# A study on the Clustering Properties of Radio-Selected sources in the
Lockman Hole Region at 325 MHz

Aishrila Mazumder,1 Arnab Chakraborty,1,2,3 Abhirup Datta1
1Department of Astronomy, Astrophysics and Space Engineering, Indian Institute
of Technology Indore, Indore 453552, India
2Department of Physics, McGill University, 3600 rue University, Montreal, QC
H3A 2T8, Canada
3McGill Space Institute, McGill University, 3550 rue University, Montreal, QC
H3A 2A7, Canada E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Studying the spatial distribution of extragalactic source populations is vital
in understanding the matter distribution in the Universe. It also enables
understanding the cosmological evolution of dark matter density fields and the
relationship between dark matter and luminous matter. Clustering studies are
also required for EoR foreground studies since it affects the relevant angular
scales. This paper investigates the angular and spatial clustering properties
and the bias parameter of radio-selected sources in the Lockman Hole field at
325 MHz. The data probes sources with fluxes $\gtrsim$0.3 mJy within a radius
of 1.8∘ around the phase center of a $6^{\circ}\times 6^{\circ}$ mosaic. Based
on their radio luminosity, the sources are classified into Active Galactic
Nuclei (AGNs) and Star-Forming Galaxies (SFGs). Clustering and bias parameters
are determined for the combined populations and the classified sources. The
spatial correlation length and the bias of AGNs are greater than SFGs-
indicating that more massive haloes host the former. This study is the first
reported estimate of the clustering property of sources at 325 MHz,
intermediate between the preexisting studies at high and low-frequency bands.
It also probes a well studied deep field at an unexplored frequency with
moderate depth and area. Clustering studies require such observations along
different lines of sight, with various fields and data sets across frequencies
to avoid cosmic variance and systematics. Thus, an extragalactic deep field
has been studied in this work to contribute to this knowledge.

###### keywords:

Galaxies - galaxies: active$<$ Galaxies- cosmology: large-scale structure of
Universe$<$Cosmology - cosmology: observations$<$Cosmology - radio continuum:
galaxies$<$ Resolved and unresolved sources as a function of wavelength

††pubyear: 2015††pagerange: A study on the Clustering Properties of Radio-
Selected sources in the Lockman Hole Region at 325 MHz–A study on the
Clustering Properties of Radio-Selected sources in the Lockman Hole Region at
325 MHz

##  1 Introduction

Observations of the extragalactic sky at radio frequencies are essential for
the study of both large-scale structures (LSS) and different populations of
sources present in the Universe. The initial research on LSS using clustering
was performed with the reporting of slight clustering signals from nearby
sources (Seldner & Peebles, 1981; Shaver & Pierre, 1989). With the advent of
large-area surveys like FIRST (Faint Images of the Radio Sky at Twenty-
Centimeters, Becker et al. 1995) and NVSS (NRAO VLA Sky Survey, Condon et al.
1998), the studies became more precise due to the large number of sources
detected in these surveys.

The extragalactic sky at radio frequencies is dominated by sources below mJy
flux densities (at frequencies from MHz to a few GHz, see for example Simpson
et al. 2006; Mignano, A. et al. 2008; Seymour et al. 2008; Smolčić et al.
2008; Prandoni et al. 2018). The source population can be divided into Active
Galactic Nuclei (AGNs) and Star-Forming Galaxies (SFGs) (Condon, 1989; Afonso
et al., 2005; Simpson et al., 2006; Bonzini et al., 2013; Padovani et al.,
2015; Vernstrom et al., 2016). The dominant sources at these fluxes are SFGs,
AGNs of Fanaroff-Riley type I (FR I, Fanaroff & Riley 1974), and radio-quiet
quasars (Padovani, 2016). Emission mechanism dominating populations at low
frequencies ($\lesssim$10 GHz) is synchrotron emission, modeled as a power law
of the form $\textrm{S}_{\nu}\propto\nu^{-\alpha}$, where $\alpha$ is the
spectral index. Study of the extragalactic population using synchrotron
emission can help trace the evolution of the LSS in the Universe. It also
helps to map their dependence on various astrophysical and cosmological
parameters (Blake & Wall, 2002b; Lindsay et al., 2014; Hale et al., 2018).
Radio continuum surveys, both wide and deep, help constrain the overall
behavior of cosmological parameters and study their evolution and relation to
the environment (Best & Heckman, 2012; Ineson et al., 2015; Hardcastle et al.,
2016; Calistro Rivera et al., 2017; Williams et al., 2018). The clustering
pattern of radio sources (AGNs and SFGs) can be studied to analyze the
evolution of matter density distribution. Clustering measurements for these
sources also provide a tool for tracing the underlying dark matter
distribution (Press & Schechter, 1974; Lacey & Cole, 1993, 1994; Sheth &
Tormen, 1999; Mo et al., 2010). The distribution of radio sources derived from
clustering is related to the matter power spectrum and thus provides insights
for constraining cosmological parameters that define the Universe. The
relationship of the various galaxy populations with the underlying dark matter
distribution also helps assess the influence of the environment on their
evolution. Clustering studies are also required for extragalactic foreground
characterization for EoR and post-EoR science. Spatial clustering of
extragalactic sources with flux density greater than the sub-mJy range
(around$\sim$150 MHz) dominate fluctuations at angular scales of arcminute
range. Thus, their modeling and removal allow one to detect fluctuation of the
21-cm signal on the relevant angular scales.

The definition of clustering is the probability excess above a certain random
distribution (taken to be Poisson for astrophysical sources) of finding a
galaxy within a certain scale of a randomly selected galaxy. This is known as
the two-point correlation function (Peebles, 1980). The angular two-point
correlation function has been studied in optical surveys like the 2dF Galaxy
Redshift Survey (Peacock et al., 2001; Percival et al., 2001; Norberg et al.,
2001), Sloan Digital Sky Survey (Eisenstein et al., 2005; Wang et al., 2013;
de Simoni et al., 2013; Shi et al., 2016; Carvalho et al., 2016) and the Dark
Energy Survey (Camacho et al., 2019). Optical surveys provide redshift
information for sources either through photometry or spectroscopy. This
information can be used to obtain the spatial correlation function and the
bias parameter (Ross et al., 2007; Heinis et al., 2009; Salazar-Albornoz et
al., 2017). But for optical surveys, observations of a large fraction of the
sky is expensive in terms of cost and time. Additionally, optical surveys
suffer the limitation of being dust-obscured for high redshift sources.
However, at radio wavelengths, the incoming radiation from these sources do
not suffer dust attenuation and thus can be used as a mean to probe such high
z sources (Hao et al., 2011; Cucciati et al., 2012; Singh et al., 2014; Jarvis
et al., 2016; Saxena et al., 2017). The highly sensitive radio telescopes like
GMRT (Swarup et al., 1991), ASKAP (DeBoer et al., 2009), LOFAR (van Haarlem,
M. P. et al., 2013) are also able to survey larger areas of the sky
significantly faster. They are thus efficient for conducting large-area
surveys in lesser time than the old systems while detecting lower flux
densities. Therefore, radio surveys provide an efficient method for
investigation of the clustering for the different AGN populations.
Additionally, at low-frequencies ($\lesssim$ 1.4 GHz), synchrotron radiation
from SFGs provide insight into their star-formation rates (Bell, 2003; Jarvis
et al., 2010; Davies et al., 2016; Delhaize et al., 2017; Gürkan et al.,
2018). These insights have lead to clustering studies of SFGs as well at low
frequencies (Magliocchetti et al., 2017; Chakraborty et al., 2020). Through
clustering studies of radio sources, deep radio surveys help trace how the
underlying dark matter distribution is traced by luminous matter distribution.
In addition to this, the two-point correlation functions can also provide
other information relevant for cosmology by fitting parameterized models to
the data to obtain acceptable ranges of parameters. These include the bias
parameter, dark energy equations of state, and $\Omega_{m}$ (total density of
matter), to name a few (Peebles, 1980; Camera et al., 2012; Raccanelli et al.,
2012; Planck Collaboration et al., 2014; Allison et al., 2015).

Extensive observations at multiple frequencies can help understand the
relationship of the various source populations with their host haloes and
individual structures (stars) present. It has been inferred from clustering
observations that AGNs are primarily hosted in more massive haloes than SFGs
and are also more strongly clustered (Gilli et al., 2009; Donoso et al., 2014;
Magliocchetti et al., 2017; Hale et al., 2018). While AGNs are more clustered
than SFGs, for the latter, the clustering appears to be dependent on the rate
of star formation. SFGs with higher star formation rates are more clustered
than the ones with a lower rate (since star formation rate is correlated to
stellar mass, which in turn is strongly correlated to the mass of the host
halo, see Magnelli et al. (2015); Delvecchio et al. (2021); Bonato et al.
(2021) and references therein). Studying the large-scale distribution of dark
matter by studying the clustering pattern of luminous baryonic matter is vital
for understanding structure formation. From linear perturbation theory,
galaxies are ”biased” tracers of the underlying matter density field since
they are mostly formed at the peak of the matter distribution (Peebles, 1980).
Bias parameter (b) traces the relationship between overdensity of a tracer
$\delta$ and the underlying dark matter overdensity ($\delta_{DM}$), given by
$\delta=\textrm{b}\delta_{DM}$. The linear bias parameter is the ratio between
the dark matter correlation function and the galaxy correlation function
(Peebles (1980); Kaiser (1984); Bardeen et al. (1986), also see Desjacques et
al. (2018) for a recent review). Measurement of the bias parameter from radio
surveys will allow measurements which probe the underlying cosmology governing
the LSS, and probe dark energy, modified gravity, and non-Gaussianity of the
primordial density fluctuations (Blake et al., 2004; Carilli & Rawlings, 2004;
Seljak, 2009; Raccanelli et al., 2015; Abdalla et al., 2015).

