ON THE ORIGIN OF THE SCATTER BROADENING OF FAST RADIO BURST PULSES AND ASTROPHYSICAL IMPLICATIONS

A power-law spectrum of the plasma density irregularities is commonly applied in studies on radio wave propagation (Lee & Jokipii 1976; Rickett 1977, 1990), which is also reinforced by growing observational evidence of interstellar density fluctuations (Armstrong et al. 1995; Chepurnov & Lazarian 2010). We assume that the scattering effect is introduced by electron density fluctuations that arise from a turbulent cascade and the relevant spectrum takes the form (Rickett 1977; Coles et al. 1987)

$$\begin{eqnarray}&&P(k)={C}_{N}^{2}{k}^{-\beta }{e}^{-{({{kl}}_{0})}^{2}},\,\,k\gt {L}^{-1},\end{eqnarray} \tag{ 1 }$$

which is cast as a power-law spectrum in the inertial range of turbulence,

$$\begin{eqnarray}&&P(k)={C}_{N}^{2}{k}^{-\beta },\,\,{L}^{-1}\lt k\lt {l}_{0}^{-1},\end{eqnarray} \tag{ 2 }$$

where L and l₀ are the outer and inner scales, corresponding to the injection and dissipation scales of turbulent energy. The spectral index β is suggested to be within the range 2 < β < 4 on observational grounds (e.g., Lee & Jokipii 1975; Rickett 1977; Romani et al. 1986). Intuitive insight to the properties of turbulence can be gained from the value of β. At the critical index β = 3, density fluctuations, which scale as δn_e ∝ k^{(3 − β)/2}, are scale-independent. That is, the density fluctuations with the same amplitude exist at all scales. Notice that δn_e represents the rms amplitude of density fluctuations. Following the power-law statistics studied in, e.g., Lazarian & Pogosyan (2000, 2004, 2006) and Esquivel & Lazarian (2005), we consider the density spectrum in both the long-wave-dominated regime with β > 3 and the short-wave-dominated regime with β < 3. The density field in the former case is dominated by large-scale fluctuations, but in the latter case is localized in small-scale structures.

Both long- and short-wave-dominated density spectra are a confirmed reality in compressible MHD turbulence (Beresnyak et al. 2005; Kowal et al. 2007). In the WIM phase of Galactic ISM, which corresponds to the transonic turbulence, the power-law spectrum of electron density fluctuations has been convincingly demonstrated to have a unique slope consistent with the Kolmogorov spectrum (β = 11/3) on scales spanning from 10⁶ to 10¹⁷ m (Armstrong et al. 1995; Chepurnov & Lazarian 2010). On the other hand, in colder and denser phases of the ISM in the Galactic plane, the turbulence becomes highly supersonic and shocks are inevitable, which produce large density contrasts and a short-wave-dominated density spectrum (Kim & Ryu 2005). The density spectra with β < 3 have been extracted from ample observations by using different tracers and techniques (e.g., Stutzki et al. 1998; Deshpande et al. 2000; Swift 2006; Xu & Zhang 2016a; also see Table 5 in the review by Lazarian 2009 and Figure 10 in the review by Hennebelle & Falgarone 2012). In addition, the Kolmogorov density spectrum also fails to reconcile with the observationally measured scaling relation between scatter-broadening time and frequency for highly dispersed pulsars (e.g., Löhmer et al. 2001, 2004; Bhat et al. 2004; Lewandowski et al. 2013). In view of the diversity of ISM phases and properties of the associated turbulence, it is necessary to perform a general analysis incorporating both long- and short-wave-dominated spectra of density fluctuations.

The normalization of the power spectrum depends on the steepness of the slope. From the density variance

$$\begin{eqnarray}&&\langle {(\delta {n}_{e})}^{2}\rangle =\int P(k){d}^{3}{\boldsymbol{k}}\end{eqnarray} \tag{ 3 }$$

and assuming L ≫ l₀, we find

$$\begin{eqnarray}&&{C}_{N}^{2}\sim \displaystyle \frac{\beta -3}{2{(2\pi )}^{4-\beta }}{(\delta {n}_{e})}^{2}{L}^{3-\beta },\quad \beta \gt 3,\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{C}_{N}^{2}\sim \displaystyle \frac{3-\beta }{2{(2\pi )}^{4-\beta }}{(\delta {n}_{e})}^{2}{l}_{0}^{3-\beta },\quad \beta \lt 3.\end{eqnarray} \tag{ 4b }$$

It shows that the turbulent power characterized by density perturbation δn_e concentrates at L for β > 3 and l₀ for β < 3. Thus δn_e is the density perturbation at the correlation scale of turbulence, which is L for a long-wave-dominated spectrum and l₀ for a short-wave-dominated spectrum.

As the radio waves propagate through a turbulent plasma, multi-path scattering causes temporal broadening of a transient pulse (e.g., Rickett 1990; Cordes & Rickett 1998). On a straight-line path of length D through the scattering medium, the integrated phase structure function is defined as the mean square phase difference between a pair of LOSs with a separation r on the plane transverse to the propagation direction, ${D}_{{\rm{\Phi }}}=\langle {({\rm{\Delta }}{\rm{\Phi }})}^{2}\rangle$. Given the spectral form of Equation (1) with 2 < β < 4, and under the condition r ≪ L ≪ D, D_Φ has expressions (Coles et al. 1987; Rickett 1990)

$$\begin{eqnarray}&&{D}_{{\rm{\Phi }}}\sim \pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{l}_{0}^{\beta -4}{r}^{2},\quad r\lt {l}_{0},\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&{D}_{{\rm{\Phi }}}\sim \pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{r}^{\beta -2},\quad r\gt {l}_{0},\end{eqnarray} \tag{ 5b }$$

where r_e is the classical electron radius and λ is the wavelength. The scattering measure SM is the integral of C_N² along the LOS path through the scattering region, and characterizes the scattering strength. Here we consider a statistically uniform turbulent medium, with the turbulence properties independent of the path length. Thus the SM is simplified as

$$\begin{eqnarray}&&{\rm{SM}}\sim {C}_{N}^{2}D.\end{eqnarray} \tag{ 6 }$$

By applying C_N² expressed in Equation (4) in the SM, the structure function D_Φ is applicable for both a long-wave-dominated spectrum of turbulence on scales below the correlation scale (L) and a short-wave-dominated spectrum on scales above the correlation scale (l₀).⁶

The transverse separation across which the rms phase perturbation is equal to 1 rad is defined as the diffractive length scale (e.g., Rickett 1990). By using Equation (5), it is expressed as

$$\begin{eqnarray}&&{r}_{{\rm{diff}}}\sim {(\pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{l}_{0}^{\beta -4})}^{-\tfrac{1}{2}},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&{r}_{{\rm{diff}}}\sim {(\pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}})}^{\tfrac{1}{2-\beta }},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 7b }$$

In a particular case when r_diff coincides with l₀, equaling r_diff from the above equation and l₀ yields

$$\begin{eqnarray}&&\pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{l}_{0}^{\beta -2}=1.\end{eqnarray} \tag{ 8 }$$

