11institutetext: Hamburger Sternwarte, University of Hamburg, Gojenbergsweg
112, 21029 Hamburg, Germany<EMAIL_ADDRESS>22institutetext:
Thüringer Landessternwarte, Sternwarte 5, D-07778 Tautenburg, Germany
33institutetext: Depto. de Astronomía, Univ. de Guanajuato, Callejón de
Jalisco s/n, Guanajuato, C.P. 36023, GTO, Mexico<EMAIL_ADDRESS>44institutetext: Sterrewacht Leiden, University of Leiden, 2300 RA, Leiden,
The Netherlands 55institutetext: Kapteyn Astronomical Institute, University of
Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands

# Giant radio galaxies in the LOFAR deep fields⋆

M. Simonte 11 H. Andernach 2233 M. Brüggen 11 G. K. Miley 44 P. Barthel 55

(Accepted ???. Received ???; in original form ???)

###### Abstract

Context. The reason why some radio galaxies (RGs) grow to form so-called giant
radio galaxies (GRGs) with sizes $>$ 700 kpc, is still unknown.

Aims. In this study, we compare the radio, optical and environmental
properties of GRGs with those of a control sample of smaller RGs we found in
the three LOw-Frequency ARray (LOFAR) deep fields, namely the Boötes,
ELAIS-N1, Lockman Hole, for a total area of $\approx$ 95 deg2.

Methods. We inspected the LOFAR deep fields and created a catalogue of 1609
extended radio galaxies (ERGs). By visual inspection, we identified their host
galaxies and spectroscopically or photometrically classified 280 of these as
GRGs. We studied their properties, such as their accretion state, stellar mass
and star formation rate (SFR) using deep optical and infrared survey data.
Moreover, we explored the environment in terms of the surface number density
of neighbouring galaxies within these surveys. Integrated flux densities and
radio luminosities were also determined for a subset of ERGs through available
survey images at 50, 150, 610, and 1400 MHz to compute integrated spectral
indices.

Results. Considering the fraction of GRGs displaying an FRII morphology
alongside the host galaxy properties, we suggest that GRGs consistently
possess sufficient power to overcome jet frustration caused by the
interstellar medium. Moreover, clear differences emerge in the environmental
densities between GRGs and smaller RGs, using the number of neighbouring
galaxies within 10 Mpc from the host galaxy as a proxy. GRGs preferentially
reside in sparser environments compared to their smaller counterparts. In
particular, only 3.6% of the GRGs reside within a 3D comoving distance of 5
Mpc from a previously reported galaxy cluster. We found that larger sources
exhibit steeper integrated spectral indices, suggesting that GRGs are late-
stage versions of RGs. These results suggest that GRGs are amongst the oldest
radio sources with the most stable nuclear activity that reside in sparse
environments.

###### Key Words.:

galaxies: active – galaxies: jets – radio continuum: galaxies

##  1 Introduction

††⋆ The full catalogue of the 1609 ERGs is only available in electronic form
at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/

The study of radio galaxies (RGs) can provide insights into the intricate
interplay between active galactic nuclei (AGN), their host galaxies and the
surrounding intergalactic medium (Magliocchetti 2022). Among the diverse
population of RGs, giant radio galaxies (GRGs) have a linear extent larger
than 700 kpc (Willis et al. 1974; Barthel et al. 1985; Kuźmicz et al. 2018).

The advent of many radio surveys in the past, such as the Faint Images of the
Radio Sky at Twenty-cm (FIRST, Becker et al. 1995), Westerbork Northern Sky
Survey (WENSS, Rengelink et al. 1997), National Radio Astronomy Observatory
(NRAO) VLA Sky Survey (NVSS, Condon et al. 1998), Sydney University Molonglo
Sky Survey (SUMSS, Mauch et al. 2003) and the recent Rapid ASKAP Continuum
Survey (RACS, McConnell et al. 2020) led to the discovery of about 1000 GRGs
(Ishwara-Chandra & Saikia 1999a; Lara et al. 2001; Schoenmakers et al. 2000b;
Machalski et al. 2001; Saripalli et al. 2005; Kuźmicz & Jamrozy 2012, 2021;
Kuźmicz et al. 2018; Dabhade et al. 2017, 2020a; Brüggen et al. 2021;
Andernach et al. 2021; Gürkan et al. 2022; Mahato et al. 2022). Furthermore,
the LOw-Frequency ARray (LOFAR, van Haarlem et al. 2013), with its relatively
high resolution and sensitivity to very low surface brightness sources,
heralds a new era in the study of very large and high redshift radio galaxies.
The LOFAR Two-metre Sky Survey (LoTSS Shimwell et al. 2019, 2022) has yielded
the discovery of approximately 10000 GRGs (Dabhade et al. 2020b; Simonte et
al. 2022; Oei et al. 2023a, Mostert et al. 2023). Nevertheless, an explanation
for the Mpc size of the GRGs is still missing.

Oei et al. (2023a) carried out a detailed study of the distribution of the
linear size of about 2000 GRGs, mainly located at z $<$ 0.4, revealing a
relatively steep (power index = $3.5\pm 0.5$) cumulative power-law
distribution. Based on this finding, the authors estimated the number density
of GRGs to be $5\pm 2(100\rm\leavevmode\nobreak\ Mpc)^{-3}$, emphasising they
are truly rare in the local universe. More comprehensive studies of GRGs,
extending beyond the local universe, found that GRGs are less rare (Simonte et
al. 2022, Mostert et al. 2023.) than previously thought. Simonte et al. (2022)
compared the distribution of the linear size of 74 GRGs with that of a sample
of RGs from Miraghaei & Best (2017). The analysis indicated that the
exponential distribution provides an acceptable description of both samples.
However, none of these studies could firmly conclude whether GRGs represent
the tail of the distribution of the RGs largest linear size.

Models of RGs predict that only very few combinations of jet power,
environment, properties of the host galaxy and age of the source can reproduce
the large size of GRGs (Turner & Shabala 2015; Hardcastle 2018; Turner et al.
2023). The goal of this paper is to constrain the relative abundance of GRGs
compared to smaller RGs. Moreover, we study the properties of GRGs and their
host galaxies in order to refine models that describe the evolution of RGs.
The comparison of the properties of GRGs and smaller RGs can highlight
similarities and differences between the two populations. Like smaller RGs,
GRGs are hosted by massive elliptical galaxies (Lara et al. 2001; Dabhade et
al. 2017). The host of the RGs can be classified as High-Excitation (or
radiatively efficient) Radio Galaxies (HERG) and Low-Excitation (or
radiatively inefficient) Radio Galaxies (LERG) according to their optical
spectra (Best & Heckman 2012). LERG is the dominant population, both, for GRGs
and smaller RGs (Dabhade et al. 2017; Mingo et al. 2022; Simonte et al. 2022;
Best et al. 2023).

The combination of multi-frequency radio observations allows us to study the
radio spectrum and thus the radiative age of the emitting electrons in RGs.
Previous studies showed that GRGs exhibit radiative ages up to $\approx$ 80
Myr, similar to smaller RGs (Schoenmakers et al. 2000a; Konar et al. 2004;
Jamrozy et al. 2008; Harwood et al. 2017; Sebastian et al. 2018; Mhlahlo &
Jamrozy 2021). More recent studies used LOFAR to investigate the GRG
properties at lower frequencies and derive spectral index maps using data from
the literature such as Very Large Array (VLA), Westerbork Synthesis Radio
Telescope (WSRT), Effelsberg, and Giant Metrowave Radio Telescope (GMRT)
observations (Shulevski et al. 2019; Cantwell et al. 2020; Dabhade et al.
2022). These studies identified GRGs with a radiative age exceeding 100 Myr,
hinting at their status as aged and evolved RGs.

Lan & Prochaska (2021), compared the statistics of the environment of small
and large radio sources. They demonstrated that the properties and the surface
number density of surrounding galaxies of GRGs and RGs are very similar.
Nevertheless, GRGs are found preferably, but not exclusively, outside of
galaxy clusters (Dabhade et al. 2020b; Andernach et al. 2021) and asymmetries
in the morphology of the radio lobes of GRGs appear to be influenced by
inhomogeneities of the surrounding medium (Subrahmanyan et al. 2008; Safouris
et al. 2009; Machalski 2011; Malarecki et al. 2013, 2015).

Previous studies mostly focused on the properties of GRGs alone and only very
few presented a comparative analysis of the properties of GRGs and smaller RGs
(e.g., Subrahmanyan et al. 1996; Dabhade et al. 2020b; Lan & Prochaska 2021).
In this paper, we extend our work on the GRGs in the LOFAR Boötes deep field
(Simonte et al. 2022) by adding a number of RGs and GRGs we found in the
Lockman Hole and ELAIS-N1 deep fields. Using radio data for our sources at
other frequencies and optical and infrared wavelengths, we compare the multi-
wavelength properties of GRGs with those of the control sample of smaller RGs.

The outline of this paper is as follows: in Sec. 2 we explain how we built our
catalogue of (G)RGs and how we carried out the analyses of the radio and
optical data. In Sec. 3 we present the results of our analysis and compare the
properties of RGs and GRGs. We draw our conclusions in Sec. 4. Throughout this
work we adopt a flat $\rm\Lambda CDM$ cosmology with $H_{0}$ = 70
$\rm\leavevmode\nobreak\ km\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\
Mpc^{-1},\leavevmode\nobreak\ \Omega_{m}=0.3,\leavevmode\nobreak\
\Omega_{\Lambda}=0.7$ and a radio source spectral index $\alpha$ defined as
$S_{\nu}\propto\nu^{-\alpha}$.

##  2 Methods

Figure 1: Distribution of the largest linear size in our LDF-RG sample. The
red line marks the value of 0.7 Mpc. Figure 2: Distribution of the redshifts
(spectroscopic and photometric) in our LDF-RG sample (blue histogram). The
orange histogram shows the distribution of the spectroscopic redshifts.

We visually inspected the images of the three LOFAR deep fields: ELAIS-N1,
Boötes, which cover an area of $\approx 25\rm\leavevmode\nobreak\ deg^{2}$
each (Sabater et al. 2021), and Lockman Hole, which covers an area of $\approx
45\rm\leavevmode\nobreak\ deg^{2}$ (Tasse et al. 2021). These observations
were performed using the LOFAR High-Band Antennas (HBA) which operate in the
range between 120-168 MHz. With an effective observing time longer than 80
hours for each field, the observations reach a root mean square noise (rms)
level at 150 MHz lower than 30 $\rm\mu Jy\leavevmode\nobreak\ beam^{-1}$
across the inner $10\rm\leavevmode\nobreak\ deg^{2}$. The sensitivity to a
wide range of angular scales of such observations makes these deep fields
ideal for the detection of the diffuse radio emission that characterizes GRGs.

