Forecasts for Helium Reionization Detection with Fast Radio Bursts in the Era of Square Kilometre Array

Thanks to their extremely high event rate (Thornton et al., 2013), abundant FRBs are expected to be detected by the upcoming telescopes. Note that uncertainties in the fluence distribution and spectral index of the FRB population would make the estimate of the event detection rate for a specific telescope uncertain by several orders of magnitude (Vedantham et al., 2016; James et al., 2019; Zhang et al., 2023). However, it is feasible to give a simple calculation on the event detection rate at operating frequencies close to those at which FRB detections are conducted (i.e., 1.2–1.7 GHz, as probed by the SKA’s mid-frequency instrument), based on reasonable assumptions on the slope of the differential fluence distribution (i.e., the $\log N-\log F$ distribution). The cumulative all-sky event rate $N_{\rm sky}$ of FRBs above a given fluence threshold $F_{\nu}$ is generally described as a power-law model (Macquart & Ekers, 2018),

where $N^{\prime}_{\rm sky}$ is the known event rate above the fluence threshold $F^{\prime}_{\nu}$ estimated by existing FRB surveys, and the power-law index $\alpha=-1.5$ is in line with the expectation of a non-evolving population in Euclidean space (Vedantham et al., 2016).

The mid-frequency aperture array of the first phase of SKA (SKA1-MID) would reach a $10\sigma$ sensitivity of $\mathrm{14\,mJy}$ from 0.9 to 1.67 $\mathrm{GHz}$ for an integration time of 1 ms (Fender et al., 2015). While the actual scope of the second phase of SKA (SKA2) will eventually be subject to further discussions, the performance of SKA2 may be 10 times the SKA1 sensitivity in the frequency range of 0.35–24 $\mathrm{GHz}$.¹¹1https://www.skao.int/en/science-users/118/ska-telescope-specifications That is, the adopted $10\sigma$ fluence threshold of SKA2-MID is $F_{\nu}=1.4$ $\mathrm{mJy\,ms}$. With Equation (7), we can estimate the all-sky event rate observed by SKA2-MID based on the rate measurements of Parkes and ASKAP telescopes. Bhandari et al. (2018) obtained $N^{\prime}_{\rm sky}=(1.7^{+1.5}_{-0.9})\times 10^{3}\,\mathrm{sky^{-1}\,day^{-1}}$ above the fluence threshold of $F^{\prime}_{\nu}=2$ $\mathrm{Jy\,ms}$ from Parkes FRB surveys. Shannon et al. (2018) obtained $N^{\prime}_{\rm sky}=(37\pm 8)\,\mathrm{sky^{-1}\,day^{-1}}$ above the fluence threshold of $F^{\prime}_{\nu}=26$ $\mathrm{Jy\,ms}$ from ASKAP FRB surveys. Based on the Parkes result, the estimated all-sky event rate for SKA2-MID is

and according to the ASKAP result, we have

It is obvious that the two estimates are in good agreement with each other.

Considering the solid angle $\Delta\Omega$ covered on the sky by the SKA instantaneous observation, the expected number of FRBs can be estimated by

where $T$ is the exposure time on source. In general, the overall sky coverage of SKA2-MID would reach $\sim 25,000-30,000\,\mathrm{deg^{2}}$ (Torchinsky et al., 2016). Since FRBs are transient phenomena, here we conservatively take $\Delta\Omega\approx 20\,\mathrm{deg^{2}}$ $(\sim 0.006\,\mathrm{sr})$ as the effective instantaneous field of view of SKA2-MID (Macquart et al., 2010). Following Zhang et al. (2023), we assume that SKA2-MID would spend an average of 20% of observing time per year on FRB search, and take $T=20\%\times 365\,\mathrm{day\,yr^{-1}}$. On the basis of the Parkes and ASKAP results, the expected detection rates of FRBs by SKA2-MID are then $\sim(3.3^{+2.9}_{-1.7})\times 10^{6}\,\mathrm{events\,yr^{-1}}$ and $\sim(3.3\pm 0.7)\times 10^{6}\,\mathrm{events\,yr^{-1}}$, respectively. Thus, we conclude that $\mathcal{O}(10^{6})$ FRBs can be detected by SKA2-MID in a one-year observation.

