Submitted **to** ApJ

**Faraday** **Rotation** **Measure** **due** **to** **the** **Intergalactic** **Magnetic** **Field**

Takuya Akahori 1 and Dongsu Ryu 2,3

1Research Institute of Basic Science, Chungnam National University, Daejeon, Korea:

akataku@canopus.cnu.ac.kr

2Department of Astronomy and Space Science, Chungnam National University, Daejeon,

Korea: ryu@canopus.cnu.ac.kr

ABSTRACT

Studying **the** nature and origin of **the** intergalactic magnetic field (IGMF) is

an outstanding problem of cosmology. Measuring **Faraday** rotation would be a

promising method **to** explore **the** IGMF in **the** large-scale structure (LSS) of **the**

universe. We investigated **the** **Faraday** rotation measure (RM) **due** **to** **the** IGMF

in filaments of galaxies using simulations for cosmological structure formation.

We employed a model IGMF based on turbulence dynamo in **the** LSS of **the**

universe; it has an average strength of ⟨B⟩ ∼ 10 nG and a coherence length of

several × 100 h −1 kpc in filaments. With **the** coherence length smaller than path

length, **the** inducement of RM would be a random walk process, and we found

that **the** resultant RM is dominantly contributed by **the** density peak along line

of sight. The rms of RM through filaments at **the** present universe was predicted

**to** be ∼ 1 rad m −2 . In addition, we predicted that **the** probability distribution

function of |RM| through filaments follows **the** log-normal distribution, and **the**

power spectrum of RM in **the** local universe peaks at a scale of ∼ 1 h −1 Mpc.

Our prediction of RM could be tested with future instruments.

Subject headings: intergalactic medium — large-scale structure of universe —

magnetic fields — polarization

3 Author **to** whom any correspondence should be addressed.

– 2 –

1. Introduction

The intergalactic medium (IGM) contains gas, which was heated mostly by cosmological

shocks (Ryu et al. 2003), along with dark matter; **the** hot gas with T > 10 7 K is found inside

and around clusters/groups of galaxies and **the** warm-hot intergalactic medium (WHIM)

with T = 10 5 − 10 7 K resides mostly in filaments of galaxies, while lower temperature gas

is distributed mostly as sheetlike structures or in voids (Cen & Ostriker 1999; Kang et al.

2005). As **the** gas in **the** interstellar medium, **the** gas in **the** intracluster medium (ICM) and

filaments is expected **to** be permeated with magnetic fields. Measuring **Faraday** rotation,

**the** rotation of **the** plane of linearly-polarized light **due** **to** **the** birefringence of magne**to**-ionic

medium, has been one of a few methods **to** explore **the** intergalactic magnetic field (IGMF).

Observational exploration of **the** IGMF using **Faraday** rotation measure (RM) was

started with **the** investigation of **the** intracluster magnetic field (ICMF) (see Carilli & Taylor

2002, for a review). An RM study of **the** Coma cluster, for instance, revealed **the** ICMF of

**the** strength of order ∼ µG for **the** coherent length of order ∼ 10 kpc (Kim et al. 1990).

For Abell clusters, **the** RM of typically ∼ 100 − 200 rad m −2 was observed, indicating an

average strength of **the** ICMF **to** be ∼ 5–10 µG (Clarke et al. 2001; Clarke 2004). RM maps

of clusters were analyzed **to** study **the** power spectrum of turbulent magnetic fields in **the**

ICM; for instance, a Kolmogorov-like spectrum with a bending at a few kpc scale was found

in **the** cooled core region of **the** Hydra cluster (Vogt & Enßlin 2005), and spectra consistent

with **the** Kolmogorov spectrum were reported in **the** wider ICM for **the** Abell 2382 cluster

(Guidetti et al. 2008) and for **the** Coma cluster (Bonafede et al. 2010).

The nature of **the** IGMF in filaments, on **the** contrary, remains largely unknown, because

**the** study of RM outside clusters is still scarce (e.g., Xu et al. 2006); detecting **the** RM

**due** **to** **the** IGMF in filaments is difficult with current facilities, and also removing **the**

galactic foreground is not a trivial task. The next generation radio interferometers including

**the** Square Kilometer Array (SKA), and upcoming SKA pathfinders, **the** Australian SKA

Pathfinder (ASKAP) and **the** South African Karoo Array Telescope (MeerKAT), as well

**the** Low Frequency Array (LOFAR), however, are expected **to** be used **to** study **the** RM.

Particularly, **the** SKA could measure RM for ∼ 10 8 polarized extragalactic sources across

**the** sky with an average spacing of ∼ 60 arcsec between lines of sight (LOS’s) (see, e.g., Carilli

& Rawlings 2004; Krause et al. 2009, and references **the**rein), enabling us **to** investigate **the**

IGMF in **the** large-scale structure (LSS) of **the** universe.

