Abstract
Type II supernovae (SNe II) originate from the explosion of hydrogenrich supergiant massive stars. Their first electromagnetic signature is the shock breakout (SBO), a shortlived phenomenon that can last for hours to days depending on the density at shock emergence. We present 26 rising optical light curves of SN II candidates discovered shortly after explosion by the High Cadence Transient Survey and derive physical parameters based on hydrodynamical models using a Bayesian approach. We observe a steep rise of a few days in 24 out of 26 SN II candidates, indicating the systematic detection of SBOs in a dense circumstellar matter consistent with a mass loss rate of \(\dot M\) > 10^{−4}M_{⊙} yr^{−1} or a dense atmosphere. This implies that the characteristic hourtimescale signature of stellar envelope SBOs may be rare in nature and could be delayed into longerlived circumstellar material SBOs in most SNe II.
Main
With a new generation of large etendue facilities, such as the Intermediate Palomar Transient Factory^{1}, SkyMapper^{2}, Panoramic Survey Telescope and Rapid Response System^{3}, Korea Microlensing Telescope Network^{4}, Asteroid Terrestrialimpact Last Alert System^{5}, Dark Energy Camera (DECam)^{6}, Hyper SuprimeCam^{7}, Zwicky Transient Facility^{8} or Large Synoptic Survey Telescope^{9}, the study of rare and shortlived phenomena in large volumes of the Universe is becoming possible. This enables us not only to find new classes of events, but also to systematically study shortlived phases of evolution in known astrophysical phenomena, such as supernova (SN) explosions. In this work, we present 26 rising optical light curves from type IIP/L SN (SNe II) candidates discovered shortly after explosion by the High Cadence Transient Survey (HiTS) (ref. ^{10}, hereafter F16) and a systematic study of their physical properties.
SNe II are thought to originate after the core collapse of massive red supergiant stars (RSGs) with mainsequence masses between 8 and 16.5M_{⊙}^{11} (but see ref. ^{12}). In the currently favoured scenario—the neutrinodriven mechanism (see ref. ^{13} and the references therein)—a small fraction of the gravitational energy lost during collapse is transferred to the outer layers of the star via neutrino heating. This triggers a shock wave that, depending on the mass of the progenitor and the total input energy, can sometimes unbind the envelope of the star in a SN explosion. This shock wave is radiation dominated, Compton scattering mediated and propagates supersonically, typically at tens of thousands of km s^{−1} (ref. ^{14}). The shock precursor has a characteristic optical depth—estimated by equating the timescales for photons to diffuse out of the shock front (the diffusion timescale) and timescale for the shock to move into a new region of the star (the advection timescale)—of τ ≈ c/v_{shock}, where c is the speed of light and v_{shock} is the shock propagation speed^{15}. This can be written as a distance of about 30 times the mean free path of a photon scattering off electrons, and is therefore proportional to the electron density.
In RSGs, the shock is expected to travel for about a day until it reaches an optical depth from infinity small enough for the shock to emerge. For example, a shock travelling at 10,000 km s^{−1} would take 19 h to traverse 1,000R_{⊙}. The emergence of the shock itself, or shock breakout (SBO) (see review in ref. ^{16}), should typically last for about one hour if it happens from the envelope of an RSG—a timescale determined by the width of the shock precursor (proportional to the density), the shock velocity and some form factors. We call this scenario an envelope SBO (for example, ref. ^{17}).
If a dense wind or atmosphere is present, the ejecta’s photosphere at shock emergence can extend beyond the typical RSG sizes predicted by stellar evolution theory. This can push shock emergence radially outwards into lower electron densities, where the shock’s radiative precursor will be significantly more extended for the same optical depth and will therefore have a longerlived emergence. Thus, apart from powering the early light curve via conversion of kinetic energy into radiation, this dense circumstellar material (CSM) can delay the shock emergence and extend its duration to several days due to both the lower electron density and light travel time effects, replacing the much shorterlived envelope SBO. We call this scenario a wind SBO^{18,19,20,21,22} or extended atmosphere SBO^{23}.
After shock emergence, the envelope of RSGs should adiabatically cool until recombination of hydrogen starts. As the RSG envelope expands and cools, its typical temperature enters the optical range at a timescale of several days, making SNe II optical light curves rise to maximum light at a timescale of about seven days^{24}, with possible subclasses of relatively slow and fast rising SNe II^{25}. This timescale has been used to derive typical RSG stellar radii using analytical approximations or hydrodynamical models. The observed rise timescales are shorter than expected and have led researchers to conclude that either the radii of RSGs are much smaller than predicted by stellar evolution theory^{24,26}, or perhaps a dense wind SBO is responsible for the fast rise. The dense wind SBO theory is supported by the early spectroscopic detection of narrow optical emission lines with broad electronscattering wings in SN2013fs^{22}, which are suggestive of slowly moving shocked material^{23}, and by a recent analysis of SN II light curves around maximum light^{27}. Thus, it is important to understand the innermost regions of the wind or extended atmospheres of RSGs.
There have been several observational efforts to discover RSG envelope SBOs using highcadence observations from space^{28,29,30,31} and groundbased observatories^{32} from ultraviolet to optical wavelengths, but only marginal detections have been achieved (a notable exception was the serendipitous discovery of the envelopestripped SN IIb 2016gkg^{33}). HiTS (F16) is a survey that uses DECam to explore the transient sky at timescales from hours to weeks, monitoring the hourtimescale transient sky during three observational campaigns in 2013, 2014 and 2015.
