RADIO OBSERVATIONS OF A SAMPLE OF BROAD-LINE TYPE IC SUPERNOVAE DISCOVERED BY PTF/IPTF: A SEARCH FOR RELATIVISTIC EXPLOSIONS

Long-duration (${T}_{\gamma }\gtrsim 2$ s) γ-ray bursts (GRBs) are extremely energetic explosions (typically, ≈10⁵² erg released in ≈10 s, also referred to as collapsars) marking the deaths of massive stars (Galama et al. 1998; Woosley & Bloom 2006). According to the popular fireball model (Piran 2004; Mészáros 2006), the explosion launches relativistic jets in which magnetic fields are amplified and particles are accelerated (Rhoads 1999). Observers located within the initial jets' opening angles (${\theta }_{j}\gtrsim {\theta }_{\mathrm{obs}};$ "on-axis" observers) see an intense flash of γ-rays. Subsequent emission from the decelerating jets produces a (slowly) decaying broadband afterglow. If the fireball model is correct, then off-axis GRBs should exist and be $\approx 2/{\theta }_{j}^{2}$ times more common than the ones we see in γ-rays (Granot et al. 2002). While γ-ray emission from off-axis GRBs cannot be observed, their longer-wavelength afterglow emission is expected to become observable at later times, once the jet decelerates and starts spreading (Nakar et al. 2002).

Off-axis GRBs have not been discovered so far, but in the light of the well-established connection between long-duration GRBs and core-collapse supernovae (SNe) of spectral type Ic with broad-lines (BL-Ic; Woosley & Bloom 2006), a natural way to search for off-axis events is to observe this type of SNe and wait for the decelerating jet to emerge. While the SN optical emission traces the slower explosion debris (v ≈ 0.03−0.1 c), synchrotron emission from the fastest ejecta peaks in the radio band. There can be two major sources of radio emission associated with GRB–SNe:

• i.  
  The SN shock, whose radio emission is brighter and earlier-peaking the faster the SN ejecta, with an expected luminosity and peak time of ≈10²⁹ erg cm⁻² s⁻¹ Hz⁻¹ and ≈10–30 days since explosion, respectively, for relativistic events like SN 1998bw (Galama et al. 1998; Kulkarni et al. 1998; Berger et al. 2003a);
• ii.  
  The GRB jet which, if off-axis, would be observed only when the SN ejecta decelerate to mildly or sub-relativistic speeds, thus producing a delayed and nearly isotropized radio emission.

Radio is indeed the best wavelength range for identifying relativistic events such as SN 1998bw, and/or off-axis GRBs (Paczynski 2001; Granot & Loeb 2003). In the past, hundreds of SNe Ib/c have been targeted with the Karl G. Jansky Very Large Array (VLA; Berger et al. 2003a; Soderberg et al. 2006b; Bietenholz et al. 2014) and the fraction of SNe Ib/c associated with GRBs has been constrained to ≲1%–3%. However, only a very small fraction of the Ic SNe targeted by these past studies were broad-lined, the only type of SNe observationally linked to GRBs. Moreover, many of these SNe Ib/c were located in large, massive, and metal-rich hosts, while GRBs are rarely seen in such galaxies (e.g., Modjaz et al. 2008; Levesque et al. 2010; Hjorth et al. 2012; Graham & Fruchter 2013; Perley et al. 2013; Xu et al. 2013; Kelly et al. 2014; Krühler et al. 2015; Perley et al. 2016). Thus, the fraction of purely BL-Ic SNe harboring relativistic jets remained, observationally, largely unconstrained.

Here, we present a sample of 15 SNe discovered by the Palomar Transient Factory and/or intermediate Palomar Transient Factory (PTF/iPTF, hereafter we use PTF for simplicity; Law et al. 2009; Rau et al. 2009), optimized to search for off-axis GRBs, namely, a sample of BL-Ics selected blindly in random galaxies (mostly dwarfs). While all (long-soft) GRBs may be accompanied by BL-Ic SNe, not all of these SNe make GRBs (e.g., Ofek et al. 2007; Soderberg et al. 2010; Milisavljevic et al. 2015). Why should some stars follow a different path to death, ending their lives as collapsars rather than as "ordinary" SNe, is still a mystery. This study aims to provide additional clues to help gain deeper insight into the nature of collapsar events.

After collecting photometric data (Section 2.1) and classifying the SNe in our sample as belonging to the family of BL-Ic SNe (Section 2.2), we performed X-ray follow-up observations for some of the events (Section 2.3), searching for X-ray signatures (not accompanied by γ-rays) from GRBs observed slightly off-axis (and/or "dirty" fireballs; Section 3). We performed centimeter-wavelength follow-up observations of all the SNe in our sample with the VLA (Section 2.4), searching for SN 1998bw-like radio emission (point (i) above) and/or later-time signatures of off-axis jets (point (ii) above). Our sample greatly enlarges the sample of radio-monitored BL-Ic SNe published over the last ≈10 years (Berger et al. 2003a; Soderberg et al. 2006b; Ghirlanda et al. 2013; Bietenholz et al. 2014), and our observational strategy allows us to probe a portion of the radio luminosity-time since explosion phase space that was left largely unexplored by previous studies (Section 4). We constrain the portion of the explosion energy-wind/ISM density parameter space that is excluded under the hypothesis that GRB jets significantly off-axis (θ_j ≈ 90°) are associated with the SNe in our sample (Section 5), and set an upper-limit on the fraction of BL-Ic SNe in our sample that show radio emission that is possibly compatible with off-axis GRBs expanding in media with densities in the range probed by this study (Section 6). Finally, we give our conclusions  in Section 7.

Hereafter, we adopt cosmological parameter values of H₀ = 69.6 km s⁻¹ Mpc⁻¹, Ω_M = 0.286, Ω_Λ = 0.714 (Wright 2006; Bennett et al. 2014).

2.1. P48 Discovery and Photometry

R-band (or g-band) discoveries (and follow-up) of the SNe in our sample (Table 1) were obtained using the 48-inch Samuel Oschin telescope at the Palomar Observatory (P48), which is routinely used by the PTF/iPTF. Processed images were downloaded from the Infrared Processing and Analysis Center (IPAC) PTF archive (Laher et al. 2014). Photometry was performed relative to the SDSS r-band (or g-band) magnitudes of stars in the field (York et al. 2000). We used our custom pipeline that performs image subtraction and then point-spread function photometry on stacks of PTF images extracted from the IPAC archive (Maguire et al. 2012; Ofek et al. 2012, 2013). The flux residuals from individual subtracted images were binned and converted to magnitudes. Errors were estimated from the standard deviation of the photometric measurements in each bin.

Table 1.  BL-Ic SNe with VLA Observations in Our Sample

PTF	R.A. Decl.	TP48a	z	dL	MR/gb	v(Si)	References
name	(J2000)	⋯	⋯	⋯	[AB]	⋯	⋯
⋯	(hh:mm:ss deg:mm:ss)	(MJD)		(Mpc)	(mag)	(km s−1)
10bzf	11:44:02.99+55:41:27.6	55250.504	0.0498	223	−18.3	2.6 × 104	Corsi et al. (2011)
10qts	16:41:37.60+28:58:21.1	55413.260	0.0907	418	−19.4	1.7 × 104	Walker et al. (2014)
10xem	01:47:06.88+13:56:28.8	55470.340	0.0567	255	−18.6	2.0 × 104	This paper
10aavz	11:20:13.36+03:44:45.2	55514.485	0.062	280	−19.2	1.5 × 104	This paper
11cmh	13:10:21.74+37:52:59.6	55673.336	0.1055	491	−18.6	1.6 × 104	This paper
11img	17:34:36.30+60:48:50.6	55755.408	0.158	761	−19.6	1.5 × 104	This paper
11lbm	23:48:03.20+26:44:33.5	55793.259	0.039	173	−18.0	1.5 × 104	This paper
11qcj	13:13:41.51+47:17:57.0	55866.520c	0.0287	124	−18.0	1.2 × 104	Corsi et al. (2014)
12as	10:01:34.05+00:26:58.4	55925.298	0.033	146	−17.5	2.2 × 104	This paper
13u	15:58:51.21+18:13:53.1	56324.481	0.10	463	−18.9	1.0 × 104	This paper
13alqd	11:48:02.09+54:34:38.2	56394.359	0.054	242	−18.9	2.3 × 104	Drake et al. (2013)
13ebw	08:17:15.88+56:34:41.6	56621.389	0.069	313	−18.2	2.3 × 104	This paper
14dby	15:17:06.29+25:21:11.4	56832.238	0.074	337	−17.9	1.4 × 104	This paper
14gaq	21:32:54.08+17:44:35.6	56924.213	0.0826	378	−18.0	1.9 × 104	This paper
15dlde	00:58:13.28-03:39:50.3	57318.322	0.047	210	−17.9	1.0 × 104	This paper/La Silla-QUEST

Notes. Our sample includes a total of 15 SNe, 12 of which are presented here for the first time. PTF10bzf, PTF10qts, and PTF11qcj were previously discussed in Corsi et al. (2011), Walker et al. (2014), and Corsi et al. (2014), respectively.