It means that the physical parameters involved in the scattering process should satisfy the condition

$$\begin{eqnarray}&&\pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{l}_{0}^{\beta -2}\gt 1\end{eqnarray} \tag{ 9 }$$

for r_diff to be smaller than l₀, and

$$\begin{eqnarray}&&\pi {r}_{e}^{2}{\lambda }^{2}{\rm{SM}}{l}_{0}^{\beta -2}\lt 1\end{eqnarray} \tag{ 10 }$$

for r_diff > l₀ to be realized. In terms of r_diff, D_Φ given by Equation (5) can be written in the form

$$\begin{eqnarray}&&{D}_{{\rm{\Phi }}}={\left(\displaystyle \frac{r}{{r}_{{\rm{diff}}}}\right)}^{2},\quad r\lt {l}_{0},\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&{D}_{{\rm{\Phi }}}={\left(\displaystyle \frac{r}{{r}_{{\rm{diff}}}}\right)}^{\beta -2},\quad r\gt {l}_{0}.\end{eqnarray} \tag{ 11b }$$

In the presence of the inner scale of the density power spectrum, D_Φ exhibits a break in the slope at r = l₀ and steepens at smaller scales. The quadratic scaling of D_Φ with r at r < l₀ comes from the Gaussian distribution of density fluctuations $\exp (-{k}^{2}{l}_{0}^{2})$ below the inner scale (Equation (1)).

For the multi-path propagation in the strong scattering regime, r_diff characterizes the coherent scale of the random phase fluctuations and the density perturbation on r_diff dominates the scattering strength, with the angular and temporal broadening given by (Rickett 1990; Narayan 1992)

$$\begin{eqnarray}&&{\theta }_{{\rm{sc}}}=\displaystyle \frac{\lambda }{2\pi {r}_{{\rm{diff}}}},\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{\tau }_{{\rm{sc}}}=\displaystyle \frac{D}{c}{\theta }_{{\rm{sc}}}^{2}=\displaystyle \frac{D{\lambda }^{2}}{4{\pi }^{2}c}{r}_{{\rm{diff}}}^{-2}.\end{eqnarray} \tag{ 13 }$$

The above formulae pertain to the Galactic scattering medium, but should be modified when the scattering plasma is located at a cosmological distance. In the observer's frame, the wavelength is λ₀ = λ(1 + z_q), where z_q is the redshift of the scattering material. By also taking into account the LOS weighting, which depends on the location of the scattering material along the LOS (Gwinn et al. 1993; Macquart & Koay 2013), the temporal broadening becomes

$$\begin{eqnarray}&&W={\tau }_{{\rm{sc,obs}}}=(1+{z}_{{\rm{q}}})\displaystyle \frac{{D}_{{\rm{eff}}}}{c}{\theta }_{{\rm{sc}}}^{2}=\displaystyle \frac{{D}_{{\rm{eff}}}{\lambda }_{0}^{2}}{4{\pi }^{2}c(1+{z}_{{\rm{q}}})}{r}_{{\rm{diff}}}^{-2}.\end{eqnarray} \tag{ 14 }$$

Here D in Equation (13) is replaced by the effective scattering distance D_eff = D_qD_qp/D_p, with D_p, D_qp, and D_q as the angular diameter distances from the observer to the source, from the source to the scattering medium, and from the observer to the scattering medium. Accordingly, SM is also replaced with the weighted SM as adopted in Cordes & Lazio (2002),

$$\begin{eqnarray}&&{\rm{SM}}\sim {C}_{N}^{2}{D}_{{\rm{eff}}}.\end{eqnarray} \tag{ 15 }$$

D_eff is comparable to D_q in the case of Galactic scattering, and comparable to D_qp when the scattering medium is close to the source. In both cases, D_eff serves as a good approximation of the path length through the scattering region, and thus Equation (15) is appropriate for estimating the actual SM. But we caution that for a thin scattering screen located somewhere between the source and the observer, its thickness, i.e., the path length that should be used for calculating SM, is in fact far smaller than the value of D_eff.

In combination with Equations 4(a), (7), and (15), the approximate expression of W in the case of β > 3 can be obtained from Equation (14),

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{D}_{{\rm{eff}}}^{2}{r}_{e}^{2}{\lambda }_{0}^{4}}{4\pi c{(1+{z}_{{\rm{q}}})}^{3}}{(\delta {n}_{e})}^{2}{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{\beta -4}{L}^{-1},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{D}_{{\rm{eff}}}^{\tfrac{\beta }{\beta -2}}{r}_{e}^{\tfrac{4}{\beta -2}}{\lambda }_{0}^{\tfrac{2\beta }{\beta -2}}}{4{\pi }^{\tfrac{2(\beta -3)}{\beta -2}}c{(1+{z}_{{\rm{q}}})}^{\tfrac{\beta +2}{\beta -2}}}{(\delta {n}_{e})}^{\tfrac{4}{\beta -2}}{L}^{\tfrac{2(3-\beta )}{\beta -2}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 16b }$$

The observationally measured wavelength dependence of the pulse width can make a distinction between the scenarios with r_diff below or exceeding l₀, which, however, is limited by the insufficient accuracy of the current data (Thornton et al. 2013; Luan & Goldreich 2014). Nevertheless, it is evident that in both situations W decreases with increasing L. A given pulse width imposes a constraint on the outer scale of turbulence. In particular, when r_diff < l₀, W also decreases with increasing l₀. Moreover, in terms of the dispersion measure DM = n_e D_eff of the scattering medium, where n_e is the electron density averaged along the LOS passing through the scattering region, W in Equation (16) is rewritten as

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{r}_{e}^{2}{\lambda }_{0}^{4}}{4\pi c{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\right)}^{2}{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{\beta -4}{L}^{-1}{{\rm{DM}}}^{2},{r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}W & \sim & \displaystyle \frac{{r}_{e}^{\tfrac{4}{\beta -2}}{\lambda }_{0}^{\tfrac{2\beta }{\beta -2}}}{4{\pi }^{\tfrac{2(\beta -3)}{\beta -2}}c{(1+{z}_{{\rm{q}}})}^{\tfrac{\beta +2}{\beta -2}}}{\left(\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\right)}^{\tfrac{\beta }{\beta -2}}\\ & & \times {(\delta {n}_{e})}^{\tfrac{4-\beta }{\beta -2}}{L}^{\tfrac{2(3-\beta )}{\beta -2}}{{\rm{DM}}}^{\tfrac{\beta }{\beta -2}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 17b }$$

In the case of β < 3, from Equations 4(b), (7), (14), and (15), W can be estimated as

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{D}_{{\rm{eff}}}^{2}{r}_{e}^{2}{\lambda }_{0}^{4}}{4\pi c{(1+{z}_{{\rm{q}}})}^{3}}{(\delta {n}_{e})}^{2}{l}_{0}^{-1},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 18a }$$

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{D}_{{\rm{eff}}}^{\tfrac{\beta }{\beta -2}}{r}_{e}^{\tfrac{4}{\beta -2}}{\lambda }_{0}^{\tfrac{2\beta }{\beta -2}}}{4{\pi }^{\tfrac{2(\beta -3)}{\beta -2}}c{(1+{z}_{{\rm{q}}})}^{\tfrac{\beta +2}{\beta -2}}}{(\delta {n}_{e})}^{\tfrac{4}{\beta -2}}{l}_{0}^{\tfrac{2(3-\beta )}{\beta -2}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 18b }$$

