# A maximum likelihood estimate of the parameters of the FRB population

Siddhartha Bhattacharyya1, Himanshu Tiwari2, Somnath Bharadwaj1 and Suman
Majumdar2,3
1Department of Physics, Indian Institute of Technology, Kharagpur, India
2Department of Astronomy, Astrophysics and Space Engineering, Indian Institute
of Technology, Indore, India
3Department of Physics, Blackett Laboratory, Imperial College, London SW7 2AZ,
U.K<EMAIL_ADDRESS>
###### Abstract

We consider a sample of $82$ non-repeating FRBs detected at Parkes, ASKAP,
CHIME and UTMOST each of which operates over a different frequency range and
has a different detection criteria. Using simulations, we perform a maximum
likelihood analysis to determine the FRB population model which best fits this
data. Our analysis shows that models where the pulse scatter broadening
increases moderately with redshift ($z$) are preferred over those where this
increases very sharply or where scattering is absent. Further, models where
the comoving event rate density is constant over $z$ are preferred over those
where it follows the cosmological star formation rate. Two models for the host
dispersion measure ($DM_{\rm host}$) distribution (a fixed and a random
$DM_{\rm host}$) are found to predict comparable results. We obtain the best
fit parameter values $\alpha=-1.53^{+0.29}_{-0.19}$,
$\overline{E}_{33}=1.55^{+0.26}_{-0.22}$ and $\gamma=0.77\pm 0.24$. Here
$\alpha$ is the spectral index, $\gamma$ is the exponent of the Schechter
luminosity function and $\overline{E}_{33}$ is the mean FRB energy in units of
$10^{33}\,{\rm J}$ across $2128-2848\;{\rm MHz}$ in the FRB rest frame.

###### keywords:

transients: fast radio bursts, scattering.

††pagerange: A maximum likelihood estimate of the parameters of the FRB
population–References

##  1 Introduction

Fast radio bursts (FRBs) are milli-second duration, highly energetic ($\sim
10^{32}-10^{34}\,{\rm J}$) radio transients (Lorimer et al., 2007; Keane et
al., 2011; Thornton et al., 2013). Several radio telescopes including Parkes (
eg. Price et al. 2018), ASKAP (eg. Bhandari et al. 2020), CHIME (eg. Amiri et
al. 2021) and UTMOST (eg. Gupta et al. 2020) have each detected a considerable
number of FRBs. Several of the detected FRBs are found to repeat (eg. Spitler
et al. 2014; Amiri et al. 2021), however the non-repeating FRBs possibly form
a separate population (Palaniswamy et al., 2018; Caleb et al., 2018; Lu &
Piro, 2019) . Here we only consider the non-repeating FRBs. The large
dispersion measures (DMs), greater than the expected Milky Way contribution,
strongly suggests that FRBs are extragalactic events. Direct redshift
estimates are available only for a few of the observed FRBs which have been
localised on the sky (eg. Macquart et al. 2020; Heintz et al. 2020). For most
FRBs the redshifts are inferred from the observed DMs.

Several models have been proposed for the physical origin of the FRB emission,
unfortunately there is no clear picture as yet. For example, the recently
detected FRB $200428$, which coincided with an X-ray burst from the Galactic
magnetar SGR $J1935+2154$ (Ridnaia et al., 2021; Li et al., 2020; Tavani et
al., 2021), suggests active magnetars as a source for some of the FRBs
(Bochenek et al., 2020; Margalit et al., 2020). Platts et al. (2019) provides
a summary of the different FRB models.

The spectral index $\alpha$ of the FRBs is not very well constrained at
present. Macquart et al. (2019) have determined a mean value of
$\alpha=-1.5^{+0.2}_{-0.3}$ for the sample of $23$ FRBs detected at ASKAP.
Houben et al. (2019) have proposed a lower limit $\alpha>-1.2\pm-0.4$
considering the dearth of simultaneous detection of FRB $121102$ at $1.4\,{\rm
GHz}$ and $150\,{\rm MHz}$ respectively. The energy and redshift distribution
of the FRBs is also not well understood. James et al. (2021) have modelled the
FRB energy distribution using a simple power law . Zhang et al. (2021) have
used the FRBs detected at Parkes and ASKAP to constrain the exponent for the
energy distribution to a value $-1.8$ .