Analysis of the clustering pattern for extragalactic sources is also important
for observations targeting the 21-cm signal of neutral hydrogen (HI) from the
early Universe. These weak signals from high redshifts have their observations
hindered by many orders of magnitude brighter foregrounds - namely diffuse
galactic synchrotron emission (Shaver et al., 1999), free-free emission from
both within the Galaxy as well as extragalactic sources (Cooray & Furlanetto,
2004), faint radio-loud quasars (Matteo et al., 2002) and extragalactic point
sources (Di Matteo et al., 2004). Di Matteo et al. (2004) showed that spatial
clustering of extragalactic sources with flux density $\gtrsim$0.1 mJy at 150
MHz (the equivalent flux density at 325MHz is $\sim$0.05 mJy) dominate
fluctuations at angular scale $\theta\gtrsim$1$\arcmin$. Thus, their modeling
and removal allow one to detect fluctuation of the 21-cm signal on relevant
angular scales. So their statistical modeling is necessary to understand and
quantify the effects of bright foregrounds. Many studies have modeled the
extragalactic source counts as single power-law or smooth polynomial (Intema
et al., 2017; Franzen et al., 2019) and the spatial distribution of sources as
Poissonian (Ali et al., 2008) or having a simple power-law clustering.
However, a Poisson distribution of foreground sources is very simplistic and
may affect signal recovery for sensitive observations like those targeting the
EoR signal (Ali et al., 2008; Trott et al., 2016). Thus more observations are
required for low-frequency estimates of the clustering pattern of compact
sources.

A number of studies have been done in recent years for observational
determination of the clustering of radio selected sources (for instance Cress
et al. (1996); Overzier et al. (2003); Lindsay et al. (2014); Magliocchetti et
al. (2017); Hale et al. (2018, 2019); Rana & Bagla (2019); Chakraborty et al.
(2020); Siewert et al. (2020)). However, more such studies are required for
modeling the influence of different processes on the formation and evolution
of LSS in the Universe. The sample used for such analyses should not be
limited to small deep fields, since the limited number of samples makes
clustering studies of different populations (AGNs/ SFGs) sample variance
limited. Studies on the statistics of the source distribution are also
essential for understanding the matter distribution across space. Thus,
observations using sensitive instruments are required to conduct more detailed
studies. At 1.4 GHz and above, many clustering studies are present (for
instance Cress et al. (1996); Overzier et al. (2003); Lindsay et al. (2014);
Magliocchetti et al. (2017); Hale et al. (2018); Bonato et al. (2020));
however there extensive studies at low frequencies (and wider areas) are still
required. The TIFR GMRT Sky Survey (TGSS) (Intema et al., 2017) is a wide-area
survey of the northern sky at 150 MHz. But the available catalog from the
TGSS- Alternate Data Release (TGSS-ADR) suggests that the data is systematics
limited. Thus it is unsuitable for large-scale clustering measurements (Tiwari
et al., 2019). The ongoing LOFAR Two-metre Sky Survey (LoTSS Shimwell, T. W.
et al. 2017) at a central frequency of 144 MHz is expected to have very high
sensitivity and cover a very wide area and thus provide excellent data for
studying source distribution statistics at low frequencies (Siewert et al.,
2020). However, to constrain cosmological parameters, consensus for the
overall behavior of sources along different lines of sight and across
frequencies is also required, and there data sets like the one analyzed here
become important (Blake et al., 2004; Carilli & Rawlings, 2004; Norris et al.,
2013). Radio data has the advantage that even flux-limited samples contain
high-z sources (Dunlop & Peacock, 1990). Thus, using the entire radio band
provides insights into physical processes driving the evolution of different
galaxy populations and helps create a coherent picture of the matter
distribution in the Universe. Therefore, studies at radio frequencies would
help constrain the cosmology underlying structure formation and evolution.

The recent study of the clustering of the ELAIS-N1 field centered at 400 MHz
using uGMRT by Chakraborty et al. (2020) was extremely sensitive, with an RMS
($\sigma_{400}$)111Unless otherwise stated, $\sigma_{frequency}$ is the RMS
sensitivity at the quoted frequency throughout the text. of 15 $\mu$Jy
$\textrm{b}eam^{-1}$. But the area covered was significantly smaller
($\sim$1.8 deg 2) than this work. This smaller field of view makes measurement
of clustering properties on large angular scales impossible. Smaller areas
also lead to smaller sample sizes for statistics, resulting in studies limited
by cosmic variance. Another study of the HETDEX spring field at 144 MHz (using
the data release 1 of LOFAR Two meter Sky Survey) by Siewert et al. (2020) has
a sky coverage of $\sim$350 square degree, but the mean $\sigma_{150}$ is
$\sim$91 $\mu$Jy beam-1. However, despite the sensitivity achieved in the
survey, the analysis by Siewert et al. (2020) is limited to flux densities
above 2 mJy. Motivated by the requirement for a study in the intermediate
range (in terms of flux density, area covered, and frequency), this work aims
to quantify the clustering of the sources detected in the Lockman Hole field.
The data analyzed here fall in the intermediate category, with a survey area
$\sim$6 deg2 with $\sigma_{325}\sim$ 50 $\mu$Jy beam -1. It is thus ideal for
clustering studies with a sizeable area of the sky covered (thus large angular
scales can be probed) and moderately deep flux threshold (catalogue will have
fluxes reliable to a lower value). Additionally, the Lockman Hole region has
excellent optical coverage through surveys like SDSS and SWIRE; thus,
associated redshift information is available to study spatial clustering and
bias parameters. This frequency also has the additional advantage of having
lesser systematics than the 150 MHz band while still being sensitive to the
low-frequency characteristics of sources. New data releases for the LoTSS
surveys promise greater sensitivity and source characterization over various
deep fields targeted by these observations (Shimwell, T. W. et al., 2019;
Tasse et al., 2021); all these observational data at multiple frequencies will
put more precise constraints on the various parameters governing the structure
formation and evolution.

This work uses archival GMRT data at 325 MHz covering a field of view of
$6^{\circ}\times 6^{\circ}$ through multiple pointings. In Mazumder et al.
(2020), data reduction procedure is described in detail. This work used the
source catalogue obtained there for clustering analyses. However, the entire
dataset could not be used due to limiting residual systematics at large
angular scales. The clustering pattern and linear bias parameter are
determined for the whole population and sub-populations, i.e., AGNs and SFGs,
separately. The previous work by Mazumder et al. (2020) had determined the
flux distribution of sources (i.e., differential source count) and
characterized the spatial property and the angular power spectrum of the
diffuse galactic synchrotron emission using the same data.

This paper is arranged in the following manner: In section 2, a brief outline
of the radio data as well as various optical data used is discussed; the
classification into source sub-populations is also using radio luminosity of
sources is also shown. The following section, i.e., Section 3 shows the
clustering quantification - both in spatial and angular scales and calculation
of linear bias for all the detected sources. Section 4 discusses the
clustering property and bias for classified population, with a brief
discussion on the choice of the field of view for this analysis discussed in
Section 5. Finally, the paper is concluded in Section 6.

For this work, the best fitting cosmological parameters obtained from the
Planck 2018 data (Planck Collaboration et al., 2019) has been used. The values
are $\mathrm{\Omega_{M}}$ = 0.31, $\mathrm{\Omega_{\Lambda}}$ = 0.68,
$\mathrm{\sigma_{8}}$ = 0.811, & $H_{0}$ = 67.36 km s-1 Mpc -1. The spectral
index used for scaling the flux densities between frequencies is taken as
$\alpha$=0.8.

##  2 Observations and Source Catalogues

This work uses 325 MHz GMRT archival data of the Lockman Hole region. The
details of the data reduction procedure have been described in Mazumder et al.
(2020), here it is discussed very briefly. The data were reduced using the
SPAM pipeline (Intema et al., 2009; Intema, 2014; Intema et al., 2017), which
performs direction-independent as well as direction-dependent calibration
techniques. The observation had 23 separate pointings , centered at
($\alpha_{2000}=10^{h}48^{m}00^{s},\delta_{2000}=58^{\circ}08^{\prime}00\arcsec$),
each of which was reduced separately. The final image is a $6^{\circ}\times
6^{\circ}$ mosaic having off-source RMS of 50$\mathrm{\mu Jybeam^{-1}}$ at the
central frequency. Figure 1 shows the primary beam corrected final mosaic
image of the observed region. This image was used to extract a source
catalogue using Python Blob Detection and Source Finder
222https://www.astron.nl/citt/pybdsf/(PYBDSF, Mohan & Rafferty (2015)) above a
minimum flux density $\textrm{S}^{cut}_{325}$ 0.3mJy (i.e., above
6$\sigma_{325}$). A total of 6186 sources were detected and cataloged. The
readers are referred to Mazumder et al. (2020) for details on catalogue
creation and subsequent comparison with previous observations.

Figure 1: Primary beam corrected mosaic of the Lockman Hole region at 325MHz.
The off source RMS at the center is $\sim$ 50 $\mu$Jy $\mathrm{beam}^{-1}$ and
beam size is $9.0\arcsec\times 9.0\arcsec$. This image is a reproduction of
Figure 1 of Mazumder et al. (2020)

.

The redshift information for the sources are derived by matching with optical
data from the Sloan Digital Sky Survey (SDSS)333https://www.sdss.org/ and the
Herschel Extragalactic Legacy Project (HELP)
444http://herschel.sussex.ac.uk/,555https://github.com/H-E-L-P (Shirley et
al., 2019). The SDSS (York et al., 2000; Eisenstein et al., 2011) has been
mapping the northern sky in the form of optical images as well as optical and
near-infrared spectroscopy since 1998. The latest data release (DR16) is from
the fourth phase of the survey (SDSS-IV, Blanton et al. (2017)). It includes
the results for various survey components like the extended Baryon Oscillation
Spectroscopic Survey eBOSS, SPectroscopic identification of ERosita Sources
SPIDERS, Apache Point Observatory Galaxy Evolution Experiment 2 APOGEE-2, etc.
The surveys have measured redshifts of a few million galaxies and have also
obtained the highest precision value of the Hubble parameter $H(z)$ to date
(Alam et al., 2021). An SQL query was run in the CasJobs
666https://skyserver.sdss.org/casjobs/ server to obtain the optical data
corresponding to the radio catalogue, and the catalogue thus obtained was used
for further analysis.

