PROBING THE INTERGALACTIC MEDIUM WITH FAST RADIO BURSTS

Variability of cosmological radio sources has long been proposed to probe the properties of the intergalactic medium (IGM). Haddock & Sciama (1965) suggested to detect IGM dispersion measure (DM) through the variability of the radio signal from quasars, aiming at using the inferred DM to distinguish different cosmological models. Weinberg (1972) and Ginzburg (1973) suggested the use of radio flares to measure the DM and thus probe the IGM density. Later, radio emission from gamma-ray bursts (GRBs) and their afterglows were proposed as means to determine distances to GRBs and to probe the IGM (Palmer 1993), to study the prehistory of GRBs (Lipunova et al. 1997), and to constrain the hydrogen reionization history of the universe (Ioka 2003; Inoue 2004). However, radio quasars and GRBs, despite the aspirations of the authors, simply lack sharp features that allow their signals to be easily used to probe the intervening electrons in the IGM.

The situation can completely change with the recently discovered short radio bursts. The first such burst was reported by Lorimer et al. (2007), which is an intense (30 Jy) and short-duration (5 ms) burst at 1.4  GHz (named as the Sparker in Kulkarni et al. 2014). Following the above discovery, Thornton et al. (2013) reported the finding of four short-duration bursts with an estimated all-sky rate of 10⁴ events day⁻¹ (denoted as "fast radio bursts" or FRBs). Spitler et al. (2014) discovered one FRB in the Arecibo Pulsar ALFA Survey. These short radio bursts show DMs of order of a few hundred to a thousand, suggesting a substantial contribution from electrons in the IGM. Kulkarni et al. (2014) performed a thorough investigation of the Sparker and FRBs to explore possible constraints on sites or processes to explain such high DMs and concluded that they are of extragalactic origin, provided that the inferred DM arises as a result of propagation through cold plasma. A variety of models have been proposed for the progenitors of such short bursts (e.g., Popov & Postnov 2010; Vachaspati 2008; Falcke & Rezzolla 2014; Totani 2013; Zhang 2014; Lasky et al. 2014; Kashiyama et al. 2013; Kulkarni et al. 2014).

The pulse nature, the high rate, and the extragalactic origin make the Sparker-like events and FRBs well suited for being used as a potentially powerful probe to the IGM. Since the discovery of the short-duration bursts, DM measurements have been proposed to probe missing baryons around halos of galaxies (McQuinn 2014), study the baryon content in the IGM (Deng & Zhang 2014), and constrain cosmology and the equation of state of dark energy (Gao et al. 2014; Zhou et al. 2014).

In this paper, we explore further potential applications of a Sparker-like population or FRBs in probing the IGM (for simplicity, hereafter we refer to the Sparker-like events and FRBs collectively as FRBs). Specifically, we first point out the use of them to probe the era of He ii reionizationand intergalactic magnetic field (Section 2) and then comment about the potential use of FRBs for detecting a cosmological population of massive compact halo objects, i.e., MACHOs (Section 3). Finally, we give a summary in Section 4.

In this section, we specifically focus on the potential use of FRBs to probe the era of He ii reionization and IGM magnetic field.

He ii reionization can be regarded as the last phase transition in the universe, after the major one related to hydrogen and He i reionization above z ∼ 6. Stars are not hot enough to ionize He ii with an ionization potential of 54.4 eV. However, there is some evidence that He ii reionization occurred at z ∼ 3 from the transmission of the He ii Lyα forest and the temperature change of the IGM (see Furlaneto & Oh 2008a, 2008b and references therein). The ionization can be caused by soft X-ray emission from activate galactic nuclei (AGNs) and hard ionizing photons from quasars. Thus, observations of He ii reionization provide a new diagnostic of the buildup of the AGN and quasar population. The same observations also provide important clues to the structure and thermal evolution of the IGM, which is related to the missing baryons problem (e.g., Gnat 2011). Clearly, there is great value in using FRBs in our study of cosmology and AGNs/quasars.

