# The restframe ultraviolet of superluminous supernovae – I. Potential as
cosmological probes

Nandita Khetan,1,2Jeff Cooke,3,4 Marica Branchesi2,5
1DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200
Copenhagen Ø, Denmark
2Gran Sasso Science Institute, L’aquila (AQ), 67100, Italy
3Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, Victoria 3122, Australia
4ARC Centre of Excellence for Gravitational Wave Discovery (OzGrav)
5 INFN, Laboratori Nazionali del Gran Sasso, Assergi, Italy
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Superluminous supernovae (SLSNe) have been detected to $z\sim 4$ and can be
detected to $z\gtrsim 15$ using current and upcoming facilities. SLSNe are
extremely UV luminous, and hence objects at $z\gtrsim 7$ are detected
exclusively via their rest-frame UV using optical and infrared facilities.
SLSNe have great utility in multiple areas of stellar and galactic evolution.
Here, we explore the potential use of SLSNe type-I as high-redshift
cosmological distance indicators in their rest-frame UV. Using a SLSNe-I
sample in the redshift range $1\lesssim z\lesssim 3$, we investigate
correlations between the peak absolute magnitude in a synthetic UV filter
centered at $250\text{\,}\mathrm{nm}$ and rise time, colour and decline rate
of SLSNe-I light curves. We observe a linear correlation between $M_{0}(250)$
and the rise time with an intrinsic scatter of 0.29. Interestingly, this
correlation is further tightened ($\sigma_{int}\approx 0.2$) by eliminating
those SLSNe which show a pre-peak bump in their light curve. This result hints
at the possibility that the “bumpy” SLSNe could belong to a different
population. Weak correlations are observed between the peak luminosity and
colour indices. No relationship is found between UV peak magnitude and the
decline rate in contrast to what is typically found in optical band. The
correlations found here are promising, and give encouraging insights for the
use of SLSNe as cosmological probes at high redshifts using standardising
relations in the UV. We also highlight the importance of early, and
consistent, photometric data for constraining the light curve properties.

###### keywords:

supernovae: general – cosmology: distance scale – ultraviolet: general

††pubyear: 2020††pagerange: The restframe ultraviolet of superluminous
supernovae – I. Potential as cosmological probes–The restframe ultraviolet of
superluminous supernovae – I. Potential as cosmological probes

##  1 Introduction

The newer generation of wide-format time-domain surveys over the past 15 years
have discovered a rare class of highly luminous transients termed
“superluminous supernovae” (SLSNe). These events are 10–100 times brighter at
peak compared to classical type Ia and core-collapse supernova events, with
total radiated energies of about $10^{51}$ ergs (e.g., Smith & McCray, 2007;
Pastorello et al., 2010; Gal-Yam, 2012; Quimby et al., 2018; Angus et al.,
2019). SLSNe are exceptionally blue events and are characterised by slowly
evolving light curves that remain optically detectable, within several
magnitudes from peak, for 100s of days. SLSNe (type I) have been observed to
have a preference to occur in low-metallicity, star-forming dwarf galaxies
(Lunnan et al., 2014; Leloudas et al., 2015; Perley et al., 2016; Schulze et
al., 2018; Hatsukade et al., 2018). A recent comprehensive review of SLSNe is
given by Gal-Yam (2019).

Previously, SLSNe were primarily defined by an arbitrary peak absolute
magnitude cutoff of $M<-21$ mag in optical filters (Gal-Yam, 2012), however,
fainter events have since been discovered which show similar spectroscopic and
photometric behaviour (De Cia et al., 2018; Lunnan et al., 2018; Angus et al.,
2019). Therefore, this limit has been relaxed and SLSNe are now identified
based on their unique spectral properties (Quimby et al., 2018). SLSNe are UV-
luminous explosions, with majority of their spectral energy distribution (SED)
emitted in the UV. This, combined with their extreme luminosities, makes SLSNe
detectable upto very high redshifts (to $z\sim 20$) with current and upcoming
optical and near-infrared space- and ground-based telescopes. Moreover,
studies detecting SLSN at $z\sim$ 1.5–4 suggest that their rate is higher than
that at lower redshifts (Neill et al., 2011; Cooke et al., 2012; Howell et
al., 2013; Prajs et al., 2017). Therefore SLSNe offer an appealing tool to
study the high redshift Universe.

Investigations of SLSN rest-frame UV light curves and key features found in
their UV spectra help to understand their progenitors, explosion mechanisms,
and physics behind their enormous energies. Moreover, high redshift SLSNe and
their rest-frame UV emission enable to investigate both stellar and galactic
evolution and physics. SLSNe are typically brighter near peak magnitude than
their host galaxies and are one of our only means to probe the $z\gtrsim 10$
Universe. As bright background beacons, SLSNe provide internal probes of their
host proto-galaxies and the intervening material in absorption in the
circumgalactic medium (CGM) and the intergalactic medium (IGM) along the line
of sight. SLSN detections can trace star formation in high redshift dwarf
galaxies (and potentially arrest star formation), which are believed to have
contributed the most to cosmic reionisation. SLSN number-counts in well-
defined volumes, as is done with SLSN detection, can place direct constraints
on the high-mass end of the stellar initial mass function. SLSN ejecta and
pre-explosion mass loss lend insight into the chemical enrichment of their
host galaxies and the CGM and IGM over cosmic time. Finally, the very high
redshifts at which SLSNe can be detected enable the study of the deaths of
Population III stars, while providing our best chance to detect pair-
instability supernovae. Here, we explore another potential utility of SLSNe as
standardisable candles to probe the Universe from $z\sim$ 0–20.

Reviewing the diversity in the population to date, SLSNe have been broadly
classified into two classes (Gal-Yam, 2012) based on their optical
spectroscopic and photometric properties. SLSNe type I (SLSNe-I) are hydrogen
poor and exhibit a blue continuum, with a distinctive “W"-shaped or
‘comb’-shaped feature from OII absorption around $\sim 4200$ Å during early
epochs. At later times, SLSNe-I spectrum transforms to a SN-Ic like spectrum
(Pastorello et al., 2010; Quimby et al., 2011). SLSNe type II (SLSNe-II) on
the other hand show hydrogen emission lines and are likely related to Type IIn
supernovae. The energy source for most SLSNe-II has been modelled as ejecta
interaction with hydrogen-rich circumstellar material (CSM; Smith & McCray,
2007; Ofek et al., 2014; Benetti et al., 2014; Inserra et al., 2018a).
However, the power engine of SLSNe-I remains under debate, as radioactive
decay of several solar masses of nickel fails to fully explain their light
curve evolution and points to additional central energy input. Some proposed
mechanisms include; central engine models, such as a magnetar spin-down (e.g;
Kasen & Bildsten, 2010; Woosley, 2010), the pair-instability process for stars
with massive cores (Kasen et al., 2011; Kozyreva et al., 2017), and fallback
accretion into a black hole that can also explain observed light curve
undulations (Dexter & Kasen, 2013; Kasen et al., 2016). Another interesting
feature that has recently been brought into light by Nicholl & Smartt (2016)
is the presence of a small ‘bump’ before the main peak in the light curves of
some of the observed SLSNe-I. (e.g.; Leloudas et al., 2012; Nicholl et al.,
2015b; Smith et al., 2016; Angus et al., 2019). In this work, we focus on
SLSNe type I including those events with a pre-peak bump.

Although the physical understanding of the SLSNe-I explosion mechanisms and
progenitor scenarios is still emerging, their fairly homogeneous observational
behaviour has attracted significant attention for their potential use as
cosmological probes for the local to high redshift universe (King et al.,
2014; Inserra & Smartt, 2014; Wei et al., 2015; Scovacricchi et al., 2016;
Inserra et al., 2020). Driven by the fact that they show a relatively small
dispersion in their peak optical magnitudes (Quimby et al., 2013), SLSNe-I
have recently been proposed as standardisable distance indicators to constrain
cosmological parameters. Inserra & Smartt (2014) [hereafter, IS14] studied
this prospect for the first time with a sample of 13 SLSNe-I over the redshift
range of $0.1<z<1.2$ to develop a method of standardisation analogous to SNe
Ia. They find a linear relationship between the peak absolute magnitude and
decline rate of the light curves (over 10, 20, and 30 days after peak),
measured in a synthetic filter centred at 400 $\mathrm{nm}$. This correlation
reduce the scatter in peak magnitudes from $\sim 0.4$ mag to around 0.25 mag
(see also Papadopoulos et al., 2015). They also find a similar relation with
the change in colour of SLSNe over 30 days after maximum. This work gives a
promising proof-of-concept that SLSNe-I could be standardised for measuring
distances and larger data samples could increase the accuracy to be
competitive with SNe Ia. However, De Cia et al. (2018) explored similar
correlations with various light curve properties of their sample of SLSNe-I
but they did not confirm the above results. Recently, Inserra et al. (2020)
[hereafter, I20] built an updated sample and used a novel technique that
classifies SLSNe-I based on their photometric properties in a 4-dimensional
parameter space (Inserra et al., 2018c). With this more homogenised sample,
I20 obtained similar scatters as IS14 for the decline rate-magnitude and
colour-magnitude relationships, thus further encouraging the exploration of
SLSNe as standardisable candles.

Overall, attempts to standardise SLSNe have been promising, however, current
small data samples greatly hamper these efforts. Larger data sets are
necessary not only to reduce statistical errors but also to understand the
population diversity and their physics to consequently reduce systematic
uncertainties. Scovacricchi et al. (2016) showed that even an addition of
$\sim$100 SLSNe-I to current SNe Ia samples could significantly improve the
cosmological constraints by extending the Hubble diagram into the deceleration
epoch of the Universe (i.e. $z>1$). Upcoming transient surveys are expected to
significantly increase the numbers and the redshift range of the detected
SLSNe. I20 predict detection of $\sim$900 SLSNe-I with the Vera C. Rubin
Observatory111Legacy Survey of Space and Time (LSST) of the Vera Rubin
Observatory is a $8.2$ m ($\sim$6.7 m effective) diameter telescope with
$9.6\mathrm{deg}^{2}$ field-of-view and will conduct several 10-year wide-
field surveys across the Southern hemisphere in the _ugrizy_ filters. in
optical-NIR filters, (_ugrizy_) to redshift $z\sim 4$, with a majority of the
detections around the redshift 2. Villar et al. (2018) also did simulations of
SLSNe-I for LSST and predict a 10 times larger number ($10^{4}$ SLSNe per
year) with most (90%) detections at $z\lesssim 3$. It is expected that such
large samples would constrain $\Omega_{M}$ and $w$ to 2% and and 4%
respectively (Scovacricchi et al., 2016). Inserra et al. (2018b) show that the
_Euclid_ satellite222 _Euclid_ is a 1.2 m optical and near-infrared (NIR)
satellite (550 - 2000 $\mathrm{nm}$) designed to probe the early Universe
(Laureijs et al., 2011). Euclid Deep Survey (EDS) is particularly suited for
long SLSNe light curves. should detect approximately 140 (lower limit) high-
quality SLSNe-I to $z\sim 3.5$ over the first five years of the mission. The
Nancy Grace Roman Space Telescope333Roman Space Telescope is an infrared
observatory based on a 2.4 m primary mirror. One of its two instruments is the
Wide-Field Instrument (WFI) that has a 300-megapixel infrared camera giving it
a field of view which is a hundred times larger than the Hubble Space
Telescope will also perform deep wide IR surveys that can detect SLSNe to
$z\sim$ 13\. Finally, the high sensitivity and long wavelength range of the
_James Webb Space Telescope_ (JWST) could conduct a powerful survey for high-
redshift transients and enable SLSNe detections to $z\sim 20$ in the near- and
mid-IR bands (e.g., Wang et al., 2017). However, since JWST has a relatively
small FOV, it will be highly useful for acquisition of spectra of high
redshift SLSNe (and their host galaxies) detected by other facilities.

For the redshifts investigated here, i.e., $z\sim$ 1–3, that span the peak of
cosmic star formation, current and future optical and NIR surveys will observe
the SLSN rest-frame far-UV (FUV) and near-UV (NUV) emission. For example, the
optical filters _g, r_ and _i_ of the Rubin Observatory will detect $\sim$ 160
$\mathrm{nm}$, 200 $\mathrm{nm}$ and 250 $\mathrm{nm}$ central rest-frame
wavelengths, respectively, for an object at $z=2$. At higher redshifts, even
NIR telescopes will sample rest-frame UV wavelengths for $z\gtrsim 6$ SLSNe.
In terms of using SLSNe as cosmological probes, the major advantage is their
detectability at redshifts beyond those possible for SNe Ia, i.e., redshifts
$\gtrsim 1.5$. Thus, not only would SLSNe be used as a secondary check on SNe
Ia results at $z\lesssim 2$, but SLSNe would complement SNe Ia by extending
the Hubble diagram to potentially $z\sim 10$ and higher, enabling a
distinction between various dark energy models well past the deceleration
epoch. Therefore, perhaps the most powerful use of SLSNe for cosmology, using
data from future instruments, lies at higher redshifts where they will be
observed at their rest-frame FUV/NUV wavelengths. In summary, in order to
exploit SLSNe for studying stellar and galaxy formation and evolution, and for
their potential use as cosmological probes, it is crucial to characterise
their UV behaviour.