HELP has produced optical to near-infrared astronomical catalogs from 23
extragalactic fields, including the Lockman Hole field. The final catalogue
consists of $\sim$170 million objects obtained from the positional cross-match
with 51 surveys (Shirley et al., 2019). The performance of various templates
and methods used for getting the photometric redshift is described in Duncan
et al. (2017); Duncan et al. (2018). Each of the individual fields is provided
separate database in the Herschel Database in Marseille site
777https://hedam.lam.fr/HELP/ where various products, field-wise and category
wise are made available via ”data management unit (DMU)”. For the Lockman Hole
field, the total area covered by various surveys is 22.41 square degrees with
1377139 photometric redshift objects. The Lockman Hole field is covered well
in the Spitzer Wide-area InfraRed Extragalactic Legacy Survey (SWIRE) with
photometric redshifts obtained as discussed in Rowan-Robinson et al. (2008);
Rowan-Robinson et al. (2012). However, additional data from other survey
catalogues like Isaac Newton Telescope - Wide Field Camera (INT-WFC, Lewis et
al. (2000)), Red Cluster Sequence Lensing Survey (RCSLenS, Hildebrandt et al.
(2016)) catalogues, Panoramic Survey Telescope and Rapid Response System 3pi
Steradian Survey (PanSTARRS-3SS, Chambers et al. (2019)), Spitzer Adaptation
of the Red-sequence Cluster Survey (SpARCS, Muzzin et al. (2007)), UKIRT
Infrared Deep Sky Survey - Deep Extragalactic Survey (UKIDSS-DXS, Lawrence et
al. (2007)), Spitzer Extragalactic Representative Volume Survey (SERVS,
Mauduit et al. (2012)) and UKIRT Hemisphere Survey (UHS, Dye et al. (2017))
resulted in more sources being detected and better photometric determination.
The publicly available photometric catalogue for the Lockman Hole region was
used to determine the redshift information for matched sources. The source
catalogue derived from the 325 MHz observation is pre-processed, matched to
add redshift information, and then further analysis is done. The following
subsections describe these steps in detail.

###  2.1 Merging multi-component sources

The final map produced has a resolution of 9″. The source finder might resolve
an extended source into multiple components for such high-resolution maps.
Such sources are predominantly radio galaxies that have a core at the center
and hotspots that extend along the direction of the jet(s) or at their ends;
these structures may be classified as separate sources (Magliocchetti et al.,
1998; Prandoni et al., 2018; Williams, W. L. et al., 2019; Galvin et al.,
2020). Using the NVSS catalogue, it has been shown in Blake & Wall (2002a)
that large radio sources with unmarked components can significantly alter
clustering measurements. Thus, for unbiased estimation of source clustering,
such sources need to be identified and merged properly. A strong correlation
between the angular extent of radio sources and their fluxes has been
discovered by Oort (1987). The angular extent ($\theta$) of a source is
related to its flux density (S) by the $\theta$-S relation,
$\theta\propto\sqrt{\textrm{S}}$. This relation was used to identify resolved
components of multi-component sources in surveys like the FIRST survey
(Magliocchetti et al., 1998).

Identification of multi-component sources in the Lockman Hole catalogue
resolved as separate sources are made using two criteria. The maximum
separation between pairs of sources (using the $\theta$-S relation) is given
by $\mathrm{\theta_{max}=20\sqrt{S_{total}}}$, where $\mathrm{S_{total}}$ is
the summed flux of the source pairs (Huynh et al., 2005; Prandoni et al.,
2018; Chakraborty et al., 2020). Sources identified by the above criteria have
been considered as the same source if their flux densities differ by less than
a factor of 4 (Huynh et al., 2005).

Figure 2: Sum of nearest neighbour flux densities from the 325 MHz catalogue
as a function of separation between nearest neighbours in the catalogue are
shown by light blue circles. Sources above the black doted line (having an
angular separation less than $\mathrm{\theta_{max}}$) shown by blue triangles
have their flux densities differing by a less than a factor of 4.

Figure 2 shows the separation between the nearest neighbour pairs from the 325
MHz catalogue as a function of the separation between them. Above the black
dotted line, the sources have separation less than $\mathrm{\theta_{max}}$ as
mentioned above. Blue triangles are sources that have flux density differences
less than a factor of 4. The two criteria mentioned gave a sample of 683
sources (out of 6186 total) to have two or more components. After merging
multi-component sources and filtering out random associations, 5489 sources
are obtained in the revised catalogue. The position of the merged sources are
the flux weighted mean position for their components.

###  2.2 Adding Redshift Information

As already mentioned, optical cross-identification have for the sources
detected has been done using the HELP and SDSS catalogues. A positional cross-
match with 9″matching radius (which is the resolution for this observation)
was used for optical cross-matching. Since the positional accuracy of the
catalogue is better than 1″(Mazumder et al., 2020), a nearest neighbour search
algorithm was used to cross-match sources with the optical catalogue with a
search radius $r_{s}$. The rate of contamination expected due to proximity to
optical sources is given by (Lindsay et al., 2014):

$P_{c}=\pi r_{s}^{2}\sigma_{opt}$

where $\sigma_{opt}$ is the surface density of the optical catalogue. For
surface density of 1.4$\times$104 deg-2, a matching radius $r_{s}$ = 9″gives a
contamination of <10%. This radius was thus used to ensure valid optical
identification of a large number of radio sources.

Figure 3: Sources with optical cross matches within 1.8∘ radius of phase
center for the full mosaic. The area considered is represented by the black
dot-dashed circle, and the sources in this region are represented by “x”
marks. The blue circles represent the sources without any redshift matches.

Large FoVs are helpful for observational studies of like the one used here is
useful for studying LSS, since the presence of a large number of sources
provides statistically robust results and also reduces the effect of cosmic
variance. Accordingly, the data used for this work had an Fov of $\rm
6^{\circ}\times 6^{\circ}$. However, cross-matching with optical catalogue
produced matches with only 70% total sources over entire FoV, and 30% sources
remained unclassified. Further investigation also revealed the presence of
unknown systematics, which resulted in excess correlation and deviation from
power-law behavior at large angles. The most probable cause for such a
deviation seems to be either the presence of many sources with no redshifts at
the field edge or the presence of artifacts at large distances from the phase
center. Analysis done by reducing the area of the field increased percentage
of optical matches and reduced the observed deviation from the power-law
nature. Thus, the cause of such a deviation has been attributed to the former
one.

Hence, the clustering properties of sources at large angular scales are not
reliable for this observation. Thus, the analysis was restricted to a smaller
area of the Lockman Hole region around the phase center; large-scale
clustering properties could not be estimated. Taking a cut-off with 1.8∘
radius around the phase center resulted in $\sim$95% sources having an optical
counter-part. Hence, it is expected that the unclassified sources present
would not affect the signal significantly (a detailed discussion on the choice
of the FoV cut-off is discussed in Section 5). This FoV cut-off yielded 2555
sources in the radio catalogue, out of which 2424 sources have optical matches
within the aforementioned match radius. This is shown in Figure 3, where the
area considered is represented with black dot-dashed circle, and the ”x” marks
denote the sources in radio catalogue; the blue circles represent the sources
without any optical cross-matches in either photometry or spectroscopy. A
total of 2415 photometric and 664 spectroscopic matches were obtained after
the cross-match with optical catalogues. Out of these, 650 sources had both
photometric and spectroscopic detection. For such cases, the spectroscopic
identifications were taken. Combined photometry and spectroscopic
identifications were obtained for a total of 2424 sources, of which 27 sources
were discarded from this analysis since they were nearby objects with 0 or
negative redshifts (Lindsay et al., 2014). The final sample thus had 2397
sources, which is $\sim$94% of the total catalogued radio sources within 1.8∘
radius of the phase center. The redshift matching information for both the
full and restricted catalogues have been summarised in Table 1. The redshift
information from the optical catalogues was incorporated for these sources and
was used for further analysis. Figure 4 shows the distribution of redshifts
for the sources detected in both HELP and SDSS. In the left panel, the
photometric redshifts are plotted as a function of the spectroscopic
redshifts. As can be seen, the two values are in reasonable agreement with
each other for most cases. Additionally to check for the reliability of
obtained photometric redshifts, following Duncan et al. (2018), the outlier
fraction defined by $\mathrm{\frac{\mid z_{phot}-z_{spec}\mid}{1+z_{spec}}}$
>0.2, is plotted as a function of the spectroscopic value (right panel of
Figure 4). For this work, the drastic outliers are the points with values
>0.5. The fraction of outliers with drastically different values between
photometric and spectroscopic redshifts is $\sim$10%. While a detailed
investigation is beyond the scope of this work, the outliers may be present
due to the combination of uncertainties in the different surveys used in the
HELP catalogue. As can be seen, the outlier fraction is not very drastic
except for some cases; however, the reason for deviations in these sources is
unknown. The median redshift for all the sources with redshift information
comes out to be 0.78. The top panel of Figure 5 shows the distribution N($z$)
as a function of source redshift, with the black dashed line indicating median
redshift.

Table 1: Summary of number of sources with redshift information Area | Number of sources | Redshift matches | Percentage of matches | AGNs | SFGs
---|---|---|---|---|---
$6^{\circ}\times 6^{\circ}$ | 5489 | 3628 | 66 | 2149 | 1479
$3.6^{\circ}$ diameter around phase center | 2555 | 2397 | 95 | 1821 | 576

Figure 4: Comparison of photometric and spectroscopic redshifts for the
sources with matches for both. (left panel) Photometric redshift plotted as a
function of the spectroscopic redshift, with the black dashed line indicating
equal values for both. The redshift distribution is shown via the histograms
for photometric (right subplot) and spectroscopic (top subplot). (right panel)
Distribution of $\mathrm{\frac{z_{phot}-z_{spec}}{1+z_{spec}}}$ plotted as a
function of $\mathrm{z_{spec}}$. The black dashed line represents 0 deviation
between the photometric and spectroscopic redshifts.

