Dispersion Measure Variation of Repeating Fast Radio Burst Sources

The largest-scale effect of the DM variation may be due to the expansion of the universe. The mean DM caused by the IGM is given by (e.g., Deng & Zhang 2014; Yang & Zhang 2016)⁶

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{IGM}}=\displaystyle \frac{3{{cH}}_{0}{{\rm{\Omega }}}_{{\rm{b}}}{f}_{\mathrm{IGM}}}{8\pi {{Gm}}_{p}}{\int }_{0}^{z}\displaystyle \frac{{f}_{e}({z}^{{\prime} })(1+{z}^{{\prime} })}{\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{(1+{z}^{{\prime} })}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}}{{dz}}^{{\prime} },\end{eqnarray} \tag{ 2 }$$

where H₀ is the current Hubble constant, ${{\rm{\Omega }}}_{{\rm{b}}}$, ${{\rm{\Omega }}}_{{\rm{m}}}$, and ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$ are the current mass fractions of baryon, matter, and dark energy, respectively, ${f}_{\mathrm{IGM}}$ is the fraction of baryon mass of the IGM that is adopted as ∼0.83 (Fukugita et al. 1998; Shull et al. 2012), and ${f}_{e}={n}_{e,\mathrm{free}}/({n}_{p}+{n}_{n})$ is the number ratio between free electrons and baryons (including protons and neutrons) in the IGM. For nearby FRBs at $z\lt 3$, since both hydrogen and helium are fully ionized, one has ${n}_{e,\mathrm{free}}\simeq {n}_{e}={n}_{p}$. On average, in the universe ${n}_{p}:{n}_{n}\simeq 7:1$, so one has ${f}_{e}\simeq 7/8$. According to Equation (2) and the redshift–time relation, e.g., ${dz}/{dt}=-{H}_{0}(1+z)\sqrt{{{\rm{\Omega }}}_{{\rm{m}}}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}$, the DM variation contributed by the Hubble expansion is therefore given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{IGM}}}{{dt}} & = & \displaystyle \frac{d{\mathrm{DM}}_{\mathrm{IGM}}}{{dz}}\displaystyle \frac{{dz}}{{dt}}=-\displaystyle \frac{3{{cH}}_{0}^{2}{{\rm{\Omega }}}_{{\rm{b}}}{f}_{\mathrm{IGM}}{f}_{e}}{8\pi {{Gm}}_{p}}{(1+z)}^{2}\\ & \simeq & -5.6\times {10}^{-8}{(1+z)}^{2}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 3 }$$

Here, we have adopted the flat ΛCDM parameters derived from the Planck data: ${H}_{0}=67.7\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1},\,{{\rm{\Omega }}}_{{\rm{m}}}=0.31,{{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.69$, and ${{\rm{\Omega }}}_{{\rm{b}}}=0.049$ (Planck Collaboration et al. 2016). Therefore, the DM variation due to Hubble expansion is very small. Interestingly, we note that Equation (3) does not depend on the cosmological parameters $({{\rm{\Omega }}}_{{\rm{m}}},{{\rm{\Omega }}}_{{\rm{\Lambda }}})$.

2.2.1. LSS Gravitational Potential Fluctuation

We consider the DM variation resulting from the fluctuations of the gravitational potential of the LSS. According to general relativity, the Shapiro delay time in a gravitational potential $U(r,t)$ is given by⁷

$$\begin{eqnarray}&&\delta {t}_{\mathrm{gra}}=-\displaystyle \frac{2}{{c}^{3}}{\int }_{{r}_{e}}^{{r}_{o}}U(r,t){dr},\end{eqnarray} \tag{ 4 }$$

where the integration is along the path of the photon emitted at r_e and received at r_o. As shown in Equation (4), for a localized FRB a fluctuation of the LSS gravitational potential along line of sight would cause a slight variation of the optical path, leading to a DM variation.

The gravitational potential of LSS can be calculated via the observed peculiar velocities, v_p, of galaxies (e.g., Nusser 2016). The gravitational acceleration produced by the mass distribution fluctuations may be approximated as $g\sim {v}_{p}/{t}_{\mathrm{acc}}\sim {v}_{p}{H}_{0}$, where ${t}_{\mathrm{acc}}$ is the acceleration timescale caused by the LSS, which corresponds to the only possible cosmological timescale, e.g., ${H}_{0}^{-1}$. For a sphere with a radius $R\sim 100\,\mathrm{Mpc}$ around us, the typical bulk peculiar velocity is ${v}_{p}\sim 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Thus, the LSS gravitational potential may be given by ${U}_{\mathrm{LSS}}\sim {gR}\sim {v}_{p}{{RH}}_{0}\sim 2.2\times {10}^{16}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-2}$ (Nusser 2016), which is of the same order of magnitude as the potential fluctuations inferred from the fluctuations of the cosmic microwave background (Bennett et al. 1994). The difference of the Shapiro delay time during the observed time T may be given by

$$\begin{eqnarray}| {\rm{\Delta }}t| & \simeq & \left|\displaystyle \frac{\partial (\delta {t}_{\mathrm{gra}})}{\partial t}\right|T\simeq \displaystyle \frac{2T}{{c}^{3}}{\displaystyle \int }_{{r}_{e}}^{{r}_{o}}\left|\displaystyle \frac{\partial U(r,t)}{\partial t}\right|{dr}\\ & \simeq & \displaystyle \frac{2{H}_{0}}{{c}^{3}}{U}_{\mathrm{LSS}}{DT}\\ & \simeq & 0.1\,\mathrm{hr}\left(\displaystyle \frac{{U}_{\mathrm{LSS}}}{2.2\times {10}^{16}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-2}}\right)\left(\displaystyle \frac{D}{1\,\mathrm{Gpc}}\right)\left(\displaystyle \frac{T}{1\,\mathrm{year}}\right),\end{eqnarray} \tag{ 5 }$$

where D is the distance between an FRB source and the observer. The observed time T corresponds to the time interval between the first and the second FRB detection over which the difference in DM is being tested. The evolution timescale of the LSS gravitation potential is approximately ${H}_{0}^{-1}$. The DM variation is estimated as

$$\begin{eqnarray}&&\left|\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{LSS},{\rm{G}}}}{{dt}}\right|\sim \displaystyle \frac{{n}_{e}c| {\rm{\Delta }}t| }{T}\simeq 3.5\times {10}^{-13}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ &&\quad \times \,\left(\displaystyle \frac{{n}_{e}}{{10}^{-7}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{{U}_{\mathrm{LSS}}}{2.2\times {10}^{16}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-2}}\right)\,\left(\displaystyle \frac{D}{1\,\mathrm{Gpc}}\right).\end{eqnarray} \tag{ 6 }$$

Here, we have assumed that the mean number density of the free electrons does not significantly change during the gravitational potential fluctuation. One can see that this effect is even less important than Hubble expansion.

2.2.2. LSS Density Fluctuation

Next, we consider gas density fluctuation in the LSS. As discussed above, for LSS, the only possible fluctuation timescale is ${H}_{0}^{-1}$. Thus, the DM variation caused by the LSS gas density fluctuation could be estimated as

$$\begin{eqnarray}\left|\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{LSS},{\rm{D}}}}{{dt}}\right| & \sim & | \delta {n}_{e}| {{DH}}_{0}\simeq 6.9\times {10}^{-9}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & \times & \left(\displaystyle \frac{| \delta {n}_{e}| }{{10}^{-7}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{D}{1\,\mathrm{Gpc}}\right),\end{eqnarray} \tag{ 7 }$$

where $\delta {n}_{e}$ is the number density fluctuation of the free electrons in IGM, and D is the distance between an FRB source and the observer. Note that even if $\delta {n}_{e}$ is taken to be of the same order of n_e, i.e., $| \delta {n}_{e}| \sim {n}_{e}$, the DM variation is negligibly small.