Observations and data processing
The observational strategy of HiTS is described in F16. In 2013, we surveyed 120 deg^{2} in the u band with a cadence of 2 h during 4 consecutive nights. In 2014 and 2015, we surveyed 120 and 150 deg^{2}, with cadences of 2 and 1.6 h during 5 and 6 consecutive nights, respectively. In 2015, the highcadence phase was followed by a few observations days later, mostly in the g band, but also in the r band. The 2014 and 2015 data were analysed in real time, with candidates being generated and filtered only minutes after the end of every exposure. Thanks to these realtime capabilities, in 2014 and 2015, we were able to trigger spectroscopic followup observations for a few of the most nearby candidates, with hourtimescale spectroscopic followup capabilities in 2015 using the Very Large Telescope (VLT). No clear signatures of envelope SBOs were found and the very fast spectroscopic observations were never triggered, but we obtained spectra for 18 objects after the main highcadence phase of the observations for classification purposes, using the New Technology Telescope (provided by the Public ESO Spectroscopic Survey for Transient Objects Survey (PESSTO) collaboration^{34}), Southern Astrophysical Research telescope (SOAR) and VLT (see Methods). These spectra were used for direct classification, but also for testing a lightcurvebased classifier. In total, 11 SN candidates were spectroscopically classified as SNe Ia and 7 SN candidates were either classified as SNe II or showed a blue continuum consistent with SNe II.
The DECam data processing from raw image to light curve creation is discussed in the Methods. Although the resulting light curves did not show the signature of envelope SBOs, it was evident that the first few days of evolution of synthetic light curves, which do not include CSM and which show an hourtimescale SBO signature, did not resemble our observations. To investigate this further, we used models from refs ^{20,21} (hereafter M18), which include CSM, and developed tools that can be used to aid the classification and physical interpretation of these light curves.
Lightcurvebased classification
When no spectral information could be acquired, we used the SN early light curves for classification. Since our light curves are based on image differences, which sometimes contain a very recent template, our observables are light curve differences between two points in time (that is, we take into account that our templates can contain some SN flux). Comparing these observables against model predictions for a class of events, we can perform model selection and classification. For the RSG SNe II, we use the family of models from M18. The most important remaining classes are thermonuclear SNe (Ia) and envelopestripped corecollapse SNe (Ib/c), which in most cases are explosions whose rise to maximum is dominated by the deposition of energy from ^{56}Ni starting from a compact configuration^{35}. Therefore, as a simple approximation, we use SN Ia spectral templates from ref. ^{36} and allow for a broad range of stretch and scale factors to account for greater diversity. Then, using a Markov chain Monte Carlo (MCMC) sampler (as explained in the Methods), we compute a median loglikelihood for these two families of models and select between them using the Bayesian information criterion (BIC). With this method, we correctly classify all 18 SNe with spectroscopic classification (7 SNe II and 11 SNe Ia; see Fig. 1), highlighting the power of using earlytime photometry for classification.
We removed SNe that had a maximum time span of less than one week, which eliminated all candidates from 2013A and one SN with spectra from 2014A. Also, since we are most interested in the effect of mass loss and wind acceleration, which mostly affect the initial rise after emergence, we removed those SNe with a poor sampling at rise. We define a poor sampling at rise as those cases for which three or more continuous days during rise have no data. This includes: (1) SNe that have gaps in the data of two or more consecutive nights during the initial rise (five SNe discovered at the end of the 15A campaign, which had a logarithmic cadence); and (2) SNe for which we cannot rule out that they were seen only three or more days after their first light. For case (2), we removed those SNe that, according to our posterior distribution, had more than 10% probability of having been observed only three or more days after first light, defining the first light as the moment when the absolute magnitude in the g band reaches −13 mag. This additional filter resulted in the removal of two SNe.
Early SN II light curves and models
The resulting sample of 26 earlytime SN II candidate light curves discovered by the HiTS survey is shown in Fig. 2. There is no similar sample of SN II early light curves in the literature in terms of cadence (that is, recorded every 2 or 1.6 h during the night for several consecutive nights) and size. We include five additional SNe II from the literature in the analysis, all with wellsampled optical early light curves: SN2006bp^{37}, PS113arp^{19}, SN2013fs^{22}, KSN2011a and KSN2011d^{30}. The SN II candidates show a fast rise with a timescale of a few days, which is suggestive of the wind/extended atmosphere SBO scenario^{19,20,23}. For comparison, we also show 100 light curves sampled from the posterior distribution of physical parameters using a grid of SN II explosion models from M18. Briefly, we sampled the posterior distribution of physical parameters, such as the explosion time, attenuation, redshift, progenitor mass, explosion energy, mass loss rate (\(\dot M\)) and wind acceleration parameter (β), using an MCMC sampler as explained in the Supplementary Information. For reference, we also show one g band light curve sampled randomly from the posterior where we set the mass loss to zero. It can be seen that for most light curves only models with a significant mass loss rate match the fast rise observed in the data.
Synthetic models
Models from M18 assume confined steadystate winds with an extended acceleration length scale. They form a grid of 518 sets of spectral time series spanning different mainsequence masses and explosion energies, but also different mass loss properties. The RSG SN progenitors and CSM around them are constructed in the same way as in M18. We provide a short summary of the progenitors and CSM.
