Constraints on the Rate of Supernovae Lasting for More Than a Year from Subaru/Hyper Suprime-Cam

The past decade saw the dawn of large-scale time-domain astronomy. Many transient surveys, such as the Palomar Transient Factory (PTF; Law et al. 2009), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; Kaiser et al. 2010), the Dark Energy Survey (DES; Flaugher et al. 2015), the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018), and the Zwicky Transient Facility (ZTF; Bellm et al. 2019), have made astronomers realize that the universe is far more dynamic in time than previously assumed. Many transient surveys in the past decade pushed the frontier of short-timescale transients to search for, e.g., shock breakout of supernovae (SNe, e.g., Förster et al. 2016, 2018). They discovered many short-timescale transients that were not known a decade ago (e.g., Drout et al. 2014; Arcavi et al. 2016; Tanaka et al. 2016; Pursiainen et al. 2018; Prentice et al. 2018; Rest et al. 2018; Margutti et al. 2019; Ho et al. 2020; Tampo et al. 2020) and provided a constraint on the rates of the short-timescale transient phenomena (e.g., Berger et al. 2013). Even a transient survey with the cadence of 0.5 s has been recently performed (Richmond et al. 2020).

However, the frontier of the time-domain astronomy is not limited to the short-timescale phenomena. Indeed, many transients are observed to have durations of years. Some transients have intrinsically long timescales. For example, Type IIn SN (SN IIn) 2008iy (Miller et al. 2010) and HSC16aayt (Moriya et al. 2019a) had rise times of more than 100 days and their luminosity decline rates were similarly slow. Another SN IIn 2003ma had a quick (≃15 days) rise but it kept its luminosity for about 1000 days after the rise (Rest et al. 2011). iPTF14hls had spectra similar to those of the standard SNe IIP but its plateau phase lasted for more than 600 days (Arcavi et al. 2017; Sollerman et al. 2019).

A transient timescale becomes even longer if it appears at high redshifts thanks to the time dilation. Superluminous SNe (SLSNe; see Moriya et al. 2018; Gal-Yam 2019 for recent reviews), for example, can be observed up to z ≃ 5 with deep optical transient surveys (Tanaka et al. 2012, 2013) and last for several years in the observer frame (Cooke 2008; Cooke et al. 2012; Villar et al. 2018; Nicholl et al. 2020). Another type of interesting SNe predicted to last for several years in the observer frame is pair-instability SNe (PISNe). The existence of PISNe was predicted in the 1960s (Barkat et al. 1967; Rakavy & Shaviv 1967) but no conclusive PISNe have been discovered (see Terreran et al. 2017 for a recent candidate). Some PISNe are predicted to have luminous long-lasting light curves (LCs; e.g., Scannapieco et al. 2005; Kasen et al. 2011; Dessart et al. 2013; Kozyreva et al. 2014; Whalen et al. 2014; Gilmer et al. 2017). Thus, deep and wide transient surveys lasting for many years are ideal for searching for the long sought-after PISNe. The discovery of high-redshift PISNe will provide precious information on massive stars in the early universe.

Unlike the case of the short-timescale transient surveys, long-term patient monitoring of the same field is required to explore the long-timescale transient phenomena. There have been some transient surveys monitoring the same fields for more than a decade. For example, the Catalina Real-time Transient Survey (CRTS) has been observing the same field since 2007 (Drake et al. 2009, 2019). The limiting magnitudes of such long-lasting transient surveys are, however, rather shallow (e.g., V ≲ 20 mag for CRTS; Drake et al. 2009) and they are not suitable for searching for long-lasting faint transients such as high-redshift SLSNe and PISNe. In particular, PISNe are expected to appear preferentially at high redshifts because they require low-metallicity environments and we require deep transient surveys to discover them.

