A&A 622, A16 (2019) 
https://doi.org/10.1051/0004-6361/201833832 
 cESO 2019 
LOFAR Surveys: a new window on the Universe 
Astronomy&Astrophysics Special issue The intergalactic magnetic feldprobedby a giant radio galaxy 
S.P. O’Sullivan1,J. Machalski2,C.L.Van Eck3,G. Heald4,M. Brgen1,J.P.U. Fynbo5,K.E. Heintz6, 
M.A. 
Lara-Lopez7,V.Vacca8,M.J. Hardcastle9,T.W. Shimwell10,11,C.Tasse12,13,F.Vazza14,1,19,H. Andernach15, 

M. 
Birkinshaw16,M.Haverkorn17,C. Horellou18,W.L.Williams9,J.J. Harwood9,G. Brunetti19,J.M. Anderson20, 


S.A. Mao21,B. Nikiel-Wroczy´, nski2,K.Takahashi22,E. Carretti19,T.Vernstrom23,R.J.vanWeeren11,E. Orr10 
L. K. Morabito24, and J. R. Callingham10 
(Aﬃliations can be found after the references) 
Received 12 July 2018 / Accepted9October 2018 
ABSTRACT 
Cosmological simulations predict that an intergalactic magnetic feld (IGMF) pervades the large scale structure (LSS) of the Universe. Measuring the IGMF is important to determine its origin (i.e. primordial or otherwise). Using data from the LOFARTwo Metre SkySurvey(LoTSS), we present theFaraday rotation measure (RM) and depolarisation properties of the giant radiogalaxy J1235+5317, at a redshift of z = 0.34 and 
3.38Mpcin size.We fnda meanRMdiﬀerence between the lobesof2.5± 0.1radm−2, in addition to small scale RM variations of ∼0.1radm−2. FromacatalogueofLSS flamentsbasedonoptical spectroscopicobservationsinthelocaluniverse,wefndanexcessof flaments intersectingthe line of sight to only one of the lobes. Associating the entire RM diﬀerencetotheseLSS flamentsleadstoagas density-weightedIGMF strength of ∼0.3 µG.However, direct comparisonwith cosmological simulationsoftheRM contributionfromLSS flamentsgivesalow probability(∼5%) foranRM contributionaslargeas2.5radm−2, for the case of IGMF strengths of 10–50 nG. It is likely that variations in the RM from the Milky Way (on 110 scales) contribute signifcantly to the mean RM diﬀerence, and a denser RM grid is required to better constrain this contribution. In general, this work demonstrates the potential of the LOFAR telescope to probe the weak signature of the IGMF. Future studies, with thousands of sources with high accuracyRMs from LoTSS, will enable more stringent constraints on the nature of the IGMF. 
Keywords. galaxies: active– radio continuum:galaxies –galaxies: magnetic felds–galaxies: jets –techniques: polarimetric – 
galaxies: clusters: individual: J1235+5317 
1. Introduction 
Diﬀusegasisexpectedtopermeatethelarge-scale structure(LSS) of the Universeaway fromgalaxy groups and clusters. Detecting and characterising this intergalacticgas is challenging due to theexpectedlow particle number density(∼10−5–10−6cm−3) and temperature (105–107K). Although diﬀuse, this warm-hot intergalactic medium(WHIM; Davéetal.2001;Cen&Ostriker 2006) potentially contains half the total baryon content of the local Universe (Bregman 2007; Nicastro et al. 2018). In addition, accretion shocks along these LSS flaments are predicted to accelerate particles to relativistic energies and to amplify magnetic felds. Thus, detecting this flamentary structure in synchrotron emission using radio telescopes is a promising avenue for studying the WHIM (e.g. Vazza et al. 2015a). Recent statistical studies based on the cross-correlation of diffuse radio synchrotron emission and the underlying galaxy distribution havederived upper limits on the magnetisation of fl-amentsofthe orderof0.1µG(Vernstrom et al. 2017;Brown et al. 2017). Furthermore, Vaccaetal.(2018)foundafaint population of sources which mightbe thetipof the icebergofa classof diffuse large-scale synchrotron sources associated with the WHIM connected to a large-scale flament of the cosmic web. An alter-native approach is to measure the Faraday rotation properties of the magnetised WHIM using many bright, polarised, background radio sources (e.g. Stasyszyn et al. 2010; Akahori et al. 2014;Vaccaetal. 2016). 
From simulations, the feld strength of the intergalactic mag-netic feld (IGMF)isexpectedtobeinthe rangeof 1–100nG (e.g. Dolag et al. 1999; Brgen et al. 2005; Ryu et al. 2008; Vazza et al. 2017). It is important to constrain the magnetic feld in the WHIM in order to determine the unknown origin of the large scale magnetic feld in the Universe(Zweibel 2006). While large scale felds are commonly detected ingalaxies andgalaxy clusters, the strong modifcation of these felds erases the signature of their origin (e.g. Vazza et al. 2015b). This may not be the case in the WHIM, as the amplifcation of primordial magnetic felds in these flamentary regions are likely primarily due to compressive and shearinggas motions, in addition to small-scale shocks, such that the observed level of magnetisation could be connected to the seeding process (e.g. Ryu et al. 2008;Vazza et al. 2014). TheAGN and star formation activity in galaxies can also drive powerful outfows that may signifcantly magnetise the intergalactic medium on large scales (e.g. Furlanetto&Loeb 2001;Donnert et al. 2009;Beck et al. 2013). Therefore, distinguishing betweena primordial origin anda later injectionofmagneticfeldthatwas initially generatedon smaller scalesbygalaxiesand starsisakeygoalfor studiesofthe IGMF (see Akahori et al. 2018, and references therein). 
It has also been proposed to study the WHIM using large or “giant” radiogalaxies (GRGs) whose linear size canextend beyond1Mpc, with the largest such example being 4.7Mpc in extent(Machalski et al. 2008). GRGs are usually FRII type radio galaxies (e.g. Dabhade et al. 2017), although some giant FRI also exist (e.g. Heesen et al. 2018; Horellou et al. 2018), that extend well beyond their host galaxy and local environments, into the surrounding intergalactic medium. Asymmetries in the GRG morphology can be used as a probe of the ambient 
Article publishedby EDP Sciences A16, page1of 12 
gas density (Subrahmanyan et al. 2008; Safouris et al. 2009; Pirya et al. 2012; Malarecki et al. 2015)and the Faraday rota-tion properties of the polarised emission from the lobes can be usedtostudythemagneticfeld propertiesofthe surroundinggas onMpc scales(Xuetal. 2006;O’Sullivanetal. 2018). Another potential approach to studying the magnetised WHIM in cluster outskirtsisby usingFaraday rotation observationsof the highly polarised emission from radio relics (e.g. Kierdorf et al. 2017; Loi et al. 2017). 