The public stellar evolution code MESA^{38,39,40} is used to compute the RSG SN progenitors of the zeroage mainsequence masses of 12, 14 and 16M_{⊙} with solar metallicity. We set the model maximum mass to 16M_{⊙} because the maximum mass of SN II progenitors is estimated to be at around 16.5M_{⊙} based on the RSG progenitor detections^{11}. The progenitors are evolved using the Ledoux criterion for convection with a mixinglength parameter of 2.0 and a semiconvection parameter of 0.01. Overshooting is taken into account on top of the hydrogenburning convective core with a step function using the overshoot parameter of 0.3H_{P}, where H_{P} is the pressure scale height. We use the ‘Dutch’ mass loss prescription in MESA without scaling for both hot and cool stars. The RSG progentiors are evolved to the core oxygen burning, from which the hydrogenrich envelope structure hardly changes until the core collapses. The final progenitor properties are summarized in Table 1.
A CSM structure with density ρ_{CSM}(r) = \(\dot M{\mathrm{/}}\left( {4\pi v_{{\mathrm{wind}}}} \right)r^{  2}\) is attached on top of these progenitors. Here, \(\dot M\) is the progenitor’s mass loss rate, r is the distance from the centre of the star and v_{wind} is the wind velocity. We do not change \(\dot M\) with r in our models. Instead of \(\dot M\), we take the radial change of v_{wind} caused by the wind acceleration into account. The following simple β velocity law for the wind velocity is adopted:
where v_{0} is the initial wind velocity, v_{∞} is the terminal wind velocity and R_{0} is the wind launching radius that we set at the stellar surface. v_{∞} = 10 km s^{−1} is fixed in our models. v_{0} is chosen so that the CSM density is smoothly connected from the surface of the progenitors and is less than 10^{−2} km s^{−1}. We take β between 1 and 5 because OB stars have β ≃ 0.5–1.0 (ref. ^{41}) and RSGs are known to experience slower wind acceleration than OB stars (that is, β > 1; for example, refs ^{42,43}). For instance, an RSG ζ Aurigae is known to have β ≃ 3.5 (ref. ^{44}).
RSG mass loss rates are known to be dependent on the luminosity of the progenitor star, but estimations differ greatly in the literature. For the typical luminosities observed in RSG SN II progenitors (between 10^{4} and 10^{5}L_{⊙}; ref. ^{11}), derived mass loss rates range from between 10^{−7} and 10^{−6}M_{⊙} yr^{−1} (ref. ^{45}) to between 10^{−5} and 10^{−4}M_{⊙} yr^{−1} (ref. ^{46}). The M18 models assume a wider range of mass loss rates, including zero mass loss and a range of values between 10^{−5} and 10^{−2}M_{⊙} yr^{−1}, always assuming enhanced mass loss episodes before explosion extending up to 10^{15} cm from the progenitor. Note that a shock travelling at 10,000 km s^{−1} would take about 11 days to cover 10^{15} cm. Recent optical and Xray modelling of the SN IIP 2013ej suggests that the dense CSM radius may be relatively small (10^{14} cm), although there are some model uncertainties^{47}.
Hydrodynamics and radiative transfer
The light curves from the explosions of the abovementioned progenitors with CSM are numerically obtained using the onedimensional multigroup radiation hydrodynamics code STELLA^{48,49,50}. STELLA follows the evolution of the spectral energy distributions (SEDs) at each time step, and the multicoloured light curves from the explosions are obtained by convolving the filter functions to the numerical SEDs. We use the standard setup for the SED resolution (that is, 100 bins that are distributed between 1 and 50,000 Å on a logscale). The code initiates the explosions as thermal bombs. All the models have 0.1M_{⊙} of the radioactive ^{56}Ni at the centre, but this does not affect the early light curves we are interested in. Every progenitor model with CSM is exploded with 4 different explosion energies to investigate the dependence on the explosion energy: 5 × 10^{50} erg, 10^{51} erg, 1.5 × 10^{51} erg and 2 × 10^{51} erg. The effects of mass, energy, mass loss rate, wind acceleration parameter, attenuation and redshift on the optical light curves can be seen in Fig. 3.
Results
Earlytime SN II optical light curves are affected by many physical parameters (see Fig. 3). These can change one or more of the following: the light curve normalization, its characteristic rise time and general shape, or the ratio between bands. The parameters that affect the normalization the most are the redshift and A_{V}, followed by the mass loss rate and energy (a tenfold difference in the peak apparent g mag range for the full grid of values tested). The parameters that affect the light curve shape the most are the mass loss rate and redshift, with somewhat similar effects but very different normalizations that help break their degeneracy. Mass and β also affect the normalization, light curve shape and colour, but not as strongly as the other variables.
In this work, we have concentrated on those variables constraining the CSM density profile, which in the M18 models used is parametrized via two numbers: \(\dot M\) and β, assuming a terminal wind velocity of 10 km s^{−1} and a maximum CSM radius of 10^{15} cm. Therefore, we focus the discussion on these two quantities.
The posterior distributions of \(\dot M\) and β for the entire sample of HiTS SN II candidates, as well as five additional SNe II from the literature with wellsampled optical early light curves, are summarized in Fig. 4. On the x and y axes, we show the 5th, 50th and 95th percentiles of the marginalized posterior distributions over all variables except \(\dot M\) or β, respectively (error bars cover the 5th and 95th percentiles and intersect at the 50th percentile). Mass loss rates smaller than \(\dot M\) = 10^{−4}M_{⊙} yr^{−1} are allowed in only 2 out of 26 HiTS SNe II candidates, and in none of the 5 SNe II from the literature. We also constrain β within the range of values covered by our grid of models (1 < β < 5). Although we find significantly large values of β for some SNe, we do not find a significant excess of large values within the interval 1 < β < 5 in the sample. The comparative light curves from the literature and their posterior sampled synthetic light curves are shown in Fig. 5. The complete list of inferred parameters is shown in Supplementary Tables 1 and 2.
