Luminosity Function and Event Rate Density of XMM-Newton-selected Supernova Shock Breakout Candidates

We follow the formalism in Sun et al. (2015) to derive the LF for fast transients that is based on the $1/{V}_{\max }$ method (Schmidt 1968), where ${V}_{\max }$ is the maximum volume within which a source can be detected for a specific telescope or survey. The method is further improved in this work by deriving ${V}_{\max }$ from simulations of more realistic detection of transient sources using the data sets of the real observations. We define the redshift-dependent LF ϕ(L, z) as the number density of transient events per unit volume, per unit time, per decade of peak luminosity, i.e., in units of Gpc⁻³ yr⁻¹ dex⁻¹. In this work, we assume that the cosmological evolution is due to the evolution of the number density and that the shape of the LF (luminosity and index) does not evolve with redshift. ⁸ Therefore, ϕ(L, z) can be written as

$$\begin{eqnarray}&&\phi (L,z)=f(z){\rm{\Phi }}(L),\end{eqnarray} \tag{ 1 }$$

where the nondimensional factor f(z) measures the z-dependence of the number density. It is defined as unity at the local universe, i.e., f(z = 0) = 1. And Φ(L) is the z-independent LF, which can be taken as the local LF since it is denoted by ϕ(L∣z = 0) = Φ(L).

The event rate density at redshift z above a minimum luminosity ${L}_{\min }$, in units of Gpc⁻³ yr⁻¹, is

$$\begin{eqnarray}\begin{array}{rcl}\rho (z) & = & {\displaystyle \int }_{\mathrm{log}{L}_{\min }}^{\infty }\phi (L,z)d\mathrm{log}L\\ & = & f(z){\displaystyle \int }_{\mathrm{log}{L}_{\min }}^{\infty }{\rm{\Phi }}(L)d\mathrm{log}L\\ & = & f(z){\rho }_{0}(\gt {L}_{\min }),\end{array}\end{eqnarray} \tag{ 2 }$$

where ρ₀ is defined as the local event rate density, i.e.,

$$\begin{eqnarray}&&{\rho }_{0}(\gt {L}_{\min })={\int }_{\mathrm{log}{L}_{\min }}^{\infty }{\rm{\Phi }}(L)d\mathrm{log}L,\end{eqnarray} \tag{ 3 }$$

and $\mathrm{log}$ denotes log₁₀ here and throughout the paper.

For a given survey, the expected detectable number of transients in the luminosity range from $\mathrm{log}L$ to $\mathrm{log}L+d\mathrm{log}L$ and in the redshift range from z to z + dz can be expressed as

$$\begin{eqnarray}&&{dN}(L,z)=\displaystyle \frac{{\rm{\Omega }}}{4\pi }\cdot {dV}(z)\cdot \phi (L,z)d\mathrm{log}L\cdot T\cdot \displaystyle \frac{1}{1+z},\end{eqnarray} \tag{ 4 }$$

where Ω is the FOV of the telescope/survey, the volume element dV over the 4π solid angle is given by ${dV}=\tfrac{{dV}}{{dz}}\cdot {dz}$, and T is the time duration that this volume element is monitored. The $\tfrac{1}{1+z}$ factor corrects for the effect of time dilation, as the observation time T is measured in the local frame of the observer. In the standard ΛCDM cosmology, the comoving volume is given by

$$\begin{eqnarray}&&\displaystyle \frac{{dV}(z)}{{dz}}=\displaystyle \frac{c}{{H}_{0}}\displaystyle \frac{4\pi {D}_{L}{\left(z\right)}^{2}}{{\left(1+z\right)}^{2}{\left[{{\rm{\Omega }}}_{M}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}\right]}^{1/2}},\end{eqnarray} \tag{ 5 }$$

where D_L (z) is the luminosity distance at the corresponding redshift z.

The f(z) factor is model dependent. By assuming that CCSNe trace the star formation history (SFH), we adopt the analytical formula from Yüksel et al. (2008),

$$\begin{eqnarray}&&f(z)={\left({\left(1+z\right)}^{3.4\eta }+{\left(\displaystyle \frac{1+z}{5000}\right)}^{-0.3\eta }+{\left(\displaystyle \frac{1+z}{9}\right)}^{-3.5\eta }\right)}^{\tfrac{1}{\eta }},\end{eqnarray} \tag{ 6 }$$

where η = −10. At z < 4, this function measures the SFH from the UV and far-IR galaxy data (Hopkins & Beacom 2006; Li 2008). At redshifts beyond z > 4, the SFH is enhanced taking into account the high-redshift GRBs by the Swift data, which exceed the SFH directly inferred from the galaxy observation as far as z ∼ 7.

