model parameters. To
measure this, we compute the error fractions associated with each of the model
parameters for the original and reconstructed fit. The error fractions for the
three Willingale parameters are given by:

$EF_{\log_{10}(T_{a})}=\left|\frac{\Delta{\log_{10}(T_{a})}}{\log_{10}(T_{a})}\right|,$
(6)

$EF_{\log_{10}(F_{a})}=\left|\frac{\Delta{\log_{10}(F_{a})}}{\log_{10}(F_{a})}\right|,$
(7)

$EF_{\alpha_{a}}=\left|\frac{\Delta{\alpha_{a}}}{\alpha_{a}}\right|.$ (8)

Here, $EF_{X}$ is the error fraction associated with the parameter $X$, and
$\Delta{X}$ is the uncertainty associated with the parameter $X$. The $X$ and
$\Delta{X}$ for $\log_{10}(T_{a})$, $\log_{10}(F_{a})$ and $\alpha_{a}$ before
reconstruction are taken from Srinivasaragavan et al. (2020a), as previously
indicated.

The computed error fractions before and after reconstruction for the W07 model
is performed for the full sample of 218 Good GRBs for two noise levels, 10%
and 20%, in Table 2. The first three columns show the error fraction for each
W07 parameter from the original fitting. The second three columns show the
error fraction for each W07 from the new fitting after reconstruction. These
are averages of 100 iterations, as mentioned previously.

We calculate the percentage decrease in the error fractions for each noise
level to analyze the improvement of the fit following the reconstruction. The
formula used for the percentage decrease is given as:

$\%_{DEC}=\frac{\left|EF^{\rm{after}}_{X}\right|-\left|EF^{\rm{before}}_{X}\right|}{\left|EF^{\rm{before}}_{X}\right|}\times
100$ (9)

The last three columns of Table 2 show the percent change in the error
fraction after reconstruction for the two noise levels. From our analysis, we
see that in all cases, the error fraction on the reconstructed parameters is
lower than the error fraction on the original parameters. To better understand
how this applies to the sample as a whole, we compute the average percentage
decrease in error fraction for all Good GRBs tested at each noise level. For
the 10% noise level, we obtain a 33.33% decrease in error fraction on the
$\log(T_{a})$ parameter, a 35.03% decrease in error fraction on the
$\log(F_{a})$ parameter, and a 43.32% decrease in error fraction on the
$\alpha$ parameter. For the 20% noise level, we obtain a 29.49% decrease in
error fraction on the $\log(T_{a})$ parameter, a 31.24% decrease in error
fraction on the $\log(F_{a})$ parameter, and a 40.57% decrease in error
fraction on the $\alpha$ parameter. As expected, when we introduce a larger
noise level, the decrease on the uncertainty also decreases since the spread
around the best-fit line increases.

A visual comparison between two noise levels for a sample GRB, GRB121217A, for
10% and 20% of the W07 fit is shown in the upper panels of Fig. 5. We observe
that lower noise levels generate points somewhat close to the W07 fit, and
these points are further spread out as the noise level increases. The
reconstructed LC plots are picked randomly from the 100 reconstructions for
each GRB. As expected, we can approximate the flux uncertainties with a
Gaussian distribution as shown in Fig. 6. The histograms of Fig. 13 show the
distribution of the percentage decrease (upper panel shows the 10% noise
level, while the bottom panel shows the 20% noise level) for all three of the
W07 parameters. As is evident, the reconstruction leads to a decrease in the
errors on the W07 parameters for all the GRBs. After fitting the model, we
expect a decrease in the uncertainty since we are refitting more points with
the same model. However, the point of this approach is to provide a first
simple solution to the lightcurve reconstruction. This toy model for the
reconstruction is indeed model dependent, but in principle this method can
work on any model.

To assess the method’s dependency on a particular model, we also fit with the
BPL model and check the differences regarding the decrease in the
uncertainties. To test the BPL, we perform the same methodology as used on the
W07 fits. Again, we see an overall decrease in error fraction on the
reconstructed parameters as compared to the error fraction on the original
parameters. We again compute the average percentage decrease in error fraction
for all Good GRBs tested at each noise level. For the 10% noise level, we
obtain a 33.31% decrease in error fraction on the $\log(T_{a})$ parameter, a
30.79% decrease in error fraction on the $\log(F_{a})$ parameter, a 14.76%
decrease in error fraction on the $\alpha_{1}$ parameter, and a 43.98%
decrease in error fraction on the $\alpha_{2}$ parameter. For the 20% noise
level, we obtain a 29.88% decrease in error fraction on the $\log(T_{a})$
parameter, a 27.20% decrease in error fraction on the $\log(F_{a})$ parameter,
a 1.78% decrease in error fraction on the $\alpha_{1}$ parameter, and a 41.1%
decrease in error fraction on the $\alpha_{2}$ parameter.