We have found evidence for a density profile consistent with large mass loss rates \(\dot M\) > 10^{−4}M_{⊙} yr^{−1} in 29 out of 31 SNe II, and evidence for a density profile consistent with relatively slow wind acceleration (large β) for some SNe. These constraints suggest that CSM densities in the vicinity of RSGs before explosion are larger than 10^{−13} g cm^{−3} at 10^{14} cm from the centre of the star (about 1,400R_{⊙}). Assuming an electronscattering opacity consistent with fully ionized CSM made of hydrogen and helium at solar abundance (κ = 0.34 cm^{2} g^{−1}), the optical depth τ ≈ 1 would occur at typical radii between 10^{14} and 10^{15} cm and at typical densities between 10^{−12} and 10^{−14} g cm^{−3}. The typical densities implied from this analysis are in very good agreement with those derived spectroscopically for SN2013fs in ref. ^{22}, and the mass loss rates are consistent with those derived using nearultraviolet data for PS113arp.
The large mass loss rates derived in this work imply total CSM masses typically between 0.1 and 1.0M_{⊙}, and typically between 0.01 and 0.1M_{⊙} above the breakout radius (τ ≤ c/v). These mass loss rates could not be sustained for large periods of time before explosion, as noted by ref. ^{22}, otherwise we would see type IInlike features at late times (narrow optical emission lines with electronscattering wings) and the progenitor star would lose its hydrogenrich envelope before explosion. This favours the interpretation of dense atmospheres or confined accelerated wind SBOs. However, it is interesting to note that the large derived mass loss rates appear to match those from the type IIb SN2013cu^{51,52,53}—a probable postRSG star^{53}. The fact that most of this CSM would be optically thick at breakout implies that, apart from delaying the SBO, most of the shocked material would contribute to the optical light curve via its cooling after breakout. In fact, the differences between light curves with varying CSM radii in ref. ^{21} are in part explained by this effect.
The range of wind acceleration parameter β favoured by these observations is consistent with those derived for many normal RSGs^{54}. However, models assume enhanced mass loss before explosion and our observations cannot be used to discard whether such enhanced mass loss is required to produce the highdensity CSM being shocked. The large mass loss rates suggest that either RSGs undergo this enhanced mass loss before explosion or that we are probing complex RSG atmospheres and not an accelerated wind. The fact that highresolution observations of nearby RSGs show complex CSM environments^{55} suggests that large CSM densities may in fact be possible without requiring accelerated or enhanced mass loss.
An important implication from this work is that if high CSM densities are present in most SNe II, the hourtimescale, highdensity RSG envelope SBOs may be rare in nature. With the exception of two slowrising SNe out of 26 SN II candidates, large CSM densities were always inferred (see Methods for the significance of this result). The photons produced when the shock crosses the outer envelope of these stars and enters the highdensity CSM would not be able to diffuse in a timescale of hours, but would instead take days to emerge in the wind SBO. Only if the CSM radii are much smaller than the values tested in this work could a short timescale signature be observed (see Fig. 7 in ref. ^{21}). Given that we have found breakout densities to be similar for different CSM density profiles, measuring the lag between the time of explosion (via neutrinos or gravitational waves in a nearby SNe II) and shock emergence (from 1 to 12 days after explosion in the parameter space probed by M18) could greatly help us to understand the true extent of the CSM.
These results also suggest that preexplosion radii cannot be derived using the early light curve of SNe II—at least at the optical bands—because the rise to maximum will not be dominated by the adiabatic cooling of the envelope, but by the wind SBO instead. Furthermore, we have shown that it is possible to accurately separate compact (SNe I) from extended (SNe II) explosions using earlytime information, opening the possibility of using highcadence observations as a tool for the detection and classification of SNe, including potentially distant SNe II, which can be used for cosmological applications, where precise explosion time determinations are required^{56}.
Finally, we conclude that it may be difficult for future highcadence surveys such as the Zwicky Transient Facility to systematically discover RSG envelope SBOs. Instead, they will be able to increase the sample of wind SBO events significantly. This also implies that the deep drilling fields from the Large Synoptic Survey Telescope will be an important source of wind SBO events at all redshifts, allowing for the study of the CSM around RSG stars with cosmic age. We expect that the analysis of the physical parameters inferred from populations of events, such as the one presented in this work, will become part of the standard set of tools for both the classification and characterization of large volumes of transients in the future.
Methods
Classification spectra
The following telegrams were sent to report the spectroscopic observations and classifications performed for this project, separated by telescope: New Technology Telescope^{57,58,59,60}, Southern Astrophysical Research^{61,62,63,64} and VLT^{65,66}. Two example classification spectra are shown in Supplementary Fig. 1 compared with SN II spectra using SNID^{67}.
Light curves
The DECam data calibration included preprocessing, image difference, candidate filtering and light curve generation. To preprocess the data, we used a modified version of the DECam community pipeline, including electronic bias calibration, crosstalk correction, saturation masking, bad pixel masking and interpolation, bias calibration, linearity correction, flatfield gain calibration, fringe pattern subtraction, bleed trail and edge bleed masking, and interpolation. We removed cosmic rays using CRBlaster^{68}. After preprocessing, we used as templates the first goodquality images (that is, those with photometric conditions and good seeing) of the fast cadence phase of observations when no other templates were available. For fields that overlapped between 2014 and 2015, we used goodquality images from 2014, which were deeper than in 2015. We aligned all the science images to the template using Lanczos interpolation and performed difference imaging using a variable size pixel kernel as described in F16. Candidates were selected using machine learning filters over stamps centred at pixels that reached an integrated flux signaltonoise ratio (S/N) above five.
