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 60/90 cm Schmidt telescope on Piszkéstető Mountain Station of Konkoly Observatory in BVRI bands, the 1.3 m 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 & Gruendl 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 (Magnier et al. 2020a, 2020b, 2020c; Waters et al. 2020), which includes standard reductions, astrometric solution, stacking of nightly images, source detection, and photometry. The photometry of all images was 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 five days after explosion, making it the first SN Ia that has both a high-cadence Kepler light curve and ground-based multiband observations in its earliest phase. DECam g-band images from well before explosion and ∼20 days after peak are shown in Figure 1(a), and the full multiband 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 hr "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 detail. 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 × 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 3σ cut, as shown in Figure 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 modeling 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 the K2 band occurs at MJD${}_{max}^{{\rm{K}}2}=\text{58,203.78}\pm 0.02$. To estimate noise, we compute the rms 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.

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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); four 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); two 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 (SOAR) Telescope (2018A-0277, PI: Foley); five 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 two spectra with the Low Resolution Spectrograph-2 (LRS2) (Chonis et al. 2016) on the 10 m 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 and 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 2.5 m 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 removal of telluric lines were performed using spectrophotometric spectra of standard stars.

Spectra from Gemini-North, Shane KAST, Keck I LRIS, and SOAR were reduced using standard IRAF/PYRAF ⁵⁶ and Python routines for bias/overscan subtractions and flat-fielding. The wavelength solution was derived using arc lamps while the final flux calibration and removal of telluric lines were performed using spectrophotometric spectra of standard stars. The EFOSC2 spectra were reduced in a similar manner, with the aid of the PESSTO pipeline. ⁵⁷ 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 the Appendix summarizes our spectroscopic observations and Figure 3 shows the evolution of the spectra.

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We triggered Hubble Space Telescope (HST) follow-up of SN2018agk on 2018 March 16 (GO-15274, PI Garnavich) and the first exposure began March 19 17:09 (UT). However, due to an 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 02 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 to 1010 nm. The spectra were extracted, wavelength-corrected, and flux-calibrated through the standard STIS pipeline, and are plotted in Figure 7 in contrast to the HST UV spectra of SN 2011fe taken around a 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 λ6355 and Si ii λ5972) and the velocity of the Si ii λ6355 line around peak can be used to classify SNe Ia into different branches. Here we applied this classification scheme to SN 2018agk. To estimate pEW, we first define the pseudo-continuum f_c (λ) by the linear curve connecting local maxima at the edge of the absorption features that does not intersect the spectral features. Then we integrate the flux normalized by the pseudo-continuum through the formula

$$\begin{eqnarray}&&{\rm{pEW}}=\int \displaystyle \frac{{f}_{c}({\lambda }_{i})-f({\lambda }_{i})}{{f}_{c}({\lambda }_{i})}{dx}\end{eqnarray} \tag{ 1 }$$

where f(λ_i ) is the measured flux. We perform the integration using scipy.integrate.quad. We also measure the velocity of the Si ii 6355 Å line by the blueshift of the absorption minimum in the spectrum normalized by the pseudo-continuum.

We do the calculation on the HST spectrum taken on MJD 58,201.90, ∼2.2 days prior to the B-band maximum, and measure the pEW of Si ii λ6355 to be 92.621 ± 0.009 Å and that of Si ii λ5972 to be 24.1 ± 0.1 Å. The velocity of Si ii λ6355 is −10,200 ± 100 km s⁻¹. The Branch diagram is plotted in Figure 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.

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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 2021) 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 SN 2018agk. We picked two spectra with the highest signal-to-noise ratio (S/N) around peak, plotted in red, to build the projections, which are plotted as black lines. The remaining spectra with high S/N plotted in green are used to test the fidelity of the neural network projections. 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 are shown in the color bars on the right. It can be seen that the spectral predictions match 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 (days −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 λ6355 features, as shown in the right panel of Figure 5. The evolution trend of Si ii λ6355 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 the high-velocity group (dashed lines). SN 2018agk shows a strong similarity with the group of normal SNe Ia and validates our conclusion that it is a typical Branch-normal SN Ia.

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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 (λ5890) doublet and measure equivalent width EW = 0.75 ± 0.15 Å. Using Equation (9) from Poznanski et al. (2012), we estimate $E{\left(B-V\right)}_{\mathrm{host}}=0.11\pm 0.05$.

Differences in SN Ia progenitor metallicity are theoretically expected to affect both the peak luminosity and UV spectral energy distribution (SED) of an SNe 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 SNe Ia data (Foley & Kirshner 2013; Graham et al. 2015; Foley et al. 2016, 2020) and show 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 to phase-matched spectra of SN 2011fe, an SN Ia with a similar light-curve shape to SN 2018agk (Figure 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 and variations generally <50%, suggesting good agreement given differences in the line features. Considering that SNe 2011fe and 2018agk have similar evolutions of ejecta velocity as shown in Figure 6, similar Δm₁₅ and rise time (see Sections 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.

