PROBING THE COSMIC GAMMA-RAY BURST RATE WITH TRIGGER SIMULATIONS OF THE SWIFT BURST ALERT TELESCOPE

We create a sample of mock GRB light curves in the rest frame based on several input GRB properties, which include the following.

• 1.  
  Cosmic GRB rate. We adopt the functional form in Wanderman & Piran (2010), which assumes a simple broken power law shape that generally follows the shape of the cosmic SFR, with adjustable parameters. That is, the cosmic GRB rate increases to some redshift z₁, and decreases afterward, as shown in Equation (1).

  $$\begin{eqnarray} R_{\rm GRB}(z) =\! R_{\rm GRB}(z=0) \left\lbrace \begin{array}{@{} ll}(1+z)^{n_1}, &\!\! z \le z_1\!,\\ (1+z_1)^{n_1-n_2} \ (1+z)^{n_2}\!, &\!\! z>z_1.\end{array} \right. \nonumber\\ \end{eqnarray} \tag{ 1 }$$

  R_GRB(z) is the comoving rate dN/dV_comov dt_src, where dV_comov is the comoving volume and dt_src is the time interval in the source frame. R_GRB(z) can be converted to the observed rate $R_{\rm GRB;{\rm dz}} = {\it dN}/d\Omega dz dt_{\rm obs}$ in units of number per solid angle (dΩ), per redshift interval (dz), and per time interval in the observed frame (dt_obs) by

  $$\begin{equation} R_{\rm GRB;{\rm dz}}(z) = R_{\rm GRB}(z) \ \frac{dV_{\rm comov} dt_{\rm src}}{d\Omega dz dt_{\rm obs}} = \frac{R_{\rm GRB}(z)}{(1+z)} \ \frac{dV_{\rm comov}}{d\Omega dz}. \end{equation} \tag{ 2 }$$
• 2.  
  GRB luminosity function. Again, we adopt the functional form in Wanderman & Piran (2010), which is also a simple broken power law function,

  $$\begin{equation} \phi (L) = \frac{\it dN}{\it dL} = \ \left\lbrace \begin{array}{@{} ll} \left(\frac{L}{L_{\star }} \right)^{x}, & L < L_{\star },\\ \left(\frac{L}{L_{\star }} \right)^{y}, & L > L_{\star },\end{array} \right. \end{equation} \tag{ 3 }$$

  where x, y, and L_⋆ are adjustable parameters. Note that the luminosity L refers to the peak luminosity, not the average luminosity. Also, the integral of the luminosity function is normalized to one.
• 3.  
  GRB pulse shape. Using the observed GRB light curves from the real BAT-detected GRBs with known redshifts, we create a library of rest-frame GRB light curves that contains 139 different GRB pulse shapes. Data with signal-to-noise ratios higher than 3σ are considered as part of the GRB pulses, while data with signal-to-noise ratios below 3σ are considered as noise and thus are ignored when constructing the GRB pulse shapes. The light curves range from duration T₉₀ = 2.16 s to 658.2 s. We randomly choose the GRB pulse shape from this library to create simulated light curves in the rest frame.

We convert the GRB light curve in the rest frame created in part 1 (Section 3.1) into the observational light curve in units of photon count rate. The conversion is calculated using XSPEC, ⁸ a program that is capable of converting flux into photon count rate based on the energy response of the instrument with an input GRB spectrum (Arnaud 1996). We adopt the commonly used Band function as the GRB spectrum (Band et al. 1993), which has the form

$$\begin{eqnarray} \frac{\it dN}{\it dE} =\ \left\lbrace \begin{array}{@{} ll}K_1 \ E^{\alpha } \ \rm exp \,\left[-{\it E}(2+\alpha)/{\it E}^{\rm obs}_{\rm peak} \right], & E < \frac{(\alpha - \beta) \ E^{\rm obs}_{\rm peak}}{2+\alpha },\\ K_2 \ E^{\beta }, & E \ge \frac{(\alpha - \beta) \ E^{\rm obs}_{\rm peak}}{2+\alpha }.\end{array} \right. \nonumber\\ \end{eqnarray} \tag{ 4 }$$

$E^{\rm obs}_{\rm peak}$ is the peak energy in the νF_ν spectrum in the observer frame. The normalization factors K₁ and K₂ are decided by the GRB luminosity after redshifting into the observer frame. We assign different α and β in our mock GRB sample based on the observed spectral distribution reported in Sakamoto et al. (2009). We leave the $E^{\rm obs}_{\rm peak}$ distribution to be one of the adjustable functions to explore possible consequences when different distributions are used.

We also take into account different energy responses of bursts coming from different angles relative to the detector plane. In other words, for the same burst, the simulation will create a light curve with a higher photon count rate if the burst is on-axis, and a lower photon count rate if the burst is off-axis. Moreover, BAT separates the detector into four quadrants and can trigger an event based on the light curve recorded in only some quadrants of the detector. The reason for doing this is to reduce noise for those GRBs coming from off-axis angles that illuminate only part of the detector. By calculating the signal-to-noise ratio of only the illuminated part of the detector, BAT can get a higher signal-to-noise ratio for these bursts and hence increase the chance of triggering these off-axis bursts.

We label the BAT detector plan with some grid ID numbers. GRBs with different incident angles will fall on different locations on the detector plane with a different "grid ID," and different calibrations are used based on the "grid ID" of the burst. Figure 1 plots the relative location of each grid ID on the detector plane. Table 1 shows how the incident angle corresponds to each grid ID. An incoming angle of zero degrees corresponds to an on-axis burst, that is, the burst that is directly above the detector plane. A larger incoming angle indicates that the burst is more off-axis. Grid IDs 6 and 28 do not exist due to historical naming reasons. To simulate the actual trigger algorithm, our program also simulates the partial illumination factor based on the "grid ID" (i.e., incident angle) of the burst, and creates light curves for four different quadrants of the detector accordingly. Figure 2 shows an example of simulated GRB light curves of the same burst with different grid IDs. One can see from the figure that the incident angle has a significant effect on the detector sensitivity. An off-axis burst can be ∼20 times dimmer than an on-axis burst, and hence becomes undetectable.

