Light Curves of Hydrogen-poor Superluminous Supernovae from the Palomar Transient Factory

Although most normal SNe are observed in the nearby universe (z ≲ 0.2), the most luminous ones can be detected out to higher redshift. SNe Type Ia, for instance, are currently discovered out to z ∼ 2 in deep imaging (Jones et al. 2013). A few SLSNe have been studied out to z ∼ 4 in the deepest surveys (Cooke et al. 2012), although in limited detail compared to nearby targets. Recently, a small sample of z ∼ 2 SLSNe has been studied by Moriya et al. (2018) and Curtin et al. (2018). In the future, the James Webb Space Telescope is expected to be able to detect SLSNe out to z ∼ 20 (Abbott et al. 2017). The PTF survey typically discovers SLSNe below z ≲ 1.

The redshifts in our SLSN sample are all measured spectroscopically and normally measured from narrow Mg ii absorption lines in the SN spectra. The typical uncertainties on the redshift estimates are of the order of 0.0005, given the typical resolution of the follow-up spectra (described in Quimby et al. 2018). The redshifts in Table 1 are taken from Quimby et al. (2018) for all events up to 2012, except for PTF 10vwg, for which we adopt the slightly more accurate redshift of Perley et al. (2016). We adopt the redshifts of Vreeswijk et al. (2014) for PTF 13ajg and of Yan et al. (2015) for PTF 13ehe. For the other 2013 events, we directly measure the redshifts from the Mg ii narrow absorption lines in the spectra. In PTF 13ehe, the most common spectral features are very weak. The redshift measurement is based on a weak O iii 5007 emission line, and its uncertainty is of the order of 0.001.

Figure. 2 shows the redshift distribution of our sample, where the mean redshift is $\langle z\rangle =0.27$ with standard deviation σ_z = 0.15. The volume-weighted mean is $\langle z\rangle =0.33$.²⁴

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The mean redshift of the PTF H-poor SLSN sample presented here is comparable to the "golden" SLSN sample of Nicholl et al. (2015a; $\langle z\rangle =0.22$, while $\langle z\rangle =0.63$ for their "silver" sample), and the SLSN host sample of Leloudas et al. (2015; $\langle z\rangle =0.34$ with a standard deviation of 0.2). On the other hand, SLSNe discovered by PS1 tend to be at higher redshifts, typically z > 0.5 (McCrum et al. 2015), and in particular 0.3 < z < 1.6 (Lunnan et al. 2018).

The drop of the redshift distribution above z ∼ 0.5 in our sample is an observational selection effect due to the limiting magnitude of the PTF survey (${m}_{{\rm{r}},\mathrm{lim}}\sim 20.5$ mag; Cao et al. 2016). This limit hampers further investigations of the evolution of the sample properties with redshift.

As part of standard PTF operations, SN candidates are discovered in P48 images using image subtraction in Mould-R (r) or the Sloan Digital Sky Survey (SDSS; York et al. 2000; SDSS Collaboration et al. 2017) $g^{\prime}$ filter. The best SN candidates are then classified spectroscopically and followed-up with other telescopes. The raw P48 images are initially processed by the Infrared Processing and Analysis Center (IPAC; Laher et al. 2014). The photometric calibration and system are described in Ofek et al. (2012). Image-subtraction point-spread function (PSF) photometry is performed with a custom routine (a pipeline written by one of us (M.S.) and used extensively in PTF; e.g., Sullivan et al. 2006; Ofek et al. 2014a; Firth et al. 2015; Dimitriadis et al. 2017). This pipeline constructs deep reference images—either before the SN explosion or after the SN has faded—and astrometrically aligns the images using the Automated Astrometry described in Hogg et al. (2008) and the Naval Observatory Mergered Astrometric Dataset (NOMAD; Zacharias et al. 2004). The image PSFs are then matched in order to perform the image subtraction and then to extract the PSF photometry of the SN only (where the contribution of the reference image has been subtracted). The fluxes are calibrated against SDSS Data Release 10 (Ahn et al. 2014) when available, otherwise against the photometric catalog of Ofek et al. (2012), and making no assumption on the SLSN colors.

The formal uncertainties derived with the MS pipeline only include statistical uncertainties, but not uncertainties from poor image subtraction or calibration. As a result, the formal uncertainties are slightly underestimated. For example, under excellent data coverage, we can observe a larger scatter than that accounted for by the formal uncertainties. The best example is for iPTF 13ajg, which shows a scatter of ∼0.5 mag around peak. We quantify the additional source of uncertainty by assuming ${\chi }_{\nu }^{2}=1$ for the light-curve fit around the peak of iPTF 13ajg (see Section 4.1). The additional required uncertainty is 0.05 mag, which we add to all formal errors derived with the MS pipeline (i.e., for the data taken with the P48, P60, and LT telescopes; see below), to account for poor image subtraction or calibration.

Nondetections, and in particular the last nondetection limits before the SN discoveries, are not included in our analysis. The reason for this is that nondetections are largely dominated by noisy data and are uninformative. In addition, in most cases the SLSNe-I were discovered long after explosions. For the case of PTF 12dam, co-adds of the prediscovery nondetections are presented in Vreeswijk et al. (2017). The analysis presented in this paper is independent of the nondetection limits. Thus, we leave the treatment of prediscovery limits, which is beyond the scope of this paper, for future case-by-case studies.

Follow-up imaging was obtained with the Palomar 60-inch telescope (P60; Cenko et al. 2006). The filters employed for our observations are Johnson B (Bessell 1990), Kron R (similar to Cousins R_C, Bessell 1990), Sloan $i^{\prime}$ and $z^{\prime}$ (Fukugita et al. 1996), and Gunn g (Thuan & Gunn 1976). The SN photometry is extracted with the same routine described above for the P48 data processing, but calibrated using the AAVSO Photometric All-Sky Survey (APASS; Henden et al. 2009) for the B filter or for fields that are not covered by the SDSS footprint.

We observed PTF SLSNe at late times using the Low-Resolution Imaging Spectrometer (LRIS) on the Keck I telescope to monitor the late-time evolution of the light curve or to produce a deep reference image (after the SN has faded) for image subtraction or host galaxy study (Perley et al. 2016). Images were processed using standard techniques via the custom pipeline LPIPE²⁵ and co-added using SWarp. Photometry was performed after image subtraction of the reference image (taken from Perley et al. 2016), with the custom-made IRAF routine mkdifflc (Gal-Yam et al. 2004, 2008).

Follow-up imaging was also obtained with the 2 m robotic Liverpool Telescope (LT, Steele et al. 2004) at the Roque de los Muchachos Observatory on La Palma, Spain, with the RATCAM and IO:O optical imagers in g, r, and i filters (similar to SDSS). The images were processed following Maguire et al. (2014) and using the image-subtraction PSF photometry custom routine described above for the P48 Telescope.

The LCO (previously known as LCOGT; Brown et al. 2013) data have been reduced using a custom pipeline (Valenti et al. 2016). The pipeline employs standard procedures (PYRAF, DAOPHOT) in a Python framework. Host galaxy flux was removed using image subtraction technique (High Order Transform of PSF ANd Template Subtraction, HOTPANTS²⁶ ). PSF magnitudes were computed on the subtracted images and transformed to the standard SDSS filter system (for gri) via standard star observations taken during clear nights.

We imaged several of the SLSNe in our sample with the Large Monolithic Imager (LMI) mounted on the 4.3 m Discovery Channel Telescope (DCT) in Happy Jack, AZ. The LMI images were processed using a custom IRAF pipeline for basic detrending (bias subtraction and flat fielding), and individual dithered images were combined using SWarp (Bertin et al. 2002). SN magnitudes were measured using aperture photometry with the inclusion radius matched to the FWHM of the image PSF. Photometric calibration was performed relative to point sources from the SDSS (York et al. 2000; SDSS Collaboration et al. 2017).

No DCT reference images are available for image subtraction, so we account for the contribution of the host galaxies by subtracting the host magnitudes from the observed fluxes, and we include this into the photometric uncertainty budget. The SLSNe iPTF 13dcc and 13ehe were observed with the DCT. The host galaxy of iPTF 13dcc has $B=26.3\pm 0.2$ and i = 25.0 ± 0.2 (D. Perley et al. 2018, in preparation), and we assume g = 26.3 ± 0.2 and r = 25.0 ± 0.2 for the subtraction of the host galaxy contribution to the r-band data point, which is a reasonable assumption given typical host galaxy colors (Perley et al. 2016). The host galaxy of iPTF 13ehe has B = 25.0 ± 0.1 and R = 24.0 ± 0.1. (D. Perley et al. 2018, in preparation), which we use to subtract the host-galaxy contribution. For both SLSNe, this host-galaxy correction affects significantly (by 0.2 mag) only the last r-band epoch of their light curves. In both cases, the DCT data points are consistent with the photometry from other facilities, including late-time Hubble Space Telescope (HST) photometry (Section 3.9).