In an earlier work Bera et al. (2016) (hereafter Paper I) have modelled the
FRB population and used this to make predictions for FRB detection at
different telescopes. In a recent work Bhattacharyya & Bharadwaj (2021)
(hereafter Paper II) have used the two-dimensional Kolmogorov-Smirnov (KS)
test to compare the FRBs observed at Parkes, ASKAP, CHIME and UTMOST with
simulated predictions for different FRB population models. It is shown there
that the parameter range $\alpha>4$ and $\overline{E}_{33}>60$ is ruled out
with $95\%$ confidence, here $\overline{E}_{33}$ is the mean energy of the
FRBs population in units of $10^{33}\,{\rm J}$. Paper II also predicts that
"CHIME is unlikely to detect an FRB with extra-galactic dispersion measure
$DM_{Ex}$ exceeding $3700\,{\rm pc\,cm^{-3}}$", a prediction which is borne
out in the recently released CHIME catalogue of $492$ FRBs where the maximum
value is $DM_{Ex}=3006.7\,{\rm pc\,cm^{-3}}$. The modelling of the FRB
population and simulations of Paper II have also been used in the present
work, and these are summarized in the next section. In the present paper we
have used a maximum likelihood analysis to estimate the parameters of the FRB
population for which the predictions best match the observed FRB distribution.

##  2 Methodology

For our analysis we use $82$ non-repeating FRBs detected by Parkes, ASKAP,
CHIME and UTMOST which have each detected more than $10$ non-repeating FRBs.
The frequency range, limiting fluence and number of FRBs for these four
telescopes are summarized in the Table 1. The recent CHIME data for $492$ FRBs
(Rafiei-Ravandi et al., 2021; Amiri et al., 2021) was released while this
paper was being written, and we have not considered these here. Non-repeating
FRBs have also been detected at several other telescopes, however the number
of events at each of these telescopes is less than $10$ which is not adequate
for the statistical analysis performed here.

Each observed FRB is characterized by its dispersion measure $(DM)$, fluence
$(F)$ and the pulse width $(w)$. Here we use the extragalactic component
$DM_{\rm Ex}=DM-DM_{\rm MW}$ where the Milky Way contribution is calculated
for each FRB using the NE$2001$ model (Cordes & Lazio, 2003). For the present
work we have analysed the observed distribution of $DM_{\rm Ex}$ and $F$
values. For each telescope, we have gridded the the observed $DM_{\rm Ex}$ and
$F$ range and calculated the number of observed FRBs $N_{a}$ at each grid
point (Figure 1), the grid points here are labelled using $a$. Given the
limited number of FRBs, it is necessary to use a very coarse grid for the
present analysis.

Table 1: Considering the four telescopes Parkes; (Burke-Spolaor & Bannister, 2014; Zhang et al., 2020; Zhang et al., 2019; Keane et al., 2011; Lorimer et al., 2007; Champion et al., 2016; Petroff et al., 2019; Thornton et al., 2013; Ravi et al., 2015; Petroff et al., 2015, 2017; Keane et al., 2016; Bhandari et al., 2018a; Ravi et al., 2016; Price et al., 2018; Bhandari et al., 2018b) ASKAP; (Bannister et al., 2017; Shannon et al., 2018; Macquart et al., 2019; Agarwal et al., 2019; Qiu et al., 2019; Bhandari et al., 2019; Bannister et al., 2019; Prochaska et al., 2019; Shannon et al., 2019) CHIME; (Amiri et al., 2019) and UTMOST; (Caleb et al., 2017; Farah et al., 2018, 2019; Gupta et al., 2020), this shows the respective frequency range, limiting fluence $F_{l}$ and the number of FRBs detected there. Note that we have not included the 492 FRBs recently reported from CHIME. The value of $F_{l}$ depends on the threshold signal to noise ratio $(S/N)_{\rm th}$ which is different for the different FRB surveys.To keep the analysis simple, we consider a fixed $(S/N)_{\rm th}=10$. Further, ASKAP’s sensitivity also differs based on the particular survey in which the FRB was detected. For simplicity, we have used a fixed value of $F_{l}$ for ASKAP. Telescope | Frequency range | $F_{l}$ | Number of
---|---|---|---
Name | $({\rm MHz})$ | $({\rm Jy\,ms})$ | non-repeating FRBs
Parkes | $1157-1546$ | $0.5$ | $29$
ASKAP | $1129-1465$ | $4.1$ | $31$
CHIME | $400-800$ | $0.64$ | $11$
UTMOST | $827-859$ | $3.25$ | $11$

