Constraints on the Physical Properties of GW190814 through Simulations Based on DECam Follow-up Observations by the Dark Energy Survey

We performed ToO follow-up observations of GW190814 0, 1, 2, 3, 6, and 16 nights following the LVC alert. The early nights were chosen to look for rapidly evolving transients immediately following the merger, and the observations 16 nights after the merger were used to exclude persisting supernovae. The observing conditions for each night are displayed in Table 1.

The moon was full on the first night of the observations, so we opted to use the redder i and z bands to minimize the effect of moon brightness on our imaging depth. On the night of the merger, we tiled 99% percent of the 38 sq deg localization region using 60 s exposures in i and 90 s exposures in z. The z exposures were offset by half the width of a DECam CCD to fill in chip gaps. We tiled the area a second time in i to identify moving objects. On the following observing nights, because the LVC had published a smaller localization region, we lengthened our exposures to 100 s in i and 200 s in z. Throughout the real-time observations, we coadded images that shared the same night and filter to increase the search depth. The i-band DECam pointings are shown atop the LVC localization probability contours in Figure 1. All DECam images were immediately made public and available for download from the National Science Foundation's NOIRLab.

The DECam images were processed by the DES Difference Imaging Pipeline, an updated version of the DES Supernova Program's Pipeline described in Kessler et al. (2015), using coadded DES wide-field survey images as templates. The updated pipeline is described in detail in Herner et al. (2020) and has been used in a variety of multimessenger applications (Soares-Santos et al. 2016, 2017; Doctor et al. 2019; Morgan et al. 2019). We briefly describe our pipeline below.

We apply standard image correction and astrometric calibration to our DECam images (Morganson et al. 2018), and subtract them from existing template images. We utilize the GAIA-DR2 catalog (Gaia Collaboration et al. 2016) to perform astrometric calibration in order to reach astrometric uncertainties smaller than 003. After image subtraction, we use SExtractor (Bertin & Arnouts 1996) to locate sources in the difference images, which correspond to objects with varying brightness between the times of the template observations and the recent images. We then obtain forced photometry at the locations of detected sources in the difference images using point-spread function (PSF) fitting in which previous or future epochs have no detections. Last, we apply an ML-based image artifact identification tool, autoscan (Goldstein et al. 2015), to the difference images in order to assign each detection a probability of being real as opposed to an artifact created by astrometric misalignment, hot/dead pixels, unidentified cosmic rays, etc. All candidates found by the DESGW pipeline are listed in Table 2.

We match each candidate to a host galaxy from the DES Y3 galaxy catalog. After removing contaminants (subtraction artifacts, variable stars, moving objects, etc.) from our sample using criteria 1–5 described in Section 4, every candidate is able to be matched to a host in the DES Y3 galaxy catalog. Properties and redshifts of the hosts are reported in Table 3. Photometric redshifts have been computed using Directional Neighborhood Fitting (DNF; De Vicente et al. 2016), while the galaxy properties have been computed using the method described in Palmese et al. (2020). The DNF method is known to be inaccurate at the redshifts relevant in this analysis, due to the characteristics of the galaxy sample upon which the algorithm was trained. The inaccuracy manifests in our analysis as underestimated host-galaxy photometric redshift uncertainty. We therefore add a minimum uncertainty of 0.02 for galaxies with host-galaxy photometric redshift less than 0.1, following the prescription of Soares-Santos et al. (2019d). The galaxies have been ranked from highest to lowest probability per unit volume based on their angular position and redshift as prescribed in Singer et al. (2016), assuming a flat ΛCDM cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹ and Ω_m = 0.3.

Table 3.  Host Galaxy Properties of the Two Objects Passing All Selection Criteria Prior to Final ML Classification

DESGW ID	Host Gal. Name	Angular Sep.	Physical Sep.	Redshift	$\mathrm{log}{M}_{* }$	$\mathrm{log}\,\mathrm{SFR}$	Mi
		(arcsec)	(kpc)		$(\mathrm{log}{M}_{\odot })$	$(\mathrm{log}M\odot \,{\mathrm{yr}}^{-1})$
666914	DES J013624.60-344557.72	3.345	6.24	0.10 ± 0.02a	9.90	−0.0386	−20.70
661188	DES J005431.17-241713.08	4.700	9.53	0.11 ± 0.02a	10.07	0.0438	−20.94

Note.

^aThe minimum host-galaxy photometric redshift uncertainty value has been utilized.

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The following selection criteria assure satisfactory detection and image-subtraction quality in all remaining candidates. We introduce two definitions to expedite discussion. A Type 2 detection is a SExtractor detection in a single filter that does not contain any image processing errors, such as an inability to measure a fitted flux, the R.A. or decl. of an object not being on a CCD, masking of bright objects overlapping the transient object, the inability to fit the PSF of the object, the inability to make a stamp in the difference image, a large number of pixels with negative flux values, or a 5σ difference between PSF flux fitting and aperture flux fitting. A Type 1 detection is a Type 2 detection that has also been given an autoscan score of 0.7 or larger.

Criterion 1. We require candidates to have at least one Type 1 detection. This criterion ensures a high-purity sample of real objects with little contamination from image processing artifacts.