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We use the sncosmo package to fit Swope and DECam data with the SALT2 model. ⁵⁸ For the Swope B and u bands, it is difficult to obtain as accurate a photometric calibration using the PS1 griz as for other bands, due to the lack of wavelength overlap and poor match of filter shape. Therefore we exclude the light curves for the Swope B and u bands from the SALT2 fit. When fitting, we fix parameters that are already determined via other methods: redshift, and Milky Way and host extinction. The results are listed in Table 1. The time of the peak in the B band is ${{\rm{MJD}}}_{{\rm{peak}}}^{B}=\text{58,204.178}$, ∼0.2 days earlier than the Kepler maximum. The absolute peak magnitude in the B band is measured to be M_B = −18.950 ± 0.002, and the decline rate in the B band is Δm₁₅ = 1.074 ± 0.003. All the photometric data are plotted in Figure 8, and part of the SALT2 light curve is 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).

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Kepler features a broadband filter that covers the g, r, and i bands with a wavelength range of 4183.66–9050.23 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 58,194.45, 58,196.26, 58,201.26, and 58,229.17 (see Figure 9). Since there are few ground-based observations that coincide with the spectra, we normalize each spectrum 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 spectrum using the Kepler bandpass available on the 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 because at late times the Kepler data reduction method breaks down, leading to unrealistic count levels (see Figure 9). To avoid biasing from residual sawtooth variability in the Kepler light curve we heavily smooth the light curve with the Savitzky–Golay smoothing method, using a third-order polynomial and a window size of 201 frames (100.5 hr). We calculate the zero-point according to

$$\begin{eqnarray}&&{zp}={{Kp}}_{\mathrm{syn}}+2.5\,\mathrm{log}(C),\end{eqnarray} \tag{ 2 }$$

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 ± 0.045.

Kepler provided a single broadband light curve covering the pre-explosion and rising phase of SN 2018agk and starting from MJD 58,179.05. 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 ${\mathrm{MJD}}_{\max }^{{\rm{K}}2}$ as the quiescent background and calculate the 3σ clipped weighted average and uncertainty σ 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 1σ above the median flux of the background. We calculate ${{\rm{MJD}}}_{det}^{{\rm{K}}2}=\text{58,186.63}$ (17.14 days before t^peak _Kepler) as the result. In comparison, the first detection from ground was at MJD = 58,186.29 ± 0.02 in the DECam g band, ∼0.3 days earlier than MJD${}_{\det }^{{\rm{K}}2}$.

The early rise of our flux-calibrated light curve for SN 2018agk is shown in Figure 10. We also include the rise of SN 2018oh, which has a clear excess in the first ∼5 days after first light (Shappee et al. 2018; Li et al. 2019; Dimitriadis et al. 2019a). 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

$$\begin{eqnarray}&&f(t)=H({t}_{{\rm{fl}}})A{\left(t-{t}_{{\rm{fl}}}\right)}^{\alpha },\end{eqnarray} \tag{ 3 }$$

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) and Dimitriadis et al. (2019a) and fit to data in the range from 25 days before peak until the Kepler flux reaches 35% of the peak flux (∼MJD 58,193.18 or ∼11 days before peak), using the Python package scipy.optimize.least_squares.

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To test whether SN 2018agk follows the traditional "expanding fireball" model or a more general power-law rise, and to examine whether a bump feature exists in SN 2018agk, we fit the Kepler light curve with three models: a fixed single power-law, a t² rise leaving t_fl as a free parameter, and two power laws with different t_fl, ${t}_{{\rm{fl}}}^{{\rm{{\prime} }}}$ and α, ${\alpha }^{{\prime} }$. In order to judge the goodness of power-law fits with different numbers of parameters, we apply the Bayesian information criterion (BIC), which is a model selection technique based on the goodness of fit and number of model parameters. As presented in Priestley (1981), the BIC for a given fit can be calculated as

$$\begin{eqnarray}&&{\mathtt{BIC}}=k\mathrm{ln}(n)+n\mathrm{ln}({\hat{\sigma }}^{2})\end{eqnarray} \tag{ 4 }$$

where k is the number of parameters in the model, n is the number of data points in the sample, and ${\hat{\sigma }}^{2}={\chi }^{2}/n$ is the variance, which is defined as the normalized mean of the squared differences between the data and the model. A model with smaller BIC is preferred in terms of balance between the agreement with data and the complexity of the model. Normally, a difference of >6 between BIC values indicates a statistically significant advantage.