2.2.3. Density Fluctuation in Galaxy Groups and Clusters

The LSS density fluctuation leads to the formations of galaxy clusters or groups. Galaxy clusters and groups are the largest known self-gravitationally bound systems in the process of cosmic structure formation, which form in the densest part of the LSS of the universe (e.g., Bahcall 1999). Galaxy groups are the small aggregates of galaxies. There are 3–30 bright galaxies with a radius of $R\simeq (0.1\mbox{--}1){h}^{-1}\,\mathrm{Mpc}$. The typical random peculiar velocity of the galaxies in a group is ${v}_{p}\sim (100\mbox{--}500)\,\mathrm{km}\,{{\rm{s}}}^{-1}$, which corresponds to a typical group mass of $\sim ({10}^{12.5}\mbox{--}{10}^{14}){h}^{-1}\,{M}_{\odot }$. The groups exhibit a spatial number density of ${n}_{\mathrm{group}}\sim ({10}^{-3}\mbox{--}{10}^{-5}){h}^{3}\,{\mathrm{Mpc}}^{-3}$, and the fraction of galaxies in groups is $\sim 55 \%$ (e.g., Bahcall 1999). X-ray observations show that the central free electron density is about ${n}_{e,c}\lesssim {10}^{-3}\,{\mathrm{cm}}^{-3}$ for the groups (e.g., Kravtsov & Borgani 2012). Clusters are larger structures than groups, but there is no significant dividing line between clusters and groups. In general, there are 30–300 bright galaxies with a radius of $R\simeq (1\mbox{--}2){h}^{-1}\,\mathrm{Mpc}$ in a cluster. The galaxies in a cluster have larger random peculiar velocities, e.g., ${v}_{p}\sim (400\mbox{--}1400)\,\mathrm{km}\,{{\rm{s}}}^{-1}$, implying a typical cluster mass of $\sim ({10}^{14}\mbox{--}2\times {10}^{15}){h}^{-1}\,{M}_{\odot }$. The spatial number density of the clusters is ${n}_{\mathrm{cluster}}\sim ({10}^{-5}\mbox{--}{10}^{-6}){h}^{3}\,{\mathrm{Mpc}}^{-3}$, and the galaxy fraction is only $\sim 5 \%$ (Bahcall 1999). The central free electron density is about ${n}_{e,c}\sim {10}^{-3}\,{\mathrm{cm}}^{-3}$ for clusters.

In order to calculate the DM variation from the gas density fluctuation in groups or clusters, we first consider the probability of an FRB passing through groups or clusters. We assume that the distance between the FRB source and the observer is D. The "optical depth" (or chance probability) is approximately

$$\begin{eqnarray}&&{\tau }_{\mathrm{group}}\sim {n}_{\mathrm{group}}(\pi {R}^{2})D\\ &&\,\simeq \,0.03\left(\displaystyle \frac{{n}_{\mathrm{group}}}{{10}^{-3}\,{h}^{3}\,{\mathrm{Mpc}}^{-3}}\right)\,{\left(\displaystyle \frac{R}{0.1{h}^{-1}\mathrm{Mpc}}\right)}^{2}\,\left(\displaystyle \frac{D}{1\,\mathrm{Gpc}}\right)\end{eqnarray} \tag{ 8 }$$

to intersect a foreground group, and

$$\begin{eqnarray}&&{\tau }_{\mathrm{cluster}}\sim {n}_{\mathrm{cluster}}(\pi {R}^{2})D\\ &&\,\simeq \,0.03\left(\displaystyle \frac{{n}_{\mathrm{cluster}}}{{10}^{-5}{h}^{3}\,{\mathrm{Mpc}}^{-3}}\right)\,{\left(\displaystyle \frac{R}{1{h}^{-1}\mathrm{Mpc}}\right)}^{2}\,\left(\displaystyle \frac{D}{1\,\mathrm{Gpc}}\right)\end{eqnarray} \tag{ 9 }$$

to intersect a foreground cluster. Both probabilities are small. However, it is possible that FRBs are born in groups or clusters, and any extragalactic light always pass through our Local Group.

The DM variation contributed by proper motion of galaxies in a galaxy group or cluster can be estimated as

$$\begin{eqnarray}\left|\displaystyle \frac{d\mathrm{DM}}{{dt}}\right| & \sim & \displaystyle \frac{| \delta {n}_{e}| {v}_{\mathrm{flu}}R}{{l}_{\mathrm{flu}}}\\ & \sim & | \delta {n}_{e}| {v}_{p}\simeq 5.1\times {10}^{-9}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & \times & \left(\displaystyle \frac{| \delta {n}_{e}| }{{10}^{-5}\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{{v}_{p}}{500\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right),\end{eqnarray} \tag{ 10 }$$

where $\delta {n}_{e}$ is the density fluctuation of the electron gas, ${l}_{\mathrm{flu}}$ is the characteristic fluctuation scale in a cluster or group, and ${v}_{\mathrm{flu}}$ is the characteristic fluctuation velocity of the hot gas in the cluster or group. The hydrodynamic simulations showed that ${l}_{\mathrm{flu}}\lesssim R$ and ${v}_{\mathrm{flu}}\sim {v}_{p}$ (e.g., Gaspari et al. 2014). Since the electron gas density profile in a group or cluster is approximately ${n}_{e}\propto {r}^{-2}$ for $0.1\,{h}^{-1}\,{\mathrm{Mpc}}^{-3}\lt r\lt 1.5\,{h}^{-1}\,{\mathrm{Mpc}}^{-3}$, the mean electron number density is ${n}_{e}\sim 0.03{n}_{e,c}$. Even if $\delta {n}_{e}$ is taken to be of the same order of n_e, i.e., $| \delta {n}_{e}| \sim {n}_{e}$, the DM variation is negligibly small.

The components of the ISM include dense molecular gas, diffuse molecular gas, cold neutral medium, warm neutral medium, H ii gas (including warm ionized medium and H ii region), and coronal gas (e.g., Draine 2011). Among them, H ii gas and coronal gas are ionized components. H ii gas has a volume filling factor of $\sim 10 \%$ and a gas density spanning a wide range, $(0.3\mbox{--}{10}^{4})\,{\mathrm{cm}}^{-3}$ (e.g., Draine 2011). An H ii region is a dense cloud where hydrogen has been photoionized by ultraviolet photons from hot stars embedded in the cloud, and the extended low-density photoionized regions are referred to as the warm ionized medium. Coronal gas has a large fraction volume of $\sim 50 \%$ but an extremely low gas density of $\sim 0.004\,{\mathrm{cm}}^{-3}$ (e.g., Draine 2011). Therefore, the DM contribution in the ISM is mainly from the H ii gas.

Similar to the discussion about the hot gas in clusters or groups, the DM variation from the ISM can be estimated as

$$\begin{eqnarray}\left|\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{ISM}}}{{dt}}\right|\sim | \delta {n}_{e}{v}_{p}| & \simeq & 2\times {10}^{-5}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & \times & \left(\displaystyle \frac{| \delta {n}_{e}| }{0.1\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{| {v}_{p}| }{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 11 }$$

where ${v}_{p}\sim 200\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is taken as the typical rotation velocity of the Milky Way. And even if $\delta {n}_{e}$ is taken to be of the same order of the mean ionized gas density in the ISM, e.g., $| \delta {n}_{e}| \sim 0.1\,{\mathrm{cm}}^{-3}$ (e.g., Draine 2011), the DM variation is negligibly small. One can see that the DM variations due to a large-scale plasma, from the cosmological scale to the galaxy scale, cannot significantly change the DM of an FRB during an observable time.

In the following, we consider the DM variation caused by the plasma local to the FRB source, which includes an SNR, a pulsar wind nebula (PWN), and a local H ii region.

A supernova leaves behind an SNR, which has three phases: (1) the free-expansion phase, during which the SNR ejecta moves with roughly the initial velocity, and the SNR mass is dominated by the ejecta; (2) the Sedov–Taylor phase, during which the ejecta begins to slow down due to the interaction with the ambient medium, and the SNR mass is dominated by the swept ambient medium; and (3) the snowplow phase, during which the shock wave undergoes significant radiative cooling.

Note that the observed DM variation has two contributions: the first one is from the ionized medium (including ejecta and swept ambient medium) in SNR, e.g., $d{\mathrm{DM}}_{\mathrm{SNR}}/{dt};$ the second one is due to the decrease of the ionized ambient medium in the upstream of the shock, e.g., $d{\mathrm{DM}}_{\mathrm{AM}}/{dt}$. Thus the observed DM satisfies

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{obs}}}{{dt}}=\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR}}}{{dt}}+\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{AM}}}{{dt}},\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{AM}}}{{dt}}=-{\eta }_{0}{nv},\end{eqnarray} \tag{ 13 }$$

n is the number density of the ambient medium, v is the SNR velocity, and ${\eta }_{0}$ is the ionization fraction in the ambient medium. For a fully ionized medium, ${\eta }_{0}=1;$ for neutral medium, ${\eta }_{0}=0$. We assume that hydrogen is the dominant component of the ambient medium.