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In addition, the active number of BAT's detector changes with time. BAT has a total number of 32,768 detectors. However, during the 8 yr of operation, many detectors are getting noisier and are automatically or manually turned off. Therefore, the average number of active detectors decreases each year, as shown in Figure 3. This factor is taken into account in our program by simulating light curves with different energy responses according to different numbers of active detectors. A burst would have a simulated light curve with higher photon count rates if we set a larger number of active detectors. Figure 4 shows an example of how the simulated light curve changes with different numbers of active detectors.

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Furthermore, many observed GRBs show time-evolving spectra. Therefore, we implement the option to include spectral evolution in our simulation. A majority of well-studied GRBs show a hard-to-soft spectral evolution as bursts become dimmer (e.g., Liang & Kargatis 1996; Crider et al. 1999; Ryde & Svensson 2000; Zhang et al. 2007; Racusin et al. 2008; Starling et al. 2008; Yonetoku et al. 2008; Page et al. 2009; Filgas et al. 2011). Some studies proposed that the spectral evolution is related to the pulse shape (Liang & Kargatis 1996; Crider et al. 1999; Ryde & Svensson 2000). However, there are also bursts that do not show a strong spectral evolution nor monotonic behavior (Zhang et al. 2007). For simplicity, we adopt the assumption that $E^{\rm obs}_{\rm peak}$ evolves with the pulse shape. In other words, $E^{\rm obs}_{\rm peak}$ increases as a GRB becomes brighter, and decreases as the burst fades. Figure 5 demonstrates an example of adding $E^{\rm obs}_{\rm peak}$ evolution in the simulated light curve.

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Finally, we add some background noise to the simulated GRB light curves. We create a library of 371 background levels from the light curves of real BAT-detected GRBs. The backgrounds added to the simulations are randomly chosen from this library and are fluctuated with Gaussian noise. The lower four panels of Figure 6 show several examples of the simulated light curves with background noise in the observational frame with units of photon count. The light curves shown here are in the 25–100 keV energy band, and are the summations of all four detector quadrants. Each panel shows the same burst with different incident angles, corresponding to the original light curve in Figure 2 (without background noise). The example shows that the 55° off-axis burst is now completely buried under the background noise, and thus is undetectable. The gray- and blue-shaded regions indicate the foreground and background periods from the trigger criterion that detects the burst in our simulation (see Section 3.3.1 for more discussion about the trigger simulator). For comparison, the top panel of Figure 6 shows an example of a real GRB light curve (GRB120701A). The real GRB light curve is also a summation of light curves from all four detector quadrants, and in the energy band 25–100 keV. Note that real GRB light curves might not have flat backgrounds like those in our simulations, as shown in this example.

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The code we developed is capable of simulating both the rate trigger and image trigger processes. The following sections describe, in detail, the methods we adopt to simulate the BAT-trigger algorithm.

3.3.1. Simulating the First Part of the Trigger Process: The Rate Trigger

Our program follows the same procedure adopted by BAT for the rate trigger. That is, we move through the light curve and calculate the signal-to-noise ratio of each bracketed time period (τ_bracket) specified by a rate trigger criterion. Each bracketed time period contains one foreground period, one or two background periods (depending on the trigger criterion), and an elapse time period between the foreground and background periods. For those trigger criteria that contain two background periods, the background periods τ_b1 and τ_b2 are placed before and after the foreground period τ_f, respectively. Both of the background periods are separated from the foreground period by the elapse period τ_e, to make sure the background light curve does not include photon counts from the source. For the trigger criteria that only use one background period τ_b1, it is placed before the foreground period. An elapse period τ_e is also placed between τ_b1 and τ_f for these one-background criteria.

We follow Graziani (2003) in defining the signal-to-noise ratio,

$$\begin{eqnarray} &\mu _{\rm f} = \frac{\tau _{\rm b1} + \tau _{\rm b2}}{\tau _{\rm f}} \ \Sigma ^2, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} &\Sigma ^2 = \tau ^2_{\rm f} \ \left(\frac{\tau ^2_{\rm b1}}{n_{\rm b1}} + \frac{\tau ^2_{\rm b2}}{n_{\rm b2}} \right)^{-1}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &\mbox{Signal-to-noise ratio} = \frac{n_{\rm f} - \mu _{\rm f}}{(\mu _{\rm f} + \Sigma ^2)^{1/2}}. \end{eqnarray} \tag{ 7 }$$

The signal-to-noise ratio is calculated by Equation (7), using all the time periods described above and the corresponding photon counts in those periods. In the equations, n_b1, n_b2, and n_f represent the photon counts in τ_b1, τ_b2, and τ_f, respectively. For the criteria with only one background, the signal-to-noise ratio is calculated by this same equation, with all the terms related to the second background (i.e., τ_b2, n_b2) dropped.

After the signal-to-noise ratio is calculated, the program moves to the next time step, which in our simulation is set to be the next light curve bin, and calculates the signal-to-noise ratio in that time period. The program moves through the whole light curve and records the time periods where the signal-to-noise ratios are higher than the threshold determined by the specific trigger criterion.

Each rate trigger criterion has different τ_b1, τ_b2, τ_f, τ_e, and signal-to-noise ratio threshold for triggering. Additionally, each trigger criterion uses light curves from different energy bands, and light curves from different summations of the four detector quadrants. The program thus goes through the light curves multiple times with settings specified by each criterion. The trigger criteria we adopt are identical to those used by the BAT flight software.