A number of supernovae in our sample were observed with the UltraViolet/Optical Telescope (UVOT; Roming et al. 2005) on board the Swift Gamma-Ray Burst Explorer (Gehrels et al. 2004). Data were processed using the standard UVOT pipeline, and photometry was extracted at the supernova location using a 3'' radius. Photometric calibration was calculated using the zero point measurements from Poole et al. (2008) and Breeveld et al. (2010). The magnitudes reported in Tables 11 and 12 are all on the AB system. No attempt has been made to correct for underlying contributions from the host galaxy emission. At these redshifts, the host galaxy contribution should not significantly affect the observed UV flux in most cases. This may not be true for some cases, in particular for PTF 12dam and its luminous underlying starburst host galaxy (Chen et al. 2015; Thöne et al. 2015; Perley et al. 2016; Cikota et al. 2017). However, even in this case, the host galaxy brightness in the $F225W$ filter is 19.94 ± 0.17 (Perley et al. 2016), which is 1–2 mag fainter that the unsubtracted SN photometry (Nicholl et al. 2013; Chen et al. 2015).

Follow-up imaging was obtained with the Palomar 200-inch Hale Telescope with the LFC.²⁷ The data reduction was performed with standard IRAF tasks. The SN magnitudes were derived by extracting aperture photometry at different radii, for the SN images and the reference image, and subtracting the host contribution. For the case of PTF 09cnd, we measure the photometry using both the image-subtraction routine mkdifflc and aperture photometry, and take the average between the two results. For the case of PTF 10cwr, no reference image is available, so we extract aperture photometry with a 3'' radius and subtract the host magnitude reported by Perley et al. (2016) and include this into the photometric uncertainty budget.

The SLSNe iPTF 13dcc and iPTF 13ehe were observed with the Advanced Camera for Surveys in the Wide Field Channel on board the HST with the F625W filter, as part of the GO-13858 program (PI A. De Cia). The data were reduced using the CALACS software, which contains corrections for degradation of the charge transfer efficiency and electronic artifacts (bias-shift and -striping effects). Cosmic rays were removed using the LA Cosmic routine (van Dokkum 2001). The images were then processed with DrizzlePac 2.0,²⁸ with inverse variance map (IVM) weighting and assuming a pixel scale of 0033 and a pixel fraction of 0.6.

The SN PSF is resolved from the more extended host galaxies. For iPTF 13dcc, the PSF of the host has an FWHM of 3.2 pixels (011), while field stars have 2.4 pixels (008). iPTF 13ehe is separated from its host galaxy. The SN PSFs were fitted and thus isolated from their host galaxies using a custom IDL routine. The PSF SN photometry was then extracted assuming HST zeropoints and applying a correction for an aperture of 05 radius.

We complement the photometric data set of the SLSNe in our sample with the data published in Quimby et al. (2011), Pastorello et al. (2010), Inserra et al. (2013), Nicholl et al. (2013), Chen et al. (2015), and Vreeswijk et al. (2014). The literature photometry is showed in Figures 18(a)–(e). The purpose of including the literature data in this paper is to collect the most complete available light curves for the SLSNe in our sample. The r-band photometry is used to calculate the rest-frame g-band photometry, which is reported in Table 12. Because the sources of our observations are already diverse, the inclusion of literature data does not affect significantly the quality of our data set.

The full light curve of PTF 10nmn will be presented by O. Yaron et al. (2018, in preparation), including a wider coverage of the SN peak, which is not presented in this paper.

We exclude from the analysis a couple of published photometry data points in cases of disagreement with the photometry secured with other (multiple) telescopes, namely for iPTF 13ajg (P60 R-band data at MJD 56429 from Vreeswijk et al. 2014, excluded) and for PTF 09cnd (Wise R-band data at MJD 55089 from Quimby et al. 2011, excluded). The new measurements supersede the earlier ones.

The data set used in this paper was collected from a diversity of facilities, and the photometry is derived with different pipelines and methods. The quoted uncertainties assess the quality of the photometry for each facility or measurement method. The contribution of the host galaxy light to the SN measured flux is taken into account and reflected in the quoted uncertainties. An exception to this is for the UV photometry (Swift), for which the contribution from the host galaxy is not subtracted, but should be minimal (Section 3.7). Often the filter transmission curves of similar filters are different for different facilities or catalogs for calibration, such as r, R, and R_c, for example. However, we did not correct for these differences because they depend on the source spectra and their evolution, and these differences are typically very small, normally well below 0.1 mag.

In Figures 18(a)–(e), all photometry are shown together. When enough data are available, the photometry from different telescopes can be directly cross-checked, and we do not find evident discrepancies. Further corrections to the photometry, such as foreground extinction and k-corrections, and their uncertainties, are described below.

We derive Galactic foreground optical extinction A_V using the maps of Schlafly & Finkbeiner (2011) through the Galactic Dust Reddening and Extinction Service at the NASA/IPAC Infrared Science Archive, assuming a standard extinction law and an extinction to reddening ratio ${A}_{V}/E(B-V)=3.1$.²⁹ The mean uncertainty in the Galactic foreground extinction A_V for our sample is 0.009 mag, and we do not include this uncertainty in the photometric budget. The adopted A_V values are listed in Table 1. We calculate the extinction A_λ at the central wavelength of each filter using the reddening curve of Cardelli et al. (1989) and including the update for the near-UV given by O'Donnell (1994). Both apparent and absolute magnitudes reported in this paper are corrected for Galactic foreground extinction.

Host-galaxy extinction is not considered. SLSN host galaxies tend to be faint and have low metallicity (Neill et al. 2011; Leloudas et al. 2015; Lunnan et al. 2015; Perley et al. 2016; Chen et al. 2017; Schulze et al. 2018), and therefore we expect them to have negligible dust extinction in the red bands, with possible regions that may be locally more dusty, affecting mostly the UV (e.g., Cikota et al. 2017). On the other hand, SN Ib/c host galaxies can show significant extinction (mean $\langle E(B-V)\rangle \sim 0.2$ and <0.6 mag for ∼80% of Type Ic, Ib, and Ic-BL SNe; Taddia et al. 2015; Prentice et al. 2016), but determining it case by case for our comparison sample is often not possible and is beyond the scope of this paper.

We calculate the k-corrections K_gr for the SLSN sample from observed PTF r to rest-frame g (SDSS filter system) using spectral series of PTF 12dam and iPTF 13ajg (Vreeswijk et al. 2014, 2017; Quimby et al. 2018) and following Hogg et al. (2002). Using individual spectra for each SLSNe was not possible here, due to the paucity of sufficient spectral coverage at all epochs for the SLSNe in our sample. The spectral coverage of PTF 12dam is frequently sampled and spans from −25 to 321 rest-frame days after the peak, while the spectra of iPTF 13ajg are reliable until 60 days after peak. In the overlapping interval, there is good agreement between the k-corrections calculated from the two series of spectra. This indicates some level of similarity between the spectra, which is also confirmed by the spectral analysis of the PTF SLSN-I sample (Quimby et al. 2018). The spectra of the more slowly evolving SLSNe-I change more slowly. However, the k-corrections based on PTF 12dam and iPTF 13ajg are similar, so the differences in k-corrections for faster and slower SLSNe-I should be small. Given this similarity, and due to the general lack of spectral series as complete as those for PTF 12dam, we apply the k-corrections derived from the spectra of PTF 12dam and their evolution to all of the SLSNe in our sample.

The spectra were not warped to match the observed photometry of the individual SLSNe. This could have led to more accurate k-corrections. However, the uncertainty from the fact that we use the spectrum of PTF 12dam as a reference for the k-correction for all individual SLSNe is likely larger than the precision that could be gained by such a procedure. In addition, to make a reliable warping, photometry in at least two bands (and much preferably three) would be necessary, and this was not always available. The spectra of PTF 12dam were carefully flux calibrated. To ensure a smooth evolution of the k-correction with time, we interpolate the individual k-correction values and obtain a smooth k-correction evolution in time for each SLSN, through a third-order polynomial fit of the individual k-correction values.

