An Embedded X-Ray Source Shines through the Aspherical AT 2018cow: Revealing the Inner Workings of the Most Luminous Fast-evolving Optical Transients

The UV Optical Telescope (UVOT; Roming et al. 2005) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004) started observing AT 2018cow on 2018 June 19 (∼3 days since discovery) with six filters, v, b, u, w1, w2, and m2, in the wavelength range λ_c = 1928 Å (w2 filter)—λ_c = 5468 Å (v filter, central wavelength). We extracted aperture photometry following standard prescriptions by Brown et al. (2009), with the updated calibration files and revised zero points by Breeveld et al. (2011). Each individual frame has been visually inspected and quality flagged. Observations with insufficient exposure time have been merged to obtain higher signal-to-noise ratio images from which we extracted the final photometry. We used a 3'' source region of extraction to minimize the effects of the contamination from the underlying host-galaxy flux, and we manually corrected for imperfections of the astrometric solution of the automatic UVOT pipeline realigning the frames. In the absence of template images, we estimated the host-galaxy contribution by measuring the host-galaxy emission at a similar distance from the nucleus. The results from our method are in excellent agreement with Perley et al. (2019). We note that at δt > 50 days, this method is likely to overestimate the UV flux of the transient, as the images show the presence of a bright knot of UV emission underlying AT 2018cow that can only be properly accounted for with template images obtained in the future.

Ground-based optical photometry has been obtained from ANDICAM, mounted on the 1.3 m telescope³⁴ at Cerro Tololo Inter-American Observatory (CTIO), the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995), and the DEep Imaging Multi-Object Spectrograph (DEIMOS; Faber et al. 2003), mounted on the Keck telescopes. Images from the latter were reduced following standard bias and flat-field corrections. Data from ANDICAM, instead, came already reduced by their custom pipeline.³⁵ Instrumental magnitudes were extracted using the point-spread-function (PSF) fitting technique, performed using the SNOoPY³⁶ package. Absolute calibrations were achieved measuring zero points and color terms for each night, estimated using as reference the magnitudes of field stars, retrieved from the Sloan Digital Sky Survey³⁷ (SDSS; York et al. 2000) catalog (DR9). SDSS magnitudes of the field stars were then converted to Johnson/Cousins, following Chonis & Gaskell (2008). Our BVRI PSF photometry agrees well with the host-galaxy-subtracted photometry presented by Perley et al. (2019).

We obtained near-IR imaging observations in the JHK bands with the Wide-field Camera (WFCAM; Casali et al. 2007) mounted on the 3.8 m United Kingdom Infrared Telescope (UKIRT) spanning δt ∼ 10–42 days. We obtained preprocessed images from the WFCAM Science Archive (Hamly et al. 2008), which are corrected for bias, flat-field, and dark current by the Cambridge Astronomical Survey Unit.³⁸ For each epoch and filter, we co-add the images and perform astrometry relative to 2MASS using a combination of tasks in Starlink³⁹ and IRAF. For the J-band, we obtain a template image from the UKIRT Hemispheres Survey DR1 (Dye et al. 2018) and use the HOTPANTS software package (Becker 2015) to perform image subtraction against this template to produce residual images. We perform aperture photometry using standard tasks in IRAF, photometrically calibrated to 2MASS. In the absence of a template image in the H- and K-bands, we performed aperture photometry of the transient and host-galaxy complex centered on the core of the host galaxy. We used standard procedures in IRAF and 2.5 FWHM apertures. At δt < 15 days, the host-galaxy contribution is negligible but dominates the HK photometry at δt > 30 days. Single epochs of JHK-band photometry were obtained on 2018 June 26 (δt ∼ 9.86 days) using the WIYN High-resolution Infrared Camera (WHIRC; Meixner et al. 2010) mounted on the 3.5 m WIYN telescope, and on 2018 August 31 (δt ∼ 75.7 days) with the MMT and Magellan Infrared Spectrograph (MMIRS; McLeod et al. 2012), mounted on the MMT telescope. These data were processed using similar methods. AT 2018cow is not detected against the host-galaxy NIR background in our final observation. After subtracting the bright sky contribution, we estimated the instrumental NIR magnitudes via PSF fitting. We calibrate our NIR photometry relative to 2MASS⁴⁰ (Skrutskie et al. 2006). No color term correction was applied to the NIR data.

UV, optical, and NIR photometry have been corrected for Galactic extinction with E(B − V) = 0.07 mag (Schlafly & Finkbeiner 2011) and no extinction in the host galaxy. Our final photometry is presented in Tables 9–12. The UV/optical/NIR emission from AT 2018cow is shown in Figure 1.

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We obtained five spectra of AT 2018cow using the Goodman spectrograph (Clemens et al. 2004) mounted on the SOAR telescope in the time range δt ∼ 4.6–34.2 days. We used the red camera and the 400 lines mm⁻¹ and 600 lines mm⁻¹ gratings, providing a resolution of ∼5 Å and ∼3 Å at 7000 Å, respectively. We reduced Goodman data following the usual steps, including bias subtraction, flat-fielding, cosmic-ray rejection (see van Dokkum 2001), wavelength calibration, flux calibration, and telluric correction using our own custom IRAF⁴¹ routines.

On 2018 July 9 (δt ∼ 21.4 days), we acquired a spectrum with the Low Dispersion Survey Spectrograph (LDSS3) mounted on the 6.5 m Magellan Clay telescope using the VPH-all grism and a 1'' slit. We obtained a spectrum with the Inamori-Magellan Areal Camera and Spectrograph (IMACS) mounted on the 6.5 m Magellan Baade telescope on 2018 August 6 (δt ∼ 51 days), using the f/4 camera and 300 lines mm⁻¹ grating with a 09 slit. The data were reduced using standard procedures in IRAF and PyRAF to bias-correct, flat-field, and extract the spectrum. Wavelength calibration was achieved using HeNeAr comparison lamps, and relative flux calibration was applied using a standard star observed with the same setup.

We observed AT 2018cow on 2018 August 29 (δt ∼ 74 days) with DEIMOS. We used a 07 slit and the 600 lines mm⁻¹ grating with the GG400 filter, resulting in a ∼3 Å resolution over the range 4500–8500 Å. We acquired a spectrum with LRIS on 2018 September 9 (δt ∼ 85.8 days). We used the 10 slit with the 400 lines mm⁻¹ grating, achieving a resolution of ∼6 Å and spectral coverage of 3200–9000 Å. Due to readout issues, we lost a portion of the spectrum between 5800 and 6150 Å. Reduction of these spectra was done using standard IRAF routines for bias subtraction and flat-fielding. Wavelength and flux calibration were performed comparing the data to arc lamps and standard stars, respectively, acquired during the night and using the same setups. A final epoch of BVRI photometry was acquired with LRIS on 2018 October 5 (δt ∼ 112 days).

We acquired one epoch of low-resolution NIR spectroscopy spanning 0.98–2.31 μm with the MMT using MMIRS on 2018 July 3 (δt ∼ 16.6 days). Observations were performed using a 1'' slit width in two configurations: zJ filter (0.95–1.50 μm) + J grism (R ∼ 2000), and HK3 filter (1.35–2.3 μm) + HK grism (R ∼ 1400). For each of the configurations, the total exposure time was 1800 s, and the slit was dithered between individual 300 s exposures. We used the standard MMIRS pipeline (Chilingarian et al. 2015) to process the data and to develop wavelength-calibrated 2D frames from which 1D extractions were made.

Figures 2–3 show our spectral series. These figures show the drastic evolution of AT 2018cow from an almost featureless spectrum around the optical peak with very broad features, to the clear emergence of H and He emission with asymmetric line profiles skewed to the red and significantly smaller velocities of a few 1000 km s⁻¹. In Table 3, we summarize our NIR/optical spectroscopic observations.

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The X-Ray Telescope (XRT) on board the Neil Gehrels Swift Observatory (Gehrels et al. 2004; Burrows et al. 2005) started observing AT 2018cow on 2018 June 19 (∼3 days following discovery). We reduced the Swift-XRT data with HEAsoft v.6.24 and corresponding calibration files, applying standard data filtering as in Margutti et al. (2013a). A bright X-ray source is detected at the location of the optical transient, with clear evidence for persistent X-ray flaring activity on timescales of a few days (Section 2.9), superimposed on an overall fading of the emission (Figure 4).

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A time-resolved spectral analysis reveals limited spectral evolution. Fitting the 0.3–10 keV data with an absorbed power-law model within XSPEC, we find that the XRT spectra are well described by a photon index ${\rm{\Gamma }}\sim 1.5$ and no evidence for intrinsic neutral hydrogen absorption (Figure 4, upper panel). We employ Cash statistics and derive the parameter uncertainties from a series of MCMC simulations. We adopt a Galactic neutral hydrogen column density in the direction of AT 2018cow, ${N}_{{\rm{H}},\mathrm{MW}}\,=0.05\times {10}^{22}\,{\mathrm{cm}}^{-2}$ (Kalberla et al. 2005). With a different method based on X-ray afterglows of GRBs, Willingale et al. (2013) estimated ${N}_{{\rm{H}},\mathrm{MW}}=0.07\times {10}^{22}\,{\mathrm{cm}}^{-2}$. In particular, the earliest XRT spectrum extracted between 3 and 5 days since discovery can be fitted with Γ = 1.55 ± 0.05 and can be used to put stringent constraints on the amount of neutral material in front of the emitting region, which is ${N}_{{\rm{H}},\mathrm{int}}\lt 6\times {10}^{20}\,{\mathrm{cm}}^{-2}$ (we adopt solar abundances from Asplund et al. 2009 within XSPEC). The material is thus either fully ionized or absent (Section 3.3.2). The results from the time-resolved Swift-XRT analysis are reported in Table 4. The total XRT spectrum collecting data in the time interval 3–60 days can be fitted with an absorbed power law with Γ = 1.55 ± 0.04 and ${N}_{{\rm{H}},\mathrm{int}}\lt 0.03\times {10}^{22}\,{\mathrm{cm}}^{-2}$. From this spectrum, we infer a 0.3–10 keV count-to-flux conversion factor of $4.3\times {10}^{-11}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{\mathrm{ct}}^{-1}$ (absorbed), $4.6\,\times {10}^{-11}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{\mathrm{ct}}^{-1}$ (unabsorbed), which we use to flux-calibrate the XRT light curve (Figure 4). At the distance of ∼60 Mpc, the inferred 0.3–10 keV isotropic X-ray luminosity at 3 days is L_X ∼ 10⁴³ erg s⁻¹. AT 2018cow is significantly more luminous than normal SNe and shows a luminosity similar to that of low-luminosity GRBs (Figure 5). The spectrum also shows evidence for positive residuals above ∼8 keV, which are connected to the hard X-ray component of emission revealed by NuSTAR and INTEGRAL (Section 2.4).

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We triggered deep XMM-Newton observations of AT 2018cow on 2018 July 23 ($\delta t\sim 36.5$ days, exposure time ∼32 ks, imaging mode; PI Margutti), in coordination with our NuSTAR monitoring. We reduced and analyzed the data of the European Photon Imaging Camera (EPIC)-pn data using standard routines in the Scientific Analysis System (SAS version 17.0.0) and the relative calibration files, and used MOS1 data as a validation check. After filtering data for high background contamination, the net exposure times are 24.0 and 31.5 ks for pn and MOS1, respectively. An X-ray source is clearly detected at the position of the optical transient. We extracted a spectrum from a circular region of 30'' radius centered at the source position. Pile-up effects are negligible as we verified with the task epatplot. The background was extracted from a source-free region on the same chip. We estimate a 0.3–10 keV net count rate of 0.519 ± 0.005 c/s. The X-ray data were grouped to a minimum of 15 counts per bin. The 0.3–10 keV spectrum is well fitted by an absorbed power-law model with best-fitting Γ = 1.70 ± 0.02 and marginal evidence for ${N}_{{\rm{H}},\mathrm{int}}\sim 0.02\times {10}^{22}\,{\mathrm{cm}}^{-2}$ at the 3σ c.l. for ${N}_{{\rm{H}},\mathrm{MW}}=0.05\times {10}^{22}\,{\mathrm{cm}}^{-2}$. Given that the uncertainty on ${N}_{{\rm{H}},\mathrm{MW}}$ is also ∼0.02 × 10²² cm⁻², we consider this value as an upper limit on ${N}_{{\rm{H}},\mathrm{int}}$ at 36.5 days.

