FARADAY ROTATION MEASURE DUE TO THE INTERGALACTIC MAGNETIC FIELD. II. THE COSMOLOGICAL CONTRIBUTION

For the LSS of the universe, we used structure formation simulations of a ΛCDM universe with the following values of cosmological parameters: Ω_b0 = 0.043, Ω_m0 = 0.27, Ω_Λ0 = 0.73, h ≡ H₀/(100 km s⁻¹ Mpc⁻¹) = 0.7, n = 1, and σ₈ = 0.8. The simulations were performed using a particle-mesh/Eulerian cosmological hydrodynamic code (Ryu et al. 1993). A cubic region of comoving volume (100 h⁻¹Mpc)³ was reproduced with 512³ uniform grid zones for gas and gravity and 256³ particles for dark matter, so the spatial resolution is 195 h⁻¹ kpc. Sixteen simulations with different realizations of the initial condition were used to compensate cosmic variance. They are the same set of simulations used in AR10.

As in AR10, we employed the model described in R08. It assumes that turbulent-flow motions are induced via the cascade of the vorticity generated due to cosmological shocks during the formation of LSS, and the IGMF is produced as a consequence of the amplification of weak seed fields of any origin through the stretching of field lines by the flow motions. Then, it can be modeled that a fraction of the turbulent-flow energy, ε_turb, is converted to the magnetic energy, ε_B, as

$$\begin{equation} \varepsilon _B = \phi \left(\frac{t}{t_{\rm eddy}}\right) \varepsilon _{\rm turb}, \end{equation} \tag{ 1 }$$

where the conversion factor, ϕ, depends only on the eddy turnover number, t/t_eddy. Here, the eddy turnover time is defined as the reciprocal of the vorticity at driving scales, t_eddy ≡ 1/ω_driving (${\vec{\omega }} \equiv {\vec{\nabla }}\times {\vec{v}}$). The local vorticity and turbulent-flow energy density were calculated from the data of the structure formation simulations described above, and the age of the universe at the redshift z was used for t. The functional form for the conversion factor was derived from a separate, incompressible, MHD simulation of a turbulence dynamo (R08). Then, the magnetic energy density was calculated according to Equation (1), and the strength of the IGMF is $B = \sqrt{8\pi \varepsilon _B}$. The average strength of the resulting model IGMF for the WHIM is 〈B〉 ∼ 10 nG or 〈ρB〉/〈ρ〉 ∼ 0.1 μG at z = 0. For the direction of the IGMF, we used that of the passive fields from structure formation simulations, in which weak seed fields were generated through the Biermann battery mechanism (Biermann 1950) at cosmological shocks and evolved passively, ignoring the back-reaction, along with flow motions (Kulsrud et al. 1997; Ryu et al. 1998).

The validity of our model IGMF was discussed in details in AR10. It was argued that our model IGMF would produce reasonable results for the RM through the cosmic web in the local universe. The simulated passive fields reproduce the directions that show the expected correlation with those of vorticity. The RM coherence length for the IGMF in filaments is estimated to be a few to several × 100 kpc at z = 0, which agrees with the estimation of CR09. It is a few times larger than the grid resolution of our simulations, 195 h⁻¹ kpc. On the other hand, the coherence length for the IGMF in clusters is expected to be a few × 10 kpc (CR09), which is smaller than the grid resolution. So the application of our model IGMF to clusters would results in an erroneous estimation of RM. However, here we study the RM through the cosmic web outside clusters, and so we will remove the contribution from clusters to RM (see Section 2.8).

Our model predicts that the IGMF for the WHIM was a bit stronger in the past; for instance, 〈B〉 ∼ 30 nG for the gas with 10⁵ K <T < 10⁷ K at z = 5 (see Figure 2 of R08). This is because the density of the WHIM was higher in the past, although the magnitudes of vorticity and the vortical component of velocity were smaller. Our model also predicts stronger IGMF for the hot gas with T > 10⁷ K in the past. However, the IGMF averaged over the entire computational volume was weaker in the past, because the volume and mass fractions of strong-field regions were smaller.

