Systematic Investigation of the Fallback Accretion-powered Model for Hydrogen-poor Superluminous Supernovae

Superluminous supernovae (SLSNe) are the most intrinsically luminous supernovae (SNe) currently known (see Moriya et al. 2018a for a review). Despite significant interest in studying these events, the power source of hydrogen-poor (Type I) SLSNe⁴ (e.g., Quimby et al. 2011) is still debated. Major suggested power sources are the nuclear decay of ${}^{56}\mathrm{Ni}$ (e.g., Gal-Yam et al. 2009; Moriya et al. 2010; Kozyreva et al. 2017), the interaction between SN ejecta and dense circumstellar media (e.g., Chevalier & Irwin 2011; Moriya & Maeda 2012; Sorokina et al. 2016), and prolonged heating by some sort of central engine (e.g., Kasen & Bildsten 2010; Woosley 2010; Metzger et al. 2015). It is also possible that several energy sources are active at the same time in SLSNe (e.g., Chen et al. 2017b; Tolstov et al. 2017).

Late-phase observations of SLSNe have revealed that their nebular phase spectra resemble those of broad-line Type Ic SNe which are often associated with long gamma-ray bursts (e.g., Pastorello et al. 2010; Nicholl et al. 2016a; Jerkstrand et al. 2017), and the host galaxies of these two classes are also similar (e.g., Lunnan et al. 2014; Angus et al. 2016; Perley et al. 2016; Chen et al. 2017a; Schulze et al. 2018). The similarity of SLSNe to broad-line Type Ic SNe implies the possible existence of a central engine in SLSNe. The most popular central engine proposed to account for the huge luminosity of SLSNe is a strongly magnetized, rapidly rotating neutron star (NS) called a "magnetar." Strong magnetic fields allow NSs to spin down quickly and convert their rotational energy to radiation (Shapiro & Teukolsky 1983). Magnetars have long been suggested as a potential power source in SNe (e.g., Ostriker & Gunn 1971; Maeda et al. 2007), and they are now intensively applied for SLSNe (e.g., Kasen & Bildsten 2010; Woosley 2010; Dessart et al. 2012; Chatzopoulos et al. 2013; Inserra et al. 2013; Nicholl et al. 2013; Metzger et al. 2015; Wang et al. 2015; Bersten et al. 2016; Moriya et al. 2017; Liu et al. 2017a; Yu et al. 2017). The recent statistical study by Nicholl et al. (2017b), which used the Bayesian light-curve (LC) fitting code MOSFiT (Guillochon et al. 2018), has found that magnetars with initial spin periods of 1.2–4 ms and magnetic field strengths of (0.2–1.8) × 10¹⁴ G can explain the overall properties of SLSNe.

However, the central engines that can be activated in SNe are not limited to magnetars. One alternative is fallback accretion (e.g., Dexter & Kasen 2013). A part of the SN ejecta that does not acquire enough energy to escape eventually falls back (Michel 1988; Chevalier 1989); this "fallback" material would ultimately accrete onto the central compact remnant. Such an accretion can result in outflows that provide additional energy to increase the energy and luminosity of the SN (Dexter & Kasen 2013; Moriya et al. 2018b). Short-term accretion onto a black hole caused by the direct collapse of a massive star may result in long gamma-ray bursts and their accompanying broad-line Type Ic SNe (Woosley 1993; see Hayakawa & Maeda 2018; Barnes et al. 2018 for recent studies), while longer-term accretion may be able to power the excess luminosity seen in SLSNe (Dexter & Kasen 2013). Metzger et al. (2018) recently point out that fallback accretion could affect the energy input from magnetars as well.

In this paper, we systematically investigate the LCs of SLSNe assuming fallback accretion as the central power source. By fitting SLSN LCs using MOSFiT (Modular Open Source Fitter for Transients), we study whether the fallback accretion powered model can satisfactorily reproduce the known SLSN LCs, and whether the required fallback accretion parameters are feasible. We first introduce our method in Section 2. The results of the fallback LC fitting are presented in Section 3 and discussed in Section 4. We conclude the paper in Section 5.

2.1. MOSFiT

We use the Python-based LC fitting code to apply the fallback accretion model to SLSNe. We briefly summarize the procedure here, and defer to Guillochon et al. (2018) and Nicholl et al. (2017b) for the details of the code. In short, MOSFiT adopts a Markov chain Monte Carlo (MCMC) approach to fit multi-band LCs, and provides the posterior probability distributions for the free parameters in the model. We perform maximum likelihood analysis, since using a full Gaussian process regression is found to have negligible impact on the fit parameters in the case of a magnetar model applied to the same SLSN sample we use here (Nicholl et al. 2017b). As in that paper, we use the first 10,000 iterations in the MCMC algorithm to burn in the ensemble (see Guillochon et al. 2018 for details of the burning process) and at least 25,000 total iterations are performed before judging whether the fitting is converged. The convergence is checked by evaluating the potential scale reduction factor (PSRF; Gelman & Rubin 1992). Reliable convergence is obtained when the PSRF is below 1.2 (Brooks & Gelman 1998) and we terminate our iterations when it is below 1.1.

The parameters and priors in the fitting procedure are essentially the same as in Nicholl et al. (2017b), but there are some differences. The central engine is, of course, changed to fallback accretion as described in the next section. We do not set the gamma-ray opacity as a free parameter. Nicholl et al. (2017b) consider that the magnetar spin-down energy is released in the form of gamma-rays, and take the gamma-ray opacity into account in heating the SN ejecta. In the fallback model, the source of energy is the kinetic energy of outflows, and dynamical interaction between the outflows and the SN ejecta provides the heat to power the LCs. During the fitting procedure, we assume the central energy input from fallback (L_fallback) is 100% thermalized, since the conversion efficiency is fully degenerate with L_fallback. We consider the importance of using a realistic efficiency in converting accretion to thermal energy in Section 4. All free parameters used in the fits are summarized in Table 1.