In this work, we investigate correlations among SLSNe-I UV light curve
properties and explore their use as standardisable candles for the high
redshift Universe. Past works (; ) investigated SLSN peak magnitude
correlations with their decline rates and colours with a
$400\text{\,}\mathrm{nm}$ filter. This work primarily focuses on the rising
properties of SLSNe light curves, such as the rise time and colour evolution
during rise, and explores corresponding luminosity correlations in the rest
frame UV synthetic filters.

There are a few physical and practical motivations to explore the rising part
of the light curve instead of (or in addition to) the decline for
standardising relations. Firstly, with the assumption that there is some
uniformity in the underlying physics of SLSNe-I, one might expect more
consistent evolution of the light curve at epochs right after the explosion
and expansion compared to post-peak magnitude, where potential interaction
with circumstellar material and unknown mechanisms (e.g., the SLSNe exhibiting
post-peak bumps and undulations), effects/efficiency of magnetar energy
transfer, and other aspects could make the light curve more complicated.
Secondly, since we attempt to study the evolution properties of SLSN at bluer
wavelengths, the effects from dust creation might be smaller at early times.
Thirdly, work on SNe Ia standardisation have examined the peak magnitude
correlations with the rising part of the light curve and have indicated its
importance for cosmological measurements (e.g Hayden et al., 2010; Firth et
al., 2015; Zheng et al., 2018; Hayden et al., 2019), making it worthwhile to
explore similar relations for SLSNe-I standardisation. Fourthly, we aim to use
a phenomenological approach because of the lack of understanding behind the
explosion mechanisms and details behind SLSNe. Finally, the limited non-
uniform sampling and fragmented nature of the light curve data does not permit
a study of the decline beyond $\sim$15 d for most cases for SLSNe at high
redshift. Nevertheless, we do explore the decline part with the limited data,
thus enabling a search for relationships between rise and decline with respect
to peak magnitudes.

Besides the reasons presented earlier for exploring SLSNe in the UV for their
cosmological use, characterising their blue wavelength behaviour is also
important more generally for their detection, to help understand their
explosion physics, their nature, and classification. For example, I20 measured
the pseudo equivalent width (pEW) of the C III/ C II/Ti III and Mg II/C II
blended lines at $\sim$ $2200\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$\sim$ $2800\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ respectively, to search
for a more quantitative way of distinguishing between Fast and Slow SLSNe
detected at high redshifts.

In this work, we assemble all the available UV and NUV SLSNe-I data from the
literature, and include all events with data coverage on the rising part for
our analysis. We measure their light curve properties and determine peak
magnitude correlations in order to probe their potential for cosmological use.
Given the presence of SLSNe with pre-peak bumps in their light curves (Nicholl
& Smartt, 2016) and the debate over their ubiquity, we identify them
separately within our data sample to look for differences, if any, in their
trends from those of a general SLSNe-I sample. The intention is to help
determine if there is different physics driving these explosions and whether
or not they are a distinct population, providing insight on their nature.
Additionally, since these events can be identified via photometry alone, we
explore the prospect of using them as standardisable candles, or a population
that can be easily eliminated from the full SLSN population, to enable ‘clean’
photometric events for standardisation. Although the number of objects
involved in this study is statistically small and the available data is
sparse, this work provides an important first investigation of the
evolutionary properties and potential cosmological use of the UV light curves
of SLSNe.

We describe the SLSN data sample used in this work in Section 2. In Section 3,
we outline the methodology for light curve fitting and present the estimated
light curve parameters along with explaining various analytical techniques
used to determine the correlation relations. Section 4 presents the observed
relationships for various light curve properties of the two data samples.
Finally we discuss our findings and draw the conclusions in Section 5

##  2 Data

The primary goal of this work is to investigate the rest-frame UV behaviour of
SLSNe-I light curves. The subset of existing high redshift SLSN data which
includes rest-frame UV wavelengths is relatively small. From among the
published events, we select all the SLSNe-I having rest-frame FUV/NUV
photometric coverage with sufficient data to reliably measure their light
curve properties (such as peak magnitude, rise time, colour evolution, etc.).
This data set is referred to as the ‘Literature’ sample because it includes
all objects published to date that have photometric data in the rest-frame UV.
Among this sample, we select SLSNe which pass certain data completeness and
quality cuts in order to define a data set which allows us to measure the
light curve properties with no, or negligible, extrapolations. This sub sample
is called the ‘SLSN-UV test’ sample and all correlations presented in this
work are measured using only this sub-sample. The literature sample objects
are shown only for completeness and comparison purposes. Additionally, as
mentioned earlier, a lesser understood phenomenon observed in some SLSNe is
the presence of a pre-peak bump in their light curves. We separately identify
and mark such objects in our data sample, and refer to them as ‘Bumpy’ SLSNe.
Below, we describe in detail our data set.

###  2.1 The Literature Sample

We take all published SLSNe type I from the literature having $z\gtrsim 1$
redshifts in order to assemble all SLSNe with available FUV/NUV data. This
redshift cut is chosen such that the observed optical filters are blue-shifted
to UV filters in the rest frame of the SLSN
($\lambda_{eff}=\lambda_{obs}/(z+1)$; for example, a SLSN at the lowest
redshift of $z=1$, the _g_ band effectively samples the spectral region around
$2400\text{\,}\mathrm{\SIUnitSymbolAngstrom}$). Exceptions here are the lower
redshift events that have UV data coverage with space-based telescopes such as
the _Neil Gehrels Swift Observatory (Swift)_. Secondly, in order to explore
the rising phase of the light curve, we require photometric coverage from
several days before the maximum to post maximum so as to efficiently constrain
the rise time and the peak magnitude/epoch. Thirdly, often the data is not
available in all optical filters, so we require data specifically in those
observer-frame filters that coincide with UV/NUV bands in the SLSN rest frame.

Finding a statistically significant sample which passes all these three
criteria is challenging since there are only a few tens ($\sim 30$) of
detected SLSNe-I at high redshift, and even fewer with adequate photometric
coverage during the rising phase. Although acquiring high-cadence light curves
for events at higher redshifts may be easier than low redshift owing to their
slow evolving nature combined with the effect of time dilation, surprisingly
few objects have coverage to later times (+30 days and beyond). Detecting
these events on their rise is demanding because of their faintness at such
high redshifts, and the fact that the temporal coverage of surveys with
sufficiently deep and wide fields is comparatively sparse. High resource
outlay also often prevents getting early spectral confirmation of these high
redshift events since exposures of several hours on highly competitive
8m-class telescopes are generally required for the m${}_{r}\gtrsim$ 24–25
near-peak magnitudes of SLSNe at $z\gtrsim 2$ (Smith et al., 2018; Curtin et
al., 2019).

Another challenge associated with SLSNe cosmology, as pointed out by I20, is
to find a uniform classification scheme which relies on SLSNe progenitor
scenarios and explosion mechanisms. Building a homogeneous sample is important
in context of using SLSNe as standardisable candles because variations in the
underlying physics of SLSNe may increase the intrinsic scatter in the
correlations. In the absence of a robust definition of SLSNe-I sub-classes and
understanding of their physics and explosion mechanisms, we do not make any
distinctions in our data sample based on the light curve evolution of the
SLSNe, unlike I20, where the objects are categorised into Fast and Slow types.
We build the sample used in this work based solely on the availability of the
photometric data and the required redshift cut ($z\gtrsim 1$ to reach the
rest-frame UV/NUV), and study their light curve properties as a whole. The
SLSN-UV test sample (see below) is also selected based only on the data
quality and cadence. Furthermore, objects with early peaks (Nicholl et al.,
2015b) are not excluded if they pass our redshift and data quality criteria,
instead, we identify them as a separate sub-set (see Section 2.1.2).

Among the $\sim 30$ published high redshift events, there are 22 SLSNe which
pass our three filtering criteria: (1) spectroscopic redshifts of $z\gtrsim
1$, (2) data coverage on the rising phase of the light curve, and (3)
photometric coverage in the rest-frame UV/NUV filters, and hence these form
our Literature Sample. An exception to the imposed redshift cut are the two
events SN2017egm and SN2015bn at $z=0.030$ and $z=0.114$, respectively. These
two objects have UV data coverage with the _Swift_ Ultraviolet and Optical
Telescope (UVOT). Figure 1 shows the redshift distribution for the Literature
Sample. The correlations presented in this work are built using only the SLSN-
UV test sample which is a sub-set of the Literature Sample (Section 2.1.1).

The Literature Sample is composed of events discovered and followed-up from
several surveys. Of the 22 SLSNe-I in the Literature Sample, 8 are discovered
with the Dark Energy Survey (DES, Dark Energy Survey Collaboration et al.,
2016), an optical imaging survey using the Dark Energy Camera (DECam, Flaugher
et al., 2015) on the $4\text{\,}\mathrm{m}$ Blanco Telescope at the Cerro
Tololo Inter-American Observatory (CTIO) in Chile. These SLSNe were discovered
during the DES-SN programme (Kessler et al., 2015; Diehl et al., 2018) which
surveyed 10 DECam pointings, imaging 27 deg2 in $g,r,i$ and $z$ filters with
an approximate 7-day cadence.These objects are reported and analysed in Angus
et al. (2019).

Another 6 objects come from the Pan-STARRS1 Medium Deep Survey (PS1 MDS,
Chambers & Pan-STARRS Team, 2016), using the PS1 telescope on the summit of
Haleakala in Hawaii, a wide-field survey instrument with a
$1.8\text{\,}\mathrm{m}$ primary mirror. PS1 MDS observed in
$g_{P1},r_{P1},i_{P1},z_{P1}$ filters with a typical 3-day cadence. These PS1
SLSNe are presented in various papers including Chomiuk et al. (2011); McCrum
et al. (2015) and Lunnan et al. (2018).

Three of the very distant SLSNe ($z\gtrsim 2$) in our sample are discovered
with the Subaru HIgh-Z sUpernova CAmpaign (SHIZUCA) that uses the Hyper-
SuprimeCam (HSC, Miyazaki et al., 2018; Kawanomoto et al., 2018) on the
$8.2\text{\,}\mathrm{m}$ Subaru telescope on the summit of Maunakea in Hawaii.
HSC has a 1.8 deg2 field of view and the necessary sensitivity required to
find high-redshift SNe. Type-I classification of the three HSC SLSNe is
uncertain owing to their low signal-to-noise spectra and have been assumed to
be SLSNe-I for this work. These objects have been reported in Moriya et al.
(2019).

Two SLSNe-I are included from the Supernova Legacy Survey (SNLS, Perrett et
al., 2010) that was based on the Deep Survey of the Canada-France-Hawaii
Telescope Legacy Survey 444Canada-France-Hawaii Telescope Legacy Survey (CFHT-
LS). CFHT-LS Deep Fields imaged four fields in $g,r,i$ and $z$ filters with a
cadence of 3-5 days. These are presented and analysed in Howell et al. (2013).
Additionally, another CFHT object in the sample is SLSNe SN2213-1745
($z=2.046$), discovered by Cooke et al. (2012) in the CFHT-LS Deep Fields
using an image stacking technique. We note that this work discovered another
SLSN, SN1000+0213, at $z\approx 4$ which is the highest redshift SLSN detected
to date and also exhibits a pre-peak bump. However, this object is excluded
from the SLSN-UV sample because the ground-based optical date probes too blue
(rest-frame gri data coverage of $850\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
to $1700\text{\,}\mathrm{\SIUnitSymbolAngstrom}$; z-band data are too shallow
for this work) compared to the rest of the sample.

Finally, two low redshift SLSN events; SN2017egm at $z=0.03$ and SN2015bn at
$z=0.114$. SN2017egm was discovered by the _Gaia_ Satellite (presented in
Nicholl et al. (2017)) and SN2015bn (presented in Nicholl et al. (2015a)) was
first discovered by the Catalina Sky Survey. These SLSNe are included in our
sample as they have rising phase data coverage with the _Swift_ -UVOT UV
filters.