The fitting results are listed in Table 2. The BIC of a single power-law fit is smaller than that of t² fit by >8 and thus is significantly preferable. Compared with the single power-law fit, a double power-law fit does not significantly improve the fitting quality, because χ² of the two fittings are similar, and because its BIC is much larger as a result of a larger degree of freedom. 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 58,185.22, ∼1 day before the first detection in DECam and Kepler as calculated in Section 3.2.

Table 2. Different Results of Power-law Fitting to the Kepler Light Curve of SN 2018agk

Model	tfl	${t}_{{\rm{fl}}}^{{\rm{{\prime} }}}$	α	${\alpha }^{{\prime} }$	A (× 103)	${A}^{{\rm{{\prime} }}}\,(\,\times {10}^{3})$	f0 (× 104)	χ2	BIC
t2 rise	−17.53 ± 0.03	⋯	2	⋯	6.79 ± 0.03	⋯	−3.2 ± 4.2	497.0	−3880.5
tα rise	−18.09 ± 0.16	⋯	2.25 ± 0.07	⋯	3.53 ± 0.06	⋯	−7.6 ± 4.3	478.5	−3896.7
Double tα rise	−14.94 ± 0.31	−17.05 ± 0.18	1.79 ± 0.11	1.14 ± 0.20	11.4 ± 2.8	20.3 ± 4.6	−7.3 ± 4.1	462.6	2257.3

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This result can be further validated when we compare the fit of SN 2018agk with that of SN 2018oh, as shown in Figure 10. For SN 2018oh we used the result of a 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 makes 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 (∼10 days after t_fl) are plotted in Figure 11, along with DECam and Swope measurements in both g and i bands made ≲4 days and ≳7 days relative to t_fl respectively. Overall, the color of SN 2018agk in these three bands is relatively constant in the first ∼10 days, with variance <0.1 mag, excluding the first observation due to the large uncertainty in the g − i band. A linear fit to g − i of SN 2018agk gives a rate of change of 0.007 ± 0.003 mag day⁻¹. Qualitatively speaking, this result agrees with the assumption of the fireball model that the temperature of the photosphere remains approximately constant when it expands.

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We also compare SN 2018agk with other SNe Ia that have early photometric measurements or series of spectra. 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), and SN 2018oh (Shappee et al. 2018; Li et al. 2019; Dimitriadis et al. 2019a). 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 (≲5 days after time of first light), and iPTF16abc has a nearly linear rise in the g band with a relatively limited amount of data. SN 2018gv has photometric data in the ${g}^{{\prime} }$ and ${i}^{{\prime} }$ bands from the Sinistro camera at the Las Cumbres Observatory from very early phases. SN 2018oh has measurements in g and i from Pan-STARRS within the first day after explosion, but due to the large uncertainty (≳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 the g, i, and Kepler bands is calculated through the same routine used in Section 3.3.

The color evolution in 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 template. ⁵⁹

To reveal the prototypical color evolution of normal SNe Ia without excess flux and distinguish them from those 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 in our sample (SN 2011fe, SN 2015F, SN 2018gv, and SN 2018agk) have similar colors, with only a small color evolution toward the blue. In g − i, the color changes less than ∼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 to g − i; they are bluer than SN 2011fe. On the other hand, iPTF16abc (with a linear rise in the g band) seems to have a similar g − i color curve to normal SNe Ia without excess. However, iPTF16abc shows very different behavior 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 ∼ 1 day after first light is significantly bluer than those of SN 2018agk, while the synthetic colors of SN 2011fe and SN 2017cbv (empty symbols in Figure 11) have a 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 SNe Ia sample from the Zwicky Transient Facility in g and i bands found relative homogeneity in early color (g − i ∼ 0.15 mag), along with a large scatter of color slope in g − r for individual events (Bulla et al. 2020; Miller et al. 2020). Stritzinger et al. (2018) analyzed a sample of 13 SNe Ia with early measurements in B and V bands, and found two populations with different color evolution in B − V within the first ∼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, at a rate of ∼0.5 mag in the first 10 days, and has a typical luminosity and decline rate among SNe Ia. The "early blue" group is ∼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 the "early blue" group tend to have weaker Si ii absorption features and lie within or close to the shallow silicon (SS) subtype in the Branch diagram (see Figure 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 CN class (see Section 3.1), and it has an 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 the "early blue" group. Meanwhile, all the SNe Ia in our sample evolve at a slow rate of no more than ∼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, an 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 change in B − V color is not caused by a change in the broad wavelength continuum. Instead, the early spectra of SN 2009ig during this period show an Si ii 4000 Å feature that is so blueshifted and broad that it merges with the Ca H and 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 predictions on the early color excess in SNe Ia. Overall the comparison between the 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 a bump. Still, the exact prediction may vary with different model parameters, e.g., the fit to SN 2018oh has a redder color ∼3–8 days after explosion than SN 2017cbv in Magee et al. (2021). Uncertainties in the fiducial models at the early time also limit our ability to fit models with high fidelity (e.g., see the discussion in Magee et al. 2021).

t_fl of SNe Ia in our sample likely faces systematic uncertainties. They are inferred based on the t^α fit with slightly different schemes in separate papers, with varying lengths of the dark phase. Nonetheless, as shown in Figure 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 a big influence on our conclusion.

