Constraints on Large-Scale Magnetic Fields in the Intergalactic Medium Using Cross-Correlation Methods
MMNRAS , 1–15 (2015) Preprint 25 February 2021 Compiled using MNRAS L A TEX style file v3.0
Constraints on Large-Scale Magnetic Fields in the IntergalacticMedium Using Cross-Correlation Methods
A.D. Amaral, , ∗ T. Vernstrom, , and B.M. Gaensler , David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, ON, M5S 3H4, Canada Dunlap Institute for Astronomy and Astrophysics, University of Toronto, Toronto, ON, M5S 3H4, Canada CSIRO Astronomy and Space Science, PO Box 1130, Bentley, WA 6102, Australia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Large-scale coherent magnetic fields in the intergalactic medium are presumed to play akey role in the formation and evolution of the cosmic web, and in large scale feedback mech-anisms. However, they are theorized to be extremely weak, in the nano-Gauss regime. Tosearch for a statistical signature of these weak magnetic fields we perform a cross-correlationbetween the Faraday rotation measures of 1742 radio galaxies at 𝑧 > . . < 𝑧 < .
5, as traced by 18 million optical and infrared foreground galax-ies. No significant correlation signal was detected within the uncertainty limits. We areable to determine model-dependent 3 𝜎 upper limits on the parallel component of the meanmagnetic field strength of filaments in the intergalactic medium of ∼
30 nG for coherencescales between 1 and 2 . ∼ . / m due to filaments along all probed sight-lines. These upper bounds are consis-tent with upper bounds found previously using other techniques. Our method can be used tofurther constrain intergalactic magnetic fields with upcoming future radio polarization sur-veys. Key words: magnetic fields – intergalactic medium: intergalactic filaments – large-scalestructure of the universe – methods: statistical – radio continuum: galaxies
Astronomers have been able to detect the presence of magneticfields across many scales, both Galactic and extra-galactic (seeVallée 1997, 1998, 2004 for reviews). However, magnetic fieldson the largest scales in the Universe remain largely unconstrained(Widrow 2002). These scales are occupied by filaments, voids,and galaxy clusters, where over-dense filaments connect to formthe Universe’s large-scale structure known as the cosmic web(Springel et al. 2006).The dominant gas phase of filaments is in the form of the warmhot ionized medium (WHIM), which contains baryons at hightemperatures (10 K ≤ 𝑇 ≤ K, Ryu et al. 2008) due to shockheating. At low redshifts, the WHIM is thought to contain ∼ ∗ E-mail: [email protected] 𝛼 − Ω dynamo, see Kulsrud & Zweibel 2008 for a review) require thepresence of an initial weak seed field within the IGM at earlytimes.It is certain that magnetic fields existed on small-scales in the pri-mordial Universe, due to the presence of currents, though theirpresence on larger cosmic scales remains theoretical (Grasso &Rubinstein 2001). These primordial magnetic fields (PMFs) mayhave been generated during early phase transitions along phasebubbles (such as the quantum chromodynamic and electroweakphase transitions, Widrow 2002), or shocks present during thesetransitional phase boundaries could have amplified and gener-ated magnetic fields with coherence scales on the order of thephase bubbles (Kahniashvili et al. 2013). Big bang nucleosyn-thesis chemical abundances place bounds on PMFs from nG to 𝜇 G (Grasso & Rubinstein 1995; Cheng et al. 1996; Kawasaki &Kusakabe 2012). Bounds on the presence of PMFs from the tem-perature and polarization maps of the cosmic microwave back-ground (CMB) are ≤
10 nG (Planck Collaboration et al. 2016b).Cosmic-scale magnetic fields also may have been seeded from as-trophysical sources that release large amounts of magnetic fluxinto the IGM. Examples of such sources include the highly mag-netized coherent outflows from active galactic nuclei (AGN), © a r X i v : . [ a s t r o - ph . C O ] F e b Amaral et al. which contain charged particles undergoing acceleration in jets(Furlanetto & Loeb 2001). The strength of the outflows can stretchmagnetic field lines, causing large coherence scales and spread-ing outwards into the IGM (Furlanetto & Loeb 2001). In a similarmanner, the first starburst galaxies could have generated signifi-cant magnetized winds due to rapid star formation, injecting mag-netic flux into the surrounding IGM (Kronberg et al. 1999).The presence of magnetic fields have been inferred observation-ally for galaxies at high redshifts (Kronberg & Perry 1982; Wolfeet al. 1992; Oren & Wolfe 1995; Bernet et al. 2008; Mao et al.2017), indicating that generation mechanisms must be present inthe early universe. Additionally, large-scale 𝜇 G fields have beendetected in the intra-cluster medium (ICM) between galaxies (Kimet al. 1991; Feretti et al. 1995; Clarke et al. 2001; Bonafede et al.2010, 2013), and in galaxy cluster haloes on the outskirts of theICM (Roland 1981; Kim et al. 1990).The difficulty detecting intergalactic magnetic fields (IGMFs) isdue to the fact that these fields are theorized to be extremely weak,in the nG regime, and contain magnetic field reversals (Ryu et al.2008; Cho & Ryu 2009; Akahori & Ryu 2010; Vazza et al. 2015b;Akahori et al. 2016). Additionally, the density of relativistic parti-cles is low in these environments (Cen & Ostriker 2006; Bregman2007), making detection via synchrotron emission difficult (Brown2011; Vazza et al. 2015a).Direct detections of IGMFs may only be possible through datafrom future telescopes (such as the upcoming Square KilometreArray and its pathfinders, Gaensler 2009). Until then, statisticaland indirect methods must be used in order to infer their pres-ence. Measured properties from these techniques, such as the fieldstrength and coherence scale, can differentiate between magneto-genesis models (Donnert et al. 2009; Vacca et al. 2015).Statistical methods using Faraday rotation data can be used toplace upper bounds on line-of-sight magnetic fields permeatingthe IGM. Magneto-ionic media are birefringent, thus as linearlypolarized light pass through magnetic fields it undergoes Faradayrotation, in which the polarization angle of the radiation rotatesalong the line-of-sight. The degree of observed rotation is:
ΔΦ =
𝑅𝑀𝜆 , (1)where 𝜆 is the observed wavelength, and RM is a proportionalityconstant called the rotation measure. This RM can be used to inferinformation about the line-of-sight component of the magneticfields causing the Faraday rotation. The RM is given by: 𝑅𝑀 = ∫ 𝑧 𝑠 ( + 𝑧 ) − 𝑛 𝑒 ( 𝑧 ) 𝐵 (cid:107) ( 𝑧 ) 𝑑𝑧 · 𝑑𝑙𝑑𝑧 rad m − , (2)where 𝑧 𝑠 is the redshift of the emitting polarized source, 𝑛 𝑒 isthe column density of free electrons along the line-of-sight tothe source measured in 𝑐𝑚 − , 𝐵 (cid:107) is the line-of-sight componentof the magnetic field measured in 𝜇𝐺 , and the line-of-sight 𝑑𝑙 ismeasured in kpc.Because the RM is an integrated effect along the line-of-sightfrom a distant source towards Earth, it contains the rotation mea-sure induced within the source itself (RM intrinsic ), an interveningextra-galactic component (RM intervening ), a Galactic component(RM Galactic ), and a component due to noise (RM noise ):RM = RM intrinsic + RM intervening + RM Galactic + RM noise . (3)Various statistical methods have been applied to place upper bounds on large-scale IGMFs, such as quantifying RM growthover redshift for large samples (Blasi et al. 1999; Kronberg et al.2008; Bernet et al. 2013; Hammond et al. 2012; Xu & Han 2014;Pshirkov et al. 2016; Neronov et al. 2013), and quantifying RMdifferences between physical pairs vs. sources that are close on thesky but occupy different redshifts (Vernstrom et al. 2019). In caseswhere redshift information is unavailable, studies looked at non-physical pairs assuming that z would not be identical (O’Sullivanet al. 2020; Stuardi et al. 2020). These techniques have placed up-per bounds on IGMFs ranging from ∼ nG to ∼ 𝜇 G, depending onthe method and assumptions used.