^aDiscovery times (T_P48) are from P48 observations in the R-band, except for the case of PTF14gaq, which was discovered and observed with P48 in the g-band. ^bM_R/g is the maximum absolute magnitude in the P48 R-band for all of the SNe but PTF14gaq, for which M_R/g is measured in the P48 g-band. These magnitudes are corrected for galactic extinction (Schlafly & Finkbeiner 2011). ^cThe SN was visible in a previous g-band image taken on 2011 October 23. ^dAlso known as CSS130415:114802+543439/SN 2013bn: PTF13alq was also discovered by the CRTS (Drake et al. 2009) and classified as a 1998bw-like type Ic SN by the Copernico Telescope in Asiago (Tomasella et al. 2013). ^ePTF15dld/LSQ15bfp was also discovered by the La Silla-QUEST variability survey (Hadjiyska et al. 2012) and classified by the Public ESO Spectroscopic Survey of Transient Objects (PESSTO; Smartt et al. 2015).

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The R-band (or g-band) light curves of the SNe in our sample are shown in Figure 1. PTF10bzf, PTF10qts, and PTF11qcj photometry were discussed previously in Corsi et al. (2011), Walker et al. (2014), and Corsi et al. (2014), respectively, so we do not present their photometry here (we refer the reader to these papers). The P48 discovery time (T_P48), and the maximum R-band (or g-band) absolute magnitudes (M_R/g) as measured by our P48 monitoring and corrected for Galactic extinction (Schlafly & Finkbeiner 2011), are reported in Table 1. Note that our M_R/g is different from the SN light curve peak for cases in which the peak emission was not observed by P48. We do not take into account k-corrections when measuring M_r/g, but refer the reader to Prentice et al. (2016) and F. Taddia et al. (2015, in preparation) for a discussion of these corrections.

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2.2. Spectral Classification

After the discovery with P48, we triggered a spectroscopic follow-up campaign¹⁶ of all the SNe in our sample. PTF10bzf, PTF10qts, and PTF11qcj spectral properties were previously discussed in Corsi et al. (2011), Walker et al. (2014), and Corsi et al. (2014), respectively, so we do not present their spectral analysis here, but we refer the reader to these papers. For the rest of the SNe in our sample, details of the observations are reported in what follows. In Table 1, we also report the estimated redshifts, and the velocities corresponding to the P-Cygni absorption minimum of the Si II 6355 Å lines, which trace reasonably closely the position of the photosphere (Mazzali et al. 2000; see also F. Taddia et al. 2015, in preparation).

2.2.1. PTF10xem

On 2010 October 10 UT (≈9 days since optical discovery), we observed PTF10xem with the dual-arm Kast spectrograph (Miller & Stone 1993) on the 3 m Shane telescope at Lick Observatory. We used a 2'' wide slit, a 600/4310 grism on the blue side, and a 300/7500 grating on the red side. The exposure time and airmass were 3600 s and 1.09, respectively. The derived spectrum shows a good match with the Ic/BL-Ic SN 2004aw (e.g., Taubenberger et al. 2006) at an epoch of about 15 days since explosion, and with the BL-Ic SN 2002ap at 6 days since explosion (Figure 2), so we classify PTF10xem as a BL-Ic SN.

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2.2.2. PTF10aavz

On 2010 November 30 UT (≈16 days since optical discovery) we observed PTF10aavz using ISIS on the William Herschel Telescope (WHT), with a 199 wide slit, the R300B grating set at a central wavelength of ≈4500 Å on the blue side, and the R158R grating set at a central wavelength of ≈7500 Å on the red side. The exposure time was 1800 s, and the mean airmass was 1.17. This spectrum of PTF10aavz is most similar to that of the BL-Ic/hyper-energetic and asymmetric SN 2003jd (e.g., Valenti et al. 2008; Mazzali et al. 2005) at an epoch of about 24 days since explosion (Figure 2).

2.2.3. PTF11cmh

We observed PTF11cmh using ISIS on the WHT on 2011 May 2 UT (≈10 days since optical discovery), with a 102 wide slit, the R300B grating set at a central wavelength of ≈4500 Å on the blue side, and the R158R grating set at a central wavelength of ≈7500 Å on the red side. For both the blue and red side observations, the exposure time was 900 s and the mean airmass was 1.01. The derived spectrum shows a good match with the Ic/BL-Ic SN 2004aw (e.g., Taubenberger et al. 2006) at an epoch of about 15 days since explosion, and with the BL-Ic SN 2002ap at 6 days since explosion (Figure 2), so we classify PTF11cmh as a BL-Ic SN.

2.2.4. PTF11img

We observed PTF11img on 2011 August 2 UT (≈20 days since optical discovery), using the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) mounted on the Keck I 10 m telescope. The spectrum was taken using a 1'' wide slit, with the 400/8500 grating set at a central wavelength of ≈7800 Å on the red side, and the 600/4000 grism on the blue side. For both sides, the exposure time and airmass were 600 s and 1.46, respectively. The derived spectrum shows a good match with both the BL-Ic SN 2002ap (e.g., Gal-Yam et al. 2002; Mazzali et al. 2002) at 13 days since explosion, and the type Ic hypernova SN 1997ef (e.g., Iwamoto et al. 2000) at 35 days since explosion (Figure 2). We thus classify PTF11img as a BL-Ic SN.

2.2.5. PTF11lbm

We observed PTF11lbm using ISIS on the WHT on 2011 August 31 UT (≈11 days since optical discovery), with a 102 wide slit, the R300B grating set at a central wavelength of ≈4500 Å on the blue side, and the R158R grating set at a central wavelength of ≈7500 Å on the red side. For both the blue and red side observations, the exposure time was 900 s. The mean airmass was about 1.01. The derived spectrum shows a good match with both the BL-Ic SN 2002ap (e.g., Gal-Yam et al. 2002; Mazzali et al. 2002) at 6 days since explosion, and the type Ic hypernova SN 1997ef (e.g., Iwamoto et al. 2000) at 35 days since explosion (Figure 2). We thus classify PTF11lbm as a BL-Ic SN.

2.2.6. PTF12as

We observed PTF12as on 2012 January 2 UT (≈2 days since optical discovery), using the Dual Imaging Spectrograph mounted on the 3.5 m telescope at the Apache Point Observatory. The spectrum was taken using a 15 wide slit, with a B400/R300 grating setup. The exposure time and airmass were 1000 s and 1.50, respectively. The derived spectrum shows a good match with the GRB-associated BL-Ic SN 2002ap (e.g., Gal-Yam et al. 2002; Mazzali et al. 2002) at 6 days since explosion (Figure 2).

2.2.7. PTF13u

On 2013 February 18 UT (≈17 days since optical discovery) we observed PTF13u with the Double Beam Spectrograph (DBSP; Oke & Gunn 1982) on the Palomar 200-inch telescope (P200). We used the 316/7500 and 600/4000 gratings for the red and blue cameras, respectively, with a D55 dichroic, resulting in a spectral coverage of ≈(3500–9500) Å. The exposure times and airmass were of 590 s and 1.06, respectively. This spectrum of PTF13u matches that of the BL-Ic/hyper-energetic and asymmetric SN 2003jd (e.g., Mazzali et al. 2005; Valenti et al. 2008) at ≈29 days since explosion (Figure 2).

2.2.8. PTF13alq

On 2013 April 13 UT (≈1 days since optical discovery) we observed PTF13alq with the DBSP (Oke & Gunn 1982) on P200. We used the 316/7500 and 600/4000 gratings for the red and blue cameras, respectively, with a D55 dichroic, resulting in a spectral coverage of ≈(3500–9500) Å. The exposure times and airmass were of 300 s and 1.1, respectively. The derived spectrum shows a good match with the GRB-associated BL-Ic SN 1998bw (e.g., Patat et al. 2001) at 9 days since explosion (Figure 2).

2.2.9. PTF13ebw

We observed PTF13ebw on 2013 December 4 UT (≈9 days since optical discovery), using LRIS mounted on the Keck I 10 m telescope. The spectrum was taken using a 1'' wide slit, with the 400/8500 grating set at a central wavelength of ≈7800 Å on the red side, and the 600/4000 grism on the blue side. The exposure time and airmass were 500 s and 1.36, respectively. The derived spectrum shows a good match with the relativistic BL-Ic SN 2009bb (Soderberg et al. 2010) at 15 days since explosion (Figure 2).

