SN 2018agk: A prototypical Type Ia Supernova with a smooth power-law rise in Kepler (K2)

We obtained ground-based photometry with the CTIO 4-m Blanco telescope with DECam in $g$ and $i$ bands, the Swope 1.0-m telescope at Las Campanas Observatory in $uBVgri$ bands, the 60/90cm Schmidt-telescope on Piszkéstető Mountain Station of Konkoly Observatory in $BVRI$ bands, the 1.3m telescope at the Cerro Tololo Inter-American Observatory (CTIO) in $RIJHK$ bands, and the PS1 1.8 m telescope at Haleakala on Maui, Hawai’i in $gri$ bands (Chambers et al., 2016). We also include photometric data from the Global Supernova Project taken with Las Cumbres Observatory previously published in Baltay et al. (2021). We downloaded the data products using the NSF NOIRLab DECam Community Pipeline (Valdes et al., 2014). The standard reduction of the Swope images is described in Kilpatrick et al. (2018). We reduced the PS1 images using the standard PS1 Image Processing Pipeline (IPP) (Magnier et al., 2020c, b, a; Waters et al., 2020) which includes standard reductions, astrometric solution, stacking of nightly images, source detection, and photometry. The photometry of all images were calibrated using standard sources from the Pan-STARRS DR1 catalog (Flewelling et al., 2020) in the same field as SN 2018agk and transformed following the Supercal method (Scolnic et al., 2015). After standard reductions, the photpipe pipeline (Rest et al., 2005, 2014) performs difference imaging and transient identification. DECam observed SN 2018agk during the crucial first 5 days after explosion, making it the first SN Ia that has both a high cadence Kepler light curve and ground-based multi-band observations in its earliest phase. DECam $g$-band images from well before explosion and $\sim 20$ days after peak are shown in Figure 1(a), and the full multi-band light curves are plotted in Figure 8.

IC 0855 was included as a Campaign 17 target through “The K2 ExtraGalactic Survey (KEGS) for Transients” (PI Rest) and the “Multi-Observatory Monitoring of K2 Supernovae” (PI Foley) programs as part of the K2 SCE (internal Kepler EPIC ID 228682548). We retrieved the IC 0855 Kepler data through the Mikulski Archive for Space Telescopes (MAST) after the end of K2 Campaign 17.

The unstable pointing of Kepler throughout K2 required special treatment to reduce the data. The Kepler/K2 mission is characterized by a unique observing strategy that uses spacecraft geometry in conjunction with periodic thruster resets to maintain pointing. This strategy induced a short scale periodic 6 hour ‘sawtooth pattern’, alongside long-term sensitivity trends, due to differential heating on the spacecraft body and zodiacal background throughout a campaign.

To correct for short and long term trends, we first reduce the data using our K2 reduction pipeline which is described in Shaya et al. (2015) in more details. This method removes the sawtooth pattern by fitting a third order polynomial in both spatial dimensions of the centroid of the image to the pattern in the light curve of a fixed 5 x 5 pixel aperture. Long term trends are removed making use of the vectors from our principal-component analysis (PCA) that characterize common simultaneous trends seen in the light curves of all the (assumed) non-varying galaxies observed on the same channel (chip). There were 293 galaxies on the same channel as SN2018agk. For supernovae, the applicable coefficients of the PCA vectors and the sawtooth function are estimated using only the times of no supernova flux at the beginning and/or end of the light curve. In this case, we do not have an end anchor point, so the reduced light curve may become unreliable for times significantly past the peak. This is not a significant hindrance to our analysis, as we only use the K2 light curve during the rise. After the long term trends are divided out, we subtract the sawtooth pattern of the galaxy throughout the light curve. The remainder is the supernova light curve which will generally have a sawtooth pattern of lower amplitude because it is a point source and thus has less of its light shifting in and out of the aperture than the galaxy. The sawtooth pattern can be scaled down to fit this and removed, but, for this supernova, this was not needed.

Finally, to remove the few outliers, we further model the smooth light curve with a Gaussian Process Regression (GPR) through the celerite package (Foreman-Mackey et al., 2017) and apply a $3-\sigma$ cut, as shown in Fig. 2. We used a Matérn 32 kernel with length scales of a few days, which will be insensitive to the short-timescale excursions caused by the K2 thruster resets. In the modelling we explored parameters of the GPR in a reasonable range to balance between rejecting obvious outliers and keeping reliable data in the rise phase for further analysis. From the smoothed light curve we find the peak in K2 band occurs at MJD${}^{K2}_{max}=58203.78\pm 0.02$. To estimate noise, we compute the root-mean-squared variation of the background flux before the explosion and then scale it by the square root of the galaxy flux plus the SN flux in the aperture.

