THE COSMIC CORE-COLLAPSE SUPERNOVA RATE DOES NOT MATCH THE MASSIVE-STAR FORMATION RATE

We address the scatter in and check the normalization of the cosmic SFR of Hopkins & Beacom (2006). In particular, the scatter mainly originates from the different indicators used to derive the SFR, while it also comes from the intrinsic uncertainty in the calibration of each indicator. The normalization is strongly influenced by the choice of IMF as well as dust corrections. Below, we discuss these in turn.

The SFR compilation includes measurements from Hα, UV, FIR, and radio indicators. The scatter in the SFR fit is dominated by the scatter in the measurements from these different groups; it is typically ±30% in the redshift range 0 ≲ z ≲ 1 (Hopkins & Beacom 2006). At the lowest redshift, the scatter approaches the intrinsic uncertainties in the calibrations of each indicators. For the bulk of the galaxies the continuous star formation approximation (i.e., star formation remains constant on timescales longer than the lifetimes of the dominant UV-emitting massive stars, ∼60 Myr; Kennicutt 1998) holds, so that the calibration uncertainties are ±10%–20%. If the assumption is incorrectly applied, as would be the case for young starburst galaxies, the SFR would be underestimated by ∼30%. Therefore, the scatter in the SFR derived from a single indicator and the scatter among different indicators are generally at the tens of percent level.

The choice of IMF plays an important role in the normalization of the SFR, because the IMF is used to scale the observed massive-SFR to the total SFR. Adopting instead of the Salpeter A IMF a classic Salpeter IMF or a flatter Baldry-Glazebrook IMF, the total SFR normalization varies at the ±30% level (e.g., Hopkins & Beacom 2006). However, it is important to stress that this does not affect the predicted SNR normalization anywhere near as strongly, because the massive stars that are used to make the SFR measurements are close in mass to the progenitors of CC SNe. In other words, an increase or decrease of the SFR due to steeper or shallower IMF is nearly completely canceled by a respective decrease or increase in the fraction of massive stars. The difference between the predicted SNR using the Salpeter A and the Baldry–Glazebrook IMF is less than 10% (Horiuchi et al. 2009).

While extinction by dust raises the possibility for the SFR to be overcorrected for dust, it is not realistically possible that the SFR in the most important redshift range (0 < z < 1) is overestimated by a factor of ∼2. Since interstellar dust absorbs UV emission and re-emits in the FIR, the UV-derived SFR must be corrected. In our adopted SFR compilation, the UV-derived SFR are dust-corrected by adding the FIR-derived SFR of Le Floc'h et al. (2005) at their respective redshifts. The accuracy of this extinction correction has been discussed using individual galaxies (Iglesias-Páramo et al. 2006) and by comparisons with independent Hα-derived SFR (Bell 2003); see also Hopkins (2004) and Hopkins & Beacom (2006). The SFR corrected for dust in this way are consistent with recent SFR measurements in many wavebands and dust-correction methods (see Section 2). In the range z ≲ 0.4, where the SNR normalization discrepancy is most significant, dust correction typically increases the UV-derived SFR by a factor of ∼2. Thus, to explain the supernova rate problem by lowering the SFR, one would require almost no dust at these distances, which is inconsistent with our understanding of the FIR universe (Hauser & Dwek 2001; Lagache et al. 2005).

At higher redshifts (0.5 ≲ z ≲ 1) the dust correction to the UV data becomes larger. By z ≈ 1, more than 80% of the total SFR comes from the FIR. Eventually, the dust opacity in galaxies becomes high enough that the FIR luminosity itself measures the bolometric luminosity of the star formation activity. To reduce the SFR by a factor of ∼2 under these conditions requires one to either have severe contamination of the FIR emission from non-star-forming sources or that the FIR to SFR calibration is incorrect. While it is known that there is a "cirrus" component in the FIR that is associated with more extended dust heated by a population of older stars, their relative contribution varies substantially from galaxy to galaxy. For spirals it can be as high as 50%–70%, but for the star-forming galaxies that dominate at high redshifts it is expected to be much smaller (Lonsdale Persson & Helou 1987; Kewley et al. 2002). The calibration uncertainties are ±10%–20% as described above.

Finally, the integrated SFR normalization can be checked against observable quantities (Madau et al. 1998), for example the stellar mass density and the extragalactic background light (EBL). The stellar mass density has been measured up to redshifts of a few and are typically a factor of ∼2 smaller than those predicted from the SFR assuming the Salpeter A IMF (see, e.g., Hopkins & Beacom 2006; Wilkins et al. 2008a). While a potential explanation of this discrepancy is to reduce the cosmic SFR normalization by a factor of ∼2, i.e., at the same time also solving the supernova rate problem, this is unlikely to be the correct explanation. First, the stellar mass density is dominated by low-mass stars rather than the high-mass stars constrained by the SFR, leaving the comparison to reveal more about the IMF slope rather than the SFR normalization. For example, an IMF with a high-mass slope shallower (−2.15; equivalent to the Baldry–Glazebrook IMF) than the Salpeter slope (−2.35) reconciles the discrepancy at z < 0.4 (Wilkins et al. 2008b). Second, it has been argued that observations could be underestimating the stellar mass density (e.g., Nagamine et al. 2004; Somerville et al. 2001). Third, as we discuss below, such an explanation creates a tension in explaining the EBL.

The EBL is powered by moderately high-mass stars (half of the stellar EBL is powered by stars with masses M ≳ 3 M_☉), so that the IMF dependency is modest. Approximately 70% of the EBL arises from the z < 1 range we are most interested in, with only a few percent contribution from non-nucleosynthesis energy sources such as active galactic nuclei (AGNs; Hopkins et al. 2006). This makes the EBL a useful probe of the integrated (0 < z < 1) SFR normalization. The minimum EBL has been derived indirectly by counting visible galaxies, while the maximum EBL has been directly measured (see, e.g., Hauser & Dwek 2001). We summarize the findings of Horiuchi et al. (2009) here. Integrating from the FIR to UV bands yields a total EBL in the range 50–100 nW m⁻² sr⁻¹, comparable with other previously evaluated estimates (e.g., Fardal et al. 2007). The EBL has recently been constrained by the observations of gamma rays from distant blazars to be between these extremes (Aharonian et al. 2006; Albert et al. 2008; Abdo et al. 2010; Orr et al. 2011). We adopt a nominal total EBL of 73 nW m⁻² sr⁻¹ that respects the gamma-ray constraints. If secondary gamma rays produced en route by beamed cosmic rays are important (Essey & Kusenko 2010; Essey et al. 2011), the EBL could be even higher. In comparison, the result calculated from the SFR of Hopkins & Beacom (2006), using the PEGASE.2 population synthesis code (Fioc & Rocca-Volmerange 1997), is 88⁺³⁶₋₂₈ nW m⁻² sr⁻¹ for the Salpeter A IMF and 78⁺³¹₋₂₄ nW m⁻² sr⁻¹ for the Baldry–Glazebrook IMF. Therefore, the predicted and measured total EBLs do not allow a factor of ∼2 reduction of the SFR.

In summary, the uncertainties associated with SFR measurements are generally at the tens of percent level. The integrated cosmic SFR normalization has been cross-checked with the EBL at a similar precision and does not allow the cosmic SFR to be decreased enough to explain the supernova rate problem.