To reproduce the cosmic space, we stacked simulation boxes up to z = 5, basically following the conventional manner of cosmological data stacking (e.g., da Silva et al. 2000). We took outputs at z_out = 0, 0.2, 0.5, 1.0, 1.5, 2.0, 3.0, and 5.0, and put the outputs of the closest redshift for stacking boxes; 56 boxes were required to reach z = 5. The stacking boxes were randomly selected from 16 simulations and randomly rotated to avoid any artificial coherent structure along the LOS. If an LOS went out of the boundary of a box, we applied the periodic boundary condition in which we replicated the box across the LOS. We also carried out the calculation with the open boundary condition in which we did not replicate the box but put different one from 16 simulation outputs, and confirmed that the statistical properties of RM we examined are not sensitive to the boundary conditions.

Our Galaxy is located in the Local Group. Since the Local Group is probably filled with the magnetized WHIM, the observed RM could contain a contribution from the IGMF of the Local Group (see Section 3.1). It would have been ideal to place the "observer" where the IGMF and gas density are similar to those in the Local Group. Unfortunately, however, not much is known about the physical states of the Local Group. Hence, following Das et al. (2008), instead, we selected groups of galaxies identified in the simulation data that have similar halo gas temperatures to the Local Group. We considered two cases: groups at z = 0 with 0.05 keV < kT_X < 0.15 keV and groups at z = 0 with 0.05 keV < kT_X < 0.5 keV (Rasmussen & Pedersen 2001). Here, T_X is the X-ray temperature (see Section 2.7). Assuming that these groups are located in a physical environment similar to that of the Local Group, we placed "mock observers" at the center of the groups.

We aim to simulate wide-field, high-sensitivity RM surveys that will be achieved with future radio interferometers. For instance, the upcoming POSSUM (Polarization Sky Survey of the Universe's Magnetism) project, one of the major surveys planned for the ASKAP, will produce an RM map that will be a dramatic improvement over existing maps. The RM map from the POSSUM will have a field of view (FOV) of θ² ≃ 30 deg² and an average source separation of 〈Δθ〉 ≃ 01 (∼6 arcmin) corresponding to an RM grid of ∼100 RM deg⁻² (see, e.g., Gaensler et al. 2010). The SKA will survey ∼10⁷ polarized radio sources over the whole sky with 〈Δθ〉 ≃ 1 arcmin (see, e.g., Beck 2009). Intending to produce a simulated map of RM, we adopted θ² = 200 deg² = (14.14)² deg² and $\Delta \theta = \sqrt{200\ {\rm deg^2 / 2048^2}} = 0.414$ arcmin with one source in each of 2048² equally spaced pixels. Specific simulations for the SKA survey and the ASKAP POSSUM that consider the feasibility of observations such as the effects of the measurement error, coarser spacing, randomly placed sources, intrinsic RM, and so on, as well as the galactic foreground will be discussed in separate papers.

For the redshift distribution of radio sources, we employed a distribution based on the estimation of the detectable radio galaxies by the SKA and ASKAP (Willman et al. 2008, see also Figure 1), in which the observed FR I and FR II galaxies are taken into consideration. The number of available sources drops substantially at z ≳ 5, and that is the reason why the data stacking was made up to z = 5 (see Section 2.3).

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RM is a measure of the Faraday rotation of polarized radio emission against a background source, defined by

$$\begin{equation} {\rm RM}\equiv \frac{\psi }{\lambda _{\rm obs}^2} = \frac{e^3}{2\pi m_{\rm e}^2 c^4} \int _{0}^{l_{\rm s}} n_{\rm e}B_\parallel \frac{\lambda (l)^2}{\lambda _{\rm obs}^2}dl, \end{equation} \tag{ 2 }$$