We acquired a second epoch of deep X-ray observations with XMM-Newton on 2018 September 6 ($\delta t\sim 82$ days; PI Margutti). The net exposure times are 30.5 and 36.8 ks for pn and MOS1, respectively. AT 2018cow is clearly detected with a net 0.3–10 keV count rate of (6.0 ± 0.7) × 10⁻³ c s⁻¹. We used a source region of 20'' to avoid contamination by a faint unrelated source located 368 southwest from our target (at earlier times, AT 2018cow is significantly brighter and the contamination is negligible). The spectrum of AT 2018cow is well fitted by a power-law model with Γ = 1.62 ± 0.20 with unabsorbed 0.3–10 keV flux ∼2 × 10⁻¹⁴ erg cm⁻² s⁻¹. We find no evidence for intrinsic neutral hydrogen absorption. Finally, we note that comparing the two XMM-Newton observations, we find no evidence for a shift of the X-ray centroid, from which we conclude that X-ray emission from the host-galaxy nucleus, if present, is subdominant and does not represent a significant source of contamination. The complete 0.3–10 keV X-ray light curve of AT 2018cow is shown in Figure 4.

INTEGRAL started observing AT 2018cow on 2018 June 22 18:38:00 UT until July 8 04:50:00 UT (δt ∼ 6–22 days) as part of a public target of opportunity observation. The total on-source time is ∼900 ks (details are provided in Table 5). A source of hard X-rays is clearly detected at the location of AT 2018cow at energies ∼30–100 keV with significance 7.2σ at δt ∼ 6 days. The source is no longer detected at δt ≳ 24 days (Figure 6). After reconstructing the incident photon energies with the latest available calibration files, we extracted the hard X-ray spectrum from the ISGRI detector (Lebrun et al. 2003) on the IBIS instrument (Ubertini et al. 2003) of INTEGRAL (Winkler et al. 2003) for each of the ∼2 ks long individual pointing of the telescope dithering around the source. We used the Off-line Scientific Analysis Software (OSA) with a sky model comprising only AT 2018cow, which is the only significant source in the field of view. The energy binning was chosen to have 10 equally spaced logarithmic bins between 25 and 250 keV, the former being the current lower boundary of the ISGRI energy window. We combined the spectra acquired in the same INTEGRAL orbit. We use these spectra in Section 2.5 to perform a time-resolved broadband X-ray spectral analysis of AT 2018cow.

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We acquired a detailed view of the hard X-ray properties of AT 2018cow between 3 and 80 keV with a sequence of four NuSTAR observations obtained between 7.7 and 36.5 days (PI Margutti; Table 6). The NuSTAR observations were processed using NuSTARDAS v1.8.0 along with the NuSTAR CALDB released on 2018 March 12. We extracted source spectra and light curves for each epoch using the nuproducts FTOOL using a 30'' extraction region centroided on the peak source emission. For the background spectra and light curves, we extracted the data from a larger region (∼85'') located on the same part of the focal plane. We produced response files (RMFs and ARFs) for each FPM and for each epoch using the standard nuproducts flags for a point source.

AT 2018cow is well detected at all epochs. The first NuSTAR spectrum at 7.7 days shows a clear deviation from a pure absorbed power-law model with Γ ∼ 1.5 and reveals instead the presence of a prominent excess of emission above ∼15 keV, which matches the level of the emission captured by INTEGRAL, together with spectral features around 7–9 keV. By day 16.5, the hard X-ray bump of emission disappeared, and the spectrum is well modeled by an absorbed power law (Figure 6). We model the evolution of the broadband X-ray spectrum as detected by Swift-XRT, XMM-Newton, NuSTAR, and INTEGRAL in Section 2.5.

Our coordinated Swift-XRT, XMM-Newton, NuSTAR, and INTEGRAL monitoring of AT 2018cow allows us to extract five epochs of broadband X-ray spectroscopy (∼0.3–100 keV) from 7.7 days to 36.5 days. We performed joint fits of data acquired around the same time, as detailed in Table 7. Our results are shown in Figure 6. We find that the soft X-rays at energies ≲7 keV are always well described by an absorbed simple power-law model with photon index Γ ≈ 1.5–1.7 with no evidence for absorption from neutral material in addition to the Galactic value. Our most constraining limits from the time-resolved analysis are ${N}_{{\rm{H}},\mathrm{int}}\lt (0.03-0.04)\times {10}^{22}\,{\mathrm{cm}}^{-2}$ (Table 7).

Remarkably, at ∼7.7 days, NuSTAR and INTEGRAL data at E > 15 keV reveal the presence of a prominent component of emission of hard X-rays that dominates over the power-law component.⁴² We model the hard X-ray emission component with a strongly absorbed cutoff power-law model (light-blue dashed line in Figure 6, top panel). This is a purely phenomenological model that we use to quantify the observed properties of the hard emission component. A cutoff power law is preferred to a simple power-law model, as a simple power law would overpredict the highest energy data points at 7.7 days. From this analysis, the luminosity of the hard X-ray component at δt ∼ 7.7 days is ${L}_{{\rm{x}},\mathrm{hard}}\sim {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (20–200 keV). A joint analysis of Swift-XRT+INTEGRAL data at δt ∼ 10.1 days indicates that the component of hard X-ray emission became less prominent, and then disappeared below the level of the soft X-ray power law by δt ∼ 16.5 days, as revealed by the coordinated Swift-XRT, XMM, and NuSTAR monitoring (Figure 6). We derive upper limits on the luminosity of the undetected hard X-ray emission component at δt ≥ 16.5 days assuming a similar spectral shape to the one observed at δt ∼ 7–10 days. As shown in Figure 4, the hard X-ray component fades quickly below the level of the power-law component that dominates the soft X-rays, which at this time evolves as L_x ∝ t⁻¹. The hard and soft X-ray emission components clearly show a distinct temporal evolution, suggesting that they originate from different emitting regions. Table 1 lists the energy radiated by each component of emission.

Table 1.  Energy Radiated by AT 2018cow at 3 < δt < 60 days

Component	Band	Radiated Energy (erg)
Power law	0.3–10 keV	${9.8}_{-0.1}^{+0.2}\times {10}^{48}$
Power law	0.3–50 keV	${2.5}_{-0.3}^{+0.4}\times {10}^{49}$
Hard X-ray bump	20–200 keV	∼1049
Blackbody	UVOIR	${1.0}_{-0.2}^{+0.2}\times {10}^{50}$
Non-thermala	UVOIR	∼5 × 1048
Total		∼1.4 × 1050 erg

Note.

^aBased on the analysis from Perley et al. (2019).

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We note that the first spectrum at 7.7 days shows positive residuals around ∼6–9 keV (Figure 6, inset). Typical spectral features observed in accretion disks (both around X-ray binaries and active galactic nuclei, AGNs) and interacting SNe are Fe Kα emission (between 6.4 and 6.97 keV depending on the ionization state) and the Fe K-band absorption edge. Typical interpretations of blueshifted iron-line profiles include edge-on (or highly inclined) accretion disks or highly ionized absorption (e.g., Reeves et al. 2004). As a reference, interpreting the spectral feature detected in AT 2018cow at $E\sim 8\,\mathrm{keV}$ with width ∼1 keV as Fe emission would require a blueshift corresponding to v ∼ 0.1c and Doppler broadening with similar velocity. We discuss possible physical implications in Section 3.3.3.

We observed AT 2018cow with the Karl G. Jansky Very Large Array (VLA) starting from δt ∼ 82 days until δt ∼ 150 days. The data were taken in the VLA's D configuration under program VLA/18A-123 (PI Coppejans). We reduced the data using the pwkit package (Williams et al. 2017), using 3C 286 as the bandpass calibrator and VCS1 J1609+2641 as the phase calibrator. We imaged the data using standard routines in CASA (McMullin et al. 2007) and determined the flux density of the source at each frequency by fitting a point-source model using the imtool package within pwkit. This package uses a Levenberg–Marquardt least-squares optimizer to fit a small region in the image plane centered on the source coordinates with an elliptical Gaussian corresponding to the CLEAN beam. These data are shown together with the rest of our radio observations in Figure 7 and Table 8.

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We also obtained 22.3 GHz VLBI observations of AT 2018cow with the High Sensitivity Array of the National Radio Astronomy Observatory (NRAO) on 2018 July 7 (Bietenholz et al. 2018). The array consisted of the NRAO Very Long Baseline Array with the exception of the North Liberty station (9 × 25 m diameter) and the Effelsberg antenna (100 m diameter). We recorded both senses of circular polarization at a total bit rate of 2048 Mbit s⁻¹. The observations were phase-referenced to the nearby compact source QSO J1619+2247, with a cycle time of ∼100 s. The amplitude gains were calibrated using the system temperature measurements made by the VLBA online system and refined by self-calibration on the calibrator sources, with the gains normalized to a mean amplitude of unity to preserve the flux-density scale as much as possible. We will report on the VLBI results in more detail in a future paper, but we include the total flux density observed with VLBI here. On 2018 July 7.96 (UT), we found 5850 ± 610 μJy at 22.3 GHz. The value was obtained by fitting a circular Gaussian directly to the visibilities by least squares. As the source is not resolved, the nature of the model does not affect the flux density, and a value well within our stated uncertainties is obtained if, for example, a circular disk model is used. The array has good u–v coverage down to baselines of length <30 Mλ; therefore, only flux density on angular scales >8 mas would be resolved out. At this epoch, the projected angular size of AT 2018cow is <8 mas even in the case of relativistic expansion, which implies that our measurement of the radio flux density from AT 2018cow is reliable. The uncertainty is the statistical one with a 10% systematic one added in quadrature. Although at our observing frequency of 22.3 GHz some correlation losses might be expected, the visibility phases were consistent from scan to scan, and we do not expect correlation losses larger than our stated uncertainty.

From our modeling of the VLA measurements at t > 80 days, we find that the radio SEDs are well described by a smoothed broken power law with decreasing spectral peak flux F_pk ∝ t^−1.7±0.1 and peak frequency ν_pk ∝ t^2.2±0.1 (thick lines in Figure 7). At ν > ν_pk, the optically thin part of the spectrum scales as F_ν ∝ ν^−1.4±0.1, while below the spectral peak we find F_ν ∝ ν^1.2±0.1. We place our VLBA and VLA measurements in the context of radio observations from the literature (An 2018; Bright et al. 2018; de Ugarte Postigo et al. 2018a; Dobie et al. 2018a, 2018b, 2018c; Ho et al. 2019; Horesh et al. 2018; Nayana & Chandra 2018; Smith et al. 2018a). An extrapolation of this model back in time shows that this model adequately represents the evolution of AT 2018cow at t > 35 days (Figure 7). At earlier times, the optically thin spectrum scales as F_ν ∼ ν^−1.0, and we find evidence for a steeper optically thick spectrum F_ν ∝ ν². A detailed discussion of the modeling of the entire data set will be presented in D. L. Coppejans et al. (2018, in preparation).

The radio luminosity and temporal behavior of AT 2018cow at ∼9 GHz are more similar to those of the most luminous normal SNe, while AT 2018cow is significantly less luminous than cosmological GRBs (Figure 8). However, differently from radio SNe that show a constant F_pk with time and ν_pk ∝ t⁻¹ (e.g., Chevalier 1998; Soderberg et al. 2005, 2012), in AT 2018cow the peak flux rapidly decreases with time. The rising light curve until $\delta t\sim 100$ days also makes it distinct from low-energy GRBs in the local universe (Figure 8). The radio flux-density measurements of AT 2018cow are presented in Table 8.

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The large X-ray luminosity of AT 2018cow initially suggested a connection with long GRBs. Thus motivated, we searched for bursts of prompt γ-ray emission between the time of the last optical non-detection and the first optical detection of AT 2018cow (i.e., between 2018 June 15 03:08 UT and June 16 10:35 UT; Prentice et al. 2018). During this time interval, one burst was detected on 2018 June 15 11:05:56 UT by the spacecraft of the Inter-Planetary Network (IPN; Mars Odyssey, Konus/Wind, INTEGRAL SPI-ACS, Swift-BAT, and Fermi-GBM). The burst localization by the IPN, INTEGRAL, and the GBM, however, excludes at high confidence the location of AT 2018cow, from which we conclude that there is no evidence for a burst of γ-rays associated with AT 2018cow down to the IPN threshold (i.e., 10 keV–10 MeV 3 s peak flux <3 × 10⁻⁷ erg cm⁻² s⁻¹ for a typical long GRB spectrum with Band parameters α = −1, β = −2.5, and E_p = 300 keV; e.g., Band et al. 1993). For the time interval of interest, the IPN duty cycle was ∼97%. For AT 2018cow, the IPN thus rules out at 97% c.l. bursts of γ-rays with peak luminosity >10⁴⁷ erg s⁻¹, which is the level of the lowest luminosity GRBs detected (e.g., Nava et al. 2012).

