# The Challenge of Detecting Intracluster Filaments with Faraday Rotation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{−3}) and T∼ 1–10 keV, respectively, as supported by numerical simulations (e.g., [3,4,5]).

^{−3}and temperature ∼10

^{5}–10

^{7}K), is instead still largely unconstrained, despite the fact that this gas phase should contain a large fraction (∼50–60%) of the baryonic mass of the Universe. Measuring the magnetic field intensity and morphology in the IGM is crucial to constrain the primordial origin of extragalactic fields, out of which present day magnetic structures might have evolved (e.g., [6]). Unluckily, direct observations of magnetic fields in the IGM are made challenging by the very high sensitivity required for the imaging at most wavelengths.

^{5}–10

^{7}K) phase of the IGM, called the Warm Hot Intergalactic Medium (WHIM, see Nicastro [22] for a recent review), have been detected by X-ray observations of cluster outskirts (e.g., [23] for A2744). Complementary to this, also microwave observations of the Sunyaev-Zeldovich (SZ) effect from single objects (e.g., [24,25]) or via stacking (e.g., [26]) proved to be effective in detecting gas in filaments connecting closely interacting clusters.

#### Predictions of Magnetic Fields in Filaments

## 2. Methods and Materials

#### 2.1. Simulations of Extragalactic Magnetic Fields

^{3}(comoving), and is simulated starting from a root grid ${256}^{3}$ cells and using ${256}^{3}$ dark matter particles. The initial density perturbation field is taken from a suite of existing cluster simulations (e.g., [42] and other works derived from this). The innermost ∼25 Mpc

^{3}volume, centred on where each cluster forms, has been further refined 5 times (${2}^{5}$)using AMR. Mesh refinements are initiated wherever the gas density is ≥1% higher than its surroundings. This give us a maximum spatial resolution of $\Delta {x}_{\mathrm{max}}\approx 31\phantom{\rule{3.33333pt}{0ex}}\mathrm{kpc}$. The mass resolution for dark matter particle in the high resolution region is ${m}_{\mathrm{DM}}=9.1\xb7{10}^{10}{M}_{\odot}$ for all clusters.

#### 2.2. Filament Selection and Properties

#### 2.3. Mock Rotation Measure observations

- A fixed contribution to RM from the Galactic foreground. We restricted ourselves to targets at high galactic latitude (≥80°), for which the RM contribution is in general of $|R{M}_{\mathrm{Gal}}|\le 10\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/{\mathrm{m}}^{2}$ [9]. In particular, we assumed here for simplicity a fixed $+6.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/{\mathrm{m}}^{2}$ contribution to each field, noticing that this contribution should in general be the easiest to tell apart in real observations, because the extent of the typical size of filaments around galaxy clusters we consider here is ≤0.1–0.5°, i.e., much smaller than the typical angular scales of variations of the Galactic foreground. For example, based on Equation (20) in Anderson et al. [45], we can estimate a typical RM fluctuation of ≤0.5 rad/m
^{2}across 0.5° from the Galactic foreground. - a residual contribution to the RM, RM
_{res}, which includes an internal contribution to each background source RM_{src}; a contribution from other extragalactic sources as intervening MgII absorbers RM_{MgII}[46]; a residual RM after Galactic foreground subtraction which can be present on scales smaller than the one used to fit the Galactic contribute ${\mathrm{RM}}_{\mathrm{MW},\mathrm{res}}$. This value has been estimated by Schnitzeler [47] to be normally distributed as ${\sigma}_{\mathrm{res}}\le 6$ rad/m^{2}and has been confirmed by Banfield et al. [48] (${\sigma}_{\mathrm{ERS}}$ in their work). We follow their treatment and keep RM_{src}, RM_{MgII}and ${\mathrm{RM}}_{\mathrm{MW},\mathrm{res}}$ together in the one factor RM_{res}. However Banfield et al. [48] note that the value is dependent on the background source population. At 1.4 GHz, the WISE-AGN population defined in Jarrett et al. [49] biases the estimate of ${\sigma}_{\mathrm{res}}$ and shows a larger ${\sigma}_{\mathrm{res}}=12\pm 0.2$ rad/m^{2}. We thus consider the latter as a more conservative case, while we considered ${\sigma}_{\mathrm{res}}=6$ rad/m^{2}as a standard case based on literature works [47,48] that can be optimized e.g., using only star-forming nearby galaxies for background studies.