Instead of L, W in this case only places a constraint on l₀. When r_diff < l₀, an excess of temporal broadening requires l₀ to be comparable to r_diff, so l₀ should be sufficiently small, while when r_diff > l₀, a larger l₀ is more favorable. The relation between W and DM can be also established:

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{r}_{e}^{2}{\lambda }_{0}^{4}}{4\pi c{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\right)}^{2}{l}_{0}^{-1}{{\rm{DM}}}^{2},\,{r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray}&&W\sim \displaystyle \frac{{r}_{e}^{\tfrac{4}{\beta -2}}{\lambda }_{0}^{\tfrac{2\beta }{\beta -2}}}{4{\pi }^{\tfrac{2(\beta -3)}{\beta -2}}c{(1+{z}_{{\rm{q}}})}^{\tfrac{\beta +2}{\beta -2}}}{\left(\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\right)}^{\tfrac{\beta }{\beta -2}}\\ &&\quad \times {(\delta {n}_{e})}^{\tfrac{4-\beta }{\beta -2}}{l}_{0}^{\tfrac{2(3-\beta )}{\beta -2}}{{\rm{DM}}}^{\tfrac{\beta }{\beta -2}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 19b }$$

By comparing Equations 17(a), (b), 19(a), and (b), one can see that the dependence of W on DM is determined by both the relation between r_diff and l₀, and the spectral properties of density fluctuations. In general, W increases more drastically with DM at a smaller β in the case of r_diff > l₀, and has its mildest dependence on DM as W ∝ DM² in the case of r_diff < l₀, irrespective of the value of β. Also, the density perturbation δn_e at L for a long-wave-dominated spectrum of density fluctuations is close to n_e averaged over a large scale, while δn_e at l₀ for a short-wave-dominated spectrum can considerably exceed the background n_e due to turbulent compression in shock-dominated flows. More exactly, following the power-law behavior, the ratio of the density perturbation at l₀ to that at L when β < 3 is

$$\begin{eqnarray}&&\displaystyle \frac{\delta {n}_{e}({l}_{0})}{\delta {n}_{e}(L)}=\displaystyle \frac{\delta {n}_{e}}{\delta {n}_{e}(L)}={\left(\displaystyle \frac{L}{{l}_{0}}\right)}^{\tfrac{3-\beta }{2}}.\end{eqnarray} \tag{ 20 }$$

Therefore, with a higher density perturbation and a smaller scale l₀ instead of L involved, a short-wave-dominated spectrum of density fluctuations provides much stronger scattering than a long-wave-dominated one when the DMs are the same.

To elucidate the millisecond scattering tail observed for some FRBs (Lorimer et al. 2007; Thornton et al. 2013), we next consider the IGM and the FRB host galaxy as two possible sources responsible for the scattering timescale.

(1) Scattering in the IGM: Growing observational evidence supports the presence of the IGM turbulence (e.g., Rauch et al. 2001; Zheng et al. 2004; Meiksin 2009; Lu et al. 2010) and the Kolmogorov-type turbulence in clusters of galaxies (Murgia et al. 2004; Schuecker et al. 2004; Vogt & Enßlin 2005). Supercomputer simulations show that the turbulent motions inside clusters of galaxies are subsonic, and are transonic or mildly supersonic in filaments (Ryu et al. 2008), which agrees with the observational detection of subsonic turbulence in, e.g., the Coma cluster (Schuecker et al. 2004) and the core of the Perseus cluster (Churazov et al. 2004). Down to small scales, theoretical studies suggest the existence of Alfvénic turbulence with a spectrum dictated by the Kolmogorov scaling (Schekochihin & Cowley 2006), which is supported by the observed spectrum of magnetic energy in the core region of the Hydra cluster (Vogt & Enßlin 2005). Based on these signatures obtained so far, the IGM turbulence is unlikely to be highly supersonic and thus unlikely to possess a short-wave-dominated density spectrum, especially on scales small enough to be important for diffractive scattering. Therefore, to numerically evaluate the temporal broadening for propagation of radio waves through the diffuse IGM, we consider a long-wave-dominated spectrum (β > 3) of turbulent density and adopt the generally accepted Kolmogorov turbulence model with β = 11/3. Meanwhile, the choice of parameters should also be made to fulfill the conditions indicated by Equations (9) and (10) in cases of r_diff < l₀ and r_diff > l₀, respectively. Inserting Equations 4(a), (15), and β = 11/3 into Equations (9) and (10) yields

$$\begin{eqnarray}&&L{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{\tfrac{5}{3}}\gt {(\pi {r}_{e}^{2}{D}_{{\rm{eff}}}{\lambda }^{2}{(\delta {n}_{e})}^{2})}^{-1},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&L{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{\tfrac{5}{3}}\lt {(\pi {r}_{e}^{2}{D}_{{\rm{eff}}}{\lambda }^{2}{(\delta {n}_{e})}^{2})}^{-1},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 21b }$$

We now rewrite W from Equation (16) in terms of typical parameters for the IGM,

$$\begin{eqnarray}W & \sim & \displaystyle \frac{0.065}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{2}{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{-\tfrac{1}{3}}{\left(\displaystyle \frac{L}{{10}^{-2}\mathrm{pc}}\right)}^{-1}\,{\rm{ms}},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}W & \sim & \displaystyle \frac{4.9}{{(1+{z}_{{\rm{q}}})}^{3.4}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{2.2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4.4}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{2.4}{\left(\displaystyle \frac{L}{{10}^{-2}\mathrm{pc}}\right)}^{-0.8}\,{\rm{ms}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 22b }$$

The value of W at r_diff < l₀ depends on the disparity between L and l₀, according to Equation 21(a), which satisfies

$$\begin{eqnarray}\displaystyle \frac{{l}_{0}}{L} & \gt & 2.4\times {10}^{-6}{(1+{z}_{{\rm{q}}})}^{1.2}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{-0.6}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-1.2}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-1.2}{\left(\displaystyle \frac{L}{{10}^{-2}\mathrm{pc}}\right)}^{-0.6}.\end{eqnarray} \tag{ 23 }$$

With the lower limit of l₀/L in the above expression adopted, we get the same result in both cases that for a low-redshift source the outer scale L on the order of 10⁻² pc can lead to the pulse duration of ∼5 ms at 0.3 GHz frequency (λ = 1 m). The derived outer scale of turbulence seems unreasonably small compared with the expected injection scale (>100 kpc) of turbulence induced by cluster mergers (Subramanian et al. 2006) or cosmological shocks (Ryu et al. 2008, 2010). Also, as pointed out by Luan & Goldreich (2014), a serious difficulty is that such a small outer scale is accompanied by a turbulent heating rate at

$$\begin{eqnarray}&&{\tau }_{{\rm{heat}}}^{-1}\sim \displaystyle \frac{{c}_{s}}{L}=0.005{\left(\displaystyle \frac{L}{{10}^{-2}\mathrm{pc}}\right)}^{-1}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{\tfrac{1}{2}}\,{{\rm{yr}}}^{-1},\end{eqnarray} \tag{ 24 }$$

where c_s is the sound speed. The typical IGM temperature T ranges from 10⁵ to 10⁷ K (Bykov et al. 2008; Ryu et al. 2008). The heating rate is so high that it is incompatible with the cooling rate, which is comparable to the inverse Hubble time. With regards to a Kolmogorov cascade with the turbulent energy injected at a scale considerably larger than ∼10⁻² pc, the resulting electron density fluctuations in the IGM make a negligible contribution to the observed temporal scattering.