The eﬀect of Faraday rotation is measured through its infuence on the linear polarisation vector as a function of wavelength-squared. The observed Faraday rotation measure, RM [radm−2], depends on the line-of-sight magnetic feld, B|| [µG], threadinga regionof ionisedgas with electron density, ne [cm−3], along a path length, l [pc], following 
Z telescope 
RM = 0.812 ne Bk dl. (1) 
source 
In this paper, we present an analysis of the linear polarisa-tion and Faraday rotation properties of an FRII radio galaxy (J1235+5317) with a linear size of 3.4 Mpc. The observa<BR>tions were done with the LOw Frequency Array (LOFAR; van Haarlem et al. 2013) which provides excellent sensitivity to diﬀuse extended structures due to the presence of numer-ous short baselines and exceptional Faraday rotation mea-sure (RM) accuracy, which depends on the total coverage in wavelength-squared. While low frequency radio telescopes provide the best RM accuracy, sources at these frequencies are most strongly aﬀected by Faraday depolarisation (e.g. Burn 1966), which decreases the degree of linear polarisation below the detection limit for many sources(Farnsworth et al. 2011). Despite this there is a growing number of polarised sources being found at low frequencies (e.g. Bernardi et al. 2013; Mulcahyet al. 2014; Jeli´c et al. 2015; Orret al. 2015; Lenc et al. 2016; Van Eck et al. 2018; O’Sullivan et al. 2018; Neld et al. 2018;Riseleyet al. 2018). J1235+5317 was discovered to be polarised at 144 MHz byVanEcketal. (2018),inLOFAR data imagedatan angu-lar resolution of 4.30 . The source was frst reported by Schoenmakers et al. (2001), and the frst optical identifcation (SDSS J123458.46+531851.3) was proposed by Banfeld et al. (2015). However, our new observations show that the previously assumed host galaxy is accidentally located close to the geometric centre between the two lobes and that the real host galaxy is actually connected to the south east (SE) lobe by a faint jet. The radio core is coincident with the galaxy SDSS J123501.52+531755.0, which is identifed as PSO J123501.519+531754.911(Flewelling et al. 2016)for the radio source ILT J123459.82+531851.0 in Williams et al. (2019). Estimates of the photometric redshift of this galaxy are 0.349(Bilicki et al. 2016), 0.41(Beck et al. 2016)and 0.44 (Brescia et al. 2014;Duncan et al. 2019). 
The host galaxy is identifed in Hao et al. (2010) as a red-sequence galaxy and a cluster candidate, GMBCG J188.75636+53.29864. This is intriguing as GRGs are often thoughttoevolvein underdensegalaxyenvironments 
(e.g. Mack et al. 1998), however, recent work indicates that they are most likely the oldest sources in the general population of powerful radio galaxies (Hardcastle et al. 2019). In addition, Hao et al. (2010)estimate a total of ∼9 galaxies within 
0.5 Mpc with luminosities L > 0.4L∗, using a weak-lensing scaling relation, which suggests a poor cluster environment. There is also no evidence for a massive cluster at this location 
A16, page2of 12 
in the sky in the Planck thermal Sunyaev-Zeldovich map (Planck Collaboration XXII 2016). 
This paper presentsafollow-up study using the same LOFAR data as Van Eck et al.(2018),but imaging at higher angular res-olution.We also confrm the new optical host identifcation and determine its spectroscopic redshift as z ∼ 0.34, giving the pro-jected linear sizeof 3.4Mpc.In Sect. 2, we describe the radio polarisation and optical spectroscopic observations. Section 3 presentsthephysical propertiesof J1235+5317, the inference on the properties of its environment based on dynamical modelling of the jets, and the RM and depolarisation behaviour. In Sect. 4 we discuss the results in the context of the study of the intergalactic medium and its magnetisation. The conclusions are listed in Sect.5. Throughout this paper, we assumea ΛCDM cosmology withH0 = 67.8kms−1Mpc−1, ΩM = 0.308 and ΩΛ = 0.692 (Planck Collaboration XIII 2016). At the redshift of the source, 100 correspondstoa linearsizeof5.04kpc.We defnethetotal intensity spectral index, α, such that the observed total intensity (I)at frequencyν follows the relation Iν ∝ ν+α . 
2. Observations and data analysis 
2.1. Radio observations 
The target source J1235+5317 was observed as part of the LOFARTwo-Metre SkySurvey(LoTSS; Shimwell et al. 2017, 2019), which is observing the whole northern sky with the LOFAR High-Band Antenna (HBA) from 120 to 168 MHz. The data relevant to our target were observed in full polarisation for 8hon 26 June 2014, as part of the observing program LC2_038 and with a pointing centre of J2000 12h38m06s.7, +52◦0701900 . This gives a distance of ∼1.26◦ of the target J1235+5317 from the pointing centre (the FWHM of the primary beam is ∼4◦). Direction-independent calibrationwas performed using the prefactor pipeline1, as described in detail in Shimwell et al. (2017) and de Gasperin et al. (2018), which includes the ionospheric RM correction using rmextract2. Residual ionospheric RM cor<BR>rection errors of ∼0.05 radm−2 are estimated between observations (Van Eck et al. 2018), while slightly larger errors of ∼0.1–0.3 radm−2 are estimated across a single 8h observation (Sotomayor-Beltran et al. 2013). 
The resulting measurement set, after the prefactor pipeline, has a time resolution of 8s and a frequency resolution of 
97.6 kHz. The direction-independent calibrated data are used throughout for the polarisation and rotation measure analysis, while the direction-dependent calibrated total intensity image (Shimwell et al. 2019)is used to determine the source morphological properties with high precision and for the identifcation ofthehostgalaxy location. Analysisof polarisationand rotation measure data products after direction-dependent calibration will be presented in future work. 
2.2. Polarisation and Faraday rotation imaging 
To analyse the polarisation and Faraday rotation properties of the target, we phase-shifted the calibrated uv-data to the coordinates of the host galaxy (12h35m01s.5, +53◦1705500), which lies almost at the centre of the extended emission. We cali-brated the data for short-timescale phase variations caused by the ionosphere, then averaged to 32s to reduce the data size and to help speed up the subsequent imaging, while avoiding 
1 https://github.com/lofar-astron/prefactor <BR>2 https://github.com/lofar-astron/RMextract <BR>
S.P. O’Sullivanet al.: The intergalactic magnetic feld probedbya giant radiogalaxy 
any signifcant time smearing (e.g. Neld et al. 2018). Both the phase-shifting and time-averaging were done using NDPPP (van Diepen&Dijkema 2011)3. The imaging software wsclean (Oﬀringa&McKinley 2014)4 was used to create I, Q, U, V channel images at 97.6 kHz resolution, for a 250 feld of view (∼twice the linear size of J1235+5317).A minimum uv-range of 150 λ was usedtoavoid sensitivityto Galactic polarised emission on scales of &250.The maximum uv-rangewassetto18kλ, and combined with a Briggs weighting of 0, resulted in a beam size of 2600 × 1800, sampled with300 × 300 pixels. The diﬀerential beam correction per channel was applied using wsclean, as the correction for the LOFAR beam gain at the pointing centre was already applied during the initial calibration of the data. All channel images with Q or U noise higher than fve times the average noise level were removed from subsequent analysis, leaving a total of 404 images covering 120–167 MHz (with a central frequencyof 143.5 MHz). 
RM synthesis and rmclean (Brentjens&de Bruyn 2005; Heald et al. 2009)were then applied to theQ and U images using pyrmsynth5. The data have an RM resolution of 1.16radm−2, are sensitive to polarised emission from Faraday thick regions up to ∼0.98 radm−2, and |RM| values forFaraday thin regions as high as 450radm−2 can be detected. An RM cube with a Faraday depth(φ)axis covering ±500 radm−2 and sampled at 
0.5 radm−2 intervalswas constructedfor initial inspectionofthe data. The concept ofFaraday depth(Burn 1966)can be useful to introduce here to describe regions with complicated distributions ofFaraday rotation along the line of sight, such as multiple distinct regions of polarised emission experiencing diﬀerent amounts ofFaraday rotation, which could be identifed through multiple peaksinaFaraday depth spectrum orFaraday dispersion function (FDF). As no signifcant emission was found at largeFaraday depths,the fnalRMand polarisation images were constructed from FDFs witharangeof ±150 radm−2,sampled at 
0.15 radm−2.To identify peaksin the FDF,a thresholdof8σQU was used, where σQU is calculated from the outer 20% of the Faraday depth range in thermclean Q and U spectra. The mean σQU across the feld was ∼90 µJybeam−1. Since no correction was made for the instrumental polarisation, peaks in theFaraday dispersion function appears near φ ∼ 0radm−2 at a typi-cal level of ∼1.5% of the Stokes I emission. This instrumental polarisation signal is also smeared out by the ionospheric RM correction making it diﬃcult to identify real polarised emission at lowFaraday depths(. ± 3radm−2). Thus, when identifying real polarised emission peaks in the FDF,the range ±3radm−2 is excluded. RM and polarised intensity images are created from the brightest, real polarised peak above 8σQU at each pixel, after ftting a parabola around the peak to obtain the best-ftting RM and polarised intensity. In the case of the polarised inten-sity image, a correction for the polarisation bias was also made following George et al. (2012). The error in the RM at each pixel was calculated in the standard way as the RM resolution divided by twice the signal to noise ratio of the detection (Brentjens&de Bruyn 2005). 