At first, we consider the dispersion of an electromagnetic wave in the SNR in the early stage. Since the number density of the ionized medium in the ejecta is very large at this moment, the frequency-dependent delay time, ${\rm{\Delta }}t\propto {\nu }^{-\alpha }$, might deviate from ${\rm{\Delta }}t\propto {\nu }^{-2}$. Once α is measured, the electron number density can be constrained via (e.g., Dennison 2014; Tuntsov 2014; Katz 2016b)

$$\begin{eqnarray}{n}_{e} & \lt & \displaystyle \frac{2}{3}| \alpha -2| \displaystyle \frac{{m}_{e}{\omega }^{2}}{4\pi {e}^{2}}\simeq 2.5\times {10}^{7}\,{\mathrm{cm}}^{-3}\\ & & \times \left(\displaystyle \frac{| \alpha -2| }{0.003}\right){\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{2}.\end{eqnarray} \tag{ 14 }$$

The tightest observational upper bound on $| \alpha -2|$ is 0.003 (e.g., Katz 2016b). During the free-expansion phase, the mass contribution from the swept ambient medium can be neglected. For the thin shell approximation, the number density of the electron in the shell is given by

$$\begin{eqnarray}{n}_{e} & \simeq & \displaystyle \frac{\eta M}{4\pi {\mu }_{m}{m}_{p}{R}^{2}{\rm{\Delta }}R}=\displaystyle \frac{\eta {M}^{5/2}}{8\sqrt{2}\pi {\mu }_{m}{m}_{p}{E}_{0}^{3/2}{t}^{3}}{\left(\displaystyle \frac{{\rm{\Delta }}R}{R}\right)}^{-1}\\ & \simeq & 2.5\times {10}^{7}\,{\mathrm{cm}}^{-3}\eta {\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{5/2}{\left(\displaystyle \frac{t}{1\mathrm{year}}\right)}^{-3}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{-3/2}{\left(\displaystyle \frac{{\rm{\Delta }}R}{0.1R}\right)}^{-1},\end{eqnarray} \tag{ 15 }$$

where M is the SNR mass, E₀ is the kinetic energy of the SNR, t is the age of the SNR, η is the ionization fraction of the medium in the SNR, and ${\mu }_{m}=1.2$ is the mean molecular weight for a solar composition in the SNR ejecta. The free-expansion velocity of the SNR is estimated as v ≃ (2E₀/M)^1/2 ≃ 10⁴ km s⁻¹(M/M_⊙)^−1/2(E₀/10⁵¹ erg)^1/2, and the SNR radius satisfies R ∼ vt ≃ 0.01 pc (M/M_⊙)^−1/2(E₀/10⁵¹ erg)^1/2(t/1 year). According to Equations (14) and (15), one can constrain the lower limit of SNR age via the observed frequency-dependent delay time (${\rm{\Delta }}t\propto {\nu }^{-\alpha }$), e.g.,

$$\begin{eqnarray}t & \gt & 1\,\mathrm{year}\,{\eta }^{1/3}{\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{5/6}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{-1/2}{\left(\displaystyle \frac{{\rm{\Delta }}R}{0.1R}\right)}^{-1/3}\\ & & \times {\left(\displaystyle \frac{| \alpha -2| }{0.003}\right)}^{-1/3}{\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-2/3}.\end{eqnarray} \tag{ 16 }$$

The DM variation of an SNR in the free-expansion phase has been studied by various authors (e.g., Katz 2016a, 2017; Piro 2016; Metzger et al. 2017; Piro & Burke-Spolaor 2017; Yang et al. 2017). A time-dependent DM provided by a young SNR in the free-expansion phase can be estimated as

$$\begin{eqnarray}{\mathrm{DM}}_{\mathrm{SNR},\mathrm{FE}} & \simeq & {n}_{e}{\rm{\Delta }}R=\displaystyle \frac{\eta {M}^{2}}{8\pi {\mu }_{m}{m}_{p}{E}_{0}{t}^{2}}\simeq 260\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\\ & & \times \eta {\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{t}{10\mathrm{year}}\right)}^{-2}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{-1},\end{eqnarray} \tag{ 17 }$$

And the DM variation of the SNR is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{FE}}}{{dt}} & = & -\displaystyle \frac{\eta {M}^{2}}{4\pi {\mu }_{m}{m}_{p}{t}^{3}{E}_{0}}=-52\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times \eta {\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{t}{10\mathrm{year}}\right)}^{-3}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{-1}.\end{eqnarray} \tag{ 18 }$$

On the other hand, due to $| d{\mathrm{DM}}_{\mathrm{AM}}/{dt}| \ll | d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{FE}}/{dt}|$ during the free-expansion phase, the observed DM variation is dominated by the SNR contribution. One can see that a young SNR with an age $\lesssim 10\,\mathrm{year}$ can give a significant decreasing DM with time.

As more and more ambient medium is swept into the shock, the shock starts to decelerate at a radius

$$\begin{eqnarray}{R}_{\mathrm{dec}} & = & {\left(\displaystyle \frac{3M}{4\pi {m}_{p}n}\right)}^{1/3}\\ & \simeq & 0.46\,\mathrm{pc}{\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-1/3}.\end{eqnarray} \tag{ 19 }$$

The corresponding decelerated time is given by

$$\begin{eqnarray}{t}_{\mathrm{dec}} & \simeq & \displaystyle \frac{{R}_{\mathrm{dec}}}{v}\simeq 45\,\mathrm{year}{\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{5/6}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{-1/2}\\ & & \times {\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-1/3}.\end{eqnarray} \tag{ 20 }$$

When $t\sim {t}_{\mathrm{dec}}$, the free-expansion phase is over, and the SNR enters the Sedov–Taylor phase. The SNR radius during the Sedov–Taylor phase satisfies the self-similar solution, which is given by (e.g., Taylor 1950; Sedov 1959; Draine 2011)

$$\begin{eqnarray}{R}_{\mathrm{ST}} & = & 1.15{E}_{0}^{1/5}{\rho }^{-1/5}{t}^{2/5}\simeq 0.84\,\mathrm{pc}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{1/5}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-1/5}{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{2/5},\end{eqnarray} \tag{ 21 }$$

for $t\gg {t}_{\mathrm{dec}}$. The corresponding SNR velocity during the Sedov–Taylor phase reads

$$\begin{eqnarray}{v}_{\mathrm{ST}} & = & \displaystyle \frac{{{dR}}_{\mathrm{ST}}}{{dt}}\simeq 3280\,\mathrm{km}\,{{\rm{s}}}^{-1}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{1/5}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-1/5}{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{-3/5}.\end{eqnarray} \tag{ 22 }$$

According to the strong shock jump condition, the temperature in the downstream medium of the shock is given by (e.g., Draine 2011)

$$\begin{eqnarray}{T}_{\mathrm{ST}} & \simeq & \displaystyle \frac{2(\gamma -1)}{{(\gamma +1)}^{2}}\displaystyle \frac{{m}_{p}{v}_{\mathrm{ST}}^{2}}{k}=\displaystyle \frac{3}{16}\displaystyle \frac{{m}_{p}{v}_{\mathrm{ST}}^{2}}{k}\simeq 2.4\times {10}^{8}\,{\rm{K}}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{2/5}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-2/5}{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{-6/5},\end{eqnarray} \tag{ 23 }$$

for the adiabatic index $\gamma =5/3$. Due to ${T}_{\mathrm{ST}}\gg {T}_{{\rm{H}}}\sim 1.6\,\times {10}^{5}\,{\rm{K}}$, where ${T}_{{\rm{H}}}$ is the ionization temperature of hydrogen, all media in the downstream of the shock would be ionized. The corresponding DM can be estimated as

$$\begin{eqnarray}{\mathrm{DM}}_{\mathrm{SNR},\mathrm{ST}} & \simeq & {\displaystyle \int }_{0}^{{R}_{\mathrm{ST}}}n(r){dr}={\displaystyle \int }_{0}^{1}\displaystyle \frac{\gamma +1}{\gamma -1}{nf}(x){R}_{\mathrm{ST}}{dx}\\ & = & \zeta {{nR}}_{\mathrm{ST}}\simeq 34\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{1/5}\\ & & \times {\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{4/5}\,{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{2/5},\end{eqnarray} \tag{ 24 }$$

where $x=r/{R}_{\mathrm{ST}}$ is the dimensionless radius, $n(r)$ is the density profile in the shock downstream, $\zeta \equiv [(\gamma +1)/(\gamma -1)]\int f(x){dx}\simeq 0.4$ for the adiabatic index $\gamma =5/3$, and $f(x)$ is the density dimensionless function of the Sedov–Taylor solution (e.g., Taylor 1950; Sedov 1959; Draine 2011). The DM variation of the SNR is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{ST}}}{{dt}} & \simeq & +0.14\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{1/5}\,{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{4/5}\\ & & \times {\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{-3/5}.\end{eqnarray} \tag{ 25 }$$

On the other hand, with the expansion of the SNR, the ambient medium outside ${R}_{\mathrm{ST}}$ decreases. The DM variation contributed by the ambient medium is given by

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{AM}}}{{dt}}=-{\eta }_{0}{{nv}}_{\mathrm{ST}}=-\displaystyle \frac{{\eta }_{0}}{\zeta }\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{ST}}}{{dt}}.\end{eqnarray} \tag{ 26 }$$

Therefore, according to Equation (12), the observed DM variation is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{obs}}}{{dt}} & = & 0.35(0.4-{\eta }_{0})\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times {\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{1/5}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{4/5}{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{-3/5}.\end{eqnarray} \tag{ 27 }$$

Interestingly, differing from the free-expansion phase, the DM evolution during the Sedov–Taylor phase depends on the ionization fraction ${\eta }_{0}$ of the ambient medium. If ${\eta }_{0}\lt 0.4$, the DM will increase with time. The reason is as follows: as more and more ambient media are swept into the shock, they will be ionized in the downstream, leading to the column density of free electrons in the SNR increasing. However, the DM variation contributed by the ambient medium in the upstream is small, because the ambient medium is near-neutral. On the other hand, if ${\eta }_{0}\gt 0.4$, the DM will decrease with time, since the density profile of all the ionized ambient media (including swept and non-swept media) significantly changes after the SNR shock-sweeping.⁸ Finally, we note that the DM variation could be observable during a long observational period, if the ambient medium is dense with $n\gtrsim 100\,{\mathrm{cm}}^{-3}$.