3.3.2. Mimicking the Second Part of Trigger Process: The Image Threshold

After an event is triggered by one or more rate trigger criteria, BAT creates real sky images for further confirmation and localization. However, creating images is very time consuming and requires substantial computational power. Fortunately, it is possible to mimic the image threshold without creating real images in our simulations. Instead, we use information from the light curves and results of the rate triggers. This method allows us to calculate the image threshold without high demands of computational resources, and thus to be able to explore a larger parameter space in our search of the GRB characteristics.

This approximation is feasible in our simulation because of the following reasons. For a real burst observation, images are created to obtain better localization information and clarify that the event is not from a known stellar object. However, in our simulation all the sources generated are GRBs. Hence, there is no confusion with other sources and all we need to know is whether the burst is triggered or not. Another reason for BAT to create a real image is to determine the real background level at the time when the burst happens. In reality, the background is constantly changing with time. Therefore, the background level when the burst happens is likely to be different than the background level before or after the burst. However, we assume a flat background with Gaussian noise in our simulation, and thus the background level should be time independent and the photon count rate calculated from the background period in the rate trigger should be the same as that obtained by an real image. Therefore, we can mimic the image threshold based on information gathered from the rate trigger.

When a rate trigger criterion successfully triggers an event, BAT uses the accumulated photon counts in the foreground period of that rate trigger criterion to create an image. The photon counts used to create this image are added from all four quadrants of the detector, even if the rate trigger criterion that triggered this event only uses light curves from part of the detector. Therefore, when mimicking the image threshold, we use the foreground and background periods from the successful rate trigger criterion, but use the light curves from all four quadrants to calculate the signal-to-noise ratio.

Determining the corresponding signal-to-noise ratio of the image threshold using only information from the light curve can be a little bit complicated. In reality, the image signal-to-noise ratio is calculated from the real image by fitting a point spread function. Therefore, the signal-to-noise ratio might not be identical to the signal-to-noise ratio calculated using light curves by Equation (7). In an ideal case of a flat background and only one source (i.e., the burst) per image, the signal-to-noise ratio calculated from the point spread function fitting should have a one-to-one correlation to the signal-to-noise ratio calculated from the light curve. Thus, we can approximate the image threshold if we can find the corresponding signal-to-noise ratio calculated from a light curve to that estimated from an image. That is, we need to know how to convert the image signal-to-noise threshold to a threshold in the light curve domain.

To determine the corresponding signal-to-noise ratio of the image threshold, we apply the BAT-trigger simulator to 121 light curves of real BAT-detected GRBs. Using the foreground and background periods of the successful trigger criteria, we calculate the signal-to-noise ratio using light curves from all four quadrants. Independently, we create real images for these real GRBs using the foreground period determined by the successful trigger criteria, and calculate the image signal-to-noise ratio using the ground software of BAT (HEASoft ⁹ ). We plot the signal-to-noise ratios calculated by these two methods to find the correlation in Figure 7. As discussed earlier, the scatter of the points shown in Figure 7 is likely due to the large fluctuation of the backgrounds of these real GRB light curves, and/or from contamination of other sources in the light curves and images. For a first-order approximation, we simply fit a linear function to the correlation of the image signal-to-noise ratio calculated from the BAT software HEASoft (SNR_image) and the signal-to-noise ratio calculated from photon count rate using all four quadrant light curves (SNR_4quad). This fitted line has the functional form of $\rm {\rm SNR}_{\rm image} = 2.47 + 0.44 \ \rm {\rm SNR}_{\rm 4quad}$, with reduced χ² =11.12 (degree-of-freedom = 118). The signal-to-noise ratio for the real image threshold is ∼6.5 to 7.0, which corresponds to a signal-to-noise ratio of ∼10 calculated from light curves. Thus, we adopt the signal-to-noise ratio = 10 to be our threshold for mimicking the image threshold.

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3.3.3. The Image Trigger

3.3.4. Comparing our Simulation with BAT's Sensitivity

To test whether our program correctly simulates the complex BAT-trigger algorithm, we compare the GRB peak fluxes of the "triggered" bursts in our simulation to those measured from the real GRBs detected by BAT.

Panel (a) of Figure 8 shows the peak fluxes of real BAT-detected GRBs with respect to the grid ID of the detector plane. As described in Section 3.2, the grid IDs are simply the number labels on the detector plane, and thus correspond to the incoming angles of the bursts relative to the normal axis of the detector. Each point in the figure represents one burst. The fluxes of these bursts are adopted from Sakamoto et al. (2011). Blue dots in the plot indicate GRBs detected by rate triggers, while red crosses represent bursts found by image triggers.

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Panels (b) and (c) of Figure 8 plot the fluxes from the mock "triggered" bursts in our simulations with 20,000 and 30,000 active detectors, respectively. Again, blue dots show the rate-triggered bursts, while red crosses indicate the image-triggered events. Because the number of active detectors decreases with time, the sensitivities using two extreme numbers of active detectors are plotted for comparison. Results show that decreasing the number of active detectors from 30,000 to 20,000 reduces the sensitivity by a factor of ∼3, and has a more significant impact on off-axis bursts than on-axis events. Moreover, the sensitivity of the image trigger is less affected by reducing the number of active detectors. The input GRB characteristics of this mock sample are based on the best result in our search, which is summarized in Table 2, Section 5. This sample creates plenty of bursts that have fluxes in the range of 10⁻⁹ erg s⁻¹ cm⁻² to 10⁻⁶ erg s⁻¹ cm⁻². Therefore, the non-detection of low flux bursts is due to the sensitivity of our trigger-simulation program, instead of the lack of low flux events.