Attempts **to** **the**oretically predict **the** RM **due** **to** **the** IGMF have been made: for instance,

Ryu et al. (1998) and Dolag et al. (2005) used hydrodynamic simulations for cosmological

structure formation **to** study RM in **the** LSS, and more recently Dubios & Teyssier

(2008) used MHD simulations **to** study RM for clusters. However, **the** properties of **the**

– 3 –

IGMF, especially in filaments, such as **the** strength and coherence length as well as **the** spatial

distribution, are largely unknown, hindering **the** **the**oretical study of RM in **the** LSS of

**the** universe.

Recently, Ryu et al. (2008) proposed a physically motivated model for **the** IGMF, in

which a part of **the** gravitational energy released during structure formation is transferred **to**

**the** magnetic field energy as a result of **the** turbulent dynamo amplification of weak seed fields

in **the** LSS of **the** universe. In **the** model, **the** IGMF follows largely **the** matter distribution

in **the** cosmic web and **the** strength is predicted **to** be ⟨B⟩ ∼ 10 nG in filaments. Cho & Ryu

(2009) studied various characteristic length scales of magnetic fields in turbulence with very

weak or zero mean magnetic field, and showed that **the** coherence length defined for RM is

3/4 times **the** integral scale in **the** incompressible limit. They predicted that in filaments,

**the** coherence length for RM would be a few × 100 h −1 kpc with **the** IGMF of Ryu et al.

(2008) and **the** RM **due** **to** **the** magnetic field would be of order ∼ 1 rad m −2 .

In this paper, we study RM in **the** LSS of **the** universe, focusing on RM through filaments,

using simulations for cosmological structure formation along with **the** model IGMF

of Ryu et al. (2008) and Cho & Ryu (2009). Specifically, we present **the** spatial distribution,

probability distribution function (PDF) and power spectrum of **the** RM, and discuss **the**

prospect of possible observations of **the** RM. In sections 2 and 3, we describe our model and

**the** results. Discussion is in Section 4, and Summary and Conclusion follows in Section 4.

2. Model

To investigate RM in **the** LSS of **the** universe, we used structure formation simulations

for a concordance ΛCDM universe with **the** following values of cosmological parameters:

ΩBM = 0.043, ΩDM = 0.227, ΩΛ = 0.73, h ≡ H0/(100 km s −1 Mpc −1 ) = 0.7, n = 1, and

σ8 = 0.8 (same as in Ryu et al. 2008). They were performed using a particle-mesh/Eulerian,

cosmological hydrodynamic code (Ryu et al. 1993). A cubic region of comoving volume

(100 h −1 Mpc) 3 was reproduced with 512 3 uniform grid zones for gas and gravity and 256 3

particles for dark matter, so **the** spatial resolution is 195 h −1 kpc. Sixteen simulations with

different realizations of initial condition were used **to** compensate cosmic variance.

For **the** IGMF, we employed **the** model of Ryu et al. (2008); it proposes that turbulentflow

motions are induced via **the** cascade of **the** vorticity generated at cosmological shocks

during **the** formation of **the** LSS of **the** universe, and **the** IGMF is produced as a consequence

of **the** amplification of weak seed fields of any origin by **the** turbulence. Then, **the** energy

density (or **the** strength) of **the** IGMF can be estimated with **the** eddy turnover number and

– 4 –

**the** turbulent energy density as follow:

(

t

εB = ϕ

teddy

)

εturb. (1)

Here, **the** eddy turnover time is defined as **the** reciprocal of **the** vorticity at driving scales,

teddy ≡ 1/ωdriving (⃗ω ≡ ⃗ ∇ × ⃗v), and ϕ is **the** conversion fac**to**r from turbulent **to** magnetic

energy that depends on **the** eddy turnover number t/teddy. The eddy turnover number

was estimated as **the** age of universe times **the** magnitude of **the** local vorticity, that is,

tage ω. The local vorticity and turbulent energy density were calculated from simulations for

cosmological structure formation described above. A functional form for **the** conversion fac**to**r

was derived from a separate, incompressible, magne**to**hydrodynamic (MHD) simulation of

turbulence dynamo. For **the** direction of **the** IGMF, we used that of **the** passive fields from

simulations for cosmological structure formation, in which weak seed fields were evolved

passively, ignoring **the** back-reaction, along with flow motions (Kulsrud et al. 1997; Ryu et

al. 1998).

