# Are stripped envelope supernovae really deficient in 56Ni?

Ryoma Ouchi Keiichi Maeda Department of Astronomy, Kyoto University,
Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan  Joseph P. Anderson
European Southern Observatory, Alonso de Córdova 3107, Casilla 19, Santiago,
Chile  Ryo Sawada Department of Earth Science and Astronomy, Graduate School
of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo
153-8902, Japan

###### Abstract

Recent works have indicated that the 56Ni masses estimated for Stripped
Envelope SNe (SESNe) are systematically higher than those estimated for SNe
II. Although this may suggest a distinct progenitor structure between these
types of SNe, the possibility remains that this may be caused by observational
bias. One important possible bias is that SESNe with low 56Ni mass are dim,
and therefore they are more likely to escape detection. By investigating the
distributions of the 56Ni mass and distance for the samples collected from the
literature, we find that the current literature SESN sample indeed suffers
from a significant observational bias, i.e., objects with low 56Ni mass - if
they exist - will be missed, especially at larger distances. Note, however,
that those distant objects in our sample are mostly SNe Ic-BL. We also
conducted mock observations assuming that the 56Ni mass distribution for SESNe
is intrinsically the same with that for SNe II. We find that the 56Ni mass
distribution of the detected SESNe samples moves toward higher mass than the
assumed intrinsic distribution, because of the difficulty in detecting the
low-56Ni mass SESNe. These results could explain the general trend of the
higher 56Ni mass distribution (than SNe II) of SESNe found thus far in the
literature. However, further finding clear examples of low-56Ni mass SESNe
($\leq 0.01M_{\odot}$) is required to add weight to this hypothesis. Also, the
objects with high 56Ni mass ($\gtrsim 0.2M_{\odot}$) are not explained by our
model, which may require an additional explanation.

stars: massive — supernovae: general

††journal: ApJ

##  1 Introduction

Core collapse supernovae (SNe) are the explosions of massive stars, marking
the termination of their lives. A small fraction of the gravitational energy
of the collapsing iron core is converted into the kinetic and thermal energy
of the ejected matter (Woosley et al., 2002). Core collapse SNe are classified
into several categories, based on their spectra and light curves. SNe with
hydrogen lines in their spectra are classified as Type II SNe (SNe II), while
those lacking hydrogen lines are called Type I SNe (SNe I). Among SNe I, those
having He lines are called Type Ib SNe (SNe Ib) and those lacking He lines are
called SNe Ic. Type IIb supernovae (SNe IIb) are characterized by hydrogen
lines in their early phase spectra which gradually disappear, and by the He
lines which become increasingly strong at later phases (Filippenko, 1997). SNe
IIb, Ib and Ic are considered to originate from massive stars that have lost a
significant fraction of the envelope during their evolution, and thus they are
collectively called Stripped-Envelope SNe (SESNe) (Smartt et al., 2009).

It has been established that SNe IIP are the explosions of red supergiants
based on light curve models (Falk & Arnett, 1977; Elmhamdi et al., 2003;
Bersten et al., 2011) and also from the direct detection of the progenitors on
pre-SN images (Smartt, 2009, 2015). On the contrary, the progenitors of SESNe
are more uncertain. For SNe IIb/Ib/Ic, two possible progenitor channels have
been proposed. One is a massive WR star (with the main-sequence mass
$M_{\mathrm{ms}}\gtrsim 25M_{\odot}$) that has blown off the H-rich envelope
by its own stellar wind (Begelman & Sarazin, 1986; Georgy, 2012; Gräfener &
Vink, 2016). The other is a relatively low mass star which loses its envelope
by mass transfer to a binary companion (Podsiadlowski et al., 1992; Stancliffe
& Eldridge, 2009). Recent observational evidence favors the latter scenario.
The light curve modeling and direct progenitor detection indicate that the
progenitors are relatively low mass stars ($M_{\mathrm{ms}}\lesssim
18M_{\odot}$), being consistent with the binary scenario (Maund et al., 2011;
Bersten et al., 2014; Van Dyk et al., 2014; Folatelli et al., 2015). Also, for
some SESNe, companion star candidates have been detected, which indicates a
binary origin (Maund et al., 2004; Folatelli et al., 2014).

One of the most important power sources of SNe is newly synthesized 56Ni.
${}^{56}\mathrm{Ni}$ decays into ${}^{56}\mathrm{Co}$, and then into
${}^{56}\mathrm{Fe}$. This nuclear decay chain powers the tail phase of SNe II
and most of the light curve of SESNe. The 56Ni masses of SNe have been
estimated using several methods (Anderson, 2019). For SNe II, the tail
luminosity has mostly been used to estimate the 56Ni mass, assuming the
complete trapping of $\gamma$-rays produced from the nuclear decay. For SESNe,
on the contrary, the tail luminosity cannot be easily used due to the
incomplete trapping of the $\gamma$-ray photons, and the ‘Arnett-rule’ has
often been used instead (Arnett, 1982; Wheeler et al., 2015). This rule
dictates that the peak luminosity of SESNe should be equal to the
instantaneous energy deposition rate by the nuclear decay. For both types of
SNe, the mass of synthesized 56Ni has also been estimated from light curve
modeling (e.g. Utrobin & Chugai, 2011; Bersten et al., 2014).

---
Figure 1:  Left: The time to peak ($t_{p}$) as a function of the 56Ni mass for
the sample of Meza-SESNe. The filled circles denote the samples excluding Type
Ic-BL, while open circles denote SNe Ic-BL. A solid line is the result of the
linear regression (i.e. equation 2), while dashed lines show the standard
error. Here, the standard error is estimated as
$\Sigma_{i=1}^{i=N}(t_{p,\mathrm{data},i}-t_{p,\mathrm{fit},i})^{2}/(N_{\mathrm{smaples}}-2)$.
Right: The peak luminosity ($L_{p}$) as a function of the 56Ni mass for the
sample of Meza-SESNe. Solid line is the prediction from the equations 1 and 2,
while dashed lines show the standard error.

Interestingly, mounting evidence has been accumulating to show that the masses
of synthesized 56Ni of the observed SESNe are systematically higher than those
of SNe II. This result was first formally outlined by Kushnir (2015). After
that, Anderson (2019) collected the 56Ni masses for 258 SNe from the published
literature and compared the 56Ni mass distributions for various types of SNe.
He found that the 56Ni masses estimated for SNe II are systematically lower
than SESNe; the median of the 56Ni masses is $0.032M_{\odot}$ for SNe II,
$0.102M_{\odot}$ for SNe IIb, $0.163M_{\odot}$ for SNe Ib, $0.155M_{\odot}$
for SNe Ic, and $0.369M_{\odot}$ for SNe Ic-broad line (SNe Ic-BL).

This result has important implications. The production of 56Ni is sensitive to
the explosion mechanism (Maeda & Tominaga, 2009; Suwa & Tominaga, 2015; Sawada
& Maeda, 2019) and the progenitor mass (Suwa et al., 2019). Indeed, this may
be qualitatively consistent with some indications that the progenitors of
SESNe may be more massive than SNe II either as an entire class or for the
particular SN Ic class (e.g. Anderson et al., 2012; Valenti et al., 2012; Fang
et al., 2019). The possible difference in the nature of the progenitors
between SNe II and SESNe may introduce some tension to the popular suggestion
for a binary origin for SESN progenitors, since the core structure should be
similar between SESNe and SNe II; the binarity mainly affects the outer
envelope but not the core structure (Yoon et al., 2010, 2017; Ouchi & Maeda,
2017). However, this picture may also be an oversimplification, since there
are several factors that can affect the nature of the progenitor even in the
binary scenario. For example, massive stars are claimed to be preferentially
formed in close binary systems (Moe & Di Stefano, 2017). The angular momentum
transfer may also have effects on the core structure and boost the synthesized
56Ni mass (Schneider et al., 2021). In any case, understanding the origin of
different 56Ni masses between SESNe and SNe II should help to clarify the
progenitors of SESNe.

Before concluding that the systematically different 56Ni mass between SESNe
and SNe II may be caused by a different structure in the progenitor cores,
systematic errors in calculating the 56Ni masses should be addressed
(Anderson, 2019). Indeed, the ‘Arnett-rule’, which has been widely used for
SESNe have been claimed to overestimate the 56Ni mass (Dessart et al., 2015,
2016; Khatami & Kasen, 2019). However, several works have concluded that even
by taking into account the different methods to derive the 56Ni mass and
various observational errors, a difference in 56Ni masses between SNeII and
SESNe remains (Afsariardchi et al., 2020; Meza & Anderson, 2020; Sharon &
Kushnir, 2020). Meza & Anderson (2020) further noted the possibility that
SESNe with a small amount of 56Ni might have been missed by the existing
surveys. Since the luminosity of SESNe is mostly powered by the radioactive
decay of 56Ni, the SESNe with the lowest 56Ni masses are the faintest (Lyman
et al., 2016). Thus, they can possibly escape from detection depending on the
survey depth. On the contrary, SNe II with a small amount of 56Ni can still
power themselves by diffusion of the thermal energy coming from the explosion
energy. Thus, SNe II can more easily be detected than SESNe, even if the 56Ni
mass is small. Indeed, several Ni-poor SESNe ($M_{\mathrm{Ni}}\lesssim
0.02M_{\odot}$) have been detected (Kasliwal et al., 2010; Shivvers et al.,
2016; Nakaoka et al., 2019). However, it should also be noted that none of
these examples are just a low-luminosity version of canonical SESNe as they
all show unusual properties.

The aim of this paper is to investigate how much observational bias may lie in
the 56Ni mass distribution of the samples collected from the published
literature. In section 2, we define the samples that are used throughout the
paper. Section 3 describes some equations that are used in this paper. In
section 4, we investigate whether there is an observational bias in the 56Ni
mass distribution of our data samples by examining the relation between
distance, luminosity and 56Ni mass. In section 5 and section 6, we conduct
mock observations of SESNe and theoretically investigate the effect of
observational bias on the ‘observed’ 56Ni mass distribution. We discuss the
results in section 7 and finally conclude the paper in section 8.

##  2 Data sample

In this section, we describe the observational samples used in this paper.
Throughout the paper, we use the samples of 56Ni estimates collected from the
published literature both for SESNe and SNe II. Anderson (2019) recently
compiled such samples, including 143 SESNe and 115 SNe II. Specifically, he
used the SAO/NASA ADS astronomy query
form111https://ui.adsabs.harvard.edu/classic-form, searching for articles with
“supernova” and “type II” that were published until August 2018, then
“supernova” and “type IIb” and so forth in manuscript abstracts. Then, he
identified those publications with published 56Ni mass estimates. In addition
to this sample, we add newly published objects between August 2018 and
November 2020. The newly added reference list can be found at the end of this
manuscript. Note that we do not include 56Ni estimates that are derived from
combined models, such as ‘magnetar + 56Ni model’ or ‘circumstellar interaction
+ 56Ni model’ (e.g. Gangopadhyay et al., 2020). We name these final samples
‘LS-SESNe (Large Sample SESNe)’ and ‘LS-SNeII (Large Sample SNe II)’,
respectively. The sample sizes are 187 and 115222The size of LS-SNeII is the
same as the sample of Anderson (2019). This occurred because this time we
excluded objects with only upper or lower limits for the 56Ni mass. The number
of thus removed events was by chance equal to that of the newly added events.
for LS-SESNe and LS-SNeII, respectively.