Criterion 2. We require a second detection in the light curve of Type 2 or Type 1, and we require this secondary detection to be on a different night from the detection in Criterion 1. By ensuring a second detection that is separated in time from the first detection, we remove all moving objects from our sample. This temporal separation could in principle be shortened to ∼1 hr, but because we coadded our images from the same night and band, this time separation requirement is effectively a multinight requirement. In these observations, we find that fast-fading transients such as KNe have a high efficiency of 93% for this multinight requirement, based on the simulations discussed in Section 5. We also relax the required autoscan score of the second detection, as the first Type 1 detection from Criterion 1 has already yielded a high-purity sample.

After the level 1 quality criteria, we are left with 2192 candidates in our sample. This sample is mostly composed of astrophysical objects with observed variable brightness as a result of the quality criteria. At this stage, there remains a large population of artifacts that passed the selection criteria, but these are removed by Criterion 5.

With the exception of artifacts, we expect the remaining sample to be dominated by three main contaminants at this stage: variable foreground stars, AGN, and bright galactic centers. The latter is a known problem in difference imaging; see Kessler et al. (2015) or Doctor et al. (2017) for context.

Criterion 3. We require that each object is separated from known foreground objects. This requirement has two components: each object must be separated from objects in a high-purity sample of well-measured stars in the DES Y3 Gold catalog by at least 05, and each object must be separated at least 8' from NGC 288 and 3' from HD4398. The globular cluster NGC 288 has a high density of bright stars, and HD4398 itself is a very bright star. Both of these factors led to large numbers of subtraction artifacts and variable star detections by our Search and Discovery Pipeline.

Criterion 4. We require that each object is at least 02 from objects in the DES Y3 Gold catalog that are not flagged as well-measured stars, which were addressed in Criterion 3. This criterion aims to remove AGN and bright galactic centers. Section 5.2 gives physical and empirical motivations for expecting KNe to be highly likely to satisfy this requirement.

Criterion 5. We visually inspect images of the 1872 remaining candidates. We remove candidates that have an imaging artifact from a misaligned subtraction or from inadequate masking, and we also remove all candidates that contained a point-like light source in the template image at the location of the detected transient. In the application of this criterion in general, the seeing of the observations can limit the efficiency of real transients, since extremely poor seeing could potentially make a bright host galaxy center appear as a point source. Our average seeing in these observations, shown by the PSF FWHM column in Table 1, is less than 13 on more than half of the nights. We therefore expect this behavior to be rare in our data.

After the level 2 catalog criteria, we are left with 116 candidates in our sample. At this stage, we expect that our data are almost entirely constituted by real astrophysical transients.

The following selection criteria are designed to remove supernovae by assuring the distance of the candidates is consistent with the LVC distance posterior distribution, requiring the light curves of the candidates are fading, and triggering spectroscopic follow-up observations.

Criterion 6. We require each object to have a host-galaxy photometric redshift consistent with the mean and standard deviation of the LVC distance posterior at the 3σ confidence level. All objects were able to matched with a host galaxy at this stage, so the criterion can be straightforwardly applied. The criterion is satisfied when

$$\begin{eqnarray}&&\displaystyle \frac{| {z}_{\mathrm{LVC}}-{z}_{\mathrm{DES}}| }{\sqrt{{\sigma }_{z,\mathrm{LVC}}^{2}+{\sigma }_{z,\mathrm{DES}}^{2}}}\lt 3,\end{eqnarray} \tag{ 1 }$$

where z_LVC = 0.06 is the redshift of GW190814, z_DES is the redshift of a candidate's host galaxy, ${\sigma }_{z,\mathrm{LVC}}=0.005$ is the uncertainty on the redshift of GW190814, and ${\sigma }_{z,\mathrm{DES}}$ is the uncertainty on the redshift of a candidate's host galaxy. To implement this criterion, we utilize the assumed cosmology in this analysis. In the case of an available spectroscopic redshift of the host galaxy, we utilize the spectroscopic information instead. Since supernovae could be detectable out to large redshifts in these observations, we seek to remove contaminants in galaxies too distant to be associated with the GW signal.

Criterion 7. If an object is detected on the final night of observations (16 nights post-merger) we require that it be fainter than 22.5 mag in at least one band. If an object is not detected on the 16th night, it passes this criterion. This criterion removes rising and flat light curves from our candidate list.

Criterion 8. We trigger spectroscopic follow-up observations from the Southern Astrophysical Research (SOAR; Sebring et al. 2003) telescope on as many of the eight remaining candidates as possible. We also incorporate real-time spectroscopic classifications from other instruments during the follow-up based on circulars posted to the GCN. The spectroscopic instruments were triggered in real time, as opposed to after the selection criteria had been refined in the offline analysis, so there is not perfect overlap between the targeted objects and the remaining candidates presented in this work. All targeted candidates were spectroscopically confirmed as SNe.