Table 1 lists all the 22 objects that comprise the Literature Sample along
with their redshifts and photometric data references. Redshifts of all the
SLSNe-I used in this work were determined spectroscopically, with the
exception of HSC16apuo, where the host galaxy redshift is estimated
photometrically as a distribution between $z\simeq$ 2.8 and 3.5 (Moriya et
al., 2019), with the most probable value from that work, $3.22$, adopted here.

Table 1: Literature Sample consisting of 22 SLSNe-I. Column 3 lists the observed filters used for calculating the magnitudes in the $250\text{\,}\mathrm{nm}$ and $310\text{\,}\mathrm{nm}$ synthetic filters. Column 4 and Column 5 give the rest frame coverage of the filters used for $250\text{\,}\mathrm{nm}$ band and $310\text{\,}\mathrm{nm}$ band respectively. Column 6 specifies whether or not the SLSN is part of the SLSN-UV test sample (i.e. the sub-sample used for measuring the correlations), and Column 7 tells if the SLSN shows a pre-peak bump in their light curves. Literature reference – a) Angus et al. (2019), b) Chomiuk et al. (2011), c) Lunnan et al. (2018), d) McCrum et al. (2015), e) Moriya et al. (2019), f) Howell et al. (2013), g) Cooke et al. (2012), h) Nicholl et al. (2017), i) Nicholl & Smartt (2016) ID | $z$ | Filters used for | Rest frame coverage | Rest frame coverage | SLSN-UV test | Bumpy | Literature
---|---|---|---|---|---|---|---
|  | 250, 310 nm |  for $250\text{\,}\mathrm{nm}$ ( Å) |  for $310\text{\,}\mathrm{nm}$ ( Å) | sample |  | reference
DES15E2mlf | 1.860 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $1983\rightarrow 2502$ | $2475\rightarrow 2993$ | Yes | may-be bumpy | a
DES14S2qri | 1.500 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2268\rightarrow 2862$ | $2832\rightarrow 3424$ | Yes | No | a
DES14C1fi | 1.302 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2463\rightarrow 3108$ | $3075\rightarrow 3719$ | No | No | a
DES15X1noe | 1.188 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2592\rightarrow 3270$ | $2592\rightarrow 3270$ | No | may-be bumpy | a
DES17X1amf | 0.920 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2174\rightarrow 2851$ | $2953\rightarrow 3726$ | Yes | Bumpy | a
DES14X2byo | 0.868 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2235\rightarrow 2930$ | $3036\rightarrow 3830$ | Yes | No | a
DES16C3ggu | 0.949 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2142\rightarrow 2809$ | $2909\rightarrow 3671$ | Yes | No | a
DES15X3hm | 0.860 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2244\rightarrow 2943$ | $3049\rightarrow 3847$ | No | No | a
PS1-10awh | 0.908 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2229\rightarrow 2887$ | $2880\rightarrow 3616$ | Yes | may-be bumpy | b
PS1-13or | 1.520 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2181\rightarrow 2738$ | $2738\rightarrow 3253$ | No | No | c
PS1-10pm | 1.206 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2491\rightarrow 3128$ | $3128\rightarrow 3716$ | Yes | Bumpy | d
PS1-11afv | 1.407 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2283\rightarrow 2866$ | $2867\rightarrow 3405$ | No | No | c
PS1-10ahf | 1.100 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2617\rightarrow 3286$ | $2617\rightarrow 3286$ | No | may-be bumpy | d
PS1-11aib | 0.997 |  _g_ $\rightarrow$ 250, _r_ $\rightarrow$ 310 | $2129\rightarrow 2758$ | $2752\rightarrow 3455$ | No | No | c
HSC17auzg | 1.965 |  _i_ $\rightarrow$ 250, _z_ $\rightarrow$ 310 | $2353\rightarrow 2877$ | $2876\rightarrow 3137$ | Yes | No | e
HSC16adga | 2.399 |  _i_ $\rightarrow$ 250, _z_ $\rightarrow$ 310 | $2052\rightarrow 2509$ | $2509\rightarrow 2737$ | Yes | No | e
HSC16apuo | 3.220 |  _z_ $\rightarrow$ 250, _z_ $\rightarrow$ 310 | $2021\rightarrow 2204$ | $2021\rightarrow 2204$ | Yes | No | e
SNLS06D4eu | 1.588 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2176\rightarrow 2658$ | $2712\rightarrow 3240$ | Yes | Bumpy | f
SNLS07D2bv | 1.500 |  _r_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2252\rightarrow 2752$ | $2808\rightarrow 3354$ | Yes | may-be bumpy | f
SN2213-1745 | 2.046 |  _i_ $\rightarrow$ 250, _i_ $\rightarrow$ 310 | $2305\rightarrow 2753$ | $2305\rightarrow 2753$ | No | No | g
SN2017egm | 0.030 |  _uvw1_ $\rightarrow$ 250, _u_ $\rightarrow$ 310 | $2187\rightarrow 2860$ | $2989\rightarrow 3745$ | Yes | may-be bumpy | h
SN2015bn | 0.114 |  _uvw1_ $\rightarrow$ 250, _u_ $\rightarrow$ 310 | $2023\rightarrow 2645$ | $2765\rightarrow 3464$ | No | Bumpy | i

####  2.1.1 The SLSN-UV test sample: SLSNe used here for testing correlations

Given the poor physical understanding of SLSNe and their classifications,
coupled with the paucity of UV data, selection criteria adopted here to build
the literature data sample rely more on the availability of the UV data than
on the physical properties of the SLSNe. However, not all the SLSNe-I
available in the literature have data quality which allows us to measure light
curve properties with small uncertainties and without biases on the light
curve shape. For example, no early data to measure the rise time or missing
peak data because of limited observing seasons visibility during the year.
This happens particularly for high redshift SNe where time dilation stretches
the transient’s visibility in the observer frame such that the full light
curve evolution (rise an decline) is not covered in one observing season.

Taking into account the data quality and cadence, we select SLSNe from the
Literature sample that have (1) a well defined main peak with at least one
data point before and after maximum and (2) rising light curve data from at
least 1 magnitude before maximum so that the rise time can be measured
reliably. These criteria are used to enable peak magnitude and rise time
measurements with negligible extrapolations. Thirteen of 22 SLSNe from the
Literature sample pass the above criteria and comprise the SLSN-UV test sample
(Table 1), which is used for all the cosmological correlations explored in
this work.

####  2.1.2 Bumpy SLSNe

Several SLSNe detected to date have been observed to exhibit multiple peaks in
their light curves. Some objects show re-brightening at later times during the
decline of the main peak, and others (with sufficient early and deep data)
have been observed to exhibit bumps prior to the main peak of the light curve
(e.g., Leloudas et al., 2012; Nicholl et al., 2015c; Smith et al., 2016;
Anderson et al., 2018). While some SLSNe show confirmed pre-peak excess in
their photometry, it is unclear whether this feature is common among the full
SLSNe-I population because pre-peak bumps are faint and may fall below the
detection limits of the surveys or individual observations. While Nicholl &
Smartt (2016) argue that bumps could be ubiquitous, Angus et al. (2019)
concluded contrary with DES SLSNe data analysis. The physical mechanism behind
the early bumps also remains unclear due to the lack of spectroscopic
information during the bump epoch. Studying the nature of _bumpy_ 555 _Bumpy_
in this works refers to bumps before the main peak of the light curve and not
to the late-time undulations SLSNe may provide the key toward understanding
the initial explosion conditions and progenitor scenarios of SLSNe and whether
these events are a separate population.

The physical understanding of pre-peak bumps notwithstanding, the distinct
photometric signature of the these SLSNe would make them powerful tools for
cosmological use. As such, a distinguishing feature enables SLSN
identification without spectroscopy, in particular at high redshift, where
spectroscopy is very time-expensive. Homogeneity in the underlying physics
could help explain their light curves and help their standardisation.
Therefore, owing to their characteristic light curve behaviour, we separately
identify events in our Literature sample which have a bump (or a possible
bump) before the main peak in their light curve and refer to them as _‘bumpy’_
SLSNe (or _‘may-be bumpy’_ SLSNe for objects with hints of a possible early
bump). We note that ‘bumpy’ here refers to the objects which have a confirmed
pre-peak bump in their light curve data, while ‘may-be-bumpy’ refers to the
objects where the data suggests the presence of pre-peak excess however not
deep and early enough to confirm it. This separate sub-identification may help
discern any trends associated with these bumpy SLSNe in the various
correlations explored here. Objects that do have neither any signature nor a
possibility of a pre-peak bump in their data are not included in either of the
bumpy or may-be-bumpy categories and are simply assumed to have no pre-peak
bumps. They are called ’non-bumpy’ to differentiate them from the former.

There are 10 SLSNe among the full Literature Sample of 22 which either have a
confirmed pre-peak bump (i.e. 4 bumpy SLSNe), or the possibility of a pre-peak
bump in their light curves could not be excluded (i.e. 6 may-be bumpy SLSNe)
and are listed in Table 1.

Figure 1: Redshift distributions of the SLSNe in the Literature Sample (22
SLSNe) and the sub-set SLSN-UV Test sample (13 SLSNe).

##  3 Method

This work aims at testing standardisation of SLSNe in the UV to enable their
use as cosmological probes from low to high redshift, in particular for $z\geq
1$. As such, for SLSNe at redshifts $z\approx$ 1–6, standard optical filters
probe the rest-frame from $\sim$ $2500\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
to the far-UV. An immediate challenge is the small number of high redshift
SLSNe-I detected to date with well-sampled light curves in a specific optical
filter. With this in mind, we attempt to develop an optimised framework that
interpolates the observed SLSNe light curves at any epoch, even if they are
sparse, and measures their properties and evolution. We aim to estimate the
peak magnitude, the epoch of peak magnitude, and to characterise their colour
and light curve behaviour. Furthermore, in order to compare peak magnitudes in
a synthetic filter, we also explore a viable solution for _K_ -correcting the
estimated peak magnitudes in the observed filters into a synthetic band in UV
(described in section 3.3), in the absence of UV spectra at peak for each
SLSN.

Apparent magnitudes at all epochs are converted to absolute magnitudes using
flat $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology with $H_{0}=70\
\mathrm{km\ s^{-1}\ Mpc^{-1}},\Omega_{\Lambda}=0.7$ and
$\Omega_{\mathrm{M}}=0.3$. Correction for cosmological expansion is applied to
all the absolute magnitudes, hence absolute magnitude $M$ at any observed
epoch is given by $M=m-5\log_{10}(d_{L}/10pc)+2.5\log_{10}(1+z)$. Light curves
are corrected for time dilation and all timescales are given in the SLSN
restframe throughout the paper. Photometric data has been corrected for Milky
Way extinction (Schlafly & Finkbeiner, 2011), however, no correction is
applied for the extinction in SLSN host galaxies, which is assumed to be small
(Nicholl et al., 2015a; Leloudas et al., 2015; Inserra et al., 2020). Several
factors support this assumption, including that SLSNe-I have been observed to
typically occur in dwarf, low metallicity hosts (Lunnan et al., 2014; Chen et
al., 2017; Perley et al., 2016; Angus et al., 2016; Izzo et al., 2018;
Hatsukade et al., 2018) and often on the outskirts of the galaxies in space-
based imaging (Curtin et al., 2019, see also Angus et al. 2016 and Lunnan et
al. 2015). Furthermore, low scatter has been observed in the SLSNe UV peak
magnitude distribution (Smith et al., 2018) and in the colour distribution in
the optical and UV (Inserra et al., 2018c; Smith et al., 2018).

The uncertainties on the absolute magnitudes are directly propagated from the
uncertainties on the observed apparent magnitudes. We do not include any
errors from the distance estimates with the redshift and the assumed
cosmological model, as they are negligible.

###  3.1 Synthetic photometric bands

The intention here is to explore SLSN-I FUV/NUV peak magnitude correlations,
however, the details of the spectra of SLSNe-I in the UV are poorly known.
Currently, _Gaia16apd_ is one of only two SLSNe-I that have UV spectral data
at/near the peak light (Yan et al., 2017; Yan et al., 2018) and we use it here
as a template for the near-peak spectral behaviour of SLSNe-I to evaluate
K-correction (See section 3.3). Strong absorption features are observed in the
rest-frame FUV and, although these absorption features are key to
understanding SLSN physical processes, they could also introduce additional
photometric scatter in SLSN light curves. One would ideally want to explore
light curve correlations in a continuum region devoid of strong absorption
features, however, this becomes difficult blueward of $\sim$3000Å (Figure 2).
Below, we motivate our choice to use synthetic photometric bands, keeping in
mind the above considerations and to maximise the sample size.