2.2.10. PTF14dby

On 2014 June 29 (≈5 days since optical discovery), we observed PTF14dby with LRIS mounted on the Keck I 10 m telescope. The spectrum was taken using a 1'' wide slit, with the 400/8500 grating set at a central wavelength of ≈7800 Å on the red side, and the 400/3400 grism on the blue side. The exposure time and airmass were 300 s and 1.19, respectively. This spectrum of PTF14dby reveals a good match with the GRB-associated BL-Ic SN 1998bw (e.g., Patat et al. 2001) at ≈18 days since explosion (Figure 2).

2.2.11. PTF14gaq

We observed PTF14gaq on 2014 October 1 UT (≈7 days since optical discovery) with the DBSP on P200. We used the 316/7500 and 600/4000 gratings for the red and blue cameras, respectively, with a D55 dichroic, resulting in a spectral coverage of ≈(3500–9500) Å. The exposure times and airmass were of 600 s and 1.05, respectively. This spectrum of PTF14gaq shows a good match with the GRB-associated SN 1998bw (e.g., Patat et al. 2001) at ≈18 days since explosion (Figure 2).

2.2.12. PTF15dld

We observed PTF15dld on 2015 November 7 UT (≈15 days since the P48 optical discovery) with the Deep Extragalactic Imaging Multi-Object Spectrograph mounted on the Keck II 10 m telescope. The spectrum was taken using the 600ZD grating and GG455 filter. The exposure time and airmass were 600 s and 1.11, respectively. The derived spectrum shows a good match with both the BL-Ic SN 2002ap (e.g., Gal-Yam et al. 2002; Mazzali et al. 2002) at 13 days since explosion, and with the Ic/BL-Ic SN 2004aw (e.g., Taubenberger et al. 2006) at an epoch of about 15 days since explosion (Figure 2).

2.3. Swift/XRT Follow-up and Data Reduction

None of the BL-Ic SNe in our sample was found to be spatially coincident with any of the well-localized GRB in the Swift (Gehrels et al. 2004) catalog. For some of the events, we triggered Swift/XRT (Burrows et al. 2005) follow-up observations via our approved Target of Opportunity Programs¹⁷ in order to further exclude the presence of a GRB X-ray afterglow with no associated γ-rays (as would be the case for a GRB jet observed slightly off-axis, or for a so-called "dirty" fireball; see, e.g., Rhoads 2003).

We downloaded the Swift-XRT data from the archive¹⁸ . None of the SNe in our sample yielded a detection with Swift/XRT, so we calculated 3σ upper limits on the 0.3–10.0 keV count rate using standard analysis procedures. The upper limits are reported in Table 2, where we have converted the 0.3–10 keV XRT count rates into fluxes, assuming a photon index of Γ_X = 2 and correcting for Galactic absorption.

Table 2.  3σ X-Ray Upper Limits or Detections for Some of the SNe in Our Sample

PTF	Date	Δ TXa	Instrument	Band	Exp.	NHb	Count Rate	Flux (unabs)c	References
name	(MJD)	(days)	⋯	(keV)	(ks)	(1020 cm−2)	(10−4 s−1)	(10−14 erg cm−2 s−1)
10bzf	55259.290	9	Swift-XRT	0.3-10	5.0	0.88	$\lt 3.7$	$\lt 1.3$	Kasliwal & Cenko (2010)
"	55263.303	13	Swift-XRT	0.3-10	5.0	"	$\lt 7.7$	$\lt 2.7$	Kasliwal & Cenko (2010)
10qts	55426.126	13	Swift-XRT	0.3-10	4.9	2.7	$\lt 4.2$	$\lt 1.6$	This paper
11qcj	55920.045	78	Swift-XRT	0.3-10	31.4	1.0	$\lt 4.1$	$\lt 1.4$	Corsi et al. (2014)
"	55939.05	97	Chandra-ACIS	0.3-8.0	9.8	"	8.8 ± 3.3d	0.76 ± 0.26	Corsi et al. (2014)
12as	55932.340	7	Swift-XRT	0.3-10	4.6	2.5	$\lt 33$	$\lt 13$	This paper
"	55962.150	37	Swift-XRT	0.3-10	4.9	"	$\lt 21$	$\lt 8.0$	This paper
13ebw	56631.719	10	Swift-XRT	0.3-10	4.8	4.6	$\lt 24$	$\lt 9.8$	This paper
"	56672.486	51	Swift-XRT	0.3-10	2.4	"	$\lt 48$	$\lt 20$	This paper
14dby	56839.043	7	Swift-XRT	0.3-10	9.5	4.2	$\lt 11$	$\lt 4.4$	This paper
14gaq	56936.081	12	Swift-XRT	0.3-10	7.8	6.9	$\lt 14$	$\lt 6.1$	This paper
15dld	57332.973	15	Swift-XRT	0.3-10	9.9	3.2	$\lt 9.8$	$\lt 3.8$	This paper

Notes.

^aEpoch in days since the P48 discovery (see Table 1), not corrected for redshift effects. ^bHydrogen column densities are weighted averages from the Leiden/Argentine/Bonn (LAB) Survey of Galactic H I (Kalberla et al. 2005). ^cThe count-rate-to-flux conversion assumes a photon index of ${{\rm{\Gamma }}}_{X}=2$. ^dChandra observations of PTF11qcj yielded detections that we attribute to the presence of strong CSM interaction rather than to a GRB X-ray afterglow. Some host galaxy contamination to the measured X-ray flux might also be present. See Corsi et al. (2014) for a complete discussion.

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X-ray observations of PTF11qcj obtained with Swift/XRT and Chandra/ACIS (Garmire et al. 2003) were previously presented in Corsi et al. (2014). We include some of these observations (the most significant Chandra/ACIS detection and the deepest Swift/XRT upper limit) in Table 2 for completeness.

2.4. VLA Follow-up Observations and Data Reduction

We observed all of the SNe in our sample, along with the necessary calibrators, with the VLA¹⁹ (Perley et al. 2009) under our Target of Opportunity programs.²⁰ VLA data were reduced and imaged using the Common Astronomy Software Applications package.

The VLA flux measurements and/or upper limits are reported in Table 3. Measurement errors are calculated by adding in quadrature the rms map error, and a basic fractional error (≈5%), which accounts for inaccuracies of the flux density calibration (Weiler et al. 1986; Ofek et al. 2011).

