11institutetext:  Leiden Observatory, Leiden University, Niels Bohrweg 2, 2300
RA Leiden, The Netherlands 22institutetext: Cahill Center for Astronomy and
Astrophysics, California Institute of Technology, 1216 E California Blvd, CA
91125 Pasadena, USA 33institutetext: Centre for Astrophysics Research,
University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
44institutetext: Somerville College, University of Oxford, Woodstock Road,
Oxford OX2 6HD, UK 55institutetext: CNRS & Sorbonne Université, UMR 7095,
Institut d’Astrophysique de Paris, 98 bis Boulevard Arago, 75014 Paris, France
66institutetext: Instituto de Astrofísica de Canarias, Calle Vía Láctea, s/n,
E-38205, La Laguna, Tenerife, Spain 77institutetext: Universidad de La Laguna,
Departamento de Astrofisica, E-38206, Tenerife, Spain 88institutetext: ASTRON,
the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD
Dwingeloo, The Netherlands 99institutetext: INAF–IRA, Via P. Gobetti 101,
40129 Bologna, Italy

# Luminous giants populate the dense Cosmic Web

The radio luminosity–environmental density relation for radio galaxies in
action††thanks:  Table 7’s extension to all selected GRGs, and an analogous
table for all selected general RGs, are available in Flexible Image Transport
System (FITS) format through the Centre de Données astronomiques de Strasbourg
(CDS). One can either use anonymous File Transfer Protocol (FTP) to
ftp://cdsarc.cds.unistra.fr (130.79.128.5) or visit
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/.

Martijn S. S. L. Oei 1122 Reinout J. van Weeren 11 Martin J. Hardcastle 33
Aivin R. D. J. G. I. B. Gast 44 Florent Leclercq 55 Huub J. A. Röttgering 11
Pratik Dabhade 6677 Tim W. Shimwell 1188 Andrea Botteon 99

(Received 6 June 2023 / Accepted 3 November 2023)

###### Abstract

Context.  Giant radio galaxies (GRGs, giant RGs, or giants) are megaparsec-
scale, jet-driven outflows from accretion disks of supermassive black holes,
and represent the most extreme pathway by which galaxies can impact the Cosmic
Web around them. A long-standing but unresolved question is why giants are so
much larger than other radio galaxies.

Aims.  It has been proposed that, in addition to having higher jet powers than
most RGs, giants might live in especially low-density Cosmic Web environments.
In this work, we aim to test this hypothesis by pinpointing Local Universe
giants and other RGs in physically principled, Bayesian large-scale structure
reconstructions.

Methods.  More specifically, we localised a LOFAR Two-metre Sky Survey (LoTSS)
DR2–dominated sample of luminous ($l_{\nu}(\nu=150\ \mathrm{MHz})\geq 10^{24}\
\mathrm{W\ Hz^{-1}}$) giants and a control sample of LoTSS DR1 RGs, both with
spectroscopic redshifts up to $z_{\mathrm{max}}=0.16$, in the BORG SDSS Cosmic
Web reconstructions. We measured the Cosmic Web density on a smoothing scale
of ${\sim}2.9\ \mathrm{Mpc}\ h^{-1}$ for each RG; for the control sample, we
then quantified the relation between RG radio luminosity and Cosmic Web
density. With the BORG SDSS tidal tensor, we also measured for each RG whether
the gravitational dynamics of its Cosmic Web environment resemble those of
clusters, filaments, sheets, or voids.

Results.  For both luminous giants and general RGs, the Cosmic Web density
distribution is gamma-like. Luminous giants populate large-scale environments
that tend to be denser than those of general RGs. This result is corroborated
by gravitational dynamics classification and a cluster catalogue crossmatching
analysis. We find that the Cosmic Web density around RGs with 150 MHz radio
luminosity $l_{\nu}$ is distributed as $1+\Delta_{\mathrm{RG}}\ |\
L_{\nu}=l_{\nu}\sim\Gamma(k,\theta)$, where
$k=4.8+0.2\cdot\textphnc{\Alamed}$, $\theta=1.4+0.02\cdot\textphnc{\Alamed}$,
and $\textphnc{\Alamed}\coloneqq\mathrm{log}_{10}(l_{\nu}\ (10^{23}\
\mathrm{W\ Hz^{-1}})^{-1})$.

Conclusions. This work presents more than a thousand inferred megaparsec-scale
densities around radio galaxies, which may be correct up to a factor of order
unity — except in clusters of galaxies, where the densities can be more than
an order of magnitude too low. We pave the way to a future in which
megaparsec-scale densities around RGs are common inferred quantities, which
help to better understand their dynamics, morphology, and interaction with the
enveloping Cosmic Web. Our data demonstrate that luminous giants inhabit
denser environments than general RGs. This shows that — at least at high jet
powers — low-density environments are no prerequisite for giant growth. Using
general RGs, we quantified the relation between radio luminosity at 150 MHz
and Cosmic Web density on a smoothing scale of ${\sim}2.9\ \mathrm{Mpc}\
h^{-1}$. This positive relation, combined with the discrepancy in radio
luminosity between known giants and general RGs, reproduces the discrepancy in
Cosmic Web density between known giants and general RGs. Our findings are
consistent with the view that giants are regular, rather than mechanistically
special, members of the radio galaxy population.

###### Key Words.:

radio continuum: galaxies – galaxies: active – jets – inter-galactic medium –
large-scale structure of Universe

##  1 Introduction

Supermassive Kerr black holes are key building blocks of the Universe, on
galactic and cosmological scales alike. During episodes of baryon accretion,
they turn into active galactic nuclei (AGN), launching winds and jets that
warm and rarefy the interstellar medium (e.g. Fabian 2012; King & Pounds
2015). These energy flows generally suppress the formation of new stars,
although local star formation enhancement can occur within expanding
kiloparsec-radius rings (e.g. Dugan et al. 2017). Meanwhile, jet-mediated AGN
outflows — also known as radio galaxies (RGs) — can have a vast, megaparsec-
scale reach, protruding from both the galaxy and its circumgalactic medium.
The synchrotron radiation from RGs illuminates their jets, lobes, cocoons, and
threads (e.g. Ramatsoku et al. 2020), and dominates the known extragalactic
radio sky. Because the behaviour of RGs is linked to the physics of black hole
accretion and galactic winds, the pressure field and magnetisation history
(e.g. Vazza et al. 2017) of the warm–hot intergalactic medium (warm–hot IGM,
or WHIM), as well as shocks (e.g. Nolting et al. 2019a), vorticity (e.g.
Nolting et al. 2019b), and cooling flows (e.g. Fabian et al. 1984) in the
intracluster medium, a precise understanding of the RG phenomenon is
indispensable to modern astrophysics.

A key goal of the study of RGs is to identify the main factors that determine
their dynamics and to formulate models (e.g. Scheuer 1974; Turner & Shabala
2015; Hardcastle 2018) that describe them quantitatively. One way to
investigate the growth of RGs is to search for particularly large examples and
to analyse what internal or external traits set them apart from the rest.
Following this logic, Oei et al. (2022) presented and studied Alcyoneus, a
giant radio galaxy (GRG, giant RG, or simply giant) whose proper length
component in the plane of the sky $l_{\mathrm{p}}=4.99\pm 0.04\ \mathrm{Mpc}$.
In general, giants are members of the RG population for which
$l_{\mathrm{p}}\geq l_{\mathrm{p,GRG}}$ — where the latter is some fixed
megaparsec-scale threshold — and that therefore rank among the largest RGs in
existence. However, despite being one of the largest RGs known, Alcyoneus is
not particularly luminous, and its host galaxy does not feature a particularly
massive central black hole or stellar population — at least, when compared to
other giants and their hosts (Oei et al. 2022). The question naturally arises
whether external properties, rather than those internal to the host galaxy,
are the most important drivers of RG growth.

It is well established that the IGM resists the growth of RGs by forcing jets
to convert a part of their kinetic energy into work spent to form lobe
cavities (e.g. Hardcastle & Croston 2020). The work needed to free up a cavity
is the product of its volume and the local pressure.111As the pressure field
in modern large-scale structure is not constant, as predicted by approximate
hydrostatic equilibrium, this expression only holds for sufficiently small
volumes. Following this line of reasoning, an RG with fixed intrinsic
properties should reach a larger end-of-life extent in more tenuous (and
colder) Cosmic Web (CW) environments. It has therefore been proposed that the
astonishing growth of giants might be explained by their presumptive tendency
to reside in tenuous parts of the Cosmic Web.

Several previous works have investigated the role of the enveloping Cosmic Web
density field on GRG growth. In all cases, the authors traced Cosmic Web
environments through three-dimensional galaxy positions estimated with
photometric or spectroscopic redshifts. The pioneering work of Subrahmanyan et
al. (2008) presented a case study of GRG MSH 05-22 and its Cosmic Web
environment as traced by 6dF data (Jones et al. 2004). Using a sample of 12
giants and environments traced by 2dF/AAOmega data (Sharp et al. 2006),
Malarecki et al. (2015) concluded that the lobes of giants grow in directions
that avoid denser regions of the Cosmic Web. In the most comprehensive study
yet, Lan & Prochaska (2021) used a sample of 110 giants and environments
traced by DESI Legacy Imaging Surveys Data Release (DR) 9 data (Dey et al.
2019). They did not find evidence that giants occur in more dilute
environments than non-giant RGs. A limitation of this latest study is the use
of photometric redshifts to probe the Cosmic Web density field.

In order to determine decisively how GRG growth and the Cosmic Web relate,
analyses with both more giants and more accurate Cosmic Web reconstructions
seem necessary. In this work, we present major developments in both regards.
First, the Low-Frequency Array (LOFAR; van Haarlem et al. 2013) Two-metre Sky
Survey (LoTSS; Shimwell et al. 2017) has made possible the discovery of
thousands of previously unknown giants in its Northern Sky imagery at the
observing frequency $\nu_{\mathrm{obs}}=144\ \mathrm{MHz}$ and at resolutions
$\theta_{\mathrm{FWHM}}\in\\{6^{\prime\prime},20^{\prime\prime},60^{\prime\prime},90^{\prime\prime}\\}$.
In particular, the joint search efforts of Dabhade et al. (2020) in the LoTSS
DR1 (Shimwell et al. 2019), those of Simonte et al. (2022) in the LoTSS Boötes
Deep Field, and those of Oei et al. (2023a) in the LoTSS DR2 (Shimwell et al.
2022) have tripled the total number of known giants, which now stands at
${\sim}3\cdot 10^{3}$. The number of known giants in the part of the Local
Universe covered by the Sloan Digital Sky Survey (SDSS; York et al. 2000) DR7
(Abazajian et al. 2009) has even quintupled — a fact whose relevance becomes
clear in Sect. 2.3. Second, the last two decades have seen the development of
principled, physics-based Bayesian inference techniques through which the
three-dimensional total (i.e. baryonic plus dark) matter density field of the
Cosmic Web can be reconstructed (e.g. Wandelt et al. 2004; Kitaura & Enßlin
2008; Jasche et al. 2010; Jasche & Kitaura 2010; Jasche & Wandelt 2013; Jasche
& Lavaux 2019). These techniques make use of the fact that the statistical
behaviour of the Early Universe’s total matter density field is known, as is
the dominant process by which this field evolved over cosmic time: gravity. By
simulating gravity acting on Early Universe density fields and comparing the
evolved fields to the observed spatial distribution of galaxies, the late-time
density field can be inferred. These late-time density fields subsequently
enable megaparsec-scale density measurements with uncertainties around
individual (giant) RGs. We shall use these measurements as an improved probe
of the Cosmic Web density field.

Section 2 presents the data: radio galaxy observables and late-time density
field reconstructions via which we probed the influence of the Cosmic Web on
the growth of giants. In Sect. 3, we explain how we combined these data to
determine megaparsec-scale densities and large-scale structure–type
probability distributions for giants and general RGs. Section 4 presents these
results, alongside a quantification of the relation between RG radio
luminosity and Cosmic Web density. We finally evaluate evidence for the claim
that the Cosmic Web affects RG growth. In Sect. 5, we explain how we tested
the reliability of RG Cosmic Web density measurements and dynamical
classifications and discuss caveats of and promising future extensions to the
current work, before we present conclusions in Sect. 6.

Figure 1:  Mollweide view of the sky showing the locations of all giants and
LoTSS DR1 RGs in the Local Universe for which we inferred Cosmic Web densities
and dynamical states. The background shows the Milky Way at $150\
\mathrm{MHz}$ (Zheng et al. 2017), on which we overlaid the LoTSS DR1
footprint (hatched dark rectangle). All RGs are drawn as lemniscates of
Bernoulli; we did not attempt to portray realistic shapes or position angles.
The colours represent redshifts $z\in(0,z_{\mathrm{max}}\coloneqq 0.16)$,
whilst the diameters are proportional to projected proper lengths
$l_{\mathrm{p}}\in({\sim}1\ \mathrm{kpc},4.6\ \mathrm{Mpc})$. Giants are
translucent. Upon zooming in, the reader can appreciate the wide variety of
observed radio galaxy sizes.

For consistency with Oei et al. (2022), Oei et al. (2023a), and Mostert & Oei
(2024), we assumed a flat, inflationary $\Lambda$CDM model with parameters
from Planck Collaboration et al. (2020): $h=0.6766$,
$\Omega_{\mathrm{BM,0}}=0.0490$, $\Omega_{\mathrm{M,0}}=0.3111$, and
$\Omega_{\Lambda,0}=0.6889$, where
$\Omega_{\mathrm{DM,0}}\coloneqq\Omega_{\mathrm{M,0}}-\Omega_{\mathrm{BM,0}}=0.2621$
and $H_{0}\coloneqq h\cdot 100\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$. With ‘Local
Universe’, we refer to the spherical region of space observed to have redshift
$z<z_{\mathrm{max}}\coloneqq 0.16$. All reported redshifts are heliocentric.
Terminology-wise, we strictly distinguish an RG (a radio-bright structure of
plasma and magnetic fields, consisting of a core, jets, hotspots, lobes, a
cocoon, and collimated synchrotron threads) from the host galaxy that has
generated it. As in our previous works, we define giants to be RGs with
projected proper lengths $l_{\mathrm{p}}\geq l_{\mathrm{p,GRG}}\coloneqq 0.7\
\mathrm{Mpc}$.222In Cosmic Web filament environments, where giants appear most
common (Sect. 4), lobes may expand along the Hubble flow, rendering their
proper and comoving extents different. To avoid ambiguity, we stress that our
projected lengths are proper instead of comoving. A less precise synonym for
‘projected proper length’ often found in the literature is ‘largest linear
size’ (LLS). We define the spectral index $\alpha$ so that it relates to flux
density $F_{\nu}$ at frequency $\nu$ as $F_{\nu}\propto\nu^{\alpha}$. Under
this convention, and when synchrotron self-absorption is negligible, radio
spectral indices are typically negative.

##  2 Data

To compare the Cosmic Web environments of Local Universe giants to those of
Local Universe radio galaxies, we combined a GRG catalogue, an RG catalogue,
and a Cosmic Web density field reconstruction. All three data sets are
publicly available.

###  2.1 Giant radio galaxies

As our source of giants we used the catalogue aggregated by Oei et al.
(2023a), which contains 3341 giants with $l_{\mathrm{p}}\geq
l_{\mathrm{p,GRG}}\coloneqq 0.7\ \mathrm{Mpc}$.333Oei et al. (2023a) provide
references to all discovery articles. This catalogue is intended to be
complete up to (and including) September 2022. Thanks to the steradian-scale
Northern Sky coverage of the LoTSS, combined with its arcsecond-scale
resolution and sensitivity up to degree scales, LoTSS-discovered giants
(Dabhade et al. 2020; Simonte et al. 2022; Oei et al. 2023a) dominate the
catalogue.

In their LoTSS DR2 manual search for giants, Oei et al. (2023a) used an
angular length threshold of
$\frac{1}{2}\left(\phi_{\mathrm{max}}-\phi_{\mathrm{min}}\right)=5^{\prime}$
to limit the duration of their search to a manageable few hundred hours. At
the same time, this $5^{\prime}$-threshold ensured that the search would yield
most giants with sufficient surface brightness in the Local Universe.444Oei et
al. (2023a) illustrate this point in their Fig. 9. This has been by design:
Oei et al. (2023a) aimed to build a surface brightness–limited, but otherwise
complete census of Local Universe giants in the LoTSS DR2 footprint, with the
intent of localising them within the Cosmic Web density field reconstructions
of Sect. 2.3.

As elaborated upon in Sect. 3.1, we retained 281 giants in the part of the
Local Universe where Cosmic Web analysis is possible. Of these, 260 have
spectroscopic redshifts; only these giants could be reliably localised. We
show their sky locations in Fig. 1. LoTSS DR2 discoveries make up 208 of the
final 260 giants (80%). Through six example LoTSS DR2 giants, Fig. 2 provides
the reader a sense of the quality of the radio imagery underpinning this work,
from which angular lengths $\phi$ have been inferred, a sense of the
reliability of our host galaxy identification, from which spectroscopic
redshifts $z_{\mathrm{s}}$ have inferred, and a sense of the morphological and
surface brightness diversity of the objects under consideration in this study.