We carried out a systematic search of extended radio galaxies (ERGs) with a
largest angular size ($LAS$) $\gtrsim 20^{\prime\prime}$. Given the range of
radio morphologies, we employed a variety of methods for estimating the $LAS$.
In the case of bright FRII sources, we measured the $LAS$ as the distance
between the two opposite hotspots, if detected in the VLA Sky Survey (VLASS,
Lacy et al. 2020), unless the LOFAR image showed evidence for emission beyond
the hotspots. For FRI and more diffuse sources, we measured the $LAS$ with a
straight line between the two opposite ends of the source outlined by the
3-$\sigma$ contours. This approach was also used for bent sources because,
otherwise, we would have to make assumptions about the (unknown) 3D structure
of the source. For the most diffuse sources, we extended the LAS measurement
beyond the 3-sigma contours when we found genuine radio emission beyond those
contours.

In order to identify the host galaxies for our ERGs, we made use of a variety
of multiple optical and infrared surveys: the Sloan Digital Sky Survey (SDSS,
York et al. 2000), the Wide-Field Infrared Survey Explorer (WISE, Wright et
al. 2010) and its more recent versions of both images and catalogues such as
AllWISE (Cutri et al. 2021), unWISE (Schlafly et al. 2019) and CWISE (Marocco
et al. 2021), the Dark Energy Survey Imaging (DESI, Dey et al. 2019; Zhou et
al. 2021) and Panoramic Survey Telescope and Rapid Response System (Pan-
STARRS, Flewelling et al. 2020). The identification of the host galaxy can be
challenging for widely separated radio components without a detected radio
core between them. The identification method was extensively explained in
previous papers (see Andernach et al. 2021; Simonte et al. 2022, 2023). Here,
we emphasise the key concerns that may arise during this procedure.

We first check whether one of the outer radio components has a convincing host
itself. In such cases, we recorded the radio source as a separate RG whenever
they are larger than $\approx 20^{\prime\prime}$. On the other hand, the
presence of a host galaxy with AGN colours (Mateos et al. 2012; Assef et al.
2013) near the geometrical centre of the ERG is a convincing sign of the
genuineness of the ERG. In asymmetric ERGs the shorter lobe is often the
brightest, perhaps caused by the denser environment encountered during the jet
expansion (Pirya et al. 2012; Malarecki et al. 2015), though relativistic
effects may also play a role (e.g., Hocuk & Barthel 2010). When faced with
uncertainty, we relied on this information to locate the host galaxy closer to
the brighter lobe for these sources. Moreover, the VLASS images with a
resolution of $2.5^{\prime\prime}$ were helpful in spotting the core in a
small fraction of ERGs. If neither asymmetry of the source nor the AGN colour
were enough to recognise the most likely host galaxies, we chose the brighter
or lower redshift host. Hence, the largest linear size (LLS) derived from the
LAS and redshift should serve as a lower limit in this case. For very few
sources ($\approx$ 10) we could not find an obvious host either in the optical
or in the infrared surveys. Hence, these sources were discarded.

We cross-matched the optical position of the host galaxy with various
spectroscopic (Kirshner et al. 1987; Wegner et al. 2003; Trouille et al. 2008;
Kochanek et al. 2012; Lacy et al. 2013; Pâris et al. 2018; Luo et al. 2018;
Ahumada et al. 2020; Wu & Shen 2022; Liu et al. 2022b; DESI Collaboration et
al. 2023) and photometric redshift catalogues (Rowan-Robinson et al. 2013;
Brescia et al. 2014; Bilicki et al. 2016; Xu et al. 2020; Beck et al. 2021;
Duncan et al. 2021; Zhou et al. 2021; Wen & Han 2021; Duncan 2022; Zhang et
al. 2022; Zou et al. 2022). For the latter, we computed the average redshift
and used as associated error the standard deviation of the multiple redshifts
of the host galaxy reported in the different catalogues. We do not quote the
errors on the spectroscopic redshifts since they are usually more accurate
(typical errors are about 0.00015) than the precision we can achieve in the
flux density and LAS measurements.

We found 1609 ERGs (which we refer to as LDF-RG or LDF-GRG sample) with an
angular size larger than $20^{\prime\prime}$, of which 280 are classified as
GRGs (i.e., with an $LLS\geq 0.7$ Mpc) and 134 have an $LLS\geq 1$ Mpc. In
total, 42% of the RGs in our sample have a spectroscopic redshift, 36% when
considering only GRGs. 82 host galaxies do not have a redshift estimate in any
of the aforementioned catalogues. The Zhou et al. (2021) DESI DR9 photometric
redshift catalogue is the deepest full-sky catalogue available (except for the
inner regions of the three deep fields where a deeper catalogue from Duncan et
al. 2021 exists). In Zhou et al. (2021), the faintest galaxies have a maximum
photometric redshift of around 1.3. Therefore, we assumed a redshift of 1.5
for those host galaxies without redshift listed in the literature. However,
the precise redshift is not essential when converting LAS to LLS since, for a
given LAS, the LLS is almost constant in the redshift range from about 1.2 to
2. (see Fig. 3 in Simonte et al. 2022). Moreover, we did not use ERGs without
redshift estimates for our environmental analysis. Nevertheless, the
uncertainties on the redshift estimation also affect the radio power
calculation and thus the location of the ERGs in the radio power-linear size
diagram (Baldwin 1982).

We classified the ERGs into FRI and FRII sources according to their radio
morphology, without considering the radio luminosity of these sources. In our
LDF-GRG sample, 84% of the GRGs exhibit an FRII morphology, 8% are classified
as FRI, and 24 GRGs are labeled as ’complex’ (cpx) due to the ambiguity in
their morphology, preventing a robust classification. Considering only the
smaller RGs, 74%, 15%, 11% of the ERGs are classified as FRII, FRI and
complex, respectively.

In Fig. 1 and 2, we show the distribution of the LLS and redshift in our LDF-
RG sample. These plots encompass all ERGs, including those for which a
redshift estimate is not provided in the literature; in such cases, given
their faintness we assigned a redshift value of 1.5.

###  2.1 Flux density and spectral index

We measured the total radio flux of the 280 GRGs using the full 6′′ resolution
images of the LOFAR deep fields (Sabater et al. 2021; Tasse et al. 2021). We
prepared cutouts with sizes equal to three times the angular size of the
source on a side. While integrating the flux, we considered only those pixels
whose intensity is larger than $3\sigma_{\rm rms}$. Here, $\sigma_{\rm rms}$
is the noise level calculated within each cutout, since the noise level varies
significantly across the deep fields, and it is measured through an iterative
approach. In each iteration, we calculated the root mean square (rms) value,
removed pixels with intensities exceeding five times the rms, and subsequently
recalculated the rms. The convergence criterion was defined based on the
difference between two consecutive rms measurements, with a threshold set to
1%. Therefore, we scaled the noise measurement by the square root of the area
of the integrated flux, measured in terms of beam areas. The final flux error
was calculated as $\sqrt{\sigma_{\rm rms}^{2}+\sigma_{\rm cal}^{2}}$, where
$\sigma_{\rm cal}$ is the uncertainty on the calibration of the flux scale
which is assumed to be 10% (Sabater et al. 2021; Tasse et al. 2021). The total
flux of some of the largest, in terms of angular size, and faintest ERGs is
underestimated with the 3-$\sigma$ clipping method which misses part of the
emission coming from the faintest part of the ERG (such as bridges). Thus, we
integrated the flux of these 28 sources considering all the pixels belonging
to the region of the radio emission.

We calculated the radio power at 150 MHz following:

$P_{150}=4\pi D_{L}^{2}S_{150}(1+z)^{\alpha-1},$ (1)

where $D_{L}$ is the luminosity distance, $S_{150}$ is the measured radio flux
density at 150 MHz, $(1+z)^{\alpha-1}$ is the standard k-correction used in
radio astronomy and $\alpha$ is the radio spectral index for which we adopted
a typical value of 0.7. This is a reasonable value considering that the radio
spectrum of GRGs is often dominated by the emission from the hotspots and the
core (see, Dabhade et al. 2020b, for a distribution of the integrated spectral
indices in a sample of GRGs). We should also note that we found slightly
steeper radio spectra in our analysis (see LLS-spectra index plots) with an
integrated spectral index. Thus, some of the radio luminosities at 150 MHz can
be slightly underestimated. Nevertheless, the exact value of $\alpha$ has a
relatively small impact on the final estimate of the radio power.

In order to estimate the core fraction, we used a similar approach to
calculate the flux of the GRG cores. We only used GRGs in which the core was
not blended with the emission coming from the inner part of the jets. We found
that 176 GRGs show a clear core emission. 33 GRGs (12% of the total sample) do
not show any core emission above the $3-\sigma$ level. For these sources, we
calculated the core flux as 3$\sigma_{rms}$, where $\sigma_{rms}$ is the rms
noise calculated in the vicinity of the source. Hence, the core fraction
derived from the core flux should serve as an upper limit. The full table with
all the properties of the full LDF-RG sample will be made available at the CDS
and through the VizieR service†† https://vizier.cds.unistra.fr (Ochsenbein et
al. 2000).

Furthermore, we used images at other radio frequencies to investigate the
spectral properties of both GRGs and smaller RGs. The LOFAR Low-Band Antenna
(LBA) Sky Survey (LoLSS, de Gasperin et al. 2021, 2023) is a wide-area survey
at 41-66 MHz. A dedicated deep observation of the Boötes field at these
frequencies has been performed at a central frequency of 50 MHz, and the
resulting image has an rms noise level of 0.7 mJy beam-1 and a resolution of
15′′ (Williams et al. 2021). Moreover, the APERture Tile In Focus array
(Apertif, van Cappellen et al. 2022) survey, carried out with the Westerbork
Synthesis Radio Telescope (WSRT), observed the same field at 1400 MHz (Kutkin
et al. 2023). The mosaicked image has an angular resolution of
$27^{\prime\prime}\times 11^{\prime\prime}$ and a median background noise of
40 $\mu$Jy beam-1. The ELAIS-N1 field was observed with the Giant Metrewave
Radio Telescope (GMRT) and imaged at 610 MHz by Garn et al. (2008a). The
resulting radio image has a resolution of $6^{\prime\prime}\times
5^{\prime\prime}$ and a rms noise level of 40-60 $\mu$Jy beam-1. The Lockman
hole field was observed both at 1400 MHz with Apertif (Morganti et al. 2021)
and 610 MHz with GMRT (Garn et al. 2008b). However, the region of overlap
between the LOFAR and the Apertif image in this region of the sky is quite
small and the few sources detected at multiple frequencies are too faint to
obtain good-quality spectral indices.

