A Fast Radio Burst Occurs Every Second throughout the Observable Universe

The fact that FRB 121102 resides in a metal-poor dwarf galaxy indicates that transients of its type do not trace star formation at all metallicities (in which case a typical FRB would occur in more massive galaxies). It might also be an indication that FRBs are more common in the high-redshift universe, which is filled with metal-poor low-mass galaxies.²

We consider two prescriptions for the hosts. First, we assume that all host galaxies have similar properties to the one in which FRB 121102 is located. In particular, we require galaxies to have a stellar mass in the range between ${M}_{* ,1}={\overline{M}}_{* }/\sqrt{10}$ and ${M}_{* ,2}={\overline{M}}_{* }\sqrt{10}$, where ${\overline{M}}_{* }=5.3\times {10}^{7}$ ${M}_{\odot }$ is the stellar mass measured for FRB 121102. The observed metallicity of the host galaxy of FRB 121102 agrees well with theoretical predictions for galaxies of its mass and does not add any constraints (Hunt et al. 2016; Ma et al. 2016). Second, we assume high-mass hosts with ${\overline{M}}_{* }=5\times {10}^{9}$ ${M}_{\odot }$ for comparison. Such galaxies are much more common in the low-redshift universe than the metal-poor dwarfs.

The number of host galaxies per comoving volume with stellar masses spanning the range between ${M}_{* ,1}$ and ${M}_{* ,2}$ is set by the number of dark matter halos that contain such galaxies and can be computed using the excursion set formalism. To infer this number, we (i) use the Sheth–Tormen prescription (Sheth & Tormen 1999) for the number density of dark matter halos versus halo mass, dn/dM_h, and (ii) populate halos by stars using abundance matching and a star formation efficiency that varies with halo mass and redshift (Behroozi et al. 2013). We select halos hosting galaxies with stellar mass in the range between ${M}_{* ,1}$ and ${M}_{* ,2}$ as FRB hosts, yielding the observed number of FRBs, per unit time

$$\begin{eqnarray}&&{N}_{\mathrm{FRB}}={\int }_{V}{\int }_{{M}_{h}={M}_{h}({M}_{* ,1})}^{{M}_{h}={M}_{h}({M}_{* ,2})}\displaystyle \frac{{dn}}{{{dM}}_{h}}\displaystyle \frac{{R}_{\mathrm{int}}}{(1+z)}{{dM}}_{h}{dV},\end{eqnarray} \tag{ 1 }$$

where dV is the comoving volume element, the factor $(1+z)$ accounts for the time dilation effect, and R_int is the intrinsic rate of FRBs produced by each galaxy in the direction of observer, into which we absorb the birth rate of each emitter, the lifetime of an emitter, the rate at which each source emits repeated FRB signals, and the beaming factor (see Section 2.3 for details). In its current form, Equation (1) assumes that all produced FRBs are observable. When calculating the number of observed FRBs with peak fluxes, S_peak, above a given flux limit, S_lim, we modify the equation to include a selection rule ${S}_{\mathrm{peak}}\gt {S}_{\mathrm{lim}}$ with S_peak dependent on the redshift and the intrinsic FRB luminosity, as discussed in the next subsection.

While the bright end of the FRB population is somewhat constrained by existing observations, the faint end is highly unconstrained and will be probed by future sensitive telescopes. We consider several options for assigning a luminosity to each event, thus bracketing a wide range of possible LFs for the entire FRB population:

Model A (the lower limit on faint FRBs): FRBs are standard candles with all events having the same intrinsic luminosity, L_int, and a Gaussian-like spectral profile. Using FRB 121102 as a prototype, we construct an intrinsic spectral profile with the observed flux fitted by

$$\begin{eqnarray}&&{S}_{\mathrm{obs}}({\nu }_{\mathrm{obs}})={S}_{\mathrm{peak}}\exp \left[-\displaystyle \frac{{({\nu }_{\mathrm{obs}}-a)}^{2}}{2{c}^{2}}\right].\end{eqnarray} \tag{ 2 }$$

We normalize L_int to yield ${S}_{\mathrm{peak}}=0.9017$ Jy for a source at z = 0.19, adopting $a=2.986\,\mathrm{GHz}$ and $c=0.1984\,\mathrm{GHz}$ (mean values for the repeater; see Table 2 of Law et al. 2017). This model gives a delta LF and provides a lower limit on the number of faint FRBs that are those located at large cosmological distances.

Model B: to investigate the effect of the spectral shape, we compare the results of Model A to a case with a flat spectrum normalized to the same value of S_peak.

