Statistical Measurements of Dispersion Measure Fluctuations in Fast Radio Bursts

Figure 1 shows the sky distribution of the FRBs from the CHIME Catalog and from FRBCAT. Some FRBs have their sky locations overlapped. The CHIME sample has a smaller sky coverage, but contains more FRB pairs with smaller angular separations.

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The SF of FRB DMs is defined as

$$\begin{eqnarray}&&D(\theta )=\left\langle {\left(\mathrm{DM}({{\boldsymbol{X}}}_{1})-\mathrm{DM}({{\boldsymbol{X}}}_{2})\right)}^{2}\right\rangle ,\end{eqnarray} \tag{ 1 }$$

where X ₁ and X ₂ are the sky locations of a pair of FRBs with the angular separation θ, and 〈...〉 denotes the average over all pairs with the same θ. The CF of DM fluctuations is

$$\begin{eqnarray}&&\xi (\theta )=\langle \delta \mathrm{DM}({{\boldsymbol{X}}}_{1})\delta \mathrm{DM}({{\boldsymbol{X}}}_{2})\rangle ,\end{eqnarray} \tag{ 2 }$$

where $\delta \mathrm{DM}=\mathrm{DM}-\overline{\mathrm{DM}}$ is the DM fluctuation, and $\overline{\mathrm{DM}}$ is the mean DM. We note that the SF of DMs and the SF of DM fluctuations are the same. For a stationary or a spatially homogeneous random process, the SF and CF contain equivalent information, and they are related by

$$\begin{eqnarray}&&D(\theta )=\mathrm{const}-2\xi (\theta ).\end{eqnarray} \tag{ 3 }$$

In a more general case with low-frequency noise or large-scale fluctuations, the statistical characteristics of SF and CF are not mutually interchangeable. It was found in statistical mechanics that the SF is less distorted than the CF by large-scale inhomogeneities (Monin & Yaglom 1965). To reach a similar accuracy, the CF requires a much larger (by 1 to 2 orders of magnitude) data set than the SF (Schulz-Dubois & Rehberg 1981).

Figure 2(a) presents D(θ) measured by XZ20 with 112 FRBs from FRBCAT, in comparison with that measured with 491 FRBs (in this work) with distinct coordinates from CHIME. The uncertainty of the latter is much smaller due to the significantly larger sample size. We note that the error bars show 95% confidence intervals, which are calculated at each θ using the Student's t-distribution when there are < 30 FRB pairs and the normal distribution for more FRB pairs. The power-law trend indicated by the dashed line at small θ is not seen for the CHIME sample. We further combine the two catalogs, and the corresponding D(θ) of a total of 603 FRBs ⁷ is displayed in Figure 2(b). We see that D(θ) is dominated by CHIME sample, so the inclusion of FRBCAT FRBs does not significantly change the result, except for the very large θ end.

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We believe that the different D(θ) measured with the FRBCAT and CHIME catalogs at small θ is mainly caused by the different sample sizes. The latter measurements are within the uncertainty range of the former. The effect of sky coverage is unclear and thus cannot be completely excluded.

To further examine the effect of sample size on D(θ), we randomly select 112 FRBs, i.e., the same number of FRBs as in FRBCAT, from the total CHIME sample as a subsample. As an example, D(θ) measured from three subsamples is shown in Figure 3(a). We see that different subsamples can lead to different D(θ) at both small and large θ. As an alternative test, we randomly select the same number of FRB pairs from the total CHIME sample in each θ bin as that for FRBCAT sample. D(θ) corresponding to three realizations is shown in Figure 3(b) as an example. In both tests, we find large statistical uncertainties of D(θ) at small θ because of the small sample size. Occasionally, a power-law trend can be seen, e.g., blue circles in Figure 3(b). It suggests that the power-law trend of D(θ) seen at small θ with a small sample of FRBs from FRBCAT is likely a coincidence.

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Both D(θ) and ξ(θ) of DM fluctuations are 2D projected statistical measurements of electron density fluctuations. The larger the dispersion of the distances to FRBs, the greater the difference between the 2D and 3D statistics is expected. Depending on the thickness of the 2D sample, the correlation information can be partly or completely lost (Einasto et al. 2021). In addition, the host DMs of FRBs can largely distort the measured D(θ) and ξ(θ). In particular, some FRBs can have very large host DMs (∼400 pc cm⁻³), as suggested by Rafiei-Ravandi et al. (2021). A repeating FRB with the host DM ≈ 902 pc cm⁻³ is recently found by Niu et al. (2021). As an attempt to constrain the thickness of the 2D sample and exclude FRBs with exceptionally large host DMs, we explore the possibility of using a subset of the FRB sample by applying a DM cut for the statistical analysis. Figure 4 shows different D(θ) corresponding to different DM cuts. Naturally, the DM fluctuation of a subset of FRBs becomes smaller at a smaller DM cut value.

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There is a trade-off between the thickness of the 2D sample and the sample size. A higher cut value leads to a thicker 2D sample, and a lower one brings larger statistical uncertainties. We apply here a tentative DM cut at 500 pc cm⁻³ (see Figure 5 for the corresponding DM distribution). According to the DM–redshift relation (Deng & Zhang 2014; Zhang 2018),

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{IGM}}\approx 807\,\mathrm{pc}\ {\mathrm{cm}}^{-3}{\int }_{0}^{z}\displaystyle \frac{(1+z){dz}}{{[{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}]}^{\tfrac{1}{2}}},\end{eqnarray} \tag{ 4 }$$

where Ω_m = 0.3089 ± 0.0062 and Ω_Λ = 0.6911 ± 0.0062 are the matter density parameter and dark energy density parameter (Planck Collaboration et al. 2016). We approximately sample the nearby density structures within redshift z ≈ 0.57 (≈2 × 10³ Mpc as the line of sight (LOS) comoving distance) by assuming the DM is dominated by its intergalactic component. The optimized selection cut should be determined based on the accurate modeling of distances and host DMs of FRBs, as well as the sample size.

