Redshift Estimation and Constraints on Intergalactic and Interstellar Media from Dispersion and Scattering of Fast Radio Bursts

The "disk" contribution to DM from the MW is obtained by integrating the direction-dependent NE2001 model (Cordes & Lazio 2002) through the entire Galaxy to give DM_mw,disk(l, b). The NE2001 model actually comprises two disk components, spiral arms, and localized regions. The differences between the NE2001 model and the alternative YMW16 model (Yao et al. 2017) are negligible for FRBs at Galactic latitudes ≳20° but NE2001 is more accurate for FRB 20121102A in the Galactic anticenter direction (Ocker et al. 2021). Also, the YMW16 model does not properly estimate scattering observables and thus cannot be used in our analysis of scattering.

For high-latitude LoSs, the spread in estimated values for DM_mw is several tens of pc cm⁻³, primarily from uncertainties in the contribution from the Galactic halo. For the two low-latitude cases, FRB 20121102A and FRB 20180916B, the uncertainty in DM_mw could be substantially larger. However, for FRB 20121102A, the measured redshift and the independent constraint on DM_h from Balmer-line measurements (Tendulkar et al. 2017) provide tighter ranges for the host-galaxy and IGM contributions. For FRB 20180916B, the total DM is small enough that a substantially larger DM_mw than provided by the NE2001 model for DM_mw,disk is not allowed, particularly for a larger estimated Galactic halo contribution, DM_mw,halo.

To include uncertainties in the disk DM estimate from NE2001, we employ a flat PDF f_mw,d(DM) with a 40% spread (i.e., ±20% deviation from the mean) centered on the NE2001 estimate. While larger departures from NE2001 (or YMW16) estimates are seen for some individual Galactic pulsars due to unmodeled H ii regions, estimates at Galactic latitudes ∣b∣ ≳ 20° appear to have much less estimation error, gauged in part by the near agreement of the NE2001 and YMW16 models and also by consistency (in the mean) with parallax distances of high-latitude pulsars (Deller et al. 2019). As a test, using a smaller 20% spread on NE2001 DM values yielded very little change in the final results.

Estimates in the literature for the MW's halo contribution to DM range from 25 pc cm⁻³ to ∼80 pc cm⁻³ (Prochaska & Neeleman 2018; Shull & Danforth 2018; Prochaska & Zheng 2019; Yamasaki & Totani 2020), large enough to impact estimates of extragalactic contributions. Though it has been argued that the MW halo could contribute as little as 10 pc cm⁻³ (Keating & Pen 2020), we conservatively use a flat distribution f_mw,h(DM) extending from 25 to 80 pc cm⁻³. We note however that FRB 20200120E in the direction of M81 (Bhardwaj et al. 2021) shows a total DM = 87.8 pc cm⁻³ in the direction (l, b) = (14219, 412). With estimates of DM_mw,disk ∼ 40 and 35 pc cm⁻³ for the NE2001 and YMW16 models, respectively, only 48–53 pc cm⁻³ is allowed for DM_mw,halo + DM_igm + DM_M81.

Recent work has shown that the burst source is coincident with a globular cluster in the M81 system (Kirsten et al. 2022), so the disk of M81 and the globular cluster make no or little contribution to the DM. If we take the assumed minimum MW halo contribution of DM_mw,halo = 25 pc cm⁻³, only 23–28 pc cm⁻³ are contributed by M81's halo along with a minimal contribution from the IGM. An alternative reckoning is to attribute DM_igm ≲ 1 pc cm⁻³ using the mean cosmic density ${n}_{{e}_{0}}$ (next subsection) and the 3.6 Mpc distance to M81, leaving a total of ≲53 pc cm⁻³ for the summed contributions of the MW and M81 halos. While the halo of M81 may be smaller and less dense than that of the MW, FRB 20200120E provides constraints that are not inconsistent with our adoption of 25 pc cm⁻³ as the minimum of the MW's halo contribution.

In our analysis we marginalize over the total MW DM contribution using the PDF for the sum DM_mw,disk +DM_mw,halo, which is the convolution of the disk and halo PDFs, f_DM,mw(DM) = f_mw,d ∗ f_mw,h, which is trapezoidal in form.

The FRBs analyzed in this paper have redshifts z < 1, so it is reasonable to consider the IGM to be almost completely ionized. We calculate the IGM term using a nominal electron density for the diffuse IGM at z = 0 given by a fraction f_igm of the baryonic contribution to the closure density, ${n}_{{e}_{0}}=2.2\times {10}^{-7}$ cm⁻³ f_igm, evaluated using cosmological parameters from the Planck 2018 analysis implemented in Astropy. Shull & Danforth (2018) specify a fiducial range f_igm ≈ 0.6 ± 0.1 for the baryon fraction although Yamasaki & Totani (2020) adopt a range [0.6, 0.9] consistent with an earlier conclusion that f_igm > 0.5 (Shull et al. 2012). Measurements of the kinematic Sunyaev–Zel'dovich effect (e.g., Hill et al. 2016; Kusiak et al. 2021) demonstrate consistency of the baryon fraction with Big Bang nucleosynthesis by attributing the apparent deficit of baryons near galaxies to the presence of ionized gas. Those results suggest f_igm ∼ 0.8 according to the baryon budget presented in Shull et al. (2012, Figure 10) in agreement with Zhang (2018). In this paper, we consider a range of values 0.4 ≤ f_igm ≤ 1 for most of the analysis but adopt f_igm = 0.85 when a specific nominal value is needed. We also show that a value of ∼0.85 minimizes the bias and minimum error for a combined dispersion–scattering redshift predictor for the sample of FRBs that have both redshift and scattering measurements.

For a constant comoving density, the IGM makes a mean contribution

$$\begin{eqnarray}\begin{array}{rcl}{\overline{\mathrm{DM}}}_{\mathrm{igm}}(z) & = & {n}_{{e}_{0}}{D}_{{\rm{H}}}{\displaystyle \int }_{0}^{z}{dz}\,^{\prime} \displaystyle \frac{(1+z\,^{\prime} )}{E(z\,^{\prime} )}\\ & \equiv & {n}_{{e}_{0}}{D}_{{\rm{H}}}{\widetilde{r}}_{1}(z)\approx 972\,\mathrm{pc}\,{\mathrm{cm}}^{-3}{f}_{\mathrm{igm}}{\widetilde{r}}_{1}(z),\end{array}\end{eqnarray} \tag{ 2 }$$

where D_H = c/H₀ is the Hubble distance and $E{(z)={[{{\rm{\Omega }}}_{{\rm{m}}}(1+z)}^{3}+1-{{\rm{\Omega }}}_{{\rm{m}}}]}^{1/2}$ for a flat ΛCDM universe with a matter density Ω_m. The second equality defines the integral ${\widetilde{r}}_{1}(z)$, where ${\widetilde{r}}_{1}\simeq z$ for z ≪ 1.

The cosmic variance of the IGM density (e.g., McQuinn 2014) produces variations in DM characterized as a zero-mean process δDM_igm with a distance-dependent rms, ${\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}(z)$. We approximate the results of cosmological simulations by adopting a simple scaling law,

$$\begin{eqnarray}&&{\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}(z)={\left[{\overline{\mathrm{DM}}}_{\mathrm{igm}}(z){\mathrm{DM}}_{{\rm{c}}}\right]}^{1/2},\end{eqnarray} \tag{ 3 }$$

where DM_c = 50 pc cm⁻³. For z = 1 this gives ${\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}(1)=233{f}_{\mathrm{igm}}^{\mathrm{1/2}}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$. We obtained results by increasing DM_c to 100 pc cm⁻³, i.e., a 41% increase in ${\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}$, and found little change in the net results described in the rest of the paper.

Our scaling law implies a decrease in the fractional variation of ${\overline{\mathrm{DM}}}_{\mathrm{igm}}$ with increasing redshift as ${\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}(z)/{\overline{\mathrm{DM}}}_{\mathrm{igm}}(z)\,={[{\mathrm{DM}}_{{\rm{c}}}/{\overline{\mathrm{DM}}}_{\mathrm{igm}}(z)]}^{1/2}$, which is consistent with simulation results reported by Ioka (2003), Inoue (2004), McQuinn (2014), and Dolag et al. (2015), although there is considerable uncertainty related to the number of halos encountered along an LoS and their sizes. This is exemplified in Pol et al. (2019), who report substantially different DM distributions between uniform weighting and matter-weighted LoS integrals. Simulations also indicate a substantial skewness of DM_igm toward larger values.

The cosmic variance in DM_igm is implemented using a redshift-dependent PDF f_DM,igm(DM_igm; z, f_igm) that is log-normal in form, ${ \mathcal N }(\mu ,\sigma )$ with parameters

$$\begin{eqnarray}&&\sigma ={\left\{\mathrm{ln}\left[1+{\left({\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}/{\overline{\mathrm{DM}}}_{\mathrm{igm}}\right)}^{2}\right]\right\}}^{1/2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mu =\mathrm{ln}{\overline{\mathrm{DM}}}_{\mathrm{igm}}-{\sigma }^{2}/2.\end{eqnarray} \tag{ 5 }$$

The skewness of the distribution, $\gamma =({e}^{{\sigma }^{2}}+2)\sqrt{{e}^{{\sigma }^{2}}-1}$, decreases with redshift and so is at least qualitatively consistent with the published simulations cited above.