Due to the radiative cooling, the SNR will leave the Sedov–Taylor phase and enter the snowplow phase, when the radiative energy is of the same order as the SNR kinetic energy. The radiative cooling time is approximately given by (e.g., Draine 2011)

$$\begin{eqnarray}&&{t}_{\mathrm{rad}}\simeq 3920\,\mathrm{year}\,{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.22}\,{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-0.55}.\end{eqnarray} \tag{ 28 }$$

The SNR radius during the snowplow phase is given by (e.g., Draine 2011)

$$\begin{eqnarray}{R}_{\mathrm{SP}} & \simeq & {R}_{\mathrm{ST}}({t}_{\mathrm{rad}}){\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{2/7}\\ & \simeq & 3.6\,\mathrm{pc}{\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{2/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.288}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-0.42}.\end{eqnarray} \tag{ 29 }$$

The corresponding SNR velocity reads

$$\begin{eqnarray}{v}_{\mathrm{SP}} & = & \displaystyle \frac{{{dR}}_{\mathrm{SP}}}{{dt}}=260\,\mathrm{km}\,{{\rm{s}}}^{-1}\\ & & \times {\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{-5/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.288}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-0.42}.\end{eqnarray} \tag{ 30 }$$

Similar to Equation (23), one can obtain the temperature in the downstream medium during the snowplow phase, e.g.,

$$\begin{eqnarray}{T}_{\mathrm{SP}} & \simeq & \displaystyle \frac{3}{16}\displaystyle \frac{{m}_{p}{v}_{\mathrm{SP}}^{2}}{k}=1.5\times {10}^{6}\,{\rm{K}}\\ & & \times {\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{-10/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.576}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-0.84}.\end{eqnarray} \tag{ 31 }$$

Note that $T\gt {T}_{{\rm{H}}}$ for ${t}_{\mathrm{rad}}\lt t\lt 5{t}_{\mathrm{rad}}$ and $T\lt {T}_{{\rm{H}}}$ for $t\gt 5{t}_{\mathrm{rad}}$. During the snowplow phase, the ionization fraction η will depend on the temperature in the SNR and the injection rate of ionizing photon from other stars. Since the mass in the SNR is dominated by the swept ambient medium, the DM of SNR may be given by

$$\begin{eqnarray}{\mathrm{DM}}_{\mathrm{SNR},\mathrm{SP}} & \simeq & \,\displaystyle \frac{\eta {M}_{\mathrm{sw}}}{4\pi {m}_{p}{R}_{\mathrm{SP}}^{2}}\simeq \displaystyle \frac{1}{3}\eta {{nR}}_{\mathrm{SP}}=120\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\\ & & \times \eta {\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{2/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.288}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{0.58},\end{eqnarray} \tag{ 32 }$$

where ${M}_{\mathrm{sw}}=(4\pi /3){{nm}}_{p}{R}_{\mathrm{SP}}^{3}$ is the mass of the swept ambient medium, which is much larger than the original mass of the ejecta. The DM variation is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{SP}}}{{dt}} & \simeq & +0.009\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times {\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{-5/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.068}\\ & & \times {\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{1.13}.\end{eqnarray} \tag{ 33 }$$

On the other hand, the DM variation contributed by the ambient medium is given by

$$\begin{eqnarray}&&\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{AM}}}{{dt}}=-{\eta }_{0}{{nv}}_{\mathrm{SP}}=-\displaystyle \frac{3{\eta }_{0}}{\eta }\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{SNR},\mathrm{SP}}}{{dt}}.\end{eqnarray} \tag{ 34 }$$

Therefore, the observed DM variation is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{obs}}}{{dt}} & \simeq & 0.009\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\,\left(1-\displaystyle \frac{3{\eta }_{0}}{\eta }\right)\\ & & \times {\left(\displaystyle \frac{t}{{t}_{\mathrm{rad}}}\right)}^{-5/7}{\left(\displaystyle \frac{{E}_{0}}{{10}^{51}\mathrm{erg}}\right)}^{0.068}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{1.13}.\end{eqnarray} \tag{ 35 }$$

Similar to the Sedov–Taylor phase, the DM evolution during the snowplow phase depends on the relation between ${\eta }_{0}$ and η of the ambient medium. If ${\eta }_{0}\lt \eta /3$, the DM will increase with time, otherwise the DM will decrease with time.

Many FRB models invoke a young pulsar or magnetar as the source of the bursts (Popov & Postnov 2010; Kulkarni et al. 2014; Connor et al. 2016; Cordes & Wasserman 2016; Murase et al. 2016, 2017; Piro 2016; Yang et al. 2016; Beloborodov 2017; Cao et al. 2017; Dai et al. 2017; Metzger et al. 2017; Nicholl et al. 2017; Waxman 2017). The pulsar/magnetar parameters have been constrained with the available observations for FRB 121102 (Tendulkar et al. 2016; Lyutikov 2017; Zhang & Zhang 2017), but generally the data require a relatively young pulsar, which is supposed to have a surrounding PWN. We now consider the DM contribution from a PWN. In a PWN, the local magnetic field in the wind may not be neglected, since the electron cyclotron frequency is larger than the wave frequency. One should consider wave dispersion in a magnetized plasma. In a magnetized plasma, there are four modes of electromagnetic waves (e.g., Mészáros 1992): O-mode (${\boldsymbol{k}}\perp {\boldsymbol{B}},{{\boldsymbol{E}}}_{{\rm{w}}}\parallel {\boldsymbol{B}}$), X-mode (${\boldsymbol{k}}\perp {\boldsymbol{B}},{{\boldsymbol{E}}}_{{\rm{w}}}\perp {\boldsymbol{B}}$), R-mode (${\boldsymbol{k}}\parallel {\boldsymbol{B}}$, right circular polarization), and L-mode (${\boldsymbol{k}}\parallel {\boldsymbol{B}}$, left circular polarization), where ${\boldsymbol{k}}$ is the wave vector of the electromagnetic wave, ${\boldsymbol{B}}$ is the local magnetic field, and ${{\boldsymbol{E}}}_{{\rm{w}}}$ is the electric field vector of the electromagnetic wave. Their dispersion relations are as follows (e.g., Mészáros 1992):

$$\begin{eqnarray}&&1-\displaystyle \frac{{\omega }_{p}^{2}}{{\omega }^{2}}=\displaystyle \frac{{c}^{2}{k}^{2}}{{\omega }^{2}},\,\,\,\,\,\,\,\mathrm{for}\,{\rm{O}}\,\mathrm{mode};\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}&&1-\displaystyle \frac{{\omega }_{p}^{2}}{{\omega }^{2}}\displaystyle \frac{{\omega }^{2}-{\omega }_{p}^{2}}{{\omega }^{2}-{\omega }_{h}^{2}}=\displaystyle \frac{{c}^{2}{k}^{2}}{{\omega }^{2}},\,\,\,\,\mathrm{for}\,{\rm{X}}\,\mathrm{mode};\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&1-\displaystyle \frac{{\omega }_{p}^{2}{/\omega }^{2}}{1-{\omega }_{c}/\omega }=\displaystyle \frac{{c}^{2}{k}^{2}}{{\omega }^{2}},\,\,\,\,\,\mathrm{for}\,{\rm{R}}\,\mathrm{mode};\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&1-\displaystyle \frac{{\omega }_{p}^{2}/{\omega }^{2}}{1+{\omega }_{c}/\omega }=\displaystyle \frac{{c}^{2}{k}^{2}}{{\omega }^{2}},\,\,\,\,\,\mathrm{for}\,{\rm{L}}\,\mathrm{mode};\end{eqnarray} \tag{ 39 }$$

respectively, where ${\omega }_{c}={eB}/{m}_{e}c$ is the electron cyclotron frequency, and ${\omega }_{h}={({\omega }_{p}^{2}+{\omega }_{c}^{2})}^{1/2}$ is the upper hybrid frequency. Here, we neglect the motion of ions due to their large masses.