We now examine which limits we can place on the probability of detection of weaker bursts, which would only trigger Fermi-GBM and/or Swift-BAT in the same time interval considerd above. Taking Earth-blocking and duty cycle into account, the joint non-detection probability by Fermi-GBM (8–1000 keV fluence limit of $\sim 4\times {10}^{-8}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}$) and Swift-BAT (15–150 keV fluence limit of ∼6 × 10⁻⁹ erg cm⁻² based on the weakest burst detected within the coded field of view, FOV) is ∼29%. The non-detection probability of weaker bursts by Swift-BAT is ∼71% (within the coded FOV) and ∼46% (outside the coded FOV).

Performing a self-consistent flux calibration of the UVOT photometry and applying a dynamical count-to-flux conversion that accounts for the extremely blue colors of the transient, we find that the UV+UBV emission from AT 2018cow is well modeled by a blackbody function at all times. We infer an initial temperature T_bb ∼ 30,000 K and radius R_bb ∼ 8 × 10¹⁴ cm, consistent with Perley et al. (2019). R_bb and T_bb show a peculiar temporal evolution, with R_bb monotonically decreasing with time (with a clear steepening around 20 days), while the temperature plateaus at ∼15,000 K, with no evidence for cooling at δt > 20 days (Figure 9). Indeed, δt ∼ 20 days marks an important transition in the evolution of AT 2018cow: H and He features emerge in the spectra, the hard X-ray hump disappears, L_X approaches the level of the optical emission and later starts a steeper decline, while the soft X-ray variability becomes more pronounced with respect to the continuum.

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After the optical peak, we find that the resulting UV/optical bolometric emission is well modeled by a power-law decay t^−α with best-fitting α = 2.50 ± 0.06 (Figure 9), in agreement with Perley et al. (2019). Also consistent with Perley et al. (2019), we find that the R, I, and NIR data from AT 2018cow are in clear excess of the thermal blackbody emission and represent a different component. In Figure 9, we show that the combined energy release of the thermal UV/optical emission and the soft X-rays (0.3–50 keV) follows a decay ∝t^−α with α = 1.94 ± 0.04. This result is relevant if the thermal optical/UV and the soft X-rays are manifestations of the same physical component, like energy release from a central engine (Section 3.1.1). Table 1 lists the energy radiated by each component of emission.

We examined the 0.3–10 keV emission at δt = 3.5–55 days for evidence of periodicity using the Lomb–Scargle periodogram (LSP; Lomb 1976; Scargle 1982), and the Fast χ² algorithm⁴³ by Palmer (2009), both of which are suitable for unevenly spaced series. As a first step, we removed the overall trend of the time series, which was found to be best modeled by a simple exponential $\exp (k-t/{t}^{* })$, with k = −0.82 ± 0.08 and ${t}^{* }=14.7\pm 0.9$ days. We applied the LSP and the Fast χ² techniques to the resulting residuals.

We calculated the LSP using the Numerical Recipes implementation (Press et al. 1992), exploring the frequency range 0.005–0.65 day⁻¹, corresponding to $1/(4T)$ and to an average Nyquist frequency, respectively, where T is the total duration. To assess the significance of the peaks we detected, one must consider two issues: (i) the presence of red noise and (ii) the number of independent frequencies (e.g., Horne & Baliunas 1986). We addressed (i) through a number of Monte Carlo simulations. We generated 5 × 10³ time series with the same sequence of observing times, t_i, every time shuffling the observed count rates and associated uncertainties so as to keep the same rate distribution and the same variance. For each of the simulated series, we calculated the LSP under the same prescriptions as for the real one. We addressed (ii) through the identification of the peaks in all of the LSPs (both the true one and the ones from the shuffled data) by means of the peak-search algorithm mepsa (Guidorzi 2015): given that only separate peaks are identified as independent structures, this properly accounts for the power associated with correlated frequencies. The significance of the peaks found in the real LSP was then compared against the distribution of peaks in the LSPs from the shuffled data. The result is shown in Figure 10.

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We further apply the Fast χ² algorithm to detect periodic harmonic signals within unevenly spaced data affected by variable uncertainties. For a given number of harmonics and any given trial frequency, the Fast χ² algorithm determines the solution that minimizes the χ². This way, for a given number of harmonics, the best fundamental frequency along with its harmonics is given by the global minimum χ². We applied the method to the observed X-ray detrended light curve, each time allowing for one to six harmonics, as a trade-off between the need to provide a relatively simple modeling and the possibility of a rather complex periodic signal involving several harmonics. The explored frequencies are in the range 0.05–1 day⁻¹. To assess the significance of our results, we applied Fast χ² to the sample of synthetic light curves and compared the results from the real time series. We find significant power (∼3σ Gaussian) on a modulation timescale of ∼4 days in the first 40 days, consistent with the results from the LSP. We note that a similar variability timescale was also independently reported by Kuin et al. (2019).

Finally, we investigate whether there is correlated temporal variability between the X-ray and the UV/optical in AT 2018cow. We fit a third-order polynomial to the soft X-ray, w1, w2, and m2 light curves in log–log space to remove the overall temporal decay trend. Our time series consists of the ratios of the observed fluxes over the best-fitting "continuum," where uncertainties have been propagated following standard practice. We find that all of the UV light curves show a high degree of correlation with p-values <0.01% for either the Spearman Rank test or the Kendall Tau test. We also find a hint for correlated behavior between the UV bands and the X-rays with limited significance corresponding to P ≳ 5% (Spearman Rank test). The correlation is stronger at δt < 30 days.

The key observational results are: (i) a very short rise time to peak, t_rise ∼ few days (Perley et al. 2019; Prentice et al. 2018). (ii) Large bolometric peak luminosity, ${L}_{\mathrm{pk},\mathrm{bol}}\sim 4\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, significantly more luminous than normal SNe and more luminous than some SLSNe (Figure 1). (iii) Persistent blue colors, with lack of evidence for cooling at δt ≳ 30 days (the effective temperature remains > 15,000 K). (iv) Large blackbody radius R_bb ∼ 8 × 10¹⁴ cm inferred at δt ∼ 3 days (Figure 9). (v) Persistent optically thick UV/optical emission with no evidence for transition into a nebular phase at δt < 90 days (Figure 2). (vi) The spectra evolve from a hot, blue, and featureless continuum around the optical peak, to very broad features with v ∼ 0.1 c at δt ∼ 4–15 days (Figure 2). (vii) Redshifted H and He features emerge at δt > 20 days with significantly lower velocities v ∼ 4000 km s⁻¹ (Figure 2), implying an abrupt change of the velocity of the material which dominates the emission. The centroid of the line emission is offset to the red with v ∼ 1000 km s⁻¹ (Figure 2). (viii) There is evidence for an NIR excess of the emission with respect to a blackbody model from early to late times, as pointed out by Perley et al. (2019; Figure 11).

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3.1.1. Engine-powered Transient

For optical/UV emission powered by the diffusion of thermal radiation from an initially compact opaque source, the light curve rises and peaks on the diffusion timescale (Arnett 1982),

$$\begin{eqnarray}&&{t}_{\mathrm{pk}}\approx {\left(\displaystyle \frac{{M}_{\mathrm{ej}}\kappa }{4\pi {v}_{\mathrm{ej}}c}\right)}^{1/2}\approx 2.7\,\mathrm{days}{\left(\displaystyle \frac{{M}_{\mathrm{ej}}}{0.3{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{v}_{\mathrm{ej}}}{0.1c}\right)}^{-1/2},\end{eqnarray} \tag{ 1 }$$

where we use κ = 0.1 cm² g⁻¹ as an estimate of the effective opacity due to electron scattering or Doppler-broadened atomic lines. For t_pk ∼ 2–4 days (Figure 1), Equation (1) implies a low ejecta mass M_ej ∼ 0.1–0.5 M_⊙ and high ejecta velocity v_ej ∼ 0.05–0.1c to match the inferred blackbody radius at early times, corresponding to a kinetic energy of the optical/UV-emitting material E_k ∼ 10^50.5–10^51.5 erg (consistent with the inferences by Perley et al. 2019 and Prentice et al. 2019). The low ejecta mass immediately excludes light-curve models powered by ⁵⁶Ni decay, which would require ${M}_{\mathrm{Ni}}\gt 6\,{M}_{\odot }$ to reproduce the large peak luminosity of AT 2018cow, and instead demands another energy source. We note that M_ej ∼ 0.1–0.5M_⊙ should be viewed as a constraint on the fast-moving ejecta mass that participates in the production of the high-luminosity peak. As we will see in the next sections, the phenomenology of AT 2018cow requires the presence of additional, slower material preferentially distributed in an equatorial belt.

One potential central energy source is an "engine," such as a millisecond magnetar or accreting black hole, which releases a total energy ${E}_{{\rm{e}}}$ over its characteristic lifetime, t_e. The engine deposits energy into a nebula behind the ejecta at a rate

$$\begin{eqnarray}&&{L}_{{\rm{e}}}(t)=\displaystyle \frac{{E}_{{\rm{e}}}}{{t}_{{\rm{e}}}}\displaystyle \frac{(\alpha -1)}{{\left(1+t/{t}_{{\rm{e}}}\right)}^{\alpha }},\end{eqnarray} \tag{ 2 }$$

where α = 2 for an isolated magnetar (Spitkovsky 2006), α = 2.38 for an accreting magnetar (Metzger et al. 2018), α = 5/3 for fallback in a TDE (e.g., Rees 1988; Phinney 1989), and α < 5/3 in some supernova fallback models (e.g., Coughlin et al. 2018) or viscously spreading disk accretion scenarios (e.g., Cannizzo et al. 1989). In engine models, the late-time decay of the bolometric luminosity obeys L_opt ∝ L_e ∝ t^−α, such that the measured value of α ≈ 2–2.5 (Section 2.8, Figure 9) would be consistent with a magnetar engine.⁴⁴ However, given the uncertainties in the bolometric correction, the TDE/supernova fallback case (α = 5/3) may also be allowed. Finally, the "engine" may not be a compact object at all, but rather a deeply embedded radiative shock, produced as the ejecta interact with a dense medium.

As a concrete example, consider an isolated magnetar (α = 2), which at times t ≫ t_e obeys L_e ≈ (E_et_e)/t². Assuming that most of the energy is released over t_e ≪ t_pk (as justified by the narrowly peaked light-curve shape) and that most of the engine energy is not radiated, but instead used to accelerate the ejecta to its final velocity v_ej, then ${E}_{{\rm{e}}}\lesssim {M}_{\mathrm{ej}}{v}_{\mathrm{ej}}^{2}/2$, with ${E}_{{\rm{rad}}}\approx {\int }_{{t}_{{\rm{pk}}}}^{\infty }{L}_{{\rm{e}}}{dt}\approx {E}_{{\rm{e}}}{t}_{{\rm{e}}}/{t}_{{\rm{pk}}}$, from which it follows that

$$\begin{eqnarray}&&{t}_{{\rm{e}}}\gtrsim \displaystyle \frac{2{E}_{\mathrm{rad}}}{{M}_{\mathrm{ej}}{v}_{\mathrm{ej}}^{2}}{t}_{\mathrm{pk}}\sim 3\times {10}^{3}{\rm{s}}{\left(\displaystyle \frac{{v}_{\mathrm{ej}}}{0.1c}\right)}^{-2}{\left(\displaystyle \frac{{M}_{\mathrm{ej}}}{0.3{M}_{\odot }}\right)}^{-1}.\end{eqnarray} \tag{ 3 }$$

Here we used the total radiated UVOIR energy, E_rad ∼ 10⁵⁰ erg from Table 1. The engine is thus relatively constrained: it must release a total energy E_e that varies from E_rad ∼ 10⁵⁰ erg to E_k ∼ 10^50.5–10^51.5 erg, most of it over a characteristic lifetime t_e ∼ 10³–10⁵ s.

We conclude with constraints on the properties of a $\mathrm{Ni}$-powered transient that might be hiding within the central-engine-dominated emission. In the context of the standard Arnett (1982) modeling, modified following Valenti et al. (2008), and assuming a standard SN-like explosion with M_ej ∼ a few M_☉ and E_k ∼ 10⁵¹–10⁵² erg, we place a limit M_Ni < 0.06 M_☉ to not overproduce the observed optical–UV thermal emission, consistent with Perley et al. (2019). The limit becomes significantly less constraining, M_Ni < 0.2–0.4 M_☉, if we allow for smaller M_ej ∼ 0.5 M_☉, as in this case the diffusion timescale is shorter (i.e., the transient emission peaks earlier), the γ-rays from the ⁵⁶Ni decay are less efficiently trapped and thermalized within the ejecta, and the transient enters the nebular phase earlier. However, the large M_Ni/M_ej would result in red colors as the UV emission would be heavily suppressed via iron-line blanketing, which is not observed. The observed blue colors (Figure 1) indicate instead that emission from a Ni-powered transient with small ejecta is never dominant, which implies M_Ni ≲ 0.1 M_☉.