Hyper Suprime-Cam (HSC; Furusawa et al. 2018; Kawanomoto et al. 2018; Komiyama et al. 2018; Miyazaki et al. 2018), which is on the 8.2 m Subaru Telescope and has a field of view of 1.75 deg², is currently the best ground-based instrument to obtain deep and wide optical images (see, e.g., a list of etendue in Figure 1 of Förster et al. 2020). We previously reported our results of the half-year deep and wide transient survey conducted with HSC at the COSMOS field (Capak et al. 2007) in 2016–2017 (Yasuda et al. 2019). The half-year survey led to the discovery of many high-redshift SNe Ia at z ≳ 1 (Yasuda et al. 2019) as well as high-redshift SLSNe (Curtin et al. 2019; Moriya et al. 2019b). Even after the completion of the half-year survey, we have been monitoring the 1.75 deg² COSMOS UltraDeep (UD) field for more than 3 yr to explore the long-timescale transient phenomena. In this paper, we introduce the long-term time-domain data and present the result of our search for SNe lasting for more than a year. Although we did not obtain spectroscopic information for some long-lasting SNe we discovered, the long duration itself can be used to exclude contamination from normal SNe by searching for targets with durations much longer than normal SNe, allowing us to constrain the PISN and other long-lasting SN rates.

The rest of this paper is organized as follows. We first describe our HSC transient survey data in Section 2. We introduce SNe detected for more than a year in Section 3. We constrain the observational rate of long-lasting SNe in Section 4. Given all the long-lasting SNe we discovered are likely SNe IIn, we constrain the fraction of long-lasting SNe IIn among SNe IIn in Section 5. We constrain PISN and SLSN rates based on our survey data in Section 6. We conclude this paper in Section 7. We adopt the standard ΛCDM cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.3, and Ω_Λ = 0.7 throughout this paper.

The long-term monitoring observations with Subaru/HSC have been conducted in the COSMOS UD field for four seasons from late 2016 to early 2020, covering 3.3 yr. The COSMOS UD field is centered at (R.A., decl.) =(10^h00^m2860, +02°12'2100) and has the area of 1.75 deg², which is the field of view of HSC (Aihara et al. 2018a). The g-, r-, i-, and z-band filters are used for the long-term monitoring observations.

Our observational data are listed in Table A1. Figure 1 summarizes the limiting magnitudes of our transient survey. The first season observations (Season 1) were conducted as a part of the Subaru Strategic Program (SSP) with HSC ("Wide-field imaging with Hyper Suprime-Cam: Cosmology and Galaxy Evolution," PI: S. Miyazaki, Aihara et al. 2018a). The observations started on 2016 November 23 (UT dates are used in this paper) and ended on 2017 June 20. The overview of this transient survey is presented in Yasuda et al. (2019). The second season (Season 2) observations were conducted as the intensive program "HSC Supernova Cosmology Legacy Survey with Hubble Space Telescope" (S17B-055I, PI: N. Suzuki) from 2018 January 10 to 2018 April 21. No g-band data were obtained in Season 2. The third season observations (Season 3) were conducted again in the SSP as a back-up target when the sky conditions did not match the SSP criteria. We have the data from 2018 December 14 to 2019 May 9 in Season 3. No r-band data were obtained in Season 3. The fourth season data (Season 4) were primarily taken during the normal open-use programs "Variability-based AGN selection with extended COSMOS time-domain survey" (S20A-073, PI: M. Kokubo) and "Exploring the long-timescale transient frontier with HSC" (S20A-042, PI: T. Moriya). Some data were again taken in the SSP as a back-up target. The data in Season 4 were taken from 2019 October 26 to 2020 February 28.

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The reference images used to search for transients were taken in 2015 during the SSP as shown in Table A1. Because the r- and i-band filters on HSC were updated in 2016, the reference images for the r- and i-band filters were taken using the different filters from those used during the transient survey. However, the difference in the transmission is small ¹⁶ and no significant effects have appeared due to the filter difference.

The data reduction is performed with the same method as described in Yasuda et al. (2019). In short, the data are reduced by hscPipe (Bosch et al. 2018), which is a version of the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) stack (Jurić et al. 2017; Ivezić et al. 2019). The astrometry and photometry are calibrated relative to the Pan-STARRS1 (PS1) 3π catalog (Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013; Chambers et al. 2016). The astrometric accuracy is 0.04 arcsec (Aihara et al. 2018b). The image subtraction was performed with the method described in Alard & Lupton (1998) and Alard (2000).

The difference images obtained after the image subtraction were used to identify transient sources. The transient candidates were classified as real or bogus through a machine-learning technique adopting a convolutional neural network (CNN) as described in Yasuda et al. (2019). If a transient candidate is identified as real in two epochs after the CNN screening, it is regarded as a real transient.