In practice, the survey may be composed of a number of observations. Equation (4) can be written as

$$\begin{eqnarray}&&{dN}(L,z)=\sum _{i=1}^{{N}_{\mathrm{obs}}}{dN}{\left(L,z\right)}_{i},\end{eqnarray} \tag{ 7 }$$

where ${dN}{(L,z)}_{i}$ is the expected number for the ith observation and N_obs is the total number of the observations. The detectable source number with L is the integral from z = 0 to ${z}_{\max ,{\rm{i}}}$ for the ith observation, i.e.,

$$\begin{eqnarray}&&{dN}{\left(L\right)}_{i}={\rm{\Phi }}(L)d\mathrm{log}L\cdot {T}_{i}\cdot {\int }_{0}^{{z}_{\max ,{\rm{i}}}}\displaystyle \frac{{\rm{\Omega }}}{4\pi }\cdot \displaystyle \frac{f(z)}{1+z}\displaystyle \frac{{dV}(z)}{{dz}}{dz},\end{eqnarray} \tag{ 8 }$$

where ${z}_{\max ,{\rm{i}}}$ is the maximum redshift at which the source can be detected by the observation and T_i is the exposure time of the observation.

Therefore, the total detectable number of sources with L is the sum over all the observations,

$$\begin{eqnarray}&&{dN}(L)={\rm{\Phi }}(L)d\mathrm{log}L\cdot \sum _{i=1}^{{N}_{\mathrm{obs}}}{T}_{i}\cdot {V}_{\max ,{\rm{i}}}^{{\prime} },\end{eqnarray} \tag{ 9 }$$

where ${V}_{\max ,{\rm{i}}}^{{\prime} }$ is the effective maximum volume that is monitored by the ith observation weighted by the density evolution and time dilation, i.e.,

$$\begin{eqnarray}&&{V}_{\max ,{\rm{i}}}^{{\prime} }={\int }_{0}^{{z}_{\max ,{\rm{i}}}}\displaystyle \frac{{\rm{\Omega }}}{4\pi }\cdot \displaystyle \frac{f(z)}{1+z}\displaystyle \frac{{dV}(z)}{{dz}}{dz}.\end{eqnarray} \tag{ 10 }$$

We denote the parameter ${ \mathcal M }$ as the total monitored volume-time (MVT) of all the observations, i.e.,

$$\begin{eqnarray}&&{ \mathcal M }=\sum _{i=1}^{{N}_{\mathrm{obs}}}{T}_{i}\cdot {V}_{\max ,{\rm{i}}}^{{\prime} }.\end{eqnarray} \tag{ 11 }$$

For a homogeneous population, i.e., sources differ only in luminosity and redshift, the maximum redshift depends mainly on the source peak luminosity. If the observed sample comprises objects that differ in other quantities, such as light curves, spectral shape (e.g., blackbody temperatures), etc., they can be considered as being drawn from a mixture of several populations. The LF Φ(L) is the sum of the individual contributions by each source type (j) in the range from $\mathrm{log}L$ to $\mathrm{log}L+d\mathrm{log}L$ (with total ΔN_L sources), i.e.,

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}(L)d\mathrm{log}L & = & \displaystyle \sum _{j=1}^{{\rm{\Delta }}{N}_{L}}{\left[{\rm{\Phi }}(L)d\mathrm{log}L\right]}_{j}\\ & = & \displaystyle \sum _{j=1}^{{\rm{\Delta }}{N}_{L}}{\displaystyle \frac{{dN}{\left(L\right)}_{j}}{\left[\sum _{i=1}^{{N}_{\mathrm{obs}}}{T}_{i}\cdot {V}_{\max ,{\rm{i}}}^{{\prime} }\right]}}_{j}\\ & = & \displaystyle \sum _{j=1}^{{\rm{\Delta }}{N}_{L}}{\displaystyle \frac{1}{{ \mathcal M }}}_{j}.\end{array}\end{eqnarray} \tag{ 12 }$$

Our work is based on the assumption that the search method in AL20 resulted in a complete sample of SN SBOs from the XMM-Newton data used. These include all publicly available archival data as of 2019 November in HEASARC and from 3XMM-DR8. The total good exposure time ⁹ is 163 Ms (5.2 yr). Of this, only 123 Ms (3.9 yr) is outside of the Galactic plane (∣b∣ > 15°). The effective FOV of XMM-Newton observations is taken as 0.2 deg².