Exploiting this multi-frequency coverage we computed the integrated spectral
index (i.e., estimated from the total flux of the source) of 25 GRGs and 74
RGs between 150 and 1400 MHz, three GRGs and 32 RGs between 150 and 610 MHz
and 24 GRGs and 21 RGs between 50 and 150 MHz. For each pair of images, we
convolved the images to the angular resolution of the image with the coarsest
resolution. We excluded those ERGs that were blended with other sources after
the convolution. Then, we calculated the integrated flux for each source
within the pair of images at two distinct frequencies. This was achieved by
applying sigma-clipping at a level of 3$\sigma_{\rm rms}$, where $\sigma_{\rm
rms}$ is the rms noise of the individual convolved image. Hence, we calculated
the spectral index of the radio spectrum of the source using the linear least-
squares method in log-space. The errors were estimated according to the
Gaussian propagation of uncertainties. Note that the calculation of the
integrated flux is dominated by the brightest part of the radio galaxy, such
as the core and the hot spots, which host the youngest electrons and with a
spectral index of about 0.4-0.7.

###  2.2 Optical analysis

Kondapally et al. (2021) and Best et al. (2023) identified the host galaxies
for all the radio sources in the inner $\approx 7\rm\leavevmode\nobreak\
deg^{2}$ of the three LOFAR deep fields and estimated their stellar masses,
star formation rates (SFRs) and accretion modes by fitting the spectral energy
distribution. We cross-matched our sample of ERG hosts with these catalogues
employing a crossmatching radius of 5′′. We found 465 matches, of which 362
have stellar mass and SFR reported in the catalogue. 61 are GRG hosts. We
checked the number of ERGs that do not have a match in the Kondapally et al.
(2021) and Best et al. (2023) catalogues within a central area of about 9 $\rm
deg^{2}$ of each deep field. We could retrieve 415 of the total 465 matches in
the three deep fields. 198 ERGs that are located within the central 9 $\rm
deg^{2}$ of each deep field do not have a host galaxy reported in Kondapally
et al. (2021) and Best et al. (2023). This subset represents RGs that may not
be recognized in these catalogues or cases where we chose a different and more
likely host galaxy.

The properties of the host galaxies reported in Kondapally et al. (2021) and
Best et al. (2023) are derived using the redshift estimates found by Duncan et
al. (2021). In contrast, we determined the redshifts of our sources by
averaging the photometric redshifts from multiple catalogues. We conducted a
comparison between our estimated photometric redshifts with those reported in
Duncan et al. (2021). Our analysis reveals that the redshift difference is
consistently below or equal to 0.05 for 75% of the ERGs and the difference is
below 0.2 for 90% of the host galaxies. While the analysis of the SFR is
likely not affected by this discrepancy, it is important to note that this
difference affects the discussion on the redshift distribution of GRGs and RGs
within this specific analysis.

###  2.3 Environmental analysis

We used the DESI DR9 photometric redshift catalogue (Zhou et al. 2021) to
study the environment of the ERGs on a scale between 1 and 10 Mpc. This
catalogue includes observations from the Beijing-Arizona Sky Survey (BASS, Zou
et al. 2017), DECam Legacy Survey (DECaLS) and Mayall z-band Legacy Survey
(MzLS) (Dey et al. 2019) and it is the deepest large-area catalogue available.
However, due to the depth limit of the DESI survey we restricted the analysis
to $z\leq 0.7$. Hence, we included in our study 810 ERGs of which 121 have a
linear size larger than 0.7 Mpc.

First, we created a volume-limited sample of potential neighbouring galaxies.
This involves excluding galaxies with an absolute magnitude (luminosity) in
the $r$ band that is fainter than the absolute magnitude (luminosity) of a
galaxy with an apparent magnitude equal to the flux limit in the $r$ band,
observed at the maximum considered redshift ($z=0.7$). Implementing this cut-
off imposes a restriction on our analysis as it focuses on the most luminous
galaxies, which are also the least common. As a consequence, we might miss
faint galaxies residing in galaxy clusters and underestimate the
overdensities. In this analysis, we looked at the surface number density of
galaxies by calculating the number of galaxies within a sphere of radius of 10
Mpc centred on the ERG hosts. We used as photometric redshift of the host
galaxy the value obtained by averaging the redshifts of different catalogues
as explained in Sec.2. We did not consider the calculated error of the
photometric redshift in this analysis. Moreover, we used spectroscopic
redshifts for, both, the host and neighbouring galaxies when available. We
should point out that more than 99% of neighbouring galaxies are located at a
distance larger than 1 Mpc from the ERG host. Thus, this analysis ignores the
information on the environment within $\approx$ 1 Mpc which may play a crucial
role in the evolution and expansion of the radio jets and lobes (Konar et al.
2008; Subrahmanyan et al. 2008; Machalski 2011; Pirya et al. 2012; Malarecki
et al. 2015).

Moreover, we used multiple catalogues of galaxy clusters compiled using
observations at different wavelengths (Koester et al. 2007; Yoon et al. 2008;
Hao et al. 2010; Rykoff et al. 2014; Takey et al. 2014; Wen & Han 2015; Planck
Collaboration et al. 2016; Clerc et al. 2016; Burenin 2017; Wen et al. 2018;
Wen & Han 2018; Abdullah et al. 2020; Koulouridis et al. 2021) to study the
distance of the RGs in our sample from known galaxy clusters, but we
restricted these clusters to those with a spectroscopic redshift. We matched
the RG hosts with the galaxy cluster having the smallest 3D comoving distance
from the host. Due to the sensitivity limitations of the surveys, the
detection of galaxy clusters at high redshift is challenging and not many of
them are detected beyond $z\approx 0.5$ (e.g Wen & Han 2018). As a
consequence, there may be undetected, high-redshift galaxy clusters closer to
the ERG host than the current matched galaxy cluster. Moreover, we may miss
the membership of the ERG host to a cluster at high redshift. Thus, we
considered only the RGs with a galaxy cluster within a comoving distance of 50
Mpc for the analysis. This value was chosen to study the distribution of the
distance of ERGs from galaxy clusters, besides their membership to a specific
cluster. The final sample counts 674 RGs and 120 GRGs.

Finally, we looked for a possible cosmological evolution of the linear size of
RGs (e.g., Kapahi 1989) by dividing our sample into five redshift bins of
width $\Delta z=0.3$ and calculated the median LLS in the bin. We included all
the RGs with a redshift larger than 1.5 in the 1.2-1.5 redshift bin. We
excluded those radio galaxies with an LLS smaller than the minimum physical
size required to be resolved at any redshift. We considered an ERG as resolved
when its angular size was $\geq 20^{\prime\prime}$ at any redshift. Thus, the
minimum LLS required for an ERG to be resolved at any redshift is about 170
kpc.

##  3 Results

With a sample size of 1609 RGs, we looked for differences in the host galaxy
properties of RGs and GRGs and compared the environments in which they reside.

###  3.1 $P-D$ diagram

Figure 3: $P-D$ diagram that shows the radio power at 150 MHz against the
linear size (LLS) of the LDF-GRG sample. Other previous GRG samples are
included in the plot for a comparison: Dabhade et al. (2017, 2020a, 2020b);
Kuźmicz & Jamrozy (2021); Andernach et al. (2021).

We used the flux densities and radio powers of the GRGs as derived in Sec. 2.1
to locate our GRGs in the radio power-linear size diagram ($P-D$ diagram,
Baldwin 1982). Such a plot is often used to study the evolution of radio
galaxies. Every radio source has a specific evolutionary track in this diagram
which depends on multiple properties such as jet power, environment and
redshift (see, Ishwara-Chandra & Saikia 1999b; Machalski et al. 2004;
Hardcastle 2018). We show the position of our LDF-GRGs sample in the $P-D$
diagram with the black points, along with GRGs from recent samples (Dabhade et
al. 2017, 2020a, 2020b; Kuźmicz & Jamrozy 2021; Andernach et al. 2021) in
which flux densities and radio power of the listed GRGs are reported. For
those GRGs with a flux estimated at other frequencies, we extrapolated the
radio power at 150 MHz by using a standard spectral index, $\alpha=0.7$. This
is the typical value found in the hotspots of RGs which often dominates the
radio emission of the GRGs, especially at higher frequencies (see also Sec.
2.1, for a discussion on the value of $\alpha$).

According to the RG models, after an initial phase and once the jet activity
stops, its luminosity decreases owing to synchrotron, inverse Compton and
adiabatic expansion losses in the latest stages (e.g., Hardcastle 2018).
Hence, if GRGs are aged RGs, it is expected that the largest RGs tend to be
less powerful compared to the smaller radio sources. This effect likely drives
the lack of GRGs in the upper-right corner of Fig. 3, where very powerful and
large RGs should reside. Figs. 8 and 14 in Hardcastle (2018) show that only
the most powerful ($P_{150MHz}>10^{26}\rm\leavevmode\nobreak\
W\leavevmode\nobreak\ Hz^{-1}$ and aged ($>100\rm\leavevmode\nobreak\ Myr$)
RGs embedded in an environment of $10^{13}-10^{15}\rm\leavevmode\nobreak\
M_{\odot}$ can grow up to Mpc sizes. Nevertheless, GRGs display a wide range
of radiative ages between 10-100 Myr (Schoenmakers et al. 2000b; Lara et al.
2000; Jamrozy et al. 2008; Pinjarkar et al. 2023). Future deep radio
observation at higher frequencies will certainly help to accurately estimate
the age of GRGs.

On the other hand, there is a lack of very large GRGs with a small radio power
as well, likely due to sensitivity limitations of the current radio
facilities. The large angular size of such RGs makes their surface brightness
rather low, challenging their detection in the lower power regime. As a matter
of fact, we have been starting to observe large ($>1\rm Mpc$) RGs of low radio
luminosity ($<10^{25}\rm\leavevmode\nobreak\ W\leavevmode\nobreak\ Hz^{-1}$)
only with the most recent and sensitive observation with LOFAR and ASKAP
(Andernach et al. 2021).

