Foreground Emission Randomization Due to Dynamics of Magnetized Interstellar Medium: WMAP and Planck Frequency Bands

Abstract

Using the results of numerical simulations and astrophysical observations (mainly in the WMAP and Planck frequency bands), it is shown that Galactic foreground emission becomes more sensitive to the mean magnetic field with the frequency, resulting in the appearance of two levels of its randomization due to the chaotic/turbulent dynamics of a magnetized interstellar medium dominated by magnetic helicity. The galactic foreground emission is more randomized at higher frequencies. The Galactic synchrotron and polarized dust emissions have been studied in detail. It is shown that the magnetic field imposes its level of randomization on the synchrotron and dust emission. The main method for the theoretical consideration used in this study is the Kolmogorov–Iroshnikov phenomenology in the frames of distributed chaos notion. Despite the vast differences in the values of physical parameters and spatio-temporal scales between the numerical simulations and the astrophysical observations, there is a quantitative agreement between the results of the astrophysical observations and the numerical simulations in the frames of the distributed chaos notion.

1. Introduction

The impact of Galactic foreground emission in astrophysics is two-fold. On the one hand, different types of Galactic emission contributing to the Galactic foreground are indispensable sources of information about the physical processes in magnetized interstellar medium. On the other hand, the Galactic foreground is the main obstacle to obtaining a clean cosmic microwave background (CMB) radiation map (all observed CMB pixels have non-primordial radiation flux contribution). Therefore, investigation of the Galactic foreground emission (and its components) is necessary to solve these two important problems of modern astrophysics. The recent major satellite missions, WMAP and Planck, were designed mainly to solve the second problem. However, to solve this problem, we need an effective method for separating the CMB and the foreground (mainly of Galactic origin). Such techniques were developed in the last decades. The maps of the most important components of the Galactic foreground (such as synchrotron and dust emission) were also obtained as a necessary by-product of this activity.

The main difficulty in interpreting and understanding these results is due to a weakness in the theory of the chaotic/turbulent processes in the interstellar magnetized medium. These processes are supposed to be the main physical source of the apparently random character of the foreground maps. The scaling (power-law) approach, widely used for the interpretation of the power spectra corresponding to the maps, requires a wide range of scales for its validation, which is rarely achievable in practice.

The conception of smoothness can be used instead to quantify the levels of randomness of the chaotic/turbulent dynamical regimes. Indeed, the stretched exponential spectrum

$$E\left(k\right)\propto exp-{(k/k_{\beta })}^{\beta }$$

(1)

is a characteristic feature of smooth chaotic dynamics. Here $1\ge \beta >0$ and k is the wavenumber.

The value of $\beta =1$ characterizes deterministic chaos (see, for instance, Refs. [1,2,3,4] and references therein):

$$E\left(k\right)\propto exp(-k/k_c).$$

(2)

When $1>\beta$ the smooth chaotic dynamics can be already non-deterministic; this type of smooth dynamics can be called “distributed chaos” (the term will be clarified below). Another term, “soft turbulence” (suggested in Ref. [5]), can also be appropriate.

The parameter $\beta$ could be used as an informative measure of randomization. Specifically, the further the value of $\beta$ is from $\beta =1$ (which corresponds to the deterministic chaos), the more significant the system’s randomization. The smaller parameter $\beta$ values are considered a precursor of hard turbulence. The scaling power spectrum is a characteristic feature of non-smooth random dynamics (the hard turbulence in terms of Ref. [5]).

Figure 1 and Figure 2 (adapted from Figures B1 and B5 of a paper [6]) show the full-sky Galactic foreground maps computed using the data measured by the probes onboard the WMAP satellite for the K and W frequency bands.

Figure 3 shows the power spectra corresponding to the full-sky Galactic foreground maps for K—central frequency 23 GHz, Ka—central frequency 33 GHz, Q—central frequency 41 GHz, V—central frequency 61 GHz, and W—central frequency 94 GHz frequency bands. The spectral data for Figure 3 were taken from Figure 8 of the paper [6]. In the original figure in the paper [6], the angular power spectra $C_l$ are shown in comparison to the multipole l. The spherical multipole l can be related to a wavenumber $k_l=\sqrt{l(l+1)}/R$, where R is the sphere’s radius, and the azimuthally averaged 2D power spectral density $E\left(k_l\right)\approx R^2{C}_l/\pi$ [7].

