# Model-independent distance calibration of high-redshift gamma-ray bursts and
constrain on the $\Lambda$CDM model

Hai-Nan Lin1, Xin Li1,3, Zhe Chang2
1Department of Physics, Chongqing University, Chongqing 401331, China
2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
3State Key Laboratory Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted xxxx; Received xxxx; in original form xxxx)

###### Abstract

Gamma-ray bursts (GRBs) are luminous enough to be detectable up to redshift
$z\sim 10$. They are often proposed as complementary tools to type-Ia
supernovae (SNe Ia) in tracing the Hubble diagram of the Universe. The
distance calibrations of GRBs usually make use one or some of the empirical
luminosity correlations, such as $\tau_{\rm lag}-L$, $V-L$, $E_{p}-L$,
$E_{p}-E_{\gamma}$, $\tau_{\rm RT}-L$ and $E_{p}-E_{\rm iso}$ relations. These
calibrating methods are based on the underling assumption that the empirical
luminosity correlations are universal over all redshift range. In this paper,
we test the possible redshift dependence of six luminosity correlations by
dividing GRBs into low-$z$ and high-$z$ classes according to their redshift
smaller or larger than 1.4. It is shown that the $E_{p}-E_{\gamma}$ relation
for low-$z$ GRBs is consistent with that for high-$z$ GRBs within $1\sigma$
uncertainty. The intrinsic scatter of $V-L$ relation is too larger to make a
convincing conclusion. For the rest four correlations, however, low-$z$ GRBs
differ from high-$z$ GRBs at more than $3\sigma$ confidence level. As such, we
calibrate GRBs using the $E_{p}-E_{\gamma}$ relation in a model-independent
way. The constraint of high-$z$ GRBs on the $\Lambda$CDM model gives
$\Omega_{M}=0.302\pm 0.142(1\sigma)$, well consistent with the Planck 2015
results.

###### keywords:

cosmological parameters – gamma-ray burst: general

††pagerange: Model-independent distance calibration of high-redshift gamma-ray
bursts and constrain on the $\Lambda$CDM model–Model-independent distance
calibration of high-redshift gamma-ray bursts and constrain on the
$\Lambda$CDM model††pubyear: 2015

##  1 Introduction

The consistent luminosity of type-Ia supernovae (SNe Ia) makes them to be
standard candles in probing the expansion history of the Universe (Riess et
al., 1998; Perlmutter et al., 1999). However, the redshift of SNe Ia is
usually less than 1.4. Gamma-ray bursts (GRBs), as the most energetic
explosions in the Universe, are bright enough to be detectable up to redshift
$z\sim 10$ (Salvaterra et al., 2009; Tanvir et al., 2009; Cucchiara et al.,
2011; Tanvir, 2013). Therefore, they are often proposed as complementary tools
to SNe Ia in tracing the Hubble diagram of the high-redshift Universe.
Actually, GRBs have already been widely used, either alone or in combination
with other data such as SNe Ia, to constrain the cosmological parameters
(Schaefer, 2003; Bloom, Frail & Kulkarni, 2003; Xu, Dai & Liang, 2005; Firmani
et al., 2005; Liang & Zhang, 2005; Firmani et al., 2006; Schaefer, 2007; Liang
et al., 2008; Liang & Zhang, 2008; Wei & Zhang, 2009; Wei, 2010; Demianski &
Piedipalumbo, 2011; Wang, Qi & Dai, 2011; Capozziello et al., 2012; Amati &
Della Valle, 2013; Wei, Wu & Melia, 2013; Velten, Montiel & Carneiro, 2013;
Cai et al., 2013; Bretón & Montiel, 2013; Chang et al., 2014; Cano &
Jakobsson, 2014; Cuzinatto, Medeiros & de Morais, 2014; Wang & Wang, 2014;
Wang, Dai & Liang, 2015; Li, Ding & Zhu, 2015). Unfortunately, since the
explosion mechanism of GRBs is still not clearly known, the distance
calibration of GRBs is not as easy as that of SNe Ia.

Many methods have been proposed to calibrate GRBs (Dai, Liang & Xu, 2004;
Ghirlanda et al., 2004; Liang & Zhang, 2005; Firmani et al., 2005; Schaefer,
2007; Liang et al., 2008; Liang & Zhang, 2008; Wei & Zhang, 2009; Wei, 2010;
Liu & Wei, 2014). Most calibrating methods rely on the empirical luminosity
correlations found in long GRBs. At least six luminosity correlations can be
used in the calibration. Norris, Marani & Bonnell (2000) found a correlation
between spectrum lag and isotropic peak luminosity ($\tau_{\rm lag}-L$
relation). Fenimore & Ramirez-Ruiz (2000) found a correlation between time
variability and isotropic peak luminosity ($V-L$ relation). Amati et al.
(2002) found a tight correlation between the peak energy of $\nu F_{\nu}$
spectrum and isotropic equivalent energy ($E_{p}-E_{\rm iso}$ relation).
Ghirlanda, Ghisellini & Lazzati (2004) found a similar correlation between
peak energy and collimation-corrected energy ($E_{p}-E_{\gamma}$ relation).
Yonetoku et al. (2004) found a correlation between peak energy and isotropic
peak luminosity ($E_{p}-L$ relation). Schaefer (2007) found a correlation
between minimum rise time of light curve and isotropic peak luminosity
($\tau_{\rm RT}-L$ relation).

All of the calibrating methods based on the empirical luminosity correlations
have an underlying assumption, that is, the luminosity correlations do not
evolve with redshift. If the luminosity correlations is not universal over the
whole redshift range, these calibrating methods will fail. In fact, the
possible redshift dependence of luminosity correlations has already been
tested by some authors. Basilakos & Perivolaropoulos (2008) investigated the
above six empirical luminosity correlations in four redshift bins, and showed
that the slopes of all six correlations differs between redshift bins,
although the intercepts do not vary significantly. Since the GRB sample is not
large enough in each bin, the statistical uncertainty is large. Therefore,
they concluded that no statistically significant evidence for the redshift
evolution of the luminosity correlations was found. With the updated data,
Wang, Qi & Dai (2011) got a similar conclusion. However, Li (2007)
investigated the Amati relation in four redshift bins and showed that the
slope and intercept varies with redshift systematically and significantly.
Recently, Lin et al. (2015) divided GRBs into two redshift bins, and found
that the Amati relation (especially the slope parameter) of low-$z$ GRBs
differs from that of high-$z$ GRBs at more than $3\sigma$ confidence level.
Dainotti et al. (2013) investigated the slope evolution of GRB correlations
and showed that correlation slope that differs from the intrinsic one may
overestimate or underestimate the cosmological parameters.

In this paper, we recheck the possible redshift dependence of six luminosity
correlations. We divide GRBs into low-$z$ and high-$z$ classes according to
their redshift smaller or larger than 1.4, and test the luminosity
correlations for low-$z$ and high-$z$ GRBs, respectively. The main difference
between our work and Wang, Qi & Dai (2011)’s is that we just divide GRBs into
two redshift bins, so that the number of GRBs in each bin is large enough to
do statistical analysis. We choose $z=1.4$ as the threshold because the
redshift of SNe Ia is usually smaller than 1.4, and the Universe below this
redshift has already been tightly constrained. We find that, among the six
luminosity correlations, only the $E_{p}-E_{\gamma}$ relation is consistent
between low-$z$ and high-$z$ GRBs within $1\sigma$ uncertainty. As such, we
can calibrate GRBs through the $E_{p}-E_{\gamma}$ relation using the Padé
approximation proposed by Liu & Wei (2014), and the Hubble diagram of GRBs can
be constructed.

The rest of the paper is arranged as follows: In section 2, we test the
redshift dependence of six luminosity correlations. In section 3, we calibrate
the distance of high-$z$ GRBs using the $E_{p}-E_{\gamma}$ relation, and then
use them to constrain the $\Lambda$CDM model. Finally, a short summary is
given in section 4.

##  2 Testing the redshift dependence of luminosity correlations

All the six luminosity correlations mentioned above have the exponential form
$R=AQ^{b}$, which can be linearized by taking the logarithm, i.e.,

$y=a+bx\qquad(y\equiv\log R,\leavevmode\nobreak\ x\equiv\log
Q,\leavevmode\nobreak\ a\equiv\log A),$ (1)

where “ $\log$” represents the logarithm of base 10. For the sake of clarity,
we write the six luminosity correlations explicitly here:

$\displaystyle\log\frac{L}{{\rm erg\leavevmode\nobreak\ s}^{-1}}$
$\displaystyle=a_{1}+b_{1}\log\frac{\tau_{{\rm
lag},i}}{0.1\leavevmode\nobreak\ {\rm s}},$ (2)
$\displaystyle\log\frac{L}{{\rm erg\leavevmode\nobreak\ s}^{-1}}$
$\displaystyle=a_{2}+b_{2}\log\frac{V_{i}}{0.02},$ (3)
$\displaystyle\log\frac{L}{{\rm erg\leavevmode\nobreak\ s}^{-1}}$
$\displaystyle=a_{3}+b_{3}\log\frac{E_{p,i}}{300\leavevmode\nobreak\ {\rm
keV}},$ (4) $\displaystyle\log\frac{E_{\gamma}}{{\rm erg}}$
$\displaystyle=a_{4}+b_{4}\log\frac{E_{p,i}}{300\leavevmode\nobreak\ {\rm
keV}},$ (5) $\displaystyle\log\frac{L}{{\rm erg\leavevmode\nobreak\ s}^{-1}}$
$\displaystyle=a_{5}+b_{5}\log\frac{\tau_{{\rm RT},i}}{0.1\leavevmode\nobreak\
{\rm s}},$ (6) $\displaystyle\log\frac{E_{\rm iso}}{{\rm erg}}$
$\displaystyle=a_{6}+b_{6}\log\frac{E_{p,i}}{300\leavevmode\nobreak\ {\rm
keV}},$ (7)

where quantities with a subscript “ $i$ ” represent the quantities in the
comoving frame, which can be transformed to the observer frame by $\tau_{{\rm
lag},i}=\tau_{\rm lag}(1+z)^{-1}$, $\tau_{{\rm RT},i}=\tau_{\rm
RT}(1+z)^{-1}$, $V_{i}=V(1+z)$, and $E_{p,i}=E_{p}(1+z)$.