###  2.3 Classification using Radio Luminosity Function

The catalogued sources with optical counterparts were divided into AGNs and
SFGs using their respective radio luminosities. Assuming pure luminosity
evolution, the luminosity function evolves approximately as
$\mathrm{(1+z)^{2.5}}$ and $\mathrm{(1+z)^{1.2}}$ for SFGs and AGNs
respectively (McAlpine et al., 2013). The value for AGNs differ slightly from
those of Smolci´c et al. (2017) and Ocran et al. (2020) for the COSMOS field
at 3 GHz and ELAIS N1 field at 610 MHz respectively. However, they are
consistent with those of Prescott et al. (2016) for the GAMA fields. The
values for redshift evolution of SFGs also agree broadly for the GAMA fields
(Prescott et al., 2016) and the ELAIS N1 field (Ocran et al., 2019).

It has been shown in Magliocchetti et al. (2014); Magliocchetti et al. (2015,
2017) that radio selected galaxies powered by AGNs dominate for radio powers
beyond a radio power $\mathrm{P_{cross}}(z)$ which is related with the
redshift $z$ as:

$\mathrm{log_{10}P_{cross}=log_{10}P_{0,cross}}+z$ (1)

upto $z\sim$1.8, with P (at 1.4 GHz) in W $\mathrm{Hz^{-1}sr^{-1}}$. In the
local Universe, the value of $\mathrm{P_{cross}}$ is $\mathrm{10^{21.7}}$ (W
$\mathrm{Hz^{-1}sr^{-1}}$), coinciding with the observed break in the radio
luminosity functions of SFGs (Magliocchetti et al., 2002), beyond which their
luminosity functions decrease rapidly and the numbers are also reduced
greatly. Thus contamination possibility between the two population of radio
sources is very low using the radio luminosity based selection criterion
(Magliocchetti et al., 2014; Magliocchetti et al., 2017).

The radio luminosity has been calculated for the sources from their flux as
(Magliocchetti et al., 2014):

$\mathrm{P_{1.4GHZ}=4\pi S_{1.4GHz}D^{2}(1+z)^{3+\alpha}}$ (2)

where D is the angular diameter distance, and $\alpha$ is the spectral index
of the sources in the catalogue. The individual spectral index for the sources
was not used (since all sources do not have the measured values). The median
value of 0.8 for $\alpha$ was derived by matching with high-frequency
catalogues in Mazumder et al. (2020). Since the probability of finding a large
number of bright, flat-spectrum sources is very low (Magliocchetti et al.,
2017), the median value of 0.8 was used to determine the luminosity functions
of the sources in the Lockman Hole field detected here.

Besides the radio luminosity criterion described above, there are several
other methods to classify sources into AGNs and SFGs. X-ray luminosity can
also be used to identify AGNs since it can directly probe their high energy
emissions (Szokoly et al., 2004). Color-color diagnostics from optical data
(like IRAC) can also be used for identifying AGNs (Donley et al., 2012).
Classification can also be done using the q24 parameter, which is the ratio of
24 $\mu$m flux density to the effective 1.4 GHz flux density (Bonzini et al.,
2013). Based on the resuts of McAlpine et al. (2013), it was shown by
Magliocchetti et al. (2014) that the radio luminosity function for SFGs fall
of in a much steeper manner than AGNs for all redshifts, and this reduces the
chances of contamination in the two samples. Additionally, these different
multi-wavelength methods are not always consistent with each other, and a
detailed investigation into any such discrepancy is beyond the scope of this
work. Hence, only the radio luminosity criterion has been used for
classification.

The sources with redshifts up to 1.8 were classified into AGNs and SFGs
according to whether their luminosity is greater than or less than the
threshold in Equation 1 (with $\mathrm{P_{cross}}$ determined using Equation
2). At higher redshifts (i.e. >1.8), $\mathrm{P_{0,cross}}$ is fixed to 1023.5
[$\mathrm{WHz^{-1}sr^{-1}}$] (McAlpine et al., 2013). Of the 2397 sources,
1821 were classified as AGNs and 576 as SFGs using the radio luminosity
criteria. The median redshifts for AGNs and SFGs are 1.02 and 0.2
respectively.

Figure 5: Redshift distribution (N($z$)) for sources with either photometric
(HELP) or spectroscopic (SDSS) redshifts. The top panel shows N($z$)
distribution for all sources with the black dashed line indicating the median
redshift for all sources. The bottom shows the same for sources classified
into AGNs (blue curve) and SFGs (red curve) using the radio luminosity
criterion discussed in the text, with the cyan and magenta dashed lines
indicating the respective median redshifts.

##  3 Estimation of Correlation Function: Combined Sources

###  3.1 The Angular Correlation Function

The angular two-point correlation function $w(\theta)$ is used to quantify
clustering in the sky on angular scales. While several estimators have been
proposed in literature (for a comparison of the different types of estimators
see Kerscher et al. (2000) and Appendix B. of Siewert et al. (2020)), this
work uses the LS estimator proposed by Landy & Szalay (1993). It is defined as
:

$w(\theta)=\mathrm{\frac{DD(\theta)-2DR(\theta)+RR(\theta)}{RR(\theta)}}$ (3)

Here DD($\theta$) and RR($\theta$) are the normalised average pair count for
objects at separation $\theta$ in the original and random catalogues,
respectively. Catalogue realizations generated by randomly distributing
sources in the same field of view as the real observations have been used to
calculate RR($\theta$). The LS estimator also includes the normalized cross-
pair separation counts DR($\theta$) between original and random catalogue,
which has the advantage of effectively reducing the large-scale uncertainty in
the source density (Landy & Szalay, 1993; Hamilton, 1993; Blake & Wall, 2002a;
Overzier et al., 2003). The uncertainty in the determination of $w(\theta)$ is
calculated using the bootstrap resampling method (Ling et al., 1986), where
100 bootstrap samples are generated to quote the 16th and 84th percentile
errors in determination of $w(\theta)$.

####  3.1.1 Random Catalogue

The random catalogues generated should be such that any bias due to noise does
not affect the obtained values of the correlation function. The noise across
the entire $\mathrm{6^{\circ}\times 6^{\circ}}$ mosaic of the field is not
uniform (see Figure 3 of Mazumder et al. (2020)). This can introduce a bias in
estimating the angular two-point correlation function since the non-uniform
noise leads to the non-detection of fainter sources in the regions with higher
noise.

PYBDSF was used for obtaining the noise map of the image. Assuming the sources
follow a flux distribution of the form dN/dS $\propto$ $S^{-1.6}$ (Intema et
al., 2011; Williams et al., 2013), random samples of 3000 sources were
generated in the given flux range (with lower limit corresponding to 2 times
the background RMS of the image) and assigned random positions to distribute
them in the entire FoV. The sources constitute a mixture of 70% unresolved
sources and 30% extended sources, which is roughly in the same ratio as the
actual source catalogue (Mazumder et al., 2020). These were injected into the
residual map, and using the same parameters in PYBDSF as the ones used in the
extraction of the original sources (see Mazumder et al. (2020)), the random
catalogues were extracted. 100 such statistically independent realizations
were used to reduce the associated statistical uncertainty.

For clustering analysis of AGNs and SFGs, two sets of random catalogues were
generated uing the publicly available catalogues for these source types from
the T-RECS simulation (Bonaldi et al., 2018). These catalogues have source
flux densities provided at different frequencies between 150 MHz to 20 GHz.
The flux densities at 300 MHz were considered for the randoms. They were
scaled to 325 MHz using $\alpha=0.8$, and 2000 sources were randomly chosen
within flux density limit for the radio catalogue of AGNs and SFGs. They were
assigned random positions within the RA, Dec limits of the original catalogues
and injected into the residual maps. Then using the same parameters for PYBDSF
as the original catalogue, the sources were recovered. 100 such realizations
were done for AGNs and SFGs separately. The recovered random catalogues were
used for further clustering analysis of the classified populations. It should
also be mentioned here that the lower cut-off of flux density for the random
catalogues was $\sim$0.1 mJy, which is 2 times the background RMS. As already
seen from Mazumder et al. (2020), even a flux limit of 0.2 mJy (4 times the
background RMS) takes care of effects like the Eddington bias (Eddington,
1913). Thus 0.1 mJy is taken as the limiting flux for both the combined and
the the classified random catalogues. The final random samples for AGNs and
SFGs consisted of a total of $\sim$120000 sources each, while for the combined
sample, it was $\sim$200000\. This is much higher than the number of sources
in the radio catalogue. Thus, it does not dominate the errors. As has already
been stated, the point and extended sources in the random catalogues
(generated for the whole sample and the classified sources) are taken in the
same ratio as that of the original radio catalogue. The drawback of this
assumption is that there is a chance of underestimating extended sources in
the random catalogue, which may lead to spurious clustering signals at smaller
angular scales. However, since no evidence of any spurious signal is seen,
taking point and extended sources in the same ratio as the original catalogue
seem reasonable.