High-cadence surveys including those by Kepler and the Transiting Exoplanet Survey Satellite (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, algorithms to detect model-independent excess are necessary for searching for 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 the method described in Dimitriadis et al. (2019a), which fits a power law to a time interval well after the explosion as the baseline light curve and finds the excess signal in the residual in the 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 × 10¹² cm, assuming the supernova ejecta has a mass of 1.4 M_⊙ and velocity of 8 × 10⁸ cm s⁻¹. With the assumption that the luminosity of the bump has a weak dependence on supernova luminosity, we add the excess in physical units before normalizing to the peak of SN 2018agk.

To minimize the degeneracy issue and consequent large 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 t₀ and n and a flat prior distribution for scale factor A. The mean and standard deviation of t_fl are set to be 18.7 ± 1.8 days as estimated from Miller et al. (2020), and we use a power-law index in the r band n_r = 2.0 ± 0.5 to simulate the distribution of n in the Kepler bandpass, considering the similar effective wavelengths of the Kepler and r bands.

The left panel of Figure 12 shows the power-law fit to the 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 fittings. In the right panel of Figure 12 we show the fitting with the same setting on SN 2018oh, with or without the excess flux model subtracted. All of the fits are done in the window of 13–10.5 days before ${t}_{{\rm{peak}}}^{{\rm{K}}2}$. One noticeable difference is that in both fits t₀ inferred from light curves with excess is ∼1.5 days earlier than t₀ inferred from the 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 routines to detect excess. 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 searches for 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.

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We adapt the fitting scheme from Section 4.2 to give a qualitative estimate of the detectability of the bump and constrain the potential progenitor system for SN 2018agk in the case of the SD model. We use the analytic companion-interaction model (Kasen 2009) multiplied by an angular dependence 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 a relatively late time interval with MCMC and estimate the detectability of excess in residuals within ∼5 days after explosion. The excess flux is mainly determined by four parameters: the orbital separation of the binary a_orb, the viewing angle θ, the velocity v, and the mass m_ejecta of the ejecta. While α and θ can vary over relatively large ranges, v and m_ejecta are usually well constrained for typical SNe Ia in the SD model. We set v = 8 × 10⁸ cm s⁻¹ and m_ejecta = 1.4 M_⊙ in accordance with a typical SNe Ia explosion with kinetic energy of 10⁵¹ erg, and we focus on the effect of a 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 a Gaussian window to sum the signal-to-noise ratio of the "detected excess" over its extended duration. Considering that the typical duration of the detected bump is ∼5 days, we use a Gaussian with σ = 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 Figure 13 shows a set of examples with θ = 0° and different orbital separations a_orb and the top column shows results for fixed a_orb = 2 × 10¹¹ 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 ≈ 30 as the dotted line in Figure 13, which is well above the rolling sum of baseline S/N without excess, i.e., the model with viewing angle θ = 180°. We run the same algorithm on excess for a large set of different θ and a and plot the color map of the peak of summed S/N with regard to these two parameters in Figure 14. The blue and red regions represent the regions of detectable and undetectable excess in the parameter space, and we add a black dashed line to mark the approximate detection limit for the SD progenitor system of SN 2018agk in Figure 14

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We are able to set 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 its 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)

$$\begin{eqnarray}&&{a}_{\mathrm{orb}}=\displaystyle \frac{0.6{q}^{2/3}+\mathrm{ln}(1+{q}^{1/3})}{0.49{q}^{2/3}}R.\end{eqnarray} \tag{ 5 }$$

For simplicity we estimate a typical main-sequence (MS) star in a WD–MS binary with mass ranging from 1 to 20 M_⊙ and mark the mass corresponding to certain a_orb at the top of Figure 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 subgiants or red giants (Branch et al. 1995; Hachisu et al. 1996; Wang et al. 2010). Such a binary system will have larger a_orb for the same mass range, and thus will only be more constrained than the WD–MS system analyzed here. As shown in Figure 14, a WD–MS binary progenitor system with viewing angle θ ≲ 60° will produce a prominent excess that is detectable by our rolling algorithm for a typical SN Ia such as SN 2018agk in Kepler. When θ increases to ≳120°, 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 determining whether SD systems can be a major channel for SNe Ia or not.