The residuals from the third-order polynomial fit of the k-correction values with time can be used to estimate the uncertainties on the k-corrections, which are between 0.004 and 0.05 mag in our sample, with an average of 0.02 mag. These values show the scatter around the best-fit k-correction curve. Fitting the k-correction through individual points ensures that potential outliers, e.g., due to inaccurate flux calibration of the reference spectra, become negligible because the fit is driven by the majority of the points. The dominant source of uncertainty on the k-correction is likely the fact that we use the spectrum of PTF 12dam as a reference for the k-correction for all SLSNe, but this cannot be trivially estimated. Using different SLSN spectra as a reference for the k-correction in a different but comparable sample of SLSNe, the uncertainties on the k-corrections are 0.01–0.1 mag (Wiis 2017, private communication Table 4.1.1), although these are potentially slightly overestimated because they are derived with linear fits to the k-correction data. We do not include the uncertainties on the k-correction in our photometry uncertainty budget.

Table 9 lists the values of the adopted smooth k-correction at the epochs of the PTF 12dam spectra, applied to the redshifts of the SLSNe of our sample. Our k-correction values are in agreement with those of Nicholl et al. (2015b; K_gr = −0.3 before peak for LSQ 14bdq at z = 0.347). The k-corrections that we apply rely on the assumption that the spectra of our SLSN sample are similar to those of PTF 12dam out to late epochs. At late times, the assumption of similarity among the spectra is less certain. We therefore recommend exercising caution in trusting our k-corrections at late epochs.

For the comparison with the Ib/c sample at maximum light, we calculate the k-correction from the observed r to rest-frame r using the spectrum at the peak of PTF 10tqv and following Hogg et al. (2002). We then derive the rest-frame g by applying a color correction from the observed mean g − r = 0.36 mag of a large sample of Type Ib/c SNe of Prentice et al. (2016). Where necessary, we convert B − V measurements from Prentice et al. (2016) at the V peak to g − r and adopt the weighted average, and otherwise directly use the observed g − r at the g peak. The standard deviation on the g − r distribution is 0.34 mag (0.25 and 0.23 mag for the g − r and B − V distributions). Since g − r evolves significantly for SNe Ib/c (e.g., Taddia et al. 2015; Prentice et al. 2016), the r to g conversion used here for SNe Ib/c is most reliable around SNe peaks.

We calculate the absolute magnitudes using the distance modulus for a given z (e.g., Hogg 1999). At the z considered here (∼0.3), the difference in distance modulus obtained from different cosmology models is negligible compared to the uncertainties in the observed apparent magnitudes. The redshifts of the SLSNe are derived in most cases to three decimal digits (see Table 1). In fact, here we are interested only in the relative luminosity distances between different SNe, and the relative uncertainties are even smaller. The uncertainties on the absolute magnitudes are therefore largely dominated by the uncertainties on the observed apparent magnitudes, and we do not make any attempt to include uncertainties due to the distance estimate.

We determine the peak times and magnitudes by fitting a second-order polynomial to the rest-frame g-band magnitudes around the maximum brightness, typically between −30 and 30 days around the approximate peak, or adjusting this interval to adapt to the data coverage. The fitted curves and the relevant time intervals are shown in Figures 18(a)–(e).

Figure 3, top panel, shows the peak-magnitude distribution of H-poor SLSNe (solid blue), Type Ic-BL SNe (shaded yellow), and Type Ib, Ic, and Ib/c SNe (solid orange), all from the PTF survey, for a brightness bin of 0.2 mag. Note that only SNe where the peak could be observed and constrained are included in this plot. When calculating the number of SNe for each brightness bin, it is important to consider the observational biases. Although SLSNe are bright enough to be observed at larger distances, many normal SNe could actually be exploding at those distances, but be too faint to be detected (Malmquist bias). To compare the numbers of SNe in a fair way, it is therefore necessary to normalize the numbers to the same comoving volume. We calculate the volumetric correction V_c for each SN as the ratio between the volume probed by the most luminous SLSN in our sample (${M}_{g,\max }=-22.42$ mag at peak) and the volume probed by the individual SN, given the limiting magnitude of the PTF survey of m_lim = 20.5 mag (Cao et al. 2016), i.e., the maximum luminosity distance at which each SN would have been observed with this limiting magnitude. The volumetric correction factor V_c is then expressed as follows:

$$\begin{eqnarray}&&{V}_{c}={V}_{\max }{/V}_{\max ,i}={\left(\displaystyle \frac{{D}_{L,\max }}{1+{z}_{\max }}\right)}^{3}/{\left(\displaystyle \frac{{D}_{L,\max ,i}}{1+{z}_{i}}\right)}^{3},\end{eqnarray} \tag{ 1 }$$

where the luminosity distance of the brightest SN in the sample is ${D}_{L,\max }={10}^{((({m}_{\mathrm{lim}}-{M}_{g,\max })+5.)/5.)}$, and the luminosity distance for each individual SN is ${D}_{L,\max ,i}={10}^{((({m}_{\mathrm{lim}}-{M}_{g,i})+5.)/5.)}$. Figure 3, bottom panel, shows the peak-magnitude distribution after the volumetric correction. The mean peak magnitude of the SLSN sample is $\langle {M}_{g,\mathrm{peak}}\rangle =-21.14$ mag with a standard deviation of 0.75 mag.

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The light curves of H-poor SLSNe can show a change in decay rate (e.g., Inserra et al. 2013). Here we independently characterize the postpeak early- and late-time decay rates of H-poor SLSNe with linear fits to the early-time and late-time data separately. We study the early decay with a linear fit to the rest-frame g magnitudes in a time interval between the peak and typically 60 days after peak. In some cases, this interval was adjusted to the data coverage, or to avoid changes of slope. The selected time intervals and the resulting linear fits to the data are displayed in Figures 18(a)–(e) (solid curves). We define the late-time decay as typically beyond 60 days after peak and characterize the decay rate with a linear fit to the data, in the same way as we did for the early-time decay. The linear fits to the late-time decays are displayed in Figures 18(a)–(e) (solid curves, typically beyond 60 days after peak).

Figure 4 shows the distribution of the early-time decay slopes (top panel) and the the late-time decay slopes (bottom panel). SLSNe that were originally classified as sub-type R within the PTF survey are marked separately in this figure and compared to the rest of the sample. The original criterion for being classified as an SLSN-R was either a slow decline or spectral similarity with SN 2007bi, with no quantitative threshold. We do not intend to use these criteria as a meaningful classification, but rather to test this classification scheme, because it is often used in the literature (e.g., Gal-Yam 2012; Inserra et al. 2017).

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The decay rate of most SLSNe slows down from early to late times. The decay rates and the times of transition from a faster to a slower decay (the intersections between the early- and late-time linear fits) are reported in Table 8.

We smooth the SN light curves to be able to further measure the rise and decay times more easily. We model the observed light curves with a nonparametric model, as follows. We first fit a first-order polynomial to the rest-frame g-band flux light curves locally. Then, we consider a fitting interval of 5 days (at phases until 5 days after peak), 10 days (at phases beyond 50 days after peak), and proportional to the phase (0.2 times) otherwise. For the interpolations, we use a Gaussian smoothing kernel that weights the fluxes according to their phase distance to each interpolated point. The smoothing algorithm also uses the uncertainties on the photometry to weight the data points. In order to avoid mathematical artifacts, a few auxiliary points are added to the observations. The light-curve smoothing algorithm is described in more detail in Rubin et al. (2016). In a few cases, to avoid unphysical wiggles in the smoothed light curves for poorly sampled regions, we binned scattered data during small time intervals. Namely, we binned the data for PTF 10aagc between 32 and 44 rest-frame days after peak; PTF 09cwl between 122 and 141; PTF 10vwg between 44 and 62; PTF 11rks at 55, and between 144 and 154; and PTF 12gty between 141 and 158. We adopt the formal error on the smoothed fluxes computed by the smoothing algorithm, and assume a minimum uncertainty of 10% of the flux in those cases where the formal errors are smaller.

The light-curve smoothing fits to the data in flux space, including the auxiliary points, are shown in Figure 19. The collection of all smoothed light curves, normalized by the peak magnitude, is shown in Figure 5. Even when normalized to the peak, there is a wide variety of light-curve behaviors among H-poor SLSNe, and the scatter is too large to reduce them to a single template. Remarkably, there is no clear gap between fast- and slow-decaying SLSNe.