In our model, as seed magnetic fields, we **to**ok **the** ones generated through **the** Biermann

battery mechanism (Biermann 1950) at cosmological shocks. There are, on **the** o**the**r hand,

a number of mechanisms that have been suggested **to** create seed fields in **the** early universe.

Besides various inflationary and string **the**ory mechanisms, **the** followings include a partial

list of astrophysical mechanisms. At cosmological shocks, in addition, Weibel instability can

operate and produce magnetic fields (Medvedev et al. 2006; Schlickeiser & Shukla 2003),

and streaming cosmic rays accelerated by **the** shocks can amplify weak upstream magnetic

fields via non-resonant growing mode (Bell 2004). In addition, for instance, galactic outflows

during **the** starburst phase of galactic evolution (Donnert et al. 2009) and **the** return current

induced by cosmic-rays produced by Supernovae of first stars (Miniati & Bell 2010) were

suggested **to** deposit seed fields. We point, however, that in our model **the** IGMF resulting

from turbulent amplification should be insensitive **to** **the** origin of seed fields.

The spatial distribution of **the** strength of **the** resulting IGMF is shown in Figure 4

of Ryu et al. (2008) and Figure 1 of Ryu et al. (2010). It is very well correlated with

**the** distribution of matter. The average strength of our model IGMF for **the** WHIM with

10 5 < T < 10 7 K in filaments is ⟨B⟩ ∼ 10 nG, ⟨B 2 ⟩ 1/2 ∼ a few × 10 nG, ⟨ρB⟩/⟨ρ⟩ ∼ 0.1 µG,

or ⟨(ρB) 2 ⟩ 1/2 /⟨ρ 2 ⟩ 1/2 ∼ a few × 0.1 µG.

3. Results

We calculated RM, defined as ∆χ/∆λ 2 (χ is **the** rotation angle of linearly-polarized

light at wavelength λ), in **the** local universe with z = 0 along a path length of L = 100 h −1

– 5 –

Mpc, which is **the** box size of structure formation simulations. Figure 1 shows **the** resulting

RM map of (28 h −1 Mpc) 2 area in logarithmic and linear scales. RM traces **the** large-scale

distribution of matter, and we see two clusters and a filamentary structure containing several

groups. Through **the** clusters, groups, and filament in **the** field, RM is roughly ∼ 100, ∼ 10,

and ∼ 1, respectively, while RM through sheets and voids is much less. The bot**to**m panel

of Figure 1 shows **the** mixture of positive and negative RM, reflecting **the** randomness of

magnetic fields in **the** LSS.

With **the** coherence length of magnetic fields for RM (see Discussion) expected **to** be

smaller than **the** path length which should be a cosmological scale, **the** inducement of RM is

expected be a random walk process. Figure 2 shows **the** distributions of RM as well as o**the**r

quantities along a few LOS’s through filaments; it confirms that **the** inducement of RM is

indeed a random walk process. However, we note that **the** resulting RM is dominated by **the**

contribution from **the** density peak along LOS’s.

To quantify RM in **the** LSS of **the** universe, we calculated **the** probability distribution

function (PDF) of |RM| for 5122 × 3 × 16 (projected grid zones × directions × runs) LOS’s.

Figure 3 shows **the** resulting PDF through **the** LOS’s of different ranges of **the** mean temperature

weighted with X-ray emissivity, TX. The figure also shows **the** fitting **to** **the** log-normal

distribution,

1

PDF(log10 |RM|) = √

2πσ2 exp

[

− (log10 |RM| − µ) 2

2σ2 ]

, (2)

finding that **the** PDF closely follows **the** log-normal distribution. We also calculated **the**

root mean square (rms) of RM, ⟨RM⟩rms; note that **the** mean of RM, ⟨RM⟩, is zero for our

IGMF. Through **the** WHIM, which mostly composes filaments, ⟨RM⟩rms = 1.41 rad m−2 .