Several different methods have been used to derive the 56Ni masses in the
literature. For SNe II, the tail luminosity is commonly used to measure the
56Ni mass. For SESNe, on the contrary, 56Ni mass is often derived by feeding a
peak luminosity into the ‘Arnett-relation’. It is true that tail luminosity
has also been used for SESNe to constrain their 56Ni masses. However, since
the assumption of complete $\gamma$-ray trapping is usually not valid for
SESNe, the tail luminosity underestimates 56Ni mass (unless additional
modelling is employed; see e.g. Sharon & Kushnir (2020).)

In addition to LS-SESNe, we also use a different sample of SESNe, which we
call ‘Meza-SESNe’; this is the same sample as that used by Meza & Anderson
(2020). Those authors defined a SESN sample with well-sampled photometry at
optical and near-IR wavelengths. This led to a sample of 37 events. To obtain
peak luminosities, they applied a local polynomial regression with a Gaussian
kernel, using the public modules from PyQt-fit in Python4. Note that the
integration was done in the wavelength range from the $B$ band to $H$ band
without extrapolation outside. Therefore, their resulting light curves should
be considered to be pseudo-bolometric, and are a lower limit to the true
bolometric luminosity at all times. However, the wavelength coverage is
reasonably large, and therefore the error here is probably less significant
than that coming from the different methods to derive the 56Ni mass. In the
paper, they tested three different methods to derive the 56Ni mass. In the
first method, they used ‘Arnett-rule’. In the second method, they used a tail
luminosity. The third method employed that recently proposed by Khatami &
Kasen (2019), that overcomes several limitations of Arnett-like models. Meza &
Anderson (2020) showed that using the different methods does not change the
overall trends in the derived 56Ni masses and their conclusions. In the
present work, we mostly use the Arnett rule in our analysis (§3), but we show
that our results are not affected by this choice in Appendix B.

In section 4.2, in order to compare the luminosity function between SESNe and
SNe II, we use the sample of 57 SNeII taken from Hamuy (2003); Müller et al.
(2017); Pejcha & Prieto (2015). These events have published values of the mid-
plateau phase luminosity. These papers are included in the reference list of
Anderson (2019), and thus, this is a sub-sample of LS-SNeII. We call this
small sample ‘SS-SNeII (Small Sample SNe II)’.

Finally, in section 7, we will compare our results to a sample of candidate
ultra-stripped envelope SNe (USSNe). The data for the USSN candidates are also
collected from the published literature published before November 2020 by
searching “ultra-stripped” and “supernova” in the ADS abstract form. The
references for them are listed at the end of the manuscript 333Note that these
reference lists are not covering all the USSN candidates claimed so far, since
we only take into account those objects for which 56Ni masses have been
estimated.

For all these objects, we adopt the distance to the host galaxy from the
redshift independent measurement in NED444https://ned.ipac.caltech.edu. In
case there is no redshift independent measurement of the distance to the host,
we adopt the Hubble distance on NED, which includes the correction of Virgo,
GA, and Shapley. For the cosmological parameters, $H_{0}$ = 67.8 km/sec/Mpc,
$\Omega_{\mathrm{matter}}$ = 0.308, $\Omega_{\mathrm{vacuum}}$ = 0.692 have
been used. If the host was anonymous or the distance was not found in NED, we
take the distance from the individual published literature.

Figure 2: The 56Ni masses of our samples as a function of the distance. The
red points refer to LS-SESNe, while yellow points refer to LS-SNeII. Note that
red filled circles exclude SNe Ic-BL among LS-SESNe, while red open circles
are for SNe Ic-BL among LS-SESNe. For reference, the limiting distance for a
given 56Ni mass estimated in section 3 is also shown for the case of limiting
magnitude of $V_{\mathrm{lim}}=17$ (blue) and 19 mag (green).

##  3 The relations used in this paper

---
Figure 3:  Left: The 56Ni mass distribution of LS-SESNe for the volume-limited
samples of different sizes. The red solid line represents the cumulative 56Ni
mass distribution of the samples among LS-SESNe whose distances are less than
5000 Mpc, while yellow, green and blue solid lines represent the cumulative
distributions of the objects whose distances are less than 100 Mpc, 60 Mpc and
20 Mpc, respectively. Dotted lines represent the same distributions, except
the we exclude SNe Ic-BL from the sample. Right: The same figure as the left
panel but for LS-SNeII. For both panels, the non-linear least square fit to
the LS-SNe II cumulative distribution is also shown with a gray line (See
section 5.1 for more detail).

###  3.1 The relations for the peak luminosity and the timescale of SESNe

Two important quantities that characterize the light curves of SESNe are the
the peak luminosity ($L_{p}$) and the time it takes from the explosion to the
peak ($t_{p}$). In the following analyses of this paper, we require the
relations that connect these values to the 56Ni mass.

For a given 56Ni mass, the peak luminosity is estimated from the formula shown
in Stritzinger & Leibundgut (2005), which is based on the ‘Arnett-rule’. This
rule assumes that the peak luminosity ($L_{p}$) of a SN powered by the decay
of 56Ni is equal to the instantaneous energy deposition rate by radioactive
decay at that time:

$\displaystyle L_{p}$ $\displaystyle=$ $\displaystyle
10^{43}\times(M_{\mathrm{Ni}}/M_{\odot})$ (1) $\displaystyle\times$
$\displaystyle(6.45\times e^{-t_{p}/8.8}+1.45\times e^{-t_{p}/111.3})\
[\mathrm{ergs^{-1}}].$

The timescale, $t_{p}$, is not necessarily determined by a 56Ni mass. However,
in this paper, we take a phenomenological approach using the observational
data, and express $t_{p}$ as a function of the 56Ni mass. For that purpose, we
derive a fitting formula for $t_{p}$ as a function of 56Ni mass using the
well-observed sample of Meza-SESNe taken from Meza & Anderson (2020). Since
this sample is composed of nearby objects (the median distance of their SESNe
sample, excluding Ic-BL, is 46.7 Mpc), the objects are considered to be less
affected by a possible observational bias than the other SESN samples (and
note that the observational bias we will discuss later would not much affect
this relation). Furthermore, the objects in their sample are chosen under the
condition that they contain the data around the peak. Thus, $L_{p}$ and
$t_{p}$ in their sample are considered to be relatively accurate.

As shown in the left panel of Fig. 1, $t_{p}$ and log $M_{\mathrm{Ni}}$
broadly follow a linear correlation. Thus, we conducted a linear regression to
the data, using the least square method. We did not use SNe Ic-BL samples for
the fit, since they may indeed involve a different explosion mechanism from
canonical SNe II and SESNe, and also they are taken at relatively distant
locations. The derived formula becomes:

$\displaystyle t_{p}$ $\displaystyle=$ $\displaystyle(9.41\pm
2.98)\times\mathrm{log}_{10}(M_{\mathrm{Ni}}/M_{\odot})$ (2)
$\displaystyle+(29.74\pm 3.42)\,[\mathrm{day}].$

Using this equation, together with equation 1, we can estimate $t_{p}$ for a
given 56Ni mass. In the right panel of Fig. 1, the peak luminosity calculated
for a given 56Ni mass using equation 1 and 2 are compared to the data points
of Meza-SESNe. It is seen that the data points for the peak luminosity are
well reproduced by our fitting formula.

###  3.2 Observable distance for a given luminosity

Now that we know the peak luminosity of a SESN for a given 56Ni mass, it is
also important to know out to what distance we can detect it assuming a fixed
limiting magnitude. For this purpose, we use the relation in Hamuy (2003):

$\displaystyle\mathrm{log}D_{\mathrm{lim}}[\mathrm{cm}]=\frac{1}{5}\times(2.5\
\mathrm{log}L[\mathrm{ergs}^{-1}]$
$\displaystyle+V_{\mathrm{lim}}-A_{t}+BC+8.14).$ (3)

Here, $V_{\mathrm{lim}}$ is the limiting magnitude in the $V$ band,
$D_{\mathrm{lim}}$ is the limiting distance, $A_{t}$ is the total extinction
and $BC$ is the bolometric correction. For simplicity, we assume zero both for
$A_{t}$ and $BC$. Using this relation, we can calculate the observable
distance for a given luminosity, assuming a fixed limiting magnitude.

##  4 Investigating observational bias in the data sample

In this section, we investigate whether there is an observational bias in the
56Ni mass distribution of our data samples by examining the relation between
distance, luminosity and 56Ni mass.

###  4.1 56Ni mass and distance

In order to clarify how the observational biases may affect the 56Ni mass
distribution in our samples, we look at the 56Ni mass of our samples as a
function of the distance. Figure 2 shows the 56Ni masses of our samples
plotted as a function of the distance. It can be seen that there is a strong
trend that the 56Ni mass decreases as the distance decreases for SESNe. This
suggests that the objects with low 56Ni masses (i.e. dim objects) and large
distance, if they exist, may be missed. It is, however, important to emphasize
that we are still lacking the SESNe with low 56Ni mass (log
$M_{\mathrm{Ni}}(M_{\odot})\lesssim-1.7$: i.e., $M_{\mathrm{Ni}}\lesssim
0.02M_{\odot}$) even at small distance (log distance (Mpc) $\lesssim 1$). For
SNe II, even though the 56Ni mass slightly decreases as the distance
decreases, the effect is much less significant than SESNe.

Figure 4: The average 56Ni mass in the volume-limited sample plotted as a
function of the threshold distance from 20 Mpc to 5000 Mpc. The red solid line
refers to LS-SESNe sample, while the red dotted line refers to LS-SESNe
excluding SNe Ic-BL. The yellow solid line refers to LS-SNeII.