Table 1.  Free Parameters and Priors in Our Model

Parameter	Prior	Min	Max	Mean	σ
L1 ($\mathrm{erg}\,{{\rm{s}}}^{-1}$)	log-flat	1051	1057	⋯	⋯
ttr (day)	log-flat	10−4	100	⋯	⋯
${M}_{\mathrm{ej}}$ (${\text{}}{M}_{\odot }$)	log-flat	0.1	100	⋯	⋯
vphot (103 $\mathrm{km}\,{{\rm{s}}}^{-1}$)	Gaussian	1	30	1.47	4.3
${\kappa }_{\mathrm{ej}}$ (${\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$)	flat	0.05	0.2	⋯	⋯
Tf (1000 K)	Gaussian	3	10	6	1
AV (mag)	flat	0	0.5	⋯	⋯
texp (day)	flat	−100	0	⋯	⋯
variance	log-flat	10−3	100	⋯	⋯

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2.2. Central Power Input and Constraints

The fallback accretion rate eventually follows a power law $\propto {t}^{-\tfrac{5}{3}}$, where t is time after explosion (Michel 1988; Chevalier 1989). Numerical fallback simulations show that the accretion rate is usually flat at earlier times (Zhang et al. 2008; Dexter & Kasen 2013). The initial flat accretion rate could be related to the freefall accretion but the reverse shock would also alter the early accretion rate significantly (Zhang et al. 2008; Dexter & Kasen 2013). Fallback accretion dominates at later times. We assume that the kinetic energy in the outflow from the accretion disk is proportional to the accretion rate and a fraction of the kinetic energy is thermalized by the inelastic collision between the disk outflow and SN ejecta. We therefore assume that the central energy input from the fallback accretion, proportional to the accretion rate, follows

$$\begin{eqnarray}{L}_{\mathrm{fallback}}(t)=\left\{\begin{array}{ll}{L}_{1}{\left(\displaystyle \frac{{t}_{\mathrm{tr}}}{1{\rm{s}}}\right)}^{-\tfrac{5}{3}}\equiv {L}_{\mathrm{flat}} & (t\lt {t}_{\mathrm{tr}})\\ {L}_{1}{\left(\displaystyle \frac{t}{1{\rm{s}}}\right)}^{-\tfrac{5}{3}} & (t\geqslant {t}_{\mathrm{tr}})\end{array}\right.,\end{eqnarray} \tag{ 1 }$$

where L₁ is a constant and t_tr is a transition time from the initial flat accretion to the power-law accretion. Briefly, the energy input is assumed to be constant (L_flat) until t = t_tr and then start to decline with $\propto {t}^{-\tfrac{5}{3}}$. In the fitting procedure, L₁ and t_tr are set as free parameters. We have also performed the fitting without t_tr. In this case, the fallback accretion power is ${L}_{1}{(t/1{\rm{s}})}^{-5/3}$ from the beginning. We found the fitting results without t_tr are not very different from those with t_tr and we show the latter in this paper.

The total input energy from the fallback accretion (E_total) and the possible energy brought by neutrinos (${E}_{\nu }\simeq {10}^{51}\,\mathrm{erg}$) are the only energy sources for the kinetic energy of SN ejecta in our model. Therefore, the total kinetic energy, roughly estimated as ${E}_{{\rm{K}}}\simeq {M}_{\mathrm{ej}}{v}_{\mathrm{phot}}^{2}/2$, should satisfy ${E}_{{\rm{K}}}\lt {E}_{\mathrm{total}}+{E}_{\nu }-{E}_{\mathrm{rad}}$, where E_rad is the total radiated energy. We constrain the parameters to vary within this condition. The constraint that the nebular phase should not be reached before 100 days, as in Nicholl et al. (2017b), is also kept.

2.3. SLSN Sample

Figure 1 shows representative results from fitting our fallback accretion model to the SLSN sample. We find that the overall quality of the LC fitting is good, showing that in principle, fallback accretion power can explain SLSN LCs. Indeed, the distribution of the Watanaba–Akaike information criterion (Watanabe 2010; Gelman et al. 2014) does not differ much from those obtained from the magnetar-powered model presented in Nicholl et al. (2017b) (Table 3). Therefore, the fallback accretion model is quantitatively as good as the magnetar spin-down model in fitting SLSN LCs. We will therefore investigate the derived parameters in the next section to see whether the fallback accretion model is actually physically reasonable.

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Table 3.  Medians and 1σ Bounds for the Fitted Parameters