Cross-correlation techniques between extragalactic RM-grids (re-gions of sky with large numbers of RM measurements) and tracersof large-scale structure can be used as another method to extractthe IGMF signal. Because it is difficult to directly observe largescale structure, wide-field galaxy catalogues and galaxy numberdensity grids can be used to trace large-scale structure and the cor-responding IGM. This has been done with success using variouswide field multi-wavelength galaxy catalogues (see Jarrett 2004,or Gilli et al. 2003 for examples). If large-scale magnetic fieldsexist within the IGM as traced by galaxy densities, any polarizedsources behind this IGMF will have an intervening RM contribu-tion due to this magnetic field, as described in equation 3. If thebackground RMs are correlated with the foreground galaxy den-sities on large scales, this suggests the presence of a large-scalemagnetic field within the IGM.A cross-correlation technique was used on mock foreground-subtracted extragalactic RM catalogues by Kolatt (1998) to detectfeatures of a PMF power spectrum. Xu et al. (2006) cross corre-lated RMs in galaxy superclusters (such as Hercules, Virgo, andPerseus), with galaxy densities of the clusters to search for a signalfrom the IGMF within these structures; they found an upper limitof ∼ . 𝜇 G for coherence scales of 500 kpc.More recently, Lee et al. (2009) used a cross-correlation techniqueover large regions of the sky (120 ◦ < 𝛼 < ◦ , 0 ◦ < 𝛿 < ◦ )between 7244 extragalactic RMs from Taylor et al. (2009) cata-logue, and the sixth data release of the Sloan Digital Sky Survey(SDSS) with photometric redshifts (Oyaizu et al. 2008) to tracelarge-scale structure. They found a positive correlation betweenthe two quantities at large-scales, indicating a field strength of ∼
30 nG. However, Lee et al. (2010) withdrew this result becausethe cross-correlation was found to be spurious. This points to theprospect of detecting such signals, but also demonstrates that theinterpretation can be susceptible to statistical errors. This claimedresult and its subsequent retraction motivates our study: we hererevisit this technique, but with a more careful and thorough ap-proach to systematics and uncertainties.Some work has been done on what such cross-correlations wouldlook like for large data-sets of simulated RMs of various largescale IGMF strengths, and using galaxy catalogues as tracers forstructures on large-scales (such as by Stasyszyn et al. 2010 andAkahori et al. 2014). Given sufficient sample sizes, one can dis-cern between magneto-genesis models given the shape and ampli-tude of the cross-correlation.In this work, we place upper bounds on the IGMF by using cross-correlation techniques between the largest RM-redshift cataloguecurrently available, and an all-sky galaxy catalogue with photo-metric redshifts. The paper is organized as follows; in Section 2we describe the methods used to constrain the IGMF signal; in
MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Figure 1. Left : Locations of 1742 background RM sources from Hammond et al. (2012) after making selection cuts described in Section 3.1.
Right : Distri-bution of 18 million foreground galaxies (0 . < 𝑧 < .
5) from WISExSuperCOSMOS.
Section 3 we discuss the data used in our analysis; in Section 4we present the cross-correlation and its results; and in Section5 we place corresponding upper limits on the magnetic field ofthe IGM, and discuss how future work with upcoming surveyscan improve constraints on large-scale magnetic fields in theIGM. Throughout the paper we assume a Λ 𝐶𝐷𝑀 cosmology of 𝐻 = . − Mpc − and Ω 𝑀 = .
308 (Planck Collaborationet al. 2016a).
We use the Taylor et al. (2009) RM catalogue, which derived RMsfrom the NRAO VLA Sky Survey (NVSS, Condon et al. 1998).The NVSS contains images of Stokes I, Q, U, at 1.4 GHz, at 45arcsecond resolution for 1.8 million sources at declinations greaterthan 𝛿 > − ◦ . Taylor et al. (2009) then reprocessed the origi-nal NVSS data to determine RMs for 37 543 radio sources withsignal-to-noise ratios greater than 8 𝜎 , using two frequency bandscentred at 1364.9 MHz and 1435.1 MHz. Given that only two nar-rowly spaced frequency bands were used to determine the RMs,the values derived might be prone to errors (such as the 𝑛𝜋 ambi-guity; see Section 3.4.2 for further discussion). This is currentlystill the largest single catalogue of rotation measures to date.For our work, the RM sources must be at large redshifts such thattheir radiation passes through the foreground galaxies that tracelarge-scale structure. To determine if a source is a backgroundsource, the redshift is needed - this information is not availablein Taylor et al. (2009). Hammond et al. (2012) cross-matchedthe Taylor et al. (2009) catalog with optical surveys and onlinedatabases (such as SDSS DR8; Aihara et al. 2011), and databases,such as SIMBAD (Wenger et al. 2000) and NED , to obtainspectroscopic redshifts. This yields a sample of 4003 sources withRMs and spectroscopic redshifts, which allows us to ensure thatthe RM sources are indeed background sources, or at higher red-shifts, than the galaxies (see Section 2.2) for cross-correlating. http://simbad.u-strasbg.fr/simbad/ The NASA/IPAC Extragalactic Database (NED) is funded by theNational Aeronautics and Space Administration and operated by theCalifornia Institute of Technology. More information can be found at:https://ned.ipac.caltech.edu
The left panel in Figure 1 shows the sky distribution of RMs usedfor this study.
We use the WISExSuperCOSMOS (Bilicki et al. 2016) survey asour tracer for foreground large-scale structure. This catalogue isthe product of cross-matching between the mid-IR Wide-field In-frared Survey Explorer (WISE, Wright et al. 2010) and the opticalSuperCOSMOS (Hambly et al. 2001) all-sky surveys, and con-tains 20 million cross-matched galaxies after filtering for quasarsand stars and other tests. To generate accurate photometric red-shifts, Bilicki et al. (2016) cross-matched their WISExSuperCOS-MOS catalogue with the Galaxy and Mass Assembly-II (GAMA-II) survey (Driver et al. 2009), an extragalactic (clean from starsand quasars) spectroscopic survey that includes 193,500 over-lapping sources with WISExSuperCOSMOS. They used this asa training set for the ANNz package, an artificial neural networkpackage that assigns photometric redshifts to a sample given atraining set with both photometric and spectroscopic redshifts.Bilicki et al. (2016) were thus able to determine photometric red-shifts for all 20 million sources.We require redshifts for the galaxies to ensure that they are in-deed foreground to the RM sources discussed in Section 2.1. TheWISExSuperCOSMOS catalogue was used due to the fact that itprovides redshifts necessary for this study (with a median galaxyredshift 𝑧 ≈ . To cross-correlate between RMs and large-scale structure, wemust ensure that the polarized radio sources are backgroundsources whose radiation pass through the IGM of the large-scalestructure traced by foreground galaxies. The redshift distributionof the WISE galaxies severely drops off for 𝑧 > . 𝑧 = . 𝑧 < . 𝑧 ≥ . MNRAS000
We use the WISExSuperCOSMOS (Bilicki et al. 2016) survey asour tracer for foreground large-scale structure. This catalogue isthe product of cross-matching between the mid-IR Wide-field In-frared Survey Explorer (WISE, Wright et al. 2010) and the opticalSuperCOSMOS (Hambly et al. 2001) all-sky surveys, and con-tains 20 million cross-matched galaxies after filtering for quasarsand stars and other tests. To generate accurate photometric red-shifts, Bilicki et al. (2016) cross-matched their WISExSuperCOS-MOS catalogue with the Galaxy and Mass Assembly-II (GAMA-II) survey (Driver et al. 2009), an extragalactic (clean from starsand quasars) spectroscopic survey that includes 193,500 over-lapping sources with WISExSuperCOSMOS. They used this asa training set for the ANNz package, an artificial neural networkpackage that assigns photometric redshifts to a sample given atraining set with both photometric and spectroscopic redshifts.Bilicki et al. (2016) were thus able to determine photometric red-shifts for all 20 million sources.We require redshifts for the galaxies to ensure that they are in-deed foreground to the RM sources discussed in Section 2.1. TheWISExSuperCOSMOS catalogue was used due to the fact that itprovides redshifts necessary for this study (with a median galaxyredshift 𝑧 ≈ . To cross-correlate between RMs and large-scale structure, wemust ensure that the polarized radio sources are backgroundsources whose radiation pass through the IGM of the large-scalestructure traced by foreground galaxies. The redshift distributionof the WISE galaxies severely drops off for 𝑧 > . 𝑧 = . 𝑧 < . 𝑧 ≥ . MNRAS000 , 1–15 (2015)
Amaral et al. (a)(b) (c)
Figure 2.
The distributions of the polarized background RM sources, and the foreground galaxies used in the cross-correlation: (a)
The redshift distributionof the galaxies used in this study from WISExSuperCOSMOS. (b)
The | RRM | distribution of the RM sources used, and (c) The redshift distribution of theRM sources used. of the galaxy catalogue for galaxies with 𝑧 < . . < 𝑧 < .
5, leaving18 , ,
238 foreground galaxies.Due to uncertainties associated with photometric redshifts of theforeground galaxies, we choose to use redshift bins of widths of Δ 𝑧 = . Δ 𝑧 = . . < 𝑧 < . , ,
929 galaxies), 0 . < 𝑧 < . , , . < 𝑧 < . , , , ,
238 galaxies across all threeredshift bins. The mean redshift of the galaxies in each bin is: 𝑧 mean = . , . , and 0 .
34, respectively.We use a
HEALPix (Górski et al. 2005) gridding scheme and http://healpix.sourceforge.net healpy (Zonca et al. 2019) to split-up the all-sky foregroundgalaxies into a number density grid using spherical projection pix-els of equal surface area. We use 𝑛𝑠𝑖𝑑𝑒 = .