These trends can be confirmed by looking at Figs. 3 and 4. Figure 3 shows how
the 56Ni mass distribution changes when we take different sizes of volume-
limited samples. It is expected that the 56Ni mass distribution approaches to
the intrinsic distribution as we take the volume-limited sample at a closer
location. It is seen that the 56Ni mass distribution of SESNe significantly
shifts to the lower mass when we take the smaller volume-limited
sample555Note, however, that the distributions for the lowest 20% of the 56Ni
masses are nearly the same for these different sizes of volume-limited
samples. This may indicate that the lack of canonical SESNe with relatively
low 56Ni mass ($\lesssim 0.02M_{\odot}$) is real.. On the contrary, the 56Ni
mass distribution of SNe II does not change notably for the different sizes of
volume-limited sample. From this, we can infer that the LS-SESNe may not trace
the intrinsic 56Ni mass distribution, while LS-SNeII nearly do. Figure 4 shows
the average 56Ni mass in the volume-limited samples plotted as a function of
the threshold distance. This figure, again, shows that LS-SESNe suffer from a
significant observational bias and the discrepancy between SESNe and SNe II
becomes smaller as we take the smaller volume-limited sample, and finally
becomes within a factor of three.

Note, however, that if we remove SNe Ic-BL from LS-SESNe, then, the trend that
56Ni mass decreases with distance is much weakened. This may indicate that SNe
Ic-BL, whose distances are larger than the other types of SNe, are heavily
affected by an observational bias, while other types of SESNe (e.g. SNe IIb,
Ib and Ic) are less affected by it.

Figure 5: The comparison of the distance distributions of LS-SESNe (red) and
LS-SNeII (yellow). The blue solid histogram represents LS-SESNe excluding SNe
Ic-BL, while the dotted one is for SNe Ic-BL among LS-SESNe.

Figure 5 compares the distance distribution between LS-SESNe and LS-SNeII. We
can see that the distance distribution is closer for SNe II than SESNe. Meza &
Anderson (2020) showed that their 35 SESNe sample, excluding two Ic-GRB
objects, have the mean distance (46.7 Mpc) similar to that of their SNe II
sample (42.7 Mpc). However, our significantly larger sample of LS-SESNe has
the larger mean distance of 226.6 Mpc, while LS-SNeII has the mean distance of
41.5 Mpc. Even if we remove Ic-BL and Ic-GRB from the SESNe sample, the mean
distance is 99.8 Mpc, which is more than twice the value of LS-SNeII. This
indicates that the SESNe samples are collected at more distant locations than
SNe II, where the objects suffer from more significant observational bias,
supporting the results derived above 666Although the samples in Meza &
Anderson (2020) were taken at small distances ($\approx 40-50$ Mpc), the SESNe
with low 56Ni mass ($\lesssim 0.02M_{\odot}$) were still lacking. We will
further discuss this issue in section 7.2.

###  4.2 Luminosity distribution

Figure 6: Left: The distributions of the peak luminosity of SESNe (blue
points) and mid-plateau luminosity of SNe II (yellow points) plotted as a
function of distance. For reference, the limiting luminosity as a function of
distance is plotted for the case of limiting magnitude of
$V_{\mathrm{lim}}=17$ (blue line) and 19 mag (green line), respectively.
Right: The comparison of the luminosity function of SESNe (blue) and SNe II
(yellow) sample. For SESNe we show the peak luminosity, while for SNeII we
show the mid-plateau luminosity.

In this section, we investigate the luminosity distribution of our samples. We
emphasise that the analysis in this section is not affected by the assumption
about the relation between the 56Ni mass and the peak luminosity of SESNe. As
noted in section 2, here, we only use the sample of 57 SNeII taken from Hamuy
(2003); Müller et al. (2017); Pejcha & Prieto (2015), which we call SS-SNeII.
Note that Hamuy (2003) only published the V-band magnitude, so, we convert it
to the bolometric luminosity assuming the bolometric correction to be zero,
following Goldberg et al. (2019). For SESNe, we use the sample of 37 from
Meza-SESNe.

The left panel of Fig. 6 shows the luminosity distribution as a function of
distance for these samples. For SESNe we show the peak luminosity, while we
show the mid-plateau luminosity for SNe II. There is a positive correlation
between the luminosity and distance both for SESNe and SNe II. Also, the
minimum luminosity for a fixed distance is similar between SNe II and SESNe.
This indicates that the ‘luminosity’ distributions seen in both of our samples
(SESNe and SNe II) may be suffering from the same observational selection
effect. The right panel of Fig. 6 compares the luminosity functions of SESNe
and SNe II. We notice that there is a luminosity cut off for both types at
around log $L$[erg s-1] $\sim 41.7$. The SNe II plateau phase and the SESNe
peak phase are powered by the different physical mechanisms, with the former
powered by the explosion energy and the latter powered by the radioactive
decay of 56Ni. It is true the plateau luminosity and the 56Ni mass of SNe II
are known to be positively correlated (Pejcha & Prieto, 2015), but it is
unlikely that the lower luminosity cut off are the same between the two groups
of SNe just in terms of physics. Thus, we speculate that this simultaneous cut
off of the luminosity functions for both types of SNe is caused by an
observational selection effect. This selection effect will introduce a bias in
the 56Ni mass distribution for SESNe, as the 56Ni mass is closely connected to
their peak luminosities.

In summary, the results derived in section 4 all point to the following
interpretation: The 56Ni masses of SESNe samples collected from the published
literature suffer from notable observational bias, i.e. the distant objects
with relatively low 56Ni mass are missed, meaning that the samples are biased
towards the luminous objects. On the contrary, the 56Ni masses of SNe II
samples suffer much less from such bias. We, again, emphasise that the
analyses in this section are not affected by the assumption about the relation
between the 56Ni mass and the peak luminosity of SESNe777We note, however,
that Meza & Anderson (2020) have shown that the statistical difference of 56Ni
mass between SESNe and SNe II remains even if they take relatively close
samples ($\approx 40-50$ Mpc), which is also confirmed by our analyses (Figs.
3 & 4). Thus, the observational bias alone may not be sufficient to explain
all of the statistical difference in the 56Ni mass (section 7)..

##  5 Method of Mock observations

In the following sections, we conduct mock observations of SESNe and
investigate the effect of observational bias on the 56Ni mass distribution of
detected SESNe. In the previous sections, we have found that the 56Ni mass
distribution in our SNe II sample is not notably suffering from the
observational bias. Therefore, below, we start additional analysis based on
the following two working hypotheses: (1) The 56Ni mass distribution in our
SNe II sample (LS-SNeII) represents the intrinsic 56Ni mass distribution of
SNe II, and (2) SESNe have the same intrinsic 56Ni mass distribution as that
of SNe II. The second assumption is based on the hypothesis that assuming a
binary origin for SESNe, progenitors of SESNe and SNe II are expected to share
the similar range in the initial mass (see section 1)888Note, however, that
there are indications that the progenitors of SESNe may be more massive than
SNe II either as an entire class or for the particular SN Ic class (e.g.
Anderson et al., 2012; Valenti et al., 2012; Fang et al., 2019).. Based on
these hypotheses, we conduct mock observations of SESNe. In the rest of this
section, we describe the procedure of the mock observation in more detail.

###  5.1 56Ni mass distribution

As noted above, here we assume that the intrinsic 56Ni mass distribution of
SESNe is the same as the 56Ni mass distribution of LS-SNe II. To simplify the
numerical analyses, we fit the cumulative histogram of 56Ni mass (denoted here
as $f(x)$) by the function of $f(x)=$tanh$(a_{0}\times x)$, using non-linear
least squares. We obtained $a_{0}=14.60$ as the best fit parameter. The
comparison of our fitted curve with our sample is shown in Fig. 7. Below, we
use the function of $f(x)$= tanh$(14.60\times x)$ to represent the assumed
intrinsic 56Ni mass distribution of SESNe.

Figure 7: The cumulative 56Ni mass distribution of LS-SNeII and the non-linear
least squares fit to it, assuming the function of $f(x)=$tanh$(a_{0}\times
x)$.

###  5.2 Simulating the observations

We simulate one SESN by selecting the 56Ni mass and distance from the given
probability distributions. We select a value of 56Ni mass from the
distribution derived in section 5.1. Then, for each 56Ni mass thus derived,
the distance is randomly chosen following the probability function of
$p\propto$ (distance)3, i.e., the volume size. The range of distance is set
from zero up to the limiting distance corresponding to the peak luminosity of
SESNe with a 56Ni mass of 1.0$M_{\odot}$, unless otherwise noted.

For each object with a given 56Ni mass and distance, we decide whether to add
it to the ‘detected’ sample or not based on the following procedure. First,
from the given 56Ni mass, we randomly pick up a value of $t_{p}$ based on the
distribution, taking into account the dispersion, derived from the fit in
section 3. Combining this value of $t_{p}$ with the chosen value of 56Ni mass,
we can estimate a peak luminosity of SESNe (see, equation 1). Next, we
estimate the limiting distance using equation 3, for the peak luminosity
derived above. If the selected distance is within the observable distance
corresponding to its peak luminosity, we consider it to be detected and add it
to the detected sample. Otherwise, we consider that the object escapes
detection and do not add it to the detected sample. Once the number of
detections reaches a specified number, we stop one iteration of the mock
observation. Below, the number of detections is set to be 100, unless
otherwise noted. The number of 100 is chosen to be consistent with the order
of magnitude of our LS-SESNe sample size. We iterate the procedure described
above $10^{3}$ times, in order to clarify the possible range of the
distributions by taking into account the statistical fluctuation due to the
limited sample size.

##  6 Results of mock observation

###  6.1 Luminosity function

Figure 8: The luminosity functions derived from the mock observation are
compared to the luminosity function of Meza-SESNe. The green histogram is the
luminosity function of the detected sample in the mock observation, assuming
the limiting magnitude of 19 mag. Among the green region, the solid line and
shaded region represent the mean distribution and standard deviation,
respectively, computed for the $10^{3}$ iterations. For reference, the yellow-
solid line represents the input intrinsic luminosity function of SESNe in the
mock observations (assuming that this is the same as that of SNeII).

Here, we show the results of the mock observation described in section 5. We
assume a fixed limiting magnitude of 19 mag in this section. Figure 8 compares
the luminosity functions derived from the mock observations to the luminosity
function of Meza-SESNe. Here, in order to make the direct comparison to Meza-
SESNe, we stop one iteration of mock observation when the number of detected
objects reaches 37 (i.e., the Meza-SESNe sample size), not 100. Then, we
repeat this $10^{3}$ times to clarify the possible range of the distributions.

Interestingly, we can see that the luminosity function in the ‘detected’
samples is shifted to high luminosity compared to the model intrinsic
luminosity function. As seen from Fig. 1, the objects with higher 56Ni mass,
in general, have higher peak luminosity. Thus, they have the larger observable
volume and dominates the detected sample. It is also worthwhile to note that
the luminosity function of our detected samples in the mock observation
roughly explains the observed luminosity function of Meza-SESNe. Especially,
the lower cut-off at around log $L\sim 41.7$ is naturally obtained.