The precise localization of FRBs is crucial for FRB cosmology. To obtain their host galaxy redshifts via optical follow-up, the positions of FRBs should be localized to within one arcsecond (Macquart et al., 2015). When complete, the SKA will have the ability of localizing FRBs to an accuracy of sub-arcseconds, and thus it is reasonable to expect that all the host galaxies of the SKA-detected FRBs would be identified and their redshifts would be measured.

In this work, we perform Monte Carlo simulations to explore the prospect of He ii reionization detection with future FRB measurements in the SKA era. The fiducial flat $\Lambda$CDM model with cosmological parameters derived from Planck observations ($H_{0}=67.4$ km $\rm s^{-1}$ $\rm Mpc^{-1}$, $\Omega_{\rm m}=0.315$, and $\Omega_{b}=0.0493$; Planck Collaboration et al. 2020) is adopted to simulate FRB data. The fiducial values of the reionization parameters are taken to be $A_{\rm He}=1$ and $z_{\rm re}=3$, though we will discuss the effect of different fiducial $z_{\rm re}$ in the following subsection.

The redshift distribution of FRBs is expressed as

where $D_{C}(z)=c\int_{0}^{z}1/H(z)\mathrm{d}z$ is the comoving distance, $H(z)$ is the Hubble parameter, and $R(z)$ denotes the intrinsic event rate density distribution. So far, the actual $R(z)$ distribution of FRBs is poorly known. Qiang & Wei (2021) stressed that the assumed redshift distributions (i.e., the assumed $R(z)$ distributions) have significant effect on cosmological parameter estimation. The true redshift distribution of FRBs is inextricably linked to the corresponding progenitor system(s) of FRBs, and it is commonly assumed that the FRB rate follows the cosmic star formation history (SFH) or compact star mergers. In the merger model, a compact star binary system needs to experience a long inspiral phase before the final merger, so there is a significant time delay with respect to star formation. In the literature, three types of merger delay timescale models have been discussed (Virgili et al., 2011; Sun et al., 2015; Wanderman & Piran, 2015): Gaussian delay model, lognormal delay model, and power-law delay model. Thus, we investigate four FRB event rate density models in detail:

• •

  SFH model: we adopt the approximate analytical form derived by Yüksel et al. (2008) using a wide range of the star formation rate data

  $$R(z)=\left[\left(1+z\right)^{3.4\eta}+\left(\frac{1+z}{5000}\right)^{-0.3\eta}+\left(\frac{1+z}{9}\right)^{-3.5\eta}\right]^{1/\eta}\;,$$

  (12)

  where $\eta=-10$.

• •

  Compact star merger models: for the Gaussian delay model, we adopt the empirical formula (Virgili et al., 2011)

  	$\displaystyle R(z)=$	$\displaystyle\left[\left(1+z\right)^{5.0\eta}+\left(\frac{1+z}{0.17}\right)^{0.87\eta}\right.$
  		$\displaystyle\left.+\left(\frac{1+z}{4.12}\right)^{-8.0\eta}+\left(\frac{1+z}{4.05}\right)^{-20.5\eta}\right]^{1/\eta}\;,$		(13)

  where $\eta=-2$.

• •

  Compact star merger models: for the lognormal delay model (Wanderman & Piran, 2015), the empirical formula of $R(z)$ is

  	$\displaystyle R(z)=$	$\displaystyle\left[\left(1+z\right)^{5.7\eta}+\left(\frac{1+z}{0.36}\right)^{1.3\eta}\right.$
  		$\displaystyle\left.+\left(\frac{1+z}{3.3}\right)^{-9.5\eta}+\left(\frac{1+z}{3.3}\right)^{-24.5\eta}\right]^{1/\eta}\;,$		(14)

  where $\eta=-2$.

• •

  Compact star merger models: for the power-law delay model (Wanderman & Piran, 2015), one has

  	$\displaystyle R(z)=$	$\displaystyle\left[\left(1+z\right)^{1.9\eta}+\left(\frac{1+z}{2.5}\right)^{-1.2\eta}\right.$
  		$\displaystyle\left.+\left(\frac{1+z}{3.8}\right)^{-4.4\eta}+\left(\frac{1+z}{7.7}\right)^{-11\eta}\right]^{1/\eta}\;,$		(15)

  where $\eta=-2.6$.