We obtained a total of 17 spectra for SN 2018agk from ground-based observatories: one spectrum with the Gemini Multi-Object Spectrograph (GMOS; Hook et al., 2004) on the Gemini-North telescope (GN-2018A-LP-13, PI: Garnavich); 4 spectra with ESO Faint Object Spectrograph and Camera (EFOSC2; Buzzoni et al., 1984) on the ESO New Technology Telescope (as part of the ePESSTO survey, Smartt et al. 2015); one spectrum with the Kast spectrograph (KAST; Miller & Stone, 1993) on the Lick Shane telescope (2018A-S023, PI: Foley); 2 spectra with the Low-Resolution Imaging Spectrometer (LRIS; Oke et al., 1995) on the Keck I telescope (U240, PI: Foley); one spectrum with the Goodman High Throughput Spectrograph (Clemens et al., 2004) at the Southern Astrophysical Research Telescope (2018A-0277, PI: Foley); 5 optical spectra with the twin FLOYDS spectrographs mounted on Las Cumbres Observatory’s 2-m Faulkes Telescopes North (FTN) at Haleakala Observatory in Hawai’i and Faulkes Telescopes South (FTS) at Siding Spring Observatory in Australia; and 2 spectra with the Low Resolution Spectrograph-2 (LRS2) (Chonis et al., 2016) on the 10m Hobby-Eberly Telescope (HET) at McDonald Observatory, where the UV and Orange arms of the Blue Integral Field Unit were applied resulting in a spectrum between 3640 - 6970 Å. We additionally use the publicly available classification spectrum, posted on the Transient Name Server (TNS), obtained with the Wide-Field CCD (WFCCD) of the 2.5m du Pont Telescope at Las Campanas Observatory (Bose et al., 2018).

The spectra were reduced using standard IRAF/PYRAF and python routines for bias/overscan subtractions and flat fielding. The wavelength solution was derived using arc lamps while the final flux calibration and telluric lines removal were performed using spectro-photometric standard star spectra.

Spectra from Gemini-North, Shane KAST, Keck I LRIS, and SOAR were reduced using standard IRAF/PYRAF¹¹1IRAF was distributed by the National Optical Astronomy Observatory, which was managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation and python routines for bias/overscan subtractions and flat fielding. The wavelength solution was derived using arc lamps while the final flux calibration and telluric lines removal were performed using spectro-photometric standard star spectra. The EFOSC2 spectra were reduced in a similar manner, with the aid of the PESSTO pipeline²²2https://github.com/svalenti/pessto. The HET LRS2 spectra were reduced through custom-developed IRAF scripts, see Yang et al. (2020) for more details on the LRS2 IFU spectrograph and the reduction process. The FLOYDS spectra from Las Cumbres Observatory’s FTS and FTN were reduced using standard IRAF tasks as described in Valenti et al. (2014). Table  3 in Appendix summarizes our spectroscopic observations and Figure 3 shows the evolution of the spectra.

We triggered Hubble Space Telescope ($HST$) followup of SN2018agk on 2018 March 16 (GO-15274, PI Garnavich) and the first exposure began March 19 17:09 (UT). However, due to a $HST$ gyro glitch, the visit failed after the first exposure. A full complement of observations were obtained on the second visit five days later. During both visits, Space Telescope Imaging Spectrograph (STIS) spectra were obtained using the G230L grating and 0.2 arcsec wide slit using the NUV-MAMA detector. On the second visit, CCD spectra using the G430L and G750L gratings were obtained providing wavelength coverage from 170 nm to 1010 nm. The spectra were extracted, wavelength corrected and flux calibrated through the standard STIS pipeline, and are plotted in Fig. 7 in contrast to the HST UV spectra of SN 2011fe taken around the similar phase.