For our data sample, we build a synthetic passband with a width of
$500\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ centred at
$2500\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. This band contains two
moderately strong absorption features that appear relatively consistent in
SLSNe-I spectra obtained to date. This band is referred to as the
$250\text{\,}\mathrm{nm}$ band throughout this work and is shown in Figure 2.
We define another similar synthetic band centred at
$3100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (Fig.2), referred to as the
$310\text{\,}\mathrm{nm}$ band, and it is used to compute colour (250-310) of
the SLSNe. This synthetic band probes a relatively featureless portion of the
spectrum and has good observed photometric coverage in the data, while still
remaining in the NUV spectral region. Table 1 presents the observed filters
which are _K_ -corrected to corresponding synthetic filters for the whole
Literature Sample.

As the aim here is to explore SLSNe in the rest-frame UV also for higher
redshift objects for cosmology, we define a bluer synthetic filter with a
width of $500\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ centred at
$1900\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (see Figure 2). This
$190\text{\,}\mathrm{nm}$ band has large absorption features and only 6 of the
highest redshift SLSNe have the required data.

Figure 2: Gaia16apd spectrum at peak (Yan et al., 2017) and the blackbody
function at 15000K. Also shown are the four synthetic filters,
$190\text{\,}\mathrm{nm}$, $250\text{\,}\mathrm{nm}$,
$310\text{\,}\mathrm{nm}$ and $400\text{\,}\mathrm{nm}$ used in this work.

###  3.2 Light Curve fitting

In order to estimate the peak brightness of SLSNe-I and interpolate the
photometric observations to characterise their light curve behaviour, we
employ Gaussian Process Regression (GPR) technique (Rasmussen & Williams,
2006; Bishop, 2006). Being a non-parametric Bayesian approach, this method
removes any assumptions on the SLSN light curve shape, which may be introduced
with polynomial fitting. Comparing polynomial fitting with Gaussian Process,
Inserra et al. (2018c) show that a Gaussian Process fit represents the
observed data better than a polynomial fit. GPR works well on small data sets
common in transient astronomy and has the ability to provide uncertainty
measurement intrinsically on the predicted values, which is particularly
important in this work.

This technique has the advantage over other methods by including uncertainty
information of the observed data, thus producing less-biased interpolated
values. Additionally, this method is very powerful for SN light curves having
incomplete or noisy photometric data. Therefore, GPs are being successfully
used in astronomy (Mahabal et al., 2008; Way et al., 2009; Gibson et al.,
2012) and supernovae analyses (Mandel et al., 2009; Kim et al., 2013; Scalzo
et al., 2014; Lochner et al., 2016). The use of Gaussian Process techniques
allows the user to marginalise over systematic sources of noise within a data,
which might otherwise not be captured in an astrophysical model.

Gaussian Process (GP) is a probability distribution over all admissible
functions that can model the correlated noise within a set of temporal or
spatial data. It is a collection of possibly infinite random variables and
hence termed non-parametric. Among these infinite parameters, any finite
number set has a multivariate Gaussian distribution. Just like a Gaussian
distribution, a GP is defined by (1) a mean function $\mu(x)$ which determines
the mean at any point of the input space, and (2) a covariance function,
a.k.a., ‘kernel’ $K(x,x^{\prime})$, which sets the covariance between data
points $x$ and $x^{\prime}$.

$f(x)=GP(\mu(x),K(x,x^{\prime}))$ (1)

For any input point $x$, the reconstruction function $f(x)$ has a normal
distribution with its mean value given by the mean function $\mu(x)$ and a
covariance between two points by the kernel function $K(x,x^{\prime})$. The
kernel determines the kind of relationship between adjacent data points making
one point dependent on the other. Both $\mu$ and $K$ may be parameterised and
the parameters of the latter are referred to as hyper-parameters since they
describe the function scatter rather than the function itself. There are two
hyper-parameters; the vertical scale $\sigma$ that describes how much the
function can span vertically, and the horizontal scale $l$ that tells how
quickly the correlation between two points drops as the distance between them
increases. A high $l$ gives a smooth function, while a lower $l$ results in a
wiggly function. For time-series data as dealt with here, these two hyper-
parameters respectively become the uncertainties in the measured magnitudes
and the timescale over which significant changes occur within the data. The
functional form of the covariance function or ‘kernel’ used can be
selected/constructed such that it represent any periodic tendencies within the
data.

The likelihood function of a GP is a multivariate Gaussian distribution of
dimension equal to the number of measurements $n$. The functional form of the
covariance function defines the relationship between the measurements and it
absorbs any intrinsic systematics which are unknown to the user. The kernel
hyper-parameters may be optimised to reach the best convergence of the
distribution.

We implement GPR to interpolate the observed light curves of all the SLSNe in
our sample. We use a flexible python library called GEORGE (Ambikasaran et
al., 2015) and employ a Matern 3/2 kernel already implemented in this library.
A Matern 3/2 kernel is mathematically similar to squared exponential function
and can be written as:

$k(r)=\sigma^{2}\left(1+\frac{\sqrt{3}r}{l}\right)\exp\left(\frac{-\sqrt{3}r}{l}\right)$
(2)

where $\sigma$ is the vertical span i.e., error in an observation, $l$ is the
horizontal scale over which variations happen in the data, and $r$ is the
separation between observations.

This kernel provides a greater flexibility in fluctuations over short time
scales and has been shown to best represent the form of SLSN light curves
(Inserra et al., 2018c; Angus et al., 2019). Our GPR fitting method determines
the best fitting hyper-parameters for each light curve via a gradient based
optimisation. We interpolate the light curves over the full temporal range of
measurements and estimate their evolution parameters. The fitting is done for
the chosen observed band(s) for each SLSN in the Literature Sample.

Figure 3 presents the mean and 1 $\sigma$ uncertainties for the GP fits of all
22 objects in the Literature Sample. The 13 SLSNe constituting the SLSN-UV
test sample are marked with an orange star on their light curve plot. The
figure shows photometric measurements for all observed bands for each SLSN,
while the GP fit is shown only for one chosen filter which is closest to the
$250\text{\,}\mathrm{nm}$ band (as given in Table 1) based on the SLSN
redshift. For those SLSNe which have an early bump in their light curve, the
data points in the bump phase (where present) are not included in the fitting
unless the absence of sufficient data characterising an early bump does not
skew the fitting process. For all GP interpolated fits, we observe that the
light curves are less flexible or ‘tighter’ in areas with high cadence data,
while interpolations over large data gaps are more uncertain or ‘loose’. The
optimised kernel defines the relationship between successive points and
greater data density results in fewer degrees of freedom for the fit, while
sparsely spaced data generate more uncertainty.

Figure 3: Light curve fits for the 22 SLSNe in the Literature Sample. Objects
marked with an orange star denote they are part of the SLSNe-UV test sample.
All phases are given in SLSN rest-frame days relative to the peak magnitude in
the fitted observed band. The bands are chosen such that they are closest to
the $250\text{\,}\mathrm{mm}$ filter in rest-frame as given in Table 1.
_(continued on the next page)_ Figure 3 Cont: Continued Figure 3.

####  3.2.1 Light Curve properties

We can estimate SLSN magnitudes and uncertainties at any epoch from the GP
fitted light curves, regardless of the data cadence. For each SLSN, we
quantify its peak and evolutionary behaviour using certain quantities measured
with the interpolated light curve. Below, we define each of these quantities
along with their notation which are used throughout this work.

  1. 1.

Peak magnitude \- The maximum absolute magnitude in a photometric band $X$
written as $M_{0}(X)$, where $0$ indicates the peak epoch. For example, peak
magnitude in $250\text{\,}\mathrm{nm}$ band is written as $M_{0}(250)$.

  2. 2.

Rise time (1 mag to peak) \- The time in rest frame days that a SLSN takes to
rise from 1 magnitude below peak to the peak magnitude. It is denoted as
$\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$. Rise time here is measured in the
$250\text{\,}\mathrm{mm}$ rest frame filter except in Figure 15 where it is
measured in $310\text{\,}\mathrm{mm}$ filter.

  3. 3.

Rise rate \- The change in magnitude from 15 rest-frame days before the peak
magnitude to the peak magnitude. The rise rate is denoted as $\Delta M_{-15}$.

  4. 4.

Decline rate \- The change in magnitude from peak magnitude to 15 rest-frame
days after the peak magnitude. It is denoted as $\Delta M_{15}$

  5. 5.

Colour at peak \- The colour index of a SLSN at the epoch of peak magnitude
and is defined as $M_{0}(X)-M_{0}(Y)$, written as $(X-Y)_{0}$. For example,
the colour at peak magnitude in the $250\text{\,}\mathrm{nm}$ and
$310\text{\,}\mathrm{nm}$ bands is denoted as $(250-310)_{0}$.

  6. 6.

Delta colour \- The change in colour from 15 days before peak magnitude to the
peak magnitude, denoted as $\Delta(X-Y)_{-15}$. For example,
$\Delta(250-310)_{-15}$.

All quantities are measured using the interpolated light curves. To measure
the changes, the epoch of 15 days before peak is chosen because many SLSNe in
our sample do not have very early data and estimating/extrapolating to earlier
epochs would incur assumptions on the light curve shape. In few cases where
there was no data 15 days before/after the peak in the required observer band
(for example, HSC16apuo, SN2015bn), we use extrapolated values with associated
uncertainties that are larger than those of interpolated values, and those
objects are not included in the SLSN-UV test sample used to determine the peak
magnitude correlations. Table 2 lists the measured light curve properties for
the full Literature Sample.