3.1.2. Shock Breakout

Alternatively, we consider the possibility that the high luminosity and rapid evolution of AT 2018cow result from a shock breakout from a radially extended progenitor star, an inflated progenitor star, or thick medium (i.e., if the star experiences enhanced mass loss just before stellar death). Shock breakout scenarios have been invoked to explain some fast-rising optical transients (e.g., Ofek et al. 2010; Drout et al. 2014; Arcavi et al. 2016; Shivvers et al. 2016; Tanaka et al. 2016).

For a typical SN shock velocity v_sh ≈ 10⁴ km s⁻¹ and the observed peak time of AT 2018cow, the inferred stellar radius is R_⋆ ≈ v_sht_pk ≳ 10¹⁴ cm ∼10 au, much larger than red supergiant stars. Furthermore, the explosion of such a massive star is expected to be followed by a longer plateau phase not observed in the monotonically declining light curve of AT 2018cow. We conclude that shock breakout from a stellar progenitor is not a viable mechanism for AT 2018cow.

The effective radius of a massive star could be increased just prior to its explosion by envelope inflation or enhanced mass loss timed with stellar death, as observed in a variety of SNe (e.g., Smith 2014). Assuming an external medium with a wind-like density profile ${\rho }_{w}={\dot{M}}_{w}/(4\pi {v}_{w}{r}^{2})=A/{r}^{2}$ and radial optical depth ${\tau }_{w}={\int }_{r}^{\infty }{\rho }_{{\rm{w}}}\kappa {dr}\simeq {\rho }_{w}r\kappa$, the photon diffusion timescale is

$$\begin{eqnarray}&&{t}_{\mathrm{pk},{\rm{w}}}\sim {\tau }_{w}\displaystyle \frac{r}{c}\approx \displaystyle \frac{A\kappa }{c}\approx 1.9\,\mathrm{day}\left(\displaystyle \frac{A}{{10}^{5}{A}_{\star }}\right),\end{eqnarray} \tag{ 4 }$$

where A_⋆ = 5 × 10¹¹ g cm⁻¹ (i.e., A = A_⋆ for the standard mass-loss rate ${\dot{M}}_{w}={10}^{-5}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and wind velocity v_w = 10³ km s⁻¹; e.g., Chevalier & Li 2000). The luminosity of the radiative shock ${L}_{\mathrm{sh}}=(9\pi /8){\rho }_{{\rm{w}}}{v}_{\mathrm{sh}}^{3}{r}^{2}$ at the breakout radius $r={{ct}}_{\mathrm{pk},{\rm{w}}}/{\tau }_{w}$ is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{sh}}({t}_{\mathrm{pk},{\rm{w}}}) & \simeq & \displaystyle \frac{9\pi }{8}{v}_{\mathrm{sh}}^{3}\displaystyle \frac{{{ct}}_{\mathrm{pk}}}{{\kappa }_{\mathrm{opt}}}\\ & \approx & 2\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\left(\displaystyle \frac{{t}_{\mathrm{pk}}}{2\,\mathrm{days}}\right){\left(\displaystyle \frac{{v}_{\mathrm{sh}}}{{10}^{4}\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3}.\end{array}\end{eqnarray} \tag{ 5 }$$

From Equations (5) and (4), we conclude that a shock breakout from an extended medium with density structure corresponding to an effective mass-loss rate A ∼ 10⁵A_⋆ can explain both the timescale and peak luminosity of the optical emission from AT 2018cow. Following the initial breakout, radiation from deeper layers of the expanding shocked wind ejecta would continue to produce emission. However, such a cooling envelope is predicted to redden substantially in time (e.g., Nakar & Piro 2014), in tension with the observed persistently blue optical/UV colors and lack of cooling at δt > 20 days (Figure 9). We conclude that, even if a shock breakout is responsible for the earliest phases of the optical emission and for accelerating the fastest ejecta layers, a separate, more deeply embedded energy source is needed at late times to explain the properties of AT 2018cow.

3.1.3. Reprocessing by Dense Ejecta and the Spectral Slope of the Optical Continuum Emission

We argued in previous sections that the sustained blue emission from AT 2018cow is likely powered by the reprocessing of radiation from a centrally located X-ray source embedded within the ejecta. Here we discuss details of the reprocessing picture and what can be learned about the ejecta structure of AT 2018cow.

Late-time optical spectra at δt > 20 days (Figure 2) show line widths of ∼4000 km s⁻¹ (∼0.01c), indicating substantially lower outflow velocities than at earlier times (when v ∼ 0.1c), and an abrupt transition from very high velocity to lower velocity emitting material (Figure 9). Although it might be possible to explain this phenomenology in a spherically symmetric model with a complex density profile, a more natural explanation is that the ejecta/circumstellar medium (CSM) of AT 2018cow is aspherical, e.g., with fast-expanding material along the polar direction and slower expanding dense matter in the equatorial plane (Figure 12). This picture is independently supported by the observed properties of X-ray emission discussed and by the emission-line profiles, as discussed in Sections 3.3 and 3.1.4.⁴⁵

Although we approximated the UV/optical spectral energy distribution (SED) with a blackbody function in Section 2.8, the true SED is likely to deviate from a single blackbody spectrum. The observed slope of the early optical SED, L_ν ∝ ν^1.2, can be used to constrain the ejecta stratification in reprocessing models. Ignoring Gaunt factors and using the result from radiative transfer calculations that do not assume local thermodynamic equilibrium (LTE), which indicate that in the outer layers of the reprocessor the free electron temperature T_e tend to level off to a constant value much greater than the effective temperature (e.g., Hubeny et al. 2000; Roth et al. 2016), the free–free emissivity in the optical to infrared is ${j}_{\nu }^{\mathrm{ff}}\propto {n}_{e}{n}_{i}\propto {\rho }^{2}$ (where n_e and n_i are the number density of electrons and positive ions, respectively, and we used the fact that hν ≪ kT_e). This result holds as long as the material is highly ionized.

While the bound electrons are coupled to the radiation field and are likely to be out of thermal equilibrium, the free electrons should be in LTE, so that ${j}_{\nu }^{\mathrm{ff}}={\alpha }_{\nu }^{\mathrm{ff}}{B}_{\nu }({T}_{e})$. For hν ≪ kT_e, we find ${\alpha }_{\nu }^{\mathrm{ff}}\propto {\rho }^{2}{\nu }^{-2}$. We assume that near the surface of the emitting material the density can be locally modeled by a power law in radius ρ ∝ r⁻ⁿ, for some n > 1. Due to the ionization from the engine, electron scattering dominates the opacity. The total optical depth (integrated from the outside in) is then wavelength-independent and ${\tau }_{{\rm{es}}}(r)\sim \left(1/(n-1)\right){\rho }_{0}{\kappa }_{{\rm{es}}}\,{r}_{0}^{n}\,{r}^{1-n}$, where r₀ is some reference radius within the region where the power-law expression for the density holds, and ρ₀ ≡ ρ(r₀). Let α^es and ${\alpha }_{\nu }^{\mathrm{abs}}$ denote the opacity coefficients from electron scattering and continuum absorption, respectively. We define an opacity ratio ${\epsilon }_{\nu }={\alpha }_{\nu }^{{\rm{abs}}}/({\alpha }^{{\rm{es}}}+{\alpha }_{\nu }^{{\rm{abs}}})\approx {\alpha }_{\nu }^{{\rm{abs}}}/{\alpha }^{{\rm{es}}}\propto \rho {\nu }^{-2}$.

The effective optical depth to absorption is ${\tau }_{\mathrm{eff}}(\nu )\sim \sqrt{{\epsilon }_{\nu }}{\tau }_{\mathrm{es}}$, where τ_es is measured from the outside in, and we evaluate _ν at the thermalization depth ${r}_{\nu ,\mathrm{therm}}$, which is the radius where τ_eff(ν) = 1. We define ${\tau }_{\mathrm{es}}({r}_{\nu ,\mathrm{therm}})\equiv {\tau }_{\nu ,\mathrm{therm}}$. It follows that ${\tau }_{\nu ,{\rm{therm}}}\approx 1/\sqrt{{\epsilon }_{\nu }}\propto {\rho }^{-1/2}{\nu }^{1}$, which implies that the thermalization radius scales with ν as ${r}_{\nu ,\mathrm{therm}}\,\propto \,{\nu }^{\tfrac{2}{2-3n}}$. Following Roth et al. (2016), we can approximate the observed spectrum as ${L}_{\nu }\approx 4\pi {\int }_{{r}_{\nu ,\mathrm{therm}}}^{\infty }{j}_{\nu }\,4\pi {r}^{2}{dr}$. Substituting the scalings above, we find⁴⁶

$$\begin{eqnarray}&&{L}_{\nu }\approx \displaystyle \frac{{\left(4\pi \right)}^{2}}{2n-3}\,{j}_{\nu }\left({r}_{\nu ,\mathrm{therm}}\right){r}_{\nu ,\mathrm{therm}}^{3}\propto {r}_{\nu ,\mathrm{therm}}^{3-2n}\propto {\nu }^{\tfrac{4n-6}{3n-2}}.\end{eqnarray} \tag{ 6 }$$

For n = 2, we have ${L}_{\nu }\,\propto \,{\nu }^{1/2}$, in reasonable agreement with the results by Roth et al. (2016). For large n, this tends toward an asymptotic scaling L_ν ∝ ν^4/3, which is similar to the measured slope of the optical continuum of AT 2018cow.

We conclude that in AT 2018cow optical continuum radiation is reprocessed in a layer with a steep density gradient n ≫ 1. Our derivation also indicates that the spectral slope should be roughly independent of the luminosity of the engine, as is observed, as long as the high ionization state is maintained.

3.1.4. Spectral Line Formation

The key observational results are (i) an optically thick spectrum with F_ν ∝ ν² at ν < 100 GHz at t < 35 days. The spectrum later develops an excess of emission at low frequencies, which was noted by Ho et al. (2019), and that makes the optically thick spectrum less steep, F_ν ∝ ν^1.2. (ii) The spectral peak frequency cascades down with time. Differently from normal radio SNe, the spectral peak flux also markedly evolves with time, with F_pk ∝ t^−1.7±0.1 at t > 80 days. (iii) With L_ν ∼ 4 × 10²⁷ erg s⁻¹ Hz⁻¹ at ν ∼ 9 GHz around δt ∼ 20 days and a rising emission with time until t ∼ 100 days, the radio emission from AT 2018cow is also markedly different from GRBs in the local universe and cosmological GRBs (Figure 8).

3.2.1. Radio Emission from External Shock Interaction

The observed radio emission from AT 2018cow in the first weeks (F_ν ∝ ν² at ν < 100 GHz; Figure 7) is consistent with being self-absorbed synchrotron radiation likely produced from an external shock generated as the ejecta interacts with a dense external medium, as observed in radio SNe (e.g., Soderberg et al. 2005, 2012). While we leave the detailed modeling of the broadband radio emission from AT 2018cow to a dedicated paper (D. L. Coppejans et al. 2018, in preparation), here we present basic inferences on the physical properties of its radio-emitting material obtained by adopting the standard modeling of self-absorbed synchrotron emission in SNe by Chevalier (1998).

In the context of self-absorbed synchrotron emission from a freely expanding blast wave propagating into a wind medium, the observed peak luminosity ${L}_{\nu ,\mathrm{pk}}$, spectral peak frequency ν_pk, and peak time t_pk directly constrain the environment density, blast wave velocity, magnetic field, and energy. Following Chevalier (1998) and Soderberg et al. (2005), in Figure 13 we show that the measured peak flux and peak frequency of the radio SEDs at t ∼ 80–150 days (corresponding to ${L}_{\nu ,\mathrm{pk}}\leqslant 4\times {10}^{28}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}$ and ν_pk < 20 GHz) imply a shock/outer ejecta velocity ${v}_{\mathrm{sh}}\sim 0.1c$ interacting with a dense medium with an effective $A\sim 10\mbox{--}100{A}_{\star }$ (depending on _B = 0.1–0.01). The corresponding wind density is

$$\begin{eqnarray}&&{n}_{w}=\displaystyle \frac{{\rho }_{w}}{{m}_{p}}=\displaystyle \frac{\dot{M}}{4\pi {v}_{w}{r}^{2}{m}_{p}}\approx 9\times {10}^{6}{\mathrm{cm}}^{-3}\left(\displaystyle \frac{A}{100{A}_{\star }}\right){v}_{0.1}^{-2}{t}_{\mathrm{wk}}^{-2},\end{eqnarray} \tag{ 7 }$$

where we have taken r = vt as the shock radius, ${v}_{0.1}\,\equiv {v}_{\mathrm{sh}}/0.1c,{t}_{\mathrm{wk}}\equiv t/\mathrm{week}$. For these parameters, the equipartition energy (which is a lower limit to the kinetic energy of radio-emitting material) is E_eq ∼ 2 × 10⁴⁸ erg and the magnetic field decreases from ∼1 G to ∼0.4 G from t ∼ 80 days to 150 days. From the radio SED around 20 days presented in Ho et al. (2019; point with the highest luminosity in Figure 13), we infer a similar shock velocity of ${v}_{\mathrm{sh}}\sim 0.1c$, indicating limited deceleration of the blast wave into the environment.