_{res}value from a Gaussian distribution with standard deviation ${\sigma}_{\mathrm{res}}=12$ or $6\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/{\mathrm{m}}^{2}$, depending on the assumed background source population.

_{P}is the signal-to-noise ratio of the source in the polarized image and $\Delta {\lambda}_{\mathrm{max}}^{2}$ is the difference between the largest and smallest observed ${\lambda}^{2}$. Also for the latter we considered two possible different values: (a) JVLA-like observations in which ${\delta}_{\mathrm{RM}}=8\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/{\mathrm{m}}^{2}$ (assuming wide total bandwidth $\Delta \nu \simeq 1$ GHz, L-band observations of background sources with ${\mathrm{SNR}}_{P}>3$); (b) SKA-MID-like observations in which ${\delta}_{\mathrm{RM}}=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{rad}/{\mathrm{m}}^{2}$ which corresponds to current estimates for SKA-MID performances (e.g., [16]).

#### 2.4. Non-Parametric Tests of RM Distributions

## 3. Results

#### Detectability of Intracluster Filaments Using RM

^{2}(Figure 3, lower panels) our test show that a tentative detection of a few prominent filaments becomes possible for ${N}_{S}\ge 50$ sources.

- On average, with our procedure we estimate that galaxy clusters have $3.3\pm 0.9$ projected filaments which can be identified by X-ray inspection. This result is consistent with the results by Colberg et al. [35], who found that ∼80% of clusters have 1 to 4 projected connections between them.
- Filaments selected in our procedure are on average ∼10–50 times denser that the smooth environment around galaxy clusters, have a mean temperature of $T\sim $ 1–5 $\xb7{10}^{7}\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$ and an average magnetic field of ${B}_{\mathrm{rms}}\sim $ 10–50 nG (see Section 4.1).
- The typical RM in filaments is in the range ∼0.2–2 rad/m
^{2}for the primordial seeding scenario considered here, i.e., a factor ∼10^{2}larger than the average RM distribution in our control fields for the detectable ones, ∼30 times larger for filaments in general. However, the distributions of RMs can reach up to ∼10 rad/m^{2}in a few % of the cells, and the chances of confirming the presence of magnetic fields in filaments rely on the detection of such rare fluctuations. - The rejection fraction has been fitted with a power-law trend with respect to the number of detected sources ${N}_{S}$. Considering the 7 filaments with the highest rejection fraction in the most favorable case (i.e., extragalactic residual RM noise is small or absent), Figure 2, right panel), the best fit gives rf ∝ ${N}_{S}^{0.55\pm 0.05}$ before saturation. To this end, just the filaments showing an improved rejection fraction above detection threshold even for small samples (${N}_{S}=5,10,15$) were considered. Including the other filaments would affect the trend with random low rejection rates.
- Limited to the most favorable objects and for low contribution from residual RM on the sources, the increased sensitivity that will be provided by SKA-MID compared to JVLA-like observations improves the rejection fractions distribution by a factor ∼3(1.5) at ${N}_{S}=100\left(20\right)$, while the number of observations with a rejection fraction larger than 0.5 increases from 0(0) to 9(3) over 29 objects.
- The actual limiting factor for the detection of filaments is the extragalactic residual RM scatter ${\sigma}_{\mathrm{res}}$, more than the number of detected sources N
_{S}throughout the field. Going from ${\sigma}_{\mathrm{res}}=0$ rad/m^{2}to ${\sigma}_{\mathrm{res}}=6\left(12\right)$ rad/m^{2}the rejection fraction median drops down by a factor 3(3.3) even for the SKA-MID-like observation with 100 sources per set, and the observations with a rejection fraction larger than 0.5 falls from 31% to 0% even for this large ${N}_{S}$ value.