21 deg on the sky. The resulting all-sky number density grid of the remaining foreground galaxies canbe seen in Figure 1.Blank cells in the right panel of Figure 1 are due to the pres-ence of the Galactic foreground in the survey coverage, or dueto empty grid cells that contain no galaxies. We then removed anyRM sources that fell into empty grid cells, and any RM sourcewithin 2.5 co-moving Mpc of an empty cell or edge of the WISEgalaxy number density map - leaving us with 2229 backgroundRM sources.Akahori et al. (2016) found that magnetic fields due to galaxyclusters tend to dominate the RM along the line-of-sight, whichhence drown out any signal due to weaker magnetic fields, such asthose in filaments. Therefore, RMs that intersect cluster sight-lines http://github.com/healpy/healpy/ MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Figure 3.
An illustration of our cross-correlation technique. An extra-galactic polarized background source ( 𝑧 > .
5) whose radiation passes through ornear foreground galaxies (0 . < 𝑧 < .
5, separated into three redshift bins) is observed by a telescope on Earth. In order to probe correlations on large-scales, we use 8 linearly spaced radial bins shaped as annuli proceeding outward from the line-of-sight of each RM source, out to an impact parameter of2.5 Mpc. Only 3 of the 8 radial bins, 𝑟 , are shown. All images used in this illustration are covered under creative commons: "Quasar" by Anthony Ledouxis licensed under CC BY 3.0, "70-m aerial P-2500 (RT-70 radio telescope)" by S. Korotkiy is licensed under CC BY-SA, and "HST image of galaxy clusterRXC J0032.1+1808" by ESA/Hubble & NASA, RELICS is licensed under CC BY 4.0. must be removed from such studies. We used the Planck Collab-oration et al. (2016c) Sunyaev-Zeldovich (SZ) cluster catalogueto check for RM sources that pass within sight-lines of clusters.This cluster catalogue is the largest all sky-catalogue, and is wellsampled at 𝑧 <
1. We found that only one RM source from oursample fell within 2 𝑅 within a cluster line-of-sight, where 𝑅 is defined as the radius for which the density of the cluster is 500times the critical density and is typically taken to be a proxy forthe radius of a galaxy cluster. We removed this source.Ma et al. (2019b) found that the Taylor et al. (2009) RMs werenot properly calibrated for off-axis polarization leakage. Fromre-observing 23 sources Taylor et al. (2009) sources, and usingsimulations to quantify this effect, they found that off-axis leakagemostly affects sources with low fractional polarization ( 𝑝 ), andrecommend discarding sources with 𝑝 < 𝑝 < .
5% (334 sources) to account for off-axisleakage effects. This left 1894 sources. We checked using cuts onfractional polarization for 1%, 1 . 𝑝 < .
5% allowed us to balance cutting out sources with uncer-tain off-axis leakage, while also preserving a larger sample size toperform adequate statistics.
As per equation 3, the rotation measure is a cumulative effectalong the line-of-sight, which contains a component due to Galac-tic rotation measure (GRM) foregrounds. The GRM can be sub-tracted to obtain a residual RM (RRM). From equation 3, theRRM gives a better representation of RM intervening , the value wewish to extract for this study, although it still also contains compo- nents due to RM intrinsic and RM noise . Because RM
Gal is typicallycharacterized by smooth RMs over large regions of sky and tendsto dominate the RM component of the source (Simard-Normandin& Kronberg 1980; Leahy 1987; Schnitzeler 2010), if not removedit could affect large-scale correlations. RM noise and RM intrinsic areunique to each RM source so we do not expect these values to af-fect our correlation, allowing us to isolate RM intervening in our cor-relations. Oppermann et al. (2015) generated an all-sky catalogueof Galactic rotation measures using simulations to differentiatethe Galactic component from previously known extragalactic RMsources to obtain the GRM for all sight-lines.In removing the Galactic foreground contribution, we want toavoid using sight-lines where the uncertainty in the estimate ofGRM is worse than the uncertainty in the RM from other contri-butions. We therefore only use sight-lines that meet the require-ment: ( . 𝜎 𝑅𝑀 ) + 𝜎 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 ≥ 𝜎 𝐺𝑅𝑀 , (4)where 𝜎 𝐺𝑅𝑀 is the uncertainty in the GRM as computed by Op-permann et al. (2015), 𝜎 𝑅𝑀 is the observed RM uncertainty fromTaylor et al. (2009), 𝜎 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = . − is the standard de-viation on RM intrinsic estimated by Oppermann et al. (2015), andwe correct published values of 𝜎 𝑅𝑀 by a factor of 1.22 as recom-mended in §4.2.1 of Stil et al. (2011).We subtract the GRM derived by Oppermann et al. (2015) fromthe 1894 Taylor et al. (2009) RMs to obtain the RRMs for eachsource only if equation 4 is satisfied: 152 sources did not meet thiscriteria. We therefore removed these sources for a final sample of1742 background RM sources.The sign attributed to RMs is due to the orientation of the mag-netic field causing the rotation. Because we are not concernedwith the direction of the field, the values we use for cross- MNRAS000
Gal is typicallycharacterized by smooth RMs over large regions of sky and tendsto dominate the RM component of the source (Simard-Normandin& Kronberg 1980; Leahy 1987; Schnitzeler 2010), if not removedit could affect large-scale correlations. RM noise and RM intrinsic areunique to each RM source so we do not expect these values to af-fect our correlation, allowing us to isolate RM intervening in our cor-relations. Oppermann et al. (2015) generated an all-sky catalogueof Galactic rotation measures using simulations to differentiatethe Galactic component from previously known extragalactic RMsources to obtain the GRM for all sight-lines.In removing the Galactic foreground contribution, we want toavoid using sight-lines where the uncertainty in the estimate ofGRM is worse than the uncertainty in the RM from other contri-butions. We therefore only use sight-lines that meet the require-ment: ( . 𝜎 𝑅𝑀 ) + 𝜎 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 ≥ 𝜎 𝐺𝑅𝑀 , (4)where 𝜎 𝐺𝑅𝑀 is the uncertainty in the GRM as computed by Op-permann et al. (2015), 𝜎 𝑅𝑀 is the observed RM uncertainty fromTaylor et al. (2009), 𝜎 𝑖𝑛𝑡𝑟𝑖𝑛𝑠𝑖𝑐 = . − is the standard de-viation on RM intrinsic estimated by Oppermann et al. (2015), andwe correct published values of 𝜎 𝑅𝑀 by a factor of 1.22 as recom-mended in §4.2.1 of Stil et al. (2011).We subtract the GRM derived by Oppermann et al. (2015) fromthe 1894 Taylor et al. (2009) RMs to obtain the RRMs for eachsource only if equation 4 is satisfied: 152 sources did not meet thiscriteria. We therefore removed these sources for a final sample of1742 background RM sources.The sign attributed to RMs is due to the orientation of the mag-netic field causing the rotation. Because we are not concernedwith the direction of the field, the values we use for cross- MNRAS000 , 1–15 (2015)
Amaral et al. correlation are | 𝑅𝑅𝑀 | . The | 𝑅𝑅𝑀 | distribution of the 1742 back-ground RM sources can be found in Figure 2b, and the corre-sponding redshift distribution of these sources can be found inFigure 2c. The sky distributions of the RM catalogue and galaxynumber density grid after applying our selection criteria areshown in Figure 1. A large-scale magnetic field within the filamentary IGM (astraced by the foreground galaxy distribution) will create an en-hanced RM along that line-of-sight (extragalactic component,RM intervening , in equation 3) for each polarized backgroundsource. Probing impact parameters at various distances from theline-of-sight of the RM source allows us to quantify the extent ofinfluence of the field present in the foreground galaxy field.We wish to probe correlations at large-scales, and thus use 8 lin-early spaced radial impact parameter bins extending from thepolarized background source between ∼
500 kpc to 2 . ∼ 𝑧 = .