Figure 1 shows DM_igm(z) for f_igm = 0.8 ± 0.1 using the parameterization of Equations (2) and (3). The inset shows the PDFs of DM_igm at z = 1 for the same values of f_igm. The cosmic variance in DM_igm implies considerable variations in DM-derived values of redshift even if the baryonic fraction f_igm is known. Likewise, uncertainties in f_igm exacerbate those of DM-derived redshifts. In a later section, we demonstrate that scattering measurements can further improve redshift estimates as well as constrain the value of f_igm.

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A frequent assumption that appears in the FRB literature is a constant host-galaxy contribution to DM, often with a value ${\mathrm{DM}}_{{\rm{h}}}^{(\mathrm{assumed})}=50\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$ (e.g., Arcus et al. 2021 and references therein) accompanied by a statement that a range of values does not matter in an analysis of mostly large FRB DMs. We find that this is not the case for published FRBs with associated galaxy redshifts. Indeed, our conclusion is underscored by the discovery of the low-redshift FRB 190520 with a large total DM (z = 0.241, DM = 1202 pc cm⁻³), which requires a large DM_h (Niu et al. 2021). In this paper, a necessary step is to calculate the Bayesian posterior PDF for each FRB.

We wish to estimate the DM_h contributed by a host galaxy (in its rest frame) taking into account uncertainties in the MW and IGM contributions. We assume that measurements of the DM and redshift z have negligible error. From Equation (1) the conditional PDF for DM_h is

$$\begin{eqnarray}&&\begin{array}{l}{f}_{{\mathrm{DM}}_{{\rm{h}}}}({\mathrm{DM}}_{{\rm{h}}}| \mathrm{DM},{\mathrm{DM}}_{\mathrm{mw}},{z}_{{\rm{h}}})\\ \ =\,{\left(1+{z}_{{\rm{h}}}\right)}^{-1}{f}_{\mathrm{DM},\mathrm{igm}}(\mathrm{DM}-{\mathrm{DM}}_{\mathrm{mw}}-{\mathrm{DM}}_{{\rm{h}}}/(1+{z}_{{\rm{h}}})).\end{array}\end{eqnarray} \tag{ 6 }$$

Marginalization over the PDF f_DM,mw for the MW contribution DM_mw then gives ${f}_{{\mathrm{DM}}_{{\rm{h}}}}({\mathrm{DM}}_{{\rm{h}}}| \mathrm{DM},{z}_{{\rm{h}}})$.

Figure 2 shows the PDF ${f}_{{\mathrm{DM}}_{{\rm{h}}}}$ and the corresponding cumulative distribution function (CDF) for four selected objects and four different values for the baryonic fraction, f_igm. The FRBs include FRB 20200120E with a very small total DM compared to another with a potentially large host-galaxy DM, FRB 20190523A. Due to the proximity of FRB 20200120E, different assumed values for the baryonic fraction yield negligible changes in the estimates of DM_h because the IGM contributes very little to DM. However, changes in DM_igm and thus DM_h are of order 100–200 pc cm⁻³ for FRB 20121102A and FRB 20200430A and several hundred pc cm⁻³ for FRB 20190523A for different values of f_igm. The range of DM_h for FRB 20121102A is consistent with that found from the analysis of Balmer lines (Bassa et al. 2017; Tendulkar et al. 2017), designated by the shaded band in the figure.

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Table 2 gives the DM inventory for all of the FRBs from Table 1. Columns (1)–(3) give the FRB name, measured DM, and host redshift. The next eight columns give the DM and credible range for the MW, IGM, and host-galaxy contributions. The last three columns give redshift estimates using only the DM inventory, as discussed in Section 5.1. In that section, the DM-based redshifts are compared with those obtained using a combined scattering–DM redshift estimator.

Table 2. FRB DM Inventory and Redshift Estimates from the DM

FRB	DM	zh	DMmw a		DMigm b (figm= 0.85)			DMh c (figm= 0.85)			$\widehat{z}(\mathrm{DM},{f}_{\mathrm{igm}}=0.85)$
	(pc cm−3)		(pc cm−3)		(pc cm−3)			(pc cm−3)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
20121102A	557	0.193	241	±27	152	−59	+97	215	−83	+69	0.373	−0.117	+0.125
20180916B	349	0.034	252	±28	24	−12	+36	82	−33	+33	0.103	−0.048	+0.066
20180924A	361	0.321	93	±17	268	−87	+129	99	−52	+67	0.319	−0.105	+0.113
20181112A	589	0.475	94	±17	411	−114	+159	206	−118	+128	0.571	−0.153	+0.158
20190102B	364	0.291	110	±17	240	−81	+122	100	−53	+64	0.302	−0.101	+0.110
20190523A	761	0.660	90	±16	585	−142	+188	261	−155	+178	0.762	−0.183	+0.188
20190608B	339	0.118	90	±16	87	−39	+72	190	−68	+45	0.297	−0.100	+0.108
20190611B	321	0.378	110	±17	320	−97	+141	58	−26	+43	0.253	−0.089	+0.099
20190711A	593	0.522	109	±17	455	−122	+167	171	−100	+123	0.559	−0.151	+0.157
20190714A	504	0.236	91	±16	191	−69	+109	289	−116	+83	0.481	−0.138	+0.143
20191001A	507	0.234	97	±17	188	−68	+109	287	−115	+82	0.477	−0.137	+0.143
20200430A	380	0.160	80	±16	123	−50	+87	217	−84	+58	0.356	−0.113	+0.120
FRB20200120E	88	⋯	93	±17	<1	⋯	⋯	13	−6	+7	⋯	⋯	⋯
20201124A	414	0.098	192	±23	71	−33	+64	172	−60	+44	0.305	−0.100	+0.107

Notes.

^a DM_mw includes contributions from the disk using the NE2001 model and halo described in Section 2.1. ^b DM_igm is calculated using the redshift z_h and the log-normal model of Section 2.2. ^c DM_h values are in the frame of the host galaxy at redshift z_h and are calculated by integrating over the PDFs for DM_mw and DM_igm using Equation (6).

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The LoS to FRB 20200120E, the FRB in a globular cluster associated with M81, evidently does not sample the disk of M81. Using the PDF for the MW contribution from the disk and halo and the negligible contribution from the diffuse IGM, the halo of M81 is found to contribute a DM of ${\mathrm{DM}}_{{\rm{M}}81,\mathrm{halo}}={13}_{-8}^{+21}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$.

The posterior PDFs for host-galaxy DMs are combined in Figure 3, which shows the global PDF and CDF for 13 objects (excluding FRB 20200120E) using five values for f_igm, including a value of 1.2 that exceeds the nominal limit for the diffuse IGM. Note again that DM_h is defined in the host-galaxy frame, not the observer's frame. The PDF shifts to larger values of DM_h for larger f_igm. We find the range f_igm ≃ 0.85 ± 0.05 to be a good representation of our overall results (see Section 5.2). For f_igm = 0.85, the 68% credible interval is ${\mathrm{DM}}_{{\rm{h}}}={166}_{-100}^{+122}\,\mathrm{pc}\,{\mathrm{cm}}^{-3}$, a result that is not inconsistent with those of James et al. (2022). The CDF implies that about 5% of FRBs will show DM_h ≳ 400 pc cm⁻³. The discovery of FRB 20190520B with an implied DM_h well in excess of 400 pc cm⁻³ (Niu et al. 2021) will extend the tail of the global PDF further but is not overly inconsistent with the statistics of the sample we have analyzed in Table 1. A detailed analysis of FRB 20190520B is given in Ocker et al. (2022a).

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We incorporate much of what has been learned about temporal scattering from Galactic pulsars. Figure 4 shows scattering times plotted against DM for 568 pulsars, including upper limits, using data from the literature. Scattering times from different radio frequencies have been scaled to 1 GHz using a scaling law with x_τ = 4.