To produce the light curve, we first removed all flux differences that had a science airmass larger than 1.7. Then, we improved the SN candidates’ central position using all the available image differences. We projected the empirical point spread function to this new central position and performed optimal photometry to estimate image difference fluxes and their associated variances, even when the object was not significantly detected. Absolute calibrations were performed using photometric nights as references, and the zero points were precomputed for DECam at different filters and chargecoupled detectors. We independently validated whether these nights achieved photometric conditions using the PanSTARRS1 public catalogues^{69}.
Spectraltimeseriestolight curves
To ingest time series of synthetic spectra and produce synthetic light curves for any redshift, attenuation and explosion time, we first assume a standard Λ cold dark matter cosmology. We light curves and attenuate the spectral series with distance and assume a Cardelli law with R_{V} = 3.1 for the dust attenuation. We integrate the resulting spectra in the DECam bands and generate synthetics light curves that can be interpolated into a given time array. To speed up the model evaluation, we precompute light curves for all DECam filters and all available models in a logarithmically spaced time array, a linearly spaced attenuation array and a logarithmically spaced redshift array. From these different time, attenuation and redshift arrays, we can interpolate into given observational times assuming an explosion time, an attenuation and a redshift.
Model interpolation
Apart from the previous interpolations, we must be able to interpolate quickly between models with different physical parameters. To do this, we first find the closest values in all the intrinsic physical dimensions; for example, the mass, energy, mass loss rate \(\dot M\) and wind acceleration parameter β, and find all the models that have combinations of these values, which we call \(\vec \theta _{{\mathrm{close}}}\). The final light curve will be a weighted combination of all these models:
where \(m\left( {t,t_{{\mathrm{exp}}},z,A_{\rm{V}},\vec \theta } \right)\) is the magnitude of the model at a given observation time t, explosion time t_{exp}, redshift z, attenuation A_{V} and vector of model parameters \(\vec \theta\), and the normalized weights \(\hat w\left( {\vec \theta ,\vec \theta _i} \right)\) are defined as:
where the weights are a function of a pair of parameter vectors \(\vec \theta\) and \(\vec \theta _i\). To avoid having to define a metric to compare values in the different dimensions of the vector of physical parameters, we define the weights to be inversely proportional to the product of the differences in all the dimensions of the vector of physical parameters \(\vec \theta\); that is:
where \(\vec \delta\) is a vector with the same physical units as the parameters, but much smaller than the typical separation in the grid of models. This avoids the divergence of the weights when a given coordinate matches the coordinates of known models.
For the attenuation, redshift and mass loss rate, we use an internal logarithmic representation. Since we also include models with \(\dot M\) = 0, we assume them to correspond to a mass loss rate of 10^{−8}M_{⊙} yr^{−1}.
The main advantages of the previous weighting scheme are that it does not require defining a metric and it allows for possible missing models in the grid. We show examples of interpolated models where we vary the physical parameters smoothly; for example, the mass loss rate continuously between 10^{−8} and 10^{−2}M_{⊙} yr^{−1} (Fig. 3).
MCMC sampler
Having the capability to quickly generate interpolated light curves for any combination of explosion time, redshift, attenuation and vector of physical parameters, we can approach the problem of inferring physical parameters from a Bayesian perspective; that is, computing the posterior probability of the model parameters given the data and assuming prior distributions.
To do this, we sample the posterior of the joint distribution of parameters using an MCMC sampler that uses an affine invariant approach^{70}. This method uses parallel Markov chains that sample the posterior distribution moving randomly in directions parallel to the relative positions of the samplers, following acceptance rules that satisfy the condition of detailed balance for reversible Markov chains. This is implemented in Python via emcee^{71}.
We run the MCMC sampler using predefined priors that are relatively flat distributions for most variables at the linear (mass, energy and β) or logarithmic (redshift, A_{V} and \(\dot M\)) scales (see Supplementary Table 3). For the explosion time, we require a first guess, for which we run an interactive fitting routine for all the SN light curves using tools found in our public repository (https://github.com/fforster/surveysim), and use a Gaussian prior around this value with a standard deviation of four days. We also allow for a variable scale parameter to allow for errors in absolute calibrations, for which we use a Gaussian prior centred at 1.0 and with a standard deviation of 0.01 (1% errors). We use 400 parallel samplers (or walkers) and 900 steps per sampler, with a burnin period of 450 steps in all cases. These numbers were set via trial and error, trying to reach the detailed balance condition while reducing the number of multiple disjoint solutions in the sampler. The Markov chains reached acceptance ratios between 0.2 and 0.5 in more than 95% of the cases. An example joint distribution can be visualized in a corner plot (as shown in Supplementary Fig. 2), and example light curves drawn from the posterior distribution are shown for the HiTS SN II sample (Fig. 2) and SNe II in the literature with wellsampled optical early light curves (Fig. 5).
To extract posterior probabilities for a given parameter, we can marginalize the joint distribution integrating over certain dimensions; for example, the explosion time, redshift when not available, attenuation A_{V}, mass and energy. This is important to derive conclusions about the true distributions of \(\dot M\) and β.
Photometric classifier examples
In Supplementary Fig. 3, we show two observed light curves compared with posterior sampled light curves assuming the SN II or scaled SN Ia models discussed in the text.