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We derive the times to rise (and decay) by 1 mag to (from) the peak, ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ (${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$) by inspecting the smooth light curves (Section 4.3). Figure 20 displays the time intervals within 1 mag from the peak. Table 8 lists the resulting rise and decay times. The errors are estimated starting from the errors on the smoothed light curves (Section 4.3). We create a pseudo-random normal distribution of the smoothed flux errors around the smoothed light curves, through n Monte Carlo realizations, and we estimate n rise (and decay) times. We finally derive the uncertainties on the rise (decay) times from the standard deviation of the distribution of rise (and decay) times and assuming a minimum uncertainty of 2 days. We test for convergence of our results by varying the number of Monte Carlo realizations n between 10, 100, 1000, and 10,000, and eventually use n = 1000.

In Figure 6, we investigate the cross-correlations between ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$, ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$, their sum, and the peak magnitude. There is a clear correlation between ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$.

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We fit this correlation linearly, assuming ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}=A\,+B\times {t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and including the observed uncertainties in both x and y axes, for each SN type. SNe where the data are not sufficient to constrain ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ are excluded from this fit, as reported in Table 8. The results of this fit are shown in Figure 6 (dotted curves) and reported in Table 2. We also compute the Pearson and Spearman correlation coefficients, which measure the strength (tightness and monotonicity) of a correlation not taking the observed uncertainty into account, and their null probabilities. These are listed in Table 2.

Table 2.  Normalizations and Slopes of the Linear Fits of the Correlations between Rise Times and Decay Times (Figures 6 and 7)

Type	A	B	r	pr	ρ	pρ
${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}=A+B\times {t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$
SLSN	12.93 ± 6.55	0.36 ± 0.15	0.69	0.057	0.76	0.028
${t}_{\mathrm{rise},1/2}=A+B\times {t}_{\mathrm{fall},1/2}$
SLSN	17.32 ± 6.22	0.21 ± 2.06	0.39	0.270	0.44	0.206

Note. r and ρ are the Pearson and Spearman correlation coefficients, respectively, and are listed with their respective null probability (p_r and p_ρ).

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We also find a trend between the peak magnitudes and ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$, ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$, and the peak width.

We further characterize the rise and decay times using the method of Prentice et al. (2016), who measured the time required to double or halve the flux with respect to the peak flux of a sample of stripped-envelope SNe (Ib/c). We derive ${t}_{\mathrm{rise},1/2}$ and ${t}_{\mathrm{fall},1/2}$, the time to double and halve the flux, respectively, for the PTF SN sample considered in this paper, using the smoothed flux light curves (Section 4.3). The resulting ${t}_{\mathrm{rise},1/2}$ and ${t}_{\mathrm{fall},1/2}$ are reported in Table 8. We calculated the uncertainties in the same way as for the rise and decay times by 1 mag (Section 4.4).

Figure 7 shows the comparison of ${t}_{\mathrm{rise},1/2}$ and ${t}_{\mathrm{fall},1/2}$ among the different samples and SN types, and compares it with the results of Prentice et al. (2016). We linearly fit the correlations between rise and decay times for each SN type in the same way as for the rise and decay times by 1 mag (Section 4.4). The results of the fit are shown in Figure 7 (dotted curves) and reported in Table 2.

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Figure 3, top panel, shows the distribution of the rest-frame g-band absolute peak magnitudes of SLSN, Ib/c, and Ic-BL. These SNe are all discovered by the PTF survey and separated into these three classes through spectroscopic classification (Quimby et al. 2018; Schulze et al. 2018). The k-correction has been applied as described in Section 3.12. Clearly, not all peak magnitudes of SLSNe are brighter than < −21 mag. This threshold was an operational definition that was used to start characterizing SLSNe in the early days of their discovery (Quimby et al. 2011; Gal-Yam 2012). In fact, spectroscopically classified SLSNe-I from PTF span a wider range in absolute peak magnitudes, $-22.5\lesssim {M}_{g,\mathrm{peak}}\lesssim -20$ mag. The mean absolute peak magnitude in the PTF sample is $\langle {M}_{g,\mathrm{peak}}\rangle =-21.14$ mag, with a standard deviation of 0.75 mag, which is about 2 and 4 mag brighter than the mean for SNe Ic-BL and Ib/c in our sample, respectively. The SLSN mean peak magnitude in the PTF sample is similar to what Lunnan et al. (2018) found for the higher-z sample from PS1, and thus we confirm no evidence for evolution of the SLSN peak luminosities with z on the currently available data.

Furthermore, the peak magnitudes of SLSNe-I are all brighter than SNe Ib/c. The gap between the brightest SN Ibc and the faintest SLSN-I is of about 0.5 mag, although somewhat uncertain given the limited size of the samples and the uncertainty on the host-galaxy extinction for the SNe Ib/c. SNe Ic-BL are typically brighter than SNe Ib/c and fill this gap. The distribution of peak magnitudes is continuous from SNe Ib/c to SNe Ic-BL and SLSNe. There is very little overlap between the SLSN population and SNe Ic-BL.

It is crucial to take into consideration the fact that fainter SNe can be observed and counted only out to smaller distances. When applying the volumetric correction to compensate for this bias (Figure 3, bottom panel), the peak magnitude distribution decays smoothly and exponentially from SNe Ib/c to Ic-BL and to SLSNe. Another important bias to keep in mind is spectroscopic completeness. Because SNe Ib/c exist in the same parameter space as Type Ia or IIp SNe, some of them may be not selected for spectroscopic classification and therefore missing from those that we sample.

We conclude that the peak magnitudes of SLSNe are brighter than those of SNe Ib/c and Ic-BL. However, there is no evidence for SLSNe being drawn from a separate population when considering only the peak-magnitude distribution and taking the volumetric correction into consideration. Further evidence for the difference between SLSNe and SNe Ib/c comes from the rise and decay timescales, which we discuss in Section 5.4.

Figure 8 shows the evolution of the observed g − r color for individual SLSNe. The g − r color seems to increase at early epochs, until a few tens of days after peak. The mean observed $\langle g-r\rangle$ at peak is 0.24 mag, with a standard deviation of 0.37 mag. At later times, the g − r color evolution seems to stabilize at around ∼0.5 mag, and perhaps higher.

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This g − r evolution in SLSNe is overall similar to that in SN Ib/c (see Figures 22–25 of Prentice et al. 2016). The g − r color in SLSNe may, however, rise for a longer time (up to ∼50 days after peak, while SNe Ib/c reach a plateau at 10–20 days after peak), and to lower g − r (the color plateau in SNe Ib/c spans roughly between 0.5–1 mag). However, we caution against a direct comparison of the observed g − r between nearby SNe and SLSN, given their higher redshift. Indeed, after k-correction, the rest-frame g − r colors at peak in SLSNe span roughly between −0.6 and 0.0 mag. The observed-frame colors are reported here only as an observational reference. In Section 5.9, we further discuss rest-frame g − r colors at peak in SLSNe.

Figure 4 shows the postpeak decay rates at early times (top panel, typically before 60 days after peak) and late times (bottom panel, typically beyond 60 days after peak), as derived in Section 4.2. The two distributions are quite different, indicating that at early times, SLSNe decay faster than at late times. The mean SLSN decay rates are 0.04 and 0.013 mag day⁻¹ at early and late times, respectively, with standard deviations of 0.02 and 0.005 mag day⁻¹. All SLSNe with available data in our sample slow down their decay rate from early to late times. Moreover, their late-time decay rate settles around ∼1 mag decay per 100 days. This rate is similar to the radioactive decay of ⁵⁶Co to stable ⁵⁶Fe (more specifically, 102.3 days mag⁻¹; Nadyozhin 1994; Wheeler & Benetti 2000). While at late times all SLSNe-I with available data show this slow decay, some selection biases may be present, because fast decays at late times may fall below the detection thresholds and not be measurable.

The late-time decays expected within the magnetar scenario can, under certain circumstances, mimic the radioactive ⁵⁶Co decay (e.g., Moriya et al. 2017). We further discuss this in Section 5.8.

The SLSN PTF 12hni is excluded from the late-time decay distribution because it exhibits a clear rebrightening (in all covered filters) and thus a negative decay rate, starting at about 75 days after peak, as reported in Table 8. This could represent a case where interaction with the CSM re-energizes the light curve at late times, through the transformation of kinetic energy into luminosity. This typically requires a high optical depth, and one may naively not expect to observe broad lines and absorption features in this case (see, however, Moriya & Tominaga 2012). The rebrightening of PTF 12hni was not covered by spectral observations (Quimby et al. 2018).