###  6.2 56Ni mass distribution

Figure 9: Left: The 56Ni mass distribution in the detected sample of our mock observation (green) is compared to the different samples, excluding SNe Ic-BL. The green-solid line represents the mean distribution of the mock observation with the $10^{3}$ iterations (and 100 detected objects in one interaction), assuming the limiting magnitude of 19 mag. The shaded region represents the standard deviation obtained with $10^{3}$ iterations. Blue- and cyan-dotted lines denote the ‘Arnett mass’ and ‘Tail mass’ in Meza-SESNe, while a red-dotted line denotes the LS-SESNe. Right: the 56Ni mass distribution, excluding SNe Ic-BL (dotted) and including SNe Ic-BL (solid) are compared for the different samples. SN distributions ($N$: Number of samples) | $D$ | $p$
---|---|---
LS-SNe II (115), LS-SESNe (187) | 0.690 | 3.4 $\times 10^{-34}$
LS-SESNe (187), Meza-SESNe (37) | 0.453 | 2.6 $\times 10^{-6}$
LS-SNe II (115), Mock | 0.472 | 1.5 $\times 10^{-9}$
LS-SENe (187), Mock | 0.433 | 5.6 $\times 10^{-13}$
LS-SENe (w/o Ic-BL; 131), Mock | 0.317 | 5.9 $\times 10^{-5}$
Meza (Arnett, 37), Mock | 0.182 | 0.60
Meza (Arnett, w/o Ic-BL; 33), Mock | 0.220 | 0.46
Meza (Tail; 20), Mock | 0.462 | 6.7 $\times 10^{-2}$
Meza (Tail, w/o Ic-BL; 18), Mock | 0.523 | 3.9 $\times 10^{-2}$
Table 1: The KS statistical differences between the different 56Ni mass
distributions. In the first column, the two distributions being compared are
listed together with the number of samples in brackets. In the second column,
the D parameter is given, while in the last column the p value is presented.
‘Mock’ means the result of the mock observation assuming a limiting magnitude
of $V_{\mathrm{lim}}=$ 19 mag. Here, the number of the simulated objects in
one iteration is set to be the same as the size of the sample with which the
mock observation is compared. The $D$ and $p$ for ‘Mock’ are the mean values
computed with the $10^{3}$ iterations.

Figure 9 compares the cumulative 56Ni mass distribution in the detected
samples to different data samples. Note that, below, we stop one iteration of
mock observation when the number of detect objects reaches 100, unless
otherwise noted. The 56Ni mass distribution in the detected sample of our mock
observation is skewed to higher mass compared to the assumed intrinsic
distribution. This is due to the observational bias, being consistent with the
shift of the luminosity function discussed in the previous section.

To make quantitative comparisons of the different 56Ni mass distributions, we
conducted K-S statistical test as shown in Table 1. We used the library,
scipy.stats.kstest, for conducting a K-S statistical test. When we compared
the result of mock observation to a data sample, we set the number of
detections for one iteration to be the same as the number of samples being
compared to. Then, we took the mean of the $D$ parameter and the $p$ value for
$10^{3}$ iterations. The $p$ value between the result of our mock observation
and Meza-SESNe (Arnet) is quite high, being 0.60. This indicates that the 56Ni
mass distribution in Meza-SESNe may be explained by taking into account for an
observational bias on the intrinsic distribution similar to that of LS-SNe II.
Also, this result is consistent with the reasonable match of the 56Ni mass
distribution and luminosity function of our mock observation to Meza-SESNe
(Figs. 8 and 10).

However, the assumption that the 56Ni mass distributions from the mock
observations and LS-SESNe originate from the same distribution is rejected,
with a quite low $p$ value of $5.6\times 10^{-13}$. Even if we exclude SNe Ic-
BL from the sample, the $p$ value is still low, being $5.9\times 10^{-5}$. One
of the possible reasons for this is that our assumption for the intrinsic 56Ni
distribution (section 5.1) may have been too simplistic. Indeed, from the
intrinsic 56Ni mass distribution we assumed, a high value of 56Ni mass
($\gtrsim 0.2M_{\odot}$) is rarely produced, while such high values are found
in LS-SESNe (Fig. 9). Another possible reason is that, while our mock
observation is based on the ‘Arnett-rule’ (section 3.1), which is the same
method used in Meza-SESNe (Arnett), the LS-SESNe consists of the 56Ni masses
estimated by various kinds of methods. Thus, direct comparison of our mock
observation to LS-SESNe may not be appropriate.

In short, the 56Ni mass distribution taking into account an observational bias
is consistent with the well-observed sample of Meza-SESNe. However, further
explanation is needed for the discrepancy between our mock observation and a
larger sample of LS-SESNe. Indeed, the difference between the two samples,
Meza-SESNe and LS-SESNe, is intriguing. It suggests that the 56Ni mass
distribution of SESNe is dependent on how the sample is constructed; the
different samples may thus be contaminated by different degrees of possible
observational biases.

###  6.3 Effect of different limiting magnitudes

Figure 10: Left: The cumulative 56Ni mass distribution of the detected samples
in the mock observation for the different limiting magnitudes. Blue, green,
orange lines refer to the limiting magnitude of 17 mag, 19 mag, and 21 mag,
respectively. Each line is the mean of the distributions derived with $10^{3}$
iterations. For reference, the fit to the 56Ni mass distribution of LS-SNe II
is shown with a gray line. Right: The same figure for the result of the mock
observation, assuming the outer boundary as $80$ Mpc (i.e. volume limited
samples).

In the previous section, we have assumed a limiting magnitude of
$V_{\mathrm{lim}}=$ 19 mag. Next, we will see how the different values of the
limiting magnitudes affect our results. In the left panel of Fig. 10, we show
the 56Ni mass distribution of the detected samples in the mock observation for
the different limiting magnitudes. We can see that the 56Ni mass distribution
is quite insensitive to the different values of the limiting magnitudes. This
can be explained as follows. The 56Ni mass distribution in the observed sample
can be derived by multiplying the assumed intrinsic distribution of 56Ni mass
by the observable volume, i.e., $D_{\mathrm{lim}}(M_{\mathrm{Ni}})^{3}$. Here,
$D_{\mathrm{lim}}(M_{\mathrm{Ni}})$ is the limiting distance calculated using
equation 3. It is seen that the term of $V_{\mathrm{lim}}$ only changes the
scale of the 56Ni mass distribution, but does not affect the normalized
distribution. Note that this apparently counterintuitive result is obtained,
because we consider a magnitude-limited sample here. For comparison, we show
the result of mock observations, assuming the outer boundary of all the events
is 80Mpc, in the right panel of Fig. 10. This constructs the volume-limited
samples. In this case, the cumulative distribution of 56Ni mass approaches to
the intrinsic one as the limiting magnitude is set larger, which is consistent
with our intuition.

Figure 11: The distance distribution of the detected samples in the mock
observation for the different limiting magnitudes. Blue, green, orange lines
refer to the limiting magnitude of 17 mag, 19 mag, and 21 mag, respectively.
Each line is the mean of the distributions derived with $10^{3}$ iterations.

In Fig. 11, we show the distance distribution of the detected samples in the
mock observation for the different limiting magnitudes. We can see that the
distance distribution is sensitive to the different value of the limiting
magnitude. The higher the limiting magnitude is, the observable volume becomes
larger. Thus, the more distant objects dominates the observed sample.

---

Figure 12: Left: The 56Ni mass and distance of the detected samples in one
iteration (i.e., 100 detections) for the limiting magnitudes of 19 mag. Gray
points are all the events that were randomly picked up until the number of
detections reached 100. For reference, the limiting distance for a given 56Ni
mass estimated as in section 3 are also shown assuming the limiting magnitudes
of 19 (green) and 25 (red) mag. The latter represents the limiting magnitude
for the single-visit depth in LSST (Ivezić et al., 2019). Right: The 56Ni mass
and distance of the detected samples in one iteration for the different
limiting magnitudes. Blue, green, orange points refer to the case of limiting
magnitude of 17 mag, 19 mag, and 21 mag, respectively. We add the ultra-
stripped envelope SNe (USSNe) candidates with magenta-star symbols (see
section 7 for discussion of these events). The references for USSNe are listed
at the end of the manuscript.

The limiting distance for a given 56Ni mass estimated as in section 3 are also
shown assuming the different limiting magnitudes.

Figure 13: Left: The mean cumulative 56Ni mass distributions for the different
distance cuts are compared. For making this figure, we assumed a limiting
magnitude of $V_{\mathrm{lim}}=19$ mag, and set the number of detected samples
to be $10^{3}$. This panel is similar to Fig. 3. Right: The average 56Ni mass
for the detected samples inside a given distance are plotted as a function of
distance cuts. This panel is similar to Fig. 4 for our mock observation.

In the right panel of Fig. 12, the 56Ni masses and distances of the detected
samples for the different limiting magnitudes are over-plotted onto Fig.2. As
discussed above, the objects with low 56Ni mass are lacking compared to the
assumed intrinsic distribution, which is consistent with the data samples.
However, our predictions from the mock observations fail to explain the high
56Ni masses of $\gtrsim 0.2M_{\odot}$ (mostly SNe Ic-BL) that exist in the
data samples collected from the published literature, as already noted in
section 6.2. We will discuss this issue in section 7.3.

In order to compare our results to Fig. 4, i.e., the average 56Ni mass for the
volume-limited samples of different sizes, we conduct an additional analysis
as follows. We set the number of detections to be $10^{3}$ (not 100) and
repeat the mock observation conducted above. For this larger sample, we
investigate how the cumulative 56Ni mass distribution changes for the
different values of distance cuts. In Fig. 13, we show the results of such an
analysis. The right panel of Fig. 13 shows that the average 56Ni mass
decreases as the distance cut is decreased, just like Fig. 4. This behaviour
can be understood as follows. As seen in Figs. 8 and 10, the average peak
luminosity of the detected samples in a magnitude-limiting sample is $\sim$ a
few $10^{42}$ erg s-1 (i.e., $M_{\mathrm{Ni}}\sim 0.1M_{\odot}$), irrespective
of the values of limiting magnitude. However, those dominant objects are
detected at different distances depending on the limiting magnitudes.
Actually, Fig. 11 shows that such dominant objects are found at $\sim 30-40$
Mpc for $V_{\mathrm{lim}}=17$ mag, while they are found at $\sim 100$ Mpc for
$V_{\mathrm{lim}}=19$ mag. Thus, if we consider a sufficiently large volume-
limited sample, then, the average 56Ni mass becomes $\sim 0.1M_{\odot}$.
However, if we consider a distance cut smaller than a value (e.g., $\sim 100$
Mpc for $V_{\mathrm{lim}}=19$ mag), then, part of the dominant objects in a
magnitude-limited sample would be missed and the average 56Ni mass starts to
decrease.