Figure 2 presents the FRB redshift distributions expected from all four assumed models. It is clear that the SFH model has the widest $z$ distribution. Due to the merger delay, the other three merger models have narrower $z$ distributions, with the overall redshift gradually shifting to lower redshifts in the order of power-law, Gaussian, and lognormal models (Zhang et al., 2021). Note that He ii reionization is generally expected to occur at $z\sim 3$ (e.g., Becker et al. 2011). In order to effectively detect the signature of He ii reionization, many high-$z$ FRBs on either side of the expected reionization redshift are required. However, only low-$z$ ($z\leq 3$) FRBs are predicted by the Gaussian and lognormal delay models (see Figure 2). For the rest of this paper, we shall therefore only consider two cases: the FRB redshift distribution tracing the SFH and power-law delay models, respectively.

In our simulations, the true FRB host galaxy redshifts $z^{\rm true}$ are randomly generated from the probability density functions (PDFs) of Equation (11) for the SFH and power-law delay cases. After generating $z^{\rm true}$, we simulate the measured redshifts $z^{\rm mea}$ by assuming an uncertainty of 10% on the spectroscopic data of host galaxies (as expected from the James Webb Space Telescope). That is, we draw $z^{\rm mea}$ from a Gaussian distribution $\mathcal{N}(\mu_{z}=z^{\rm true},\;\sigma_{z}=0.1z^{\rm true})$ (Heimersheim et al., 2022; Maity, 2024).

With the true source redshift $z^{\rm true}$, we infer the fiducial value of $\mathrm{DM^{fid}_{IGM}}$ from Equation (6). Following previous works (Kumar & Linder, 2019; Linder, 2020), the line-of-sight fluctuation of $\mathrm{DM_{IGM}}$ ($\sigma_{\mathrm{IGM}}$) is randomly added to the fiducial value of $\mathrm{DM^{fid}_{IGM}}$, assuming

That is, we sample the $\mathrm{DM^{mea}_{IGM}}$ measurement according to the Gaussian distribution ${\rm DM_{IGM}^{mea}}=\mathcal{N}({\rm DM_{IGM}^{fid}},\;\sigma_{\rm IGM})$. We note that the observed $\mathrm{DM_{obs}}$ has contributions from the interstellar medium ($\mathrm{DM_{MW}}$) and dark matter halo ($\mathrm{DM_{halo}}$) in the Milky Way, the IGM ($\mathrm{DM_{IGM}}$), and the FRB host galaxy ($\mathrm{DM_{host}}$). The $\mathrm{DM_{IGM}}$ value can then be derived by deducting $\mathrm{DM_{MW}}$, $\mathrm{DM_{halo}}$, and $\mathrm{DM_{host}}$ to $\mathrm{DM_{obs}}$. While the corresponding uncertainty of $\mathrm{DM_{IGM}}$ should include the scatter due to the inhomogeneous IGM and the scatter caused by distinctive host properties and halo, the latter scatter (order of $\mathrm{10\,pc\,cm^{-3}}$) is much smaller than the former one (order of $\mathrm{10^{2}\,pc\,cm^{-3}}$) at the reionization epoch (e.g., Prochaska & Zheng 2019). Figure 3 shows the evolution of the fractional uncertainty on a measured $\mathrm{DM_{IGM}}$, accounting for the line-of-sight fluctuation of $\mathrm{DM_{IGM}}$ ($\sigma_{\mathrm{IGM}}$) and the uncertainty of both $\mathrm{DM_{halo}}$ and $\mathrm{DM_{host}}$ ($\sigma_{\mathrm{halo,\,host}}$), i.e., $\sigma_{\mathrm{tot}}=(\sigma_{\mathrm{IGM}}^{2}+\sigma^{2}_{\mathrm{halo,\,host}})^{1/2}$. At low redshift, $\mathrm{DM_{IGM}}$ is small, so the fractional uncertainty $\sigma_{\mathrm{tot}}/\mathrm{DM_{IGM}}$ is relatively large. Since lines of sight average over IGM inhomogeneity and $\mathrm{DM_{IGM}}$ increases at higher redshift, $\sigma_{\mathrm{tot}}/\mathrm{DM_{IGM}}$ gradually decreases. Moreover, one can see from Figure 3 that adding the halo and host uncertainty (with the value of $\sigma_{\mathrm{halo,\,host}}=0,\,50,$ or $100$ $\mathrm{pc\,cm^{-3}}$) in quadrature gives negligible effect on $\sigma_{\mathrm{tot}}/\mathrm{DM_{IGM}}$, especially around the expected reionization redshift $z_{\rm re}\gtrsim 3$. Therefore, we neglect the uncertainty of both $\mathrm{DM_{halo}}$ and $\mathrm{DM_{host}}$ subtractions from $\mathrm{DM_{obs}}$ to derive $\mathrm{DM_{IGM}}$ observationally.