Branch et al. (2006) and Wang et al. (2009) showed that the pseudo-equivalent width (pEW) of the Si absorption lines (Si II $\lambda$6355 and Si II $\lambda$5972) and the velocity of Si II $\lambda$6355 line around peak can be used to classify SNe Ia into different branches. Here we applied this classification schema to SN 2018agk. To estimate pEW, we first define the pseudo-continuum $f_{c}(\lambda)$ by the linear curve connecting local maxima at the edge of the absorption features that doesn’t intersect the spectral features. Then we integrate the flux normalized by the pseudo-continuum through the formula

where $f(\lambda_{i})$ is the measured flux. We perform the integration using scipy.integrate.quad. We also measure the velocity of Si II 6355 Å line by the blueshift of absorption minimum in the spectrum normalized by pseudo-continuum.

We do the calculation on the HST spectrum taken on MJD58201.90, $\sim 2.2$ days prior to the B-band maximum, and measure the pEW of Si II $\lambda$6355 to be $92.621\pm 0.009$Å and pEW of Si II $\lambda$5972 to be $24.1\pm 0.1$Å. The velocity of Si II $\lambda$6355 is $-10200\pm 100$km/s. The Branch diagram is plotted in Fig. 4, in comparison to the sample from Blondin et al. (2012). It clearly shows that SN 2018agk matches the features of branch normal SNe Ia and does not have any peculiar features.

The spectroscopic evolution of SNe Ia forms a rather homogeneous family which allows any individual SN Ia to be studied based on knowledge of the entire SN Ia sample. The data-driven model of the SNe Ia developed by Hu et al. (in press) is applied to the spectroscopic data of SN 2018agk. This model is based on the Long Short-Term Memory (LSTM) neural networks. The model can construct a complete spectral sequence of an SN Ia using spectroscopic data around optical maximum. For normal SNe Ia, the flux level of the entire spectral sequence can be reconstructed to a precision of better than 7%. The evolution of spectral features can also be predicted accurately. The left panel of Figure 5 shows the LSTM neural network projection of the spectral sequence of the SN 2018agk. We picked two spectra with highest S/N around peak, plotted in red color, to build the projections which are plotted as black lines. The rest of spectra with high S/N plotted in green are used to test the fidelity of the neural network projects. The bottom panel shows the ratios of the observations to the neural network projections. The predictions are good to a few percent in general. The phases of the spectra were shown in the color bars on the right. It can be seen that the spectral matches up to the earliest spectroscopic observations (day -9.5 from optical maximum) and down to around 1 month after the optical maximum can be very well reconstructed using only the spectra taken around and after optical maximum (day -2.8 and 9.8 from optical maximum). The success of this reconstruction again supports the notion that SN 2018agk is a member of well-observed SNe Ia that are used to build the LSTM neural network models. We further utilize the LSTM neural network projections to study the evolution of the blueshifts of the Si II $\lambda 6355$ features, as shown in the right panel of Figure 5. The evolution trend of Si II $\lambda 6355$ measured from the projected spectra of SN 2018agk has been plotted in Figure 6 against the projections of other SNe Ia from the same LSTM neural networks. The black line shows the velocity of SN 2018agk measured from the projected spectra and the black solid dots show the measured velocity from the observed spectra. In this Figure we also plot normal SNe Ia (dotted lines) and high velocity group (dashed lines). SN 2018agk shows strong similarity with normal SNe Ia group and validates our conclusion that SN 2018agk is a typical normal-branch SN Ia.

We estimate the host extinction in the line of sight using the method from Poznanski et al. (2012). After correcting our highest resolution spectra for redshift, we fit the Na I D ($\lambda 5890$) doublet and measure equivalent width $EW=0.75\pm 0.15$ Å. Using equation 9 from Poznanski et al. (2012) we estimate $E(B-V)_{\mathrm{host}}=0.11\pm 0.05$.