where ψ is the rotated angle of the plane of polarized radio emission, l_s is the path length up to the source, n_e is the thermal electron density, B_∥ is the LOS component of the magnetic field, λ_obs is the observed wavelength, and λ(l) is the wavelength along the path length of polarized radio emission (e.g., Rybicki & Lightman 1979). For a source at cosmological distance of redshift z_s, it is modified to

$$\begin{equation} {\rm RM}= \frac{e^3}{2\pi m_{\rm e}^2 c^4} \int _{0}^{l_{\rm s}(z_{\rm s})} (1+z)^{-2} n_{\rm e}(z) B_\parallel (z) dl(z), \end{equation} \tag{ 3 }$$

where n_e(z), B_∥(z), and l(z) are the quantities in proper coordinates. The path length is given as

$$\begin{equation} dl(z) = \frac{c\ dz}{H_0(1+z)\sqrt{\Omega _{\rm m0}(1+z)^3+\Omega _{\rm \Lambda 0}}}, \end{equation} \tag{ 4 }$$

for the flat universe with Ω_m0 + Ω_Λ0 = 1 (e.g., Peebles 1993).

We calculated Equation (3) by counting the contribution from computational grid zones along stacked boxes up to mock sources as

$$\begin{eqnarray} {\rm RM} ({\rm rad\, m^{-2}}) &=& 8.12\times 10^5 \sum _{i=1}^{N_s(z_s)} (1+z_i)^{-2} \cdot n_{\rm e}(z_i)\nonumber\\ && \cdot B_\parallel (z_i) \cdot \Delta l (z_i), \end{eqnarray} \tag{ 5 }$$

where N_s(z_s) is the number of grid zones up to sources at z_s. The density and magnetic field strength are in units of cm⁻³ and μG, respectively. For Δl(z), we used the proper size of grid zone, which is 0.195 h⁻¹(1 + z)⁻¹ Mpc. With the simulation FOV of θ² = (14.14)² deg², the maximum tilt of LOSs relative to the coordinates of simulation box is cos (θ/2) = 0.99, so we ignored the effect on Δl(z).

In the hot gas with T ≳ 10⁷ K, X-ray emission is produced mainly by thermal bremsstrahlung. We hence considered only the bremsstrahlung emission from electrons and neglected line emissions from ions. The emissivity of thermal bremsstrahlung can be expressed as

$$\begin{eqnarray} \varepsilon ^{\rm ff} ({\rm erg \,s^{-1}\, cm^{-3}}) &=&7.77\times 10^{-38}T^{-1/2} n_e^2 \cdot \int _{\nu _1}^{\nu _2} \bar{g}(T,\nu)\nonumber\\ &&\times\, \exp \left(-\frac{h_{\rm p}\nu }{k_{\rm B}T}\right) d\nu, \end{eqnarray} \tag{ 6 }$$

where T, n_e, and ν are in units of K, cm⁻³, and Hz, respectively, and h_p and k_B are the Planck and Boltzmann constants, respectively (Rybicki & Lightman 1979). We adopted the approximate Gaunt factor, $\bar{g} \approx 0.9(h_{\rm p}\nu /k_{\rm B}T)^{-0.3}$, and used the bolometric emissivity, that is, ν₁ = 0 and ν₂ = ∞.

For the X-ray temperature of clusters and groups, we calculated the X-ray emissivity-weighted temperature as

$$\begin{equation} T_X=\int \varepsilon ^{\rm ff} T dV\ \Bigg /\ \int \varepsilon ^{\rm ff} dV, \end{equation} \tag{ 7 }$$

over a spherical volume of comoving radius 0.5 h⁻¹ Mpc.