Name	log L1	$\mathrm{log}{t}_{\mathrm{tr}}$	$\mathrm{log}{E}_{\mathrm{total}}$	${M}_{\mathrm{ej}}$	vphot	EK	${\kappa }_{\mathrm{ej}}$	Tf	AV	σ	WAIC
	($\mathrm{erg}\,{{\rm{s}}}^{-1}$)	(day)	(${\text{}}{M}_{\odot }{c}^{2}$)	(${\text{}}{M}_{\odot }$)	(1000 $\mathrm{km}\,{{\rm{s}}}^{-1}$)	(1051 erg)	(${\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$)	(1000 K)	(mag)
SN 2005ap	${54.84}_{-0.10}^{+0.09}$	$-{2.43}_{-1.16}^{+1.13}$	$-{0.69}_{-0.85}^{+0.86}$	${5.37}_{-1.65}^{+4.40}$	${19.17}_{-2.14}^{+2.73}$	${19.73}_{-8.96}^{+27.14}$	${0.12}_{-0.05}^{+0.05}$	${4.84}_{-0.75}^{+0.62}$	${0.21}_{-0.14}^{+0.18}$	${0.01}_{-0.00}^{+0.01}$	56.22
SN 2006oz	${55.49}_{-0.30}^{+0.36}$	${1.54}_{-0.21}^{+0.31}$	$-{2.69}_{-0.51}^{+0.50}$	${2.63}_{-1.01}^{+2.16}$	${9.19}_{-1.01}^{+1.25}$	${2.22}_{-1.14}^{+2.99}$	${0.14}_{-0.05}^{+0.05}$	${5.92}_{-1.11}^{+1.02}$	${0.16}_{-0.11}^{+0.20}$	${0.01}_{-0.00}^{+0.01}$	75.30
SN 2007bi	${54.93}_{-0.04}^{+0.06}$	$-{2.75}_{-0.81}^{+0.74}$	$-{0.39}_{-0.53}^{+0.60}$	${67.61}_{-15.13}^{+23.59}$	${12.96}_{-2.51}^{+3.22}$	${113.53}_{-56.23}^{+125.14}$	${0.16}_{-0.04}^{+0.03}$	${8.47}_{-0.29}^{+0.37}$	${0.10}_{-0.08}^{+0.12}$	${0.14}_{-0.01}^{+0.01}$	162.35
SN 2010gx	${54.69}_{-0.01}^{+0.01}$	$-{1.98}_{-1.38}^{+1.07}$	$-{1.14}_{-0.72}^{+0.93}$	${8.13}_{-1.96}^{+6.33}$	${13.19}_{-0.28}^{+0.38}$	${14.14}_{-3.87}^{+12.46}$	${0.14}_{-0.06}^{+0.05}$	${3.62}_{-0.22}^{+0.24}$	${0.01}_{-0.01}^{+0.02}$	${0.14}_{-0.01}^{+0.01}$	247.49
SN 2011ke	${54.52}_{-0.01}^{+0.01}$	$-{3.15}_{-0.59}^{+0.87}$	$-{0.53}_{-0.59}^{+0.40}$	${6.76}_{-1.97}^{+5.26}$	${8.80}_{-0.27}^{+0.30}$	${5.24}_{-1.76}^{+4.72}$	${0.12}_{-0.05}^{+0.05}$	${3.89}_{-0.18}^{+0.16}$	${0.01}_{-0.01}^{+0.01}$	${0.31}_{-0.01}^{+0.01}$	95.59
SN 2011kf	${54.76}_{-0.05}^{+0.05}$	$-{2.84}_{-0.82}^{+0.73}$	$-{0.50}_{-0.54}^{+0.60}$	${24.55}_{-6.76}^{+16.19}$	${18.39}_{-1.64}^{+2.12}$	${83.00}_{-33.10}^{+88.38}$	${0.13}_{-0.05}^{+0.05}$	${5.48}_{-0.11}^{+0.13}$	${0.03}_{-0.03}^{+0.05}$	${0.07}_{-0.02}^{+0.02}$	65.76
SN 2012il	${54.62}_{-0.02}^{+0.03}$	$-{2.69}_{-0.85}^{+1.24}$	$-{0.74}_{-0.85}^{+0.60}$	${3.24}_{-0.95}^{+1.66}$	${8.23}_{-0.49}^{+0.48}$	${2.19}_{-0.82}^{+1.53}$	${0.11}_{-0.04}^{+0.04}$	${6.02}_{-0.15}^{+0.20}$	${0.05}_{-0.03}^{+0.06}$	${0.10}_{-0.01}^{+0.02}$	60.57
SN 2013dg	${54.65}_{-0.02}^{+0.02}$	$-{2.51}_{-0.93}^{+1.31}$	$-{0.83}_{-0.89}^{+0.64}$	${4.68}_{-1.37}^{+3.09}$	${8.65}_{-0.22}^{+0.22}$	${3.50}_{-1.15}^{+2.62}$	${0.12}_{-0.05}^{+0.05}$	${3.02}_{-0.02}^{+0.04}$	${0.06}_{-0.04}^{+0.07}$	${0.01}_{-0.01}^{+0.02}$	128.50
SN 2013hy	${54.98}_{-0.04}^{+0.07}$	${1.71}_{-0.04}^{+0.03}$	$-{3.31}_{-0.06}^{+0.10}$	${5.89}_{-0.88}^{+2.05}$	${6.00}_{-0.48}^{+0.63}$	${2.12}_{-0.60}^{+1.37}$	${0.16}_{-0.05}^{+0.03}$	${7.65}_{-0.32}^{+0.35}$	${0.08}_{-0.06}^{+0.13}$	${0.12}_{-0.02}^{+0.01}$	124.29
SN 2015bn	${55.38}_{-0.02}^{+0.03}$	$-{3.23}_{-0.60}^{+0.88}$	${0.38}_{-0.61}^{+0.43}$	${31.62}_{-8.71}^{+19.66}$	${6.38}_{-0.17}^{+0.19}$	${12.87}_{-4.04}^{+9.28}$	${0.13}_{-0.05}^{+0.04}$	${8.89}_{-0.19}^{+0.25}$	${0.04}_{-0.03}^{+0.05}$	${0.30}_{-0.01}^{+0.01}$	313.28
PTF09atu	${55.58}_{-0.07}^{+0.06}$	$-{0.74}_{-1.68}^{+0.88}$	$-{1.08}_{-0.66}^{+1.18}$	${33.88}_{-9.90}^{+22.35}$	${8.30}_{-0.41}^{+0.36}$	${23.35}_{-8.40}^{+18.79}$	${0.12}_{-0.04}^{+0.05}$	${5.10}_{-0.65}^{+0.39}$	${0.38}_{-0.13}^{+0.09}$	${0.11}_{-0.01}^{+0.01}$	157.03
PTF09cnd	${55.64}_{-0.01}^{+0.02}$	${1.49}_{-0.04}^{+0.04}$	$-{2.50}_{-0.04}^{+0.05}$	${7.24}_{-1.22}^{+1.47}$	${7.52}_{-0.18}^{+0.14}$	${4.10}_{-0.85}^{+1.02}$	${0.16}_{-0.03}^{+0.03}$	${3.97}_{-0.32}^{+0.26}$	${0.01}_{-0.00}^{+0.01}$	${0.13}_{-0.01}^{+0.01}$	262.10
PTF09cwl	${55.34}_{-0.14}^{+0.12}$	$-{2.35}_{-1.04}^{+1.06}$	$-{0.24}_{-0.85}^{+0.81}$	${35.48}_{-16.43}^{+23.40}$	${9.67}_{-1.84}^{+1.83}$	${33.15}_{-21.47}^{+44.69}$	${0.13}_{-0.04}^{+0.05}$	${4.71}_{-1.03}^{+1.55}$	${0.20}_{-0.13}^{+0.18}$	${0.65}_{-0.11}^{+0.11}$	−9.39
PTF10hgi	${54.49}_{-0.03}^{+0.04}$	$-{0.32}_{-0.35}^{+0.34}$	$-{2.45}_{-0.26}^{+0.27}$	${5.50}_{-1.51}^{+3.02}$	${5.37}_{-0.20}^{+0.14}$	${1.59}_{-0.52}^{+1.00}$	${0.12}_{-0.05}^{+0.04}$	${6.43}_{-0.20}^{+0.27}$	${0.06}_{-0.05}^{+0.10}$	${0.16}_{-0.01}^{+0.02}$	167.33
PTF11rks	${54.74}_{-0.09}^{+0.07}$	${0.90}_{-0.76}^{+0.25}$	$-{3.01}_{-0.26}^{+0.58}$	${3.89}_{-0.80}^{+1.12}$	${11.18}_{-0.58}^{+0.57}$	${4.87}_{-1.39}^{+2.06}$	${0.19}_{-0.02}^{+0.01}$	${7.88}_{-0.42}^{+0.33}$	${0.47}_{-0.07}^{+0.02}$	${0.28}_{-0.02}^{+0.02}$	124.50
PTF12dam	${55.28}_{-0.02}^{+0.02}$	$-{0.61}_{-1.18}^{+0.60}$	$-{1.46}_{-0.42}^{+0.81}$	${33.88}_{-8.18}^{+22.35}$	${6.58}_{-0.29}^{+0.26}$	${14.69}_{-4.50}^{+11.68}$	${0.13}_{-0.05}^{+0.04}$	${9.98}_{-0.05}^{+0.02}$	${0.01}_{-0.01}^{+0.01}$	${0.60}_{-0.01}^{+0.01}$	−30.70
iPTF13ajg	${55.44}_{-0.09}^{+0.05}$	${0.34}_{-1.09}^{+0.30}$	$-{1.94}_{-0.29}^{+0.78}$	${7.76}_{-2.01}^{+4.83}$	${13.97}_{-0.52}^{+1.22}$	${15.14}_{-4.74}^{+13.88}$	${0.17}_{-0.04}^{+0.02}$	${4.73}_{-0.58}^{+0.30}$	${0.47}_{-0.04}^{+0.03}$	${0.14}_{-0.02}^{+0.02}$	132.27
iPTF13dcc	${55.46}_{-0.05}^{+0.07}$	$-{2.30}_{-1.07}^{+1.09}$	$-{0.16}_{-0.78}^{+0.78}$	${22.91}_{-6.31}^{+10.20}$	${5.29}_{-0.48}^{+0.49}$	${6.42}_{-2.57}^{+4.65}$	${0.09}_{-0.03}^{+0.05}$	${4.67}_{-0.29}^{+0.34}$	${0.06}_{-0.04}^{+0.10}$	${0.23}_{-0.03}^{+0.02}$	69.82