Table 3.  VLA Observations

PTF	TVLAa	Δ TVLAb	Conf.	ν	BW	Flux	Reference
Name	(MJD)	(d)		(GHz)	(MHz)	(μJy)
10bzf	55268.222	18	D	5.0	256	$\lesssim 33$	Chomiuk & Soderberg (2010)
"	55337.221	87	D	6.0	1024	$\lesssim 36$	Corsi et al. (2011)
"	55527.490	277	C	5.0	256	$\lesssim 35$	Corsi et al. (2011)
10qts	55426.028	13	D	8.5	256	$\lesssim 86$	Gal-Yam et al. (2010)
"	55948.524	535	DnC	6.2	2048	$\lesssim 39$	This paper
"	55950.435	537	DnC	6.2	2048	$\lesssim 28$	This paper
10xem	55767.647	297	A	5.0	256	$\lesssim 66$	This paper
10aavz	55566.527	52	C	4.9	256	$\lesssim 31$	Soderberg & Chomiuk (2011)
"	55770.969	256	A	4.9	256	$\lesssim 105$	This paper
11cmh	55692.145	19	B	8.4	256	$\lesssim 72$	This paper
"	55949.356	276	DnC	6.3	2048	159 ± 11	This paper
"	56766.099	1093	A	6.2	2048	17.4 ± 5.4	This paper
"	56781.411	1108	"	1.7	"	57 ± 13	This paper
"	56808.981	1136	"	2.9	"	18.0 ± 4.2	This paper
11img	55781.981	31	A	5.0	256	$\lesssim 48$	This paper
"	55950.478	199	DnC	5.0	2048	$\lesssim 66$	This paper
11lbm	55812.283	19	A	5.0	256	$\lesssim 78$	This paper
"	55948.043	155	DnC	6.2	2048	$\lesssim 28$	This paper
12as	55933.555	8	DnC	6.2	2048	$\lesssim 87$	This paper
"	55947.382	22	DnC	6.2	2048	$\lesssim 69$	This paper
"	56999.567	1074	C	6.2	2048	$\lesssim 75$ b	This paper
13u	56391.376	67	D	6.2	2048	$\lesssim 95$ d	This paper
"	56767.366	443	A	6.2	2048	$\lesssim 14$	This paper
"	57311.028	987	D	6.2	2048	$\lesssim 100$ e	This paper
13alq	56401.120	7	D	4.8	2048	$\lesssim 30$	Kamble & Soderberg (2013)
13alq	56423.080	29	DnC	6.2	2048	$\lesssim 60$ f	This paper
13alq	57336.833	942	D	6.3	2048	$\lesssim 33$	This paper
13ebw	56667.566	46	B	6.2	2048	$\lesssim 15$	This paper
"	57001.417	380	C	6.2	2048	$\lesssim 41$ g	This paper
14dby	56838.197	6	D	5.2	1024	$\lesssim 33$	This paper
"	"	"	"	7.5	"	40 ± 12	"
"	56853.994	22	"	5.2	"	40.1 ± 9.8	"
"	"	"	"	7.5	"	117 ± 12	"
"	56879.117	47	"	5.0	"	194 ± 12	"
"	"	"	"	7.4	"	279 ± 15	"
"	56889.108	57	"	2.5	"	164 ± 54	"
"	"	"	"	3.3	"	125 ± 18	"
"	"	"	"	8.5	"	176 ± 14	"
"	"	"	"	9.5	"	159 ± 14	"
"	"	"	"	13.5	"	107 ± 13	"
"	"	"	"	14.5	"	105 ± 13	"
"	56902.138	70	"	5.0	"	72 ± 19	"
"	"	"	"	7.4	"	73 ± 17	"
"	56903.045	71	"	2.7	"	121 ± 40	"
"	"	"	"	3.2	"	79 ± 16	"
"	"	"	"	8.5	"	108 ± 11	"
"	"	"	"	9.5	"	102 ± 11	"
"	"	"	"	13.5	"	56.5 ± 9.6	"
"	"	"	"	14.5	"	55.5 ± 9.7	"
"	56913.144	81	"	5.1	"	73 ± 18	"
"	"	"	"	7.5	"	60 ± 16	"
"	56914.770	83	"	8.5	"	61 ± 14	"
"	"	"	"	9.5	"	80 ± 15	"
"	"	"	"	13.5	"	38 ± 12	"
"	"	"	"	14.5	"	41 ± 13	"
"	56946.949	115	DnC	2.5	"	101 ± 28	"
"	"	"	"	3.5	"	138 ± 16	"
"	"	"	"	5.0	"	84 ± 12	"
"	"	"	"	6.0	"	79 ± 15	"
"	"	"	"	8.5	"	48 ± 10	"
"	"	"	"	9.5	"	59 ± 11	"
"	56998.800	167	C	2.5	"	$\lesssim 111$	"
"	"	"	"	3.4	"	48 ± 13	"
"	"	"	"	5.0	"	53 ± 10	"
"	"	"	"	6.0	"	44 ± 14	"
"	"	"	"	8.5	"	28.9 ± 9.3	"
"	"	"	"	9.5	"	40 ± 11	"
PTF14gaq	56932.496	8	DnC	6.3	2048	$\lesssim 28$	This paper
"	56998.857	75	C	6.2	2048	$\lesssim 25$	This paper
"	57335.919	411	D	6.3	2048	$\lesssim 19$	This paper
PTF15dld	57336.049	18	D	5.0	2048	$\lesssim 110$ h	This paper

Notes. For non-detections, the quoted UL are at $3\sigma$ (where σ is the image rms) unless otherwise stated.

^aThe VLA observation time ${T}_{{\rm{VLA}}}$ is the time at the midpoint of the VLA observation. ^bVLA observation epoch in days since PTF discovery, not corrected for redshift effects. ^c5σ UL corresponding to the brightness of the host galaxy. ^d10σ UL corresponding to the brightness of the host galaxy. ^e10σ UL corresponding to the brightness of the host galaxy. ^f3.6σ UL. ^g7σ UL corresponding to the brightness of the host galaxy. ^h10σ UL corresponding to the brightness of the host galaxy.

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As mentioned in the previous Section, none of the SNe in our sample are spatially coincident with any well-localized GRB. For a limited number of these SNe, our observations with the Swift/XRT allow us to make some comparisons with the X-ray light curve that would be expected from an accompanying GRB 980425-like event, or from a high-luminosity GRB observed slightly off-axis. For the off-axis GRB case, we use the numerical model by van Eerten & MacFadyen (2011) and van Eerten et al. (2012), which considers a relativistic GRB fireball expanding in a uniform density medium.

As evident from Figure 3, while Swift/XRT upper limits can exclude X-ray afterglows associated with high-luminosity (high-energy) GRBs observed slightly off-axis (up to ${\theta }_{\mathrm{obs}}\lesssim (2\mbox{--}3){\theta }_{j}$), X-ray emission as faint as the afterglow of the low-luminosity GRB 980425 cannot be excluded. As we discuss in Section 4, radio data collected with the VLA enable us to exclude 1998bw-like emission for most of the SNe in our sample.

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For PTF11qcj, Chandra observations yielded a detection but we attribute this X-ray emission to the presence of strong circumstellar medium (CSM) interaction rather than to a GRB X-ray afterglow. See Corsi et al. (2014) for a complete discussion.

Here, we aim at observationally constraining the fraction of BL-Ic SNe with radio luminosities comparable to that of the GRB-associated SN 1998bw (Kulkarni et al. 1998) to ultimately constrain the fraction of BL-Ic SNe harboring low-luminosity GRBs. Indeed, most of the GRBs with an associated and spectroscopically confirmed SN are low-luminosity bursts (${E}_{\gamma ,\mathrm{iso}}\lesssim {10}^{50}$ erg), although notable exceptions are GRB 030329 (Stanek et al. 2003) and GRB 130427A (e.g., Melandri et al. 2014; Perley et al. 2014). The fraction of BL-Ic SNe harboring low-luminosity GRBs was left largely unconstrained by previous efforts (Berger et al. 2003a; Soderberg et al. 2006b; Bietenholz et al. 2014) due to the very small number of BL- Ic events with radio follow-up available to the community.

Theoretical studies have indirectly constrained the fraction of BL-Ic SNe harboring low-luminosity GRBs by constraining the local rate of low-luminosity GRBs (via luminosity function fitting) and then comparing this estimated local rate with the rate of BL-Ic SNe collected via optical surveys. Following this approach, Guetta & Della Valle (2007) derived that $\gtrsim 10 \%$ of BL-Ic SNe are accompanied by low-luminosity GRBs. This is consistent with the earlier results by Podsiadlowski et al. (2004), who found that the rates of GRBs and BL-Ic SNe are comparable to within the uncertainties, and their ratio likely $\gtrsim 30 \%$ (see Table 1 in Podsiadlowski et al. 2004). More recently, following a statistical approach inspired by the Drake equation, Graham & Schady (2016) estimated that there are 4000 ± 2000 BL-Ic SNe in low-metallicity environments for every (long) GRB aligned in our direction. This number is a composite of the fraction of such SNe that produce GRBs and the fraction that are beamed in our direction, and would imply that $\lesssim 5 \%$ of the BL-Ic SNe are associated with a GRB.

With our PTF discoveries, we now have a sample of 15 BL-Ic SNe discovered independently of a GRB trigger, with at least one radio follow-up observation on timescales $\lesssim 300$ days since explosion (as measured in the SN rest frame; Figures 4 and 5). Of these 15 SNe, 12 have been uniquely observed via our VLA programs. Our observations have greatly enlarged the sample of 8 BL-Ic SNe with radio follow-up at $\lesssim 300$ days collected via independent efforts during the last decade (Figure 4, yellow; Figure 5, green, yellow, and magenta asterisks; Berger et al. 2003a; Chomiuk & Soderberg 2010; Soderberg et al. 2010; Soderberg & Chomiuk 2011; Drake et al. 2013; Kamble & Soderberg 2013; Salas et al. 2013; Chakraborti et al. 2015; Milisavljevic et al. 2015). We are thus in a position to start constraining the theoretical expectations for the low-luminosity GRB-to-BL-Ic SN ratio using a direct observational signature: the presence (or absence) of 1998bw-like radio emission.

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In Figures 4 and 5, the 5 GHz radio light curve of SN 1998bw is compared with the upper limits and detections obtained for the SNe in our sample (4.8–6.3 GHz; See Table 3). As evident from Figure 5, we have three SNe (PTF11cmh, PTF11qcj, and PTF14dby) that show bright radio emission, much brighter than the ordinary BL-Ic SN 2002ap and almost at the level of SN 1998bw, but their radio peak occurs $\gtrsim 5\times$ later than for SN 1998bw. Thus, as we explain in the following section, we consider these SNe as observationally different from SN 1998bw, likely related to events on the dividing line between ordinary SNe and GRBs, although an interpretation as off-axis GRB jets might also be possible. For the remaining 12 SNe in our sample, we detect no radio emission and set upper limits (Figure 4, black downward-pointing triangles). For 10 of these 12 SNe (all but PTF10xem and PTF13u), we have at least one upper limit that constrains the radio emission to be dimmer than the emission of SN 1998bw at a similar epoch, thus excluding a radio light curve observationally similar to that of the prototype relativistic BL-Ic SN 1998bw. (We note that 8 out of the 12 SNe also exclude radio emission similar to SN 2009bb, a relativistic BL-Ic SN with no associated GRB; Soderberg et al. 2010).