For the X-ray surface brightness and surface temperature up to redshift z, we first sought all the grid zones, k, and their proper volume, V_k, which enters within the angular beam of (Δθ)². Note that Δθ = 0.414 arcmin corresponds to the comoving size of a grid zone 195 h⁻¹ kpc at z ≃ 0.6; hence perpendicular to an LOS, less than one zone enters at lower redshifts, but more than one at higher redshifts within (Δθ)². For each grid at redshift z_k, the X-ray luminosity was calculated as

$$\begin{equation} L_k = \int _{V_k} \varepsilon ^{\rm ff} dV. \end{equation} \tag{ 8 }$$

Then, the X-ray surface brightness was calculated as

$$\begin{equation} S_X^* = \frac{1}{(\Delta \theta)^2} \sum _k^{{\rm up\ to\ }z} \frac{L_k}{4\pi d_c(z_k)^2(1+z_k)^2}, \end{equation} \tag{ 9 }$$

and the X-ray emissivity-weighted surface temperature was calculated as

$$\begin{equation} T_X^* = \frac{1}{S_X^*(\Delta \theta)^2} \sum _k^{{\rm up\ to\ }z} \frac{T_{Xk} L_k}{4\pi d_c(z_k)^2(1+z_k)^2}, \end{equation} \tag{ 10 }$$

where T_Xk is the temperature of grid zone k. Here, d_c(z_k) is the comoving distance up to z_k, which is given as

$$\begin{equation} d_c(z_k) = \frac{c}{H_0} \int _0^{z_k} \frac{dz}{\sqrt{\Omega _{\rm m0}(1+z)^3+\Omega _{\rm \Lambda 0}}}. \end{equation} \tag{ 11 }$$

The contribution from clusters was subtracted from calculated RM based on different criteria using X-ray brightness and temperature. The simplest approach would be to exclude the volume associated with clusters in the cosmic space. We employed the following two criteria. In the first model, labeled "TM7," all the hot gas with T > 10⁷ K was excluded. In the second model, labeled "CLS," the spherical region of comoving radius 1 h⁻¹ Mpc around the clusters and groups with T_X > 2 keV was excluded.

For simulations of RM surveys, however, a better approach would be the exclusion of the pixels that include clusters in the projected sky map. We would like to subtract the contribution from all clusters, but it is still hard to detect faint X-ray clusters. So we tried the following criteria. In the model labeled "TS8," all the pixels with T*_X > 10⁷ K and S*_X > 10⁻⁸ erg s⁻¹ cm⁻² sr⁻¹ were excluded; 10⁻⁸ erg s⁻¹ cm⁻² sr⁻¹ is close to the detection limit of current X-ray facilities (e.g., Hoshino et al. 2010). And in another model labeled "TS0," the pixels with T*_X > 10⁷ K and S*_X > 10⁻¹⁰ erg s⁻¹ cm⁻² sr⁻¹ were excluded; 10⁻¹⁰ erg s⁻¹ cm⁻² sr⁻¹ is intended to mimic the improved detection limit of future X-ray facilities. Here, S*_X and T*_X up to z = 5 were used. Finally, we also considered the case where no volume or pixel was excluded, labeled "ALL," for comparison.

We first examine the contribution to RM from the local IGMF around observers. For the two temperature range of groups containing observers, 0.05 keV < kT_X < 0.15 and 0.05 keV < kT_X < 0.5 keV (see Section 2.4), we set up 200 mock observers and calculated RM by integrating along LOSs up to ∼0.5 h⁻¹ Mpc (21/2 grid zones) from observers. Figure 2 plots the PDF of |RM|, the absolute value of resulting RM. The PDF peaks around ∼2 × 10⁻³ rad m⁻² and ∼6 × 10⁻³ rad m⁻² for 0.05 keV < kT_X < 0.15 keV and 0.05 keV < kT_X < 0.5 keV, respectively. The rms values are ∼5 × 10⁻² rad m⁻² and ∼6 × 10⁻¹ rad m⁻², respectively. The peak values can be estimated roughly as follows. Our model IGMF predicts B_peak ∼ 10⁻³ μG for groups (see Das et al. 2008). With n_e ∼ 10⁻⁵–10⁻⁴ cm⁻³ for groups, and the coherence length of magnetic fields comparable to the size of grid zones, i.e., l_coherence ∼ 195 h⁻¹ kpc, we get RM_peak ∼ 10⁻³–10⁻² rad m⁻². This may indicate that the contribution from the local IGMF is affected by the grid resolution. But as we will see in Section 3.3, this contribution makes only a small, negligible fraction of the RM through the cosmic web. Below we show only the results with 0.05 keV < kT_X < 0.15; those with 0.05 keV < kT_X < 0.5 keV are statistically indistinguishable.