To search for SNe lasting for more than a year, we first checked the results of the CNN screening. If a transient is classified as real at any time in one season, it is regarded as a detection in the season. From the transients detected in multiple seasons, we first excluded "negative" candidates that have negative flux because they exist in the reference images and we aim at discovering long-lasting SNe that appeared after the reference images were taken. Then, we excluded those identified on top of a point source in the reference image to avoid variable stars. We also exclude those located at the center (within 0.1 arcsec) of their host galaxy to avoid active galactic nucleus (AGN) activities. The value of 0.1 arcsec corresponds to 0.6 kpc at z = 0.5 and 0.8 kpc at z = 1–3. Most SLSNe and SNe IIn discovered in the local transient surveys have more off-sets from the host galaxy center (e.g., Schulze et al. 2020) and this criterion is not likely to miss many long-lasting SNe. Transients within 0.1 arcsec are independently studied to investigate AGN activities and any peculiar long-lasting transients can also be identified in the separate AGN study. After this screening, 2212 long-lasting SN candidates remained. The remaining candidates were mostly bogus caused by the failure of the image subtraction, but they were not excluded by the initial CNN screening. We visually checked all the candidates and identified three SNe that were detected for more than a year. These candidates are listed in Table 1 and their images are presented in Figure 2. We introduce them in the next section.

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Table 1. List of Long-lasting SNe

HSC Name	IAU Name	R.A.	Decl.	Redshift	Note
3 yr (4 seasons)
none
2 yr (3 seasons)
HSC16aayt	SN 2016jhm	10h02m2012	+02°48'433	0.6814 a	SN IIn
1 yr (2 seasons)
HSC19edgb	AT 2019aadg	10h01m3613	+02°42'254	${0.226}_{-0.01}^{+0.06}$ b	SN IIn?
HSC19edge	AT 2018lto	10h01m3155	+02°48'265	0.33094 c	SN IIn?

Notes.

^a Spectroscopic redshift from the SN spectra (Moriya et al. 2019a). ^b Photometric redshift of the host galaxy in the COSMOS2015 catalog (Laigle et al. 2016). Another photometric redshift solution at z = 2.7 is derived based solely on the HSC photometry (see Section 3.3.1). ^c Spectroscopic redshift of the host galaxy in the COSMOS2015 catalog (Laigle et al. 2016).

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We introduce our long-lasting SNe in this section. We discuss the discovery rates of long-lasting SNe based on our discovery. When we present the rest-frame magnitudes of SNe, the simple K correction of $2.5\mathrm{log}(1+z)$ is applied.

3.1. 3 yr-long SNe

3.2. 2 yr-long SNe

One SN (HSC16aayt) is detected for 2 yr (three seasons).

3.2.1. HSC16aayt

HSC16aayt (SN 2016jhm) is discovered at the beginning of our long-term HSC transient survey and identified as an SN IIn at z = 0.6814 through spectroscopic follow-up observations (Moriya et al. 2019a). The SN is located 0.71 arcsec away from the host galaxy center to the southwest (Figure 2). The first two seasons' data including spectra have been published in Moriya et al. (2019a) and we refer to their paper for the full details of HSC16aayt. It continued to be detected in Season 3 but was below the detection limit in Season 4. The LC of HSC16aayt is shown in Figure 3. The LC decline rate is similar to those of other SNe IIn (Figure 3).

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3.3. 1 yr-long SNe

Two SNe (HSC19edgb and HSC19edge) are detected for two seasons and they are 1 yr-long SNe in the observer frame. Although we do not have spectroscopic information from the two SNe, their location (offset from the host galaxy center) and LC similarity to those of SNe indicate that they are most likely SNe.

3.3.1. HSC19edgb

HSC19edgb (AT 2019aadg) was first detected on 9 January 2019 in the z band. It was not detected in the z band on 14 December 2018 when the first data in Season 3 were obtained. Therefore, it was discovered within 26 days after the explosion in the observer frame. The full LC data are presented in Figure 4.