Overall, we see a similar trend to the W07 fitting. The results for the
average change in error fraction for the BPL fitting are given in Table 3. The
comparison between two noise levels for a sample GRB, GRB121217A, for 10% and
20% of the BPL fit is shown in the lower panels of Fig. 5. Figure 22 shows the
distribution of the percentage decrease (upper panel shows the 10% noise
level, while the bottom panel shows the 20% noise level) for all four of the
BPL parameters. Again, the reconstruction leads to a decrease in the
uncertainties on the BPL parameters for all the GRBs. The average percentage
decrease on all parameters for both the W07 and BPL fits are summarized in
Table 4.

We also check the differences between the Gaussian processes (a completely
model-independent approach) and the BPL (a model-dependent approach) to assess
the differences between these methods.

W07 reconstruction

BPL reconstruction

Figure 5: Reconstructed LCs for GRB121217A at two noise levels, using the
Willingale (W07) and Broken Power Law (BPL) fits. The first row shows the
reconstructed LCs using the W07 model at 10% and 20% noise, and the bottom row
show the reconstructed LCs using the BPL model at 10% and 20% noise. These
data points are generated from the noise distribution produced by fitting a
Gaussian distribution on the flux residual histograms. Figure 6: Histogram of
the error bars of the observed fluxes, shown to be approximated with a
Gaussian distribution.

Figure 7: (a) Distribution of the 10% noise level for the $\log F_{a}$ with
W07. Figure 8: (b) Distribution of the 10% noise level for the $\log T_{a}$
with W07

Figure 9: (c) Distribution of the 10% noise level for the $\alpha_{a}$ with
W07 Figure 10: (d) Distribution of the 20% noise level for the $\log F_{a}$
with the W07

Figure 11: (e) Distribution of the 20% noise level for the $\log T_{a}$ with
W07 Figure 12: (f) Distribution of the 20% noise level for the $\alpha$ with
W07

Figure 13: Distribution of Relative Percentage Decrease for the parameters of
the W07 function when we apply the W07 for the reconstruction.

Figure 14: (a) Distribution of the 10% noise level for the $\log(F_{a})$ with
BPL Figure 15: (b) Distribution of the 10% noise level for the $\log(T_{a})$
with BPL Figure 16: (c) Distribution of the 10% noise level for the
$\alpha_{1}$ with BPL  Figure 17: (d) Distribution of the 10% noise level for
the $\alpha_{2}$ with BPL Figure 18: (e) Distribution of the 20% noise level
for the $\log(F_{a})$ with BPL Figure 19: (f) Distribution of the 20% noise
level for the $\log(T_{a})$ with BPL Figure 20: (g) Distribution of the 20%
noise level for the $\alpha_{1}$ with BPL Figure 21: (h) Distribution of the
20% noise level for the $\alpha_{2}$ with BPL

Figure 22: Distribution of Relative Percentage Decrease for the parameters of
the BPL fit when we apply the BPL for the reconstruction.

###  3.2 Results from Gaussian process reconstruction

Gaussian Process-based reconstruction provides a model-independent way of
achieving our aim. The reconstructed LC of GRB121217A is presented in Fig. 23.
Following the reconstruction, the full LC is fitted with both the W07 and the
BPL model to obtain new estimates of the parameters and their errors.

For the parameter $\log(T_{a})$ we observe an average decrease in the error
fraction of 25%. Similarly, for $\log(F_{a})$ and $\alpha$ the average
decreases in error fraction are 27.9% and 41.5%, respectively. Note that in
this case also the error fractions are calculated according to Eq. 9. The
histogram distributions of Fig. 27 show the distribution of the relative
percentage decrease of the three W07 parameters. The results of the GP for the
W07 are given for each GRB in Table 5, while the average percentage decrease
on the uncertainties of the parameters for all GRBs are presented in Table 4.

For the BPL fits, we see that for the $\alpha_{1}$ parameter, there is
occasionally an increase in error fraction from the original parameters to the
reconstructed parameters. This is because the slope of the LC before the break
in the BPL fitting can be very flat, which will cause the uncertainty to be
high compared to the $\alpha_{1}$ value and cause the error fraction to be
$>1$. Because of this, we remove the GRBs which exhibit a flat $\alpha_{1}$,
or $|\alpha_{1}|<0.1$. We see a percentage decrease in the parameters of
15.02% for $\log(T_{a})$, 11.91% for $\log(F_{a})$, 25.10% for $\alpha_{1}$,
and 35.92% for $\alpha_{2}$. Again, the BPL fitting results show a similar
trend to the W07 fits. The results of the GP for each GRB are given in Table
LABEL:tab:tableBPL, while the average percentage decrease on the uncertainties
of the parameters for all GRBs are given in Table 4.

Figure 23: Gaussian Process for the GRB 121217A. The left panel shows the
Gaussian process fit, while the right panel shows the reconstructed LCs.

Figure 24: (a) Distribution of error fraction decrease for $\log F_{a}$.
Figure 25: (b) Distribution of error fraction decrease for $\log T_{a}$.

Figure 26: (c) Distribution of error fraction decrease for $\alpha_{a}$.

Figure 27: Distribution of Relative Percentage Decrease for the parameters of
the W07 function following Gaussian Process reconstruction.