The total sample inferred from AL20 is shown in Table 1. The silver sample differs from the golden one owing to the insufficient redshift estimate or possible Galactic foreground contamination. Following AL20, a pseudo-redshift of 0.3 is assumed for the candidates without redshift measurements. The SN SBO peak bolometric luminosities and the temperatures derived assuming the blackbody model are shown in Table 1. In Figure 1, we show the distribution of the peak bolometric luminosities and redshifts for XMM-Newton SN SBO candidates and LL-GRBs. The majority of the LL-GRB sample is summarized in Sun et al. (2015), supplemented with the recently reported event LL-GRB 171205A (D'Elia et al. 2018). The bolometric luminosities of SN SBOs normally peak in the soft X-ray band, with the blackbody temperatures in the range of $\left[0.1,1\right]\,\mathrm{keV}$. The peak bolometric luminosities of the newly reported SN SBO candidates generally reside in the range of $\left[{10}^{42},{10}^{47}\right]\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. The peak luminosity of XRO 080109/SN 2008D is located within the range of the XMM-Newton sample. GRB 060218/SN 2006aj has a higher luminosity than the entire XMM-Newton sample.

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For illustration purposes, we also show how the limiting luminosity corresponding to the XMM-Newton detection threshold varies as a function of redshift for an exposure time of ∼ 1000 s (roughly the timescale for a typical SN SBO) in soft X-rays (corresponding to F_th = 2 × 10⁻¹⁴ erg cm⁻² s⁻¹ in the range of 0.5–2 keV; Watson et al. 2001) with the solid line. The minimum peak flux of the sample is higher than this sensitivity limit, as shown with the dashed green line (${F}_{\mathrm{peak},\min }=4.5\times {10}^{-14}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$). All SN SBOs lie above the F_th curve, and therefore the sample is generally considered as a complete SN SBO sample above this flux limit.

From Equation (12), to derive the LF, one needs to quantify the total MVT for each SN SBO of the XMM-Newton sample. The key is to determine the maximum redshift ${z}_{\max }$ for each of the observations up to which a given source could be detected. Its value depends not only on the source property but also on the several observational factors that impact the detection of the source, including the Galactic absorption column density (N_H,gal) on the line of sight (LOS), the detector filter used, the off-axis angle of the source in the FOV, and the background. In this work we examine the detectability of an SN SBO by simulating observations with XMM-Newton and analyzing the resulting data, as follows.

For a hypothesized SN SBO with a given redshift, peak luminosity, X-ray spectrum, light curve, and intrinsic absorption (if any), we simulate the detection of such a source for each of the XMM-Newton observations (with ∣b∣ > 15° in the large FOV imaging mode) considered. First, its flux spectrum model after Galactic absorption is calculated in the observer's frame using the N_H,gal value on the LOS of that pointed observation. This model is then convolved with the energy response and effective area of the three EPIC detectors, pn, MOS1, and MOS2, to generate detected source counts, respectively. For this, an averaged effective area, weighted by off-axis angle, is calculated and adopted to account for the vignetting effect of the telescope, assuming that the source off-axis angle is randomly distributed within the FOV. The resultant source counts of the three cameras are co-added to form a single source light curve. The background count rate is randomly sampled from the XMM-Newton observations, where background flares are excluded (Carter & Read 2007), and then a background light curve is generated following a Poisson distribution. This transient source is considered to be detected if all the following criteria are met:

• 1.  
  The signal-to-noise ratio of the detected source counts is above 5.
• 2.  
  The total net source counts are no less than ${N}_{\mathrm{ph},\min }=62$, the minimum counts of the XMM-Newton SN SBOs in the AL20 sample.
• 3.  
  The ratio of peak flux to average flux is at least above 1.9, which is the minimum of the observed sample.