Due to the high heating rate at small scales in the IGM, any small-scale density enhancement would be rapidly erased by the thermal streaming motions in the IGM (Cordes et al. 2016). For this reason, the scenario in which the scattering medium is concentrated and localized in a thin layer in the IGM may not reflect the reality. Based on this questionable assumption, one tends to overestimate the contribution to the pulse broadening from the IGM.

(2) Scattering in the host galaxy ISM: In the multiphase ISM of the Galaxy, the distribution of the electron density fluctuations throughout the diffuse WIM is described by a Kolmogorov spectrum (Armstrong et al. 1995; Chepurnov & Lazarian 2010), but exhibits a much shallower spectrum in the supersonic turbulence prevalent in inner regions of the Galaxy (e.g., Lazarian 2009; Hennebelle & Falgarone 2012). By assuming that the host galaxy of an FRB is similar to the Galaxy and the general properties of turbulence are applicable, we next attribute the strong scattering to the propagation of radio waves through the ISM of the host galaxy and analyze the scattering effects from the Kolmogorov and short-wave-dominated density spectra, respectively.

We again start with the Kolmogorov power law of turbulence. The resulting W from Equation (16) is

$$\begin{eqnarray}W & \sim & \displaystyle \frac{0.0065}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-2}{\mathrm{cm}}^{-3}}\right)}^{2}{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{-\tfrac{1}{3}}{\left(\displaystyle \frac{L}{{10}^{-3}\mathrm{pc}}\right)}^{-1}\,{\rm{ms}},\quad {r}_{{\rm{diff}}}\lt {l}_{0},\end{eqnarray} \tag{ 25a }$$

$$\begin{eqnarray}W & \sim & \displaystyle \frac{1.9}{{(1+{z}_{{\rm{q}}})}^{3.4}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{2.2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4.4}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-2}{\mathrm{cm}}^{-3}}\right)}^{2.4}{\left(\displaystyle \frac{L}{{10}^{-3}\mathrm{pc}}\right)}^{-0.8}\,{\rm{ms}},\quad {r}_{{\rm{diff}}}\gt {l}_{0}.\end{eqnarray} \tag{ 25b }$$

Here the normalization of D_eff is assigned a typical galaxy size and δn_e the electron density in diffuse ISM, i.e., δn_e ∼ n_e. At r_diff < l₀, using Equation 21(a), we have

$$\begin{eqnarray}\displaystyle \frac{{l}_{0}}{L} & \gt & 3.8\times {10}^{-8}{(1+{z}_{{\rm{q}}})}^{1.2}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{-0.6}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-1.2}\\ & & \times {\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-2}{\mathrm{cm}}^{-3}}\right)}^{-1.2}{\left(\displaystyle \frac{L}{{10}^{-3}\mathrm{pc}}\right)}^{-0.6}.\end{eqnarray} \tag{ 26 }$$

Substituting the lower limit of the ratio l₀/L into Equation 25(a) yields the consistent result on the value of W as in the case of r_diff > l₀. We see that L inferred from the millisecond pulse broadening is far smaller than the injection scale of the turbulence throughout the Galactic WIM, which is suggested to be on the order of ∼100 pc by measuring the spectra of interstellar density fluctuations (Armstrong et al. 1995; Chepurnov & Lazarian 2010). It is also below the smaller outer scale of a few parsecs of the turbulence found in the Galactic spiral arms (Minter & Spangler 1996; Haverkorn et al. 2004). This heightens the challenge to interpreting the driving mechanism of the Kolmogorov turbulence as well as the cooling efficiency in the host galaxy.

A plausible solution is that a short-wave-dominated spectrum of electron density fluctuations, which is extracted from the observations of the inner Galaxy, also applies in the ISM of the host galaxy. As the density power spectrum becomes flat in supersonic turbulence, if the turbulent ISM of the host galaxy through which the LOS traverses contains highly supersonic turbulent motions and as a result is characterized by numerous small-scale clumpy density structures, we expect that the spectrum of electron density fluctuations deviates from the Kolmogorov power law and has β < 3.

In the above calculations, we assume that the volume filling factor f of the scattering material is comparable to unity, which is valid for a long-wave-dominated density spectrum characterized by large-scale density fluctuations. For small-scale clumpy density structures described by a short-wave-dominated density spectrum, however, it is necessary to consider that only a fraction of volume is filled by the overdense regions and replace δn_e with $\sqrt{f}\delta {n}_{e}$. In the case of the Galactic ISM, the WIM phase where the Kolmogorov density spectrum is present has f ∼ 25%. In contrast, the filling factors of the cold neutral medium and molecular clouds are as low as 1% and 0.05% (Tielens 2005; Haverkorn & Spangler 2013). In these colder and denser phases, which only fill a small fraction of the volume, the short-wave-dominated density spectrum gives rise to small-scale density structures with the spatial profile of the density field characterized by peaks of mass as a result of strong shocks (see Figure 2 in Kim & Ryu 2005). Therefore, the small-scale density structures created within these phases have an even smaller value of f. Accordingly, we include the effect of a small filling factor in the case of a short-wave-dominated density spectrum, so as to reach a more realistic evaluation of the scattering produced by the supersonic turbulence in the host ISM.

In the case of r_diff < l₀, by inserting Equations 4(b) and (15) into Equation (9), we find

$$\begin{eqnarray}{l}_{0} & \gt & {(\pi {r}_{e}^{2}{D}_{{\rm{eff}}}{\lambda }^{2}f{(\delta {n}_{e})}^{2})}^{-1}\\ & = & 1.3\times {10}^{7}{(1+{z}_{{\rm{q}}})}^{2}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{-1}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-2}\\ & & \times {\left(\displaystyle \frac{f}{{10}^{-6}}\right)}^{-1}{\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-1}{\mathrm{cm}}^{-3}}\right)}^{-2}\,{\rm{cm}}.\end{eqnarray} \tag{ 27 }$$

Given the parameters adopted in the above expression, the minimum l₀ is comparable to the inner scale of the density spectrum in the Galactic ISM inferred from observations (Spangler & Gwinn 1990; Armstrong et al. 1995; Bhat et al. 2004). By using a larger value of l₀ and substituting the normalization parameters into Equation 18(a) for a short-wave-dominated spectrum of density fluctuations, we get

$$\begin{eqnarray}W & \sim & \displaystyle \frac{6.5}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}\\ & & \times \left(\displaystyle \frac{f}{{10}^{-6}}\right){\left(\displaystyle \frac{\delta {n}_{e}}{{10}^{-1}{\mathrm{cm}}^{-3}}\right)}^{2}{\left(\displaystyle \frac{{l}_{0}}{{10}^{-10}\mathrm{pc}}\right)}^{-1}\,{\rm{ms}}.\end{eqnarray} \tag{ 28 }$$

As r_diff is below the inner scale of density power spectrum, the scaling presented in the above equation is independent of the spectral slope β of density fluctuations. It shows that clumps of electron density 0.1 cm⁻³ and size 10⁻¹⁰ pc (∼10⁸ cm), which occupy a small fraction of the volume of the host galaxy, would be adequate to produce the observed scattering delay.