After the previous eight criteria have been enforced, we have two remaining candidates as shown in Figure 2. As described in Appendix, we apply light-curve-based ML classification to determine the probability that any of these objects are potentially a KN. Briefly, we fit a large set of simulated SNe (both SNe-Ia and SNe-CC) and KNe (from the Kasen et al. (2017) models) light curves that pass Criteria 1 through 7 with a Bayesian SN template fitting tool PSNID (Sako et al. 2011), select the template features and goodness-of-fit metrics with the largest difference in mean value for SN and KN samples, and build a random forest classifier (Breiman 2001) using those best-fit parameters as features. This PSNID+RFC approach shows a significant improvement in classification power when using the KN false-positive rate and KN true-positive rate as diagnostics. A similar version of this method is described in Morgan et al. (2019). Figure 3 shows the performance of this machine-learning approach and the resulting probabilities of each remaining candidate being a KN. DESGW-666914 has a 0.92 probability of being an SN and DESGW-661188 has a 0.86 probability of being an SN from our PSNID+RFC approach, both of which are classified as SNe based on our choice of operating threshold. Six additional candidates that made it to this stage and were later spectroscopically typed as SNe were correctly classified as SNe by our PSNID + RFC approach.

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Table 4 shows the number of candidates remaining after each criterion. After all selection criteria have been applied and the remaining candidates have been photometrically classified, zero candidates remain. We therefore use our data to set upper limits on KN properties given a nondetection and to inform future follow-up observations.

SNANA enables the simulation of light curves of SNe, KNe, and other transients as they would be measured by DECam during observations. This process uses a measured or theoretical time-evolving spectrum for the transient object and then accounts for cosmological redshift, Milky Way dust extinction, and the measured observing conditions of the DECam observations, such as sky brightness, zero points, the point-spread function of the imager, and CCD noise in the camera. The corrected time-evolving spectra are then multiplied by the transmission of the DECam filters, and light curves are sampled at epochs matching the cadence of the observations.

The KN models used in the simulations are from spectral energy distributions derived in Kasen et al. (2017), and parameterize the optical light from a KN by the mass ejected in the explosion, the abundance of lanthanide elements in the ejecta, and the velocity of the ejecta (hereafter M_ej, X_lan, and v_ej). These models were chosen because they characterize the optical behavior based on physical properties of the NS ejecta, rather than having a dependence on the geometry or dynamics of the merger itself. While other models for KNe and models specific to NSBH mergers exist (Barbieri et al. 2019; Bulla 2019; Hotokezaka & Nakar 2020, among multiple others), we find this simple, agnostic, three-component model based on observable properties of NS ejecta to apply well to GW190814. In the simplicity of this approach, we make the assumption of either spherically symmetric emission or that the particular component being considered is directed toward Earth. We utilize 329 total models, which discretize the parameter space in the ranges 0.001 M_⊙ ≤ M_ej ≤ 0.1 M_⊙, 0.03c ≤ v_ej ≤ 0.3c, and $1\times {10}^{-9}\leqslant {X}_{\mathrm{lan}}\leqslant 1\times {10}^{-2}$. The simulated KNe are uniformly drawn from this population of models, though the grid that discretizes the parameter space is nonuniform, as shown in Figure 4. This nonuniform grid is not believed to have an effect on the physical constraints, insofar as the interpolation of model efficiencies between points in the grid is smooth and monotonic. As a proxy for the observer-frame explosion time of the simulated KNe, we fix the time the KNe fluxes reach 1% of their peak flux to the time of the LVC GW alert and note that this approximation is justified by the rapid rise times of the KNe. The simulated KNe are also distributed in redshift according to a polynomial fit of the LVC distance posterior and the cosmology used in this analysis. The redshift distribution is constructed independent of spatial information on the sky. This approximation is based on the small localization area of GW190814; however, for future events with larger localization areas, the volume-rendered luminosity distance distribution should be utilized.

Because our selection criteria effectively remove all moving objects, known foreground variable stars, and AGN, the most likely remaining contaminants in our data are SNe. We therefore use SN simulations to understand the types of SNe passing our selection criteria, as well as the number expected be present in our final candidate sample. We simulate Type Ia SNe (SNe-Ia) using templates from Guy et al. (2010) and measured volumetric rates from Dilday et al. (2008). We also simulate core-collapse SNe (SNe-CC) using templates from Kessler et al. (2010) and volumetric rates from Li et al. (2011). The SNe-CC population includes Type Ib, Type Ic, Type Ibc, Type IIP, Type IIN, and Type IIL SNe, and we weight the different subtypes according to their measured volumetric rates. Unlike the KN sample, we allow the SNe to have a random observer-frame explosion time that would make them bright enough to observe with DECam during our observing window. This explosion time range is implemented by requiring the date of peak flux to be greater than 60 days prior to the LVC GW alert and less than 30 days after it, because the explosion time itself is not well-measured.

SNANA produces catalog-level photometric fluxes for transient objects by correcting model spectral energy distributions and multiplying them with the DECam filters, and this approach bypasses several image processing and catalog matching steps that we apply to the real DECam data. We therefore take additional steps to impute information necessary for modeling the selection criteria in this analysis on the simulated light curves.