The dashed curves in Figure 3 indicate the best fit by the stretched exponential spectrum in Equation (1) (the distributed chaos). One can see that $\beta =1/4$ for the frequency bands K and Ka, whereas $\beta =1/5$ for the frequency bands Q, V, and W. These values of $\beta$ will be explained below. Now we would like to emphasize that there is a two-level randomization in the WMAP foreground maps (the two values of $\beta$), and the higher level of randomization (smaller $\beta$) corresponds to higher frequencies.

The present paper will relate the apparent two-level randomization to the Kolmogorov–Iroshnikov phenomenology [8,9] applied to the magnetic helicity-dominated chaotic/ turbulent motion of the magnetized interstellar medium in the frames of distributed chaos (the magneto-inertial range of scales [10]). It will be shown that the foreground emissions become more sensitive to the large-scale magnetic field (and more randomized) with frequency. In this sense, the role of the large-scale magnetic field increases with frequency (cf. Figure 3). There exists a vast amount of literature devoted to the large-scale magnetic fields in the spiral galaxies and in the Milky Way in particular (for some relevant recent papers, see [11,12,13,14,15,16] and references therein), but its crucial influence on the foreground emissions is still not fully understood (the corresponding discussion will be presented in Section 7).

It should be noted that the background emission (cosmic microwave background- CMB) is considerably less randomized $\beta =1/2$ than the foreground ones, and its randomization can be related to a parity violation at baryogenesis preceding the recombination epoch [17]. This difference can be used, in particular, for a more effective separation between the CMB and the foreground.

2. Distributed Chaos and Magnetic Helicity

2.1. Deterministic Chaos in Magnetized Plasma

The estimates of the values of the Galactic magnetic field obtained by the observations turned out to be considerably larger than the values predicted for the primordial magnetic field. Therefore, certain mechanisms of amplification of the magnetic fields by the intense chaotic/turbulent motion of the electrically conducting Galactic plasma (dynamo) were suggested in the existing literature (see [11,18,19,20,21] and references therein). The Galactic differential rotation or the supernova explosions could produce this motion.

In a recent paper [22] a numerical simulation of a small-scale (fluctuating) dynamo with parameters favorable to deterministic chaos was performed using magnetohydrodynamic (MHD) equations

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}& +\nabla ·\left(\rho \mathbf{u}\right)=0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathbf{b}}{\partial t}}& =\nabla \times (\mathbf{u}\times \mathbf{b})+\eta {\nabla }^2\mathbf{b},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathbf{u}}{\partial t}}& +(\mathbf{u}·\nabla )\mathbf{u}=-{\displaystyle \frac{\nabla p}{\rho }}+{\displaystyle \frac{\mathbf{j}\times \mathbf{b}}{c\rho }}\hfill \\ \hfill & +\nu \left({\nabla }^2\mathbf{u}+{\displaystyle \frac{1}{3}}\nabla (\nabla ·\mathbf{u})+2\mathbf{S}·\nabla ln\rho \right)+\mathbf{F},\hfill \end{array}$$

(5)

in a triply periodic cubic domain. In these equations $\mathbf{u}$ is the plasma velocity field, $\mathbf{b}$ is the divergence-free magnetic field, $\rho$ is the plasma density, p is the plasma pressure, $\nu$ is the plasma viscosity, $\eta$ is the plasma magnetic diffusivity, $\mathbf{j}=(c/4\pi )\nabla \times \mathbf{b}$ was taken for electric current density, c is the speed of light, $S_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}-{\displaystyle \frac{2}{3}}{\delta }_{ij}\nabla ·\mathbf{u}\right)$ was taken for the rate-of-strain tensor, and $\mathbf{F}$ is a random delta-correlated in time solenoidal forcing function. An isothermal equation of state, $p=c_s^2\rho$, was assumed with constant sound speed $c_s$.

In this simulation the following nondenominational parameters were used: the Reynolds and magnetic Reynolds numbers $Re=R{e}_m=1122$, the Mach number $M\approx 0.11$, and the magnetic Prandtl number $P{r}_m=1$. A weak random magnetic field (with zero net flux across the computational domain) was used as an initial seed field.