Afull-band StokesI image was made using the same image parameters as the channel images specifed above, with multiscale cleaning applied for an automatic threshold of 3σ and deeper cleaning (to 0.3σ)within an automatic masked region created from the clean components. The degree-of-polarisation image was created by dividing the band-averaged polarised 
3 https://support.astron.nl/LOFARImagingCookbook/ <BR>4 https://sourceforge.net/projects/wsclean <BR>5 https://github.com/mrbell/pyrmsynth <BR>

Fig. <BR>1. <BR>Optical spectrum of the host galaxy SDSS J123501.52+ 531755.0 taken with AIFOSC instrument on the Nordic Optical Telescope, which shows emission linesHα,[Oii]and[Oiii]at a redshift of 0.34. 
intensity image from RM synthesis (with a cutoﬀ at8σQU )by the full-band Stokes I image (with a cutoﬀ at3 times the local noise level). 
2.3. Optical spectroscopic observations 
SDSS J123501.52+531755.0 was observed with the Nordic OpticalTelescope on March25 and March26 2018 fora total integration timeof 5400s.We used the AndaluciaFaint Object Spectrograph and Camera (AlFOSC) and a 1.3 arcsec wide longslit and grism4 with 300 rules per millimetre providinga spectral resolution of 280 and a useful spectral range of 3800– 9100Å. The slit was placed at a parallactic angle of 60◦ east of north on both nights at the onset of integration. The airmass ranged from 1.20 to 1.15. The observing conditions were poor withavariable seeing above2arcsec and with passing clouds. Despite this we clearly detected several emission lines (Fig. 1) consistent with a mean redshift of 0.3448 ± 0.0003 (1-sigma error). The [Oii]and [Oiii]images have a peculiar morphology extending away from the continuum source to the northern side of the galaxy. In particular [Oiii],λ5008Å can be traced over 4arcsec below the continuum trace (20 kpc at z = 0.34). This indicates the presence of an extended emission line region. 
3. Results 
3.1. Radio morphology of J1235+5317 
Figure2 shows the total intensity image at600 resolution from the LoTSS direction-dependent calibrated data(Shimwell et al. 2019). This provides the best radio image to date for this source, enabling an unambiguous host galaxy identifcation with SDSS J123501.52+531755.0. The noise level in the image ranges from ∼70 µJybeam−1 in areas away from bright sources to ∼100 µJybeam−1 near the hotspots/lobes. 
The core of this FRII radio galaxy, located at J2000 12h35m01s.5, +53◦1705500, has an integrated fux density of ∼1.1 mJy at 144 MHz and 1.4 GHz (FIRST; Becker et al. 1995) suggesting a fat spectrum. However, the core is also detected in the VLASS6 Quick-Look (QL) imageat3GHz(∼2.9 mJy) and the 9C catalogue(Waldram et al. 2010)at 15 GHz(∼4mJy) indi-cating an inverted spectral index of αcore ∼ +0.3when combined with the LoTSS core fux density. As the LoTSS, VLASS and 9C observations are closest in time, we consider the core to have 
6 https://archive-new.nrao.edu/vlass/ <BR>
A16, page3of 12 

600 
Fig. <BR>2. <BR>LoTSS total intensity image at 144 MHz at resolution (after direction-dependent calibration). The contours start at 300 µJybeam−1 and increasebyfactorsof2(with one negative contour at −300 µJybeam−1). The greyscale image is tuned to show the noise variation across the image(∼70 µJybeam−1 away from bright sources and ∼100 µJybeam−1 near the hotspots), as well asafaint hintof the south-east jet. The radiogalaxy core coincident with the hostgalaxy SDSS J123501.52+531755.0 is indicated by the horizontal arrow. The synthesised beam size is shown in the bottom left hand corner of image. 
an inverted spectral index, with time variability explaining the lower than expected fux density from FIRST at 1.4 GHz. There isalsoafainthintofajet connectingthehostwiththe south-east (SE) lobe. If this is real, then it suggests that the SE jet and lobe are orientated slightly towards us on the sky. 
Using the3σ contour to defne the lobe edges, we fnd the lobes have a width of ∼8300 and ∼9400, giving an axial ratio of ∼4.4 for the north-west (NW) lobe and ∼3.3 for the SE lobe, respectively. This is consistent with the typical axial ratios from 2to7forthelobesofmost (smaller)GRGs(e.g.Machalskietal. 2006).InTable 1,wecompiletheintegratedfux densitiesofthe NW and SE lobes and hotspots from both current and archival data. The integrated fux densities of the NW lobe and hotspot are slightly higher than the SE lobe and hotspot at 144 MHz, with both having spectral index values of αlobe ∼−0.8. The NW hotspot is resolved into primary and secondary hotspot regions inthe VLASSat3GHz(2.400×2.100 beam), while the SE hotspot maintains a single component. 
The straight-line distance from the core to the NW hotspot is ∼36500 (1.84 Mpc), compared to ∼31100 (1.56 Mpc) from the core totheSE hotspot,givinga lobe length ratioof 1.17.The inferred jet-misalignment (from co-linearity) of ∼13.6◦ is most likely due to bending of the NW and/or SE jets on large scales, as is sometimes observed in other FRII radio sources (Black et al. 1992). We expect that the lobe-length asymmetry and jet-misalignment are caused by interactions between the jet and the external environment on large scales, as opposed to light travel time eﬀects (Longair&Riley1979).Asymmetriesinthejetandlobelengths of GRGs are often attributed to interactions with the large scale structure environment(Pirya et al. 2012;Malarecki et al. 2015). The advancing NW jet may be infuenced by a nearby flament (see Sect. 4.4.1 and the flament in the z ∼ 0.335 slice), although deeper optical spectroscopic observations would be required to determine whether or not this flament is indeed close enough in redshifttothatofthehostgalaxytohavean infuence. 
3.2. Faraday rotation measure distribution 
Figure3 shows the RM distribution for J1235+5317, using an 8σQU threshold,overlaidby Stokes I contours at the same angu-lar resolution. TheFaraday dispersion functions for the brightest pixelin polarised intensityineachlobearealsoshown,withared cross marking the peak polarisation at which the RM was found. Other peaks in the spectrum are either noise peaks or related to the instrumental polarisation near RM ∼ 0radm−2. The RM distributionsofeachlobeareshowninFig. 4.The meanand stan<BR>dard deviation of the RM are +7.42 radm−2 and 0.07radm−2 for theNW lobe,and +9.92 radm−2and 0.11radm−2 for the SE lobe, respectively. The median RM errors for the NW and SE lobe regions are 0.04radm−2 and 0.06radm−2. The mean RM diﬀerence betweenthe lobesof2.5± 0.1radm−2 is thus highly signifcant.Atthe angular separationofthe lobes(110),systematic errors inthe ionosphericRM correctionwouldaﬀect both lobes equally andthusdonot contributetotheRMdiﬀerence between the lobes. Wecan estimatethe signifcanceofthesmallRMvariationswithin each lobe accounting for the number of pixels in each synthesised beam following Leahyet al. (1986),whereareduced-chi-squared of ∼1is expected if noise errors dominate the RM fuctuations. We fnd no evidence for the detection of signifcant RM varia-tions acrosstheNWlobe,withareduced-chi-squaredof1.1.However,areduced-chi-squaredof1.8providesevidence,atalevelof ∼1.35σ,forRMvariations acrosstheSE lobeof ∼0.1 radm−2. 