We now briefly discuss our model for the FRB population. The model for the FRB
population is presented in Paper I, and the reader is referred there for
details. The intrinsic properties of each FRB event are characterised by three
quantities $\alpha$, $E_{33}$ and $w_{i}$. Here the energy of the FRB pulse at
any frequency $\nu$ is assumed to be proportional to $\nu^{\alpha}$ where
$\alpha$ is the spectral index. For the present analysis we have assumed that
all the FRBs have the same value of the spectral index $\alpha$. Here $E_{33}$
is the energy of the FRB (in units of $10^{33}\,{\rm J}$) emitted in the
frequency interval $2128\,{\rm MHz}$ to $2848\,{\rm MHz}$ at the rest frame of
the source. $w_{i}$ here is the intrinsic pulse width of the FRB. Our earlier
work (Paper II) shows that the results do not change much if we vary $w_{i}$
in the range $0.1\,{\rm ms}$ to $2.0\,{\rm ms}$, and here we have used a fixed
value $w_{i}=1\,{\rm ms}$ for the entire analysis. In addition, each FRB also
has a redshift $z$ and an angular position $\mathbf{\theta}$ on the sky.

Figure 1: Considering the FRB observations at Parkes, ASKAP, CHIME and UTMOST,
this shows the number of observed FRBs $N_{a}$ at each grid point. The color
scale shows the number of FRBs at each grid.

The FRB energy distribution is currently unknown. It is reasonable to assume
that FRBs have a characteristic energy $\overline{E}_{33}$, and the
distribution falls of rapidly beyond $\overline{E}_{33}$. For the analysis
presented here we have assumed a Schechter luminosity function (Schechter,
1976) for the energy distribution expressed as:

$n(E_{33},z)=\begin{cases}\frac{n_{0}(z)}{\overline{E_{33}}}\;\exp\left(-\frac{E_{33}}{\overline{E}_{33}}\right)&{\rm
for}\;\gamma=0\\\
\frac{n_{0}(z)}{\overline{E}_{33}}\left(\frac{\gamma}{\Gamma(1+\gamma)}\right)\left(\frac{\gamma
E_{33}}{\overline{E}_{33}}\right)^{\gamma}\exp\left(-\frac{\gamma
E_{33}}{\overline{E}_{33}}\right)&{\rm for}\;\gamma>0\end{cases}$ (1)

where $\overline{E}_{33}$ is the mean energy of the population, $\gamma$ is
the exponent and $\Gamma(n)$ is the Gamma function. Here $n(E_{33},z)$ is the
FRB event rate per unit comoving volume per energy interval $dE_{33}$ and
$n_{0}(z)$ is the FRB event rate per unit comoving volume. In our analysis, we
have considered two possibilities for the event rate evolution with redshift,
namely (a.) CER - constant event rate where $n_{0}(z)$ is independent of $z$
over the redshift range of our interest and (b.) SFR - where
$n_{0}(z)\propto(0.015(1+z)^{2.7})/(1+((1+z)/2.9)^{5.6})$ traces the star
formation rate over cosmic time (Madau & Dickinson, 2014).

Considering an FRB located at redshift $z$, the observed pulse width
$w=\sqrt{w_{\rm cos}^{2}+w_{\rm DM}^{2}+w_{\rm sc}^{2}}$ mainly has three
contributions, namely (a) the cosmic expansion $w_{\rm cos}=w_{i}\,(1+z)$ \-
the intrinsic pulse width $w_{i}$, expanded by the cosmological expansion, (b)
the dispersion broadening $w_{\rm DM}=(8.3\times 10^{6}\,{\rm
DM}\,\Delta\nu_{c})/\nu_{0}^{3}$ \- the residual dispersion broadening after
the incoherent dedispersion of the FRB signal where $\nu_{0}$ is the
observational frequency and $\Delta\nu_{c}$ is the channel width of the
telescope, and (c) the scatter broadening $w_{\rm sc}$ \- this is not well
understood to date. Here $w$ is expressed in ${\rm ms}$, and both $\nu_{0}$
and $\Delta\nu_{c}$ are expressed in $\rm MHz$. In this work we have
considered three scattering models, namely (a) Sc-I - based on the empirical
fit of a large number of Galactic pulsar data provided by Bhat et al. (2004)
and we have extrapolated this for the IGM, (b) Sc-II - this is a pure
analytical model proposed by Macquart & Koay (2013) considering the turbulent
IGM, and (c) No-Sc - where there is no scattering and thus $w_{\rm sc}=0$. Our
earlier work (Figure 1 of Paper-I) shows that for both Sc-I and -II,
scattering dominates the total pulse width at redshifts $z>0.5$. Further, the
pulse width increases very sharply with increasing redshift for Sc-I relative
to Sc-II. The pulse width increases very slowly with redshift for No-Sc.