Differences in SN Ia progenitor metallicity are theoretically expected to affect both the peak luminosity and UV spectral-energy distribution (SED) of a SN Ia while having minimal impact on the optical SED (e.g., Höflich et al., 1999; Lentz et al., 2000). The predicted trends have been observed in SN Ia data (Foley & Kirshner, 2013; Foley et al., 2016, 2020; Graham et al., 2015), showing a difference in UV continuum that correlates with shape-corrected luminosity (Foley et al., 2020) and host-galaxy metallicity (Pan et al., 2020). Since the exact model SED depends on the exact progenitor and explosion model, Foley & Kirshner (2013) used the flux ratio between Lentz et al. (2000) models with different metallicities compared to the flux ratio between different SNe to determine the relative metallicity of the SNe. We compare the reddening-corrected UV spectra of SN 2018agk at two epochs compared to phase-matched spectra of SN 2011fe, a SN Ia with a similar light-curve shape as SN 2018agk (Fig. 7). In general, the UV spectra of two SNe Ia share similar continuum levels and spectral features, with the flux ratio being close to unity at both epochs with variations generally $<$50%, suggesting good agreement given differences in the line features. Considering that SNe 2011fe and 2018agk have similar ejecta velocity evolutions as shown in Fig. 6, similar $\Delta m_{15}$ and rise time (see Section 3.2 and 3.4) and the similarity in their UV SEDs, we can qualitatively conclude that the progenitors of the two SNe Ia had similar metallicity.

We use the sncosmo package to fit Swope and DECam data with the SALT2 model³³3Due to the limited wavelength range, SALT2 can not fit to Kepler light curve of nearby SNe Ia within certain redshift range, including SN 2018agk.. For the Swope B and u band, it is difficult to obtain an accurate photometric calibration using the PS1 $griz$ as other bands, due to the lack of wavelength overlap and poor filter shape match. Therefore we exclude the Swope B and u band light curves from the SALT2 fit. When fitting, we fix parameters that are already determined via other methods: redshift, Milky Way and host extinction. The results are listed in Table 1. The time of peak in B-band is MJD${}^{B}_{peak}=58204.178$, $\sim 0.2$ days earlier than the Kepler maximum. The absolute peak magnitude in B-band is measured to be $M_{B}=-18.950\pm 0.002$, and the decline rate in B-band is $\Delta m_{15}=1.074\pm 0.003$. All the photometric data are plotted in Figure 8 and part of the SALT2 light curve are plotted in Figure 9 . All of SN 2018agk’s SALT2 parameters are well within the range of a normal SN Ia, reinforcing that it is a typical normal SN Ia (Scolnic et al., 2018).

Kepler features a broadband filter that covers the $g,r$ and $i$ bands with a wavelength range of 4183.66 to 9050.23 Å. We calibrate the observed Kepler counts to physical AB magnitudes with the SN 2018agk spectra that fully cover the Kepler bandpass taken on MJD 58194.45, 58196.26, 58201.26 and 58229.17 (see Figure 9). Since there are few ground based observations that coincide with the spectra, we normalise the spectra by mangling it to the SALT2 model $gri$ magnitudes using the mangle_spectrum2 routine from the SNooPy package (Burns et al., 2011). We calculate the synthetic Kepler magnitude for each spectra using the Kepler bandpass available on Spanish Virtual Observatory (SVO) (Rodrigo et al., 2012; Rodrigo & Solano, 2020) and algorithms in the pysynphot package (STScI Development Team, 2013).

With the synthetic Kepler magnitudes, we then calculate the Kepler zero-point using the first three values from rise to peak. We limit the sample to these three points as at late times the Kepler data reduction method breaks down leading to unrealistic count levels (see Figure 9). To avoid biasing from residual saw tooth variability in the Kepler light curve we heavily smooth the light curve with the Savitzky-Golay smoothing method, using a 3rd order polynomial and a window size of 201 frames (100.5 hours). We calculate the zero-point according to:

where $Kp_{syn}$ is the synthetic Kepler magnitude and $C$ is the observed smoothed counts. We find all three points are consistent with $zp=25.219\pm 0.045$.

Kepler provided a single broad-band light curve covering the pre-explosion and rising phase of SN 2018agk starting from MJD 58179.05. With these calibrated data, we determine the onset of the supernova light curve as follows: we define the interval from the start of Kepler observation to $22$ days before $MJD^{K2}_{max}$ as the quiescent background and calculate $3-\sigma$ clipped weighted average and uncertainty $\sigma$ in this interval, and then we mark the time of K2 detection as the time when the GPR smoothed flux (see Section 2.2) rises to $1-\sigma$ above the median flux of the background. We calculate MJD${}_{det}^{K2}=58186.63$($17.14$ days before $t^{peak}_{Kepler}$)as the result. In comparison, the first detection from ground was at MJD$=58186.29\pm 0.02$ in DECam-g band, $\sim 0.3$ days earlier than MJD${}_{det}^{K2}$.