In our real-time analysis, we applied selection requirements on the autoscan score and SExtractor detection flag. Both of these programs run at the image level, so their information is not present in SNANA-simulated light curves. We adopt the empirical approach from Doctor et al. (2017) to determine realistic values for autoscan and SExtractor quantities in the simulations. This process involves inserting simulated point-source objects of known brightness (hereafter "fakes") into the real DECam images, and applying our image processing pipeline to the images to record the autoscan score and SExtractor detection flag. From the processed fake objects, we extract the probability mass functions (pmfs) for the autoscan score and SExtractor detection flag at discrete levels of signal-to-noise ratio ranging from 0.5 to 50.0. Each filter is treated independently when extracting the pmfs. In the process of generating simulations, based on the signal-to-noise ratio of each observation, values for the autoscan score and SExtractor detection flag are drawn from the corresponding empirically derived pmf. We also introduce a reduced correlation coefficient of 0.1 to the drawn autoscan scores for observations of the same object, determined so that the simulations accurately reflect the fake data.

Level 2 of our selection criteria rules out known objects by matching to the DES Y3 Gold Catalog. When matching to DES stars, globular clusters NGC 288, and the star HD4398, we estimate the sky area masked by our criteria using a Monte Carlo sampling of position space. We find that a 05 radius around DES stars masks 0.11% of the sky area covered in our follow-up observations, and an 8' radius around NGC 288 and a 3' radius around HD4398 each mask 0.01%. In the simulations, we use these percentages of the sky masked by these selection criteria as the probability for a simulated object to be removed by the criterion.

We take a slightly different approach to modeling the criterion of removing known galactic centers from our sample, since these objects are not in the foreground of our observations. Here, we model the transient–galaxy separation empirically and impute that separation into the simulations. We extract a probability distribution function of SN–host galaxy center separation in units of physical distance from the DES 3 yr spectroscopic SNe sample (DES Collaboration 2018). This sample is dominated by SN-Ia for cosmological analyses, which makes it more applicable to KN–host galaxy separation than a balanced SNe sample: the progenitors of SN-Ia are thought to be white dwarf stars in binary systems (Woosley & Weaver 1986; Hillebrandt & Niemeyer 2000; Maoz et al. 2014), meaning to first order they would be similar in age and hence host separation to other binary systems of stellar remnants (Bloom et al. 2006; Prochaska et al. 2006). We believe this assumption to be conservative, given that supernova (or sometimes called "prenatal") kicks during the evolution of binary massive star systems into BNS or NSBH systems are expected to cause an increase in the separation from the host-galaxy center (Stairs et al. 2006). We therefore apply the same transient–galaxy separation probability distribution function (pdf) to both the KNe and SNe simulations. In the application of the selection criterion, we draw a separation from the pdf and remove the object if the separation is less than 02. When testing this criterion on the DES 3 yr SNe sample, we estimate 97% of transients will be recovered while effectively removing all time-varying galactic centers.

Our real-time candidate reduction also relied on visual inspection of the images to remove artifacts and point-like light sources without a host galaxy. We assume near-perfect efficiency in the simulations, with one exception stemming from the fact that this criterion has a dependence on the seeing of the observations. A bright galaxy center in poor seeing conditions can hide real transients in the image or appear like a point source itself, resulting in it being removed from the sample. For the simulations, if the imputed host separation is less than half of the seeing, we reject the simulated object.

The final pieces of additional information that were necessary to add to the SNANA simulations were photometric redshifts and photometric redshift errors. Here, we take the i-band galaxy magnitudes of all galaxies in the DES Y3 Gold catalog also in the LVC 90% containment region in order to empirically determine the i-band magnitude pdf in several redshift bins. Using the true simulated redshift of our SNe and KNe, we select the corresponding host galaxy i-band magnitude pdf and draw a random value. With a chosen i-band host magnitude, we determine the expected value of the photometric redshift error from the validation of the Gold catalog. We define a Gaussian distribution centered on the true simulated redshift with a standard deviation of the photometric redshift error. We account for known underestimations of low-redshift galaxies' photometric redshift uncertainty using the same treatment discussed in Section 3.3. Thus, after drawing a photometric redshift from this distribution, each simulated transient will have a photometric redshift and photometric redshift error to match the candidates in our observations.

We model spectroscopic targeting and classification by implementing the ratio of the number of objects targeted by spectroscopic instruments to the number of candidates remaining at that stage in the follow-up as the probability of an SN being rejected.

Here, we present the results of applying our real-time selection criteria to SNANA-simulated SNe and KNe light curves. We stress that this approach of representing the expected signal and background samples by applying selection criteria to the light curves provides our best understanding of the characteristics of the objects present in the final candidate sample. We utilize our simulated light curves to quantify the expected number of remaining SNe in the final candidate sample, to determine the detection efficiencies of all available KN models, to understand the light curves of objects passing our selection criteria, to place upper limits on the physical properties of the merger, and to forecast our discovery potential in future follow-up observations. In Section 6, we use all these pieces of information to inform a discussion of efficient follow-up strategy and the dynamics of the merger.