The contribution $DM_{\rm Host}$ from the host galaxy is an unknown factor
that enters FRB observations for most of the unlocalized FRBs. The value of
$DM_{\rm Host}$ is expected to vary from FRB to FRB depending on the host
galaxy and the location of the FRB within it. However, the FRB detections
suggest that $DM_{\rm Host}$ may not exceed the value $100\,{\rm pc\,cm}^{-3}$
(Macquart et al., 2020). In addition to this, the observed DM will have
another contribution $(DM_{\rm Halo}\sim 50-80\,{\rm pc\,cm}^{-3})$ from the
Galactic halo (Prochaska & Zheng, 2019). Here we have absorbed the $DM_{\rm
Halo}$ contribution in $DM_{\rm Host}$. In this work we consider two scenarios
for $DM_{\rm Host}$ namely (a) DM120 - where all FRBs have fixed $DM_{\rm
Host}=120\,{\rm pc\,cm}^{-3}$, and (b.) DMRand - where $DM_{\rm Host}$ values
are randomly drawn from a Gaussian distribution with mean $\overline{DM}_{\rm
Host}=120\,{\rm pc\,cm}^{-3}$ and root mean square value $\Delta{DM}_{\rm
Host}=15\,{\rm pc\,cm}^{-3}$. The $DM_{\rm Host}$ distribution is truncated at
$0$.

In summary, our model for the FRB population has three parameters $\alpha$,
$\overline{E}$ and $\gamma$. Further, we have two models for the event rate
distribution namely SFR and CER, three models for the scattering namely Sc-I,
Sc-II and No-Sc, and two models for $DM_{\rm Host}$ namely DM120 and DMRand.

We now briefly discuss how we use simulations to calculate $\mu_{a}$ (Figure
2) which is the model prediction for the mean number of FRBs expected to be
detected at each grid point for any given telescope. These simulations are
discussed in Paper II, and the interested readers are requested to refer to it
for more details. We consider a comoving volume which extends up to $z_{\rm
max}=5$, which is considerably higher than the highest redshift inferred for
any of the FRB events included in our analysis. The angular extent of this
comoving volume is equal to the full width half maxima (FWHM) of the primary
beam of the telescope and this varies from telescope to telescope. We populate
this comoving volume with $10^{6}$ randomly located FRBs whose mean comoving
number density follow $n_{0}(z)$. This provides the comoving distance $r$ and
angular position $\mathbf{\theta}$ for each simulated FRB. We consider the
$\Lambda CDM$ cosmology (Aghanim et al., 2020) to calculate $z$ from $r$. The
energy $E_{33}$ of each FRB is randomly drawn from the distribution in eq.
(1). We calculate $DM_{\rm Ex}$, $F$ and $w$ for each simulated FRB. The
telescope can only detect an FRB if $F\times\sqrt{1{\rm ms}/w}\geq F_{l}$
where $F_{l}$ is the limiting fluence of the telescope which depends on the
threshold signal to noise ratio $(S/N)_{\rm th}$. Considering $(S/N)_{\rm
th}=10$, Table 1 lists the value of $F_{l}$ for the four telescopes considered
here. We determine the fraction of observable events corresponding to each
($DM_{\rm Ex},F$) grid point and multiply this with the total number of FRBs
actually observed by the telescope to calculate $\mu_{a}$ which is the mean
number of FRB’s expected in each grid point for the particular model under
consideration. Figure 2 shows $\mu_{a}$ predicted for a particular model
$\alpha=-1.53$, $\overline{E}_{33}=1.55$ and $\gamma=0.77$ with Sc-II, CER and
DM120. We have compared the simulated predictions ($\mu_{a}$) with the actual
observation ($N_{a}$) to identify preferred models for the FRB population.