For the purpose of computing the DM from the IGM, there are three effects on the propagation time, t_p , of a photon traveling through the IGM to reach the observer from a cosmological distance: the continuous change of the photon's frequency, ω, due to the redshift of light; the change of the plasma frequency, $\omega _p^2=4\pi n_e(z)e^2/m_e$, due to the change in the IGM electron density, n_e (z), with redshift; and the time dilation effect. The first two effects lead to a change in the group velocity, $v_g=c(1-\omega _p^2/\omega ^2)^{1/2}$, with redshift. The propagation time of a photon emitted at redshift z seen by an observer at redshift 0 is then

$$\begin{eqnarray} t_p&=&\int _0^z dz\frac{dl}{dz} \frac{1}{v_g} (1+z), \cr &=&\int _0^z \frac{c dz}{(1+z)H(z)} \frac{1}{c}\left(1+\frac{1}{2}\frac{\omega _p^2}{\omega ^2}\right)(1+z), \end{eqnarray} \tag{ 1 }$$

where $H(z)=H_0[\Omega _m(1+z)^3+\Omega _\Lambda ]^{1/2}$ is the Hubble constant at z, with Ω_m the matter density parameter and $\Omega _\Lambda =1-\Omega _m$ (assuming a spatially flat universe), and the last (1 + z) factor accounts for time dilation. The frequency, ω, is related to the observed frequency, ω_obs, through ω = (1 + z)ω_obs.

The IGM electron density n_e (z) can be expressed as

$$\begin{eqnarray} n_e(z) &=& n_0(1+z)^3\left[(1-Y)f_{{\rm H\,\scriptsize{II}}}+\frac{1}{4}Y(f_{{\rm He\,\scriptsize{II}}}+2f_{{\rm He\,\scriptsize{III}}})\right],\cr &=& n_0(1+z)^3f_e(z), \end{eqnarray} \tag{ 2 }$$

where

$$\begin{equation} n_0 =\frac{\Omega _b\rho _c}{m_H}=2.475\times 10^{-7} \left(\frac{\Omega _b h^2}{0.022}\right) {\rm cm}^{-3} \end{equation} \tag{ 3 }$$

is the mean number density of nucleons at z = 0. Here Ω_b is the baryon density in units of the z = 0 critical density ρ_c and h is the z = 0 Hubble constant in units of 100 km s⁻¹ Mpc⁻¹. Since we assume to observe a large number of FRBs at each redshift, it is appropriate to use the mean density of the IGM in the calculation without worrying about the density fluctuations. In the expression, Y ≃ 0.25 is the mass fraction of helium, $f_{{\rm H\,\scriptsize{II}}}$ is the ionization fraction of hydrogen, and $f_{{\rm He\,\scriptsize{II}}}$ and $f_{{\rm He\,\scriptsize{III}}}$ are the ionization fractions of singly and doubly ionized helium. After helium reionization (z ∼ 2–3), we essentially have $f_{{\rm H\,\scriptsize{II}}}=1$, $f_{{\rm He\,\scriptsize{II}}}=0$, and $f_{{\rm He\,\scriptsize{III}}}=1$, which gives f_e ≃ 0.88 at low redshifts.

The observed DM is defined as

$$\begin{equation} \frac{dt_p}{d\omega _{\rm obs}} =-\frac{4\pi e^2}{cm_e\omega _{\rm obs}^3} {\rm DM}. \end{equation} \tag{ 4 }$$

In combination with Equation (1), we have

$$\begin{eqnarray} {\rm DM}& = & n_0\frac{c}{H_0} \int _0^z\frac{dz (1+z) f_e(z)}{\sqrt{\Omega _m(1+z)^3+\Omega _\Lambda }} \nonumber, \\ & = & 1060\, {\rm cm}^{-3}\ {\rm pc} \left(\frac{\Omega _b h^2}{0.022}\right) \left(\frac{h}{0.7}\right)^{-1}\nonumber\\ & & \times \int _0^z\frac{dz (1+z) f_e(z)}{\sqrt{\Omega _m(1+z)^3+\Omega _\Lambda }}. \end{eqnarray} \tag{ 5 }$$