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We produced the simulated sky map of RM (using Equation (5)) with different cluster subtraction models, as well as the accompanying map of S*_X (using Equation (9)) and T*_X (using Equation (10)), for 200 different mock observers with different stacks, each of which has 2048² pixels in an FOV of 200 deg², as described in Section 2.5. The statistics below were obtained with the maps. In this section, we show maps for a typical case.

Figure 3 shows RM maps for ALL and cluster subtraction models CLS and TS8, and S*_X and T*_X maps. To see the dependence on the redshift depth, the maps integrated up to five different epochs are presented. With TS8, about ∼15% of pixels were subtracted. At the lowest redshift depth, X-ray bright sources, which can be identified as clusters or groups, are apparent. They have the angular size of up to a couple of degrees and the peak value of |RM| ≳ 100 rad m⁻². With CLS, some, but not all, contribution to RM from those clusters and groups was excluded. With TS8, most contribution from clusters and groups was removed. On the other hand, the contribution from filamentary structures was kept; filamentary structures with |RM| ∼ 0.1 − a few rad m⁻² are seen in CLS and TS8 as well as in ALL. At higher redshift depths, the value of |RM| is larger. This is attributed to the fact that the inducement of RM is a random walk process (AR10); the value increases with the increasing path length. Also at higher redshift depths, the angular size of RM structures is smaller, since the distance is larger and the proper size of filaments is smaller. At the depth of z = 5, most of nearby filamentary structures are smeared out, and the sky distribution of RM appears close to be random.

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We quantified the RM in 2048² pixels of 200 maps, so in total 2048² × 200 pixels, first with the PDF and rms value. Figure 4 shows the PDF of |RM| for three different redshift depths, z = 0.05, 0.3, and 5, for ALL, CLS, and TS8. As for the |RM| of the local universe (AR10), the PDF follows the lognormal distribution. For TS8, the peak value shifts from a few × 10⁻² rad m⁻² at the depth of z = 0.05 to ∼1 rad m⁻² at z = 5. Compared to CLS and TS8, ALL has a high-value tail due to the contribution from clusters and groups.

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Figure 5 shows RM_rms, the rms value of RM, as a function of the redshift depth for ALL and all cluster subtraction models. RM_rms increases steeply for z ≲ 1 and saturates for z ≳ 1. The saturated value of RM_rms is ∼10 rad m⁻² for CLS and ∼7–8 rad m⁻² for TM7, TS8, and TS0, while it is ∼40 rad m⁻² for ALL. Again the high value for ALL is attributed to clusters and groups. So if we observe the RM through the cosmic web outside of clusters, we would get ∼7–8 rad m⁻² for RM_rms.

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We note that the saturated value depends on the redshift distribution of radio sources, but only weakly. In our model the redshift distribution peaks at z ≲ 1 (see Figure 1). We carried out comparison runs, in which we put all radio sources at z = 5. Then, the saturated value of RM_rms increased slightly; specifically, for TM7, the value increased from ∼8 to ∼9 rad m⁻².

The saturation for z ≳ 1 is mostly due to the convergence of the path length at z ∼ 1 and the (1 + z)⁻² dependence of RM (see Equation (5)). To see this, we calculated the path length and also the column density across the WHIM with 10⁵ K <T < 10⁷ K as a function of the redshift depth. Figure 6 shows the resulting path length and column density. The path length saturates at ∼100 Mpc for z ≳ 1. On the other hand, up to z = 1, the column density reaches ∼40% of the saturated value. But the contribution to RM from higher redshift should diminish due to the (1 + z)⁻² dependence.