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HSC19edgb was located 0.58 arcsec away to the east of the host galaxy center (Figure 2). The redshift of the host galaxy is uncertain. The photometric redshift estimated by the COSMOS2015 data is $z={0.226}_{-0.01}^{+0.06}$ (Laigle et al. 2016). The photometric redshifts obtained based on the HSC photometry in the reference images are split into two solutions at z ≃ 0.23 and 2.7 depending on the method of the photometric redshift estimation (Tanaka et al. 2018). Because the COSMOS2015 photometric redshift is based on the multiwavelength data ranging from radio to X-ray, the lower redshift solution at z ≃ 0.23 is more reliable. The COSMOS2015 photometric redshift of the extended galaxy at 1.9 arcsec toward east from the transient location is $z={0.227}_{-0.01}^{+0.009}$ and it also supports the low redshift solution.

Figure 5 shows the LCs of HSC19edgb at the two photometric redshifts at z = 0.23 and 2.7. In the case of z = 0.23, both luminosity and LC evolution are consistent with those of the slowly evolving SN IIn 1988Z (Turatto et al. 1993). HSC19edgb is naturally explained as an SN IIn if it is at z ≃ 0.23. The peak luminosity is consistent with low-mass PISN models, but their rise times are much longer than that of HSC19edgb (Kasen et al. 2011). Thus, it is not likely a PISN at z ≃ 0.23.

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If HSC19edgb is at z ≃ 2.7, it is an ultraviolet-bright transient (Figure 5). Some SLSNe, such as Gaia16apd (Kangas et al. 2017; Nicholl et al. 2017a; Yan et al. 2017), are known to become very bright in ultraviolet and the peak magnitude of HSC19edgb at z = 2.7 in ultraviolet is consistent with that observed for Gaia16apd. However, the LC decline rate of HSC19edgb in ultraviolet is much slower than that of Gaia16apd. The rise time of HSC19edgb at z = 2.7 becomes 7 days in the rest frame. Such a fast rise has never been observed in SLSNe, although the ultraviolet LC information in SLSNe is still limited. HSC19edgb is not likely a PISN even at z ≃ 2.7 because they are not bright in ultraviolet (Kasen et al. 2011; Dessart et al. 2013).

To summarize, HSC19edgb is more likely an SN IIn at z ≃ 0.23 because (i) the photometric redshift based on broad-frequency information prefers z ≃ 0.23 and (ii) the LC evolution is naturally explained as a slowly evolving SN IIn at z ≃ 0.23. However, the possibility that HSC19edgb is an ultraviolet-bright SLSN at z ≃ 2.7 is not excluded. HSC19edgb is not likely a PISN at either z ≃ 0.23 or 2.7.

3.3.2. HSC19edge

HSC19edge (AT 2018lto) was detected from the beginning of Season 3. Nothing was detected at the position in Seasons 1 and 2. It was located 0.67 arcsec away to the north of the host galaxy center (Figure 2). The spectroscopic redshift of the host galaxy, z = 0.33094, is available in the COSMSO2015 catalog (Laigle et al. 2016). We assign the same redshift to HSC19edge. Figure 6 shows the LC of HSC19edge.

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HSC19edge has a flux excess in the z band (Figure 6). In Figure 7, we plot the spectral energy distribution (SED) of HSC19edge based on the g-, i-, and z-band photometry in Season 3. We find that the rest-frame wavelength of the z-band filter, which is the reddest band we have, matches with the Hα wavelength in the rest frame. Therefore, the z-band flux excess is likely caused by the strong Hα emission from HSC19edge. A strong Hα emission is a generic feature of SNe II. Among SNe II, SNe IIn and some peculiar SNe II such as iPTF14hls are known to have LCs with a long luminous phase as seen in HSC19edge. Figure 7 compares the SED of HSC19edge with the spectra of SN IIn 2010jl and iPTF14hls. The two spectra match well up to around 4000 Å, but the blue SN IIn spectrum matches the SED of HSC19edge better at the shorter wavelengths than that of iPTF14hls. Therefore, HSC19edge is more likely to be an SN IIn, although the possibility of it being an iPTF14hls-like SN is not excluded.

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Based on the long-lasting (more than a year) SN discoveries reported in the previous section, we estimate the observational rate of long-lasting SNe with a survey depth of around 26 mag.