Here, it is crucial to mention that the right panel of Fig. 13 does not
perfectly match to Fig. 4. It is true that the decrease of average 56Ni mass
for SESNe at $\sim 100$ Mpc is seen in Fig. 4, just like Fig. 13. However,
Fig. 13 implies that the average 56Ni mass should decrease down to $\sim
0.05M_{\odot}$ at sufficiently small distance ($\lesssim 100$ Mpc), which is
not the case in Fig. 4. These results imply either that an observational bias
alone may not be sufficient to explain the different 56Ni mass between SESNe
and SNe II or that the objects with low 56Ni mass may escape detection due to
other reasons (see section 7.2). Still, our result here suggests that an
observational bias is likely to be present at least for high 56Ni masses, and
one needs to consider this when discussing the 56Ni mass distribution of
SESNe.

###  6.4 Effects of different observational cadences

So far, we implicitly assumed an infinitely small observational cadence in the
mock observation. This means that an object is always detected as long as its
peak luminosity exceeds the observational limiting magnitudes. However,
existing surveys have a wide range of observational cadence from hours to a
few tens of days, depending on their scientific aims. Therefore, some objects
may be missed due to infrequent observations, even if the peak luminosity
exceeds the observational limiting magnitudes. Therefore, here we attempt to
take this into account. Following this, we investigate how the different
observational cadences affect our results. We fix the limiting magnitude as
19.0 mag in this section for simplicity.

To proceed with this investigation, we take a simplified approach. We assume
that the peak luminosity is maintained for the duration of $t_{p}$ calculated
using equation 2. In the mock observation, we add a procedure as follows, in
order to decide whether an object is detected or not: if the duration of the
event is less than the observational cadence, we add it to the observed sample
with the probability of $p=t_{p}/t_{\mathrm{cadence}}$. If the duration of the
event is longer than the observational cadence, we consider the object is
detected and add it to the observed sample.

Figure 14 compares the 56Ni mass distribution of the detected samples for the
different observational cadences. It is seen that the cumulative distribution
shifts to higher mass, as the observational cadence is set larger. This
happens because an object with relatively low 56Ni mass tends to escape the
detection for a large observational cadence, due to its short timescale.
However, the difference seen in the 56Ni mass distributions is almost
negligible. Thus, we conclude that our results are robust to the different
assumptions about the observational cadences.

Note, however, that our assumption that a peak luminosity maintains for
$t_{p}$ is quite simplistic. Thus, although our discussion clarifies the
qualitative effect of observational cadence, we do not consider that it has a
quantitative predictive power. Also, this investigation is based on the linear
fitting equations 2. This fitting is done using the samples with 56Ni masses
above $0.03M_{\odot}$, and the validity of the linear extrapolation to the
lower 56Ni masses is not trivial (see section 7).

##  7 Discussion

In this section, we firstly discuss some caveats about our analyses conducted
above. Next, we discuss the lack of low 56Ni mass objects ($\lesssim
0.02M_{\odot}$) in the literature. Then, we discuss the high 56Ni mass
($\gtrsim 0.2M_{\odot}$) objects, which are not explained by our model.

###  7.1 Caveats

In section 4, we found that the 56Ni masses of SESNe samples collected from
the published literature suffer from notable observational bias, while those
of SNe II samples suffer from much less bias. This may be because: (1) SNe II
samples are collected at closer distances compared to SESNe samples (Fig. 5),
meaning that the former suffer less bias; or (2) the luminosity of SESNe have
higher dependence on the 56Ni mass than SNe II. Indeed, the peak luminosity of
SESNe is theoretically expected to follow $L_{\mathrm{peak}}\propto
M_{\mathrm{Ni}}$ (Arnett, 1982), while the mid-plateau luminosity of SNe II is
phenomenologically known to follow $L_{\mathrm{plateau}}\propto
M_{\mathrm{Ni}}^{0.65}$ (Pejcha & Prieto, 2015). Moreover, at the early phase,
SNe II generally have higher luminosity than the mid-plateau phase. Thus, the
detectability of SNe II is affected much less by the 56Ni mass than SESNe.

We have assumed that SESNe share the same 56Ni mass distribution as SNe II.
However, as noted in section 1, there are several indications that at least a
fraction of SESNe progenitors may be more massive than those of SNe II (e.g.
Anderson et al., 2012; Maund, 2018; Fang et al., 2019). This allows a
possibility that SESNe indeed have higher 56Ni mass than SNe II in general. If
this is the case, then the 56Ni mass distribution from the mock observation
would shift to even higher mass with the observational bias effect found in
this paper. It may then match with the LS-SESN samples better.

The results derived in section 6 are based on the fitting relation for $t_{p}$
as a function of 56Ni mass (equation 2). These 56Ni masses have been derived
using ‘Arnett-rule’, which has been claimed to overestimate the value by a
factor of few (Dessart et al., 2016; Khatami & Kasen, 2019). In appendix B, we
have shown that even if we use the 56Ni mass derived from tail luminosity for
deriving the $t_{p}-M_{\mathrm{Ni}}$ relation, our main results do not change
notably. Thus, our results in section 6 is not affected by the methods for
deriving the 56Ni mass.

For the mock observation, we assumed zero both for $A_{t}$ and $BC$. In
reality, non-zero values of $A_{t}$ and $BC$ would shift the observable
distance for a given limiting magnitude. While the effect of the $BC$ is
probably not large given that their colors are similar around the peak999The
colors of SNe II in the plateau (e.g. de Jaeger et al., 2019) are similar to
those of SESNe around the peak (e.g. Drout et al., 2011), and BC is nearly the
same for a given color between SNe II and SESNe (Lyman et al., 2016)., the
extinction may be systematically different between SNe II and SESNe; the
latter are typically associated with a more active star-forming region
(Anderson et al., 2012). If we would assume a larger value for $A_{t}$ for
SESNe, the limiting distance for SESNe will be decreased and the effect of the
observational bias investigated in this paper will become even more
substantial.

Throughout the paper, we have contrasted the SNe II to SESNe in general.
Anderson (2019) has suggested that there may be difference in the 56Ni mass
distribution even among the different types of SESNe. Especially, SNe IIb seem
to have smaller 56Ni masses than SNe Ib/Ic. One possible observational bias
that might explain this behavior is that SNe IIb can be detected more easily
than SNe Ib/Ic due to their cooling emission. However, quantitatively
investigating this possibility is beyond the scope of this paper.

###  7.2 low 56Ni mass objects

We have conducted mock observations and shown that if we assume that the
intrinsic 56Ni mass distribution of SESNe is the same as that of LS-SNe II,
the 56Ni mass distribution of SESNe in the detected samples becomes more
massive compared to the assumed intrinsic distribution; the resulting
distribution is found to be very close to the distribution of Meza-SESNe. This
indicates that even if a significant number of SESNe with low 56Ni masses
(i.e. similar to those found in the SNe II samples) would exist, we would find
a difficulty in detecting them and thus they would be significantly
underrepresented in the current literature samples.

However, some problems still remain to be solved. It is true that our mock
observations predict that the detection of SESNe is dominated by relatively
luminous objects. This would predict that there should be at least a few SESNe
with a low 56Ni mass, $M_{\mathrm{Ni}}\lesssim 0.02M_{\odot}$, especially at
small distances, considering that many SNe II with such low 56Ni masses have
been detected and that the observed fraction of SESNe to that of SNe II is
0.52 (Li et al., 2011). However, in our samples, very few SESNe have been
found with such a low 56Ni mass. Of course, it may indicate that SESNe with
such low 56Ni masses actually would not exist and the statistical difference
of 56Ni mass between SESNe and SNe II is real. However, it is also possible
that the SESNe with low 56Ni masses would not appear as canonical SESNe but
instead appear as peculiar objects, and therefore they may not be labeled as
SESNe (and thus missing in the present ‘SESN’ samples).

First, such low-56Ni mass SESNe may be related to the so called rapidly
evolving transients. As shown in Fig. 15, there is a hint that the 56Ni mass
and the timescale of SESNe are positively correlated. Thus, SESNe with lower
56Ni mass are expected to have shorter timescales. Also shown in Fig. 15 are
USSNe candidates. Taking these objects into account, the timescale of SESNe
may decrease more rapidly than our prediction (Fig. 15). Thus, our linear fit
(section 3.1) may not be valid at small 56Ni masses, and it is possible that
SESNe with a low 56Ni mass ($\lesssim 0.02M_{\odot}$) are observed as rapidly
evolving transients with timescales shorter than 10 days101010Note, that some
of the rapidly evolving transient are known to be difficult to explain by only
considering the radioactive decay model (e.g. Drout et al., 2014). However,
the properties of the rapidly evolving the transients are diverse (Pursiainen
et al., 2018) and there are many that are compatible with the radioactive
decay scenario. Indeed, a recent compilation of the rapid transients found by
the ZTF shows that this population is indeed largely contaminated by rapidly
evolving SESNe (Ho et al., 2021). Actually, SN 2017czd in our sample is an SN
IIb with very small 56Ni mass of $0.003M_{\odot}$. This object was classified
as a rapidly evolving transient (Nakaoka et al., 2019). Drout et al. (2014)
have estimated that the rate of rapidly evolving transients is 4-7 % of the
core collapse SNe rate. Since the fraction of SESNe in the core collapse SNe
is 36.6 % (Smith et al., 2011), the rapidly evolving transients occupy 11-19 %
of SESNe. This number is comparable to the fraction of SESNe with
$M_{\mathrm{Ni}}\lesssim 0.01M_{\odot}$ assuming the same 56Ni mass
distribution as LS-SNeII. Since the events with short timescales ($\lesssim
10$ days) can be easily missed, this hypothesis may be consistent with the
lack of SESNe with low 56Ni masses ($\lesssim 0.02M_{\odot}$)111111Note, that
most of the rapidly evolving transient discovered so far have 56Ni mass of
$\gtrsim 0.03M_{\odot}$ (Drout et al., 2014; Pursiainen et al., 2018; Tampo et
al., 2020). However, considering that the number of samples detected so far is
limited ($\approx 100$) (Pursiainen et al., 2018), it is natural that they are
dominated by the relatively luminous objects as we have shown in section 6.

The SESNe with low 56Ni mass may also originate from the so-called ultra-
stripped envelope SNe (USSNe). Actually, the ejecta mass and 56Ni mass of
SESNe are known to be positively correlated (Lyman et al., 2016). Thus, the
ejecta mass of the SESNe with low 56Ni mass are expected to be small. In Fig.
12, the USSNe candidates are also shown. They have 56Ni masses lower than most
of our SESNe sample. Theoretical calculations also indicate that USSNe should
synthesize quite low 56Ni of $\sim 0.01M_{\odot}$ (Suwa et al., 2015; Moriya
et al., 2017). Specifically, SN2019dge, an USSNe candidate, has an estimated
56Ni mass of 0.017$M_{\odot}$ (Yao et al., 2020), which is quite low. The rate
of such events is estimated as 2-12% of core collapse supernova: i.e.,
5.6-33.3% of SESNe (Smith et al., 2011). This number is consistent with the
fraction of SESNe with $M_{\mathrm{Ni}}\lesssim 0.02M_{\odot}$ under the
distribution we assumed. Furthermore, the timescale of USSNe candidates are
known to be short ($\lesssim 10$ days), which is much less than our prediction
(Fig. 15). Such short-timescale objects may be systematically non-detected in
the existing surveys as noted in the previous paragraph. Note, however, that
the 56Ni masses of USSNe candidates discovered so far are in general not too
low: i.e., many SNe II have been detected with 56Ni mass lower than these
objects. Therefore, these objects alone may not be sufficient to explain the
deficit of SESNe with low 56Ni mass.