Figure 2:  LoTSS DR2 cutouts at central observing frequency
$\nu_{\mathrm{obs}}=144\ \mathrm{MHz}$ and resolution
$\theta_{\mathrm{FWHM}}=6^{\prime\prime}$, centred around giant-generating
BCGs of Local Universe clusters. All giants shown are discoveries of Oei et
al. (2023a) for which no previous images have been published. Each cutout
covers a solid angle of $15^{\prime}\times 15^{\prime}$. Contours signify 3,
5, and 10 sigma-clipped standard deviations above the sigma-clipped median.
For scale, we show the stellar Milky Way disk (with a diameter of 50 kpc)
generated using the Ringermacher & Mead (2009) formula, alongside a 3 times
inflated version. Each DESI Legacy Imaging Surveys DR9 $(g,r,z)$ inset shows
the central $3^{\prime}\times 3^{\prime}$ region.

###  2.2 General radio galaxies

In order to determine whether the Cosmic Web environments of giants are
exceptional, we must create a sample of reference Cosmic Web environments. For
this reason, we also localised within the Cosmic Web a sample of general RGs,
selected without regard for their length.

In particular, our starting point for this sample is the radio-bright active
galactic nucleus (RLAGN) sub-sample described in Hardcastle et al. (2019),
which contains 23,344 of the 318,520 sources (7%) in the LoTSS DR1 value-added
catalogue of Williams et al. (2019). To construct their sub-sample, Hardcastle
et al. (2019) combined multiple criteria that separate RLAGN from star-forming
galaxies (SFGs). At the low redshifts considered in this work
($z<z_{\mathrm{max}}\coloneqq 0.16$), most RLAGN are identified through SDSS
spectroscopy, and we therefore expect high RLAGN completeness and minimal
contamination from SFGs.

As elaborated upon in Sect. 3.1, we retained 1870 LoTSS DR1 RGs in the part of
the Local Universe where Cosmic Web analysis is possible. Of these, 1443 have
spectroscopic redshifts; only these RGs could be reliably localised. We show
also their sky locations in Fig. 1. The LoTSS DR1 image quality is very
similar to the LoTSS DR2 image quality, which Fig. 2 illustrates.

###  2.3 Cosmic Web late-time density field

To localise RGs within the Cosmic Web, we used data products from the Bayesian
Origin Reconstruction from Galaxies (BORG; Jasche & Wandelt 2013) SDSS run
(Jasche et al. 2015). The BORG SDSS uses second-order Lagrangian perturbation
theory (2LPT; Bouchet et al. 1995) to forward model structure formation and
evaluates the plausibility of a proposed structure formation history by
comparing its late-time density field to the three-dimensional positions of
galaxies in the SDSS DR7 Main Galaxy Sample (MGS; Abazajian et al. 2009). As a
result, the BORG SDSS provides a late-time total555The BORG algorithm does not
differentiate between baryonic and dark matter, assuming identical behaviour
on the multi-megaparsec scale. matter density field posterior for the part of
the Local Universe covered by the SDSS DR7 footprint. The posterior is
represented by a Hamiltonian Monte Carlo (HMC; Duane et al. 1987) Markov chain
of approximately ten thousand samples after the burn-in phase. Each sample
covers the same volume of $(750\ \mathrm{Mpc}\ h^{-1})^{3}$ extent with a
cubical grid of $256^{3}$ voxels. Thus, the side length of a BORG SDSS voxel
$L=\frac{1}{256}\cdot 750\ \mathrm{Mpc}\ h^{-1}\approx 2.9\ \mathrm{Mpc}\
h^{-1}$. The late-time posterior can be compactly summarised by taking the
mean and standard deviation (SD) of the samples on a per-voxel basis. In this
work, we use both the individual samples and the summary cubes — the latter of
which we call the BORG SDSS mean and SD. However, by using these summary
cubes, information contained in higher moments of single-voxel posteriors and
inter-voxel correlations remains unused.

Leclercq et al. (2015) have extended the BORG SDSS by calculating, for each
sample, the Cosmic Web classification as stipulated by the $T$-web definition
(Hahn et al. 2007). The result is a probabilistic classification with a
marginal distribution for each voxel. As one aspect of a broader information
theoretic analysis, Leclercq et al. (2016) demonstrated how these
classifications can be used to predict galaxy properties, such as $g-r$
colours. We used these classifications to characterise the dynamical
environments of luminous giants and general RGs.

Both the BORG SDSS posterior and a posterior from another BORG run, the BORG
2M++ (Jasche & Lavaux 2019), have been used before to relate properties of
active galaxies to the density of the enveloping Cosmic Web (Frank et al.
2016; Porqueres et al. 2018). Our work is the first to relate properties of
radio galaxies to BORG (or BORG-like) Cosmic Web reconstructions.

##  3 Methods

As pointed out in Sect. 1, the number of known giants has increased
substantially in recent years. For example, within the Local Universe covered
by the SDSS DR7, the manual search of Oei et al. (2023a) alone has quintupled
the number of known giants with spectroscopic redshifts — from 52 to 260.
Meanwhile, the BORG SDSS now offers the first physically principled,
probabilistic reconstruction of the total matter density field over this
volume. In this section, we explain how we combined both advances by
localising giants and general RGs within the BORG SDSS.

###  3.1 BORG SDSS localisation procedure

For each RG in our two samples (be it a giant or a general RG), we first
transformed its right ascension, declination, and redshift into a vector with
comoving coordinates $\mathbf{r}\coloneqq\left[x,y,z\right]^{\top}$ following
Jasche et al. (2015)’s coordinate system convention. This transformation is
cosmology-dependent. The BORG SDSS adopts the cosmological parameters of
Jasche et al. (2015) to convert SDSS DR7 MGS redshifts into radial comoving
distances and subsequently infer the structure formation history of the Local
Universe. In order to obtain valid RG localisations, it was imperative that we
used the same conversion between redshift and radial comoving distance. We
therefore adopted the Jasche et al. (2015) cosmology for this particular step.
Afterwards, we simply associated each RG to the BORG SDSS voxel nearest to
$\mathbf{r}$.

After associating RGs to voxels, we checked whether the late-time BORG SDSS
density field at these locations is sufficiently constrained. More precisely,
we evaluated whether — at each RG’s voxel — the survey response operator of
Jasche et al. (2015)’s lowest $r$-band absolute magnitude bin
($-21.00<M_{r}<-20.33$), $R^{0}$, equals or exceeds some threshold
$R^{0}_{\mathrm{min}}$. If indeed $R^{0}(\mathbf{r})\geq
R^{0}_{\mathrm{min}}$, we retained the RG for Cosmic Web analysis; if not, we
discarded it. In this work, we chose $R^{0}_{\mathrm{min}}=0.1$. Higher
choices for $R^{0}_{\mathrm{min}}$ rid the sample of comparatively
uninformative (that is, more prior-dominated) density and dynamical state
measurements, but come at the obvious cost of a reduced sample size.

Figure 3:  Example localisations of giant radio galaxies within the large-
scale structure of the Local Universe. The top row shows two cluster giants,
the middle row shows two filament giants, and the bottom row shows two sheet
giants. For each giant, we show a slice of constant Cartesian comoving $z$
through the late-time BORG SDSS posterior mean total matter density field and
a LoTSS DR2 6′′ image at $\nu_{\mathrm{obs}}=144\ \mathrm{MHz}$ (inset).
Outside of the SDSS DR7–constrained volume, the posterior mean tends to the
Universe’s late-time mean total matter density $\bar{\rho}_{0}$. The locations
of the giants are marked by white circles.

In Fig. 3, we show six example localisations of giants in the Local Universe.
We pinpointed the giants in the upper two panels to galaxy clusters, the
giants in the middle two panels to filaments, and the giants in the bottom two
panels to sheets. Next, in order to obtain density distributions for each
retained RG, we explored two methods.

####  3.1.1 Fixed voxel method

In the ‘fixed voxel method’, we considered for each RG the marginal posterior
density distribution at its voxel — that is the posterior density distribution
for that voxel in isolation — despite the fact that the BORG SDSS provides
reconstructions with complex inter-voxel density correlations. This method is
the simplest of the two methods we have used.

####  3.1.2 Flexible voxel method

The fixed voxel method has at least two disadvantages. One issue is the fact
that the BORG SDSS has inferred the Cosmic Web with a limited set of bright
SDSS DR7 galaxies, which (amongst other factors) causes reconstruction
uncertainty. In practice, this means that a given cluster or filament may
morph and wiggle around in different BORG SDSS samples. If one does not wiggle
around the voxel to sample from accordingly (but sticks with the same voxel
all the time), one regularly samples outside of the cluster or filament in
which the RG of interest resides. This, of course, biases the inferred
densities low.

On top of Cosmic Web reconstruction uncertainty, host galaxy peculiar motion
uncertainty leads to additional difficulty in the determination of an RG’s
Cosmic Web density. In Appendix A, we show that spectroscopic redshift
uncertainties generally lead to sub-voxel localisation uncertainty, even after
taking low-mass galaxy cluster–like peculiar motion into account. For high-
mass galaxy clusters however, peculiar motion can cause multi-voxel
localisation errors. We also show that photometric redshift uncertainties
cause localisation uncertainties of $10^{1}$–$10^{2}\ \mathrm{Mpc}$, or up to
tens of voxels, that are unworkably large. Cosmic Web localisation therefore
only seems possible for RGs with spectroscopically detected hosts.

To counteract the fixed voxel method’s tendency to sample densities outside of
the clusters and filaments in which the RGs truly reside, we propose a
flexible voxel method. In this method, the voxel considered hitherto serves as
a reference voxel, around which we search for the most likely correct voxel to
sample from. More precisely, we iterate over $1000$ BORG SDSS samples equally
spaced within the MCMC, and consider for each sample all voxels in a sphere of
radius $5\ \mathrm{Mpc}\ h^{-1}$ around the reference voxel. We then simply
adopt the voxel with the highest density as the most likely correct voxel for
that sample. This procedure encapsulates our prior belief that the massive
ellipticals which give rise to RGs are more likely to occur in a high-density
region than in a low-density region of the same volume.

The search radius of $5\ \mathrm{Mpc}\ h^{-1}$ chosen here is arbitrary to
some degree. Clearly, for the flexible voxel method to be any different from
the fixed voxel method, this radius must exceed the side length of a single
voxel, $2.9\ \mathrm{Mpc}\ h^{-1}$. For larger radii, we are able to correct
for larger peculiar motion errors, and thus provide more accurate densities
for RGs located in massive galaxy clusters. At the same time, for larger
radii, we are at risk of straying too far from the reference voxel; for
example, this could lead to sampling cluster-like densities for RGs that
actually reside in an adjacent filament. Because the majority of RGs appears
to reside in filaments, and because the gravity solver of the BORG SDSS
significantly limits the usefulness of cluster densities (see Sect. 5), we
chose a relatively ‘small’ search radius of $5\ \mathrm{Mpc}\ h^{-1}$ — less
than two voxels in each direction.

In Fig. 4, we compare the mean Cosmic Web densities inferred through the two
methods for all RGs considered in this work. Flexible voxel–based relative
densities are typically a factor two higher. The methods agree most often at
the lowest redshifts, where the BORG SDSS Cosmic Web reconstructions are least
uncertain.

Figure 4:  Comparison between Cosmic Web densities of giants and LoTSS DR1 RGs
in the Local Universe, inferred via two variations of our BORG SDSS–based
approach. For each RG, we show the mean density measured through Sect. 3.1.1’s
fixed voxel method (horizontal axis) and through Sect. 3.1.2’s flexible voxel
method (vertical axis).

###  3.2 BORG SDSS localisation in practice

As described in Sect. 3.1, for each giant and general RG in our samples, we
measured the marginal posterior density RV. Of 281 giants that lie within the
part of the BORG SDSS volume where $R^{0}(\mathbf{r})\geq
R^{0}_{\mathrm{min}}$, there are $260$ with a spectroscopic redshift (93%),
which are therefore suitable for Cosmic Web analysis. Of these, 208 (80%) are
LoTSS DR2 discoveries (Oei et al. 2023a). In exactly the same way, of the 1870
LoTSS DR1 RGs that lie within the constrained BORG SDSS volume, we retained
1443 specimina with spectroscopic redshifts (77%), which we selected for
Cosmic Web analysis.

At the BORG SDSS resolution of $2.9\ \mathrm{Mpc}\ h^{-1}$ per voxel side, the
baryonic matter density field approximately equals the dark matter density
field scaled down by a factor
$\frac{\Omega_{\mathrm{BM,0}}}{\Omega_{\mathrm{DM,0}}}$. The BORG SDSS does
not distinguish between these two fields. Instead, it provides the sum of the
baryonic and dark matter density field: the total matter density field. In
this article, we exclusively mention relative total matter densities — total
matter densities divided by today’s cosmic mean total matter density
$\Omega_{\mathrm{M,0}}\rho_{\mathrm{c},0}$, where $\rho_{\mathrm{c},0}$ is
today’s critical density. To be less verbose, we refer to just ‘relative
densities’, and write $1+\delta$, where $\delta$ denotes overdensity in the
usual sense.

Figure 5: Distributions for the measured relative total matter density RVs
$1+\Delta_{\mathrm{GRG,obs}}\ |\ 1+\Delta_{\mathrm{GRG}}=1+\delta$ and
$1+\Delta_{\mathrm{RG,obs}}\ |\ 1+\Delta_{\mathrm{RG}}=1+\delta$ for an
individual giant (top row; solid), and for an individual RG (bottom row;
hatched). Fixed voxel method densities (left column; green) are lower than
flexible voxel method densities (right column; orange). The MLE-fitted
lognormal PDFs (solid curves) demonstrate that the distributions are almost
lognormal.  Figure 6:  Probability density functions (PDFs) of the relative
total matter density RVs of 260 Local Universe giants (top row), and of 1443
Local Universe RGs (bottom row), determined through the fixed voxel method
(left column) and flexible voxel method (right column). These RVs correspond
to the density field smoothed to a scale of $2.9\ \mathrm{Mpc}\ h^{-1}$. We
also show PDFs of gamma-distributed RVs with parameters obtained via MLE, both
before (solid lines) and after (dash-dotted lines) a heteroskedasticity
correction. We warn that these distributions are affected by surface
brightness selection, and thus represent known populations only.

##  4 Results

In this section, we present the first empirical distributions of the Cosmic
Web density and dynamical state around luminous giants and general RGs. We
also determine the RG radio luminosity–Cosmic Web density relation, and test
whether it can cause the density distribution discrepancy between luminous
giants and general RGs.

###  4.1 Cosmic Web density distributions

The relative density posterior distributions of individual RGs resemble
lognormal distributions, irrespective of whether the fixed or flexible voxel
method was used to determine them. Figure 5 demonstrates this through example
marginals for a typical luminous giant and a typical LoTSS DR1 RG. Indeed
modelling $1+\Delta_{\mathrm{GRG,obs}}\ |\
1+\Delta_{\mathrm{GRG}}=1+\delta\sim\mathrm{Lognormal}(\mu,\sigma^{2})$ and
$1+\Delta_{\mathrm{RG,obs}}\ |\
1+\Delta_{\mathrm{RG}}=1+\delta\sim\mathrm{Lognormal}(\mu,\sigma^{2})$, we can
succinctly summarise each RG’s measured density distribution with two
parameters. We provide these, for 50 out of 260 giants, in Table 7. (For
access to such data for all giants, and for similar data on LoTSS DR1 giants,
see the table’s footnote.)