Based on the above results, we conclude that of the 10 + 3 PTF SNe whose radio observations can set constraints on SN 1998bw-like emission, none of them where in fact like SN 1998bw in the radio, i.e., they were all observationally different. Adding to this sample the BL-Ic SN 2002ap (Gal-Yam et al. 2002; Mazzali et al. 2002) and SN 2002bl (Armstrong 2002; Berger et al. 2003a), and the CSM-interacting BL-Ic SN 2007bg (Salas et al. 2013), we have a total of 16 BL-Ic SNe for which radio emission observationally similar to SN 1998bw is excluded. Because the 99.865% confidence (3σ Gaussian equivalent for a single-sided distribution) Poisson upper limit on zero SNe compatible with SN 1998bw is ≈6.61, we conclude that the rate of BL-Ic SNe that are observationally similar to SN 1998bw is $\lesssim 6.61/16\approx 41 \%$.

We note that for these 16 SNe we also exclude on-axis radio emission that is typical of long GRB afterglows at cosmological distances (blue line and shaded area in Figure 4; see Chandra & Frail 2012), and radio emission observationally similar to that of the low-luminosity GRB 031203. However, none of our upper limits exclude radio emission similar to GRB 060218. This is not surprising since the afterglow of this low-luminosity GRB faded on timescales much faster than the ones our VLA monitoring campaign was designed to target (peak timescales of ≈20–30 d). Finally, only some of our upper limits exclude radio afterglow emission similar to that of the low-luminosity GRB 100316D (although a more quantitative comparison with this burst is hampered by its poorly sampled radio light curve).

Low-luminosity GRBs (such as GRB 980425 associated with SN 1998bw) are believed to be intrinsically less energetic events (when compared to high-luminosity ones) with jet opening angles ≳30° (e.g., Liang et al. 2007). However, the possibility that low-luminosity GRBs are higher-energy events observed off-axis has also been discussed (e.g., Waxman 2004b; Ramirez-Ruiz et al. 2005). Indeed, most (high-luminosity) GRBs are believed to have opening angles of the order of ∼10° (e.g., Frail et al. 2001; Liang et al. 2008; Racusin et al. 2009; Zhang et al. 2015). Here, we aim at answering the question of whether the BL-Ic SNe in our sample that do not show evidence for radio emission observationally similar to that of SN 1998bw (Section 4) could still be accompanied by an off-axis (θ_obs ≈ 90°) GRB afterglow that would become visible in the radio band long past the explosion (at timescales of the order of ∼1 years; Levinson et al. 2002; Waxman 2004b; Gal-Yam et al. 2006), when the relativistic fireball enters the sub-relativistic phase and starts spreading, rapidly intersecting the viewer's line of sight while approaching spherical symmetry. Our upper limits add to the late-time ones that have been collected in the past ($t\gtrsim 500$ days since explosion, not plotted in Figures 4–5; see Soderberg et al. 2006b; Bietenholz et al. 2014).

We model the late-time radio emission from an off-axis GRB following the works by Livio & Waxman (2000) and Waxman (2004b), modified to account for the results of more recent numerical simulations (Zhang & MacFadyen 2009; van Eerten & MacFadyen 2012). The last have shown that at the end of the Blandford–McKee (BM) phase, the fireball becomes non-relativistic, but differing from what was previously thought, the transition to the spherical Sedov–Neumann–Taylor (SNT) blast wave takes a rather long time. Thus, accurate modeling of the fireball evolution over timescales in between the BM and SNT phases (which are relevant for this study) requires numerical simulations. However, Zhang & MacFadyen (2009) have shown that for fireballs expanding in a medium of constant density n_0,ISM (in units of cm⁻³) and at timescales

$$\begin{eqnarray}&&t\gtrsim (1+z)\times {t}_{\mathrm{SNT}}/2,\end{eqnarray} \tag{ 1 }$$

where

$$\begin{eqnarray}&&{t}_{\mathrm{SNT}}\approx 92\text{days}{\left(\displaystyle \frac{{E}_{51}}{{n}_{0,\mathrm{ISM}}}\right)}^{1/3},\end{eqnarray} \tag{ 2 }$$

an acceptable analytical approximation to the afterglow flux is given by:

$$\begin{eqnarray}{F}_{\nu }(t) & \approx & 0.16\,{d}_{L,28}^{-2}{(1+z)}^{\tfrac{1}{2}}\left(\displaystyle \frac{{\epsilon }_{e}}{0.1}\right){\left(\displaystyle \frac{{\epsilon }_{B}}{0.1}\right)}^{3/4}{n}_{0,\mathrm{ISM}}^{\tfrac{9}{20}}{E}_{51}^{\tfrac{13}{10}}\,\,\,\\ & & \times {\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-1/2}{\left(\displaystyle \frac{t}{92\text{days}(1+z)}\right)}^{-9/10}\,{\rm{mJy}}.\,\,\,\end{eqnarray} \tag{ 3 }$$

Here E₅₁ is the beaming-corrected ejecta energy in units of 10⁵¹ erg; ${\epsilon }_{e}$ and ${\epsilon }_{B}$ are the fraction of ejecta energy density going into electrons and magnetic fields, respectively; d_L,28 is the luminosity distance of the source in units of 10²⁸ cm; z is the source redshift; and the power-law index of the electron energy distribution has been set to p ≈ 2.

Equation (3) corresponds to Equation (15) in Waxman (2004b) where, however, the dependence on t_SNT has been eliminated by using our Equation (2) (or, equivalently, Equation (11) in Waxman 2004b). Using Equation (3) to constrain the fireball parameters by comparison with observations taken at timescales t that satisfy Equation (1) yields constraints on the (beaming-corrected) energy that are accurate to within a factor of ≈2 (see Figure 10 in Zhang & MacFadyen 2009).

By imposing

$$\begin{eqnarray}&&t\gtrsim {t}_{\mathrm{SNT}}\quad {F}_{\nu }(t)\gtrsim {F}_{{obs},\nu }(t),\end{eqnarray} \tag{ 4 }$$

we thus calculate, for the SNe in our sample (Table 3), the values of (beaming-corrected) energy and medium density that would give a radio luminosity above our upper limit F_obs,ν(t), at the time t of our observation. The exclusion regions obtained in this way are shown in Figure 6. In this Figure we have set the microphysics parameters equal to their median values as estimated by Santana et al. (2014) i.e., ${\epsilon }_{e}\approx 0.22$ and ${\epsilon }_{B}\approx 0.01$. However, these parameters (and especially ${\epsilon }_{B}$) vary within large ranges, $0.02\lesssim {\epsilon }_{e}\lesssim 0.6$ and $3\times {10}^{-5}\lesssim {\epsilon }_{B}\,\lesssim 0.33$ (Santana et al. 2014). As evident from Equation (3), for a given upper limit on the flux at a certain epoch, the smaller the value of the product ${\epsilon }_{e}{\epsilon }_{B}^{3/4}$, the larger the minimum E₅₁ excluded for each value of n₀ (thus, off-axis emission from lower-energy fireballs / low-luminosity GRBs is less constrained).

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In Figure 7 we use our upper limits to set similar constraints on off-axis GRBs expanding in a wind medium. In this case, the expected radio flux is approximated as (Waxman 2004b; Soderberg et al. 2006b):

$$\begin{eqnarray}{F}_{\nu }(t) & \approx & 0.053\,{d}_{L,28}^{-2}{(1+z)}^{1/2}\left(\displaystyle \frac{{\epsilon }_{{\rm{e}}}}{0.1}\right){\left(\displaystyle \frac{{\epsilon }_{{\rm{B}}}}{0.1}\right)}^{3/4}{A}_{* }^{3/4}{E}_{51}\\ & & \times {\left(\displaystyle \frac{\nu }{1{\rm{GHz}}}\right)}^{-1/2}{\left(\displaystyle \frac{t}{92\text{days}(1+z)}\right)}^{-3/2}\,\mathrm{mJy},\,\,\,\end{eqnarray} \tag{ 5 }$$

where A_* is the circumstellar density, which is related to progenitor mass-loss rate $\dot{M}$ and wind velocity v_w as ${A}_{* }=(\dot{M}/{10}^{-5}{M}_{\odot }\,{{\rm{yr}}}^{-1})/({{\rm{v}}}_{{\rm{w}}}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})$; and where we have used (Waxman 2004b; Soderberg et al. 2006b):

$$\begin{eqnarray}&&{t}_{{\rm{SNT}}}\approx 92\,{\rm{days}}\left(\displaystyle \frac{{E}_{51}}{{A}_{* }}\right).\end{eqnarray} \tag{ 6 }$$

We note that Equation (5) corresponds to Equation (14) in Waxman (2004b), where the dependence on t_SNT is eliminated by using our Equation (6) (or, equivalently, Equation (8) in Waxman 2004b). Here we are assuming that the conclusions reached by Zhang & MacFadyen (2009) for the constant density case are also valid for a fireball expanding in a wind medium, namely, that Equation (5) provides an estimate of the fireball parameters that is good to within a factor of ≈2 when compared to flux measurements carried out at epochs $t\gtrsim {t}_{\mathrm{SNT}}$.