The isotropic peak luminosity $L$ can be calculated from the bolometric peak
flux $P_{\rm bolo}$ as (Schaefer, 2007)

$L=4\pi d_{L}^{2}P_{\rm bolo},$ (8)

where $d_{L}$ is the luminosity distance. The bolometric peak flux $P_{\rm
bolo}$ is calculated from the observed peak photon flux in the rest frame
$1-10,000$ keV energy band by assuming the Band spectrum (Band et al., 1993).
The luminosity distance depends on a specific cosmological model. In the
concordance $\Lambda$CDM model, it is given as

$d_{L}(z)=(1+z)\frac{c}{H_{0}}\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{M}(1+z)^{3}+(1-\Omega_{M})}},$
(9)

where $\Omega_{M}$ is the mater density, $H_{0}$ is the Hubble constant, and
$c$ is the light speed. Here we take $\Omega_{M}=0.280$ and
$H_{0}=70.0\leavevmode\nobreak\ \rm{km}\leavevmode\nobreak\
\rm{s}^{-1}\leavevmode\nobreak\ \rm{Mpc}^{-1}$ from fitting to the Union2.1
dataset (Lin et al., 2015). The uncertainty of $L$ propagates from the
uncertainty of $P_{\rm bolo}$, while that from $d_{L}$ is absorbed into the
intrinsic scatter. The isotropic equivalent energy $E_{\rm iso}$ can be
calculated from the bolometric fluence $S_{\rm bolo}$ as (Schaefer, 2007)

$E_{\rm iso}=4\pi d_{L}^{2}S_{\rm bolo}(1+z)^{-1}.$ (10)

Similar to the bolometric peak flux, the bolometric fluence $S_{\rm bolo}$
also corresponds to the rest frame $1-10,000$ keV energy band. For $E_{\rm
iso}$, we also only consider the error propagation from $S_{\rm bolo}$. The
collimation-corrected energy, $E_{\gamma}$, is the isotropic equivalent energy
multiplied by a beaming factor $F_{\rm beam}\equiv 1-\cos\theta_{\rm jet}$,
where $\theta_{\rm jet}$ is the jet opening angle, i.e,

$E_{\gamma}\equiv E_{\rm iso}F_{\rm beam}=4\pi d_{L}^{2}S_{\rm bolo}F_{\rm
beam}(1+z)^{-1}.$ (11)

The uncertainty of $E_{\gamma}$ propagates from the uncertainties of both
$S_{\rm bolo}$ and $F_{\rm beam}$. The error propagation from $Q$ to $\log Q$
is given as

$\sigma_{\log Q}=\frac{1}{\ln 10}\frac{\sigma_{Q}}{Q},$ (12)

where “ $\ln$” represents the natural logarithm. If $Q$ has nonsymmetric
error, we symmetrize it by taking the average, i.e.,
$\sigma_{Q}=(\sigma_{Q}^{+}+\sigma_{Q}^{-})/2$.

To test the possible redshift dependence of luminosity correlations, we
analyze the GRB sample taken from Wang, Qi & Dai (2011). This sample consists
of 116 long GRBs in the redshift range $z\in[0.17,8.2]$. This dataset is a
collection of GRBs with well-measured spectra properties from various
instruments, such as BATSE, Konus, Swift, etc.. We divide GRBs into two
subsamples according to their redshift smaller or larger than 1.4, and call
them low-$z$ and high-$z$ subsamples, respectively. We choose $z=1.4$ as the
threshold because the redshift of SNe Ia is usually smaller than 1.4. The
Universe below this redshift has already been well studied using SNe Ia
(Amanullah et al., 2010; Suzuki et al., 2012; Betoule et al., 2014). Low-$z$
and high-$z$ subsamples consist of 50 and 66 GRBs, respectively. We fit each
luminosity correlation to the two subsamples separately. Since the plot of
each correlation in the $xy$ plane show large error bars in both the
horizontal and vertical axes, and intrinsic scatter dominates over the
measurement error, the ordinary least-$\chi^{2}$ method does not work well. We
apply the fitting method presented in D’Agostini (2005). The best-fit
parameters ($a,b,\sigma_{\rm int}$) can be derived by maximizing the
D’Agostini’s likelihood,

$\mathcal{L}_{D}(\sigma_{\rm
int},a,b)\propto\prod_{i}\frac{1}{\sqrt{\sigma_{\rm
int}^{2}+\sigma_{y_{i}}^{2}+b^{2}\sigma_{x_{i}}^{2}}}\times\exp\left[-\frac{(y_{i}-a-bx_{i})^{2}}{2(\sigma_{\rm
int}^{2}+\sigma_{y_{i}}^{2}+b^{2}\sigma_{x_{i}}^{2})}\right],$ (13)

where the intrinsic scatter $\sigma_{\rm int}$ represents any other unknown
errors except for the measurement error. Equivalently, we can minimizing the
$\chi^{2}$,

$\chi_{D}^{2}(\sigma_{\rm int},a,b)=\sum_{i}\ln(\sigma_{\rm
int}^{2}+\sigma_{y_{i}}^{2}+b^{2}\sigma_{x_{i}}^{2})+\sum_{i}\frac{(y_{i}-a-bx_{i})^{2}}{\sigma_{\rm
int}^{2}+\sigma_{y_{i}}^{2}+b^{2}\sigma_{x_{i}}^{2}}.$ (14)

We use the publicly available Matlab package
FMINUIT111http://www.fis.unipr.it/$\sim$giuseppe.allodi/Fminuit/Fminuit¯intro.html
to derive the best-fit parameters and their uncertainties. The results are
listed in the fourth to sixth columns in Table 1. This table gives the mean
values of the best-fit parameters and their $1\sigma$ uncertainties.

Table 1: The intrinsic scatters ($\sigma_{\rm int}$), intercepts ($a$) and slopes ($b$) of six luminosity correlations for low-$z$ and high-$z$ GRBs, derived from maximizing the D’Agostini’s likelihood. The quoted errors are of $1\sigma$. $N$ is the number of GRBs available in the fitting. (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---
correlation | subsample | $N$ | $\sigma_{\rm int}$ | $a$ | $b$
$\tau_{\rm lag}-L$: | low-$z$ | 27 | $0.475\pm 0.066$ | $52.103\pm 0.093$ | $-0.783\pm 0.141$
| high-$z$ | 32 | $0.321\pm 0.049$ | $52.486\pm 0.064$ | $-0.633\pm 0.111$
$V-L$: | low-$z$ | 47 | $0.875\pm 0.101$ | $51.550\pm 0.179$ | $0.441\pm 0.266$
| high-$z$ | 57 | $0.546\pm 0.057$ | $52.262\pm 0.121$ | $0.221\pm 0.117$
$E_{p}-L$: | low-$z$ | 50 | $0.577\pm 0.063$ | $51.880\pm 0.087$ | $1.461\pm 0.174$
| high-$z$ | 66 | $0.386\pm 0.039$ | $52.363\pm 0.054$ | $1.102\pm 0.143$
$E_{p}-E_{\gamma}$: | low-$z$ | 12 | $0.159\pm 0.059$ | $50.637\pm 0.061$ | $1.539\pm 0.144$
| high-$z$ | 12 | $0.261\pm 0.105$ | $50.649\pm 0.096$ | $1.354\pm 0.275$
$\tau_{\rm RT}-L$: | low-$z$ | 39 | $0.470\pm 0.058$ | $52.685\pm 0.122$ | $-1.318\pm 0.181$
| high-$z$ | 40 | $0.395\pm 0.052$ | $52.747\pm 0.078$ | $-0.784\pm 0.158$
$E_{p}-E_{\rm iso}$: | low-$z$ | 40 | $0.561\pm 0.069$ | $52.561\pm 0.095$ | $1.586\pm 0.190$
| high-$z$ | 61 | $0.365\pm 0.040$ | $52.874\pm 0.053$ | $1.243\pm 0.134$

Note that not all GRBs are available in the analysis of each luminosity
correlation. For example, GRBs without measurement of the jet opening angle is
unavailable in the $E_{p}-E_{\gamma}$ analysis, while GRBs having no spectrum
lag measurement are invalid in the $\tau_{\rm lag}-L$ analysis. For this
reason, we also list the number of available GRBs in each fitting in the third
column of Table 1. All the six luminosity correlations are plotted in Figure 1
in logarithmic coordinates. Low-$z$ and high-$z$ GRBs are denoted by black and
red dots, respectively. The error bars represent $1\sigma$ uncertainties.
Since the Swift/BAT instrument is only sensitive in a narrow energy band
($\sim 15-150$ keV), the uncertainties of peak energy of some GRBs are
extremely large. The lines stand for the best-fit results (black line for
low-$z$ GRBs and red line for high-$z$ GRBs).