Figure 28: (a) Distribution of error fraction decrease for $\log(T_{a})$
Figure 29: (b) Distribution of error fraction decrease for $\log(F_{a})$
Figure 30: (c) Distribution of error fraction decrease for $\alpha_{1}$.
Figure 31: (d) Distribution of error fraction decrease for $\alpha_{2}$.

Figure 32: Distribution of Relative Percentage Decrease for the parameters of
the BPL function following Gaussian Process reconstruction.

##  4 Summary and Conclusion

We propose a relatively simple methodology for GRB LCR for LCs with X-ray
plateaus observed by Swift. This case study involves two functional form toy-
models, the W07 and the BPL model and a limited GRB sample, as the main aim is
to open the way to a new methodology that can be tested with more fitting
functions and different statistics (e.g., Bayesian) in the future.

Our method assumes the W07 and the BPL models as the starting model for
reconstruction, but the methodology presented here is versatile and can be
extended to any other model (smoothly BPL, double BPL, etc). Even models such
as the pulse model, which includes the prompt emission, can be explored in the
future. We also took a further step in the analysis adopting Gaussian
processes with the aim of being completely model-independent. We find that
after testing our methodology on a sample of 218 Good GRBs, we obtain a
reduction in the error bars of the plateau parameters of the W07 fitting - in
this case, we have a reduction of 37.22% on average for all parameters when
the noise is set at the 10% level, and a 33.77% reduction, on average, for all
parameters with a 20% noise level. Regarding the BPL, we have a reduction of
30.69% in average for the 10% noise level and a reduction of 24.97% for the
20% noise level. In average for GP we have a decrease of 31.43% for the W07
and 21.99% for the BPL. The reduction of uncertainties on plateau parameters
can be crucial for many cosmological studies that benefit from precise
measurements of these parameters, including the investigation of Population
III stars or the use of GRBs as standard candles to study the early universe.

There are many advantages to the community from our study, as filling in the
temporal gaps of observed LCs with a toy model solves the problem of the lack
of data and thus allows us to use the reconstructed LC for:

  * •

discovering plateau features in GRB LCs, which otherwise may remain
undetected,

  * •

increasing the sample size for building GRB correlations among the plateau
parameters,

  * •

testing theoretical models which would use the reconstructed LCs, since the
method is general and can be used with many different functions,

  * •

cosmological studies to reduce the scatter on the cosmological parameters.
This will be achieved using the enhanced GRB correlations involving the
plateau emission built with LCR,

  * •

estimating the redshift information with machine learning analysis which uses
the new, more accurate estimates of the plateau parameters. This can allow us
to determine the redshift of the high-z GRBs and thus further enhance the
studies of Population III stars,

  * •

classification of GRBs according to their morphology with increased accuracy.

As a final remark, this reconstruction is based on the LCs provided by Swift,
but it can be extended to any current or future missions and across different
wavelengths as well.