Table 2. Summary of the Set of Parameters That Generates Results That Match the Best with the Observed GRB Characteristics

RGRB(z = 0) (Gpc−3 yr−1)	0.84
z1	3.6
n1	2.07
n2	−0.70
L⋆ (erg s−1)	1052.05
x	−0.65
y	−3.00
$E^{\rm src}_{\rm peak}$ distribution	Modified Yonetoku relation (Equation (8))
KS Test (D) for z Distribution	4.77 × 10−2
Significance for z Distribution	99.79%
KS Test (D) for Peak Flux Distribution	$3.09^{+4.12}_{-1.04} \times 10^{-2}$
Significance for Peak Flux Distribution	$85.59^{+14.10}_{-81.93}\%$

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Results show that the rate trigger process in our trigger simulator can detect GRBs with fluxes ∼10⁻⁸ erg s⁻¹ cm⁻² for directly on-axis bursts, and fluxes ∼10⁻⁷ erg s⁻¹ cm⁻² for extremely off-axis events, which is very similar to the real BAT sensitivity. It is harder to compare the sensitivity of the image trigger process, due to the very low number of statistics in real BAT detections. However, in general, the image trigger process in our simulations detects bursts with fluxes that are slightly lower than those found by the rate trigger algorithm, as expected. A similar trend is also seen in the real BAT-detected GRBs.

Around 30% of all the BAT-detected GRBs have measured redshifts (Gehrels & Mészáros 2012). These redshifts are usually measured from the burst afterglows and/or host galaxies observed by follow-up ground observations. Therefore, the GRB redshift sample suffers greatly from observational biases, and is highly skewed toward lower redshifts. As there are many unpredictable factors that affect the redshift measurements, such as the local weather conditions and the number of telescopes that are available for follow-up observations, it is difficult to construct a complete redshift sample of the observed GRBs.

Despite the complications, many studies have discussed the observational selection biases and their effect on the GRB redshift distribution, and explored possibilities to construct a complete GRB sample (e.g., Coward et al. 2008, 2013; Fynbo et al. 2009; Jakobsson et al. 2012; Hjorth et al. 2012; Salvaterra et al. 2012). In particular, Fynbo et al. (2009) carefully consider possible observational constraints on performing follow-up observations and create a GRB sample that is less affected by the observational biases. These authors set up a number of criteria to select GRBs that have optimal conditions for follow-up observations. For example, bursts that are too close to the Sun or the Moon, or have high Galactic extinction, are removed from the sample, because it is harder to perform follow-up observations for these bursts and thus these bursts need to be extremely bright to have measured redshifts.

There are 79 GRBs with redshift measurements (either from afterglows, host galaxies, or both) in the statistical GRB sample compiled by Fynbo et al. (2009, Table 73 in their paper). These authors note that bursts with redshift measurements from the optical afterglows are not representative for all the Swift bursts in their statistical sample. The GRBs without optical spectroscopy in their sample are likely to suffer from dust obscuration other than the Galactic extinction, such as dust in the GRB host galaxies. A similar conclusion has also been obtained by The Optically Unbiased Gamma-Ray Burst Host (TOUGH) survey, which performs a systematic search for the GRB host galaxies, and found that the host galaxies of GRBs with no optical and/or near infrared afterglows are significantly brighter and redder than those with optical and/or near infrared afterglows (Hjorth et al. 2012). Although it is possible to use host galaxy detections to recover these missing redshifts of those bursts without optical afterglows, this approach introduces other biases due to host galaxy brightness and the optical survey sensitivity.

We therefore adopt the GRBs from the statistical sample constructed in Fynbo et al. (2009) that have redshift measurements either from only afterglows, or from both afterglows and host galaxies. We exclude bursts with redshift measurements only from host galaxies, because low redshift bursts have a higher probability of having detectable host galaxies. We also exclude GRBs with photometric redshifts due to their large uncertainties. Additionally, we remove one burst that does not have E_iso information (see Section 4.4). There are 66 bursts in this redshift sample we adopt.

The GRB peak fluxes in the BAT energy range 15–150 keV can be measured directly from the BAT observations using the fewest assumptions about the burst characteristics. Therefore, the 15–150 keV peak flux is the least uncertain among all the GRB properties we consider here. We adopt the 15–150 keV peak fluxes of the real BAT-detected GRBs reported in Sakamoto et al. (2011); a total number of 402 bursts are given.

Not all of the BAT-detected GRBs have measured $E^{\rm obs}_{\rm peak}$. Therefore, we use the $E^{\rm obs}_{\rm peak}$ estimator found in Sakamoto et al. (2009) to estimate the burst $E^{\rm obs}_{\rm peak}$ based on the power-law index (Γ) of the burst spectra when fitted by a simple power-law model. The power-law indices are reported in Sakamoto et al. (2011).

The $E^{\rm obs}_{\rm peak}$ estimator in Sakamoto et al. (2009) can only calculate $E^{\rm obs}_{\rm peak}$ in a limited range of power-law indices (1.3 ⩽ Γ ⩽ 2.3), which corresponds to $E^{\rm obs}_{\rm peak}$ values inside the detectable energy range of the BAT. In the cases where the power-law indices are out of the range, we use the power-law indices to estimate whether $E^{\rm obs}_{\rm peak}$ lies below or above the BAT detectable energy range. A small power-law index (Γ < 1.3) indicates that $E^{\rm obs}_{\rm peak}$ falls above the BAT detectable energy range ($E^{\rm obs}_{\rm peak} \gtrsim 150$ keV), while a large power-law index (Γ > 2.3) implies that $E^{\rm obs}_{\rm peak}$ is lower than the BAT detectable energy range ($E^{\rm obs}_{\rm peak} \lesssim 15$ keV). There are 414 bursts in the $E^{\rm obs}_{\rm peak}$ sample we adopt, in which 26 bursts are below the BAT detectable energy range, and 80 bursts are above the BAT detectable energy range.

The total energy output (i.e., the fluence) E_iso of an observed GRB is difficult to measure. Both the burst spectrum and redshift are required to calculate E_iso. However, most of the bursts are lacking redshift measurements (see Section 4.1) and/or well-constrained $E^{\rm obs}_{\rm peak}$, leading to a poor characterization of the spectra, especially when $E^{\rm obs}_{\rm peak}$ lies outside of the BAT energy range. Therefore, even with a burst that has an observed redshift, estimating E_iso requires one to extrapolate the measured spectrum to an energy out of the detectable range, and hence introduces uncertainties. In addition, an observation might not capture the complete light curve of a burst due to the background noise. In order words, it is likely that BAT only detects the tip of the burst and thus underestimates the total energy output of the event.