^{2}in the sky, which is approximately the putative SKA-MID field-of-view (≃0.49 deg

^{2}). A JVLA obseravation with similar settings (1 GHz total bandwidth centered at 1.4 GHz, A-configuration array with resolution ${1.5}^{\u2033}$, 40% flagged data) would cover the target with a 4 pointings mosaic, requiring 68 h of total observing time to reach the required sensitivity of 3 μJy per beam (17 h per single pointing).

## 4. Discussion

#### 4.1. Properties of Most Detectable Filaments

^{−5}of the kinetic energy in the undetectable filament. The visual inspection further suggests that the sub-volume around the detectable filament contains more gas substructures, which are likely responsible for enhanced density fluctuations and for the mixing of magnetic field lines on small-scales. While the clumpiest part of this volume is excised from our RM analysis (see Section 2.2), a higher degree of structures within the filament implies that the environment has surely been subject to a higher dynamical activity in the recent past, which significantly boosted the magnetic field beyond compression.

#### 4.2. Alternative Models of Magnetic Fields in Filaments

^{2}) in the ICM and filaments, and lower RM values in the control field (which produces a larger contrast between them), telling the two models apart based on the statistical analysis of observable RMs becomes more difficult (see Figure 7). Virtually no filaments would be detectable in this case, not even by considering no contamination from residual RM (${\sigma}_{\mathrm{res}}=0$ rad/m

^{2}), and even for a SKA-MID-like observation (Figure 7, right panel).

^{2}makes JVLA-like observations of filaments to reach a rejection fraction larger than 0.5 in ∼34% of the sample with ${N}_{S}=100$ detected polarized sources, and ∼7% with ${N}_{S}=20$ sources. With the SKA-MID more than a half of our objects will be significantly detectable with ${N}_{S}=100$ sources. Even for a small number of available polarized sources ${N}_{S}=5$ about ∼10(17)% of filaments have a chance of ≥1/5 to be distinguished from the control field in a JVLA(SKA-MID)-like observation.

## 5. Conclusions

^{2}is key to probe magnetic fields in filaments, due to the higher statistics of detectable polarised sources, which will allow us to distinguishing filaments from nearby control fields in a statistically robust way. However, our analysis shows that strong limitations to this search arise because of the spurious contributions to the RM on the polarized radio sources (e.g., [48,57]).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RM | Rotation Measure |