Table 1:  Cosmic Web properties for 50 out of 260 BORG SDSS–constrained
giants, sorted by right ascension.777 We provide MLE parameters of lognormal
fits to total matter (i.e. both baryonic and dark matter) relative density
distributions. We report parameters for both the fixed and flexible voxel
methods. Densities span the BORG SDSS comoving voxel volume of $(2.9\
\mathrm{Mpc}\ h^{-1})^{3}$ and are relative to today’s cosmic mean total
matter density. The mean and variance of a density distribution are
$\mathbb{E}[1+\Delta_{\mathrm{GRG,obs}}\ |\
1+\Delta_{\mathrm{GRG}}=1+\delta_{i}]=\exp{(\mu+\frac{1}{2}\sigma^{2})}$ and
$\mathbb{V}[1+\Delta_{\mathrm{GRG,obs}}\ |\
1+\Delta_{\mathrm{GRG}}=1+\delta_{i}]=(\exp{(\sigma^{2})}-1)\exp{(2\mu+\sigma^{2})}$.
The components of the $T$-web probability vector
$\@vec{p}=(p_{1},p_{2},p_{3},p_{4})$ correspond to clusters, filaments,
sheets, and voids, respectively. Cluster masses stem from crossmatching with
Wen & Han (2015). We share a full table (with all 260 entries), alongside an
analogous table for our 1443 selected LoTSS DR1 RGs, via the CDS.

rank | host coordinates | spectroscopic | Cosmic Web density | Cosmic Web density | Cosmic Web $T$-web | cluster mass | host
---|---|---|---|---|---|---|---
$\downarrow$ | equatorial J2000 $(\degree)$ | redshift $z_{\mathrm{s}}\ (1)$ | fixed voxel $\mu,\sigma^{2}\ (1)$ | flexible voxel $\mu,\sigma^{2}\ (1)$ | probabilities $\@vec{p}\ (1)$ | $M_{500}\ (10^{14}\ M_{\odot})$ | BCG
1 | $111.57678,\ 38.63317$ |  $0.15387$ $\pm$ $3\cdot 10^{-5}$ |  $-0.49$, $0.60$ |  +$0.60$, $0.45$ | $(0.0,$ | $0.2,$ | $0.7,$ | $0.1)$ | - | -
2 | $113.77185,\ 41.97432$ |  $0.08732$ $\pm$ $2\cdot 10^{-5}$ |  +$1.94$, $0.06$ |  +$2.43$, $0.03$ | $(0.1,$ | $0.9,$ | $0.0,$ | $0.0)$ | - | -
3 | $114.98866,\ 43.98326$ |  $0.14887$ $\pm$ $2\cdot 10^{-5}$ |  $-0.73$, $0.52$ |  +$0.42$, $0.42$ | $(0.0,$ | $0.2,$ | $0.6,$ | $0.2)$ | - | -
4 | $115.96358,\ 28.35779$ |  $0.10633$ $\pm$ $2\cdot 10^{-5}$ |  +$0.88$, $0.53$ |  +$1.87$, $0.20$ | $(0.1,$ | $0.7,$ | $0.2,$ | $0.0)$ | - | -
5 | $116.03381,\ 43.99169$ |  $0.13484$ $\pm$ $2\cdot 10^{-5}$ |  +$0.95$, $0.23$ |  +$2.41$, $0.05$ | $(0.0,$ | $0.9,$ | $0.1,$ | $0.0)$ | - | -
6 | $117.53962,\ 26.73550$ |  $0.13042$ $\pm$ $2\cdot 10^{-5}$ |  +$1.59$, $0.17$ |  +$2.16$, $0.06$ | $(0.2,$ | $0.8,$ | $0.0,$ | $0.0)$ | - | -
7 | $118.14466,\ 35.83983$ |  $0.13650$ $\pm$ $3\cdot 10^{-5}$ |  +$0.95$, $0.25$ |  +$2.09$, $0.10$ | $(0.0,$ | $0.8,$ | $0.2,$ | $0.0)$ | - | -
8 | $119.15943,\ 32.46314$ |  $0.14620$ $\pm$ $2\cdot 10^{-5}$ |  +$0.47$, $0.50$ |  +$1.28$, $0.23$ | $(0.1,$ | $0.5,$ | $0.4,$ | $0.0)$ | - | -
9 | $119.26227,\ 36.62242$ |  $0.13948$ $\pm$ $3\cdot 10^{-5}$ |  $-0.02$, $0.42$ |  +$0.94$, $0.22$ | $(0.0,$ | $0.2,$ | $0.8,$ | $0.0)$ | - | -
10 | $119.27937,\ 35.97967$ |  $0.13296$ $\pm$ $2\cdot 10^{-5}$ |  +$1.15$, $0.28$ |  +$2.08$, $0.07$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | $1.3$ | $y$
11 | $119.47158,\ 36.67279$ |  $0.12844$ $\pm$ $2\cdot 10^{-5}$ |  +$1.26$, $0.15$ |  +$1.66$, $0.08$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | $0.8$ | $y$
12 | $120.25565,\ 13.83117$ |  $0.10872$ $\pm$ $2\cdot 10^{-5}$ |  +$2.46$, $0.03$ |  +$2.58$, $0.02$ | $(0.7,$ | $0.3,$ | $0.0,$ | $0.0)$ | $3.2$ | $y$
13 | $120.30535,\ 34.67522$ |  $0.08269$ $\pm$ $2\cdot 10^{-5}$ |  +$1.56$, $0.08$ |  +$1.93$, $0.04$ | $(0.2,$ | $0.8,$ | $0.0,$ | $0.0)$ | - | -
14 | $120.38318,\ 47.60446$ |  $0.15684$ $\pm$ $3\cdot 10^{-5}$ |  $-0.30$, $0.56$ |  +$1.14$, $0.32$ | $(0.0,$ | $0.4,$ | $0.6,$ | $0.0)$ | - | -
15 | $120.80515,\ 51.93263$ |  $0.06947$ $\pm$ $2\cdot 10^{-5}$ |  +$2.00$, $0.04$ |  +$2.21$, $0.03$ | $(0.3,$ | $0.7,$ | $0.0,$ | $0.0)$ | - | -
16 | $121.01419,\ 40.80261$ |  $0.12617$ $\pm$ $1\cdot 10^{-5}$ |  +$0.85$, $0.20$ |  +$1.59$, $0.11$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | $2.5$ | $y$
17 | $121.32789,\ 28.62417$ |  $0.14256$ $\pm$ $2\cdot 10^{-5}$ |  +$1.18$, $0.39$ |  +$1.78$, $0.17$ | $(0.1,$ | $0.6,$ | $0.3,$ | $0.0)$ | - | -
18 | $121.38042,\ 25.80317$ |  $0.13698$ $\pm$ $2\cdot 10^{-5}$ |  +$1.02$, $0.31$ |  +$1.89$, $0.09$ | $(0.0,$ | $0.8,$ | $0.2,$ | $0.0)$ | - | -
19 | $121.42954,\ 16.23223$ |  $0.10002$ $\pm$ $2\cdot 10^{-5}$ |  +$2.46$, $0.03$ |  +$2.67$, $0.02$ | $(0.6,$ | $0.4,$ | $0.0,$ | $0.0)$ | $0.9$ | $y$
20 | $122.14848,\ 38.91450$ |  $0.04081$ $\pm$ $1\cdot 10^{-5}$ |  +$1.79$, $0.04$ |  +$2.26$, $0.02$ | $(0.1,$ | $0.9,$ | $0.0,$ | $0.0)$ | - | -
21 | $122.28630,\ 29.67904$ |  $0.12572$ $\pm$ $2\cdot 10^{-5}$ |  +$1.18$, $0.18$ |  +$2.03$, $0.06$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | - | -
22 | $122.31280,\ 41.28897$ |  $0.13344$ $\pm$ $3\cdot 10^{-5}$ |  +$0.86$, $0.38$ |  +$1.48$, $0.14$ | $(0.2,$ | $0.6,$ | $0.2,$ | $0.0)$ | - | -
23 | $123.41209,\ 41.36503$ |  $0.09984$ $\pm$ $2\cdot 10^{-5}$ |  +$1.05$, $0.15$ |  +$2.27$, $0.04$ | $(0.0,$ | $0.7,$ | $0.3,$ | $0.0)$ | $0.8$ | $y$
24 | $123.91600,\ 50.54063$ |  $0.13803$ $\pm$ $3\cdot 10^{-5}$ |  +$1.16$, $0.29$ |  +$1.85$, $0.08$ | $(0.2,$ | $0.6,$ | $0.2,$ | $0.0)$ | - | -
25 | $124.44458,\ 54.70087$ |  $0.11867$ $\pm$ $2\cdot 10^{-5}$ |  +$2.22$, $0.07$ |  +$2.63$, $0.03$ | $(0.2,$ | $0.8,$ | $0.0,$ | $0.0)$ | $4.9$ | $n$
26 | $125.78050,\ 10.59828$ |  $0.06605$ $\pm$ $1\cdot 10^{-4}$ |  +$0.95$, $0.14$ |  +$1.53$, $0.05$ | $(0.2,$ | $0.6,$ | $0.2,$ | $0.0)$ | - | -
27 | $126.47417,\ 41.60254$ |  $0.15359$ $\pm$ $2\cdot 10^{-5}$ |  +$1.38$, $0.38$ |  +$2.03$, $0.15$ | $(0.2,$ | $0.7,$ | $0.1,$ | $0.0)$ | - | -
28 | $127.86456,\ 32.32412$ |  $0.05120$ $\pm$ $1\cdot 10^{-5}$ |  +$0.69$, $0.12$ |  +$1.78$, $0.04$ | $(0.0,$ | $1.0,$ | $0.0,$ | $0.0)$ | - | -
29 | $127.99873,\ 30.65853$ |  $0.10704$ $\pm$ $2\cdot 10^{-5}$ |  +$1.80$, $0.08$ |  +$2.07$, $0.04$ | $(0.4,$ | $0.6,$ | $0.0,$ | $0.0)$ | - | -
30 | $128.14208,\ 4.41000$ |  $0.10600$ $\pm$ $1\cdot 10^{-5}$ |  +$0.47$, $0.24$ |  +$1.34$, $0.10$ | $(0.0,$ | $0.3,$ | $0.7,$ | $0.0)$ | - | -
31 | $129.03256,\ 26.81206$ |  $0.08780$ $\pm$ $2\cdot 10^{-5}$ |  +$1.22$, $0.12$ |  +$1.91$, $0.04$ | $(0.1,$ | $0.7,$ | $0.2,$ | $0.0)$ | - | -
32 | $130.18516,\ 58.69709$ |  $0.14392$ $\pm$ $3\cdot 10^{-5}$ |  +$0.15$, $0.40$ |  +$1.21$, $0.19$ | $(0.0,$ | $0.4,$ | $0.6,$ | $0.0)$ | $1.2$ | $y$
33 | $130.50142,\ 38.93753$ |  $0.11977$ $\pm$ $2\cdot 10^{-5}$ |  +$1.82$, $0.09$ |  +$2.23$, $0.04$ | $(0.2,$ | $0.8,$ | $0.0,$ | $0.0)$ | $0.9$ | $n$
34 | $130.53983,\ 41.94039$ |  $0.12565$ $\pm$ $2\cdot 10^{-5}$ |  +$0.67$, $0.31$ |  +$1.93$, $0.08$ | $(0.0,$ | $0.8,$ | $0.2,$ | $0.0)$ | - | -
35 | $131.24330,\ 42.07739$ |  $0.14932$ $\pm$ $2\cdot 10^{-5}$ |  +$1.57$, $0.30$ |  +$2.40$, $0.08$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | - | -
36 | $131.36289,\ 44.92399$ |  $0.15062$ $\pm$ $3\cdot 10^{-5}$ |  +$1.70$, $0.27$ |  +$2.19$, $0.10$ | $(0.3,$ | $0.6,$ | $0.1,$ | $0.0)$ | $0.8$ | $n$
37 | $132.73665,\ 42.80464$ |  $0.09238$ $\pm$ $1\cdot 10^{-5}$ |  +$1.05$, $0.17$ |  +$1.68$, $0.05$ | $(0.2,$ | $0.8,$ | $0.0,$ | $0.0)$ | $1.3$ | $y$
38 | $133.45742,\ 14.87390$ |  $0.06933$ $\pm$ $2\cdot 10^{-5}$ |  +$2.00$, $0.04$ |  +$2.24$, $0.02$ | $(0.8,$ | $0.2,$ | $0.0,$ | $0.0)$ | $0.7$ | $y$
39 | $133.80972,\ 49.19334$ |  $0.11773$ $\pm$ $1\cdot 10^{-5}$ |  +$0.99$, $0.36$ |  +$2.14$, $0.09$ | $(0.0,$ | $0.8,$ | $0.2,$ | $0.0)$ | - | -
40 | $134.23524,\ 47.95594$ |  $0.14927$ $\pm$ $2\cdot 10^{-5}$ |  +$1.20$, $0.44$ |  +$2.00$, $0.17$ | $(0.1,$ | $0.7,$ | $0.2,$ | $0.0)$ | - | -
41 | $134.58152,\ 46.37086$ |  $0.11711$ $\pm$ $1\cdot 10^{-5}$ |  +$1.19$, $0.24$ |  +$1.54$, $0.11$ | $(0.2,$ | $0.6,$ | $0.2,$ | $0.0)$ | - | -
42 | $135.36742,\ 55.04455$ | $0.04602$ |  +$2.15$, $0.03$ |  +$2.35$, $0.01$ | $(0.4,$ | $0.6,$ | $0.0,$ | $0.0)$ | - | -
43 | $135.45964,\ 55.92429$ |  $0.14090$ $\pm$ $2\cdot 10^{-5}$ |  +$1.60$, $0.21$ |  +$2.06$, $0.09$ | $(0.4,$ | $0.6,$ | $0.0,$ | $0.0)$ | - | -
44 | $136.09149,\ 19.72455$ |  $0.09953$ $\pm$ $2\cdot 10^{-5}$ |  +$1.30$, $0.10$ |  +$1.72$, $0.06$ | $(0.1,$ | $0.9,$ | $0.0,$ | $0.0)$ | - | -
45 | $137.13529,\ 18.28034$ |  $0.11554$ $\pm$ $3\cdot 10^{-5}$ |  +$1.17$, $0.17$ |  +$1.96$, $0.06$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | - | -
46 | $137.23745,\ 58.38413$ |  $0.14364$ $\pm$ $3\cdot 10^{-5}$ |  +$0.17$, $0.52$ |  +$1.33$, $0.20$ | $(0.0,$ | $0.6,$ | $0.4,$ | $0.0)$ | - | -
47 | $137.87830,\ 48.54753$ |  $0.10865$ $\pm$ $2\cdot 10^{-5}$ |  +$0.32$, $0.30$ |  +$1.45$, $0.09$ | $(0.0,$ | $0.6,$ | $0.4,$ | $0.0)$ | - | -
48 | $139.74751,\ 31.86128$ |  $0.06194$ $\pm$ $1\cdot 10^{-5}$ |  +$1.65$, $0.08$ |  +$1.78$, $0.04$ | $(0.6,$ | $0.4,$ | $0.0,$ | $0.0)$ | - | -
49 | $139.95191,\ 57.84889$ |  $0.13695$ $\pm$ $3\cdot 10^{-5}$ |  +$1.21$, $0.17$ |  +$1.67$, $0.09$ | $(0.1,$ | $0.8,$ | $0.1,$ | $0.0)$ | $0.8$ | $n$
50 | $140.34212,\ 54.86499$ |  $0.04469$ $\pm$ $1\cdot 10^{-5}$ |  +$1.90$, $0.02$ |  +$2.31$, $0.02$ | $(0.0,$ | $0.6,$ | $0.4,$ | $0.0)$ | - | -

By aggregating just the means of these marginal relative density
distributions, we could analyse the distributions for our observed populations
as a whole. In green, Fig. 6 shows kernel density estimated (KDE) relative
density distributions for both luminous giants (top panels) and the broader
population of RGs (bottom panels) in the Local Universe. The panels in the
left column represent the fixed voxel method, whilst the panels in the right
column represent the flexible voxel method. More precisely, these KDE
distributions approximate the distributions of the observed GRG relative
density RV $1+\Delta_{\mathrm{GRG,obs}}$ and the observed RG relative density
RV $1+\Delta_{\mathrm{RG,obs}}$.

We sought to summarise these distributions parametrically. After testing
various two-parameter distributions for continuous, non-negative RVs (such as
the gamma distribution and the lognormal distribution), the KDE distributions
of $1+\Delta_{\mathrm{RG,obs}}$ in the bottom panels — which are based on all
1443 selected LoTSS DR1 RGs — appeared best approximated by a gamma
distribution.888In Appendix B, we provide an astrophysical–statistical
argument that could explain the gamma distribution’s emergence. Maximum
likelihood estimation (MLE) suggested
$1+\Delta_{\mathrm{RG,obs}}\sim\Gamma(k=2.1,\theta=1.8)$ for the fixed voxel
method and $1+\Delta_{\mathrm{RG,obs}}\sim\Gamma(k=3.6,\theta=1.8)$ for the
flexible voxel method. Similarly, fitting a gamma distribution to
$1+\Delta_{\mathrm{GRG,obs}}$ through MLE gave
$1+\Delta_{\mathrm{GRG,obs}}\sim\Gamma(k=2.5,\theta=1.9)$ for the fixed voxel
method and $1+\Delta_{\mathrm{GRG,obs}}\sim\Gamma(k=4.7,\theta=1.6)$ for the
flexible voxel method, although the latter does not provide a tight fit. These
gamma distributions constitute practical, two-parameter representations of the
underlying data and are drawn as solid lines in Fig. 6.

The mean of a gamma-distributed RV $1+\Delta\sim\Gamma(k,\theta)$ is
$\mathbb{E}[1+\Delta]=k\theta$. Thus, for the fixed voxel method,
$\mathbb{E}[1+\Delta_{\mathrm{GRG,obs}}]=4.8$ and
$\mathbb{E}[1+\Delta_{\mathrm{RG,obs}}]=3.8$. Similarly, for the flexible
voxel method, $\mathbb{E}[1+\Delta_{\mathrm{GRG,obs}}]=7.5$ and
$\mathbb{E}[1+\Delta_{\mathrm{RG,obs}}]=6.5$. Naively, it appears that we can
conclude that, in a statistical sense, giants occupy denser regions of the
Cosmic Web than radio galaxies in general. Using a two-sample
Kolmogorov–Smirnov (KS) test, we formally tested the null hypothesis that the
observed giant and general RG relative density distributions of Fig. 6 share a
common underlying distribution. For both the fixed and flexible voxel methods,
the $p$-value $p\lesssim 10^{-6}$. The null hypothesis is thus rejected (for
typical significance levels). These distributions come with at least two major
caveats, however.