1) is larger than our largest radial bin of 2.5Mpc. The coherence scale of the magnetic field is considered tobe the scale within which the magnetic field remains uniform withno field reversals. If a signal is observed at a given scale, 𝑟 , thissuggests that the magnetic field is coherent at this scale.We then can calculate the cross-correlation function, 𝜉 ( 𝑟 ) , be-tween the background RRMs and the foreground galaxies using, 𝜉 ( 𝑟 ) = ∑︁ 𝑖 ∑︁ 𝑗 ( + 𝑧 𝑗 ) (cid:32) 𝑁 𝑔𝑎𝑙,𝑖 ( 𝑟, 𝑧 𝑗 ) − (cid:10) 𝑁 𝑔𝑎𝑙 ( 𝑟, 𝑧 𝑗 ) (cid:11) 𝐴 ( 𝑟 ) (cid:33) · (| 𝑅𝑅𝑀 | 𝑖 − (cid:104)| 𝑅𝑅𝑀 |(cid:105)) . (5)The index i represents each individual polarized backgroundsource with RRM of | 𝑅𝑅𝑀 𝑖 | , and index 𝑗 represents the 3 red-shift bins of the foreground. (cid:104)| 𝑅𝑅𝑀 |(cid:105) is the average residual ro-tation measure of all of the RRM sources, where (cid:104)|
𝑅𝑅𝑀 |(cid:105) = . / m . Here, 𝑁 𝑔𝑎𝑙,𝑖 ( 𝑟, 𝑧 𝑗 ) is the number of galaxies in red-shift bin 𝑧 𝑗 and at radial impact distance 𝑟 from the line-of-sightof 𝑅𝑅𝑀 𝑖 , while (cid:10) 𝑁 𝑔𝑎𝑙 ( 𝑟, 𝑧 𝑗 ) (cid:11) is the predicted number of galax-ies in each redshift and radial bin around each RRM source. Thepredicted numbers are determined by assuming a Poisson distri-bution in each redshift bin and calculating the expected numberof sources given the on-the-sky area, 𝐴 ( 𝑟 ) , of each radial bin, 𝑟 .As seen in equation 2, there is a redshift contribution that affectsthe RM. The foreground galaxies span a substantial redshift range,thus we un-weight this redshift dependence in equation 5 by in-cluding the term (cid:0) + 𝑧 𝑗 (cid:1) − , where 𝑧 𝑗 is the average redshift ofthat bin.We are interested in quantifying if any deviations from the aver-age number of galaxies around each RM source correlates withdeviations from (cid:104)| 𝑅𝑅𝑀 |(cid:105) . A visual representation of our cross-correlation method can be found in Figure 3.
We use equation 5 to calculate the cross-correlation function be-tween background RRM sources and foreground galaxies. Theresulting value of 𝜉 ( 𝑟 ) as a function of r is shown in Figure 4. Nostrong signal is seen at any value of 𝑟 . There is a slight positivesignal in the lowest radial bins, with a decreasing trend with in-creasing 𝑟 . In order to determine if any significant detection ismade, we determined uncertainties as detailed in the followingsubsection. In this section, we calculate the confidence and significance levelsof the cross-correlation result from section 3.4.1. We define theconfidence levels as the intervals in which we are confident thatthe resultant cross-correlation falls within, whereas the signifi-cance levels define how significant the cross-correlation is above anull signal.To determine the confidence levels on our cross-correlation re-sult, we use a boot-strap method and randomly re-sample the 1742RRM sources, with replacement. We do this for 1000 realizationscomputing a cross-correlation for each realization, then generating1 𝜎 , 2 𝜎 , and 3 𝜎 percentiles as the confidence levels on the cross-correlation.To obtain the significance levels of the cross-correlation, weproduce a null cross-correlation signal. To do this, we gener-ate random background radio sources randomly located on thecelestial sphere such that the correlation between them and theforeground galaxy density should average to zero in all radialbins. We mask out any values that fall within the Galactic plane( − ◦ < 𝑏 < ◦ ).Using the method described in the preceding paragraphs, we cal-culate random locations on the celestial sphere for 2,674,800 newsight-lines. From these new locations, we select 1742 and assigneach a random RRM from Hammond et al. (2012) catalogue, us-ing bootstrap resampling to form a set of random locations andRRMs. We then use these bootstrapped 1742 RRMs to computethe cross-correlation function for 1000 different realizations. Wethen take the 1 𝜎 , 2 𝜎 , and 3 𝜎 percentiles from the 1000 randomcross-correlation realizations to obtain the significance of the re-sult.The final cross-correlation function can be seen in Figure 4, wherethe error bars for various confidence levels are plotted in green,and the 3 𝜎 significance level of the result (3 𝜎 confidence level forthe null signal) is plotted as red dashed lines. The cross-correlation between the foreground galaxies and back-ground RM sources (Figure 4) is consistent with zero within 3 𝜎 .Although no significant signal is detected, the cross-correlationfunction can still be used to obtain upper limits on the magneticfield of the WHIM within the IGM. We use the significance levels(see Section 3.4.2) on the cross-correlation function determinedfrom the data to constrain an upper bound on the parallel compo-nent on large-scale magnetic fields contained in the WHIM. MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Figure 4.
The cross-correlation function (CCF) between the | 𝑅𝑅𝑀 | s of 1742 background sources from Hammond et al. (2012) and 18 million foregroundgalaxies selected from Bilicki et al. (2016). The observed CCF is shown as the solid blue line. The red dashed lines mark the 3 𝜎 significant level of thecross-correlation, while the green coloured regions mark the confidence intervals of the cross-correlation. To place an upper bound on the magnetic field in the WHIM fromour cross-correlation, we first model the rotation measure gen-erated by the WHIM within the foreground galaxy density field, 𝑅𝑀 𝑓 𝑖𝑙 . We adopt a model for 𝑅𝑀 𝑓 𝑖𝑙 that is derived from the RMdefinition (equation 2): 𝑅𝑀 𝑓 𝑖𝑙,𝑖 = ∑︁ 𝑗 ( + 𝑧 𝑗 ) 𝑛 𝑒, 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) 𝐵 (cid:107) , 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) √︁ 𝑁 𝑅 ( 𝑧 𝑗 ) Δ 𝑙 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) , (6)where 𝑅𝑀 𝑓 𝑖𝑙,𝑖 is the observed RM induced by foreground fila-ments over line-of-sight, 𝑖 . We now explain the components of thismodel.The line-of-sight integral in the RM equation is represented by asummation Σ 𝑗 over each redshift bin 𝑗 . Therefore, the differentialpath length 𝑑𝑙 ( 𝑧 𝑗 ) is approximated by Δ 𝑙 ( 𝑧 𝑗 ) = 𝑙 ( 𝑧 𝑗 + ) − 𝑙 ( 𝑧 𝑗 ) ,where 𝑙 ( 𝑧 𝑗 ) is the co-moving distance to redshift bin 𝑧 𝑗 : 𝑙 ( 𝑧 𝑗 ) = 𝑐𝐻 ( + 𝑧 𝑗 ) √︃ Ω 𝑀 ( + 𝑧 𝑗 ) + Ω Λ + ( − Ω 𝑀 − Ω Λ )( + 𝑧 𝑗 ) . (7) The foreground galaxy densities trace filaments, although fila-ments do not occupy the entire line-of-sight. To account for thiswe introduce a line-of-sight volume filling factor for filaments, 𝑓 , which can range between 0 and 1, and is not a function of z.The line-of-sight path length passing through filaments is then Δ 𝑙 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) = 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) .We include magnetic field reversals in our model, where the mag-netic field changes direction over a coherence scale, 𝐿 𝑐 . We modelfield reversals as a random walk process (Cho & Ryu 2009). Thishas the effect of reducing the observed RM by 1 /√ 𝑁 𝑅 , where 𝑁 𝑅 is the number of field reversals over the line-of-sight. Assumingthat 𝐿 𝑐 < 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) , where 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) is the total line-of-sight cov-ered by filaments within redshift bin 𝑧 𝑗 , the total number of fieldreversals along the line-of-sight passing through filaments in red-shift bin 𝑧 𝑗 is then: 𝑁 𝑅 ( 𝑧 𝑗 ) = 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) 𝐿 𝑐 . (8)The 𝑅𝑀 𝑓 𝑖𝑙 of the sources will be affected by the number offree electrons contained in filaments along the line-of-sight, MNRAS000