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Multifrequency observations yield a range of roughly 3 ≲ x_τ ≲ 4.5 for the power-law index x_τ , whereas idealized models of diffraction from small-scale density fluctuations in the interstellar plasma indicate x_τ = 4 (e.g., Scheuer 1968; Rickett 1990) or x_τ = 2β/(β − 2) = 4.4 for the simplest form of Kolmogorov fluctuations with a wavenumber spectral slope β = 11/3. Departures from x_τ = 4.4 are expected if the inner scale for the fluctuations is larger than the diffraction scale (Spangler & Gwinn 1990; Bhat et al. 2004; Rickett et al. 2009), or if scattering is anisotropic (e.g., Brisken et al. 2010). Scattering regions that are finite in size transverse to the LoS also alter the scaling law (Cordes & Lazio 2001). These effects invariably reduce x_τ from the simple Kolmogorov value. Keeping this variety of scaling exponents in mind, we adopt x_τ = 4 as a fiducial value. This value is also consistent with the multifrequency analysis of Bhat et al. (2004) and Krishnakumar et al. (2015). Fitting a function $\widehat{\tau }(\mathrm{DM})=A\times {\mathrm{DM}}^{a}(1+B\times {\mathrm{DM}}^{b})$ (Ramachandran et al. 1997) to the pulsar data yields the scattering–DM relation for Galactic pulsars at frequencies ν in GHz,

$$\begin{eqnarray}\begin{array}{rcl}{\left[\widehat{\tau }(\mathrm{DM},\nu )\right]}_{\mathrm{mw},\mathrm{psr}} & = & 1.90\times {10}^{-7}\,\mathrm{ms}\times {\nu }^{-{x}_{\tau }}{\mathrm{DM}}^{1.5}\\ & & \times (1+3.55\times {10}^{-5}\,{\mathrm{DM}}^{3.0}),\end{array}\end{eqnarray} \tag{ 8 }$$

with scatter ${\sigma }_{\mathrm{log}\tau }=0.76$ (Bhat et al. 2004; Cordes & Chatterjee 2019). The fit is shown in Figure 4 as a red band with a centroid line given by Equation (8) and the upper and lower boundaries corresponding to $\pm 1\,{\sigma }_{\mathrm{log}\tau }$. The band steepens significantly at large DMs, a feature that is due to the larger density fluctuations in the inner Galaxy, where large-DM pulsars are located, compared to those near the solar system or in the outer galaxy (Cordes et al. 1991; Cordes & Lazio 2002).

The measured scattering times necessarily include the fact that Galactic pulsars are embedded in the interstellar scattering medium. The same medium will scatter FRBs but by larger amounts because of their much larger distances. The difference between spherical wavefronts from Galactic pulsars and plane waves from distant extragalactic FRBs amounts to an increase by a factor g_sw→pw = 3 in the scattering time. The same holds true for FRBs scattered by their host galaxies by reciprocity (or by the time reversal of propagation).

Figure 5 shows the distribution of τ versus DM for Galactic pulsars after applying this geometrical correction. The cyan band in Figure 5 is a schematic depiction of the fit to Galactic pulsars. Also shown are scattering times from FRBs in the first CHIME catalog of 535 distinct FRBs and a measurement of the largest measured scattering for FRB 20191221A (The CHIME/FRB Collaboration et al. 2021), τ(0.6 GHz) = 340 ± 10 ms or τ(1 GHz) ≃ 44 ± 1.3 ms. This latter point is included to show the wide range of values for FRB scattering.

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The abscissa in Figure 5 should in principle stand for the DM of the relevant extragalactic scattering medium with any redshift correction due, but of course, we do not know the redshifts of most FRBs. Using the nominal total DM values shows a long-recognized feature (e.g., Cordes & Chatterjee 2019) of FRB scattering that they are "underscattered" compared to Galactic pulsars. This signifies that the scattering properties of a significant portion of the total DMs are deficient in scattering strength.

We further analyze FRB scattering in terms of a parameterized cloudlet model for the scattering medium.

As implied by the τ–DM relation for Galactic pulsars, free electrons both disperse and scatter pulses and bursts. However, while all nonrelativistic free electrons cause dispersion, scattering requires small-scale density fluctuations that are likely very different in warm ∼ 10⁴ K plasma and hot-phase gas (>10⁶ K). Consequently, we expect the τ–DM relation to differ greatly between interstellar media in galaxies and hot, tenuous plasma in galaxy halos and in the IGM. This is demonstrated to be the case using existing scattering measurements.

To model the scattering medium in any one component (e.g., the MW, a host or intervening galaxy, or subregions within galaxies, such as H ii complexes), we use a population of small clouds of ionized gas, each with internal density fluctuations. Following the formalism first presented in Cordes et al. (1991) and further developed by Taylor & Cordes (1993), Cordes & Lazio (2002), Cordes et al. (2016), Macquart & Koay (2013), and Ocker et al. (2020, 2021), cloudlets have internal electron densities ${\overline{n}}_{{\rm{e}}}$ and fractional rms density fluctuations $\varepsilon ={\sigma }_{{n}_{{\rm{e}}}}/\langle {\overline{n}}_{{\rm{e}}}\rangle \leqslant 1$ (angular brackets denote ensemble average). Variations between cloudlets are given by $\zeta =\langle {\overline{n}}_{{\rm{e}}}^{2}\rangle /\langle {\overline{n}}_{{\rm{e}}}{\rangle }^{2}\geqslant 1$. Cloudlets have a volume-filling factor f. We assume internal fluctuations follow a power-law spectrum $\propto {C}_{n}^{2}{q}^{-\beta }\exp [-{(2\pi q/{l}_{{\rm{i}}})}^{2}]$ for wavenumbers 2π/l_o ≤ q ≲ 2π/l_i, where l_o and l_i ≪ l_o are the outer and inner scales, respectively. We use a Kolmogorov spectrum with β = 11/3 as a reference spectrum.

The resulting broadening time from a layer with dispersion depth DM_ℓ is derived in the Appendix,

$$\begin{eqnarray}\begin{array}{rcl}\tau ({\mathrm{DM}}_{{\ell }},\nu ) & = & {C}_{\tau }{\nu }^{-4}{A}_{\tau }\widetilde{F}G\,{\mathrm{DM}}_{{\ell }}^{2}\\ & \simeq & 0.48\,\mathrm{ms}\times {\nu }^{-4}{A}_{\tau }\widetilde{F}G\,\times {\mathrm{DM}}_{100}^{2},\end{array}\end{eqnarray} \tag{ 9 }$$

with ν in GHz and DM₁₀₀ ≡ DM_ℓ /(100 pc cm⁻³). The quantity C_τ is a numerical constant defined in the Appendix. The quantity A_τ depends on the inner scale l_i and spectral index β and accounts for the shape of the pulse-broadening function, as described in the Appendix. It can range from ∼ 1/6 to unity. Other parameters that characterize density fluctuations combine into the quantity

$$\begin{eqnarray}&&\widetilde{F}=\displaystyle \frac{\zeta {\varepsilon }^{2}}{f{\left({l}_{{\rm{o}}}^{2}{l}_{{\rm{i}}}\right)}^{1/3}},\end{eqnarray} \tag{ 10 }$$

which has units of ${({\mathrm{pc}}^{2}\,\mathrm{km})}^{-1/3}$ for the outer scale in parsecs and the inner scale in kilometers. The location of the scattering layer relative to the source strongly affects τ and determines the geometric factor, G (see next section). In most of our analysis, the composite quantity ${A}_{\tau }\widetilde{F}G$ is constrained by observations, though we expect G = 1 for the LoSs considered in this paper.

For cosmological distances, a source at redshift z_s and a scattering region in a host or intervening galaxy at z_ℓ gives

$$\begin{eqnarray}&&\begin{array}{l}\tau ({\mathrm{DM}}_{{\ell }},\nu ,{z}_{{\ell }},{z}_{{\rm{s}}})\\ \simeq \,0.48\,\mathrm{ms}\times \displaystyle \frac{{A}_{\tau }\widetilde{F}G({z}_{{\ell }},{z}_{{\rm{s}}}){\mathrm{DM}}_{l,100}^{2}}{{\nu }^{4}{\left(1+{z}_{{\ell }}\right)}^{3}}.\end{array}\end{eqnarray} \tag{ 11 }$$

Here DM_ℓ is in the rest frame of the scattering layer (i.e., a host or intervening galaxy or halo), which contributes to the measured DM as DM_ℓ /(1 + z_ℓ ).

3.2.1. Scattering Geometries

We define the dimensionless geometric factor G so that it is unity for a source embedded in the scattering medium, such as its host galaxy, and the source distance is much larger than the thickness of the scattering medium. Generally, G is a strong function of the LoS distribution of scattering electrons and can exceed unity by many orders of magnitude. In Euclidean space,

$$\begin{eqnarray}&&G=\displaystyle \frac{{\int }_{\mathrm{layer}}{ds}\,s(1-s/d)}{{\int }_{\mathrm{host}}{ds}\,s(1-s/d)}.\end{eqnarray} \tag{ 12 }$$

For scattering within the MW or in a distant FRB host galaxy G = 1, but it is ≫1 for an intervening galaxy or halo.