###  3.2 Properties of the host galaxies

Combining our LDF-RG sample with the catalogues provided by Kondapally et al.
(2021) and Best et al. (2023) for the LOFAR deep fields we compared the star
formation rate (SFR), stellar mass and accretion state of the host galaxies in
GRGs and RGs. The final sample for this analysis counts 385 RGs
($LLS<0.7\rm\leavevmode\nobreak\ Mpc$) and 68 GRGs. Fig. 4 shows that the
distribution of the stellar mass in the host of RGs (red) and GRGs (blue) are
very similar with both populations being hosted by massive early-type
galaxies. The result is in line with previous work (Lara et al. 2001; Dabhade
et al. 2017; Simonte et al. 2022). However, very rare, massive spiral galaxies
with radio lobes extending up to Mpc size have been found as well (Bagchi et
al. 2014; Oei et al. 2023b). Moreover, previous studies (Zovaro et al. 2022;
Kuźmicz et al. 2019) found a young ($<10^{7}\rm\leavevmode\nobreak\ yr$) and
’intermediate’ ($\approx 10^{9}\rm\leavevmode\nobreak\ yr$) stellar
population, besides the population of evolved stars with an age larger than
$10^{10}\rm\leavevmode\nobreak\ yr$, in the host of some GRGs. These results
suggest that the star formation in some GRG hosts is still ongoing. In our
sample, we found that 40% of GRG hosts have a SFR $>10\rm\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$, while only 20% of the RG hosts have a
SFR above this threshold. Interestingly, in our sample, about one-third of the
galaxies with a SFR $>10\rm\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}$ host a GRG, while this number drop to 12% for host galaxies with a
SFR $<10\rm\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}$. We
compared the SFR distribution of GRGs and smaller RGs by conducting a
Kolmogorov-Smirnov (KS) test and we obtained a p-value of 0.004. This result
provides a strong evidence to reject the null hypothesis that the two samples
were drawn from the same distribution. To validate this result, we performed a
Mann-Whitney U test as well (Mann & Whitney 1947). This is a non-parametric
test which uses the rank sums of the two samples to determine whether they
come from the same distribution. The resulting p-value of 0.006 further
validates the rejection of the null hypothesis. In Fig. 5, we show the
distribution of the specific star formation rate (sSFR) calculated as the
ratio between the SFR and the stellar mass of the host galaxy. For comparison,
star-forming galaxies are known to span a wide range of sSFR (Gürkan et al.
2018) with significant redshift dependence and typical values are larger than
0.1 $\rm Gyr^{-1}$ (Damen et al. 2009) across a range extending from the
nearby Universe to the median redshift of the sample employed in our analysis,
which is 0.7. While the number of RGs decreases at larger sSFR in Fig. 5, an
excess of hosts with moderate and high sSFR ($\gtrsim
0.1\rm\leavevmode\nobreak\ Gyr^{-1}$) is visible in the sSFR distribution of
the GRG hosts with respect to the RG distribution. The returned p-value from
both the KS and Mann-Whitney U tests applied to the sSFR distribution is
smaller than 0.05, showing that the differences between GRGs and RGs persist
even when normalising the SFR by the stellar mass of the galaxy.

Figure 4: Distribution of the stellar mass of the host galaxies in RGs (red
histogram) and GRGs (blue histogram). Figure 5: Distribution of the specific
star formation rate of the host galaxies in RGs (red histogram) and GRGs (blue
histogram).

While both FRI and FRII RGs are commonly hosted by passive early-type host
galaxies, the latter tend to be more star-forming (Mingo et al. 2022). The
fraction of FRI sources in the GRG and RG samples used for this analysis is 9%
and 14%. Moreover, half of the GRG hosts with SFR $>10\rm\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$ are classified as HERGs, which commonly
exhibit larger SFR compared to LERGs. The HERG-LERG classification is taken
from Best et al. (2023). The percentage of HERGs in the GRG and RG samples
used for this analysis is slightly different: 22% and 13%, respectively. Thus,
the relatively higher presence of HERGs, coupled with the lower fraction of
FRI in the GRG sample compared to the RG sample, likely drives the difference
in the distribution of the (s)SFR of the two populations. When shifting the
linear size threshold used for the definition of GRGs to lower values (e.g.,
500 kpc), the percentage of star-forming optical galaxies (SFR
$>10\rm\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}$) hosting a
GRG increases to 35%. However, the percentage of optical galaxies with SFR
$<10\rm\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}$ hosting a
GRG increases to almost 50% . As a consequence, both, the KS test and the
Mann-Whitney U test returns a p-value $>0.1$. We noticed that the percentage
of FRI RGs and HERGs is similar in the GRG and RG samples when employing a
threshold of 500 kpc. Mingo et al. (2022) found that the FR morphology of the
RG depends on the combination of the radio luminosity of the RG and the
stellar mass of the host galaxy. Specifically, a higher galaxy mass or a less
powerful radio source increases the likelihood of FRI production. The fraction
of FRI in previous GRG samples as well as in the LDF-GRG sample is around 5%
(e.g., Dabhade et al. 2020b; Andernach et al. 2021). Such a low fraction
implies that there is a mechanism ensuring that GRGs always possess sufficient
power to avoid being frustrated by the host galaxy’s environment. It should be
noted that in our LDF-GRG sample, there are a few FRIs with a radio luminosity
$<10^{26}\rm\leavevmode\nobreak\ WHz^{-1}$ and stellar masses larger than
$10^{11}\rm\leavevmode\nobreak\ M_{\odot}$. However, all these FRI GRGs, for
which an estimate of the stellar mass is available, exhibit a bent morphology
typical of RGs residing in dense environments, such as galaxy clusters. Within
these environments, the lobes of RGs can be advected and stirred by the
velocity field of the intracluster medium, resulting in larger and more
diffuse lobes (see also Sec. 3.3). Larger samples of GRGs with available host
galaxy mass and SFR are needed to confirm these results.

Figure 6: Redshift distributions of the GRG (blue) and RG (red) host galaxies
used for the analysis of the (specific) star formation rate.

Previous studies have indicated an evolutionary trend in the properties of the
RG hosts, revealing that the number density of LERGs exhibits a weak
dependence on redshift, whereas HERGs have a higher abundance at higher
redshifts (Best et al. 2014; Pracy et al. 2016; Williams et al. 2018; Butler
et al. 2019; Best et al. 2023). Moreover, a large population of star-forming
LERGs has been found at higher redshift (Delvecchio et al. 2017; Kondapally et
al. 2022). Thus, the different redshift distributions for RGs and GRGs may
drive the differences in the distribution of the (s)SFR. We show the redshift
distribution of the host galaxies of RGs (red histogram) and GRGs (blue
histogram) used for this analysis in Fig. 6. We examined the similarity
between the redshift distributions of the GRG and RG hosts in the present
work, for which SFRs are available in Best et al. (2023). A KS test was
performed, yielding a p-value of 0.22. It is worth noticing that while the
SFRs reported in Best et al. (2023) are obtained using the redshift catalogue
of Duncan et al. (2021), we averaged the photometric redshifts across multiple
catalogues. For this reason, we also performed a KS test and a Mann-Whitney U
test on the redshift distributions of GRGs and RGs using the photometric
redshift in Duncan et al. (2021). We found a p-value $>$ 0.05 in both cases.

The tendency of GRGs to exhibit higher (s)SFRs suggests that their host
galaxies undergo multiple star-forming phases, possibly resulting in a burst
of heightened nuclear activity which can trigger variability in the luminosity
of the central AGN (e.g., Miley 1980; Miley & De Breuck 2008) or a new jet
activity (Gürkan et al. 2015; Kozieł-Wierzbowska et al. 2017; Shabala et al.
2017; Toba et al. 2019). However, we did not find any correlation between the
core fraction and the SFR or core radio power and SFR of the host galaxy,
possibly due to inherent time delays between star-forming processes and AGN
activity. While the star formation observed in a galaxy is contemporaneous
with our observations, AGN radio activity likely occurred $10^{7}-10^{8}$
years ago. Therefore, establishing a direct link between the radio emission of
RGs and the current SFR of their host galaxies is non-trivial, especially
considering the uncertainty of whether the present star-forming activity is a
result of the same processes that might have triggered AGN activity in the
past.

Figure 7: $\rm sSFR-P_{150MHz}$ diagram for LDF-GRG sample. The orange points
are radio galaxy hosts classified as LERGs, while the blue points are HERGs.
The red line represents the traditional boundary used to distinguish between
FRI and FRII (Fanaroff & Riley 1974; Ledlow & Owen 1996). The filled and empty
black diamonds are HERGs and LERGs from the Mingo et al. (2022) sample
respectively.

In Fig. 7, we show the sSFR-radio power ($P_{\rm 150MHz}$) diagram for our
LDF-GRG sample (orange and blue points) along with the RG sample from Mingo et
al. (2022) (black and empty diamonds). The RGs are colour-coded according to
their accretion status: black filled diamons and blue points for radiatively
efficient and orange and empty diamonds for radiatively inefficient RGs. The
threshold value of $10^{26}\rm\leavevmode\nobreak\ W\leavevmode\nobreak\
Hz^{-1}$ (Ledlow & Owen 1996; Mingo et al. 2022) is represented by the red
line. We notice that RGs with a radiatively efficient accretion have also high
radio luminosities ($>10^{26}\rm\leavevmode\nobreak\ W\leavevmode\nobreak\
Hz^{-1})$, while they are rare at lower luminosities, as also reported by
Mingo et al. (2022), showing that this dichotomy encompasses ERGs of any size,
including GRGs. As expected, LERGs have commonly lower sSFR ($\lesssim
0.1\rm\leavevmode\nobreak\ Gyr^{-1}$) compared to HERGs. However, there is a
population of LERGs with moderate or high sSFR which is driven by a subgroup
of FRII sources, both, above and below (but close to) the traditional
luminosity boundary.

It is worth noticing that the apparent relation between the SFR and radio
power is not real. As mentioned before, we expect an evolution of the SFR of
the host galaxies with cosmic epoch, with a peak of the SFR around z=2 (Madau
& Dickinson 2014). Furthermore, in brightness-limited surveys such as the
LOFAR deep fields, more luminous sources can be observed at a greater
redshift. We calculated the partial correlation coefficient (Baba et al. 2004)
which measures the strength of the correlation between two variables while
controlling for the effect of one or more other variables (which is the
redshift in our case). The resulting coefficient of R=0.11 suggests that the
correlation between the sSFR and the radio power is very mild.

Figure 8: Distribution of the galaxy surface density, within 10 Mpc, around
the hosts of RGs smaller (grey) and larger (blue) than 500 kpc. The labels on
the top x-axis have the same meaning as those on the lower one, only converted
to the number of galaxies within a sphere of 10 Mpc radius.