In Figure 4, we distinguish between classical H-poor SLSNe and those SNe originally classified as SLSNe-R within the PTF survey due to their slow decay early after the peak (consistent with radioactive decay, or spectrally similar to SN 2007bi). We stress again that we do not intend to use these criteria as a meaningful classification, but rather to test this classification scheme. At late times, the decay rates of SLSNe-R indeed cluster around the radioactive nickel decay rate, which is expected given the way SLSNe-R were originally selected. However, at early times, a couple of SLSNe-R have steeper decay slopes. Moreover, there is no evidence for a bimodal distribution in the decay properties in Figure 5, where all smoothed light curves are plotted together, after being normalized around the peak. Therefore, there is no clear separation between SLSNe-R and classical SLSNe-I. This casts doubt on the existence of SLSNe-R as a separate class, as initially suggested by Gal-Yam (2012).

Nevertheless, we note that early-time light-curve features are more common in SLSNe-R than in classical SLSNe-I. While none of the classical SLSNe-I show these features, three out of five SLSNe-R with early-time coverage (10nmn, O. Yaron et al. 2018, in preparation; 12dam and 13dcc, Vreeswijk et al. 2017) and possibly a fourth case (13ehe, Yan et al. 2015) have an early plateau or bumps of different strengths. At late times, the light-curve decay in SLSNe-R shows wiggles and bumps in virtually all events. Similar conclusions have also been drawn by Nicholl & Smartt (2016) and Inserra et al. (2017). In the case of the hybrid SLSN iPTF 15esb (late-time emergence of H emission), Liu et al. (2017) showed that the double peak of the light curves could be explained with a multiple-shell CSM interaction model. On the other hand, classical SLSNe-I might show fewer late-time features, the only clear example being PTF 12hni and perhaps PTF 12gty, which are the least luminous among our sample. However, the paucity of late-time data for such events prevents us from drawing firm conclusions. For this reason, we keep open the possibility that two separate subclasses of SLSNe-I exist (slowly/rapidly declining), until further evidence is collected. The light curves of all potential SLSNe-R from PTF are shown together in Figure 9. The presence of bumps in the light curves indicate that either CSM interaction or multiple sources are responsible for powering the light curve, as also found by Vreeswijk et al. (2017) and Inserra et al. (2017).

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In Figure 6, we compare the rise and decay timescales (${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$; Section 4.4) and peak magnitudes for PTF SLSNe-I and PTF Type Ib/c and Ic-BL SNe.

The peak magnitudes are brighter for SLSNe-I than for SNe Ib/c and Ic-BL, as discussed in Section 5.1. We find possible mild trends between M_g and ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and between M_g and ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$, albeit with a very large scatter, and mostly when all SNe are considered. Indeed, neither luminous and fast-evolving events nor faint and slow-evolving ones are observed. A correlation between peak luminosity and rise time was observed for SNe IIn by Ofek et al. (2014a) and is consistent with the explanation of CSM interaction. The predictions for this correlation are in Ofek et al. (2014b).

The SLSNe in our sample tend to have longer rise timescales than SNe Ib/c; see below. Most SNe Ib/c and Ic-BL also tend to decay faster than SLSNe, although there are a few exceptions of slow-decaying, long-lived SNe Ib/c (e.g., PTF 11bov, also known as SN 2011bm; Valenti et al. 2012).

We find a correlation between ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ and ${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ for SLSNe (Figure 6, top left panel). The parameters and strength of this correlation are reported in Table 2. Such a correlation is expected for both magnetar and nickel decay models (e.g., Nicholl et al. 2015a). This correlation is continuous and does not show two separate classes of SLSNe-I, in contrast to the findings of Nicholl et al. (2015a). Interestingly, SLSNe and "normal" SNe Ibc follow separate ${t}_{\mathrm{rise}}^{{\rm{\Delta }}1\,\mathrm{mag}}$–${t}_{\mathrm{fall}}^{{\rm{\Delta }}1\,\mathrm{mag}}$ correlations. Although the correlation is not strong, there is an evident offset toward longer rise times for SLSNe with respect to SNe Ib/c. The offset is roughly a 10 day longer rise timescale for SLSNe. Finally, this correlation has a large scatter, and moreover, in some cases (PTF 11dij and PTF 11rks), the measured rise and decay times are similar to those of SNe Ib/c. Therefore, the above-mentioned correlation cannot be used to distinguish between SLSNe-I and SNe Ib/c, but only as an indicator of the average properties of the two populations.

As a sanity check, we further study the rise and decay times with a different approach, by considering the time to rise and decay by half-flux from the peak (${t}_{\mathrm{rise},1/2}$ and ${t}_{\mathrm{fall},1/2}$, Section 4.5). Again, the rise and decay times correlate continuously for SLSNe-I and the correlations are different for the three different classes of SNe. (The parameters and strength of this correlation are reported in Table 2.)

The difference between the correlations of the rise with the decay timescales for SLSNe and SNe Ib/c (and Ic-BL) is evident using both independent methods (Δ1 mag and half-flux). This suggests that SLSNe have longer rise timescales than SNe Ib/c, even for similar decay timescales. Observationally, we conclude that SLSNe show overall different light-curve properties from SNe Ib/c and Ic-BL. Therefore, SLSNe can be considered as a separate population, not only from a spectroscopic (Quimby et al. 2018) but also from a photometric perspective.

We use the spectral information derived from a well-observed event to estimate the bolometric luminosity L_bol from single-band photometry. We adopt the conversion from the rest-frame g band to the bolometric luminosity derived for PTF 12dam by Vreeswijk et al. (2017) up to 334 rest-frame days after peak. The bolometric light curve of PTF 12dam was constructed from the observed spectral series and blackbody models of the UV/optical data. More details on this derivation are explained in Vreeswijk et al. (2017). We apply the bolometric correction from absolute magnitudes to the bolometric luminosity of PTF 12dam to all SLSNe in our sample, i.e., by basically adding a constant to the rest-frame g-band absolute magnitudes, where this constant evolves with the SN phase. This is valid under the assumption of spectral similarity among SLSNe-I. While there are strong indications for such similarity in our sample (Quimby et al. 2018), this is not always guaranteed, especially at late times when the spectral coverage is typically poorer than around peak. A solid case-by-case bolometric correction can in principle only be attempted for the few best observed cases with sufficient spectral coverage. Due to the paucity of such complete data sets, this cannot be done for the full sample and is beyond the scope of the current paper. Nevertheless, given the overall similarity among the spectra in our sample, it is still informative, as a first approximation for the study of the energetics, to use a simple bolometric correction to derive the bolometric luminosities. Because the relative shapes of the light curves does not change between different SLSNe in bolometric luminosity, we do not show the individual light curves. We report the bolometric luminosities at peak in Table 3. The total radiated energy is then derived by integrating the bolometric light curves. Because the bolometric light curves are defined over a limited time interval, the derived radiated energies are lower limits.

Table 3.  Radiated Energy and Nickel Mass Estimates from the Peak Luminosity and the Late-time Decay

PTF ID	$\mathrm{log}{L}_{\mathrm{bol},\mathrm{peak}}$	$\mathrm{log}{E}_{\mathrm{rad}}$	${M}_{\mathrm{Ni},\mathrm{peak}}$	${M}_{\mathrm{Ni},\mathrm{decay}}$
	(erg s−1)	(erg)	$({M}_{\odot })$	$({M}_{\odot })$
09as	43.4	≥49.3	≤3.4	⋯
09atu	44.3	≥51.2	≤28.3	⋯
09cnd	44.5	≥51.3	≤37.3	≤5.5
09cwl	44.4	≥51.2	≤33.7	≤5.5
10aagc	43.7	≥50.0	≤6.5	⋯
10bfz	44.0	≥50.4	≤12.1	⋯
10bjp	43.8	≥50.5	≤8.8	⋯
10cwr	44.3	≥50.8	≤23.5	≤1.0
10hgi	43.7	≥50.5	≤7.2	≤1.7
10nmn	43.7	≥50.6	≤8.3	≤4.5
10uhf	43.9	≥50.3	≤9.8	⋯
10vqv	44.2	≥50.8	≤21.4	≤6.0
10vwg	44.4	≥51.2	≤32.0	≤5.0
11dij	44.2	≥50.8	≤21.0	≤0.9
11hrq	43.6	≥50.2	≤6.2	≤3.1
11rks	43.9	≥50.6	≤10.7	≤2.7
12dam	44.3	≥51.2	≤24.6	≤9.7
12gty	43.7	≥50.5	≤5.9	≤2.4
12hni	43.6	≥50.1	≤4.8	≤0.8
12mxx	44.2	≥50.8	≤22.6	⋯
13ajg	44.6	≥51.2	≤49.5	≤5.4
13bdl	43.9	≥50.6	≤9.6	⋯
13bjz	44.0	≥50.0	≤12.1	⋯
13cjq	44.1	≥50.7	≤15.2	≤3.4
13dcc	44.5	≥51.3	≤48.6	≤5.0
13ehe	44.2	≥51.2	≤18.3	≤6.5