This agrees well with **the** value predicted with **the** mean strength and coherence length of

**the** IGMF in filaments by Cho & Ryu (2009). However, this is an order of magnitude smaller

than **the** values of |RM| **to**ward **the** Hercules and Perseus-Pisces superclusters reported in

Xu et al. (2006). The difference is mostly **due** **to** **the** mass-weighted path length; **the** value

quoted by Xu et al. (2006) is about two orders of magnitude larger than ours. Through

**the** hot gas with 107 < T < 108 K, ⟨RM⟩rms = 108 rad m−2 , which is in good agreement

with RM observations of galaxy clusters (Clarke et al. 2001; Clarke 2004). Through **the** hot

gas, however, we found RM of up **to** � 1000 rad m−2 . This should be an artifact of limited

resolution (see Discussion). So **the** values for **the** hot gas in our work should not be taken

seriously.

Finally, we calculated **the** two-dimensional power spectrum of RM on 3×16 (directions×

runs) projected planes; PRM(k) ∼ |RM( ⃗ k)| 2 k, where RM( ⃗ k) is **the** Fourier transform of

RM(⃗x) on planes. Figure 4 shows **the** resulting power spectrum along with **the** power spectra

of electron density, magnetic fields, and **the** curl component of flow motions, ⃗vcurl, which

– 6 –

satisfies **the** relation ⃗ ∇ × ⃗vcurl ≡ ⃗ ∇ × ⃗v. The power spectrum of RM peaks at k ∼ 100,

which corresponds **to** ∼ 1 h −1 Mpc. Cosmic variance is not significant around **the** peak,

although it is larger at smaller k, as expected. The power spectrum of RM reflects **the**

spatial distributions of electron density, ne, and LOS magnetic field, B∥. The power spectra of

projected ne and projected B∥, have peaks at ∼ 3 h −1 Mpc and ∼ 1.5 h −1 Mpc, respectively.

The shape of **the** power spectrum of RM follows that of projected B∥ ra**the**r than that of

projected ne, implying that **the** statistics of RM would primarily carry **the** statistics of **the**

IGMF.

4. Discussion

Our results depend of RM on **the** strength and coherence length of **the** IGMF. We

employed a model where **the** strength of **the** local IGMF was estimated based on turbulence

dynamo, while **the** direction was gripped from structure formation simulations with passive

fields (see Section 2). In principle, if we had performed full MHD simulations, we could have

followed **the** amplification of **the** IGMF through turbulence dynamo along with its direction.

In practice, however, **the** currently available computational resources do not allow a numerical

resolution high enough **to** reproduce **the** full development of MHD turbulence. Since **the**

numerical resistivity is larger than **the** physical resistivity by many orders of magnitude, **the**

growth of magnetic fields is expected **to** be saturated before dynamo action becomes fully

operative (see, e.g., Kulsrud et al. 1997). In such situation, **the** state of magnetic fields in

full MHD, including, for instance, **the** power spectrum, is expected **to** mimic that of passive

fields. This is **the** reason why we adopted **the** model of Ryu et al. (2008) **to** estimate **the**

strength of **the** IGMF, but we still used passive fields from structure formation simulations

**to** model **the** field direction.

The validity of our model IGMF was checked as follows:

1) In MHD turbulence, **the** distribution of magnetic fields, including **the** direction, is expected

**to** correlate with that of vorticity, since magnetic fields and vorticity are described by

similar equations except **the** baroclinity term in **the** equation for vorticity (if dissipative processes

are ignored) (see, e.g., Kulsrud et al. 1997). Such a correlation can be clearly seen in

Figure 5, in which we depicts **the** distributions of our IGMF and vorticity in two-dimensional

slices.

2) Full MHD turbulence simulations suggest that **the** peak of magnetic field spectrum occurs

∼ 1/2 of **the** energy injection scale, or **the** peak scale of velocity power spectrum, at saturation;

in **the** linear growth stage, **the** peak scale of magnetic field spectrum grows as ∼ t 1.5

or so (Cho & Ryu 2009). With our model IGMF, **the** peak scale of magnetic field spectrum

– 7 –

is ∼ 1 h −1 Mpc (**the** third panel of Figure 4); on **the** o**the**r hand, **the** curl component of

flow motions has **the** peak of power spectrum at ∼ 4 h −1 Mpc (**the** bot**to**m panel of Figure

4). That is, **the** peak scale of magnetic field spectrum is ∼ 1/4 of **the** energy injection scale

in our model IGMF. By considering **the** turbulence in **the** LSS of **the** universe has not yet

reached **the** fully saturated stage (see, e.g., Ryu et al. 2008), **the** ratio of **the** two scales seems

**to** be feasible.

These suggest that our model IGMF would produce reasonable results, although eventually

it needs **to** be replaced with that from full MHD simulations for cosmological structure

formation when computational resources allow such simulations in future.