First, we derive the geometric factor G with reference to the geometry shown in Figure 6 for a statistically homogeneous (i.e., constant ${C}_{n}^{2}$) layer of thickness L that is offset from the FRB source by Δd. Letting x = L/d_so and y = Δd/d_so for a source–observer distance d_so and defining $g(a,b)={\int }_{a}^{b}{ds}\,s(1-s)$, the geometric factor is defined so that G = 1 for a slab representing a host galaxy (y = 0) or the MW (y = 1 − x),

$$\begin{eqnarray}\begin{array}{rcl}G(x,y) & = & \displaystyle \frac{g(y,x+y)}{g(0,x)}\\ & = & \displaystyle \frac{1-2x/3+(2y/x)(1-y-x)}{(1-2x/3)}.\end{array}\end{eqnarray} \tag{ 13 }$$

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Treating intervening galaxies and halos as thin slabs (x ≪ 1) that are close to neither the source nor observer, we have G ≃ (2y/x)(1 − y) ≫ 1, illustrating that scattering from an intermediately positioned slab yields much greater pulse broadening, all else being equal. The strong dependence of G on y/x suggests that some of the scatter at a fixed DM in the cyan band shown in Figure 4 derives from different pulsars having different concentrations of scattering regions along their LoS. This "Galactic variance" yields different values of G and thus τ for objects with identical values of DM.

Figure 7 shows four scattering configurations involving the MW, a host galaxy, and an intervening galaxy that are likely to be encountered in FRB observations. Two cases apply to an FRB source that is unaffiliated with or on the near side of a host galaxy, and cases are shown with and without an intervening galaxy. While objects discussed in this paper involve only the first two cases in the figure, we also need to dismiss the possibility that the other two cases do not apply to the current sample, as discussed below.

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For nonnegligible redshifts, the expression for G is replaced by one involving angular diameter distances, yielding G(z_ℓ , z_s) = 2d_sl d_lo/Ld_so, where d_sl and d_lo are distances from the source to the scattering layer and from layer to observer, respectively, and d_so is the source distance. Figure 8 shows G versus redshift ratio for several values of the source redshift, z_s, which we take to be the redshift of a host galaxy (though generally a source need not be associated with a galaxy). For low redshifts, G is symmetric about the midpoint where z_ℓ /z_s = 1/2. However, for large source redshifts, G maximizes at progressively smaller values of z_ℓ /z_s, though at intervening redshifts z_ℓ that are still cosmological. This effect enters into any consideration of scattering of high-redshift FRBs, which we defer to another paper in progress (Ocker et al. 2022b).

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For scattering in a host galaxy, d_lo/d_so → 1 and d_sl → L/2, yielding G → 1 as with the Euclidean expression. For Gpc distances (d_sl, d_lo, d_so) and L = 1 kpc, G ∼ Gpc/kpc ∼ 4 × 10⁵. However, unless a galaxy disk is encountered with edge-on geometry, the DM_ℓ may be small. Nonetheless, even with DM_ℓ = 10 pc cm⁻³, the scattering time for an intervening galaxy would be τ ≃ 2 s if $\widetilde{F}$ is similar to Galactic values. This fact can be used to rule out whether any observed scattering occurs in an intervening galaxy instead of a host galaxy if an FRB can be detected along an LoS that intersects a galaxy disk. More likely, given the large implied scattering for nominal parameters, FRB detections would be strongly suppressed along any such LoS, such as those in the two cases shown on the right in Figure 7.

Our results indicate that FRB LoSs that pierce an intervening galaxy disk are unlikely to be seen at frequencies ν ≲ 1.5 GHz because the scattering is much larger than the intrinsic burst width. If only a halo is intersected, the FRB's DM will be enhanced but the scattering will not increase significantly.

3.2.2. Required Values of $\widetilde{F}G$ for Galactic Pulsars

On both observational and theoretical grounds, the IGM's contribution to scattering is likely negligible in comparison with contributions from the interstellar media of galaxy disks, including the MW, host, and intervening galaxies. Not all FRBs with large measured DMs ≳ 10³ pc cm⁻³ show large scattering times, which might have been expected if scattering were IGM dominated even with cosmic variance taken into account.

The τ(DM) relation for galaxy disks does not change qualitatively in a cosmological context once redshift dependencies are included, as in Equation (7) (see also Macquart & Koay 2013). Given that the IGM contributes DM values comparable to those of galaxy disks, one might expect scattering to also be similar. However, the $\widetilde{F}G$ factor is likely to be quite different. Assume the product ζ ε² is the same because it measures fractional fluctuations that are of order unity in the ISM, and consider equal contributions to the total DM. Ignoring redshift factors, which are close to unity for low-z objects, the ratio of scattering times from the IGM and from a galaxy's ISM is

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{\mathrm{IGM}}}{{\tau }_{\mathrm{ISM}}}\approx \displaystyle \frac{{\left[f{\left({l}_{{\rm{o}}}^{2}{l}_{{\rm{i}}}\right)}^{1/3}\right]}_{\mathrm{ISM}}}{{\left[f{\left({l}_{{\rm{o}}}^{2}{l}_{{\rm{i}}}\right)}^{1/3}\right]}_{\mathrm{IGM}}},\end{eqnarray} \tag{ 14 }$$

where G = 1 applies to a source embedded in the ISM of a host galaxy.

The outer scales alone are probably very different because in standard turbulence pictures, they correspond to the scales on which energy is injected. ISM scales are ≲kpc and IGM scales ≳Mpc, giving τ_IGM/τ_ISM ≲ (kpc/Mpc)^2/3 ≲ 10⁻².

The filling factor and inner scale are also likely larger for the IGM, further reducing the ratio. For example, the inner scale for the solar wind may be linked to the thermal proton gyroradius r_g,p = v_p (T)/Ω_g,p (where v_p (T) is the rms thermal speed and Ω_g,p is the gyrofrequency) or to the proton inertial length, ℓ_i,p = c/ω_p,p , where ω_p,p is the proton plasma frequency (Goldstein et al. 2015). These would imply l_i ∝ T^1/2/B or ${l}_{{\rm{i}}}\propto {n}_{{\rm{e}}}^{-1/2}$, respectively. Given the higher temperature, smaller magnetic field, and smaller plasma density of the IGM compared to an ISM, the ratio τ_IGM/τ_ISM might be reduced by another order of magnitude.

Luan & Goldreich (2014) argue similarly that the outer scale for the IGM must be comparable to Galactic values to allow a significant contribution to τ, but they also point out that the resultant turbulent heating would cause the IGM to be hotter than inferred from observations. In the following we therefore exclude any contribution from the diffuse IGM to scattering.

Galactic pulsars show a strong Galactic latitude dependence for scattering times indicative of contributions from a strongly scattering thin disk (Cordes & Lazio 2002; Yao et al. 2017) and a thick disk with a scattering scale height of one-half the scale height ${H}_{{n}_{{\rm{e}}}}\sim 1.6\,\mathrm{kpc}$ for the electron density (Ocker et al. 2020). Comparison with the scattering of AGNs, which sample the entire MW halo (unlike pulsars in or near the thick disk or pulsars in the Magellanic clouds), shows no increase in scattering over that provided by the disk components. This implies a modest DM contribution from the Galactic halo along with a small value of $\widetilde{F}G$. We therefore exclude contributions to scattering from the Galactic halo. This may be true for the halos of other galaxies. A specific case is FRB 20200120E in a globular cluster near M81 that likely samples only the halos of M81 and the MW along with the MW disk components. Burst amplitude substructure is seen down to tens of microseconds (combined with shot pulses at the resolution limit of 31.25 ns) that shows no hint of scattering from outside the MW disk (Nimmo et al. 2022).

Nonetheless, given the dynamic processes involved with halo evolution (e.g., Smercina et al. 2020) that might also drive turbulence and the prospects for there being substructure in halos that might also cause radio scattering (Vedantham & Phinney 2019), the possibility is still open that some halos may contribute to scattering.

All FRBs have been found from directions where pulse broadening from the MW is too small to detect at observation frequencies larger than 1 GHz but it has been measured at 0.15 GHz for FRB 190816 (Pastor-Marazuela et al. 2021). MW scattering has also been measured in the form of intensity variations with a characteristic scintillation bandwidth Δν_d for several objects (e.g., Masui et al. 2015; Hessels et al. 2019; Bhandari et al. 2020; Marcote et al. 2020). For the low-latitude FRB 20121102A (b = −02), the implied scattering is only about τ ∼ 1/2πΔν_d ≃ 20 μs at 1 GHz for its Galactic anticenter direction, in agreement with the NE2001 prediction within a factor of 2 (Ocker et al. 2021). Future observations will likely probe a wide range of scattering strengths as more bursts are found at low frequencies and low Galactic latitudes.

For the remainder of the paper, we ignore contributions to τ from the MW disk along with those from the halo and from the IGM. As with the DM, we also ignore for now any scattering from intervening galaxies and their halos and also from any intercluster medium, leaving only host galaxies as the main contributor to pulse broadening. The observed scattering time τ_obs is then given by Equation (11) using DM_h and z_g = z_s = z_h. Future studies are likely to include FRBs with significant scattering from the MW. For these cases, the pulse broadening is simply the sum of the contributions from the host galaxy and the MW.

The interplay between dispersion, scattering, and redshift is shown in Figure 9 for two cases, FRB 20121102A and FRB 20190523A. In the top panel of each frame, DM_h is plotted against redshift using Equations (1) and (2) and taking into account cosmic variance in DM_igm as described in Section 2.2. If the redshift were unknown and only the measured DM is available (along with a model for the MW's contribution), a wide range of redshifts is allowed, roughly a factor of 2 in both cases. The actual redshifts shown as vertical red lines indicate a somewhat narrow range for DM_h for FRB 20121102A but a much wider range for FRB 20190523A.