Model C (the upper limit on the faint FRBs): each event has an average Gaussian-like profile as in Model A, but the LF has the Schechter form

$$\begin{eqnarray*}&&\displaystyle \frac{{dN}}{{dL}}\propto {\left(\displaystyle \frac{L}{{L}_{\star }}\right)}^{\alpha }{e}^{-L/{L}_{\star }},\end{eqnarray*}$$

where ${L}_{\star }$ is set so that FRB 121102 will match the average luminosity at z = 0.19 (with no dependence of ${L}_{\star }$ on redshift). We integrate over the LF when calculating the number of observed FRBs from Equation (1). This model gives the maximal possible number of intrinsically faint FRBs for $\alpha =-2$, the steepest slope for which the luminosity density does not diverge. The observed faint population includes both the intrinsically faint sources and those at large cosmological distances.

Model D: FRBs have a flat spectrum as in Model B, but their LF has the Schechter form.

Assuming a constant intrinsic FRB rate per galaxy, we can find R_int in Equation (1) by matching the predicted N_FRB to available observations. The projected all-sky FRB rate ${R}_{p}=2\times {10}^{3}$ sky⁻¹ day⁻¹ measures the total number of events observed from $z\lesssim 1$ with a fluence sensitivity of 1 Jy ms (Law et al. 2017; Nicholl et al. 2017) and yields a different value of R_int for each one of the considered models. To extract R_int from Equation (1), we assume a survey with a flux limit of ${S}_{\mathrm{lim}}=1$ Jy observing all the events out to z = 1 (assuming a 1 ms duration). Naturally, the inferred rate depends on the properties of the host galaxies. In the case of low-mass galactic hosts, the intrinsic rate is ${R}_{\mathrm{int}}\approx 2\times {10}^{-5}$ per galaxy per day for Models A and B and ${R}_{\mathrm{int}}\approx 3\times {10}^{-3}$ for Models C and D. The normalization in the case of high-mass hosts is ${R}_{\mathrm{int}}\approx 3\times {10}^{-4}$ for Models A and B and ${R}_{\mathrm{int}}\approx 4\times {10}^{-2}$ for Models C and D.

If more details on the nature of FRBs were known, their intrinsic rate could be inferred from the relation ${R}_{\mathrm{int}}={R}_{b}\tau {R}_{s}{{\rm{\Omega }}}_{b}$, where R_b is the birth rate of FRB emitters, τ is their lifetime, R_s is the rate at which each source emits repeated FRB signals, and ${{\rm{\Omega }}}_{b}$ is the beaming factor.

We compare model predictions to the surveys conducted by Parkes, Arecibo, and ASKAP (which found 19 different FRBs in total) based on the online FRB catalog. We ignore FRBs observed by GBT and UTMOST because they probe a different frequency band (0.8 GHz compared to 1.4 GHz). The top panel of Figure 1 shows the number counts of FRBs observed at 1.4 GHz as a function of flux limit in the band. We rescale the observed numbers to match the total rate of $2\times {10}^{3}$ sky⁻¹ day⁻¹ at ${S}_{\mathrm{lim}}=1$ Jy. As shown in Figure 1, the predicted FRB number counts in all models, as well as in the case where the sources are uniformly distributed in comoving volume (dotted line) overpredict the observed numbers at the faint end and underpredict them at the bright end. For all models, including Model A, which is designed to yield the lower limit on faint FRB numbers, our curves are steeper than observed.

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The discrepancy between our models and observations might indicate that above the flux limit the surveys of faint FRBs are not complete and miss many events, while the bright end is most likely highly biased. An additional possibility is that low-luminosity FRBs do not exist or that the LF has a positive spectral index (which, however, is not typical). The deficit of FRBs at low frequencies might also be caused by self-absorption around the source if the line of sight goes through the emitting region of the quiescent radio source (Metzger et al. 2017). Cordes et al. (2017) argued that if a burst passes through a galactic disk where the gas density is large enough, it could be scattered into non-detectability. However, the probability of having such a small impact parameter from an unrelated galactic center along a random line of sight through the universe is very small, even for sources at $z\sim 10$. Such events would also be strongly lensed, and Barkana & Loeb (2000) have shown that the lensing optical depth is still much smaller than unity even for high-redshift sources.

The expected numbers rise at low fluxes until saturated. Our models predict that including all faint events (high-redshift events in Models A and B and high-z plus low-luminosity events in Models C and D), we expect more than $3.5\times {10}^{5}$ events per day from the entire universe (i.e., more than one burst per second).