Figure 4 shows the one-dimensional (shell-averaged) magnetic energy spectrum at the saturated stage of the dynamo (the spectral data were taken from Figure 2 of the paper [22]). The dashed curve is the best fit corresponding to Equation (2) (deterministic chaos).

2.2. Magnetic Helicity

The ideal MHD has three fundamental quadratic invariants: total energy, cross helicity, and magnetic helicity [23]. The validity of magnetic helicity conservation increases with the magnetic Reynolds number value $R{e}_m$. The Galaxy magnetized plasma is characterized by very large magnetic Reynolds numbers [24].

The average magnetic helicity density is

$$h_m=\langle \mathbf{a}·\mathbf{b}\rangle$$

(6)

here $\mathbf{a}$ is the vector potential, $\mathbf{b}=\left[\nabla \times \mathbf{a}\right]$ is the fluctuating magnetic field, and $\langle ...\rangle$ means spatial average (for the fluctuating variables $\langle \mathbf{a}\rangle =\langle \mathbf{b}\rangle =0$).

The magnetic helicity is not invariant in the uniform mean magnetic field ${\mathbf{B}}_{\mathbf{0}}$. However, a generalized average magnetic helicity density

$${\widehat{h}}_m=h_m+2{\mathbf{B}}_{\mathbf{0}}·\langle \mathbf{A}\rangle$$

(7)

where $\mathbf{B}={\mathbf{B}}_{\mathbf{0}}+\mathbf{b}$, $\mathbf{A}={\mathbf{A}}_{\mathbf{0}}+\mathbf{a}$, is still an ideal invariant [25]

$${\displaystyle \frac{d{\widehat{h}}_m}{dt}}=0$$

(8)

(see also Ref. [26]).

The magnetic helicity can be considered as an adiabatic invariant not only in the ideal MHD but also in a weakly dissipative magnetized plasma (see, for instance, Ref. [24]), which makes it especially interesting for the interstellar media.

2.3. Distributed Chaos Dominated by Magnetic Helicity

The transition from deterministic chaos to a distributed one can be considered as a randomization. Namely, the change in physical parameters can result in the random fluctuations of the characteristic scale $k_c$ in Equation (2). One has to take this phenomenon into account. It can be performed using an ensemble averaging

$$E\left(k\right)\propto {\int }_0^{\infty }P\left(k_c\right)exp-(k/k_c)d{k}_c$$

(9)

Here a probability distribution $P\left(k_c\right)$ describes the random fluctuations of $k_c$. This is the rationale behind the name “distributed chaos”.

For the magnetic field dynamics dominated by the magnetic helicity the scaling relationship between characteristic values of $k_c$ and $B_c$ based on dimensional considerations

$$B_c\propto {\left|h_m\right|}^{1/2}{k}_c^{1/2}$$

(10)

can be used to find the probability distribution $P\left(k_c\right)$.

The value of $B_c$ can be taken half-normally distributed $P\left(B_c\right)\propto exp-(B_c^2/2{\sigma }^2)$ [8]. It is a normal distribution with zero mean and is truncated to have a nonzero probability density function for positive values of its argument only. For instance, if B is a normally distributed random variable, then the variable $B_c=\left|B\right|$ is half-normally distributed [27].

From Equation (10) we then obtain

$$P\left(k_c\right)\propto k_c^{-1/2}exp-(k_c/4k_{\beta })$$

(11)

It is the chi-squared probability distribution where $k_{\beta }$ is a new constant.

Substituting Equation (11) into Equation (9) one obtains

$$E\left(k\right)\propto exp-{(k/k_{\beta })}^{1/2}$$

(12)

2.4. Spontaneous Breaking of Local Reflectional Symmetry

For chaotic/turbulent flows with net reflectional symmetry, the net magnetic helicity is equal to zero, whereas the point-wise magnetic helicity is not (because of the spontaneous breaking of the local reflectional symmetry). The spontaneous local symmetry breaking in such flows is accompanied by the emergence of the blobs with nonzero helicity [23,28,29,30,31,32]. The magnetic surfaces of these blobs can be defined by the boundary conditions: ${\mathbf{b}}_{\mathbf{n}}·\mathbf{n}=0$, where $\mathbf{n}$ is a unit normal to the boundary of the blob.