Selection effects
Given the different filters applied to the data, it is reasonable to ask whether they play any role in the observed excess of SNe II with \(\dot M\) > 10^{−4}M_{⊙} yr^{−1}. Two cases need to be considered: (1) contamination of other SN types at large \(\dot M\) values; and (2) a lower detection/classification efficiency of SNe II with low \(\dot M\). Contamination of other SN types at large \(\dot M\) values seems unlikely, since no other SN types show the very fast rise times observed in our sample (contamination from other SN types at low \(\dot M\) values seems more likely, but this would increase the number of low \(\dot M\) SN II candidates in the opposite direction to what we observe), but a lower detection/classification efficiency of SNe II with low \(\dot M\) cannot be discarded a priori. Low detection/classification efficiencies at low \(\dot M\) could be due to four possibilities: (1) a lower detection efficiency of low \(\dot M\) SNe II (that is, relatively fewer SNe II with low \(\dot M\) reach the necessary S/N for detection); (2) a lower classification efficiency (that is, low \(\dot M\) SNe II are more often missclassified as type I SNe by our classifier than large \(\dot M\) SNe II); (3) a lower selection efficiency when removing SNe with a poorly sampled rise (that is, removing SNe with either gaps in the data or a nonnegligible probability of having been seen three or more days after their first light favours large \(\dot M\) SNe); or (4) an inference process biased towards large \(\dot M\) values.
We have investigated all of these possibilities via simulations assuming a uniform logarithmic distribution of \(\dot M\) values between 10^{−8} and 10^{−2}M_{⊙} yr^{−1} and testing whether there is an excess in the recovery fractions of the samples below and above the median value of 10^{−5}M_{⊙} yr^{−1}, which we call the low and large \(\dot M\) samples, respectively. First, for case (1), we simulate 150,000 synthetic SNe II with a uniform distribution of \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\), β, mass, energy and explosion time, and an exponential distribution of attenuations, taking into account cosmology, the star formation history, the efficiency at converting stars into SNe II and the actual cadence and depth of the survey (see F16 for more details). We found that the number of SNe with at least two detections (with S/N ≥ 5) and that have at least one observation (with S/N ≥ 2) within the first three days after emergence should be distributed in a proportion of 28 to 72% between the low and large \(\dot M\) samples; that is, there is a significant bias towards large \(\dot M\) values. However, using a binomial distribution with the expected fractions of low and large \(\dot M\) SNe II and the total number of SNe in the sample, we infer that the probability of having two or fewer SNe in the low \(\dot M\) sample is only 1%. Moreover, the probability of having two or fewer SNe below 10^{−4}M_{⊙} yr^{−1} is only 2 × 10^{−4}; that is, we can discard that the relative absence of low \(\dot M\) SNe II is due to a statistical fluctuation. With these simulations, we can also compute the detection efficiency as a function of \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\) (Supplementary Fig. 4), which we will use to correct the sample distribution of \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\) values.
To test case (2), we simulated 300 SNe whose parameters are drawn from the inferred distribution of parameters in our sample (except for the mass loss rate which is drawn from a uniform logarithmic distribution) and ran MCMC to get median loglikelihoods and the BIC classifier on them. We have found that more than 95% of the SNe in the low \(\dot M\) sample are correctly classified as SNe II; therefore, this effect cannot explain the relative absence of low \(\dot M\) SN II candidates. For case (3), we expect that the presence of gaps in the data during the initial rise should not be related to the value of \(\dot M\), but perhaps SNe with low \(\dot M\) values have a more uncertain time of emergence (due to their shallower rise), which would lead to larger reported probabilities of having missed the initial three days after first light. We tested this case by measuring the relative uncertainties (50th percentile minus 10th percentile) in the inferred time of first light in the low and large \(\dot M\) samples, finding that the large \(\dot M\) case uncertainties tend to be 50% smaller. Therefore, there could be some low \(\dot M\) SNe occurring at the beginning of the survey, which would be preferentially removed. Consider the case when a low \(\dot M\) SN had its first light at the time of first observation, but its uncertainties allowed for a first light more than three days before explosion (for comparison, the median and maximum difference between the 50th and 5th percentiles of the inferred explosion times are 1.8 and 2.9 days, respectively). A similarly observed large \(\dot M\) SN having its first light at the same time would not have been removed because its uncertainties were 50% smaller, giving this family of explosions up to one additional effective survey day, or a 10% relative excess considering Supplementary Fig. 5. This cannot explain the large observed excess of large \(\dot M\) SN II candidates either. Finally, for case (4), we performed similar simulations with a flat distribution of log mass loss rates and found that if anything there is a bias towards small \(\dot M\) values (see discussion in the following paragraphs). Thus, from the discussion above, we conclude that the excess of large \(\dot M\) SN II candidates is not due to a selection effect and that it is significant.
Parameter inference tests
In Supplementary Fig. 6, we show a kernel density estimation of the sum of posterior distributions marginalized over \(\dot M\) and β for the 26 SNe in the sample. To combine distributions with very different variances, we use Silverman’s rule to estimate a kernel width^{72}:
where h is the kernel width, x is the random variable whose distribution we want to estimate, var is the variance, IQR is the interquartile range and N is the number of sampled values of x. In our case, we use the sampled posterior as our random variable, marginalizing over all other variables first. We correct these distributions by the detection efficiencies derived previously, confirming that \(\dot M\) < 10^{−4}M_{⊙} yr^{−1} is not favoured by the data. Although we find some significantly large values of β for individual SNe, we do not find a strong preference for large β values in the sample.