###  3.2 Angular Clustering Pattern at 325 MHz

The angular correlation function of the sources detected in this observation
is calculated using the publicly available code TreeCorr
888https://github.com/rmjarvis/TreeCorr(Jarvis et al., 2004). The 325 MHz
catalogue was divided into 15 equispaced logarithmic bins between $\theta\sim
36\arcsec$ to 2∘. The lower limit corresponds to the four times the PSF at 325
MHz, and the upper limit is the half-power beamwidth at this frequency. Figure
6 shows the angular correlation function of the 325 MHz in red circles; the
error bars are estimated using the bootstrap method as discussed earlier. A
power law of the form $w(\theta)=A\theta^{1-\gamma}$ is also fitted. The power
law index, $\gamma$ is kept fixed at the theoretical value of 1.8. The
parameter estimation for this fit is done using Markov chain Monte Carlo
(MCMC) simulation by generating 106 data points by applying the Metropolis-
Hastings algorithm in the $A$ parameter space. The first 102 samples have been
removed from the generated chains to avoid the burn-in phase. From the sampled
parameter space, $\chi^{2}$ is used to estimate the most likely values of the
parameters. The best fit parameters are
$\mathrm{log(A)=-2.73^{+0.11}_{-0.15}}$, with the error bars being the
1-$\sigma$ error bars from the 16th and 84th percentiles of the chain points.

Figure 6: Angular correlation function for all sources in the Lockman Hole
region at 325 MHz. The dashed black line is the best-fitting power law to
$w(\theta)$. Comparison has been done with best fitted power laws obtained
from previous studies of LoTSS (blue triangle, Siewert et al. 2020), TGSS-ADR
(green diamonds, Rana & Bagla 2019), FIRST (magenta thin diamonds, Lindsay et
al. 2014), ELAIS-N1 (cyan inverted triangles, Chakraborty et al. 2020), XMM-
LSS (orange thin diamonds, Hale et al. 2019), COSMOS (purple squares, Hale et
al. 2018), & Lockman Hole 1.4 GHz observation (maroon pentagons, Bonato et al.
2020). Different studies mentioned here have different flux density limits
(see Table 2).

####  3.2.1 Comparison with previous Observations

The best fit values obtained for parameters A and $\gamma$ of the 325 MHz
catalogue have been compared with those for other observations at radio
frequencies. The parameters obtained for different radio surveys, namely from
Lindsay et al. (2014); Hale et al. (2018, 2019); Rana & Bagla (2019);
Chakraborty et al. (2020); Bonato et al. (2020); Siewert et al. (2020) have
been summarised in Table 2. The best-fit estimate of the slope $\gamma$ for
the correlation function is found to be in reasonable agreement with the
theoretical prediction of Peebles (1980) and previous observations (for
example see Bonato et al. (2020); Siewert et al. (2020)). The scaled flux
limit at 325 MHz for the Bonato et al. (2020) catalogue at (originally 1.4
GHz, scaled using a spectral index of 0.8) is $\sim$0.4 mJy, very close to the
flux limit for this work. However, their estimates are higher than all
previous estimates (they particularly compare with (Magliocchetti et al.,
2017)), which they assign partly to the presence of sample variance. While the
area probed by (Bonato et al., 2020) is also included within the region this
work probes, the area covered are different, the one covered here being
larger. This might be the reason for differences between the estimates in this
work and (Bonato et al., 2020), despite both having similar flux density cut-
offs. The clustering amplitude for this work is similar to Hale et al. (2018)
at almost all the angular scales. One possible reason is that the flux limit
for the study at 3 GHz was 5.5 times the 2.3 $\mu$Jy $\mathrm{beam}^{-1}$
limit corresponding to a flux of $\sim$0.1 mJy at 325 MHz, which is near the
flux cut-off for this work (0.3 mJy), and thus can trace similar halo masses
and hence clustering amplitudes.

Table 2: Clustering Parameters for Observed Data. The columns indicate the name of the survey (Observation), observing frequency in MHz (Frequency), the flux density cut-off at the observing frequency (Scut,ν), the equivalent 325 MHz flux-density (Scut,325), best fit clustering amplitude ($\mathrm{log_{10}(A)}$) and best fit power-law index ($\gamma$) respectively. Observation | Frequency | S${}_{\mathrm{cut,\nu}}^{\dagger}$ | S${}_{\mathrm{cut,325}}^{*}$ | $\mathrm{log_{10}(A)}$ | $\gamma$ | Reference
---|---|---|---|---|---|---
| (MHz) | (mJy) |  |  |  |
FIRST | 1400 | 1.00 | 3.21 | -2.30${}^{\mathrm{{}^{+0.70}_{-0.90}}}$ | 1.82$\pm$.02 | Lindsay et al. (2014)
COSMOS | 3000 | 0.013 | 0.08 | -2.83${}^{\mathrm{{}^{+0.10}_{-0.10}}}$ | 1.80 | Hale et al. (2018)
XMM-LSS | 144 | 1.40 | 0.73 | -2.08${}^{\mathrm{{}^{+0.05}_{-0.04}}}$ | 1.80 | Hale et al. (2019)
TGSS-ADR | 150 | 50 | 26.9 | -2.11${}^{\mathrm{{}^{+0.30}_{-0.30}}}$ | 1.82$\pm$.07 | Rana & Bagla (2019)
ELAIS-N1 | 400 | 0.10 | 0.12 | -2.03${}^{\mathrm{{}^{+0.10}_{-0.08}}}$ | 1.75$\pm$0.06 | Chakraborty et al. (2020)
Lockman Hole | 1400 | 0.12 | 0.39 | -1.95${}^{\mathrm{{}^{+0.005}_{-0.005}}}$ | 1.96$\pm$.15 | Bonato et al. (2020)
LoTSS | 144 | 2.00 | 1.04 | -2.29${}^{\mathrm{{}^{+0.6}_{-0.6}}}$ | 1.74$\pm$.16 | Siewert et al. (2020)
Lockman Hole | 325 | 0.30 | 0.30 | -2.73$\mathrm{{}^{+0.11}_{-0.15}}$ | 1.80 | This work

The clustering properties of the radio sources in the VLA-FIRST survey (Becker
et al., 1995) has been reported in Lindsay et al. (2014), where
$\mathrm{log(A)}$ is -2.30${}^{\mathrm{{}^{+0.70}_{-0.90}}}$. Hale et al.
(2018) and Hale et al. (2019) have reported $\mathrm{log(A)}$ value of -2.83
and -2.08 for the COSMOS and XMM-LSS fields, respectively, by fixing $\gamma$
at the theoretical value of 1.80. The clustering amplitude of the 150 MHz
TGSS-ADR (Intema et al., 2017) has been shown by Rana & Bagla (2019) for a
large fraction of the sky and at different flux density cut-offs. In the
recent deep surveys of the ELAIS-N1 field at 400 MHz (Chakraborty et al.,
2020), $\mathrm{log(A)}$ and the best fit power law index have the values
-2.03${}^{\mathrm{{}^{+0.10}_{-0.08}}}$ and 1.75$\pm$0.06 respectively.

Comparison has also been made with the wide-area survey of LoTSS data release
1 (Siewert et al., 2020). This study (with data obtained at a central
frequency of 144 MHz) employed various masks on the data to obtain the angular
clustering values. The survey covers a wider area, but the flux cut-off
threshold is above 1 mJy for all of the masks due to systematic uncertainties.
A wide range of angles, 0.1${}^{\circ}\leq\theta\leq$ 32∘ was fixed to
determine the angular clustering. Taking three different flux density limits-
at 1, 2 and 4 mJy and different masks, the values of $\mathrm{log(A)}$ and
power-law index were obtained(the fitting for the power-law form was done for
0.2${}^{\circ}\leq\theta\leq$ 2∘). Siewert et al. (2020) have applied various
flux density cuts and masks to their sample for obtaining the angular
clustering parameters. They have concluded that the flux density cut-off of 2
mJy provides the best estimate for the angular clustering parameters, and the
same has been used here for comparison. Comparison of the present work with
LoTSS 2 mJy flux cut shows that the values of $\mathrm{log(A)}$ agree well.
The best fit power-law index is also consistent within 1$\sigma$ error bars.
Hence, it is seen that the angular correlation function obtained in the
present work gives values for the parameters $\mathrm{log(A)}$ and $\gamma$
consistent with those reported in previous surveys. Additionally, since this
survey has both wider coverage than the recent EN1 data and a lower flux
density threshold than the LoTSS data used by Siewert et al. (2020), it
provides an intermediate data set along a different line of sight to probe
cosmology.

###  3.3 The Spatial Correlation Function at 325 MHz

For known angular clustering $w(\theta)$, the spatial clustering of sources is
quantified by the two-point correlation function $\xi(r)$. Using the Limber
inversion (Limber, 1953), $\xi(r)$ can be estimated for known redshift
distribution. Gravitational clustering causes the spatial clustering to vary
with redshift, and thus a redshift dependent power-law spatial correlation
function can be defined as (Limber, 1953; Overzier et al., 2003):

$\mathrm{\xi(r,z)}=\mathrm{(r_{0}/r)^{\gamma}(1+z)^{\gamma-(3+\epsilon)}}$ (4)

where the clustering length r is in comoving units, $\epsilon$ specifies
clustering models (Overzier et al., 2003) and $\mathrm{r_{0}}$ is the
clustering length at z=0. For this work, comoving clustering model, in which
the correlation function is unchanged in the comoving coordinate system and
with $\epsilon$ = $\gamma$-3, is used. The comoving cluster size is constant.
The correlation length is calculated using (Peebles, 1980):

$\mathrm{A}=\mathrm{r_{0}^{\gamma}H_{\gamma}(H_{0}/c)\frac{\int_{0}^{\infty}N^{2}(z)(1+z)^{\gamma-(3+\epsilon)}\chi^{1-\gamma}(z)E(z)dz}{[\int_{0}^{\infty}N(z)dz]^{2}}}$
(5)

where
$\mathrm{H_{\gamma}=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{\gamma-1}{2})}{\Gamma(\frac{\gamma}{2})}}$,
$\mathrm{E(z)=\sqrt{\Omega_{m,0}(1+z)^{3}+\Omega_{k,0}(1+z)^{2}+\Omega_{\Lambda,0}}}$
is the cosmological factor, N(z) is the redshift distribution of the sources
and $\mathrm{\chi(z)}$ is the line of sight comoving distance. Equation 5 can
be used to estimate $\mathrm{r_{0}}$ using the angular clustering amplitude A
and the redshift distribution shown in Figure 5.