Figure 7:  Mean lobe surface brightness estimates $B_{\nu}$ at
$\nu_{\mathrm{ref}}=150\ \mathrm{MHz}$ for LoTSS DR1 RGs at $z<0.2$, shown as
a function of projected proper length $l_{\mathrm{p}}$. During the majority
(${\sim}90\%$) of their lifetime, RGs grow while maintaining a roughly
constant radio luminosity. If, additionally, growth is shape-preserving,
surface brightness decreases quadratically with $l_{\mathrm{p}}$. The seven
golden lines therefore denote approximate evolutionary tracks of RGs at $z=0$
with end-of-life radio luminosities $l_{\nu}\in\\{10^{21}\ \mathrm{W\
Hz^{-1}},...,10^{27}\ \mathrm{W\ Hz^{-1}}\\}$. The region of parameter space
where RGs can become detectable giants, $l_{\nu}\geq
l_{\nu,\mathrm{min}}\coloneqq 10^{24}\ \mathrm{W\ Hz^{-1}}$, is shaded gold.
The region of parameter space occupied by giants, $l_{\mathrm{p}}\geq
l_{\mathrm{p,GRG}}\coloneqq 0.7\ \mathrm{Mpc}$, is shaded green. The region of
parameter space inaccessible to the LoTSS, $B_{\nu}<\sigma_{I_{\nu}}=25\
\mathrm{Jy\ deg^{-2}}$, is shaded red.

####  4.1.1 Heteroskedasticity

First, BORG SDSS density measurements are heteroskedastic: higher densities
have higher measurement errors than lower densities (see e.g. Fig. 6 of Jasche
et al. 2015). Heteroskedasticity causes the observed distributions to differ
from the intrinsic distributions; not only by widening them (as also occurs in
the more familiar homoskedastic setting), but also by systematically shifting
the distributions towards lower densities. Appendix C presents a simple method
to infer distributions corrected for this effect. We thus distinguish between
$1+\Delta_{\mathrm{GRG,obs}}$ and $1+\Delta_{\mathrm{GRG}}$, and between
$1+\Delta_{\mathrm{RG,obs}}$ and $1+\Delta_{\mathrm{RG}}$; in both cases, the
latter RV is heteroskedasticity-corrected. As the effect induces minor shifts
only, we assumed that the heteroskedasticity-free distributions are also
approximately gamma. We then applied the method of Appendix C to both our GRG
and RG data. Having no closed-form MLE expressions at our disposal, we
obtained the MLE parameters of the heteroskedasticity-free gamma distributions
by evaluating the likelihood function over an exhaustive grid of $(k,\theta)$
values. For the fixed voxel method, this yielded
$1+\Delta_{\mathrm{GRG}}\sim\Gamma(k=3.7,\theta=1.3)$ and
$1+\Delta_{\mathrm{RG}}\sim\Gamma(k=3.0,\theta=1.3)$;
$\mathbb{E}[1+\Delta_{\mathrm{GRG}}]=4.8$ and
$\mathbb{E}[1+\Delta_{\mathrm{RG}}]=3.9$. For the flexible voxel method, this
yielded $1+\Delta_{\mathrm{GRG}}\sim\Gamma(k=6.7,\theta=1.1)$ and
$1+\Delta_{\mathrm{RG}}\sim\Gamma(k=4.8,\theta=1.4)$;
$\mathbb{E}[1+\Delta_{\mathrm{GRG}}]=7.4$ and
$\mathbb{E}[1+\Delta_{\mathrm{RG}}]=6.7$. We draw the corresponding PDFs as
grey, dash-dotted lines in Fig. 6.

Figure 8:  PDFs of the observed RG relative total matter density RV given a
radio luminosity at $\nu_{\mathrm{ref}}=150\ \mathrm{MHz}$, using MLE
parameter values for the model described in Sect. 4.2. The black contours
denote CDF values. We overplot all 1443 selected LoTSS DR1 RGs (dots), with
those above the empirical median density coloured red, and those below
coloured blue. We used flexible voxel method densities. For fixed voxel method
densities, see Fig. 19.

####  4.1.2 Surface brightness selection

The second — and plausibly most important — caveat is the imprint of selection
effects, and in particular the surface brightness selection effect induced by
the LoTSS noise level $\sigma_{I_{\nu}}=25\ \mathrm{Jy\ deg^{-2}}$ at
$\theta_{\mathrm{FWHM}}=6^{\prime\prime}$. This effect causes RGs whose lobe
surface brightnesses are below the noise level to evade sample inclusion. Lobe
surface brightness depends on the RG’s projected proper length
$l_{\mathrm{p}}$, morphological parameters $\textphnc{\Af}_{l}$ and
$\textphnc{\Af}_{l_{\nu}}$, radio luminosity $l_{\nu}$, redshift $z$, and lobe
spectral index $\alpha$. Equation 40 of Oei et al. (2023a) presents an
approximate formula for the mean lobe surface brightness $B_{\nu}$:

$\displaystyle
B_{\nu}(\nu_{\mathrm{ref}})=\frac{2\cdot\textphnc{\Af}_{l_{\nu}}\cdot
l_{\nu}(\nu_{\mathrm{ref}})}{\pi^{2}\cdot\mathbb{E}[D](\eta(\textphnc{\Af}_{l}))\cdot\textphnc{\Af}^{2}_{l}\cdot
l_{\mathrm{p}}^{2}\cdot(1+z)^{3-\alpha}},$ (1)

where $\textphnc{\Af}_{l_{\nu}}$ is the fraction of the total radio luminosity
that belongs to the lobes, $\textphnc{\Af}_{l}$ is the fraction of the RG’s
axis that lies inside the lobes,
$\eta(\textphnc{\Af}_{l})=\frac{\textphnc{\Af}_{l}}{2-\textphnc{\Af}_{l}}$,
and $\mathbb{E}[D]$ is the mean deprojection factor as given by Eq. A.29 of
Oei et al. (2023a):

$\displaystyle\mathbb{E}[D](\eta)=(1+\eta)\left(\frac{\pi}{2}-\frac{\eta}{\sqrt{1-\eta^{2}}}\ln{\left(\frac{\sqrt{1-\eta^{2}}}{\eta}+\frac{1}{\eta}\right)}\right).$
(2)

To explicitly demonstrate that surface brightness selection biases our Local
Universe GRG and RG samples towards short lengths and high radio luminosities,
we approximated $B_{\nu}(\nu_{\mathrm{ref}}\coloneqq 150\ \mathrm{MHz})$ for
all LoTSS DR1 RGs with $z<0.2$, assuming
$\textphnc{\Af}_{l}=\textphnc{\Af}_{l_{\nu}}=0.3$ (Oei et al. 2022) and
$\alpha=-0.7$. Figure 7 shows the results. As mature RGs grow, they trace out
a curve approximately parallel to the golden lines (which represent constant
radio luminosities), directed towards the bottom-right of the plot. If their
lives last long enough, they therefore reach a length — which depends on their
radio luminosity — beyond which they disappear into the noise of the LoTSS. In
particular, our calculations suggest that the only giants that can be detected
through the LoTSS are those with $l_{\nu}(\nu_{\mathrm{ref}})\gtrsim 10^{24}\
\mathrm{W\ Hz^{-1}}$.999In Fig. 7, the approximate evolutionary track for
$l_{\nu}=10^{24}\ \mathrm{W\ Hz^{-1}}$ shows that when RGs with this radio
luminosity turn into giants (i.e. when the fourth golden line intersects the
green line), they disappear in the LoTSS noise (this golden line intersects
the red line). Indeed, the lowest–radio luminosity giant in the LoTSS DR1 GRG
sample of Dabhade et al. (2020) — which consists of 239 giants — has
$l_{\nu}=1\cdot 10^{24}\ \mathrm{W\ Hz^{-1}}$. Clearly, if RGs with
$l_{\nu}\sim 10^{23}\ \mathrm{W\ Hz^{-1}}$ can also form giants, then this
subpopulation will be missing in LoTSS GRG samples. Equivalently, the giants
that _are_ included in our LoTSS GRG sample necessarily have radio
luminosities $l_{\nu}\gtrsim 10^{24}\ \mathrm{W\ Hz^{-1}}$. It is
overwhelmingly likely that the GRG sample’s radio luminosities are biased
high. Concerningly, Croston et al. (2019) have shown that, for RGs at $z<0.4$,
radio luminosity and environmental richness correlate positively. This gives
rise to the possibility that the trend shown in Fig. 6, where giants appear to
reside in denser Cosmic Web environments than RGs in general, is a result of
surface brightness selection: a GRG sample biased high in radio luminosity
will be biased high in Cosmic Web density.

Figure 9:  Posterior distribution over $k(0)$, $\theta(0)$, $k^{\prime}(0)$,
and $\theta^{\prime}(0)$, based on selected Local Universe LoTSS DR1 RGs. We
show all two-parameter marginals of the likelihood function, with contours
enclosing $50\%$ and $90\%$ of total probability. We mark the MLE (grey dot)
and the maximum a posteriori (MAP; grey cross). The one-parameter marginals
again show the MLE (dash-dotted line), a mean-centred interval of standard
deviation–sized half-width (hashed region), and a median-centred $90\%$
credible interval (shaded region). We used flexible voxel method densities.
For fixed voxel method densities, see Fig. 20.

###  4.2 Radio luminosity–Cosmic Web density relation

Interestingly, our data allow us to revisit the radio luminosity–environmental
richness correlation that Croston et al. (2019) have found. In Fig. 8, we plot
all 1443 selected LoTSS DR1 RGs in Cosmic Web density–radio luminosity
parameter space. By eye, it appears that for higher radio luminosities, the
Cosmic Web density distribution shifts upwards. To quantify the apparent
tendency that more luminous radio galaxies occupy denser regions of the Cosmic
Web, we extended the gamma distribution fitting discussed in Sect. 4.1. In
particular, we assumed that the RG relative density RV at some fixed radio
luminosity, $1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}=l_{\nu}$, is gamma
distributed:

$\displaystyle 1+\Delta_{\mathrm{RG}}\ |\
L_{\nu}=l_{\nu}\sim\Gamma(k(\textphnc{\Alamed}),\theta(\textphnc{\Alamed})).$
(3)

Instead of assuming constant parameters $k$ and $\theta$, we assumed that $k$
and $\theta$ depend on radio luminosity:

$\displaystyle\textphnc{\Alamed}=\textphnc{\Alamed}(l_{\nu})\coloneqq\mathrm{log}_{\mathrm{10}}\left(\frac{l_{\nu}}{l_{\nu,\mathrm{ref}}}\right).$
(4)

The natural minimal extension to assuming that $k$ and $\theta$ remain
constant as $\textphnc{\Alamed}$ changes, is to consider Taylor polynomial
approximations of degree one (around $\textphnc{\Alamed}=0$ — i.e. Maclaurin
polynomials):

$\displaystyle k(\textphnc{\Alamed})\approx
k(0)+k^{\prime}\left(0\right)\cdot\textphnc{\Alamed},\ \ \ \ \ \ \ \
\theta(\textphnc{\Alamed})\approx\theta(0)+\theta^{\prime}\left(0\right)\cdot\textphnc{\Alamed}.$
(5)

However, this model does not allow for a direct comparison to our data, which
are samples from $1+\Delta_{\mathrm{RG,obs}}\ |\ L_{\nu}=l_{\nu}$ rather than
from $1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}=l_{\nu}$. Therefore, we had to extend
the model by applying heteroskedastic measurement errors. In line with
Appendix C, we assumed

$\displaystyle 1+\Delta_{\mathrm{RG,obs}}\ |\
1+\Delta_{\mathrm{RG}}=1+\delta\sim\mathrm{Lognormal}(\mu,\sigma^{2}),$ (6)

where the parameters $\mu=\mu(a,b)$ and $\sigma^{2}=\sigma^{2}(a,b)$ are given
by

$\displaystyle\mu=\ln{\left(1+\delta\right)}-\frac{1}{2}\ln{\left(1+a\left(1+\delta\right)^{b-2}\right)};$
(7)
$\displaystyle\sigma^{2}=\ln{\left(1+a\left(1+\delta\right)^{b-2}\right)}.$
(8)

The parameters $a$ and $b$ characterise the heteroskedasticity. By considering
the means and SDs of the relative densities of all selected LoTSS DR1 RGs, we
found $a=0.4$ and $b=1.1$ for the fixed voxel method and $a=0.4$ and $b=0.9$
for the flexible voxel method.

In order to obtain a fit that is valid over many orders of magnitude in radio
luminosity, rather than only over the order of magnitude in which most data
are concentrated, we calculated the likelihood in such a way that all decades
in radio luminosity that contain at least $20$ RGs receive equal weight. We
simply determined the likelihood function, and thus the posterior distribution
for a flat prior, through a grid search over the four parameters $k(0)$,
$\theta(0)$, $k^{\prime}(0)$, and $\theta^{\prime}(0)$. We defined
$l_{\nu,\mathrm{ref}}\coloneqq 10^{23}\ \mathrm{W\ Hz^{-1}}$. Figure 9
visualises this flat-prior posterior. In addition, Fig. 8 shows percentile
scores of $10\%$, $50\%$, $90\%$, $95\%$, and $99\%$ corresponding to the MAP
model. Figures 19 and 20 demonstrate that, although the densities are lower,
the results remain qualitatively the same upon switching to the fixed voxel
method. We summarise the posterior for both methods in Table 4.2.

Table 2: MAP and posterior mean and SD of the free parameters in our Cosmic
Web density–RG radio luminosity inference.111111As we assume a flat prior, the
MAP parameters are also the MLE parameters.

fixed voxel method:

parameter | MAP | posterior mean and SD
---|---|---
$k(0)$ | $2.9$ | $2.9\pm 0.1$
$\theta(0)$ | $1.4$ | $1.4\pm 0.1$
$k^{\prime}(0)$ | $0.04$ | $0.04\pm 0.08$
$\theta^{\prime}(0)$ | $0.08$ | $0.08\pm 0.04$

flexible voxel method:

parameter | MAP | posterior mean and SD
---|---|---
$k(0)$ | $4.8$ | $4.7\pm 0.2$
$\theta(0)$ | $1.4$ | $1.4\pm 0.1$
$k^{\prime}(0)$ | $0.2$ | $0.2\pm 0.1$
$\theta^{\prime}(0)$ | $0.02$ | $0.02\pm 0.04$

As values of $k^{\prime}(0)$ and especially $\theta^{\prime}(0)$ tend to be
positive, we find evidence for a positive radio luminosity–environmental
richness relation, in line with the results of Croston et al. (2019). More
quantitatively, we call the relation positive (at
$l_{\nu}=l_{\nu,\mathrm{ref}}$) when

$\displaystyle\left.\frac{\mathrm{d}\mathbb{E}[1+\Delta_{\mathrm{RG}}\ |\
L_{\nu}=l_{\nu}]}{\mathrm{d}\textphnc{\Alamed}}\right|_{\textphnc{\Alamed}=0}=k(0)\theta^{\prime}(0)+k^{\prime}(0)\theta(0)>0.$
(9)

We combined the posterior with Eq. 9 to obtain the posterior probability that
the radio luminosity–Cosmic Web density relation is positive — that is to say
that more luminous RGs tend to live in denser regions of the Cosmic Web. For
both the fixed and flexible voxel methods, this probability exceeds $99\%$.

####  4.2.1 Surface brightness selection

However, we must again be mindful of a possible surface brightness selection
effect. RGs of a given radio luminosity may be able to grow larger in more
tenuous Cosmic Web environments, as radio luminosity traces jet power (in
close-to-linear fashion; e.g. Willott et al. 1999; Shabala & Godfrey 2013;
Hardcastle 2018). If this is indeed the case, then the lobe surface
brightnesses of RGs in tenuous environments are likely lower than those of RGs
with the same radio luminosity in dense environments. As a result, of all RGs
with the same radio luminosity, those in tenuous environments are more likely
to be missing from observational samples. This selection effect, that favours
higher relative densities, holds for all radio luminosities, and could only
explain the trend of Fig. 8 if the effect becomes more severe for higher radio
luminosities. On the one hand, RGs with higher radio luminosities at a given
relative density are expected to grow longer. On the other hand, as is clear
from Fig. 7, the projected proper length range over which RGs retain
detectable lobe surface brightnesses increases with radio luminosity. In other
words, at higher radio luminosities, larger RGs remain detectable.