Figure 1: The luminosity correlations for low-$z$ (black) and high-$z$ (red)
GRBs. Error bars represent the $1\sigma$ uncertainties. The lines are the
best-fit results, which are derived from maximizing the D’Agostini’s
likelihood.

Besides, we also plot the $1\sigma$, $2\sigma$ and $3\sigma$ contours in the
$(a,b)$ plane for low-$z$ (black curves) and high-$z$ (red curves) GRBs in
Figure 2. The best-fit central values are denoted by dots.

Figure 2: The $1\sigma$, $2\sigma$ and $3\sigma$ contours in the $(a,b)$ plane
for low-$z$ (black curves) and high-$z$ (red curves) GRBs derived from the
D’Agostini’s likelihood. The central values are denoted by dots.

From Table 1 and Figure 1, we can see that among six luminosity correlations,
the $V-L$ relation has the largest intrinsic scatter, while the
$E_{p}-E_{\gamma}$ relation has the smallest intrinsic scatter. The intrinsic
scatter of the $V-L$ relation is so large that it is unreasonable to fit it
with a line. For all the six luminosity correlations, high-$z$ GRBs have
larger intercept, but smaller absolute slope than low-$z$ GRBs, although the
difference of intercepts between low-$z$ and high-$z$ GRBs is not as
significant as that of slopes. The slope difference of the $\tau_{\rm RT}-L$
relation is especially evident. This can be seen more clearly from the contour
plots in the ($a,b$) plane in Figure 2. The $E_{p}-E_{\gamma}$ relation of
low-$z$ GRBs is consistent with that of high-$z$ GRBs within $1\sigma$
uncertainty. However, for the rest five luminosity correlations, low-$z$ GRBs
differ from high-$z$ GRBs at more than $3\sigma$ confidence level. Especially,
there is no overlap between the $3\sigma$ contours of two subsamples for the
$E_{p}-L$ relation. As for the Amati relation, we recover the results of Lin
et al. (2015).

The results above are derived using D’Agostini’s likelihood. Since the
observed data points have significant errors on both the $x$-axis and
$y$-axis, there is no unique method to determine the best-fit parameters.
Reichart (2001) has constructed a likelihood which is slightly different from
D’Agostini’s one. To test whether the above results depend on the choice of a
specific best-fit method, we also do a similar calculation using Reichart’s
likelihood. The Reichart’s likelihood is written as (Reichart, 2001)

$\mathcal{L}_{R}(\sigma_{x},\sigma_{y},a,b)\propto\prod_{i}\frac{\sqrt{1+b^{2}}}{\sqrt{\sigma_{y}^{2}+\sigma_{y_{i}}^{2}+b^{2}(\sigma_{x}^{2}+\sigma_{x_{i}}^{2})}}\times\exp\left[-\frac{(y_{i}-a-bx_{i})^{2}}{2[\sigma_{y}^{2}+\sigma_{y_{i}}^{2}+b^{2}(\sigma_{x}^{2}+\sigma_{x_{i}}^{2})]}\right],$
(15)

where $\sigma_{x}$ and $\sigma_{y}$ are the intrinsic scatters along the
$x$-axis and $y$-axis, respectively. The corresponding $\chi^{2}$ is given as

$\chi_{R}^{2}(\sigma_{x},\sigma_{y},a,b)=\sum_{i}\ln[\sigma_{y}^{2}+\sigma_{y_{i}}^{2}+b^{2}(\sigma_{x}^{2}+\sigma_{x_{i}}^{2})]+\sum_{i}\frac{(y_{i}-a-bx_{i})^{2}}{\sigma_{y}^{2}+\sigma_{y_{i}}^{2}+b^{2}(\sigma_{x}^{2}+\sigma_{x_{i}}^{2})}-N\ln(1+b^{2}),$
(16)

where $N$ is the number of data points. The best-fit parameters are the one
which can minimize the right-hand-side of Eq.(16).

The best-fit parameters and their $1\sigma$ uncertainties are listed in Table
2.

Table 2: The intrinsic scatters along the $x$-axis ($\sigma_{x}$) and $y$-axis ($\sigma_{y}$), intercepts ($a$) and slopes ($b$) of six luminosity correlations for low-$z$ and high-$z$ GRBs, derived from maximizing the Reichart’s likelihood. The quoted errors are of $1\sigma$. $N$ is the number of GRBs available in the fitting. The last column gives the total intrinsic scatter $\sigma_{\rm int}\equiv(\sigma_{y}^{2}+b^{2}\sigma_{x}^{2})^{1/2}$. (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
---|---|---|---|---|---|---|---
correlation | subsample | $N$ | $\sigma_{x}$ | $\sigma_{y}$ | $a$ | $b$ | $\sigma_{\rm int}$
$\tau_{\rm lag}-L$: | low-$z$ | 27 | $0.271\pm 1.757$ | $0.417\pm 1.385$ | $52.136\pm 0.102$ | $-1.101\pm 0.183$ | $0.513\pm 1.593$
| high-$z$ | 32 | $0.236\pm 1.390$ | $0.255\pm 0.855$ | $52.471\pm 0.065$ | $-0.815\pm 0.115$ | $0.319\pm 0.966$
$V-L$: | low-$z$ | 47 | $0.395\pm 0.055$ | $0.010\pm 29.430$ | $47.945\pm 2.413$ | $6.135\pm 3.571$ | $2.425\pm 1.457$
| high-$z$ | 57 | $0.390\pm 0.754$ | $0.236\pm 8.803$ | $49.730\pm 1.324$ | $2.661\pm 1.206$ | $1.064\pm 2.804$
$E_{p}-L$: | low-$z$ | 50 | $0.042\pm 3.396$ | $0.666\pm 1.053$ | $51.950\pm 0.102$ | $2.235\pm 0.252$ | $0.672\pm 1.479$
| high-$z$ | 66 | $0.091\pm 1.036$ | $0.396\pm 0.829$ | $52.309\pm 0.065$ | $1.871\pm 0.208$ | $0.431\pm 1.078$
$E_{p}-E_{\gamma}$: | low-$z$ | 12 | $0.047\pm 1.121$ | $0.152\pm 0.951$ | $50.634\pm 0.064$ | $1.663\pm 0.163$ | $0.171\pm 1.198$
| high-$z$ | 12 | $0.159\pm 0.931$ | $0.073\pm 6.867$ | $50.598\pm 0.119$ | $1.836\pm 0.399$ | $0.300\pm 2.347$
$\tau_{\rm RT}-L$: | low-$z$ | 39 | $0.073\pm 1.438$ | $0.551\pm 0.880$ | $53.088\pm 0.182$ | $-2.144\pm 0.317$ | $0.573\pm 1.196$
| high-$z$ | 40 | $0.222\pm 0.778$ | $0.293\pm 1.291$ | $52.873\pm 0.094$ | $-1.481\pm 0.270$ | $0.440\pm 1.217$
$E_{p}-E_{\rm iso}$: | low-$z$ | 40 | $0.225\pm 0.658$ | $0.373\pm 2.042$ | $52.631\pm 0.108$ | $2.270\pm 0.255$ | $0.632\pm 1.705$
| high-$z$ | 61 | $0.160\pm 0.474$ | $0.275\pm 0.955$ | $52.838\pm 0.063$ | $1.863\pm 0.193$ | $0.405\pm 0.917$

The last column gives the “equivalent” total intrinsic scatter, which is
calculated from $\sigma_{\rm
int}\equiv(\sigma_{y}^{2}+b^{2}\sigma_{x}^{2})^{1/2}$. Comparing to Table 1,
we can see from Table 2 that the $E_{p}-E_{\gamma}$ relation has the smallest
(while the $V-L$ relation has the largest) intrinsic scatters, although the
uncertainties of intrinsic scatters in Table 2 are much larger. Using
Reichart’s likelihood, the parameters (especially the intrinsic scatter)
cannot be well constrained. Reichart’s likelihood leads to larger absolute
slope parameters compared to D’Agostini’s likelihood. Figure 3 is the contour
plot in the $(a,b)$ plane.

Figure 3: The $1\sigma$, $2\sigma$ and $3\sigma$ contours in the $(a,b)$ plane
for low-$z$ (black curves) and high-$z$ (red curves) GRBs derived from the
Reichart’s likelihood. The central values are denoted by dots.