Figure 2: Considering Parkes, ASKAP, CHIME and UTMOST, this shows $\mu_{a}$
the mean number of FRBs predicted at each grid point for a model with
$\alpha=-1.53$, $\overline{E}_{33}=1.55\,{\rm J}$ and $\gamma=0.77$ with Sc-
II, CER and DM120. The color scale shows the number of FRBs at each grid.

We have used a Bayesian inference framework to constrain the parameters
$\alpha$, $\overline{E}_{33}$ and $\gamma$ for all FRB models discussed here.
The likelihood is estimated using the Poisson statistic. Considering the
$a^{\rm th}$ grid the probability of getting $N_{a}$ for a given $\mu_{a}$ is
expressed as

$P_{a}=\frac{\mu_{a}^{N_{a}}\exp{(-\mu_{a})}}{N_{a}!}$ (2)

where we assume that the likelihood $l_{a}$ is proportional to the
probability, i.e. $l_{a}\propto P_{a}$. In this analysis we choose the
proportionality constant to be $1$. Considering all the grid points, the
logarithmic value of the total likelihood is given by

$\log(\mathcal{L})=\sum_{a}\log(l_{a})=\sum_{a}\left[N_{a}\log(\mu_{a})-\mu_{a}-\log(N_{a}!)\right]$
(3)

where the summation is taken over all the grid points. Our aim here is to
probe efficiently the maximum likely region of the parameter space using a
Markov-Chain-Monte-Carlo (MCMC) algorithm. It has been demonstrated by Autcha
(2014) that in case of a Poissonian likelihood this can done efficiently by
estimating the loss function:

$L(\mu_{a},N_{a})=-\sum_{a}\left[\mu_{a}-N_{a}\right]^{T}.\left[\mu_{a}-N_{a}\right]$
(4)

at each step of the random walker in the parameter space. The evaluation of
this loss function at each step of the random walk can efficiently guide the
random walker towards the most likely region of the parameter space. For a
detailed discussion on this, the interested readers are referred to Autcha
(2014). In this work we have used the Python-based package Emcee111Publicly
available at: https://pypi.org/project/emcee/ (Foreman-Mackey et al., 2013)–
an Affine invariant MCMC ensemble sampler (Goodman & Weare, 2010) to perform
the exploration of the parameter space. For a specific FRB model and the given
set of model parameters $(\alpha,\overline{E}_{33},\gamma)$ we first predict
$\mu_{a}$ and then using $N_{a}$ and $\mu_{a}$ we estimate the loss function
$L(\mu_{a},N_{a})$ that eventually leads to estimation of the
$\log(\mathcal{L})$ of the region around the maximum likelihood. For this
analysis, we have used a standard MCMC chain of $20,000$ samples with $40$
random walkers. We discarded the initial $10$ per cent of the $20,000$ samples
as the burn-in steps. We have varied our parameter search between the range
$\alpha\rightarrow(-30.0,+30.0),\overline{E}_{33}\rightarrow(0.01,20.00),\gamma\rightarrow(0.0,6.0)$,
and assumed a uniform prior while doing the estimations.

##  3 Results

Table 2 shows the best fit parameter values for all the models which we have
considered here, the maximum values of log-likelihood $\log(\mathcal{L})$ are
also shown alongside for reference. Considering the models with Sc-I,
comparing the $\log(\mathcal{L})$ values we see that the models with CER are
preferred over the SFR models. The same feature is also seen for the other
scattering models. Guided by this, we exclude the SFR models and focus
entirely on the CER models for the subsequent discussion. Comparing the
different scattering models next, we find that Sc-II is preferred over both
Sc-I and No-Sc. Restricting our attention to Sc-II with CER, we find that
DM120 and DMRand have comparable $\log(\mathcal{L})$ values. Although the
$\log(\mathcal{L})$ value is slightly larger for DM120, the difference between
DM120 and DMRand is very small in comparison to the difference with all the
other models.