Using the method described above, one can generate a catalog of the simulated FRB events with $z^{\rm mea}$, $\mathrm{DM^{mea}_{IGM}}$, and $\sigma_{\mathrm{IGM}}$. Due to the lack of computational efficiency, we first simulate a population of $10^{4}$ such events, assuming that the FRB redshift distribution follows the SFH model or the power-law delay model separately.

For a set of $10^{4}$ simulated FRBs, we only employ those FRBs with $z<6$ for He ii reionization parameter estimation, since the ionization fractions of hydrogen $X_{e,\mathrm{H\,II}}(z)$ at $z\geq 6$ are not well understood and we take all of the hydrogen to be ionized at $z=6$. We then utilize the standard Bayes theorem to explore the posterior probability distribution $\mathcal{P}(\mathbf{\theta}|\mathcal{D})$ of the model parameters $\mathbf{\theta}$, provided the observed dataset $\mathcal{D}$, i.e.,

where $\mathcal{L}(\mathcal{D}|\mathbf{\theta})$ is the likelihood of data conditional on the hypothetical model, $\pi(\mathbf{\theta})$ denotes some prior knowledge about the model parameters, and $\mathcal{P}(\mathcal{D})$ is the Bayesian evidence (which can be regarded as the normalized parameter and has no effect on our analysis). In our study, $\mathcal{D}$ is the mock dataset consisting of $\mathrm{DM^{mea}_{IGM}}$ and $z^{\rm mea}$, and the free parameters to be constrained are $\mathbf{\theta}=\{A_{\rm He},\,z_{\rm re}\}$. The likelihood is given by (Maity, 2024)

where $\mathrm{DM^{th}_{IGM}}(z_{i};\,\mathbf{\theta})$ is the theoretical value of $\mathrm{DM_{IGM}}$ calculated from the set of He ii reionization parameters $\mathbf{\theta}$ (see Equation 6). To ensure the final resulting constraints are unbiased, the simulation process is repeated 100 times for each FRB data set by using different noise seeds.

Figure 4 displays the resulting constraints on the reionization parameters $A_{\rm He}$ and $z_{\rm re}$ from $10^{4}$ simulated FRBs. For the case of $z$ distribution tracing the SFH model (solid contours), we find the marginalized parameter uncertainties are $\sigma(A_{\rm He})=0.20$ and $\sigma(z_{\rm re})=0.22$. That is, the detection $\mathrm{S/N}$ for He ii reionization is $\mathrm{S/N}=A_{\rm He}/\sigma(A_{\rm He})=1.00/0.20=5.0$. Besides, the sudden reionization redshift can be inferred to be $z_{\rm re}=3.00\pm 0.22$. To investigate the effect of $z$ distribution, in Figure 4 we also plot those constraints obtained from the case of $z$ distribution tracing the power-law delay model (dashed contours). Now we have $A_{\rm He}=0.99\pm 0.31$ and $z_{\rm re}=2.99\pm 0.31$. The detection $\mathrm{S/N}$ decreases to $\mathrm{S/N}=A_{\rm He}/\sigma(A_{\rm He})=0.99/0.31=3.2$. We can see that the constraint precisions of $A_{\rm He}$ and $z_{\rm re}$ obtained from the power-law delay model are slightly worse than those from the SFH model. This is mainly because reionization parameters are more sensitive to high-$z$ FRB data, and more high-$z$ mock events, especially around the expected reionization redshift $z_{\rm re}\gtrsim 3$, are generated in the SFH model (see Figure 2).