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We investigate whether the peaks and light curve decays of SLSNe-I could be powered by Ni decay, using two independent methods. First, we derive a very rough estimate of what the required nickel masses would be if the SN peaks were completely powered by nickel. We use the relation ${L}_{\mathrm{peak}}=\alpha {E}_{\mathrm{Ni}}$ = $(6.45\,\times {10}^{43}{e}^{-({t}_{\mathrm{peak}}/{\tau }_{\mathrm{Ni}})}$ + $1.43\times {10}^{43}{e}^{-({t}_{\mathrm{peak}}/{\tau }_{\mathrm{Co}})})$ × ${M}_{\mathrm{Ni}}/{M}_{\odot }$ (Nadyozhin 1994; Stritzinger & Leibundgut 2005), where τ_Ni = 8.8 days and τ_Co = 111.3 days, and assume no deviation from the Arnett rule (α = 1, Arnett 1979). The time of explosion is quite uncertain in our sample, because the rise times are often not well-covered.³⁰ Thus, we use a representative t_peak = 70 days. For a few SLSNe-I that show indications for a longer rise time, we assume an explosion time of 100 days before peak (namely for PTF 10nmn, PTF 11hrq, and PTF 13dcc), and we assume the literature explosion time of 66 days before the peak for PTF 12dam (Vreeswijk et al. 2017). The uncertainties in this Ni mass calculation are of about 20% for an uncertainty in explosion date of about 30%. The Ni mass that we derive with this method for PTF 12dam is similar to what has been derived by Vreeswijk et al. (2017) with a more detailed Ni decay model of the full light curve. In this exercise, the main assumption is that the light curves are totally powered by radioactive Ni decay, while in fact there may be a significant contribution from CSM interaction, magnetars, or other sources. The derived nickel masses are therefore upper limits of the true values, for a peak time of 70 days after explosion. The results are reported in Table 3.

Second, we derive what the required nickel masses would roughly be if the SN late-time decay would be completely powered by nickel. In this case, we compare the SLSN decay, if sufficient photometry is available, to the decay of SN 1987A. For this SN, the well-studied decay is thought to be powered by the radioactive decay of 0.07 M_⊙ of ${}^{56}\mathrm{Ni}$ (Fransson & Kozma 2002; Seitenzahl et al. 2014). To compare the SLSNe light curves with SN 1987A, we assume an explosion date for the SLSNe and shift the bolometric light curves to that of SN 1987A (taken from Pun et al. 1995), using the same method as Gal-Yam et al. (2009). We shifted the light curve of SN 1987A to match the potential transition from the diffusive phase to the radioactive decay in the SLSN light curves or to the late-time decay. The explosion dates are quite uncertain. We assume the same explosion dates as discussed above. While the assumption on the explosion dates are not secure, here we are only interested in a zero-order estimate of the Ni masses from the tails. Figure 10 shows the comparison between the SLSN light curves and SN 1987A. The Ni masses derived from the light-curve decay are labeled on the figure and reported in Table 3. These masses are upper limits, because they are calculated assuming that the late-time light curves are powered only by radioactive Ni, with no other contribution. The typical uncertainties on these nickel masses are large, roughly of the order of 50% (accounting for a shift in explosion date of up to 100 days). Despite the large uncertainties, these estimates are useful for the comparison with the nickel masses derived from the peaks.

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In Table 3, we compare the nickel masses that we derived above with the two methods, one based purely on the peak luminosity and one based purely on the late-time decay. It is evident that the nickel masses derived from the peak luminosities are much higher than the nickel masses derived from the SN decay, both for fast- and slow-evolving SLSNe-I. This suggests that the SLSN peaks are not powered by nickel. This confirms the results of Inserra et al. (2013). In addition, the large ejecta masses required for powering the SLSN-I peaks with Ni radioactivity would increase the diffusion times, and therefore the light curves would show broader peaks than what is observed. A different component, such as magnetar spin-down or CSM interaction, is likely causing the high peak luminosities of most SLSNe-I. A factor of 5 discrepancy between the Ni masses required for the peak and the late-time decay was also found for SN 1998bw, which was possible to reconcile only with asymmetry of the ejecta (Dessart et al. 2017). Strong evidence for asphericity of this event, based on the spectra, was found by Mazzali et al. (2001) and Maeda et al. (2002, 2006). If asymmetry is relevant for SLSNe-I as well, the discrepancy between the nickel masses derived from the peak and the late-time decay may be partly mitigated (perhaps by a factor of 5 for asymmetries levels similar to SN 1998bw).

When data coverage is available, we observe a late-time decay of SLSNe-I, which is close to the radioactive decay of ⁵⁶Co to stable ⁵⁶Fe, as observed from the late-time decay slopes (Section 4.2). The nickel masses derived from this late-time decay are between ≤1 and ≤10 M_⊙. This suggests that while nickel production is not the main source powering the light curve peaks, a nickel component could be important, and perhaps dominant, at late times. While the derived Ni masses are upper limits, producing up to 10 M_⊙ of Ni is challenging in classical SN models. The PISN model can produce 1–10 M_⊙ of nickel from progenitor stars with cores of 90–105 M_⊙ (Heger & Woosley 2002).

In the case of PTF 10hgi, the only data point at late times seems fainter than what would be predicted from the decay rate of ⁵⁶Co (Figure 10). The current ⁵⁶Ni mass estimated from the light-curve decay is M_Ni ∼ 2 M_⊙. However, estimating the Ni mass directly by scaling the SN 1987A light curve to the fainter data point at late times would provide M_Ni ∼ 0.2 M_⊙.

One potential power source of SLSN-I light curves is radioactive decay of ⁵⁶Co to stable ⁵⁶Fe (e.g., Gal-Yam et al. 2009). The half-life time of the ⁵⁶Co decay is 77.2 days (Junde et al. 2011). As we discussed above, it is rather unlikely that the SLSN peaks are powered by radioactive decay because of the discrepancy between the Ni masses required by the peak luminosities and those required by the late-time decays. One possibility is that the late-time light curves are powered by radioactive decay. Indeed, we showed in Section 4.2 that whenever observable, the SLSN-I light curves tend to slow down, and at late times settle around the 0.01 mag day^–1 decay rate, which is typical of radioactive ⁵⁶Co decay with full trapping. On the other hand, Inserra et al. (2017) argued that SLSNe tend to decay faster than the radioactive rate, and therefore could not easily be associated with ⁵⁶Co decay. However, the escape of γ-rays can increase the decay rate. In this section, we investigate under which conditions γ-ray escape can efficiently induce a light-curve decline that is faster than the nominal radioactive decay rate.

The radioactive decay energy (RDE) deposition is the heating/excitation/ionization of the SN ejecta because of radioactive emission of γ-rays (and e⁺) and the subsequent acceleration of electrons through Compton scattering (Jeffery 1999). This phenomenon is important for SNe Ia as well as for core-collapse SNe. After a diffusion phase when the γ-rays are fully trapped, a transition to a quasi-steady state marks the beginning of a regime where the decay is dominated by RDE deposition (and the diffusion timescale is much larger than the dynamical and radioactive timescales). At the transition point, the SN luminosity is purely determined by the total amount of radioactive material. The quasi-steady state decay is then exponential, starting with a radioactive slope that corresponds to full trapping. In time, the γ-rays can start to escape, and the decay can appear faster. We investigate here whether γ-ray escape is important for massive star progenitors.

We simulate the quasi-steady state decay from pure RDE deposition for stars with a density profile that has an inner plateau and decays exponentially (similar to the "s25e12" profile of Dessart & Hillier 2011), where the ⁵⁶Ni mass is distributed in the inner ejecta. We consider total ejecta masses between 25 and 100 M_⊙, ⁵⁶Ni masses between 5 and 20 M_⊙, and maximum expansion velocity between 10,000 and 20,000 km s⁻¹. The total ejecta mass and maximum expansion velocity determine the absolute value of the density profile at each point. These simulations cannot treat the diffusive phase, but only the light-curve decay beyond maximum light and beyond the transition to the quasi-steady state. As a sanity check, we reproduce the observed radioactive light-curve phase of SN 1987A given an expansion velocity of 6000 km s⁻¹, the same density profile as we used for SLSNe, total ejecta mass of 10 M_⊙, and $M{(}^{56}\mathrm{Ni})=0.07$ M_⊙.