For a constant f_e , the above integral can be approximated as

$$\begin{eqnarray} {\rm DM} & \cong & 933\, {\rm cm}^{-3}\ {\rm pc} \left(\frac{f_e}{0.88}\right) \left(\frac{\Omega _b h^2}{0.022}\right) \left(\frac{h}{0.7}\right)^{-1} \nonumber\\ & & \times \left [ \left(\frac{\Omega _m}{0.25}\right)^{0.1} a_1 (x-1) \ + \left(\frac{\Omega _m}{0.25}\right) a_2 (x^{2.5}-1)\right. \nonumber\\ &&\left. + \left(\frac{\Omega _m}{0.25}\right)^{1.5} a_3 (x^4-1) \right ], \end{eqnarray} \tag{ 6 }$$

with x = 1 + z, a₁ = 0.5372, a₂ = −0.0189, and a₃ = 0.00052. The accuracy of this approximation is better than ∼2% for z < 5. At low redshifts, one can use the following approximation:

$$\begin{eqnarray} {\rm DM} &\cong & 933\ {\rm cm^{-3}\ pc} [z+(0.5-0.75\Omega _m)z^2 ] \nonumber\\ & & \times \left(\frac{f_e}{0.88}\right) \left(\frac{\Omega _b h^2}{0.022}\right) \left(\frac{h}{0.7}\right)^{-1}, \end{eqnarray} \tag{ 7 }$$

which has a 5% accuracy up to z = 0.6. For a constant f_e , the integral in Equation (5) shares some similarity with the expression of the luminosity distance D_L , with the (1 + z) factor pulled out of the integral in the latter. In terms of D_L , the integral can be approximated as

$$\begin{equation} {\rm DM} \cong n_0 f_e D_L \left[1+0.932z+(0.16\Omega _m-0.078)z^2\right]^{-0.5}, \end{equation} \tag{ 8 }$$

which has an accuracy ≲0.5% for 0 < z < 3 with 0.25 < Ω_m < 0.35.

As an illustration, the DM as a function of z is displayed in Figure 1. This is an idealized plot since we assume a sharp He ii reionization at z ∼ 3. The reionization is better seen in the slope or derivative of the DM curve. The jump is about 8%. Whether this jump will be seen or not will depend very strongly on the contribution to the DM of FRBs by the electrons in the host galaxies and whether FRBs can be found to redshifts as high as z ∼ 3.

Zoom In Zoom Out Reset image size

Similarly, we can obtain the rotation measure (RM). The Faraday rotation is

$$\begin{equation} \Delta \theta = \frac{2\pi e^3}{m_e^2 c^2\omega _{\rm obs}^2} n_0B_0\frac{c}{H_0} \int _0^z\frac{f_e(z)b_\parallel (z) dz}{\sqrt{\Omega _m(1+z)^3+\Omega _\Lambda }}, \end{equation} \tag{ 9 }$$

where b_∥(z) ≡ B_∥(z)/B₀ is the line-of-sight magnetic field, B_∥(z), in units of the local IGM magnetic field, B₀. We then have

$$\begin{eqnarray} {\rm RM} &=& 8.61 \,{\rm rad}\,{\rm m}^{-2}\left(\frac{\Omega _b h^2}{0.022}\right) \left(\frac{h}{0.7}\right)^{-1} \left(\frac{B_0}{\rm 10\ nG}\right) \nonumber\\ & & \times \int _0^z\frac{f_e(z)b_\parallel (z) dz}{\sqrt{\Omega _m(1+z)^3+\Omega _\Lambda }}. \end{eqnarray} \tag{ 10 }$$

At low redshifts, where we approximate f_e = 0.88 and b_∥ = 1, the RM can be written as

$$\begin{eqnarray} {\rm RM} &\cong & 7.57(z-0.75\Omega _m z^2)\,{\rm rad}\,{\rm m}^{-2} \nonumber\\ & & \times \left(\frac{f_e}{0.88}\right) \left(\frac{\Omega _b h^2}{0.022}\right) \left(\frac{h}{0.7}\right)^{-1} \left(\frac{B_0}{\rm 10\ nG}\right). \end{eqnarray} \tag{ 11 }$$

So far, there are no accurate measurements for the magnetic field in the IGM with densities of the order of the mean density (see Kronberg et al. 2008). We note that the local IGM magnetic field would have a strength of 4(T_IGM/10⁴ K) nG if energy equipartition is assumed. Radio-synchrotron radiation has been detected in the Coma supercluster (Kim et al. 1989), implying a field strength of 0.3–0.6 μG. RM measurements of FRBs can constraint the magnitude of IGM magnetic field and its evolution, providing clues on its origin.