In order to comprehensively test what role the number of simulated FRBs ($N$) could play in He ii reionization detection, we show the best-fit reionization parameters and their corresponding $1\sigma$ uncertainties as a function of $N$ in Figure 5 and Table 1. We find that the uncertainties of these constraints from the FRB data are well consistent with the $1/\sqrt{N}$ behavior, independent of what kind of $z$ distribution is assumed. According to our estimation of the FRB detection rate by SKA2-MID in Section 3.1, the one-year SKA2 observation would detect $\mathcal{O}(10^{6})$ FRBs. In principle, it would be ideal to show the analysis with $10^{6}$ mock FRBs. However, due to the lack of computational efficiency, we estimate the capacity of $10^{6}$ mocks by extrapolating from present results of $10^{4}$ mocks. Since all uncertainties will scale in proportion to the $1/\sqrt{N}$ behavior, $10^{6}$ FRB data can give the reionization parameter uncertainties $\sigma(A_{\rm He})=0.020$ and $\sigma(z_{\rm re})=0.022$ ($\sigma(A_{\rm He})=0.031$ and $\sigma(z_{\rm re})=0.031$) for the case of $z$ distribution tracing the SFH (power-law delay) model. Hence, for a one-year SKA2 observation with $10^{6}$ FRBs, the detection $\mathrm{S/N}$ can approach 32–50.

In our above simulations, the fiducial reionization redshift is set to be $z_{\rm re}=3$. To study the effect of different fiducial $z_{\rm re}$, we estimate the sensitivity as we vary $z_{\rm re}$. Note that here we also mock a population of $10^{4}$ FRBs with the $z$ distribution following the SFH and power-law delay models, respectively. Figure 6 shows that no matter which $z$ distribution is assumed, the $\mathrm{S/N}$ gets better rapidly with lower $z_{\rm re}$, but falls below $\mathrm{S/N}=1$ for $z_{\rm re}\gtrsim 4$. As the fiducial value of $z_{\rm re}$ grows, the reionization redshift uncertainty $\sigma(z_{\rm re})$ becomes larger. These are pretty easy to understand, because the number of FRBs at higher $z_{\rm re}$ is smaller (see the blue and red lines in Figure 2). Over the main range of the expected reionization redshifts, say $z_{\rm re}\approx 2.5-3.5$, the $\mathrm{S/N}$ varies from 7.4 (5.0) to 3.2 (2.1), and $\sigma(z_{\rm re})$ varies from 0.16 (0.21) to 0.29 (0.49) for the SFH (power-law delay) model.

In our analysis, we have neglected the uncertainty of the DM contributions from both the Milky Way halo and the FRB host ($\sigma_{\mathrm{halo,\,host}}$), as described in Section 3.2. Although it seems to be a reasonable treatment as suggested by Figure 3, it would be good to add an analysis incorporating $\sigma_{\mathrm{halo,\,host}}$ and check if there is any bias in the posterior recovery. Especially, when He ii reionization is assumed to occur at $z=2$, adding the halo and host uncertainty (for the maximum case of $\sigma_{\mathrm{halo,\,host}}=100$ $\mathrm{pc\,cm^{-3}}$) in quadrature gives a slight increasement of $\sigma_{\mathrm{tot}}/\mathrm{DM_{IGM}}$ (see Figure 3). To investigate how sensitive our results on the uncertainties of $A_{\rm He}$ and $z_{\rm re}$ are on the choice of $\sigma_{\mathrm{halo,\,host}}$, we also perform two parallel comparative analyses of $10^{4}$ simulated FRBs using $\sigma_{\mathrm{halo,\,host}}=0$ $\mathrm{pc\,cm^{-3}}$ and $100$ $\mathrm{pc\,cm^{-3}}$, assuming that the fiducial reionization redshift is $z_{\rm re}=2$ and the $z$ distribution traces the power-law delay model. If we change $\sigma_{\mathrm{halo,\,host}}$ from $0$ $\mathrm{pc\,cm^{-3}}$ to $100$ $\mathrm{pc\,cm^{-3}}$, then $\sigma(A_{\rm He})$ goes from $0.135$ to $0.144$, and $\sigma(z_{\rm re})$ from $0.145$ to $0.154$. So there is no concern regarding a possibly large dependence on $\sigma_{\mathrm{halo,\,host}}$.

We emphasize that all our results are based on the assumption of sudden He ii reionization. However, in reality, it is expected that the reionization phenomenon should evolve gradually with $z$. Assuming a linear evolution in redshift for the He ii reionization within a range of $z_{\rm re,min}$ to $z_{\rm re,max}$, Bhattacharya et al. (2021) found that the difference between results for sudden reionization and gradual reionization is generally not significant. This is a positive outcome in the sense that our results on He ii reionization detection are robust to the assumption about suddenness, but does imply that the duration of the reionization process is difficult to determine.