Table 2: Light curve properties of the 22 SLSNe in the full Literature sample, measured with the interpolated GP fits. The peak magnitudes $M_{0}(250)$ given here have been _K_ -corrected using the _Gaia16apd_ spectrum at peak (see Section 3.3). The errors on all properties are estimated intrinsically with GPR, while errors on the rise time are computed using data resampling (see Section 3.2.2). The last column gives the SLSNe numbers used on some plots as aliases for their names. ID | z | $M_{0}(250)$ | $\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ | $\Delta M_{-15}$ | $\Delta M_{15}$ | $\Delta M_{30}$ | $(250-310)_{0}$ | $(250-310)_{-15}$ | $\Delta(250-310)_{-15}$ | Plotting
---|---|---|---|---|---|---|---|---|---|---
|  |  | (days) |  |  |  |  |  |  | Number
DES15E2mlf | 1.860 | $-22.32\pm 0.05$ | $9.45\pm 1.95$ | $2.40\pm 0.11$ | $0.40\pm 0.06$ | $1.65\pm 0.41$ | $-0.25\pm 0.08$ | $0.87\pm 0.50$ | $1.01\pm 0.51$ | 1
DES14S2qri | 1.500 | $-21.42\pm 0.06$ | $10.33\pm 1.11$ | $1.89\pm 0.09$ | $0.75\pm 0.17$ | $1.75\pm 0.54$ | $0.31\pm 0.08$ | $0.95\pm 0.33$ | $0.68\pm 0.34$ | 2
DES14C1fi | 1.302 | $-21.43\pm 0.03$ | $22.16\pm 2.42$ | $0.58\pm 0.03$ | $0.37\pm 0.03$ | $0.93\pm 0.03$ | $0.32\pm 0.04$ | $0.18\pm 0.08$ | $0.06\pm 0.09$ | 3
DES15X1noe | 1.188 | $-21.63\pm 0.20$ | $27.52\pm 3.16$ | $0.29\pm 0.20$ | $0.22\pm 0.23$ | $0.76\pm 0.25$ | $0.39\pm 0.26$ | $0.12\pm 0.08$ | $0.02\pm 0.28$ | 4
DES17X1amf | 0.920 | $-21.14\pm 0.04$ | $13.87\pm 1.99$ | $1.18\pm 0.05$ | $0.20\pm 0.04$ | $0.90\pm 0.04$ | $0.63\pm 0.04$ | $0.83\pm 0.10$ | $0.19\pm 0.11$ | 5
DES14X2byo | 0.868 | $-21.43\pm 0.03$ | $7.37\pm 0.47$ | $3.40\pm 0.11$ | $0.50\pm 0.03$ | $2.42\pm 0.11$ | $-0.07\pm 0.04$ | $0.24\pm 0.67$ | $0.38\pm 0.67$ | 6
DES16C3ggu | 0.949 | $-21.02\pm 0.09$ | $11.83\pm 2.41$ | $1.78\pm 0.10$ | $0.28\pm 0.30$ | $0.99\pm 0.76$ | $0.37\pm 0.11$ | $0.55\pm 0.18$ | $0.13\pm 0.22$ | 7
DES15X3hm | 0.860 | $-21.64\pm 0.02$ | $26.90\pm 6.40$ | $0.48\pm 0.06$ | $0.62\pm 0.02$ | $2.32\pm 0.03$ | $0.13\pm 0.02$ | $-0.38\pm 0.28$ | $-0.43\pm 0.28$ | 8
PS1-10awh | 0.908 | $-21.50\pm 0.05$ | $15.29\pm 3.36$ | $0.94\pm 0.06$ | $1.03\pm 0.08$ | $2.65\pm 0.38$ | $0.26\pm 0.08$ | $0.08\pm 0.15$ | $-0.13\pm 0.17$ | 9
PS1-13or | 1.520 | $-22.41\pm 0.03$ | $24.03\pm 1.60$ | $0.42\pm 0.03$ | $0.50\pm 0.05$ | $1.15\pm 0.18$ | $0.06\pm 0.04$ | $-0.23\pm 0.08$ | $-0.32\pm 0.09$ | 10
PS1-10pm | 1.206 | $-21.29\pm 0.06$ | $18.57\pm 4.19$ | $0.76\pm 0.08$ | $0.45\pm 0.06$ | $1.64\pm 0.08$ | $0.51\pm 0.08$ | $0.39\pm 0.23$ | $0.15\pm 0.24$ | 11
PS1-11afv | 1.407 | $-21.57\pm 0.12$ | $29.41\pm 10.96$ | $0.27\pm 0.13$ | $0.06\pm 0.17$ | $0.18\pm 0.27$ | $0.44\pm 0.39$ | $0.29\pm 0.08$ | $-0.11\pm 0.40$ | 12
PS1-10ahf | 1.100 | $-20.46\pm 0.11$ | $49.00\pm 4.78$ | $0.18\pm 0.11$ | $0.19\pm 0.14$ | $0.31\pm 0.23$ | $0.59\pm 0.14$ | $0.41\pm 0.07$ | $0.13\pm 0.16$ | 13
PS1-11aib | 0.997 | $-21.10\pm 0.04$ | $61.36\pm 14.06$ | $0.20\pm 0.04$ | $0.22\pm 0.05$ | $0.54\pm 0.11$ | $0.67\pm 0.05$ | $0.74\pm 0.06$ | $-0.00\pm 0.08$ | 14
HSC17auzg | 1.965 | $-20.36\pm 0.03$ | $21.31\pm 2.23$ | $0.30\pm 0.03$ | $0.81\pm 0.03$ | $2.08\pm 0.16$ | $0.37\pm 0.04$ | $0.24\pm 0.04$ | $-0.07\pm 0.06$ | 15
HSC16adga | 2.399 | $-20.43\pm 0.04$ | $17.36\pm 3.17$ | $0.82\pm 0.06$ | $0.67\pm 0.04$ | $1.97\pm 0.11$ | $0.25\pm 0.07$ | $-0.04\pm 0.28$ | $-0.35\pm 0.29$ | 16
HSC16apuo | 3.220 | $-21.94\pm 0.07$ | $5.51\pm 0.65$ | $4.02\pm 0.38$ | $1.62\pm 0.15$ | $3.18\pm 0.87$ | $-0.01\pm 0.14$ | $-0.69\pm 2.55$ | $-0.72\pm 2.56$ | 17
SNLS06D4eu | 1.588 | $-21.83\pm 0.03$ | $10.79\pm 0.89$ | $1.62\pm 0.07$ | $0.65\pm 0.03$ | $2.40\pm 0.11$ | $0.24\pm 0.04$ | $0.36\pm 0.35$ | $0.06\pm 0.35$ | 18
SNLS07D2bv | 1.500 | $-20.79\pm 0.05$ | $18.18\pm 1.65$ | $0.57\pm 0.05$ | $0.67\pm 0.05$ | $1.55\pm 0.16$ | $0.52\pm 0.06$ | $0.28\pm 0.07$ | $-0.24\pm 0.09$ | 19
SN2213-1745 | 2.046 | $-21.17\pm 0.03$ | $39.28\pm 3.96$ | $0.15\pm 0.03$ | $0.10\pm 0.05$ | $0.35\pm 0.08$ | $0.24\pm 0.04$ | $0.02\pm 0.05$ | $-0.11\pm 0.07$ | 20
SN2017egm | 0.030 | $-20.84\pm 0.03$ | $18.53\pm 1.48$ | $0.78\pm 0.03$ | $0.79\pm 0.03$ | $1.62\pm 0.16$ | $0.32\pm 0.04$ | $0.10\pm 0.07$ | $-0.21\pm 0.08$ | 21
SN2015bn | 0.114 | $-20.77\pm 0.05$ | $41.32\pm 12.67$ | $0.26\pm 0.05$ | $0.28\pm 0.05$ | $1.05\pm 0.05$ | $0.93\pm 0.06$ | $1.09\pm 0.09$ | $0.03\pm 0.10$ | 22

####  3.2.2 Error on Rise time

While the errors on interpolated magnitudes at peak or any other epoch are
directly estimated by GP fitting depending on the cadence and uncertainties in
observed data, the time stamp of any observed data has essentially zero error.
Though the gaps in photometric data cause uncertainty in predicting the time
of maximum, this uncertainty in peak epoch is not explicitly measured with the
GPR. Quantifying the error on the estimated peak epoch is important since we
want to measure the rise time (here, time taken to rise 1 magnitude below peak
to the peak). We derive the uncertainties in peak epoch and the rise time by
using a data resampling technique with Monte Carlo method (See e.g., Burtscher
et al., 2009).

This approach is intuitive to error estimation in the sense that it simulates
repeated measurements of the light curve. We assume that the error
distribution of the measured data is Gaussian. For each data point $x_{n}$ on
the light curve, we invoke a Gaussian with mean $x_{n}$ and standard deviation
$\sigma_{n}$ as the measured uncertainty. We randomly sample a new data point
$x^{\prime}_{n}$ from this distribution. Similarly, for all data points, we
generate an alternative light curve with the same temporal sampling as the
observed light curve. We then estimate the evolution parameters for this
resampled light curve, namely peak magnitude, peak epoch and the epoch for 1
magnitude before peak using GPR. Repeating this resampling process 500 times,
and estimating the light curve parameters each time, we get a frequency
distribution of each parameter. We can then infer the uncertainty in that
parameter as the spread of the frequency distribution. Figure 4 presents an
example of the resampled light curves (left panel) and the resulting
distribution of the time of maximum (right panel) for SLSN DES15E2mlf. We
estimate the errors on the peak epoch and on the epoch for 1 magnitude before
the peak for each SLSNe in our data sample. These two errors are then added in
quadrature to give the final error on the rise time.

This method provides an upper limit of the uncertainty, because even though we
use the measured error as the standard deviation of the Gaussian, we centre
this error distribution on the measured value instead of the unknown true
value. This introduces additional scatter and leads us to overestimate the
uncertainty. Nevertheless, it still provides a conservative estimate of the
error which is useful for the purpose of this work.

Figure 4: Resampled light curves simulated from the measured data of SLSN
DES15E2mlf are shown in blue (left panel) along with the density distribution
for the measured peak epoch for each one these light curves (right panel). The
error on peak epoch is estimated as the standard deviation of the
distribution.

###  3.3 Cross-filter K corrections

A comparative study of peak magnitudes for a set of objects having a wide
redshift range (See Figure 1) requires K-correcting the peak magnitudes to a
single common synthetic band. The method of K-correction requires the spectrum
of each object at or around the peak epoch. As described in section 3.1, peak
absolute magnitudes for our data sample are determined in the
$250\text{\,}\mathrm{nm}$ and $310\text{\,}\mathrm{nm}$ (used for calculating
colour) rest-frame bands. Additionally, in order to reach further blue in the
spectrum, we compute the peak magnitudes also in the $190\text{\,}\mathrm{nm}$
band for higher redshift SLSNe.

In order to K-correct the estimated peak magnitude of a SLSN in an observed
filter into a fiducial UV filter, one requires the SLSN spectrum taken at or
near peak epoch with coverage in the UV. We intend to use spectra at peak or
within $\pm 5$ days from peak to calculate K-corrections. However, we found
only 5 objects out of 13 in our SLSN-UV test sample ($<40\%$) which have a
UV/NUV spectrum taken within $\pm 5$ days with none of them at peak. This
number significantly limits any analysis. Therefore, we resort to K-correcting
our peak magnitudes by adopting either a ‘standard’ template spectrum or a
blackbody (BB) curve depending on the synthetic filter in question. SLSN
spectra have been observed to be largely featureless redwards of $\lesssim
3000\mbox{\ \AA}$, and are well represented by a blackbody at those
wavelengths. However, they show significant absorption features at UV
wavelengths below $\lesssim 3000\mbox{\ \AA}$, that become even stronger
shortward of $\sim 2000\mbox{\ \AA}$. The strength of these UV absorption
features is generally highest around SLSN peak magnitude (i.e., when the
photosphere is hottest), and decreases as the supernova cools down (Angus et
al., 2019).

As mentioned before in sec 3.1, _Gaia16apd_ is one of two SLSNe-I that have UV
spectra at/near the peak (Yan et al., 2017; Yan et al., 2018). We use this
spectrum for K-correcting our peak absolute magnitudes in the UV filters with
the assumption that it represents a ‘standard’ SLSN-I spectrum over this
wavelength range (Smith et al., 2018). Besides _Gaia16apd_ being the only UV
spectrum at peak, using it as a template spectrum provides a baseline against
which future measurements can be compared. Additionally, we minimize the
K-corrections as much as possible by choosing observer-frame filters closest
to the synthetic-filters (as given in Table 1). We further test this
assumption as given below to make sure that it does not compromise any
results. Figure 2 shows the _Gaia16apd_ spectrum at peak, the BB fit to the
spectrum, and the three synthetic bands used in this work.

Among our synthetic filters, the $310\text{\,}\mathrm{nm}$ band is the bluest
wavelength range that has few absorption features and follows a BB function
well. The $250\text{\,}\mathrm{nm}$ has two relatively moderate-strength broad
absorption features. Shortward of $\sim$2200 Å, there are a number of broad
absorption features, including those near $\sim$2100, 1700, and 1400 Å. We
note that although the absorption features strongly affect the flux as
compared to a BB spectrum, measuring magnitudes over these features is useful
if the features are shown to be consistent amongst SLSNe-I. Moreover, with a
larger sample of near-peak spectra, the flux scatter in these UV bands can be
computed and tested for their usefulness. With the discovery of higher
redshift SLSNe-I, one or more synthetic bands blueward of
$190\text{\,}\mathrm{nm}$ and redward of Lyman-$\alpha$ may likely be
required.

We adopt a hybrid approach for calculating K-corrections to our synthetic
filters. Since the $310\text{\,}\mathrm{nm}$ band (used for calculating color
at peak) is continuum dominated and closely follows a blackbody, the
K-corrections to the this band have been computed using a constant temperature
BB spectrum of $15\,000\text{\,}\mathrm{K}$. This value was adopted as a mean
value among the various SLSN-I peak temperatures evaluated in the literature.
To verify this, we also measure the peak temperatures of all SLSNe in our
sample using the available photometry and found them to be scattered around
$15\,000\text{\,}\mathrm{K}$. Comparing K-corrections computed using
individual temperature BB curves with those computed using 15000K BB curve, we
find the differences to be smaller than the individual errors on the peak
absolute magnitude. Additionally, owing to the high redshifts of the objects,
majority of the photometry is in NUV rest frame bands (
$200\text{\,}\mathrm{nm}$ – $450\text{\,}\mathrm{nm}$) where spectral region
bluer than $300\text{\,}\mathrm{nm}$ has strong absorption features. Hence,
fitting BB to this blue photometry does not give accurate results. Arbitrarily
removing bluer bands data from the BB fits often leave us with two or even one
data which is not enough for fitting. Hence, for consistency and minimising
arbitrary assumptions, we decide to use a constant temperature BB curve at
$15\,000\text{\,}\mathrm{K}$ for calculating the K-corrections to the
$310\text{\,}\mathrm{nm}$ band.

However, $250\text{\,}\mathrm{nm}$ filter contains moderate-strength features
and therefore we use the _Gaia16apd_ spectrum at peak (Yan et al., 2017) for
calculating those magnitudes. We test this by comparing the 5 available UV
near-peak spectra with the _Gaia16apd_ spectrum. Figure 5 shows normalised
near-peak spectra of the 5 SLSNe along with the Gaia16apd peak spectrum and
the 250nm synthetic band used in the analysis. Visually, we find that the
existing data are sufficiently consistent with the assumption of a template
spectra at this wavelength. Furthermore, we calculated the K-corrections for
these 5 objects using their spectra and found that the mean difference between
them and those from the _Gaia16apd_ spectrum is about 0.07 mag. As explained
later in sec 4.2, we add an error floor of 0.15 mag to the computed errors on
peak absolute magnitude from GPR, to account for uncertainties in the
K-correction. Hence, using _Gaia16apd_ spectrum as template enables us to have
a statistically significant number of objects, provides a baseline for future
comparison, and does not alter the results. We note that this work does not
aim to make a precise cosmological measurement, but rather explores the
possible use of high redshift SLSNe-I as standardisable candles in the UV
wavelengths. We also use the _Gaia16apd_ spectrum for the
$190\text{\,}\mathrm{nm}$ filter used in our higher redshift exploratory test.