The sign-defined magnetic helicity of the j-blob can be defined as

$$H_j^{\pm }={\int }_{V_j}(\mathbf{a}(\mathbf{x},t)·\mathbf{b}(\mathbf{x},t))d\mathbf{x}$$

(13)

where (‘+’ or ‘−’) denotes the blob’s helicity sign. The $H_j^{\pm }$ is an adiabatic invariant [23] (see also above).

Then we can consider the total sign-defined adiabatic invariant

$${\mathrm{I}}^{\pm }=\underset{V\to \infty }{lim}{\displaystyle \frac{1}{V}}\sum _j{H}_j^{\pm }$$

(14)

The summation takes into account the blobs with a certain sign only (‘+’ or ‘−’), and V is the total volume of the blobs taken into account.

The adiabatic invariant ${\mathrm{I}}^{\pm }$ defined by Equation (14) can be used instead of the averaged magnetic helicity density $h_m$ in the above estimate Equation (10) for the special case of the local reflectional symmetry breaking

$$B_c\propto {\left|{\mathrm{I}}^{\pm }\right|}^{1/2}{k}_c^{1/2}$$

(15)

and spectrum Equation (12) can also be obtained for this case.

When a nonzero mean magnetic field ${\mathbf{B}}_0$ is significant, the $h_m$ should be replaced by the generalized averaged magnetic helicity density ${\widehat{h}}_m$ Equation (7) in the estimate given in Equation (10). The ${\widehat{h}}_m$ has the same dimensionality as $h_m$, and therefore, the magnetic energy spectrum will be given by the same equation, Equation (12) (the case of the local reflectional symmetry breaking can be treated analogously).

In recent papers [33,34] numerical simulations similar to that considered in Section 2.1 were performed, but in these simulations, the large Mach number $M=10$ was achieved. Figure 5 shows the magnetic energy spectra computed in these numerical simulations. The spectral data shown in this figure were taken from Figure C4 of Ref. [34]. While for the small Mach number $M=0.1$, the bottom dashed curve in Figure 5 indicates the exponential spectrum Equation (2) (deterministic chaos, cf. Figure 1), for the large Mach number $M=10$, the top dashed curve indicates a stretched exponential spectrum Equation (12) with $\beta =1/2$ (i.e., distributed chaos dominated by magnetic helicity).

3. Magneto-Inertial Range of Scales

For high Reynolds numbers the so-called inertial range of scales, dominated by the kinetic energy dissipation rate $\epsilon$ only, is conventionally considered in hydrodynamic turbulence. The Kolmogorov phenomenology assumes that energy is transferred through this range with negligible dissipation to the sufficiently small scales where it is dissipated [35] (see also Ref. [8] and references therein). In magnetohydrodynamics (and also at the kinetic scales), a magneto-inertial range of scales has been recently introduced in [10]. The two parameters, the magnetic helicity dissipation rate ${\epsilon }_h$ and total energy dissipation rate $\epsilon$ determine the magnetic field dynamics in this range.

In the case of a considerable mean magnetic field, the energy dissipation rate $\epsilon$ can be replaced by the parameter $\left(\epsilon {\tilde{B}}_0\right)$ [9]. Here ${\tilde{B}}_0=B_0/\sqrt{{\mu }_0\rho }$ is normalized mean magnetic field. In the Alfvén units the ${\tilde{B}}_0$ has the same dimension as velocity. In Iroshnikov’s phenomenology [9] the eddies, considered in the Kolmogorov phenomenology for hydrodynamics, are replaced by the Alfvénic wave-packets which propagate in opposite directions along the mean magnetic field. The applicability of the Kolmogorov-like and Iroshnikov phenomenologies to magnetohydrodynamics was discussed for decades, and some modifications were suggested. However, their main idea, using $\epsilon$ or $\left(\epsilon {\tilde{B}}_0\right)$ as the dominant dimensional parameters in the inertial-like range of scales, remained.

There is an analogy between the magneto-inertial range approach and the Corrsin–Obukhov inertial–convective range approach to the passive scalar, where the two governing parameters, the passive scalar dissipation rate and energy dissipation rate, dominate the inertial–convective range [8]) (see also Ref. [36]).