The last two are adopted to ensure that a simple transient-like light curve can be constructed and identified, as in AL20.

For each SN SBO source of the observed sample, we derive the ${z}_{\max }$ for each of the XMM-Newton observations considered. We assume the source to be at a set of redshifts starting from z = 0, and we calculate its light-curve models using the source properties taken from AL20, i.e., the peak luminosity, blackbody temperature, light curve, and intrinsic absorption column density. Then, each of the light-curve models is taken as input to the above simulation to examine whether the source at the assumed redshift can be detected or not, as described above, for each of the XMM-Newton observations considered. The simulation is carried out for light-curve models with increasing redshift values until a ${z}_{\max }$ is reached, beyond which the source can no longer be detected.

In this way, for a given source in the SN SBO sample, the ${z}_{\max }$ values are derived for all the observations at ∣b∣ > 15° during low background intervals, and the total MVT is calculated using Equations (10) and (11). Since it is expected that a source will statistically have the same ${z}_{\max }$ values from the above simulations for observations with the same N_H,gal, in practice, to save computing time, we group all the observations considered into 10 bins according to their N_H,gal values in logarithm. For each N_H,gal, the above process is repeated 10 times by randomly sampling the background, and the average value of ${z}_{\max }$ is taken as the final maximum redshift. In calculating Equation (11), the exposure times are first summed up for observations within the same N_H,gal bin. Figure 2 shows the distribution of the summed exposure times in different N_H,gal bins. Repeating the procedures above, the parameter ${ \mathcal M }$ is calculated for all 12 sample objects. The LF Φ(L) is derived by summing up the contribution of each individual source in the same luminosity bins (Equation (12)).

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The LF of SN SBOs derived above based on the full sample of candidates is shown in Figure 3. The horizontal error bars represent the width in logarithmic luminosity (${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.6$). The vertical error bars represent the 1σ Gaussian errors calculated following Gehrels (1986) using the number of events in each luminosity bin. We fit the LFs of SN SBOs with both a single power law (SPL) and a broken power law (BPL) using astropy curve-fit. A linear least-squares algorithm has been applied to perform the fitting. We use the Python package Uncertainties to derive the standard deviation of the fitting parameters.

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The fitted results are summarized in Table 2. It is shown that the LF generally decreases with increasing luminosity, followed by a steep decay around L ∼ 10⁴⁵ erg s⁻¹, initiating a motivation of fitting with a BPL. We derive the BPL fit as

$$\begin{eqnarray}\mathrm{log}{{\rm{\Phi }}}_{\mathrm{BPL}}(L)=\left\{\begin{array}{ll}-{\alpha }_{1}(\mathrm{log}L-\mathrm{log}{L}_{b})+{c}_{b} & \mathrm{log}L\lt \mathrm{log}{L}_{b}\\ -{\alpha }_{2}(\mathrm{log}L-\mathrm{log}{L}_{b})+{c}_{b} & \mathrm{log}L\gt \mathrm{log}{L}_{b}\end{array}\right.\end{eqnarray} \tag{ 13 }$$

with the indices (at the 68% confidence level, same as the following) α₁ = 0.48 ± 0.28 and α₂ = 2.11 ± 1.27, c_b = 3.47 ± 0.65, and the break luminosity $\mathrm{log}({L}_{b}/\mathrm{erg}\,{{\rm{s}}}^{-1})$ = 45.32 ± 0.55. In this model, the local event rate density above the minimum luminosity ${L}_{\min }=5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is derived from Equation (3), i.e.,

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{0,\mathrm{BPL}}(\gt {L}_{\min }) & = & {\displaystyle \int }_{\mathrm{log}{L}_{\min }}^{\infty }{{\rm{\Phi }}}_{\mathrm{BPL}}(L)d\mathrm{log}L\\ & \simeq & {4.6}_{-1.3}^{+1.7}\times {10}^{4}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}.\end{array}\end{eqnarray} \tag{ 14 }$$