Considering Sc-II with CER, Figure 3 shows a corner plot of the parameters
$\alpha$, $\overline{E}_{33}$ and $\gamma$ for both DM120 and DMRand. We first
consider the three plots which show the joint distribution of pairs of these
parameters. A visual inspection shows that in all cases the $68\%$ and $95\%$
confidence intervals for DM120 largely overlap with those for DMRand. Further,
the orientation of the confidence intervals indicates that the constraints on
the three parameters $\alpha$, $\overline{E}_{33}$ and $\gamma$ are largely
uncorrelated. For DMRand for ($\alpha,\gamma$) we notice that the $68\%$
confidence interval is divided into two disconnected regions both of which are
enclosed within the $95\%$ confidence interval. Here also one of the region
dominates whereas the other has a very low probability associated with it.
Considering the two dimensional parameter plots, we see that in all cases our
analysis imposes tight constraints on the joint distribution of the parameters
$\alpha$, $\overline{E}_{33}$ and $\gamma$. We now consider the marginalised
one dimensional plots which correspond to the best fit parameter values and
the $68\%$ confidence intervals reported in Table 2. We find that $\alpha$ is
well constrained by our analysis with $\alpha=-1.53^{+0.29}_{-0.19}$ and
$-1.57^{+0.26}_{-0.24}$ for DM120 and DMRand respectively. Similarly, we find
that $\overline{E}_{33}$ is well constrained with
$\overline{E}_{33}=1.55^{+0.26}_{-0.22}$ and $1.46^{+0.25}_{-0.18}$ for DM120
and DMRand respectively. We also find that $\gamma$ is well constrained with
$\gamma=0.77\pm 0.24$ and $0.988^{+0.058}_{-0.102}$ for DM120 and DMRand
respectively. For $\alpha$, $\overline{E}_{33}$ and $\gamma$, the probability
distributions are dominated by a single peak. Further, we see that the
probability distributions for DM120 and DMRand have considerable overlap which
indicates that the $\alpha$, $\overline{E}_{33}$ and $\gamma$ values estimated
for these two models are consistent with one another.

##  4 Conclusions

In this analysis we have considered two distinct scenarios, one where the
comoving FRB event rate evolves with redshift following the star formation
rate and another where it is constant independent of redshift. Our analysis
shows that the models with a constant event rate are preferred over the models
where the event rate follows the star formation rate. The pulse broadening due
to scattering in the IGM is not clearly understood at present. Here we have
considered three possibilities. In the first model, based on Bhat et al.
(2004) which provides an empirical fit of a large number of Galactic pulsar
data, the scattering pulse width increases very sharply with increasing
redshift. The second one is a purely theoretical model proposed by Macquart &
Koay (2013) considering the turbulent IGM. In this model the scattering pulse
has a modest increase with increasing redshift. The third model assumes no
scattering. Our analysis shows that the second model where there is a modest
increase in pulse width with increasing redshift is preferred to the model
where this increase is much steeper or the model where there is no scattering.
The contribution to the dispersion measure from the host galaxies $DM_{\rm
Host}$ is another quantity which affects FRB detections. Here we consider two
models, one where all the FRBs have fixed $DM_{\rm Host}=120\,{\rm
pc\,cm}^{-3}$, and another where the $DM_{\rm Host}$ values are randomly drawn
from a truncated Gaussian distribution with mean $\overline{DM}_{\rm
Host}=120\,{\rm pc\,cm}^{-3}$ and root mean square value $\Delta{DM}_{\rm
Host}=15\,{\rm pc\,cm}^{-3}$. Our analysis shows that the model with a fixed
$DM_{\rm Host}$ is slightly preferred, however we find comparable results from
both the models. Considering the preferred combination of the event rate
distribution and scattering, we have
$(\alpha,\overline{E}_{33},\gamma)=(-1.53^{+0.29}_{-0.19},1.55^{+0.26}_{-0.22},0.77\pm
0.24)$ and
$(-1.57^{+0.26}_{-0.24},1.46^{+0.25}_{-0.18},0.988^{+0.058}_{-0.102})$ for
DM120 and DMRand respectively. The Schechter luminosity function approaches a
Dirac-delta function as the value of $\gamma$ is increased. The relatively low
value of $\gamma$ indicates that there is a considerable spread in the values
of the FRB energies. We have assumed $w_{i}=1\,{\rm ms}$ for the entire
analysis presented here. The analysis was repeated for $w_{i}=0.5\,{\rm ms}$
for which the change in the best fit values was found to be less than $10\%$.

An earlier work (Paper II) had used the KS test to rule out the parameter
range $\alpha>4$ and $\overline{E}_{33}>60$ with $95\%$ confidence. We have
checked that the best fit parameter values obtained here are well within the
allowed parameter range identified in Paper II.