In the magnetized plasma, the two important properties are the presence of resonances and cutoffs, which occur at $k\to \infty$ and $k\to 0$, respectively. In O-mode, the wave dispersion relation is the same as that in an unmagnetized plasma and its cutoff occurs at $\omega ={\omega }_{p}$. Only the O wave with $\omega \gt {\omega }_{p}$ can propagate through the medium. In X-mode, the wave has a more complicated dispersion relation. The resonance occurs at $\omega ={\omega }_{h}$ and the cutoffs occur at $\omega ={\omega }_{L}$ and $\omega ={\omega }_{R}$, where ${\omega }_{R,L}=[\pm {\omega }_{c}+{({\omega }_{c}^{2}+4{\omega }_{p}^{2})}^{1/2}]/2$. Only the X wave with ${\omega }_{L}\lt \omega \lt {\omega }_{h}$ or $\omega \gt {\omega }_{R}$ can propagate through the medium. In R-mode, the wave is right circularly polarized, and its cutoff occurs at $\omega ={\omega }_{R}$ and resonance occurs at $\omega ={\omega }_{c}$. Only the R wave with $\omega \gt {\omega }_{R}$ or $0\lt \omega \lt {\omega }_{c}$ can propagate through the medium. In L-mode, the wave is left circularly polarized, and the cutoff occurs at $\omega ={\omega }_{L}$ and it has no resonance. Only the L wave with $\omega \gt {\omega }_{L}$ can propagate through the medium. The dispersion relations of the R-mode and L-mode approach the dispersion relation in an unmagnetized plasma only when $\omega \gg {\omega }_{c}$.

The radiation mechanism of the coherent radio emission from pulsars is still not identified. Observations of pulsars show that there might be at least two locations of radio emission, one in the open field line region near the polar cap, and the other near the light cylinder (e.g., Melrose 2017). It is not clear whether any of those mechanisms might produce FRBs with even much higher brightness temperatures than pulsar radio emission. In any case, it is safe to integrate the DM contribution of a PWN starting from the light cylinder. In the pulsar magnetosphere, the dispersion relation of an electromagnetic wave is very complex (e.g., Arons & Barnard 1986). More importantly, the radio emission is likely still in the growth mode within the magnetosphere. This is certainly the case for the models invoking coherent emission near the light cylinder or the outer gap, but is valid for some polar cap models as well (e.g., Melrose 2017). In the following, we only consider the DM contribution outside the light cylinder, which corresponds to a lower limit but should be close to the total DM near a pulsar.

For a magnetized rapidly rotating neutron star, a relativistic electron–positron pair wind is expected to stream out from the magnetosphere (Murase et al. 2016; Cao et al. 2017; Dai et al. 2017). The pair number density of the wind at r is given by (e.g., Yu 2014)

$$\begin{eqnarray}&&{n}_{{\rm{w}}}(r)\simeq \displaystyle \frac{{\dot{N}}_{{\rm{w}}}}{4\pi {r}^{2}c}={\mu }_{\pm }{n}_{\mathrm{GJ}}({R}_{\mathrm{LC}}){\left(\displaystyle \frac{r}{{R}_{\mathrm{LC}}}\right)}^{-2},\end{eqnarray} \tag{ 40 }$$

where ${\mu }_{\pm }$ is the multiplicity parameter of the electron–positron pairs, ${R}_{\mathrm{LC}}\equiv c/{\rm{\Omega }}$ is the radius of the light cylinder, ${n}_{\mathrm{GJ}}={({\rm{\Omega }}{B}_{p}/2\pi {ec})(r/R)}^{-3}$ is the Goldreich-Julian number density, ${\dot{N}}_{{\rm{w}}}\simeq 4\pi {R}_{\mathrm{LC}}^{2}{\mu }_{\pm }{n}_{\mathrm{GJ}}({R}_{\mathrm{LC}})c$ is the particle number flux, R is the radius of the neutron star, and B_p is magnetic field strength at the polar cap. In the PWN, the plasma frequency is given by

$$\begin{eqnarray}{\omega }_{p} & = & {\left(\displaystyle \frac{4\pi {e}^{2}{n}_{{\rm{w}}}}{{m}_{e}}\right)}^{1/2}\simeq 4.5\times {10}^{9}\,\mathrm{rad}\,{{\rm{s}}}^{-1}\\ & & \times {\left(\displaystyle \frac{{\mu }_{\pm }}{{10}^{4}}\right)}^{1/2}{\left(\displaystyle \frac{{B}_{p}}{{10}^{14}{\rm{G}}}\right)}^{1/2}{\left(\displaystyle \frac{P}{0.1{\rm{s}}}\right)}^{-2}{\left(\displaystyle \frac{r}{{R}_{\mathrm{LC}}}\right)}^{-1},\end{eqnarray} \tag{ 41 }$$

and the electron cyclotron frequency is given by

$$\begin{eqnarray}{\omega }_{c} & \simeq & \displaystyle \frac{{{eB}}_{p}}{{m}_{e}c}{\left(\displaystyle \frac{{R}_{\mathrm{LC}}}{R}\right)}^{-3}{\left(\displaystyle \frac{r}{{R}_{\mathrm{LC}}}\right)}^{-1}\simeq 1.6\times {10}^{13}\,\mathrm{rad}\,{{\rm{s}}}^{-1}\\ & & \times \left(\displaystyle \frac{{B}_{p}}{{10}^{14}\,{\rm{G}}}\right){\left(\displaystyle \frac{P}{0.1{\rm{s}}}\right)}^{-3}{\left(\displaystyle \frac{r}{{R}_{\mathrm{LC}}}\right)}^{-1}.\end{eqnarray} \tag{ 42 }$$

Note that at $r\gt {R}_{\mathrm{LC}}$, $B\propto {r}^{-1}$, so the factor of ${\omega }_{p}/{\omega }_{c}$ is independent of radius. Near the magnetosphere, the compositions of wave modes and the conversions among them are complex, depending on the radiation mechanism, magnetic field configuration, and the plasma environment in strong magnetic fields. For simplification, we assume that the angles between the wave polarization and ${\boldsymbol{k}}$ and ${\boldsymbol{B}}$ are essentially unchanged during the wave propagation, and ignore any conversion among modes. Considering more complicated situations does not alter the basic picture discussed below.

First, the O-mode wave satisfies the classical unmagnetized plasma dispersion relation, so the corresponding DM contributed by the pulsar/magnetar wind may be given by (Yu 2014; Cao et al. 2017; Yang et al. 2017)

$$\begin{eqnarray}{\mathrm{DM}}_{{\rm{w}},{\rm{O}}} & \simeq & {\displaystyle \int }_{{R}_{\mathrm{LC}}}^{{R}_{\mathrm{sh}}}2{\rm{\Gamma }}(r){n}_{{\rm{w}}}(r){dr}\simeq 3{{\rm{\Gamma }}}_{\mathrm{LC}}{\mu }_{\pm }{n}_{\mathrm{GJ}}({R}_{\mathrm{LC}}){R}_{\mathrm{LC}}\\ & \simeq & 380\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\left(\displaystyle \frac{{\mu }_{\pm }}{{10}^{4}}\right)}^{2/3}{\left(\displaystyle \frac{{B}_{p}}{{10}^{14}{\rm{G}}}\right)}^{4/3}{\left(\displaystyle \frac{P}{0.1{\rm{s}}}\right)}^{-11/3}\end{eqnarray} \tag{ 43 }$$

for ${R}_{\mathrm{sh}}\gg {R}_{\mathrm{LC}}$, where P is the rotation period of the neutron star, ${R}_{\mathrm{sh}}$ is the radius of the shock (where the pulsar wind collides with the SNR (Murase et al. 2016, 2017; Kashiyama & Murase 2017) or an ambient ISM (Dai et al. 2017)), ${{\rm{\Gamma }}}_{\mathrm{LC}}\sim {({L}_{\mathrm{sd}}/{\dot{N}}_{{\rm{w}}}{m}_{e}{c}^{2})}^{1/3}$ is the Lorentz factor of the relativistic wind at ${R}_{\mathrm{LC}}$, and ${L}_{\mathrm{sd}}={B}_{p}^{2}{R}^{6}{{\rm{\Omega }}}^{4}/6{c}^{3}$ is the spin-down luminosity of the pulsar/magnetar. The wind Lorentz factor at radius r may be given by ${\rm{\Gamma }}{(r)\sim {{\rm{\Gamma }}}_{\mathrm{LC}}L(r/{R}_{\mathrm{LC}})}^{1/3}$ (Drenkhahn 2002). In fact, the contribution of ${\mathrm{DM}}_{{\rm{w}},{\rm{O}}}$ is mainly from the inner region, due to the larger number density of the electrons.