iPTF13ehe	${55.39}_{-0.02}^{+0.03}$	$-{2.21}_{-1.12}^{+1.14}$	$-{0.29}_{-0.78}^{+0.78}$	${77.62}_{-10.02}^{+11.50}$	${8.39}_{-0.37}^{+0.23}$	${54.61}_{-11.19}^{+11.64}$	${0.16}_{-0.02}^{+0.03}$	${5.00}_{-0.09}^{+0.13}$	${0.04}_{-0.04}^{+0.06}$	${0.06}_{-0.01}^{+0.01}$	116.75
iPTF16bad	${54.45}_{-0.03}^{+0.03}$	$-{1.17}_{-1.77}^{+0.89}$	$-{1.92}_{-0.62}^{+1.21}$	${2.69}_{-0.55}^{+1.11}$	${7.09}_{-0.29}^{+0.47}$	${1.35}_{-0.36}^{+0.82}$	${0.08}_{-0.02}^{+0.03}$	${5.69}_{-0.57}^{+0.54}$	${0.05}_{-0.04}^{+0.07}$	${0.07}_{-0.01}^{+0.02}$	99.17
PS1-10ahf	${55.83}_{-0.12}^{+0.09}$	${1.96}_{-0.03}^{+0.03}$	$-{2.63}_{-0.14}^{+0.11}$	${7.24}_{-2.00}^{+3.72}$	${6.93}_{-1.20}^{+0.80}$	${3.48}_{-1.76}^{+3.06}$	${0.17}_{-0.04}^{+0.02}$	${7.78}_{-0.59}^{+0.54}$	${0.30}_{-0.19}^{+0.14}$	${0.19}_{-0.02}^{+0.02}$	81.79
PS1-10awh	${54.91}_{-0.04}^{+0.07}$	$-{1.06}_{-1.42}^{+0.87}$	$-{1.53}_{-0.62}^{+1.02}$	${7.24}_{-2.00}^{+4.24}$	${10.79}_{-0.89}^{+0.71}$	${8.43}_{-3.29}^{+6.76}$	${0.14}_{-0.06}^{+0.04}$	${6.14}_{-1.29}^{+1.02}$	${0.04}_{-0.03}^{+0.06}$	${0.05}_{-0.01}^{+0.01}$	228.56
PS1-10bzj	${54.87}_{-0.05}^{+0.06}$	${1.26}_{-0.15}^{+0.11}$	$-{3.12}_{-0.12}^{+0.16}$	${1.70}_{-0.38}^{+0.59}$	${10.72}_{-0.56}^{+0.55}$	${1.95}_{-0.59}^{+0.96}$	${0.15}_{-0.03}^{+0.03}$	${6.74}_{-0.31}^{+0.33}$	${0.18}_{-0.13}^{+0.16}$	${0.14}_{-0.02}^{+0.02}$	87.90
PS1-10ky	${55.13}_{-0.05}^{+0.05}$	$-{0.86}_{-1.66}^{+1.08}$	$-{1.45}_{-0.77}^{+1.16}$	${5.50}_{-1.42}^{+2.82}$	${10.30}_{-0.80}^{+1.30}$	${5.83}_{-2.15}^{+5.36}$	${0.12}_{-0.04}^{+0.05}$	${7.64}_{-0.30}^{+0.45}$	${0.11}_{-0.07}^{+0.15}$	${0.13}_{-0.02}^{+0.02}$	143.36
PS1-10pm	${55.06}_{-0.04}^{+0.06}$	$-{2.64}_{-1.03}^{+1.01}$	$-{0.33}_{-0.71}^{+0.75}$	${28.84}_{-9.79}^{+22.45}$	${22.36}_{-2.11}^{+1.51}$	${144.18}_{-66.06}^{+148.07}$	${0.15}_{-0.06}^{+0.04}$	${7.40}_{-0.72}^{+0.63}$	${0.23}_{-0.14}^{+0.10}$	${0.09}_{-0.02}^{+0.03}$	53.64
PS1-11ap	${55.53}_{-0.07}^{+0.06}$	${1.55}_{-0.09}^{+0.05}$	$-{2.65}_{-0.10}^{+0.12}$	${5.62}_{-1.82}^{+3.09}$	${5.98}_{-0.34}^{+0.27}$	${2.01}_{-0.80}^{+1.39}$	${0.11}_{-0.04}^{+0.05}$	${7.62}_{-1.55}^{+0.53}$	${0.34}_{-0.16}^{+0.13}$	${0.28}_{-0.01}^{+0.01}$	265.23
PS1-11bam	${55.25}_{-0.16}^{+0.13}$	$-{0.38}_{-2.34}^{+1.61}$	$-{1.65}_{-1.23}^{+1.69}$	${5.01}_{-1.78}^{+5.22}$	${8.67}_{-0.81}^{+0.94}$	${3.77}_{-1.77}^{+5.69}$	${0.10}_{-0.04}^{+0.07}$	${5.91}_{-1.08}^{+0.94}$	${0.08}_{-0.06}^{+0.10}$	${0.09}_{-0.03}^{+0.03}$	69.73
PS1-14bj	${55.53}_{-0.12}^{+0.06}$	${1.80}_{-0.65}^{+0.17}$	$-{2.82}_{-0.23}^{+0.49}$	${32.36}_{-7.24}^{+11.29}$	${3.39}_{-0.34}^{+0.37}$	${3.72}_{-1.38}^{+2.44}$	${0.17}_{-0.03}^{+0.02}$	${9.14}_{-0.29}^{+0.35}$	${0.06}_{-0.04}^{+0.08}$	${0.23}_{-0.03}^{+0.03}$	56.58
LSQ12dlf	${54.99}_{-0.03}^{+0.04}$	${0.70}_{-0.42}^{+0.27}$	$-{2.63}_{-0.21}^{+0.32}$	${3.80}_{-0.92}^{+1.82}$	${8.47}_{-0.31}^{+0.22}$	${2.73}_{-0.81}^{+1.52}$	${0.13}_{-0.04}^{+0.04}$	${3.59}_{-0.12}^{+0.13}$	${0.24}_{-0.14}^{+0.11}$	${0.08}_{-0.01}^{+0.01}$	145.27
LSQ14mo	${54.40}_{-0.05}^{+0.04}$	$-{2.52}_{-1.14}^{+0.93}$	$-{1.07}_{-0.67}^{+0.80}$	${6.92}_{-1.67}^{+4.56}$	${14.06}_{-0.48}^{+0.46}$	${13.68}_{-4.00}^{+10.54}$	${0.14}_{-0.06}^{+0.04}$	${4.71}_{-0.11}^{+0.09}$	${0.16}_{-0.08}^{+0.08}$	${0.02}_{-0.02}^{+0.02}$	158.16
LSQ14bdq	${55.33}_{-0.05}^{+0.11}$	$-{2.70}_{-0.99}^{+1.12}$	$-{0.02}_{-0.80}^{+0.77}$	${52.48}_{-14.46}^{+21.65}$	${12.12}_{-1.66}^{+2.20}$	${77.13}_{-35.48}^{+74.98}$	${0.15}_{-0.05}^{+0.04}$	${6.12}_{-0.53}^{+0.81}$	${0.15}_{-0.11}^{+0.20}$	${0.37}_{-0.05}^{+0.05}$	14.49
Gaia16apd	${55.01}_{-0.03}^{+0.04}$	${0.22}_{-0.51}^{+0.25}$	$-{2.29}_{-0.20}^{+0.38}$	${5.75}_{-1.39}^{+3.16}$	${9.13}_{-0.35}^{+0.27}$	${4.80}_{-1.44}^{+3.09}$	${0.15}_{-0.06}^{+0.04}$	${8.33}_{-0.19}^{+0.21}$	${0.09}_{-0.05}^{+0.05}$	${0.21}_{-0.01}^{+0.01}$	379.84
DES14X3taz	${55.01}_{-0.08}^{+0.04}$	$-{2.11}_{-1.12}^{+0.87}$	$-{0.73}_{-0.66}^{+0.79}$	${16.60}_{-5.63}^{+12.92}$	${12.66}_{-1.65}^{+1.79}$	${26.60}_{-13.30}^{+35.06}$	${0.13}_{-0.06}^{+0.05}$	${5.95}_{-0.34}^{+0.36}$	${0.41}_{-0.13}^{+0.07}$	${0.28}_{-0.03}^{+0.04}$	29.87
SCP-06F6	${55.73}_{-0.11}^{+0.16}$	${1.69}_{-0.12}^{+0.11}$	$-{2.55}_{-0.18}^{+0.24}$	${7.76}_{-2.51}^{+3.20}$	${7.15}_{-1.04}^{+1.27}$	${3.97}_{-2.01}^{+3.80}$	${0.15}_{-0.04}^{+0.04}$	${5.84}_{-1.03}^{+1.10}$	${0.24}_{-0.17}^{+0.15}$	${0.60}_{-0.11}^{+0.17}$	−10.58
SNLS06D4eu	${55.18}_{-0.07}^{+0.07}$	${1.29}_{-0.15}^{+0.11}$	$-{2.83}_{-0.14}^{+0.17}$	${2.40}_{-0.74}^{+0.69}$	${11.93}_{-0.56}^{+0.97}$	${3.42}_{-1.27}^{+1.73}$	${0.18}_{-0.04}^{+0.02}$	${6.12}_{-1.13}^{+0.91}$	${0.09}_{-0.06}^{+0.10}$	${0.14}_{-0.02}^{+0.02}$	52.85
SNLS07D2bv	${54.90}_{-0.04}^{+0.05}$	$-{1.28}_{-1.40}^{+0.27}$	$-{1.40}_{-0.22}^{+0.98}$	${21.38}_{-5.16}^{+10.98}$	${14.52}_{-0.60}^{+0.56}$	${45.11}_{-13.67}^{+28.58}$	${0.15}_{-0.04}^{+0.03}$	${6.12}_{-1.16}^{+0.85}$	${0.25}_{-0.08}^{+0.06}$	${0.07}_{-0.00}^{+0.01}$	156.92
SSS120810	${54.64}_{-0.05}^{+0.04}$	$-{1.34}_{-1.59}^{+1.56}$	$-{1.62}_{-1.09}^{+1.10}$	${2.88}_{-0.75}^{+1.69}$	${11.01}_{-0.81}^{+1.12}$	${3.49}_{-1.27}^{+3.23}$	${0.12}_{-0.04}^{+0.06}$	${3.49}_{-0.11}^{+0.11}$	${0.23}_{-0.12}^{+0.20}$	${0.21}_{-0.03}^{+0.05}$	28.85