The SESNe with low 56Ni masses may also have the possible link to SNe Ibn,
which are not included in our samples. SNe Ibn are characterized by He
emission lines that are considered to originate from the interaction with the
He-rich circumstellar material. These objects are shown to eject less 56Ni
than the bulk of other SESNe (Moriya & Maeda, 2016). Further, Ho et al. (2021)
recently showed that SNe Ibn with short time scale do contaminate the ZTF
rapid transient sample substantially, together with the rapid (non-
interacting) SESNe.

When deep surveys like LSST are deployed in the future, we can test our
hypotheses. In the left panel of Fig. 12, we show the detection limit for a
limiting magnitude of 25 mag, representing the single-visit depth in LSST
(Ivezić et al., 2019). We can see that basically all the SESNe with low 56Ni
masses ($\lesssim 0.02M_{\odot}$) are detected if they occur closer than
$\approx 100$ Mpc121212Although there are many objects with
$M_{\mathrm{Ni}}\lesssim 10^{-3}M_{\odot}$ in the left panel of Fig.12, they
are considered to be an artifact caused by an analytical fitting to the
distribution, considering that there are no such objects in LS-SNe II (section
7). Thus, we will be able to construct a complete sample of SESNe in the local
universe. With such a sample, we can test whether the lack of SESNe with
$\lesssim 0.02M_{\odot}$ is real or not.

Figure 14: The 56Ni mass distribution of the detected samples for the
different observational cadences. Green, blue, yellow, and red points refer to
the cadence of 0, 10, 20, 30 days, respectively. Each line is the mean of the
distributions derived from $10^{3}$ iterations. For reference, the fit to the
56Ni mass distribution of LS-SNe II is shown with a gray line.

###  7.3 High 56Ni mass objects

As previously noted (Fig. 9 and 12), our predictions from the mock
observations fail to explain the objects with high 56Ni masses ($\gtrsim
0.2M_{\odot}$). One of the possible reasons for this is that such objects may
indeed represent a different population from other SESNe which does not have a
counterpart in SNe II. Indeed, most of those objects are SNe Ic-BL, for which
the natures of the progenitor and the explosion have been proposed to be
different from canonical SESNe. Thus, our analyses may not be applicable to
these objects 131313Among those objects with high 56Ni masses ($\gtrsim
0.2M_{\odot}$), there are several objects that are not SNe Ic-BL (Fig. 12).
The possible reason why they are not classified as SNe Ic-BL is that they are
highly off-axis, considering that SNe Ic-BL often have a jet-like structure
(Valenti et al., 2008). . Another possible reason is that our assumption about
the intrinsic 56Ni mass distribution (section 5.1) may be too simplistic.
Indeed, the intrinsic 56Ni mass distribution we assumed rarely produce such a
high value of 56Ni mass ($\gtrsim 0.2M_{\odot}$). Yet another possibility is
that the different amounts of the hydrogen-rich envelope may indeed affect the
56Ni production, even though the core structure would be similar between SNe
II and SESNe; SNe II have the thick hydrogen envelope outside the He core, and
the shock is decelerated while it is propagating through the envelope. Thus,
it is expected that SNe II suffer from a fall back of the inner material,
including 56Ni, more substantially than SESNe. In this case, the 56Ni mass
distribution of SNe II we have used may provide a lower limit for SESNe
(Sawada et al. in prep).

Figure 15: The estimated time to peak from the explosion ($t_{p}$)

of SESNe using equation 2 plotted as a function of 56Ni mass. For reference,
the observational data taken from Lyman et al. (2016) and Prentice et al.
(2019) are also shown with blue points. Among them, open circles are the SNe
Ic-BL, while the filled circles are the other types of SESNe. Also, the data
of USSNe candidates are also shown with magenta-star symbols, which are taken
from the references listed at the end of the manuscript.

##  8 Conclusions

The nuclear decay of 56Ni is one of the most important power sources of
supernovae (SNe). Recent works have indicated that the 56Ni masses estimated
for SESNe are systematically higher than those estimated for SNe II. Although
this may indicate a distinct progenitor structure or explosion mechanism
between these types of SNe, the possibility remains that this may be caused by
observational biases.

By investigating the distributions of 56Ni mass and distance for the data
samples collected from the literature, we have found that SESNe samples suffer
from significant observational bias; objects with low 56Ni masses may be
systematically missed, especially at larger distances. Thus, this work has
elucidated that the observational bias must be taken into account in
discussing the different 56Ni masses between SNe II and SESNe.

We also conducted mock observations assuming that the intrinsic 56Ni mass
distribution of SESNe is the same as the 56Ni mass distribution of SNe II
collected from the literature. We have found that the 56Ni distribution for
the detected samples of SESNe becomes more massive compared to the assumed
intrinsic distribution due to the observational bias. This result may, at
least partially, explain the lack of low 56Ni mass objects in the SESNe data
sample collected from the literature. Although this result relies on the
assumption noted above, this supports that at least a part of the
systematically different 56Ni masses between these types of SNe are due to the
observational bias.

We emphasize, however, that the SESNe with low 56Ni mass ($\lesssim
0.02M_{\odot}$) are still lacking even at small distances ($\lesssim$ 30 Mpc).
This may indicate that the observational bias alone may not be sufficient to
explain all of the statistical difference between SESNe and SNe II. Another
possibility is that the SESNe with low 56Ni mass appear as either rapidly
evolving transients or ultra-stripped SNe, which are difficult to detect due
to their short timescales.

##  9 acknowledgement

R.O. acknowledges support provided by Japan Society for the Promotion of
Science (JSPS) through KAKENHI grant (19J14158). K.M. acknowledges support
provided by Japan Society for the Promotion of Science (JSPS) through KAKENHI
grant (18H05223, 20H00174, and 20H04737). This work is partly supported by the
JSPS Open Partnership Bilateral Joint Research Project between Japan and
Chile.

##  Appendix A Newly added reference list for 56Ni masses

Below, the newly added references to the reference list in Anderson (2019) are
listed.

SNe II: Utrobin & Chugai (2011), Bose et al. (2018), Lisakov (2018), Singh et
al. (2018, 2019a, 2019b), Afsariardchi et al. (2019), Andrews et al. (2019),
Buta & Keel (2019), Dastidar et al. (2019), Hillier & Dessart (2019), Martinez
& Bersten (2019); Martinez et al. (2020), Meza et al. (2019), Ricks &
Dwarkadas (2019), Szalai et al. (2019), Van Dyk et al. (2019), Bostroem et al.
(2019, 2020), Gutiérrez et al. (2020a, b), Jäger et al. (2020), Sharon &
Kushnir (2020), Müller-Bravo et al. (2020), Rodríguez et al. (2020), Zhang et
al. (2020).

SESNe: Margutti et al. (2017), Ashall et al. (2019), Fremling et al. (2019),
Ho et al. (2019, 2020a, 2020b), Nakaoka et al. (2019), Taddia et al. (2018,
2019a, 2019b), Prentice et al. (2018, 2019, 2020b), Singh et al. (2019), Wang
et al. (2019), Xiang et al. (2019), Moriya et al. (2020), Sharon & Kushnir
(2020), Stritzinger et al. (2020).

Ultra-stripped envelope SESNe candidates: Kasliwal et al. (2012) De et al.
(2018), De et al. (2018), Yao et al. (2020), Prentice et al. (2020a).

##  Appendix B The case when using tail luminosity for deriving 56Ni masses

The results derived in section 6 are based on the fitting relation for $t_{p}$
as a function of 56Ni mass (equation 2). The 56Ni mass used for the fit has
been derived using the ‘Arnett-rule’ (below, the ‘Arnett mass’), which has
been claimed to have an uncertainty of a factor of a few (Dessart et al.,
2015, 2016; Khatami & Kasen, 2019). An alternative method to derive a 56Ni
mass is to use a tail luminosity (below, the ‘Tail mass’). This gives a lower
limit to the 56Ni mass of SESNe. In this Appendix, we investigate how these
different methods of deriving 56Ni mass affect the results in section 6. For
that purpose, we repeat the mock observations in section 5 and 6 using the
‘Tail mass’ when fitting $t_{p}$. In Meza & Anderson (2020), both ‘Tail mass’
and ‘Arnett mass’ have been measured for the same sample. Thus, we adopt the
‘Tail mass’ from this literature.

---
Figure 16: The same figure as Fig. 1, except that we use the ‘Tail mass’ for
the fit and plots.

Fig. 16 is the same figure as Fig. 1, except that we use the ‘Tail mass’ for
the fit and plots. That is, we first use the least square method and linearly
fit to $t_{p}$ as a function of log ‘Tail mass’, as shown in the left panel of
Fig. 16. Then, from this relation thus derived, we use equation 1 and
calculate the peak luminosity as a function of 56Ni mass, propagating the
errors. Since the ‘Tail mass’ gives the lower limit to the actual value, the
peak luminosity calculated from it should be lower than the observed value.
This is indeed the case as shown in the right panel of Fig. 16.

---
Figure 17: The same as Fig. 8 but using the ‘tail mass’. Right: The same Fig.
9 but using the ‘tail mass’.

In Fig. 17, we compare the luminosity function and 56Ni mass distribution for
the cases of using the ‘Tail mass’ and the ‘Arnett mass’. The 56Ni masses for
the ‘Tail mass’ case are slightly higher than the case of the ‘Arnett mass’.
However, it is seen that the difference in the luminosity function and the
56Ni mass distribution between these two cases are nearly indistinguishable.

From these analyses, we conclude that our results in section 6 are robust to
the different methods of deriving 56Ni mass. It is true that several works
model the incomplete $\gamma$-ray trapping when using the tail luminosity of
SESNe (Afsariardchi et al., 2020; Sharon & Kushnir, 2020). Since the 56Ni
masses derived from such methods typically lie between ‘Tail mass’ and ‘Arnett
mass’ (Sharon & Kushnir, 2020), the effect of using such methods on the mock
observation is covered by our discussion on the two cases above (i.e., ‘Arnett
mass’ and ‘Tail mass’).

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# Generalization of Weinstein’s Morphism

Andrés Pedroza  Facultad de Ciencias
Universidad de Colima
Bernal Díaz del Castillo No. 340
Colima, Col., Mexico 28045<EMAIL_ADDRESS>
###### Abstract.