The expected number of events N during our survey can be expressed as

$$\begin{eqnarray}&&N=\epsilon {{Rt}}_{{\rm{s}}}{A}_{{\rm{s}}},\end{eqnarray} \tag{ 1 }$$

where is the discovery efficiency, R is the event rate in units of event deg⁻² yr⁻¹, t_s is the survey duration in units of yr, and A_s is the survey area in units of deg². Our survey conducted with HSC has A_s = 1.75 deg².

In order to be identified as a multiseason transient during our survey, the transient needs to appear by Season 3 so that we can confirm their long-lasting nature in Season 4. It is not yet clear whether or not the transients identified only in Season 4 last for more than one season. Therefore, we set t_s = 2.5 yr, which is from the beginning of Season 1 to the end of Season 3 (Figure 1), to estimate the long-lasting SN rate.

The discovery efficiency is an uncertain parameter. We performed a mock transient survey simulation to estimate . We assumed the same observational epochs as in our survey and randomly generated 1 yr-long SNe from Season 1 to Season 3 (t_s = 2.5 yr) that are brighter than the detection limits. We generated 10⁶ 1 yr-long SNe in the mock survey simulation and found that 54% of them are observed in multiple seasons. For instance, if a 1 yr-long SN appears shortly after Season 1, it disappears before Season 3, and it is only detected in Season 2. Such a long-lasting transient is missed in our survey. Discovery efficiency for transients lasting for two or more years are likely larger because our observational gaps are about 0.5 yr. In addition to the effects caused by the observational epochs, some transients can be missed during the image subtraction process and the CNN screening (Yasuda et al. 2019). The exclusion of the transients from the galaxy center to avoid contamination by AGN activities can also reduce the efficiency, although its effect is likely not significant (Section 2). Overall, we set ≃ 0.5 ± 0.1 based on our mock survey simulation. For reference, we show the rates within 20% of the assumed efficiency of 0.5, which is 0.1, to present the possible systematic uncertainty.

Under the assumptions discussed so far, the discovery of three long-lasting SNe in our survey sets the observational rate of SNe lasting for more than a year with a survey depth of around 26 mag to be

$$\begin{eqnarray}&&R={1.4}_{-0.7}^{+1.3}(\mathrm{stat}.){}_{-0.3}^{+0.2}(\mathrm{sys}.)\,\mathrm{events}\,{\deg }^{-2}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 2 }$$

The statistical error corresponds to the 84% confidence limits assuming the Poisson statistics.

The three long-lasting SNe we discovered are likely all SNe IIn. The LCs of SNe IIn are heterogeneous (e.g., Kiewe et al. 2012; Taddia et al. 2013; Nyholm et al. 2020). Some evolve very quickly and others last for many years. In this section, we constrain the fraction of long-lasting SNe IIn among SNe IIn with our survey data, assuming that the three long-lasting SNe we discovered are all SNe IIn.

Based on the discovered long-lasting SNe IIn, the volumetric rate of long-lasting SNe IIn can be roughly estimated as ∑_i (1 + z_i )/ Vt_s , where V is the survey volume and z_i is the redshift of the individual object. We use ≃ 0.5 and t_s = 2.5 yr as in the previous section.

The peak optical magnitude of most SNe IIn are brighter than − 17 mag (Nyholm et al. 2020; Richardson et al. 2014). Allowing two magnitudes to identify long-lasting SNe IIn for more than a year in our survey (Nyholm et al. 2020), we assume that our survey is complete for the discovery of long-lasting SNe IIn up to z ≃ 0.35 at which the limiting magnitude of 26 mag roughly corresponds to −15 mag. The corresponding survey volume is V ≃ 4.6 × 10⁻⁴ Gpc³ with the field of view of 1.75 deg². Because we limit z < 0.35, we exclude HSC16aayt at z = 0.68 in this analysis.

Given all the assumptions discussed so far, we obtain the long-lasting SN rate of ∼ 3000 SNe Gpc⁻³ yr⁻¹. We note that we do not take the effect of dust extinction into account in our estimate. SNe IIn do not tend to appear at the center of the host galaxies (Habergham et al. 2014; Schulze et al. 2020) and our exclusion of transients from the host galaxy centers would not affect the rate estimate significantly. The total SN IIn rate in our search volume is ∼7000 SNe Gpc⁻³ yr⁻¹ in which we assume that the total core-collapse SN rate is 10⁵ SNe Gpc⁻³ yr⁻¹ (Li et al. 2011; Dahlen et al. 2012) and the fraction of SNe IIn in core-collapse SNe is 7% (Shivvers et al. 2017). Thus, roughly 40% of SNe IIn are estimated to be long-lasting SNe IIn if we take ≃ 0.5. About a half of SNe IIn are found to last long in nearby transient surveys (Nyholm et al. 2020) and our estimate with ≃ 0.5 is not far from the local fraction.