The local (≲ 100 Mpc) SFR sets the birth rate of stars nearby and has important implications for our study of the local SNR. Here, we discuss whether the local SFR measurements are different from the cosmic SFR. Since the SFR are densities, they should not depend on distance, except for evolution with redshift, which is modest within 100 Mpc. We address the validity of the comparison in light of expected cosmic variance and bias due to missing massive galaxies in small volumes, before discussing the results.

Cosmic variance falls approximately as distance to the power of −2 (Peacock & Dodds 1994) and is minimal by several tens of Mpc. Normalizing the density power spectrum to the seven-year WMAP result of σ₈ = 0.809 (Jarosik et al. 2011), the density fluctuations within 30 Mpc should only be ∼0.1 of the mean density. Variations in smaller volumes are more likely. For example, it is not surprising that the galaxy B-band luminosity density in the 8 Mpc volume is a factor of 1.7–2.0 times the global luminosity density (as derived from the SDSS and the Millennium Galaxy Catalog; Karachentsev et al. 2004).

Given the smaller volume, the rarest and most massive galaxies would be missing. However, the effect of missing their star formation is minimal. While massive galaxies have higher SFR, massive galaxies are rarer and have lower star formation per mass, so that the contribution to the total SFR peaks at galaxy mass ∼6 × 10¹⁰ M_☉. Consider the volume within 40 Mpc. Integrating the SDSS galaxy stellar mass function, the number of galaxies more massive than 4 × 10¹¹ M_☉ is expected to be ≲ 1. The fraction of the SFR missed by excluding galaxies above 4 × 10¹¹ M_☉ is less than 10% (Brinchmann et al. 2004). Therefore, the volume within several tens of Mpc is a fair representation of the global SFR.

In Figure 2, we show SFR measurements in the local volume, derived from Hα (Gallego et al. 1995; Pérez-González et al. 2003; Hanish et al. 2006; James et al. 2008), FIR (Yun et al. 2001), radio (Serjeant et al. 2002; Condon et al. 2002), and a combination of UV and IR (Martin et al. 2005; Bothwell et al. 2011). The measurements are corrected for dust and have been scaled to our chosen cosmology and IMF. At distances just beyond those shown, a recent thorough study using a combined UV and IR data set find 0.0193 ± 0.0012 h₇₃ M_☉ Mpc⁻³ yr⁻¹ (at z ≈ 0.05), in excellent agreement with the cosmic SFR and with small uncertainty (Bothwell et al. 2011).

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The SFR within 11 Mpc has been measured by the 11 Mpc Hα and Ultraviolet Galaxy Survey (11HUGS; Kennicutt et al. 2008; Bothwell et al. 2011). The value, 0.0262^+0.0044_−0.0035 h₇₃ M_☉ Mpc⁻³ yr⁻¹, shows an overdensity with respect to the cosmic SFR. The degree of overdensity is approximately a factor of1.7 and is within expectations of cosmic variance. It is smaller than but similar to the enhancement seen in the galaxy B-band luminosity density. The remaining measurements at larger distances are compatible within errors with the cosmic SFR. The mean of the measurements, omitting the 11HUGS measurement because of the known overdensity, is 0.017^+0.007_−0.005 M_☉ Mpc⁻³ yr⁻¹. This is in good agreement with the cosmic SFR. The uncertainty range has been conservatively taken to be the range of the SFR measurements, because some of the measurements are in disagreement with each other, suggesting the true uncertainties are larger than reported for some individual measurements. We conclude that the local SFR data do not support large deviations from the cosmic SFR.

In summary, SFR measurements in the local volume are in agreement within uncertainties with the better-measured cosmic SFR, except for the volume within ≲ 10 Mpc which has a factor of ≲ 2 enhancement. These results provide the important basis on which we interpret the local SNR in Section 3.6.

The predicted SNR depends on the number of stars that become CC SNe per unit stellar mass formed. To a first approximation, this depends on the mass range of stars that make CC SNe. There is observational evidence that a wide range of massive stars yield optically luminous CC SNe. Conceptually, the mass range is controlled by two parameters, the lower mass limit that is the boundary between the formation of a white dwarf and a neutron star, and the upper mass limit that is less well defined but could be the boundary between forming a neutron star and a stellar mass black hole. This is only approximate, since stellar evolution and SN simulations indicate that parameters such as rotation and metallicity also affect the outcome of core collapse, and prompt black hole formation may be accompanied by some transient phenomena, even if dim. For example, a rare group of highly rotating stars that collapse to black holes form the current paradigm for gamma-ray burst central engines (Woosley 1993). We assume that parameters other than mass are averaged out in integrated populations and that the mass is the primary parameter for the CC SNe outcome. The existence of CC SNe progenitors of a given mass does not require or prove that all progenitors of that mass produce similar CC SNe.

The lower mass limit is newly supported by direct observations of CC SNe progenitors. Using archival images, it has been possible to identify the stellar progenitors of some nearby CC SNe (see, e.g., Smartt 2009, for a review). Using the stars' luminosities and other information, the stellar masses can be measured or limited. In Smartt et al. (2009), the authors reviewed 20 progenitors of Type IIP SNe and statistically found the progenitor mass limit M^IIP_min = 8.5⁺¹_−1.5 M_☉. This is consistent with the highest masses estimated for white dwarf progenitors, ∼7 M_☉ (Kalirai et al. 2008; Williams et al. 2009). Thus, two different approaches seem to be converging to M_min ≈ 8 ± 1 M_☉. This uncertainty affects the predicted SNR at the ±20% level.

On the other hand, the upper mass limit is less certain. Fortunately, as long as the upper limit is large, it does not strongly affect the predicted SNR. For example, to fully explain the normalization discrepancy in Figure 1 would require M_max ≈ 13 M_☉ for M_min = 8 M_☉. Theoretically, the upper mass limit is M_max ≈ 40 M_☉ for a sub-solar metallicity star (Fryer 1999). Higher metallicity generates stronger mass loss prior to collapse and predicts the formation of a Wolf–Rayet (WR) star, which can extend M_max to 100 M_☉ (Heger et al. 2003). The upper mass limit derived from the analysis of 20 Type IIP SN progenitors is M^IIP_max = 16.5 ± 1.5 M_☉ (Smartt et al. 2009). No progenitors of Type Ibc SNe have been directly identified, but the spatial distribution of CC SNe shows that Type Ibc SNe progenitors must be more massive than those of Type IIP SNe (Anderson & James 2008, 2009). It is widely expected that Type Ibc originate from evolved massive WR stars that have shed their envelopes. From Galactic stellar clusters and stellar clusters in the LMC, such WR stars are estimated to have evolved from main-sequence masses 25 M_☉ and above (Massey et al. 2000, 2001). Very luminous Type IIn have been observed to arise from the core collapse of very massive stars of the luminous blue variable (LBV) type (Gal-Yam et al. 2007; Gal-Yam & Leonard 2009; Smith et al. 2011), whose masses are ≳ 20 M_☉. Recent studies of the peculiar SN 1961V support its nature as a true CC SNe, implying its progenitor, a massive ≳ 80 M_☉ star, produced an optically luminous CC SNe (Kochanek et al. 2010; Smith et al. 2010). Therefore, there is substantial evidence for the upper mass limit being high.