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We can roughly estimate the saturated value of RM_rms, as follows. Since the inducement of RM is a random walk process, we may take the value as ∼RM_{rms, filament} × (N_filaments)^1/2, where RM_{rms, filament} is the rms RM induced by a single filament and N_filaments is the number of encounters of filaments. AR10 estimated that RM_{rms, filament} ∼ 1.5 rad m⁻². Since the typical width of filaments is several h⁻¹ Mpc (see, e.g., Ryu et al. 2003; Kang et al. 2005), the path length of ∼100 Mpc corresponds to N_filament ∼ 25. Then, we get ${\rm RM}_{\rm rms} \sim 1.5 \times \sqrt{25} \sim 7.5$ rad m⁻², which agrees well with those for TM7, TS8, and TS0.

We also calculated the two-dimensional power spectrum with the sky map of RM. Figure 7 shows the resulting spectrum, averaged over those from 200 maps, for redshift depths z = 0.05, 0.3, and 5. For our best models TS8 and TS0, at the depth of z = 0.05, the power spectrum peaks at ∼1°, which corresponds to the proper length of ∼2.5 h⁻¹ Mpc at that redshift. As the redshift depth increases, the power adds up mostly at smaller scales. At z = 5, the power spectrum shows a broad plateau over ∼1–01 with peak around ∼015. The peak angle corresponds to the proper length of ∼3 h⁻¹ Mpc at z = 1 and ∼2.5 h⁻¹ Mpc at z = 5. With the typical radius of filaments of a few h⁻¹ Mpc, we argue that the peak in the power spectrum reflects the scale of filaments.

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The point that the power spectrum peaks at a small angular scale of ∼015 could be the key to the search of the RM due to the IGMF in future RM surveys such as the SKA survey and the ASKAP POSSUM; it can be used to remove the galactic foreground. Toward the galactic poles, the galactic foreground is expected to have the peak of the power spectrum at a much larger scale of ∼10° or so (e.g., Frick et al. 2001). Then, it would be possible to separate the extragalactic component from the galactic foreground in the Fourier space, when RM surveys become available. The issue of the galactic foreground, along with other uncertainties, in the study of the extragalactic component of RM will be discussed in separate papers.

From the observer's point of view, the structure function (SF) is a statistical quantity which is easier to obtain. The nth order SF is defined as

$$\begin{equation} S_n(r)=\langle |RM(\vec{x}+\vec{r})-RM(\vec{x})|^n\rangle _{\vec{x}} \end{equation} \tag{ 12 }$$

with $r=|\vec{r}|$, where the subscript indicates the averaging over the data domain. It can be easily calculated for the data of an arbitrary domain with irregular sampling intervals. Figure 8 shows the second-order SF (n = 2) for the depth of z = 5 in our model, again averaged over those from 200 maps. The SF increases slightly at small angular separations, ≲ 02, and saturates and stays flat at larger separations. The second-order SF is basically the Fourier transform of the power spectrum. The saturation at ∼02 reflects the fact that the power spectrum has the peak around that angular scale, which is indeed the case as shown in Figure 7.

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Recently, for instance, Mao et al. (2010) and Stil et al. (2011) reported the second-order SF from RM surveys toward the galactic poles; their data are also shown in Figure 8. The SF toward the south pole is relatively flat on the angular scale of ∼1°–10°, while it decreases with decreasing angular separation at smaller scales. On the other hand, the SF toward the north pole decreases with decreasing angular separation on the scale of ∼1°–10°, while it is very noisy but looks flat at smaller scales (the noise is partly attributed to the Coma cluster in the field). The SF toward the south pole is somewhat larger than that toward the north pole. As Schnitzeler (2010) and Stil et al. (2011) noted, however, the error in the observed SFs is still too large to allow any concrete conclusion. Nevertheless, our results suggest that a substantial fraction of the RM toward the galactic poles may be attributed to the IGMF. As a matter of fact, our estimation of the extragalactic contribution to RM, ∼7–8 rad m⁻², agrees with the estimation of ∼6 rad m⁻² from a survey by Schnitzeler (2010).