In addition, if the DM and RM have a large contribution from a scattering screen, measurement of RM will provide a strong clue to the location of the scattering (and thus dispersion) screen. If the scattering arises in the IGM, then the RM from the IGM is less than 30 rad m⁻² for z ≲ 0.3 (Kronberg et al. 2008). A much larger RM can be produced if the screen is located in the host galaxy.

Another potential use of FRBs is to constrain the existence of floating MACHO-like objects in the IGM via gravitational lensing (e.g., Gould 1992; Stanek et al. 1993; Marani et al. 1999).

A fortunate alignment of an intervening point-mass object of mass M with an FRB will result in two images. The time delay for each image, with respect to a ray that arrives by the shortest path, is the sum of a geometric term and a gravitational delay term and is given by Narayan & Bartelmann (1996) as

$$\begin{equation} t(\theta)=\frac{1+z_l}{c}\frac{4GM}{c^2} \left(\frac{1}{2}\frac{\theta ^2}{\theta _{\rm E}^2}-\ln |\theta |\right), \end{equation} \tag{ 12 }$$

where z_l is the redshift of the lens, θ is the angular distance between the lens and the image, and θ_E is the annular radius of the Einstein ring,

$$\begin{equation} \theta _{\rm E} = \sqrt{\frac{4GM}{c^{2}} \frac{D_{ls}}{D_lD_s}}. \end{equation} \tag{ 13 }$$

Here D_l , D_s , and D_ls are the observer-lens, observer-source, and lens-source angular diameter distances, respectively. The positions of the two images are

$$\begin{equation} \theta _\pm =\frac{1}{2}\left(b\pm \sqrt{b^2+4\theta _{\rm E}^2}\,\right), \end{equation} \tag{ 14 }$$

where b is the impact parameter (i.e., source-lens angular distance).

From Equations (12) and (14), the lensing time delay between the two images is

$$\begin{eqnarray} \Delta t_l&\equiv & t(\theta _-)-t(\theta _+) \nonumber\\ &=&\frac{1+z_l}{c}\frac{4GM}{c^2} \left[\frac{1}{2}u\sqrt{u^2+4} +\ln \left(\frac{\sqrt{u^2+4}+u}{\sqrt{u^2+4}-u}\right)\right],\qquad \end{eqnarray} \tag{ 15 }$$

where u ≡ b/θ_E is the impact parameter in units of the Einstein ring radius. For a typical value of u = 1,

$$\begin{equation} \Delta t_l=41(1+z_l)(M/1\, M_\odot)\ \mu {\rm s}. \end{equation} \tag{ 16 }$$

The short timescales of FRBs make the effect of relative motions between the source, lens, and observer completely negligible (e.g., the fractional change in u is of the order of 10⁻¹² during a lensed millisecond FRB event at cosmological distances with relative motion of 100 km s⁻¹). So unlike the case for Galactic MACHOs, we do not expect to observe microlensing light curves for MACHO-like objects in the IGM. However, the two images have different amplifications, $A_\pm =1/2\pm (u^2+2)/(2u\sqrt{u^2+4})$, and the time delay ensures they also have different phases. Therefore, the two images undergo constructive and destructive interference. The total amplification is

$$\begin{equation} A(\omega)=\frac{u^2+2}{u\sqrt{u^2+4}}+ \frac{2}{u\sqrt{u^2+4}}\cos (\omega \Delta t_l), \end{equation} \tag{ 17 }$$

where ν = ω/(2π) is the frequency in the observer's frame. The spectrum of the lensed FRB will thus consist of maxima and minima with the separation of two maxima (or minima) being $\Delta \nu =\Delta t_l^{-1}$, which is tens of kHz for M ∼ 1 M_☉ and u ∼ 1 – well within the reach of current technology. Note that Δν is independent of the observing frequency, if the time delay is purely caused by lensing. However, in general Δt_l in Equation (17) should be replaced by the sum of all possible time delays. There could be dispersive delay due to the rays suffering different DM or delay due to multipath propagation from a scattering screen. For FRBs, the latter effect destroys the coherence ⁶ of the rays. For this reason, observing at higher frequencies (i.e., ≳ 5 GHz) is highly desirable. Furthermore, a matched-filter approach ⁷ can additionally improve the detection.