For the X-mode wave, near the light cylinder $\omega \ll {\omega }_{p}\ll {\omega }_{c}$ is satisfied. According to Equation (37), the dispersion relation reads ${c}^{2}{k}^{2}/{\omega }^{2}\simeq 1-{\omega }_{p,\mathrm{eff}}^{2}/{\omega }^{2}$, where ${\omega }_{p,\mathrm{eff}}={\omega }_{p}^{2}/{\omega }_{c}\sim {\omega }_{L}\ll {\omega }_{p}$ is the effective plasma frequency. In this case, the dispersion relation of the X wave is similar to that in an unmagnetized plasma (with ${\omega }_{p,\mathrm{eff}}$ replaced by ${\omega }_{p}$), and the condition for the propagation of the X-mode wave is $\omega \gt {\omega }_{p,\mathrm{eff}}$. Therefore, the apparent DM is corrected by a factor ${({\omega }_{p}/{\omega }_{c})}^{2}$, i.e.,

$$\begin{eqnarray}{\mathrm{DM}}_{{\rm{w}},{\rm{X}}} & \simeq & {\left(\displaystyle \frac{{\omega }_{p}}{{\omega }_{c}}\right)}^{2}{\mathrm{DM}}_{{\rm{w}},{\rm{O}}}\simeq 3\times {10}^{-5}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\\ & & \times {\left(\displaystyle \frac{{\mu }_{\pm }}{{10}^{4}}\right)}^{5/3}\,{\left(\displaystyle \frac{{B}_{p}}{{10}^{14}{\rm{G}}}\right)}^{1/3}\,{\left(\displaystyle \frac{P}{0.1{\rm{s}}}\right)}^{-5/3}.\end{eqnarray} \tag{ 44 }$$

Therefore, the GHz X-mode wave can still propagate through the medium even though $\omega \lt {\omega }_{p}$ (e.g., Kumar et al. 2017), and the apparent DM contribution is extremely small. However, as the X wave propagates further, e.g., $r\gtrsim {r}_{p}$, where r_p is defined by ${\omega }_{p}(r)=\omega$, the dispersion relation of X wave would not keep the unmagnetized form (see Equation (37)). In particular, when ${\omega }_{p}\ll \omega \ll {\omega }_{c}$, the dispersion relation is approximately ${c}^{2}{k}^{2}/{\omega }^{2}\simeq 1+{\omega }_{p}^{2}/{\omega }_{c}^{2}$, leading to the group velocity satisfying ${v}_{g}=\partial \omega /\partial k\simeq c/{(1+{\omega }_{p}^{2}/{\omega }_{c}^{2})}^{1/2}$. Thus, the electromagnetic wave with ${\omega }_{p}\ll \omega \ll {\omega }_{c}$ does not show significantly frequency-dependent delay time, i.e., there is no contribution to the DM. Finally, as the X wave propagates further, the resonance occurs when $\omega ={\omega }_{h}\sim {\omega }_{c}$ is satisfied.

For the R-mode and L-mode waves, near the light cylinder $\omega \ll {\omega }_{c}$ is satisfied. Their dispersion relations could be approximated as ${c}^{2}{k}^{2}/{\omega }^{2}=1\pm {\omega }_{p}^{2}/({\omega }_{c}\omega )$, where "+" and "−" correspond to R-mode and L-mode, respectively. The group velocity is ${v}_{g}=c{(1\pm {\omega }_{p,\mathrm{eff}}/\omega )}^{1/2}/(1\pm {\omega }_{p,\mathrm{eff}}/2\omega )$, where ${\omega }_{p,\mathrm{eff}}={\omega }_{p}^{2}/{\omega }_{c}$. Due to $\omega \gg {\omega }_{p,\mathrm{eff}}$, the group velocities of both modes are extremely close to the speed of light, i.e., ${v}_{g}\simeq c$. There is no significant frequency-dependent delay time. As the waves propagate further, both ${\omega }_{p}$ and ${\omega }_{c}$ decrease. For the R-mode wave, the resonance finally occurs at $\omega ={\omega }_{c};$ for the L-mode wave, its dispersion relation approaches that in the unmagnetized plasma when $\omega \gg {\omega }_{c}$, but the DM contribution from the PWN is very small at that time.

In addition to the wind contribution, the shocked electron–positron pairs may contribute to a part of the DM. These thermalized electron–positron pairs would undergo cooling and become non-relativistic (if the radiative cooling time is much less than that of the dynamical time of the shock). In this case, due to $\omega \gg {\omega }_{c}$, the dispersion relations of all modes keep the classical unmagnetized form. The DM from these thermalized particles is given by (Yang et al. 2017)

$$\begin{eqnarray}{\mathrm{DM}}_{\mathrm{sh}} & \simeq & \displaystyle \frac{{\dot{N}}_{{\rm{w}}}{T}_{\mathrm{sd}}}{4\pi {R}_{\mathrm{sh}}^{2}}\simeq \displaystyle \frac{3{c}^{2}{\mu }_{\pm }I}{2\pi {{eB}}_{p}{R}^{3}{R}_{\mathrm{sh}}^{2}}\simeq 3\times {10}^{-7}\mathrm{pc}\,{\mathrm{cm}}^{-3}\\ & \times & \left(\displaystyle \frac{{\mu }_{\pm }}{{10}^{4}}\right){\left(\displaystyle \frac{{B}_{p}}{{10}^{14}{\rm{G}}}\right)}^{-1}{\left(\displaystyle \frac{{R}_{\mathrm{sh}}}{0.1\mathrm{pc}}\right)}^{-2},\end{eqnarray} \tag{ 45 }$$

for $T\gg {T}_{\mathrm{sd}}$, where T is the PWN age, ${T}_{\mathrm{sd}}$ is the spin-down time of the pulsar, $I\simeq {10}^{45}\,{\rm{g}}\,{\mathrm{cm}}^{2}$ is the moment of inertia, and ${\dot{N}}_{{\rm{w}}}{T}_{\mathrm{sd}}$ is the total number of electron–positron pairs ejected over ${T}_{\mathrm{sd}}$, which does not depend on Ω. Due to ${\mathrm{DM}}_{\mathrm{sh}}\ll {\mathrm{DM}}_{{\rm{w}},{\rm{O}}}$, one has ${\mathrm{DM}}_{\mathrm{PWN}}\simeq {\mathrm{DM}}_{{\rm{w}},{\rm{O}}}$ (Yang et al. 2017). Therefore, the DM variation of the PWN wind may be given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{{\rm{w}}}}{{dt}} & \simeq & \displaystyle \frac{d{\mathrm{DM}}_{{\rm{w}},{\rm{O}}}}{{dP}}\displaystyle \frac{{dP}}{{dt}}\simeq -11\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times {\left(\displaystyle \frac{{\mu }_{\pm }}{{10}^{4}}\right)}^{2/3}{\left(\displaystyle \frac{{B}_{p}}{{10}^{14}{\rm{G}}}\right)}^{10/3}{\left(\displaystyle \frac{P}{0.1{\rm{s}}}\right)}^{-17/3},\end{eqnarray} \tag{ 46 }$$

where ${dP}/{dt}=(2{\pi }^{2}/3{c}^{3})({B}^{2}{R}^{6}/{IP})$. Here, we only consider the contribution from the O-mode waves, since the DMs of the other modes are extremely small. One can see that a young pulsar/magnetar wind may provide a significant contribution to the DM and to its variation.

Next, we consider an FRB embedded in an H ii region in its host galaxy. If the FRB source is related to recent star formation (as indicated by the host galaxy type of FRB 121102), there are two (exotic) mechanisms that can cause DM variation. One is that there might be a newborn young star in the H ii region, whose ionization front still evolves with time. The other possibility is that the FRB source may be rapidly moving within the H ii region due to, e.g., a supernova kick. In general, the evolution of the H ii region radius R may be estimated as

$$\begin{eqnarray}&&{N}_{u}=4\pi {{nR}}^{2}\displaystyle \frac{{dR}}{{dt}}+{\alpha }_{{\rm{B}}}{n}^{2}\displaystyle \frac{4\pi }{3}{R}^{3},\end{eqnarray} \tag{ 47 }$$

where N_u is the emission rate of the ionizing photons from a star, n is the gas number density in the H ii region, and ${\alpha }_{{\rm{B}}}$ is the recombination rate. The first term represents ionization of the surrounding neutral gas that absorbs ionizing photon. The second term represents recombination of the ionized gas. Here we have assumed that the neutral gas is neutral atomic hydrogen. Equation (47) has the solution

$$\begin{eqnarray}&&R(t)={R}_{\mathrm{str}}{(1-{e}^{-{\alpha }_{{\rm{B}}}{nt}})}^{1/3}.\end{eqnarray} \tag{ 48 }$$

Once $t\gg 1/({\alpha }_{B}n)$, the radius of the H ii region reaches the maximum value $R\simeq {R}_{\mathrm{str}}\equiv {(3{N}_{u}/4\pi {\alpha }_{{\rm{B}}}{n}^{2})}^{1/3}$, where ${R}_{\mathrm{str}}$ is defined as the Strömgren radius (Strömgren 1939). In this case, the ionization and recombination are balanced.