However, the outcomes of the mass range from 17 to 25–30 M_☉ remain uncertain. Smartt et al. (2009) noted that stars in this mass range are not found as CC SNe progenitors, and termed this the "red supergiant problem." Future progenitor studies may discover CC SNe progenitors in this mass range. For example, the progenitor of the recent Type IIL SN 2009kr has been suggested to be ∼20 M_☉ (Elias-Rosa et al. 2010; although Fraser et al. 2010a find ∼15 M_☉). It is also possible that the majority of these stars form black holes at core collapse and lead to dim or dark CC SNe (Smartt et al. 2009). In Section 5, we discuss how monitoring for the disappearance of high-mass stars will allow observation of optically dark core collapses (Kochanek et al. 2008).

In summary, we adopt the nominal mass range for optically luminous CC SNe of 8–40 M_☉, based on stellar and supernova simulations; the maximum mass range 8–100 M_☉, and the conservative minimum mass range the combination of 8.5–16.5 M_☉ and 25–40 M_☉. Compared to the nominal 8–40 M_☉, the maximum and minimum mass ranges affect the predicted cosmic SNR by +10% and −30%, which are insufficient to explain the supernova rate problem.

Luminous CC SNe (we henceforth define luminous CC SNe as CC SNe more luminous than −15 mag) that fall within the sensitivity of SN surveys can still be missed. Here, we discuss the potential impact of these missed luminous CC SNe.

Flux-limited surveys targeting a pre-selected sample of galaxies are naturally biased against small galaxies, and inevitably result in some CC SNe being missed. Even the LOSS survey views only 45% of the total stellar light accessible from the survey position (Leaman et al. 2011). In the early works of Cappellaro et al. (1999) and updates by Cappellaro et al. (2005) and Botticella et al. (2008), this has been corrected for by assuming that the CC SN rate per galaxy size (mass or luminosity) remains constant in galaxies having the same Hubble type or B–K color. Therefore, the volumetric SNR could be determined even if the faint end of the galaxy luminosity function is not fully sampled, by measuring the CC SN rate in large galaxies and multiplying by an independently measured galaxy luminosity density that has been integrated to include small galaxies. However, more careful sampling by LOSS helped reveal the rate–size relation, where the SN frequency is not linearly proportional to the galaxy size but rather size to the power of 0.4–0.6, the exact number depending on the SN Type and galaxy Hubble type (Li et al. 2011a). The rate–size relation is comparable to the trend of specific-SFR with galaxy mass (Salim et al. 2007; Schiminovich et al. 2007) and implies more emphasis on small galaxies.

However, the rate–size relation has been measured with galaxies down to L_K ≈ 2 × 10¹⁰ L_☉ (M_K ≈ −22.4 mag, approximately 1.3 mag fainter than the Milky Way) and additional effects in smaller galaxies may not addressed. In particular, there is mounting evidence that CC SNe in small galaxies are unlike those in large normal galaxies. Some types of rare CC SNe occur only in low-metallicity dwarf galaxies (Le Floc'h et al. 2003; Fruchter et al. 2006) and results from the Palomar Transient Factory (PTF), which remedy the bias against small galaxies by discovering the SNe first and the galaxy second, suggest systematic differences in CC SNe in small and normal galaxies. For example, more CC SNe have been discovered in dwarf galaxies than predicted in Young et al. (2008) from SFR and a CC SN mass range of 8–50 M_☉ (Arcavi et al. 2010). Also, the distributions of CC SNe types are different (Arcavi et al. 2010).

It is instructive to compare the distributions of CC SNe and SFR as functions of host galaxy properties. The fraction of CC SNe in dwarfs (defined as galaxies with M_R > −18 mag in the PTF) is 15/69 ≈ 0.22 (Figure 3 of Arcavi et al. 2010), even though the share of the total SFR in such galaxies is only 0.1 (Figure 15 of Brinchmann et al. 2004). For a cut of M_R > −19 mag, the fractions are 0.33 and 0.20, respectively. K corrections are insufficient to account for this difference, and the difference could be interpreted as an increased efficiency of forming CC SNe in small galaxies. The increase is a factor of ∼2 over that in normal galaxies, and the extra CC SNe would increase current SNR measurements. Since 20% of the SFR (in small galaxies) yields twice as many CC SNe, the increase in the SNR measurements is approximately 20%. As an extreme, we adopt an increase of 2 × 0.2/0.8 ≈ 50%, i.e., assuming that all CC SNe in small galaxies are unaccounted for at present. A detailed comparison of the distributions of CC SNe and SFR requires more data and is beyond the scope of this paper. The importance of small galaxies will become clearer with time, as SN surveys continue to discover more CC SNe in small galaxies. In particular, if the efficiency is enhanced over a wider range of galaxies, the increase in the SNR could be larger than discussed here.

Finally, in all SN surveys, a short cadence is desirable for discovering SN during the most luminous epochs, as well as for obtaining a well sampled light curve for efficiency calculation purposes. The cadences of surveys, shown in Table 1, are generally shorter than the peak durations of CC SNe. For example, Type Ibc typically drop by a magnitude in less than 1 month, while Type IIP evolve more slowly, with a plateau remaining within 1 mag of the peak for about 2–3 months.

In summary, incomplete surveying of small galaxies may lead to SNR measurements underestimating the true SNR. Based on current data on dwarf galaxies, the effect may be a 20%–50% increase in the SNR measurements. The importance of small galaxies remains to be investigated further with more data, including its caveats (Section 4.4).

Host dust extinction makes CC SNe appear dimmer and correcting for missing obscured CC SNe is often the most uncertain ingredient in SNR measurements. Here we review dust corrections performed in SNR measurements.

The dust correction applied to SNR measurements ranges from a few tens of percent at low redshifts to a factor of ∼2 at high redshifts. According to the semi-analytic model of Hatano et al. (1998), the distribution of CC SN extinction due to host galaxy inclination peaks at low extinction but has an extended tail reaching many magnitudes (19.8 mag for a perfectly edge-on galaxy). This model was applied in SNLS, who found that it increased their measured SNR by ≈15% (Bazin et al. 2009). Botticella et al. (2008) used a statistical approach similar to Hatano et al. (1998) and found an increase of a factor of ∼2. Although the authors also present an extreme dust scheme which would in fact find agreement with the SNR predicted from the cosmic SFR, they point out that such an extreme dust scenario is not strongly motivated and is meant as an upper limit only. At higher redshift, Dahlen et al. (2004) combined the model of Hatano et al. (1998) with a starburst extinction law, reporting increases of 60% and a factor of ∼2 for their two measurements at z ≈ 0.3 and z ≈ 0.7, respectively.

As pointed out in Mannucci et al. (2007), at high redshift there is additional extinction due to starburst galaxies and highly star-forming galaxies (luminous and Ultraluminous IR Galaxies, which we collectively call ULIRG). Identified by their strong FIR emission and high SFR, ULIRG should house many more CC SNe per galaxy, but we only detect a small fraction of CC SNe because of the higher dust obscuration. It has been estimated that as many as 60%–90% of the CC SNe could go undiscovered (Mannucci et al. 2003), although multi-wavelength studies are starting to discover some of these (Kankare et al. 2008). Because the importance of ULIRG increases with redshift, the fraction of CC SNe that are expected to be missing increases from ∼5% at z ≈ 0 to 20%–40% at z ≈ 1 (Mannucci et al. 2007).