We can see an important common feature between the results derived from two
different likelihoods: only for the $E_{p}-E_{\gamma}$ relation, low-$z$
subsample is consistent with high-$z$ subsample within $1\sigma$ uncertainty.
For the $\tau_{\rm lag}-L$ and $E_{p}-L$ relations, low-$z$ subsample still
differs from high-$z$ subsample at more than $3\sigma$ confidence level. As
for the $\tau_{\rm RT}-L$ and $E_{p}-E_{\rm iso}$ relations, low-$z$ subsample
differs from high-$z$ subsample at more than $2\sigma$ confidence level. The
relatively lower significance is due to the larger uncertainties of the best-
fit parameters. The uncertainties of slope parameters of $V-L$ relation
derived from Reichart’s likelihood are extremely large. In a word, only
$E_{p}-E_{\gamma}$ relation shows no significant evidence for the redshift
evolution. This conclusion does not depend on the choice of the best-fit
methods.

Reichart’s likelihood differs from D’Agostini’s one by an extra factor
$(1+b^{2})^{1/2}$. Otherwise, these two likelihoods are identical (if we set
$\sigma_{\rm int}^{2}\equiv\sigma_{y}^{2}+b^{2}\sigma_{x}^{2}$). D’Agostini
(2005) pointed out that Reichart’s likelihood has a problem: $m^{2}$ cannot be
added tout court to 1, since $m^{2}$ is in general dimensional (although in
our case it is dimensionless). The factor $(1+b^{2})^{1/2}$ has the net effect
of overestimating $m$. This is one reason why Reichart’s likelihood leads to a
larger slope parameters relative to D’Agostini’s likelihood. Therefore, we use
the results of D’Agostini’s likelihood when calibrating the distance of GRBs
in the next section.

##  3 Distance calibration and cosmological implications

As we have shown that the $E_{p}-E_{\gamma}$ relation does not significantly
evolve with redshift, we can use it to calibrate GRBs. To avoid the
circularity problem, the Padé method proposed by Liu & Wei (2014) is applied.
The main calibrating procedures are as follows: Firstly, derive the distance-
redshift relation of SNe Ia (here we use the Union2.1 (Suzuki et al., 2012)
dataset) using the Padé approximation of order (3,2), i.e.,

$\mu(z)=\frac{\alpha_{0}+\alpha_{1}z+\alpha_{2}z^{2}+\alpha_{3}z^{3}}{1+\beta_{1}z+\beta_{2}z^{2}},$
(17)

where the coefficients
$(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3},\beta_{1},\beta_{2})$ and the
corresponding covariance matrix are derived by fitting Eq.(17) to the Union2.1
dataset (see Liu & Wei (2014) for details). Assuming that the low-$z$ GRBs
trace the same Hubble diagram to SNe Ia, we can calculate the distance moduli
of low-$z$ GRBs directly from Eq.(17). The uncertainty of $\mu$ propagates
from the uncertainties of the coefficients $(\alpha_{i},\beta_{i})$. Then the
luminosity distance of low-$z$ GRBs can be obtained using the relation

$\mu(z)=5\log\frac{d_{L}(z)}{\rm{Mpc}}+25.$ (18)

As $d_{L}$ is known, the collimation-corrected energy can be further
calculated from Eq.(11). Note that there are only 12 low-$z$ GRBs and 12
high-$z$ GRBs available since the others have no measurement of jet opening
angle. Then we fit the $E_{p}-E_{\gamma}$ relation (i.e., Eq.(5)) to the 12
low-$z$ GRBs, which gives the best-fit parameters

$\sigma_{\rm int}=0.161\pm 0.059,\leavevmode\nobreak\ \leavevmode\nobreak\
a=50.632\pm 0.062,\leavevmode\nobreak\ \leavevmode\nobreak\ b=1.537\pm 0.145.$
(19)

By directly extrapolating the $E_{p}-E_{\gamma}$ relation to high-$z$ GRBs, we
can inversely obtain the collimation-corrected energy for 12 high-$z$ GRBs
from Eq.(5). Finally, calculate the luminosity distance of high-$z$ GRBs from
Eq.(11), and then the distance moduli from Eq.(18). The uncertainty of
distance moduli propagates from the uncertainties of $E_{\gamma}$, $S_{\rm
bolo}$ and $F_{\rm beam}$, i.e. (Schaefer, 2007),

$\sigma_{\mu}^{2}=\left(\frac{5}{2\ln 10}\right)^{2}\left[(\ln
10)^{2}\sigma_{\log E_{\gamma}}^{2}+\frac{\sigma_{S_{\rm bolo}}^{2}}{S_{\rm
bolo}^{2}}+\frac{\sigma_{F_{\rm beam}}^{2}}{F_{\rm beam}^{2}}\right],$ (20)

where

$\sigma_{\log
E_{\gamma}}^{2}=\sigma_{a}^{2}+\left(\sigma_{b}\log\frac{E_{p,i}}{300\leavevmode\nobreak\
{\rm keV}}\right)^{2}+\left(\frac{b}{\ln
10}\frac{\sigma_{E_{p,i}}}{E_{p,i}}\right)^{2}+\sigma_{\rm int}^{2}.$ (21)

The distance moduli of 12 high-$z$ GRBs and their $1\sigma$ uncertainties
calibrated through the $E_{p}-E_{\gamma}$ relation are listed in Table 3.

Table 3: The distance moduli of 12 high-$z$ GRBs calibrated through the $E_{p}-E_{\gamma}$ relation. GRB | $z$ | $\mu$ | $\sigma_{\mu}$
---|---|---|---
010222 | 1.48 | 45.0990 | 0.4684
030328 | 1.52 | 44.8357 | 0.5404
990123 | 1.61 | 45.5094 | 0.6244
990510 | 1.62 | 45.6337 | 0.4921
030226 | 1.98 | 46.6219 | 0.6476
021004 | 2.32 | 46.0496 | 0.9760
050820A | 2.61 | 47.4273 | 0.7877
030429 | 2.66 | 46.8414 | 0.8608
050401 | 2.9 | 47.1522 | 0.6272
020124 | 3.2 | 46.5993 | 0.6032
060526 | 3.21 | 45.4623 | 0.6342
060605 | 3.8 | 50.9127 | 1.0548

We also plot the 12 high-$z$ GRBs in $z-\mu$ plane in Figure 4, where the
black curve is the best-fit result to $\Lambda$CDM model.

Figure 4: The Hubble diagram of 12 long GRBs calibrated using the
$E_{p}-E_{\gamma}$ relation. The black curve is the best-fit result to
$\Lambda$CDM model. The best-fit parameter is $\Omega_{M}=0.302\pm 0.142$.

The fit of 12 high-$z$ GRBs to the $\Lambda$CDM model gives
$\Omega_{M}=0.302\pm 0.142$, well consistent with the Planck 2015 results (Ade
et al., 2015). From Figure 4, we can see that the distance of GRB 060605 is
much overestimated. This is because the $E_{p}-E_{\gamma}$ relation
overestimates the energy of GRB 060605 (see also the $E_{p}-E_{\gamma}$ plot
in Figure 1, where the red star represents this burst). On the contrary, the
distance of 060526 is underestimated because the $E_{p}-E_{\gamma}$ relation
underestimates its energy. The rest 10 GRBs are consistent with the
$\Lambda$CDM model within $1\sigma$ uncertainties.

For comparison, we also calibrate GRBs through the $E_{p}-E_{\rm iso}$
relation (the so called Amati relation). In this case, 40 low-$z$ GRBs and 61
high-$z$ GRBs are available. The constraint of 61 high-$z$ GRBs on the
$\Lambda$CDM model gives $\Omega_{M}=0.805\pm 0.144$, which is much larger
than the Planck 2015 results. The reason for this can be easily understood.
From the $E_{p}-E_{\rm iso}$ plot in Figure 1, we can see that high-$z$ GRBs
have in average larger isotropic equivalent energy than low-$z$ GRBs at the
same $E_{p}$ value. Therefore, when extrapolating the Amati relation from
low-$z$ GRBs to high-$z$ GRBs, the energy (so the distance) of most high-$z$
GRBs is underestimated. The underestimation of distance further leads to the
overestimation of $\Omega_{M}$. For this reason, we can predict that GRBs
calibrated through the rest four luminosity correlations ($\tau_{\rm lag}-L$,
$V-L$, $E_{p}-L$ and $\tau_{\rm RT}-L$) may also overestimate the value of
$\Omega_{M}$.