SLSNe and some PISNe are intrinsically luminous and long-lasting SNe. Thus, they can be observed at high redshifts and their duration can be even longer because of the time dilation. In this section, we conduct the transient survey simulations assuming the same conditions as our HSC transient survey and constrain the PISN and SLSN rates by comparing the survey simulation results and our actual survey results.

Our survey simulations are conducted in the following way. We first make the redshift bin with the interval of 0.1 from z = 0 to 6. At a given time, we judge whether or not an SN occurs based on the assumed SN rate at each redshift bin using the SN rate and the volume in the redshift bin. If an SN occurs, we take the SN LC model from the SN redshift in the observer frame and check if the SN satisfies our detection criteria. If it does, the SN is marked as discovered. We adopt the same limiting magnitudes we had in the real survey presented in Figure 1 and summarized in Table A1. The survey simulations start long before the observation epochs in order not to miss any SNe in the simulations. If the simulated SN is above the limiting magnitude when we take the reference images, it is excluded because we only searched for long-lasting SNe that became bright after the reference images were taken. The transient survey simulations with the same conditions are performed 1000 times. In each simulation, we count the number of the SN discoveries. We present the average discovery numbers of the 1000 simulations below. The average number is compared with our actual observational results to see the adopted SN model and SN rate are consistent with the observations.

We use PISN LC models calculated by Kasen et al. (2011) in our survey simulations. Based on the SED time evolution obtained by Kasen et al. (2011), we calculate the LCs of high-redshift PISNe at the observer frame. We use the R250, R225, R200, R175, R150, B250, B200, He130, He100, and He80 models. This model set includes both faint and luminous PISNe. The number in the model names shows the PISN progenitor mass in units of M_⊙. The "R" models are red supergiant (RSG) PISN models, the "B" models are blue supergiant (BSG) PISN models, and the "He" models are helium star PISN models. The SLSN templates we adopt are introduced in Appendix A. We consider SLSNe I (Quimby et al. 2011) and refer to them as SLSNe here.

All the SN models are assumed to have the same event rate in our simulations. The SN rate is also assumed to be the same at all redshifts for simplicity. We conduct the survey simulations for the event rates of 100 Gpc⁻³ yr⁻¹ and 10 Gpc⁻³ yr⁻¹. The local (z ≃ 0.2) SLSN rate is around 30 Gpc⁻³ yr⁻¹ (Quimby et al. 2013) and it increases to around 100 Gpc⁻³ yr⁻¹ at z ≃ 1 (Prajs et al. 2017).

The predicted numbers of SN discoveries from our survey simulations with 100 Gpc⁻³ yr⁻¹ are summarized in Table 2. The discovery numbers with the simulations with 10 Gpc⁻³ yr⁻¹ are reduced by a factor of 10 and the discovery numbers are proportional to the SN rate. We find that the expected numbers of SNe discovered are mostly proportional to the SN rate. No models are detected for 3 yr in our simulations. We only find one SN IIn, which is not a PISN or SLSN, lasting for 2 yr in our survey (Section 3.2). On average, one 2 yr-long discovery for both R250 and R225 models is predicted in our survey simulations. Given the small expected number of the detection, we constrain their rates to be of the order of 100 Gpc⁻³ yr⁻¹ at most.

We find two 1 yr-long SNe during our survey, HSC19edgb and HSC19edge. HSC19edge is likely an SN IIn or a peculiar SN II (Section 3.3.2). HSC19edgb is most likely to be an SN IIn, but we cannot exclude the possibility that it is an SLSN (Section 3.3.1). They are not likely PISNe as discussed in the previous sections. The survey simulations predict that three to five luminous PISN models would be discovered during our survey if their rates are 100 Gpc⁻³ yr⁻¹. The typical redshifts of PISNe discovered in the simulations are z ≃ 1–3 (Figure 8). Thus, the rates of the luminous PISNe, which come from the massive PISN progenitors at z ≃ 1–3 are constrained to be less than 100 Gpc⁻³ yr⁻¹ in this case.