To test whether the preference for large \(\dot M\) could be due to some bias in the posterior sampling, we simulate 300 SNe from a uniform distribution of \(\dot M\) in a logarithmic scale, a uniform distribution of β and a uniform distribution of progenitor masses. We use the inferred explosion times from the observed SN sample to mimic the same time coverage, and the inferred energy, redshift and attenuation to mimic the observed apparent magnitudes. We then run the same posterior sampling algorithm and test whether the sum of the posterior distributions resembles the input distribution. For \(\dot M\), we see a relatively flat recovered distribution with a median value at 10^{−5.4}M_{⊙} yr^{−1}; that is, below 10^{−6}M_{⊙} yr^{−1} or only a slight bias towards small mass loss rates. For β, we also see a relatively flat distribution with a median β of 2.8, below 3.0 or a slight bias towards small values as well. For comparison, the posterior distribution of \(\dot M\) in the observed sample has a median value of 10^{−2.8}M_{⊙} yr^{−1} and the posterior distribution of β in the observed sample has a median value of 3.5.
Then, to test whether there is a bias in β or the progenitor masses at the range of high \(\dot M_{}^{}\) suggested by our observations, we simulate another 300 SNe where we instead use the inferred distribution of \(\dot M\), but uniform distributions of β and progenitor mass. Again, we do not find a bias in β, with a relatively flat distribution and a median at 3.0, exactly in the middle of the distribution. However, we detect a large bias in the progenitor mass distribution towards small values, with a median inferred progenitor mass in both the simulated and observed sample of 12.8M_{⊙} (instead of 14.0M_{⊙}). This means that we are not able to derive meaningful conclusions about the distribution of progenitor masses in the sample. For the case of energy, where we expect a strong degeneracy with redshift, we also simulate a random sample of 300 SNe using inferred values for all the physical quantities, except for the energy, which is drawn from a uniform distribution between 0.5 and 2. The median of the inferred distribution is at 1.30, slightly above 1.25, which means that there is only a slight bias towards larger energies. However, the standard deviation of the differences between the medians of the inferred energy distribution and the simulated energies is 0.44, almost the same as the value one would obtain assuming flat energy posteriors (0.43). Thus, the energy is poorly constrained by our observations, but we detect no significant bias. Note that although the energy is poorly constrained, it is important to marginalize over its possible values to learn about the distributions of the other variables.
Using simulated light curves, we can also test how well we recover each parameter. For this, we measure the root mean square difference between simulated values and the median of the posterior distribution for each SNe. We obtain a root mean square of 1.2 for \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\), 1.4 for β, 1.2M_{⊙} for the progenitor mass, 0.03 for the redshift, 0.44 for the energy (see previous paragraph) and 0.14 for the attenuation. Interestingly, we find that the three SNe with host galaxy redshifts have values consistent with those inferred from the light curves alone; that is, allowing for the redshift to vary during Bayesian inference. The host galaxy redshifts for SNHiTS15C, SNHiTS15aq and SNHiTS15aw are 0.08, 0.11 and 0.07, respectively, while the 5th, 50th and 95th redshift percentiles inferred from the light curves alone are 0.08, 0.10 and 0.12, 0.10, 0.11 and 0.13, and 0.06, 0.07 and 0.08, respectively.
We also test whether some of the inferred variables are correlated. For this, we build a correlation matrix using the median values of the posterior distributions. We find that the only two variables that are significantly correlated are A_{V} and \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\), with a correlation coefficient of −0.79, which points to a degeneracy in the normalization of the light curves. If we remove the two low mass loss rate SNe in the sample (SNHiTS15G and SNHiTS15Q), this correlation weakens to −0.5 and the stronger correlations become redshift with \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\) (0.68) and redshift with β (−0.66), possibly pointing to a degeneracy in the shape of the light curves (see Fig. 3). No other offdiagonal correlation coefficients are larger than 0.6 in absolute value. This suggests that having independent redshift determinations for the SNe in the sample would significantly improve the quality of our results, as exemplified by SNHiTS15aw.
To test for biases related to the observational strategy, we study the distribution of explosion times for the HiTS14A and HiTS15A campaigns (see F16), which we show in Supplementary Fig. 5. For the 2014A campaign, the posterior of explosion times span a shorter time than the survey itself, which suggests some bias towards explosions happening at the beginning of the survey. For the 2015A campaign, we detect a possible bias for detections during the initial highcadence phase of observations as well. The difference between the maximum and minimum median explosion time is 9.5 days, and the median of these values differs by 1.3 days from the midpoint between the minimum and maximum values. We observe a similar behaviour when estimating detection efficiencies as a function of explosion time.
We also show the distribution of inferred redshifts and attenuations in Supplementary Fig. 7. We can see that the survey efficiency starts decreasing at z = 0.3, and that we may be able to detect SNe up to z = 0.4–0.5, which highlights the potential for similar surveys to be used for highredshift SNe II studies. The apparent bimodality is not significant: we ran a Hartigans’ dip test of unimodality^{73} using the inferred median redshift values and we could not reject unimodality with a P value of 0.84. The distribution of attenuations appears to follow an exponential distribution with a characteristic scale of 0.07. It is worth noting that in our simulations most SNe above z = 0.3 have \(\dot M\) > 10^{−3}M_{⊙} yr^{−1}, which explains the previously described selection effects.
Finally, we show the distribution of favoured \(\dot M\) values as a function of the redshift in Supplementary Fig. 8, where a lack of low \(\dot M\) models at high redshifts is observed. In the same figure, we show the predicted distribution of \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\) and redshift as linearly spaced density contours, assuming a flat distribution of \({\mathrm{log}}_{10}{\kern 1pt}{\lbrack}\dot M{\rbrack}\). These simulations predict more SNe II than are observed at both very large mass loss rates (only two SNe with \(\dot M\) > 10^{−2.45}M_{⊙} yr^{−1}) and at low mass loss rates (only two SNe with \(\dot M\) < 10^{−4}M_{⊙} yr^{−1}), with probabilities of 0.07 and 0.0002, respectively; that is, they are unlikely to be due to a statistical fluctuation.