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Table 3 summarizes the parameters and their standard deviations constrained by our fitting. Figure 2 presents the combined constraints from all SLSN fitting results. Several posteriors are bimodal in a way that is not found in the magnetar model (Nicholl et al. 2017b). The parameters related to the fallback accretion are constrained to be $\mathrm{log}({L}_{1}/\mathrm{erg}\,{{\rm{s}}}^{-1})\,={55.04}_{-0.39}^{+0.47}$ and $\mathrm{log}({t}_{\mathrm{tr}}/\mathrm{day})=-{0.99}_{-2.09}^{+2.45}$. Figure 3 presents the two fallback parameters; we find no correlation between them. We also present the first constant luminosity ${L}_{\mathrm{flat}}\,={L}_{1}{({t}_{\mathrm{tr}}/1{\rm{s}})}^{-\tfrac{5}{3}}$ in Figure 3. The apparent correlation between t_tr and L_flat originates from the fact that L₁ is well constrained with little diversity. Because L₁ in ${L}_{\mathrm{flat}}={L}_{1}{({t}_{\mathrm{tr}}/1\sec )}^{-\tfrac{5}{3}}$ is not diverse, L_flat appears to have a correlation with t_tr as in Figure 3. The uncertainties in t_tr make L_flat uncertain. L₁ is better constrained and L₁ is chosen as a free parameter instead of L_flat. From L₁ and t_tr, we can derive the total input energy ${E}_{\mathrm{total}}\,=\int {L}_{\mathrm{fallback}}(t){dt}=2.5{L}_{1}{t}_{\mathrm{tr}}^{-2/3}$ which is also shown in Figure 2. The total input energy from the fitting is $\mathrm{log}{E}_{\mathrm{total}}/{\text{}}{M}_{\odot }{c}^{2}\,=-{1.56}_{-1.16}^{+1.41}$.