3.3. Faraday depolarisation 
The polarised intensity and degree of polarisation distributions are shown in Fig. 3. The NW lobe is much brighter with a peak polarised intensity of 6.5 mJy beam−1 (coincident with the hotspot) and a degree of polarisation of 4.9% at that loca-tion (ranging from 1.2% to 5.1% across the detected emission). TheSE lobeisfainter witha peak polarised intensityof 
1.1 mJy beam−1. The degree of polarisation at that location is 2.8%, and it ranges from 1.1 to 3.3% across the lobe. The non-detection of polarised emission from the SE hotspot is likely due to intrinsic non-uniform feld structures andFaraday depolarisation on scales smaller than the resolution of our observations. The fainter, extended lobe emission would have to be &10% polarised to be detected in these observations. 
In order to estimate the amount of depolarisation between 
1.4 GHz and 144 MHz, the LoTSS data were compared with those of the NRAO VLA Sky Survey (NVSS; Condon et al. 1998).To determinethedegreeof polarisationatthe same angu-lar resolution as the NVSS survey, the RM pipeline was re-applied to the LoTSS data imaged at a lower angular resolution of ∼4500 . 
At the peak polarised intensity location in the NW lobe of the LOFAR image, matched to the NVSS resolution, the degree of polarisationis4.0± 0.3%. At the same location in the NVSS imageat1.4GHz,thedegreeof polarisationis6.4± 1.4%. This givesa depolarisationfactorofDP144 ∼ 0.6, where DP144 
1400 1400 is the degree of polarisation at 144 MHz divided by the degree of 
polarisation at 1.4 GHz. Assuming the commonly used external 
−2σ2 
Faraday dispersion model for depolarisation, p(λ) ∝ eRMλ4 (Burn 1966), provides a value of σRM ∼ 0.1radm−2. 
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S.P. O’Sullivanet al.: The intergalactic magnetic feld probedbya giant radiogalaxy 

Fig. <BR>3. <BR>Left <BR>image:main image:Faraday rotation measure distribution (colour scale)of the north-west (NW) and south-east (SE) lobe regions that are detected above the thresholdof8σQU ,overlaidby the total intensity contours starting at5mJy beam−1and increasinginfactorsoftwo. Insets: The absolute value of the rmclean Faraday dispersion function for the brightest polarised pixel in the NW lobe (top) and SE lobe (bottom).Right image:main image: polarised intensity greyscale, in mJy beam−1,overlaidbythe total intensity contours. Insets:degreeof polarisation colourscale (in per cent) from zoomed in regions of the NW and SE lobes. 

For the SE lobe, the degree of polarisation at the peak polarised intensity at 144 MHz is 1.8 ± 0.7% (at 4500 resolution) and 10.1±2.1%atthe same locationat1.4GHz. Thisgives 
DP144 
1400 ∼ 0.2, corresponding to larger amounts of depolarisation thanin theNW lobe.In the caseofexternalFaraday dispersion, this corresponds to σRM ∼ 0.2radm−2. 
The observed diﬀerence in depolarisation between the NW and SE lobes may be due to the diﬀerent location within each lobe from which the polarised emission arises. In the case of the NW lobe, the peak polarised emission is coincident with the hotspot location, whereas in the SE lobe, the peak polarised emission is signifcantly oﬀset from the hotspot(∼4000 away, in the bridge emission, with the oﬀset also present in the NVSS images). Furthermore, from the non-detection of polarisation in the SE hotspot at 144 MHz, with a degree of polarisation <0.35%, we can placealower limit on theFaraday depolarisation at this location of σRM ∼ 0.25 radm−2,based on comparison with the NVSS degree of polarisation of ∼5% at this location. 
From inspection of the VLASS QL image at 3GHz, the physical extent of the NW hotspot (∼2.400) is smaller com-pared to the SE lobe region (of order 2000 in size) and thus less aﬀected by depolarisation caused by RM variations within the synthesised beam at 144 MHz. Since the amount of depo-larisation scales roughly as the square-root of the number of Faraday rotation cells, this could reasonably explain the diﬀerence in the observed depolarisation between the lobes. However, the enhanced depolarisation at the location of the SE hotspot is more diﬃcult to explain and may indicate a signifcant inter-action between the hotspot/lobe magnetic feld and the ambient medium. This warrants further investigation with more sensitive observations at low frequencies. 
Overall,giventhe small amountof observedFaraday depo-larisation, it is important to consider the accuracy of the correction forFaraday rotation from the ionosphere. Van Eck et al. (2018)estimate a residual error in the ionosphere RM correction between observations of 0.05radm−2. As the ionosphere RM corrections across an observation (i.e.8h) are linearly inter-polated in time between direct estimates every2h, a rough 
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Table 1. Archival and measured fux densities, as well as the best-ft fux densities (in the self-consistent, s.c., fts) for the north-west and south-east lobes of J1235+5317. 
N-lobe S-lobe 
Freq. Entire Lobe Hotspots s.c. ft Entire Lobe Hotspots s.c. ft (MHz) (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) (1) (2) (3) (4) (5) (6) (7) 
143.6i 403 ± 40 151 ± 21 356.6 378 ± 40 132 ± 25 345.3 151a 350 ± 52 344.4 320 ± 52 333.3 151b 375 ± 32 344.4 302 ± 31 333.3 325c 177 ± 36 193.0 149 ± 36 185.1 325i 154 ± 58 193.0 153 ± 58 185.1 408d 160 ± 40 160.6 145 ± 34 153.2 1400e 59 ± 4 55.9 50 ± 2 51.0 1400i 55 ± 19 36 ± 4 55.9 47 ± 19 33±5 51.0 2980g 21 ± 3 20 ± 3 4850 f 21 ± 4 18.2 18.4± 4 15.6 15 200h (5.2± 2) 5.2± 1 6.3 (6.6± 2) 6.6± 1 5.1 
References. (a)6C3 (Hales et al. 1990); (b)7Cn (Rileyet al. 1999); (c)WENSS (Rengelink et al. 1997); (d)B3.3(Pedani&Grueﬀ <BR>1999); (e)NVSS(Condon et al. 1998);(f )GB6(Gregory et al. 1996);(g)VLASS (Lacyetal.in prep.); (h)9Cc(Waldram et al. 2010);(i)this paper. 
estimate can be made for the residual error within the observa-
√ tion of ∼0.05 4∼ 0.1radm−2. This means that most (or all) of the observed depolarisation in the NW hotspot is possibly due to residual errors in the ionospheric RM correction. However, the diﬀerence in depolarisation between the NW hotspot and SE lobe cannot be explained by ionosphere RM errors. Therefore, a σRM of at least ∼0.1 radm−2 in the SE lobe can be considered astrophysically meaningful. This is comparable to the RM variations across the SE lobe of ∼0.1radm−2 found in Sect. 3.2. 
3.4. Dynamical modelling 
In order to decouple the properties of the electron density and magnetic feld along the line of sight in the measuredFaraday rotation and depolarisation, additional information is required on the physical characteristics of J1235+5317 (i.e. the magnetic feld strength of the emission region) and the properties of its surrounding environment (i.e. the ambientgas density). These properties can be estimated through dynamical modelling of the radio lobes, while simultaneously accounting for energy losses of relativistic particles (electrons and positrons) injected into the expanding lobes by the relativistic jets (e.g. Machalski et al. 2011, 2016, and references therein). This is important because we lack X-ray data that could constrain the properties of the external medium (e.g. Ineson et al. 2017)and/or the magnetic feld strength of the hotspot and lobes, without the need for the assumption of equipartition between the radiating particles and magnetic feld (e.g. Mingo et al. 2017).Therefore, here we apply theevolutionaryDYNAGE codeof Machalskietal. (2007)to the radio lobes of J1235+5317, primarily to obtain an estimate of theexternalgas density, as well as estimates for the magnetic feld strength of the lobes. The ftting procedure is performed separately for each lobe using the observational data given in Sect. 3.1, together with the radio luminosities calculated from the fux densities listed inTable 1. The input model parameters that are assumed aregiveninTable 2. 