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Figure 6 presents the D(θ) of the subset of FRBs, with the total DM used in Figure 6(a), and the DM excess between DM determined by fitburst (The CHIME/FRB Collaboration et al. 2021) and Milky Way DM modeled by NE2001 (Cordes & Lazio 2002), i.e., DM (NE2001), in Figure 6(b) and YMW16 (Yao et al. 2017), i.e., DM (YMW16), in Figure 6(c). We see that D(θ) is not sensitive to the modeled Milky Way DMs. In all cases, it shows nonflat D(θ) at small θ. Nonflat SF indicates a possible correlation of electron density fluctuations within scales ≈2000 Mpc × 10° ≈350 Mpc. To evaluate the statistical uncertainty, we perform Monte Carlo simulations by randomizing DMs while holding the locations of FRBs fixed. As shown by D(θ) measured from 40 Monte Carlo realizations, the uncertainty becomes larger toward smaller θ with fewer pairs of FRBs available.

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The measured ξ(θ) is shown in Figure 7. It is more sensitive to the modeled Milky Way DMs than D(θ). For comparison, we also present the analytically modeled ξ(θ; z_s ) by Takahashi et al. (2021). In their model the free-electron abundance and the power spectrum of its spatial fluctuations are measured from hydrodynamic cosmological simulations, IllustrisTNG300 (Nelson et al. 2018). The cyan line shows its analytical approximation at θ ≳ 1°,

$$\begin{eqnarray}&&\xi (\theta ;{z}_{s})\approx 2400{\left(\displaystyle \frac{\theta }{\deg }\right)}^{-1}{\mathrm{pc}}^{2}{\mathrm{cm}}^{-6},\end{eqnarray} \tag{ 5 }$$

where z_s is the source redshift. Takahashi et al. (2021) found that the above expression is insensitive to z_s at z_s ≳ 0.3 as a large-scale signal is dominated by nearby structures. This further justifies our use of a subset of FRBs with relatively small DMs. Our measured ξ(θ), with a large uncertainty, shows some trends of increasing ξ(θ) toward smaller θ at θ < 10°. Toward larger θ, ξ(θ) gradually approaches zero as theoretically expected. We see that as the theoretically modeled ξ(θ; z_s ) is small, a more accurate comparison between the model and observational measurements requires a larger sample size and a higher angular resolution.

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To examine whether D(θ) and ξ(θ) of FRBs provide similar information on the correlation of electron density fluctuations, in Figure 8, we compare the measured D(θ) with that derived from ξ(θ) by using the relation in Equation (3). The constant value in Equation (3) is determined by visually matching the two at large θ, which is 2.0 × 10⁴ pc² cm⁻⁶ for the case with total DMs, and slightly larger for modeled extragalactic DMs, i.e., 2.1 × 10⁴ pc² cm⁻⁶ for DM (NE2001) and 2.3 × 10⁴ pc² cm⁻⁶ for DM (YMW16). We see that the measured D(θ) and that derived from ξ(θ) have a better agreement at θ ≳ 10°, indicative of the homogeneity of the large-scale density distribution. The discrepancy seen at smaller θ can be caused by the limited sample size, as smaller-θ measurements suffer larger statistical uncertainties. It can also be caused by intermediate-scale (on the order of 100 Mpc) inhomogeneities of intergalactic density distribution. Possible inhomogeneities in the universe on such length scales were suggested in, e.g., Kopylov et al. (1988), Sylos Labini et al. (2009), Sylos Labini (2011), and Perivolaropoulos (2014), based on the observed galaxy distribution and other effects, though these claims are controversial (e.g., Hogg et al. 2005; Ntelis et al. 2017). In the latter situation, we do not expect that D(θ) and ξ(θ) can be determined from each other at small θ even with a larger sample of FRBs. As the SF is less sensitive to large-scale fluctuations and has a higher accuracy using less data compared with the CF (Schulz-Dubois & Rehberg 1981), the statistical measurements of DM fluctuations at small θ with SF can be more reliable and informative.

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Obviously, the quantitative comparison between the simulated and observationally measured statistics of DM fluctuations requires a sufficiently large sample of FRBs. By using the rejection sampling method (Mackay 2003), we generate a large sample of mock FRBs from the target DM distribution of the CHIME FRBs with DM < 500 pc cm⁻³ (see Figure 9(a)). Under the consideration that future radio telescopes, such as the Square Kilometre Array, will enable full sky coverage, we have the mock FRBs randomly distributed across the entire sky. The measured D(θ) and that derived from ξ(θ) of the mock samples are presented in Figures 9(b) and (c). It shows that at least several thousands of FRBs are needed to have a clean comparison with the expectation from cosmological simulations. A larger angular resolution is achieved with the increase of sample size. We note that as the DM fluctuations of mock FRBs are uncorrelated and homogeneous, the flatness of D(θ) and the good match between D(θ) and that converted from ξ(θ) are expected for a sufficiently large sample (see Figure 9(c)).

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