##  4 Summary

In this paper, we checked the possible redshift dependence of six luminosity
correlation in long GRBs. We divided GRBs into low-$z$ and high-$z$ subsamples
according to their redshift smaller or larger than 1.4. The slope and
intercept parameters of six luminosity correlations are derived by maximizing
the D’Agostini’s likelihood. For all the six luminosity correlations, high-$z$
GRBs seem to have larger intercept, but smaller absolute slope than low-$z$
GRBs. It was shown that the intrinsic scatter of $V-L$ relation is to large to
make a convincing conclusion. The $E_{p}-E_{\gamma}$ relation has the smallest
intrinsic scatter among the six, although the number of available GRBs is
small. Most importantly, the $E_{p}-E_{\gamma}$ relation shows weak redshift
dependence. Strong evidence ($>3\sigma$) for the redshift evolution was found
in the rest four correlations. Similar features can be seen when we use
Reichart’s likelihood instead of D’Agostini’s, although the statistical
significance is lower. We calibrated high-$z$ GRBs using the
$E_{p}-E_{\gamma}$ relation in a model independent way and reconstruct the
Hubble diagram. The constraint of high-$z$ GRBs on the $\Lambda$CDM model
gives matter density $\Omega_{M}=0.302\pm 0.142$, which is well consistent
with the Planck 2015 results, although the error bar is large. Calibrating
GRBs using the Amati relation, as was done by Liu & Wei (2014), in some cases
may overestimate $\Omega_{M}$. One of the disadvantage in using the
$E_{p}-E_{\gamma}$ relation, of course, is that only a small number of GRBs
are available since most GRBs have no measurement of jet opening angle. We
hope that the future observation will enlarge the GRB sample so as to improve
the statistical significance.

## Acknowledgements

We are grateful to Y. Sang, P. Wang, S. Wang and D. Zhao for useful
discussions. We thanks the anonymous referee for his/her useful comments and
suggestions. We also appreciate G. Allodi for making the FMINUIT publicly
available. This work has been funded by the National Natural Science Fund of
China under grants Nos. 11375203, 11305181.

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# Systematic performance of the ASKAP Fast Radio Burst search algorithm

Hao Qiu (邱昊)1, Evan F. Keane2, Keith W. Bannister3, Clancy W. James4 & Ryan M.
Shannon5
1 SKA Observatory, Jodrell Bank, Lower Withington, Macclesfield, SK11 9FT, UK
2School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
3Australia Telescope National Facility, CSIRO Space & Astronomy, P.O. Box 76,
Epping, NSW 1710, Australia
4International Centre for Radio Astronomy Research, Curtin Institute of Radio
Astronomy, Curtin University, WA 6845, Australia
5Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn,VIC 3122, Australia E-mail<EMAIL_ADDRESS>0000-0002-9586-7904 0000-0002-4553-655X 0000-0003-2149-0363
0000-0002-6437-6176 0000-0002-7285-6348

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Detecting fast radio bursts (FRBs) requires software pipelines to search for
dispersed single pulses of emission in radio telescope data. In order to
enable an unbiased estimation of the underlying FRB population, it is
important to understand the algorithm efficiency with respect to the search
parameter space and thus the survey completeness. The Fast Real-time Engine
for Dedispersing Amplitudes (fredda) search pipeline is a single pulse
detection pipeline designed to identify radio pulses over a large range of
dispersion measures (DM) with low latency. It is used on the Australian Square
Kilometre Array Pathfinder (ASKAP) for the Commensal Real-time ASKAP Fast
Transients (CRAFT) project . We utilise simulated single pulses in the low-
and high-frequency observation bands of ASKAP to analyse the performance of
the pipeline and infer the underlying FRB population. The simulation explores
the Signal-to-Noise Ratio (S/N) recovery as a function of DM and the temporal
duration of FRB pulses in comparison to injected values. The effects of intra-
channel broadening caused by dispersion are also carefully studied in this
work using control datasets. Our results show that for Gaussian-like single
pulses, $>85\%$ of the injected signal is recovered by pipelines such as
fredda at DM < 3000 pc cm-3using standard boxcar filters compared to an ideal
incoherent dedispersion match filter. Further calculations with sensitivity
implies at least $\sim 10\%$ of FRBs in a Euclidean universe at target
sensitivity will be missed by fredda and heimdall, another common pipeline, in
ideal radio environments at 1.1 GHz.

###### keywords:

fast radio bursts – methods: data analysis – software: simulations

††pubyear: 2021††pagerange: Systematic performance of the ASKAP Fast Radio
Burst search algorithm–C

##  1 Introduction

Fast Radio Bursts (FRBs; Lorimer et al., 2007) are dispersed bright single
pulses of extragalactic origin (Chatterjee et al., 2017; Bannister et al.,
2019b). Like pulses observed from pulsars, FRBs are subject to propagation
effects: for FRBs this arises when passing through the extragalactic and
interstellar media. Their most distinctive feature is a frequency-dependent
time delay caused by dispersion when travelling through cold ionised media.
The magnitude of this effect is given by the integral of the free electron
column density along the line of sight, known as the dispersion measure (DM).
A key feature of FRBs, and perhaps their working observational definition
(Thornton et al., 2013; Keane, 2016), is that their DMs exceed the maximum
contribution from the Milky Way, indicating an extragalactic origin. In
support of this, there is an strong observed correlation between the
extragalactic DM and luminosity distance (Macquart et al., 2020). Additional
propagation effects such as scatter broadening and Faraday rotation caused by
turbulent or clumping plasma structures and magnetic fields along the line of
sight can have a significant effect on the pulse morphology of FRBs (Macquart
& Koay, 2013; Prochaska et al., 2019; Chittidi et al., 2021). These
propagation effects have shown to be useful probes of the extragalactic medium
(Simha et al., 2020) and, for example, can be used as an independent
measurement to reveal the density and magnetic fields of foreground galaxy
haloes (Prochaska et al., 2019).

The search for FRBs is performed using single pulse search pipelines for both
real-time observations and post-facto offline processing; typically there is a
trade-off between processing speed and the degree of thoroughness in terms of
data cleaning (Keane et al., 2018). The core algorithm of single pulse
detection is dedispersing the data to a large number of trial DM values, and
then identifying high signal-to-noise ratio (S/N) pulses within these data for
a range of pulse durations. Many algorithms have been used to detect FRBs,
including but not limited to: seek (Lorimer et al., 2007), destroy (Keane et
al., 2012), dedisperse-all (Burke-Spolaor & Bannister, 2014), presto (Spitler
et al., 2016), heimdall (Barsdell et al., 2012), bonsai (CHIME/FRB
Collaboration et al., 2018), fredda (Bannister et al., 2017; Bannister et al.,
2019a), amber (Sclocco et al., 2020) and astroaccelerate (Armour et al., 2011;
Carels et al., 2019; Adámek & Armour, 2020). For the typical case where rapid
follow-up is needed, it is the real-time pipelines that make the discoveries.
These real-time pipelines are often optimised for low latency while
maintaining as complete a search sensitivity as affordable.

The ideal pipeline should have a stable uniform characterized performance
across the parameter space during interference-free observations. The
characteristic response of the algorithm or pipeline, across its searched
parameter space, impacts how one interprets the output results. Pipeline
performance can be understood through the analysis of detection rates and also
the S/N reported by the algorithm. Incomplete signal recovery of pulses and
other systematic biases during single pulse searches will cause lower S/N and
fewer pulses to be detected above the threshold; these systematic errors are
often related to the DM and pulse width of the pulse. The underlying
population distribution estimated from the observed sample is crucial to
cosmological parameter estimations using FRBs (Connor, 2019; Luo et al., 2020;
James et al., 2022). This paper focuses on the identification of this S/N
response function by using simulated pulses injected in observational format
data for two FRB search pipelines: fredda and heimdall.

Mock FRB injections systems have been deployed in major FRB observing
facilities such as CHIME (CHIME/FRB Collaboration et al., 2018), UTMOST (Farah
et al., 2019) and the GBT (Agarwal et al., 2020). The greenburst system for
GBT developed in Agarwal et al. (2020) measured the system recall curve,
measuring a 100% recovery of injected bursts at $\mathrm{S/N}\gtrsim 12$.
Recent systematic injection tests by Gupta et al. (2021) showed that heimdall
used on UTMOST was able to recover over 90% of the synthetic injections above
a S/N threshold of 9.

The Fast Real-time Engine for Dedispersing Amplitudes (fredda; Bannister et
al., 2017; Bannister et al., 2019a) is the search pipeline used on the
Australian Square Kilometre Array Pathfinder (ASKAP) for the Commensal Real-
time ASKAP Fast Transients (CRAFT) project. It is a GPU-based implementation
of the Fast Dispersion Measure Transform (FDMT; Zackay & Ofek, 2017), a rapid
dedispersion trial algorithm to detect dispersed single pulses. fredda aims to
detect FRBs from the incoherent data stream of ASKAP in low latency to trigger
the download of baseband ring buffer data for interferometry.

In this paper, we examine the performance of fredda using dispersed pulses
simulated over a large range of dispersion measures and pulse widths. The aim
is to understand the signal recovery fraction from the detection pipeline to
infer the real sensitivity threshold of ASKAP FRBs and other major FRB
surveys. We use heimdall, a widely-used effective search pipeline, to perform
a comparison search on the simulated data. This allows us to verify the
simulated bursts and compare the S/N algorithms between these two software.
The simulation methods are described in § 2. The data processing setup and
results are presented in § 3. We then, in § 4, analyse and interpret these
results, discussing possible implications for how to interpret the observed
FRB samples.

##  2 Modelling

###  2.1 Simulation data format

For most detection pipelines acting in a blind survey mode, the input data are
the Stokes I dynamic spectra, i.e. sigproc filterbank
files111https://sigproc.sourceforge.net/. The data are usually stored with
millisecond time resolution with no coherent dedispersion applied. As a
result, the microsecond fine structure seen in some FRBs (e.g Farah et al.
2018; Day et al. 2020; Michilli et al. 2018; Nimmo et al. 2020) cannot be
resolved in the initial detection. If a pulse is detected with low latency and
a higher resolution data product is in temporary storage then analysis at
higher resolution is possible post-facto.