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The bottom panels show how the scattering time estimate depends on redshift (using Equations (7) and (9)), again taking into account cosmic variance in DM_igm, but including a wide range for ${A}_{\tau }\widetilde{F}G$ in the host galaxy. This "interstellar variance" expands the range of possible scattering times. The upper bound on τ for FRB 20121102A is compatible with this range while the measured τ for FRB 20190523A is at the high end of the range of ${A}_{\tau }\widetilde{F}G$ at the measured redshift but overall is not inconsistent with the predicted ranges when the cosmic variance of DM_igm is also taken into account.

Next we compare scattering in host galaxies with that in intervening galaxies. As shown in Figure 8, the geometric factor used in Equation (11) that enhances scattering is orders of magnitude larger for an intervening galaxy compared to G = 1 in a host galaxy. A consequence is that $\widetilde{F}$ needs to be proportionately smaller in intervening galaxies if they are not to cause scattering times vastly exceeding measured values. These very small values of $\widetilde{F}$ would imply that intervening galaxies can produce significant contributions to DM without corresponding scattering times like those derived from pulsars in the MW. This in turn would require an explanation for why FRBs sample dispersive gas in intervening galaxies with significantly different turbulence properties. A simpler hypothesis is that extragalactic scattering occurs in host galaxies, not in any intervening galaxies in the sample we have analyzed.

To compare host and intervening galaxies, we define a scaled scattering time,

$$\begin{eqnarray}&&\widetilde{\tau }=\displaystyle \frac{{\nu }^{4}}{{[{\mathrm{DM}}_{100}^{\mathrm{obs}}]}^{2}}\left(\displaystyle \frac{\tau }{0.48\,\mathrm{ms}}\right)=\displaystyle \frac{{A}_{\tau }\widetilde{F}G({z}_{{\ell }},{z}_{{\rm{s}}})}{1+{z}_{{\ell }}},\end{eqnarray} \tag{ 15 }$$

which involves observable quantities after the first equality and unknown quantities after the second. The redshift of the scattering layer z_ℓ is either that of an intervening galaxy or a region in a host galaxy (with z_ℓ very slightly smaller than z_h so that d_sl = L/2). The DM contributed by the layer ${\mathrm{DM}}_{100}^{\mathrm{obs}}$ is expressed in the observer's frame in units of 100 pc cm⁻³. Figure 10 shows $\widetilde{\tau }$ versus ${A}_{\tau }\widetilde{F}G({z}_{{\ell }},{z}_{{\rm{s}}})/(1+{z}_{{\ell }})$ for two values of the geometric factor, G = 1, for host galaxies, and G = 5 × 10⁵, which is typical for an intervening galaxy at a redshift that maximizes G (see Figure 8). In both cases we have used the inferred DM_h as the DM contributed by the layer expressed in the observer's frame. Measurements and upper limits on $\widetilde{\tau }$ yield a range for the abscissa of ${A}_{\tau }\widetilde{F}/(1+{z}_{{\rm{h}}})\sim 0.018$ to 7.9 $\,{({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$ if scattering occurs in host galaxies with G = 1 (blue band along the horizontal axis). However, if scattering were to occur in intervening galaxies, the values would be smaller by a factor ${(5\times {10}^{5})}^{-1}$ or ${A}_{\tau }\widetilde{F}/(1+{z}_{{\rm{h}}})\sim 3.6\times {10}^{-8}$ to $1.6\times {10}^{-5}\,{({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$ (orange band along the horizontal axis). These values are significantly smaller than those that apply to the ISM of the MW, which range from about 10⁻³ to 10 ${({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$. We conclude that scattering of FRBs with known redshifts occurs in host galaxies with interstellar media similar to those in the MW as gauged by $\widetilde{F}$. Based on this, in Section 5 we adopt a flat prior for ${A}_{\tau }\widetilde{F}$ over the range 0.01 to 10 ${({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$. This is consistent with values in Figure 2 of Ocker et al. (2021) based on measurements of Galactic pulsars. ²

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The PDF of τ given the redshift and host-galaxy DM_h is $\delta (\tau -\widehat{\tau })$ with $\widehat{\tau }={C}_{\tau }{\nu }^{-4}{A}_{\tau }\widetilde{F}G\,{\mathrm{DM}}_{{\rm{h}}}^{2}/{(1+z)}^{3}$. If the redshift is known to high precision but the measured broadening has an error distribution f_{δ τ} (τ − τ_obs, σ_τ ), where τ_obs is a nominal value and σ_τ is the uncertainty, we calculate the likelihood function for x ≡ DMP_h and $\phi \equiv {A}_{\tau }\widetilde{F}G$, the two parameters we use to characterize the interstellar medium of a host galaxy,

$$\begin{eqnarray}&&\begin{array}{l}{\mathscr{L}}(x,\phi \,| \,\mathrm{DM},{z}_{{\rm{h}}},\tau )\\ \quad =\,{f}_{\tau }(\widehat{\tau }\,| \,{\mathrm{DM}}_{{\rm{h}}},{z}_{{\rm{h}}})={f}_{\delta \tau }(\widehat{\tau }-{\tau }_{\mathrm{obs}}).\end{array}\end{eqnarray} \tag{ 16 }$$

Combined with the MW-marginalized PDF for DM_h in Equation (6) and assuming an uninformative flat prior for ϕ, the posterior PDF is

$$\begin{eqnarray}&&\begin{array}{l}{f}_{{\mathrm{DM}}_{{\rm{h}}},\phi }(x,\phi | \mathrm{DM},{z}_{{\rm{h}}},\tau )\\ \ \propto \,{f}_{{\mathrm{DM}}_{{\rm{h}}}}(x| \mathrm{DM},{z}_{{\rm{h}}}){f}_{\tau }(\tau -{A}_{\tau }{C}_{\tau }{\nu }^{-4}\phi \,{x}^{2}/{\left(1+z\right)}^{3}).\end{array}\end{eqnarray} \tag{ 17 }$$

Figure 11 gives two examples of the joint posterior PDFs for x and ϕ using f_igm = 0.85 for FRB 20121102A, an object with only an upper limit on scattering, and FRB 20190523A, which has a significant measurement of the scattering time. The flat prior for DM_h extends from 50 pc cm⁻³ in both cases up to different maximum values, 500 pc cm⁻³ and 1000 pc cm⁻³, respectively.

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For FRB 20121102A, the marginalized PDF for ${A}_{\tau }\widetilde{F}G$ in the side panel includes a tail to values of ${A}_{\tau }\widetilde{F}G$ larger than unity that are still consistent with the upper limit on τ because the corresponding values of DM_h are very small. These small values are strongly disfavored by Balmer-line measurements that indicate DM_h ∼ 55 to 380 pc cm⁻³ (as indicated in Figure 2). The marginalized PDF for DM_h in the upper panel (black curve) indicates these values along with a wider range extending to ∼400 pc cm⁻³. The same frame shows in red the posterior PDF for DM_h resulting from the DM inventory also shown in Figure 2.

FRB 20190523A, by comparison, shows a joint PDF with a shape determined by the relationship $\phi ={A}_{\tau }\widetilde{F}G\propto \tau {\mathrm{DM}}_{{\rm{h}}}^{-2}$. Without other constraints, the LoS through the host galaxy can encounter ionized gas with values for DM_h and ${A}_{\tau }\widetilde{F}G$ anywhere along the curved ridge of high probability density. Formally, the broad prior in DM_h allows a solution with small DM_h and a corresponding value for ${A}_{\tau }\widetilde{F}G$ much larger (by one or two orders of magnitude) than is encountered in the MW. While it is conceivable there could be such regions along the LoS to an FRB, their required properties run counter to those in the ISM of the MW and other galaxies.

A simpler conclusion is that the actual range of values for DM_h and thus also for ${A}_{\tau }\widetilde{F}G$ are significantly smaller than the plotted ranges in Figure 11. In particular, the blue horizontal band in Figure 10 corresponds to values for ${A}_{\tau }\widetilde{F}G$ in the interval $[0.02,6]\,{({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$, and in the next section, we will use a slightly larger range, $[0.01,10]\,{({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$ as one of the flat priors used in redshift estimation.