Figure 11 shows the resulting light curves of our RDE deposition simulations for different initial parameters. In all cases, we observe ∼100% gamma-ray trapping at ∼150 days after the explosion. At later times the luminosity can decrease more rapidly because of the reduced trapping due to lower densities. This effect is stronger for high expansion velocities and high ${M}_{\mathrm{Ni}}/{M}_{\mathrm{ej}}$ ejected masses. Within this set of simulations, the deviation from a pure radioactive ⁵⁶Co decay ranges from 0.01 (still fully trapped) to a maximum of 0.27 mag (∼50% escape fraction) at about 450 days after explosion, with a decay rate of 1.33 mag in 100 days.

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The contours in Figure 12 show the energy deposition fraction (where 100% means full trapping) from our RDE deposition simulations for the cases of 5 and 10 M_⊙ of nickel. These results confirm that massive star progenitors, with high SN ejecta velocities and high Ni fraction in the ejecta, have limited trapping and therefore can decay faster than the radioactive exponential decline. Figure 12 can also be used to roughly estimate the total ejecta masses in case the expansion velocity and trapping are known from the spectra and light curves, respectively.

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An additional factor that can efficiently limit the trapping (and induce faster decays) is the potential mixing of ⁵⁶Ni in the outer layers. While this can be a dominant effect, we do not attempt to model it here, because this is highly dependent on the geometry of the mixing, which cannot be constrained.

We investigate whether the SLSN-I light curves could be powered by the spin-down of a magnetar. We consider an analytic magnetar model sourced from Arnett (1982) and Kasen & Bildsten (2010). The fitting technique is described in detail in A. Rubin et al. (2018, in preparation). The main model parameters are the initial pulsar spin P, the magnetic field B, the diffusion timescale ${t}_{m}\approxeq {M}_{\mathrm{ej}}^{3/4}{E}_{k}^{1/4}$ (where M_ej and E_k are the ejected mass and kinetic energy, respectively), and the explosion time t_exp. This is a basic modeling that includes neither photon escape nor multiple components. The uncertainties are derived with a Monte Carlo treatment and shown in Figure 22. The treatment of the opacity is the same as in Inserra et al. (2013).

Figure 21 shows a fit of the magnetar model described above to the bolometric SLSN-I light curves. The best-fit parameters are reported in Table 4 and displayed in Figure 13. In most cases, we obtain a satisfactory overall description of the light curves, with the exception of PTF 10hgi, PTF 10vwg, and PTF 11rks, for which we observe a different decay than predicted from our magnetar fit. In addition, the magnetar model does not describe well the light curve of PTF 11dij, both for the late-time decay and the early rise.³¹ The confidence levels of the best-fit parameters are shown in Figure 22.

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Table 4.  Resulting Parameter from a Magnetar Fit to the Bolometric Light Curves

ID	B	P	τm	texp	12 + log[O/H]
	(1014 G)	(ms)	(days)	(days)	(PP04 N2)
09cnd	1.56 + 0.12−0.11	${1.30}_{-0.18}^{+0.16}$	${63.0}_{-3.0}^{+3.2}$	$-{45.0}_{-1.0}^{+1.0}$	${8.22}_{-0.15}^{+0.09}$
09cwl	${2.24}_{-0.20}^{+0.27}$	${1.31}_{-0.39}^{+0.42}$	${57.7}_{-8.1}^{+6.2}$	$-{37.1}_{-2.2}^{+4.4}$	⋯
10bjp	${2.40}_{-0.55}^{+0.87}$	${4.63}_{-0.84}^{+0.29}$	${38.3}_{-7.0}^{+8.8}$	$-{46.6}_{-2.6}^{+3.0}$	${8.14}_{-0.10}^{+0.06}$
10cwr	${9.41}_{-0.90}^{+0.67}$	${0.06}_{-0.03}^{+0.10}$	${34.8}_{-1.0}^{+0.9}$	$-{12.6}_{-0.8}^{+0.6}$	${7.96}_{-0.24}^{+0.12}$
10hgi	${3.64}_{-0.13}^{+0.13}$	${4.82}_{-0.21}^{+0.21}$	${39.0}_{-2.1}^{+1.9}$	$-{42.0}_{-1.3}^{+1.1}$	${8.27}_{-0.06}^{+0.05}$
10nmn	${1.78}_{-0.04}^{+0.04}$	${0.89}_{-0.22}^{+0.40}$	${48.5}_{-1.9}^{+2.4}$	$-{104.7}_{-1.0}^{+0.4}$	${8.16}_{-0.04}^{+0.03}$
10vqv	${2.10}_{-0.14}^{+0.17}$	${3.22}_{-0.39}^{+0.30}$	${29.9}_{-4.9}^{+6.1}$	$-{28.5}_{-2.3}^{+2.3}$	${8.28}_{-0.05}^{+0.04}$
10vwg	${2.60}_{-0.43}^{+0.55}$	${2.72}_{-0.48}^{+0.43}$	${29.1}_{-4.8}^{+6.5}$	$-{34.1}_{-7.1}^{+2.8}$	⋯
11dij	${5.18}_{-0.46}^{+0.65}$	${0.93}_{-0.41}^{+0.39}$	${39.2}_{-1.3}^{+1.4}$	$-{27.6}_{-0.8}^{+0.4}$	${7.93}_{-0.20}^{+0.10}$
11rks	${4.15}_{-0.43}^{+0.51}$	${4.40}_{-0.43}^{+0.28}$	${26.7}_{-1.7}^{+1.7}$	$-{22.8}_{-1.3}^{+1.1}$	${8.14}_{-0.18}^{+0.13}$
12dam	${1.38}_{-0.05}^{+0.05}$	${1.68}_{-0.09}^{+0.08}$	${76.8}_{-2.2}^{+2.1}$	$-{56.0}_{-0.8}^{+0.8}$	${8.07}_{-0.01}^{+0.01}$
12gty	${2.62}_{-0.25}^{+0.27}$	${5.36}_{-0.32}^{+0.25}$	${54.0}_{-4.2}^{+4.1}$	$-{58.5}_{-2.3}^{+2.0}$	⋯
12mxx	${1.13}_{-0.19}^{+0.26}$	${2.59}_{-0.16}^{+0.08}$	${44.3}_{-4.2}^{+5.0}$	$-{48.4}_{-1.9}^{+2.1}$	${8.19}_{-0.19}^{+0.13}$
13ajg	${1.98}_{-0.08}^{+0.08}$	${1.73}_{-0.09}^{+0.09}$	${33.8}_{-1.3}^{+1.3}$	$-{28.7}_{-0.8}^{+1.4}$	⋯

Note. The last column lists the host-galaxy metallicity from Perley et al. (2016).

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Recently, Nicholl et al. (2017) have modeled a large literature SLSN-I sample with a magnetar model, including several published (i)PTF objects. The spin values P that we obtain are mostly consistent with the values of Nicholl et al. (2017). For PTF 12dam, we find a spin value $P={1.87}_{-0.08}^{+0.07}$ ms, which is a bit lower than the ∼2.3 ms values that were derived by Nicholl et al. (2017) and Vreeswijk et al. (2017), and also lower than that by Chen et al. (2015). The magnetic fields B that we obtain are in all cases higher than what was derived by Nicholl et al. (2017). For PTF 13ajg, our B value is more similar to that found by Vreeswijk et al. (2014). For PTF 12dam, our B value is more similar to those found by Chen et al. (2015) and Vreeswijk et al. (2017).

The late-time luminosities expected from the spin-down of a magnetar decline as t⁻² (e.g., Woosley 2010). In principle, such model may also be able to mimic the ⁵⁶Co decay of about 1 mag per 100 days, which we observe in SLSN-I late-time light curves. In the case of pure dipole radiation, the magnetar light curves start to have the same decay rate around 200 days postpeak. At later times, e.g., 400 days postpeak, the magnetar model is expected to have a decay rate that is noticeably slower than 1 mag/100 days (e.g., Inserra et al. 2013). However, the capability of a magnetar to mimic radioactive decay would require a pure dipole radiation and a narrow set of fine-tuned parameters, in particular for the magnetic field and Ni masses (Moriya et al. 2017). In Figure 14, we display the space parameter where a magnetar (in the dipole case) can mimic the radioactive ⁵⁶Co decay from Moriya et al. (2017), and compare it with the results from our magnetar fit on the SLSN light curves. The reference time intervals are derived from the observed times after peak where the SLSN decay follows the radioactive rate (Figure 10) and using the explosion times from the magnetar fits to the data (Figure 21). The magnetic field B is taken from the magnetar fit, while the Ni masses are taken from scaling the late-time light-curve decays (Section 5.6). The SLSNe for which these measurements are available are lying in the parameter space where the magnetar decay mimics the radioactive decay of ⁵⁶Co, with the marginal exception of PTF 11rks, for which a magnetar model does not describe the data well. Given that we do observe radiative-like light-curve decays at late times in SLSNe, it is perhaps not surprising that most of the derived magnetar parameters that we derive fulfill the radiative-mimicking criteria. These results indicate that we cannot disentangle between the magnetar and the radioactive decay models at these epochs, up to 400 days after explosion, but that observations at later times can be extremely powerful in disentangling between the two models.