In this work, we use single Gaussian pulses to mimic the smeared pulse
appearance of most observed FRBs (Pleunis et al., 2021). The simulated pulses
have a constant injected S/N of $50$. The data are generated at time and
frequency resolution of $0.1$ ms and $0.1$ MHz respectively. The data are then
down-scaled by a factor of $10$ in both dimensions. This is done so as to
introduce various smearing effects and create a precise pulse profile. The
reduced resolutions aim to match closely those of ASKAP data (see Table 1).
The pulses are injected into a background of Gaussian white noise. Together
these steps simulate ideal observation conditions.

###  2.2 Simulation format

Table 1: Simulated Dataset properties Dataset | Gaussian | Scattering
---|---|---
DM ( pc cm-3) | 0–3000 | 0–3000
DM step | 50 | 500
Intrinsic Width (ms) | 0.5–11.0 | 1 & 5
$\sigma_{\rm intrinsic}$ step (ms) | 0.5 | –
Scattering time $\tau_{\rm{scat}}$ (ms) | 0 | 0.5–10
$\tau_{\rm{scat}}$ step | – | 0.5
$\rm{N{bits}}$ | 8
$\rm{N{chan}}$ | 336
$\rm{t_{samp}}$ (ms) | 1
$\Delta\nu$ (ms) | 1
High Freq Band (GHz) | 1.1–1.436
Low Freq Band (GHz) | 0.764–1.1
Injected S/N | 50
Number of pulses | 50

The data recorded for ASKAP FRB searches are also filterbanks. ASKAP observes
at radio frequencies between 0.7 and 1.8 GHz. The ASKAP beamformers create
$36$ Stokes I beams per dish, i.e. $36\times N_{\rm dish}$ data streams
(Clarke et al., 2014). The CRAFT backend adds these incoherently to create
$36\times 1$ data streams. The CRAFT pipeline is designed to search such
filterbank data generated with a time resolution typical between 0.7-1.7 ms
and with a bandwidth of 336 MHz. In this work we use the two standard
frequency bands employed by CRAFT (see Table 1). The ASKAP data streams are
typically channelized to $336\times 1\mathrm{MHz}$ channels with a time
resolution of 1.26 ms (before 2019, later changed to 1.73 ms for lower
frequency band searches). The resolution of the simulated dynamic spectrum in
this work is intended to be similar to the resolution of the current
incoherent sum data stream during CRAFT observations. The format is $336\times
1$ MHz channels and a time resolution of 1 ms recorded in 8-bit data(see Table
1).

###  2.3 Model of injected pulses

We assume a single Gaussian as the underlying profile of each pulse, i.e. in
$0.1$-MHz $0.1$-ms resolution with the following equation. For channel $i$ the
flux of the pulse is:

$S_{i}(t)=\frac{A}{\sqrt{2\pi\sigma_{i}^{2}}}\exp\left[\frac{-(t-t_{0}-t_{\rm
DM}(\nu_{\rm i})^{2}}{2\sigma_{i}^{2}}\right],$ (1)

where $A$ is an amplitude scale factor, $t_{0}$ is the time reference of the
burst at the reference frequency which we take to be the top of the band, i.e.
the centre of the highest frequency channel. The standard deviation of the
intrinsic Gaussian is $\sigma_{\rm i}$ and the dispersion delay time, $t_{\rm
DM}(\nu_{\rm i})$ relative to the reference frequency and is proportional to
the DM according to:

$t_{\rm DM}(\nu_{\rm i})=4.15\
\mathrm{DM}(\nu_{\mathrm{top}}^{-2}-\nu_{i}^{-2})\;\mathrm{ms},$ (2)

where $\nu_{\rm i}$ is the centre frequency of the ith channel and
$\nu_{\rm{top}}$ is the centre frequency of the top channel. The intra-channel
dispersion smearing is the dispersion delay time within one channel, causing
the pulse to be broadened within the channel (Clarke et al., 2013). The
dispersion smearing in the ith channel is:

$\Delta t_{\rm DM}=(8.3\times 10^{-3})\ \mathrm{\Delta\nu\ {DM}}\
\nu_{i}^{-3}\rm{ms}.$ (3)

Here $\Delta\nu$ is the channel bandwidth in units of MHz and $\nu_{\rm i}$ is
the channel frequency in GHz. We interpret $\Delta t_{\rm DM}$ as the full-
width half maximum (FWHM) of a Gaussian with width $\sigma_{\rm DM}=\Delta
t_{\rm DM}/(2\sqrt{2\ln 2})$. For our simulated pulses we define the standard
deviation in Eq. 1 as the quadrature sum:

$\sigma_{i}=(\sigma_{\rm{intrinsic}}^{2}+\sigma_{\mathrm{DM}}^{2})^{1/2}.$ (4)

This generates a Gaussian pulse profile with a typically very small amount of
dispersion smearing across the $0.1$ MHz channels. Then reducing the
resolution in both frequency and time by a factor of $10$ in each produces
dispersion smearing in each channel that is $10$ times higher accounting for
the majority of the effect. An example of a resultant pulse is shown in Figure
1.

Figure 1: A simulated Gaussian pulse with an intrinsic width of $\sigma$=0.5
ms, a dispersion time delay of $\rm{DM}=100$ pc cm-3, a scattering time of
$\tau_{1\rm{GHz}}=10$ ms, scattering index $\alpha=4$ and flat spectral index.
The time resolution of the filterbank is 1 ms, with no background noise. The
slight variability in the intensity of the pulse shows the pulse energy
distributed over different number of time samples in each channel caused by
intra-channel dispersion and different pulse arrival time.

For this simulation the software we have developed allows for the addition of
scatter broadening to the pulse profile. This is achieved by the convolution
with a one-sided frequency-dependent exponential decay function. The
scattering index is set as $\alpha=4$ based on measurements from scattering in
pulsar observations (Bhat et al., 2004):

$S_{i}(t)=\begin{cases}\exp\left[-\frac{(t-t_{p})}{\tau({\nu_{i}}/{{1\rm{GHz}}})^{-\alpha}}\right],&(t\geq
t_{p}),\\\ 0,&(t<t_{p}).\end{cases}$ (5)

where $1$ GHz is the reference frequency and
$\rm{t_{p}=t_{0}+t_{DM}(\nu_{i})}$.

###  2.4 S/N rescaling

The pulse is independently modelled in each channel based on equation 1. Down-
sampling causes a DM smearing effect on the pulse profile, broadening its
width. The S/N of an injected pulse is calculated using a match filter on the
dedispersed time series in units of the standard deviation of the noise
background. All of the simulation steps are done using 32-bit floats; the very
last step in our pipeline is to write out 8-bit files (again to match the
ASKAP output). This calculation standardizes the S/N of the simulated burst so
that we can compare with the S/N measured from the pipelines. We define the
match-filter S/N in the time series as:

$\rm{S/N_{filter}=\sqrt{\sum S_{i}^{2}}},$ (6)

where $\rm{S_{i}}$ is the signal intensity of each sample in units of the
standard deviation.

After being time averaged, the pulse fluence (in units of $\mathrm{Jy\;ms}$)
remains constant, but S/N is not conserved. This is shown in Figure 2: for
narrow Gaussian pulses with FWHM less than 1 coarse time sample (the final
data time resolution, here $1$ ms), the S/N decreases with burst width. The
pulse S/N is again re-adjusted, by scaling, to the intended value so that all
injected pulses have consistent S/N for the purpose of searching. Gaussian
background noise with a standard deviation of 1 is then added to the array and
the data are written to disk in sigproc filterbank format. Using the match
filter equation (Eq.6), it is known that the final readout S/N of the pulse
under the added noise background will have an approximate uncertainty of 1
standard deviation unit, our output results also match this result.

Figure 2:  The theoretical S/N scaling after resolution downscaling measured
by boxcar matchfilters for pulses in the high frequency band.

The bursts are chosen to be scaled to S/N=50 during this process. The response
of the algorithm is independent of this choice. This exact value is arbitrary,
we choose to inject bright bursts to guarantee we do not miss candidates below
the minimum S/N threshold, as would occur if using injections at S/N=10. (See
§ A)

###  2.5 Dataset setup

The parameters describing the datasets are shown in Table 1. The experiment is
conducted using two frequency bands that are commonly observed by ASKAP. We
refer to the two frequency bands in this work as the high frequency band: 1100
MHz – 1436 MHz and the low frequency band: 764 MHz – 1100 MHz. For each band,
we generated Gaussian single pulses over a large range in DM and intrinsic
pulse width. For each sample in DM and width space, $50$ pulses were injected
to effect $50$ trial iterations to sample the noise. Scattered pulses were
created as a separate dataset for additional comparison studies (see § B).