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The ejecta mass (${M}_{\mathrm{ej}}$) estimates are ${7.8}_{-4.4}^{+27.7}\,{\text{}}{M}_{\odot }$. They are slightly larger than those found in the magnetar model (${4.8}_{-2.6}^{+8.1}\,{\text{}}{M}_{\odot }$; Nicholl et al. 2017b), but overall they are not very different within uncertainties in the two models. We sometimes find a large discrepancy in the two mass estimates. For example, SN 2007bi is estimated to have ${M}_{\mathrm{ej}}={67.6}_{-15.1}^{+23.6}\,{\text{}}{M}_{\odot }$ in our fallback model while the magnetar model results in ${M}_{\mathrm{ej}}\,={3.8}_{-1.1}^{+1.4}\,{\text{}}{M}_{\odot }$ (Nicholl et al. 2017b). The results of the LC fitting are not very different from each other in the two models (Figure 4). We found that a large mass discrepancy tends to appear when the spin-down timescale in the magnetar model is larger than the diffusion timescale. In such a case, the rise time of the magnetar model is strongly affected by the spin-down timescale, not by the diffusion timescale which strongly affects the ejecta mass estimate (e.g., Kasen & Bildsten 2010).

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To have an idea of the kinetic energy of the ejecta, we assume ${E}_{{\rm{K}}}={M}_{\mathrm{ej}}{v}_{\mathrm{phot}}^{2}/2$. Because v_phot is just a photospheric velocity, E_K is unlikely to be the true kinetic energy of SN ejecta, but it provides a rough approximation (e.g., Arnett 1982). Figure 5 shows the relation between the ejecta mass and the kinetic energy. We find that SLSNe requiring higher ejecta masses tend to have higher kinetic energies. Spectral modeling of SLSNe indicates that $({E}_{{\rm{K}}}/{10}^{51}\,\mathrm{erg})/({M}_{\mathrm{ej}}/{\text{}}{M}_{\odot })\sim 1$ in SLSNe (Howell et al. 2013; Mazzali et al. 2016; Liu et al. 2017b) and a similar ratio was found from LC modeling using a magnetar engine (Nicholl et al. 2017b). We find our results also follow this trend. In the fallback accretion scenario, the initial explosion energy should be relatively small, in order to achieve significant fallback. However, the ejecta that do escape should gain additional energy from the accretion power after the explosion, such that they eventually reach $({E}_{{\rm{K}}}/{10}^{51}\,\mathrm{erg})/({M}_{\mathrm{ej}}/{\text{}}{M}_{\odot })\sim 1$.