Characteristic of almost all FRII sources is a modest asym-metry in the length and radio luminosity of the lobes. Therefore, as mightbeexpected, theDYNAGE results for thejetpower Qj, 
Table 2. Dynamical modelling input model parameters. 
Parameter Symbol Value (1) (2) (3) 
Set: Adiabatic index of the lobes’ material Γlb 4/3 Adiabatic index of the ambient medium Γx5/3 Adiabatic index of the lobes’ magnetic feld ΓB4/3 Minimum electron Lorentzfactor (injected) γmin 1 Maximum electron Lorentzfactor (injected) γmax 107 Core radius of power-law ambient density distribution a0 10 kpc Initial slope of power-law ambient density distribution β 1.5 Thermal particles within the lobes k 0 Jet viewing angle θ 90◦ 
Free: Jet power Qj(ergs−1) External density at core radius ρ0(gcm−3) Exponent of initial power-law energy distribution of relativistic particles p = 1+ 2αinj Source (lobe) age t(Myr) 
the central density of the external medium ρ0, and other physi-cal parameters can appear diﬀerent for the two lobes of the same source. This aspect has been analysedby Machalski et al. (2009, 2011)for a sample of thirty GRGs. While some of the diﬀerences were within the uncertainties of the ftted values for the model parameters, signifcant diﬀerences were possible in cases where the evolution of the magnetic feld and/or various energy losses and acceleration processes of the relativistic particles are diﬀerent at the hotspots of the opposite lobes. Alternatively,such diﬀerences, especially in GRGs, may refect diﬀerent external conditions well beyond the hostgalaxy and cluster/group environment. 
Following Machalski et al. (2009), we averaged the values of Qj and ρ0 initially found in the “independent solution” and treated them as fxed parameters in the “self-consistent” model, hQji and hρ0i, respectively. New values of the slope of the ambient density distribution(β)and the age(t)for the NW and SE lobes, are denoted as βs.c. and ts.c. (Table 3). The DYNAGE fts to the observed data points are shown with solid lines in Fig. 5. Table 3 presents the derived physical properties of the lobes, including a minimum-energy magnetic feld strength in the lobes of Bme ∼ 1µG and an external density of ∼2 × 10−31gcm−3 (i.e. ne ∼ 10−7cm−3). This density is similar to the mean density of the Universe assuming half the baryons are in the WHIM(Machalski et al. 2011), and implies that the radio lobesarelikely propagatingintoalow-densityregionoftheUni-verse. 
We also used the synchrotron minimum energy (equipartition) magnetic feld formulation in Worrall et al.(2006)to esti-mate the lobe magnetic feld strength. From this we fnd an equipartition magnetic feld strength that is 2.6 times higher than the1 µGderived from the dynamical modelling (forγmin = 10). When calculated in this manner the lobe equipartition feld strength is usually found to be overestimated, by a factor of 2 to 3, compared to that found from X-ray Inverse Compton observations of lobes (e.g. Ineson et al. 2017; Mingo et al. 2017). This highlights some of the uncertainties in the calcula<BR>tion of equipartition magnetic feld strengths in radio galaxies 
(e.g. Beck&Krause 2005;Konaretal. 2008). Here we adopt the lobe magnetic feld strength obtained from the dynamical 
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S.P. O’Sullivanet al.: The intergalactic magnetic feld probedbya giant radiogalaxy 

Fig. <BR>5. <BR>DYNAGE fts (solid lines) to the total intensity spectra of the north-west and south-east lobes (open circles), and the spectral points of the hotspot regions (flled dots; not used in the fts). Note that the north-west lobe fux density scale is shifted one decade up in relation to the given ordinate scale. 
modelling as it takes into account more physical eﬀects, such as the jet power, adiabatic expansion and age of the lobes. 
4. Interpretation 
The diﬀerence in the mean RM between the NW and SE lobes is 2.5± 0.1radm−2. This may be due to variations in the Galactic RM (GRM) on scales of ∼110, diﬀerences in the magnetoionic material of the intergalactic medium on large scales, and/or line-of-sight path length diﬀerences towards either lobe. The observedFaraday depolarisationof σRM ∼ 0.1radm−2 associated with the SE lobe could be due to small scale fuctuations of the magnetic feld in the local external medium and/or from Faraday rotation internal to the source. Constraining the likelihood of these possibilities requires some considerations of the expectedvariationsintheGRM,knowledgeofthe geometryand physical properties of the radio lobes, and details of the environment surrounding the radiogalaxy andin the foreground. 
4.1. Galactic RM variations 
The reconstruction of the GRM by Oppermann et al. (2012, 2015)gives+14.8 ± 4.5radm−2 across both the NW and SE lobe (the Galactic coordinates of J1235+5317 are l = 128.46◦ , b = 63.65◦). This is higher than the mean RMs of +7.4 and +9.9radm−2 found for the NW and SE lobes, respectively. However,it shouldbekeptin mind thatthe LoTSSRMvalueshave been corrected for the time-variable ionosphere RM(+1.6 to +1.9radm−2), while the catalogue from which the GRM map is mainly made(Taylor et al. 2009)does not have this correction applied. Thus, the RM of the NW and SE lobe are within the 1-sigma and 2-sigma errors in the GRM, respectively. 
The variation in the GRM map for three adjacent pixels (in the direction of the largest gradient) across the source is 
Table 3. Fitted values of the model free-parameters in the “selfconsistent” dynamical modelling solution. 
Parameter Symbol Value forN-lobe Value forS-lobe (1) (2) (3) (4) 
Initial eﬀective spectral index αinj −0.45 ± 0.05 −0.52 ± 0.0 Source (lobe) age (Myr) ts.c 95 ± 23 80 ± 16 Jet power(×1045ergs−1) hQji 1.1± 0.11.1± 0.1 Core density(×10−28gcm−3) hρ0i 4.7± 0.44.7± 0.4 Slope of ambient density distribution βs.c. 1.431 1.613 External density(×10−31gcm−3) ρ(D) 2.8 ± 1.1 1.4 ± 0.7 Lobe pressure(×10−14dyn cm−2) plb 3.0 ± 0.1 3.1 ± 0.1 Minimum energy magnetic feld(µG) Bme 1.0 ± 0.2 1.0 ± 0.2 Longitudinal expansion speed vh/c 0.05 ± 0.02 0.06 ± 0.02 
∼2.2radm−2 (on a scale of ∼1◦). As the GRM map has a resolution of ∼1◦, which is the typical spacing of extragalactic sources in the Tayloretal. (2009)catalogue,it cannotbe used to probe RM variations on smaller scales. The true GRM variation on smaller scales at this location is unknown, but RM structure function analyses for GRM variations at high Galactic latitudes have probed scales smaller than 1◦ in both observations (e.g. Mao et al. 2010;Stil et al. 2011)and simula<BR>tions (e.g. Sun&Reich 2009). In particular, using the results from Stil et al. (2011), we fnd that GRM variations rang-ing from approximately 3radm−2 to 13radm−2 are possible on angular scales of ∼110, depending on the highly uncertain slope of the RM structure function on angular scales less than1◦ . 
Better estimates of the GRM are required to reliably remove the GRM and its variation across the extent of J1235+5317. 