Table 4 lists the number of candidates surviving each criterion enforced during our real-time analysis. We also show that the number of candidates remaining after all selection criteria is consistent with the expected background SNe in these follow-up observations. The ML classification of our candidates found no potential KNe remaining in our final sample. Furthermore, because the PSNID+FRC classifier performed with a false-positive rate of 0.01, a remaining candidate would be identified as a KN at the 3σ confidence level. This low false-positive rate of the classifier effectively reduces the SNe background to zero objects, which will prove to be essential for claiming an association between a GW signal and a candidate counterpart in subsequent optical follow-up observations.

A second result of this analysis is the detection efficiency of 329 independent KN models as they would appear in our DECam observations. The top panel of Figure 4 shows the efficiency of each model after Criterion 1 was placed. Criterion 1, which requires a single Type 1 detection, can be thought of as assuring the maximum brightness of the objects is greater than the 5σ limiting magnitude of the observations. The middle panel displays the detection efficiencies after all criteria up to the ML classification, and the bottom panel of the figure shows the detection efficiencies after the ML classification occurs. The blue and red boxes in the panels identify the best-fit model components of the emission from AT2017gfo (Drout et al. 2017; Kilpatrick et al. 2017), the optical counterpart for GW170817, at distances consistent with GW190814 and accounting for the environmental conditions of our follow-up observations. We find that low M_ej and high X_lan yield an optical signature that would be difficult to detect in our DECam observations. At the same time, we note that our selection criteria limit our ability to detect KNe models with low v_ej and high M_ej. Physically, the light curves of these models fade more slowly than other KN models and are more similar to some SN models, which leads to class confusion at the ML stage.

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A third product of this analysis is a prediction of the average light curves of the objects that pass our selection criteria. In Figure 5, we overlay the measured i-band magnitudes of our candidates on the average light curves of simulated objects passing the same selection criteria. In the top row, we consider our candidates in the context of SNe. The high-redshift SNe pass our selection criteria because their photometric host-galaxy redshift and uncertainty are consistent with the LVC distance posterior at the 3σ level. As shown in Figure 5, these high-redshift SNe very closely resemble our candidates in terms of light-curve properties: the fading rates of the light curves over the 16 nights and the apparent magnitudes are quite similar. The bottom panel compares our candidates to KN models. Each KN light curve is the average across the full range of X_lan, since this parameter was found to have the smallest effect on light-curve shape—it does, however, affect the color, but we only show monochromatic light curves in the figure. This averaging is subject to the nonuniform grid of models and the parameterization of X_lan in log space; however, we reiterate that this parameter has the smallest effect on the light-curve shapes shown in the figure.

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The KN models as a class fade much more quickly than our candidates, which results in many of them becoming too faint to detect in our observations 16 nights after the merger. An understanding of the light curves for a potential KN and for the expected background is essential for choosing an efficient observing strategy, which we will discuss in Section 6. We note again that KN light curves from models with low v_ej and high M_ej fade the slowest out of all KN models, and at a rate comparable to the faster-fading SN models in the top panel of Figure 5. This observation only applies to optical emission in the i and z filters, and we are unable to speculate on the generalization of this behavior to other wavelength ranges. This behavior of this subset of KN models poses the greatest confusion to our ML classifier, as a result of the light-curve similarities.

Using the fact that a KN was not detected in these observations, we can translate our KN detection efficiencies and expected background rates into upper limits on merger properties. We estimate the properties of a KN that would go undetected from a Bayesian standpoint:

$$\begin{eqnarray}&&P({\mathrm{KN}}_{i}| {n}_{\mathrm{cand}})=\displaystyle \frac{P({n}_{\mathrm{cand}}| {\mathrm{KN}}_{i})\times P({\mathrm{KN}}_{i})}{P({n}_{\mathrm{cand}})}.\end{eqnarray} \tag{ 2 }$$

In Equation (2), KN_i refers to an individual KN model and n_cand is the number of candidates detected in the observations. In this analysis, n_cand = 0, though we present the generalized formalism. The likelihood in Equation (2) can be explicitly written as

$$\begin{eqnarray}\begin{array}{rcl}P({n}_{\mathrm{cand}}| {\mathrm{KN}}_{i}) & = & {\varepsilon }_{i}\times \mathrm{Poisson}({n}_{\mathrm{cand}}-1| B)\\ & & +(1-{\varepsilon }_{i})\times \mathrm{Poisson}({n}_{\mathrm{cand}}| B),\end{array}\end{eqnarray} \tag{ 3 }$$

where ε_i represents the detection efficiency of KN_i and B represents the expected SN background, both of which are determined after all selection criteria have been applied. The Poisson distribution utilized in Equation (3) has an expectation value of B objects and yields the probability of detecting n_cand − 1 or n_cand objects. This formulation is motivated by summing the probability that a KN is detected and the remainder of the candidates are a realization of the predicted SN background with the probability that a KN is not detected and all detected candidates are a realization of the predicted SN background. In Equation (2), P(KN_i) is the prior distribution of KN models, which we make uninformative by assigning equal probability to each model in the nonuniform grid. The denominator can be evaluated directly by computing

$$\begin{eqnarray}&&P({n}_{\mathrm{cand}})=\displaystyle \sum _{i\in \mathrm{KN}\ \mathrm{models}}P({n}_{\mathrm{cand}}| {\mathrm{KN}}_{i})\times P({\mathrm{KN}}_{i}),\end{eqnarray} \tag{ 4 }$$

which can be interpreted as a probability normalization constant. Thus, the posterior distribution of KN models, given the nondetection in this analysis, can be estimated by setting n_cand = 0 and sampling the likelihood space. The results of this sampling are displayed in Figure 6.