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Shock velocities ∼0.1 c are common among normal stripped-envelope radio SNe (Figure 13). The values $A\sim 10\mbox{--}100{A}_{\star }$ needed to explain the luminosity of the radio emission of AT 2018cow are similar to those inferred in previous radio-bright SNe (e.g., SN 2003L; Chevalier & Fransson 2006), but substantially smaller than the values $A\sim {10}^{5}{A}_{\star }$ needed on smaller radial scales to explain the early optical peak if the latter is powered by shock breakout from a wind (Equations (4), (5)). The velocity of the fastest ejecta inferred from the radio ∼0.1c is consistent with that needed to explain the rapid optical rise time (Equation (1)).

We can further constrain the environment density using the lack of evidence for a low-frequency cutoff in the radio spectrum (Figure 7) due to free–free absorption (e.g., Weiler et al. 2002), as follows. The optical depth of the forward shock to Thomson scattering and free–free absorption is given, respectively, by

$$\begin{eqnarray}&&{\tau }_{{\rm{T}}}\simeq \displaystyle \frac{\dot{M}{\kappa }_{\mathrm{es}}}{4\pi {v}_{w}r}\approx 0.01\left(\displaystyle \frac{A}{100{A}_{\star }}\right){v}_{0.1}^{-1}{t}_{\mathrm{wk}}^{-1},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\tau }_{\mathrm{ff}}\simeq \displaystyle \frac{{\alpha }_{\mathrm{ff}}r}{3}\approx 10{\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-2}{\left(\displaystyle \frac{{T}_{{\rm{g}}}}{{10}^{6}{\rm{K}}}\right)}^{-3/2}{\left(\displaystyle \frac{A}{100{A}_{\star }}\right)}^{2}{v}_{0.1}^{-3}{t}_{\mathrm{wk}}^{-3},\end{eqnarray} \tag{ 9 }$$

where we have taken κ_es = 0.38 cm² g⁻¹ for fully ionized solar-composition ejecta and ${\alpha }_{\mathrm{ff}}\approx 0.03{n}_{w}^{2}{\nu }^{-2}{T}_{{\rm{g}}}^{-3/2}$ cm⁻¹ as the free–free absorption coefficient, where T_g is the temperature of the gas, normalized to a value T_g ≳ 10⁶ K, following the argument by Ho et al. (2019).

From Figure 7, the optically thick part of the radio spectrum, which scales as F_ν ∝ ν², without any evidence for free–free absorption, demands τ_ff(15 GHz) ≪1 at t = 6.5 days and τ_ff(5 GHz) ≪ 1 at t = 12 days. These limits translate into similar upper limits on the environment density:

$$\begin{eqnarray}&&\displaystyle \frac{A}{{A}_{\star }}\lt 1000{v}_{0.1}^{3/2}{\left(\displaystyle \frac{{T}_{{\rm{g}}}}{{10}^{6}{\rm{K}}}\right)}^{3/4},\end{eqnarray} \tag{ 10 }$$

consistent with the values $A\sim 10\mbox{--}100{A}_{\star }$ for ${v}_{\mathrm{sh}}\sim 0.1c$ (Figure 13).

We conclude with considerations of the shock microphysics parameters and the properties of the distribution of electrons responsible for the radio emission. The radio emission is produced by relativistic electrons accelerated into a power-law distribution at the forward shock, e.g., ${dN}/d{\gamma }_{e}\propto {\gamma }_{e}^{-p}$ with p ≥ 2. The Lorentz factor γ_e = γ_ν of the electrons which contribute at the radio frequency $\nu =0.3({eB}{\gamma }_{e}^{2}/2\pi {m}_{e}c)$ is

$$\begin{eqnarray}\begin{array}{rcl}{\gamma }_{\nu } & \simeq & 4.6{\left(\displaystyle \frac{{m}_{e}c\nu }{{eB}}\right)}^{1/2}\\ & \approx & 153{\epsilon }_{{\rm{B}},-2}^{-1/4}\,{t}_{\mathrm{wk}}^{1/2}{\left(\displaystyle \frac{\nu }{100\mathrm{GHz}}\right)}^{1/2}{\left(\displaystyle \frac{A}{100{A}_{\star }}\right)}^{-1/4},\end{array}\end{eqnarray} \tag{ 11 }$$

where we have estimated the magnetic field behind the shock to be

$$\begin{eqnarray}&&{B}_{\mathrm{sh}}={\left(6\pi {\epsilon }_{B}{\rho }_{{\rm{w}}}{v}_{\mathrm{sh}}^{2}\right)}^{1/2}\approx 5.1{\rm{G}}{\epsilon }_{{\rm{B}},-2}^{1/2}\,{t}_{\mathrm{wk}}^{-1}{\left(\displaystyle \frac{A}{100{A}_{\star }}\right)}^{1/2}.\end{eqnarray} \tag{ 12 }$$

From Figure 13, for _B ≪ 0.01, we would require larger values $A\gtrsim 100{A}_{\star }$, incompatible with the observed lack of free–free absorption (Equation (10)).

As shown in Section 3.3.1, electrons with γ_e ∼ γ_ν responsible for the optically thin radio emission ν ≳ ν_sa cool rapidly due to inverse-Compton (IC) emission in the optical/UV radiation of AT 2018cow. Therefore, above ν ≳ ν_sa, the fast-cooling scaling holds (e.g., Granot & Sari 2002):

$$\begin{eqnarray}&&{F}_{\nu }={\nu }^{-p/2}\mathop{\approx }\limits_{p=2-3}{\nu }^{-1}-{\nu }^{-1.5}.\end{eqnarray} \tag{ 13 }$$

Our radio SEDs at t > 80 days, with F_ν ∝ ν^−1.4±0.1 above ν_sa confirms this inference (Figure 7). The predicted luminosity in the optical/NIR band (ν ≳ 10¹⁴ Hz ≫ν_sa) is thus similar to, or smaller than, the radio luminosity νL_ν ∼ a few ×10⁴⁰ erg s⁻¹ and thus is insufficient to explain the possible non-thermal NIR excess of luminosity ∼10⁴² erg s⁻¹ identified by Perley et al. (2019). Any non-thermal component in the IR/optical range cannot be from the same synchrotron source as the forward shock that produces the radio emission at ν < 100 GHz.

The radio emission at ν > 100 GHz reported at early times is also more luminous than predicted by our model (Figure 7). This observation might suggest that the very early radio data at ν > 300 GHz might be dominated by a separate emission component (e.g., reverse shock) if it is physically associated with AT 2018cow (Figure 7).

3.2.2. Constraints on Off-axis Relativistic Jets

The observed radio emission is consistent with arising from non-relativistic ejecta with velocity similar to that of normal SNe (v_sh ∼ 0.1c) interacting with dense CSM with A ∼ 10–100A_⋆. No high-energy prompt emission was detected in association with AT 2018cow (Section 2.7). However, AT 2018cow showed evidence for broad spectral features in the optical emission, with velocities comparable to and even larger than those seen in broad-lined SNe Ic associated with GRBs (Modjaz et al. 2016, Figure 2). In this section, we constrain the properties of an off-axis jet in AT 2018cow. Emission from a collimated outflow originally pointed away from our line of sight becomes detectable as the blast wave decelerates into the environment and relativistic beaming of the radiation becomes less severe with time (e.g., Granot et al. 2002).

The observed radio emission from an off-axis jet primarily depends on the jet-opening angle (${\theta }_{j}$), off-axis angle (θ_obs), jet isotropic equivalent kinetic energy (${E}_{{\rm{k}},\mathrm{iso}}$), environment density (parametrized as n and $\dot{M}$ for an ISM and wind-like medium, respectively), and shock microphysical parameters (_B, _e). We employ realistic simulations of relativistic jets propagating into an ISM and wind-like medium to capture the effects of lateral jet spreading with time, finite jet-opening angle, and transition into the non-relativistic regime. We ran the code BOXFIT (v2; van Eerten et al. 2010, 2012) at 1.4 GHz, 9 GHz, 15 GHz, and 34 GHz for a range of representative parameters of long GRB jets (${E}_{{\rm{k}},\mathrm{iso}}\,={10}^{50}-{10}^{55}\,\mathrm{erg}$, ${\epsilon }_{B}={10}^{-4}-{10}^{-2},{\epsilon }_{e}=0.1$) and environment densities ($n={10}^{-3}-{10}^{2}\,{\mathrm{cm}}^{-3},\dot{M}={10}^{-8}-{10}^{-3}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$). We use p = 2.5 and explore the parameter space for two jets of θ_j = 5° and θ_j = 30°, representative of strongly collimated and less collimated outflows, respectively (as found for normal GRBs and low-energy GRBs; e.g., Racusin et al. 2009; Margutti et al. 2013b; Ryan et al. 2015).

With reference to Figure 14, we find that less collimated outflows with θ_j = 30° are ruled out in the ISM case for large densities n > 1 cm⁻3. For a wind-type medium with ${\dot{M}}_{w}\sim {10}^{-3}-{10}^{-4}{M}_{\odot }\,{\mathrm{yr}}^{-1}$, consistent with the values A ∼ 10–100A_⋆ inferred for the forward shock radio emission, jets with ${E}_{{\rm{k}},\mathrm{iso}}\geqslant {10}^{52}$ erg are presently ruled out for _e = 0.1, _B = 0.01, and jet-opening angles θ_j ≈ 5°–30°, corresponding to beaming-corrected jet energies E_k ≥ 4 × 10⁴⁹ erg (for θ_j = 5°) or E_k ≥ 10⁵¹ erg (for θ_j = 30°). Successful jets with E_j < 4 × 10⁴⁹ erg (θ_j = 5°) and E_j < 10⁵¹ erg (θ_j = 30°) propagating into a wind medium with $\dot{M}\lt {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ are allowed.

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The key observational results are as follows: (i) luminous X-ray emission discovered at the level of L_X ∼ 10⁴³ erg s⁻¹. L_x is significantly larger than seen in normal SNe and is similar to the values seen in GRBs in the local universe (Figure 5). (ii) Persistent X-ray flaring with short variability timescales of a few days superimposed on a secular decay, which is initially gradual ∝t⁻¹ but then steepens around δt ∼ 25 days to a faster decay ∝t⁻⁴ around the same time as the appearance of narrow optical features. (iii) Presence of two X-ray components of emission with distinct temporal evolution and spectral properties: a persistent source in the $\gt 0.1\,\mathrm{keV}$ range, as well as a transient component of hard X-ray emission at energies >10 keV detected at δt ∼ 8 days and which disappears by δt ∼ 17 days (Figure 6). (iv) The persistent X-ray spectral component of the emission is well modeled by F_ν ∝ ν^−β with β ∼ 0.5 with no evidence for intrinsic neutral hydrogen absorption (Figure 4).

Below we discuss the physical origin of the X-ray emission associated with AT 2018cow.

3.3.1. X-Ray Emission from External Shock Interaction

We first consider the possibility that the X-rays originate from the same forward shock responsible for the radio synchrotron emission (Section 3.2.1). The kinetic luminosity of the radio-emitting forward shock,

$$\begin{eqnarray}&&{L}_{\mathrm{sh}}=4\pi {r}^{2}\displaystyle \frac{9}{32}{v}_{\mathrm{sh}}^{3}{\rho }_{{\rm{w}}}\approx 5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}{v}_{0.1}^{3}\left(\displaystyle \frac{A}{100{A}_{\star }}\right),\end{eqnarray} \tag{ 14 }$$

is close to the X-ray luminosity of AT 2018cow (Figure 4). This suggests a picture in which the X-rays are IC emission from optical/UV photons upscattered by relativistic electrons accelerated at the forward shock. Further supporting this scenario, the radio-to-X-ray luminosity ratio ${\nu }_{{\rm{s}}{\rm{a}}}{L}_{{\nu }_{{\rm{s}}{\rm{a}}}}/{L}_{{\rm{X}}}\sim 0.02$ is comparable to the ratio of the magnetic energy density ${U}_{B}={B}_{\mathrm{sh}}^{2}/8\pi$ (Equation (12)) to the optical/UV photon energy density ${U}_{\gamma }={L}_{\mathrm{opt}}/(4\pi {{cr}}_{\mathrm{sh}}^{2})$, where r_sh = v_sht,

$$\begin{eqnarray}&&\displaystyle \frac{{U}_{{\rm{B}}}}{{U}_{\gamma }}\simeq 0.02{\epsilon }_{{\rm{B}},-2}{v}_{0.1}^{2}{t}_{\mathrm{wk}}^{2}\left(\displaystyle \frac{A}{100{A}_{\star }}\right),\end{eqnarray} \tag{ 15 }$$

where ${L}_{\mathrm{opt}}=8\times {10}^{43}{t}_{\mathrm{wk}}^{-2}$ erg s⁻¹.