Table 2. Results of Luminosity Function and Local Event Rate Densities

Models	Parameters	Full Sample		Golden Sample
		${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.6$	${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.9$	${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.9$
BPL	α1	0.48 ± 0.28	−0.11 ± 0.77	0.00 ± 1.01
	α2	2.11 ± 1.27	1.27 ± 0.35	1.59 ± 0.40
	$\mathrm{log}({L}_{b}/\mathrm{erg}\,{{\rm{s}}}^{-1})$	45.32 ± 0.55	44.15 ± 0.53	44.17 ± 0.79
	cb	3.47 ± 0.65	4.60 ± 0.57	4.44 ± 1.17
	ρ0,BPL a	${4.6}_{-1.3}^{+1.7}\times {10}^{4}$	${6.2}_{-1.8}^{+2.4}\times {10}^{4}$	${5.6}_{-1.6}^{+2.1}\times {10}^{4}$
SPL	α0	0.80 ± 0.16	0.75 ± 0.17	0.95 ± 0.21
	c0	4.78 ± 0.27	4.94 ± 0.28	4.84 ± 0.41
	ρ0,SPL	${4.9}_{-1.4}^{+1.9}\times {10}^{4}$	${5.2}_{-1.5}^{+2.0}\times {10}^{4}$	${8.0}_{-2.3}^{+3.0}\times {10}^{4}$

Note. The errors are given at the 68% confidence level.

^a The local event rate density, in units of Gpc⁻³ yr⁻¹, is the one above 5 × 10⁴² erg s⁻¹ in this table, and the 1σ Gaussian errors are derived from Gehrels (1986).

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In general, one can also use an SPL model to fit the LF, which is given as

$$\begin{eqnarray}&&\mathrm{log}{{\rm{\Phi }}}_{\mathrm{SPL}}(L)=-{\alpha }_{0}(\mathrm{log}L-\mathrm{log}{L}_{0})+{c}_{0}.\end{eqnarray} \tag{ 15 }$$

By normalizing to $\mathrm{log}({L}_{0}/\mathrm{erg}\,{{\rm{s}}}^{-1})=43$, we obtain α₀ = 0.80 ± 0.16 and c₀ = 4.78 ± 0.27. The local event rate density above ${L}_{\min }=5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is

$$\begin{eqnarray}\begin{array}{rcl}{\rho }_{0,\mathrm{SPL}}(\gt {L}_{\min }) & = & {\displaystyle \int }_{\mathrm{log}{L}_{\min }}^{\infty }{{\rm{\Phi }}}_{\mathrm{SPL}}(L)d\mathrm{log}L\\ & \simeq & {4.9}_{-1.4}^{+1.9}\times {10}^{4}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1},\end{array}\end{eqnarray} \tag{ 16 }$$

which is close to ρ_0,BPL. It should be noted that the LF in this work is the one for logarithmic luminosity. In the case of the LF in linear luminosity space, the absolute value of the index (β) should be steeper by 1 than the index used here (α), i.e., β = α + 1.

To estimate the uncertainty induced by the width of bins, we also perform the fitting for the case of choosing ${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.9$. The results are also summarized in Table 2. Due to the small number of the sample, the best-fit values vary for different width choices. Within the uncertainties, the results from different bin widths are consistent for both BPL and SPL models. We take ${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.6$ as a representative for analysis in the following context.

The results are based on the assumption that the cosmological evolution of SN SBOs follows the description in Equation (6). The effect of this assumption on the result should be limited since the majority of the sample is distributed at low redshift (except one candidate that has a redshift of 1.17). We also test the SFH model in Li (2008) and find that the differences in the results are negligible.

It should also be noted that the LF is based on the assumption that the observed 12 sample members are the representatives of all X-ray SN SBO populations. In AL20, four of the candidates, XT 050925, XT 160220, XT 140811, and XT 040610, are possibly misidentified Galactic foreground sources. Besides, one more event, XT 060207, is without a host redshift estimate. To estimate the uncertainty caused by the confidence of the sample being an SN SBO and the redshift measurement, we perform a further fitting for the "Golden" sample by excluding these five events. The rest of the sample is not enough for a bin width of ${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.6$. By choosing a bin width of ${\rm{\Delta }}\mathrm{log}{L}_{\mathrm{SN}\ \mathrm{SBO}}=0.9$, the BPL fitting gave the power-law indices of ${\alpha }_{1}^{{\prime} }=0.00\pm 1.01$, ${\alpha }_{2}^{{\prime} }=1.59\pm 0.40$ before and after the break luminosity at $\mathrm{log}({L}_{b}^{{\prime} }/\mathrm{erg}\,{{\rm{s}}}^{-1})=44.17\pm 0.79$ and SPL fitting gave the slope of ${\alpha }_{0}^{{\prime} }=0.95\pm 0.21$. The results derived for the "Golden" sample are in agreement with those for the full sample with the same width of binning, as shown in Table 2. A larger sample is needed in the future to improve the LF measurement.