According to this analogy, one can replace the estimate Equation (10) with the estimate

$$B_c\propto {\epsilon }_{h_m}^{1/2}{\epsilon }^{-1/6}{k}_c^{1/6}$$

(16)

for the magneto-inertial range without mean magnetic field and with the estimate

$$B_c\propto {\epsilon }_{{\widehat{h}}_m}^{1/2}{\left(\epsilon {\tilde{B}}_0\right)}^{-1/8}{k}_c^{1/8}$$

(17)

when a substantial mean (large-scale) magnetic field ${\tilde{B}}_0$ is present.

The specific estimates Equations (10) and (15)–(17) can be generalized

$$B_c\propto k_c^{\alpha }$$

(18)

In the asymptotic of large $k_c$, the stretched exponential form of the distributed chaos spectra

$${\int }_0^{\infty }P\left(k_c\right)exp-(k/k_c)d{k}_c\propto exp-{(k/k_{\beta })}^{\beta }$$

(19)

results in the probability distribution [37]

$$P\left(k_c\right)\propto k_c^{-1+\beta /\left[2\right(1-\beta \left)\right]}exp(-\gamma k_c^{\beta /(1-\beta )})$$

(20)

In the case of the half-normally distributed $B_c$, a relationship between $\alpha$ and $\beta$ can be obtained from Equations (18) and (20)

$$\beta ={\displaystyle \frac{2\alpha }{1+2\alpha }}$$

(21)

For $\alpha =1/6$ Equation (16), we obtain from Equation (21)

$$E\left(k\right)\propto exp-{(k/k_{\beta })}^{1/4},$$

(22)

and for $\alpha =1/8$ Equation (17), we obtain from Equation (21)

$$E\left(k\right)\propto exp-{(k/k_{\beta })}^{1/5}$$

(23)

A numerical simulation of a MHD dynamo in a Milky Way-like galaxy was performed in a recent article [38] using the following equations:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \left(\rho \mathbf{u}\right)& =& 0\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \rho \mathbf{u}}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}-\mathbf{B}\mathbf{B}\right)+\nabla P_{tot}& =& -\rho \nabla \varphi \hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial E_{tot}}{\partial t}}+\nabla \left[(E_{tot}+P_{tot})\mathbf{u}-(\mathbf{u}·\mathbf{B})·\mathbf{B}\right]& =& -\mathbf{u}·\nabla \varphi -\rho \Lambda +\Gamma \hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \mathbf{B}}{\partial t}}-\nabla \times (\mathbf{u}\times \mathbf{B})& =& 0\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \nabla ·\mathbf{B}& =& 0\hfill \end{array}$$

(28)

The functions of density $\rho$ and temperature T: $\Gamma =\Gamma (\rho ,T)$ and $\Lambda =\Lambda (\rho ,T)$, represent the heating and cooling rates of the intragalactic plasma, $\varphi$ is the gravitational potential, the total pressure was taken as

$$P_{tot}=p+{\displaystyle \frac{\mathbf{B}·\mathbf{B}}{2}}$$

(29)

and the total energy was taken as

$$E_{tot}=E_{int}+\rho {\displaystyle \frac{\mathbf{u}·\mathbf{u}}{2}}++{\displaystyle \frac{\mathbf{B}·\mathbf{B}}{2}}$$

(30)

with $E_{int}$ being the internal energy of the fluid. The equation of state was taken as $P=(\gamma -1)E_{int}$.

The model also takes into account the stars, their feedback, and the dark matter halo. The configuration of the intragalactic medium simulates a Milky Way-like galaxy. A weak toroidal magnetic field was taken as an initial seed condition. It should be noted in this respect that the seed field can be generated by the Biermann battery mechanism, which produces fields directed along the galactic rotation axis. Toroidal fields are produced with large-scale dynamos, which work at the next stages (see, for instance, a recent paper [39] and references therein).

The chaotic motion of the intragalactic plasma was generated (simulated) by the galactic differential rotation and supernova explosions. At some stage of the simulation, the generated magnetic field was smoothed to separate the mean field and chaotic residual fluctuations.

Figure 6 shows the magnetic energy spectrum of chaotic magnetic field fluctuations computed at this stage. The spectral data were taken from Figure 5 of Ref. [38]. The dashed curve indicates the best fit by the stretched exponential Equation (22) (the magneto-inertial range of scales).