In summary, correction for host galaxy obscuration remains uncertain. Extreme corrections that would explain the supernova rate problem are not prohibited, in particular at high redshifts where the excess extinction due to starburst galaxies is considered. However, this does not easily explain the supernova rate problem at low redshifts, where starburst galaxies provide only a small fraction of the SFR. We discuss missing dim CC SNe in more detail next.

Type IIP SNe are the most common type of CC SNe and also the most varied in terms of luminosity (e.g., Richardson et al. 2002; Hamuy 2003). The luminosity of a typical Type IIP SN remains nearly constant for a relatively long duration of ∼100 days (the plateau phase), after which it drops sharply, marking the transition to the nebular phase (e.g., Patat et al. 1994). It is expected that the radius of the progenitor plays a key role in shaping both the plateau length and luminosity (Arnett 1980). For example, the unusually dim light curve of SN 1987A is attributed to its compact progenitor (Arnett 1987). Recently, SN searches have revealed a class of Type IIP SNe that have peak and tail luminosities that are even lower than those of SN 1987A (Zampieri et al. 2003; Pastorello et al. 2004). The prototype of these ⁵⁶Ni-poor (≲ 0.01 M_☉) low-velocity Type IIP SNe is SN 1997D (M_V ≈ −14.7 mag; Turatto et al. 1998), and there are now more than a dozen similar CC SNe (see, e.g., Pastorello et al. 2004, 2006; Fraser et al. 2010b). The dimmest are SN 1999br (M_R ≈ −13.5 mag; Pastorello et al. 2004) and PTF10vdl/SN 2010id (M_R ≈ −14.0 mag; Gal-Yam et al. 2011), although even dimmer suspects exist, e.g., the transient in M85 (M_R ≈ −12 mag; Pastorello et al. 2007; see also Kulkarni et al. 2007 for an alternative scenario). Additionally, CC SNe dimmer than M_R ≈ −13.5 mag have been inferred from the non-observation of SNe associated with several low-redshift long-GRBs (Fynbo et al. 2006; Della Valle et al. 2006). There is a population of dim (M_V ≲ −12 mag) Type IIn SNe, but these are suspected to be LBV outbursts and not core collapses (e.g., Maund et al. 2006; Smith et al. 2010).

In addition to the cadence and galaxy completeness effects mentioned in Section 3.4, SN surveys are biased against low-luminosity CC SNe like SN 1997D. The limiting absolute magnitudes of SN surveys are typically M ≈ −15 to −16 (Table 1). Although SN surveys can discover dimmer CC SNe, the rapidly falling detection efficiency and the smaller volume from which dim CC SNe (we define CC SNe that are dimmer than M = −15 mag as "dim" CC SNe) may be confidently discovered makes collecting dim CC SNe challenging. To correct for missing dim CC SNe, an SN luminosity function that is complete to dimmer CC SNe must be adopted. Here, we revisit the importance of dim CC SNe. We make use of SN discoveries recorded in SN catalogs. The SN catalog combines results from SN surveys and amateur discoveries, resulting in a non-uniform sample of CC SNe of various qualities, bands, and uncertainties. However, it is illustrative because it sets a strong lower bound on dim CC SNe. We first discuss the local (≲ 100 Mpc) volume for a qualitative analysis in Section 3.6.1. Then, we focus on the very local (≲ 10 Mpc) volume for a more quantitative analysis in Section 3.6.2.

3.6.1. How Ubiquitous are Dim CC SNe?

We start with the Sternberg Astronomical Institute (SAI) SN Catalog (Bartunov et al. 2007), selecting CC SNe of Types IIP, IIL, IIn, IIb, and Ibc. We select only the most recent decade, from 2000 to 2009, since previous decades have severe incompleteness problems (Horiuchi & Beacom 2010). We cross check the classification with the Harvard SN database,¹⁰ which results in tagging some unclassified SNe as CC SNe. We update magnitudes to peak magnitudes where possible by investigating the literature and circulars.¹¹ We update SN host galaxy distances with redshift-independent distance measures from the Extragalactic Distance Database (EDD; Tully et al. 2009). When these are not available, we used distance estimates from NED.¹² The uncertainties in the distances, which can reach ±10%–20%, affect the absolute magnitudes. As we discuss in Section 4.3, this does not strongly affect our results.

From the SAI catalog we discard CC SNe that seem to clearly be LBV transients (e.g., SN 2000ch, SN 2001ac, SN 2002kg, SN 2003gm, SN 2006fp, SN 2007sv, SN 2009ip). On the other hand, we include the possible SN impostors SN 2008S, NGC 300-OT, and SN 2002bu, whose nature as either true CC SNe or extreme stellar outbursts remains subject to debate; we discuss the alternative assumption in Section 4.4. Next, we check the discovery phases of CC SNe dimmer than −16 mag to identify CC SNe that are dim due to being discovered post-peak. For this identification we adopt the following criteria: SN IIP discovered more than 2 months post-peak, SN IIL over 1.5 months, and all SN IIn, IIb, Ibc over 1 month. These are the typical time scales over which the nominal light curves remain within 1 mag of the peak (Li et al. 2011b). We identify 36 CC SNe that fit this category during 2000–2009 within 100 Mpc.

From the modified SAI sample we construct three sets of "catalog" SNR. The first is the definitely luminous catalog SNR. For this we select CC SNe with absolute magnitudes below −15 mag. The second set is the definitely dim catalog SNR. For this we select CC SNe above −15 mag. We do not include dim CC SNe that have been discovered in their late phases, as the true peak magnitude is uncertain. The third and final set is the total catalog SNR. This includes all CC SNe, including definitely luminous CC SNe, definitely dim CC SNe, and late-time discoveries. Thus, the total catalog SNR is larger than the sum of the definitely luminous and definitely dim catalog SNR.

All catalog SNR should be considered lower limits since they are derived from simple counting of likely incomplete SN discoveries. In addition to increasing incompleteness in older data (Horiuchi & Beacom 2010), there is a strong observational bias against CC SNe in the southern hemisphere. In the latest decade and within 100 Mpc, there are almost 1.9 times more CC SNe discovered in the northern hemisphere than the southern hemisphere. There is also a strong bias against CC SNe in small galaxies. Considerations of the observational efficiency and coverage will only increase the catalog SNR.

If the SN discoveries are sufficiently complete, the catalog SNR should be flat with distance. This is because they are number densities, and because the local SFR did not reveal significant cosmic variance, except in the very local volume within ∼10 Mpc (Section 3.2). In other words, any significant decrease of the catalog SNR at any distance signifies an incompleteness of the SN discoveries at that distance.

In Figure 3, the catalog SNR are binned in distance and compared to the cosmic SNR predicted from the SFR and directly measured. Neither host nor Galactic extinction corrections have been applied to the catalog SNR. Correcting would shift a small number of dim CC SNe to luminous CC SNe, but would not change qualitative conclusions; in Section 3.6.2, where we discuss quantitative results, we apply Galactic extinction corrections. Each bin contains 20 luminous CC SNe, except for the smallest-distance bin, which contains 10 luminous CC SNe.

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The luminous catalog SNR should be compared to the measured cosmic SNR. The luminous catalog SNR is generally decreasing with distance but it is flat out to ∼25 Mpc, suggesting reasonable completeness to that distance. The normalization of the flat section is comparable to the measured cosmic SNR on larger scales by LOSS, supporting this interpretation. There are 50 luminous CC SNe in the first three distance bins. The smallest-distance bin shows an enhancement as expected from the SFR.