An incoherently dedispersed ‘zero-DM’ dataset was also created as a control
sample for the Gaussian pulses. The pulses in this dataset are the exact same
pulses as the original dataset except that the pulse time is aligned across
the frequency band. This dataset was created to avoid the effects of incorrect
dedispersion that is unavoidable for blind searches with finite numbers of DM
trials. Thus it examined only the basic boxcar S/N retrieval efficiency of the
pipelines.

##  3 Pipeline setup

###  3.1 Pipeline Settings

For fredda, we use a basic setup only defining two parameters beyond default
values. These are the block size of the data stream to ingest at any one time,
and the number of DM trials (these are the -t and -d input flags for fredda).
We use a block size of $8192$ ($16384$) samples for the high (low) frequency
bands to ensure that the widest pulses at the highest DMs are not split across
blocks. By default fredda is configured so that boxcar widths searches range
from $1$ up to a maximum of 32 samples in steps of $1$ (Bannister et al.,
2017).

For heimdall we also use mostly default parameters. We set the baseline length
parameter (-baseline_length flag) to be $20$ s, $10$ times the default as we
would need to do when observing a strong pulsar, so as to avoid incorrect
statistical estimates for the noise. We further turned down a setting to the
friends-of-friends algorithm (the -cand_sep flag) to ensure adjacent pulses
did not get erroneously associated. In the case of our zero-DM pulses, we
remove (using the -rfi_no_broad flag) the usual zero-DM filter (Eatough et
al., 2009). Finally we set the DM range and set the maximum boxcar width to 32
samples, however in the case of heimdall this parameter is searched in
logarithmic steps of a factor of $2$.

One parameter for heimdall can drastically change its output response; this is
the DM tolerance parameter (-dm_tol flag). This parameter defines the DM step
size, the lower the tolerance the smaller the step size. A range of tolerances
are used at different observatories (see Table 2); the default value is
$1.25$. fredda uses a constant (frequency-dependent) DM step size for the full
DM range, and this feature is not configurable. We perform analyses using
$1.25$ and $1.01$ tolerances. The former represents a typical value for
searches which have discovered FRBs; the latter is in some sense a fairer
comparison to fredda. We show results for a DM tolerance of $1.01$ for
heimdall. Coarser tolerance results are worse and can be examined in the
supplementary online material for this paper.

Survey/Telescope | dm_tol | Reference
---|---|---
Parkes-SUPERB | 1.20 | Keane et al. (2018)
UTMOST | 1.20 | Gupta et al. (2021)
DSA-10 | 1.15 | Kocz et al. (2019)
STARE2 | 1.25 | Bochenek et al. (2020)
Sardinia (Targeted) | 1.01 | Pilia et al. (2020)
Table 2: Example heimdall set up for the -dm_tol input flag parameter in
different real-time blind surveys and targeted searches

###  3.2 Known boxcar efficiency and scallop responses

To measure the dispersion of the candidates, fredda uses the fast dispersion
measure transform (FDMT) algorithm (Zackay & Ofek, 2017), while heimdall uses
a brute force dedispersion tree. In order to understand the performance of the
pipeline, we investigate the single pulse search algorithms in isolation from
the dedispersion algorithms.

Both pipelines use boxcar filters. We first calculate the theoretical response
on single channel boxcar pulses, where the search filter perfectly matches the
shape of the signal.

For the results shown here we ensured that the boxcars were aligned in phase
with the time samples, but verified that the S/N fall-off when pulses are out-
of-phase with the sampling is as expected. We use filter templates with widths
of 1, 2, 4, 8 samples to measure the S/N of boxcar signals (injected S/N of
$50$, no noise) with boxcar width between $1-10$ ms. The result in Figure 3
shows how using a set of fixed-width match filters, the S/N between the exact
filter widths dips. The response is sharply peaked at perfect recovery; the
falls off are $(W/B)^{-1/2}$ ($(B/W)^{-1/2}$) when the boxcar width $B$ is
greater (less) than the injected pulse width $W$. The overall response shape
taken from the maximum achivable S/N drops between matching widths and is
commonly known as ‘scalloping’.

Figure 3: Theoretical S/N recovery of S/N=50 boxcar signals using boxcar match
filters with a width of 1, 2, 3, 4 and 8 samples.

The response is different when we inject Gaussian pulses but again use the
boxcar filters of the same widths, as deployed by heimdall and fredda. The
peak response is not perfect but reaches the theoretical maximum of $94.3\%$
recovery (Keane & Petroff, 2015; Morello et al., 2020), with a smoother
falloff than for boxcar pulses. Using a set of boxcar filters with a power of
2 step size, we can observe that heimdall suffers between boxcar widths from
the scalloping in Figure 4. Due to the low time resolution of the data
ingested by the CRAFT pipeline, fredda is designed to search consecutive
sample widths as this gives a more optimal smooth response curve for wider
pulses as shown in Figure 5.

It should be noted that both fredda and heimdall are specifically designed to
lower the computation costs and latency when searching for pulses across a
large range of width. For Parkes radio telescope data, where heimdall was
first applied, the time resolution could be as high as 64 $\mu$s. Hence a
wider boxcar width range is needed to search for wide pulses and heimdall
enables a faster search across the range of DM and width in real-time for
these observations. On the other hand, fredda was designed specifically for
ASKAP data at coarse time resolution, the FDMT algorithm also optimised
processing for a large ranges of dedispersion trials.

Figure 4: The match filter S/N of Gaussian pulses in 1-ms time resolution
using Log-spaced boxcar widths. The upper figure shows the response of each
respective match filter while the lower figure shows the resulting response
curve by taking the maxima at each width.

Figure 5: Match filter S/N of Gaussian pulses in 1-ms time resolution using
linearly spaced boxcar filters in comparison to Figure 4. By using a set of
uniformly spaced and consecutive width boxcar match filters up to 32 samples,
it is possible to achieve a more uniform response towards wider pulses.

##  4 Results

###  4.1 Reported properties of simulated single pulses

We first process the pulses in their original dispersed form. The average S/N
values reported over our $50$ iterations as a function of $\sigma_{\rm
intrinsic}$ and DM are shown in Figure 6. The results confirm that fredda
maintains a high S/N recovery across the entire parameter range. This is
especially important for high DM searches where a drop in S/N is expected due
to broadening caused by DM smearing. The low SN at higher DM and widths of
FREDDA low band are caused by the maximum limit of 32 sample boxcars. The
scalloping effect can be seen across the parameter space for heimdall in
Figure 6. The response is the one shown in Figure 4, but with the
peaks/troughs of the scallops at constant observed pulse width, i.e. ever
lower $\sigma_{\rm i}$ as DM increases.

The average boxcar values reported over $50$ iterations are displayed in
Figure 7. This shows that heimdall, as expected, is using less steps of
boxcars compared to fredda. The total pulse width which consists of a
contribution from both the intrinsic pulse width and the smearing width (which
correlates with DM). This produces the ripple-like S/N response pattern of
heimdall as it applies the same fixed boxcars over wide width ranges.

Figure 6: FREDDA and HEIMDALL reported/injected S/N of single pulses in the
high and low frequency band.

###  4.2 Zero-DM pulses

For comparison, we have a dataset that is incoherently dedispersed to the
exact DM (intrachannel smearing remains). The results of processing these data
are shown in Figure 8. The results are a direct examination of the S/N boxcar
algorithm applied by the software when the pulse is correctly dedispersed but
smeared, hence this is the theoretical best S/N achievable for each pipeline.
It can be seen that the response is consistent to the simulated estimates in
Figure 4 and Figure 5 as DM and $\sigma_{\rm intrinsic}$ correlates with the
pulse FWHM.

Figure 7: Average reported boxcars of injected single pulses in the high and
low frequency band from FREDDA and HEIMDALL. FREDDA benefits from consecutive
boxcar filters and provides a very smooth boxcar measurement.

Figure 8: The S/N reported/injected by HEIMDALL and FREDDA from the perfectly
incoherently dedispersed dataset.

Figure 9: Reported boxcar from measuring dedispersed pulses with HEIMDALL and
FREDDA.

###  4.3 FREDDA high S/N at intermediate DM

We find that for some parts of the parameter space the S/N reported by FREDDA
from dispersed pulses is nominally better than the theoretical maximum, and
even exceeds 100% of that injected. For example, in the high frequency range,
the maximum S/N reported by fredda at around $\rm{DM\sim 750\ pc\ cm^{-3}}$
reaches 102% of the injected S/N. For the low frequency dataset, the S/N bump
shifts to a lower range at $\rm{DM\sim 350\ pc\ cm^{-3}}$ and is less
emphatic. This result greatly affects the statistics of FRBs discovered by
fredda. For these data there are no islands in the parameter space where
excessively high S/N values are obtained.

We deem it to be important that this issue only appears for dispersed pulses,
i.e. not for the perfectly dedispersed pulses. This would indicate that the
underlying issue is related to the dedispersion process, not the pulse search
part of the algorithm. We note that the channel smearing width at the top of
the band for a burst at these DM ranges in each frequency band is between
$1-2$ ms (corresponding to $1-2$ time sample in this work) as shown in Figure
10. This indicates that the S/N reported is affected by how the algorithm
collects the dispersed signal for the S/N calculation, e.g. if the samples
occupied by the dispersive sweep were under-estimated so too would the rms
noise, resulting in an over-estimated S/N.