Individual clumps of excess electrons have been included for modeling the Galactic distribution of electrons and scattering properties of Galactic ISM (Pynzar' & Shishov 1999; Cordes & Lazio 2003; Cordes et al. 2016). The clumpy component of the ionized plasma introduced in these studies are associated with discrete H ii regions or supernova remnants with a characteristic scale of ∼1 pc (Haverkorn et al. 2004). However, based on Equation (28) we note that the density fluctuations appearing on parsec scales, unless the local density is extraordinarily high, are unable to cause the intense scattering related with some FRBs. In contrast, we consider much smaller-scale density structures corresponding to a short-wave-dominated density spectrum with a sufficiently small inner scale. If the host galaxy medium is dominated by supersonic turbulence, in accordance with the concentrated density distribution induced by shock compression, the spectral form is dominated by the formation of small-scale density fluctuations and exhibits a rather shallow slope.

Compared with the above situation with r_diff < l₀ (Equation (28)), the density spectrum in the case of r_diff > l₀ can lead to a significantly larger degree of scattering due to the stronger dependence of W on the physical parameters involved (see Equations 18(b) and 19(b)). When the β value can be determined, the scaling relations presented in Equation 18(b) (or Equation 19(b)) can be used to constrain the turbulence properties. This small-scale properties of turbulent density can account for more pronounced scattering observed for some FRBs, and can also provide a plausible scattering source for the Galactic pulsars with high DMs (Xu & Zhang 2016b). It implies that, with similar properties to that of the Galaxy, the host galaxy is adequate to provide the observed scattering strength for an FRB.

(3) Locations of scattering and dispersion: The above results inform us that a long-wave-dominated power-law spectrum, e.g., the Kolmogorov spectrum, of electron density fluctuations with a reasonably large outer scale of turbulence in both the diffuse IGM and the host galaxy medium are incapable of producing the millisecond scattering tail. A short-wave-dominated electron density spectrum with β < 3 from the ISM of the host galaxy can easily render the host galaxy a strong scatterer. The excess fluctuation power at small scales characterized by a short-wave-dominated density spectrum gives rise to enhanced diffractive scattering and thus strong temporal broadening of a transient pulse.

A short-wave-dominated spectrum of density fluctuations in Galactic ISM can also produce the desired amount of scattering for FRBs. However, most of the known FRBs were discovered at high Galactic latitudes in directions through the WIM component of the ISM, where the turbulence is transonic (Haffner et al. 1999; Kim & Ryu 2005; Hill et al. 2008) and the density fluctuations follow a Kolmogorov spectrum (Armstrong et al. 1995; Chepurnov & Lazarian 2010) with little scattering effect. Comparisons with the Galactic pulsars detected at comparable latitudes confirm the negligible Galactic contribution to the temporal broadening of FRBs (Cordes et al. 2016). In fact, the heavy scattering from the supersonic turbulence that pervades the inner Galaxy prevents the detection of FRBs (Cordes et al. 2016).

It is commonly accepted that the diffuse IGM makes an unimportant contribution to scattering. Instead, intervening galactic halos along the LOS are appealed to for explaining the observed scattering (Yao et al. 2016). Indeed, if the intervening ISM happens to be in a state of supersonic turbulence, and located close to us with a small reduction factor, which depends on redshift, the intervening galaxy would dominate the scattering. However, we regard this scenario as implausible because for a source at a cosmological distance, the probability for the LOS to intersect with an intervening galaxy is very low, e.g., ≤5% within z_q ∼ 1.5 (Macquart & Koay 2013; Roeder & Verreault 1969), and the probability for the intervening ISM to be supersonically turbulent is further lower. This is in contradiction with the fact that around half of the known FRBs have detectable scattering tails (Petroff et al. 2016).

After identifying the host galaxy medium as the most promising candidate for dominating the observed scattering, we see from Equation (28) that

$$\begin{eqnarray}W & \sim & \displaystyle \frac{6.5}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}\left(\displaystyle \frac{f}{{10}^{-6}}\right){\left(\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\right)}^{2}\\ & & \times {\left(\displaystyle \frac{{l}_{0}}{{10}^{-10}\mathrm{pc}}\right)}^{-1}{\left(\displaystyle \frac{{\rm{DM}}}{100{{\rm{pc}}{\rm{cm}}}^{-3}}\right)}^{2}\,{\rm{ms}}.\end{eqnarray} \tag{ 29 }$$

The dependence of W on DM is affected by the turbulence properties in the surrounding ISM of the source. Under the condition of a short-wave-dominated spectrum of density fluctuations, strong scattering does not entail large DM in the host medium. As the Galactic contribution to the total DM is minor compared with its extragalactic component (Cordes et al. 2016), the IGM is most likely the dominant location for the observed DMs of FRBs.

The FRB data exhibit considerable scatter around any modeled (Caleb et al. 2016) or fitted (Yao et al. 2016) scattering time–DM relation. After considering an order-of-magnitude scatter similar to the case of Galactic pulsars (Bhat et al. 2004), one still cannot reach a satisfactory fit of the intergalactic scattering model to the FRB data (Caleb et al. 2016). As suggested in Caleb et al. (2016), the LOS-dependent inhomogeneity in the Galactic ISM (Johnston et al. 1998; Cordes & Lazio 2002) may not apply to the IGM, which further poses difficulty for the IGM scattering scenario. Besides, by plotting the scattering time versus DM for high-Galactic-latitude FRBs, Katz (2016a) claimed that no correlation between the two variables can be seen. More plausibly, scattering and dispersion are separately dominated by the host galaxy and the IGM. As shown above, the scattering time is largely affected by the turbulence properties (e.g., β > 3 or β < 3) and scattering regimes (r_diff < l₀ or r_diff > l₀). Therefore, the variation of the scattering time for FRBs can be attributable to the diverse interstellar environments of their host galaxies. From the observational point of view, it is also necessary to point out that the estimated scattering time is subject to effects such as the signal-to-noise ratio and limited temporal resolution due to dispersion smearing, leading to non-negligible uncertainties in the observationally measured scattering time–DM relation.

For intergalactic plasmas, the ion collision frequency ν_ii is much lower than the cyclotron frequency Ω_i, and accordingly, the mean free path of ions (Braginskii 1965)

$$\begin{eqnarray}{\lambda }_{{\rm{mfp}}} & = & \displaystyle \frac{{v}_{\mathrm{th},i}}{{\nu }_{{ii}}}=\displaystyle \frac{3\sqrt{2}{({k}_{{\rm{B}}}T)}^{2}}{4\sqrt{\pi }\mathrm{ln}{\rm{\Lambda }}{e}^{4}{n}_{i}}\\ & = & 2.15\times {10}^{21}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-1}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{2}{\left(\displaystyle \frac{{n}_{i}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-1}\,{\rm{cm}}\end{eqnarray} \tag{ 35 }$$

is significantly larger than the ion gyroradius

$$\begin{eqnarray}{l}_{i} & = & \displaystyle \frac{{v}_{\mathrm{th},i}}{{{\rm{\Omega }}}_{i}}=\displaystyle \frac{{v}_{\mathrm{th},i}{m}_{i}c}{{eB}}\\ & = & 4.2\times {10}^{9}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{\tfrac{1}{2}}{\left(\displaystyle \frac{B}{0.1\mu {\rm{G}}}\right)}^{-1}\,{\rm{cm}},\end{eqnarray} \tag{ 36 }$$

where ${v}_{\mathrm{th},i}=\sqrt{2{k}_{{\rm{B}}}T/{m}_{i}}$ is the ion thermal speed, and k_B, $\mathrm{ln}{\rm{\Lambda }}$, n_i, and c are the Boltzmann constant, Coulomb logarithm, ion number density, and speed of light, respectively. The magnetic field strength B is taken as the inferred value from the Faraday rotation measures of polarized extragalactic sources (Ryu et al. 1998; Xu et al. 2006). We also treat the IGM as a fully ionized hydrogen plasma, so ions have the same charge e and mass m_i = m_H as protons.