GRB ID | $EF_{\log_{10}(T_{i})}$ | $EF_{\log_{10}(F_{i})}$ | $EF_{\alpha_{i}}$ |  $EF_{\log_{10}(T_{i})}$ RC |  $EF_{\log_{10}(F_{i})}$ RC |  $EF_{\alpha_{i}}$ RC | $\%_{\log_{10}(T_{i})}$ | $\%_{\log_{10}(F_{i})}$ | $\%_{\alpha_{i}}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | 10% noise |  |  |  |  |
050712 | 0.019 | 0.005 | 0.044 | 0.014 | 0.004 | 0.027 | -26.67 | -24.96 | -38.04
050318 | 0.011 | 0.006 | 0.046 | 0.008 | 0.004 | 0.033 | -22.91 | -24.05 | -28.62
050416A | 0.024 | 0.005 | 0.018 | 0.016 | 0.003 | 0.012 | -31.24 | -33.54 | -34.85
050607 | 0.021 | 0.005 | 0.044 | 0.016 | 0.004 | 0.027 | -22.62 | -22.79 | -39.18
050713A | 0.01 | 0.003 | 0.018 | 0.008 | 0.002 | 0.011 | -16.21 | -14.31 | -36.35
050822 | 0.011 | 0.003 | 0.026 | 0.007 | 0.002 | 0.015 | -31.06 | -35.46 | -43.07
050824 | 0.025 | 0.006 | 0.094 | 0.015 | 0.003 | 0.056 | -40.7 | -38.45 | -40.27
050826 | 0.029 | 0.019 | 0.131 | 0.026 | 0.016 | 0.196 | -9.54 | -15.21 | 49.99
050915B | 0.036 | 0.008 | 0.115 | 0.025 | 0.005 | 0.068 | -29.03 | -34.25 | -40.92
051016A | 0.033 | 0.006 | 0.051 | 0.021 | 0.004 | 0.024 | -36.11 | -28.49 | -53
051109A | 0.012 | 0.005 | 0.016 | 0.006 | 0.002 | 0.01 | -51.32 | -58.89 | -33.99
051221A | 0.02 | 0.005 | 0.051 | 0.015 | 0.004 | 0.033 | -27.39 | -28.33 | -34.6
060105 | 0.004 | 0.001 | 0.007 | 0.003 | 0.001 | 0.003 | -16.2 | -14.1 | -54.34
060108 | 0.024 | 0.006 | 0.071 | 0.018 | 0.004 | 0.047 | -26.39 | -30.58 | -33.59
060109 | 0.014 | 0.005 | 0.057 | 0.008 | 0.003 | 0.025 | -40.63 | -45.09 | -56.2
060124 | 0.008 | 0.004 | 0.012 | 0.006 | 0.003 | 0.007 | -28.27 | -29.34 | -42.7
060218 | 0.028 | 0.014 | 0.082 | 0.014 | 0.005 | 0.065 | -50.59 | -63 | -20.77
060306 | 0.012 | 0.003 | 0.024 | 0.009 | 0.002 | 0.017 | -22.76 | -28.7 | -29.23
060418 | 0.018 | 0.005 | 0.03 | 0.01 | 0.003 | 0.01 | -43.68 | -32.48 | -66.37
060421 | 0.041 | 0.01 | 0.087 | 0.022 | 0.006 | 0.039 | -46.03 | -40.58 | -55.27
|  |  |  | 20% noise |  |  |  |  |
050712 | 0.019 | 0.005 | 0.044 | 0.014 | 0.004 | 0.028 | -24.37 | -22.28 | -35.56
050318 | 0.011 | 0.006 | 0.046 | 0.009 | 0.004 | 0.034 | -18.58 | -19.74 | -24.97
050416A | 0.024 | 0.005 | 0.018 | 0.017 | 0.003 | 0.012 | -27.39 | -29.91 | -31.97
050607 | 0.021 | 0.005 | 0.044 | 0.017 | 0.004 | 0.028 | -17.04 | -17.37 | -36.04
050713A | 0.01 | 0.003 | 0.018 | 0.009 | 0.003 | 0.012 | -12.34 | -10.16 | -33.05
050822 | 0.011 | 0.003 | 0.026 | 0.008 | 0.002 | 0.015 | -27.43 | -32.13 | -40.11
050824 | 0.025 | 0.006 | 0.094 | 0.016 | 0.004 | 0.059 | -37.67 | -35.27 | -36.93
050826 | 0.029 | 0.019 | 0.131 | 0.025 | 0.014 | 0.172 | -13.33 | -28.04 | 31.46
050915B | 0.036 | 0.008 | 0.115 | 0.027 | 0.006 | 0.073 | -23.99 | -30.54 | -37.03
051016A | 0.033 | 0.006 | 0.051 | 0.022 | 0.004 | 0.025 | -32.78 | -24.82 | -50.61
051109A | 0.012 | 0.005 | 0.016 | 0.006 | 0.002 | 0.011 | -48.9 | -56.82 | -30.45
051221A | 0.02 | 0.005 | 0.051 | 0.016 | 0.004 | 0.035 | -23.51 | -24.37 | -31.17
060105 | 0.004 | 0.001 | 0.007 | 0.003 | 0.001 | 0.003 | -11.94 | -9.72 | -51.99
060108 | 0.024 | 0.006 | 0.071 | 0.018 | 0.004 | 0.049 | -23.82 | -27.76 | -30.82
060109 | 0.014 | 0.005 | 0.057 | 0.009 | 0.003 | 0.026 | -37.84 | -42.4 | -54.15
060124 | 0.008 | 0.004 | 0.012 | 0.006 | 0.003 | 0.007 | -23.7 | -24.55 | -40.13
060218 | 0.028 | 0.014 | 0.082 | 0.014 | 0.005 | 0.067 | -48.31 | -61.5 | -18.12
060306 | 0.012 | 0.003 | 0.024 | 0.01 | 0.002 | 0.018 | -20.13 | -25.98 | -26.29
060418 | 0.018 | 0.005 | 0.03 | 0.011 | 0.003 | 0.011 | -41.52 | -29.94 | -64.95
060421 | 0.041 | 0.01 | 0.087 | 0.024 | 0.006 | 0.041 | -41.96 | -36.37 | -52.87
Table 2: The error fractions of $\log_{10}(T_{a})$, $\log_{10}(F_{a})$ and
$\alpha_{a}$ before and after reconstruction (with relative percentage
decrease in error for all three parameters) at 10% noise ($n=0.1$; top) and
20% noise ($n=0.2$; bottom). First three columns give error fraction for
original W07 fit, while second three columns give error fraction for new W07
fit after reconstruction. Final three columns give percentage decrease in
error fraction after reconstruction. Full table is available online.