The total catalog SNR should be compared to the predicted cosmic SNR. The total catalog SNR is flat out to ∼25 Mpc and shows a normalization that is only slightly lower than expectations from the cosmic SFR. This supports our earlier claims that the true cosmic SNR is as large as expected (e.g., Horiuchi et al. 2009; Beacom 2010). There are 75 CC SNe within the first three distance bins. Again, the smallest-distance bin shows an enhancement with respect to the predicted cosmic SNR, as expected from the SFR.

Finally, the dim catalog SNR falls with distance at all distances, a sign of incompleteness. At the smallest-distance bin, where dim CC SNe are least likely to be missed, the dim catalog SNR is, surprisingly, just as large as the luminous SNR measured by LOSS. There are 4 dim CC SNe in this bin. This normalization is large enough to help solve the supernova rate problem, at least at z ≈ 0, although the enhancement of the SFR in this distance range needs to be taken into account.

Catalog CC SNe and Type Ia supernovae (SNe Ia) can be used to make an independent test of the normalization of the cosmic CC SNe rate (Horiuchi et al. 2009). SNe Ia are luminous and have been intensively sought. Their cosmic rate history is better measured than that of CC SNe, and the distance out to which their catalog is apparently complete is larger (Horiuchi & Beacom 2010). We consider the SN Ia to CC SN ratio, comparing rates from cosmic surveys near z ≈ 0 (for larger z, the effect of SN Ia progenitor delays changes the ratio) and SN catalogs. From our fits to the measured cosmic rates, we find a SN Ia to CC SNe ratio at z ≈ 0 of ∼0.18. For comparison, the same ratio for the LOSS results is ∼0.43. This difference of a factor of ∼2 in the ratios is another statement of the supernova rate problem. The same ratio for the SN catalog should largely cancel incompleteness or local enhancement effects and is independent of the assumed cosmic SFR. We use a maximum distance large enough to contain enough SNe Ia and small enough that the CC SNe are reasonably complete. For SNe within 20 Mpc, we find a ratio of 0.25; this increases to 0.30 by 25 Mpc and rises to 0.35 at larger distances. We interpret this as confirming that cosmic CC SNe are as common as predicted and that many dim ones are missed at increasing distances in catalogs and cosmic surveys.

In summary, we analyze the local (≲ 100 Mpc) volume to study dim CC SNe. The SNR follows the trend expected from the SFR out to ∼25 Mpc. We find that dim CC SNe are severely incomplete. Despite this, dim CC SNe are numerous in the very local (≲ 10 Mpc) volume. In the next section, we discuss the fraction of dim CC SNe in more detail.

3.6.2. What is the Fraction of Dim CC SNe within ∼10 Mpc?

We use the very local (≲ 10 Mpc) volume to quantify the importance of dim CC SNe. It is not feasible to extend to greater distances because of the severe incompleteness of dim CC SNe. We choose a nominal distance of 10 Mpc, which yields sufficient statistics, and discuss how halving and doubling the volume changes the results. As discussed in the previous section, the normalization of the dim catalog SNR within 10 Mpc is high. However, the very local volume has an enhanced absolute SFR. Therefore, we conservatively use the fraction of dim CC SNe, f_dim = N_dim/(N_dim + N_luminous), the ratio of definitely dim CC SNe over the sum of definitely dim and definitely luminous CC SNe. Here, the subscript "dim" refers to objects fainter than the dim-th magnitude. We will focus on a dim-luminous magnitude cut of −15 mag, but also show −16 mag for illustration.

The very local CC SNe are shown in Table 2. Here, the magnitudes are corrected for Galactic extinction following Schlegel et al. (1998) and assuming a Cardelli et al. (1989) Galactic reddening law. Distance estimates for CC SNe host galaxies vary among sources, and we adopt the estimates used in dedicated papers on the CC SNe when this is available. Otherwise, we continue to use the distances from EDD or NED as explained in Section 3.6.

Table 2. Very Local CC SNe During 2000–2009 Inclusive

SN	Galaxy	Type	D (Mpc)	E(B − V)	Absolute Magnitudea	Discovery Phase	References
SN 2002bu	NGC 4242	IInb	5.8	0.012	$\bf M_R\approx -14.1$	Early	Hornoch (2002); Smith et al. (2010)
SN 2002hh	NGC 6946	IIP	5.9	0.342	$\bf M_R\approx -14.3$	Early	Pozzo et al. (2006)
SN 2002kg	NGC 2403	LBV	3.2	0.04	MV ≈ −9	Not CC SN	Maund et al. (2006)
SN 2004am	NGC 3034 (M82)	IIP	3.5	0.159	MR < −12.2	3 months	Singer et al. (2004)
SN 2004dj	NGC 2403	IIP	3.2	0.04	MR ∼ −16.0	1 month	Vinkó et al. (2006); Zhang et al. (2006)
SN 2004et	NGC 6946	IIP	5.9	0.342	MR ≈ −17.6	Early	Sahu et al. (2006)
SN 2005af	NGC 4945	IIP	3.6	0.177	$\bf M_R\sim -15.4$	1 month	Jacques & Pimentel (2005)
SN 2005at	NGC 6744	Ic	7.1	0.043	$\bf M_R\sim -15.1$	2 weeks	Martin et al. (2005b)
SN 2008bk	NGC 7793	IIP	4.1	0.019	$\bf M_R\sim -15.5$	1 month	Monard (2008a); Morrell & Stritzinger (2008)
SN 2008iz	NGC 3034 (M82)	II?	3.5	0.159	no optical	Radio only	Brunthaler et al. (2010)
SN 2008S	NGC 6946	IInb	5.9	0.342	$\bf M_R\approx -13.3$	Early	Botticella et al. (2009); Smith et al. (2009)
NGC 300-OT	NGC 300	IInb	1.9	0.013	$\bf M_V \sim -12.3$	1 month	Bond et al. (2009)
SN 2002ap	NGC 0628	IcPec	9.0	0.07	MR ≈ −17.8	Early	Gal-Yam et al. (2002)
SN 2003gd	NGC 0628	IIP	9.0	0.07	MR ∼ −16.7	2 months	McNaught (2003)
SN 2005cs	NGC 5194 (M51)	IIP	8.4	0.035	$\bf M_R\approx -15.4$	1 month	Pastorello et al. (2006)
SN 2007gr	NGC 1058	Ic	9.9	0.062	MR ≈ −17.4	Early	Valenti et al. (2008)
SN 2008ax	NGC 4490	IIb	9.6	0.022	MR ≈ −16.6	2 weeks	Pastorello et al. (2008)
SN 2009hd	NGC 3627 (M66)	IIP	8.3	0.032	$\bf M_R\approx -13.9$	Early	Monard (2009)
SN 2001ig	NGC 7424	IIb	11.5	0.011	MR ≈ −17.3	Early	Bembrick et al. (2002)
SN 2003ie	NGC 4051	II	12.2	0.013	MR < −15.6	Uncertain	Arbour & Boles (2003)
SN 2003jg	NGC 2997	Ibc	11.3	0.109	$\bf M_R\sim -14.1$	Few weeks	Biggs et al. (2003)
SN 2007it	NGC 5530	IIP	11.7	0.116	MV ≈ −18.7	Early	Pojmanski (2007)
SN 2008eh	NGC 2997	Ibc?	11.3	0.109	$\bf M_R \sim -15.3$	1 month	Monard (2008b)
SN 2009ib	NGC 1559	IIP	12.6	0.03	$\bf M_R\approx -15.9$	Early	Pignata (2008)

Notes. CC SNe that are definitely dimmer than −16 mag are shown in bold type. CC SNe with magnitude limits may be dim or luminous CC SNe, but cannot be convincingly identified with the data available. The list has been divided into blocks by distance (7.9, 10, and 12.6 Mpc). The first block and second block contain equal volumes, while the third block contains twice that. ^aShows the peak magnitudes for pre-peak discoveries, and the discovery or inferred peak magnitude for others. Values have been corrected for Galactic extinction. Unfiltered images are categorized as R band. ^bPossible SN impostors, which may be CC SNe or stellar outbursts.