4.2. Local environment RM contribution 
The hotgas in rich groups and clusters is known to be magnetised from observations of synchrotron radio halos and relics, as well as Faraday rotation observations of embedded and background radio sources (see Carilli&Taylor 2002, and refer<BR>ences therein). For radio galaxy lobes that have not expanded signifcantly beyond their host galaxy or cluster/group environment, the Laing-Garrington eﬀect is often present(Laing 1988;Garrington et al. 1988;Garrington&Conway 1991). This is where the polarised emission from the counter-lobe travels through a greater amount of magnetoionic material and thus incurs a larger amount ofFaraday depolarisation. However, as the lobes of J1235+5317 are expected to be orientated close to the plane of the sky and extend well outside the infuence of the group/cluster environment, the Laing-Garrington eﬀect is notexpectedtobe strong (e.g. Laing&Bridle 2014). Addition<BR>ally, if the faint collimated emission SE of the host is indeed a jet, then the larger amount of depolarisation towards the SE lobe is opposite to that expected for the Laing-Garrington eﬀect. 
Models of the variations in RM across radio galaxies in groups and clusters are typically constructed assuming turbu-lent magnetic feld fuctuations over a range of scales embedded in a spherically-symmetric gas halo whose radial density profle is derived from X-ray observations (e.g. Guidetti et al. 2008). For J1235+5317 we do not have X-ray data to con-strain the properties of the hotgas environment, although it is likely that the red-sequence host galaxy is close to the centre of a poor cluster(Hao et al. 2010). Therefore, we attempt to estimate the required density and feld strength to selfconsistently explain the mean RM and depolarisation (e.g. 
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Murgia et al. 2004), for a single-scale model of a randomly ori<BR>entated feld structure(Felten et al. 1996). In reality, the mag-netic feld will fuctuate on a range of scales, from an inner scale to an outer scale(Enßlin&Vogt 2003), a single-scale model can provide a reasonable approximation to the RM varia-tions if the scale length is interpreted as the correlation length of the magnetic feld (see Murgia et al. 2004, Sect. 4.4 for details). 
An appropriategas density profle, n(r), foragalaxy group 
2)−3β/2 
or cluster is a “beta-profle”, where n(r) = n0(1 + r2/r. 
c 
We assume that the magnetic feld strength scales linearly with the gas density, B(r) = B0n(r)/n0, where B0 is the central magnetic feld strength (e.g. Dolag et al. 2001;Laing et al. 2006; Vacca et al. 2012; Govoni et al. 2017). Values of n0 ∼ 
−3 
10−3cm, rc ∼ 100 kpc and β ∼ 0.5 are not unreasonable for a poor cluster (e.g. Laing et al. 2008;Bonafede et al. 2010; Guidetti et al. 2012). The choice of these parameters is arbitrary given our limited information about the environment of the host galaxy (Sect.1)but we use them simply asa plausibleexample. Following(Murgiaetal.2004,Eq. (15)),wefndaFaradaydis<BR>persion of σRM ∼ 0.1radm−2 at r ∼ 1.5Mpc requiresB0 ∼ 5µG witha magnetic feld correlation lengthof ∼25kpc. This implies an ambient density of ∼1.7× 10−5cm−3 and feld strength B ∼ 
0.09 µGat the location of the hotspots7. Using these values and a large outer scale for the magnetic feld fuctuations of 500 kpc (Vacca et al. 2010)gives a mean|RM| of ∼0.4 radm−2. Therefore, while we can reasonably explain σRM ∼ 0.1radm−2 at r ∼ 1.5Mpc, we cannot self-consistently explain the large mean RM excess of ∼2.5 radm−2, even for a large outer scale of tur-bulence in the magnetic feld power spectrum(Enßlin&Vogt 2003; Murgia et al. 2004). Note that the outer scale is mainly responsible for the observed mean RM and the inner scale for the value of σRM.Weusedalarge outer scaleheretoshowthatthis model cannot self-consistently explain both σRM and the mean RM. 
Draping of the ambient feld in addition to compression of the ambient magnetoionicgas could enhance the mean RM near the surface of the lobes(Guidetti et al. 2011, 2012), and may also help explain the higher depolarisation of σRM & 
0.15 radm−2 at the location of the SE hotspot. Enhancements in the feld strength and gas density by factors of 4 over a path length of ∼50 kpc outside the lobes could produce an additional |RM| of ∼0.5 radm−2. More sensitive observations at high angu-lar resolution are required to determine if such ordered feld structures are indeed present. 
We note that the external gas density used here is two orders of magnitude higher than estimated from the dynam-ical modelling. This means that either the observed depolarisation does not occur in the external medium local to the source or that the dynamical modelling is severely underestimating the external density. Such low density gas may be challenging to detect in X-rays, but extrapolation of an X-ray profle from the inner region would be very instructive. In general, comparison with simulations of the propagation of large scale jets within a realistic cosmological environment may provide the best avenue for progress in this area 
(e.g. Huarte-Espinosa et al. 2011; Hardcastle&Krause 2014; Turner&Shabala 2015;English et al. 2016;Vazza et al. 2017). 
For comparison, usingasimple model withaconstant electron number density of ne ∼ 10−5 cm−3 and constant magnetic feld strength of B|| ∼ 0.1µG, with a magnetic feld reversal scale of l ∼ 20kpc 
√ over a total path length of L ∼ 1Mpc gives σRM ∼ 0.81ne B|| lL ∼ 0.1radm−2. 
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4.3. Internal Faraday depolarisation 
Our observations are insensitive to polarised emission from RM structures broader than ∼1radm−2 (Sect. 2.2). Therefore, the large amounts of internal Faraday rotation required to explain the mean RM excess are ruled out. However, it is worth con-sidering if the small amount ofFaraday depolarisation(σRM ∼ 0.1radm−2) can be explained by Faraday rotating material mixed with the synchrotron emitting material in the lobes. 
One of the most commonly used magnetic feld models for the lobes of extragalactic sources is one where the feld is highly tangled on small scales, with the observed appreciable degrees of polarisation produced due to stretching and com-pression(Laing 1980). Given the equipartition magnetic feld strength of ∼1µG within the lobes (Sect. 3.4), and as an illus<BR>trativeexample,we choosea thermalgas density internaltothe lobes of ne ∼ 10−5cm−3, with 500 feld reversals through a lobe depth of ∼500 kpc, to produce σRM ∼ 0.1radm−2 (using Eq.(1) 
√ 
and assuming B|| = B/ 3). Observations at even lower frequencieswouldbe requiredto resolveaFaraday depth widthof 
0.1 radm−2 in theFaraday spectrum (e.g. using LOFAR observations down to at least 30 MHz, in combination with the data in this paper). In addition, broadband polarisation modelling would be needed to distinguish between internal and externalFaraday depolarisation scenarios (e.g. Anderson et al. 2018; O’Sullivan et al. 2018). Using the LOFAR international base-lines to obtain sub-arcsecond resolution would further enhance the ability to isolate diﬀerent contributions by resolving the external RM variations across the emission region. 
For now, we can assess the likelihood of this scenario in terms of the implied energetics. For expected internal ther-mal gas temperatures of &10keV(Gitti et al. 2007), the lobe thermal gas pressure is pth ∼ 2nekT ∼ 3 × 10−13 dyn cm−2, which is an order of magnitude larger than the pressure from the synchrotron-emitting plasma in the lobes(plb in Table 3). Thisis inconsistent withexpectations from studiesof other FRII lobes(Croston et al. 2005;Ineson et al. 2017),and thus unlikely, unlessthe internal thermalgasismuchcoolerthan assumedhere. 
4.4. RM contribution from large-scale structure 
Signifcant asymmetries in the magnetoionic material in the foreground IGM,far from the local source environment, could also contribute to the observed mean RM diﬀerence between the lobes. Such variations could be caused by the magnetised com-ponent of the large scale structure (LSS) at low redshift, as Ryuetal. (2008),Cho&Ryu (2009)andAkahori&Ryu (2010) predict a root-mean-square RM (RMrms)through LSS flaments of order1radm−2. In our case, the polarised emission of one lobe needs to pass through more foreground flaments than the other toexplainthe observedRMdiﬀerenceof 2.5radm−2. Therefore, information is required on the location of LSS flaments with respect to the lines of sight probed by the polarised emission from the lobes of J1235+5317. 