###  3.3 Environment

We tested the conjecture that GRGs reside in rather sparse environments
following the method described in Sec. 2.3. In Fig. 8 we show the distribution
of the surface number density of galaxies, $\Sigma_{gal}$, within 10 Mpc. In
order to increase the sample of the largest RGs and the significance of our
results, we relaxed the definition of GRGs by considering a threshold of 500
kpc rather than the usual 700 kpc. Our sample of 810 RGs used for this
analysis consists of 255 RGs with an LLS $>$ 500 kpc with redshift up to
$z=0.7$. The distribution shows that only the smaller RGs reside in the
densest environment while only very few GRGs larger than 500 kpc have a number
of neighbouring galaxies larger than $\approx$ 200 ($\rm Log(\Sigma_{\rm
gal})\gtrsim-0.2$).

Figure 9: Distribution of the minimum distance of ERG hosts from known galaxy
clusters. The blue and grey histograms show the distribution for GRGs
($LLS\geq 700\rm\leavevmode\nobreak\ kpc$) and RGs ($LLS\leq
700\rm\leavevmode\nobreak\ kpc$), respectively.

We performed a KS test to compare the distribution of the two which resulted
in a p-value $<$ 0.001. Thus, we can reject the hypothesis that the two
distributions come from the same distribution with a confidence level $>$99%.
The Mann-Whitney U test returns a similar result with a p-value $<$ 0.001. The
result of the KS test does not hold when keeping a threshold of 500 kpc for
the smaller galaxies while using the widely accepted definition of GRGs (700
kpc). However, this discrepancy is likely a consequence of the reduced sample
size (121 GRGs with a linear size larger than 700 kpc) resulting from the more
stringent threshold. Furthermore, the Mann-Whitney U test, with a p-value of
0.014, continues to support a difference in the distribution even with these
new thresholds.

Whenever available, we employed spectroscopic redshifts for both the
neighbouring and host galaxies to calculate the comoving distances in our
analysis. However, the majority of redshifts used in this analysis are
photometric. The average error of the latter is 0.09 for the host galaxies and
0.14 for the neighbouring galaxies with a slight tendency to increase with the
redshift estimate. To validate our results we performed the same analysis
considering only the RGs whose hosts have a spectroscopic redshift (i.e. the
uncertainties on the linear size are only due to the uncertainties on the
LAS), up to $z=0.7$, while all the neighbouring galaxies were retained,
without consideration for the type of redshift. The employed sample counts 364
RGs (LLS$<$500 kpc) and 157 GRGs (LLS$\geq$500) kpc with spectroscopic
redshifts. We obtained a p-value of 0.01 for both the KS and the Mann-Whitney
U test. Moreover, we performed such an analysis by using the Early Data
Release of the Dark Energy Spectroscopic Instrument (DESI Collaboration et al.
2023), referred to as DESI EDR in what follows, which provides a catalogue of
spectroscopic redshifts obtained with the DESI survey. Thus, we only used
spectroscopic redshifts for both the sample of RGs and the sample of
neighbouring galaxies used to calculate the galaxy density within 10 Mpc. In
this catalogue, there is no dedicated column for the optical magnitude of the
astrophysical object. Thus, we did not apply any cut in magnitude. DESI EDR
does not cover the Lockman Hole, which limits the number of RGs and
neighbouring galaxies we can use in the analysis. The retrieved p-values are
larger than 0.10. However, the distribution still shows that only RGs ($<$ 700
kpc) reside in environments with a $\rm Log(\Sigma_{\rm gal})\gtrsim-0.2$.

Oei et al. (2022), using data from the SDSS DR7 (Abazajian et al. 2009), found
that ”Alcyoneus”, the largest GRG published yet, has only five neighbouring
galaxies with similar r-band magnitude to its host within 10 Mpc. This result
positions Alcyoneus on the far left end of the distribution shown in Fig. 8.
Komberg & Pashchenko (2009), using SDSS data, demonstrated that GRGs with
redshifts up to 0.1 can be found in various environments, ranging from small
groups to rich clusters, although they tend to predominately inhabit sparsely
populated environments, in agreement with our study. Moreover, in their study,
Lan & Prochaska (2021) reported no difference in the environment within a 1
Mpc radius from the host when comparing a sample of GRGs to smaller RGs. It is
worth noting that, although our sample of GRGs is of a similar size, our study
focused on the galaxies within 10 Mpc, but is less reliable on scales smaller
than 1 Mpc as our algorithm likely misses faint satellites around the main
galaxies.

Concerning the association of ERGs with catalogued clusters, we show the
distribution of the minimum comoving distance (D) of ERG hosts from known
clusters with spectroscopic redshift in Fig. 9. While for RGs smaller than 700
kpc the number of host galaxies per bin remains almost constant in the range
of 0-25 Mpc, the GRG distribution shows a lack of host galaxies per bin in the
range 0-15 Mpc with respect to the range 15-30 Mpc. In particular, the GRG
hosts within 15 Mpc are almost half of the GRGs with the nearest galaxy
cluster in the range 15-30 Mpc. This indicates that GRGs tend to avoid the
densest environments such as galaxy groups and clusters. Nevertheless, the KS
test and Mann-Whitney U test reveal that the differences in the two
distributions are not very significant, showing p-values between 0.02 to 0.15,
dependent on the specific test and size threshold employed for defining GRGs.

We looked at the 109 RGs that fall within a 3D distance of $<$ 5 Mpc from a
reported cluster (from references cited in Sec. 2.3) and found seven GRGs
larger than 1 Mpc plus another three larger than 0.7 Mpc. A closer inspection
of these ten revealed that seven of them are in fact the brightest members of
their clusters (BrClG in what follows), of which five are wide-angled tailed
RGs (WATs) which is a very common radio morphology for BrClGs. The most
prominent BrClG in our sample is HB 13 (GRG J1032+5644), also known as CGCG
290$-$048, the second-largest cluster-associated ERG in our sample (LLS=2.4
Mpc). It is the BrClG of SDSS–C4 3047 (Von Der Linden et al. 2007; Abdullah et
al. 2020) and GalWCat19 cluster 1279 (Abdullah et al. 2020). Interestingly,
the cluster-associated GRG with the largest linear size, J1057+5357, and BrClG
of WHL J105727.9+535756 is of FR II radio morphology, which is thought to be
rare in clusters. However, we find that more than half of the 109 RG hosts
within 5 Mpc from clusters in fact coincide (to within 2′′) with the listed
cluster centre and are thus likely to be the BrClG, since these are most often
taken as the cluster centre position. Out of 60 such coincidences between ERG
host and cluster centre, more than half (36) are of type FRII with a median
LLs of 280 kpc, showing that FRII type sources are not that rare among BrClG.

The presence of ten GRGs in our sample that reside within a comoving distance
of 5 Mpc from a galaxy cluster challenges the idea that GRGs reside only in
sparse environments. A hypothesis is that they reside only in small clusters
or groups with a lower central density of the gas. However, using LOFAR and
ASKAP data, Pasini et al. (2022) and Böckmann et al. (2023) did not find any
relation between the linear size of RGs and the central density of their host
galaxy clusters detected within the eROSITA survey (Merloni et al. 2012;
Predehl et al. 2021; Liu et al. 2022a) suggesting that radio power is more
prominent than ambient density in determining the size of the radio galaxy in
clusters. However, as emphasised by Fig. 3, a relation between the size and
the radio power in our sample is not obvious. Nevertheless, variations of
density and velocity in the intracluster medium may account for the stirring
of the plasma injected by the jets and lead to Mpc size WAT and Narrow-Angled
Tail (NAT) RGs with complex morphologies (Srivastava & Singal 2020; Lusetti et
al. 2024). Thus, the physical mechanisms responsible for the shape and extent
of the diffuse emission of GRGs may be different for those residing in galaxy
clusters.

Figure 10: Relation between the integrated spectral index, calculated in the
range 50-150 MHz (upper panel), 150-610 MHz (middle panel) and 150-1400 MHz
(lower panel), and the largest linear size of RGs. The colorbar highlights the
redshift of the sources, while the errorbars represent the error on the
spectral index.

###  3.4 Cosmological size evolution of RGs

Former studies have suggested an evolution of the linear sizes of RGs with
redshift, which might be explained by the redshift evolution of the
intergalactic medium (Kapahi 1989; Machalski et al. 2007; Onah et al. 2018).
For this analysis, we used both smaller RGs and GRGs, leading to a sample of
1423 ERGs with a linear size larger than 170 kpc. We employed this threshold
to remove those ERGs with a linear size smaller than the minimum physical size
required to be resolved at any redshift. We found that binning the ERGs in
redshift bins of width $\Delta z=0.3$ (see Sec. 3.3), the median LLS in each
bin does not show evolution with redshift. Furthermore, we computed the KS and
Mann-Whitney U tests to compare the LLS distribution in different bins of
$\Delta z$. We found no evidence that they are drawn from different
populations and hence no evidence for cosmological size evolution of RGs.
These results are in agreement with Brüggen et al. (2021) who found no
dependence of the median linear size or the median radio luminosity on the
redshift and hence no evidence for cosmological evolution of the population of
GRGs.