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Several CC SNe are dim due to being highly extinguished by host dust, e.g., SN 2002hh with A_V ≈ 5 mag and SN 2008ax with A_R ∼ 1 mag (Pozzo et al. 2006; Mattila et al. 2004). The extreme case is SN 2008iz, which was discovered in radio observations and is not seen in the optical, suggesting an explosion behind a large gas or dust cloud in the central part of the host galaxy (Brunthaler et al. 2010). We assume SN 2008eh to have been a CC SN even though it lacks a spectroscopic classification. Despite its relatively early discovery (it was not observed in the field 1 month prior to discovery), it was dim (M_R ∼ −15.3; Monard 2008b). From archival Spitzer images of its host galaxy NGC 2997, we locate the SN position in a spiral arm, next to a star forming region. Its light curve most closely resembles a dim Type Ibc CC SNe rather than a SN 2002cx-like peculiar Type Ia.

It is not feasible to estimate the peak magnitudes of some CC SNe. For example, SN 2004am was discovered in the nebular phase, and the discovery phase of SN 2003ie is unknown. We collectively treat these CC SNe as poor-quality CC SNe. We exclude these CC SNe from estimates of f_dim, since we want to estimate the dim fraction from a clean sample of SNe.

Within 10 Mpc, there are 18 CC SNe and 1 LBV (SN 2002kg). Excluding poor-quality CC SNe leaves 16 CC SNe, of which 6 are more luminous than −16 mag (SN 2002ap, SN 2003gd, SN 2004dj, SN 2004et, SN 2007gr, SN 2008ax) and 10 are dimmer than −16 mag. Of the latter, 3 are highly reddened (SN 2002hh, SN 2008iz, SN 2009hd), 4 are intrinsically dim (SN 2005af, SN 2005at, SN 2005cs, SN 2008bk), and 3 are possible SN impostors (SN 2002bu, SN 2008S, NGC 300-OT). The number of CC SNe dimmer than −15 mag is 6, so the dim fraction is f_>−15 = 6/16 (f_>−16 = 10/16). If we include poor-quality CC SNe and allow them to be either luminous or dim CC SNe, the dim fraction varies between f_>−15 = 6/18 and f_>−15 = 8/18.

We expect the dim fraction to decrease with distance, since dim CC SNe are more easily missed at larger distances. Going out to twice the volume (i.e., within 12.6 Mpc) we find 6 more CC SNe (5 after excluding poor-quality CC SNe). There is 1 dim CC SNe (SN 2003jg), bringing the dim fraction to f_>−15 = 7/21 (f_>−16 = 13/21). Considering half the volume (i.e., CC SNe within 7.9 Mpc) we find 12 CC SNe (10 after excluding poor-quality CC SNe), with f_>−15 = 5/10 (f_>−16 = 8/10). The CC SNe tallies are summarized in Table 3.

Table 3. Summary of Very Local CC SNe

Number of CC SNe	0–7.9 Mpc	7.9–10 Mpc	10–12.6 Mpc
Expected total (cosmic rate)	2.9 ± 0.5	2.9 ± 0.5	5.8 ± 1
Expected (LOSS rate)	1.5 ± 0.4	1.5 ± 0.4	3.0 ± 0.9
Total observed CC SNe	12a	6	6
M < −16	2	4	2
−16 < M < −15	3	1	2
−15 < M (dim)	2	1	1
Possible SN impostor	3	0	0
Poor-quality CC SNe	2	0	1

Notes. Expected number of CC SNe are shown for comparison. They are derived from measurements at larger distances, and should be multiplied by ≲ 2 within ∼10 Mpc to reflect the overdensity of SFR at these distances. The uncertainty is from the 1σ uncertainty on the cosmic SFR or the uncertainty on the SNR measurements. ^a9 CC SNe, if possible SN impostors are not included.

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In Figure 4, we plot the probability density function of f_dim given the observed number of total and dim CC SNe, for different distance cuts. We use the binomial distribution in N_dim for a fixed N_tot and varying f_dim, using a flat prior in f_dim over the interval 0–1. On the top panel, we show a magnitude cut of −16 mag for illustration. Due to the broad distribution of f_dim, we are not able to derive a precise value for f_dim with the present data: the fractions are f_>−15 ∼ 30%–50% (f_>−16 ∼ 60%–80%), but lower and higher values are not improbable. However, our estimates are clearly larger than previous studies based on more systematic samples at higher distances which were not as sensitive to the dim end of the CC SNe luminosity function (f_>−15 ∼ 3%–20%; Section 4.3). Our estimates decrease to f_>−15 ∼ 20%–40% (f_>−16 ∼ 50%–70%) if possible SN impostors are excluded from our list of CC SN. This is still larger than previous studies.

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In summary, the very local 10 Mpc volume shows the dim fraction could be as high as f_>−15 ∼ 50%. This is significantly higher than previous estimates, which range between f_>−15 ∼ 3%–20%. Thus the SNR measurements are scaled up by (1 − 0.2)/(1 − 0.5) = 1.6 (a factor of 1.9 if the original dim fraction is 5%), largely solving the supernova rate problem. While the very local 10 Mpc volume is special in its absolute SFR, this should not affect the fraction of dim CC SNe.

First, to investigate whether the predicted SNR is too large, we explore the intrinsic uncertainties in the SFR measurements (IMF, obscuration, and conversion factor) and the additional factors when the SFR is converted to a SNR (IMF and CC SN mass range). The results are shown in Figure 5 by dashed arrows.

• 1.  
  IMF. We vary the IMF shape from the Salpeter A IMF to a classic Salpeter IMF and a flatter Baldry–Galzebrook IMF. This has only a small effect on the predicted SNR.
• 2.  
  Obscuration. Obscuration correction is redshift dependent. Our adopted SFR compilation is corrected for obscuration by adding the FIR-derived SFR to the UV-derived SFR. We consider uncertainty due to two potential FIR contaminants: first, from AGN, which has been estimated to be ≲ 10% of the FIR (Silva et al. 2004), and second, from cirrus emission, which we assume to be ≲ 50% of the FIR (Lonsdale Persson & Helou 1987). We thus assume a reduction to the predicted SNR of 25%. Extinction uncertainties are unlikely to reduce the SNR prediction normalization by a full factor of two (see Section 3.1 for integral constraints on the low-redshift SFR normalization).
• 3.  
  SFR conversion factor. We assume formal uncertainties of ±15%. Although conversion uncertainties for young star-forming galaxies are larger, such galaxies are rare in the low-z universe (Soifer & Neugebauer 1991) and can be safely neglected.
• 4.  
  CC SN mass range. The nominal mass range is 8–40 M_☉. We use a minimum mass range taken from observations of CC SN progenitors: 8.5–16.5 M_☉ and 25–40 M_☉. We stress that this range does not account for the progenitors of all CC SNe Types. Further SN observations are needed to clarify this situation. We adopt the maximum mass range of 8–100 M_☉.