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The PISN rate is predicted to be of the order of 0.1% of the core-collapse SN rate at z ≲ 3 based on the estimated amount of low-metallicity massive stars (e.g., Langer et al. 2007; du Buisson et al. 2020). Given the core-collapse SN rate of ∼ 10^5–6 Gpc⁻³ yr⁻¹ at these redshifts (Li et al. 2011; Dahlen et al. 2012), the nearby PISN rate is predicted to be ∼ 100–1000 Gpc⁻³ yr⁻¹. Our survey simulations with 100 Gpc⁻³ yr⁻¹ predict a couple of PISN discoveries. Given the small number statistics, the PISN rate is constrained to be of the order of 100 Gpc⁻³ yr⁻¹ at most. Therefore, we conclude that the theoretical prediction is still consistent with our survey results. It is interesting to note that we expect a couple of PISNe lasting only for one season during our survey if the PISN rate is ∼100 Gpc⁻³ yr⁻¹. The one-season data must be investigated carefully because most SNe, including abundant SNe Ia, last only for a season. We present our investigation of the one-season data in our forthcoming paper.

We discovered only one potential SLSN lasting for 1 yr (HSC19edgb), although it is more likely to be an SN IIn. If we take the Fast SLSN model, the number of discoveries (one at most) is consistent with the local SLSN rate of 100 Gpc⁻³ yr⁻¹ with which we expect 0.4 1 yr-long SLSN discoveries (Table 2). The Slow SLSN model predicts around three SLSN discoveries in our survey data, which is higher than the actual number of discoveries but nondetection is still consistent with the results given the small number statistics. The Fast model reproduces the u-band LC behavior better than the Slow model (Appendix A). Thus, it is possible that the ultraviolet LC evolution may be generally fast in SLSNe. The SLSN discoveries in the first season data reported in Moriya et al. (2019b) and Curtin et al. (2019) are consistent with the single season discovery expectation (∼2 SLSNe per season).

We present the results of our survey for SNe lasting for more than a year by using long-baseline deep (around 26 mag) and wide (1.75 deg²) HSC time-domain data obtained for four seasons from late 2016 to early 2020 in the g, r, i, and z bands. We discovered no SNe lasting for 3 yr, one SNe lasting for 2 yr (HSC16aayt), and two SNe lasting for 1 year (HSC19edgb and HSC19edge). Therefore, the discovery rates of SNe lasting for more than a year in the transient surveys with a typical limiting magnitude of 26 mag are estimated to be ${1.4}_{-0.7}^{+1.3}(\mathrm{stat}.){}_{-0.3}^{+0.2}(\mathrm{sys}.)\,\mathrm{events}\,{\deg }^{-2}\,{\mathrm{yr}}^{-1}$. The statistical error corresponds to the 84% confidence limits assuming the Poisson statistics and the systematic uncertainty is from the uncertainty in the discovery efficiency.

The three long-lasting SNe we found are all consistent with being SNe IIn. Assuming that they are all SNe IIn, we estimate that about 40% of SNe IIn have long-lasting LCs. No plausible PISN candidates lasting for more than a year were discovered. By comparing survey simulations and the survey results, we constrain that the PISN rate up to z ≃ 3 is less than 100 Gpc⁻³ yr⁻¹. In other words, the PISN rate is less than 0.01–0.1% of the core-collapse SN rate at these redshifts.

The exploration of the long-timescale (years or more) transient phenomena requires a patient long-term monitoring of the same field. Our HSC data currently have a baseline of around 1000 days. We discovered a couple of long-lasting SNe but longer monitoring of the same field is required to explore the frontier of the long-lasting transients. There likely exist many long-lasting rare transients such as PISNe that require longer persistent monitoring of the same field to discover. Our exploration has just started and it is important to keep the monitoring observations for even longer.

We thank the anonymous referee for constructive comments that improved this paper. T.J.M. is supported by the Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (JP17H02864, JP18K13585, JP20H00174). J.C. would like to acknowledge funding by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), CE170100004. L.G. was funded by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 839090. This work has been partially supported by the Spanish grant PGC2018-095317-B-C21 within the European Funds for Regional Development (FEDER). G.P. acknowledge support from the Ministry of Economy, Development, and Tourisms Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS.