SKA | Square Kilometer Array |

MeerKAT | Karoo Array Telescope |

JVLA | Jansky Very Large Array |

K-S | Kolmogorov Smirnov |

M-W | Mann Whitney |

ΛCDM | Lambda Cold Dark Matter |

AGN | Active Galactic Nucleus |

CMB | Cosmic Microwave Background |

AMR | Adaptive Mesh Refinement |

MHD | Magneto Hydro Dynamics |

ICM | Intra Cluster Medium |

IGM | Inter Galactic Medium |

WHIM | Warm Hot Interagalactic Medium |

LOS | Line of Sight |

## References

- Feretti, L.; Giovannini, G.; Govoni, F.; Murgia, M. Clusters of galaxies: Observational properties of the diffuse radio emission. Astron. Astrophys. Rev.
**2012**, 20, 54. [Google Scholar] [CrossRef] - Ryu, D.; Kang, H.; Cho, J.; Das, S. Turbulence and Magnetic Fields in the Large-Scale Structure of the Universe. Science
**2008**, 320, 909–912. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dolag, K.; Bartelmann, M.; Lesch, H. SPH simulations of magnetic fields in galaxy clusters. Astron. Astrophys.
**1999**, 348, 351–363. [Google Scholar] - Vazza, F.; Brunetti, G.; Brüggen, M.; Bonafede, A. Resolved magnetic dynamo action in the simulated intracluster medium. Mon. Not. R. Astron. Soc.
**2018**, 474, 1672–1687. [Google Scholar] [CrossRef] - Donnert, J.; Vazza, F.; Brüggen, M.; ZuHone, J. Magnetic Field Amplification in Galaxy Clusters and its Simulation. arXiv, 2018; arXiv:1810.09783. [Google Scholar]
- Widrow, L.M.; Ryu, D.; Schleicher, D.R.G.; Subramanian, K.; Tsagas, C.G.; Treumann, R.A. The First Magnetic Fields. Sci. Space Rev.
**2012**, 166, 37–70. [Google Scholar] [CrossRef] - Brentjens, M.A.; de Bruyn, A.G. Faraday rotation measure synthesis. Astron. Astrophys.
**2005**, 441, 1217–1228. [Google Scholar] [CrossRef][Green Version] - Xu, Y.; Kronberg, P.P.; Habib, S.; Dufton, Q.W. A Faraday Rotation Search for Magnetic Fields in Large-scale Structure. Astrophys. J.
**2006**, 637, 19–26. [Google Scholar] [CrossRef] - Oppermann, N.; Junklewitz, H.; Greiner, M.; Enßlin, T.A.; Akahori, T.; Carretti, E.; Gaensler, B.M.; Goobar, A.; Harvey-Smith, L.; Johnston-Hollitt, M.; et al. Estimating extragalactic Faraday rotation. Astron. Astrophys.
**2015**, 575, A118. [Google Scholar] [CrossRef] - Vernstrom, T.; Gaensler, B.M.; Brown, S.; Lenc, E.; Norris, R.P. Low-frequency radio constraints on the synchrotron cosmic web. Mon. Not. R. Astron. Soc.
**2017**, 467, 4914–4936. [Google Scholar] [CrossRef] - Brown, S.; Vernstrom, T.; Carretti, E.; Dolag, K.; Gaensler, B.M.; Staveley-Smith, L.; Bernardi, G.; Haverkorn, M.; Kesteven, M.; Poppi, S. Limiting magnetic fields in the cosmic web with diffuse radio emission. Mon. Not. R. Astron. Soc.
**2017**, 468, 4246–4253. [Google Scholar] [CrossRef] - O’Sullivan, S.P.; Machalski, J.; Van Eck, C.L.; Heald, G.; Brueggen, M.; Fynbo, J.P.U.; Heintz, K.E.; Lara-Lopez, M.A.; Vacca, V.; Hardcastle, M.J.; et al. The intergalactic magnetic field probed by a giant radio galaxy. arXiv, 2018; arXiv:1811.07934. [Google Scholar]
- Govoni, F.; Murgia, M.; Xu, H.; Li, H.; Norman, M.L.; Feretti, L.; Giovannini, G.; Vacca, V. Polarization of cluster radio halos with upcoming radio interferometers. Astron. Astrophys.