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Intriguingly, a correlation between the magnetar initial spin and the host galaxy metallicity was found by Chen et al. (2017). In Figure 15, we show these quantities for our SLSN data using the host metallicities derived in Perley et al. (2016; PP04 N2 scale) and compare them to the correlation of Chen et al. (2017). We cannot confirm the existence of such a correlation in the PTF sample, or at least we find a large scatter (more than a factor of 10) in the derived spin periods, for a similar metallicity range.

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Because SLSNe can be observed out to large distances, as far out to z ∼ 4 (Cooke et al. 2012) and likely well beyond with future facilities, the prospects of using SLSNe for cosmological distance determinations is of primary interest. Indeed, Inserra & Smartt (2014) suggested that SLSNe-I may be standardizable, based on (i) the narrowness of the peak magnitude distribution, (ii) a weak correlation between the peak magnitude (at rest-frame 400 nm) and its decay after a certain time, and (iii) the dependence of the peak magnitude (at rest-frame 400 nm) on the SLSN-I color (rest-frame 400–520 nm). We test similar correlations in the current sample.

(i) We use the rest-frame g band as a proxy for the rest-frame 400 nm band of Inserra & Smartt (2014). The peak magnitudes of the SLSNe in our sample are widely distributed around their mean value ($\langle {M}_{g}\rangle =-21.14$ mag), with a standard deviation of 0.75 mag, which is almost twice as in the sample of Inserra & Smartt (2014).

(ii) Figure 16 shows the distribution of the rest-frame g peak magnitude, ${M}_{g,\mathrm{peak}}$, with its decay, ΔM_g, at 10, 20, and 30 days after the peak. These were all calculated from the smoothed light curves (Section 4.3) and can therefore be slightly different from the ${M}_{g,\mathrm{peak}}$ calculated with the second-order polynomial around the peak (Section 4.1). As a comparison, SLSNe-I and also Type Ic and Ic-BL SNe from PTF are shown in Figure 16. There are very weak correlations, highlighted by the linear fits to the data. However, none of these trends are significant correlations. In every case, the Pearson correlation coefficient is ∣r∣ < 0.3, and the intrinsic scatter is up to ∼0.8 mag for SLSNe-I; see Figure 16. The intrinsic scatter of the correlation is the scatter required for the correlation to have χ² ∼ 1 (Bedregal et al. 2006; Williams et al. 2010), and it is a way of discriminating the observational scatter from what is intrinsic to the correlation. The data and fit results for SLSNe-I are reported in Tables 5 and 6, respectively.

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Table 5.  Rest-frame g Magnitude at Peak and Its Decay from the Peak after 10, 20, and 30 days, Calculated from the Smoothed Light Curves (Section 4.3)

PTF ID	${M}_{g,\mathrm{peak}}$	${\rm{\Delta }}{M}_{g10}$	${\rm{\Delta }}{M}_{g20}$	${\rm{\Delta }}{M}_{g30}$
09atu	−21.8 ± 0.1	0.1 ± 0.1	0.2 ± 0.2	0.3 ± 0.2
09cnd	−22.1 ± 0.1	0.1 ± 0.1	0.3 ± 0.1	0.4 ± 0.1
09cwl	−22.0 ± 0.1	0.1 ± 0.1	0.2 ± 0.1	0.3 ± 0.2
10aagc	−20.1 ± 0.1	0.7 ± 0.5	1.2 ± 0.5	2.0 ± 0.6
10cwr	−21.6 ± 0.1	0.2 ± 0.1	0.7 ± 0.1	1.2 ± 0.1
10hgi	−20.3 ± 0.1	0.1 ± 0.1	0.4 ± 0.1	1.1 ± 0.2
10vqv	−21.5 ± 0.1	0.2 ± 0.2	0.5 ± 0.2	0.8 ± 0.2
10vwg	−21.9 ± 0.1	0.2 ± 0.1	0.5 ± 0.2	1.0 ± 0.3
11dij	−21.4 ± 0.1	0.0 ± 0.1	0.4 ± 0.1	0.9 ± 0.1
11rks	−20.9 ± 0.1	0.2 ± 0.1	0.5 ± 0.1	1.2 ± 0.1
12dam	−21.7 ± 0.1	0.3 ± 0.1	0.4 ± 0.1	0.5 ± 0.1
12gty	−20.1 ± 0.1	0.3 ± 0.1	0.5 ± 0.1	0.6 ± 0.2
12hni	−19.9 ± 0.1	0.1 ± 0.1	0.4 ± 0.1	1.0 ± 0.2
13ajg	−22.4 ± 0.1	0.3 ± 0.1	0.8 ± 0.1	1.2 ± 0.1
13cjq	−21.1 ± 0.1	0.1 ± 0.2	0.5 ± 0.2	1.3 ± 0.4
13dcc	−22.0 ± 0.1	0.4 ± 0.1	0.6 ± 0.1	1.1 ± 0.1
13ehe	−21.3 ± 0.1	0.1 ± 0.2	0.1 ± 0.2	0.2 ± 0.2

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(iii) In Figure 17 and Table 7, we compare the rest-frame g peak magnitude with the rest-frame g − r color at peak. The rest-frame r was derived from the observed i-band photometry using the same techniques as described for the r to rest-frame g conversion. The k-correction values are listed in Table 10. Both the rest-frame g- and r-band peaks were estimated by fitting a second-order polynomial to the data around peak. We could then constrain the rest-frame g − r for a subsample of SLSNe-I, as listed in Table 7.

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Table 7.  Rest-frame g and Rest-frame g − r Magnitudes at Peak for the SLSNe in Our Sample (see Figure 17)

PTF ID	${M}_{g,\mathrm{peak}}$	$g-r(\mathrm{peak})$
09atu	−21.78 ± 0.04	−0.24 ± 0.15
09cnd	−22.09 ± 0.03	−0.37 ± 0.07
10cwr	−21.59 ± 0.06	−0.25 ± 0.14
10hgi	−20.31 ± 0.02	−0.29 ± 0.16
11dij	−21.39 ± 0.19	−0.41 ± 0.19
11rks	−20.88 ± 0.13	−0.06 ± 0.19
12dam	−21.61 ± 0.05	−0.46 ± 0.15
12gty	−19.90 ± 0.15	0.11 ± 0.23
12mxx	−21.60 ± 0.08	−0.06 ± 0.16
13ajg	−22.42 ± 0.16	−0.34 ± 0.19
13dcc	−21.88 ± 0.15	−0.64 ± 0.47
13ehe	−21.26 ± 0.20	−0.09 ± 0.22

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The correlation ${M}_{g,\mathrm{peak}}=A+B\times {(g-r)}_{\mathrm{peak}}$ is weak, with a Pearson correlation coefficient of 0.59 (null probability p_r = 0.04) and a Spearman correlation coefficient of 0.57 (null probability p_ρ = 0.04). The normalization and slope are A = −19.5 ± 1.2 and B = 7.5 ± 3.8. In this case, the intrinsic scatter is consistent with zero. While this may suggest a potentially real correlation, it may simply be the consequence of the large error bars that we measure for g − r. A larger statistical sample is needed to further investigate the solidity of this relation.

The possible dust reddening $E(B-V)$ of the SLSN-I host galaxies is not taken into account here. While we expect the reddening to be small (e.g., Perley et al. 2016), there may be regions that are locally more dusty (e.g., Cikota et al. 2017). However, a case-to-case characterization of the host-galaxies' $E(B-V)$ is not possible here and is beyond the scope of this paper.

While we find the same overall trends as Inserra & Smartt (2014), the diagnostics above show mostly weaker correlations than reported by their work. One exception is the relation between the peak magnitude and g − r at peak, for which we find a similar Pearson correlation coefficient to Inserra & Smartt (2014).

We further investigate possible correlations of M_g with other variables, such as the early decay slope or host metallicity, with no convincing results. The weak trends of M_g with the rise and decay times are shown in Figure 6. We therefore cannot strengthen the claim that SLSNe might be standardizable candles with the current data. Future transient surveys may clarify this issue with much improved statistics (e.g., Scovacricchi et al. 2016).