Apart from our model for **the** IGMF, **the** finite numerical resolution of simulations

could affect our results. The average strength of our model IGMF is ⟨B⟩ ∼ a few µG in

clusters/groups, ∼ 0.1µG around clusters/groups, and ∼ 10 nG in filaments. Ryu et al.

(2008) tested **the** numerical convergence of **the** estimation. With simulations of different

numerical resolutions for cosmological structure formation, it was shown that ⟨B⟩ of our

model IGMF for **the** WHIM with 10 5 < T < 10 7 K would approach **the** convergence value

within a fac**to**r ∼ 2 − 3 at **the** resolution of 512 3 grids (see Figure S5 of SOM of Ryu et al.

(2008)).

It is, on **the** o**the**r hand, ra**the**r tricky **to** assess **the** effect of finite resolution on **the**

coherence length of our model IGMF, because **the** definition of coherence length for RM is

not completely clear and **the** estimation of coherence length, for instance, for **the** filament

IGMF alone is not trivial. We tried **to** quantify coherence length in **the** following three ways:

1) We directly calculated **the** coherence length of B∥, that is, **the** length with **the** same sign

of B∥, along LOS’s. Figure 6 shows **the** PDF of **the** resulting coherence length through **the**

WHIM, which composes mostly filaments. It peaks at **the** length of 3 zones corresponding

**to** 586 h−1 kpc. 2) We calculated 3/4 times **the** integral scale,

∫

3D

3 PB (k)/k dk

× 2π ∫ , (3)

3D

4 PB (k) dk

which is **the** coherence length defined for RM in **the** incompressible limit (see Introduction),

for **the** IGMF inside **the** whole computational box of (100 h −1 Mpc) 3 volume. Here, P 3D

B (k)

is **the** three-dimensional power spectrum of magnetic fields (**the** third panel of Figure 4).

We found **the** value **to** be ∼ 800 h −1 kpc for our model IGMF. 3) We also calculated **the**

largest energy containing scale in **the** whole computational box, which is **the** peak scale of

kP 3D

B (k) (not shown). It is ∼ 900 h−1 kpc for our model IGMF. Note that **the** latter two

values include contributions from **the** IGMF in filaments as well as in clusters, sheets, and

voids. All **the** three scales are comparable. These length scales are ∼ 3 **to** 5 times larger

than **the** grid resolution of our simulations, 195 h −1 kpc.

– 8 –

Cho & Ryu (2009) studied characteristic lengths in incompressible simulations of MHD

turbulence (see Introduction); based on it, **the**y predicted that **the** coherence length for RM

would be a few × 100 h −1 kpc in filaments, while a few × 10 h −1 kpc in clusters. With our

grid resolution of 195 h −1 kpc, **the** coherence length of **the** IGMF in clusters should not be

resolved and so our estimation of RM for clusters should be resolution-affected, as pointed

in Section 3. On **the** o**the**r hand, while **the** predicted coherence length for **the** IGMF in

filaments is still larger than **the** grid resolution, **the** estimated coherence length of B∥ for **the**

WHIM is a couple of times larger than **the** prediction for filaments. It could be partly **due** **to**

**the** limited resolution in our simulations. However, as noted in Section 3, RM is dominantly

contributed by **the** density peak along LOS’s (Figure 2).

The above statements indicate that our estimate of **the** RM through filaments is expected

**to** have a uncertainty, especially **due** **to** **the** limited resolution of our simulations; it could be

up **to** a fac**to**r of several.

5. Summary and Conclusion

We studied RM in **the** LSS of **the** universe, focusing on RM through filaments; simulations

for cosmological structure formation were used and **the** model IGMF of Ryu et al.

(2008) and Cho & Ryu (2009) based on turbulence dynamo was employed. Our findings are

summarized as follows. 1) With our model IGMF, **the** rms of RM through filaments at **the**

present universe is ∼ 1 rad m −2 . 2) The PDF of |RM| through filaments follows **the** lognormal

distribution. 3) The power spectrum of RM **due** **to** **the** IGMF in **the** local universe

peaks at a scale of ∼ 1 h −1 Mpc. 4) Within **the** frame of our mode IGMF, we expect that

**the** uncertainty in our estimation for **the** rms of RM through filaments, **due** **to** **the** finite

numerical resolution of simulations, would be a fac**to**r of a few.