**2013**, 554, A102. [Google Scholar] [CrossRef][Green Version] - Akahori, T.; Gaensler, B.M.; Ryu, D. Statistical Techniques for Detecting the Intergalactic Magnetic Field from Large Samples of Extragalactic Faraday Rotation Data. Astrophys. J.
**2014**, 790, 123. [Google Scholar] [CrossRef] - Johnston-Hollitt, M.; Govoni, F.; Beck, R.; Dehghan, S.; Pratley, L.; Akahori, T.; Heald, G.; Agudo, I.; Bonafede, A.; Carretti, E.; et al. Using SKA Rotation Measures to Reveal the Mysteries of the Magnetised Universe. In Proceedings of the Advancing Astrophysics with the Square Kilometre Array (AASKA14), Giardini Naxos, Italy, 9–13 June 2015; p. 92. [Google Scholar]
- Govoni, F.; Murgia, M.; Xu, H.; Li, H.; Norman, M.; Feretti, L.; Giovannini, G.; Vacca, V.; Bernardi, G.; Bonafede, A.; et al. Cluster magnetic fields through the study of polarized radio halos in the SKA era. In Proceedings of the Advancing Astrophysics with the Square Kilometre Array (AASKA14), Giardini Naxos, Italy, 9–13 June 2015; p. 105. [Google Scholar]
- Bonafede, A.; Vazza, F.; Brüggen, M.; Akahori, T.; Carretti, E.; Colafrancesco, S.; Feretti, L.; Ferrari, C.; Giovannini, G.; Govoni, F.; et al. Unravelling the origin of large-scale magnetic fields in galaxy clusters and beyond through Faraday Rotation Measures with the SKA. In Proceedings of the Advancing Astrophysics with the Square Kilometre Array (AASKA14), Giardini Naxos, Italy, 9–13 June 2015; p. 95. [Google Scholar]
- Vacca, V.; Oppermann, N.; Enßlin, T.; Jasche, J.; Selig, M.; Greiner, M.; Junklewitz, H.; Reinecke, M.; Brüggen, M.; Carretti, E.; et al. Using rotation measure grids to detect cosmological magnetic fields: A Bayesian approach. Astron. Astrophys.
**2016**, 591, A13. [Google Scholar] [CrossRef] - Akahori, T.; Ryu, D.; Gaensler, B.M. Fast Radio Bursts as Probes of Magnetic Fields in the Intergalactic Medium. Astrophys. J.
**2016**, 824, 105. [Google Scholar] [CrossRef] - Vazza, F.; Brüggen, M.; Hinz, P.M.; Wittor, D.; Locatelli, N.; Gheller, C. Probing the origin of extragalactic magnetic fields with Fast Radio Bursts. Mon. Not. R. Astron. Soc.
**2018**, 480, 3907–3915. [Google Scholar] [CrossRef] - Akahori, T.; Ideguchi, S.; Aoki, T.; Takefuji, K.; Ujihara, H.; Takahashi, K. Optimum Frequency of Faraday Tomography to Explore the Inter-Galactic Magnetic Field in Filaments of Galaxies. arXiv, 2018; arXiv:1808.10546. [Google Scholar]
- Nicastro, F. A Decade of WHIM Search: Where Do We Stand and Where Do We Go. XMM-Newton: The Next Decade. 2016; p. 27. Available online: http://xxx.lanl.gov/abs/1611.03722 (accessed on 30 July 2018).
- Eckert, D.; Jauzac, M.; Shan, H.; Kneib, J.P.; Erben, T.; Israel, H.; Jullo, E.; Klein, M.; Massey, R.; Richard, J.; et al. Warm-hot baryons comprise 5–10 per cent of filaments in the cosmic web. Nature
**2015**, 528, 105–107. [Google Scholar] [CrossRef] [PubMed][Green Version] - Planck, C.; Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Balbi, A.; Banday, A.J.; et al. Planck intermediate results. VIII. Filaments between interacting clusters. Astron. Astrophys.
**2013**, 550, A134. [Google Scholar] [CrossRef] - Bonjean, V.; Aghanim, N.; Salomé, P.; Douspis, M.; Beelen, A. Gas and galaxies in filaments between clusters of galaxies. The study of A399–A401. Astron. Astrophys.
**2018**, 609, A49. [Google Scholar] [CrossRef] - De Graaff, A.; Cai, Y.C.; Heymans, C.; Peacock, J.A. Missing baryons in the cosmic web revealed by the Sunyaev-Zel’dovich effect. arXiv, 2017; arXiv:1709.10378. [Google Scholar]
- Brown, S.D. Synchrotron Emission on the Largest Scales: Radio Detection of the Cosmic-Web. J. Astrophys. Astron.