Butler et al. (2007, 2010) report a list of E_iso values of the BAT-detected bursts. They estimate the E_iso in the T₉₀ burst duration and an energy range of 1–10⁴ keV. Due to the difficulties in directly measuring E_iso, these authors adopt a Bayesian approach to estimate the values, with a prior $E^{\rm obs}_{\rm peak}$ distribution following results from the observations of CGRO/BATSE (Preece et al. 2000), the primary GRB instrument prior to Swift. Robertson & Ellis (2012) compile a list of BAT-detected GRBs that have measured redshifts and E_iso values. The majority of the E_iso values of their list are from Butler et al. (2007, 2010) with a few more from Sakamoto et al. (2011). In addition, these authors calculate E_iso for 29 new bursts from their fluence and redshift values based on numerous references (see Robertson & Ellis 2012). The E_iso estimations become more uncertain for bursts with $E^{\rm obs}_{\rm peak}$ lying outside of the BAT detectable energy range, because it is hard to pinpoint the turnover of the spectrum. Therefore, for the E_iso sample, we only consider bursts with $E^{\rm obs}_{\rm peak}$ in the BAT energy range and adopt the corresponding E_iso values from the list in Robertson & Ellis (2012).

We modify the parameters for the intrinsic GRB redshift distribution and luminosity function in Equations (1) and (3) to search for a set of the parameters that generates a mock-triggered burst sample that matches the best with the observed GRB characteristics. Since we follow the same functional forms for the redshift and luminosity equations as those in Wanderman & Piran (2010), we start our search around numbers reported by those authors.

In our search, we simulate 10,000 bursts for each set of parameters in order to have enough mock-triggered bursts to reduce statistical fluctuations. Our code was run on computers with four Intel quad-core Q9650 processors at 3 GHz on the Scientific Linux release 5.9 (Boron). Without including $E^{\rm obs}_{\rm peak}$ evolution, our code takes around 5 hr to simulate 10,000 bursts and to run them through the trigger simulator, for light curves with bin size = 1.6 s. If $E^{\rm obs}_{\rm peak}$ evolution is included, the simulation takes ∼3 days for a sample of 10,000 bursts with light curves binned into 1.6 s. If we decrease the light curve bin size (i.e., increase the number of light curve bins for each burst), the time required for the trigger code to go through the whole light curve range increases as the total number of light curve bins.

The equations for the redshift distribution and luminosity function we adopt contain seven parameters total (see Equations (1) and (3)). Therefore, to run a complete Monte Carlo simulation and search through the full parameter space is highly demanding of computational power and time. For example, if we explore a range of 10 values for each parameter, we will have 10⁷ combinations for the parameter set, and hence the full Monte Carlo simulations with burst light curves binned to 1.6 s will take ∼5700 yr for the simulations to finish. One possibility might be adopting the Markov Chain Monte Carlo method. However, due to the complexity in the GRB observables we are taking into account (e.g., cosmic GRB rate, luminosity function, $E^{\rm obs}_{\rm peak}$ distribution), it is difficult to find a good and efficient algorithm that guarantees speeding up the process significantly and converging to the correct answer. Therefore, instead of searching the full parameter space, here we only try to find at least one possible set of parameters that generates a good match with the observed GRB characteristics.

Among all the long rate trigger criteria of the BAT, 250 criteria have foreground periods longer than 2 s. Since we focus only on long bursts in this paper, most of the bursts that are triggered by criteria with foreground periods shorter than 2 s should also be triggered by criteria with foreground periods longer than 2 s. Therefore, to speed up the search process for the GRB properties, we only run through the 250 criteria that are longer than 2 s in our main search. There could be a few scenarios where the long GRBs are "long" by virtue of consisting of multiple "short" spikes that are well-separated (longer than the foreground period of the trigger criteria). For these bursts, it is possible that they could only be triggered by criteria with foreground durations shorter than 2 s. Hence, once we find a parameter set that matches well with observations, we will rerun the sample with all the long trigger criteria to check whether the results remain a good fit with observations.

Additionally, our main searches are done without including $E^{\rm obs}_{\rm peak}$ evolution, and with a fixed $E^{\rm src}_{\rm peak}$ distribution, for the purpose of speeding up the search process as well. To decide what kind of $E^{\rm src}_{\rm peak}$ distribution to use in our main search, we perform some test runs at the beginning using several very different functions for the $E^{\rm src}_{\rm peak}$ distribution (see Section 6.1 for a detailed discussion of the choices of functions). We find that, in general, assuming some kind of intrinsic relation between $E^{\rm src}_{\rm peak}$ and the burst energy output (e.g., luminosity, E_iso) seems to create a better match with observations. Therefore, in our main searches we adopt an $E^{\rm src}_{\rm peak}$ distribution that follows the Yonetoku relation of $E^{\rm src}_{\rm peak}$ and L_peak (Yonetoku et al. 2004; see more discussion in Section 6.1). Once we find a sample that matches well with the observations, we modify the $E^{\rm src}_{\rm peak}$ distribution using several very different functions to see whether this would cause significant changes in the results. Similarly, we also run the same set of parameters with $E^{\rm obs}_{\rm peak}$ evolution to see how the results might change. If the effect of adopting a different $E^{\rm src}_{\rm peak}$ distribution and/or $E^{\rm obs}_{\rm peak}$ evolution turns out to be significant, we adjust the parameters in the redshift and luminosity functions and repeat this process.