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From these observations and sensitivity analysis, we report our constraints on candidate counterpart ejecta properties in Table 5. To determine the likelihood of a physical KN parameter rather than an individual model in our nonuniform model grid, we perform a three-dimensional linear interpolation between the model efficiencies in the space of $\mathrm{log}({X}_{\mathrm{lan}})$, M_ej, and v_ej. We note that this linear interpolation is justified by the smoothness of adjacent points in the grid of efficiencies in Figure 4. These results are less constraining than what would be obtained using the KN efficiencies and expected backgrounds after Criterion 1, but this is only the case when n_cand = 0. In this specific case, the first term in Equation (3) is zero, which leads to a cancellation of the background in Equation (2), so the effect of the selection criteria only manifests through reducing KN detection efficiencies. In general, reducing the SN background will produce better constraints.

Table 5.  Constraints on Counterpart Ejecta Properties of the Candidate NSBH Merger GW190814

Ejecta Property	1σ Constraint	2σ Constraint
Mej	<0.016 M⊙	<0.07 M⊙
vej	$\rlap{/}{\in }[0.16c,0.26c]$	$\rlap{/}{\in }[0.18c,0.21c]$
Xlan	>10−5.92	>10−8.56

Notes. These constraints are derived by interpolating the grid of efficiencies in Figure 4 for each of the Kasen et al. (2017) KN models and applying the Bayesian formalism presented in Section 5.3. This calculation utilized an uninformative prior by assigning equal probability to each point in the KN ejecta parameter space.

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It is worth noting here that previous analyses have demonstrated that derived constraints can depend on the models employed in the analysis (Coughlin et al. 2019b), and furthermore that the discretized grid of model parameters can affect the constraints as well (Dietrich et al. 2020). For this specific event, optical light would be emitted by tidally stripped NS material, which motivated our choice of models focused on the ejecta properties. By not using a model tied to the dynamics of the system, we marginalize over the dependencies on these features of the merger and focus our analysis on directly observable characteristics. To account for the nonuniform spacing of our grid of model parameters, we performed a three-dimensional linear interpolation of the efficiencies in Figure 4 when performing the Bayesian analysis. In the Bayesian analysis, we assigned an equal probability to each point in the parameter space of our models. While not all ejecta parameter combinations may be equally likely, given the NSBH nature of GW190814, we believe our uniformed prior is well-motivated, given the mass of the lighter object involved in the merger. In the event that the object truly was a 2.6 M_⊙ NS, we believe all values in the ranges of M_ej, v_ej, and X_lan are physically accessible under the right dynamical conditions.

To show the benefit of selection criteria that reduce the SN background in GW follow-up observations, we perform simulations of follow-up observations at several points in this analysis. After each criterion, we take the expected SN background and KN detection efficiency for the blue component of a GW170817-like KN, and calculate the significance level at which that KN would be identified as the counterpart. Assuming a nearly complete coverage of the GW alert localization area, we report the fraction of follow-up observations where an association at the 1σ, 2σ, 3σ, and 4σ confidence level would be possible in Figure 7 as functions of the remaining SN background. Without placing any selection criteria, less than 3% of DECam follow-up observations can be expected to identify the counterpart at the 3σ confidence level. Conversely, with the selection criteria and ML classification developed in this analysis, approximately 95% of follow-up observations are expected to be able to identify a counterpart in the DECam observations at the 4σ confidence level.

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In this subsection, we highlight the differences between our approach and those presented in other analyses and follow-up observations of GW190814. While G20 analyzed the public DECam observations discussed in this work, multiple teams performed independent observations. V20 observed GW190814 using MegaCam/CHFT. The V20 observations utilized the g, i, and z bands reaching 5σ limiting magnitudes of ∼23 mag on nights 1, 2, 3, 4, 6, 7, 8, and 20 following the merger. The imaging covered 69% of the total integrated probability area, as the 1 sq. degree FoV of the imager limited the feasible area to cover each night. E20 utilized several observatories and filters to image the 90% localization area, including the Gravitational wave Optical Transient Observer, the Visible and Infrared Survey Telescope for Astronomy, the Very Large Telescope, the Asteroid Terrestrial-impact Last Alert System, and Pan-STARRS1. They reach limiting magnitudes comparable to DECam on a significant fraction of the localization area and distribute a cadence similar to the DECam and MegaCam cadences across their network of observatories. M20 performed a galaxy-targeted search, using the Magellan Baade telescope, within the 50% localization area on nights 1 and 2 following the merger. They reached a 3σ i-band limiting magnitude of 22.2 mag. Last, W20 utilized the DDOTI wide-field robotic imager on the first two nights covering the merger. They cover the full localization area to ∼18 mag in the $w=r+0.23(g-r)$ band.