The early rise of our flux-calibrated light curve for SN 2018agk is shown in Fig. 10. We also include the rise of SN 2018oh, which has a clear excess in the first $\sim 5$ days after first light (Dimitriadis et al., 2019a; Shappee et al., 2018; Li et al., 2019). The comparison shows that there is no large excess in SN 2018agk’s Kepler light curve. We then use a series of different power-law models to fit the light curve of SN 2018agk. The form of the basic power-law model is defined as

where $A$ is the scale parameter, $t_{fl}$ is the time of the first light, and $H(t_{fl})$ is the Heaviside function such that $H(t_{fl})=0$ for $t<t_{fl}$ and $H(t_{fl})=1$ otherwise. We set the baseline flux equal to 0 in the fitting since the background flux has been subtracted out in normalization. We replicate the method used in Olling et al. (2015) & Dimitriadis et al. (2019a) and fit to data in the range of $25$ days before peak until the Kepler flux reaches $35\%$ of the peak flux ($\sim$ MJD$58193.18$ or $\sim 11$ days before peak), using the Python package scipy.optimize.least_squares.

The fitting results are listed in Table 2. The BIC of single power-law fit is smaller than that of $t^{2}$ fit by $>8$ and thus is significantly preferrable. Compared with the single power-law fit, double power-law fit does not significantly improve the fitting quality, as $\chi^{2}$ of two fittings are similar, and because of larger degree of freedom, its BIC is much larger. Thus, we can conclude that the Kepler light curve of SN 2018agk can be well-fitted by a single power-law and there is no evidence of excess-like features at the earliest stage. The inferred time of first light is MJD $58185.22$, $\sim 1$ days before the first detection in DECam and Kepler as calculated in Section 3.2.

This result can be further validated when we compare the fit of SN 2018agk with that of SN 2018oh, as shown in Fig. 10. For SN 2018oh we used the result of two component fit (power-law component for supernova flux and skewed Gaussian profile for excess) as described in Dimitriadis et al. (2019a). Comparing the residuals to the power law fits shown in the lower panel make it clear that the excess of SN 2018oh is well beyond the uncertainty limit of the binned light curve of SN 2018agk.

Using the previous fits we calculate the range of color evolution for SN 2018agk. The results in the crucial early phase ($\sim 10$ days after $t_{fl}$) are plotted in Fig. 11, along with DECam and Swope measurement in both g- and i-band in $\lesssim 4$ days and $\gtrsim 7$ days relative to $t_{fl}$ respectively. Overall, the color of SN 2018agk in these 3 bands is relatively constant in the first $\sim 10$ days, with variance $<0.1$ mag excluding the first observation due to the large uncertainty in $g-i$ band. A linear fit to $g-i$ of SN 2018agk gives a change rate of $0.007\pm 0.003$ $\rm mag\;day^{-1}$. Qualitatively speaking, this result agrees with the assumption of the fireball model that the photospheric temperature approximately remains constant when it expands.

We also compare SN 2018agk with other SNe Ia that have early photometric measurements or spectra series. Our sample includes SN2011fe (Pereira et al., 2013), SN 2015F (Cartier et al., 2017), iPTF16abc (Miller et al., 2018), SN 2017cbv (Hosseinzadeh et al., 2017), SN 2018gv (Miller et al., 2020), SN 2018oh (Dimitriadis et al., 2019a; Li et al., 2019; Shappee et al., 2018). In this sample, SN 2011fe, SN 2015F and SN 2018gv are well-observed normal SNe Ia; whereas, SN 2017cbv and SN 2018oh are among few SNe Ia that have been detected with significant excess flux at early times ($\lesssim 5$ days after time of first light), and iPTF16abc has a nearly linear rise in $g$-band with relatively limited amount of data. SN 2018gv has photometric data in $g^{\prime}$ and $i^{\prime}$ band from Sinistro camera at the Las Cumbres Observatory from very early phases. SN 2018oh has measurements in $g$ and $i$ from PanSTARRS within the first day after explosion, but due to the large uncertainty ($\gtrsim 0.3$ mag) we do not include them in the comparison. SN 2011fe lacks $i$-band data, but Pereira et al. (2013) has produced a spectrophotometric time series with high precision, which enables us to calculate synthetic photometry in the $g$, $i$ and Kepler bands. SN 2017cbv has both photometric measurements and spectra in the early phase. The synthetic photometry of SN 2011fe and SN 2017cbv in $g$,$i$ and Kepler bands are calculated through the same routine used in Sec. 3.3.