Whether the net result of these counteracting effects is less or more bias
towards higher relative densities as radio luminosity increases, depends on
the scaling between radio luminosity and growth. Self-similar growth predicts
(Kaiser & Alexander 1997) that an RG’s projected proper length scales with jet
power $Q_{\mathrm{jet}}$, Cosmic Web density at the host galaxy
$1+\delta_{\mathrm{g}}$, and time since birth $t$ as

$\displaystyle l_{\mathrm{p}}\propto
Q_{\mathrm{jet}}^{\frac{1}{5-\beta}}\cdot(1+\delta_{\mathrm{g}})^{\frac{-1}{5-\beta}}\cdot
t^{\frac{3}{5-\beta}},$ (10)

where $-\beta$ is the exponent of the local Cosmic Web density profile:

$\displaystyle(1+\delta)(r)\propto(1+\delta_{\mathrm{g}})\cdot r^{-\beta}.$
(11)

Under the additional assumption of equipartition between the relativistic
electron–positron kinetic energy density and the magnetic field energy
density, and upon neglecting electron–positron energy losses (such as
adiabatic, synchrotron, and inverse Compton losses), the RG’s radio luminosity
obeys (Shabala & Godfrey 2013)

$\displaystyle l_{\nu}\propto
Q_{\mathrm{jet}}^{\frac{5+p}{6}}l_{\mathrm{p}}^{3-\frac{(4+\beta)(5+p)}{12}}.$
(12)

Here, $-p$ is the exponent of the initial post-acceleration electron–positron
kinetic energy distribution $n(\gamma)\propto\gamma^{-p}$, valid for Lorentz
factors between some minimum and maximum values $\gamma_{\mathrm{min}}$ and
$\gamma_{\mathrm{max}}$. Solving for $Q_{\mathrm{jet}}$, substituting the
result in Eq. 10, and solving for $l_{\mathrm{p}}$, yields — at fixed
$1+\delta_{\mathrm{g}}$ and $t$ —

$\displaystyle l_{\mathrm{p}}\propto l_{\nu}^{\frac{12}{36+(5+p)(6-3\beta)}}.$
(13)

For a constant-density environment (i.e. $\beta=0$) and a strong-shock
spectral index $p=2$, one finds $l_{\mathrm{p}}\propto
l_{\nu}^{\nicefrac{{2}}{{13}}}$. For an environment where density falls off
quadratically with distance, $\beta=2$, and therefore $l_{\mathrm{p}}\propto
l_{\nu}^{\nicefrac{{1}}{{3}}}$. Meanwhile, Eq. 1 predicts that the maximum
detectable projected proper length for RGs with a given morphology, at a given
redshift, and with given lobe spectral index, scales as
$l_{\mathrm{p,max}}\propto\sqrt{l_{\nu}}$. These arguments suggest that
$l_{\mathrm{p,max}}$ increases more rapidly with $l_{\nu}$ than
$l_{\mathrm{p}}$ does. Thus, amongst a family of RGs with varying radio
luminosity but seen at the same age (i.e. time after birth) and with the same
morphology and Cosmic Web density, the most luminous is most easily
detectable. This provides some reassurance that the trend seen in Fig. 8 is
real: surface brightness selection at a fixed relative density seems to become
less severe as radio luminosity increases.

However, there are at least two caveats here. First, more luminous RGs may
live longer than less luminous RGs, as jointly suggested by the radio
luminosity–stellar mass relation and the stellar mass–radio AGN occurrence
relation of Sabater et al. (2019). As a result, in a snapshot of the RG
population at a given instant, more luminous RGs may typically be older than
less luminous RGs. Figure 13 illustrates this point. If more luminous RGs
indeed tend to be older than less luminous RGs, then the relation between
projected length and radio luminosity at a fixed Cosmic Web density for a
snapshot population will be steeper than the one of Eq. 13, which assumes
equal ages. Secondly, we warn for the possibility — which remains subject of
debate (Mingo et al. 2019) — that the RG shape distribution at a fixed Cosmic
Web density may change with radio luminosity.

Figure 10:  Monte Carlo simulation of a population of nine galaxies (thick
stripes) with successive active and quiescent phases. We mark the active
phases (thin stripes). Each galaxy spends a different fraction
$\textphnc{\Af}_{\mathrm{a}}$ of the time in the active phase. We model the
time spent in both phases as draws from exponential distributions:
$t_{\mathrm{a}}\sim\mathrm{Exp}(\lambda_{\mathrm{a}})$ and
$t_{\mathrm{q}}\sim\mathrm{Exp}(\lambda_{\mathrm{q}})$. We assume that all
galaxies share the same quiescent phase rate parameter
$\lambda_{\mathrm{q}}=10\ \mathrm{Gyr}^{-1}$, although this may not be
entirely correct (Turner & Shabala 2015). This assumption fixes the active
phase rate parameter:
$\lambda_{\mathrm{a}}=(\textphnc{\Af}_{\mathrm{a}}^{-1}-1)\lambda_{\mathrm{q}}$.131313
We define
$\displaystyle\textphnc{\Af}_{\mathrm{a}}\coloneqq\frac{\mathbb{E}[t_{\mathrm{a}}]}{\mathbb{E}[t_{\mathrm{a}}+t_{\mathrm{q}}]}=\frac{\mathbb{E}[t_{\mathrm{a}}]}{\mathbb{E}[t_{\mathrm{a}}]+\mathbb{E}[t_{\mathrm{q}}]}=\frac{\lambda_{\mathrm{a}}^{-1}}{\lambda_{\mathrm{a}}^{-1}+\lambda_{\mathrm{q}}^{-1}}=\left(1+\frac{\lambda_{\mathrm{a}}}{\lambda_{\mathrm{q}}}\right)^{-1}.$
(14) We note that the more natural definition
$\displaystyle\textphnc{\Af}_{\mathrm{a}}\coloneqq\mathbb{E}\left[\frac{t_{\mathrm{a}}}{t_{\mathrm{a}}+t_{\mathrm{q}}}\right]=\mathbb{E}\left[\left(1+\frac{t_{\mathrm{q}}}{t_{\mathrm{a}}}\right)^{-1}\right]$
(15) gives, for $r\coloneqq\frac{\lambda_{\mathrm{a}}}{\lambda_{\mathrm{q}}}$,
$\displaystyle\textphnc{\Af}_{\mathrm{a}}=\begin{cases}\frac{1}{2}&\text{if
}r=1;\\\ \frac{r\ln{r}-r+1}{(r-1)^{2}}&\text{if }r>0,r\neq 1.\end{cases}$ (16)
To arrive at Eq. 16, we used standard identities for the PDFs of the ratio
distribution and the inverse distribution, and applied l’Hôpital’s rule twice.
When $r=1$, that is to say when $\lambda_{\mathrm{a}}=\lambda_{\mathrm{q}}$,
both definitions predict the same $\textphnc{\Af}_{\mathrm{a}}=\frac{1}{2}$.
For $r<1$, our definition predicts higher $\textphnc{\Af}_{\mathrm{a}}$ than
the natural definition does; for $r>1$, our definition predicts lower
$\textphnc{\Af}_{\mathrm{a}}$. To find $\lambda_{\mathrm{a}}$ given
$\lambda_{\mathrm{q}}$ and $\textphnc{\Af}_{\mathrm{a}}$ under the natural
definition, one must solve Eq. 16 numerically. For simplicity, we generated
Fig. 13 using Eq. 14.  On human time scales, we can access only a snapshot
view of this population. During the present snapshot (vertical line), seven
galaxies generate RGs, and could be included in an RG sample. RG ages at the
snapshot (hatched parts of the thin green stripes) generally increase with
$\textphnc{\Af}_{\mathrm{a}}$.

####  4.2.2 Predicting the density distribution of observed giants by treating
them as luminous general RGs

We then tested whether Fig. 6’s higher densities for observed giants than for
observed general RGs could be explained by these giants’ higher radio
luminosities and Fig. 8’s radio luminosity–Cosmic Web density relation. We
first calculated the RG relative total matter density RV for radio
luminosities of $l_{\nu,\mathrm{min}}$ and higher. By applying standard PDF
identities, we found

$\displaystyle f_{1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}}(1+\delta)=$
$\displaystyle\frac{\int_{l_{\nu,\mathrm{min}}}^{\infty}f_{1+\Delta_{\mathrm{RG}}\
|\ L_{\nu}=l_{\nu}}(1+\delta)\cdot f_{L_{\nu}}(l_{\nu})\
\mathrm{d}l_{\nu}}{\int_{l_{\nu,\mathrm{min}}}^{\infty}f_{L_{\nu}}(l_{\nu})\
\mathrm{d}l_{\nu}}=$
$\displaystyle\frac{\int_{l_{\nu,\mathrm{min}}}^{\infty}f_{1+\Delta_{\mathrm{RG}}\
|\ L_{\nu}=l_{\nu}}(1+\delta)\cdot f_{L_{\nu}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}}(l_{\nu})\
\mathrm{d}l_{\nu}}{\int_{l_{\nu,\mathrm{min}}}^{\infty}f_{L_{\nu}\ |\
L_{\nu}\geq l_{\nu,\mathrm{min}}}(l_{\nu})\ \mathrm{d}l_{\nu}},$ (17)

To proceed, one must specify $f_{L_{\nu}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}}(l_{\nu})$ — up to a constant, at least. We modelled
$L_{\nu}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}\sim\mathrm{Pareto}(l_{\nu,\mathrm{min}},\lambda)$, where
$\lambda$ is the tail index or rate parameter of the associated exponential
distribution.141414Equivalently, one could model
$\displaystyle\ln{\left(\frac{L_{\nu}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}}{l_{\nu,\mathrm{min}}}\right)}\sim\mathrm{Exp}(\lambda),$
(18) or  $\displaystyle\mathrm{log}_{10}\left(\frac{L_{\nu}\ |\ L_{\nu}\geq
l_{\nu,\mathrm{min}}}{l_{\nu,\mathrm{min}}}\right)\sim\mathrm{Exp}\left(\lambda\cdot\ln{10}\right).$
(19) For both $l_{\nu,\mathrm{min}}=10^{23}\ \mathrm{W\ Hz^{-1}}$ and
$l_{\nu,\mathrm{min}}=10^{24}\ \mathrm{W\ Hz^{-1}}$, we found
$\lambda_{\mathrm{MLE}}=0.6$ (based on $1\cdot 10^{3}$ and $3\cdot 10^{2}$
RGs, respectively). For higher $l_{\nu,\mathrm{min}}$,
$\lambda_{\mathrm{MLE}}$ increases somewhat, but becomes less reliable because
of decreasing sample sizes. We therefore stuck with
$\lambda_{\mathrm{MLE}}=0.6$. We then evaluated Eq. 17 for
$l_{\nu,\mathrm{min}}=10^{25}\ \mathrm{W\ Hz^{-1}}$ and added
heteroskedastistic noise in order to compare our prediction to observations.
We propagated the correlated uncertainties on the parameters $k(0)$,
$\theta(0)$, $k^{\prime}(0)$, and $\theta^{\prime}(0)$, which together
determine $f_{1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}=l_{\nu}}$, by rejection
sampling from the joint posterior distribution. We show the resulting mean
prediction with standard deviation half-width (pink) in Fig. 11.

Figure 11:  Density distribution predicted for observed giants combining Sect.
4.2’s RG radio luminosity–Cosmic Web density relation and a radio luminosity
cut-off $l_{\nu,\mathrm{min}}=10^{25}\ \mathrm{W\ Hz^{-1}}$ (pink); in
addition, we show the KDE density distributions for giants with gamma
distribution MLE fit (solid green) and for general RGs (hatched green) — all
as in the left column of Fig. 6.

Although the prediction is based solely on observations of general RGs — and
so is built without knowledge of the observed GRG distribution (solid green
KDE) — it appears to fit it better than a direct MLE gamma fit to the GRG data
(solid green curve) does. However, a better fit is not necessarily also a good
fit. We therefore tested whether the prediction is consistent with our sample
of Cosmic Web densities for observed giants. A KS test suggested an answer in
the affirmative, as it yielded a $p$-value of $0.7$.

Section 4.1 presented a paradox: known giant radio galaxies reside in denser
Cosmic Web environments than their smaller kin, in apparent violation of the
long-standing hypothesis that giants should emerge predominantly in the dilute
Cosmic Web. The modelling in this section suggests that the higher densities
found for known giants than for general RGs can be explained by a combination
of these giants’ higher radio luminosities and the general relation between RG
radio luminosity and Cosmic Web density. Fainter giants — which might inhabit
the more dilute Cosmic Web — might exist, but are currently not
observationally accessible.

###  4.3 Radio galaxy number densities

After accreting baryons, supermassive black holes (SMBHs) can launch jets that
give rise to radio galaxies. Although it is clear that SMBH jets and RGs play
an important role in galaxy evolution and cosmology, the physics of jet
launching is not yet fully understood. Moreover, because SMBHs are
astronomical unit–sized, galaxies are kiloparsec-sized (ratio
$10^{8}$–$10^{9}$), and the Cosmic Web is megaparsec-sized (ratio
$10^{11}$–$10^{12}$), it is not yet possible to build cosmological simulations
in which a realistic interplay between SMBHs, their host galaxies, and the
enveloping Cosmic Web naturally arises. Thus, for now, major advances in our
understanding of RGs on the cosmological scale must come from observations
instead of from simulations. It is therefore of considerable interest to
observationally constrain the occurrence of RGs as a function of Cosmic Web
density.

Enticingly, our relative total matter density measurements — in combination
with the BORG SDSS HMC Markov chain samples in their entirety — allow for a
determination of the RG number density as a function of Cosmic Web density:
$n_{\mathrm{RG}}(1+\delta)$. To see why, we first emphasise that the PDF
$f_{1+\Delta_{\mathrm{RG}}}$, which we have approximated through observations
and visualised in the bottom row of Fig. 6, provides a key ingredient for
$n_{\mathrm{RG}}(1+\delta)$. It shows that if a volume of cosmological extent
is surveyed, one will obtain more RGs with moderate (filament-like) densities
than with low (void-like) or high (cluster-like) densities. We remark that in
such a survey sample the number of RGs whose densities fall within a given
interval depends not only on the RG number density at the interval’s
densities, but also on the prevalence of environments with such densities. As
a result, if a sample contains more filament-inhabiting RGs than, say,
cluster-inhabiting RGs, one need not necessarily conclude that the RG number
density in filaments is higher than in clusters; in fact, the RG number
density in the latter environment could well be higher, provided that clusters
are sufficiently rare.

Conveniently, the density fields of the BORG SDSS HMC Markov chain samples
enable one to quantify the rarity of each Cosmic Web environment through a PDF
$f_{1+\Delta_{\mathrm{CW}}}$, which corresponds to the Cosmic Web relative
density RV $1+\Delta_{\mathrm{CW}}$. This RV represents the relative total
matter density at a randomly chosen point in the Local Universe (and at the
BORG SDSS resolution of $2.9\ \mathrm{Mpc}\ h^{-1}$). We determine the
distribution of $1+\Delta_{\mathrm{CW}}$ simply by binning the density fields
of a few hundred Markov chain samples. As a result,
$f_{1+\Delta_{\mathrm{CW}}}$ is best determined at low densities, which
correspond to the most common environments, and least so at high densities,
which are rare.151515At first sight, it is tempting to approximate
$f_{1+\Delta_{\mathrm{CW}}}$ by binning the BORG SDSS mean rather than
individual BORG SDSS samples. This, however, leads to incorrect results. The
mean tends to the cosmic mean wherever the structure of the Cosmic Web is not
well constrained by the SDSS DR7 MGS, causing low-density environments to be
overrepresented in it and high-density environments underrepresented. By
contrast, the samples have low-density and high-density environments
statistically correctly proportioned — that is, insofar as the gravity solver
allows. When using the BORG SDSS mean, the probability densities
$f_{1+\Delta_{\mathrm{CW}}}(1+\delta)$ for $1+\delta\gtrsim 10$ are up to an
order of magnitude lower than predicted by BORG SDSS samples. As we expect
$f_{1+\Delta_{\mathrm{CW}}}$ to be smooth, we applied a Savitzky–Golay filter
(of polynomial order 1) to the binned data; however, this step is not
essential.

Appendix E shows that $n_{\mathrm{RG}}(1+\delta)$, the RG number density as
would arise in an environment of constant density $1+\delta$, obeys

$\displaystyle\frac{n_{\mathrm{RG}}(1+\delta)}{\bar{n}_{\mathrm{RG}}}=\frac{f_{1+\Delta_{\mathrm{RG}}}(1+\delta)}{f_{1+\Delta_{\mathrm{CW}}}(1+\delta)}.$
(20)

Here, $\bar{n}_{\mathrm{RG}}$ is the cosmic mean RG number density. Thus, a
simple point-wise division of the PDFs of $1+\Delta_{\mathrm{RG}}$ and
$1+\Delta_{\mathrm{CW}}$ yields the RG number density at a given Cosmic Web
density relative to the cosmic mean value.

Figure 12:  Number densities of RGs with a given radio luminosity $l_{\nu}$,
as a function of Cosmic Web density at the $2.9\ \mathrm{Mpc}\ h^{-1}$ scale.
The number density functions reflect conditions in the Local Universe, and are
given relative to the cosmic mean number density at $l_{\nu}$. The solid
curves reflect the posterior mean, whilst the shaded areas around the mean
denote $-1$ to $+1$ posterior standard deviation ranges. We used flexible
voxel method densities.