However, the IC forward shock model cannot naturally explain the F_ν ∝ ν^−0.5 spectrum of the persistent X-ray component, offers no consistent explanation of the transient hard X-ray component, and has difficulties accounting for the observed short-timescale variability, as we detail below.

Electrons heated or accelerated at the shock cool in the optical radiation field on the expansion timescale for electron Lorentz factors above the critical value,

$$\begin{eqnarray}&&{\gamma }_{{\rm{c}}}\simeq \displaystyle \frac{3{m}_{e}c}{4{\sigma }_{{\rm{T}}}{U}_{\gamma }t}=\displaystyle \frac{3\pi {r}^{2}{m}_{e}{c}^{2}}{{\sigma }_{{\rm{T}}}{L}_{\mathrm{opt}}t}\mathop{\approx }\limits_{r={vt}}1.0{v}_{0.1}^{2}{t}_{\mathrm{wk}}^{3},\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{{\rm{T}}}\simeq 6.6\times {10}^{-25}$ cm² is the Thomson cross-section. The electrons responsible for upscattering optical/UV seed photons of energy ${\epsilon }_{\mathrm{opt}}\simeq 3{{kT}}_{\mathrm{opt}}=5\mathrm{eV}({T}_{\mathrm{opt}}/2\times {10}^{4}{\rm{K}})$ eV to X-ray energy ${E}_{{\rm{X}}}=(4/3){\epsilon }_{\mathrm{opt}}{\gamma }_{{\rm{X}}}^{2}$ must possess Lorentz factors

$$\begin{eqnarray}&&{\gamma }_{{\rm{X}}}\approx {\left(\displaystyle \frac{3{E}_{{\rm{X}}}}{4{\epsilon }_{\mathrm{opt}}}\right)}^{1/2}\approx 12{\left(\displaystyle \frac{{E}_{{\rm{X}}}}{1\mathrm{keV}}\right)}^{1/2}.\end{eqnarray} \tag{ 17 }$$

The values γ_X ≈ 6–40 needed to populate the XRT bandpass 0.3–10 keV, though lower than those producing the millimeter radio emission (Equation (11)), are in the fast-cooling regime ≳γ_c for the first few weeks of evolution. Thus, while for slow-cooling electrons the observed F_ν ∝ ν^−0.5 spectrum would match the expectation ${F}_{\nu }\propto {\nu }^{-(p-1)/2}\approx {\nu }^{-0.5}-{\nu }^{-1}$ for p = 2–3, it is incompatible with the fast-cooling expectation, F_ν ∝ ν^−p/2, which gives a much softer spectrum than observed, ${F}_{\nu }\approx {\nu }^{-1}-{\nu }^{-1.5}$, for p = 2–3.

We now consider an IC origin of the transient hard X-ray component of emission, which shows a rising slope of F_ν ∝ ν^0.5 (Figure 6). This emission is too hard to be free–free or synchrotron radiation (it violates the "synchrotron death line"; e.g., Rybicki & Lightman 1979), possibly hinting at an IC origin. In addition to accelerating electrons into a non-thermal distribution, the forward shock is also predicted to heat electrons (Sironi et al. 2015), generating a relativistic Maxwellian particle distribution with a mean thermal Lorentz factor,

$$\begin{eqnarray}&&{\gamma }_{\mathrm{th}}\approx {f}_{e}\displaystyle \frac{{m}_{p}}{2{m}_{e}}{\left(\displaystyle \frac{{v}_{\mathrm{sh}}}{c}\right)}^{2}\approx 4.6({f}_{e}/0.5){v}_{0.1}^{2},\end{eqnarray} \tag{ 18 }$$

where f_e = 0.5 is the fraction of the shock energy imparted to the electrons.⁴⁷ Thus, it may be tempting to associate the transient hard X-ray "bump" with IC emission by a relativistic Maxwellian distribution of electrons. However, the expected spectral peak would occur at an energy

$$\begin{eqnarray}&&{E}_{{\rm{X}},\mathrm{th}}=\displaystyle \frac{4}{3}{\gamma }_{\mathrm{th}}^{2}{\epsilon }_{\mathrm{opt}}\approx 0.14({f}_{e}/0.5){v}_{0.1}^{2}\left(\displaystyle \frac{{T}_{\mathrm{opt}}}{2\times {10}^{4}\,{\rm{K}}}\right)\mathrm{keV},\end{eqnarray} \tag{ 19 }$$

which is a factor of ≲100 smaller than the observed peak E_X ≈ 50 keV.

A final problem of the forward shock model is the rapid and persistent X-ray variability, which in this model has to be attributed to density inhomogeneities in the environment (e.g., a series of thin shells or "clumps"). The shortest allowed variability time Δt if the ejecta cover a large fraction of the solid angle is the light-crossing time, which for a shock of radius r_sh = v_sht with v_sh ≈ 0.1–0.2c (Section 3.2.1) is constrained to obey

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}t}{t}\gtrsim \displaystyle \frac{{r}_{\mathrm{sh}}}{{ct}}\sim \displaystyle \frac{{v}_{\mathrm{sh}}}{c}\approx 0.1\mbox{--}0.2.\end{eqnarray} \tag{ 20 }$$

We measure the properties of the X-ray flares in AT 2018cow following the same procedure as is used for long GRBs, adopting a Norris et al. (2005) profile. We find that the X-ray flares in AT 2018cow show much faster variability and violate this expectation, and furthermore show no evidence for a linear increase of their duration as the blast wave expands, contrary to expectations (Figure 15). Instead, our analysis in Section 2.9 suggests the presence of a dominant timescale of variability of a few days. In Figure 15, we show that the X-ray variability observed in AT 2018cow is also not consistent with the expectations from density fluctuations encountered by a relativistic jet (with an observation either on axis or off axis). Differently from Rivera Sandoval et al. (2018), we thus conclude that density fluctuations in the CSM environment of AT 2018cow are unlikely to be the physical cause of the observed X-ray variability. The very rapid turnoff of the X-ray emission as L_X ∝ t⁻⁴ at δt > 20 days (Figure 4) is also difficult to accommodate in models where the X-ray emission is powered by an external shock (the typical L_X decline is ∝t⁻¹ for a spherical blast wave and ∝t⁻² for a collimated outflow after "jet break," e.g., Granot & Sari 2002).⁴⁸

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3.3.2. X-Rays from a Central Hard X-Ray Source

We consider an alternative scenario in which the observed X-rays originate primarily from an internal hard radiation source (either in the form of shocks or a compact object; Figure 12), embedded within an aspherical, potentially bipolar ejecta shell. The asphericity of the ejecta is a key requirement to explain the observed X-ray properties. The high-density material at lower latitudes (blue region in Figure 12) is opaque to X-rays below ∼15 keV due to bound–free absorption. The observed X-rays in this energy range either escape directly through the highly ionized low-density polar ejecta (lighter gray shaded region in Figure 12) and/or are scattered into the line of sight by this material. The X-rays absorbed by the dense equatorial shell are reprocessed to lower frequencies and are powering the optical light curve. The polar cavity is initially narrow and grows with time as the ejecta dilutes, expands, and becomes progressively transparent.

This scenario provides a natural explanation of the transient hard X-ray spectral component of Figure 6 created by the combined effect of photoelectric absorption at soft X-ray energies ≲10 keV and Compton-downscattering of very hard X-rays at >50–100 keV. Here we assume transmission and reflection through neutral gas, and leave the discussion of Fe-line formation to the next section. The power-law spectrum that dominates at soft X-ray energies E < 10 keV is instead produced by X-ray photons that reach the observer without being absorbed or Compton-downscattered (i.e., these photons provide a direct view of the central engine). At high energies, the spectral shape of the observed spectrum is controlled by the Thomson optical depth along our line of sight τ_T. At early times, close to the optical peak at t_pk ∼ t_diff ≈ 3 days (Equation (1)), τ_T ∼ (c/v_ej)(κ_es/κ) ∼ 20–40, where ${v}_{\mathrm{ej}}\,\sim 0.1\mbox{--}0.2c$ and κ_es ∼ 4κ. At this time, nearly all of the UV/X-ray radiation of the central X-ray source is absorbed by the shell and reprocessed into optical/UV radiation. However, as the ejecta expands with time, its optical depth decreases τ_T ∝ t⁻², reducing the fraction of the central X-rays being absorbed. At the time of our first NuSTAR/INTEGRAL observations at δt ∼ 8 days, τ_T ∼ 3, resulting in moderate downscattering and attenuation of radiation at $E\geqslant 511\,\mathrm{keV}/{\tau }_{{\rm{T}}}^{2}\sim 50\,\mathrm{keV}$ (Figure 6). For the same parameters, we calculate τ_T ≲ 1 at δt ∼ 17 days, by which time Compton-downscattering plays a negligible role. This prediction is consistent with our observation of an uninterrupted power-law spectrum extending from 0.3 keV to ∼70 keV at δt ≥ 17 days (Figure 6).

The observed soft X-ray spectral shape is controlled by the bound–free optical depth τ_bf. Most opacity in the keV range is the result of photoelectric absorption by CNO elements, particularly the usually abundant oxygen. The bound–free opacity of oxygen is

$$\begin{eqnarray}&&{\kappa }_{\mathrm{bf}}=\displaystyle \frac{{f}_{{\rm{n}}}{X}_{{\rm{O}}}{\sigma }_{\mathrm{bf}}}{16{m}_{p}},\end{eqnarray} \tag{ 21 }$$

where X_O is the oxygen mass fraction, f_n is its neutral fraction, and

$$\begin{eqnarray}&&{\sigma }_{\mathrm{bf}}={\sigma }_{\mathrm{th}}{\left(\nu /{\nu }_{\mathrm{th}}\right)}^{-3},\nu \gtrsim {\nu }_{\mathrm{th}},\end{eqnarray} \tag{ 22 }$$

where for K-shell electrons of oxygen we have σ_th = (2–1) × 10⁻¹⁹ cm⁻² and hν_th = 0.74–0.87 keV. Following Metzger (2017), the neutral fraction in an ejecta shell of mass M_ej and radius R_ej = v_ejt due to photoionization by X-ray luminosity ${L}_{{\rm{X}}}\approx {\nu }_{\mathrm{th}}{L}_{{\nu }_{\mathrm{th}}}$ is

$$\begin{eqnarray}&&{f}_{{\rm{n}}}\approx \displaystyle \frac{{\alpha }_{\mathrm{rec}}{M}_{\mathrm{ej}}h{\nu }_{\mathrm{th}}}{{m}_{p}{L}_{{\rm{X}}}{v}_{\mathrm{ej}}t{\sigma }_{\mathrm{th}}}\approx \displaystyle \frac{4\pi {\alpha }_{\mathrm{rec}}{\tau }_{{\rm{T}}}({v}_{\mathrm{ej}}t)h{\nu }_{\mathrm{th}}}{{m}_{p}{L}_{{\rm{X}}}{\sigma }_{\mathrm{th}}{\kappa }_{\mathrm{es}}},\end{eqnarray} \tag{ 23 }$$

where for K-shells of oxygen α_rec ≈ 10⁻¹² cm³ s⁻¹ (Nahar & Pradhan 1997) for temperatures ∼10⁶ K characteristic of the X-ray photoheated gas (Ho et al. 2019). The X-ray optical depth at ν_th ∼ 1 keV is thus

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\mathrm{bf}} & = & {\tau }_{{\rm{T}}}\displaystyle \frac{{\kappa }_{\mathrm{bf}}}{{\kappa }_{\mathrm{es}}}\sim \displaystyle \frac{\pi }{4}\displaystyle \frac{{\alpha }_{\mathrm{rec}}}{{\kappa }_{\mathrm{es}}^{2}}\displaystyle \frac{{\tau }_{{\rm{T}}}^{2}{X}_{{\rm{O}}}}{{m}_{p}^{2}}\displaystyle \frac{h{\nu }_{\mathrm{th}}}{{L}_{{\rm{X}}}}{v}_{\mathrm{ej}}t\\ & \sim & 0.01\left(\displaystyle \frac{{X}_{{\rm{O}}}}{0.01}\right){\tau }_{{\rm{T}}}^{2}\left(\displaystyle \frac{t}{1\ \mathrm{week}}\right){v}_{0.1}{\left(\displaystyle \frac{{L}_{\mathrm{keV}}}{{10}^{43}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 24 }$$

For ejecta with an oxygen abundance close to solar abundance (X_O ∼ 0.01), ∼keV X-rays of luminosity ∼10⁴³ erg s⁻¹ ionize the ejecta sufficiently to escape unattenuated on timescales of a couple weeks when the Thomson column along the low-density polar region is also low τ_T ≲ 1. However, at earlier times, we expect τ_bf ≳ 1 when τ_T is larger, i.e., around the time of the NuSTAR/INTEGRAL spectral "hump" (Figure 6).