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We can compare the results shown in Figures 6–7 with the energy and density derived from the broadband afterglow modeling of the high-luminosity GRB 030329 (for which θ_j ≈ 5°−17°, E₅₁ = 0.67, n_0,ISM ≈ 3; Berger et al. 2003b; Soderberg et al. 2006b) and GRB 130427A (for which ${E}_{51}\gtrsim 0.5$, ${\theta }_{j}\gtrsim 5^\circ$, and $0.01\lesssim {A}_{* }\lesssim 0.05;$ Perley et al. 2014), and of the low-luminosity GRB 980425 (for which E₅₁ ≈ 0.05 and A_* ≈ 0.04; Waxman 2004a, 2004b; Soderberg et al. 2006b) and GRB 031203 (for which E₅₁ = 0.017 and n_0,ISM = 0.6; Soderberg et al. 2004; Ramirez-Ruiz et al. 2005; Soderberg et al. 2006b). From such a comparison we conclude that most of the SNe in our sample exclude GRBs with (beaming-corrected) energy and ISM density comparable to GRB 030329, observed largely off-axis and/or during the non-relativistic phase. On the other hand, our upper limits cannot exclude a GRB as sub-energetic as GRB 980425 when observed during its non-relativistic phase, nor an off-axis GRB expanding in a low-density environment such as GRB 031203 and GRB 130427A.

As discussed before, our conclusions depend somewhat on the assumed values of the microphysics parameters. Indeed, as evident from Equation (5), also in a wind environment, the smaller the value of the product ${\epsilon }_{e}{\epsilon }_{B}^{3/4}$, the larger the minimum E₅₁ excluded for each value of A_*.

Non-thermal (self-absorbed) synchrotron radiation can be emitted from SN or GRB ejecta during interaction with a CSM. The temporal and spectral evolution of the synchrotron emission are determined by the dynamics, and by the properties of the ejecta and CSM. While young, non-relativistic SNe expand freely (their ejecta are largely undecelerated), GRB relativistic blast waves expand and decelerate following the BM solution. At late enough times, both non-relativistic radio SNe and GRBs are expected to approach the non-relativistic adiabatic expansion phase (SNT dynamics, see also Section 5). In what follows, we discuss the SNe with radio detections in our sample within these two scenarios (decelerated GRB ejecta and non-relativistic radio SN).

6.1. GRB Jets Observed Off-axis?

Three SNe in our sample, PTF11cmh, PTF11qcj, and PTF14dby, were detected during our radio follow-up with the VLA. Interestingly, two out of these three SNe (PTF11qcj and PTF14dby) are found to be spectroscopically similar to SN 1998bw. Thus, the rate of radio detections for the BL-Ic SNe in our sample that are spectroscopically most similar to SN 1998bw is ≈2/4 = 50% (see Figure 2). As noted in Section 2.3, the X-ray upper limit on PTF14dby does not exclude the presence of X-ray afterglow emission comparable to that of GRB 980425; however, its radio emission appears different from SN 1998bw in the fact that it peaks at later times. Moreover, the radio emission from PTF11cmh, PTF11qcj, and PTF14dby is orders of magnitudes dimmer than that of an average long GRB observed on-axis (shaded blue curves in Figure 4), and also dimmer than most low-luminosity GRBs with well-sampled radio light curves. Thus, here we address the question of whether the radio emission from these three SNe could be associated with a GRB observed off-axis.

In Figures 8 and 10 we show a tentative comparison of the observed radio light curves of PTF14dby and PTF11cmh with numerical model light curves of off-axis low-luminosity GRBs expanding in a constant density environment of density n_ISM (dashed lines; van Eerten & MacFadyen 2011; van Eerten et al. 2012). We note that numerical models for GRB jets expanding in a wind environment are not currently available to the community (at least not in a format that can allow us to easily compare these models with our observations). Thus, hereafter we limit our discussion to the case of a constant density ISM.

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For PTF11cmh (Figure 8, dashed line), we have set: θ_j ≈ 10°, observer'(s) angle θ_obs ≈ 90°, beaming-corrected energy E₅₁ ≈ 0.1, n_ISM = 10 cm⁻³, ${\epsilon }_{B}={\epsilon }_{e}\approx 0.1$, and p ≈ 2.2. These values for the model parameters provide a model light curve that is in agreement with the (limited) 6 GHz data. We also point out that the limited data set available for PTF11cmh does leave open the possibility of a mildly relativistic event (discussed in more detail in the following section).

For PTF14dby (Figure 10, dashed line) we have set: θ_j ≈ 6°, observer'(s) angle θ_obs ≈ 70°, beaming-corrected energy E₅₁ ≈ 0.01, n_ISM = 10 cm⁻³, ${\epsilon }_{B}={\epsilon }_{e}\approx 0.1$, and p ≈ 2.4. While the simplest off-axis GRB model in a constant density ISM does not provide a perfect match, the model light curves are broadly compatible with the observations of PTF14dby, thus an off-axis GRB cannot be securely ruled out. We also note that in the PTF14dby radio light curve there is a hint for a late-time peak (or flattening) reminiscent of SN 1998bw, which may be better fit using off-axis GRB models expanding in a wind environment, and/or by invoking an energy injection episode similar to what has been proposed by Li & Chevalier (1999) for SN 1998bw.

PTF11qcj is the most difficult to interpret within the simplest off-axis GRB models (see also Corsi et al. 2014), due to the clear late-time radio re-brightening. However, this late-time re-brightening also requires modifications to the simplest non-relativistic radio SN model, such as the presence of a denser CSM shell (e.g., Salas et al. 2013).

Based on the tentative comparison with available models described in this section, and on the results described in Section 5, we can attempt to constrain the fraction of BL-Ic SNe in our sample potentially harboring off-axis GRB jets. Indeed, since $\lesssim 3$ BL-Ic SNe in our sample may be associated with off-axis (low-luminosity) GRBs expanding in an ISM with n_ISM ∼ 10 cm⁻³ (or A_* ∼ 4; compare Equations (14) and (15) in Waxman 2004b), we set a 99.865% confidence Poisson upper limit of $\lesssim 12.68/15\approx 85 \%$ on the fraction of BL-Ic SNe possibly harboring off-axis GRBs expanding in media with densities of this order. We note, however, that a comparison with numerical models for off-axis low-luminosity GRBs expanding in a wind environment would be needed to better determine the values of A_* constrained by our data set.

6.2. Non-relativistic Radio SN Emission

In what follows, we model the radio emission observed from PTF11cmh and PTF14dby within the standard radio SN model based on the interaction of non-relativistic ejecta with CSM deposited via a constant mass-loss rate, constant velocity wind (i.e., ${\rho }_{{\rm{CSM}}}=\dot{{M}_{w}}/(4\pi {{\rm{v}}}_{w}{r}^{2})$) from a massive progenitor (Chevalier 1982). We follow the formulation of this standard model given in Soderberg et al. (2005), which replaces the SNT dynamics with a general parameterization of the shock evolution, which enables us to model the early SN synchrotron emission (when the ejecta is very close to free expansion), while recovering (in the appropriate time limit) the correct behavior for GRBs transitioning to the sub-relativistic adiabatic expansion phase (Waxman 2004b).

In the standard model, synchrotron emission observed at time t is produced from an expanding spherical shell of shock-accelerated electrons with radius r and thickness r/η. The shell interacts with a smooth CSM following a self-similar evolution. The electrons, which are accelerated into a power-law energy distribution $N(\gamma )\propto {\gamma }^{-p}$ (with $\gamma \gtrsim {\gamma }_{m}$), carry a fraction ${\epsilon }_{e}$ of the energy density of the ejecta. Magnetic fields carry a fraction ${\epsilon }_{B}$ of the energy density. The temporal evolution of the shell and its properties is parameterized as (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray}&&r={r}_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{r}}\quad B={B}_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{B}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\gamma }_{m}={\gamma }_{m,0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{\gamma }}\quad \displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}},={{\mathfrak{F}}}_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{{\mathfrak{F}}}}.\end{eqnarray} \tag{ 8 }$$

In the above Equations, t₀ is an arbitrary reference time (here set to day 10 since explosion), t_e is the explosion time of the SN, ${\alpha }_{r}=(n-3)/(n-s)$ (Chevalier 1982, 1996) with n as the power-law index of the outer SN ejecta density profile (${\rho }_{{\rm{SN}}}\propto {(r/t)}^{-n}$), and s the is power-law index of the shocked CSM electrons' density profile (${n}_{e}\propto {r}^{-s}$).