GRB ID | $EF_{\log_{10}(T_{i})}$ | $EF_{\log_{10}(F_{i})}$ | $EF_{\alpha_{1}}$ | $EF_{\alpha_{2}}$ |  $EF_{\log_{10}(T_{i})}$ RC |  $EF_{\log_{10}(F_{i})}$ RC |  $EF_{\alpha_{1}}$ RC |  $EF_{\alpha_{2}}$ RC | $\%_{\log_{10}(T_{i})}$ | $\%_{\log_{10}(F_{i})}$ | $\%_{\alpha_{1}}$ | $\%_{\alpha_{2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | 10% noise |  |  |  |  |  |  |  |
GRB050712 | 0.031 | 0.031 | 0.155 | 0.056 | 0.022 | 0.008 | 0.158 | 0.033 | -30.6 | -28.48 | 2.15 | -39.97
GRB050318 | 0.011 | -0.007 | 0.075 | 0.047 | 0.009 | 0.005 | 0.058 | 0.038 | -20.09 | -23.11 | -23.14 | -19.47
GRB050416A | 0.038 | -0.01 | 0.078 | 0.023 | 0.029 | 0.007 | 0.056 | 0.013 | -23.52 | -22.5 | -28.03 | -40.54
GRB050607 | 0.036 | -0.011 | 0.15 | 0.053 | 0.03 | 0.01 | 0.166 | 0.03 | -16.39 | -11.78 | 10.48 | -43.15
GRB050713A | 0.015 | -0.006 | 0.043 | 0.019 | 0.012 | 0.004 | 0.039 | 0.013 | -18.58 | -18.89 | -9.77 | -34.12
GRB050822 | 0.012 | -0.004 | 0.223 | 0.024 | 0.01 | 0.004 | 0.164 | 0.014 | -14.58 | -14.2 | -26.57 | -42.26
GRB050824 | 0.026 | -0.006 | 0.276 | 0.105 | 0.015 | 0.004 | 0.223 | 0.057 | -40.25 | -30.63 | -19.2 | -46.1
GRB050826 | 0.008 | -0.004 | 0.328 | 0.064 | 0.005 | 0.002 | 0.226 | 0.039 | -33.14 | -35.02 | -30.9 | -38.46
GRB050915B | 0.021 | -0.006 | 0.085 | 0.13 | 0.012 | 0.004 | 0.057 | 0.069 | -43.28 | -33.53 | -32.4 | -46.8
GRB051016A | 0.089 | -0.025 | 0.327 | 0.048 | 0.047 | 0.013 | 0.2 | 0.024 | -47.27 | -48.27 | -38.99 | -49.86
GRB051109A | 0.011 | -0.005 | 0.136 | 0.015 | 0.009 | 0.003 | 0.049 | 0.01 | -21.7 | -24.17 | -63.81 | -31.43
GRB051221A | 0.021 | -0.007 | 0.082 | 0.058 | 0.016 | 0.006 | 0.067 | 0.038 | -23.99 | -23.82 | -18.86 | -34.93
GRB060105 | 0.007 | -0.002 | 0.011 | 0.011 | 0.003 | 0.002 | 0.011 | 0.003 | -51.9 | -34.29 | 8.85 | -68.73
GRB060108 | 0.026 | -0.008 | 0.147 | 0.075 | 0.019 | 0.006 | 0.113 | 0.048 | -26.83 | -25.54 | -23.34 | -36.11
GRB060109 | 0.01 | -0.003 | 0.347 | 0.047 | 0.007 | 0.003 | 0.271 | 0.022 | -26.18 | -9.85 | -22.03 | -54.26
GRB060124 | 0.014 | -0.008 | 0.034 | 0.019 | 0.011 | 0.007 | 0.027 | 0.01 | -17.7 | -16.08 | -21.58 | -47.95
GRB060218 | 0.013 | -0.005 | 0.582 | 0.064 | 0.01 | 0.004 | 0.811 | 0.042 | -22.2 | -21.99 | 39.27 | -34.22
GRB060306 | 0.075 | -0.026 | 0.395 | 0.024 | 0.018 | 0.006 | 0.107 | 0.018 | -76.3 | -78.27 | -72.97 | -24.73
GRB060418 | 0.066 | -0.034 | 0.11 | 0.029 | 0.029 | 0.014 | 0.054 | 0.011 | -56.32 | -57.75 | -51.17 | -64.14
GRB060421 | 0.092 | -0.038 | 0.557 | 0.082 | 0.04 | 0.015 | 0.249 | 0.041 | -56.87 | -59.92 | -55.33 | -50.46
|  |  |  | 20% noise |  |  |  |  |  |  |  |
GRB050712 | 0.031 | -0.011 | 0.155 | 0.056 | 0.022 | 0.008 | 0.161 | 0.035 | -28.73 | -25.69 | 3.85 | -37.46
GRB050318 | 0.011 | -0.007 | 0.075 | 0.047 | 0.009 | 0.006 | 0.061 | 0.039 | -16.43 | -19.69 | -19.51 | -17.34
GRB050416A | 0.038 | -0.01 | 0.078 | 0.023 | 0.03 | 0.008 | 0.06 | 0.014 | -20.36 | -19.71 | -23.32 | -38.02
GRB050607 | 0.036 | -0.011 | 0.15 | 0.053 | 0.031 | 0.01 | 0.175 | 0.032 | -13.92 | -9.42 | 16.3 | -40.4
GRB050713A | 0.015 | -0.006 | 0.043 | 0.019 | 0.012 | 0.005 | 0.04 | 0.013 | -15.14 | -15.83 | -7.65 | -30.87
GRB050822 | 0.012 | -0.004 | 0.223 | 0.024 | 0.011 | 0.004 | 0.176 | 0.015 | -10.2 | -9.64 | -21.26 | -39.39
GRB050824 | 0.026 | -0.006 | 0.276 | 0.105 | 0.016 | 0.005 | 0.243 | 0.06 | -36.94 | -26.72 | -11.73 | -43.25
GRB050826 | 0.008 | -0.004 | 0.328 | 0.064 | 0.005 | 0.003 | 0.228 | 0.041 | -29.66 | -32.24 | -30.38 | -35.41
GRB050915B | 0.021 | -0.006 | 0.085 | 0.13 | 0.012 | 0.004 | 0.059 | 0.071 | -42.53 | -31.23 | -30.44 | -45.47
GRB051016A | 0.089 | -0.025 | 0.327 | 0.048 | 0.049 | 0.013 | 0.199 | 0.025 | -44.35 | -45.47 | -39.16 | -48.04
GRB051109A | 0.011 | -0.005 | 0.136 | 0.015 | 0.009 | 0.004 | 0.051 | 0.011 | -18.25 | -20.91 | -62.47 | -28.61
GRB051221A | 0.021 | -0.007 | 0.082 | 0.058 | 0.017 | 0.006 | 0.071 | 0.04 | -19.03 | -18.86 | -14.41 | -31.45
GRB060105 | 0.007 | -0.002 | 0.011 | 0.011 | 0.004 | 0.002 | 0.012 | 0.004 | -49.46 | -30.98 | 14.36 | -67.13
GRB060108 | 0.026 | -0.008 | 0.147 | 0.075 | 0.02 | 0.006 | 0.12 | 0.05 | -24.59 | -22.87 | -18.53 | -33.57
GRB060109 | 0.01 | -0.003 | 0.347 | 0.047 | 0.008 | 0.003 | 0.291 | 0.023 | -22.04 | -4.53 | -16.26 | -51.69
GRB060124 | 0.014 | -0.008 | 0.034 | 0.019 | 0.012 | 0.007 | 0.029 | 0.01 | -13.29 | -11.75 | -15.88 | -45.85
GRB060218 | 0.013 | -0.005 | 0.582 | 0.064 | 0.01 | 0.004 | 0.876 | 0.045 | -17.72 | -17.19 | 50.51 | -30.2
GRB060306 | 0.075 | -0.026 | 0.395 | 0.024 | 0.018 | 0.006 | 0.108 | 0.019 | -75.59 | -77.65 | -72.61 | -20.71
GRB060418 | 0.066 | -0.034 | 0.11 | 0.029 | 0.03 | 0.015 | 0.056 | 0.011 | -53.89 | -55.46 | -49.07 | -62.74
GRB060421 | 0.092 | -0.038 | 0.557 | 0.082 | 0.041 | 0.016 | 0.262 | 0.041 | -55.29 | -58.42 | -53.06 | -50.02
Table 3: The error fractions of $\log_{10}(T_{a})$, $\log_{10}(F_{a})$,
$\alpha_{1}$ and $\alpha_{2}$ before and after reconstruction (with relative
percentage decrease in error for all three parameters) at 10% noise ($n=0.1$;
top) and 20% noise ($n=0.2$; bottom). First three columns give error fraction
for original BPL fit, while second three columns give error fraction for new
W07 fit after reconstruction. Final three columns give percentage decrease in
error fraction after reconstruction. Full table is available online.