4.4.1. Location of large-scale structure flaments 
The catalogueof Chenetal. (2015,2016)providesa cosmicfl<BR>ament reconstruction from the SDSS data for 130 redshift slices in the range0.05 < z < 0.7. In Fig. 6, we plot the location of the flaments that are in the foreground of J1235+5317 (i.e. at z < 0.34). There are fve flaments identifed in diﬀerent foreground redshift slices that pass through the feld. We assign a thicknessof1Mpcto each flament(Vazzaetal. 2015b)to 
S.P. O’Sullivanet al.: The intergalactic magnetic feld probedbya giant radiogalaxy 

Fig. <BR>6. <BR>Location of foreground large-scale-structure flaments (lines) in relation to the background radiogalaxy (contours) and itsFaraday rota-tion measure (colour scale), as described in Fig. 3. The width of the lines corresponds to ∼1Mpc at the redshift of the flament. 
determine which flaments most likely intersect lines of sight towards the polarised lobes (Fig. 6).Fora thicknessof1Mpc, there are four flaments that cover the NW lobe and one flament that covers the SE lobe. Therefore, we estimate that there is an excess of three flaments covering the NW lobe. Considering diﬀerent flament thicknesses results in diﬀerent numbers of flaments covering each lobe, with an excess of flaments covering the NW lobe remaining for flaments up to a thickness of ∼3.8 Mpc (i.e. the thickness above which the same number of flaments cover both lobes). In light of this result, we consider if the RM diﬀerence betweenthelobescanbeexplainedbymagnetisedgasin these flaments.We note that thereis noevidenceof an individual interveninggalaxy in the SDSS images that could explain the RM diﬀerence. 
4.4.2. Magnetic feld stength in flaments 
To explain the RM diﬀerence between the lobes, an RM excess of −2.5 radm−2 must be provided by the three extra fla-ments covering the NW lobe. Simulations suggest that the elec-tron number density of LSS flaments can vary from 10−6 to 10−4cm−3(Cen&Ostriker 2006;Ryu et al. 2008;Cho&Ryu 2009;Akahori&Ryu2010;Vazzaetal. 2015b),thusweadopt 
−3 
a mean electron density of 10−5cm. Akahori&Ryu (2011) found a peak in the RM power spectrum, due to their simulated IGMFin flaments,on scales correspondingtoa proper lengthof ∼3Mpc, which theyexpect to correspond to the typical line-ofsight path through LSS flaments. Therefore, using a path length (L)of3Mpc anda coherence length(l)of 300kpc(Cho&Ryu 2009)leadstoa magnetic feld strengthinthe flaments(BLSS) of approximately 
&#18; &#19;−1 ne BLSS ∼ 0.3 10−5cm−3  !−1/2 L l µG, 3(3 Mpc) 300 kpc  (2)  
√  

for B|| = BLSS/ 3. This estimate of the density-weighted IGMF strength of ∼0.3µGhas signifcant uncertaintygiven our limited knowledgeof the particle number densityof thegasin these fl-aments, as well as the observationally unconstrained coherence length of the feld and the path length though each flament. Fur-thermore, this estimate cannot be treated as an upper limit as a large Galactic RM variation across the source (Sect. 4.1)could make the diﬀerence in RM between the lobes even larger (since the RM can be positive or negative). Furthermore, much larger RM variations are observed across radio relics which cannot be explained by Galactic RM variations, indicating the presence of large scale ordered feldsin the outskirtsofgalaxy clusters (e.g. Kierdorf et al. 2017;Loi et al. 2017). 
Therefore, a better approach may be to compare directly with cosmological simulations of the RM contribution from such LSS flaments. These simulations suggest that the magnetic feld strength in flaments could range somewhere from ∼1to 100nG 
(e.g. Vazza et al. 2015b). Early hydrodynamic simulations by Ryuetal. (2008)useda prescriptionto produce magnetic felds from the kinetic energyof turbulentgas fows (guidedbyexpectations from small-scale magnetic dynamo simulations), which produced average IGMF strengths of ∼10 nG. Subsequent work by Cho&Ryu (2009)andAkahori&Ryu (2010, 2011), using the results of these simulations, provided estimates of the “typical” RM contribution from LSS flaments. The most relevant numberforFaraday rotationisthegas density( ρ)weightedaver-age of the strength of the magnetic feld through the flaments, 
i.e. h(ρB)2i1/2/hρ2i1/2, whichgaveafew ×0.1µG in the above simulations. From this, it was found that the root-mean-square RM (RMrms)through the flaments scales with the number of fl-aments(Nf)as RMrms ∼ 1.5N1/2 radm−2,uptoasaturation point 
f 
that corresponds to ∼25 flaments for z > 1. In the case of three flaments, the predicted RMrms ∼ 2.6radm−2, which is consis-tent with our observations (where we have an RM diﬀerence of 
2.5 radm−2 between only two lines of sight, in which one passes though three additional flaments). Therefore, it can be argued that our results are consistent with the expected Faraday rota-tion signature from an average magnetic feld strength in LSS flaments of ∼10 nG. 
We furtherinvestigatedtheabove fndingsby direct comparison with recent MHD cosmological simulations, as described inVazzaetal. (2014).In particular, we analysed theRM dis-tributionin thewarm-hotgas simulatedina cosmicvolumeof 503Mpc3,ata spatial resolutionof20kpc (comoving).To bet-ter compare with our observations, we generated a long integration cone for this volume, stacking several randomly oriented, mirrored replicas of the volume, covering the comoving dis-tance out to z = 0.34. In this way, we could measure the probability of havinga contribution as large as 2.5radm−2 from LSS flaments for the J1235+5317 observations at z = 0.34. We found that this occured in only 5% of cases, for typical magnetisation values of ∼10–50 nG, amplifed from an initial magnetic feld strength of 1nG, which was seeded at an early cosmological epoch and is in line with the upper limits given by the Planck satellite(Planck Collaboration XIX 2016). The probability was negligible for a signifcantly smaller seed feld of 0.1 nG. 
Lower limits on the primordial feld strength of ∼10−16 G (Neronov&Vovk 2010)and ∼10−20G(Takahashi et al. 2013) imply that the true value may indeed be much lower. However, this is not the only possible scenario, as the LSS can be mag-netised by a more “astrophysical” mechanism, such as galaxy feedback (e.g. Vazza et al. 2017, for a recent review), or pro-duced by a more eﬃcient dynamo amplifcation of primordial felds(Ryu et al. 2008)than is found in current MHD simulations. Therefore, from comparison with the MHD simulations, we consider it unlikely that the true RM contribution from the IGMFis as large as 2.5radm−2,andthatthe observedRMexcess 
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is possibly dominated by other contributions along the line of sight, such as small scale GRM variations (Sect. 4.1). 
5. Conclusions 
We have presented a linear polarisation and Faraday rotation study of a giant FRII radio galaxy, J1235+5317, using data fromtheLOFARTwo-MetreSkySurvey(Shimwellet al. 2019). After obtaining the spectroscopic redshift of the host galaxy (SDSS J123501.52+531755.0, z = 0.3448 ± 0.003), we fnd that the radiogalaxy hasa projected linearextentof 3.4Mpc. Both lobes are detected in polarisation with a mean RM diﬀerence betweenthe lobesof2.5±0.1radm−2. Small amountsofFaraday depolarisation(∼0.1radm−2)are also detected. In the absence of direct tracers of thegas density on large scales, we employ dynamical modelling of the advancing hotspots to infer a parti-cle number density of the ambientgas of ne ∼ 10−7cm−3. This implies that the radio galaxy is expanding into an underdense regionoftheUniverse.However,explainingthe observedFaraday depolarisation (that most likely occurs in the environment local to the source) requires ne ∼ 10−5cm−3 in combination with a turbulent magnetic feld strength of ∼0.09 µGat a distance of ∼1.5Mpc from the hostgalaxy. Therefore, either the dynamical modellingis underestimatingthe densityoftheexternal medium or the depolarisation does not occur in the local source environment. Simulations of the propagation of FRII jets to large scales withina realistic cosmological environment may help dis-tinguish between these scenarios. In general, the estimated mag-netic feld strength is unable to account for the observed mean Faraday rotation diﬀerenceof 2.5radm−2 between the twolobes. 