We consider the case that FRB is associated with an H ii region that is initially expanding outward due to the ionizing photon injection. For simplification, we assume that the timescale for the onset of a star is short. In this case, the star has just started producing a large flux of ionizing photons, making an expanding H ii region. According to Equation (47) or Equation (48), for $t\ll 1/{({\alpha }_{{\rm{B}}}n)\simeq 1220\mathrm{year}(n/100{\mathrm{cm}}^{-3})}^{-1}$, where ${\alpha }_{{\rm{B}}}=2.6\,\times {10}^{-13}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$ for $T={10}^{4}\,{\rm{K}}$, recombination can be neglected. The radius of the H ii region is then approximately given by

$$\begin{eqnarray}R(t) & = & {\left(\displaystyle \frac{3{N}_{u}}{4\pi n}t\right)}^{1/3}\simeq 2.3\,\mathrm{pc}\,{\left(\displaystyle \frac{{N}_{u}}{5\times {10}^{49}{{\rm{s}}}^{-1}}\right)}^{1/3}\\ & & \times {\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{-1/3}\,{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{1/3}.\end{eqnarray} \tag{ 49 }$$

Here we assume that there is an O5 star with ${N}_{u}\sim 5\times {10}^{49}\,{{\rm{s}}}^{-1}$ in the H ii region. Therefore, the time-dependent DM of a young ($t\ll 1/({\alpha }_{{\rm{B}}}n)$) H ii region is given by

$$\begin{eqnarray}{\mathrm{DM}}_{{\rm{H}}{\rm{II}}} & = & {nR}(t)\simeq 230\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\left(\displaystyle \frac{{N}_{u}}{5\times {10}^{49}{{\rm{s}}}^{-1}}\right)}^{1/3}\\ & & \times {\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{2/3}\,{\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{1/3},\end{eqnarray} \tag{ 50 }$$

and the corresponding DM variation is given by

$$\begin{eqnarray}\displaystyle \frac{d{\mathrm{DM}}_{{\rm{H}}{\rm{II}}}}{{dt}} & = & {\left(\displaystyle \frac{{N}_{u}{n}^{2}}{36\pi {t}^{2}}\right)}^{1/3}\simeq +0.78\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times {\left(\displaystyle \frac{{N}_{u}}{5\times {10}^{49}{{\rm{s}}}^{-1}}\right)}^{1/3}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{2/3}\\ & & \times {\left(\displaystyle \frac{t}{100\mathrm{year}}\right)}^{-2/3}.\end{eqnarray} \tag{ 51 }$$

One can see that a young H ii region can provide an observable positive DM variation.

For an old H ii region with $t\gg 1/({\alpha }_{{\rm{B}}}n)$, due to the balance of the ionization and recombination, the H ii region radius would keep at the Strömgren radius, i.e., $R\sim {R}_{\mathrm{str}}$ and ${dR}/{dt}\sim 0$. The DM contributed by a Strömgren sphere is time-independent, i.e., $d{\mathrm{DM}}_{{\rm{H}}{\rm{II}}}/{dt}\sim 0$, with a value (Yang et al. 2017)

$$\begin{eqnarray}{\mathrm{DM}}_{{\rm{H}}{\rm{II}}} & \simeq & {{nR}}_{\mathrm{str}}={\left(\displaystyle \frac{3{N}_{u}n}{4\pi {\alpha }_{{\rm{B}}}}\right)}^{1/3}\\ & = & 540\,\mathrm{pc}\,{\mathrm{cm}}^{-3}{\left(\displaystyle \frac{{N}_{u}}{5\times {10}^{49}{{\rm{s}}}^{-1}}\right)}^{1/3}{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{1/3}.\end{eqnarray} \tag{ 52 }$$

For an O5 star in H ii region, one obtains ${\alpha }_{{\rm{B}}}=2.6\,\times {10}^{-13}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$ for $T={10}^{4}\,{\rm{K}}$, and ${R}_{\mathrm{str}}=5.4\,\mathrm{pc}$. The scale of the ionized region is larger than the projected size of $\lesssim 0.7\,\mathrm{pc}$ of the persistent source of FRB 121102 (Marcote et al. 2017). As discussed above, the balanced H ii region cannot provide DM variation for a static source in the H ii region.

On the other hand, if the FRB source is moving in an H ii region, the plasma in the H ii region may give a change to a DM variation. There are two scenarios here. (1) An FRB source is inside the H ii region and is moving toward or away from us, which would decrease or increase the line-of-sight DM contribution;. (2) An FRB source moves in the transverse direction, so it enters or exits an H ii region and may cause a sudden change in the measured DM along the line of sight. For the first case, the DM variation is given by

$$\begin{eqnarray}\left|\displaystyle \frac{d{\mathrm{DM}}_{{\rm{H}}{\rm{II}}}}{{dt}}\right| & \sim & n| {v}_{\parallel }| \simeq 0.02\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times \left(\displaystyle \frac{n}{100\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{| {v}_{\parallel }| }{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right),\end{eqnarray} \tag{ 53 }$$

where ${v}_{\parallel }$ is the velocity along the line of sight. Note that the typical pulsar kick velocities are $v\lesssim 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Hobbs et al. 2005), and ${v}_{\parallel }\lt v$. For the second case, the DM variation may be given by

$$\begin{eqnarray}\left|\displaystyle \frac{d{\mathrm{DM}}_{{\rm{H}}{\rm{II}}}}{{dt}}\right| & \sim & \displaystyle \frac{{nR}}{{\rm{\Delta }}R/| {v}_{\perp }| }\simeq 0.02\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times \left(\displaystyle \frac{n}{100\,{\mathrm{cm}}^{-3}}\right)\left(\displaystyle \frac{| {v}_{\perp }| }{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{\rm{\Delta }}R}{R}\right)}^{-1},\end{eqnarray} \tag{ 54 }$$

where ${v}_{\perp }$ is the transverse velocity, and ${\rm{\Delta }}R$ corresponds to the thickness of the H ii region edge. Similar to the first case, ${v}_{\perp }\lt v\lesssim 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Hobbs et al. 2005).

It is worth noting that the free–free absorption optical depth in the H ii region is small, so FRBs can escape. The free–free absorption coefficient in the Rayleigh–Jeans limit is given by

$$\begin{eqnarray}{\alpha }_{\mathrm{ff}} & = & \displaystyle \frac{4}{3}{\left(\displaystyle \frac{2\pi }{3}\right)}^{1/2}\displaystyle \frac{{Z}^{2}{e}^{6}{n}_{e}{n}_{i}{\bar{g}}_{\mathrm{ff}}}{{{cm}}_{e}^{3/2}{({k}_{B}T)}^{3/2}{\nu }^{2}},\\ {\bar{g}}_{\mathrm{ff}} & = & \displaystyle \frac{\sqrt{3}}{\pi }\left[\mathrm{ln}\left(\displaystyle \frac{{(2{k}_{B}T)}^{3/2}}{\pi {e}^{2}{m}_{e}^{1/2}\nu }\right)-\displaystyle \frac{5}{2}\gamma \right],\end{eqnarray} \tag{ 55 }$$

where $\gamma =0.577$ is Euler's constant, ${\bar{g}}_{\mathrm{ff}}$ is the Gaunt factor, and n_i and n_e are the number densities of ions and electrons, respectively. Here we take ${n}_{e}={n}_{i}$ and Z = 1 for an H ii region. The optical depth for the free–free absorption is given by (e.g., Luan & Goldreich 2014; Murase et al. 2016; Metzger et al. 2017; Yang et al. 2017)

$$\begin{eqnarray}{\tau }_{\mathrm{ff}}\lesssim {\alpha }_{\mathrm{ff}}{R}_{\mathrm{str}} & \simeq & 0.018{\left(\displaystyle \frac{n}{100{\mathrm{cm}}^{-3}}\right)}^{4/3}\\ & & \times {\left(\displaystyle \frac{T}{{10}^{4}{\rm{K}}}\right)}^{-1.5}{\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-2},\end{eqnarray} \tag{ 56 }$$

where ${\alpha }_{{\rm{B}}}=2.6\times {10}^{-13}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$ and ${\bar{g}}_{\mathrm{ff}}=6.0$ for $T={10}^{4}\,{\rm{K}}$ and $\nu =1\,\mathrm{GHz}$ are taken, and the H ii region radius satisfies $R\lesssim {R}_{\mathrm{str}}$. Therefore, such an H ii region is optically thin for FRBs.