The cross-correlation function (CCF) between the | 𝑅𝑅𝑀 | s of 1742 background sources from Hammond et al. (2012) and 18 million foregroundgalaxies selected from Bilicki et al. (2016). The observed CCF is shown as the solid blue line. The red dashed lines mark the 3 𝜎 significant level of thecross-correlation, while the green coloured regions mark the confidence intervals of the cross-correlation. To place an upper bound on the magnetic field in the WHIM fromour cross-correlation, we first model the rotation measure gen-erated by the WHIM within the foreground galaxy density field, 𝑅𝑀 𝑓 𝑖𝑙 . We adopt a model for 𝑅𝑀 𝑓 𝑖𝑙 that is derived from the RMdefinition (equation 2): 𝑅𝑀 𝑓 𝑖𝑙,𝑖 = ∑︁ 𝑗 ( + 𝑧 𝑗 ) 𝑛 𝑒, 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) 𝐵 (cid:107) , 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) √︁ 𝑁 𝑅 ( 𝑧 𝑗 ) Δ 𝑙 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) , (6)where 𝑅𝑀 𝑓 𝑖𝑙,𝑖 is the observed RM induced by foreground fila-ments over line-of-sight, 𝑖 . We now explain the components of thismodel.The line-of-sight integral in the RM equation is represented by asummation Σ 𝑗 over each redshift bin 𝑗 . Therefore, the differentialpath length 𝑑𝑙 ( 𝑧 𝑗 ) is approximated by Δ 𝑙 ( 𝑧 𝑗 ) = 𝑙 ( 𝑧 𝑗 + ) − 𝑙 ( 𝑧 𝑗 ) ,where 𝑙 ( 𝑧 𝑗 ) is the co-moving distance to redshift bin 𝑧 𝑗 : 𝑙 ( 𝑧 𝑗 ) = 𝑐𝐻 ( + 𝑧 𝑗 ) √︃ Ω 𝑀 ( + 𝑧 𝑗 ) + Ω Λ + ( − Ω 𝑀 − Ω Λ )( + 𝑧 𝑗 ) . (7) The foreground galaxy densities trace filaments, although fila-ments do not occupy the entire line-of-sight. To account for thiswe introduce a line-of-sight volume filling factor for filaments, 𝑓 , which can range between 0 and 1, and is not a function of z.The line-of-sight path length passing through filaments is then Δ 𝑙 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) = 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) .We include magnetic field reversals in our model, where the mag-netic field changes direction over a coherence scale, 𝐿 𝑐 . We modelfield reversals as a random walk process (Cho & Ryu 2009). Thishas the effect of reducing the observed RM by 1 /√ 𝑁 𝑅 , where 𝑁 𝑅 is the number of field reversals over the line-of-sight. Assumingthat 𝐿 𝑐 < 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) , where 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) is the total line-of-sight cov-ered by filaments within redshift bin 𝑧 𝑗 , the total number of fieldreversals along the line-of-sight passing through filaments in red-shift bin 𝑧 𝑗 is then: 𝑁 𝑅 ( 𝑧 𝑗 ) = 𝑓 Δ 𝑙 ( 𝑧 𝑗 ) 𝐿 𝑐 . (8)The 𝑅𝑀 𝑓 𝑖𝑙 of the sources will be affected by the number offree electrons contained in filaments along the line-of-sight, MNRAS000 , 1–15 (2015)
Amaral et al. 𝑛 𝑒, 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) . We assume that the free electron density scales withthe relative number density of galaxies: 𝑛 𝑒, 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) = 𝑛 𝑒, 𝑓 𝑖𝑙 (cid:32) 𝑛 𝑔𝑎𝑙, . ,𝑖 ( 𝑧 𝑗 ) (cid:10) 𝑛 𝑔𝑎𝑙, . ( 𝑧 𝑗 ) (cid:11) (cid:33) , (9)where 𝑛 𝑒, 𝑓 𝑖𝑙 is the average number of free electrons in fila-ments. The weighted galaxy count within an impact parame-ter of 2.5 Mpc of line-of-sight 𝑖 and redshift bin 𝑧 𝑗 is given by 𝑛 𝑔𝑎𝑙, . ,𝑖 ( 𝑧 𝑗 ) = (cid:205) 𝑘 𝑁 𝑔𝑎𝑙,𝑖 ( 𝑟 𝑘 , 𝑧 𝑗 ) × 𝑤 ( 𝑟 𝑘 ) , where 𝑁 𝑔𝑎𝑙,𝑖 ( 𝑟 𝑘 , 𝑧 𝑗 ) is the number of galaxies within radial impact parameter bin 𝑟 𝑘 and redshift bin 𝑧 𝑗 , and 𝑤 ( 𝑟 𝑘 ) is the radial weighing functionevaluated at radial bin 𝑟 𝑘 . (cid:10) 𝑛 𝑔𝑎𝑙, . ( 𝑧 𝑗 ) (cid:11) is the average weightedgalaxy count within an impact parameter of 2.5 Mpc in red-shift bin 𝑧 𝑗 for all 𝑖 sight-lines. Therefore, the quotient of theseweighted galaxy count quantities represent whether line-of-sight 𝑖 is in an under- or over-dense region.We use the radial weighting of a King (1972) profile, given by: 𝑤 ( 𝑟 ) = (cid:16) + 𝑟 𝑟 𝑐 (cid:17) − 𝛽 , where 𝛽 = . 𝑟 𝑐 = 𝑅𝑀 𝑓 𝑖𝑙 along that sight-line than galaxies that are farther away.This is a reasonable assumption as the magnetic field and 𝑛 𝑒 , andhence the resulting RM, will decrease as 𝑟 decreases away fromregions of high density.We assume a simple magnetic field case, in which a large-scalecoherent magnetic field, 𝐵 𝑓 𝑖𝑙 , is already in place at redshifts 𝑧 > .
5, and at 0 . < 𝑧 < . 𝑛 𝑒 (electron number densities) present in the foreground galaxydensity. We do not consider magnetic energy injected on large-scales by compact astrophysical sources (such as AGN). This isa reasonable assumption as AGN activity becomes less commonin the 𝑧 < ℎ >> 𝑅 regime, where ℎ = height, and 𝑅 = radius of a cylindrical filament). As in Blasi et al. (1999),we assume that the magnetic fields and filaments obey both fluxconservation and mass conservation, to derive: 𝐵 (cid:107) , 𝑓 𝑖𝑙 ( 𝑧 𝑗 ) = 𝐵 , (cid:107) , 𝑓 𝑖𝑙 (cid:32) 𝑛 𝑔𝑎𝑙, . ,𝑖 ( 𝑧 𝑗 ) (cid:10) 𝑛 𝑔𝑎𝑙, . ( 𝑧 𝑗 ) (cid:11) (cid:33) / , (10)where 𝐵 , (cid:107) , 𝑓 𝑖𝑙 is the co-moving mean magnetic field strength infilaments. We assume that for scales < . 𝑅𝑀 𝑓 𝑖𝑙,𝑖 = ∑︁ 𝑗 𝑓 / ( + 𝑧 𝑗 ) 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, × (cid:32) 𝑛 𝑔𝑎𝑙, . ,𝑖 ( 𝑧 𝑗 ) (cid:10) 𝑛 𝑔𝑎𝑙, . ( 𝑧 𝑗 ) (cid:11) (cid:33) / √︄ 𝐿 𝑐 Δ 𝑙 ( 𝑧 𝑗 ) Δ 𝑙 ( 𝑧 𝑗 ) . (11) We can use equation 11 to calculate the predicted 𝑅𝑀 𝑓 𝑖𝑙 alongthe sight-lines of the 1742 observed RM sources used in thisstudy. As in Section 3.3 using equation 5, we can calculate thecross-correlation between the 1742 predicted 𝑅𝑀 𝑓 𝑖𝑙 s and theforeground galaxy densities. Since 𝐵 (cid:107) , 𝑓 𝑖𝑙, is an input to equa-tion 11, we determine the 𝐵 (cid:107) , 𝑓 𝑖𝑙, required to scale the resultingcross-correlation above the 3 𝜎 significance levels (as calculated inSection 3.4.2).The cross-correlation is measured at 8 linearly spaced radial binsextending to 2.5 Mpc. Therefore we chose to scale the resultingpredicted cross-correlation to 3 𝜎 at the radial bin closest in valueto the input coherence scale ( 𝐿 𝑐 ) of the magnetic field. The input 𝐵 (cid:107) , 𝑓 𝑖𝑙, to produce 𝑅𝑀 𝑓 𝑖𝑙 s that correlate above 3 𝜎 in the radialbin closest in value to 𝐿 𝑐 is the upper bound on the IGMF. Wethen can determine the average RM induced by the IGMF of fore-ground filaments, (cid:10) 𝑅𝑀 𝑓 𝑖𝑙 (cid:11) , by taking the ensemble average ofeach individual 𝑅𝑀 𝑓 𝑖𝑙 over all 1742 sight-lines.In order to do this calculation, we must assume values for themean electron number density in filaments ( 𝑛 𝑒, 𝑓 𝑖𝑙 ), the coherencescale of the IGMF ( 𝐿 𝑐 ), and filling factor ( 𝑓 ). Alternatively, fromthe model RM we derived in equation 11, we can place bounds onthe combination of these parameters 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, .Because the origin of these fields are unknown and no previousdirect observations exist, we chose to test a set of 5 coherencescales: 0.5, 1.0, 1.5, 2.0, and 2.5 Mpc. We cannot probe scaleslarger than 2.5 Mpc as our correlation scale does not extend fur-ther than its largest radial bin. We find that the upper bound on 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, is similar for all scales tested, all on theorder of ∼ − 𝜇 G cm − . For the largest coherence scale of2 . 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, < . × − 𝜇 G cm − , corresponding to an average RM enhance-ment of 4.7 rad / m due to filaments in the observer frame. Theupper bounds for the other coherence scales are found in column2 of Table 1, these upper bounds are plotted in Figure 6 as a func-tion of coherence scales for a visual representation.To further constrain 𝐵 (cid:107) , 𝑓 𝑖𝑙, , we make assumptions about 𝑓 and 𝑛 𝑒, 𝑓 𝑖𝑙 . The filling factor of filaments within the IGM is not wellconstrained. Simulations have shown that 𝑓 = .
06 at lower red-shifts ( 𝑧 < .