DM-based redshift estimation follows from previous work (e.g., Macquart et al. 2020) and expressions in Section 2.2. First, we express the DM-based redshift ${\hat{z}}_{\mathrm{DM}}$ in terms of an assumed value for DM_h, either an a priori value or one based on Balmer-line measurements of a host galaxy to determine an emission measure EM from which a galaxy-wide estimate for DM_h is estimated. This implies a point estimate for the IGM's contribution (see Equation (1)),

$$\begin{eqnarray}&&{\widehat{\mathrm{DM}}}_{\mathrm{igm}}=\mathrm{DM}-{\widehat{\mathrm{DM}}}_{\mathrm{mw}}-{\widehat{\mathrm{DM}}}_{{\rm{h}}}/(1+z),\end{eqnarray} \tag{ 18 }$$

that yields a redshift by inverting the function ${\widetilde{r}}_{1}(z)$ defined in Equation (2),

$$\begin{eqnarray}&&{\hat{z}}_{\mathrm{DM}}={\widetilde{r}}_{1}^{\,-1}({\widehat{\mathrm{DM}}}_{\mathrm{igm}}/{n}_{{e}_{0}}{D}_{{\rm{H}}})\simeq {\widehat{\mathrm{DM}}}_{\mathrm{igm}}/{n}_{{e}_{0}}{D}_{{\rm{H}}},\end{eqnarray} \tag{ 19 }$$

where the approximate equality is for small redshifts.

More useful is the posterior PDF for redshift based on a likelihood function,

$$\begin{eqnarray}&&{\mathscr{L}}({\mathrm{DM}}_{{\rm{h}}},z| \mathrm{DM};{f}_{\mathrm{igm}})\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&=\delta (\mathrm{DM}-{\mathrm{DM}}_{\mathrm{mw}}-{\mathrm{DM}}_{\mathrm{igm}}(z)-{\mathrm{DM}}_{{\rm{h}}}(1+z)).\end{eqnarray} \tag{ 21 }$$

Using a flat, unconstraining prior ${f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})$ for the host galaxy's redshift and integrating over prior PDFs for u = DM_mw, v = DM_h, and w = DM_igm yields a posterior redshift PDF

$$\begin{eqnarray}&&\begin{array}{l}{f}_{z}({z}_{{\rm{h}}}| \mathrm{DM};{f}_{\mathrm{igm}})\\ \ \propto \,{f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})\iiint {du}\,{dv}\,{dw}\,{f}_{\mathrm{DM},\mathrm{mw}}(u){f}_{{\mathrm{DM}}_{{\rm{h}}}}(v)\\ \quad \times \ {f}_{\mathrm{DM},\mathrm{igm}}(w,{z}_{{\rm{h}}};{f}_{\mathrm{igm}})\delta (\mathrm{DM}-v/(1+{z}_{{\rm{h}}})-u-w)\\ \ \propto \,{f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})\iint {dv}\,{dw}\,{f}_{{\mathrm{DM}}_{{\rm{h}}}}(v){f}_{\mathrm{DM},\mathrm{igm}}(w,{z}_{{\rm{h}}};{f}_{\mathrm{igm}})\\ \quad \times \,{f}_{\mathrm{DM},\mathrm{mw}}(\mathrm{DM}-v/(1+{z}_{{\rm{h}}})-w).\end{array}\end{eqnarray} \tag{ 22 }$$

In the following we hold DM_h fixed at a nominal value to compare our results with the common practice of setting DM_h = 50 pc cm⁻³. Other fixed values can also be used, and there is some tradeoff between f_igm and a choice for DM_h. However, foreshadowing later results, it is unrealistic to assume a constant value for DM_h given the wide variety of galaxies found to harbor FRB sources, as well as their different locations in those galaxies and the orientations of those galaxies relative to the LoS.

Figure 12 shows the DM-based redshift estimator plotted against true redshift for 13 objects. This sample excludes FRB 20201120E because its association with M81 makes it too close to the MW to be characterized with a redshift, which is negative. The three panels for baryonic fractions f_igm = 0.4, 0.8, and 1 demonstrate the much larger bias and scatter for f_igm = 0.4 (left panel) compared to the two larger values used for the center and right panels. The larger bias and scatter for f_igm = 0.4 arises because a larger redshift is needed to provide the IGM contribution to DM, on average, when f_igm is smaller and the cosmic variance of DM_igm is correspondingly larger. This may be seen from Equation (2), which gives ${\overline{\mathrm{DM}}}_{\mathrm{igm}}\propto {f}_{\mathrm{igm}}{\widetilde{r}}_{1}(z)\propto {f}_{\mathrm{igm}}z$ (for z ≪ 1), which implies that $\widehat{z}\propto {f}_{\mathrm{igm}}^{-1}\times (\mathrm{required}\ {\mathrm{DM}}_{\mathrm{igm}})$. The rms ${\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}\propto \sqrt{{f}_{\mathrm{igm}}z}$ translates into an error on $\widehat{z}$ that then scales as ${\sigma }_{\widehat{z}}\propto \sqrt{\widehat{z}/{f}_{\mathrm{igm}}}$, which is also larger for smaller f_igm.

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As measures of the goodness of fit, we show in Figure 13 the mean residual $\delta z=\langle \widehat{z}-z\rangle$, which measures the estimation bias, and the rms residual ${\sigma }_{\delta \widehat{z}}=\langle {(\delta z)}^{2}{\rangle }^{1/2}$ versus f_igm. We have used values for f_igm that exceed unity here (and in further analyses below) to include the possibility that FRBs reside in regions of atypically high baryon fraction (e.g., Pol et al. 2019). Angular brackets denote a weighted average using weights equal to the reciprocal of the variance of $\widehat{z}$ for each FRB (determined from the 68% probability region centered on the median of the posterior CDF). The figure shows these to be monotonically decreasing with larger f_igm. If a larger fixed value of DM_h were used instead of 50 pc cm⁻³, the bias would be reduced for the FRBs with z ≲ 0.25 but would increase for larger redshifts.

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Scattering can further constrain redshifts if it is measurable and sufficiently large to require a substantial host-galaxy DM_h. For a scattering time τ attributed to a host galaxy at redshift z_h, the host-galaxy contribution to the DM (in the host frame) is

$$\begin{eqnarray}\begin{array}{rcl}{\widehat{\mathrm{DM}}}_{{\rm{h}}}(\tau ) & = & {\left[\displaystyle \frac{{\left(1+{z}_{{\rm{h}}}\right)}^{3}{\nu }^{4}\tau (\nu )}{{C}_{\tau }{A}_{\tau }\widetilde{F}G}\right]}^{1/2}\\ & \simeq & 144\,\mathrm{pc}\,{\mathrm{cm}}^{-3}{\left[\displaystyle \frac{{\left(1+{z}_{{\rm{h}}}\right)}^{3}{\nu }^{4}{\tau }_{\mathrm{ms}}(\nu )}{{A}_{\tau }\widetilde{F}G}\right]}^{1/2},\end{array}\end{eqnarray} \tag{ 23 }$$

for ν in GHz, τ in ms, and $\widetilde{F}$ in ${\left({\mathrm{pc}}^{2}\,\mathrm{km}\right)}^{-1/3}$ in the approximate equality. This in turn yields a scattering-based point estimate for DM_igm,

$$\begin{eqnarray}&&\begin{array}{l}{\widehat{\mathrm{DM}}}_{\mathrm{igm}}(\tau ,{z}_{{\rm{h}}})\\ \quad =\,\mathrm{DM}-{\widehat{\mathrm{DM}}}_{\mathrm{mw}}-{\widehat{\mathrm{DM}}}_{{\rm{h}}}(\tau )/(1+{z}_{{\rm{h}}}),\end{array}\end{eqnarray} \tag{ 24 }$$

from which a DM–τ-based redshift is estimated by inverting the dimensionless quantity ${\widetilde{r}}_{1}(z)$ (defined in Equation (2)),

$$\begin{eqnarray}&&{\hat{z}}_{\mathrm{DM},\tau }={\widetilde{r}}_{1}^{-1}({\widehat{\mathrm{DM}}}_{\mathrm{igm}}(\tau ,{z}_{{\rm{h}}})/{n}_{{e}_{0}}{D}_{{\rm{H}}}).\end{eqnarray} \tag{ 25 }$$

To obtain the posterior redshift PDF we use the likelihood function

$$\begin{eqnarray}&&\begin{array}{l}{\mathscr{L}}(x,\phi ,z| \mathrm{DM},\tau )\\ \quad =\,{f}_{\tau }(\tau | {\mathrm{DM}}_{{\rm{h}}},z)={f}_{\delta \tau }(\widehat{\tau }-\tau ),\end{array}\end{eqnarray} \tag{ 26 }$$

where, as before, f_{δ τ} (δ τ) is the measurement error PDF for the scattering time, x ≡ DM_h, and $\phi \equiv {A}_{\tau }\widetilde{F}G$ . We marginalize over the joint PDF of x and ϕ and use a prior ${f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})$ for the host galaxy's redshift,

$$\begin{eqnarray}&&\begin{array}{l}{f}_{z}({z}_{{\rm{h}}}| \mathrm{DM},\tau )\\ \ \propto \,{f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})\iint {dx}\,d\phi \,{f}_{{\mathrm{DM}}_{{\rm{h}}},{A}_{\tau }\widetilde{F}G}(x,\phi )\\ \quad \times \ {f}_{\delta \tau }(\tau -{A}_{\tau }{C}_{\tau }{\nu }^{-4}\phi \,{x}^{2}/{\left(1+z\right)}^{3}).\end{array}\end{eqnarray} \tag{ 27 }$$

Errors in the estimates ${\hat{z}}_{\mathrm{DM}}$ and ${\hat{z}}_{\mathrm{DM},\tau }$ are due to the usual uncertainties in the MW contribution to DM, the measurement error in τ, and the astrophysical variance in $\widetilde{F}$ but mostly from cosmic variance in DM_igm and uncertainty in f_igm.