We quantitatively explore the predictions of our model by performing a series of Monte Carlo calculations where we follow the escape of photons as they propagate through a uniform shell of radius ${R}_{\mathrm{sh}}$ and thickness 2R_sh with a polar cavity of opening angle θ_sh = 30° carved into it. We assume an isotropic source with intrinsic spectrum F_ν ∝ ν^−0.5 as observed and self-consistently account for photoelectric absorption and Compton-scattering. Figure 16 shows the results for the transmitted X-rays for different lines of sight θ_obs and optical depth τ_T. Polar observing angles (i.e., small θ_obs) receive a larger fraction of "direct" X-rays (including X-rays reflected off the cavity walls; Figure 12) at any τ_T, while more equatorial views with larger θ_obs are associated with more prominent "humps," as a larger fraction of X-rays intercepts absorbing/scattering material. However, as τ_T drops with time as a result of the shell expansion, X-rays become detectable from a larger range of viewing angles while the "hump" moves to lower energies to eventually disappear.

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In our model, (i) the X-ray variability is intrinsic to the central source, rather than being a consequence of inhomogeneities in the external medium. (ii) The soft X-rays ≲10 keV, which originate directly from the engine, may show more pronounced time variability than those associated with the transient hard X-ray spectral component, which instead are diffusing through an optically thick shell. (iii) The true luminosity evolution of the central source is the sum of the optical and X-ray luminosities (L_engine in Figure 9); the fact that the X-ray light curve decreases less rapidly at early times than the optical light curve, as shown in Figure 9, is a consequence of the increasing fraction of escaping X-ray radiation with time.

3.3.3. The Connection of AT 2018cow to Other Astrophysical Sources with Compton-hump Spectra

In the previous section, we provided a proof of concept that interaction (in the form of scattering and absorption) of X-ray photons from a source located within expanding ejecta with temporally declining optical depth τ provides a natural explanation of the broadband X-ray spectrum of AT 2018cow. The model is agnostic with regard to the physical nature of both the X-ray radiation and the reprocessing medium. The incident hard X-ray radiation in AT 2018cow might originate from an embedded shock, or from a central "nebula" (similar to a pulsar-wind nebula around a young magnetar), or an X-ray "corona" around an accreting BH.

Consistent with the picture above, reflection emission from the reprocessing of inverse-Compton photons off a thick accretion flow produces a "Compton-hump" feature similar in shape to what we observed in AT 2018cow. Such emission is typical of X-ray binaries (XRBs) and AGNs (e.g., Risaliti et al. 2013; Tomsick et al. 2014 for recent examples), where a high-energy power-law component associated with the Compton-upscattering of seed thermal photons from the BH accretion disk by a hot cloud of electrons—the "corona"—interacts with cold matter in the disk. Reflection emission is typified by a ∼30 keV Compton hump along with prominent Fe Kα-band emission and Fe K-shell absorption edges (e.g., Figure 1 of Reeves et al. 2004; Risaliti et al. 2013), which can all become broadened by relativistic effects. It is tempting to associate the transient excess of emission around ∼8 keV detected in the first spectrum of AT 2018cow (Figure 6, inset) with a Fe K-shell spectral feature (emission/absorption). The ∼8 keV excess of emission disappears by 16.5 days together with the hump, which supports the idea of a physical link between the two components and motivates our attempt below to model AT 2018cow with standard disk-reflection models. While the actual geometry of AT 2018cow is likely to be more complicated than in standard accretion disks (for AT 2018cow the reprocessing material might be rapidly expanding and diluting), the same physics of hard X-ray radiation reprocessing (including reflection and partial transmission) applies.

Fitting the broadband X-ray data of AT 2018cow at δt = 7.7 days with a Comptonized disk-reflection model via simpl (Steiner et al. 2009) acting on a thermal component and relxill (Dauser et al. 2014) produces a good fit to the data (${\chi }^{2}/{\rm{d}}.{\rm{o}}.{\rm{f}}.\sim 1.0$), matching a kT_e ≥ 30 keV corona, with reflection fraction R_f ≳ 1 (Figure 17).⁴⁹ The best-fitting model predicts a moderate optical depth to the corona τ ∼ 1–2, which illuminates the walls of the reprocessing material in a funnel-like geometry. If sufficiently compact, the innermost corona may be Compton-thick, which would produce a thermal feature at the coronal temperature, partially accounting for the hard X-ray excess. In this model, the 7–9 keV excess is naturally explained as Fe–K fluorescence emission originating away from the core along the funnel, where the ionization parameter drops below $\mathrm{log}\,\xi \equiv L/{{nR}}^{2}\lesssim 4$. In this model, the Fe–K feature is primarily distorted by the orbital and thermal motion of the gas to produce the observed broad blueshifted Fe–K emission. In this scenario, the observed disappearance of the hard X-ray Compton hump and associated Fe emission can be explained by either (i) a decline in the accretion rate, which makes the funnel opening angle grow. The corona is both less confined to a compact geometry and the walls of the funnel are extended, jointly resulting in less illumination and a diminished contribution from reflection. (ii) Or, if the hard excess is supplied somewhat by a Compton-thick coronal core with kT_e ∼ 30 keV, then as $\dot{M}$ declines, the region becomes optically thin, causing the high-energy thermal feature to drastically fade.

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Finally, we address the question of super-Eddington luminosity, which is of particular relevance if the L_X ≳ 10⁴³ erg s⁻¹ in AT 2018cow is powered by a stellar-mass BH (discussed in Section 4.2). In the case of intermediate-mass BHs (IMBHs) discussed in Section 4.3, the required accretion rate would only be mildly super-Eddington. Theoretical studies of super-Eddington accretion flows have recently experienced a surge of interest thanks to the observational finding that numerous ultraluminous X-ray sources (ULXs), systems that are brighter than the Eddington limit of stellar-mass black holes, are in fact powered by pulsars (e.g., Walton et al. 2018a, 2018b). Theoretical models predict that highly super-Eddington accretion produces a funnel geometry in the central flow (Sa̧dowski & Narayan 2015), not dissimilar from our model above. The emission is collimated and generally paired with powerful outflows, including radiatively powered jets (Sa̧dowski & Narayan 2015). In these systems, the seed X-ray luminosity can be "boosted" by a factor of ∼tens by scatterings by hot coronal electrons at τ ∼ 1–2. The underlying seed X-ray emission from the disk is then required to be ∼10³ L_Edd for a BH of a few M_☉, in line with the observed super-Eddington emission in known NS ULXs.

Perley et al. (2019) identified an excess of NIR emission with respect to the UV/optical blackbody with ${F}_{\nu }\,\propto \,{\nu }^{-0.75}$, which they interpret as non-thermal synchrotron emission physically connected with the radio–millimeter emission at ν > 100 GHz. Our observations confirm the presence of the NIR excess (Figure 11). As shown in Figure 11, the extrapolation of the model that best fits the radio observations at ν < 100 GHz severely underpredicts the NIR flux. From theoretical arguments, we inferred in Section 3.2.1 that electrons radiating at >100 GHz must be fast cooling, which would predict a steeper radio-to-NIR spectral slope than what is needed to connect the radio to the NIR band on the same synchrotron SED. Extrapolating the X-ray component to the NIR frequency range produces the same result of underpredicting the observed NIR emission. We conclude that the NIR excess is unlikely to be directly related to the same populations of electrons that produce the non-thermal radio emission at ν < 100 GHz or the X-ray radiation.

Kuin et al. (2019) favored a different interpretation of the NIR excess as free–free emission from an expanding "atmosphere" with a shallow density gradient. The NIR-emitting material would be located at larger distances than that of the optical-/UV-emitting material. This process is well known to produce an NIR excess of emission in hot stars surrounded by dense winds and luminous blue variables (see, e.g., Wright & Barlow 1975) and has been invoked to explain the NIR excess in SN 2009ip (Margutti et al. 2014). In this scenario, the spectral slope is directly connected with the density gradient of the NIR-emitting material, and it is not expected to evolve with time, as observed, as long as the high ionization state is maintained. From Equation (6), the measured spectral slope F_ν ∝ ν^−0.75 suggests a medium with a shallow density gradient of ionized material ρ ∝ r⁻ⁿ with n < 2. In these conditions, matching the observed NIR luminosity requires large densities corresponding to an effective mass-loss rate $\gg 100{A}^{* }$, which is inconsistent with our findings from the radio data modeling (Section 3.2.1). More complicated geometries with a detached equatorial shell might provide a more consistent explanation. However, regardless of the geometry, this class of models does not naturally account for the NIR temporal variability reported by Perley et al. (2019).

An alternative model consists of a light echo (i.e., radiation reprocessed by a shell of material at some distance R_shell). The NIR excess is already present in the first SED at ∼3.4 days, as presented by Perley et al. (2019), which implies an upper limit to the location of the reprocessing shell through simple light-travel time arguments R_shell < 9 × 10¹⁵ cm. Following Dwek (1983), we estimated that at these distances only large dust grains with radius ≥10 μm might have been able to survive the harsh environment created by the luminous and hot radiation from AT 2018cow, which makes the light-echo interpretation contrived.

We conclude that the observed NIR excess of emission is not directly related to the non-thermal X-ray and radio emission at ν < 100 GHz and that an "extended atmosphere" or light-echo models are unlikely to offer a quantitative explanation of the observed phenomenology.

One possibility is a core-collapse event with low ejecta mass giving birth to a rapidly spinning magnetar. The presence of hydrogen in the ejecta favors low-mass stellar progenitors that have been theoretically linked to electron-capture SNe (ecSNe; e.g., Miyaji et al. 1980; Nomoto et al. 1982), rather than an ultra-stripped massive star explosion (e.g., Tauris et al. 2015). ecSNe are expected to originate from progenitors with mass ∼8–10 M_☉ and are predicted to have low explosion energy E_k ∼ 10⁵⁰ erg, small ⁵⁶Ni production (∼10⁻³ M_☉), and ejecta mass of a few M_☉. The low M_Ni and explosion E_k are consistent with our inferences for AT 2018cow, as most of the kinetic energy of the fast ejecta of AT 2018cow might have been provided by the engine (as opposed to the initial explosion). The larger M_ej of ecSN models can also be consistent with AT 2018cow, as our constraint M_ej < 0.3 M_☉ from Section 3.1.1 applies to the fastest polar ejecta only. Special circumstances are, however, required to create the aspherical ejecta distribution of AT 2018cow and to produce a magnetar remnant. We speculate that the rapid rotation of the star, needed to endow the magnetar engine with its rapid rotation, or a jet/wind bubble (see below) could impart the ejecta with the needed equatorial–polar density asymmetry, and that these special requirements naturally explain why AT 2018cow-like transients are much rarer than 8–10 M_☉ progenitor stars in the nearby universe.

To explain the required engine energy E_e ≳ 10⁵⁰–10^51.5 erg as rotational energy of the magnetar, its initial spin period should obey P₀ ∼ 3–20 ms. The duration of the engine for an isolated magnetar is given by the dipole spin-down time (e.g., Spitkovsky 2006), which for a 1.4 M_⊙ NS is given by

$$\begin{eqnarray}&&{t}_{{\rm{e}}}={t}_{\mathrm{sd}}\approx 1.4\times {10}^{4}{\rm{s}}{\left(\displaystyle \frac{{B}_{{\rm{d}}}}{{10}^{15}{\rm{G}}}\right)}^{-2}{\left(\displaystyle \frac{{P}_{0}}{10\mathrm{ms}}\right)}^{2}.\end{eqnarray} \tag{ 25 }$$

Thus, to explain the engine timescale ${t}_{e}\sim {10}^{3}-{10}^{5}$ s for P₀ ∼ 3–20 ms, we require a magnetar with B_d ≈ 10¹⁵ G, in agreement with the findings by Prentice et al. (2018). This strong B field is in the range of "magnetars" normally invoked as central engines of GRBs (Metzger et al. 2011), but larger than those inferred for SLSNe, which typically require B_d ∼ 10¹⁴ G (e.g., Kasen & Bildsten 2010; Inserra et al. 2013; Nicholl et al. 2017). Alternatively, the intrinsic dipole magnetic field of the magnetar could be weaker, but the effective spin-down rate could be enhanced due to fallback accretion from the progenitor star (Metzger et al. 2018); such a scenario predicts a spin-down luminosity L_e ∝ t^−2.38, steeper than the usual ∝t⁻² for an isolated magnetar.