The color evolution at the earliest phase of SNe Ia can be a crucial piece of evidence in distinguishing different progenitor models, as discussed above. Still, the small sample of high quality measurements in this critical time window limits the application of such studies. Currently, there are not enough SNe Ia light curves with sufficient cadence and S/N at these early epochs to build a reliable SN Ia template⁴⁴4There do exist templates at the earliest stage, e.g. SiFTO (Conley et al., 2008), though the uncertainties are very large due to the small number of objects..

To reveal the prototypical color evolution of normal SNe Ia without excess flux and distinguish them from ones with excess flux, we show the color evolution of SNe Ia with early observations in Figure 11. In general, the normal SNe Ia without early excess (SN 2011fe, SN 2015F, SN 2018gv and SN 2018agk) in our sample have similar colors, with only a small color evolution toward the blue. In $g-i$, the color changes less than $\sim 0.2$ mag in the first 10 days. Remarkably, there is a clear distinction between normal SNe Ia and SN 2017cbv in $g-i$, where the latter is significantly blue not only during the time of excess, but up to 10 days after first light. In addition, the synthetic colors of SN 2017cbv ($g-$Kepler and Kepler$-i$ ) show a similar trend than $g-i$, they are bluer than SN 2011fe. On the other hand, iPTF16abc (with a linear rise in $g$-band) seems to have a similar $g-i$ color curve as normal SNe Ia without excess. However, iPTF16abc shows very different behaviour in $B-V$ color (Stritzinger et al., 2018) and will be discussed later in more detail. Notably, in Kepler$-i$, the earliest high S/N photometric measurement of SN 2018oh at $t\sim 1$ day after first light is significantly bluer than those of SN 2018agk, while the synthetic color of SN 2011fe and SN 2017cbv (empty symbols in Fig. 11) have smaller difference in the earliest stage in this band.

This result is slightly different from previous studies in other optical bands. Previous statistical study on the ZTF SNe Ia sample in $g$ and $i$ bands found relative homogeneity in early color ($g-i\sim 0.15$ mag), along with a large scatter of color slope in $g-r$ for individual events (Miller et al., 2020; Bulla et al., 2020). Stritzinger et al. (2018) has analyzed a sample of 13 SNe Ia with early measurements in $B$ and $V$-band, and find two populations with different color evolution in $B-V$ within the first $\sim 3$ days after explosion (see Figure 2 in Stritzinger et al. (2018)). The ‘early red’ group has redder initial color and turns blue rapidly in $B-V$, with a rate of $\sim 0.5$ magnitude in the first 10 days and has typical luminosity and decline rate among SNe Ia. The ‘early blue’ group is $\sim 0.5$ mag bluer than the ‘early red’ group in $B-V$, evolves at a negligible rate and tends to be brighter than the red group. Spectroscopically, the SNe Ia in ‘early blue’ group tend to have weaker Si II absorption features and lie within or close to the shallow silicon (SS) sub-type in Branch diagram (see Fig.4), while the SNe Ia in the ‘early red’ group belong to either core normal (CN) or cool (CL) type. Qualitatively speaking, the spectral features of SN 2018agk fit into the core normal (CN) class (see Sec. 3.1), and it has intermediate peak brightness and decline rate among the normal SNe Ia sample. Combining all the characteristics, SN 2018agk aligns with the ‘early-red’ events as characterized in Stritzinger et al. (2018). In terms of early color, in $g-i$ the SNe Ia in the ‘early-red’ group (SN 2011fe, SN 2015F and SN 2018agk) are similar to iPTF16abc but are significantly redder than SN 2017cbv, both of which belong to ‘early blue’ group. Meanwhile, all the SNe Ia in our sample evolve with a slow rate no more than $\sim 0.02$ mag per day in the first 10 days in $g-i$, and the rapid drop of the ‘early-red’ group in $B-V$ is not seen.

Foley et al. (2012) analyzed SN 2009ig, a SN Ia discovered hours after explosion with significant $B-V$ color evolution at early times. The redder colors (e.g., $V-R$ and $R-I$) of SN 2009ig do not evolve quickly, indicating that the rapid $B-V$ color change is not caused by a broad wavelength continuum change. Instead, the early spectra of SN 2009ig during this period show Si II 4000 feature that is so blueshifted and broad that it merges with the Ca H&K absorption feature. As a result, the flux at 4000 Å is severely depressed. Over the next few days, corresponding to the time of rapid $B-V$ color evolution, the features become distinct and more similar to the appearance of SN Ia spectra near peak brightness. This strongly indicates that spectral features likely drive these color changes. Such a mechanism may explain the difference in color evolution in different bands seen for the full sample, although a larger and more complete sample of SNe Ia will be necessary to resolve this issue.