The theoretical value of 1.8 for $\gamma$, as predicted by Peebles (1980) is
consistent with the values across various surveys, as well as within 2
$\sigma$ of the current analysis (tabulated in Table 2). Thus the theoretical
value of $\gamma$, the distribution of A obtained from the MCMC distribution
discussed in 3.2 and the combined redshift distribution distribution discussed
in Section 2.2 are used to estimate the value of $\mathrm{r_{0}}$. Figure 7
shows the probability distribution function (PDF) of the spatial clustering
length. As already mentioned, the median redshift of the samples is
$\sim$0.78, and at this redshift, the median value of $\mathrm{r_{0}}$ is
3.50$\mathrm{{}^{+0.50}_{-0.50}}$ Mpc h-1, where the errors are the 16th and
84th percentile errors.

Figure 7: Probability distribution function of spatial clustering length
($r_{0}$) for the entire sample at 325 MHz.

###  3.4 The Bias Parameter

The bias parameter is used to quantify the relation between the clustering
property of luminous sources and the underlying dark matter distribution. The
ratio of the galaxy to the dark matter spatial correlation function is known
as the scale-independent linear bias parameter $b(z)$ (Kaiser, 1984; Bardeen
et al., 1986; Peacock & Smith, 2000). For cosmological model with dark matter
governed only by gravity, following Lindsay et al. (2014); Hale et al. (2018);
Chakraborty et al. (2020), $b(z)$ is calculated as :

$b(z)=\Bigg{(}\frac{r_{0}(z)}{8}\Bigg{)}^{\gamma/2}\frac{J_{2}^{1/2}}{\sigma_{8}D(z)/D(0)}$
(6)

where $J_{2}$ = 72/[(3-$\gamma$)(4-$\gamma$)(6-$\gamma$)2γ], $D(z)$ is the
linear growth factor, calculated from CMB and galaxy redshift information
(Eisenstein et al., 1999), and $\sigma_{8}^{2}$ is the amplitude of the linear
power spectrum on a comoving scale of 8 Mpc $\rm h^{-1}$.

For this work, the bias parameter has been calculated using the median
redshift value of the r0 distribution with the 16th and 84th percentile
errors. The value of the bias parameter $b(z)$ at $z$=0.78 is
2.22$\mathrm{{}^{+0.33}_{-0.36}}$.

##  4 Estimation of Correlation Function: AGNs and SFGs

This section discusses the angular and spatial correlation scales and the bias
parameter obtained for the two separate populations of sources (i.e., AGNs and
SFGs). The obtained values are also compared with previously reported values
using radio and other bands data. Following a similar procedure as that done
for the entire population, initially, the angular clustering was calculated,
and a power law of the form $A\theta^{1-\gamma}$ was fitted. The best value of
clustering amplitude $A$ is determined, once again keeping $\gamma$ fixed at
the theoretical value of 1.8 for both AGNs and SFGs populations. Figure 8
shows the angular correlation function of AGNs (left panel) and SFGs(right
panel). Using the MCMC simulations as discussed previously, the clustering
amplitudes, $\mathrm{log(A)}$ have values $\textrm{-}2.18^{+0.20}_{-0.20}$ and
$\textrm{-}1.69^{+0.10}_{-0.10}$ respectively for AGNs and SFGs. The results
of the fit and the subsequent values of clustering length and bias parameter
obtained here and results from previous surveys in radio wavelengths are also
tabulated in Table 3.

The spatial clustering length and bias parameter bz for the AGNs with
$z_{median}$=1.02 are $8.30^{+0.96}_{-0.91}$ Mpc $\rm h^{-1}$ and
$3.74^{+0.39}_{-0.36}$. For SFGs with $z_{median}\approx$0.20, the values are
$\mathrm{r_{0}}$= $3.22^{+0.34}_{-0.32}$ Mpc $\rm h^{-1}$ and
bz=$1.06^{+0.1}_{-0.1}$. It is seen that the spatial clustering length and
consequently the bias factor for AGNs is more than SFGs, which implies that
the latter are hosted by less massive haloes, in agreement with previous
observations (Gilli et al., 2009, 2007; Starikova et al., 2012; Dolley et al.,
2014; Magliocchetti et al., 2017; Hale et al., 2018; Chakraborty et al.,
2020).

Figure 8: Angular correlation function for sources classified as AGNs (left
panel) and SFGs (right panel). The slope of the best fit function $\gamma$ is
fixed at 1.8 and the distribution of correlation amplitude ($A$) is shown in
the inset of each panel.

###  4.1 Comparison with previous Observations

Table 3: Spatial Clustering Length and Bias Parameter from Different Observations. The columns are respectively name of the survey field, observing frequency in MHz, type of radio source (AGNs/SFGs), median redshift, angular clustering amplitude, spatial clustering length in Mpc h-1 & bias parameter value. Observation | Frequency | Source type | $\textrm{z}_{median}$ | $\mathrm{log_{10}(A)}$ | $\mathrm{r_{0}}$ | b${}_{z_{median}}$ | Reference
---|---|---|---|---|---|---|---
| (MHz) |  |  |  | (Mpc h-1) |  |
COSMOS | 3000 | AGNs | 0.70 | $-2.30^{+0.1}_{-0.1}$ | $6.9^{+0.60}_{-0.70}$ | $2.1^{+0.2}_{-0.2}$ | Hale et al. (2018)
|  | AGNs | 1.24 | $-2.60^{+0.1}_{-0.1}$ | $9.6^{+0.70}_{-0.70}$ | $3.6^{+0.2}_{-0.2}$ |
|  | AGNs | 1.77 | $-2.60^{+0.1}_{-0.1}$ | $7.3^{+0.90}_{-0.90}$ | $3.5^{+0.4}_{-0.4}$ |
|  | SFG | 0.62 | $-2.60^{+0.1}_{-0.1}$ | $5.0^{+0.50}_{-0.60}$ | $1.5^{+0.1}_{-0.2}$ |
|  | SFG | 1.07 | $-2.90^{+0.1}_{-0.1}$ | $6.1^{+0.60}_{-0.70}$ | $2.3^{+0.2}_{-0.2}$ |
VLA-COSMOS | 1400 | AGNs | 1.25 | $-2.79^{+0.1}_{-0.1}$ | $7.84^{+1.75}_{-2.31}$ | - | Magliocchetti et al. (2017)
|  | SFG | 0.50 | $-2.36^{+0.3}_{-0.3}$ | $5.46^{+1.12}_{-2.10}$ | - |
ELAIS N1 | 400 | AGNs | 0.91 | $-2.22^{+0.16}_{-0.16}$ | $7.30^{+1.4}_{-1.2}$ | $3.17^{+0.5}_{-0.5}$ | Chakraborty et al. (2020)
|  | SFG | 0.64 | $-2.16^{+0.05}_{-0.06}$ | $4.62^{+0.39}_{-0.40}$ | $1.65^{+0.14}_{-0.14}$ |
ELAIS N1 | 612 | AGNs | 0.85 | $-2.30^{+0.02}_{-0.03}$ | $6.0^{+1.5}_{-1.3}$ | $2.6^{+0.6}_{-0.5}$ | Chakraborty et al. (2020)
|  | SFG | 0.87 | $-2.19^{+0.01}_{-0.02}$ | $4.16^{+0.7}_{-0.8}$ | $1.59^{+0.2}_{-0.2}$ |
Lockman Hole | 325 | AGNs | 1.02 | $-2.18^{+0.20}_{-0.20}$ | $8.30^{+0.96}_{-0.91}$ | $3.74^{+0.39}_{-0.36}$ | This work
|  | SFG | 0.20 | $-1.65^{+0.1}_{-0.1}$ | $3.22^{+0.34}_{-0.32}$ | $1.06^{+0.10}_{-0.10}$ |

Figure 9: (left panel) Spatial correlation length ($\mathrm{r_{0}}$) for AGNs
as a function of redshift (red square). For comparison, the values for the
same obtained by Magliocchetti et al. (2004) (orange circle), Gilli et al.
(2005) (blue triangle), Gilli et al. (2009) (purple square), Magliocchetti et
al. (2017) (green diamonds), Hale et al. (2018) (cyan circles) & Chakraborty
et al. (2020) (magenta inverted triangle) have been plotted. Predictions from
SKADS (Wilman et al., 2008) have also been plotted for Radio Quiet Quasars
(RQQ), FR-I and FR-II radio galaxies. (right panel) Spatial correlation length
($\mathrm{r_{0}}$) SFGs as a function of redshift (red circle). For
comparison, values obtained by Gilli et al. (2007) (black circles), Starikova
et al. (2012)(maroon diamonds), Magliocchetti et al. (2013) (green inverted
triangles), Dolley et al. (2014) (blue diamonds), Magliocchetti et al. (2017)
(green circles), Hale et al. (2018) (cyan pentagon), & Chakraborty et al.
(2020) (magenta triangles) have been plotted. Predictions from SKADS (Wilman
et al., 2008) have also been plotted for SFGs and Starburst galaxies (SB).

Figure 9 shows the observationally determined values from surveys at various
wavebands for $\mathrm{r_{0}}$ as a function of their redshift, while Figure
10 shows the same for the bias parameter. The left and right panels of Figure
9 are for AGNs and SFGs, respectively. Table 3 summarizes the values obtained
in radio surveys only, while Figures 9, 10 show the observed values for
surveys at radio as well as other wavebands, e.g. IR and X-Ray.