The weakly collisional and magnetized IGM is subject to firehose and mirror instabilities driven by pressure anisotropies with respect to the local magnetic field direction (Fabian 1994; Carilli & Taylor 2002; Schekochihin et al. 2005; Rincon et al. 2015). The instability growth rate increases with wave numbers, resulting in fluctuating magnetic fields peaking at a plasma microscale comparable to the ion gyro-scale l_i (Schekochihin & Cowley 2006). The compressive mirror instability induces variations in density, which are anti-correlated with the magnetic field variations. The fluctuations in density and magnetic field are related as (Hall 1980)

$$\begin{eqnarray}&&\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\sim \displaystyle \frac{\delta B}{B},\end{eqnarray} \tag{ 37 }$$

where δn_e, δB and n_e, B are the fluctuating and uniform components of electron density and magnetic field strength, respectively. If the density perturbation δn_e(d) at d = l_i ∼ 4.2 × 10⁹ cm (Equation (36)) is sufficient to account for strong scattering, Equation (34) sets the lower limit of density perturbation at d,

$$\begin{eqnarray}\delta {n}_{e}(d) & \gt & 5.5\times {10}^{-9}(1+{z}_{{\rm{q}}}){\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{-\tfrac{1}{4}}{\left(\displaystyle \frac{B}{0.1\mu {\rm{G}}}\right)}^{\tfrac{1}{2}}\,{{\rm{cm}}}^{-3}.\end{eqnarray} \tag{ 38 }$$

Inserting the above expression and Equation (36) into Equation (32) results in

$$\begin{eqnarray}&&W\gt \displaystyle \frac{1.5\times {10}^{3}}{1+{z}_{{\rm{q}}}}\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\,{\rm{Gpc}}}\right){\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{2}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{-1}{\left(\displaystyle \frac{B}{0.1\mu {\rm{G}}}\right)}^{2}\,{\rm{ms}}.\end{eqnarray} \tag{ 39 }$$

The predicted timescale is obviously inconsistent with the observed FRB pulses with millisecond or shorter durations. To accommodate the observations, the saturated amplitude of the density fluctuations and the associated magnetic fluctuations generated by plasma instabilities should remain at a marginal level, so that the strong scattering cannot be realized. We can see from Equation (38) that by adopting an average electron density as n_e = 10⁻⁷ cm⁻³, a conservative estimate of the magnetic field and density perturbations near l_i is

$$\begin{eqnarray}&&\displaystyle \frac{\delta B}{B}\sim \displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\lt \displaystyle \frac{5.5\times {10}^{-9}\,{\mathrm{cm}}^{-3}}{{10}^{-7}\,{\mathrm{cm}}^{-3}}=0.055.\end{eqnarray} \tag{ 40 }$$

This result suggests that although the micro-plasma instabilities have a fast growth rate in comparison with the large-scale turbulent motions, they are mostly suppressed over the fluid timescale. As demonstrated by earlier works, the enhanced particle scattering originating from the plasma instabilities can effectively relax the pressure anisotropy and increase the collision rate. As a result, both the turbulent cascade over small scales and efficient magnetic field amplification can be facilitated (Lazarian & Beresnyak 2006; Santos-Lima et al. 2014). This naturally explains the magnetization and turbulent motions in the IGM inferred from the observations (e.g., Ryu et al. 2008). By taking into account the relaxation effect of pressure anisotropy, the collisionless MHD simulations carried out by Santos-Lima et al. (2014) exhibit the statistical properties of turbulence similar to that of collisional-MHD turbulence, which justifies a collisional-MHD description of collisionless plasmas at the intracluster medium (and IGM) conditions. The observed pulse widths of transient radio sources at cosmological distances, like the FRBs, offer a strong argument supporting the above picture of the IGM turbulence, whereas the model of nonlinear evolution of the plasma instabilities with a secular growth of small-scale magnetic field fluctuations to large amplitudes, δB/B ∼ 1, is disfavored (Schekochihin et al. 2008; Rincon et al. 2015).

Corresponding to the large mean free path of ions in the IGM, the viscosity parallel to magnetic field lines is

$$\begin{eqnarray}{\nu }_{i} & = & {\lambda }_{{\rm{mfp}}}{v}_{\mathrm{th},i}=\displaystyle \frac{3{({k}_{{\rm{B}}}T)}^{\tfrac{5}{2}}}{2\sqrt{\pi }\mathrm{ln}{\rm{\Lambda }}\sqrt{{m}_{i}}{e}^{4}{n}_{i}}\\ & = & 8.7\times {10}^{27}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-1}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{\tfrac{5}{2}}\\ & & \times {\left(\displaystyle \frac{{n}_{i}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-1}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 41 }$$

It damps the turbulent cascade at a large viscous scale, which can be obtained by equaling the turbulent cascading rate

$$\begin{eqnarray}&&{\tau }_{{\rm{cas}}}^{-1}=\displaystyle \frac{{v}_{l}}{l}={k}^{\tfrac{2}{3}}{L}^{-\tfrac{1}{3}}{V}_{L}\end{eqnarray} \tag{ 42 }$$

with the viscous damping rate k² ν_i. Here we use the Kolmogorov scaling, where v_l is the turbulent velocity at scale l, V_L is the turbulent velocity at the injection scale L, and k = 1/l is the wavenumber. The viscous scale calculated by using the parallel viscosity is

$$\begin{eqnarray}{l}_{0} & = & {L}^{\tfrac{1}{4}}{V}_{L}^{-\tfrac{3}{4}}{\nu }_{i}^{\tfrac{3}{4}}\\ & = & 3.78\times {10}^{21}{\left(\displaystyle \frac{L}{100\mathrm{kpc}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{V}_{L}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{3}{4}}\\ & & {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-\tfrac{3}{4}}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{\tfrac{15}{8}}{\left(\displaystyle \frac{{n}_{i}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{3}{4}}\,{\rm{cm}}.\end{eqnarray} \tag{ 43 }$$

The viscous-scale eddies are responsible for the random stretching of magnetic field lines that drives an exponential growth of the initially weak magnetic energy at a rate equal to the viscous-eddy turnover rate. As mentioned in Section 3.1, the particle scattering in the presence of the plasma instabilities makes the effective parallel viscosity sufficiently small, and thus the corresponding dynamo growth rate becomes fast (Schekochihin & Cowley 2006; Santos-Lima et al. 2014), so that the kinematic dynamo process can be efficient enough to generate strong magnetic fields within the cluster lifetime.