Reconstruction process | $\%^{avg}_{\log_{10}(T_{a})}$ | $\%^{avg}_{\log_{10}(F_{a})}$ | $\%^{avg}_{\alpha_{a}}$ | $\%^{avg}_{\alpha_{1}}$ | $\%^{avg}_{\alpha_{2}}$
---|---|---|---|---|---
W07 reconstruction (10%) | -33.33 | -35.03 | -43.32 | - | -
W07 reconstruction (20%) | -29.49 | -31.24 | -40.57 | - | -
BPL reconstruction (10%) | -33.3 | -30.78 | - | -14.76 | -43.9
BPL reconstruction (20%) | -29.88 | -27.2 | - | -1.7 | -41.1
Gaussian process (W07) | -24.9 | -27.9 | -41.5 | - | -
Gaussian process (BPL) | -15.02 | -11.91 | - | -25.10 | -35.92
Table 4: The average percentage decrease in the error fractions for the various parameters, following the reconstruction. GRB ID | $EF_{\log_{10}(T_{i})}$ | $EF_{\log_{10}(F_{i})}$ | $EF_{\alpha_{i}}$ |  $EF_{\log_{10}(T_{i})}$ RC |  $EF_{\log_{10}(F_{i})}$ RC |  $EF_{\alpha_{i}}$ RC | $\%_{\log_{10}(T_{i})}$ | $\%_{\log_{10}(F_{i})}$ | $\%_{\alpha_{i}}$
---|---|---|---|---|---|---|---|---|---
|  |  |  | GP (W07) |  |  |  |  |
050712 | 0.019 | 0.005 | 0.044 | 0.012 | 0.003 | 0.028 | -34.01 | -31.09 | -36.97
050318 | 0.011 | 0.006 | 0.046 | 0.008 | 0.004 | 0.031 | -26.66 | -27.68 | -32.19
050416A | 0.024 | 0.005 | 0.018 | 0.016 | 0.003 | 0.011 | -32.44 | -34.04 | -38.46
050607 | 0.021 | 0.005 | 0.044 | 0.017 | 0.004 | 0.028 | -18.58 | -19.62 | -36.98
050713A | 0.01 | 0.003 | 0.018 | 0.008 | 0.002 | 0.011 | -18.81 | -15.29 | -35.28
050822 | 0.011 | 0.003 | 0.026 | 0.008 | 0.002 | 0.015 | -29.98 | -33.14 | -43.56
050824 | 0.025 | 0.006 | 0.094 | 0.015 | 0.003 | 0.056 | -39.2 | -37.19 | -40.83
050826 | 0.029 | 0.019 | 0.131 | 0.02 | 0.012 | 0.156 | -32.22 | -36.64 | 19.63
050915B | 0.036 | 0.008 | 0.115 | 0.024 | 0.005 | 0.068 | -34.36 | -37.46 | -41.29
051016A | 0.033 | 0.006 | 0.051 | 0.024 | 0.005 | 0.025 | -25.01 | -17.02 | -51.08
051109A | 0.012 | 0.005 | 0.016 | 0.007 | 0.002 | 0.009 | -41.5 | -49.53 | -40.98
051221A | 0.02 | 0.005 | 0.051 | 0.013 | 0.004 | 0.033 | -33.79 | -32.68 | -34.76
060105 | 0.004 | 0.001 | 0.007 | 0.003 | 0.001 | 0.003 | -23.04 | -20.31 | -52.2
060108 | 0.024 | 0.006 | 0.071 | 0.018 | 0.004 | 0.047 | -23.29 | -28.84 | -33.57
060109 | 0.014 | 0.005 | 0.057 | 0.008 | 0.003 | 0.025 | -40.9 | -44.23 | -55.74
060124 | 0.008 | 0.004 | 0.012 | 0.005 | 0.003 | 0.007 | -34.32 | -34.36 | -45.13
060218 | 0.028 | 0.014 | 0.082 | 0.013 | 0.005 | 0.062 | -54.56 | -65.62 | -24.38
060306 | 0.012 | 0.003 | 0.024 | 0.01 | 0.002 | 0.018 | -19.95 | -26.66 | -26.92
060418 | 0.018 | 0.005 | 0.03 | 0.009 | 0.003 | 0.01 | -48.49 | -37.4 | -67.46
060421 | 0.041 | 0.01 | 0.087 | 0.032 | 0.008 | 0.046 | -21.65 | -20.75 | -46.81
Table 5: The W07 error fractions of $\log_{10}(T_{a})$, $\log_{10}(F_{a})$ and
$\alpha_{a}$ before and after performing the Gaussian Process (GP)
reconstruction (with relative percentage decrease in error for all three
parameters). First three columns give error fraction for original W07 fit,
while second three columns give error fraction for new W07 fit after GP
reconstruction. Final three columns give percentage decrease in error fraction
after GP reconstruction. Full table is available online. Table 6: Data for the
BPL fit of the GP-based reconstruction

