ERRATUM: "PROBING THE COSMIC GAMMA-RAY BURST RATE WITH TRIGGER SIMULATIONS OF THE SWIFT BURST ALERT TELESCOPE" (2014, ApJ, 783, 24)

Recently, we found a mistake in the code of the published article, which affects the normalization of the GRB rate, i.e., the parameter ${R}_{\mathrm{GRB}}(z=0)$ in the following equation (Equation (1) in the original paper):

$$\begin{eqnarray}{R}_{\mathrm{GRB}}(z)={R}_{\mathrm{GRB}}(z=0)\left\{\begin{array}{ll}{(1+z)}^{{n}_{1}}, & z\leqslant {z}_{1},\\ {(1+{z}_{1})}^{{n}_{1}-{n}_{2}}\ {(1+z)}^{{n}_{2}}, & z\gt {z}_{1}\end{array}\right.\end{eqnarray} \tag{ 1 }$$

The mistake was caused by incorrectly using the numerical recipe subroutine (Press et al. 1992), "qromb", recursively to perform a double integration. Because one of the routines ("trapzd") called by qromb uses a static local variable, this function should not be used recursively.

For the equation we use for GRB rate calculation, incorrectly using "qromb" recursively results in an integrated value that is a factor of two smaller than the correct number. In other words, the true integrated number is twice as high as what we calculated. Therefore, we need to lower the normalization parameter ${R}_{\mathrm{GRB}}(z=0)$ by a factor of two in order to have the same accumulated GRB number that matches with ${\text{}}{Swift}$'s detection. This affects all the numbers of ${R}_{\mathrm{GRB}}(z=0)$ reported in the paper (all of them need to be divided by 2), including the normalization parameter in our best-fit model (mentioned in Table 2 and Section 11 in the original paper), in the upper and lower limits (Tables 5 and 6 in the original paper), and in the fits we used to study the luminosity evolution (Tables 7 and 8 in the original paper). The corrected ${R}_{\mathrm{GRB}}(z=0)$ are listed in Table 1 below. In addition, the corrected version of Figure 17 (in the original paper), which shows the GRB comoving rate, is also provided in this erratum (see Figure 1 below). The Appendix in this erratum includes a more detailed explanation of where the double integration and the factor of two come from.

Table 1. Summary of the Normalization Parameters ${R}_{\mathrm{GRB}}(z=0)$ Presented in the Paper that are Affected by this Mistake. The Corrected Numbers here are Half of the Original Numbers

Location in the Original Paper	Corrected ${R}_{\mathrm{GRB}}(z=0)[{\mathrm{Gpc}}^{-3}\ {\mathrm{yr}}^{-1}]$
Table 2 and Section 11	0.42
Table 5	0.38
Table 6	0.51
Table 7	0.54
Table 8	0.56

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To double check for consistency, we use Python and the Scipy library (the "quad" subroutine; Jones et al. 2001) to perform the integration. We found the integrated GRB number (i.e., the total number of GRBs in the universe that are beamed toward us) to be ${4571}_{-1584}^{+829}$ GRBs per year. This number is very similar to the one reported in the paper, as expected. The small difference is likely due to numerical uncertainty and rounding error.

We have double checked all of our codes to make sure the normalization is the only thing affected by this problem. We sincerely apologize for this mistake, and for not noticing it earlier.

APPENDIX: DETAILED EXPLANATION OF THE PROBLEM

We intergrate Equation (2) in the original paper

$$\begin{eqnarray}&&{R}_{\mathrm{GRB};\mathrm{dz}}(z)=\displaystyle \frac{{dN}}{d{{\rm \Omega }}\ {dz}\ {{dt}}_{\mathrm{obs}}}={R}_{\mathrm{GRB}}(z)\ \displaystyle \frac{{{dV}}_{\mathrm{comov}}{{dt}}_{\mathrm{src}}}{d{{\rm \Omega }}{{dzdt}}_{\mathrm{obs}}}=\displaystyle \frac{{R}_{\mathrm{GRB}}(z)}{(1+z)}\ \displaystyle \frac{{{dV}}_{\mathrm{comov}}}{d{{\rm \Omega }}{dz}}\end{eqnarray} \tag{ 2 }$$

from redshift z = 0 to z = 10 to get the total GRB number in the universe (per solid angle per observation time), and convert this intrinsic rate to a detected rate by multiplying the detection fraction, the Swift/BAT's field of view, and the Swift survey time (see Equation (9) in the original paper). The estimated detection rate is then used for constraining the normalization factor ${R}_{\mathrm{GRB}}(z=0)$ by comparison with the true Swift detection rate. Equation (2) has an implicit integration from ${{dV}}_{\mathrm{comov}}={r}_{\mathrm{com}}^{2}\ d{{\rm \Omega }}\ {{dr}}_{\mathrm{com}}$, where ${r}_{\mathrm{com}}=\int (c/H(z)){dz}$. $H(z)$ is the Hubble parameter and c is the speed of light.

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Because of how the numerical recipe subroutine "trapzd" is set up, as long as the number of the first integration ($\int {R}_{\mathrm{GRB};\mathrm{dz}}(z){dz}$ in our case) is much greater than the second integration ($\int (1/H(z)){dz}$ in our case), incorrectly using the subroutine recursively will always produce a number about a factor of two smaller than the correct answer. Table 2 shows an example of the fraction of incorrect and correct answer converges to ∼0.5 as we integrate with a larger redshift range. The numbers in this example are calculated from the GRB parameters of Table 2 in the paper, including the original incorrect ${R}_{\mathrm{GRB}}(z=0)$.