###  3.5 Spectral analysis

According to the hypothesis that GRGs represent a later stage in the evolution
of RGs, we expect GRGs to exhibit steeper spectral indices, indicative of the
presence of older electrons compared to smaller RGs. We calculated spectral
indices for three frequency ranges: 50-150 MHz (LBA, HBA), 150-610 MHz (HBA,
GMRT), and 150-1400 MHz (HBA, Apertif). We aimed to identify differences
between GRGs and smaller RGs that may indicate the presence of older electrons
in the larger RGs. Fig. 10 shows the relation between the linear size of RGs
and the integrated spectral index for each frequency pair. The colorbar
indicates the redshift of the relevant source and we applied a redshift limit
of z=1.5 in the plots to enhance clarity and visualization. Consequently, the
redshift of the sources with the highest values in these plots is equal to or
greater than 1.5. However, it is worth noting that a few sources are located
at higher redshifts and 13 (G)RGs used in this analysis have a spectroscopic
redshift greater or equal to one. When using a combination of Apertif and
LOFAR images (lower panel), the two variables show a correlation suggesting
that the largest RGs exhibit steeper integrated spectra. We assessed the
degree of correlation by calculating the weighted Pearson’s correlation
coefficient, R. The correlation coefficient ranges between -1 and 1. These
limits represent a perfect (anti)correlation, while a value of 0 implies that
the variables are uncorrelated. We calculated the weights as the inverse of
the squared error of the spectral index. We retrieved R=0.36 corresponding to
a p-value $<$ 0.001. Such a result indicates that the largest RGs contain the
most aged electrons, pointing to a scenario in which GRGs are in the latest
stage of the evolutionary track. It is worth noticing that this correlation is
less evident when exclusively considering GRGs ($\rm LLS\gtrsim 700\rm kpc$),
also because of the smaller number of objects in this range. The result may
suggest that factors beyond the age of GRGs contribute to the final size of
GRGs. The obtained spectral indices between 150 and 1400 MHz are similar to
those reported by Dabhade et al. (2020b) who combined LOFAR DR1 and NVSS data.
Moreover, previous work suggested a correlation between the size and the
radiative age of the RGs (Parma et al. 1999; Murgia et al. 1999; Jamrozy et
al. 2008). Our results support this correlation although we did not directly
compute the age of the sources. We also observed a trend involving the
spectral index and redshift. It is known that the largest RGs are rarer
compared to their smaller counterparts (e.g., Oei et al. 2023a; Simonte et al.
2022); thus, larger cosmological volumes are needed to sample large RGs.
Consequently, the correlation between spectral index and redshift is primarily
driven by the relationship between linear size and redshift that may appear in
an incomplete sample, rather than an independent trend, in our case.
Nevertheless, high-redshift RGs have been found originally by their steep
spectral index and small angular size (Tielens et al. 1979; Blumenthal & Miley
1979) and the correlation between the spectral index and redshift is still
under discussion (e.g. Miley & De Breuck 2008). The correlation is not
significant when using the ERGs detected at 50 MHz (R=0.27,p-value=0.07) and
610 MHz (R=0.03, p-value=0.87) only. However, the lack of correlation might be
due to the smaller sample size employed for the analysis.

Using the KS and Mann-Withney U tests, we also analysed the distribution of
the integrated spectral index of GRGs and smaller RGs. These tests reject the
null hypothesis that the two distributions are drawn from the same population
with a confidence level greater or equal to 97% only when employing a
combination of LOFAR HBA-LBA and HBA-Apertif data and not using HBA-GMRT data.
The outcomes may indicate that GRGs and their smaller counterparts have a
distinct integrated spectral index distribution, even if the results are not
conclusive.

##  4 Discussion and conclusions

Even though the number of known GRGs has significantly increased in the past
few years (Oei et al. 2023a, Mostert et al. 2023), the origin and the causes
for their large sizes are still not understood. Previous studies mostly
focused only on the physics of GRGs themselves without looking for differences
between the properties of the largest and smallest ERGs. Here, we visually
inspected the three LOFAR deep fields and compiled a catalogue of 1609 ERGs,
of which 280 are classified as GRGs. We measured the integrated radio fluxes
of the GRGs and located them in the radio power-size (P-D) diagram. The result
highlights the ability of LOFAR to detect faint radio emissions from extended
RGs. In particular, we reported a few GRGs with a $\rm LLS>1$ Mpc and a radio
power smaller than $10^{25}\rm\leavevmode\nobreak\ W\leavevmode\nobreak\
Hz^{-1}$ that were not detected in previous radio surveys. The lack of very
powerful ($>10^{27}\rm\leavevmode\nobreak\ W\leavevmode\nobreak\ Hz^{-1}$) and
large ($\rm LLS>2Mpc$) GRGs indicates that the largest GRGs are likely in
their latest stage of evolution. Moreover, we used optical catalogues to
perform a comprehensive analysis to study the properties of the host galaxies
and the environment of GRGs and smaller RGs. To quantify the environmental
density, we calculated the galaxy surface density within 10 Mpc from the host
galaxy using DESI catalogues (Zhou et al. 2021; DESI Collaboration et al.
2023). We found that the distribution of the galaxy surface density for RGs
larger and smaller than 500 kpc is significantly different (p-value=0.002),
showing that GRGs statistically reside in sparser environments compared to
their smaller counterparts. The result holds when performing the same analysis
considering only the ERG hosts with a spectroscopic redshift while retaining
all the neighbouring optical galaxies, regardless of their type of redshift.
This result does not imply that the environment solely dictates the size of
the GRGs. Rather, it indicates that, statistically, within our sample, larger
RGs tend to reside in sparser environments compared to their smaller
counterparts. Much larger samples with spectroscopic redshift available are
needed to firmly conclude that GRGs reside in less densely populated
environments.

The host galaxies of RGs and GRGs share similar properties. Both populations
are hosted by massive early-type galaxies with similar stellar masses and
accretion mode (predominantly LERG). Nevertheless, the distribution of the
(s)SFR in GRG hosts shows an excess of host galaxies with a $\rm
sSFR>0.1\leavevmode\nobreak\ Gyr^{-1}$ compared to the hosts of smaller RGs.

Moreover, we looked at the radio spectra of some of the sources in the LDF-RG
sample to find differences in the spectral properties of GRGs and smaller RGs.
We calculated the integrated spectral index between multiple pairs of
frequencies (50-150 MHz, 150-610 MHz, 150-1400 MHz) and found a positive
correlation between the linear size of the ERGs and the steepness of their
radio spectrum when using a combination of LOFAR HBA (150 MHz) and Apertif
(1400 MHz) observations. On the other hand, the correlation is less evident
when considering other frequencies. This result supports the idea that the
largest ERGs are also evolved RGs hosting more aged electrons compared to
their smaller counterparts. In summary, we propose that the final linear size
of RGs is primarily shaped by the combination of jet power and host galaxy’s
mass, the properties of the large-scale environment, and the age of the
source. The power of the jets of GRGs ensures that the final lobes always
expand well beyond the host galaxy, leading to a FRII morphology (see Mingo et
al. 2022, for details). This is suggested by the very low fraction ($\approx
5\%$) of FRI GRGs. Furthermore, the FRI GRGs exhibit a bent morphology
suggesting they reside in a dense and dynamical environment that may
facilitate their growth to Mpc sizes due to the advection and stirring of
their lobes by the turbulent intracluster medium. Fig. 8 suggests that
differences in the density of the environment just outside the host galaxy and
up to 10 Mpc can explain the different linear sizes of medium-sized ($\approx$
200-500 kpc) RGs and GRGs. The more isolated environment in which host
galaxies of GRGs are embedded may also explain the differences in the sSFR
distribution between GRGs and smaller RGs. The steeper integrated spectral
index of GRGs, compared to smaller RGs, implies their advanced evolutionary
stage. This underscores the idea that RGs need considerable time to evolve
into structures extending beyond Mpc sizes. The full LoTSS coupled with deep
optical and infrared surveys, such as DESI, WEAVE (Jin et al. 2023) and Euclid
(Euclid Collaboration et al. 2022), holds the potential to provide the
necessary confirmation of these results.

###### Acknowledgements.

This research project made use of the following Python packages: APLpy
(Robitaille & Bressert 2012), Astropy (Astropy Collaboration et al. 2013) and
NumPy (van der Walt et al. 2011). MB acknowledges support from the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC 2121 ”Quantum
Universe” - 390833306 and from the BMBF ErUM-Pro grant 05A2023. LOFAR (van
Haarlem et al. 2013) is the Low Frequency Array designed and constructed by
ASTRON. It has observing, data processing, and data storage facilities in
several countries, which are owned by various parties (each with their own
funding sources), and that are collectively operated by the ILT foundation
under a joint scientific policy. The ILT resources have benefited from the
following recent major funding sources: CNRS-INSU, Observatoire de Paris and
Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation
Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI),
Ireland; NWO, The Netherlands; The Science and Technology Facilities Council,
UK; Ministry of Science and Higher Education, Poland; The Istituto Nazionale
di Astrofisica (INAF), Italy. This research made use of the Dutch national
e-infrastructure with support of the SURF Cooperative (e-infra 180169) and the
LOFAR e-infra group. The Jülich LOFAR Long Term Archive and the German LOFAR
network are both coordinated and operated by the Jülich Supercomputing Centre
(JSC), and computing resources on the supercomputer JUWELS at JSC were
provided by the Gauss Centre for Supercomputing e.V. (grant CHTB00) through
the John von Neumann Institute for Computing (NIC). This research made use of
the University of Hertfordshire high-performance computing facility and the
LOFAR-UK computing facility located at the University of Hertfordshire and
supported by STFC [ST/P000096/1], and of the Italian LOFAR IT computing
infrastructure supported and operated by INAF, and by the Physics Department
of Turin university (under an agreement with Consorzio Interuniversitario per
la Fisica Spaziale) at the C3S Supercomputing Centre, Italy.