Figure 5: Normalised _Gaia16apd_ spectra at peak compared with available near
peak spectra of 5 SLSNe from the SLSN-UV Test sample along with the
$250\text{\,}\mathrm{nm}$ synthetic band.

For each SLSN, we K-correct the estimated peak magnitude in an observer frame
filter to one of the synthetic filters. The observer frame filter is chosen
such that its central wavelength in the SLSN rest frame is closest to the
synthetic band. We calculate the average integrated flux in the rest-frame
filter (i.e., blue-shifted observed filter) and then compare it with the flux
in the Gaia16apd spectrum ($250\text{\,}\mathrm{nm}$) or blackbody function
($310\text{\,}\mathrm{nm}$) synthetic band. The difference between the two is
added as correction to the peak magnitude estimated with the GPR
interpolation. As a result, the absolute peak magnitude in synthetic filters
can be written as:

$M_{0}(250)=M_{0}(X)+K_{X\rightarrow 250}$ (3)
$M_{0}(310)=M_{0}(X)+K_{X\rightarrow 310}$ (4)

where $M_{0}(X)$ is the peak magnitude in the chosen observer frame filter $X$
for SLSNe, such that $X$ after accounting for the cosmological redshift
$(1+z)$ is closest to the target synthetic filter and $K_{X\rightarrow 250}$
is the K-correction from observed band $X$ to the synthetic band.

###  3.4 Bayesian Inference

We perform linear fits to the correlations studied here employing a Bayesian
approach for a weighted linear regression using Markov Chain Monte Carlo
(MCMC) sampling. Borrowing from Bayes’ principles, this method provides
posterior distributions of our correlation parameters and enables us to
reflect on the uncertainties in our estimates. Additionally, this method is
termed weighted because one can tailor the variance of the likelihood allowing
for the uncertainties in both the $x$ and $y$ variables along with the
intrinsic scatter.

We use linear models to correlate the peak absolute magnitude with each of the
light curve properties (Section 3.2.1). For a light curve parameter $x$, we
model the peak absolute magnitude in a synthetic filter $\gamma$
($250\text{\,}\mathrm{nm}$ or $310\text{\,}\mathrm{nm}$ band) as follows,

$M_{\gamma}=b_{0}+b_{1}(x)$ (5)

where $b_{0}$ and $b_{1}$ are the model parameters. Bayes’s theorem gives the
posterior probability distribution of the model parameters as

$P(\Theta|D)\propto P(D|\Theta)P(\Theta),$ (6)

where D is the vector for the observed SLSN light curve data and $\Theta$
denotes the vector for the model parameters ($b_{0}$, and $b_{1}$). For a
sample of $N$ SLSNe, model parameters for each SLSN are marginalised over and
the likelihood probability distribution $P(D|\Theta)$ can be written as

$P(D|\Theta)=\prod_{i=1}^{N}P(D_{i}|\Theta)$ (7)

where $i$ is the index for the $N$ SLSNe of the sample (here $N$=13 for the
key SLSN-UV sample) and $D_{i}$ is the observed light curve data for the $i$th
SLSN. Assuming normally distributed errors and treating the peak absolute
magnitude ($M_{\gamma}$) as the target variable, the log likelihood can be
written as

$\ln\mathcal{L}=-\frac{1}{2}\sum_{i=1}^{N}\frac{(M_{\gamma}^{i}-M_{\gamma}^{T})^{2}}{\sigma_{i}^{2}}-\frac{1}{2}\sum_{i=1}^{N}\ln
2\pi\sigma_{i}^{2}$ (8)

where $M_{\gamma}^{i}$ is the measured peak absolute magnitude of $i$th SLSN
in the synthetic filters $\gamma$ and $M_{\gamma}^{T}$ is the true magnitude
given by the model in equation 5. The variance $\sigma_{i}$ for a SLSN is
computed as the quadrature sum of the errors on light curve data. We also
include an additional intrinsic scatter term $\sigma_{int}$. This term is
added to the variance and is left as a free parameter in the analysis
accounting for any _"unexplained"_ dispersion observed in the peak absolute
magnitudes. The errors on peak magnitudes are estimated from the GPR fits and
the error on rise time are calculated by resampling as explained in section
3.2.2. The variance of the likelihood is then given as

$\sigma_{i}^{2}=\sigma^{2}_{M_{\gamma,i}}+(b_{1}\sigma_{\tau_{\mathrm{rise}},i})^{2}+\sigma_{int}^{2}$
(9)

The term $P(\Theta)$ in equation 6 are the priors on the model parameters. We
adopt normal priors for the correlation parameters and a Half Cauchy
distribution for the intrinsic scatter. The MCMC sampling is implemented using
the "No U-Turn Sampler" (NUTS) provided in the
PyMC3666See:https://docs.pymc.io/ (Salvatier et al., 2016), a python
probabilistic programming package, with $10^{5}$ iterations. For all linear
fits performed in this work, we use the observed light curve data parameters
as input and estimate the posterior distributions for the correlation
parameters $b_{0}$ and $b_{1}$, along with the intrinsic scatter. All best-fit
values provided in this work are the posterior means and the errors in the
parameters are the standard deviation of their posterior.

##  4 Results

We measure the peak magnitudes and the light curve evolution properties using
the GPR interpolated light curves for the literature sample. We proceed to
investigate their peak magnitude correlations modelled using Bayesian
regression. We note that all correlations are measured using only the SLSNe in
the SLSN-UV test sample

###  4.1 Peak Magnitude distribution

As a first check, we plot the distributions of uncorrected peak magnitudes
$M_{0}(250)$ for our Literature Sample and its subset, the SLSN-UV test
sample. Figure 6 shows the $M_{0}(250)$ histograms and density distributions
from the GP interpolated light curves. The Literature Sample has an absolute
magnitude mean of $-21.30$ with a standard deviation of $0.55$ and the SLSN-UV
test sample has very similar values with a mean of $-21.25$ and standard
deviation of $0.55$. We also measure the mean and spread for the bumpy (and
may-be bumpy) population in our sample. These 10 SLSNe from the Literature
Sample have a mean absolute magnitude of $-21.25$ with standard deviation of
$0.54$, and 7 of these 10 included in the SLSN-UV test sample have a mean of
$-21.39$ and standard deviation of $0.51$. These suggest that the various sub-
samples are a good representation of the whole literature set and their
selection has not introduced any arbitrary biases.

The scatter in the uncorrected $250\text{\,}\mathrm{nm}$ band peak magnitude
distributions for the full Literature Sample as well as the SLSN-UV test
sample are higher than those measured by IS14 in the $400\text{\,}\mathrm{nm}$
band. Differences between these two works include the rest-frame filters
probed, redshift range of the samples (IS14 has $0.1<z<1.2$), and the sample
selection criteria. Notably, Lunnan et al. (2018) in the PS1 SLSN sample
looked at peak magnitude distribution at $260\text{\,}\mathrm{nm}$ rest frame
and found a even higher scatter of 1.15.

Figure 6: Uncorrected peak absolute magnitude ($M_{0}(250)$) distributions of
the full Literature Sample (22 objects) and the SLSN-UV test sample (13
objects). Normal distributions with the respective means and standard
deviations are plotted in dashed lines.

###  4.2 Peak magnitude - Rise time relation

The maximum brightness of a SLSN has been shown to be dependent on the shape
of its optical light curve or, more specifically, the rate of decline of the
optical light curve. This correlation reduces the scatter observed in the
uncorrected peak magnitude distributions and has been used to demonstrate that
SLSNe have the potential to be standardised for measuring cosmological
distances (; ). As described in Section 4.1, the scatter in the raw
$250\text{\,}\mathrm{nm}$ band peak absolute magnitudes is $\sim$0.55 mag.
Here we investigate whether this scatter decreases using the correlation amomg
the peak magnitude and the rise time in the rest-frame UV.

Previous works on SLSNe-I standardisation (e.g., ; , ) use the declining part
of the SLSN light curve to characterise its shape and correlate the peak
magnitude in $400\text{\,}\mathrm{nm}$ band with the decline rate over
different time scales (10, 20, and 30 days). In this work, we choose instead
to study the rising behaviour of the SLSN, characterised by rise time
$\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ (see Section 3.2.1), and determine
the relationships in the UV bands (see Section 3.1). Rise time here is defined
as the time elapsed in the rest-frame as the SLSN rises from 1 magnitude below
peak magnitude to the peak. In order to better characterise the rising trend
of the light curve, it would have been useful also to measure the rise time
earlier from the peak, for instance 2 magnitudes before peak. However, the
data at hand do not accommodate this investigation, since few objects (a
statistically small number) have such early data.

The interpolated light curves are used to measure the peak absolute magnitudes
and $\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ for all the SLSNe. The errors
on the peak absolute magnitude are estimated with GPR and are of the order
$\sim 0.05$ mag. As this error is small compared to the photometric errors, we
add in quadrature an error floor of 0.15 mag to peak magnitude errors. This
will help account for uncertainties from the _K_ -correction method adopted
here (e.g., Smith et al., 2018). The error floor is added in the variance of
the likelihood to the peak magnitude error term. We use only the SLSN-UV test
sample (Section 2.1.1) to perform the final correlation fits. The fitting
method employs a Bayesian approach as described in Section 3.4.

Figure 7 plots the peak absolute magnitude $M_{0}(250)$ versus the rise time
$\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ for the literature sample. The
errors on both parameters are estimated as described in Section 3. Noting that
the SLSN-UV test sample is a subset pulled from the literature sample, the
SLSN-UV test sample are highlighted in dark blue while the remaining SLSNe are
shown in light blue. SLSNe which show a bump (or possible bump) are marked in
red circles (or yellow squares).

The literature sample includes SLSNe with large peak epoch uncertainties, as
seen in Figure 7, and the events with the most complete data, i.e., the SLSN-
UV test sample, are clustered in the short rise-time region of the plot.
Focusing on this sample of 13 SLSNe, we observe a correlation where the
brighter SLSNe in the $250\text{\,}\mathrm{nm}$ band rise faster. To quantify
this relation and its scatter, we fit a linear function as in equation 5 where
the $x$ is $\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ using Bayesian
inference.

The results are shown in Figure 8 that presents the linear fit to the SLSN-UV
test sample, along with the posterior lines and sigma limits. The intrinsic
scatter of the correlation is measured to be $0.29$ mag (roughly half the
scatter for uncorrected magnitudes) with root mean square error (RMSE) of
0.35. The scatter in the $250\text{\,}\mathrm{nm}$ band rise time correlation
is comparable to the scatter in the decline rate relationship in the
$400\text{\,}\mathrm{nm}$ band measured by Inserra et al. (2020). The
Pearson’s $r$ coefficient is found to be $0.80$.

Figure 8 also shows the 3 bumpy (red crosses) and 4 may-be bumpy (yellow
squares) SLSNe that are present in the SLSN-UV test sample. These seven
objects are observed to reside toward one side of the linear fit, possibly
suggesting a similar relationship for bumpy objects but with longer rise times
and/or brighter peak mags (i.e., they appear shifted in the x-axis or y-axis
(or both) compared to the ones without a pre-peak bump), perhaps due to the
added luminosity fade of the pre-peak bump. Under an assumption that bumpy
SLSNe are a different population, potentially adding extra scatter in the
relationship, we fit a linear relation excluding these 7 bumpy objects using
our Bayesian framework as shown in Figure 9. We see that while the correlation
parameters ($b^{0}$ and $b^{1}$) remain similar (annotated in Figure 8 & 9),
the intrinsic scatter reduces from 0.29 to 0.20 and the RMSE reduces from 0.35
to 0.15. This is a significant improvement if the underlying hypothesis is
true, i.e., that bumpy SLSNe are a different population than those observed
without a pre-peak bump. If so, these SLSNe can be eliminated from any
cosmological sample based on their observed photometric behaviour. Finally, we
fit a linear relation to the set of 7 Bumpy SLSNe data separately, as plotted
in Figure 10. We find that the correlation parameters are similar to the two
relations above with an intrinsic scatter of 0.22 and RMSE of 0.25, which is
tighter than the values obtained with the full SLSN-UV test sample.