**2011**, 32, 577–584. [Google Scholar] [CrossRef][Green Version] - Vazza, F.; Ferrari, C.; Brüggen, M.; Bonafede, A.; Gheller, C.; Wang, P. Forecasts for the detection of the magnetised cosmic web from cosmological simulations. Astron. Astrophys.
**2015**, 580, A119. [Google Scholar] [CrossRef][Green Version] - Cho, J.; Ryu, D. Characteristic Lengths of Magnetic Field in Magnetohydrodynamic Turbulence. Astrophys. J.
**2009**, 705, L90–L94. [Google Scholar] [CrossRef] - Brüggen, M.; Ruszkowski, M.; Simionescu, A.; Hoeft, M.; Dalla Vecchia, C. Simulations of Magnetic Fields in Filaments. Astrophys. J.
**2005**, 631, L21–L24. [Google Scholar] [CrossRef][Green Version] - Vazza, F.; Brüggen, M.; Gheller, C.; Wang, P. On the amplification of magnetic fields in cosmic filaments and galaxy clusters. Mon. Not. R. Astron. Soc.
**2014**, 445, 3706–3722. [Google Scholar] [CrossRef] - Marinacci, F.; Vogelsberger, M.; Mocz, P.; Pakmor, R. The large-scale properties of simulated cosmological magnetic fields. Mon. Not. R. Astron. Soc.
**2015**, 453, 3999–4019. [Google Scholar] [CrossRef] - Vazza, F.; Brüggen, M.; Gheller, C.; Hackstein, S.; Wittor, D.; Hinz, P.M. Simulations of extragalactic magnetic fields and of their observables. Class. Quantum Gravity
**2017**, 34, 234001. [Google Scholar] [CrossRef][Green Version] - Gheller, C.; Vazza, F.; Brüggen, M.; Alpaslan, M.; Holwerda, B.W.; Hopkins, A.M.; Liske, J. Evolution of cosmic filaments and of their galaxy population from MHD cosmological simulations. Mon. Not. R. Astron. Soc.
**2016**, 462, 448–463. [Google Scholar] [CrossRef][Green Version] - Colberg, J.M.; Krughoff, K.S.; Connolly, A.J. Intercluster filaments in a ΛCDM Universe. Mon. Not. R. Astron. Soc.
**2005**, 359, 272–282. [Google Scholar] [CrossRef] - Dolag, K.; Meneghetti, M.; Moscardini, L.; Rasia, E.; Bonaldi, A. Simulating the physical properties of dark matter and gas inside the cosmic web. Mon. Not. R. Astron. Soc.
**2006**, 370, 656–672. [Google Scholar] [CrossRef][Green Version] - Gheller, C.; Vazza, F.; Favre, J.; Brüggen, M. Properties of cosmological filaments extracted from Eulerian simulations. Mon. Not. R. Astron. Soc.
**2015**, 453, 1164–1185. [Google Scholar] [CrossRef][Green Version] - Donnert, J.; Dolag, K.; Lesch, H.; Müller, E. Cluster magnetic fields from galactic outflows. Mon. Not. R. Astron. Soc.
**2009**, 392, 1008–1021. [Google Scholar] [CrossRef][Green Version] - Xu, H.; Li, H.; Collins, D.C.; Li, S.; Norman, M.L. Turbulence and Dynamo in Galaxy Cluster Medium: Implications on the Origin of Cluster Magnetic Fields. Astrophys. J.
**2009**, 698, L14–L17. [Google Scholar] [CrossRef] - Bryan, G.L.; Norman, M.L.; O’Shea, B.W.; Abel, T.; Wise, J.H.; Turk, M.J.; Reynolds, D.R.; Collins, D.C.; Wang, P.; Skillman, S.W.; et al. ENZO: An Adaptive Mesh Refinement Code for Astrophysics. Astrophys. J. Suppl.
**2014**, 211, 19. [Google Scholar] [CrossRef] - Wang, P.; Abel, T. Magnetohydrodynamic Simulations of Disk Galaxy Formation: The Magnetization of the Cold and Warm Medium. Astrophys. J.
**2009**, 696, 96–109. [Google Scholar] [CrossRef] - Vazza, F.; Brunetti, G.; Gheller, C.; Brunino, R. Massive and refined: A sample of large galaxy clusters simulated at high resolution. I: Thermal gas and properties of shock waves. New Astron.