5) (Aragón-Calvo et al. 2010; Cautun et al. 2014;Davé et al. 2010). The average free electron density within fila-ments is also not well constrained because filaments are difficultto directly observe (Cantalupo et al. 2014); most values on theaverage electron densities come from simulations. These studiesquote values from 10 − to 10 − cm − (Cen & Ostriker 2006; Ryuet al. 2008; Akahori & Ryu 2010), as in O’Sullivan et al. (2019)here we adopt a typical value 𝑛 𝑒, 𝑓 𝑖𝑙 ≈ − cm − .The upper bounds of 𝐵 (cid:107) , 𝑓 𝑖𝑙, for the various coherence scales arefound in column 3 of Table 1. For the largest scale we test, and theabove values used for 𝑛 𝑒, 𝑓 𝑖𝑙 and 𝑓 we obtain a 3 𝜎 upper boundon the IGMF to be 31 nG for a coherence scale of 2 . 𝜎 upper bound on the IGMF forthe smallest coherence scale of 0 . 𝑅𝑀 𝑓 𝑖𝑙 s generated using equation 11 canbe seen in the left panel of Figure 5, in which we also compare MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Figure 5. Left:
Distribution of predicted RM values compared to observed | 𝑅𝑅𝑀 | values used, generated using the model given in equation 11, using 𝑛 𝑒, 𝑓 𝑖𝑙 = − cm − , 𝑓 = .
06, and 𝐿 𝑐 = Right:
The resulting cross-correlation function from the observed and predicted RMs shown in the leftpanel. The green shaded region corresponds to the red dashed lines indicating a 3 𝜎 significance level in Figure 4.Coherence Scale ( 𝐿 𝑐 ) 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, Upper Bound 𝐵 (cid:107) , 𝑓 𝑖𝑙, Upper Bound (cid:10) 𝑅𝑀 𝑓 𝑖𝑙 (cid:11) Upper Bound0 . . × − 𝜇 G cm −
44 nG 3 . ± . / m . . × − 𝜇 G cm −
32 nG 3 . ± . / m . . × − 𝜇 G cm −
30 nG 3 . ± . / m . . × − 𝜇 G cm −
27 nG 3 . ± . / m . . × − 𝜇 G cm −
31 nG 4 . ± . / m Table 1.
Table containing the results for constraints on the magnetic field within filaments, using the predicted RM model (equation 11) as derived in section4.1 for various tested coherence scales, 𝐿 𝑐 . Column 1: The coherence scales tested; Col. 2: 3 𝜎 upper bound on the combined quantity 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, ,for the associated coherence scale; Col. 3: 3 𝜎 upper bounds on 𝐵 (cid:107) , 𝑓 𝑖𝑙, assuming 𝑓 = .
06 and 𝑛 𝑒, 𝑓 𝑖𝑙 = − cm − for the associated coherence scales;Col. 4: The average upper bound RM induced by filaments in foreground large-scale structure, calculated by taking the average of each individual 𝑅𝑀 𝑓 𝑖𝑙,𝑖 .The uncertainty on this quantity is given by calculating the 1 𝜎 standard deviation. Figure 6.
Upper bounds on the IGMF for all coherence scales tested inthis study, the exact values can be found in Table 1. to the observed RRMs used to calculate the cross-correlation inSection 3.3. The right panel of Figure 5 shows the resulting cross- correlation function calculated using the predicted 𝑅𝑀 𝑓 𝑖𝑙 s, thisoutput correlation is greater than 3 𝜎 at the largest radial bin - al-lowing us to use the associated 𝐵 (cid:107) , 𝑓 𝑖𝑙, value as an upper bound. To ensure that the choice of weighting functions and parame-ters produced robust upper limits, we tested scaling length val-ues, 𝑟 𝑐 , between 0.5 Mpc and 2.5 Mpc, and found that the finalupper bound 𝐵 (cid:107) , 𝑓 𝑖𝑙, results were consistent within ±
10% for 𝐿 𝑐 = − . ±
30% for 𝐿 𝑐 = . 𝑟 𝑐 values tested. We also found that the choice of a King (1972)profile matched the upper bounds results from using a Gaussianprofile, within ± MNRAS000
30% for 𝐿 𝑐 = . 𝑟 𝑐 values tested. We also found that the choice of a King (1972)profile matched the upper bounds results from using a Gaussianprofile, within ± MNRAS000 , 1–15 (2015) Amaral et al. spherical symmetry, to derive 𝐵 ∼ 𝑛 / 𝑒 (a relation used in IGMFstudies such as Pshirkov et al. 2016; O’Sullivan et al. 2020). Whenusing the 𝐵 ∼ 𝑛 / 𝑒 scaling relation in our study, we obtain upperbound results that are ∼
15% lower than those quoted in Table 1.The scaling relation 𝐵 ∼ 𝑛 𝑘 of magnetic fields with various gasdensities has been previously explored in the literature (see Val-lée 1995 for a thorough review). Using measurements from pre-vious studies, Vallée (1995) concludes that for low gas densities 𝑛 𝑒 <
100 cm − , 𝑘 = . ± .
03. When using this scaling rela-tion in our model, we find that the upper bound results increase by ∼ 𝑓 / 𝑛 𝑒, 𝑓 𝑖𝑙 𝐵 (cid:107) , 𝑓 𝑖𝑙, aremore robust than those for 𝐵 (cid:107) , 𝑓 𝑖𝑙, because we include furtherassumptions of values of 𝑛 𝑒, 𝑓 𝑖𝑙 and 𝑓 . We tested various fillingfactors to understand the effect on our resulting upper bounds. Inour calculations, we assumed 𝑓 = .
06. For a lower value 𝑓 = .
01, and for a higher value 𝑓 = .
2, we find an increase by ∼ ∼
45% in the upper bound on all scales,respectively.As discussed in Section 4.2, simulations have shown that 𝑛 𝑒, 𝑓 𝑖𝑙 can vary between 10 − cm − and 10 − cm − . In our quoted upperbounds, we assumed 10 − cm − . Changing 𝑛 𝑒, 𝑓 𝑖𝑙 in equation 9to test the extreme range for filaments, we find that the resultingupper bounds increase and decrease by an order-of-magnitude for 𝑛 𝑒, 𝑓 𝑖𝑙 = − cm − and 𝑛 𝑒, 𝑓 𝑖𝑙 = − cm − , respectively, forall 𝐿 𝑐 values. While the changes in both 𝑓 and 𝑛 𝑒 cause a moredrastic change in the upper bound values we obtain, these are alsothe least well-constrained values used within our study. It is difficult to make direct comparisons between upper boundstudies, as the techniques and data used differ greatly. Also,knowledge of many of the values we assume ( 𝑛 𝑒 , 𝐿 𝑐 , and 𝑓 ) inour study are not robustly known, and thus values assumed varybetween studies. Techniques to place bounds come from statisti-cal methods, observations, and simulations. Furthermore, becauseof the lack of observations, simulations have been used to placebounds on the IGMF. These range from simulated observablesfor forecasting future radio polarimetric surveys, to full universemagneto-hydrodynamic (MHD) simulations.Vazza et al. (2017) used the MHD ENZO (Bryan et al. 2014) codeand incorporated galaxy formation, dark matter, gas, and chemi-cal abundances, to simulate cosmic magnetic field evolution from 𝑧 =
38 to 𝑧 =
0. Whereas most cosmic simulations focus on a sin-gle magneto-genesis model, Vazza et al. (2017) incorporate bothprimordial and astrophysical seeding models to compare outputobservables (such as Faraday rotation, synchrotron emission, cos-mic ray propagation, etc.). They found that the magnetic fields inhigher density regions such as within galaxy clusters and groupswere on the order of 𝜇 G for all scenarios, whereas in the lessdense filament environments, the magnetic field strength stronglydepends on the magneto-genesis models assumed. For purely as-trophysical phenomena, they found filaments to be magnetizedat the nG level, whereas for purely primordial fields magneticfields are found to be ∼ . − nG and allow for dynamo amplification of these fields through solenoidal turbulence within large scale structure.However, in general the magnetic field bounds we have placed inthis study are not stringent enough to be able to discern betweenpure primordial vs. pure astrophysical models.In Table 2, we list upper bounds on the IGMF from relevant obser-vational studies. We will discuss a subset of them in further detailbelow.Observationally, upper bounds placed on the IGMF are extremelylimited - this is mostly due to the estimated low electron densitiesin filaments (Cen & Ostriker 2006). Notably, observing a giantdouble-lobed radio galaxy, O’Sullivan et al. (2019) determinedthat one of the galaxy’s radio lobes passed through an IGM fil-ament, allowing them to single out the RM due to the filament.This RM corresponded to a density weighted upper bound of ∼ . 𝜇 G. This is consistent with, but less constraining then, theupper limit we have derived here.Xu et al. (2006) provided IGMF upper bounds from a large sta-tistical study using RM data derived from Simard-Normandin& Kronberg (1980) (with and without redshift information), andLSS information from galaxy counts in the Hercules and Perseus-Pisces superclusters (from the second Center for Astrophysics sur-vey, CfA2, and the Two Micron All Sky Survey, 2MASS, galaxycatalogues). They used RMs smoothed at a scale of 7 ◦ (smoothedRMs, SRM) to study very large scale variations. Xu et al. (2006)then looked for averaged correlations across known super-clusterstructures between the averaged SRMs within the region, and thegalaxy-count-weighted path-length through the structure. They usea very simple model for RMs induced from a filament, with sim-ilar but less complex components compared to our model, suchas; coherence scale ( 𝐿 𝑐 ), free electrons in filaments ( 𝑛 𝑒, 𝑓 𝑖𝑙 ), pathlength through filament structures, and 𝐵 (cid:107) , 𝑓 𝑖𝑙, .Xu et al. (2006) obtained upper bounds by comparing expec-tations between this simple RM model and the SRMs in theHercules cluster. Testing values of Lc = 200 kpc - 800 kpc and 𝑛 𝑒, 𝑓 𝑖𝑙, = . − × − 𝑐𝑚 − , they obtain upper bounds between0.4 and 0.3 𝜇 G. These bounds are larger than ours for a multi-tude of reasons such as: fewer RM sources and galaxies used, andnot subtracting off the GRM. This sort of correlation and analysiswas done looking at averaged and smoothed values over the en-tire structure, while our approach calculates a cross-correlationfunction across all RM sight-lines.Various other statistical methods have been used to place bounds,each with their own respective strengths and weaknesses. A largesource of error with these types of analyses is accurately subtract-ing the Galactic foreground contribution from RMs so that weonly obtain extragalactic signals (see equation 3). If sources oc-cupy the same region of sky, the Galactic component of the RMis comparable between the sources. Vernstrom et al. (2019) usedthis to determine limits for the IGMF by comparing statistical RMdifferences between populations of physical pairs of extragalac-tic polarized sources (found at similar redshifts and close on thesky) versus populations of non-physical pairs (occupying differentredshifts but still close together on the plane of the sky) within theHammond et al. (2012) RM catalogue. The statistical differencesbetween the RMs of these populations would be due to the extrarotation induced on the non-physical pairs from the IGMF, allow-ing upper limits of 40 nG to be placed on the parallel componentof 𝐵 𝐼 𝐺𝑀 . This value broadly agrees with our bounds at 24 nG,using a subset of the same RM catalogue though using differenttechniques.Following the same novel technique as Vernstrom et al. (2019),O’Sullivan et al. (2020) used RMs of 201 physical and 148 ran-
MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Table 2.