It has been shown that several LL-GRBs (e.g., GRB 980425, GRB 031203, GRB 061218, GRB 100316D) may have SBO origins, as they can be well explained by the relativistic breakout relation of energy, temperature, and duration (see Nakar & Sari 2012 and references therein). Previously, a joint fit of XRO 080109 and LL-GRB was described by an SPL with an index of β = 2, which is similar to the LF of LL-GRBs (Sun et al. 2015). We investigate how the SN SBOs and LL-GRBs are distributed in the LF with the new data. In fact, LL-GRBs are detected in hard X-ray/gamma-rays, and SN SBOs are detected in soft X-rays. We compare their bolometric luminosities, which peak at different energy bands. The bolometric luminosities of SN SBOs are given in Table 1, with the blackbody temperatures in the range of $\left[0.1,1\right]\,\mathrm{keV}$. The bolometric luminosities of LL-GRBs are defined in the energy range of 1–10⁴ keV and derived through k-correction assuming that the spectrum follows a standard GRB Band function (Band et al. 1993; Sun et al. 2015). The data and fitted line of LL-GRB are inferred from Sun et al. (2015) and improved with the event GRB 171205A. ¹⁰

Based on the new sample, we perform a new joint fit for SN SBOs and all the LL-GRBs and obtain an SPL index of α_j = 0.84 ± 0.08, similar to the previous one. The joint fit result is very close to the SPL fit of SN SBOs, as shown in Figure 3. There is also the controversial case emerging as LL-GRB 120422A (Zhang et al. 2012), with a duration much shorter than that predicted from the relation. ¹¹ It has been proposed that there is a luminosity threshold of 10⁴⁸ erg s⁻¹, above/below which LL-GRBs may be central engine driven or relativistic shock breakout originated. Under this assumption, we also perform another joint fit for SN SBOs and LL-GRBs below 10⁴⁸ erg s⁻¹; the SPL fit gives an index of ${\alpha }_{j}^{{\prime} }=0.76\pm 0.10$, which is also similar to α₀. From both joint fits, one can infer that a fair fraction of LL-GRBs follow the extension of SN SBO SPL LF, in agreement with their relativistic shock breakout origins.

The data of XRO 080109 and GRB 060218 that were derived independently based on the Swift detection in Sun et al. (2015) are plotted in Figure 3. For comparison, their horizontal error bars are taken with the same values as those of SN SBOs and LL-GRBs. Within the error bars, the one of XRO 080109 is consistent with the distribution of SN SBOs for both models. The one of LL-GRB 060218 is more consistent with the SPL extrapolation of SN SBOs LF.

The LFs and event rate densities of other high-energy transients, including HL-LGRBs, short GRBs (SGRBs), thermal tidal disruption events (TDEs), jetted TDEs, and binary neutron star merger magnetar-powered X-ray transients, have also been investigated (e.g., Sun et al. 2015 and references therein; Sun et al. 2017, 2019; Auchettl et al. 2018; Xue et al. 2019). In Figure 4, we display the local event rate density as a function of the peak bolometric luminosity for SN SBOs and other high-energy transients. Similar to LL-GRBs, the bolometric luminosities of other high-energy transients are also in the range of 1–10⁴ keV (Sun et al. 2015), except for CDF-S XT1 and XT2, whose luminosities are in the range of 0.3–10 keV owing to the lack of knowledge of their energy spectrum. The typical local event rate densities above the minimum luminosities are also summarized in Table 3. These high-energy transients are all promising sources for future wide-field X-ray telescopes. With threshold luminosity spanning 10⁴²–10⁵⁰ erg s⁻¹, the local event rate densities decrease from 10⁴ to order of unity in units of Gpc⁻³ yr⁻¹.