**2010**, 15, 695–711. [Google Scholar] [CrossRef][Green Version] - Wittor, D.; Vazza, F.; Brüggen, M. Testing cosmic ray acceleration with radio relics: A high-resolution study using MHD and tracers. Mon. Not. R. Astron. Soc.
**2017**, 464, 4448–4462. [Google Scholar] [CrossRef] - Subramanian, K. The origin, evolution and signatures of primordial magnetic fields. arXiv, 2015; arXiv:1504.02311. [Google Scholar]
- Anderson, C.S.; Gaensler, B.M.; Feain, I.J.; Franzen, T.M.O. Broadband Radio Polarimetry and Faraday Rotation of 563 Extragalactic Radio Sources. Astrophys. J.
**2015**, 815, 49. [Google Scholar] [CrossRef] - Joshi, R.; Chand, H. Dependence of residual rotation measure on intervening Mg II absorbers at cosmic distances. Mon. Not. R. Astron. Soc.
**2013**, 434, 3566–3571. [Google Scholar] [CrossRef] - Schnitzeler, D.H.F.M. The latitude dependence of the rotation measures of NVSS sources. Mon. Not. R. Astron. Soc.
**2010**, 409, L99–L103. [Google Scholar] [CrossRef] - Banfield, J.K.; Schnitzeler, D.H.F.M.; George, S.J.; Norris, R.P.; Jarrett, T.H.; Taylor, A.R.; Stil, J.M. Radio galaxies and their magnetic fields out to z ≤ 3. Mon. Not. R. Astron. Soc.
**2014**, 444, 700–710. [Google Scholar] [CrossRef] - Jarrett, T.H.; Masci, F.; Tsai, C.; Petty, S.; Benford, D. WISE Nearby Galaxy Atlas. In Proceedings of the American Astronomical Society Meeting, Nantes, France, 2–7 October 2011; Volume 43. [Google Scholar]
- Rudnick, L.; Owen, F.N. The Distribution of Polarized Radio Sources >15 μJy in GOODS-N. Astrophys. J.
**2014**, 785, 45. [Google Scholar] [CrossRef] - Neuhäuser, M. Wilcoxon–Mann–Whitney Test. In International Encyclopedia of Statistical Science; Lovric, M., Ed.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1656–1658. [Google Scholar]
- Connor, T.; Kelson, D.D.; Mulchaey, J.; Vikhlinin, A.; Patel, S.G.; Balogh, M.L.; Joshi, G.; Kraft, R.; Nagai, D.; Starikova, S. Wide-Field Optical Spectroscopy of Abell 133: A Search for Filaments Reported in X-ray Observations. arXiv, 2018; arXiv:1809.08241. [Google Scholar]
- Domínguez Fernández, P.; Vazza, F.; Brüggen, M. Spectral analysis of magnetic fields in simulated galaxy clusters. arXiv, 2018; arXiv:1810.08009. [Google Scholar]
- Planck, C.; Ade, P.A.R.; Aghanim, N.; Arnaud, M.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; et al. Planck 2015 results. XIX. Constraints on primordial magnetic fields. Astron. Astrophys.
**2016**, 594, A19. [Google Scholar] [CrossRef] - Cassano, R.; Bernardi, G.; Brunetti, G.; Brüggen, M.; Clarke, T.; Dallacasa, D.; Dolag, K.; Ettori, S.; Giacintucci, S.; Giocoli, C.; et al. Cluster Radio Halos at the crossroads between astrophysics and cosmology in the SKA era. arXiv, 2014; arXiv:1412.5940. [Google Scholar]
- Nuza, S.E.; Gelszinnis, J.; Hoeft, M.; Yepes, G. Can cluster merger shocks reproduce the luminosity and shape distribution of radio relics? Mon. Not. R. Astron. Soc.
**2017**, 470, 240–263. [Google Scholar] [CrossRef][Green Version] - Lamee, M.; Rudnick, L.; Farnes, J.S.; Carretti, E.; Gaensler, B.M.; Haverkorn, M.; Poppi, S. Magnetic Field Disorder and Faraday Effects on the Polarization of Extragalactic Radio Sources. Astrophys. J.
**2016**, 829, 5. [Google Scholar] [CrossRef]