The characteristics of the different data sets collected, such as imaging depth, observing cadence, sky area covered, and image quality shaped the analyses performed by the counterpart search teams. W20 was able to detect transients to ∼18 mag, meaning KN-like optical signatures at the distance of GW190814 would be too faint to detect. For this reason, they are unable to place constraints on counterpart properties that are competitive with the groups employing deeper optical imaging. M20 obtained deep imaging, but only targeted galaxies with the 50% localization area (70% of the galaxy-weighted probability). While they calculate that KN-like counterparts with more than 0.03 M_⊙ would be too faint to detect in their observations, without covering the full 90% localization area, they cannot place constraints above the 90% confidence level. We, G20, V20, and E20 covered high fractions of the 90% localization area and utilized telescopes and images powerful enough to detect potential counterparts at the distance of GW190814.

No group reports an EM counterpart, and G20, V20, and E20 use their observations to place constraints on the properties of the merger. G20 fixes the distance of the merger to the mean value of 267 Mpc and finds M_ej > 0.05 M_⊙. They also consider the viewing angle of the merger in their constraints, which enters in our analysis through the line-of-sight component of the ejecta velocity. V20 finds slightly tighter constraints on the ejecta mass of a potential EM counterpart (0.015 M_⊙), though their analysis fixes v_ej to 0.2c, which we show in this work is disfavored at the 2σ confidence level. We suggest this choice of disfavored ejecta velocity is the cause of the comparatively tighter constraints reported by V20. E20 reports that KN-like counterparts with M_ej > 0.1 M_⊙ are excluded at the 90% confidence level. They arrive at this result by utilizing the limiting magnitudes of their observations and the expected magnitudes of KN models (similar to the work of G20 and V20) at the distances distributed according to the luminosity distance posterior of GW190814 from the LVC (similar to this work).

The characteristic distinguishing the work presented here from the analyses of all other groups is the extent of the sensitivity analysis used to understand KN detection efficiencies in the observations. G20 and V20 choose a handful of representative fixed distances for the KN and assess whether the the apparent magnitude of a particular model would be brighter than the 5σ magnitude limit in the band of the observations. This approach does not consider the effect of the selection criteria applied to the candidates to rule out all objects on the KN model efficiencies, nor does it accurately marginalize over the LVC distance posterior for the merger. The simulations developed for our work fully incorporate the effects of our real-time selection criteria, as well as the full posterior of luminosity distances, and enable us to place meaningful constraints without fixing any KN parameters. Understanding the effects of the selection criteria placed during a real-time search on the set of detectable counterpart configurations is essential for accurately constraining the physical properties of the potential optical counterpart. The approach demonstrated in this work has been utilized in Doctor et al. (2017), Morgan et al. (2019), De et al. (2020), Kasliwal et al. (2020), and Garcia et al. (2020), and has been facilitated by the development of code bases such as simsurvey (Feindt et al. 2019). We advise that future analyses employ this approach in GW counterpart searches and population studies.

The optical signature from an NSBH merger is highly dependent on the dynamics of the system and the characteristics of the compact objects involved (Bauswein et al. 2013; Rosswog 2013; Radice et al. 2017, 2018). For a KN-like signature to be emitted, the NS would need to be tidally disrupted to produce light-emitting ejecta. Therefore, the spins and masses of the coalescing bodies, which determine the degree of tidal disruption of the NS, are intimately linked to the optical signature (Capano et al. 2020).

At the 2σ confidence level, we were able to exclude counterparts with M_ej > 0.07 M_⊙. Thus, only a small fraction of the NS material was ejected. We also exclude counterparts with X_lan < 10^−8.56 at the 2σ level. The constraint on this quantity is 10^−5.92 at the 1σ level, indicating that higher X_lan are favored overall, and that in the most probable case, any ejecta produced would have been rich in heavy elements. This richness could result from the small (if any) amount of NS material ejected in the merger, as the majority of the material would be synthesized into heavy elements by the gravitational potential in close proximity to the BH. A final result from this sensitivity analysis that can be used to infer properties of the merger is the nondetection of a KN-like counterpart. Since our KN detection efficiency decreases with M_ej, the lack of an observation of a KN in this merger event suggests a small or nonexistent amount of ejected material. The DECam observations are therefore consistent with the NS retaining structural integrity until it passed the radius of the last stable circular orbit.