Finally, we can estimate the optical depth for lensing (e.g., Narayan & Bartelmann 1996). For a proper number density n(z) of lenses with mass M in a spatially flat universe, the optical depth for lensing to sources at redshift z_s is

$$\begin{eqnarray} \tau (z_s) & = & \int _0^{z_s} n(z) \pi (D_l\theta _E)^2 dD_c/(1+z) \nonumber\\ & = & \frac{4\pi GM}{c^2 D_s}\int _0^{z_s} n(z) D_{ls} D_l dD_c/(1+z), \end{eqnarray} \tag{ 18 }$$

where the angular diameter distances are D_s = D_A (0, z_s ), D_l = D_A (0, z), D_ls = D_A (z, z_s ), and $D_A(z_1,z_2)=\int _{z_1}^{z_2} dD_c/(1+z_2)$, with dD_c = cdz/H(z) the comoving distance element. In the case of proper number density evolving as n₀(1 + z)² (i.e., comoving number density evolving as n₀/(1 + z)), the optical depth can be put in a form similar to the result in the static Euclidean space,

$$\begin{equation} \tau = \frac{2\pi }{3} \frac{G\rho _{l,0}}{c^{2}} D_c^{2} \simeq 0.014\, \Omega _l (D_c/ 1\ {\rm Gpc})^{2}, \end{equation} \tag{ 19 }$$

where ρ_{l, 0} = n₀ M is the mass density of lenses at z = 0 and Ω_l is this mass density in units of the z = 0 critical density of the universe. In the case of a constant comoving number density, the result is about 44% higher than that in Equation (19) for z_s = 1.

The pulse nature, the high rate, and the extragalactic origin make FRBs ideal for probing the IGM. In this paper, we present potential applications of FRBs to probe He ii reionization, IGM magnetic field, and MACHO-like objects in the IGM.

For these applications, a large population of FRBs is necessary. The He ii reionization causes a small change in the DM. It leads to a ∼8% jump in the differential DM across the He ii reionization epoch. At least a few hundred FRBs around z ∼ 2–3 are needed to detect such a change. Our Galaxy, FRB host galaxies, and any intervening galaxies or clouds with free electrons can add scatter in the DM. Clearly, more FRBs and searching for host galaxies are desired to constrain and understand such a scatter. It is likely that we need thousands of FRBs around z ∼ 2–3 to learn about the He ii reionization from the DM measurements.

The probe of IGM magnetic field with FRBs could be more challenging, given that its strength is likely of the order of nG while that inside a galaxy is of the order of μG. The RM from a typical galaxy is about 812 rad m⁻²(n_e /cm⁻³)(B_∥/μG)(l/kpc) (with l the path length). The RM from IGM to z = 1 is only 6 rad m⁻² for an IGM magnetic field of 10 nG according to Equation (11). If the scatter in the galaxy-caused RM is of the same order of ∼800 rad m⁻², tens of thousands of FRBs are needed to clearly map out the redshift evolution of RM caused by the IGM magnetic field.

For the MACHO-like objects in the IGM, the upper bound for their density parameter Ω_l is Ω_m . If they are of baryonic origin (e.g., stellar remnants), which may be more likely, the upper bound is then Ω_b . According to Equation (19), we expect the lensing optical depth to be (much) less than 6 × 10⁻⁴ to a distance of 1 Gpc, which requires at least tens of thousands of FRBs to discover the events with the lensing signal we point out.

The current estimated all-sky rate of FRBs is 10⁴ events day⁻¹ (Thornton et al. 2013). Therefore, with well-designed and dedicated FRB surveys, all the above requirements of a large statistical sample of FRBs are not demanding at all. Such surveys would provide an invaluable opportunity to advance our understanding of the IGM.

We thank Shude Mao for useful comments. Z.Z. was partially supported by NSF grant AST-1208891 and NASA grant NNX14AC89G. M.J. acknowledges the support of the Washington Research Foundation through its Data Science Chair and the University of Washington Provost's Initiative in Data-Intensive Discovery. E.O.O. is incumbent of the Arye Dissentshik career development chair and is grateful to support by grants from the Willner Family Leadership Institute Ilan Gluzman (Secaucus, NJ), Israeli Ministry of Science, Israel Science Foundation, Minerva and the I-CORE Program of the Planning and Budgeting Committee, and The Israel Science Foundation. S.R.K. thanks the hospitality of the Institute for Advanced Study (IAS). The sylvan surroundings and verdant intellectual ambiance of IAS resulted in a fecund mini-sabbatical stay (Fall 2007).