Furthermore, as discussed in Section 3.3.4, the sensitivity of BAT, and hence the sensitivity of our trigger simulator, is slightly different for different numbers of active detectors. Ideally, to simulate the decrease in BAT's sensitivity, we need to run our simulation for different numbers of active detectors and take an average of all the results. However, this can easily increase the search time to a point that it becomes impractical. Therefore, we start our search with one number of active detectors. Usually we start with ∼25, 000 active detectors, which is the medium number of detectors for the 8 yr of BAT's operation. Once we find a set of parameters that matches well with the real observations, we perform the simulations for several different numbers of active detectors and take the average. Minor adjustments of the parameters might be needed until this averaged result fits well with the observational GRB characteristics.

To see how well the mock-triggered sample matches the observed GRB characteristics, we compare several burst properties of the mock-triggered sample with those of the real BAT-detected GRBs. As discussed in Section 4, we use the redshift and peak flux distributions as our main guides. For these two distributions, we perform the Kolmogorov-Smirnov test (KS test) to quantify how good the matches are between the distributions from the mock-triggered sample and those from the real BAT-detected GRBs. We modify the parameters in the redshift and luminosity functions until the KS test (with uncertainties) gives a significance level above 90%. Additionally, we use the $E^{\rm obs}_{\rm peak}$ distribution and the $E^{\rm src}_{\rm peak}$-E_iso relation as references. We try to make these two distributions match as well as possible to the observed ones. However, due to the large and hard-to-quantify uncertainties in $E^{\rm obs}_{\rm peak}$ and E_iso (see discussion in Section 4), we do not require them to match perfectly, and do not perform statistical tests on these two distributions.

The parameter set shown in Table 2 contains our best fit parameters for Equations (1) and (3). This set of parameters generates mock-triggered bursts that have a redshift distribution and peak flux distribution that match the best with those from the real observed GRBs. Table 2 also lists the KS test values for both the redshift distribution and the peak flux distribution. Both the number "D" in the KS test and the significance level of "D" (i.e., the probability of "D" larger than the reported value) are given in the table. The value "D" is the key parameter in the KS test, which indicates the maximum distance between the cumulative distribution of the two tested samples. Smaller "D" implies a better match of the two samples. All of the real BAT-detected bursts we adopted for comparison are GRBs before 2009. Therefore, we run the simulations five times (each time generates 10,000 bursts) with a different average number of enabled detectors from the years 2005 to 2009. The results of the simulations shown in the following figures are generated from the total 50,000 bursts with different numbers of enabled detectors.

Figure 9 shows the redshift distribution from this best-fit sample. Panels (a) and (b) of Figure 9 give the input redshift and luminosity distributions, respectively, of all 50,000 bursts created. Panel (c) of the figure shows the comparison of the redshift distribution between the mock-triggered bursts and real observations. The red bars in panel (c) show the normalized numbers of the simulated bursts that are triggered by our trigger simulator. The blue dots in panel (c) show the normalized numbers of real BAT detections. The error bars along the y-axis are the statistical errors (i.e., square root of the number in each bin). The x-axis error bars simply represent the bin size. As mentioned in Section 4.1, we adopt the GRB redshift sample reported in Fynbo et al. (2009) for our comparison.

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Figure 10 shows the peak flux distribution from the best-fit sample. The top panel gives the input peak flux distribution from the whole sample of 50,000 bursts. The bottom panel shows the peak flux distribution of the simulated bursts that are triggered by our trigger simulator. Again, this mock-triggered sample is given as red bars and the distribution of the real BAT-detected GRBs is shown as blue dots. The peak fluxes of the real BAT-detected GRBs are reported in Sakamoto et al. (2011). These authors also listed the errors for each measured peak flux. Therefore, we run a quick Monte Carlo simulation to see how the match of the two distributions (i.e., the value from the KS test) changes if one allows the peak fluxes to change within their error bars. Results show that the uncertainty in the peak flux measurements can change the "D" value in the KS test from 7.21 × 10⁻² to 2.05 × 10⁻², which correspond to a significance level of 3.66% and 99.69%, respectively. These are the uncertainties we listed in Table 2.

Figure 11 plots the E_peak distribution for this best-fit sample and its comparison with the distribution from real BAT-detected GRBs. Panel (a) plots the input $E^{\rm src}_{\rm peak}$ distribution of the 50,000 simulated bursts. Panel (b) shows the comparison of the $E^{\rm obs}_{\rm peak}$ distribution between the mock-triggered bursts and the real BAT-detected GRBs. Similarly, this mock-triggered sample is plotted in red bars and the distribution of the real BAT-detected GRBs is shown as blue dots. The $E^{\rm obs}_{\rm peak}$ values of the real GRBs are estimated from Sakamoto et al. (2009, 2011), as described in Section 4.3. Because it is hard to estimate $E^{\rm obs}_{\rm peak}$ when the values fall out of the BAT energy range (∼15–150 keV), we make two large bins for bursts with $E^{\rm obs}_{\rm peak}$ outside of the BAT range. Inside the BAT energy range, the data are binned into two smaller bins.

Panel(c) of Figure 11 shows the E_iso–$E^{\rm src}_{\rm peak}$ correlation. Red dots in this plot represent values from the mock-triggered sample. Dark red dots show the bursts with $E^{\rm obs}_{\rm peak}$ values inside the BAT energy range, while light red dots are events with $E^{\rm obs}_{\rm peak}$ values outside of the BAT energy range. Blue crosses show the bursts from the real BAT detections. Due to the large uncertainties of the $E^{\rm src}_{\rm peak}$ and E_iso values for real bursts, only events with the observed $E^{\rm obs}_{\rm peak}$ inside the BAT energy range are shown in the plot (see detailed discussion in Section 4). The E_iso values in the figure are integrated over T₉₀, both for the real GRBs and the simulated bursts. The intrinsic E_iso values of the real BAT-detected bursts are unknown due to background noise. Therefore, we also restrict the E_iso to the T₉₀ range for the simulated bursts to have a fair comparison with real observations.