GP (BPL)
---
GRB ID | $EF_{\log_{10}(T_{i})}$ | $EF_{\log_{10}(F_{i})}$ | $EF_{\alpha_{1}}$ | $EF_{\alpha_{2}}$ |  $EF_{\log_{10}(T_{i})}$ RC |  $EF_{\log_{10}(F_{i})}$ RC |  $EF_{\alpha_{1}}$ RC |  $EF_{\alpha_{2}}$ RC | $\%_{\log_{10}(T_{i})}$ | $\%_{\log_{10}(F_{i})}$ | $\%_{\alpha_{1}}$ | $\%_{\alpha_{2}}$
050712 | 0.031 | -0.011 | 0.155 | 0.056 | 0.02 | 0.008 | 0.114 | 0.035 | -36.96 | -31.16 | -26.56 | -36.7
050318 | 0.011 | -0.007 | 0.075 | 0.047 | 0.009 | 0.005 | 0.057 | 0.037 | -21.44 | -24.84 | -25.07 | -22.08
050416A | 0.038 | -0.01 | 0.078 | 0.023 | 0.03 | 0.008 | 0.054 | 0.013 | -22.17 | -20.55 | -30.49 | -41.1
050607 | 0.036 | -0.011 | 0.15 | 0.053 | 0.033 | 0.01 | 0.265 | 0.03 | -9.03 | -6.18 | 76.48 | -44.07
050713A | 0.015 | -0.006 | 0.043 | 0.019 | 0.012 | 0.005 | 0.038 | 0.013 | -19.09 | -17.92 | -10.82 | -33.81
050822 | 0.012 | -0.004 | 0.223 | 0.024 | 0.011 | 0.004 | 0.126 | 0.014 | -13.12 | -10.96 | -43.37 | -40.74
050824 | 0.026 | -0.006 | 0.276 | 0.105 | 0.016 | 0.004 | 0.257 | 0.056 | -36.37 | -29.54 | -6.94 | -47.22
050826 | 0.008 | -0.004 | 0.328 | 0.064 | 0.009 | 0.004 | 0.387 | 0.068 | 18.19 | 10 | 18.16 | 6.79
050915B | 0.021 | -0.006 | 0.085 | 0.13 | 0.017 | 0.005 | 0.068 | 0.085 | -19.3 | -12.5 | -19.55 | -34.08
051016A | 0.089 | -0.025 | 0.327 | 0.048 | 0.055 | 0.016 | 0.123 | 0.029 | -38.03 | -36.85 | -62.33 | -38.73
051109A | 0.011 | -0.005 | 0.136 | 0.015 | 0.009 | 0.004 | 0.043 | 0.01 | -17.46 | -18.84 | -67.92 | -33.14
051221A | 0.021 | -0.007 | 0.082 | 0.058 | 0.015 | 0.005 | 0.062 | 0.038 | -29.47 | -27.45 | -24.21 | -34.37
060105 | 0.007 | -0.002 | 0.011 | 0.011 | 0.004 | 0.002 | 0.012 | 0.004 | -47.21 | -29.55 | 16.34 | -66.73
060108 | 0.026 | -0.008 | 0.147 | 0.075 | 0.02 | 0.006 | 0.13 | 0.05 | -21.43 | -19.39 | -11.85 | -33.82
060109 | 0.01 | -0.003 | 0.347 | 0.047 | 0.008 | 0.003 | 0.249 | 0.022 | -22.93 | -1 | -28.48 | -52.56
060124 | 0.014 | -0.008 | 0.034 | 0.019 | 0.012 | 0.007 | 0.026 | 0.009 | -15.57 | -14.13 | -24.78 | -49.47
060218 | 0.013 | -0.005 | 0.582 | 0.064 | 0.011 | 0.004 | 0.551 | 0.045 | -16.69 | -15.08 | -5.28 | -29.79
060306 | 0.075 | -0.026 | 0.395 | 0.024 | 0.017 | 0.005 | 0.092 | 0.019 | -77.62 | -79.69 | -76.75 | -20.34
060418 | 0.066 | -0.034 | 0.11 | 0.029 | 0.031 | 0.015 | 0.068 | 0.01 | -52.96 | -55.91 | -38.76 | -64.8
060421 | 0.092 | -0.038 | 0.557 | 0.082 | 0.053 | 0.02 | 0.177 | 0.051 | -43.01 | -46.22 | -68.23 | -37.96