The source of the X-ray emission in this case is the "nebula" of hot plasma and magnetic fields inflated by the magnetar behind the expanding SN ejecta (e.g., Metzger & Piro 2014). Though the details of the nebular spectrum are uncertain, the inferred F_ν ∝ ν^−0.5 would be consistent with synchrotron radiation.

It is natural to ask whether the same engine responsible for creating the nebula of luminous variable X-rays would also be expected to create a successful relativistic jet. Margalit et al. (2018) showed that the central engine jet can escape homologously expanding ejecta if the energy of the jet exceeds a critical value:

$$\begin{eqnarray}&&{E}_{{\rm{j}}}\gtrsim 0.195{\left({\gamma }_{{\rm{j}}}/2\right)}^{-4}{E}_{{\rm{k}},0}\sim {10}^{49}-{10}^{50}\,\mathrm{erg},\end{eqnarray} \tag{ 26 }$$

where ${E}_{{\rm{k}},0}\gtrsim {10}^{51}$ erg is the initial kinetic energy of the explosion and γ_j is the jet Lorentz factor while propagating through the star (values of γ_j ∼ 2–3 are required to produce jet opening angles similar to those of GRBs; Mizuta & Ioka 2013). Thus, if a modest fraction of the engine energy goes into a collimated jet, a jet could break through the star on a timescale ≲t_e ∼ 10³–10⁵ s prior to the optical peak.

Our prompt γ-ray search with the IPN in Section 2.7 led to no evidence for prompt bursts of γ-ray emission in AT 2018cow that might be associated with successful jets (in analogy with GRBs). Our analysis in Section 3.2.2 limits the allowed parameter space to successful jets with E_k,iso <10⁵² erg and _B < 0.01 propagating into a medium with $\dot{M}\lt {10}^{-4}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, corresponding to E_j < 4 × 10⁴⁹ erg (θ_j = 5°) and E_j < 10⁵¹ erg (θ_j = 30°).

Regardless of whether a tightly collimated jet is created, a wider jet or wind bubble from the engine could impart the ejecta with the needed aspherical (e.g., bipolar) structure, even starting with spherical ejecta. During this process, the secondary shock driven through the ejecta on timescale ≪t_e accelerates the outer layers of the ejecta to v ∼ 0.1–0.2c (e.g., Kasen et al. 2016; Blondin & Chevalier 2017; Suzuki & Maeda 2017), explaining the early optical rise and the non-thermal radio emission.

Another possibility is the core collapse of a massive star that initially fails to explode as a successful SN, instead creating a black hole remnant (e.g., Quataert & Kasen 2012; Dexter & Kasen 2013; Quataert et al. 2018). Blue supergiant stars have been recently invoked as progenitors of ultralong GRBs (e.g., Quataert & Kasen 2012; Gendre et al. 2013; Nakauchi et al. 2013; Wu et al. 2013; Liu et al. 2018; Perna et al. 2018; Quataert et al. 2018). Fernández et al. (2018) showed that the failed explosion of a blue supergiant star (M_⋆ ≈ 25 M_⊙; R_⋆ ≈ 70–150 R_⊙), originating from the neutrino-induced mass loss that follows the formation of the NS (Nadezhin 1980; Lovegrove & Woosley 2013; Piro 2013; Coughlin et al. 2018), results in the shock-driven ejection of $\sim 0.1\,{M}_{\odot }$ at a velocity v ∼ 0.02c. The remaining star then accretes onto the newly formed black hole.

If the envelope of the remaining bound star has sufficient angular momentum, it will form an accretion disk around the newly formed BH, possibly producing the BH accreting scenario that well explains the broadband X-ray spectrum of Section 3.3.3. The disk will also produce wind ejecta that collides with the outflowing unbound shell, thermalizing its energy and accelerating it to a higher velocity v_ej ∼ 0.1c (e.g., Dexter & Kasen 2013). The timescale of the engine in this case is set by the gravitational freefall time of the outer layers of the blue supergiant progenitor onto the central black hole,

$$\begin{eqnarray}&&{t}_{{\rm{e}}}\sim {t}_{\mathrm{ff}}\approx {\left(\displaystyle \frac{{R}_{\star }^{3}}{{{GM}}_{\star }}\right)}^{1/2}\approx 1.3{\rm{d}}{\left(\displaystyle \frac{{M}_{\star }}{25{M}_{\odot }}\right)}^{-1/2}{\left(\displaystyle \frac{{R}_{\star }}{50{R}_{\odot }}\right)}^{3/2},\end{eqnarray} \tag{ 27 }$$

consistent with the constraints on the engine lifetime t_e. The engine luminosity in this scenario would be expected to decay as the fallback rate, L_e ∝ t^−5/3 (however, see Tchekhovskoy & Giannios 2015).

We conclude by commenting on the rate of occurrence of AT 2018cow-like transients in the universe. In principle, it is possible to distinguish among the different progenitor paths based on the rate of transient events with similar properties (energetics, timescales, and environments). From the PanSTARRS sample, Drout et al. (2014) inferred an FBOT rate of 4%–7% for the core-collapse SN rate. However, although AT 2018cow shows blue featureless spectra at early times and a fast rise to peak similar to the PanSTARRS FBOTs, all of the PanSTARRS FBOTs are significantly less luminous at peak L_pk ≤ 3 × 10⁴³ erg s⁻¹ (compared to L_pk ∼ 4 × 10⁴⁴ erg s⁻¹ of AT 2018cow). It is thus unclear how representative is the PanSTARRS FBOT rate of AT 2018cow-like transients. Future observations and analysis will provide an answer to this question. In this work, we will not speculate any further on AT 2018cow-like transient rates.

Alternatively, Perley et al. (2019) and Kuin et al. (2019) suggested that AT 2018cow could have been caused by the tidal disruption of a main-sequence star by an intermediate-mass black hole (IMBH) not coincident with the host-galaxy nucleus. For an IMBH of mass M_•, the fallback time of the mostly tightly bound debris following a TDE is given by (e.g., Stone & Metzger 2016; Chen & Shen 2018)

$$\begin{eqnarray}&&{t}_{\mathrm{fb}}\approx 4.1\,\mathrm{days}{\left(\displaystyle \frac{{M}_{\bullet }}{{10}^{4}{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{1/5},\end{eqnarray} \tag{ 28 }$$

where we have assumed a mass–radius relationship ${M}_{\star }\propto {R}_{\star }^{0.8}$ appropriate for the lower main sequence. t_fb is similar to the observed rise time of AT 2018cow. Therefore, if circularization of the debris is relatively efficient, such that the light curve rises on the initial fallback time, then an IMBH mass of 10⁴ M_⊙ could explain the short rise time of the emission. We note however that the circularization timescale of the TDE debris is a highly debated point in the literature (see, e.g., Chen & Shen 2018, who claim the circularization process could take years). Efficient circularization could be particularly problematic for such a low-mass black hole, as the angle through which the stream precesses due to general relativistic effects is extremely small (unless the pericenter of the tidally disrupted star was well within the tidal radius, which is unlikely).

One complication of this scenario is that the IMBH would be accreting at a rate exceeding its Eddington luminosity L_Edd ≈ 10⁴² M_•,4 erg s⁻¹ by a large factor ≳10–100 at early times (Figure 9). However, recent radiation MHD simulations of super-Eddington flows by Jiang et al. (2014) find that radiative efficiencies of ∼5% (similar to thin disks) are possible for flows accreting up to $22{\dot{M}}_{\mathrm{Edd}}$, in the range relevant here. Sa̧dowski & Narayan (2016) find a similar result for the kinetic luminosities from the disk. If coming from the inner disk, the relevant radiation temperature would be close to that of the disk photosphere at the ISCO radius,

$$\begin{eqnarray}&&{{kT}}_{\mathrm{eff}}\approx 0.11\,\mathrm{keV}{\left(\displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}\displaystyle \frac{0.1}{\eta }\right)}^{1/4}{\left(\displaystyle \frac{{M}_{\bullet }}{{10}^{4}{M}_{\odot }}\right)}^{-1/4}.\end{eqnarray} \tag{ 29 }$$

Although this is softer than the required emission of the central X-ray source powering AT 2018cow, an additional process like inverse-Compton-scattering of soft photons from the disk by the corona electrons could create the necessary high-energy tail (as described in Section 3.3.3).

Perhaps a larger problem with the IMBH scenario is the origin of the dense external CSM responsible for the self-absorbed radio emission (Figure 13), which must then be present prior to the tidal disruption. Interpreted as a wind of velocity v_w = 10⁴ km s⁻¹, characteristic of AGN outflows, one requires a mass-loss rate of ∼10⁻²–10⁻³ M_⊙ yr⁻¹, in excess of the Eddington accretion rate of ${\dot{M}}_{\mathrm{Edd}}\approx {L}_{\mathrm{Edd}}/0.1{c}^{2}\approx {10}^{-4}{M}_{\odot }$ yr⁻¹, for a 10⁴ M_⊙ BH. Alternatively, if the CSM represents an accretion flow onto the IMBH, the required accretion rate on radial scales r ∼ 3 × 10¹⁵ cm achieved by the forward shock on timescales of weeks would be

$$\begin{eqnarray}&&\dot{M}\sim \displaystyle \frac{{M}_{\mathrm{enc}}}{{t}_{\mathrm{ff}}}\sim 3\times {10}^{-4}{M}_{\odot }{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 30 }$$

where M_enc ∼ 3 × 10⁻⁶ M_⊙ is the effective mass at radius r for $A\sim 30{A}_{\star }$ (as needed to explain the radio observations) and t_ff ∼ (r³/GM_•)^1/2 ≈ 2 × 10⁷ s is the freefall time for M_• = 10⁴ M_⊙. Therefore, unless the IMBH were already embedded in a gas-rich AGN-like environment prior to the tidal disruption event, it is challenging to explain the inferred presence of the external CSM.

Although we disfavor an external CSM shock as the origin of the observed X-ray emission from AT 2018cow (Section 3.3.1), the central source could be powered by a deeply embedded, internal shock (e.g., Andrews & Smith 2018). If the CSM were concentrated in an equatorial ring or sheet, this shock would be localized to the dense equatorial region (e.g., the central X-ray source would in fact be an X-ray ring; Figure 12, left panel). One way to explain the steeply decaying luminosity of AT 2018cow, ${L}_{{\rm{e}}}\propto {t}^{-\alpha }$ with α ≈ 2–2.5, would be if the shock were decelerating. In this class of models, the torus of dense material existed before the AT 2018cow event. However, the very short rise time of AT 2018cow is directly related to its fast polar ejecta, and it is thus an intrinsic property of AT 2018cow, independent of the denser equatorial ring.

Consider the CSM to have a radial density profile ρ ∝ r^−γ. If the shock is radiative, then its self-similar deceleration evolution is momentum conserving, and so, for a disk of vertical scale-height H with a constant aspect ratio H/r ∼ constant, we have $\rho {v}_{\mathrm{sh}}{R}_{\mathrm{sh}}^{3}\propto {R}_{\mathrm{sh}}^{4-\gamma }/t=\mathrm{constant}$, i.e., R_sh ∝ t^{1/(4 − γ)} and v_sh ∼ R_sh/t ∝ t^{(γ − 3)/(4 − γ)}. Therefore, the shock luminosity would evolve as

$$\begin{eqnarray}&&{L}_{{\rm{e}}}={L}_{\mathrm{sh}}\propto {v}_{\mathrm{sh}}^{3}{R}_{\mathrm{sh}}^{2}\rho \propto {t}^{(2\gamma -7)/(4-\gamma )}.\end{eqnarray} \tag{ 31 }$$

For a constant CSM density profile $\gamma =0$, we find L_sh ∝ t^−7/4, similar to the luminosity decay of AT 2018cow.

Not addressed in this scenario is the relatively soft intrinsic spectrum of the X-rays escaping from the shock. Given the high shock velocities, one would expect the shock-heated gas to emit through the free–free process, which for a single-temperature plasma predicts ${F}_{\nu }\propto {\nu }^{0}$ at frequencies ν ≪ kT_sh, where T_sh is the post-shock temperature, flatter than that observed for AT 2018cow. However, due to the aspherical shape of the shock front and various hydrodynamical instabilities that radiative shocks are susceptible to, the post-shock gas cannot be characterized by a single temperature (e.g., Steinberg & Metzger 2018). As the velocity of the shock decreases with time, the temperature of the post-shock gas will also decrease. This would result in a greater fraction of the shock power emerging at UV/soft X-ray frequencies, which are more easily absorbed by the ejecta, and thus could contribute to the observed rapid late-time decay in the X-ray luminosity L_x ∝ t⁻⁴. Future theoretical work is required to assess the X-ray emission emerging from shocks propagating into aspherical environments, as similar physics is at work in a variety of astrophysical transients (e.g., SNe IIn, luminous red novae, and classical novae; e.g., Andrews & Smith 2018; Metzger & Pejcha 2017).