We evaluate Equation (27) using a flat, uninformative redshift prior ${f}_{{z}_{{\rm{h}}}}({z}_{{\rm{h}}})$ and a flat PDF for x ≡ DM_h over a range DM_h = [20, 1600] pc cm⁻³. For $\phi \equiv {A}_{\tau }\widetilde{F}G$, we use a flat PDF that is sampled at logarithmic intervals over two ranges to provide two different priors for ϕ: a broad range $[{\phi }_{\min },{\phi }_{\max }]=[0.01,10]\ {({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$ and a narrow range, $[0.5,2]\ {({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$. We take this approach to illustrate the effects of alternative priors for the current limited sample of nine objects with scattering measurements and redshifts. The broad range is consistent with most of the pulsar and FRB measurements. In the future, when more redshifts and scattering times are available, we will explore the usage of an alternative prior for ${A}_{\tau }\widetilde{F}G$, such as a log-normal distribution.

Figure 14 shows the posterior PDFs for the three estimators applied to the nine FRBs with available redshifts (again excluding the nearby FRB 20200120E) and scattering times and using f_igm = 0.85, a choice that is discussed below. Scattering is constraining on the redshift if ${\hat{z}}_{\mathrm{DM},\tau }$ is substantially smaller than ${\hat{z}}_{\mathrm{DM}}$ or equivalently if ${\widehat{\mathrm{DM}}}_{{\rm{h}}}(\tau )$ is substantially larger than an a priori chosen value. The scattering-based redshift estimator is likely more accurate for cases where the resulting change in ${\widehat{\mathrm{DM}}}_{\mathrm{igm}}$ from a scattering-based estimate of DM_h is larger than one standard deviation from cosmic fluctuations, or ${\widehat{\mathrm{DM}}}_{{\rm{h}}}(\tau )\gt (1+{z}_{{\rm{h}}}){\sigma }_{{\mathrm{DM}}_{\mathrm{igm}}}({z}_{{\rm{h}}})$.

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Applying this constraint using Equations (2), (3), and (23) and approximating ${\widetilde{r}}_{1}({z}_{{\rm{h}}})\sim {z}_{{\rm{h}}}$ for redshifts z_h ≲ 1, the criterion for when scattering influences redshift estimates is

$$\begin{eqnarray}&&{\tau }_{1\,\mathrm{GHz}}\gtrsim 2\,\mathrm{ms}\times {A}_{\tau }\widetilde{F}G({f}_{\mathrm{igm}}/0.85){z}_{{\rm{h}}}.\end{eqnarray} \tag{ 28 }$$

This expression is consistent with the posterior PDFs shown in Figure 14 for FRBs with different redshifts and scattering times. Those with small scattering times yield nearly identical PDFs for the DM-only and DM+scattering estimators while those with scattering times greater than about 1 ms are clearly influenced by scattering. This unsurprising result simply underscores the consistency of the method.

Table 3 gives redshift estimates for the objects in Table 1. The columns are the FRB name, measured DM, redshift, median host-galaxy DM_h, scattering quantity ${A}_{\tau }\widetilde{F}G$, and median redshift estimates and credible ranges using narrow and wide ranges for ${A}_{\tau }\widetilde{F}G$. Estimates for ${A}_{\tau }\widetilde{F}G$ are made by inversion of Equation (11) (again with ν in GHz and τ in ms):

$$\begin{eqnarray}\begin{array}{rcl}{A}_{\tau }\widetilde{F}G & = & 2.1\,{({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}\\ & & \times \,\displaystyle \frac{{\nu }^{4}{\left(1+{z}_{{\rm{h}}}\right)}^{3}\tau (\nu )}{{\left({\mathrm{DM}}_{{\rm{h}}}/100\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\right)}^{2}}.\end{array}\end{eqnarray} \tag{ 29 }$$

The five FRBs with scattering upper limits yield upper limits on ${A}_{\tau }\widetilde{F}G$. The upper limit on ${A}_{\tau }\widetilde{F}G$ for FRB 20200120E is very small, consistent with the absence of scattering from either the halo of M81 or the halo of the MW. The values and upper limits on ${A}_{\tau }\widetilde{F}G$ are all consistent with the adopted prior that is flat between 0.01 and 10 ${({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3}$.

Table 3. FRB Host-galaxy Parameters and Redshift Estimates

FRB	DM	zh	DMh	${A}_{\tau }\widetilde{F}G$	$\widehat{z}(\mathrm{DM},\tau ,\mathrm{narrow})$			$\widehat{z}(\mathrm{DM},\tau ,\mathrm{wide})$
	(pc cm−3)		(pc cm−3)	$({({\mathrm{pc}}^{2}\ \mathrm{km})}^{-1/3})$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
20121102A	557	0.193	215	<0.46	⋯	⋯	⋯	⋯	⋯	⋯
20180916B	349	0.034	82	<0.092	⋯	⋯	⋯	⋯	⋯	⋯
20180924A	361	0.321	99	8.7	0.128	−0.066	+0.080	0.216	−0.097	+0.106
20181112A	589	0.475	206	0.094	0.566	−0.147	+0.151	0.561	−0.159	+0.159
20190102B	364	0.291	100	0.48	0.290	−0.094	+0.101	0.292	−0.111	+0.113
20190523A	761	0.660	261	2.0	0.576	−0.147	+0.153	0.623	−0.206	+0.190
20190608B	339	0.118	190	7.0	0.027	−0.005	+0.009	0.111	−0.049	+0.075
20190611B	321	0.378	58	8.3	0.170	−0.067	+0.079	0.207	−0.092	+0.099
20190711A	593	0.522	171	<8.0	⋯	⋯	⋯	⋯	⋯	⋯
20190714A	504	0.236	289	<2.5	⋯	⋯	⋯	⋯	⋯	⋯
20191001A	507	0.234	287	0.72	0.298	−0.099	+0.109	0.356	−0.152	+0.143
20200430A	380	0.160	217	3.9	0.124	−0.075	+0.141	0.205	−0.108	+0.128
20200120E	88	⋯	13	<0.014	⋯	⋯	⋯	⋯	⋯	⋯
20201124A	414	0.098	172	3.9	0.109	−0.064	+0.117	0.165	−0.088	+0.109

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For the four cases with τ ≥ 3.1 ms (at 1 GHz), the true redshift is below that of the DM-based estimator and more consistent with the scattering-based estimator using the broader range of ${A}_{\tau }\widetilde{F}G$. For small scattering times, τ ≲ 0.1 ms, the three estimators give the same result because the measured scattering does not require a large DM_h for either of the ranges for ${A}_{\tau }\widetilde{F}G$. The intermediate cases FRB 20191001A and FRB 20180924A with τ = 1.5 ms and 1.8 ms, respectively, are mixed, with the former object being more consistent with the scattering-based redshift and the latter slightly more consistent with the DM-only estimator. The outlier in this sample of nine is FRB 20190611B, where the true redshift is larger than the mode, mean, or median of any of the estimators but is not improbable for the DM-only estimator or the scattering estimator with a broad range of ${A}_{\tau }\widetilde{F}G$. Macquart et al. (2020) noted that the association of the FRB with the galaxy at z = 0.378 is tentative, and the redshift estimations here may reflect that possibility.

Figure 15 shows $\widehat{z}$ plotted against z using the three different redshift estimators with values of 0.4 and 0.8 for f_igm that allow comparison with two of the panels in Figure 12. The plotted points are the median values of the posterior PDFs. Note that here the posterior PDF for DM_h is calculated only for the 9 objects with τ measurements compared to the 13 objects used in Figures 12 and 13. The other two estimators incorporate scattering using the two different ranges for ${A}_{\tau }\widetilde{F}G$ described above The larger value f_igm = 0.8 yields much greater consistency between $\widehat{z}$ and z than the smaller value.

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To identify the plausible range for f_igm, we show the mean and rms residual $\langle \delta z\rangle =\langle \widehat{z}-z\rangle$ and ${\sigma }_{\delta \widehat{z}}=\langle {(\delta z)}^{2}{\rangle }^{1/2}$ for the three estimators in the left- and right-hand panels of Figure 16. The figure shows 〈δ z〉 to be monotonically decreasing with larger f_igm for all three estimators and ${\sigma }_{\delta \widehat{z}}$ decreasing to f_igm ∼ 0.85 ± 0.05. It is notable that all three of the redshift estimators are better for larger values of the baryonic fraction, f_igm ≳ 0.8, showing less bias and less scatter about the measured redshifts.