Following Chevalier (1996), the magnetic energy density (${U}_{B}\propto {B}^{2}$) and the relativistic electron energy density (${U}_{e}\propto {n}_{e}{\gamma }_{m}$) are assumed to be a fixed fraction (i.e., ${\alpha }_{{\mathfrak{F}}}=0$) of the total post-shock energy density ($U\propto {n}_{e}{\langle {\rm{v}}\rangle }^{2}$, where v is the velocity). Making the additional conservative assumption that the energy of the radio-emitting material is partitioned equally into accelerating electrons and amplifying magnetic fields (${\epsilon }_{e}={\epsilon }_{B}$, which implies ${{\mathfrak{F}}}_{0}=1$) and assuming s = 2 (as expected for a wind density profile), we have (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray}&&{U}_{e}\propto U\,\,\Rightarrow {\alpha }_{\gamma }=2({\alpha }_{r}-1),\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&{U}_{B}\propto U\,\,\Rightarrow {\alpha }_{B}=\displaystyle \frac{(2-s)}{2}{\alpha }_{r}-1=-1,\end{eqnarray} \tag{ 10 }$$

and, for the flux density from the uniform shell of radiating electrons (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray}{f}_{\nu } & = & {C}_{f}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(4{\alpha }_{r}+1)/2}(1-\exp (-{\tau }_{\nu }))\\ & & \times {\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{5/2}\,\times {F}_{3}(x){F}_{2}^{-1}(x)\,{\rm{mJy}},\,\,\end{eqnarray} \tag{ 11 }$$

where ${C}_{f}={C}_{f}({r}_{0},{B}_{0},p)$ (see Equation (A.13) in Soderberg et al. 2005), $x=2/3(\nu /{\nu }_{m})$, and

$$\begin{eqnarray}{\nu }_{m} & = & {\gamma }_{m}^{2}\displaystyle \frac{{eB}}{2\pi {m}_{e}c}={\gamma }_{m,0}^{2}\displaystyle \frac{{{eB}}_{0}}{2\pi {m}_{e}c}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{2{\alpha }_{\gamma }+{\alpha }_{B}}\\ & = & {\nu }_{m,0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{4{\alpha }_{r}-5}\end{eqnarray} \tag{ 12 }$$

is the characteristic synchrotron frequency of electrons with Lorentz factor γ_m. As typically assumed for radio SNe, we set ${\nu }_{m,0}\approx 1\,{\rm{GHz}}$ (which, in turn, implies that γ_m,0 is a function of B₀ only). In Equation (11), F₂ and F₃ are integrals of the modified Bessel function of order 2/3 (see Equation (A11) in Soderberg et al. 2005); and

$$\begin{eqnarray}{\tau }_{\nu }(t) & = & {C}_{\tau }{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(p-2){\alpha }_{\gamma }+(3+p/2){\alpha }_{B}+{\alpha }_{r}}\\ & & \times {\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-(p+4)/2}{F}_{2}(x)\\ & = & {C}_{\tau }{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(2p-3){\alpha }_{r}-(5p/2-1)}{\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-(p+4)/2}{F}_{2}(x)\end{eqnarray} \tag{ 13 }$$

is the optical depth (Soderberg et al. 2005, 2006a), with ${C}_{\tau }={C}_{\tau }({r}_{0},{B}_{0},{\gamma }_{m,0},p)$ (see Equation (A.14) in Soderberg et al. 2005). Thus, as evident from Equations (11) and (13), the observed spectral and temporal evolutions of the radio emission ultimately depend on the parameters $({r}_{0},{B}_{0},{t}_{e},p,{\alpha }_{r},\eta )$, which we determine by comparison with the data.

6.2.1. PTF11cmh Radio Modeling

Our VLA follow-up observations of PTF11cmh started at an epoch of about ≈20 days since optical discovery, and were carried out until more than 10³ days after (Table 3). Our first radio detection of PTF11cmh was more than 100 days since optical discovery.

Because our radio observations for PTF11cmh are very limited, we expect any model fitting to return only tentative estimates of model parameters. Within the standard synchrotron self-absorbed scenario (Section 6.2), we can set t_e=55673.336 MJD (see Table 1) and η = 5 (as typically assumed in radio SN studies). This leaves four free model parameters to be compared with four radio detections and one upper limit (see Table 3). We thus attempt a crude fit by considering the upper limit at epoch ≈20 days since explosion as a data point, with flux value equal to the maximum radio flux detected in a circular region centered on the optical position of PTF11cmh with a radius equal to half the VLA FWHP synthesized beam for the observation, and error equal to the image rms, i.e., 15 ± 24 μJy. This way our fit returns a χ² ≈ 4 for 1 degrees of freedom (dof).²¹ From the best-fit light curves shown in Figure 8 (solid lines), we also estimate ν_p ≈ 5 GHz at ≈100 days since explosion, and ${L}_{p,5{\rm{GHz}}}\approx {10}^{29}$ erg s⁻¹ Hz⁻¹. The last value is comparable to the radio spectral luminosity of the GRB-associated SN 1998bw (Kulkarni et al. 1998).

The best-fit values for the model parameters are p ≈ 3, B₀ ≈ 5 G, and a blast wave radial evolution of $R\approx 9\times {10}^{15}{[(t-{t}_{e})/10{\rm{days}}]}^{0.86}$ cm. The last value implies an average ejecta speed of $R/{\rm{\Delta }}t\approx 0.3\,c$, where c is the speed of light. This is ≈3×higher than the average speed of ordinary Ib/c SNe (≈0.1 c), but smaller than relativistic events such as SN 2009bb and SN 1998bw. In Figure 9 we show the uncertainties on the best values of the average ejecta speed ($\langle {\rm{v}}\rangle ={r}_{0}/10$ d) and power-law index α_r of the temporal evolution of the ejecta radius, as derived by mapping the difference ${\rm{\Delta }}{\chi }^{2}={\chi }_{2}^{2}-{\chi }_{4}^{2}$, where χ²₄ is the best-fit χ² value returned by our 4-parameter fit to the data (see above); and ${\chi }_{2}^{2}$ is the best-fit χ² value obtained when mapping the α_r-r₀ space over a grid of possible values and minimizing the χ² over the remaining two "non-interesting" parameters (e.g., Avni 1976). As evident from this Figure, because of the limited data set available for this event, the speed of the radio-emitting material is very poorly constrained, with the 99% confidence region extending in the range $\langle {\rm{v}}\rangle /c\approx 0.11-2/3$.

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Assuming equipartition (${\epsilon }_{e}={\epsilon }_{B}=0.33$), and using Equation (14) in Soderberg et al. (2005), we derive a minimum energy of E ≈ 6 × 10⁴⁸${({\epsilon }_{e}/0.33)}^{-1}{[(t-{t}_{e})/10{\rm{days}}]}^{0.58}$ erg coupled to the fastest radio-emitting outflow. This energy is at the higher end of the range derived for other radio Ib/c SNe (Margutti et al. 2014). However, we also note that because $E\propto {r}_{0}^{3}\propto {\langle {\rm{v}}\rangle }^{3}$, a factor of ≈6 uncertainty on $\langle {\rm{v}}\rangle$ (at 99% confidence) implies a factor of ≈200 uncertainty in the estimated ejecta energy.

Finally, the estimated progenitor mass-loss rate is $\dot{M}={10}^{-4}({{\rm{v}}}_{w}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})\,{M}_{\odot }$ yr⁻¹, where v_w is the velocity of the stellar wind and where we have assumed a nucleon-to-proton ratio of 2 (see Equation (13) in Soderberg et al. 2005). This mass-loss rate is higher than the typical range derived for low-luminosity GRBs (see e.g., Figure 7), and more similar to CSM-interacting BL-Ic SNe such as PTF11qcj (Corsi et al. 2014). Based on these results, we suggest that PTF11cmh is likely a CSM-interacting event similar to PTF 11qcj. However, this conclusion has to be taken with the caveat of being derived from a very limited data set. Indeed, since $\dot{M}\propto {r}_{0}^{2}\propto {\langle {\rm{v}}\rangle }^{2}$, a factor of ≈6 uncertainty on $\langle {\rm{v}}\rangle$ (at 99% confidence) implies a factor of ≈40 uncertainty in the estimated mass-loss rate.