The clustering length for AGNs in this work is at z${}_{median}\approx$1.02 is
$\textrm{8}.30^{+0.96}_{-0.91}$ Mpc $\rm h^{-1}$. Using X-Ray selected AGNs in
the COSMOS field, Gilli et al. 2009 obtained clustering lengths at redshifts
upto $\sim$3.0. They divided their sample into a number of bins, to obtain
$\mathrm{r_{0}}$ at different median redshifts. For their entire sample,
taking slope of the angular correlation function as 1.80, $\mathrm{r_{0}}$ was
$\textrm{8}.39^{+0.41}_{-0.39}$ Mpc $\rm h^{-1}$, for a median redshift of
0.98. It is consistent with the value obtained here at a similar redshift. The
clustering length with this work is also consistent within error bars for AGNs
at 400 MHz and 610 MHz of Chakraborty et al. (2020). For their work, they
obtain an $\mathrm{r_{0}}$ value of $\textrm{7}.30^{+1.14}_{-1.12}$ Mpc $\rm
h^{-1}$ at $z\approx 0.91$, $\textrm{6}.00^{+1.5}_{-1.3}$ Mpc $\rm h^{-1}$ at
$z_{median}=0.84$. The clustering length estimates for radio selected AGNs in
the COSMOS field at 1.4 GHz (Magliocchetti et al., 2017) and 3 GHz (Hale et
al., 2018) observed with the VLA also agree within error bars with the
estimates obtained here. Magliocchetti et al. (2017) have found a clustering
length of $\textrm{7}.84^{+1.75}_{-2.31}$ Mpc $\rm h^{-1}$ at z $\approx$1.25
while (Hale et al., 2018) obtained $\textrm{6}.90^{+0.60}_{-0.70}$ Mpc $\rm
h^{-1}$, $\textrm{9}.60^{+0.70}_{-0.70}$ Mpc $\rm h^{-1}$ and
$\textrm{7}.30^{+0.90}_{-0.90}$ Mpc $\rm h^{-1}$ at $z\approx$ 0.70, 1.24,
1.77 respectively. Using X-ray selected AGNs in the CDFS field, Gilli et al.
(2005) obtained a value of $\textrm{1}0.30^{+1.7}_{-1.7}$ Mpc $\rm h^{-1}$ at
z $\approx$ 0.84. This value though higher than the values for radio selected
AGNs, is still consistent within error bars.

For the SFGs population (right panel of Figure 9), the median redshift is
0.20. At this redshift, the clustering length is
$\textrm{3}.22^{+0.34}_{-0.32}$ Mpc $\rm h^{-1}$. This estimate is at a
redshift lower than previous observations. An extensive study at mid-IR
frequency has been done by Dolley et al. (2014) for SFGs. The lowest redshift
probed in their study is 0.31, where $\mathrm{r_{0}}$ is
$\textrm{3}.41^{+0.18}_{-0.18}$ Mpc $\rm h^{-1}$. Thus, the value is
consistent with that obtained here at a nearby redshift. Magliocchetti et al.
(2013) studied the clustering of SFGs using the Herschel PACS Evolutionary
Probe observations of the COSMOS and Extended Groth Strip fields. They found
clustering lengths for SFGs out to $z\approx$ 2\. For the ELAIS-N1 field at
400 MHz and 610 MHz, Chakraborty et al. (2020) reported clustering length of
$\textrm{4}.62^{+0.39}_{-0.40}$ Mpc $\rm h^{-1}$ and
$\textrm{4}.16^{+0.70}_{-0.80}$ Mpc $\rm h^{-1}$ at redshifts 0.64 and 0.87
respectively. The 3 GHz COSMOS field studies of Hale et al. (2018) gave
clustering lengths $\textrm{5}.00^{+0.50}_{-0.60}$ Mpc $\rm h^{-1}$ and
$\textrm{6}.1^{+0.60}_{-0.70}$ Mpc $\rm h^{-1}$ respectively at z$\approx$
0.62 and 1.07. The mid-IR selected samples for Lockman Hole give
$\mathrm{r_{0}}$ values $\textrm{4}.98^{+0.28}_{-0.28}$ Mpc and
$\textrm{8}.04^{+0.69}_{-0.69}$ Mpc $\rm h^{-1}$ at $z\approx$ 0.7 and 1.7
respectively. Similarly, the mid-IR sample for Gilli et al. (2007) has
clustering lengths $\mathrm{r_{0}}$ is $\textrm{4}.25^{+0.12}_{-0.12}$ Mpc
$\rm h^{-1}$ and $\textrm{3}.81^{+0.10}_{-0.10}$ Mpc $\rm h^{-1}$ for
$z\approx$ 0.67 and 0.73.

The results have also been compared with the assumed bias models of the semi-
empirical simulated catalogue of the extragalactic sky, the Square Kilometer
Array Design Studies (referred to as SKADS henceforth, Wilman et al. 2008).
This simulation models the large-scale cosmological distribution of radio
sources to aid the design of next-generation radio interferometers. It covers
a sky area of $20^{\circ}\times 20^{\circ}$, with sources out to a
cosmological redshift of $z\sim$20 and a minimum flux 10 nJy at 151, 610 MHz &
1.4, 4.86 and 18 GHz. The simulated sources are drawn from observed and, in
some cases, extrapolated luminosity functions on an underlying dark matter
density field with biases to reflect the measured large-scale clustering. It
uses a numerical Press–Schechter (Press & Schechter, 1974) style filtering on
the density field to identify clusters of galaxies. The SKADS catalogue has
been used here for statistical inference of the spatial and angular clustering
variations of the sources with redshift. It should be mentioned here that the
T-RECS catalogue (Bonaldi et al., 2018) incorporates more updated results from
the recent observations. However, the evolution of the bias parameter and
clustering length with redshift is not available for the same; hence SKADS has
been used.

The bias parameter for AGNs and SFGs at zmedian 1.02 and 0.20 is
$3.74^{+0.39}_{-0.36}$ and $1.06^{+0.10}_{-0.10}$ respectively. Although the
value for AGNs is slightly higher than those obtained by Hale et al. (2018)
and Chakraborty et al. (2020), it is still with reasonable agreement with the
SKADS for FR-I galaxies. Comparison with the population distribution of the
SKADS simulation of Wilman et al. (2008), in terms of both clustering length
and the bias parameter (solid magenta pentagons in Figure 10) show that the
AGN population is dominated by FR-I type galaxies hosted in massive haloes
with $\sim M_{h}$= 5$\times$10${}^{13}\rm h^{-1}M_{\odot}$. It can also seen
from Figure 10, that the mass of the haloes hosting the SFG samples of the
current sample is $\sim M_{h}$= 3$\times 10^{12}\rm h^{-1}M_{\odot}$. Thus, it
is seen that the SFGs have a lower range of halo masses compared to AGNs,
which implies that the latter inhabits more massive haloes and are more biased
tracers of the dark matter density field.

Figure 10: Bias parameter $b_{z}$ for whole population (black inverted
triangle) as well AGNs (blue pentagons) and SFGs (magenta squares). Bias
parameters from previous observations by Lindsay et al. (2014) (purple
inverted traingles), Dolley et al. (2014) (blue diamonds), Hale et al. (2018)
(cyan triangles for AGNs and cyan circles SFGs) & Chakraborty et al. (2020)
(magenta triangles and circles for AGNs and SFGs respectively) are shown.
Predictions from SKADS is also shown by the continuous curves.

##  5 Discussion

The analysis of clustering properties of radio selected sources in the Lockman
Hole region presented in this work is one of the first results reported at 325
MHz. Similar studies were previously done at 400 MHz for the ELAIS-N1 field
(Chakraborty et al., 2020), however for a much smaller area. Beside analysing
at a mostly unexplored frequency, this work also presents a comparatively
large area with a significant number of sources. Previous clustering study of
the same area using 1.4 GHz data from WSRT by Bonato et al. (2020) used 1173
sources with a flux density cut-off of 0.12 mJy at 1.4 GHz (or 0.4 mJy at 325
MHz). Their obtained clustering amplitude is slightly higher than previous
surveys, as acknowledged by the authors. However, further investigation is
required to ascertain the reason for the deviation. Clustering analysis was
also done using the recent LoTSS observation of the HETDEX spring field
(Siewert et al., 2020). The clustering analyses were produced with many flux
density cut-offs and masks, and the most reliable estimate was for a flux
density limit of 2 mJy at 150 MHz (or $\sim$1.0 mJy at 325 MHz). The
clustering amplitude estimate at 2 mJy limit for LoTSS is consistent with that
obtained here within error bars. As seen from Figure 6, the clustering
amplitude obtained here agrees with previous observations. The slightly higher
values for the bias parameter of the AGNs for this work, compared to that of
Hale et al. (2018) and Chakraborty et al. (2020), may be attributed to the
different flux limits of the studies. This implies that each of these
observations are probing slightly different populations of sources (with
slightly different luminosities as discussed later). Nevertheless, as seen
from Figure 10, the values are broadly consistent with each other.

The angular clustering amplitude for this work as shown in Figure 8 agree with
previous observations, as seen in Table 3. The angular clustering of sources
for this work are calculated with TreeCorr using the default values for most
parameters. It has been shown in Siewert et al. (2020) that using default
value of 1 for the parameter bin_slop gives less accurate results than for
values $\leq$1\. Siewert et al. (2020) obtained the most accurate values for
bin_slop=0. They also showed that angular clustering amplitudes deviate
largely from precise values (calculated from a separate brute-force algorithm,
see (Siewert et al., 2020) for details) at angular scales $\gtrsim$1∘.
However, the computation times also significantly increased for bin_slop=0.
The use of default parameters in this work might be the cause slight
oscillation of the correlation function seen around the best fit curves.
Nevertheless, since results obtained here are in reasonable agreement with the
previous observations and owing to constraints in the available computing
power, the default values have been used.