Unfortunately, as discussed in Sect. 4.1, our measurements of
$f_{1+\Delta_{\mathrm{RG}}}$ suffer from surface brightness selection. To
avoid selection effects, we formulated a more specific version of Eq. 20. In
particular, we considered the number density of RGs per unit of radio
luminosity as a function of radio luminosity and relative density:
$\textphnc{\Anun}_{\mathrm{RG}}(l_{\nu},1+\delta)$ — again, relative to a
cosmic mean value: $\bar{\textphnc{\Anun}}_{\mathrm{RG}}(l_{\nu})$. Appendix E
shows that this ratio equals

$\displaystyle\frac{\textphnc{\Anun}_{\mathrm{RG}}(l_{\nu},1+\delta)}{\bar{\textphnc{\Anun}}_{\mathrm{RG}}(l_{\nu})}=\frac{f_{1+\Delta_{\mathrm{RG}}|L_{\nu}=l_{\nu}}(1+\delta)}{f_{1+\Delta_{\mathrm{CW}}}(1+\delta)}.$
(21)

This radio luminosity–dependent relative number density is, as before, equal
to a point-wise division of PDFs. With respect to the right-hand side (RHS) of
Eq. 20, only the PDF in the numerator has changed. Section 4.2 explains how we
modelled this PDF. It depends on the four parameters $k(0)$, $k^{\prime}(0)$,
$\theta(0)$, and $\theta^{\prime}(0)$, for which the data shown in Fig. 8
induce the posterior distribution shown in Fig. 9. We evaluated the RHS of Eq.
21 by rejection sampling from the posterior, calculating
$f_{1+\Delta_{\mathrm{RG}}|L_{\nu}=l_{\nu}}$ for each sampled quartet
$(k(0),k^{\prime}(0),\theta(0),\theta^{\prime}(0))$, and dividing the
resulting PDFs point-wise by $f_{1+\Delta_{\mathrm{CW}}}$. In Fig. 12, for
four observationally accessible radio luminosities, we visualise the posterior
mean alongside a range that is two standard deviations wide.

As expected, RG number densities generally increase with Cosmic Web density.
The more luminous RGs become, the more pronounced the contrast between their
number density in clusters and their cosmic mean number density is. Our
results indicate that high-luminosity RGs, with $l_{\nu}\in 10^{25}$–$10^{27}\
\mathrm{W\ Hz^{-1}}$, attain number densities in clusters that are
${\sim}10^{2}$ times higher than average. Interestingly, for radio
luminosities $l_{\nu}\in 10^{21}$–$10^{23}\ \mathrm{W\ Hz^{-1}}$, we obtain
statistically significant, preliminary evidence that the RG number density
peaks at a cluster-like density that depends positively on $l_{\nu}$.161616For
radio luminosities $l_{\nu}\in 10^{25}$–$10^{27}\ \mathrm{W\ Hz^{-1}}$, the
uncertainty margins are too large to claim evidence for a peak. Beyond this
density, these low-luminosity RGs become less prevalent per unit of volume,
possibly because the underlying galaxy population starts generating more
luminous RGs instead.

We stress that these results are tentative, as both
$f_{1+\Delta_{\mathrm{RG}}|L_{\nu}=l_{\nu}}$ and $f_{1+\Delta_{\mathrm{CW}}}$
suffer from systematic errors at higher densities. These systematic errors are
not reflected in the uncertainties of Fig. 12. In the case of
$f_{1+\Delta_{\mathrm{RG}}|L_{\nu}=l_{\nu}}$, the gamma distribution
approximation loses validity in the tail: beyond some point, it will
consistently under- or overestimate the probability density. Our LoTSS DR1
sample contains only a modest number (${\sim}10^{2}$) of RGs at relative
densities $1+\delta\gtrsim 10$ — especially when selecting those around a
given value of $l_{\nu}$ only (in which case just ${\sim}10^{1}$ RGs remain).
As a result, in its tail the gamma PDF takes on the character of an
extrapolation function rather than of a fitting function, and should
consequently be treated with caution. In the future, larger RG samples can
resolve this situation. In the case of $f_{1+\Delta_{\mathrm{CW}}}$, the 2LPT
model of structure formation used by the BORG SDSS loses validity at cluster
densities. As a result, this work’s estimate of $f_{1+\Delta_{\mathrm{CW}}}$
underpredicts the prevalence of cluster-like densities. Newer BORG runs, such
as the BORG 2M++, employ an enhanced gravity solver (Jasche & Lavaux 2019),
relieving this limitation.

Although preliminary in nature, the inferences shown in Fig. 12 could in
principle be used to calibrate cosmological simulations that feature RG
feedback. The fact that our number densities are given with respect to a
cosmic mean value is not problematic, because such a cosmic mean number
density is straightforward to calculate from simulation snapshots. However,
apart from making sure that the RG number densities in simulated environments
of various densities are correctly proportioned, it is also important to
calibrate the absolute level of RG feedback in simulations. Interestingly,
there are variations on Eq. 20 for which an absolute number density can be
explicitly calculated with current-day data. One can consider, for example,
non-giant radio galaxies (i.e. $l_{\mathrm{p}}<l_{\mathrm{p,GRG}}\coloneqq
0.7\ \mathrm{Mpc}$) in the Local Universe with radio luminosities $l_{\nu}\geq
10^{24}\ \mathrm{W\ Hz^{-1}}$. Figure 7 suggests that this RG subpopulation is
already fully accessible by the LoTSS. Calling these RGs luminous non-giant
radio galaxies (LNGRGs), we have

$\displaystyle
n_{\mathrm{LNGRG}}(1+\delta)=\bar{n}_{\mathrm{LNGRG}}\cdot\frac{f_{1+\Delta_{\mathrm{LNGRG}}}(1+\delta)}{f_{1+\Delta_{\mathrm{CW}}}(1+\delta)}.$
(22)

We calculated $\bar{n}_{\mathrm{LNGRG}}$ by counting all LNGRGs in the LoTSS
DR1 footprint up to $z_{\mathrm{max}}$, finding $N_{\mathrm{LNGRG}}=172$.
Because of calibration problems and higher noise levels towards the edges of
the footprint, a fraction of the footprint is — artificially — devoid of
LNGRGs. We thus applied a correction factor to $N_{\mathrm{LNGRG}}$ of 1.2,
yielding $N_{\mathrm{LNGRG}}=200$. Assuming that
$N_{\mathrm{LNGRG}}\sim\mathrm{Poisson}(\lambda_{\mathrm{LNGRG}})$, maximum
likelihood estimation simply suggests
$\hat{\lambda}_{\mathrm{LNGRG,MLE}}=200$. We combined this MLE Poisson
distribution for $N_{\mathrm{LNGRG}}$ with the comoving volume $V=16\cdot
10^{6}\ \mathrm{Mpc}^{3}$ to obtain a probability distribution for
$\bar{n}_{\mathrm{LNGRG}}$. We find $\bar{n}_{\mathrm{LNGRG}}=12\pm 1\ (100\
\mathrm{Mpc})^{-3}$. Next, we approximated
$f_{1+\Delta_{\mathrm{LNGRG}}}\approx f_{1+\Delta_{\mathrm{RG}}\ |\
L_{\nu}\geq 10^{24}\ \mathrm{W\ Hz^{-1}}}$, and calculated the latter through
Eq. 17 and Sect. 4.2’s subsequent procedure. Taking
$f_{1+\Delta_{\mathrm{CW}}}$ as before, we obtained a probability distribution
over LNGRG number density functions, visualised in Fig. 13.

Figure 13: Number density of luminous non-giant radio galaxies in the Local
Universe, as a function of Cosmic Web density. We define these as RGs with
$150\ \mathrm{MHz}$ radio luminosities $l_{\nu}\geq 10^{24}\ \mathrm{W\
Hz^{-1}}$, projected proper lengths
$l_{\mathrm{p}}<l_{\mathrm{p,GRG}}\coloneqq 0.7\ \mathrm{Mpc}$, and redshifts
$z<z_{\mathrm{max}}\coloneqq 0.16$. The Cosmic Web densities encompass
baryonic and dark matter, are defined on a $2.9\ \mathrm{Mpc}\ h^{-1}$ scale,
and are given relative to the cosmic mean. We used flexible voxel method
densities here. We show the posterior mean with a shaded $-1$ to $+1$
posterior standard deviation range. We also show five individual realisations
(grey).

As it should, the LNGRG number density function follows the same trend as the
functions of Fig. 12; the novelty here is the prediction of absolute number
densities. We reiterate that systematic errors at cluster-like densities
render only the left side of the plot reliable. It predicts that, in
filaments, $n_{\mathrm{LNGRG}}\sim 10^{-1}\ (10\ \mathrm{Mpc})^{-3}$, or
$n_{\mathrm{LNGRG}}\sim 10^{-4}\ \mathrm{Mpc}^{-3}$. As major filaments have
volumes $V\in 10^{2}$–$10^{3}\ \mathrm{Mpc}^{3}$, only one in ten major
filaments harbours an LNGRG.

###  4.4 Cosmic Web dynamical state distributions

Figure 14:  Bayesian Cosmic Web reconstructions allow one to determine, for
each giant, a probability distribution $\@vec{p}$ over the four $T$-web
environment classes (clusters, filaments, sheets, and voids). Here we
summarise these distributions: each panel shows how a component of
$\@vec{p}\coloneqq(p_{1},p_{2},p_{3},p_{4})$ is itself distributed.

Figure 15:  Dynamical environment classification of giant radio galaxies
(left), radio galaxies in general (centre), and the Local Universe in its
entirety (right; from Table 3 of Leclercq et al. 2015). We define clusters,
filaments, sheets, and voids in the $T$-web sense. These distributions depend
on the scale to which the Cosmic Web density field is smoothed; in this case,
the smoothing scale is $2.9\ \mathrm{Mpc}\ h^{-1}$. If giants and RGs were
scattered uniformly throughout the Cosmic Web, their distributions would be
similar to that of the volume-filling fractions (VFFs). Instead, observed RGs
— and observed giants in particular — favour cluster and filament
environments.

For each RG in the two samples, we obtained a probability distribution over
$T$-web environment classes from the BORG SDSS extensions of Leclercq et al.
(2015). The $T$-web scheme classifies the Cosmic Web at each point on the
basis of its local gravitational dynamics. On an intuitive level, the scheme
counts the number of dimensions along which the orbit of a test particle
released at rest with respect to the environment is stable. More formally, at
any location the Hessian of the gravitational potential, or tidal tensor $T$,
has either three, two, one, or zero positive eigenvalues; the environment is
interpreted correspondingly as a cluster, filament, sheet, or void. Leclercq
et al. (2015) have determined the $T$-web environment for each voxel of each
BORG SDSS sample. By iterating over samples and counting the frequencies of
the four environment classes on a per-voxel basis, we obtained a marginal
environment class distribution $\@vec{p}=(p_{1},p_{2},p_{3},p_{4})$ for each
voxel. In Fig. 14, we show how each of $\@vec{p}$’s components is distributed
for luminous giants. We note the following.

For 99% of our giants, the most likely environment class is more than 50%
probable. For 20% of giants, it is more likely than not that they inhabit a
cluster (i.e. $p_{1}>50\%$); for 68% of giants, it is more likely than not
that they inhabit a filament (i.e. $p_{2}>50\%$); and for 11% of giants, it is
more likely than not that they inhabit a sheet (i.e. $p_{3}>50\%$). Finally,
for 0% of giants, it is more likely than not that they inhabit a void (i.e.
$p_{4}>50\%$).

Some giants inhabit sheets (e.g. those in the bottom row of Fig. 3), but
concluding this in individual cases rarely has a high degree of certainty: for
just 0.5% of giants, it is very likely that they inhabit a sheet (i.e.
$p_{3}>80\%$).

A cluster environment is very unlikely (i.e. $p_{1}<20\%$) for a clear
majority of 63% of giants, whilst a filament environment is very unlikely
(i.e. $p_{2}<20\%$) for only 9% of giants; the filament environment hypothesis
can thus seldom be excluded. A sheet environment is very unlikely (i.e.
$p_{3}<20\%$) for 71% of giants; finally, a void environment is nearly always
very unlikely (i.e. $p_{4}<20\%$) — this holds for 99% of giants.

The corresponding distributions for LoTSS DR1 RGs (not shown) are largely
similar. This is also clear from Fig. 15, in which we compare the environment
class distributions of luminous giants, LoTSS DR1 RGs, and the Local Universe
in its entirety. Evidently, luminous giants and general RGs are not scattered
uniformly throughout the Cosmic Web, but trace cluster and filament
environments. Remarkably, nearly a quarter of all luminous giants occurs in
clusters, which have a volume-filling fraction (VFF) of just 1–2%. It is easy
to derive that the ratio between a GRG environment class probability (left
pie) and the corresponding VFF (right pie) yields the GRG number density of
that class relative to the cosmic mean GRG number density. Of course, the
analogous statement for RGs also holds. In Table 3, we present relative number
densities for luminous giants and general RGs in the Local Universe.

Table 3:  Number densities of observed Local Universe giants and general RGs
relative to their cosmic means. These numbers depend on environment class
definitions and the scale to which the Cosmic Web density field is smoothed;
in this case, we use the $T$-web classification scheme and a smoothing scale
of $2.9\ \mathrm{Mpc}\ h^{-1}$. Clusters and filaments are (giant) radio
galaxy overdense; sheets and voids are underdense.

| cluster | filament | sheet | void
---|---|---|---|---
| environment | environment | environment | environment
GRGs | $15\pm 2$ | $2.4\pm 0.1$ | $0.29\pm 0.04$ | $0.06\pm 0.04$
RGs | $10.6\pm 0.6$ | $2.41\pm 0.05$ | $0.39\pm 0.02$ | $0.08\pm 0.02$

Clusters are strongly (G)RG overdense; voids are strongly (G)RG underdense.

The left and central pie charts of Fig. 15 can be regarded as prior
distributions over Cosmic Web dynamical states for luminous giants and RGs in
or near the Local Universe. This fact can be useful whenever direct BORG SDSS
environment characterisation is not possible. For a randomly picked luminous
giant, Fig. 15’s left pie chart specifies the prior probability distribution
over dynamical states to be $\@vec{p}=(23\%,59\%,17\%,1\%)$. Interestingly,
there is often additional information available to constrain the dynamical
state. For example, multiwavelength imagery demonstrates that Alcyoneus (Oei
et al. 2022), one of the longest known RGs, does not inhabit a galaxy cluster;
in addition, the number of luminous galaxies in its vicinity excludes the
possibility of a void environment.171717Characterising Alcyoneus’s Cosmic Web
environment is of scientific interest, as it could help understand how the
largest RGs form. However, at $z=0.25$, it is too far away to probe directly
with the BORG SDSS. Combining this information with the prior, we obtain —
after rounding — the posterior probability distribution
$\@vec{p}=(0\%,80\%,20\%,0\%)$. We thus estimate the probability that
Alcyoneus inhabits a filament to be 80%. Of course, this line of reasoning
generalises to any luminous giant outside of galaxy clusters that has massive
galactic neighbours in its megaparsec-scale vicinity.

##  5 Discussion

###  5.1 Cosmic Web density accuracy

In this work, we have determined megaparsec-scale total matter densities
around RGs in the Local Universe, and used these to compare the environments
of luminous giants to those of general RGs. A natural question to ask is how
accurate these densities are, and to which degree they can be used in other
analyses, such as in the fitting of dynamical models to individual RGs (e.g.
Hardcastle 2018). To quantify accuracy, one would ideally know the ground-
truth densities that the inferred densities are meant to approach.
Problematically, however, no ground-truth densities are known — especially for
filaments, the environment type which most RGs seem to inhabit. For Local
Universe clusters though, approximate masses are known. We exploited this fact
to test the accuracy of our densities for cluster RGs. We immediately remark
that our density accuracy is likely to vary with density, so that the accuracy
of cluster RG densities might not be informative of the accuracy of filament
RG densities. This is because the 2LPT gravity solver of the BORG SDSS has
particularly limited validity in the strongly non-linear cluster regime.

To test the cluster RG density accuracy, we cross-matched our samples with Wen
& Han (2015)’s SDSS DR12–based galaxy cluster catalogue. To specify each
cluster’s location, this catalogue provides the right ascension, declination,
and redshift of the brightest cluster galaxy (BCG). First, we selected all
clusters in the Local Universe ($z<z_{\mathrm{max}}$). Next, we cross-matched
in an angular sense, and considered an RG matched with a BCG if their angular
separation is less than $3^{\prime\prime}$. We thus retained 60 BCG giants and
113 BCG LoTSS DR1 RGs. Figure 2 shows six members (10%) of the BCG giants
sample. The perturbed jet and lobe shapes seen in the radio, and the massive
ellipticals and crowded environments seen in the optical, together visually
demonstrate the adequacy of our BCG giant identification. We could identify
LoTSS DR1 BCG RGs with equal confidence.

As a quality check of the cluster masses, we tested whether they are accurate
enough to contain fingerprints of known physical effects. In particular, we
tested whether clusters with RG-generating BCGs are more massive than other
Local Universe clusters. The AGN feedback paradigm suggests that this is the
case: in low-mass clusters, AGN feedback converts cool cores to non-cool cores
within a gigayear, shutting down further AGN feedback for many gigayears to
come; by contrast, high-mass clusters maintain stable AGN feedback cycles with
gigayear-scale periods (Nobels et al. 2022), and thus their BCGs have a higher
probability of being observed while generating RGs. The median mass of
clusters with RG-generating BCGs is $M_{500}=1.1\cdot 10^{14}\ M_{\odot}$,
while for other clusters it is $M_{500}=1.0\cdot 10^{14}\ M_{\odot}$. A
Kolmogorov–Smirnov test yielded a $p$-value $p=7\%$, suggesting to reject the
null hypothesis of equality in distribution only for high significance levels
(e.g. $\alpha=10\%$). We also tested whether clusters with giant-generating
BCGs are more massive than other clusters. The median mass of clusters with
giant-generating BCGs is $M_{500}=1.2\cdot 10^{14}\ M_{\odot}$, while for
other clusters it is $M_{500}=1.0\cdot 10^{14}\ M_{\odot}$. A KS test yielded
$p=5\%$, suggesting to reject equality in distribution only for rather high
significance levels (e.g. $\alpha=5\%$ or $\alpha=10\%$).