Using a catalogue of large scale structure (LSS) flaments in the local universe derived from optical spectroscopic observations, we fnd an excess of flaments intersecting lines of sight towards the polarised emission of the NW lobe. Assuming that magnetisedgasin theseLSS flamentsis responsiblefortheRM diﬀerence between the lobes,givesa density-weighted magnetic feld strength of 0.3 µG (assuming ne ∼ 10−5cm−3, a line-ofsight path length through each flament of 3Mpc, and a mag-netic feld coherence length of 300 kpc). However, we fnd that predictions from cosmological simulations of the RM contribution from LSS flamentsgivesalow probability(∼5%) for an RM contributionaslargeas2.5radm−2. This probability applies to the case of magnetic felds strengths in the LSS flaments of 10–50 nG, which are amplifed from primordial magnetic felds close to current upper limits from the CMB of ∼1nG (the probability decreases to ∼0% for weaker felds). Extrapolation of the observedvariationsinthe MilkyWayRMto110 scales (i.e. the angular size of J1235+5317) indicates that this likely contributes signifcantly to the mean RM diﬀerence, however, further observations are required to obtain better constraints. 
In the near future, large samples of RMs from radiogalaxies with known redshifts will allow more advanced statistical analysis techniques to be used, such as RM structure function analyses (e.g. Akahori et al. 2014) and cross-correlation with other tracersofLSS(e.g. Stasyszynetal.2010;Vernstrometal. 2017; Brown et al. 2017). This will enable a better separation of theFaraday rotation due to our Galaxy (e.g. Haverkorn et al. 2004;Sun&Reich 2009;Mao et al. 2010;Stil et al. 2011)from that due to the cosmic web, and put stronger constraints on the strength and structure of the intergalactic magnetic feld. 
Acknowledgements. This paper is based (in part) on data obtained with the International LOFAR Telescope (ILT) under project codes LC2_038 and LC3_008. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array 
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designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientifc policy. The ILT resources have beneftted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Edu-cation, Poland. SPO and MB acknowledge fnancial support from the Deutsche Forschungsgemeinschaft (DFG) under grant BR2026/23.Partof thisworkwas carried out on the Dutch national e-infrastructure with the support of the SURF Cooperative through grant e-infra 160022&160152. The LOFAR software and dedicated reduction packages on https://github.com/apmechev/GRID_LRT <BR>were deployed on the e-infrastructureby the LOFAR e-infragroup, consistingof 
J.B.R. 
Oonk (ASTRON&Leiden Observatory),A.P. Mechev(Leiden Observa-tory) andT. Shimwell (ASTRON) with support fromN. Danezi (SURFsara) and 

C. 
Schrijvers (SURFsara). This research has made use of data analysed using the University of Hertfordshire high-performance computingfacility(http: <BR>//uhhpc.herts.ac.uk/)and the LOFAR-UK computing facility located at the University of Hertfordshire and supported by STFC [ST/P000096/1]. This research made use of Astropy, a community-developed core Python package for astronomy(AstropyCollaboration 2013)hosted athttp://www.astropy. <BR>org/, of Matplotlib(Hunter 2007), of APLpy(Robitaille&Bressert 2012), an open-source astronomical plotting package for Python hosted at http:// <BR>aplpy.github.com/,andofTOPCAT,an interactive graphicalviewerandedi-tor for tabular data(Taylor et al. 2005). FV acknowledges fnancial support from the ERC Starting Grant “MAGCOW”, no.714196, and the usage of e usage of computational resources on the Piz-Daint supercluster at CSCS-ETHZ (Lugano, Switzerland) under project s701 and s805. Based on observations made with the Nordic OpticalTelescope, operatedby the Nordic OpticalTelescope Scientifc Association at the Observatorio del Roque de los Muchachos, LaPalma, Spain, of the Institutode Astrofsicade Canarias. KEH and JPUF acknowledge support by a Project Grant (162948-051) from The Icelandic Research Fund. The Cosmic Dawn Center is funded by the DNRF. RJvW acknowledges support from the ERC Advanced Investigator programme NewClusters 321271 and the VIDI research programme with project number 639.042.729, whichis fnancedby the Netherlands Organisation for Scientifc Research (NWO). HA benefted from grantDAIP #66/2018 of Universidad de Guanajuato. KT is partially supported by JSPS KAKENHI Grant Number 16H05999 and 17H01110, MEXT KAKENHI Grant Number 15H05896, and Bilateral Joint Research Projects of JSPS. LKM acknowledges support from Oxford Hintze Centre for Astrophysical Sur-veys whichis funded through generous support from the HintzeFamily CharitableFoundation. This publication arises from research partly fundedby the John Fell Oxford University Press (OUP) Research Fund. SPO thanks A. G. de Bruyn for stimulating discussions on the topic of this paper, and the referee for their helpful comments. 


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1 Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg112, 21029 Hamburg, Germany e-mail: shane@hs.uni-hamburg.de 
2 Astronomical Observatory, Jagiellonian University, ul. Orla 171, Krak 30-244, Poland 
3 Department of Physics and Astronomy, University of Calgary, Calgary, Alberta T2N 1N4, Canada 
4 CSIROAstronomy and Space Science,PO Box 1130, Bentley,WA 6102, Australia 
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5 The Cosmic Dawn Center, Niels Bohr Institute, University of Copenhagen, Juliane MariesVej30, 2100 CopenhagenØ, Denmark 6 Centre for Astrophysics and Cosmology, Science Institute, Univer-
sity of Iceland, Dunhagi 5, 107 Reykjavík, Iceland 7 Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane MariesVej30, 2100 CopenhagenØ, Denmark 8 INAF – Osservatorio Astronomicodi Cagliari,Via della Scienza5, 09047 Selargius (CA), Italy 
9 Centre for Astrophysics Research, School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hat-feld AL10 9AB, UK 
10 ASTRON, The Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands 
11 Leiden Observatory,Leiden University,PO Box 9513, 2300 RA Lei-den, The Netherlands 
12 GEPI & USN, Observatoire de Paris, Université PSL, CNRS, 5 Place Jules Janssen, 92190 Meudon, France 
13 Department of Physics&Electronics, Rhodes University, PO Box 94, Grahamstown 6140, South Africa 
14 Dipartimento di Fisica e Astronomia, Universitá di Bologna, Via Gobetti 93/2, 40121 Bologna, Italy 
15 Departamento de Astronomía, DCNE, Universidad de Guanajuato, Guanajuato, Mexico 
16 HHWillsPhysics Laboratory,Universityof Bristol,TyndallAvenue Bristol BS8 1TL, UK 
17 Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, the Netherlands 
18 Dept. of Space, Earth and Environment, Chalmers Univer-sity of Technology, Onsala Space Observatory, 43992 Onsala, Sweden 
19 INAF – Istituto di Radioastronomia, via P. Gobetti 101, 40129 Bologna, Italy 
20 GFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, Germany 
21 Max-Planck-Institut f Radioastronomie, Auf dem Hel 69, 53121, Bonn, Germany 
22 Department of Physics, Kumamoto University, Kumamoto 8608555, Japan 
23 Dunlap Institute for Astronomy and Astrophysics University of Toronto,Toronto,ON M5S 3H4, Canada 
24 Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, OxfordOX1 3RH,UK 
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