1. | |

2. | We notice that the density weighting in the velocity spectra ensures that the magnetic and velocity spectra have the same units and can be quantitatively compared, as in Vazza et al. [4]. |

**Figure 1.**Projected gas density (

**upper**) and RM (

**lower**) for our cluster e18b at $z=0$, for the non-radiative run with primordial magnetic fields. Each box has sides $22\times 20\phantom{\rule{3.33333pt}{0ex}}{\mathrm{Mpc}}^{2}$. In the RM map, we show with red colors pixels with $\mathrm{RM}>0$ and with blue colors the pixels with $\mathrm{RM}<0$.

**Figure 2.**Rejection fraction for an increasing number of detected RMs calculated for all the simulated filaments in the primordial seeding scenario, for an idealized scenario in which we set ${\sigma}_{\mathrm{res}}$ to 0 rad/m

^{2}for the residual RM on background sources. Different colors mark different filaments, different marks point to different simulated clusters, while the solid black line show the median for fixed ${N}_{S}$. (

**Left**) JVLA-like parameters; (

**Right**) SKA-MID-like parameters. Color and mark codes of filaments are conserved through all the plots in this work.

**Figure 3.**Rejection fraction for an increasing number of detected RM as in Figure 2, but by assuming ${\sigma}_{\mathrm{res}}=12$ rad/m

^{2}for the residual RM of sources (top panels), while the lower panels we assumed ${\sigma}_{\mathrm{res}}=6$ rad/m

^{2}.

**Figure 4.**Distributions of RMs, projected mean gas density and mean magnetic field strength for all control fields and filaments considered in our datasets (primordial model, including all analyzed lines of sight). The additional dot-dashed lines show the distributions of the same fields limited to the 4 most detectable filaments in Figure 3.

**Figure 5.**The upper thin lines show the 3-dimensional power spectra for the density weighted velocity field (${\rho}^{1/2}v$) and the lower thick lines show the 3-dimensional magnetic power spectra in three sub-volumes of the cluster e18b.

**Figure 6.**(

**Left**) RM distribution for our e18b cluster at $z=0$, simulated with the cooling and feedback model, in which magnetic fields are injected by AGN activity. The box as the meaning of colors is as in Figure 1. (

**Right**) distribution of RMs for all filament and control fields, for the primordial and the astrophysical model.

**Figure 7.**Same as Figure 2 for the AGN seeding scenario.

**Figure 8.**Same as Figure 3 (lower panels) for the primordial seeding scenario with ${B}_{\mathrm{init}}=1$ nG.

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## Share and Cite

**MDPI and ACS Style**

Locatelli, N.; Vazza, F.; Domínguez-Fernández, P. The Challenge of Detecting Intracluster Filaments with Faraday Rotation. *Galaxies* **2018**, *6*, 128.
https://doi.org/10.3390/galaxies6040128

**AMA Style**

Locatelli N, Vazza F, Domínguez-Fernández P. The Challenge of Detecting Intracluster Filaments with Faraday Rotation. *Galaxies*. 2018; 6(4):128.
https://doi.org/10.3390/galaxies6040128

**Chicago/Turabian Style**

Locatelli, Nicola, Franco Vazza, and Paola Domínguez-Fernández. 2018. "The Challenge of Detecting Intracluster Filaments with Faraday Rotation" *Galaxies* 6, no. 4: 128.
https://doi.org/10.3390/galaxies6040128