In addition, the ordinary Spitzer resistivity in the IGM is negligibly small (Spitzer 1956),

$$\begin{eqnarray}\eta & = & \displaystyle \frac{{c}^{2}\sqrt{{m}_{e}}{e}^{2}\mathrm{ln}{\rm{\Lambda }}}{4{({k}_{{\rm{B}}}T)}^{\tfrac{3}{2}}}\\ & = & 3.05\times {10}^{5}\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right){\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{-\tfrac{3}{2}}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 44 }$$

Thus the magnetic Prandtl number P_m = ν_i/η ∼ 10²² (Equations (41) and (44)) in the IGM is high, and magnetic fluctuations can be developed in the viscosity-damped regime of MHD turbulence (Cho et al. 2002, 2003; Lazarian et al. 2004). During the dynamo growth of magnetic energy, the stretched magnetic fields form a folded structure in the sub-viscous range, with the field variation along the field lines at the viscous scale l₀ and the field direction reversal at the resistive scale (Schekochihin et al. 2004; Goldreich & Sridhar 2006; Lazarian 2007; Braithwaite 2015). The folded magnetic fields compress gas into dense sheet-like structures. Such dense sheets have been invoked to explain the formation of the small ionized and neutral structures (SINS) in the partially ionized ISM (Dieter et al. 1976; Heiles 1997; Stanimirović et al. 2004) by Lazarian (2007), and is also proposed as the source of extreme diffractive scattering in the Galactic center by Goldreich & Sridhar (2006).

However, as regards the fully ionized IGM environment, the persistence of the folded structure of magnetic fields is speculative. First, the folded structure, especially its curved part, is unstable to the plasma instabilities and the resulting thickness of the fold can be much larger than the resistive scale (Schekochihin et al. 2005). Moreover, not only can the parallel viscosity be effectively reduced, but the viscosity perpendicular to magnetic field lines also substantially decreases with increasing field strength (Simon 1955),

$$\begin{eqnarray}{\nu }_{i,\perp } & = & \displaystyle \frac{3{k}_{{\rm{B}}}T{\nu }_{{ii}}}{10{{\rm{\Omega }}}_{i}^{2}{m}_{i}}=5.1\times {10}^{3}\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right){\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{-\tfrac{1}{2}}\\ & & \times \left(\displaystyle \frac{{n}_{i}}{{10}^{-7}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{B}{0.1\mu {\rm{G}}}\right)}^{-2}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 45 }$$

It implies that the turbulent motions perpendicular to magnetic field lines are undamped at the viscous scale l₀ (Equation (43)) derived from the parallel viscosity and can initiate a cascade of Alfvénic turbulence at smaller scales down to the cutoff scale determined by the much smaller perpendicular viscosity, which tends to violate the preservation of the folded structure of magnetic fields on scales below l₀.

In the following analysis, we nevertheless presume that the magnetic fields appear in folds with undetermined thickness at scales below l₀, and the local magnetic perturbation is determined by the equilibrium between the turbulent energy at l₀ and the magnetic-fluctuation energy,

$$\begin{eqnarray}&&\delta B=\sqrt{4\pi {\rho }_{i}}{v}_{0}=\sqrt{4\pi {\rho }_{i}}{V}_{L}{\left(\displaystyle \frac{{l}_{0}}{L}\right)}^{\tfrac{1}{3}},\end{eqnarray} \tag{ 46 }$$

where ρ_i = m_H n_i is the average mass density of ions. In pressure equilibrium, the density perturbation across the sheet of folded fields is approximately given by the ratio between the local magnetic and gas pressure (Lazarian 2007),

$$\begin{eqnarray}&&\displaystyle \frac{\delta {n}_{e}}{{n}_{e}}\sim \displaystyle \frac{{P}_{{\rm{B}}}}{{P}_{g}},\end{eqnarray} \tag{ 47 }$$

with the magnetic pressure P_B = (δB)²/8π, and the thermal pressure

$$\begin{eqnarray}&&{P}_{g}={P}_{i}+{P}_{e}={n}_{i}{k}_{{\rm{B}}}T+{n}_{e}{k}_{{\rm{B}}}T=2{n}_{i}{k}_{{\rm{B}}}T,\end{eqnarray} \tag{ 48 }$$

where the number density of ions n_i and electrons n_e are equal. Therefore, we can get (Equations (43), (46)–(48))

$$\begin{eqnarray}\displaystyle \frac{\delta {n}_{e}}{{n}_{e}} & = & \displaystyle \frac{{(\delta B)}^{2}}{16\pi {n}_{i}{k}_{{\rm{B}}}T}=\displaystyle \frac{{m}_{i}{V}_{L}^{2}{l}_{0}^{\tfrac{2}{3}}}{4{k}_{{\rm{B}}}{{TL}}^{\tfrac{2}{3}}}\\ & = & 0.16{\left(\displaystyle \frac{L}{100\mathrm{kpc}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{V}_{L}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{\tfrac{3}{2}}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{n}_{i}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{1}{2}}.\end{eqnarray} \tag{ 49 }$$

By taking δn_e (d)/n_e ∼ 0.16 from above expression and n_e = 10⁻⁷ cm⁻³, the condition of strong scattering requires (Equation (33))

$$\begin{eqnarray}&&d\gt 5.0\times {10}^{8}{(1+{z}_{{\rm{q}}})}^{2}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{-1}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-2}\,{\rm{cm}},\end{eqnarray} \tag{ 50 }$$

with the lower limit smaller than l_i (Equation (36)). It implies that the density perturbation we adopt for the folded structure at any sub-viscous scale can contribute to strong scattering. We have demonstrated in Section 2.2 that the intergalactic scattering is likely weak. Therefore, in accordance with the observationally determined scattering timescale W, the lower limit of the characteristic sheet thickness is set by (Equation (32)),

$$\begin{eqnarray}d & \gt & \displaystyle \frac{{D}_{{\rm{eff}}}^{2}{r}_{e}^{2}{\lambda }_{0}^{4}}{4\pi c{(1+{z}_{{\rm{q}}})}^{3}}\displaystyle \frac{{(\delta {n}_{e}(d))}^{2}}{W}\\ & = & \displaystyle \frac{5.2\times {10}^{13}}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1{\rm{Gpc}}}\right)}^{2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}{\left(\displaystyle \frac{W}{1{\rm{ms}}}\right)}^{-1}\,{\rm{cm}}.\end{eqnarray} \tag{ 51 }$$

As expected, it is larger than the resistive scale, which can be calculated from Equations (41), (43), and (44),

$$\begin{eqnarray}{l}_{R} & = & {l}_{0}{P}_{m}^{-\tfrac{1}{2}}=2.2\times {10}^{10}{\left(\displaystyle \frac{L}{100\mathrm{kpc}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{V}_{L}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{3}{4}}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{T}{{10}^{5}{\rm{K}}}\right)}^{-\tfrac{1}{8}}{\left(\displaystyle \frac{{n}_{i}}{{10}^{-7}{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{1}{4}}\,{\rm{cm}}.\end{eqnarray} \tag{ 52 }$$

For the small-scale folded magnetic fields generated by fluctuation dynamo, besides the geometrical structure that is related with the scattering effects on radiation propagation, in terms of one-dimensional magnetic energy spectrum in the viscosity-dominated regime, a distinctive spectral slope of k⁻¹ has been analytically derived by Lazarian et al. (2004) and numerically confirmed by Cho et al. (2002). The detection of such a spectral index and comparison between the measured spectral cutoff scale and the lower limit of sheet thickness in Equation (51) can verify the existence of the folded magnetic fields and provide more definite information on the properties of the viscosity-damped regime of turbulence.