Note. — The BPL error fractions of $\log_{10}(T_{a})$, $\log_{10}(F_{a})$,
$\alpha_{1}$ and $\alpha_{2}$ before and after Gaussian Process (GP)
reconstruction (with relative percentage decrease in error for all three
parameters). First four columns give error fraction for original BPL fit,
while second four columns give error fraction for new BPL fit after GP
reconstruction. Final four columns give percentage decrease in error fraction
after reconstruction. Full table is available online.

##  5 Acknowledgments

This work was made possible in part by the United States Department of Energy,
Office of Science, Office of Workforce Development for Teachers and Scientists
(WDTS) under the Science Undergraduate Laboratory Internships (SULI) program
who funded the initial help in creating the code for the reconstructed LCs of
Luke Nearhood and Sarvesh Gharat. We thank Malgorzata Bogdan for helpful
comments and discussion of our analysis. We also thank Susrestha Paul, Anish
Kalsi, Shubham Bhardwaj, and Sachin Venkatesh for their help in the GRB
classification. We are particularly grateful to Artem Poliszczuk for the
initial discussion on the manuscript. We would also like to thank Dr. J.
Xavier Prochaska for his helpful discussion of the Gaussian Processes. This
research was also support by the Visibility and Mobility module of the
Jagiellonian University (Grant number: WSPR.WSDNSP.2.5.2022.5) and the NAWA
STER Mobility Grant (Number: PPI/STE/2020/1/00029/U/00001). A. N. and M.G.D
are grateful to the Exploratory Research Grant (ERF) of the Division of
Science at NAOJ for the financial support for the visit of A. Narendra. A. N.
is also extremely grateful to NAOJ for providing all the essential amenities
during the visit to the NAOJ Mitaka Campus. D. L. is also grateful for the
support of the ERF grant. This work was also supported by the Polish National
Science Centre grant UMO-2018/30/M/ST9/00757 and by Polish Ministry of Science
and Higher Education grant DIR/WK/2018/12.

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