Results from our search suggest that a slightly modified Yonetoku relation

$$\begin{eqnarray} &&E^{\rm src}_{\rm peak} = 1.8 \times \left(2.34 \times 10^{-5} \times \frac{L_{\rm peak}}{10^{52} \ \rm erg \ s^{-1}} \right)^{-0.5} \end{eqnarray} \tag{ 8 }$$

creates a better match for the E_peak distribution, as shown in Panel (b) of Figure 11. Everything inside the bracket of Equation (8) is the original functional from Yonetoku et al. (2004), but we use peak luminosity L_peak instead of the isotropic luminosity L_iso. In our modified Yonetoku relation, $E^{\rm src}_{\rm peak}$ is 1.8 times larger than that produced by the original Yonetoku relation. The main reason for us to consider this modified Yonetoku relation is because this relation generates higher $E^{\rm src}_{\rm peak}$ bursts for the same L_peak, and thus increases the number of detections of higher $E^{\rm src}_{\rm peak}$ bursts and matches the observations better.

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This best-fit sample predicts that Swift should detect ∼96 bursts per year, which is in good agreement with the average number of ∼95 GRBs per year from 2005 to 2009 based on real BAT observations (Sakamoto et al. 2011). The predicted detection rate for Swift (R_Swift) is calculated based on the following equation,

$$\begin{eqnarray} &&R_{{\rm Swift}} = {\cal R}_{\rm GRB; dz} \times f_{\rm detect} \times \rm FOV \times {\it t}_{\rm survey}, \end{eqnarray} \tag{ 9 }$$

where ${\cal R}_{\rm GRB; dz}$ is the observed GRB rate in units of number per redshift per solid angle per time in the observed frame (Equation (2)), f_detect is the detection rate, as shown in the third column of Table 3, FOV ∼2 sr is the field-of-view of the BAT (Barthelmy et al. 2005), and t_survey ∼ 90% is the fraction of the time that BAT spends on searching for GRBs.

Table 3. Summary of the Results from Simulations without $E^{\rm obs}_{\rm peak}$ Evolution

$E^{\rm src}_{\rm peak}$	Detection	Prediction for	KS Test Significance	KS Test Significance for
Distribution	Rate	Swift (yr−1)	for z Distribution	Peak Flux Distribution
Modified Yonetoku	13.95%	95.60	99.79%	$85.59^{+14.10}_{-81.93}\%$
Flat in linear space	3.76%	25.76	14.88%	$88.81^{+10.35}_{-69.58}\%$
Flat in log space	8.54%	58.51	68.24%	$39.95^{+59.41}_{-34.21}\%$
Gaussian	9.74%	66.73	79.08%	$79.34^{+20.65}_{-76.50}\%$
Specified function	9.05%	62.00	63.18%	$96.32^{+3.68}_{-83.07}\%$

Note. The five samples shown here are based on different $E^{\rm src}_{\rm peak}$, as described in Section 6.1.

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As discussed in Section 5.1, we rerun this best-fit sample with the complete set of long trigger criteria, including those with foreground periods shorter than 2 s, in order to make sure the results remain a good fit with the observations. We use 26,884 active detectors in this run, which is the average number of active detectors from 2005 to 2009. Results show that comparing with the same sample (with 26,884 active detectors) using only the trigger criteria longer than 2 s, adopting the complete set of trigger criteria, changes the detection rate from 13.73% to 14.01%. The KS test significance for the redshift distribution changes from 99.02% to 99.32%, and the KS test significance for the flux distribution changes from $81.80^{+17.91}_{-65.55}\%$ to $89.78^{+9.41}_{-67.80}\%$. All of these changes are significantly smaller than the statistical uncertainty. Therefore, results using our best-fit parameters remain good matches with observations when adopting the full set of long trigger criteria.

The intrinsic $E^{\rm src}_{\rm peak}$ distribution remains uncertain and controversial. Many studies have suggested possible correlations between the total energy output of the burst and the peak energy of the νF_ν spectrum in the burst rest frame $E^{\rm src}_{\rm peak}$. These relations often attempt to relate $E^{\rm src}_{\rm peak}$ and E_iso, the total energy fluence of the burst, or $E^{\rm src}_{\rm peak}$ and L_iso, the total luminosity of the burst (Amati et al. 2002; Amati 2006; Ghirlanda et al. 2004; Yonetoku et al. 2004).

In order to see the consequences of adopting different $E^{\rm src}_{\rm peak}$ distributions, we test several functions that have significantly different shapes in comparison to that used in our best-fit sample (described in Section 5). The distributions we test include: (1) A flat $E^{\rm src}_{\rm peak}$ distribution in linear space. (2) A flat $E^{\rm src}_{\rm peak}$ distribution in logarithmic space. (3) A Gaussian $E^{\rm src}_{\rm peak}$ distribution in logarithmic space, with average =300 keV and σ = 1. (4) A special function

$$\begin{eqnarray} \phi \left(\rm log_{10} \left(\it E^{\rm src}_{\rm peak}\right)\right) \propto \left\lbrace \begin{array}{@{} ll}\rm \! {log}_{10} \left(\it E^{\rm src}_{\rm peak} \right) & \mbox{if} \ E^{\rm src}_{\rm peak} < 10 \ \mbox{keV}\!, \\ \! \rm {log}_{10}(10 \ \rm keV) & \mbox{otherwise.} \end{array} \right. \end{eqnarray} \tag{ 10 }$$

This function contains a lower number of bursts for $E^{\rm src}_{\rm peak} < 10$ keV and follows a flat distribution in logarithmic space for events with $E^{\rm src}_{\rm peak} > 10$ keV. $E^{\rm obs}_{\rm peak}$ evolution is not included in these simulations.

Table 3 summarizes the major results from using different $E^{\rm src}_{\rm peak}$ distributions. All the results, except our best-fit sample (the one with a modified Yonetoku $E^{\rm src}_{\rm peak}$ distribution), are based on 10,000 simulated bursts using 26,884 active detectors, which is the average number of active detectors from the years 2005 to 2009 (the time period of the real BAT-detected bursts adopted in this paper). Our best-fit result contains a total number of 50,000 simulated bursts with different numbers of enabled detectors, as described in Section 5.2.