Therefore, the uncertainties affecting the predicted SNR are generally smaller than the normalization discrepancy.

Second, to investigate whether the measured SNR is too small, we explore the contribution from missing CC SNe (incomplete galaxy sample and dim CC SNe). This is shown by solid arrows in Figure 5.

• 1.  
  Incomplete galaxy sample. This only affects SN surveys targeting a pre-selected sample of galaxies. Though careful work is done to correct for the bias against CC SNe in small galaxies, recent results suggest CC SNe in dwarf galaxies are unlike those in normal galaxies. We adopt a tentative factor of 2 enhancement in the CC SNe formation efficiency in dwarfs, which would yield a 20%–50% increase in the total SNR, as shown by stacked arrows. Detailed studies need to await more data.
• 2.  
  Dim CC SNe. There is an observational bias against the discovery and correction of dim CC SNe. In the very local (≲ 10 Mpc) volume, where dim CC SNe are least likely to be missed, the fraction of dim CC SNe is higher than in SN surveys targeting larger distances. Using the years 2000–2009, we find f_>−15 ∼ 30%–50% (if possible SN impostors are excluded from our list of CC SNe, we find 20%–40%), whereas SN surveys find 3%–20%. Therefore, we scale up the SNR measurement accordingly by a factor between (1 − 0.2)/(1 − 0.3) ≈ 1.14 and (1 − 0.03)/(1 − 0.5) ≈ 1.9, shown by stacked arrows.

Therefore, if possible SN impostors are true CC SNe, the fraction of dim CC SNe can be sufficiently high that dim CC SNe largely explain the supernova rate problem. If possible SN impostors are not true CC SNe, f_dim is smaller, and dim CC SNe do not fully bridge the normalization discrepancy.

In Figure 5 we show how the different effects compare. The supernova rate problem is likely explained by a combination of the effects shown.

The validity of using the dim fraction derived from very local (≲ 10 Mpc) CC SNe relies on how representative the very local volume is. We discuss several issues.

First, distance uncertainties for the very local galaxies can be up to ±25% in some cases. This has two effects: (1) the absolute magnitude is affected and can cause erroneous categorization of CC SNe near the dim-luminous magnitude cut and (2) CC SNe can be categorized in the wrong distance bin. We have considered the dim fraction using different magnitude cuts and different distance cuts, showing that it is consistently higher than previous estimates.

Second, the very local volume shows an enhancement in its SNR (Figure 3 and Table 3). The enhancement is significant: within 12.6 Mpc we observe 14 luminous CC SNe while the expectation is 6, an ∼0.2% occurrence. The enhancement is a factor of ∼2, which is comparable to the factor of1.7 enhancement observed in the SFR in similar distances. The enhancement is observed not only in the luminous CC SNe but also the total CC SNe: there are 23 discovered (including three possible SN impostors) while the expectation is 12. This can be interpreted as dim CC SNe are not severely incomplete within these distances.

Finally, to apply the dim fraction at cosmological distances requires some care. The IMF at high redshift remains poorly constrained (Nagashima et al. 2005; Le Delliou et al. 2006), and studies find that IMF evolution may be required at z > 0.5 (e.g., Wilkins et al. 2008b). If dim CC SNe are mass-dependent, an evolving IMF will lead to an evolving f_dim. Also, redshift dependence may arise from metallicity dependence. A more important issue is obscuration by dust, which is known to be redshift-dependent (Section 3.5). Our work in the very local volume is mainly on normal galaxies, whose importance is overtaken by ULIRG at high redshifts (Mannucci et al. 2007). The higher dust obscuration in ULIRG causes as many as 90% of the CC SNe to be missed by current SN surveys (Mannucci et al. 2003). Assuming that the dim fraction in ULIRG is 90%, the weighted dim fraction increases from ∼50% at z ≈ 0 to ∼70% at z ≈ 1.

In summary, we find that the very local (≲ 10 Mpc) volume has a high absolute SNR, which is related to an enhanced SFR. The dim fraction should be independent of the absolute normalization, and thus our estimate may be used at low redshifts. However, the dim fraction may increase with redshift.

One of the earliest discussions of dim CC SNe was based on historical CC SNe in the Milky Way and nearby galaxies (Schaefer 1996). It was suggested that the CC SN luminosity function rises toward dimmer objects, with perhaps a third or more of the CC SNe dimmer than M_B = −15. However, the completeness of historical SNe has been difficult to assess, and the SN sample contains significant extinction and type uncertainties. Some of the dim CC SNe are now known to have been LBV, e.g., SN 1954J, for which the surviving star has been observed (Van Dyk et al. 2005).

In more recent studies, the fractions are smaller. Richardson et al. (2002) report the mean and standard deviation of Type IIP, Type IIL, and Type Ibc SNe luminosity functions to be M_B ≈ −16.18 (σ ≈ 1.23), M_B ≈ −17.49 (σ ≈ 1.29), and M_B ≈ −17.37 (σ ≈ 0.88) (adjusted to our chosen cosmology). Using the type fractions of the volumetric SN sample of LOSS, the fraction of CC SNe dimmer than M_B = −15 (−16) is only 12% (34%). In Botticella et al. (2008), the authors adopt somewhat narrower luminosity functions which result in a smaller dim fraction of 3% (13%). The SNLS uses an absolute magnitude scale relative to the mean magnitude of Type Ia SNe (Bazin et al. 2009). Adopting M_V = −19 mag as the mean Type Ia SN peak magnitude, the fraction of CC SNe dimmer than M_V = −15 (−16) is 11% (30%). Here we use their luminosity function that has been corrected for the smaller target volume for dim CC SNe (data points of their Figure 12). The recently published luminosity functions of LOSS contain large numbers of dim CC SNe, including several in Table 2, e.g., SN 2002hh, SN 2004am, SN 2007gr, and SN 2008ax. From their luminosity functions, the fraction of CC SNe dimmer than M_R = −15 (−16) is 20% (40%). Here, we have summed their luminosity histograms, rather than integrate their Gaussian fits which the authors caution are poor fits to the data (Li et al. 2011a). In summary, the dim fraction in the literature is f_>−15 ∼ 3%–20% (f_>−16 ∼ 10%–40%). In Figure 4 we label the fractions according to SNLS and LOSS, the two most recent and reliable measurements.

Using the very local SN catalog data, we estimate the dim fraction to be f_>−15 ∼ 30–50% (f_>−16 ∼ 60%–80%), larger than those of previous studies. Furthermore, we have assessed the completeness of the catalog CC SNe (Section 3.6.1). The likelihood of our high f_dim being a simple statistical fluctuation is small given the sample size, e.g., the likelihood of fluctuating f_>−15 ∼ 3% (20%) to f_>−15 ≳ 50% in a sample size of 10 CC SNe is 0.0005% (3%). If possible SN impostors are excluded, f_>−15 ∼ 20%–40% (f_>−16 ∼ 50%–70%). This is in better agreement although still somewhat larger than the most recent luminosity function of LOSS. It is still significantly larger compared to other luminosity functions.