As mentioned in Section 2, the classification of three HSC SLSNe (HSC16apuo,
HSCadga, HSCauzg) as type-I is unclear. We also evaluate the results excluding
them that results in a linear fit scatter of 0.30 and a RMSE of 0.35. Hence,
the fit results do not change significantly.

Figure 7: Absolute peak magnitudes estimated in the $250\text{\,}\mathrm{nm}$
synthetic band versus rise time $\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$
for the literature sample. The SLSN-UV test sample, comprising 13 of the 22
SLSNe, are highlighted with dark blue circles. The bumpy and may-be bumpy
SLSNe are marked with red crosses and yellow squares, respectively. Figure 8:
$M_{0}(250)$ vs. $\tau_{\mathrm{rise}}^{\Delta 1\mathrm{m}}$ correlation for
the SLSN-UV test sample. The black solid line is the linear fit obtained using
Bayesian regression. The light grey lines show the posterior fits with dashed
lines representing the 1 and 2 $\sigma$ confidence intervals. The fit
parameters, RMSE and Pearson’s $r$ coefficient are given in the lower legend.
Figure 9: Same as Figure 8, but for the SLSNe without a pre-peak bump among
the SLSN-UV test sample. Figure 10: Same as Figure 8, but for the bumpy (and
may-be bumpy) SLSNe from the SLSN-UV test sample.

###  4.3 Peak magnitude - Colour relation

Past studies have shown SLSN peak magnitudes in rest-frame optical bands to be
dependent on their colour at peak and on the rate of change of colour (Inserra
& Smartt, 2014). We make measurements of the colour evolution of the SLSNe in
our data sample and investigate their correlation with the peak absolute
magnitude in the $250\text{\,}\mathrm{nm}$ band. Due to the paucity in data
cadence of the SLSNe, the colour estimation is done using the GPR fitted
curves that allows for a uniform method for measuring the colour and provides
the errors on the values. Table 2 lists the colour parameters for all the
SLSNe in the Literature Sample.

Section 3.2.1 gives the definition of colour at peak and Delta colour, a
quantity to measure the change of colour between two epochs in a light curve.
The colours here are computed as $M_{d}(250)-M_{d}(310)$ where $d$ is the
epoch ($0$ for peak). The peak magnitudes in the $310\text{\,}\mathrm{nm}$
band for our sample are estimated in the same way as for the
$250\text{\,}\mathrm{nm}$ band using the observed filter closest to
$310\text{\,}\mathrm{nm}$ in the rest frame of the SLSN. The _K_ -corrections
for the peak magnitude in $310\text{\,}\mathrm{nm}$ band ($M_{0}(310)$) are
calculated using the $15\,000\text{\,}\mathrm{K}$ blackbody curve. The colour
at peak is then measured as $M_{0}(250)-M_{0}(310)$. We also calculate colour
at 15 days before the peak $(d=-15)$ in a similar way. Furthermore, Delta
colour on the rising curve is then calculated as the difference between the
colour at peak and colour at $-15$ days; $(250-310)_{-15}-(250-310)_{0}$, and
is written as $\Delta(250-310)_{-15}$. All the objects in the literature
sample have sufficient data to calculate the colours. However, there are two
peculiar cases at high redshift ($z>2$) where the available filters were bluer
than required. HSC16apuo ($z=3.22$), and SN2213-1745 ($z=2.05$) have effective
rest-frame colour approximately as $185\mathrm{nm}-212\mathrm{nm}$ and
$205\mathrm{nm}-252\mathrm{nm}$, respectively. We discuss the results below.

Figure 11 plots the peak absolute magnitude $M_{0}(250)$ versus the colour at
peak $(250-310)_{0}$ (first panel), colour at 15 days before peak
$(250-310)_{-15}$ (second panel), and _Delta colour_ $\Delta(250-310)_{-15}$
(last panel), for the 13 SLSNe in the key SLSN-UV sample. The plots also show
the Bayesian linear fits to these objects along with their $1\sigma$ and
$2\sigma$ confidence intervals. The intrinsic scatter and RMSE of all the
three relations are annotated on their respective plots. The bumpy (may-be
bumpy) SLSNe in the sample are marked with red cross (yellow squares).

Figure 11: Absolute peak magnitude $M_{0}(250)$ vs. colour at peak
$(250-310)_{0}$ (first panel), with colour at 15 days before the peak
$(250-310)_{-15}$ (second panel), and with change in colour over 15 days
before the peak $\Delta(250-310)_{-15}$ (third panel) for the SLSN-UV Test
sample. Dashed lines show the linear fit functions estimated using Bayesian
regression and the shaded regions mark the 1 and 2 $\sigma$ confidence
intervals. The bumpy (may-be bumpy) SLSNe in the sample are marked with red
cross (yellow squares).

We observe a weak correlation between peak magnitudes $M_{0}(250)$ and colour
at peak, with a model intrinsic scatter of 0.46 mag and RMSE of 0.43. Fainter
objects in the $250\text{\,}\mathrm{nm}$ filter appear to be redder at their
peak. This is similar to the corresponding observation by IS14 in the
$400\text{\,}\mathrm{nm}$ band. For the colour at 15 days before peak (second
panel), the intrinsic scatter from fit is found to be 0.44 mag and RMSE of
0.64. The large errors in the colour data may contribute to the lower
intrinsic scatter compared to the RMSE. The colour at 15 days before peak has
larger errors because at early epochs many objects either have very sparse
data or no data leading to inflated errors on the interpolated magnitudes. The
correlation of peak magnitude with the change in colour over 15 days
$\Delta(250-310)_{-15}$ is shown in the third panel. We find that during the
rising phase, brighter objects tend to become redder faster. This fit has an
intrinsic scatter of 0.31 mag and RMSE of 0.54. For the decline phase, IS14
found an opposite relationship; fainter objects become redder faster over 30
days after peak. We note that HCS16apuo (number 17 on the plots) is not
included in the last two correlation fits because it does not have data 15
days before the peak, and hence, no reliable data points are available for the
analysis.

###  4.4 Rise rate and Decline rate

Here we measure the rate of the SLSN light curve evolution during its rising
and declining phases to study their relationship with the peak absolute
magnitude. Rise rate ($\Delta M_{-15}$) is measured as change in the magnitude
over 15 days before the maximum, and decline rates ($\Delta M_{15}$, $\Delta
M_{30}$) are measured analogously over 15 or 30 days after the peak (see
section 3.2.1). As described in section 4.2, we observed a fairly good
correlation ($\sigma_{int}=0.29$) of peak magnitudes in UV ($M_{0}(250)$) with
the rise time ($\tau_{\text{rise}}^{\Delta 1\text{m}}$) for the key SLSN-UV
sample. Therefore a correlation with the rate of rise is an expected result.
On the other hand, one of the most important SLSNe-I cosmological correlations
explored by IS14 and I20 is the dependence of $400\text{\,}\mathrm{nm}$ band
peak magnitude on the decline rate. We explore a similar relation here, but in
the $250\text{\,}\mathrm{nm}$ band.

Figure 12 plots the SLSN-UV test sample peak magnitude with the rise rate
$\Delta M_{-15}$ (first panel), with the decline rate over 15 days $\Delta
M_{15}$ (second panel), and lastly decline rate over 30 days $\Delta M_{30}$
(last panel). We observe a correlation between the rise rate and the peak
absolute magnitude for the SLSN-UV test sample. Given that we observe a linear
relationship between the rise time and the peak magnitude, and the rate of
rise has dimension $1/\mathrm{time}$, we fit a function of the form
$M=b^{0}+b^{1}\log(\Delta M_{15})$ instead of a linear fit. The best fit
function estimated is shown in Figure 12 along with $1\sigma$ and $2\sigma$
confidence intervals. The intrinsic scatter of the relation is 0.41 mag.
HSC16apuo (number 17) is not included in this fit, as it does not have
sufficient early data as mentioned earlier.

The second and third panel of Figure 12 show the decline rate plots for 15 and
and 30 days after maximum, respectively. For peak magnitude in UV
($M_{0}(250)$), we do not observe any correlation with the decline rate
measured over 15 and 30 days. This is in contrast with the tight correlation
($\sigma=0.33$) observed between peak magnitude and decline rate by IS14 and
I20 in the optical $400\text{\,}\mathrm{nm}$ band. In addition, our analysis
also doesn’t find correlation among the SLSNe-I rise and decline rates in the
UV. Figure 13 shows rise rate vs decline rate for SLSN-UV test sample.

Figure 12: Absolute peak magnitude $M_{0}(250)$ vs. rise rate over 15 days
$\Delta M_{-15}$ (first panel), decline rate over 15 days $\Delta M_{15}$
(second panel), and decline rate over 30 days $\Delta M_{30}$ (last panel) for
the SLSN-UV test sample. The dashed line in the left panel shows the best fit
function estimated using Bayesian regression and the shaded regions mark the 1
and $2\sigma$ confidence intervals. Figure 13: Rise rate ($\Delta M_{-15}$)
vs. Decline rate ($\Delta M_{15}$) plot for the SLSN-UV test sample.

###  4.5 Exploring the data for the highest redshift SLSNe-I

Following our primary aim to investigate SLSN correlations in the rest frame
UV, a sub-sample of the highest redshift SLSNe-I in our data enable an
exploration of the rise time relation to wavelengths bluer than
$250\text{\,}\mathrm{nm}$. Studying the highest energy wavelength behaviour of
SLSNe-I is vital to understand the physics behind these extreme events. A
standardisation at wavelengths as close to Lyman-$\alpha$ as possible would
enable the cosmological use of the highest redshift SLSNe-I.

As mentioned earlier, one challenge in our investigation at wavelengths
shorter than $\sim 3000\mbox{\ \AA}$ is the presence of strong broad
absorption features. These features can potentially introduce scatter in any
peak magnitude relation. In absence of a large sample of SLSN UV spectra, we
make the same assumptions as we have for the $250\text{\,}\mathrm{nm}$ band,
that all SLSNe-I exhibit similar spectral features near peak. Considering the
SLSNe-I spectral library of (Quimby et al., 2018; Yan et al., 2018), this
assumption is reasonable and provides an important first step towards
investigating high redshift SLSNe as cosmological probes in the FUV.

For our current SLSNe sample (in terms of their redshift and available data),
the bluest regions which guarantee a statistically useful number of SLSNe
cover a synthetic filter centred at $190\text{\,}\mathrm{nm}$ (see Section 3.1
for details). Six SLSNe in the literature SLSN-UV sample have photometric data
coverage close to 190 nm in their rest frame, and their data quality in the
bluest filter passes the quality criteria defined in Section 2.1.1. Similar to
other synthetic bands, we fit light curves in the observed filters that are
closest to $190\text{\,}\mathrm{nm}$ in the supernova rest-frame and employ
GPR to estimate the peak absolute magnitude $M_{0}(190)$ and the rise time
$\tau_{\text{rise}}^{\Delta 1\text{m}}$. The error on rise time is calculated
by Monte Carlo resampling (section 3.2.2). We calculate cross-filter
$K$-corrections for the peak magnitudes using the Gaia16apd spectrum as a
standard. Any results obtained here can be scaled and corrected to a better
standard in the future with a larger spectral sample.

Figure 14 shows the peak magnitude in $190\text{\,}\mathrm{nm}$ filter
$M_{0}(190$) versus the rise time ($\tau_{\text{rise}}^{\Delta 1\text{m}}$)
for these six SLSNe-I. Supernovae have been observed to typically evolve
faster at bluer wavelengths, and the $190\text{\,}\mathrm{nm}$ rise times of
about $\sim$5–10 days confirm this expectation when compared to the redder
bands, i.e., $250\text{\,}\mathrm{nm}$ where the rise times are longer
($\sim$5–20 days).

The light curve peak absolute magnitude is observed to be correlated with rise
times, however, with a steeper slope as compared to $250\text{\,}\mathrm{nm}$.
The fit function obtained with the Bayesian regression analysis is shown as
the dashed line in Figure 14 along with $1\sigma$ and $2\sigma$ confidence
interval limits. The intrinsic scatter of the fit is $0.87$ mag and the
posteriors of the correlation coefficients are very sensitive to the prior
information. We provide normal priors motivated by the values of the
correlation parameters estimated with the least square fitting method. The
slope of the relation is $0.37\pm 0.19$, a steeper value as compared to the
$0.09\pm 0.02$ estimated for the $M_{0}(250)$ \- $\tau_{\text{rise}}^{\Delta
1\text{m}}$relation. This result, along with the other results presented here,
albeit with small data sets, motivate further exploration of SLSNe as
cosmological tools at high redshift.

Figure 14: $M_{0}(190)$ versus $\tau_{\text{rise}}^{\Delta 1\text{m}}$ for
$z>1.5$ SLSNe with sufficiently blue photometry in the rest-frame. The dashed
line is the linear fit to the data obtained using Bayesian regression and the
shaded regions show the $1\sigma$ and $2\sigma$ confidence intervals. The fit
parameters and Pearson’s $r$ coefficient are given in the legend.