We also examine how fredda calculates the noise across different DMs by
processing 20 seconds of Gaussian white noise data under the same settings. We
show in Figure 11 the number of noise candidates above $3\sigma$ in comparison
to the average S/N in high frequency band data.

The number of noise candidates is expected to drop slightly as DM increases,
as the larger DM increases the minimum width and time delay of pulses which
reduces the effective search length of the data, which will be more
significant in shorter length data (20 seconds). We however see significant
fluctuations in the figure: a large increase in the number of noise candidates
at around $700-800$ pc cm-3, which correlates with the rise of reported S/N in
fredda. We also speculate that the lower-DM S/N dips may be correlated with
the dip in the number of noise candidates at 400 pc cm-3and 1400 pc
cm-3respectively. This indicates that there is an error with the noise
estimation function that may be the cause of the S/N inconsistency.

Further investigation of the algorithm is being conducted to explain the
phenomena and remove this software error in future searches. But this feature
exists in all searches to date and so the response function for the ASKAP FRB
sample, for instance, should include it as it is calculated here.

Figure 10: Dispersion smearing at different frequencies in relation to DM of
the pulse. We also draw two horizontal lines identifying width of 1 ms and 2
ms, which corresponds to 1-2 time samples in this work. We identify that the
smearing width is between 1-2 time samples at 200-400 pc cm-3at 1 GHz and
600-1000 pc cm-3at 1.3 GHz. This DM range corresponds to the unusual S/N rise
in FREDDA Figure 11: The number of noise candidates above 3 sigma in
comparison to the average S/N reported by fredda over DM in 60 seconds of high
frequency band data. The data points of the noise candidate count are the
averaged bins of 64 dm steps.

###  4.4 Search Completeness

The sensitivity of the pipelines are well characterised from this work. For
statistical studies on ASKAP FRBs and the FRB searches conducted with
heimdall, we can utilise the reported S/N results to estimate the real
S/N/fluence of these FRBs. We assess the search completeness at different
frequencies for both pipelines based on the $\log N-\log S$ relation of a
uniformly distributed source population in a Euclidean Universe with a slope
of -1.5. We relate the search flux density $\rm{S_{search}}$ with the target
search sensitivity $\rm{S_{target}}$ with
$\rm{S_{search}=S_{target}\times{\cal R}}$. Here ${\cal R}$ is the ratio
between the reported and the injected S/N (e.g Figure 6) scaled by the S/N
down-scaling response function for constant S/N pulses (see § 2.4 and Figure
2).

When the reported S/N is higher than 100% of that injected, the recovery is
considered as 100% (e.g parts of the fredda results), as it has reached target
sensitivity. The real flux density of the population actually detected is
lower than the target sensitivity of the observations. The search completeness
is therefore expressed in the terms of the cumulative
$\rm{N_{search}/N_{target}}$.

Our calculations are shown in Table 3. The reported S/N maps show that
pipelines are only probing $\sim 90\%$ of the designated parameter space in
this work due to not reaching the real target sensitivity. At detection
thresholds or for real time searches, the search completeness will be further
lowered, according to the noise statistics.

Pipeline | Band | dm_tol | $\rm{N_{search}/N_{target}}$
---|---|---|---
fredda | High | – | $92.6\%$
Low | – | $87.5\%$
heimdall | High | 1.01 | $90.2\%$
Low | 1.01 | $90.0\%$
heimdall | High | 1.25 | $86.7\%$
Low | 1.25 | $85.1\%$
Table 3: Search completeness compared to target sensitivity for both pipelines
in high and low frequency bands. heimdall shows consistency over frequency,
while fredda performs better at high frequency.

##  5 Conclusion

In this work, we utilise simulated single pulses to test the performance of
the FRB search pipeline fredda and heimdall. We simulated pulses in Gaussian
white noise with no interference across a parameter space of DM < 3000 pc
cm-3and intrinsic pulse standard deviation below < 11 ms.

Our results show that both heimdall and fredda perform effectively in
detecting single dispersed pulses under ideal conditions. When a very fine DM
tolerance of heimdall is taken, it matches the performance of fredda, with
both pipelines sensitive to $\sim 90\%$ of the search volume. However, in
exchange for computation cost and low latency, if one reduces the DM tolerance
for heimdall(as is typically done) the performance is worse at $\sim 86\%$. We
identify an issue in fredda where the reported S/N is higher than 100% of that
injected likely due to dispersion smearing dominating pulse width.

We reconfirm results from previous tests (Keane & Petroff, 2015) that
pipelines using a non-consecutive set of boxcar match filters such as heimdall
will receive a signal to noise penalty on pulses in between the boxcar widths.
We further broadened the response calculation to include DM and a number of
other additional subtle effects. We demonstrate that using consecutive boxcars
(e.g fredda) is more sensitive to search for pulses in coarse time resolution
real-time data.

We also calculate a theoretical search completeness using the S/N response
distribution over the DM and pulse width parameter space obtained in this work
for heimdall and fredda. We show that fredda achieves a search completness of
92.6% at 1.1-1.4 GHz. Depending on the settings the search completeness of
heimdall in that frequency band is between 86.7% to 90.2%.

We believe this result demonstrates the importance of understanding the
pipeline search completeness not only for intial detection but also for FRB
population and statistical studies. It is essential to the FRB community that
current and future large FRB surveys such as CHIME/FRB and DSA-2000 to
investigate such underlying effects. The data and software for this work has
been made publicly available for further simulation and analysis use on other
pipelines.

## Acknowledgements

HQ thanks David McKenna, Vivek Gupta, Wael Farah, Andrew Jameson and Adam
Deller for useful comments on FREDDA and HEIMDALL for this work. HQ and EFK
acknowledge support from Fondation MERAC (Project: SUPERHeRO, PI: Keane) and
are grateful to Aaron Golden for support when working in Galway. This work was
performed on the OzSTAR national facility at Swinburne University of
Technology and the SKA Observatory Science Computing facilities. The OzSTAR
program receives funding in part from the Astronomy National Collaborative
Research Infrastructure Strategy (NCRIS) allocation provided by the Australian
Government.

## Data Availability

The source code to simulate FRBs for this work is publicly available at the
following repository: https://github.com/hqiu-nju/simfred/. The data results
in this work, supplement material and script to reproduce the simulations are
available in the following repository : https://github.com/hqiu-nju/CRAFT-ICS-
pipeline

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##  Appendix A S/N Injection

We present the results of injecting narrow ($\sigma=0.5$ ms) pulses across DM
$=0-3000$ pc cm-3and S/N $=10-100$. Figure 12 shows that the response is the
same regardless of the input S/N except for a slight loss as the input S/N
gets very high. This latter effect is understood; it is the due to the very
high S/N pulses contaminating the background noise calculation in fredda. We
note that scallop effects also play a role in the relative S/N loss. The full
dataset is available in the Appendix A directory of the repository.

Figure 12: The relative difference of reported S/N of narrow pulses at
different dispersion measures with an S/N of 10, 50 and 100. The mean S/N loss
of each dataset is plotted as the blue dash line for comparison.

##  Appendix B Scattering

We also designed a simulation with scatter broadened pulses to test the
pipeline capability with scattered and other asymmetric pulse profiles. The
scatter broadening of pulses may affect the S/N as predicted, as the pulse
duration is extended leading to a wider boxcar.

We do not find strong evidence of differences in S/N between the scattered
pulses and Gaussian single pulses. The average S/N of scattered pulses varies
similarly to Gaussian pulses as scattering time, DM and intrinsic width both
affect the pulse duration. A longer pulse duration caused by scatter
broadening leads to larger boxcars resulting in the similar S/N decrease with
pulse width However, the correlation between DM and S/N is much more
significant.

The simulation code is available in the repository to regenerate this dataset
for further studies.

##  Appendix C Fractional Bandwidth

Many FRBs have been observed to exhibit frequency profiles concentrated into a
narrow bandwidth (Kumar et al., 2021; Pleunis et al., 2021). Additionally,
narrowband radio frequency interference (RFI) or interstellar scintillation
may reduce the effective bandwidth. Such effects may interact with the time-
frequency templates used by FREDDA to search for dispersed pulses, affecting
the S/N response.

We choose to simulate an extreme example of such spectral effects so as to
gauge the magnitude of the impact on the algorithm. We simulated a dataset of
Gaussian pulses with the same parameter settings used in the main section of
this paper but only with $1/6$ the bandwidth; this is comparable to the 65 MHz
bandwidth observed from FRB 20190711A Kumar et al. (2021). We choose band
occupancy to be concentrated into the third sextet of the band. Pulses were
scaled to S/N=20 to prevent saturation of the samples in our 8-bit data.

Figure 13: FREDDA reported/injected S/N of fractional bandwidth pulses in high
frequency band.

We show in Figure 13 the result from fredda, where the overall response
remains consistent with that of the overall performance from the broadband
pulses in Figure 6. The scalloping effect across dispersion measure is also
observed, but is more pronounced. This suggests that the DM scallop structure
is present over the entire bandwidth, but its effects are washed out in the
case of broadband pulses at high DM, due to different scalloping as a function
to frequency.