We present a generalization of Weinstein’s morphism defined on
$\pi_{2k-1}(\textup{Ham}(M,\omega))$. We use this morphism to show that for
$n\geq 2$ the Lie group $SU(2)$ induces an element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ of infinite
order.

###### Key words and phrases:

Weinstein’s morphism.

###### 1991 Mathematics Subject Classification:

Primary: 57S05, 53D35 Secondary: 57R17, 53D05

The author was supported by CONACYT-CB-2017-2018-A1-S-8830 grant

##  1\. Introduction

Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$. For $1\leq
k\leq n$ define the group $\mathcal{P}_{2k}(M,\omega)$ as the image of the
pairing $\langle\omega^{k},\cdot\rangle\colon
H_{2k}(M;\mathbb{Z})\to\mathbb{R}.$ Notice that for $k=1$,
$\mathcal{P}_{2}(M,\omega)$ is the well-known period group of the symplectic
manifold $(M,\omega)$. In [4], A. Weinstein defined a morphism

$\displaystyle\mathcal{A}\colon\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2}(M,\omega)$

based on the action functional of a Hamiltonian function
$H:M\times[0,1]\to\mathbb{R}$ associated with the loop of Hamiltonian
diffeomorphisms. Inspired by Weinstein’s morphism, in this article we define a
group morphism for each $k\in\\{2,\ldots,n\\}$

$\displaystyle\mathcal{A}\colon\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega).$

In broad terms, the value of the morphism $\mathcal{A}$ on
$[\psi_{\underline{t}}]\in\pi_{2k-1}(\textup{Ham}(M,\omega))$, is the average
over $(M,\omega)$ of the $\omega^{k}$-area of $2k$-balls whose boundary is the
topological $(2k-1)$-sphere $\\{\psi_{\underline{t}}(p)|{\underline{t}}\in
S^{2k-1}\\}$ as $p$ varies in $(M,\omega)$.

As an application, for $n\geq 2$ we compute $\mathcal{A}$ for a particular
element in $\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$. Such
element is induced by the Lie group $SU(2)$, which topologically is a
3-sphere. For every $A\in SU(2)$ we define a Hamiltonian diffeomorphism
$\psi_{A}$ on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ and using the fact that
there is a symplectic embedding of the open unit $2n$-ball in the standard
symplectic Euclidean space into $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ with
dense image, we compute the value of $\mathcal{A}([\\{\psi_{A}\\}]_{A\in
SU(2)})$ in Propositions 3.4 and 3.5. The description of such symplectic
embedding appears in the Appendix of [3]. Hence, the computation is performed
on the standard symplectic Euclidean space. Such computations give rise to the
main result of this article.

######  Theorem 1.1.

Let $n\geq 2$ and consider the $3$-sphere of Hamiltonian diffeomorphisms
$\\{\psi_{A}\\}_{A\in SU(2)}$ on $(\mathbb{C}P^{n},\omega_{\textup{FS}}).$
Then $[\\{\psi_{A}\\}_{A\in
SU(2)}]\in\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ is an
element of infinite order.

In [1], M. Gromov showed that
$\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}})$ has the homotopy type of
$PSU(3)$, thus
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}}))\simeq\mathbb{Z}$.
Therefore, the element of Theorem 1.1 is a generator of
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{2},\omega_{\textup{FS}}))$.

The fact that $\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$
contains a subgroup isomorphic to $\mathbb{Z}$, for $n\geq 4$, in not new. In
[2, Theorem 1.1], J. Kȩdra and D. McDuff showed that if $(M,\omega)$ admits a
Hamiltonian circle action that is contractible in $\textup{Ham}(M,\omega)$
then $\pi_{3}(\textup{Ham}(M,\omega))$ contains an element of infinite order.
This result applies to $(\mathbb{C}P^{n},\omega_{\textup{FS}})$. The
difference with our approach is that we present a generalization of
Weinstein’s morphism that enables us to present an explicit generator of the
element of infinite order.

##  2\. Definition of the morphism $\mathcal{A}$

Fix $1\leq k\leq n$ and assume that the symplectic manifold $(M,\omega)$ is
such that $\pi_{2k-1}(M)$ is trivial. Let $D^{2k}$ stand for the
$2k$-dimensional closed ball. Then for any $\gamma\in\Omega_{2k-1}(M)$ there
exists a smooth map $u_{\gamma}\colon D^{2k}\to M$ that maps the boundary of
$D^{2k}$ onto $\gamma$. Define the map
$A\colon\Omega_{2k-1}(M)\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$ as

(1) $\displaystyle A(\gamma):=\int_{D^{2k}}u_{\gamma}^{*}(\omega^{k})$

where $u_{\gamma}\colon D^{2k}\to M$ is as above. Since $A$ takes values in
$\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$, it follows that $A(\gamma)$ is
independent of the choice of $u_{\gamma}$.

For $\psi_{\underline{t}}\in\Omega_{2k-1}(\textup{Ham}(M,\omega))$ and $p\in
M$ let $\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p):=\\{\psi_{\underline{t}}(p)|{\underline{t}}\in S^{2k-1}\\}$ the
trace of $p$ with respect to $\psi_{\underline{t}}$. Therefore,
$\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in S^{2k-1}},p)\subset M$
is a topological $(2k-1)$-sphere and hence it induces the element
$\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)\in\Omega_{2k-1}(M)$. Define
$\mathcal{A}:\Omega_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$
as the average of $A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))$ over $(M,\omega)$,

(2)
$\displaystyle\mathcal{A}(\psi_{\underline{t}}):=\frac{1}{\textup{Vol}(M,\omega^{n}/n!)}\int_{M}A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)))\frac{\omega^{n}}{n!}.$

Next, we show that homotopic elements of
$\Omega_{2k-1}(\textup{Ham}(M,\omega))$ have the same value under
$\mathcal{A}$.

######  Lemma 2.2.

If
$\psi^{(0)}_{\underline{t}},\psi^{(1)}_{\underline{t}}\in\Omega_{2k-1}(\textup{Ham}(M,\omega))$
are such that $[\psi^{(0)}_{\underline{t}}]=[\psi^{(1)}_{\underline{t}}]$ in
$\pi_{2k-1}(\textup{Ham}(M,\omega))$, then
$\mathcal{A}(\psi^{(0)}_{\underline{t}})=\mathcal{A}(\psi^{(1)}_{\underline{t}})$.

###### Proof.

Let ${\bf\Psi}\colon[0,1]\times S^{2k-1}\to\textup{Ham}(M,\omega)$ be a
homotopy between $\psi^{(0)}_{\underline{t}}$ and
$\psi^{(1)}_{\underline{t}}$. For $p\in M$ denote by $u_{0},u_{1}\colon
D^{2k}\to M$ be the caping maps of the $(2k-1)$-spheres
$\psi^{(0)}_{\underline{t}}(p)$ and $\psi^{(1)}_{\underline{t}}(p)$
respectively.

Evaluating at $p\in M$ the homotopy ${\bf\Psi}$ induces a homotopy in $M$,
${\bf\Psi}_{p}\colon[0,1]\times S^{2k-1}\to M$, between the $(2k-1)$-spheres
$\psi^{(0)}_{\underline{t}}(p)$ and $\psi^{(1)}_{\underline{t}}(p)$.
Therefore, the image of $u_{0},\overline{u_{1}}$ and ${\bf\Psi}_{p}$
represents a $2k$-sphere in $M$ and hence

$\displaystyle\int_{D^{2k}}u_{0}^{*}(\omega^{k})+\int_{[0,1]\times
S^{2k-1}}{\bf\Psi}_{p}^{*}(\omega^{k})+\int_{D^{2k}}\overline{u_{1}}^{*}(\omega^{k})=0$

in $\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$. Here $\overline{u_{1}}$ stands for
the same map as $u_{1}$ where the domain $D^{2k}$ has the opposite
orientation. Note that the integral of ${\bf\Psi}_{p}^{*}(\omega^{k})$ is
zero, since is the trace of a $(2k-1)$-sphere under Hamiltonian
diffeomorphisms. Therefore,
${A}(\gamma(\\{\psi^{(0)}_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))={A}(\gamma(\\{\psi^{(1)}_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p))$ for any $p\in M$ and as a consequence
$\mathcal{A}(\psi^{(0)}_{\underline{t}})=\mathcal{A}(\psi^{(1)}_{\underline{t}}).$
∎

Hence there is an induced map

$\displaystyle\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega)$

that we still denote by $\mathcal{A}$. Furthermore, from the definition of $A$
in Equation (1) we have that $-A(\gamma_{1})=A(-\gamma_{1})$ and
$A(\gamma_{1}+\gamma_{2})=A(\gamma_{1})+A(\gamma_{2})$ for
$\gamma_{1},\gamma_{2}\in\Omega_{2k-1}(M)$. It then follows that $\mathcal{A}$
is a group morphism. Thus, we have a generalization of Weinstein’s morphism as
follows.

Let $(M,\omega)$ be a closed symplectic $(2n)$-manifold, fix
$k\in\\{2,\ldots,n\\}$ and assume that the group $\pi_{2k-1}(M)$ is trivial,
then define Weinstein’s morphism,
$\mathcal{A}\colon\pi_{2k-1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2k}(M,\omega),$
by

$\displaystyle\mathcal{A}(\psi_{\underline{t}}):=\frac{1}{\textup{Vol}(M,\omega^{n}/n!)}\int_{M}A(\gamma(\\{\psi_{\underline{t}}\\}_{{\underline{t}}\in
S^{2k-1}},p)))\frac{\omega^{n}}{n!}.$

for $\psi_{\underline{t}}\in\pi_{2k-1}(\textup{Ham}(M,\omega))$.

An alternative definition to

$\displaystyle\mathcal{A}\colon\pi_{1}(\textup{Ham}(M,\omega))\to\mathbb{R}/\mathcal{P}_{2}(M,\omega)$

is presented in [4]. Namely,

$\displaystyle\mathcal{A}(\psi^{H})=\int_{D^{2}}u_{x_{0}}^{*}(\omega)+\int_{0}^{1}H_{t}(x_{0})dt$

where $H_{t}$ is a normalized Hamiltonian generating the loop $\psi^{H}$,
$x_{0}\in M$ is a base point and $u_{x_{0}}:D^{2}\to M$ is a smooth function
that maps the boundary to $\\{\psi_{t}^{H}(x_{0})\\}$. This alternative
definition for $k=1$ does not admit a direct generalization for $k>1$ as the
one presented here.