A selection of relevant observational upper bounds on large scale extragalactic magnetic fields within the filamentary IGM, sorted by increasingIGMF upper bound.
Upper Bound Coherence Scale ( 𝐿 𝑐 ) Redshift Range Reference Technique Summary 𝜆 𝑗 ( 𝑧 ) ) 0 < 𝑧 < / 𝐻 / 𝐻 < 𝑧 < . 𝐿 𝑐 / 𝜆 𝑗 ( 𝑧 = ) ≤ O’Sullivan et al. (2020) Δ RM between adjacent sources21 nG ∼ Gpc 𝑧 ∼ . . < 𝑧 < . 𝜃 = ◦ 𝑧 < .
048 Brown et al. (2017) Synchrotron cross-correlation30 nG - 60 nG 2 Mpc < 𝑧 > = . ± .
01 Vernstrom et al. (2021) Statistically Stacking Filaments30 nG - 1980 nG 1 - 4 Mpc 0 < 𝑧 < .
57 Vernstrom et al. (2017) Synchrotron cross-correlation37 nG 1 Mpc 0 < 𝑧 < Δ RM between adjacent sources250 nG ∼
10 Mpc 0 . < 𝑧 < .
142 Locatelli et al. (2021) Non-detection of filament accretion shocks300 nG 300 kpc 𝑧 ∼ .
34 O’Sullivan et al. (2019) Observed intervening filament300 nG 0.5 Mpc 0 . < 𝑧 < . Xu et al. (2006) RM correlation with galaxy counts Redshift information not used Averaged over full sight-line Cross-correlation computed on an angular scale The redshift range of the Hercules super-cluster was determined by Kopylova & Kopylov (2013) dom pairs derived from the LOFAR Two-Metre Sky Survey(LoTSS) DR1 (Shimwell et al. 2019). They found an upper boundson the magnetic field of the IGM of 4nG at Mpc scales. BecauseLoTTS is observed at 144 MHz, any RMs derived from thesebands are extremely sensitive to any Faraday depolarization, there-fore the authors conclude that these sources preferentially occupyless dense and weakly magnetized regions within the IGM - suchas on the boundaries of filaments and voids. In contrast, the 1.4GHz sample used in both Vernstrom et al. (2019) and our studyare less susceptible to Faraday depolarization and the bounds onthe IGM derived from these data should be more representative ofdenser filaments. Because the bounds we derived use the same 1.4GHz RM data (Hammond et al. 2012) and obtain similar magneticfield upper limit as Vernstrom et al. (2019), we cannot disfavourthe hypothesis that LoTSS sources occupy rarefied environments.The hypothesis that LoTSS sources occupy extremely rarefiedenvironments was further confirmed by Stuardi et al. (2020),who looked at 37 polarized LoTTS giant radio galaxies (GRGs)- galaxies whose outer lobes extend well into the IGM and have alimiting size of 700 kpc. Using the same approaches as Vernstromet al. (2019) and O’Sullivan et al. (2020), they analyzed the RMdifferences between the lobes of each of the galaxies. They foundthat the RM difference between the lobes of the galaxies were con-sistent with variations in the local Milky Way Galactic foregroundmagnetic field, as was found by Vernstrom et al. (2019). Stuardiet al. (2020) also found that there were small amounts of Faradaydepolarization between 144 MHz and 1.4GHz, 𝐷 . 𝐺𝐻 𝑧
𝑀 𝐻 𝑧 > . 𝑛 𝑒 ∼ − cm − ) with magnetic fields on theorder of 0 . 𝜇 G, tangled on kpc scales. While it is unclear whetherthese sources probe the same IGM component as our studies, this study showed the capabilities of using GRGs to study intergalacticmagnetic fields as more data becomes available.On scales smaller than we probe here, Lan & Prochaska (2020)used similar cross-correlation and analysis techniques to ours,between 1100 background RRMs ( 𝑧 >
1, using redshift infor-mation from Hammond et al. 2012, RM information from Farneset al. 2014, and foreground GRM subtraction from Oppermannet al. 2015) and foreground galaxy distributions as sampled by theDESI Legacy Imaging Surveys (Dey et al. 2019), to extract thecharacteristic RM and < 𝐵 (cid:107) > of the circum-galactic medium(CGM) around galaxies. They obtained a null correlation, butused the trend between 𝜎 𝑅𝑀 and 𝑁 𝑔𝑎𝑙 (number of foregroundgalaxies within the RRM line-of-sight) to model the random fieldreversals due to intersecting magnetic fields of the CGM. Theyuse the 3 𝜎 significance levels on 𝜎 𝑅𝑅𝑀 vs. 𝑁 𝑔𝑎𝑙 and a randomwalk model to place upper bounds. They found 3 𝜎 upper boundsof 𝑅𝑀 𝐶𝐺𝑀 ∼
15 rad m − and 𝐵 (cid:107) ,𝐶𝐺𝑀 < 𝜇𝐺 within the CGMvirial radius (impact parameters less than 200 kpc).Using a combination of the cross-correlation methods from Lan& Prochaska (2020) and our study, with foreground galaxy den-sity maps and denser RM grids, we can hope to accurately probethe magnetic fields of the CGM, and the transition region betweenthe CGM and the IGM (the CGM is thought to extend between100 kpc - 200 kpc from individual galaxies, Shull 2014). Thefiner precision of denser RM grids will allow to accurately mea-sure correlations at these scales (Bernet et al. 2010, 2013; Farneset al. 2014; Prochaska & Zheng 2019). Understanding the mag-netic properties of this transition region is key to understandingthe interplay between the properties of the IGM, CGM, and galax-ies themselves (Borthakur et al. 2015). Additionally, this regionmediates the acquiring of neutral gas from the IGM to individual MNRAS000
15 rad m − and 𝐵 (cid:107) ,𝐶𝐺𝑀 < 𝜇𝐺 within the CGMvirial radius (impact parameters less than 200 kpc).Using a combination of the cross-correlation methods from Lan& Prochaska (2020) and our study, with foreground galaxy den-sity maps and denser RM grids, we can hope to accurately probethe magnetic fields of the CGM, and the transition region betweenthe CGM and the IGM (the CGM is thought to extend between100 kpc - 200 kpc from individual galaxies, Shull 2014). Thefiner precision of denser RM grids will allow to accurately mea-sure correlations at these scales (Bernet et al. 2010, 2013; Farneset al. 2014; Prochaska & Zheng 2019). Understanding the mag-netic properties of this transition region is key to understandingthe interplay between the properties of the IGM, CGM, and galax-ies themselves (Borthakur et al. 2015). Additionally, this regionmediates the acquiring of neutral gas from the IGM to individual MNRAS000 , 1–15 (2015) Amaral et al. galaxies to further fuel star formation and other galactic processes.Furthermore, exploring the gas and metal properties of the CGMand IGM may allow us to differentiate between various galaxyformation models (Tumlinson et al. 2017; Codoreanu et al. 2018).Cross-correlation techniques can also be used to detect the syn-chrotron cosmic web (which is also much too faint to directlyobserve), and hence constrain the perpendicular component ofthe IGMF in filaments. This was recently done by Brown et al.(2017) by cross-correlating S-PASS continuum data at 2.3 GHzand MHD simulations of the cosmic web (using models by Dolaget al. 2005; Donnert et al. 2009), placing bounds on the order of0 . 𝜇 G on the IGMF. The upper bound places by Brown et al.(2017) agreed with other synchrotron based bounds placed, suchas in Vernstrom et al. (2017), and is consistent with bounds placedin our study.Most recently Locatelli et al. (2021) used the non-detection of synchrotron accretion shocks between twoadjacent galaxy clusters located using LOFAR (pair 1:RXCJ1155.3+2324 /RXCJ1156.9+2415, and pair 2:RXCJ1659.7+3236/RXCJ1702.7+3403) by comparing theirobservations to MHD
ENZO (Bryan et al. 2014) simulations ofgalaxy clusters of similar properties. By doing this they wereable to place bounds on the intergalactic magnetic field presentto 𝐵 𝑓 𝑖𝑙 < . 𝜇 G for Mpc scales. Although these bounds areconsistent but less constraining than those places in our study,Locatelli et al. (2021) used only two clusters. With more clusterobservations with radio telescopes such as LOFAR this techniquecan prove to be powerful.Forecasts of future radio surveys have lead to predictions for whatsuch an IGMF signal would look like, if detected. Stasyszyn et al.(2010) found, using cross-correlations of simulated data (a simi-lar technique to our study), that direct detections of the large-scalemagnetic field that permeates the IGM in filaments will only bepossible with larger RM catalogues from future telescopes. Aka-hori et al. (2014) explored the effectiveness of using simulatedRM grids with varying sample sizes and statistical techniques todetect magnetic fields structure in filaments. They found that ba-sic properties of large-scale magnetic fields can be determinedstatistically using upcoming surveys, allowing us to distinguishbetween magneto-genesis models. For our study, it is not possibleto analyze the shape of our cross-correlation function to obtain in-formation about the coherence scale of the IGMF, nor to constrainthe generation mechanism of the field, as we did not detect a sig-nificant cross-correlation signal. Such studies must wait for futuresurveys.