## References

  * Abazajian et al. (2009) Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543
  * Abdullah et al. (2020) Abdullah, M. H., Wilson, G., Klypin, A., et al. 2020, ApJS, 246, 2
  * Ahumada et al. (2020) Ahumada, R., Allende Prieto, C., Almeida, A., et al. 2020, ApJS, 249, 3
  * Andernach et al. (2021) Andernach, H., Jiménez-Andrade, E. F., & Willis, A. G. 2021, Galaxies, 9, 99
  * Assef et al. (2013) Assef, R. J., Stern, D., Kochanek, C. S., et al. 2013, ApJ, 772, 26
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Baba et al. (2004) Baba, K., Shibata, R., & Sibuya, M. 2004, Australian & New Zealand Journal of Statistics, 46, 657
  * Bagchi et al. (2014) Bagchi, J., Vivek, M., Vikram, V., et al. 2014, ApJ, 788, 174
  * Baldwin (1982) Baldwin, J. E. 1982, in Extragalactic Radio Sources, ed. D. S. Heeschen & C. M. Wade, Vol. 97, 21–24
  * Barthel et al. (1985) Barthel, P. D., Schilizzi, R. T., Miley, G. K., Jagers, W. J., & Strom, R. G. 1985, A&A, 148, 243
  * Beck et al. (2021) Beck, R., Szapudi, I., Flewelling, H., et al. 2021, MNRAS, 500, 1633
  * Becker et al. (1995) Becker, R. H., White, R. L., & Helfand, D. J. 1995, ApJ, 450, 559
  * Best & Heckman (2012) Best, P. N. & Heckman, T. M. 2012, MNRAS, 421, 1569
  * Best et al. (2014) Best, P. N., Ker, L. M., Simpson, C., Rigby, E. E., & Sabater, J. 2014, MNRAS, 445, 955
  * Best et al. (2023) Best, P. N., Kondapally, R., Williams, W. L., et al. 2023, MNRAS, 523, 1729
  * Bilicki et al. (2016) Bilicki, M., Peacock, J. A., Jarrett, T. H., et al. 2016, ApJS, 225, 5
  * Blumenthal & Miley (1979) Blumenthal, G. & Miley, G. 1979, A&A, 80, 13
  * Böckmann et al. (2023) Böckmann, K., Brüggen, M., Koribalski, B., et al. 2023, A&A, 677, A188
  * Brescia et al. (2014) Brescia, M., Cavuoti, S., Longo, G., & De Stefano, V. 2014, A&A, 568, A126
  * Brüggen et al. (2021) Brüggen, M., Reiprich, T. H., Bulbul, E., et al. 2021, A&A, 647, A3
  * Burenin (2017) Burenin, R. A. 2017, Astronomy Letters, 43, 507
  * Butler et al. (2019) Butler, A., Huynh, M., Kapińska, A., et al. 2019, A&A, 625, A111
  * Cantwell et al. (2020) Cantwell, T. M., Bray, J. D., Croston, J. H., et al. 2020, MNRAS, 495, 143
  * Clerc et al. (2016) Clerc, N., Merloni, A., Zhang, Y. Y., et al. 2016, MNRAS, 463, 4490
  * Condon et al. (1998) Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
  * Cutri et al. (2021) Cutri, R. M., Wright, E. L., Conrow, T., et al. 2021, VizieR Online Data Catalog, II/328
  * Dabhade et al. (2017) Dabhade, P., Gaikwad, M., Bagchi, J., et al. 2017, MNRAS, 469, 2886
  * Dabhade et al. (2020a) Dabhade, P., Mahato, M., Bagchi, J., et al. 2020a, A&A, 642, A153
  * Dabhade et al. (2020b) Dabhade, P., Röttgering, H. J. A., Bagchi, J., et al. 2020b, A&A, 635, A5
  * Dabhade et al. (2022) Dabhade, P., Shimwell, T. W., Bagchi, J., et al. 2022, A&A, 668, A64
  * Damen et al. (2009) Damen, M., Labbé, I., Franx, M., et al. 2009, ApJ, 690, 937
  * de Gasperin et al. (2023) de Gasperin, F., Edler, H. W., Williams, W. L., et al. 2023, A&A, 673, A165
  * de Gasperin et al. (2021) de Gasperin, F., Williams, W. L., Best, P., et al. 2021, A&A, 648, A104
  * Delvecchio et al. (2017) Delvecchio, I., Smolčić, V., Zamorani, G., et al. 2017, A&A, 602, A3
  * DESI Collaboration et al. (2023) DESI Collaboration, Adame, A. G., Aguilar, J., et al. 2023, arXiv e-prints, arXiv:2306.06308
  * Dey et al. (2019) Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168
  * Duncan (2022) Duncan, K. J. 2022, MNRAS, 512, 3662
  * Duncan et al. (2021) Duncan, K. J., Kondapally, R., Brown, M. J. I., et al. 2021, A&A, 648, A4
  * Euclid Collaboration et al. (2022) Euclid Collaboration, Scaramella, R., Amiaux, J., et al. 2022, A&A, 662, A112
  * Fanaroff & Riley (1974) Fanaroff, B. L. & Riley, J. M. 1974, MNRAS, 167, 31P
  * Flewelling et al. (2020) Flewelling, H. A., Magnier, E. A., Chambers, K. C., et al. 2020, ApJS, 251, 7
  * Garn et al. (2008a) Garn, T., Green, D. A., Riley, J. M., & Alexander, P. 2008a, MNRAS, 383, 75
  * Garn et al. (2008b) Garn, T., Green, D. A., Riley, J. M., & Alexander, P. 2008b, MNRAS, 387, 1037
  * Gürkan et al. (2015) Gürkan, G., Hardcastle, M. J., Jarvis, M. J., et al. 2015, MNRAS, 452, 3776
  * Gürkan et al. (2018) Gürkan, G., Hardcastle, M. J., Smith, D. J. B., et al. 2018, MNRAS, 475, 3010
  * Gürkan et al. (2022) Gürkan, G., Prandoni, I., O’Brien, A., et al. 2022, MNRAS, 512, 6104
  * Hao et al. (2010) Hao, J., McKay, T. A., Koester, B. P., et al. 2010, ApJS, 191, 254
  * Hardcastle (2018) Hardcastle, M. J. 2018, MNRAS, 475, 2768
  * Harwood et al. (2017) Harwood, J. J., Hardcastle, M. J., Morganti, R., et al. 2017, MNRAS, 469, 639
  * Hocuk & Barthel (2010) Hocuk, S. & Barthel, P. D. 2010, A&A, 523, A9
  * Ishwara-Chandra & Saikia (1999a) Ishwara-Chandra, C. H. & Saikia, D. J. 1999a, MNRAS, 309, 100
  * Ishwara-Chandra & Saikia (1999b) Ishwara-Chandra, C. H. & Saikia, D. J. 1999b, MNRAS, 309, 100
  * Jamrozy et al. (2008) Jamrozy, M., Konar, C., Machalski, J., & Saikia, D. J. 2008, MNRAS, 385, 1286
  * Jin et al. (2023) Jin, S., Trager, S. C., Dalton, G. B., et al. 2023, MNRAS[arXiv:2212.03981]
  * Kapahi (1989) Kapahi, V. K. 1989, AJ, 97, 1
  * Kirshner et al. (1987) Kirshner, R. P., Oemler, Augustus, J., Schechter, P. L., & Shectman, S. A. 1987, ApJ, 314, 493
  * Kochanek et al. (2012) Kochanek, C. S., Eisenstein, D. J., Cool, R. J., et al. 2012, ApJS, 200, 8
  * Koester et al. (2007) Koester, B. P., McKay, T. A., Annis, J., et al. 2007, ApJ, 660, 239
  * Komberg & Pashchenko (2009) Komberg, B. V. & Pashchenko, I. N. 2009, Astronomy Reports, 53, 1086
  * Konar et al. (2008) Konar, C., Jamrozy, M., Saikia, D. J., & Machalski, J. 2008, MNRAS, 383, 525
  * Konar et al. (2004) Konar, C., Saikia, D. J., Ishwara-Chandra, C. H., & Kulkarni, V. K. 2004, MNRAS, 355, 845
  * Kondapally et al. (2022) Kondapally, R., Best, P. N., Cochrane, R. K., et al. 2022, MNRAS, 513, 3742
  * Kondapally et al. (2021) Kondapally, R., Best, P. N., Hardcastle, M. J., et al. 2021, A&A, 648, A3
  * Koulouridis et al. (2021) Koulouridis, E., Clerc, N., Sadibekova, T., et al. 2021, A&A, 652, A12
  * Kozieł-Wierzbowska et al. (2017) Kozieł-Wierzbowska, D., Vale Asari, N., Stasińska, G., et al. 2017, ApJ, 846, 42
  * Kutkin et al. (2023) Kutkin, A. M., Oosterloo, T. A., Morganti, R., et al. 2023, A&A, 676, A37
  * Kuźmicz et al. (2019) Kuźmicz, A., Czerny, B., & Wildy, C. 2019, A&A, 624, A91
  * Kuźmicz & Jamrozy (2012) Kuźmicz, A. & Jamrozy, M. 2012, MNRAS, 426, 851
  * Kuźmicz & Jamrozy (2021) Kuźmicz, A. & Jamrozy, M. 2021, ApJS, 253, 25
  * Kuźmicz et al. (2018) Kuźmicz, A., Jamrozy, M., Bronarska, K., Janda-Boczar, K., & Saikia, D. J. 2018, ApJS, 238, 9
  * Lacy et al. (2020) Lacy, M., Baum, S. A., Chandler, C. J., et al. 2020, PASP, 132, 035001
  * Lacy et al. (2013) Lacy, M., Ridgway, S. E., Gates, E. L., et al. 2013, ApJS, 208, 24
  * Lan & Prochaska (2021) Lan, T.-W. & Prochaska, X. J. 2021, MNRAS, 502, 5104
  * Lara et al. (2000) Lara, L., Mack, K. H., Lacy, M., et al. 2000, A&A, 356, 63
  * Lara et al. (2001) Lara, L., Márquez, I., Cotton, W. D., et al. 2001, A&A, 378, 826
  * Ledlow & Owen (1996) Ledlow, M. J. & Owen, F. N. 1996, AJ, 112, 9
  * Liu et al. (2022a) Liu, A., Bulbul, E., Ghirardini, V., et al. 2022a, A&A, 661, A2
  * Liu et al. (2022b) Liu, C., Gebhardt, K., Cooper, E. M., et al. 2022b, ApJS, 261, 24
  * Luo et al. (2018) Luo, A. L., Zhao, Y. H., Zhao, G., & et al. 2018, VizieR Online Data Catalog: LAMOST DR4 catalogs (Luo+, 2018), VizieR On-line Data Catalog: V/153. Originally published in: 2018RAA..in.prep..L
  * Lusetti et al. (2024) Lusetti, G., de Gasperin, F., Cuciti, V., et al. 2024, MNRAS, 528, 141
  * Machalski (2011) Machalski, J. 2011, MNRAS, 413, 2429
  * Machalski et al. (2004) Machalski, J., Chyzy, K. T., & Jamrozy, M. 2004, Acta Astron., 54, 391
  * Machalski et al. (2007) Machalski, J., Chyży, K. T., Stawarz, Ł., & Kozieł, D. 2007, A&A, 462, 43
  * Machalski et al. (2001) Machalski, J., Jamrozy, M., & Zola, S. 2001, A&A, 371, 445
  * Madau & Dickinson (2014) Madau, P. & Dickinson, M. 2014, ARA&A, 52, 415
  * Magliocchetti (2022) Magliocchetti, M. 2022, A&A Rev., 30, 6
  * Mahato et al. (2022) Mahato, M., Dabhade, P., Saikia, D. J., et al. 2022, A&A, 660, A59