###  4.6 Are Bumpies different?

Nicholl & Smartt (2016) first suggested that pre-peak bumps may be ubiquitous
in SLSNe light curves and may not be different populations. That is, the pre-
peak bumps for those without evidence in their light curves may be the result
of insufficient depth and/or lack of early data. However, there has not been
any conclusive study on this topic to date. Angus et al. (2019) confirmed
existence of SLSNe without any pre-peak bump within DES data. We assess the
peak magnitude vs. rise time correlation in $250\text{\,}\mathrm{nm}$ band
(Section 4.2) where we separately mark the bumpy SLSNe-I in the SLSN-UV test
sample. In Figure 8, bumpy (and may-be-bumpy) SLSNe show a similar peak
magnitude vs. rise time relationship as the non-bumpy objects in the
$250\text{\,}\mathrm{nm}$ filter. However, we note an apparent offset between
the bumpy and non-bumpy $M_{0}(250)$–$\tau_{\text{rise}}^{\Delta
1\text{m}}$relationships, where the former appear slightly shifted to the
upper right suggesting that they are either brighter, slower evolving, or
both. This is indicated in the measured scatters when the correlation is
determined separately with each of them, as shown in Figures 9 and 10. No
clear trends are found in correlations with other parameters, such as colour
or decline rate (Figures 11 and 12). A more luminous peak and/or a slower rise
time can be attributed to an addition of a separate pre-peak bump when
convolved with the main burst light curve, if indeed the main bursts follow a
standard relation.

To explore the populations further, we measure the peak absolute magnitude and
rise times in the $310\text{\,}\mathrm{nm}$ synthetic band for the SLSN-UV
test sample, owing to the data availability (Figure 15). Two objects namely
HSCadga and HSCapuo with redshifts 2.4 and 3.2 respectively are not included
here because they lack data in rest-frame $310\text{\,}\mathrm{nm}$ filter.
The bumpy (may-be bumpy) SLSNe are marked with red cross (yellow squares). In
Figure 15 SLSNe-I identified having a confirmed pre-peak bump are shifted to
the upper right similar to what was observed for the $250\text{\,}\mathrm{nm}$
correlation although the may-be-bumpy SLSNe do not stand out. Additionally,
the whole population has a relatively broader distribution compared to
$M_{0}(250)$. Given the very limited data set, one can not draw a strong
conclusion on whether or not pre-peak bumps in SLSN-I are ubiquitous, but the
present analysis suggests that they could be different sub-class.

A KS test on the two samples gives a p-value $>>0.05$ indicating that we can
not reject that both SLSN-UV and the bumpy sample SLSNe are drawn from the
same distribution. However, we do observe a significant reduction in scatter
in peak magnitude-rise time correlation when Bumpy SLSNe are treated
separately suggesting two distinct sub-populations. Larger samples by future
surveys with deep photometric sensitivities and early and consistent cadence
data are needed to properly test this hypothesis.

Figure 15: $M_{0}(310)$ versus $\tau_{\text{rise}}^{\Delta 1\text{m}}$also
measured in $310\text{\,}\mathrm{nm}$ band for SLSN-UV test sample. The bumpy
(may-be bumpy) SLSNe in the sample are marked with red cross (yellow squares)

##  5 Summary and Perspectives

High redshift SLSNe are, and will be detected by their rest-frame UV emission
with current and future optical/NIR surveys. This work presents a preliminary
attempt to explore the rest-frame UV of SLSNe type I in context of their use
as cosmological probes at high redshift. If standardisable in the UV, SLSNe
would provide an effective tool for measuring cosmological parameters from
$z\sim$ 0–20. SLSNe would provide a complementary method to SNe Ia to measure
the Hubble diagram up to $z\sim 1.2$ as well as a means to extend it beyond
that, well into the epoch of deceleration, enabling us to distinguish between
various dark energy models.

The data sample compiled for the work is a set of 22 published SLSNe-I in
redshift range $z\sim$ 1–3, referred to as the literature sample. We apply
data quality cuts to the literature sample in order to distinguish objects
with the best and/or most complete available UV/NUV data, and this sub-set of
13 SLSNe-I is called the SLSN-UV test sample. All the cosmological
correlations are determined using the SLSN-UV test sample. We also identify
SLSNe which have, or most likely have, an early pre-peak bump in their light
curve, and refer to them as bumpy SLSNe. With the aim of exploring rest-frame
UV, and given the redshift range of the sample and available data, we chose a
synthetic filter centered at $250\text{\,}\mathrm{nm}$ to analyse the peak
magnitude correlations. All the SLSNe light curves are fit using GPR to avoid
any assumption on the light curve shape (see section 3.2), and these GP
interpolated light curves are used to estimate the peak absolute magnitudes
($M_{0}(250)$) and the light curve properties (along with their
uncertainties). We apply cross-filter K-corrections to the estimated peak
magnitudes from the observer frame filter into the $250\text{\,}\mathrm{nm}$
band using the Gaia16apd UV spectrum at peak, and into the
$310\text{\,}\mathrm{nm}$ band (used for calculating colors) with a
$15\,000\text{\,}\mathrm{K}$ blackbody function assuming them as standards.
Peak magnitude correlations are modelled using a Bayesian framework which
constraints the posteriors of the correlation coefficients along with the
intrinsic scatter in the relation. The main results of the work are summarised
in the following.

  * •

We observe a linear correlation between $M_{0}(250)$ and
$\tau_{\text{rise}}^{\Delta 1\text{m}}$for the SLSN-UV test sample where
brighter objects appear to rise faster. The intrinsic scatter of the model,
$\sigma_{int}$, is found to be 0.29 mag (RMSE = 0.35). If we exclude those
SLSNe which are identified as having pre-peak bumps (bumpy SLSNe-I), the
$M_{0}(250)$ – $\tau_{\text{rise}}^{\Delta 1\text{m}}$correlation becomes
tighter with $\sigma_{int}=0.2$ and RMSE = 0.15. With or without the bumpy
SLSNe-I, this result strongly encourage further investigations into their use
as high redshift cosmological probes in the rest-frame UV.

  * •

For the colour evolution of SLSN light curves, we correlate the peak magnitude
with three quantities; colour at peak, colour at 15 days before peak, and the
rate of change of colour between these two epochs. The colour terms are
calculated using $250\text{\,}\mathrm{nm}$ and $310\text{\,}\mathrm{nm}$
magnitudes. We observe correlations, albeit weak, for the three color
quantities with the peak magnitude $M_{0}(250)$ for the SLSN-UV test sample.
Relatively stronger relationship is seen between $M_{0}(250)$ and delta color
$\Delta(250-310)_{-15}$ with a scatter of 0.37.

  * •

Peak magnitude versus rise rate relation ($\Delta M_{-15}$) reproduces the
result similar to that observed for the rise time. More interestingly though,
we do not observe any correlation of peak magnitudes in
$250\text{\,}\mathrm{nm}$ band with the decline rate over 15 and and 30 days
($\Delta M_{15}$ and $\Delta M_{30}$). This is contrary to what has been
observed for peak magnitude in $400\text{\,}\mathrm{nm}$ band ; .

  * •

Six high redshift SLSNe from the SLSN-UV test sample have photometric data
that enable an analysis at a synthetic band centred at
$190\text{\,}\mathrm{nm}$ in their rest-frame. We explore this band in order
to investigate the peak magnitude versus rise time relation at bluer
wavelengths for using SLSNe at very high redshifts. We observe a correlation
with an intrinsic scatter $\sigma_{int}$ = 0.87 (RMSE = 0.67).

  * •

We also perform peak magnitude-rise time correlation in the
$310\text{\,}\mathrm{nm}$ band for the test sample SLSNe which have the
required data. This is done to examine whether Bumpy SLSNe show any
distinguishing trend in this band similar to the $250\text{\,}\mathrm{nm}$. We
find that the three SLSNe with confirmed pre-peak bumps in their light curve
are shifted to the upper right corner of the plot similar to what was observed
for the $250\text{\,}\mathrm{nm}$ correlation. This indicates that pre-peak
bumps might not be a universal phenomena among SLSNe-I, however, it is
difficult to draw a conclusive result given the small data set. Additionally,
comparing the rise times at three different wavelength bands, we observe that
$\tau_{\text{rise}}^{\Delta 1\text{m}}$show a consistent decrease with
decreasing wavelength, with rise times of $\sim$10–30 days at 310 nm,
$\sim$10–20 days at 250 nm, and $\sim$5–10 days at 190 nm.

The results obtained in this work are highly promising for the use of SLSNe-I
as high redshift cosmological probes in their rest-frame UV. We have adopted
an unbiased approach for data selection, using all available data with
appropriate photometry. However, the data set is yet small for a very robust
analysis. Larger samples in future are strongly encouraged to test the
observed correlations. Furthermore, SLSNe-I UV spectroscopy is critical to
test for absorption-line consistency at wavelengths from 1216 (Lyman-$\alpha$)
to $\sim$3000 Å, where SLSNe-I continua show absorption.

We note that if the Bumpy and non-Bumpy populations are indeed different, the
peak magnitude vs. rise time relationship is significantly improved when
eliminating Bumpy (and may-be-bumpy) SLSNe-I and is almost comparable to SNe
Ia, as demonstrated in Section 4.2, with the important caveat of a small
sample. With sufficiently deep photometry, Bumpy SLSNe-I can be identified by
their distinct light curves and removed from cosmological samples.
Additionally, we would like to highlight an interesting inference that at UV
wavelengths, SLSNe peak magnitudes exhibit a relatively tighter correlation
with the rising phase of the light curve (here $\tau_{\text{rise}}^{\Delta
1\text{m}}$) instead of the decline which is the common parameter used in
literature for SLSNe relations in optical bands. With the present sample, we
do not observe any trend for $M_{0}(250)$ with the decline rate. Due to data
limitations, we can not perform this analysis in the $400\text{\,}\mathrm{nm}$
band for a direct comparison with ; . It would be natural to explore rise time
correlations at optical wavelengths with a larger sample in future.

Finally, a key challenge for surveys that aim to detect very high redshift
SLSNe is the very long (1 $+z$) time-dilated search baselines needed to detect
their evolution. For example, for an assumed rest-frame 50-day rise and
overall 200-day evolution at z $\sim$ 15 becomes $\sim$2 and $\sim$9 years,
respectively. Supernovae are observed to typically evolve faster at shorter
wavelengths and this work confirms this relation to the NUV and FUV, with rise
timescales increasingly shorter, and as short as $\sim$5–10 days for rise from
1 magnitude to peak at wavelengths near $\sim$1900Å. The comparatively shorter
evolution timescales for SLSNe-I in the UV compared to the optical will help
mitigate the expected extremely long duration evolution of SLSNe at the
highest redshifts (z $\sim$ 6–20), making surveys more practical and enabling
their detection in surveys with cadences designed for other types of lower
redshift supernovae.

Surveys for high redshift SLSNe require both deep imaging capability (m
$\gtrsim$ 26 per filter, per epoch) and wide fields to discover these
relatively rare events. However, their high utility outlined in Section 1,
including galaxy and stellar evolution ISM, CGM, and IGM probes, detection of
the deaths of Population III stars, searching for pair-instability events, and
cosmological probes warrant such surveys. SLSNe can be detected to $z\sim 6$
using optical facilities such as CTIO Dark Energy Camera, NAOJ Subaru Hyper-
SuprimeCam, and the future Keck Wide-Field Imager. SLSNe to $z\sim 20$ can be
detected by facilities such as the Nancy Roman Space Telescope, Euclid, the
University of Tokyo Atacama Observatory SWIMS, and the Kunlun Dark Universe
Survey Telescope (KDUST).

## Acknowledgements

NK is supported by a grant from VILLUM FONDEN (project number 16599). NK would
like to thank Charlotte Angus, Mat Smith, Luca Izzo for the very helpful
discussions. We also thank L.Yan for providing us the reduced spectra for
Gaia16apd. Part of this research was funded by the Australian Research Council
Centre of Excellence for Gravitational Wave Discovery (OzGrav), CE170100004
and the Australian Research Council Centre of Excellence for All-sky
Astrophysics in 3 Dimensions (ASTRO-3D), CE170100013. MB acknowledges
financial support from MIUR (PRIN 2017 grant 20179ZF5KS).

## Data Availability

All data underlying this article is available within the article and enlisted
in Table 1 and Table 2. In addition, individual SLSNe-I light curves and
spectra analysed in this article are available to the public via their
corresponding papers which are referenced in Table 1.

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