Our study is mostly limited by the availability of extra-galacticRM sources with redshift information. As we progress towards anera of large data availability, such as the upcoming data releasesfrom the Australian Square Kilometre Array Pathfinder (ASKAP)within the next few years, the bounds we can place on 𝐵 (cid:107) , 𝑓 𝑖𝑙 willbecome more robust. Once completed, ASKAP’s Polarization SkySurvey of the Universe’s Magnetism (POSSUM) (Gaensler et al.2010) will comprise the largest catalogue of extra-galactic rotationmeasures to date, with RMs for one million extra-galactic radiosources with a source-density coverage of ≥
25 RMs deg − .To demonstrate that using this technique with a larger sample of http://askap.org/possum Figure 7.
The 3 𝜎 significance levels of random CCFs for increasingsample sizes of RRM sources. The RRMs were generated by generatingrandomized sightlines and using RRMs from real data. RMs will obtain more stringent upper bounds, we generated sim-ulated RM data using the technique described in Section 3.4.2 toobtain sample catalogues of 1742 RMs (the size of the Hammondet al. 2012 sample used in this study, for comparison), 10,000RMs, and 100,000 RMs. We then obtain 3 𝜎 significance levelsfor 1000 realizations of sample catalogues of these sizes, plot-ted in Figure 7. For the sample sizes larger than 1742, we drawRM values (with replacement) from the full Taylor et al. (2009)catalogue, applying the same sample cuts as in section 3.1, andanalysis in section 3.2 to obtain RRMs. The error bars becomesmaller with increasing sample sizes. At the largest radial bin, 2.5Mpc, this corresponds to a ∼ ∼ 𝐵 (cid:107) , 𝑓 𝑖𝑙 .The main source of uncertainties in this analysis originates fromthe data-sets used. Hammond et al. (2012) use RMs determinedfrom NVSS, which is determined by linearly fitting the polariza-tion angle vs. 𝜆 for only 2 narrowly spaced bands (see Section2.1), which may lead to spurious RM values for sources.These spurious RMs could manifest in 𝑛𝜋 ambiguities. In this casethe phase angles wrap around at 𝜋 and thus the observed polar-ization angle is only known to modulo 𝑛𝜋 radians (Heald 2009).Sources affected by this could have erroneously high RMs offsetfrom the true RM by ± . / m (Ma et al. 2019a). Ma et al.(2019a) explored this effect in the Taylor et al. (2009) catalogueby using follow up large-bandwidth observations of suspicioussources. They concluded that the 𝑛𝜋 ambiguity would affect > ∼ 𝜆 assumed by Taylor et al. (2009). ThisFaraday thick behaviour can be caused by many situations; regions MNRAS , 1–15 (2015) ross-correlation to Constrain IGMF Figure 8.
The resulting cross-correlation functions when we include theerrors associated with the RM sources. The red dashed line is calculatedby adding a random Gaussian drawn fraction of the error to the | 𝑅𝑅𝑀 | value, while the orange line is calculated in the same manner but the errorsare subtracted from the | 𝑅𝑅𝑀 | values. simultaneously emitting and rotation, or various regions rotatingat different amounts along the line-of-sight (see a detailed discus-sion in Sokoloff et al. 1998). Because the two bands used to pro-duce the RMs of these sources are spaced close together, the RMinformation captured could be from a particular Faraday rotatingstructure and not representative over larger bandwidths.However, for both of these cases without re-observing all sourcesat larger bandwidths, it is impossible to determine which sourceswould be affected. POSSUM will observe at 800 MHz to 1088MHz at 1 MHz resolution, corresponding to Δ 𝜈 = 288 MHz, lead-ing to much more robust RM value assignments. Additionally,POSSUM will use techniques such as QU-fitting (O’Sullivan et al.2012) and RM synthesis (Brentjens & de Bruyn 2005) to extractFaraday depth information and RMs from sources. These tech-niques are much more reliable than angle fitting techniques usedfor older data-sets.We did not use the error bars associated with the Hammondet al. (2012) RM sources in our cross-correlation. To ensure thatthe RM errors would not have an effect on the resulting cross-correlation function, we recalculated the cross-correlation for val-ues of RM adjusted by a Gaussian drawn value multiplied by itserror. We found that running this for 100 cases did not change theoutcome of the cross-correlation function within 1 𝜎 . This effectcan be seen for one run in Figure 8. Because of the low gas densities in cosmic web filaments, cos-mic magnetic fields are difficult to detect, with estimated valuesin the nG regime. These weak magnetic fields require statisticalapproaches to extract their signal from large data-sets.We use a cross-correlation statistical approach between 1742background RMs (from Hammond et al. 2012, 𝑧 > .
5) and thesuperCOSMOSxWISE all-sky photometric redshift catalogue of 18 million galaxy sources (Bilicki et al. 2016) to trace largescale structure between 0 . < 𝑧 < . . < 𝑧 < . . < 𝑧 < .
3, and 0 . < 𝑧 < .
5) at impact parameters between ∼ . 𝜎 significance levels from theobserved cross-correlation to place an upper bound on the meanco-moving magnetic fields within filaments. The 3 𝜎 upper boundfor magnetic field coherence scales between 1 - 2.5 Mpc was ∼ < . ACKNOWLEDGEMENTS
We thank the anonymous referee for their useful comments. Wealso thank Lawrence Rudnick, Franco Vazza, Dongsu Ryu, YikKi (Jackie) Ma, Shane O’Sullivan, Ue-Li Pen, Cameron Van Eck,Mubdi Rahman, Norm Murray, and Matthew Young for usefuldiscussions.This research has made use of the SIMBAD database, operatedat CDS, Strasbourg, France, and the NASA/IPAC ExtragalacticDatabase (NED), which is funded by the National Aeronautics andSpace Administration and operated by the California Institute ofTechnology.The Dunlap Institute is funded through an endowment establishedby the David Dunlap family and the University of Toronto. Weacknowledge the support of the Natural Sciences and EngineeringResearch Council of Canada (NSERC) through grant RGPIN-2015-05948, and of the Canada Research Chairs program.This work was performed traditional land of the Huron-Wendat,the Seneca, and most recently, the Mississaugas of the CreditRiver.
DATA AVAILABILITY
The Hammond et al. (2012) RM-redshift data underlying thisarticle was accessed from
MNRAS000
MNRAS000 , 1–15 (2015) Amaral et al.
RMCatalogue , the Oppermann et al. (2015) Galactic RM datawas accessed from , and the Bilicki et al. (2016) galaxy catalogue wasaccessed from the Wide Field Astronomy Unit at the Institute forAstronomy, Edinburgh at http://ssa.roe.ac.uk/WISExSCOS .The derived data generated in this research will be shared on rea-sonable request to the corresponding author.
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