4. Observations in the Galactic Magnetized Plasma

The main difficulty of measurements of the interstellar magnetic field is that (unlike the interplanetary magnetic field) it cannot be measured directly. In the accessible observables, the magnetic field is usually entangled and mixed with other variables characterizing the plasma.

The Faraday effect, for instance, is a rotation of the polarization position angle propagating through the magnetized plasma. It entangles the electron density $n_e$ with the line-of-sight component of the magnetic field $B_{LoS}$ into an observable characteristic—Faraday depth:

$$\varphi ={\displaystyle \frac{e^3}{2\pi m_e^2{c}^4}}{\int }_{\mathrm{LoS}}dl{n}_{\mathrm{e}}{B}_{LoS}$$

(31)

In the paper [40] a disentangling of an all-sky map for average intragalactic $B_{LoS}$ from the Faraday effect map was reported. Some complementary tracers of the $n_e$ (the free–free map of the Planck survey [41], extra-Galactic Faraday data [42], a $H-\alpha$ map [43], pulsar data [44]) were also used for the purpose, and an all-sky map of the electron dispersion measure (the integrated electron density) was also constructed.

Figure 7 shows the power spectrum of $B_{LoS}$. The spectral data were taken from Figure 9a of the paper [40]. The dashed curve in Figure 7 indicates the best fit by the stretched exponential spectral law Equation (22).

An all-sky map of the integrated electron density (electron dispersion measure) was also obtained by this method. Figure 8 shows the power spectrum of the electron dispersion measure. The spectral data were taken from Figure 9b of the paper [40]). The dashed curve in Figure 8 indicates the best fit by the same stretched exponential spectral law Equation (22). Apparently the magnetic field imposes its degree of randomization (the $\beta =1/4$) on the electron dispersion measure.

Now it is not surprising that the magnetic field also imposes its degree of randomization on the Faraday map itself. Figure 9 shows the spectrum of the Faraday map. The spectral data were taken from Figure 6 (reconstruction II) of a paper ([45]). The dashed curve in Figure 9 indicates the best fit by the stretched exponential spectral law Equation (21).

5. Synchrotron Emission

The astroparticles (relativistic electrons and positrons) produce the synchrotron emission moving through the magnetic field. Therefore the diffuse (polarized) synchrotron emission is an effective tracer of the magnetic field in the magnetized non-thermal plasma. This emission is also one of the main components of the polarized foreground for the cosmic microwave background radiation. Therefore, studying its spectral properties is important for understanding the magnetized intragalactic plasma’s physical processes and obtaining clean CMB maps.

In a paper [46] a reanalysis of the famous so-called “Haslam” 0.408 GHz all-sky survey of the synchrotron emission was made, with a subtraction of the point sources and an averaging over different lines-of-sight. Figure 10 shows the power spectrum obtained for this reconsidered Haslam survey. The spectral data were taken from Figure 3 of Ref. [46].

The dashed curve in Figure 10 indicates the best fit by the stretched exponential spectral law Equation (22). Apparently the magnetic field imposes its degree of randomization in this case as well.

A spin-2 decomposition of the polarization tensor was suggested in a paper [47]. The polarization tensor is decomposed into two rotationally invariant quantities: scalar E and pseudo-scalar B. The B-mode of this decomposition has magnetic field-type parity.

In a paper, [48], results obtained using this technique for the Parkes 2.4 GHz survey of synchrotron polarized emission for the Southern Galactic plane were reported. Figure 11 shows the power spectrum for the B-mode. The spectral data were taken from Figure 3 of Ref. [48]. The dashed curve in Figure 11 indicates the best fit by the stretched exponential spectral law Equation (22).

6. Dust Emission

Polarized dust emission is not only one of the most reach sources of information about the Galactic magnetic field (see, for instance, a recent paper [49], and references therein) but also the main obstacle for observing (detecting) the gravitational waves in primordial CMB B-modes, especially for the high-frequency Planck’s channels [50].

Let us begin with a recent numerical simulation. Results of this simulation were reported in Ref. [51]. The ideal compressible MHD simulation was performed in a 3D periodic spatial box with an approximately isothermal equation of state. A stochastic (Gaussian) non-helical large-scale forcing with a constant energy injection rate drove the gas. The polarization of the thermal dust emission comes from the elongated grains of dust grains spinning around the local magnetic field (their long axes were perpendicular to the field). The dust-to-gas ratio was uniform and constant, the grains were aligned with the magnetic field direction, the dust cloud was optically thin, and the dust temperature coincided with the constant gas temperature. The projections perpendicular to the mean magnetic field were the main subject of the consideration. The power spectra were computed by averaging over an annulus in Fourier space.