The clustering lengths and bias parameters obtained here also agree with
previous studies, as evident from Table 3 and Figures 9, 10. Comparison of
bias parameter with SKADS simulation Wilman et al. 2008 shows that the
expected mass for dark matter haloes hosting AGNs is orders of magnitude
higher than that for SFGs. This trend is consistent with previous observations
(see for example Magliocchetti et al. 2017; Hale et al. 2018; Chakraborty et
al. 2020). Studies on the luminosity of the AGNs and SFGs suggest that it is
correlated with source clustering, and hence with the bias parameter and host
halo mass (Hale et al., 2018). Using observations of high and low luminosity
AGNs in the COSMOS field, Hale et al. (2018) showed that the luminosity and
clustering are correlated, with the higher luminosity AGNs residing in more
massive galaxies (Jarvis et al., 2001). Thus, they are hosted by more massive
haloes. Again, this points to AGNs (which are in general more luminous than
SFGs) being hosted by more massive haloes. However, it should also be
mentioned that the two population of AGNs in Hale et al. (2018) studied are at
different redshifts, with the low excitation population studied at $z\lesssim
0.65$. So it is possible that this population may evolve into higher mass
haloes at higher redshifts. Moreover, some works (for example Mendez et al.
2016) do not find any relation between clustering and luminosity.
Nevertheless, following Hale et al. (2018), studies using a larger population
of samples covering a large range of luminosities is required for probing the
relationship with clustering.

It is also observed in Figure 10 that the bias parameter for SFGs (solid blue
square) is higher than the SKADS predicted values for this population. This
trend is consistent with that observed in previous studies of Hale et al.
(2018) (cyan circles) and Chakraborty et al. (2020) (light magenta triangles).
The trend observed in Dolley et al. (2014) (light blue diamonds) is almost
similar as well. There may be two reasons that cause this variation-
contamination of the SFG sample by star-burst (SB) galaxies or underestimation
of halo mass for SKADS. If there exists a few SB samples in the SFG
population, comparison with SKADS (blue dotted curve in Figure 10) shows that
the overall value for bias (as well as $\mathrm{r_{0}}$) will be higher than
an uncontaminated sample. However, the most likely reason remains the second
one, the halo mass used in the SKADS is not a correct representation, which
has also been hinted at by comparing the values obtained from previous
observations (for instance Dolley et al. (2014); Hale et al. (2018);
Chakraborty et al. (2020)) in Figure 10. However, it is also seen from Figure
9 spatial clustering length for SFGs agrees with SKADS. The exact reason for
agreement of $\mathrm{r_{0}}$ and disagreement of $b(z)$ for SFGs in this work
with SKADS is unclear, and will be investigated in detail in later works.

Analyses like the one presented here are important for fully understanding how
bias scales with redshift as well as with source properties like luminosity.
This is important for cosmology, since the bias relates to dark matter
distribution, and is thus essential for understanding the underlying
cosmological parameters that define the Universe.

It should be mentioned here that the current study also has certain
limitations. Due to unknown systematics at large scales, and lack of optical
matches, the entire observed field could not be utilised for this study.
Additionally, the source classification is done based solely on the radio
luminosity of the sources. As has already been mentioned previously,
Magliocchetti et al. (2014) showed that the chances of contamination between
the populations is very less using this criteria. Nonetheless, there are
several other methods that can also be used to classify AGNs and SFGs in a
sample (detailed in Section 2.3). Future works will present a more detailed
analysis using the different multi-wavelength classification schemes
available.Such multi-frequency studies, combined with the present work and
similar studies with other fields will enhance the knowledge of the
extragalactic sources and provide more insights into the processes governing
their formation and evolution.

##  6 Conclusion

This work investigates the higher-order source statistics, namely the angular
and spatial clustering of the sources detected in the Lockman Hole field. The
data was observed by the legacy GMRT at 325 MHz. The details of data analysis
and catalogue extraction are discussed in Mazumder et al. (2020). The initial
step involved merging the multi-component sources present in the raw
catalogue. The resultant catalogue was cross-matched with SDSS and HELP
catalogues to identify sources with either spectroscopic or photometric
redshift information. A region of radius 1.8∘ around the phase center was
selected for optical identifications, yielding $\sim$95% matches. All the
sources with redshift distribution were separated into AGN and SFG populations
using the criterion for radio luminosity. The angular correlation function was
determined for the combined population for separation between $36\arcsec$ to
2∘. A power law fitted to this function, keeping a fixed power law index of
$\mathrm{\gamma=1.80}$, as estimated theoretically (Peebles, 1980). This gave
the value of clustering amplitude $\mathrm{log_{10}(A)}=-2.73^{0.11}_{-0.15}$.

The source population was further divided into AGNs and SFGs based on their
radio luminosity, and clustering analyses were done for these populations as
well. Using the redshift information and the clustering amplitude, spatial
correlation length was determined using Limber inversion for the AGNs and
SFGs. The correlation length and bias parameters have been obtained for the
full sample, as well as the classified AGN and SFG population. For the full
sample at z${}_{median}\approx$ 0.78, $\mathrm{r_{0}}$ =
3.50$\mathrm{{}^{+0.50}_{-0.50}}$ Mpc h${}^{\textrm{-1}}$ and $b(z)$ =
2.22$\mathrm{{}^{+0.33}_{-0.36}}$. For AGNs, the values are $\mathrm{r_{0}}$ =
8.30$\mathrm{{}^{+0.96}_{-0.91}}$ Mpc h-1 and $b(z)$ =
3.74$\mathrm{{}^{+0.39}_{-0.36}}$ at z${}_{median}\approx$ 1.02. At
z${}_{median}\approx$ 0.20, SFGs have values $\mathrm{r_{0}}$ =
3.22$\mathrm{{}^{+0.34}_{-0.32}}$ Mpc h-1 and $b(z)$ =
1.06$\mathrm{{}^{+0.10}_{-0.10}}$. The clustering length for AGNs is
reasonably consistent with Gilli et al. (2009) as well as Chakraborty et al.
(2020). For SFGs, the values are consistent with Dolley et al. (2014).

The obtained values have also been compared with the SKADS simulation of
Wilman et al. (2008). The comparative analysis suggests that the AGNs are
dominated by FR-I galaxies, with host dark matter halo masses of $M_{h}$=5-6
$\times$ 10${}^{13}h^{-1}M_{\odot}$. For SFGs, the estimated halo mass
obtained from SKADS is lower compared to the value $\sim M_{h}$=3 $\times
10^{12}h^{-1}M_{\odot}$ obtained here. The halo mass obtained here are in
agreement with previous literature (Dolley et al., 2014; Hale et al., 2018;
Chakraborty et al., 2020). It is worthwhile to mention that while the current
classifications are based on radio luminosity for each source alone, the
results are in agreement with clustering properties for populations of AGNs
and SFGs in X-Ray and mid-IR surveys as well. However, there are some
deviations from the predictions of the SKADS simulation. This deviation, seen
in other observations as well, emphasizes the need for wider and deeper low-
frequency observations. These will be able to constrain the host properties
better, leading to better models formation and evolution of the different
sources, and better understanding of the distribution of these populations
over space and time (redshift).

The study done in this work aims to characterize the clustering and bias of an
observed population of radio-selected sources, both as a combined population
and as distinct classes of sources (namely AGNs and SFGs). This work is the
first to report the clustering properties of radio-selected sources at 325
MHz. Thus, current data, being at a frequency with little previous clustering
study, bridges the gap between low frequency and high-frequency studies. It
also has the advantage of covering a wider area than many recent studies with
moderately deep RMS values, thus probing a larger number of sources with
fluxes at the sub-mJy level ($\sim$0.3 mJy). More such studies using
observational data are required for constraining cosmology and probing how
different source populations are influenced by their parent halos and how they
evolve with time (redshift). Additionally, for sensitive observations of
CD/EoR and post-EoR science, accurate and realistic models for compact source
populations that comprise a significant fraction of foregrounds are required.
Studies on the effect of imperfections in foreground modeling in the power
spectrum estimates will require detailed observational studies of source
position and flux distributions. For realistic estimates, second-order
statistics like clustering also cannot be ignored. Thus many more analyses
like the one done in this paper will be required for better understanding the
effects of the interplay between the various cosmological parameters on the
different populations of sources and putting constraints on said parameters.
Sensitive large area surveys like the MIGHTEE (Jarvis et al., 2016) with the
MeerKat telescope, LoTSS (Shimwell, T. W. et al., 2017; Tasse et al., 2021;
Sabater, J. et al., 2021) with the LOFAR, EMU (Norris et al., 2011, 2021) with
the ASKAP, as well as those to be done with the upcoming SKA-mid telescope
would provide wider as well as deeper data for doing cosmology. The current
work demonstrates that even instruments like the legacy GMRT can provide
reasonable data depth and coverage for cosmological observations.

## ACKNOWLEDGEMENTS

AM would like to thank Indian Institute of Technology Indore for supporting
this research with Teaching Assistantship. AM further acknowledges Akriti
Sinha for helpful suggestions and pointing to the HELP catalogue and Sumanjit
Chakraborty for helpful discussions. The authors thank the staff of GMRT for
making this observation possible. GMRT is run by National Centre for Radio
Astrophysics of the Tata Institute of Fundamental Research. The authors also
acknowledge SDSS for the spectroscopic redshift catalogues. Funding for SDSS-
III has been provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Science Foundation, and the U.S. Department of
Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/. The
authors thank the anonymous reviewer for their thorough review that has helped
to improve the quality of the work.

## Data Availability

The raw data for this study is available in the GMRT archive
(https://naps.ncra.tifr.res.in/goa/data/search). The spectroscopic redshift
data is available from https://skyserver.sdss.org/casjobs/, and the
photometric redshift data is available from https://hedam.lam.fr/HELP/.The
PYBDSF catalogue used here (accompanying Mazumder et al. (2020)) is available
on VizieR at https://cdsarc.cds.unistra.fr/viz-bin/cat/J/MNRAS/495/4071.

## Software

This work relies on the Python programming language (https://www.python.org/).
The packages used here are astropy (https://www.astropy.org/; Astropy
Collaboration et al. (2013); Price-Whelan et al. (2018)), numpy
(https://numpy.org/), scipy (https://www.scipy.org/), matplotlib
(https://matplotlib.org/), TreeCorr (https://github.com/rmjarvis/TreeCorr).

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