An important reason is the small distance cut adopted. In SN surveys targeting larger distances, the dimmest CC SNe are only discovered in a fraction of the volume corresponding to the smallest distance. Furthermore, a good understanding of the detection efficiency is required to correct the small statistics of dim CC SNe. For example, Richardson et al. (2002) suggests that the fraction of CC SNe dimmer than M_B = −15 is likely larger than 20% even though their nominal luminosity function suggests a value closer to 10%, because of the bias against dim SNe and small number statistics. On the other hand, we limit ourselves to a nominal distance of 10 Mpc and the past decade from 2000 to 2009 to study dim CC SNe. The majority of SN searches in this very local volume have limiting magnitudes of 18 mag, corresponding to ≈ − 12 mag. Therefore, our very local sample is only limited by cadence and incompleteness of the galaxies observed. This may also explain why the dim fraction is lower at larger distances: by 30–40 Mpc, the limiting magnitudes of most searches are already more luminous than −15 mag. Finally, we add that even so, our results should be treated as lower limits, since many of the CC SNe have been discovered by amateurs with different systematics, galaxy samples, and cadences, i.e., they are not systematically collected.

In summary, our estimate of f_dim is significantly larger than previous studies. If possible SN impostors are not true CC SNe, we estimate a value for f_dim that is somewhat larger than the SN luminosity function of LOSS.

The dimmest objects in Table 2 (SN 2002bu, SN 2008S, and NGC 300-OT) are a special kind of explosion. Their nature—whether an extremely dim CC SNe or an extremely luminous stellar outburst—remains debated. Here, we discuss implications for the supernova rate problem if they are stellar outbursts.

There is currently not enough data to conclude the true nature of possible SN impostors. Smith et al. (2009) interpret SN 2008S as a super-Eddington wind of a ∼20 M_☉ star, based on, among others, the lack of a ⁵⁶Co → ⁵⁶Fe decay tail. On the other hand, Botticella et al. (2009) identify in their pseudo-bolometric late-time light curve a decay slope consistent with ⁵⁶Co decay, concluding an electron-capture CC SNe of a ∼9 M_☉ star. Measuring the bolometric light curve is a difficult multi-wavelength task, and further observations are required to settle the case. The progenitor of SN 2008S has been identified on Spitzer pre-explosion images to be a highly dust-enshrouded ∼10 M_☉ star (Prieto et al. 2008, 2010). Such stars are extremely rare (Khan et al. 2010), suggesting it may be a common but short-lived phase prior to some kind of explosion (Thompson et al. 2009). The situations are similar for NGC 300-OT (Bond et al. 2009; Berger et al. 2009) and SN 2002bu (Thompson et al. 2009; Smith et al. 2011).

If possible SN impostors are not CC SNe, the dim fraction is only somewhat larger than the luminosity function of LOSS, and missing dim CC SNe provide only a partial explanation to the supernova rate problem. In this case, there are several interpretations of the supernova rate problem, each with their own tensions with other measurements. We detail some of them below.

First, the supernova rate problem may be interpreted as a large number of undetected CC SNe. For example, the fraction of optically dark CC SNe could be high (see, e.g., Figure 3 of Lien et al. 2010 for how the optically-dark CC SNe fraction relates to the SFR and measured SNR). Dark CC SNe as the solution to the supernova rate problem is only barely allowed by neutrino considerations (Lien et al. 2010) and may be implausible on a theoretical basis (e.g., O'Connor & Ott 2011). An alternate possibility is that more CC SNe than expected are driven below survey sensitivities by heavy host dust attenuation. However, this would appear difficult since such a change in dust properties would also increase the derived SFR and hence the predicted SNR, thus not necessarily helping solve the supernova rate problem. As pointed out by Botticella et al. (2008), the consistency of the extinction model used in SFR and SNR measurements need to be further examined.

A second possibility is that CC SNe in small galaxies play a significantly larger role than expected. As discussed in Section 3.4, the efficiency of forming CC SNe in dwarf galaxies may be higher than in normal galaxies. If the enhancement exists in a wider range of small galaxies, the effect of missing small galaxies could be larger than we estimate. However, there is a potential caveat with the CC SNe in small galaxies for helping solve the supernova rate problem. It is convenient to re-express the supernova rate problem in terms of the efficiency, or N_CCSN/M, the number of CC SNe per unit stellar mass formed. The measured cosmic SNR and the measured cosmic SFR imply 0.0041/M_☉, whereas the nominal prediction is 0.0088/M_☉ (see Section 2); the factor of ∼2 difference is another statement of the supernova rate problem. If the efficiency in small galaxies is taken to be enhanced, the observed N_CCSN is corrected upwards, so that the observed N_CCSN/M is closer to nominal predictions. However, the same enhanced efficiency will also increase the N_CCSN/M used in predictions, unless the efficiency in normal galaxies is sub-nominal. This is not seen in the LOSS galaxies, as Maoz et al. (2011) find N_CCSN/M = 0.0059–0.010/M_☉ for a subset of LOSS galaxies with SDSS spectroscopic SFR measurements. Correcting for systematic effects, their best estimate is N_CC/M = 0.010 ± 0.002/M_☉, close to the nominal value. Therefore, enhanced CC SNe formation efficiencies in small galaxies may not bridge the supernova rate normalization discrepancy. Whether this is the case remains to be investigated with more data. For example, the known aperture bias from the finite SDSS fiber diameter is most critical for low-redshift galaxies like those in the LOSS survey (Kewley et al. 2005). This would tend to underestimate the SFR in the LOSS galaxies, potentially leading to an artificially larger N_CCSN/M. This may allow some parameter space for CC SNe in small galaxies to help bridge the cosmic SNR normalization mismatch.

The final possibility is that the cosmic SFR is overestimated. In particular, the observed efficiency in the LOSS galaxies, N_CCSN/M = 0.010 ± 0.002/M_☉, implies that there is very little room for missing CC SNe in the LOSS galaxies. The supernova rate problem could then be interpreted as an overestimated cosmic SFR. However, this may not help solve the supernova rate problem. Say there was some new effect causing the cosmic SFR to be overestimated, for example contamination of SFR estimators or systematic offset in SFR calibrations. The same effect would also cause the SFR in the LOSS galaxies to be overestimated, because the same SFR estimators are used. Thus, while reducing the cosmic SFR will increase the observed efficiency closer to nominal predictions, the efficiency in the LOSS galaxies will also increase to almost twice the nominal predictions (∼0.02/M_☉). In other words, the supernova rate problem has only shifted.

Additionally, the nominal uncertainties in the SFR input physics do not easily tolerate a factor of 2 reduction in the cosmic SFR (Section 3.1). The largest uncertainty in the measured SFR is due to dust correction. However, to reduce the cosmic SFR sufficiently requires removing most of the dust correction. Apart from the need to redefine our understanding of the FIR universe, demanding no dust would also reduce the SNR measurements. The other, and perhaps more intriguing, possibility is the presence of previously unknown contaminations to the SFR indicators. However, several SFR indicators used must be systematically contaminated by the same degree, making this seem unlikely. In either case, the implications for affecting the SFR estimators would go beyond the supernova rate problem to a wide range of topics.