As a first quality check of our densities, we tested whether our BCG giants
have higher densities than our non-BCG giants. The median relative densities
for BCG giants and non-BCG giants are $1+\delta=8.5$ and $1+\delta=7.2$,
respectively. A KS test yielded $p=0.2\%$, indeed suggesting to reject
equality in distribution. Similarly, we tested whether our LoTSS DR1 BCG RGs
have higher densities than our LoTSS DR1 non-BCG RGs. The median relative
densities for BCG RGs and non-BCG RGs are $1+\delta=8.4$ and $1+\delta=5.9$,
respectively. A KS test yielded $p\ll 0.1\%$, again suggesting to reject
equality in distribution.

Figure 16:  Comparison between environmental estimates for luminous giants
(large dots) and LoTSS DR1 RGs (small dots) generated by BCGs in the Local
Universe. We compare cluster masses $M_{500}$ from Wen & Han (2015) with
relative densities $1+\delta$ from the BORG SDSS, and colour the dots based on
their cluster probability $p_{1}$ (relative to the average cluster probability
$\bar{p}_{1}$ for luminous giants and LoTSS DR1 RGs). For three randomly
chosen giants and three randomly chosen RGs, we visualise uncertainties.
Generally, as expected, RG-generating BCGs claimed to be in more massive
clusters by Wen & Han (2015) have higher BORG SDSS densities and $T$-web
cluster probabilities. The contours indicate fractions of the density expected
through $M_{500}$, and show that our relative densities fall short
significantly in the cluster regime — by a factor that ranges from less than
two to more than an order of magnitude. Densities of low-mass clusters are
less biased than densities of high-mass clusters. We used flexible voxel
method densities here.

Figure 16 shows the relation between cluster mass and relative density.
Although it is qualitatively clear that BCG RGs in more massive clusters have
higher inferred densities, the densities are all lower than one would expect
from spreading out the cluster mass over a BORG SDSS voxel. (For reference, a
cluster of mass $M\sim 10^{14}\ M_{\odot}$ should have a $2.9\ \mathrm{Mpc}\
h^{-1}$–scale relative density $1+\delta\sim 40$.) This systematic discrepancy
cannot be attributed to errors in the cluster masses, as Wen & Han (2015) have
made sure that their estimates are unbiased. The alternative is to conclude
that our densities are biased low in the cluster regime. This is unsurprising,
as the assumptions of the BORG SDSS’s 2LPT gravity solver are violated in
clusters. Our densities are least accurate for high-mass clusters, where they
can be off by an order of magnitude, and most accurate for low-mass clusters,
where they can be less than a factor two too low. It is unclear whether, and
if so how, these accuracy characterisations can be extrapolated to the
filament density regime. The increased applicability of 2LPT to the filament
density regime suggests that the accuracy will increase. We therefore expect
our filament densities to be correct up to a small factor of order unity.

Until Bayesian Cosmic Web reconstructions are further developed, we urge
caution in using our inferred densities as precision estimates: in filament
environments, they should be trusted up to a factor of order unity only; in
cluster environments, for the moment, other sources (such as cluster
catalogues) appear to provide more accurate density estimates. If one would
insist in using the cluster densities, then Fig. 16 suggests that one should
correct them by multiplying by a factor of ${\sim}5$.

###  5.2 Cosmic Web dynamical state accuracy

Similar questions of accuracy can be raised about the Cosmic Web dynamical
state distributions for luminous giants and RGs presented in the left and
central pie charts of Fig. 15. To test the cluster GRG and cluster RG
percentages, we again used the cluster catalogue of Wen & Han (2015).

Cluster giants can be subdivided into BCG giants and non-BCG cluster giants.
In Sect. 5.1, we determined that 60 out of 260 giants (23%) are BCG
giants.181818In comparison, Dabhade et al. (2020) found that only 20 out of
128 LoTSS DR1 giants with $z\leq 0.55$ (16%) were BCG giants. We hypothesise
that we find a higher occurrence of BCG giants because our study is restricted
to the Local Universe ($z<z_{\mathrm{max}}\coloneqq 0.16$), where cluster
catalogues are generally more complete. We defined non-BCG cluster giants as
giants generated by non-BCG galaxies for which the comoving distance to the
BCG is less than some megaparsec-scale threshold. We obtained approximate
comoving coordinates from right ascensions, declinations, and spectroscopic
redshifts, assuming vanishing peculiar motion. In order to allow for a fair
comparison to the BORG SDSS, we set the megaparsec-scale threshold to a BORG
SDSS voxel side length: $2.9\ \mathrm{Mpc}\ h^{-1}$. Although much larger than
a galaxy cluster virial radius, this threshold does ensure that cluster
galaxies with significant peculiar motion — a standard deviation
$\sigma_{v_{\mathrm{p}}}=10^{2}$–$10^{3}\ \mathrm{km\ s^{-1}}$ is typical —
have a reasonable chance to be linked to their cluster. For this choice of
threshold, we found 21 non-BCG cluster giants. We thus find a total of 81
cluster giants (31%), which is more than expected based on the BORG SDSS
measurement ($23\begin{subarray}{c}+3\\\ -3\end{subarray}\%$) shown in Fig.
15’s GRG pie chart. This suggests that the BORG SDSS underpredicts the
occurrence of cluster giants. Indeed, the strongly underpredicted densities of
galaxy clusters discussed in Sect. 5.1 also leave their mark on the accuracy
of our dynamical states. For example, the giant shown in the top-left panel of
Fig. 2 and tabulated on the tenth row of Table 7, clearly occupies a cluster-
like environment with an estimated $M_{500}=1.3\cdot 10^{14}\ M_{\odot}$, but
is erroneously assigned an 80% chance to inhabit a filament.

Similarly, cluster RGs can be subdivided into BCG RGs and non-BCG cluster RGs.
In Sect. 5.1, we determined that 113 out of 1443 selected LoTSS DR1 RGs (8%)
are BCG RGs. By defining non-BCG cluster RGs in a way analogous to non-BCG
cluster giants, we found 109 non-BCG cluster RGs.191919Whereas three-quarters
of all luminous cluster giants are BCG giants, only half of all general
cluster RGs are BCG RGs. Thus, whenever a luminous giant is found in a
cluster, it is most likely generated by the BCG; by contrast, whenever a
general RG is found in a cluster, a BCG origin is just as likely as a non-BCG
origin. We thus find a total of 222 cluster RGs (15%), consistent with the
BORG SDSS measurement ($16\begin{subarray}{c}+1\\\ -1\end{subarray}\%$) shown
in Fig. 15’s RG pie chart.

To assess the accuracy of the filament GRG and RG fractions shown in Fig. 15,
we cross-matched our giants and selected LoTSS DR1 RGs with the galaxy group
and cluster catalogue of Tempel et al. (2017), which provides estimates of
$M_{200}$. Wen & Han (2015) define clusters through $M_{500}\geq 0.6\cdot
10^{14}\ M_{\odot}$, which translates to $M_{200}\geq 0.9\cdot 10^{14}\
M_{\odot}$: a typical conversion is $M_{500}=0.7\ M_{200}$ (e.g. Pierpaoli et
al. 2003). To separate groups from clusters, we therefore defined groups
through $M_{200}<0.9\cdot 10^{14}\ M_{\odot}$. As with our cluster tests, we
associated RGs to groups on the basis of estimated comoving coordinates,
assuming vanishing peculiar motion. In the case of groups, this assumption is
more reasonable. We again used a $2.9\ \mathrm{Mpc}\ h^{-1}$ association
threshold. We associated 155 out of 260 luminous giants (60%) and 903 out of
1443 general RGs (63%) with galaxy groups. These percentages provide lower
bounds to the occurrence of filament giants and RGs, as there are also
filament giants and RGs that do not reside in galaxy groups. It is therefore
likely that the filament GRG and RG fractions of Fig. 15
($59\begin{subarray}{c}+3\\\ -3\end{subarray}\%$ and
$60\begin{subarray}{c}+1\\\ -1\end{subarray}\%$, respectively) are
underestimates.

Finally, as a probe of sheet and void environments, we explored the occurrence
of giants and RGs for which the closest galaxy group or cluster is more than
$10\ \mathrm{Mpc}$ away. We identified 18 out of 260 luminous giants (7%) and
115 out of 1443 general RGs (8%) as occupying sheets and voids. This suggests
that the combined sheet and void GRG and RG fractions of Fig. 15
($18\begin{subarray}{c}+2\\\ -2\end{subarray}\%$ and
$24\begin{subarray}{c}+1\\\ -1\end{subarray}\%$, respectively) are significant
overestimates.

Rounded to an accuracy of $10\%$, our ancillary environment analysis suggests
an environment class probability distribution for luminous giants of
$\@vec{p}=(30\%,60\%,10\%)$, representing clusters, filaments, and a
combination of sheets and voids, respectively. For general RGs, it suggests
$\@vec{p}=(20\%,70\%,10\%)$. Both distributions are in fair agreement with the
BORG SDSS measurement $\@vec{p}=(20\%,60\%,20\%)$. The ancillary analysis
reinforces the idea that, in the Local Universe, luminous giants detectable at
LoTSS-like image qualities occupy denser environments than general RGs.

###  5.3 Cosmic Web density resolution

All Cosmic Web density and dynamical state measurements presented in this work
correspond to a density field with a resolution, or smoothing scale, of $2.9\
\mathrm{Mpc}\ h^{-1}$. In fact, given the granular nature of matter, one must
always adopt a smoothing scale in order to define an informative notion of
density.202020As there are many situations in which there exists a natural
smoothing scale, the smoothing scale is often left unmentioned. Ideally, this
scale is chosen such that the resulting density field varies smoothly over the
physical scales of interest — avoiding both unwanted stochastic variations
over space and time (as would occur when the smoothing scale is too small),
and excessive blurring of the physical phenomena under study (as would occur
when the smoothing scale is too large). In the case of this work, the
smoothing scale of the Cosmic Web density field could be considered too large.
It is set by the limitations of the BORG SDSS 2LPT gravity solver, the
ambition to cover the entire Local Universe in the direction of the SDSS MGS
footprint, and — most practically — the relation between the number of voxels
and the numerical cost of generating the HMC Markov chain. As galaxy clusters
have typical radii of ${\sim}1\ \mathrm{Mpc}\ h^{-1}$, they are not resolved
by the BORG SDSS; the same therefore certainly holds for galaxy groups. Thus,
even if the gravitational dynamics of the reconstructions were exact, the
utility of the BORG SDSS in improving dynamical model fits to individual RGs
is limited: the smoothing scale of the density field is larger than the scale
required to resolve (beta model–like) density profiles within filaments and
clusters ($0.1$–$1\ \mathrm{Mpc}\ h^{-1}$). At best, our densities reflect the
correct multi-megaparsec-scale averages, but these are still much lower than
the, say, central $1\ \mathrm{Mpc}$–scale densities. This is because the
massive galaxies that usually host RGs tend to occupy the bottom of their
local gravitational potential wells, where the large-scale density is
typically higher than everywhere in the vicinity. Relying on simulations, Oei
et al. (2023b) have estimated the relation between the $2.9\ \mathrm{Mpc}\
h^{-1}$–scale relative total matter density $1+\delta$ (as used throughout
this work), and the $1\ \mathrm{Mpc}$–scale relative IGM density
$1+\delta_{\mathrm{IGM}}$. For galaxies with a stellar mass
$M_{\star}=10^{11}\ M_{\odot}$, their Fig. 11 suggests
$1+\delta_{\mathrm{IGM}}=10\cdot(1+\delta)^{0.75}$. For luminous giants, the
mode of the $2.9\ \mathrm{Mpc}\ h^{-1}$–scale relative total matter density RV
is $1+\delta=3.5$ for the fixed voxel method and $1+\delta=6.3$ for the
flexible voxel method (see Fig. 6); the corresponding host-centred $1\
\mathrm{Mpc}$–scale relative IGM density is $1+\delta_{\mathrm{IGM}}=23$ for
the fixed voxel method and $1+\delta_{\mathrm{IGM}}=40$ for the flexible voxel
method. This relation in principle allows one to perform Bayesian inference of
the distributions of the $1\ \mathrm{Mpc}$–scale Cosmic Web density around RG
hosts — possibly even conditioned on RG radio luminosity, as in Sect. 4.2 — by
converting parametrised high-resolution distributions to the lower BORG SDSS
resolution, applying heteroskedastic measurement errors, and finally comparing
the resulting distribution to data. We leave this to future work.

Concerning resolution, we finally warn that care must be taken when
calculating (AGN feedback–related) quantities that relate quadratically to
density, such as the local bremsstrahlung emissivity: these tend to be
underestimated if computed from low-resolution densities. To see why, we
considered $\rho$, the density at some natural smoothing scale, alongside
$\langle\rho\rangle$, the density averaged over some larger-than-natural
smoothing scale. We assumed that only the latter is known. Next, we picked a
quantity $y=a\rho^{2}$, which we aimed to compute at the larger smoothing
scale: $\langle y\rangle=\langle a\rho^{2}\rangle=a\langle\rho^{2}\rangle$.
Naively, one would be tempted to proceed in calculating $\langle y\rangle$ by
assuming $\langle\rho^{2}\rangle\approx\langle\rho\rangle^{2}$, as to use
$\langle y\rangle\approx a\langle\rho\rangle^{2}$. However, by doing so, one
risks underestimating $\langle y\rangle$, because
$\langle\rho^{2}\rangle\geq\langle\rho\rangle^{2}$ by Jensen’s inequality. In
Appendix F, we explicitly calculated the ratio between
$\langle\rho^{2}\rangle$ and $\langle\rho\rangle^{2}$ for sheets and voids,
filaments, and clusters. For sheets and voids, this ratio is at most
$\frac{4}{3}$; for filaments, it is at most a number of order unity; for
clusters, it can exceed a thousand.

###  5.4 Spectroscopic redshift selection

Figure 17: Observations underpinning the RG radio luminosity–Cosmic Web
density relation of Fig. 8, but with Pan-STARRS $r$-band Kron flux densities
$F_{\nu}$ of the host galaxies indicated. We show uncertainty ellipses whose
semi-major and semi-minor axes represent one standard deviation along each
dimension. In the top panel, we show KDE Cosmic Web density distributions for
$F_{\nu}<1\ \mathrm{nJy}$, and for $F_{\nu}\geq 1\ \mathrm{nJy}$. These
suggest that spectroscopic redshift selection biases the observed Cosmic Web
density distributions high.  Figure 18:  Overview of radio luminosity
selection resulting from spectroscopic redshift selection, radio surface
brightness limitations, and the correlation between radio and optical
luminosity. The top panel compares radio luminosity distributions at
$\nu_{\mathrm{ref}}=150\ \mathrm{MHz}$ of LoTSS DR1 RGs in the constrained
part of the BORG SDSS, distinguishing between those with spectroscopic
redshifts (green) and those without (grey).222222We perform KDE on samples of
$\mathrm{log}_{10}(L_{\nu}(\nu_{\mathrm{ref}})\cdot(\mathrm{W\
Hz^{-1}})^{-1})$ rather than of $L_{\nu}(\nu_{\mathrm{ref}})$, and thus the
probability densities on the vertical axis are dimensionless. Radio galaxies
with spectroscopic redshifts dominate (77%) the population. The bottom panel
shows the two subpopulations in radio luminosity–redshift space, with RGs
without spectroscopic redshifts marked (grey circles). We show uncertainty
ellipses whose semi-major and semi-minor axes represent half a standard
deviation along each dimension. One retrieves the top panel by collapsing the
data on the horizontal axis.

To localise RGs in the Cosmic Web and subsequently measure their relative
total matter densities, we required them to have a spectroscopic redshift. By
selecting on spectroscopic redshift availability, we implicitly selected on
both Cosmic Web density and radio luminosity. The key reason is that
spectroscopic redshifts are preferentially available for galaxies with high
optical flux densities — so galaxies that are nearby,232323There is an
exception to this rule: physically large galaxies at low redshifts, such as
ellipticals in nearby galaxy clusters, can be angularly too extended for SDSS
fibres and are therefore sometimes excluded from SDSS’s spectroscopic survey.
Excluding the RGs associated to these galaxies from our samples introduces a
bias against high density environments. This artificially vacates the right
sides of Figs. 5 and 8, and leads towards an underestimation of the occurrence
of cluster RGs in our environment analysis. and galaxies that have high
optical luminosities. Galaxies with high optical luminosities tend to inhabit
dense environments and, when active, tend to generate RGs with high radio
luminosities.