In general different models give qualitatively different prediction on the early color of excess in SNe Ia. Overall the comparison between normal SNe Ia sample with SN 2017cbv agrees with fits from the companion-interaction model in Hosseinzadeh et al. (2017) and simulations of the ⁵⁶Ni shell model from Magee & Maguire (2020), both of which can produce a bluer color than the fiducial model without bump. Still, the exact prediction may vary with different model parameters, e.g. the fit to SN 2018oh has a redder color between $\sim 3$-$8$ days after explosion compared to that of SN 2017cbv in Magee et al. (2021). Uncertainties in the fiducial models at the early time also limits our ability to fit models with high fidelity (e.g. see the discussion in Magee et al., 2021).

The $t_{fl}$ of SNe Ia in our sample likely face systematic uncertainies. They are inferred based on the $t^{\alpha}$ fit with slightly different schemes in separate papers, with varying lengths of the dark phase. Nonetheless, as shown in the Fig 11, SNe Ia in our sample only have a very low level of variability in color at the early time. Therefore, the uncertainty in $t_{fl}$ should not have big influence on our conclusion.

High cadence surveys including Kepler and TESS probe a large sample of SNe Ia soon after their explosion, making it possible to study the early light curve of SNe Ia statistically and estimate the fraction of SNe Ia with detectable early excess features. Resolving the detectability of the early excess and potential contamination will be a key component to such studies. Considering the current limitation in fitting with physical models, model-independent excess detection algorithms are necessary for searching abnormal features in the rise of SNe Ia. In this section we present the application of the rolling sum algorithm to evaluate excess flux in early SNe Ia light curves.

With the high quality light curve of SN 2018agk, we try to evaluate the factors influencing efficiency and accuracy of bump-detection quantitatively. We add the excess model onto SN 2018agk’s light curve and evaluate the significance of bump detection with method described in Dimitriadis et al. (2019a), which fits power-law to a time interval well-after the explosion as the baseline light curve and find the excess signal in the residual within days after explosion. We use an excess flux model based on a companion-interaction model fitted to SN 2018oh in Dimitriadis et al. (2019a), with a subgiant companion at a separation of $2\times 10^{12}$cm assuming the supernova ejecta has mass of $1.4M_{\odot}$ and velocity of $8\times 10^{8}$cm s^-1. With the assumption that the luminosity of the bump has a weak dependence of supernova luminosity, we add the excess in physical units before normalizing to the peak of SN 2018agk.

To minimize the degeneracy issue and induced high uncertainty in power-law fitting, we perform the Markov Chain Monte Carlo (MCMC) fitting with the emcee package (Foreman-Mackey et al., 2013) assuming a Gaussian prior distribution for $t_{0}$ and $n$ and a flat prior distribution for scale factor $A$. The mean and standard deviation of $t_{0}$ are set to be $18.7\pm 1.8$ as estimated from Miller et al. (2020), and we use power-law index in $r$-band $n_{r}=2.0\pm 0.5$ to simulate the distribution of $n$ in Kepler bandpass, considering similar effective wavelength of Kepler and $r$-band.

The left panel of Fig.12 shows the power-law fit to Kepler light curve of SN 2018agk (PL1) and the same light curve plus the excess flux model (PL2), along with the residuals of these two power-law fitting. In the right panel of Fig.12 we show the fitting with the same setting on SN 2018oh, with or without excess flux model subtracted. All of the fits are done in the window of $13$ to $10.5$ days before $t^{peak}_{K2}$. One noticeable difference is that in both fits $t_{0}$ inferred from light curves with excess is $\sim 1.5$ days earlier than $t_{0}$ inferred from original light curve. In the residual plot, we show the contrast between the excess model (purple dashed line) and the reconstructed bump after power-law fitting. Clearly, due to the extended tail of collision flux, the power-law fit overestimates the supernova flux and underestimates the excess flux by a large amount after subtraction. This is an indication of the low effectiveness of current excess detection routines. This problem will be exacerbated for ground-based surveys with much larger cadences or similar space telescopes with shallower detection limits (e.g. TESS). Therefore, establishing a robust fiducial model for early light curves of SNe Ia without excess is necessary for future search on early excess. Meanwhile, signatures in either early spectra or color evolution, as analyzed previously, would also be key evidence to facilitate bump detection and distinguish different models.