This work is supported by the Japan Society for the Promotion of Science Open Partnership Bilateral Joint Research Project between Japan and Chile (JPJSBP120209937).

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University.

This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at http://dm.lsst.org.

This paper is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg, and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queens University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.

Facility: Subaru(HSC). -

Software: hscPipe (Bosch et al. 2018).

We describe the SLSN LC templates used for the survey simulations. The SLSNe we consider in this paper are Type I SLSNe that do not have hydrogen emission features (Quimby et al. 2011). We take the similar approach to make the SLSN templates as in Prajs et al. (2017). We use the magnetar-powered model (Woosley 2010; Kasen et al. 2011) to make the SLSN LC templates. Although the magnetar model is not necessarily the complete model for SLSNe (Moriya et al. 2018, for a recent review), it reproduces the SLSN LCs at around the peak well (e.g., Inserra et al. 2013; Wang et al. 2015; Nicholl et al. 2017b).

We take the same approach described in Inserra et al. (2013) to obtain the multicolor LCs of magnetar-powered SNe. The bolometric luminosity is obtained by assuming the central energy input through the dipole radiation,

$$\begin{eqnarray}&&{L}_{\mathrm{dipole}}(t)=\displaystyle \frac{{E}_{p}}{{t}_{p}}{\left(1+\displaystyle \frac{t}{{t}_{p}}\right)}^{-2},\end{eqnarray} \tag{ A1 }$$

where E_p is the initial rotational energy of the magnetar and t_p is the spin-down timescale of the magnetar. Assuming the momentum of inertia of the magnetar is ≃ 10⁴⁵ g cm², ${E}_{p}\simeq 2\times {10}^{52}{P}_{\mathrm{ms}}^{-2}$, where P_ms is the initial rotational period scaled with 1 ms. Similarly, assuming the magnetar radius of ≃ 10 km, ${t}_{p}\simeq 4.1\times {10}^{5}{B}_{14}^{-2}{P}_{\mathrm{ms}}^{2}$, where B₁₄ is the dipole magnetic field strength of the magnetar. We adopt the semi-analytic model of Arnett (1982; see also Chatzopoulos et al. 2012) to obtain the bolometric LC for which we assume a diffusion time τ_d in the ejecta. We use our own code to calculate the magnetar bolometric LC (Moriya et al. 2017). Then, we estimate the location of the photosphere assuming the broken power-law ejecta structure (ρ_ejecta ∝ r⁻¹⁰ at the outer ejecta and ρ_ejecta ∝ r⁻¹ at the inner ejecta) and the ejecta opacity of 0.1 cm² g⁻¹. Given the bolometric luminosity and photospheric radius, we assume the blackbody function to obtain the SED. The optical SEDs of SLSNe around the peak luminosity match the blackbody function well but the SEDs below ≃3000 Å are often suppressed (e.g., Vreeswijk et al. 2014; Nicholl et al. 2017a; Yan et al. 2017). Therefore, we show the models with and without the ultraviolet suppression. We suppress the blackbody function in the same way as in Prajs et al. (2017) in which the ultraviolet suppression is estimated based on the SLSN iPTF13ajg (Vreeswijk et al. 2014).

We set P_ms = 1.5 and B₁₄ = 0.3 in making the SLSN templates. We use two τ_d : 45 days (the "Slow" models) and 25 days (the "Fast" models). These values are the typical values obtained when the SLSN LCs are fitted by the magnetar model (e.g., Nicholl et al. 2017b; Prajs et al. 2017). The "Slow" and "Fast" models are presented in Figure 9. We can find that the peak magnitudes of our models match well to those of the well-obsereved SLSN SN 2015bn (Nicholl et al. 2016). The g-, r-, i-, and z-band LC evolution is well reproduced by the Slow model, while the u-band LC evolution is well reproduced by the Fast model.

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The Slow and Fast models presented so far assume the ultraviolet suppression. We also adopt the models without the ultraviolet suppression in our simulations. They are referred as the "SlowUV" and "FastUV" models.

We provide the log of the HSC observations in Table A1. The photometry data of HSC16aayt, HSC19edgb, and HSC19edge are listed in Tables A2, A3, and A4, respectively.