As pointed out by Cordes et al. (2017), plasma lenses in FRB host galaxies can generate multiple images for one burst, which may arrive differentially by $\lt 1\,\mu {\rm{s}}$ to tens of $\mathrm{ms}$ and show an apparent DM perturbation of a few $\mathrm{pc}\,{\mathrm{cm}}^{-3}$. Here, we briefly discuss the perturbation and evolution of the DM contributed by a plasma lens. We assume that the source-lens, source-observer, and lens-observer distances are ${d}_{\mathrm{sl}}$, ${d}_{\mathrm{so}}$, and ${d}_{\mathrm{lo}}={d}_{\mathrm{so}}-{d}_{\mathrm{sl}}$, respectively. The transverse coordinates in the source, lens, and observer's planes are ${x}_{{\rm{s}}}$, x, and ${x}_{{\rm{o}}}$, respectively. We define ${u}_{{\rm{s}}}={x}_{{\rm{s}}}/a$, $u=x/a$, and ${u}_{{\rm{o}}}={x}_{{\rm{o}}}/a$, respectively, and define the effective offset as

$$\begin{eqnarray}&&{u}^{{\prime} }\equiv \displaystyle \frac{{d}_{\mathrm{lo}}}{{d}_{\mathrm{so}}}{u}_{{\rm{s}}}+\displaystyle \frac{{d}_{\mathrm{sl}}}{{d}_{\mathrm{so}}}{u}_{{\rm{o}}}.\end{eqnarray} \tag{ 57 }$$

Assuming that the lens centered on u = 0 has a characteristic scale a, and that the DM profile of the lens has the form $\mathrm{DM}(u)={\mathrm{DM}}_{{\rm{l}}}\exp (-{u}^{2})$, one can derive the lens equation in geometric optics (Clegg et al. 1998; Cordes et al. 2017), e.g.,

$$\begin{eqnarray}&&u(1+\alpha {e}^{-{u}^{2}})={u}^{{\prime} }.\end{eqnarray} \tag{ 58 }$$

Here,

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{c}^{2}{r}_{{\rm{e}}}{\mathrm{DM}}_{{\rm{l}}}}{\pi {a}^{2}{\nu }^{2}}\left(\displaystyle \frac{{d}_{\mathrm{sl}}{d}_{\mathrm{lo}}}{{d}_{\mathrm{so}}}\right)=\displaystyle \frac{3430\,{\mathrm{DM}}_{{\rm{l}}}{d}_{\mathrm{sl},\mathrm{kpc}}}{{\nu }_{\mathrm{GHz}}^{2}{a}_{\mathrm{au}}^{2}}\left(\displaystyle \frac{{d}_{\mathrm{lo},\mathrm{Gpc}}}{{d}_{\mathrm{so},\mathrm{Gpc}}}\right),\end{eqnarray} \tag{ 59 }$$

where ${\nu }_{\mathrm{GHz}}=\nu /\mathrm{GHz}$, ${d}_{\mathrm{sl},\mathrm{kpc}}={d}_{\mathrm{sl}}/\mathrm{kpc}$, ${d}_{\mathrm{lo},\mathrm{Gpc}}={d}_{\mathrm{lo}}/\mathrm{Gpc}$, ${d}_{\mathrm{so},\mathrm{Gpc}}={d}_{\mathrm{so}}/\mathrm{Gpc}$, ${a}_{\mathrm{au}}=a/\mathrm{au}$, and ${\mathrm{DM}}_{{\rm{l}}}$ has the units of $\mathrm{pc}\,{\mathrm{cm}}^{-3}$. For a given ${u}^{{\prime} }$, if $\alpha \gt {\alpha }_{{\rm{m}}}\equiv {e}^{3/2}/2$, Equation (58) would have one to three solutions to u, which correspond to one to three sub-images of an FRB.

Following Cordes et al. (2017), we consider a plasma lens with a dispersion depth ${\mathrm{DM}}_{{\rm{l}}}=10\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, and adopt ${d}_{\mathrm{sl},\mathrm{kpc}}=1$, ${d}_{\mathrm{so},\mathrm{Gpc}}\simeq {d}_{\mathrm{lo},\mathrm{Gpc}}=1$, ${a}_{\mathrm{au}}=60$. The lens parameters give $\alpha =9.5{\nu }_{\mathrm{GHz}}^{-2}$. The focal frequency is ${\nu }_{{\rm{f}}}=\nu {(\alpha /{\alpha }_{{\rm{m}}})}^{1/2}\sim 2\,\mathrm{GHz}$, which is comparable to typical FRB observation frequencies. As pointed out by Cordes et al. (2017), for a certain gain threshold, e.g., $G\gt 5$, most events have only one image or sub-image above the gain threshold. Some doublet cases occur at the same location and frequency but arrive at different times. The number of triplet cases is small, and most of them are near the focal frequency ${\nu }_{{\rm{f}}}\sim 2\,\mathrm{GHz}$ with an arrival time less than $10\,\mathrm{ms}$.

For a given ${u}^{{\prime} }$, the DM perturbation occurs at a certain narrow frequency band where the gain is $G\gg 1$, leading to the apparent DM deviating from $\mathrm{DM}(u)={\mathrm{DM}}_{{\rm{l}}}\exp (-{u}^{2})$ (Cordes et al. 2017). At another frequency, the DM perturbation could be neglected. Due to the motion of the source or the observer relative to the lens, the DM and its perturbation can show an evolution. According to Equation (57), the effective transverse velocity is given by

$$\begin{eqnarray}&&{v}_{\perp }^{{\prime} }=\displaystyle \frac{{d}_{\mathrm{lo}}}{{d}_{\mathrm{so}}}{v}_{{\rm{s}},\perp }+\displaystyle \frac{{d}_{\mathrm{sl}}}{{d}_{\mathrm{so}}}{v}_{{\rm{o}},\perp }.\end{eqnarray} \tag{ 60 }$$

In general, since ${v}_{{\rm{s}},\perp }\sim {v}_{{\rm{o}},\perp }$ and ${d}_{\mathrm{sl}}\ll {d}_{\mathrm{so}}$, one has ${v}_{\perp }^{{\prime} }\sim ({d}_{\mathrm{lo}}/{d}_{\mathrm{so}}){v}_{{\rm{s}},\perp }\sim {v}_{{\rm{s}},\perp }$. Here, we take ${v}_{\perp }^{{\prime} }\sim 200\,\mathrm{km}\,{{\rm{s}}}^{-1}$ as a typical virial velocity in a galaxy. The characteristic timescale of a caustic crossing ($G\gt 1$) is given by Cordes et al. (2017)

$$\begin{eqnarray}{t}_{\mathrm{cau}}\simeq \displaystyle \frac{a(\delta G/G)}{{v}_{\perp }^{{\prime} }{G}^{2}} & \simeq & 21\,\mathrm{day}\left(\displaystyle \frac{\delta G}{G}\right){\left(\displaystyle \frac{G}{5}\right)}^{-2}\\ & & \times \left(\displaystyle \frac{a}{60\,\mathrm{au}}\right){\left(\displaystyle \frac{{v}_{\perp }^{{\prime} }}{200\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}\end{eqnarray} \tag{ 61 }$$

where $\delta G$ is the corresponding change in gain. For the given typical parameters, one has $\delta \mathrm{DM}/{\mathrm{DM}}_{{\rm{l}}}\sim 0.1$ (Cordes et al. 2017). Therefore, the DM variation during a caustic crossing may be

$$\begin{eqnarray}\left|\displaystyle \frac{d{\mathrm{DM}}_{\mathrm{cau}}}{{dt}}\right| & \simeq & \displaystyle \frac{| \delta \mathrm{DM}| }{{t}_{\mathrm{cau}}}\simeq 17\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\left(\displaystyle \frac{{\mathrm{DM}}_{{\rm{l}}}}{10\,\mathrm{pc}\,{\mathrm{cm}}^{-3}}\right)\\ & & \times {\left(\displaystyle \frac{\delta G}{G}\right)}^{-1}{\left(\displaystyle \frac{G}{5}\right)}^{2}{\left(\displaystyle \frac{a}{60\mathrm{au}}\right)}^{-1}\left(\displaystyle \frac{{v}_{\perp }^{{\prime} }}{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 62 }$$

For the non-gained frequency or a long-term observation with ${T}_{\mathrm{obs}}\gg {t}_{\mathrm{cau}}$, the DM perturbation could be neglected, and the apparent DM only depends on the DM profile of the plasma lens. Here, we simply consider a range of ${u}^{{\prime} }\sim 2\mbox{--}10$. According to Equation (58), one has $u\sim 0.2$ at ${u}^{{\prime} }\sim 2$ and $u\sim 10$ at ${u}^{{\prime} }\sim 10$ for $\alpha \sim 9.5$. Thus, ΔDM ∼ DM(u ∼ 0.2) − DM(u ∼ 10) ∼ 9.6 pc cm⁻³(DM_l/10 pc cm⁻³). The corresponding DM variation may be approximately estimated as

$$\begin{eqnarray}\left|\displaystyle \frac{d\mathrm{DM}}{{dt}}\right| & \simeq & | {\rm{\Delta }}\mathrm{DM}| {\left(\displaystyle \frac{\delta {u}^{{\prime} }a}{{v}_{\perp }^{{\prime} }}\right)}^{-1}\simeq 0.8\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{\mathrm{yr}}^{-1}\\ & & \times \left(\displaystyle \frac{{\mathrm{DM}}_{{\rm{l}}}}{10\,\mathrm{pc}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{a}{60\mathrm{au}}\right)}^{-1}\left(\displaystyle \frac{{v}_{\perp }^{{\prime} }}{200\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 63 }$$

We note that this value is much larger than the DM variation caused by the gas density fluctuation in the ISM (cf. Equation (11)). The reason is that plasma lensing is required to be dense and concentrated.