The physical parameters of the merger most closely tied to the potential tidal disruption of the NS, and hence the optical constraints derived in this analysis, are the mass ratio $q\equiv {M}_{\mathrm{BH}}/{M}_{\mathrm{NS}}$, the magnitude of the final BH spin χ, the radius of the neutron star r_ns, and the chirp mass ${ \mathcal M }$. The ejected mass increases with decreasing M_BH, increasing χ, and r_ns (harder EOS). From numerical simulations, the upper limit to disk formation is a mass ratio of ∼3–5 (Pannarale & Ohme 2014; Foucart et al. 2018, 2019; Lattimer 2019). For a fixed BH mass, as the NS mass increases, a larger BH spin is required to produce a massive disk. The reason is that higher black hole spin decreases the last stable circular orbit radius, allowing a higher-mass, generally more compact NS to reach its disruption radius and thus leave the disrupted NS matter remaining in orbit. Holding the NS mass fixed, increasing the BH mass increases the gravitational radii, and higher spins are needed to bring the last circular orbit radius in below the disruption radius. Binaries with low mass ratios and high BH spins maximize the chance of massive disk formation. Based on our observations, the spins, masses, and alignments of the merging bodies disfavor tidal disruption of the NS. In their recently release parameter estimation of the merger, the LVC determined q = 0.11, χ = 0.28 of 0.28, and ${ \mathcal M }=6.1$ M_⊙ (Abbott et al. 2020). This spin and mass ratio would lead to small amounts of tidal disruption of the NS, and would be consistent with the lack of an accretion disk, the lack of an accompanying gamma-ray burst, and the lack of a detection a KN-like counterpart. Therefore, the constraints on NS ejecta properties derived from the DECam observations in this analysis are consistent with the LVC parameter estimation of the merger dynamics.

We find two aspects of our observing strategy for this event to be highly efficient and recommend their use in follow-up observations going forward. Namely, our observing cadence and exposure time, and specifically how we tailored them to the conditions of the event, were essential in detecting the large number of candidates published by teams using the public DECam data.

The cadence of our observations was well-suited for the detection of a KN-like optical signal. By triggering DECam immediately after the LVC alert, and by repeating observations on the next three nights following the merger, we increased our chances of detecting a rapidly changing object. Furthermore, our choice to include epochs on nights 6 and 16 after the merger enabled the characterization of light curves for longer-lived transient objects in that part of the sky. This choice proved to be essential in systematically eliminating fading objects unassociated with the GW signal. While a KN is not expected to be bright at these later epochs, the detection of any potential candidate on these nights can be used as evidence to exclude the object.

In this sensitivity analysis, the availability of light curves spanning a 16 day interval with six observing epochs enabled the development of a powerful ML-based photometric classifier. Our PSNID + RFC approach was able to effectively eradicate the SN background in these observations, and we expect a similar performance in subsequent follow-up observations with similar cadences. The benefit of devoting resources to background reduction is a key point we seek to make. Figure 7 demonstrates how reducing the SN background dramatically increases the probability that optical follow-up observations will associate a candidate with the LVC alert at a statistically significant confidence level. In this figure, we select a single KN model for simplicity and consider its detection efficiency as a function of the remaining SNe background as we apply the selection criteria in this analysis. The "Before ML" section references the real-time selection criteria of this analysis, while the "With ML" section varies the PSNID + RFC operating threshold to move along the receiver operating characteristic (ROC) curve of Figure 3 toward regions of higher purity. Again, the performance of this classifier and the possibility to reduce the SN background are primarily determined by the cadence chosen by the observing team. Reducing the SNe background using the techniques of this work leads to the possibility of associating a detected transient with the merger at the 4σ confidence level 95% of the time in identical follow-up observations.

The exposure time of the DECam images was dynamically varied in response to changing observing conditions. On the first two nights, when a potential KN-like counterpart would be expected to be near peak brightness, we opted for a shorter exposure time to quickly tile the area twice. This choice enabled us to rule out moving objects while maintaining enough depth to detect a KN-like object positioned at the estimated distance of GW190814. In subsequent nights, we increased the exposure time such that we would be sensitive to fainter objects, since a KN-like object would be expected to fade by ∼0.5 mag per day (Kasen et al. 2017). While the choice to vary the exposure time introduces nonuniformity in the image quality of the DECam data, it is useful for maximizing the probability of detecting a rapidly fading transient on each night of observation. Furthermore, we note that this variance of image quality over the course of the observations necessitates the use of detailed simulations of the follow-up observations for quantifying constraints. Our choice of exposure times resulted in the deepest optical observations of the entire 90% localization region on each night DECam was operated, compared to all follow-up teams. Thus, the observing strategy described in this work is a useful baseline for future DECam follow-up observations.

The chances of detecting a potential counterpart were greatly improved by SOAR spectra being obtained for the most interesting candidates. While we were able to achieve high-accuracy machine-learning-based photometric classification of the objects in our sample in this work, the success of that approach requires the availability of several nights of photometric observations. In the real-time portion of the observations, the spectroscopic component of the search is essential. The use of SOAR enabled us to confidently exclude ∼20% of our most promising candidates on the first few nights of the observations.

We see this spectroscopic efficiency as an aspect of gravitational-wave counterpart identification that can be dramatically improved given the resources of the astronomical community—for example, with the use of wide-field multi-object spectroscopy (Palmese et al. 2019c). In cases where the 90% localization is larger than what one telescope can cover in a single night, the fraction of sky area covered by the astronomical community is another improvable trait of the follow-up strategy. As we look forward to the increased sensitivity in Observing Run 4 and consequent increased alert frequency, synergy among follow-up teams will be integral to the association of gravitational waves with their electromagnetic counterparts. Distributed and coordinated observations among follow-up teams will be essential, and furthermore the sharing of observation metadata to improve sensitivity and forecasting studies will benefit the field as a whole. For further discussion of these topics, see Coughlin (2020)