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The total central energy input required to power SLSNe (E_total) is summarized in Figure 6. We find that ${E}_{\mathrm{total}}/{c}^{2}\,\sim {10}^{-3}-1\,{\text{}}{M}_{\odot }$ must be liberated in the accretion process. The essential question then is how much total mass needs to be accreted to the central compact remnant in order to produce this amount of energy. If we set η as the conversion efficiency from accretion, the required accretion mass becomes ${E}_{\mathrm{total}}/\eta {c}^{2}$. Dexter & Kasen (2013) estimate that the typical conversion efficiency from the fallback accretion to the large-scale outflow is η ∼ 10⁻³ by using the disk accretion models. Using this efficiency with our derived energies, we find that the total mass accreted must be ∼1–1000 ${\text{}}{M}_{\odot }$ (Figure 6). Accreting ∼1–10 ${\text{}}{M}_{\odot }$ of material from the carbon and oxygen core might be possible if the core is massive enough (e.g., Aguilera-Dena et al. 2018). However, having carbon and oxygen cores of ∼100 ${\text{}}{M}_{\odot }$ is very challenging even in low-metallicity environments (e.g., Yoshida et al. 2014). Therefore, most SLSNe likely require too great an accretion mass to be explained by fallback accretion.

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Some region of parameter space likely remains available for the fallback model as the true efficiency η is quite uncertain. Dexter & Kasen (2013) estimated η ∼ 10⁻³ with a plausible parameter set, assuming a large-scale disk outflow. Alternatively, if we assume that the major source of the outflow is a jet launched at the inner edge of the accretion disk, the efficiency might be as high as η ∼ 0.1 (e.g., McKinney 2005; Kumar et al. 2008; Gilkis et al. 2016). In this extreme, most SLSNe would require a more reasonable accreted mass of less than 10 ${\text{}}{M}_{\odot }$.

Even with an inefficient conversion efficiency, there are some SLSNe for which we need only a modest accreted mass to account for their LCs, and so could still be powered by fallback accretion. Those SLSNe that only require the accretion of E_total/c² < 0.01 ${\text{}}{M}_{\odot }$, indicated in red in Figures 3, 5, 6, and 7, tend to have low kinetic energy. However, both intermediate SLSNe (0.01 ${\text{}}{M}_{\odot }$ < E_total/c² < 0.1 ${\text{}}{M}_{\odot }$; gray in the figures) and SLSNe requiring large accretion mass (E_total/c² > 0.1 ${\text{}}{M}_{\odot }$; blue in the figures) can also sometimes have low kinetic energy, so this alone is not a robust diagnostic of fallback candidates. Figure 7 shows the rise time in the bolometric LC and the peak bolometric magnitude of the SLSNe in our sample, obtained from our fitting results. The SLSNe with lower accretion masses do not have a clear distinction from those with the higher accretion masses. The peak luminosity is found not to have a strong relation with the accreted mass, although the brightest events seem to require high accretion and it might be hard for the fallback model to explain them with physically plausible parameters (Figure 7).