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Table 3. Summary of Event Rate Densities of High-energy Transients above the Minimum Luminosities

Transients	ρ0 (Gpc−3 yr−1)	${L}_{\min }$ (erg s−1)	References
SN SBO	${4.6}_{-1.3}^{+1.7}\times {10}^{4}$	5 × 1042	This work (BPL)
LL-GRB	${160}_{-65}^{+98}$	5 × 1046	(1)
HL-GRB	${2.4}_{-0.3}^{+0.3}$	3 × 1049	(1)
SGRB	${4.2}_{-1.0}^{+1.3}$	7 × 1049	(1)
Thermal TDE	${5.0}_{-1.6}^{+2.3}\times {10}^{3}$	1043	(1)
Jetted TDE	${0.03}_{-0.02}^{+0.04}$	1048	(1)
CDF-S XT2	${1.4}_{-1.2}^{+3.3}\times {10}^{4}$	3 × 1045	(2)
CDF-S XT1	${1.1}_{-0.9}^{+2.5}\times {10}^{4}$	7 × 1046	(2)

References. (1) Sun et al. 2015; (2) Sun et al. 2019.

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We plot the SN SBO $\mathrm{log}N-\mathrm{log}S$ distribution in Figure 5, which is calculated from the LF (both BPL and SPL) and the redshift evolution f(z) from Equation (6), where N( > S) is the cumulative source counts with the peak flux above S. The numbers are given in units of sr⁻¹ yr⁻¹. The peak luminosity is assumed to be in the range from 5 × 10⁴² erg s⁻¹ to 10⁴⁷ erg s⁻¹. The $\mathrm{log}N-\mathrm{log}S$ plot shows that when the peak flux is above S > 10⁻¹³ erg cm⁻² s⁻¹, the number generally declines following the S^−3/2 law (dashed–dotted line), as predicted for a Euclidean geometry. It flattens toward the lower flux end. It should be pointed out that the peak flux cannot be directly related to the sensitivity limit of the telescopes/surveys when one estimates the detection rate.

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We compare the local event rate density of SN SBOs with that of CCSNe inferred independently from optical SN searches. The local event rate density of CCSNe is given as ∼10⁵ Gpc⁻³ yr⁻¹ (Smartt et al. 2009; Li et al. 2011). Among them, ∼ 59% of the CCSNe are Type II-P SNe, which are expected to originate from explosions of RSGs; ∼1%–3% of CCSNe are Type II SNe similar to SN 1987A, which are from the BSG collapse (Arnett et al. 1989; Pastorello et al. 2005); and ∼30% of them are Type Ib/c SNe from the explosions of W-R stars. It is suggested that the SN SBOs from RSGs (with a much larger radius and explosion energy) peak at a few eV and therefore are expected to be absorbed by the interstellar medium (Nakar & Sari 2010). In comparison, SN SBOs from the less massive BSGs are more easy to detect by soft X-rays detectors. The local event rate density of SN SBOs in this work is close to that of the Type II-P SNe but is around an order of magnitude higher than that of SN 1987A-like SNe. This can be interpreted as the estimated local event rate density being the total contribution by SN SBOs from the collapse of all three types of progenitors. In fact, two SN SBO candidates in AL20, XT 100424 and 151128, have been inferred with much larger radii (≥100 R_⊙) and softer temperature (∼0.15 keV), which are more consistent with the predictions from RSGs. Sapir et al. (2013) showed that even peaking at a lower energy band (like UV), SN SBOs from RSGs can emit a significant amount of photons in the soft X-rays, with the predicted luminosity ( ∼ 10⁴⁴ erg s⁻¹) consistent with the observations of the two events.

To perform a detailed comparison, we estimate the local event rate densities of each type of SN SBO separately. We follow the identification of AL20 and take those candidates that can be relatively well interpreted by one of the progenitor models, as summarized in Table 1. We derive the event rate densities following the above methodology. For the BSG SN SBO sample (XT 161028, XT 151219, XT 070618, and XT 060207), the event rate density is inferred as ${2.1}_{-0.6}^{+0.8}\times {10}^{3}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ above the minimum luminosity ${L}_{\min ,\mathrm{BSG}}\sim 6\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. For the RSG SN SBO sample (XT 100424 and XT 151128), one infers a rate of ${2.0}_{-0.6}^{+0.8}\times {10}^{4}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ above the minimum luminosity ${L}_{\min ,\mathrm{RSG}}\sim 5\times {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. For W-R SN SBOs, we take the less convincing candidates labeled BSG/WG for a rough estimate, i.e., both BSG and W-R are possible origins. We derive a rate of ${1.7}_{-0.5}^{+0.7}\times {10}^{4}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}$ above the minimum luminosity ${L}_{\min ,\mathrm{WR}}\sim {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. It is generally concluded that the event rate densities derived from the subsample and their relative ratios are consistent with the optically inferred rates of CCSNe. One should be cautious that these estimates are affected by the classification of the sample. Furthermore, the event rate density inferred from RSG SN SBOs is around two times lower than that of Type II-P SNe. This discrepancy may be caused by different selection effects. For example, some X-ray RSG SN SBOs that are extremely soft may not be observed owing to severe Galactic absorption. There are also some SN SBOs peaking at the UV band that will be missed by the X-ray population study, while the optical SN observations do not suffer from such selection bias.