To demonstrate that spectroscopic redshift selection indeed affects our RG
radio luminosity–Cosmic Web density relation, Fig. 17 revisits the same Local
Universe LoTSS DR1 RG data as shown in Fig. 8, but now with the Pan-STARRS
$r$-band Kron flux densities $F_{\nu}$ of the host galaxies indicated. The top
panel of Fig. 17 shows KDEs of $1+\Delta_{\mathrm{RG,obs}}$ for hosts with
$F_{\nu}<1\ \mathrm{nJy}$, and for hosts with $F_{\nu}\geq 1\ \mathrm{nJy}$.
Hosts with low optical flux densities tend to inhabit more tenuous
environments than hosts with high optical flux densities. By requiring
spectroscopic redshifts, we have discarded RGs with only photometric
redshifts, whose hosts almost exclusively have $F_{\nu}<1\ \mathrm{nJy}$. As a
result, our estimates of $1+\Delta_{\mathrm{RG,obs}}$,
$1+\Delta_{\mathrm{RG}}$, and $1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}=l_{\nu}$ are
biased high. The severity of the bias depends on the percentage of RGs that
have been selected out. As discussed in Sect. 3.2, 23% of all LoTSS DR1 RGs
within the constrained BORG SDSS volume lack a spectroscopic redshift. Our
sample of luminous giants is much less affected: only 7% of the population
within the constrained BORG SDSS volume lack spectroscopy. Thus, correcting
for spectroscopic redshift selection would shift the Cosmic Web density
distribution for general RGs to somewhat lower densities, while it would not
appreciably shift the corresponding distribution for luminous giants. The
result would be an even bigger discrepancy between the Cosmic Web density
distributions of luminous giants and general RGs than established in this
work, strengthening our results.

Spectroscopic redshift selection also leaves an imprint on the sample’s radio
luminosities. The top panel of Fig. 22 shows that the radio luminosities of
the 1870 LoTSS DR1 RGs in the constrained BORG SDSS volume are distributed
differently between the subpopulations with and without spectroscopic
redshifts. In particular, the subpopulation with spectroscopic redshifts is
more spread out over radio luminosity compared to the subpopulation without:
both very low and very high values are more probable.242424The distribution is
not only more dispersed, but is also shifted rightwards: the median and mean
radio luminosities of the spectroscopic subpopulation are 1.3 and 3.2 times
larger, respectively. To understand why, we turn to the bottom panel of Fig.
22. The LoTSS $6^{\prime\prime}$ noise level, the inverse square law, and
cosmological surface brightness dimming together determine an RG radio
luminosity detection threshold that increases with redshift. This threshold
induces the hatched region, which is inaccessible to LoTSS-like radio
observations. As a straightforward consequence, RGs with low radio
luminosities can be detected only at low redshifts. It is not immediately
clear whether the host galaxies of low–radio luminosity RGs at low redshifts
have low or high optical flux densities. On the one hand, low radio
luminosities suggest low optical luminosities: both forms of spectral
luminosity correlate positively with stellar mass (as evinced by, for example,
Figs. 2 of Mahajan et al. 2018; Sabater et al. 2019) and therefore252525There
exist cases in which this argument does not hold: correlation is not
necessarily transitive. positively with eachother. However, one must again be
cautious of selection effects, which can inject strong but spurious
luminosity–luminosity correlations in observations (Singal et al. 2019). On
the other hand, observing low-redshift galaxies is easier, again because of
the inverse square law and cosmological surface brightness dimming. The bottom
panel of Fig. 22, and in particular its lower left corner, shows that the
correlation between radio and optical luminosity is not strong enough to deny
the observational advantages of low redshifts: low–radio luminosity RGs at low
redshifts have high optical flux densities. As a result, spectroscopic
redshifts are more common for such RGs. Similarly, the correlation between
radio and optical luminosity explains that spectroscopic redshifts are more
common for high–radio luminosity RGs.

This indirect radio luminosity selection effect is not expected to change the
radio luminosity–Cosmic Web density relation inferred in this work and shown
in Figs. 8 and 9 (as well as in Figs. 19 and 20). This is because the inferred
relation concerns the Cosmic Web density distribution at given radio
luminosity. Changes in the number of RGs available in each decade of radio
luminosity — and in particular an increase in the number of RGs with $10^{22}\
\mathrm{W\ Hz^{-1}}<L_{\nu}(\nu=150\ \mathrm{MHz})<10^{23}\ \mathrm{W\
Hz^{-1}}$ — do not materially affect our analysis, as we already performed
data rebalancing that downweights our sample’s most populated
decades.262626The hatched region of Fig. 22 shows that radio surface
brightness selection causes our Local Universe RG sample to be strongly
incomplete in the radio luminosity decades $10^{21}$–$10^{22}$ and
$10^{22}$–$10^{23}\ \mathrm{W\ Hz^{-1}}$. Extending our sample in the lowest
of these decades upweights this decade’s importance, likely revealing that the
positive scaling between RG radio luminosity and Cosmic Web density is
stronger than we suggest in Sect. 4.2.

###  5.5 Future outlook

Excitingly, there appear to be many opportuntities to improve our
understanding of the large-scale environments of radio galaxies through Cosmic
Web reconstructions, both with current and near-future data.

In this work, we have used LoTSS DR1 Local Universe RLAGN with spectroscopic
redshifts from Hardcastle et al. (2019). With the release of the LoTSS DR2
radio data (Shimwell et al. 2022) and the associated optical crossmatching
results (Hardcastle et al. 2023), it will become possible to redo our analysis
on a footprint that is more than 13 times larger. Doing so would increase the
size of the general RG sample by an order of magnitude, from ${\sim}10^{3}$ to
${\sim}10^{4}$. In turn, this would allow us to revisit Sect. 4.2’s RG radio
luminosity–Cosmic Web density relation, but with all of Fig. 8’s six radio
luminosity decades fully evenly weighted. If the positive correlation between
radio luminosity and Cosmic Web density is genuine, it will emerge more
clearly upon weighting evenly. Such a LoTSS DR2 sample might even be large
enough to meaningfully investigate possible differences in the radio
luminosity–Cosmic Web density relation for different types of RGs, such as
those generated by quasars, and those generated by other AGN. In the near
future, the WEAVE–LOFAR Survey (Smith et al. 2016) will generate
${\sim}10^{6}$ spectra of LOFAR-selected galaxies, which will help alleviate
the effects of spectroscopic redshift selection described in Sect. 5.4.
Finally, the availability of optical spectra for all AGN in the current
analysis and for those in similar follow-ups means that a comprehensive
classification into HERGs and LERGs appears possible. In turn, this
classification would allow for studies into the relationships between AGN
accretion mode, RG growth, and the surrounding Cosmic Web. As these AGN reside
in the Local Universe, many could be further characterised by crossmatching
with X-ray catalogues.

Another promising extension of our work would be to use Cosmic Web
reconstructions that are complementary to or improve upon the BORG SDSS.
Notably, the BORG 2M++ (Jasche & Lavaux 2019) offers one such possibility: it
provides density reconstructions for the entire sky and at a higher spatial
resolution, and features more accurate selection functions, bias modelling,
and gravitational dynamics than the BORG SDSS. However, its reconstructions
are limited to $z_{\mathrm{max}}=0.1$, whereas those of the BORG SDSS extend
to $z_{\mathrm{max}}=0.16$. Using the BORG SDSS and the BORG 2M++ in
conjunction would allow for an extension of the number of giants for which a
density can be determined: the catalogue of giants by Oei et al. (2023a),
introduced in Sect. 2, contains 83 giants with spectroscopic redshifts
$z_{\mathrm{s}}<0.1$ that lie outside of the constrained BORG SDSS volume.
Adding these to the 260 giants for which we have already determined densities
would increase the number of BORG giants by 32%. Perhaps more importantly,
there are 89 giants for which both BORG SDSS and BORG 2M++ density estimates
are possible. Comparing these densities allows for a quality assessment of the
BORG SDSS density measurements at large. In particular, given its better
gravity solver, we expect BORG 2M++ cluster densities to perform much better
in Sect. 5.1’s Cosmic Web density–cluster mass test. In addition to the BORG
2M++, additional BORG runs are on their way. Besides, there exist comparable
Cosmic Web reconstruction frameworks, such as COSMIC BIRTH (e.g. Kitaura et
al. 2021). In any case, thanks to the combination of new spectroscopic galaxy
surveys and advances in computing power, high-fidelity Bayesian Cosmic Web
reconstructions will become available for an increasingly large fraction of
the Local Universe and beyond. We expect that Cosmic Web density $1+\delta$
will therefore become a standard RG observable, alongside properties such as
projected proper length $l_{\mathrm{p}}$ and radio luminosity $l_{\nu}$.

Through the work of Leclercq et al. (2015), we have derived probabilistic
Cosmic Web environment classifications, based on the $T$-web definition, for
260 luminous giants and 1443 LoTSS DR1 RGs. However, Leclercq et al. (2015)
provides additional environmental classifications for the BORG SDSS volume
that allow us to measure and understand the Cosmic Web environments of giants
and other RGs in complementary ways. Future work could explore and compare RG
environments in the $T$-web, DIVA, ORIGAMI, and LICH sense (Leclercq et al.
2016, 2017).

##  6 Conclusion

Using Bayesian Cosmic Web reconstructions of the Local Universe, we compared
the large-scale environments of giant radio galaxies with those of the radio
galaxy population in general. In particular, we measured multi-megaparsec-
scale Cosmic Web densities around the hosts of ${\sim}10^{2}$ luminous giants
and ${\sim}10^{3}$ general RGs. This reveals that the currently observable
population of giants inhabits denser regions of the Cosmic Web than general
RGs — contradicting the popular hypothesis that giants primarily form in the
dilute Cosmic Web. Our interpretation is that high jet powers, as implied by
the high radio luminosities of the known population of giants, enable these
RGs to overcome the IGM’s resistance to their megaparsec-scale growth.

Next, we quantified, for the first time, the relation between radio
luminosity, a proxy for jet power, and Cosmic Web density. Our radio
luminosity–conditioned Cosmic Web density distributions reinforce the idea
that AGN that generate weak jets primarily reside in the dilute Cosmic Web,
while AGN that generate powerful jets primarily reside in the dense Cosmic
Web. We subsequently showed that the radio luminosity–Cosmic Web density
relation, which we have inferred using our sample of general RGs only, can
accurately explain the density distribution observed for luminous giants.

Additional evidence corroborates that luminous giants form more often in the
dense Cosmic Web than RGs in general. Classifying Cosmic Web environments as
clusters, filaments, sheets, or voids on the basis of the local gravitational
field, we find that luminous giants occur more often in clusters than RGs in
general. We also find evidence that clusters with giant-generating BCGs are
more massive than other clusters; the evidence for the analogous claim for
clusters with RG-generating BCGs is weaker. We now summarise our main findings
in more detail.

  1. 1.

Using the BORG SDSS, we have performed the most physically principled
measurements yet of the Cosmic Web environments of giant radio galaxies and
radio galaxies in general. Our characterisations require spectroscopic
redshifts, and are currently confined to the Local Universe
($z<z_{\mathrm{max}}\coloneqq 0.16$). In particular, we have determined $2.9\
\mathrm{Mpc}\ h^{-1}$–scale total matter densities and gravitational
environment classifications for 260 luminous giants, of which 208 (80%) have
recently been discovered through the LoTSS DR2, and for 1443 LoTSS DR1 RGs.

  2. 2.

While the marginal probability distributions of the Cosmic Web density around
individual RGs are approximately lognormal, the probability distribution of
the Cosmic Web density for the RG population as a whole is approximately
gamma. We provide a physical argument that motivates why the gamma
distribution could have arisen. The probability distribution of the Cosmic Web
density for luminous giants also appears roughly gamma, but favours higher
densities. Heteroskedastic measurement errors, which are a general feature of
our densities, spuriously shift the population distributions towards lower
densities. We demonstrate a forward modelling method to correct for
heteroskedasticity.

  3. 3.

We quantified the radio luminosity–Cosmic Web density relation for general
RGs. We find that the $2.9\ \mathrm{Mpc}\ h^{-1}$–scale Cosmic Web density
distribution at a given $150\ \mathrm{MHz}$ radio luminosity is well described
by $1+\Delta_{\mathrm{RG}}\ |\ L_{\nu}=l_{\nu}\sim\Gamma(k,\theta)$, where
$k=4.8+0.2\cdot\textphnc{\Alamed}$, $\theta=1.4+0.02\cdot\textphnc{\Alamed}$,
and $\textphnc{\Alamed}\coloneqq\mathrm{log}_{10}(l_{\nu}\cdot(10^{23}\
\mathrm{W\ Hz^{-1}})^{-1})$. This shows that more luminous RGs tend to live in
denser regions of the Cosmic Web. Treating giants as ordinary RGs with
$l_{\nu}\geq 10^{25}\ \mathrm{W\ Hz^{-1}}$, we used this relation to predict
the Cosmic Web density distribution for luminous giants. The prediction is
consistent with the observed distribution for giants: a Kolmogorov–Smirnov
test yields a $p$-value of 0.7. Whether less luminous giants — assuming they
exist — also form more often in the dense Cosmic Web, remains to be seen when
surveys with increased surface brightness sensitivity commence. If such giants
obey the same radio luminosity–Cosmic Web density relation as other RGs, this
will not be the case.

  4. 4.

We show that our methodology enables the inference of radio galaxy number
densities as a function of Cosmic Web density. Whether or not the RG number
density is a strong function of Cosmic Web density, depends on the radio
luminosity considered. In clusters, the number densities of high-luminosity
RGs (e.g. $10^{25}$–$10^{27}\ \mathrm{W\ Hz^{-1}}$) are ${\sim}10^{2}$ times
higher than on average; in the same environment type, the number densities of
low-luminosity RGs (e.g. $10^{21}$–$10^{23}\ \mathrm{W\ Hz^{-1}}$) are just
${\sim}10^{1}$ times higher than on average. Furthermore, we obtain tentative
evidence that the number densities of low-luminosity RGs peak in low-mass
clusters, before decreasing again. A possible explanation could be that, as
cluster mass increases, the underlying population of active galaxies starts
favouring the generation of high-luminosity RGs over low-luminosity RGs.

  5. 5.

We used the BORG SDSS $T$-web classification to generate probability
distributions over four Cosmic Web structure types — clusters, filaments,
sheets, and voids — for both luminous giants and general RGs. Luminous giants
inhabit clusters more often ($23\begin{subarray}{c}+3\\\ -3\end{subarray}\%$
versus $16\begin{subarray}{c}+1\\\ -1\end{subarray}\%$). Independently, by
crossmatching our RGs with a cluster catalogue, we find that the former
percentage presumably is an underestimate, while confirming the latter
percentage. The same crossmatching procedure reveals tentative evidence
($p=5\%$) that clusters with giant-generating BCGs are more massive than other
clusters. At the same time, evidence that clusters with RG-generating BCGs are
more massive than other clusters is weaker ($p=7\%$), even though the sample
involved is almost twice as big.

Although RG environment characterisations with Bayesian Cosmic Web
reconstructions are still far from perfect, we have demonstrated that the
current generation of reconstructions already allows one to address open
questions in radio galaxy research. Given the pitfalls of the BORG SDSS, every
measured Cosmic Web density should be treated with skepsis, and we warn in
particular that the densities around cluster RGs are strongly underestimated.
Nevertheless, given the ever-increasing number of galaxies with spectroscopic
redshifts, a growing list of technical refinements, and the widespread
availability of computing power, exciting contemporary and near-future
opportunities exist to expand and improve upon the results of this work. It is
well possible that Cosmic Web density will soon become a staple quantity that
helps to unravel the physics of radio galaxies.

###### Acknowledgements.

M.S.S.L. Oei, R.J. van Weeren, and A. Botteon acknowledge support from the
VIDI research programme with project number 639.042.729, which is financed by
the Dutch Research Council (NWO). A. Botteon acknowledges financial support
from the European Union - Next Generation EU. M.S.S.L. Oei warmly thanks
Rafaël Mostert, Erik Osinga, and Wendy Williams for helpful discussions and
the sharing of data. LOFAR data products were provided by the LOFAR Surveys
Key Science project (LSKSP; https://lofar-surveys.org/) and were derived from
observations with the International LOFAR Telescope (ILT). 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 which are collectively operated by the ILT foundation under a
joint scientific policy. The efforts of the LSKSP have benefited from funding
from the European Research Council, NOVA, NWO, CNRS-INSU, the SURF Co-
operative, the UK Science and Technology Funding Council and the Jülich
Supercomputing Centre. Oei et al. (2023a)’s LoTSS DR2 GRG data are available
at the CDS via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/672/A163. The
LoTSS DR1 RG data with optical identifications from the value-added catalogue
of Williams et al. (2019) are available at https://lofar-
surveys.org/dr1_release.html; those interested in Hardcastle et al. (2019)’s
RLAGN subsample should contact Martin J. Hardcastle. The BORG SDSS data
release is publicly available at https://github.com/florent-
leclercq/borg_sdss_data_release. Individual BORG SDSS HMC Markov chain samples
are not available in the public domain; those interested should contact
Florent Leclercq. The cluster catalogue of Wen & Han (2015) is available at
the CDS.

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