Data availability
The SN II light curves used in this analysis will be available in the VizieR service and Open Supernova catalogue upon publication. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank L. Dessart, K. Maeda, Ph. Podsiadlowski and K. Pichara for useful discussions. F.F. and T.J.M. thank the Yukawa Institute for Theoretical Physics at Kyoto University, where part of this work was initiated during the YITPT1605 on ‘Transient Universe in the Big Survey Era: Understanding the Nature of Astrophysical Explosive Phenomena’. T.J.M. is supported by the GrantsinAid for Scientific Research of the Japan Society for the Promotion of Science (16H07413 and 17H02864). The Powered@NLHPC research was partially supported by the supercomputing infrastructure of the NLHPC (ECM02). Numerical computations were partially carried out on PC cluster at the Center for Computational Astrophysics, National Astronomical Observatory of Japan. We acknowledge support from Conicyt through the infrastructure Quimal project (number 140003). F.F., J.S.M., E.V. and S.G.G. acknowledge support from Conicyt Basal fund AFB170001. F.F., G.C.V., A.C., P.A.E., M.H., P. Huijse, H.K., J.M., G.M., F.O.E., G.P., A.R. and I.R. acknowledge support from the Ministry of Economy, Development and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics. S.C.Y. was supported by the Korea Astronomy and Space Science Institute under the research and development programme (project number 334820160002) supervised by the Ministry of Science, ICT and Future Planning and Monash Centre for Astrophysics via the distinguished visitor programme. E.Y.H. and C.A. acknowledge the support provided by the National Science Foundation under grant number AST1613472 and the Florida Space Grant Consortium. F.F., J.M., G.M. and A.R. acknowledge support from Conicyt through Fondecyt project number 11130228. J.C.M., G.CV., P.A.E., P. Huijse, H.K., G.P. and F.O.E. acknowledge support from Conicyt through Fondecyt project numbers 11170657, 3160747, 1171678, 3150460, 3140563, 1140352 and 11170953, respectively. G.M. and I.R. acknowledge support from CONICYTPCHA/Magister Nacional/201622162353 and 201622162464, respectively. G.G. is supported by the Deutsche Forschungsgemeinschaft, grant number GR 1717/5. S.G.G. acknowledges support from Comité Mixto ESO Chile project ORP 48/16. F.F., J.C.M., P. Huijse, G.C.V. and P.A.E. acknowledge support from Conicyt through the Programme of International Cooperation project DPI20140090. L.G. was supported in part by the US National Science Foundation under grant AST1311862. A.G.Y. is supported by the EU via ERC grant number 725161, the Quantum Universe ICore programme, the ISF, the BSF Transformative programme and a Kimmel award. M.F. is supported by the Royal Society–Science Foundation Ireland University Research Fellowship (reference 15/RSURF/3304). S.B. acknowledges funding from project RSCF 181200522. This study was based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programmes 292.D5042(A) and 094.D0358(A). This work is partly based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile as part of the PESSTO ESO programmes 188.D3003, 191.D0935 and 197.D1075. It is partly based on observations obtained with the Gran Telescopio Canarias telescope. This project used data obtained with DECam, which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES projects was provided by the US Department of Energy, US National Science Foundation, Ministry of Science and Education of Spain, Science and Technology Facilities Council of the United Kingdom, Higher Education Funding Council for England, National Center for Supercomputing Applications at the University of Illinois at Urbana–Champaign, Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and AstroParticle Physics at The Ohio State University, Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Ministério da Ciência, Tecnologia e Inovação, Deutsche Forschungsgemeinschaft and Collaborating Institutions in the DES. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas–Madrid, University of Chicago, University College London, DES–Brazil Consortium, University of Edinburgh, Eidgenössische Technische Hochschule Zürich, Fermi National Accelerator Laboratory, University of Illinois at Urbana–Champaign, Institut de Ciències de l’Espai, Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, Ludwig–Maximilians Universität München and the associated Excellence Cluster Universe, University of Michigan, National Optical Astronomy Observatory, University of Nottingham, Ohio State University, University of Pennsylvania, University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, University of Sussex and Texas A&M University.
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F.F. computed the SN light curves based on DECam data and performed the analyses, including writing the analysis software. T.J.M. computed the grid of SN models. F.F. and J.C.M. wrote the SN discovery pipeline. J.P.A., F.B., A.C., Th.d.J., L.G., S.G.G., E.Y.H., H.K., J.M., G.M., F.O.E., G.P., R.C.S. and A.K.V. helped with photometric and spectroscopic observations under the HiTS programme. T.J.M., S.B., G.G. and S.C.Y. computed the progenitor models. A.R. computed the light curve observations made with the du Pont telescope. G.C.V., P.A.E., P.Huentelemu, P.Huijse, I.R. and J.S.M. helped develop the SN detection algorithms, including image processing, statistical methods and machine learning. R.C.S. and E.V. coordinated the fast data access required to achieve the realtime analysis and fast spectroscopic classifications. C.A., M.F., A.G.Y., E.K., L.L.G., P.A.M., N.A.W. and D.R.Y. contributed to the PESSTO observations. A.d.U.P. contributed spectroscopic observations using the Gran Telescopio Canarias telescope. All coauthors contributed comments.
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Supplementary Tables 1–3, Supplementary Figures 1–8
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Förster, F., Moriya, T.J., Maureira, J.C. et al. The delay of shock breakout due to circumstellar material evident in most type II supernovae. Nat Astron 2, 808–818 (2018). https://doi.org/10.1038/s4155001805634
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