In addition, there seems to be a dip in the LF around the luminosity of ∼10⁴⁴ erg s⁻¹ as shown in Figure 4, which is also consistent with the hypothesis that the LF is contributed by different populations of SN explosions. However, the current sample size is too small to draw a firm conclusion. The diversity of the SN SBOs from different progenitors can be confirmed by more events detected in the future.

The EP satellite is an all-sky monitor type space mission with a wide-field X-ray telescope (WXT) on board (Yuan et al. 2018). Adopting micropore optics technology for X-ray focusing imaging, the WXT has a large FOV of 3600 deg² and operates in the soft X-ray band of 0.5–4 keV. The unprecedented Grasp (FOV × effective area) and soft X-ray bandpass make EP an ideal monitor to find transients like SN SBOs. The satellite plans to observe the nightside sky and scan half of the sky in every three orbits within 5 hr. We perform a simulation of the WXT all-sky survey (H.-W. Pan et al. 2022, in preparation) with the LF and redshift evolution models in this work and use the light curves of the 12 candidates. The results show a detection of 10 to several tens of SN SBO events per year above the minimum peak flux ∼10⁻¹⁰ erg cm⁻² s⁻¹ for a detection with signal-to-noise ratio S/N > 5, consistent with the prediction in Figure 5. The majority of the secure detection is within the redshift of 0.1. Large-area surveys in the optical band and spectroscopic follow-up observations will be necessary for identifying X-ray transients with an SN SBO origin.

In this work, we provide a realistic method of deriving the LF and event rate density for fast X-ray transients, and we apply it to the newly reported SN SBO sample discovered from XMM-Newton archival data. We derive the maximum volume (${V}_{\max }$) within which a source can be detected for a specific telescope or survey from simulations of more realistic detection of transient sources using the data sets of the real observations. It is shown that the SN SBO LF can be a BPL with indices (at the 68% confidence level) of 0.48 ± 0.28 and 2.11 ± 1.27 before and after the break luminosity $\mathrm{log}({L}_{b}/\mathrm{erg}\,{{\rm{s}}}^{-1})$ = 45.32 ± 0.55 or an SPL with index of 0.80 ± 0.16. The local event rate densities of SN SBOs above 5 × 10⁴² erg s⁻¹ are ${4.6}_{-1.3}^{+1.7}\times {10}^{4}$ Gpc⁻³ yr⁻¹ and ${4.9}_{-1.4}^{+1.9}\times {10}^{4}$ Gpc⁻³ yr⁻¹ for BPL and SPL models, respectively. We perform the joint LF for SN SBOs and LL-GRBs, and the SPL indices are similar to that of the SPL fit of SN SBO. It should be noted that the results are affected by the systematic uncertainties due to the binning choices and the possibility of misclassified SBOs. Though the LFs and event rate densities derived from different cases of bin width and from the alternative subsample are moderately different, they are mostly within each other's error range.

We interpret the local event rate density as being the total contribution by SN SBOs from the collapse of all three types of progenitors (i.e., RSGs, W-R stars, and BSGs). The LF of SN SBOs in the low luminosity range is mainly contributed by the explosion of RSGs and W-R stars. The BSG SN SBOs tend to have higher peak luminosities and an event rate density an order of magnitude lower than that of the former two. We also present a $\mathrm{log}N-\mathrm{log}S$ distribution of SN SBOs based on the LF results. It is expected that a wide-field X-ray telescope like EP can detect 10 to a few tens of SN SBOs per year.