The results of this paper are expected to provide inputs for any physical
model for the nature of the FRB sources and their cosmological distribution.
It also throws some light on the effect of scattering in the IGM. We plan to
carry out a similar analysis using the recently released catalogue of $492$
FRBs detected at CHIME. We expect tighter constraints as more FRBs get
included in the analysis.

Table 2: Considering the FRB detected at Parkes, ASKAP, CHIME and UTMOST, this shows the best fit values along with $68\%$ confidence intervals for $\alpha$, $\overline{E}_{33}$ and $\gamma$ for the different models considered here. The maximum values of log-likelihood $[\log(\mathcal{L})]_{\rm max}$ are also shown for reference. Results are shown for two different models for $DM_{\rm Host}$ namely DM120 and DMRand. |  |  | DM120 |  |  |  | DMRand |  |
---|---|---|---|---|---|---|---|---|---
| FRB | Parameter Values | Value of | Parameter Values | Value of
| Rate | $\alpha$ | $\overline{E}_{33}$ | $\gamma$ | $[\log(\mathcal{L})]_{\rm max}$ | $\alpha$ | $\overline{E}_{33}$ | $\gamma$ | $[\log(\mathcal{L})]_{\rm max}$
Sc-I | CER | $-0.50^{+0.25}_{-0.27}$ | $2.18^{+0.29}_{-0.29}$ | $0.96^{+0.04}_{-0.40}$ | $-129.64$ | $-0.58^{+0.19}_{-0.15}$ | $2.19^{+0.27}_{-0.25}$ | $0.89^{+0.12}_{-0.52}$ | $-133.08$
| SFR | $-26.0^{+1.8}_{-1.6}$ | $4.82^{+0.98}_{-0.76}$ | $0.31^{+0.38}_{-0.11}$ | $-235.57$ | $-25.8^{+1.7}_{-1.5}$ | $5.01^{+0.86}_{-0.84}$ | $0.26^{+0.40}_{-0.05}$ | $-235.39$
Sc-II | CER | $-1.53^{+0.29}_{-0.19}$ | $1.55^{+0.26}_{-0.22}$ | $0.77^{+0.24}_{-0.24}$ | $-78.39$ | $-1.57^{+0.26}_{-0.24}$ | $1.46^{+0.25}_{-0.18}$ | $0.99^{+0.06}_{-0.10}$ | $-79.95$
| SFR | $-27.5^{+1.5}_{-1.7}$ | $3.96^{+0.76}_{-0.70}$ | $0.37^{+0.41}_{-0.16}$ | $-233.31$ | $-27.5^{+1.5}_{-1.6}$ | $4.03^{+0.63}_{-0.68}$ | $0.29^{+0.43}_{-0.05}$ | $-233.18$
No-Sc | CER | $-2.26^{+0.19}_{-0.20}$ | $1.09^{+0.19}_{-0.24}$ | $0.56^{+0.34}_{-0.12}$ | $-94.82$ | $-2.25^{+0.12}_{-0.15}$ | $1.09^{+0.07}_{-0.08}$ | $0.47^{+0.33}_{-0.02}$ | $-95.29$
| SFR | $-30.0^{+1.1}_{-0.0}$ | $2.81^{+0.61}_{-0.56}$ | $0.40^{+0.49}_{-0.16}$ | $-229.16$ | $-30.0^{+1.2}_{-0.0}$ | $2.80^{+0.58}_{-0.56}$ | $0.40^{+0.50}_{-0.17}$ | $-229.34$
Figure 3: Considering the FRB observations at Parkes, ASKAP, CHIME and UTMOST,
this shows the maximum likelihood contours for the parameters
$\alpha,\overline{E}_{33}$ and $\gamma$ considering models with Sc-II and CER.
The 2D plot show both $68\%$ and $95\%$ confidence intervals. Only the $68\%$
confidence interval is shown in the 1D marginalized plots. The best fit values
of $\alpha$, $\overline{E}_{33}$ and $\gamma$ are shown in the figure with
blue color for DM120 and orange color for DMRand respectively.

## Acknowledgement

We acknowledge the Supercomputing facility ‘PARAM-Shakti’ at IIT Kharagpur
established under the National Supercomputing Mission (NSM), Government of
India and supported by Centre for Development of Advanced Computing (CDAC),
Pune. Some part of the statistical analysis for this work was done using the
computing resources available to the Cosmology with Statistical Inference
(CSI) research group at IIT Indore.

## Data availability

The data and codes underlying this article will be shared on reasonable
request to the corresponding author.

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