  * Malarecki et al. (2015) Malarecki, J. M., Jones, D. H., Saripalli, L., Staveley-Smith, L., & Subrahmanyan, R. 2015, MNRAS, 449, 955
  * Malarecki et al. (2013) Malarecki, J. M., Staveley-Smith, L., Saripalli, L., et al. 2013, MNRAS, 432, 200
  * Mann & Whitney (1947) Mann, H. B. & Whitney, D. R. 1947, The Annals of Mathematical Statistics, 18, 50
  * Marocco et al. (2021) Marocco, F., Eisenhardt, P. R. M., Fowler, J. W., et al. 2021, ApJS, 253, 8
  * Mateos et al. (2012) Mateos, S., Alonso-Herrero, A., Carrera, F. J., et al. 2012, MNRAS, 426, 3271
  * Mauch et al. (2003) Mauch, T., Murphy, T., Buttery, H. J., et al. 2003, MNRAS, 342, 1117
  * McConnell et al. (2020) McConnell, D., Hale, C. L., Lenc, E., et al. 2020, PASA, 37, e048
  * Merloni et al. (2012) Merloni, A., Predehl, P., Becker, W., et al. 2012, arXiv e-prints, arXiv:1209.3114
  * Mhlahlo & Jamrozy (2021) Mhlahlo, N. & Jamrozy, M. 2021, MNRAS, 508, 2910
  * Miley (1980) Miley, G. 1980, ARA&A, 18, 165
  * Miley & De Breuck (2008) Miley, G. & De Breuck, C. 2008, A&A Rev., 15, 67
  * Mingo et al. (2022) Mingo, B., Croston, J. H., Best, P. N., et al. 2022, MNRAS, 511, 3250
  * Miraghaei & Best (2017) Miraghaei, H. & Best, P. N. 2017, MNRAS, 466, 4346
  * Morganti et al. (2021) Morganti, R., Oosterloo, T. A., Brienza, M., et al. 2021, A&A, 648, A9
  * Mostert et al. (2023) Mostert, R. I. J., Oei, M. S. S. L., Barkus, B., et al. 2023, A&A, subm.
  * Murgia et al. (1999) Murgia, M., Fanti, C., Fanti, R., et al. 1999, A&A, 345, 769
  * Ochsenbein et al. (2000) Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23
  * Oei et al. (2023a) Oei, M. S. S. L., van Weeren, R. J., Gast, A. R. D. J. G. I. B., et al. 2023a, A&A, 672, A163
  * Oei et al. (2022) Oei, M. S. S. L., van Weeren, R. J., Hardcastle, M. J., et al. 2022, A&A, 660, A2
  * Oei et al. (2023b) Oei, M. S. S. L., van Weeren, R. J., Hardcastle, M. J., et al. 2023b, MNRAS, 518, 240
  * Onah et al. (2018) Onah, C. I., Ubachukwu, A. A., Odo, F. C., & Onuchukwu, C. C. 2018, Rev. Mexicana Astron. Astrofis., 54, 271
  * Pâris et al. (2018) Pâris, I., Petitjean, P., Aubourg, É., et al. 2018, A&A, 613, A51
  * Parma et al. (1999) Parma, P., Murgia, M., Morganti, R., et al. 1999, A&A, 344, 7
  * Pasini et al. (2022) Pasini, T., Brüggen, M., Hoang, D. N., et al. 2022, A&A, 661, A13
  * Pinjarkar et al. (2023) Pinjarkar, S., Hardcastle, M. J., Harwood, J. J., et al. 2023, MNRAS, 523, 620
  * Pirya et al. (2012) Pirya, A., Saikia, D. J., Singh, M., & Chandola, H. C. 2012, MNRAS, 426, 758
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A27
  * Pracy et al. (2016) Pracy, M. B., Ching, J. H. Y., Sadler, E. M., et al. 2016, MNRAS, 460, 2
  * Predehl et al. (2021) Predehl, P., Andritschke, R., Arefiev, V., et al. 2021, A&A, 647, A1
  * Rengelink et al. (1997) Rengelink, R. B., Tang, Y., de Bruyn, A. G., et al. 1997, A&AS, 124, 259
  * Robitaille & Bressert (2012) Robitaille, T. & Bressert, E. 2012, APLpy: Astronomical Plotting Library in Python, Astrophysics Source Code Library, record ascl:1208.017
  * Rowan-Robinson et al. (2013) Rowan-Robinson, M., Gonzalez-Solares, E., Vaccari, M., & Marchetti, L. 2013, MNRAS, 428, 1958
  * Rykoff et al. (2014) Rykoff, E. S., Rozo, E., Busha, M. T., et al. 2014, ApJ, 785, 104
  * Sabater et al. (2021) Sabater, J., Best, P. N., Tasse, C., et al. 2021, A&A, 648, A2
  * Safouris et al. (2009) Safouris, V., Subrahmanyan, R., Bicknell, G. V., & Saripalli, L. 2009, MNRAS, 393, 2
  * Saripalli et al. (2005) Saripalli, L., Hunstead, R. W., Subrahmanyan, R., & Boyce, E. 2005, AJ, 130, 896
  * Schlafly et al. (2019) Schlafly, E. F., Meisner, A. M., & Green, G. M. 2019, ApJS, 240, 30
  * Schoenmakers et al. (2000a) Schoenmakers, A. P., de Bruyn, A. G., Röttgering, H. J. A., van der Laan, H., & Kaiser, C. R. 2000a, MNRAS, 315, 371
  * Schoenmakers et al. (2000b) Schoenmakers, A. P., Mack, K. H., de Bruyn, A. G., et al. 2000b, A&AS, 146, 293
  * Sebastian et al. (2018) Sebastian, B., Ishwara-Chandra, C. H., Joshi, R., & Wadadekar, Y. 2018, MNRAS, 473, 4926
  * Shabala et al. (2017) Shabala, S. S., Deller, A., Kaviraj, S., et al. 2017, MNRAS, 464, 4706
  * Shimwell et al. (2022) Shimwell, T. W., Hardcastle, M. J., Tasse, C., et al. 2022, A&A, 659, A1
  * Shimwell et al. (2019) Shimwell, T. W., Tasse, C., Hardcastle, M. J., et al. 2019, A&A, 622, A1
  * Shulevski et al. (2019) Shulevski, A., Barthel, P. D., Morganti, R., et al. 2019, A&A, 628, A69
  * Simonte et al. (2023) Simonte, M., Andernach, H., Brüggen, M., Best, P. N., & Osinga, E. 2023, A&A, 672, A178
  * Simonte et al. (2022) Simonte, M., Andernach, H., Brüggen, M., et al. 2022, MNRAS, 515, 2032
  * Srivastava & Singal (2020) Srivastava, S. & Singal, A. K. 2020, MNRAS, 493, 3811
  * Subrahmanyan et al. (1996) Subrahmanyan, R., Saripalli, L., & Hunstead, R. W. 1996, MNRAS, 279, 257
  * Subrahmanyan et al. (2008) Subrahmanyan, R., Saripalli, L., Safouris, V., & Hunstead, R. W. 2008, ApJ, 677, 63
  * Takey et al. (2014) Takey, A., Schwope, A., & Lamer, G. 2014, A&A, 564, A54
  * Tasse et al. (2021) Tasse, C., Shimwell, T., Hardcastle, M. J., et al. 2021, A&A, 648, A1
  * Tielens et al. (1979) Tielens, A. G. G. M., Miley, G. K., & Willis, A. G. 1979, A&AS, 35, 153
  * Toba et al. (2019) Toba, Y., Yamashita, T., Nagao, T., et al. 2019, ApJS, 243, 15
  * Trouille et al. (2008) Trouille, L., Barger, A. J., Cowie, L. L., Yang, Y., & Mushotzky, R. F. 2008, ApJS, 179, 1
  * Turner & Shabala (2015) Turner, R. J. & Shabala, S. S. 2015, ApJ, 806, 59
  * Turner et al. (2023) Turner, R. J., Yates-Jones, P. M., Shabala, S. S., Quici, B., & Stewart, G. S. C. 2023, MNRAS, 518, 945
  * van Cappellen et al. (2022) van Cappellen, W. A., Oosterloo, T. A., Verheijen, M. A. W., et al. 2022, A&A, 658, A146
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22
  * van Haarlem et al. (2013) van Haarlem, M. P., Wise, M. W., Gunst, A. W., et al. 2013, A&A, 556, A2
  * Von Der Linden et al. (2007) Von Der Linden, A., Best, P. N., Kauffmann, G., & White, S. D. M. 2007, MNRAS, 379, 867
  * Wegner et al. (2003) Wegner, G., Bernardi, M., Willmer, C. N. A., et al. 2003, AJ, 126, 2268
  * Wen & Han (2015) Wen, Z. L. & Han, J. L. 2015, ApJ, 807, 178
  * Wen & Han (2018) Wen, Z. L. & Han, J. L. 2018, MNRAS, 481, 4158
  * Wen & Han (2021) Wen, Z. L. & Han, J. L. 2021, MNRAS, 500, 1003
  * Wen et al. (2018) Wen, Z. L., Han, J. L., & Yang, F. 2018, MNRAS, 475, 343
  * Williams et al. (2018) Williams, W. L., Calistro Rivera, G., Best, P. N., et al. 2018, MNRAS, 475, 3429
  * Williams et al. (2021) Williams, W. L., de Gasperin, F., Hardcastle, M. J. H., et al. 2021, A&A, 655, A40
  * Willis et al. (1974) Willis, A. G., Strom, R. G., & Wilson, A. S. 1974, Nature, 250, 625
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868
  * Wu & Shen (2022) Wu, Q. & Shen, Y. 2022, ApJS, 263, 42
  * Xu et al. (2020) Xu, H., Dai, Y. S., Huang, J.-S., et al. 2020, ApJ, 905, 103
  * Yoon et al. (2008) Yoon, J. H., Schawinski, K., Sheen, Y.-K., Ree, C. H., & Yi, S. K. 2008, ApJS, 176, 414
  * York et al. (2000) York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579
  * Zhang et al. (2022) Zhang, P., D’Amico, G., Senatore, L., Zhao, C., & Cai, Y. 2022, J. Cosmology Astropart. Phys., 2022, 036
  * Zhou et al. (2021) Zhou, R., Newman, J. A., Mao, Y.-Y., et al. 2021, MNRAS, 501, 3309
  * Zou et al. (2022) Zou, H., Sui, J., Xue, S., et al. 2022, Research in Astronomy and Astrophysics, 22, 065001
  * Zou et al. (2017) Zou, H., Zhou, X., Fan, X., et al. 2017, PASP, 129, 064101
  * Zovaro et al. (2022) Zovaro, H. R. M., Riseley, C. J., Taylor, P., et al. 2022, MNRAS, 509, 4997

##  Appendix A Images of the LDF-GRG sample

Figure 11: Cutouts of the LOFAR deep fields image around our GRGs at 150 MHz
with 3 and 24-$\sigma_{rms}$ red contours superimposed. The resolution of the
images is 6′′. The cyan circle identifies the position of the host galaxy. The
bar in the bottom-right corner represents an angular size of 1′, while the
name and the largest linear size (in Mpc) of the GRGs are reported in the top-
left and top-right corners, respectively. Figure 1: Continued. Figure 1:
Continued. Figure 1: Continued. Figure 1: Continued. Figure 1: Continued.
Figure 1: Continued. Figure 1: Continued.