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The DM-based estimator remains positive for values of f_igm ≲ 1 and is therefore biased. The wide-range ${A}_{\tau }\widetilde{F}G$ scattering estimator crosses zero at f_igm ∼ 0.85, and the narrow-range estimator crosses at f_igm ∼ 0.75. The rms residual curve for the DM-only estimator decreases monotonically and is slightly below that of the narrow-${A}_{\tau }\widetilde{F}G$ estimator at f_igm = 1 but is larger than both scattering-based estimators for f_igm ≲ 0.85. The two scattering-based estimators bottom out at f_igm ≃ 0.8 to 0.9. Considering both the bias and the minimum rms residual, a value f_igm ≃ 0.8 to 0.9 appears to give the best match. For these values, the bias of the wide-${A}_{\tau }\widetilde{F}G$ scattering estimator is $| \langle \delta \widehat{z}\rangle | \lesssim 0.02$ and the rms redshift error is ${\sigma }_{\delta \widehat{z}}\sim 0.1$.

In previous sections, scattering has been attributed to host galaxies, yielding a range for ${A}_{\tau }\widetilde{F}G$ of ∼ [0.1, 9] pc cm⁻³ (Table 3), from which estimates for the host-galaxy DM contribution and redshift were made. We now discuss individually each FRB for which there are both scattering and redshift measurements. Quoted scattering times are referenced to 1 GHz.

FRB 20180924A (τ = 1.78 ± 0.08 ms, z = 0.321): Heintz et al. (2020, hereafter H20) designate the galaxy association as high probability (A class); the FRB is only slightly offset from the galaxy center. Scattering is constraining on the redshift for smaller values of the baryon fraction, f_igm ≲ 0.6, but the DM-only estimate matches the measured redshift for f_igm ≳ 0.7. For these larger f_igm, ${A}_{\tau }\widetilde{F}G\simeq 8.7$ is needed to better match the redshift.

FRB 20181112A (τ = 0.06 ± 0.003 ms, z = 0.475): Also designated a high-probability association by H20. The scattering is too small to be constraining, in accordance with the criterion in Equation (28). The measured redshift is slightly less than the median $\widehat{z}$ for all three estimators.

FRB 20190102B (τ = 0.11 ± 0.008 ms, z = 0.291): Another high-probability association (H20). Scattering is too small to be constraining. The $\widehat{z}$ estimators favor f_igm ≳ 0.6.

FRB 20190523A (τ = 1.4 ± 0.2 ms, z = 0.66): The FRB is offset from the galaxy center by 27 kpc, and there is a 7% probability of a chance association, yielding a C classification by H20. Scattering is constraining and requires ${A}_{\tau }\widetilde{F}G\sim 2$ in order to match the measured redshift for f_igm = 0.85. The DM-only estimator requires f_igm ≳ 0.6.

Given the large offset from the galaxy center, it is possible that the required value of ${A}_{\tau }\widetilde{F}G$ receives a significant contribution from a geometric factor, G > 1. This could arise from a contribution to the DM from a galaxy halo or disk with a very small value of $\widetilde{F}$ but with a large geometric boost. An alternative is that the candidate galaxy association is incorrect. Given that the galaxy has the largest redshift in the sample, a misassociation would require reassessment of the empirical DM(z) statistics.

FRB 20190608B (τ = 8.7 ± 0.5 ms, z = 0.118): An A-class association with the FRB coincident in projection with a spiral arm (H20; see also Macquart et al. 2020; Mannings et al. 2021). Scattering is strongly constraining on redshift and requires smaller values of ${A}_{\tau }\widetilde{F}G$ in the [0.1, 10] range. The DM-only estimate is a poor match for all values of f_igm.

FRB 20190611B (τ = 0.51 ± 0.06 ms, z = 0.378): The FRB–galaxy association was called "tentative" by Macquart et al. (2020) but designated as class A by H20 in spite of a significant offset ∼11 ± 4 kpc from the galaxy center compared to an i-band radial size of ∼2 kpc.

The galaxy's redshift is within the credible region for $\widehat{z}$ only for relatively small values of f_igm ≲ 0.7 for the DM-only estimator and for f_igm ≲ 0.6 and ≲ 0.5 using the DM–τ estimators with wide and narrow ranges of ${A}_{\tau }\widetilde{F}G$, respectively. This implies that a smaller than normal IGM contribution to the DM is needed to match the DM inventory and allow the host-galaxy DM_h to be large enough to account for the measured scattering for the two ranges of ${A}_{\tau }\widetilde{F}G$ considered in the analysis.

From Table 2, the median DM_igm using the the log-normal model of Section 2.2 is 320 pc cm⁻³, nearly equal to the measured DM = 321 pc cm⁻³ without any consideration of contributions from the MW or host galaxy. When those contributions are included, the measured DM is estimated to have a total extragalactic contribution, DM_xg ≃ DM_igm +DM_h/(1 + z_h) = 211 ± 17 pc cm⁻³ and an implied IGM contribution, DM_igm = DM_xg − DM_h/(1 + z_h) ≃ 179 pc cm⁻³, with an uncertainty of about 40 pc cm⁻³ (where we have used the geometric mean of the asymmetric confidence interval values for DM_h in quadrature with the uncertainty in the MW contribution).

This IGM value is ∼141 pc cm⁻³ below the median IGM from the log-normal model of Section 2.2, or about 1.5 times the 68% confidence range, σ₋ = 97 pc cm⁻³, to smaller DM_igm values (column 7 of Table 2). While this is not overly improbable in a nine-object sample, the necessarily smaller IGM contribution implies that the LoS to this FRB needs further study.

The measured scattering requires the second largest value of ${A}_{\tau }\widetilde{F}G$ given in Table 3 (column 3), which is based on f_igm = 0.85, a value that is consistent with the entire set of objects. Using Equation (29) along with the median inferred value for DM_h = 58 pc cm⁻³, we require ${A}_{\tau }\widetilde{F}G\simeq 8.3$ (Table 3).

The summary for this source is that the DM inventory requires a lower-than average contribution from the IGM for the redshift of the proposed galaxy association. However, it is not so extreme that the association is necessarily incorrect. However an incorrect association is certainly a possibility.

FRB 20191001A (τ = 1.5 ± 0.1, z = 0.234): An A-class association (H20) with a 2:1 offset relative to the i-band radial size but with the FRB overlying a spiral arm (Mannings et al. 2021). The redshift is overestimated by the DM-only estimator but is consistent with either of the scattering-based estimators.

FRB 20200430A (τ = 5.6 ± 2.8 ms, z = 0.160): An A-class association (H20). Scattering is constraining on the redshift. The DM-only redshift estimator is disfavored compared to the DM–τ estimators, especially for smaller values of f_igm ≲ 0.6. But even for f_igm ≥ 0.9, the scattering constraint gives a better estimate.

FRB 20201124A (τ = 3.13 ± 1.7 ms, z = 0.098): The lowest-redshift galaxy in the sample, J0508+2603, has a stellar mass comparable to the two candidate galaxies for FRB 20190523A and FRB 20191001A (Ravi et al. 2022, H20) with an extended source spatially coincident with the FRB and associated with star formation activity. The scattering time is large enough to constrain the redshift with both DM–τ estimates yielding credible values superior to the DM-only result for any value of f_igm. The two DM–τ estimates are equally good for f_igm ≳ 0.7.

In summary, combined measurements of the scattering time τ and DM yield better estimates for redshift than DM-only estimates in the majority of the nine FRBs for which such measurements exist along with redshifts. The exceptions are when the scattering time is too small to constrain the host-galaxy DM using plausible values of the fluctuation-geometry parameter ${A}_{\tau }\widetilde{F}G$.

Four FRBs in Table 1 have A-class galaxy associations (H20). Three of these (FRBs 20121102A, 20180916B, and 20190711A) have scattering upper limits too large to be constraining on the host-galaxy DM or on the redshift. The fourth case, FRB 20200120E, has a very low upper limit that is informative about scattering in the halos of the MW and M81, as previously mentioned. A detailed interpretation of that case is deferred to another paper (in preparation).

FRB 20121102A warrants additional discussion because Balmer-line measurements place ancillary constraints on DM_h (Tendulkar et al. 2017). FRB 20121102A is in a dwarf, star-forming galaxy at redshift z_h = 0.193 and produces bursts with a total DM ≈ 570 pc cm⁻³ contributed to roughly equally by the MW, the IGM, and its dwarf host galaxy (Bassa et al. 2017; Kokubo et al. 2017; Tendulkar et al. 2017). Intensity scintillations imply a small Galactic contribution to temporal broadening (∼20 μs at 1 GHz), and an upper bound on extragalactic scattering is τ(1 GHz) < 0.6 ms. The dependencies of DM_h and extragalactic scattering on redshift and other quantities shown in Figure 9 (left panel) indicate that the upper bound on τ is consistent with the plausible range of DM_h (see Figure 2) combined with possible values for ${A}_{\tau }\widetilde{F}G$ that match the ranges for ${A}_{\tau }\widetilde{F}G$ implied by FRBs with measured τ and with the expectations based on Galactic pulsars.