Figure 12 shows the power spectra of sub-Alfvénic (bottom) and super-Alfvénic (top) magnetic field. The sonic Mach number $M\approx 6$ for both cases. The spectral data were taken from Figure 4 of Ref. [51].

Figure 13 and Figure 14 show the projected T-mode (temperature) and B-mode (polarization) power spectra of the polarized dust emission for the same conditions. The spectral data were taken from Figure 6 of Ref. [51].

The dashed curves in Figure 12, Figure 13 and Figure 14 indicate the best fit by the stretched exponential spectral law Equation (23), that means the importance of the mean magnetic field in this simulation. The same level of randomization of the magnetic field and the polarized emission confirms the imposition of the magnetic field level of randomization on the polarized emission (cf. previous section).

Figure 15 and Figure 16 show the T-mode (temperature) and B-mode (polarization) power spectra of the polarized dust emission computed using the GNILC-cleaned full-sky maps obtained using the Planck data at 353 GHz. The spectral data were taken from Figure 11 of a paper [52]. The dashed curves in Figure 15 and Figure 16 indicate the best fit by the stretched exponential spectral law Equation (23) (cf. Figure 13 and Figure 14).

In a recent paper [53] the third Planck public release’s maps were used to characterize the polarized dust emission at high Galactic latitudes. Using these maps the angular power spectra of the B-mode were computed at 353 GHz channel for six sky regions. The largest region covers 71% of the sky (naturally, the best measurement results were obtained in this region). Figure 17 shows the power spectrum of the B-mode for this region. The spectral data were taken from Table C.1 of Ref. [53]. The dashed curve in Figure 17 indicates the best fit by the stretched exponential spectral law Equation (23).

7. Conclusions and Discussion

One can conclude that galactic foreground emissions become more sensitive to the mean (large-scale) magnetic field with frequency. The two levels of randomization observed in Figure 3, which are lower ($\beta =1/4$) for the smaller frequencies and higher ($\beta =1/5$) for larger frequencies, can be related to the difference in the spatial locations of the Galaxy regions, which provide the main contribution to the foreground emission at different frequencies. Specifically, in the regions providing the main contribution to the K and Ka channels, the mean (large-scale) magnetic field does not play an important role (can be neglected), and therefore, the spectra should be described by Equation (22), whereas in the regions providing the main contribution to the Q, V, and W channels, the mean (large-scale) magnetic field plays an important role, and the spectra should be described by Equation (23).

Analogously, one can conclude that in the regions providing the main contribution to the all-sky map for the Faraday rotation measure, the mean (large-scale) magnetic field also does not play an important role (Figure 9). Therefore, the spectra corresponding to the magnetic field $B_{LoS}$ depicted in Figure 7 and integrated electron density (electron dispersion measure) depicted in Figure 8, obtained by disentangling the all-sky Faraday rotation map, are well fitted by Equation (22).

The difference in the levels of randomization could also be partially related to the fact that for the smaller and middle frequencies, the synchrotron and free–free emissions dominate the Galactic foreground, whereas, for the large frequencies, the dust emission takes this role (see Figure 5 of Ref. [50]). The problem is that the observed angular power spectra of the synchrotron and free–free emission for the middle and high frequencies are rare in the literature, as well as the spectra of the dust emission for small frequencies. Maybe the above considerations can be encouraging when solving this problem.

The results of the numerical simulations and the astrophysical observations indicate that the magnetic helicity-dominated magnetic field imposes its level of randomization on the synchrotron and dust emission.

An additional application of interest would be the radiation emitted from Black Hole accretion disks since a disk is a known chaotic plasma. In this case, unlike the bulk’s Galactic foreground, the level of randomization can coincide with that of the CMB ($\beta =1/2$) [17], which will make it more difficult for separation.

Despite the vast differences in the values of physical parameters and spatio-temporal scales between the numerical simulations and the astrophysical observations (see [54] for a recent review), there is a quantitative agreement between the results of the astrophysical observations and the numerical simulations in the frames of the distributed chaos approach.