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The transitional time t_tr could be physically related to the freefall accretion timescale, which is roughly ${(G\rho )}^{-1/2}$. t_tr ranges from ∼0.1 days to ∼100 days among the SLSNe that require the smallest amount of accretion (red in the figures) and, therefore, are the plausible fallback accretion-powered SLSN candidates. t_tr ∼ 0. 1 days corresponds to the main-sequence-like density $\rho \sim 0.1\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ and ${t}_{\mathrm{tr}}\sim 100\,\mathrm{days}$ corresponds to the giant-like density $\rho \sim {10}^{-6}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$. t_tr may imply such a progenitor but it may also affected by the explosion dynamics such as reverse shocks (e.g., Zhang et al. 2008; Dexter & Kasen 2013).

The necessity of a large fallback accretion rate with ${L}_{1}\sim {10}^{55}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ in fitting the SLSN LCs can be understood in a simple way. SLSNe have rise times of 20–100 days and peak luminosities of the order of ${10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Nicholl et al. 2015b; Lunnan et al. 2018; De Cia et al. 2018). A simple analytic model finds that the peak luminosity powered by a central heating source matches the energy input at the LC peak (the so-called Arnett law; Arnett 1979). Figure 8 shows a comparison between the range of the fallback accretion power and the range of the magnetar spin-down power required to fit SLSNe (Nicholl et al. 2017b). We can see in the figure that the two central energy inputs found to explain SLSN LCs occupy almost the same luminosity range, especially at 20–100 days when the LCs reach their peak. Following the Arnett law, we can say that energy input as in Figure 8 is generally required to power SLSNe with any central engines. The Arnett law is a simplified formalism, but the necessity of long-sustained high accretion rates in the fallback accretion model, as estimated by our Bayesian approach, is unavoidable.

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In this work, we treat ${M}_{\mathrm{ej}}$ as a constant parameter in the fitting procedure. However, ${M}_{\mathrm{ej}}$ can change as a function of time in the actual fallback accretion-powered SNe. One reason is that the disk outflow is the origin of the thermal energy and the flown material eventually becomes a part of the SN ejecta. In Figure 6, we draw a solid line that shows the total energy that is released with the corresponding ejecta mass assuming the typical disk outflow velocity of 0.3c. The SLSNe on this line should have outflows as massive as the ejecta mass. A large fraction of SLSNe are located well below the line and the change in the ejecta mass is expected to be rather small. Even if the ejecta mass were increased by the disk wind, the necessity of the large total ejecta mass to account for the large diffusion time would not change. Another reason is that the power source of our model is the mass accretion and the ejecta mass should be reduced with the accretion. We often find that the required accreted mass is larger than the ejecta mass with the assumption of η = 10⁻³. The actual mass reduced from the ejecta strongly depends on the uncertain η. Regardless of the uncertainty in η, our conclusion that the required accretion mass becomes too large would remain in the cases where it starts to dominate the ejecta mass.

Although we find that the fallback model is statistically as good as the magnetar model in fitting the SLSN LCs, the two models may have significant differences in their expected spectroscopic properties. The fallback accretion model often requires an accretion of ≳1 ${\text{}}{M}_{\odot }$ to the central compact remnant. This means that most of the inner core of the progenitor would be accreted. Therefore an SLSN that shows a lack of heavy element signatures in the spectrum would be a possible fallback candidate. So far, SLSNe with late-time spectra do not show a deficiency of iron-group elements relative to other SNe (Gal-Yam et al. 2009; Nicholl et al. 2016a; Jerkstrand et al. 2017), and SLSNe are even likely to produce more iron than other SNe (Nicholl et al. 2018). Another possible smoking gun is an off-axis afterglow from the jet that is likely produced when the accretion disk is highly super-Eddington. However, no such afterglows have been found from SLSNe and no afterglows have shown SN counterparts that have SLSNe luminosities, although one afterglow from an ultra-long gamma-ray burst is found to have a bright SN component (Greiner et al. 2015).

We have systematically investigated the fallback accretion central engine model for hydrogen-poor SLSNe. By using MOSFiT, we have fitted the multi-band LCs of 37 SLSNe, finding that the model provides satisfactory fits to the full ensemble. The quality of the LC fits are quantitatively as good as a similar model powered by magnetar spin-down, previously investigated using the same approach (Nicholl et al. 2017b).

However, we have found that the total energy input from the fallback accretion that needs to be provided to power the SLSNe is 0.002–0.7 ${\text{}}{M}_{\odot }$c² (Figure 6). Assuming a realistic conversion efficiency from the fallback accretion disk to the large-scale outflow (∼10⁻³), the total mass that must be accreted is 2–700 ${\text{}}{M}_{\odot }$. Therefore, this model often requires too much accretion to be achieved by massive stars. Thus we conclude that fallback is unlikely to power the majority of SLSNe.

The conversion efficiency is uncertain, and if it could reach ∼0.1 in some cases, the required accretion mass might be limited to 0.02–7 ${\text{}}{M}_{\odot }$. However, realistic simulations are needed to see if such a high value could be attained in practice.

Regardless of the uncertain efficiency, there are some SLSNe for which the required accretion is low enough that it could be compatible with massive star collapse. We find that they are difficult to distinguish by LCs (Figure 7). The lack of heavy elements in the nebular phase spectra or afterglow observations may identify fallback-powered SNe, although the recent studies show that SLSNe likely produce more iron than ordinary SNe (Nicholl et al. 2018).

We thank the referee for the constructive comments that improved this work. T.J.M.. thanks Tomohisa Kawashima and David Aguilera-Dena for very useful discussion. T.J.M. and M.N. acknowledge the support from the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe." T.J.M. is supported by the Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (16H07413, 17H02864, 18K13585). Numerical computations were in part carried out on computers at Center for Computational Astrophysics, National Astronomical Observatory of Japan.

Software: MOSFiT (Guillochon et al. 2018).