The folded structure of magnetic fields in the sub-viscous range of turbulence can also be present in the ISM of the host galaxy. We next follow the similar calculations as shown above, but use the environment parameters for the Galactic WIM (McKee & Ostriker 1977), which account for most of the ionized gas within the Galactic ISM (Haffner et al. 2009) and are taken as an example of the fully ionized phase of the host galaxy medium.

Given the parallel viscosity (Equation (41))

$$\begin{eqnarray}&&{\nu }_{i}=1.6\times {10}^{19}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-1}{\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{\tfrac{5}{2}}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-1}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 53 }$$

and the Spitzer resistivity (Equation (44))

$$\begin{eqnarray}&&\eta =1.3\times {10}^{7}\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right){\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{-\tfrac{3}{2}}\,{{\rm{cm}}}^{2}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 54 }$$

the WIM phase has a large P_m,

$$\begin{eqnarray}{P}_{m} & = & \displaystyle \frac{{\nu }_{i}}{\eta }=1.2\times {10}^{12}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-2}\\ & & \times {\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{4}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-1}.\end{eqnarray} \tag{ 55 }$$

The resulting resistive scale

$$\begin{eqnarray}{l}_{R} & = & {l}_{0}{P}_{m}^{-\tfrac{1}{2}}=7.2\times {10}^{8}{\left(\displaystyle \frac{L}{30\mathrm{pc}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{V}_{L}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{3}{4}}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{-\tfrac{1}{8}}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{1}{4}}\,{\rm{cm}}\end{eqnarray} \tag{ 56 }$$

is smaller than the ion mean free path (Equation (35))

$$\begin{eqnarray}&&{\lambda }_{{\rm{mfp}}}=1.4\times {10}^{13}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-1}{\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{2}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-1}\,{\rm{cm}},\end{eqnarray} \tag{ 57 }$$

and thus falls in the collisionless regime. Similar to the IGM plasma, the folded structure of magnetic fields can be significantly affected by the plasma instabilities and turbulent cascade at small scales. Nevertheless, to seek the possibility of enhanced scattering introduced by different structures of magnetic fields arising in the host galaxy medium, we suppose that the folded fields survive at scales below the viscous scale l₀ (Equation (43)),

$$\begin{eqnarray}{l}_{0} & = & 7.8\times {10}^{14}{\left(\displaystyle \frac{L}{30\mathrm{pc}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{V}_{L}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-\tfrac{3}{4}}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-\tfrac{3}{4}}{\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{\tfrac{15}{8}}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{3}{4}}\,{\rm{cm}}.\end{eqnarray} \tag{ 58 }$$

In the case of the WIM, the turbulent cascade along the long-wave-dominated Kolmogorov spectrum over an extended inertial range leads to small turbulent fluctuations at l₀. So the corresponding density perturbation given by Equation (49) is relatively small,

$$\begin{eqnarray}\displaystyle \frac{\delta {n}_{e}}{{n}_{e}} & = & \displaystyle \frac{{m}_{i}{V}_{L}^{2}{l}_{0}^{\tfrac{2}{3}}}{4{k}_{{\rm{B}}}{{TL}}^{\tfrac{2}{3}}}\\ & = & 1.6\times {10}^{-4}{\left(\displaystyle \frac{L}{30\mathrm{pc}}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{{V}_{L}}{10\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{\tfrac{3}{2}}\\ & & \times {\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-\tfrac{1}{2}}{\left(\displaystyle \frac{T}{8000{\rm{K}}}\right)}^{\tfrac{1}{4}}{\left(\displaystyle \frac{{n}_{i}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{1}{2}}.\end{eqnarray} \tag{ 59 }$$

It follows that to fulfill the strong scattering condition, the characteristic scale of the density fluctuations should be sufficiently large (Equation (33)),

$$\begin{eqnarray}&&d\gt 5.3\times {10}^{8}{(1+{z}_{{\rm{q}}})}^{2}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{-1}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{-2}\,{\rm{cm}},\end{eqnarray} \tag{ 60 }$$

where δn_e (d)/n_e ∼ 1.6 × 10⁻⁴ and n_e = 0.1 cm⁻³ are used. But in the meantime, as the density perturbation is rather weak, only with a small value of d can the millisecond pulse duration be reached (Equation (32))

$$\begin{eqnarray}&&d\leqslant \displaystyle \frac{4.9\times {10}^{7}}{{(1+{z}_{{\rm{q}}})}^{3}}{\left(\displaystyle \frac{{D}_{{\rm{eff}}}}{1\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{{\lambda }_{0}}{1{\rm{m}}}\right)}^{4}{\left(\displaystyle \frac{W}{1{\rm{ms}}}\right)}^{-1}\,{\rm{cm}}.\end{eqnarray} \tag{ 61 }$$

The thickness of the sheet-like structure in  the density field is expected to be larger than l_R (Equation (56)) due to the effect of plasma instabilities (Schekochihin et al. 2005), and thus larger than the value indicated from the above equation, leading to insignificant pulse broadening.

This result shows that the density fluctuations induced by the folded structure of magnetic fields in the WIM-like environment are inadequate to render the host galaxy a strong scatterer. It has been suggested earlier by Goldreich & Sridhar (2006) that the large density contrast associated with the folded fields suffices for interpreting the extreme scattering of radio waves taking place in the Galactic center. Besides different environment parameters employed, as the major difference between our analysis and their work, we use the local magnetic field fluctuations with the magnetic energy equal to the turbulent energy at the viscous scale in deriving the density perturbation, rather than the magnetic field coherent on the scale of the largest turbulent eddy taken in Goldreich & Sridhar (2006), which has a much stronger strength than the perturbed field on the scale of the smallest eddy. The scenario described in Goldreich & Sridhar (2006) can be realized when the forcing scale of turbulence is comparable to the viscous scale and the inertial range of turbulence is absent. Otherwise the folded fields only emerge in the sub-viscous region with larger-scale magnetic perturbations irrelevant in determining the local density structure.

It is also necessary to point out that we use the isotropic Kolmogorov scaling for analytical simplicity in Sections 3.2 and 3.3. But in fact, as the magnetic field becomes dynamically important, anisotropic MHD turbulence develops with the turbulent eddies more elongated along the local magnetic field direction toward smaller scales. Then the Goldreich & Sridhar (1995) scaling applies as a more appropriate description of the relation between the parallel and perpendicular scales with respect to the local magnetic field. If one takes into account the effect of turbulence anisotropy in the above calculations, the viscous damping rate k²ν_i is replaced by ${k}_{\parallel }^{2}{\nu }_{i}$, and the latter is relatively small. Here ${k}_{\parallel }$ and k_⊥ are the parallel and perpendicular components of wavevector ${\boldsymbol{k}}$. Accordingly, the viscous scale is shifted downward and the corresponding density fluctuations are further reduced (Equation (49)), leading to a less important contribution of the plasma sheets in scatter broadening.