##  3\. $\pi_{3}$ of the group of Hamiltonian diffeomorphisms of the complex
projective plane

Consider $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ where the symplectic form is
normalized so that the symplectic area of the line is $\pi$. Thus, there is a
symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ of
the open ball $(B^{2n}(1),\omega_{0})$ into the complex projective space is
given by

$\displaystyle(z_{1},\ldots,z_{n})\to\left[z_{1}:\cdots:z_{n}:\sqrt{1-\sum_{j=1}^{n}|z_{j}|^{2}}\right].$

For a proof of this claim see for instance the Appendix in [3]. Furthermore,
the complement of the embedding corresponds to the hypersurface
$\mathbb{C}P^{n-1}$. Using Weinstein’s morphism $\mathcal{A}$ defined in the
previous section, we will exhibit a nontrivial element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ for $n\geq 2.$
Recall that through the Hopf fibration, $\pi_{3}(\mathbb{C}P^{n})$ is trivial
for $n\geq 2$. Also, since
$\langle\omega^{2}_{\textup{FS}},\mathbb{C}P^{2}\rangle=\pi^{2}/2$ and
$[\mathbb{C}P^{2}]$ is the only generator of
$H_{4}(\mathbb{C}P^{n};\mathbb{Z})$ then
$\mathcal{P}_{4}(\mathbb{C}P^{n},\omega_{\textup{FS}})=\langle\pi^{2}/2\rangle$
for $n\geq 2$.

Next, the $3$-sphere of Hamiltonian diffeomorphisms on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$ that we will consider corresponds to
the Lie group $SU(2)$. For, $A\in SU(2)$ define the map $\psi_{A}$ on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$ as

(3)
$\displaystyle\psi_{A}([z_{1}:\cdots:z_{n+1}])=[A(z_{1},z_{2}):z_{3}:\cdots:z_{n+1}]$

Here, $A(z_{1},z_{2})$ corresponds to the first two homogeneous coordinates of
$\psi_{A}([z_{1}:\cdots:z_{n+1}])\in\mathbb{C}P^{n}$. Notice that since $n\geq
2$, in the definition of $\psi_{A}$ in Equation (3) we consider the group
$SU(2)$ instead of the group $PSU(2)$. Also, note $\psi_{A}$ is well-defined
and a Hamiltonian diffeomorphism. Therefore $\\{\psi_{A}\\}_{A\in SU(2)}$
induces an element in
$\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ for each $n\geq
2.$

Next, take into account the action of $SU(2)$ on the first two entries of
$\mathbb{C}^{n}$;

$\displaystyle
A.(z_{1},\ldots,z_{n})\mapsto(A(z_{1},z_{2}),z_{3},\ldots,z_{n}).$

Thus, the unit ball $B^{2n}(1)$ is an invariant set. Furthermore, with this
action the symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
$SU(2)$-equivariant. Observe that if ${\bf z}=(z_{1},\ldots,z_{n})\in
B^{2n}(1)$ is such that $(z_{1},z_{2})\neq(0,0)$, then $\\{A.{\bf z}:A\in
SU(2)\\}\subset B^{2n}(1)$ is a $3$-sphere. Also, the volume of the 4-ball in
the $(z_{1},z_{2})$-plane that is bounded by $\\{A.{\bf z}:A\in
SU(2)\\}\subset B^{2n}(1)$ is

(4) $\displaystyle\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}.$

In the case when $(z_{1},z_{2})=(0,0)$, ${\bf z}$ is fixed by the action of
$SU(2)$. Since $j$ is a $SU(2)$-equivariant embedding, the same statements
hold for the points $j({\bf z})\in\mathbb{C}P^{n}$.

As above, for $[{\bf z}]\in\mathbb{C}P^{n}$ denote by
$\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}])$ the trace of $[{\bf z}]$ under
$\\{\psi_{A}\\}_{A\in SU(2)}$.

######  Lemma 3.3.

If $n\geq 2$ and $[{\bf z}]=[z_{1}:\cdots:z_{n+1}]\in\mathbb{C}P^{n}$ is such
that $z_{n+1}\neq 0$, then

$A(\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf
z}]))=\frac{\pi^{2}}{2}\left(\frac{\left|\frac{z_{1}}{z_{n+1}}\right|^{2}+\left|\frac{z_{2}}{z_{n+1}}\right|^{2}}{1+\sum_{j=1}^{n}\left|\frac{z_{j}}{z_{n+1}}\right|^{2}}\right)^{2}.$

###### Proof.

We perform the computation in $(B^{2n}(1),\omega_{0})$, then using the
symplectic embedding $j$ we obtain the assertion of the lemma.

For ${\bf z}\in B^{2n}(1)$ let $\gamma(SU(2),{\bf z})$ be the orbit of ${\bf
z}$ under the action of $SU(2)$ on $\mathbb{C}^{n}$. If
$(z_{1},z_{2})\neq(0,0)$, then $\gamma(SU(2),{\bf z})$ is a 3-sphere inside
$B^{2n}(1)$. Define $u_{\gamma(SU(2),{\bf z})}:D^{4}\to B^{2n}(1)$ as the
rescaling of the unit closed 4-ball to the closed 4-ball of radius
$\sqrt{|z_{1}|^{2}+|z_{2}|^{2}}$ followed by the inclusion into of the
$(z_{1},z_{2})$-plane into $B^{2n}(1)$. Thus $u_{\gamma(SU(2),{\bf
z})}:D^{4}\to B^{2n}(1)$ is such that the boundary is mapped to the 3-sphere
${\gamma(SU(2),{\bf z})}$. Therefore

$A({\gamma(SU(2),{\bf z})})=\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}.$

Notice that in the case when $(z_{1},z_{2})=(0,0)$, ${\gamma(SU(2),{\bf z})}$
consists of a single point and in this case the constant map
$u_{\gamma(SU(2),{\bf z})}:D^{4}\to B^{2n}(1)$ gives $A({\gamma(SU(2),{\bf
z})})=0$.

Since the embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
$SU(2)$-equivariant it follows that $\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf
z}])=j({\gamma(SU(2),{\bf z})})$. Hence the boundary of the image of the map
$j\circ u_{\gamma(SU(2),{\bf z})}:D^{4}\to\mathbb{C}P^{n}$ is precisely
$\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}]).$ Finally, since $j$ is a
symplectic embedding the claim of the lemma follows. ∎

We have computed $A(\gamma(\\{\psi_{A}\\}_{A\in SU(2)},[{\bf z}]))$ for every
$[{\bf z}]\in\mathbb{C}P^{n}$ such that $z_{n+1}\neq 0.$ Thus we are in a
position to compute $\mathcal{A}([\psi_{A}])$. Once again we used the
symplectic embedding $j$ to perform the average on $(B^{2n}(1),\omega_{0})$
instead that on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$. Henceforth,

$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}({\mathbb{C}P^{n}},\omega_{\textup{FS}}^{n}/n!)}\int_{\mathbb{C}P^{n}}A(\psi_{A}([{\bf
z}]))\frac{\omega_{\textup{FS}}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}(B^{2n}(1),\omega_{0}^{n}/n!)}\int_{B^{2n}(1)}A(\gamma_{{\bf
z}})\frac{\omega_{0}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{\textup{Vol}(B^{2n}(1),\omega_{0}^{n}/n!)}\int_{B^{2n}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\frac{\omega_{0}^{n}}{n!}.$

By the nature of the expression $(|z_{1}|^{2}+|z_{2}|^{2})^{2}$, we must
consider two cases separately when integrating over $B^{2n}(1)$. Namely, the
cases when $n=2$ and $n\geq 3$.

######  Proposition 3.4.

Let $n\geq 3$ and $\\{\psi_{A}\\}$ be the $3$-sphere of Hamiltonian
diffeomorphisms on $(\mathbb{C}P^{n},\omega_{\textup{FS}})$ defined in (3).
Then,

$\displaystyle\mathcal{A}([\psi_{A}])=\frac{96}{n^{2}(n^{3}-4n^{2}-4n+16)}\pi^{n-1}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$

###### Proof.

The image of the symplectic embedding
$j\colon(B^{2n}(1),\omega_{0})\to(\mathbb{C}P^{n},\omega_{\textup{FS}})$ is
dense. Thus we compute $\mathcal{A}$ on $(B^{2n}(1),\omega_{0})$ instead of on
$(\mathbb{C}P^{n},\omega_{\textup{FS}})$. Recall that the volume of
$(B^{2n}(1),\omega_{0})$ is $\pi^{n}/n!$. Therefore,

$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{n!}{\pi^{n}}\int_{B^{2n}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\
\ \frac{\omega_{0}^{n}}{n!}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{n-2}}\cdot\frac{96\pi^{2}}{n^{2}(n^{3}-4n^{2}-4n+16)}\cdot\pi^{2n-6}\cdot
2\pi$ $\displaystyle=$
$\displaystyle\frac{96}{n^{2}(n^{3}-4n^{2}-4n+16)}\pi^{n-1}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$

∎

For the case $n=2$ we proceed similarly. Thus,

$\displaystyle\mathcal{A}([\psi_{A}])$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{2}}\int_{B^{4}(1)}\frac{\pi^{2}}{2}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\
\ \frac{\omega_{0}^{2}}{2}$ $\displaystyle=$
$\displaystyle\int_{B^{4}(1)}(|z_{1}|^{2}+|z_{2}|^{2})^{2}\ \
\frac{\omega_{0}^{2}}{2}$ $\displaystyle=$
$\displaystyle\frac{97\pi^{3}+8\pi^{2}}{256}=\frac{97}{256}\pi^{3}+\frac{1}{32}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$

######  Proposition 3.5.

Let $\\{\psi_{A}\\}$ be the $3$-sphere of Hamiltonian diffeomorphisms on
$(\mathbb{C}P^{2},\omega_{\textup{FS}})$ defined in (3). Then,

$\displaystyle\mathcal{A}([\psi_{A}])=\frac{97}{256}\pi^{3}+\frac{1}{32}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$

The proof of the main result is now straightforward.

###### Proof of Theorem 1.1.

Recall that
$\mathcal{P}_{4}(\mathbb{C}P^{n},\omega_{\textup{FS}})=\langle\pi^{2}/2\rangle$
for all $n\geq 2$. Thus, only the case $n=3$ deserves attention since in this
case the expression $\mathcal{A}([\psi_{A}])$ is a rational multiple of
$\pi^{2}$. For $n=3$, Proposition 3.4 takes the form

$\displaystyle\mathcal{A}([\psi_{A}])=-\frac{32}{15}\pi^{2}\in\mathbb{R}/\langle\pi^{2}/2\rangle.$

Henceforth,
$[\psi_{A}]\in\pi_{3}(\textup{Ham}(\mathbb{C}P^{n},\omega_{\textup{FS}}))$ has
infinite order for all $n\geq 2$. ∎

## References

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  * [3] McDuff, D., and Polterovich, L. Symplectic packings and algebraic geometry.  Invent. Math. 115, 3 (1994), 405–434.  With an appendix by Yael Karshon.
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