We note that our model does not include o**the**r possible contributions **to** **the** IGMF, for

instance, that from galactic black holes (AGN feedbacks) (see, e.g., Kronberg et al. 2001).

So our model may be regarded as a minimal model, providing a baseline for **the** IGMF.

With such contributions, **the** real IGMF might be somewhat stronger, resulting in somewhat

larger RM.

It has been suggested that future radio observa**to**ries such as LOFAR, ASKAP, MeerKAT

and SKA could detect **the** extragalactic RM of ∼ 1 rad m −2 we predict (see, e.g., Beck 2009).

However, it is known that **the** typical galactic foreground of RM is a few tens and of order

ten rad m −2 in **the** low and high galactic latitudes, respectively (see, e.g., Simard-Normandin

& Kronberg 1980). So **the** detection of **the** extragalactic RM of ∼ 1 rad m −2 or so could be

– 9 –

possible only after **the** galactic foreground is removed. We note, however, that with filaments

at cosmological distance, **the** peak of **the** power spectrum of **the** RM **due** **to** **the** IGMF in

filaments would occur at small angular scales; for instance, for a filament at a distance of

100 h −1 Mpc, **the** peak would occur at ∼ 0.5 degree or so. This is much smaller than **the**

expected angular scale of **the** peak of **the** galactic foreground, which would be around tens

degree (see, e.g., Frick et al. 2001). Then, it would be plausible **to** extract **the** signature of

**the** RM of ∼ 1 rad m −2 **due** **to** **the** IGMF in filaments. We leave this issue and connecting

our **the**oretical prediction of RM in **the** LSS of **the** universe **to** observation for future studies.

TA was supported in part by National Research Foundation of Korea (R01-2007-000-

20196-0). DR was supported in part by National Research Foundation of Korea (K20901001400-

09B1300-03210)

Bell, A. R. 2004, MNRAS, 353, 550

Biermann, L. 1950, Z. Naturforsch, A, 5, 65

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This preprint was prepared with **the** AAS L ATEX macros v5.2.

B|| [μG] ne [cm-3]

local RM [rad m-2]

10 −4

10 −5

10 −6

10 −7

10−3 1

0.1

0.01

10−5 10−4 10−3 10

1

0.1

0.01

10 −4

RM= 2.91

10 15

[h−1 0 5

Mpc]

RM= 0.886

– 12 –

10 15

[h−1 0 5

Mpc]

RM= 0.316

10 15

[h−1 0 5

Mpc]

Fig. 2.— Profiles of local RM, LOS magnetic field, B∥, electron density, ne, and gas temperature,

T , along a few LOS’s through filaments. The arrows indicate **the** sign of local RM

and B∥.

ne

Tgas

10 7

10 6

10 5

10 4

1

0.1

10−3 0.01

10−5 10−4 10−3 10

1

0.1

0.01

10 −4

Tgas [K] B|| [μG] local RM [rad m-2]

PDF

PDF

0.6

0.4

0.2

0.6

0.4

0.2

10 5 K < TX < 10 7 K

rms=1.41

μ=-2.47

σ=1.33

10 6 K < TX < 10 7 K

rms=2.77

μ=-0.98

σ=0.94

−10 −5 0

log |RM| [rad m−2 ]

– 13 –

10 5 K < T X < 10 6 K

rms=0.034

μ=-2.84

σ=1.05

10 7 K < T X < 10 8 K

rms=108.

μ=1.02

σ2=0.89

−10 −5 0

log |RM| [rad m−2 ]

Fig. 3.— PDF of |RM| through LOS’s of different ranges of **the** mean temperature weighted

with X-ray emissivity, TX. Thin lines, thick lines (red or black), and thick lines (blue or

gray) show **the** PDFs from 16 independent runs, **the**ir average, and **the** best-fit **to** **the** lognormal

distribution, respectively. The values of fitting parameters and **the** rms of RM are

also shown.

PDF

0.4

0.2

0

– 16 –

0 1 2 3 4

coherent scale [h−1 5 6

Mpc]

Fig. 6.— PDF of **the** coherence length of B∥ along LOS’s (**the** length with **the** same sign of

B∥) through **the** WHIM. Thin and thick lines show **the** PDFs for 16 independent runs and

**the**ir average, respectively.