6.2.2. PTF11qcj Radio Modeling

Our VLA follow-up observations of PTF11qcj started about two weeks after optical discovery, and were carried out until ≈600 days after (Corsi et al. 2014). As described in Corsi et al. (2014), modeling our radio observations in the standard synchrotron self-absorbed scenario yielded best-fit values of B₀ ≈ 5.7 G for the magnetic field, and a blast wave radial evolution of $R\approx 1.1\times {10}^{16}{[(t-{t}_{e})/10{\rm{days}}]}^{0.80}$ cm (assuming η = 5). That last value implies an average ejecta speed of $R/{\rm{\Delta }}t\approx 0.4\,c$. This is ≈4×higher than the average speed of ordinary Ib/c SNe (≈0.1 c), but smaller than relativistic events such as SN 2009bb and SN 1998bw. For η = 5, we also estimated a minimum energy of $E\approx 1.2\times {10}^{49}{({\epsilon }_{e}/0.33)}^{-1}$ ${[(t-{t}_{e})/10{\rm{days}}]}^{0.4}$ erg coupled to the fastest radio-emitting outflow, and a progenitor mass-loss rate of $\dot{M}\approx 1.5\times {10}^{-4}({{\rm{v}}}_{w}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})$ M_⊙ yr⁻¹, where v_w is the velocity of the stellar wind (and assuming a nucleon-to-proton ratio of 2).

6.2.3. PTF14dby Radio Modeling

Our VLA follow-up observations of PTF14dby started at an epoch of about ≈6 days since optical discovery, and were carried out until more than 150 days after (Table 3). The first clear ($\gtrsim 4\sigma$) VLA detection of PTF14dby at 5 GHz was obtained about 20 days since optical discovery.

We collected a total of 36 detections and 2 upper limits for PTF14dby (see Table 3). We model our radio detections in the standard synchrotron self-absorbed scenario (Section 6.2) using a χ² minimization procedure where we set η = 5, so we are left with 5 free model parameters. The fit returns a χ² ≈ 115 for 31 dof. From the best-fit light curves shown in Figure 10 (solid lines) we estimate ν_p ≈ 7.4 GHz at ≈40 days since explosion, and a spectral peak luminosity of ${L}_{p,7.4{\rm{GHz}}}\,\approx 3\times {10}^{28}$ erg s⁻¹ Hz⁻¹. The last is ≈4×smaller than the peak radio luminosity of the GRB-associated SN 1998bw, but comparable to the radio peak luminosity of the engine-drive SN 2009bb (Figure 5; Soderberg et al. 2010).

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The best-fit values for the model parameters are t_e ≈ 56832 MJD (which is consistent with the discovery date reported in Table 1), p ≈ 2.9, B₀ ≈ 1.6 G, and a blast wave radial evolution of $R\approx 9.8\times {10}^{15}{[(t-{t}_{e})/10{\rm{days}}]}^{0.78}$ cm. The last value implies an average ejecta speed of $R/{\rm{\Delta }}t\approx 0.38\,c$. This is ≈4×higher than the average for ordinary Ib/c SNe (≈0.1 c), but smaller than relativistic events such as SN 2009bb and SN 1998bw.

In Figure 9 we show the uncertainties on the best values of the average ejecta speed ($\langle {\rm{v}}\rangle ={r}_{0}/10$ d) and power-law index α_r obtained in a way similar to what was described in the previous Section. As evident from this Figure, the 99% confidence region for the average ejecta speed is $\langle {\rm{v}}\rangle /c\,\approx 0.32-0.44$.

Assuming equipartition (${\epsilon }_{e}={\epsilon }_{B}=0.33$), we derive a minimum energy of E ≈ 8 × 10⁴⁷${({\epsilon }_{e}/0.33)}^{-1}{[(t-{t}_{e})/10{\rm{days}}]}^{0.34}$ erg coupled to the fastest radio-emitting outflow.

Finally, the estimated progenitor mass-loss rate is $\dot{M}\approx 5\times {10}^{-6}({{\rm{v}}}_{w}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})$ M_⊙ yr⁻¹ (where again we have assumed a nucleon-to-proton ratio of 2). This mass-loss rate is in agreement with values derived for low-luminosity GRBs (see e.g., Figure 7), and smaller than the one derived for CSM-interacting BL-Ic SNe such as PTF11qcj ($\dot{M}\approx {10}^{-4}({{\rm{v}}}_{w}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})$ M_⊙ yr⁻¹; Corsi et al. 2014). This, together with the fact that the simplest off-axis GRB models (dashed lines in Figure 10) are in broad agreement with the radio light curve of PTF14dby, calls for a more accurate numerical modeling of this SN, which is beyond the scope of this paper, but which we hope will get the attention of the community.

We have presented the P48 photometry, spectral classification, and radio/X-ray follow-up observations of 15 BL-Ic SNe discovered by the PTF/iPTF. All of the SNe in our sample exclude radio afterglows typical of long-duration GRBs at cosmological distances observed on-axis. Thanks to deep VLA follow-up observations, we are able to exclude the presence of 1998bw-like (or 2009bb-like) radio emission for most of the SNe in our sample. Because radio emission traces the fastest moving ejecta, we conclude that events as relativistic as, and observationally similar to, SN 1998bw, are $\lesssim 41 \%$ of the BL-Ic population (99.865% confidence). None of our upper limits exclude radio emission similar to the radio afterglow of GRB 060218, which faded on timescales much faster than our VLA monitoring campaign was designed to target.

Using the X-ray upper limits collected via our programs, we rule out the presence of off-axis GRB jets observed slightly off-axis for some of the SNe in our sample. We also constrain the energy and density parameters of (largely) off-axis GRBs potentially harbored by the SNe in our sample for which 1998bw-like radio emission was excluded. While we can rule out the presence of GRBs as energetic as GRB 030329 observed at large off-axis angles and expanding in a constant ISM with density ${n}_{0,\mathrm{ISM}}\gtrsim 0.1$, we cannot rule out the presence of off-axis GRBs expanding in a low-density wind medium, such as the one found around GRB 130427A.

Finally, we presented detailed radio modeling of two radio-loud BL-Ic, PTF11cmh and PTF14dby, which add to our previous radio detection of PTF11qcj. While the ejecta speed of PTF11cmh is very poorly constrained due to the limited data set, we constrained the speed of the radio-emitting material in PTF14dby to be intermediate between that of non-relativistic BL-Ic SNe, and relativistic events such as SN 2009bb. Because we cannot securely rule out off-axis GRB models for these three events, we set an upper limit of $\lesssim 85 \%$ (99.865% confidence) on the fraction of BL-Ic SNe in our sample that could potentially harbor a GRB observed off-axis and expanding in a medium of density n_ISM ∼ 10 cm⁻³. This estimate could be improved by comparing our data with numerical models for off-axis GRBs expanding in a wind medium.

In summary, our results show that the VLA (thanks to its improved sensitivity), working in tandem with surveys like the iPTF, can help us clarify key open questions regarding the GRB-SN connection (such as, what fraction of purely BL-Ic SNe can host low-luminosity GRBs) and enable us to discover more events on the dividing line between ordinary BL-Ic and relativistic GRBs. Over the course of 5 years, we have greatly enlarged the sample of BL-Ic SNe (discovered independently of a GRB trigger) with radio follow-up within one year since discovery. We expect that the Zwicky Transient Facility will be able to boost even further the rate at which we are discovering the rare BL-Ic events (Smith et al. 2014).

A.C. thanks P. Chandra for graciously providing the data for the average long GRB radio light curve showing in Figure 4. A.C. acknowledges support from the NSF CAREER award #1455090. A.C. and N.P. acknowledge partial support from NASA/Swift Cycle 10 and 11 GI via grants NNX15AB79G and NNX16AC12G. A.G.Y.'s team is supported by the EU/FP7 via ERC grant No. 307260 the Quantum universe I-Core program by the Israeli Committee for planning and budgeting and the ISF; by Minerva and ISF grants; by the Weizmann-UK "making connections" program; and by Kimmel and ARCHES awards. K.M. acknowledges support from the STFC through an Ernest Rutherford Fellowship. M.M.K. acknowledges partial support from the GROWTH project via NSF grant 1545949. M.S. acknowledges support from the Royal Society and EU/FP7-ERC grant No. 615929. The Karl G. Jansky Very Large Array is operated by NRAO, for the NSF under cooperative agreement by Associated Universities, Inc. W. M. Keck Observatory, is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofsica de Canarias. The authors acknowledge the High Performance Computing Center (HPCC) at Texas Tech University at Lubbock (http://cmsdev.ttu.edu/hpcc) for providing HPC resources that have contributed to the research results reported within this paper. This research also used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231.