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For all four different $E^{\rm src}_{\rm peak}$ distributions we test, the resulting redshift and peak flux distributions of the mock-triggered samples remain adequate fits to the real observations. In fact, for the peak flux distributions, for which we can quantify the uncertainties of the fit, the KS tests show that all these matches based on different $E^{\rm src}_{\rm peak}$ distributions are within the calculated uncertainties of each other. In other words, all the $E^{\rm src}_{\rm peak}$ distributions in our tests give the same level of good fits to the observations, and thus comparing the results to the redshift and peak flux distributions alone would not be sufficient to distinguish different $E^{\rm src}_{\rm peak}$ distributions. However, using different $E^{\rm src}_{\rm peak}$ distributions does change the detection rates. For the four different cases we tried, the detection rate can change from ∼4% to ∼14%, and thus will affect the normalization of the GRB rate (i.e., the GRB rate at z = 0) up to ∼4 times higher.

The major distinctions of adopting different $E^{\rm src}_{\rm peak}$ distributions appear in the E_peak distributions of the mock-triggered samples. Figures 12 and 13 compare the resulting $E^{\rm obs}_{\rm peak}$ distributions and the $E_{\rm iso}-E^{\rm src}_{\rm peak}$ correlations based on these four different $E^{\rm src}_{\rm peak}$ distributions. As one can see from these figures, all these additional $E^{\rm src}_{\rm peak}$ distributions we test seem to generate worse matches to the observational sample than the modified Yonetoku sample (see Figure 11), in both the $E^{\rm obs}_{\rm peak}$ distributions and the E_iso–$E^{\rm src}_{\rm peak}$ correlations. However, due to the large and hard-to-quantify uncertainties in the $E^{\rm obs}_{\rm peak}$ and E_iso of the real observational sample, it remains ambiguous whether any of these $E^{\rm src}_{\rm peak}$ distributions can be excluded. Therefore, these plots are only shown to indicate how the distributions can change if one assumes different $E^{\rm src}_{\rm peak}$ distributions; no conclusion about the intrinsic $E^{\rm src}_{\rm peak}$ distribution can be drawn until direct measurements of $E^{\rm obs}_{\rm peak}$ and E_iso become available.

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As discussed in Section 3.2, spectral evolution has been observed in many GRBs, and thus we implement an option to include $E^{\rm obs}_{\rm peak}$ evolution in our simulation. To see how the results might change if $E^{\rm obs}_{\rm peak}$ evolution is included, we apply this option to the best-fit parameter set (Table 2) with all five different $E^{\rm src}_{\rm peak}$ distributions we test.

Results with $E^{\rm obs}_{\rm peak}$ evolution included are summarized in Table 4. Similar to those samples presented in Table 3, results are from simulations of 10,000 bursts using the average number of 26,884 active detectors from the years 2005 to 2009. These simulations show that including $E^{\rm obs}_{\rm peak}$ evolution can result in noticeable changes of the outcome, especially in the distributions of the burst characteristics. Although our best-fit sample (the one with a modified Yonetoku $E^{\rm src}_{\rm peak}$ distribution) remains a good match with both the redshift and peak flux distributions, samples with other $E^{\rm src}_{\rm peak}$ distributions show that the resulting KS test values can change considerably with $E^{\rm obs}_{\rm peak}$ evolution included. However, we noted that when actually plotting these distributions, the changes do not seem to be as significant as indicated by the KS test significance levels in the tables. The general shapes and widths of the distributions remain similar with or without $E^{\rm obs}_{\rm peak}$ evolution. This is because the KS test values are especially sensitive to the medium of the distribution. Therefore, a slight change in the medium can lead to a major difference in the significance level. Moreover, if the uncertainties in the significance levels of the peak flux distributions are taken into account, the data from real observations actually cannot distinguish the difference between the results with or without $E^{\rm obs}_{\rm peak}$ evolution.

Table 4. Summary of the Results from Simulations with $E^{\rm obs}_{\rm peak}$ Evolution

$E^{\rm src}_{\rm peak}$	Detection	Prediction for	KS Test Significance	KS Test Significance for
Distribution	Rate	Swift (yr−1)	for z Distribution	Peak Flux Distribution
Modified Yonetoku	15.05%	103.10	98.62%	$69.34^{+20.00}_{-62.60}\%$
Flat in linear space	4.65%	31.86	31.39%	$0.32^{+2.02}_{-0.32}\%$
Flat in log space	8.63%	59.12	59.15%	$47.88^{+51.11}_{-41.73}\%$
Gaussian	10.10%	69.19	76.84%	$50.68^{+48.47}_{-49.47}\%$
Specified function	9.37%	64.19	57.44%	$89.00^{+10.95}_{-80.81}\%$

Note. The five samples shown here are based on different $E^{\rm src}_{\rm peak}$, as described in Section 6.1.

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The sample using a flat $E^{\rm src}_{\rm peak}$ distribution in linear space (see description in Section 6.1) shows the most remarkable change in the KS test significance level when $E^{\rm obs}_{\rm peak}$ evolution is included. Therefore, Figure 14 plots the peak flux distributions with and without $E^{\rm obs}_{\rm peak}$ evolution of this sample as an example of how the general shapes of the distributions change.

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When including the $E^{\rm obs}_{\rm peak}$ evolution, the detection rates in these samples either stay similar or increase slightly compared to those without $E^{\rm obs}_{\rm peak}$ evolution. A possible explanation is that as the burst spectra get softer, some of the bursts might have $E^{\rm obs}_{\rm peak}$ fall into the BAT energy range, and thus become detectable. However, due to the low number of samples with $E^{\rm obs}_{\rm peak}$ evolution in our simulations, we cannot exclude the possibility that this trend is a mere coincidence.