We adapt the fitting scheme from Section 4.2 to give a qualitative estimate on detectability of the bump and constrain the potential progenitor system for SN 2018agk in the case of SD model. We use the analytic companion interaction model (Kasen, 2009) multiplied with an angular dependency term calculated in Brown et al. (2012) as a test case. We first add excess flux to its Kepler flux, and then fit a power-law to relatively late time interval with MCMC and estimate the detectability of excess in residuals within $\sim 5$ days after explosion. The excess flux is mainly determined by 4 parameters: the orbital separation of the binary $a_{orb}$; viewing angle $\theta$; the velocity $v$; and mass $m_{ejecta}$ of the ejecta. While $\alpha$ and $\theta$ can vary with relatively large ranges, $v$ and $m_{ejecta}$ are usually well constrained for typical SNe Ia in the SD model. We set $v=8\times 10^{8}$cm/s and $m_{ejecta}=1.4M_{\odot}$ in accordance with a typical SNe Ia explosion with kinetic energy of $10^{51}$ erg, and we focus on the effect of different viewing angle and orbital separation.

To evaluate the overall signal of excess on top of the power-law rise, we applied the rolling method with Gaussian window to sum signal-to-noise of the ‘detected excess’ over its extended duration. Considering that the typical duration of the detected bump is $\sim 5$ days, we use a Gaussian with $\sigma=2$ days over a $12$-day window, and then we normalize the results to the rolling sum of a constant background to remove the edge effect in the algorithm. The left column of Fig. 13 shows a set of examples with $\theta=0^{\circ}$ and different orbital separation $a_{orb}$ and the left column shows results for fixed $a_{orb}=3\times 10^{11}$cm at different viewing angles. As can be seen from the plot, this method can effectively take the accumulated signal in an extended interval into account and smooth out the random noise in the light curve at the same time. We conservatively set the detection limit as S/N $\approx 30$ as the dotted line in Fig. 13, which is well above the rolling sum of baseline S/N without excess, i.e. the model with viewing angle $\theta=180^{\circ}$. We run the same algorithm on excess for a large set of different $\theta$ and $a$ and plot the color map of the peak of summed S/N with regard to these two parameters in Fig. 14. The blue and red region represents the regions of detectable and non-detectable excess in the parameter space, and we add a black dashed line marking approximate detection limit for the SD progenitor system of SN 2018agk in Fig. 14

We are able to give a rough constraint on the potential progenitor system of SN 2018agk in the SD model and the detectability of excess for similar events with high cadence light curves. An assumption of the companion interaction model is that the companion overfills the Roche lobe, and the orbital separation $a$ is correlated with the companion radius $R$ and the mass ratio of the binary $q$, approximately following the equation (Eggleton, 1983):

For simplicity we estimate a typical MS star in the WD-MS binary with the mass ranging from $1-20M_{\odot}$ and mark the mass corresponding to certain $a_{orb}$ at the top of Fig. 14, though such binary systems are believed to likely produce less energetic explosions, e.g. nova (Prialnik & Shara, 1986; Shara et al., 1986; Michaely & Shara, 2021). In the more favorable SD models, the companions are usually believed to be sub-giants or red giants (Branch et al., 1995; Hachisu et al., 1996; Wang et al., 2010). Such binary system will have larger $a_{orb}$ for the same mass range, and thus will only be more constrained compared to the WD-MS system analyzed here. As shown in the Fig. 14, a WD-MS binary progenitor system with viewing angle $\theta\lesssim 60^{\circ}$ will produce a prominent excess that is detectable to our rolling algorithm for a typical SN Ia like SN 2018agk in Kepler. When $\theta$ increase to $\gtrsim 120^{\circ}$, the excess becomes undetectable for such WD-MS binary systems. From this perspective, viewing angle seems to be the more dominant factor in the detectability of the bump, and a statistical analysis of SNe Ia with high cadence early light curves from Kepler and TESS will be the key to determine if SD system can be a major channel for SNe Ia or not.