The Gaussian spectral profile leads to a factor of 20%–40% deficit in the expected number of faint events at 1.4 GHz and ${S}_{\mathrm{lim}}=0.1$ mJy, compared to a flat spectrum (Model A versus B and Model C versus D). This is because if the spectrum is line-like, emission from distant sources redshifts out of the band and is no longer observable. This effect is much stronger at lower frequencies.

Comparing Models A and C we find that the Schechter LF yields many more faint sources. Note, however, that one needs a much higher R_int in this case to match the low-redshift observations because a smaller fraction of the events have high observed flux.

The cumulative number of observed FRBs with ${S}_{\mathrm{lim}}=0.1$ mJy and redshifts above z is shown on the bottom panel of the Figure 1. For each observed FRB we compute the redshift of the host galaxy by subtracting from the measured DM the contribution of the Milky Way (NE2001 model of Cordes & Lazio 2002) and assuming fully ionized IGM. Based on the properties of FRB 121102, we assume that the contribution of each host to DM is 324 pc cm⁻³ (Tendulkar et al. 2017). Because of the relatively high Galactic contribution, the cosmological DM for 2 out of 19 FRBs (010621 and 150807) results in negative DM values. We exclude these 2 events, which leaves 17 FRBs in total. We re-normalize our models to yield this number of events.

Figure 1 shows that once FRBs will be observed at greater numbers, the redshift evolution of their number counts can differentiate between the Schechter LF and the standard candle scenarios. The drop with redshift is more significant in the case of the Schechter LF because a large fraction of the sources are intrinsically faint and quickly redshift below the flux detection limit even if their central frequency is still within the telescope band.

We apply the formalism discussed above to make predictions for the number of FRBs detectable with future observatories. We choose to focus our attention on the Square Kilometre Array (SKA) as this facility will be very sensitive and have two different low-frequency instruments thus being able to probe faint high-redshift FRBs. The two arrays are: the low-band array (SKA-LOW) at 50–350 MHz with a sensitivity of 2 mJy and the high-band instrument (SKA-MID) probing frequencies between $0.35\mbox{--}0.95\,\mathrm{GHz}$ (SKA-MID1) and $0.95\mbox{--}1.76\,\mathrm{GHz}$ (SKA-MID2) at a sensitivity of 0.8 mJy. Detection rates of FRBs predicted for the SKA as a function of flux limit and the minimal redshift of FRBs are shown in Figure 2. Another powerful machine to detect FRBs will be the Canadian Hydrogen Intensity Mapping Experiment (CHIME), which is expected to have a 125 mJy flux limit in the $400\mbox{--}800\,\mathrm{MHz}$ frequency range and a large collecting area (Newburgh et al. 2014; Rajwade & Lorimer 2017). Other relevant experiments (which, however, are not as sensitive as the SKA and have a smaller field of view than CHIME) include, among others, the Low-Frequency Array (LOFAR; Coenen et al. 2014) with 107 Jy sensitivity at 142 MHz and the Green Bank Northern Celestial Cap at 350 MHz with a sensitivity of 0.63 Jy to a 5 ms signal (Chawla et al. 2017).

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We find that spectral shape of the bursts is very important and strongly affects the predictions for the high-redshift and low-frequency surveys. The top panels of Figure 2 show the estimated FRB rates in the SKA bands. Models B and D (flat spectrum) yield band-independent predictions. A very sensitive survey conducted at any frequency would measure the same number of sources out to the edge of the universe (at least ∼7 FRBs sky⁻¹ s⁻¹ in total), while with its sensitivity limit of 125 mJy, CHIME will observe at least $3\times {10}^{4}$ events sky⁻¹ day⁻¹ (in agreement with numbers reported in the literature; e.g., Rajwade & Lorimer 2017).

However, the expected rate is much lower for models with the Gaussian spectral shape (Models A and C), and in this case each one of the instruments will probe an FRB population from a separate redshift range defined by the frequency band. In particular, CHIME will target the redshift range of 3.4–7.9; however, such events would be below the flux sensitivity limit of the instrument. But owing to its higher sensitivity, SKA will be able to detect the faint high-redshift events. Based on the central frequency of the spectrum, SKA-MID2 will probe FRBs from $z=1\mbox{--}2.7$, which includes the peak of star formation activity (e.g., Madau & Dickinson 2014); SKA-MID1 will probe events from $z=2.7\mbox{--}9$, thus including a large part of the epoch of reionization (EoR; e.g., Loeb & Furlanetto 2013), which extends between $z\sim 6$ and $z\sim 12$ (e.g., Planck Collaboration XLVII 2016); and SKA-LOW will have the correct band to search for all the bursts above z = 9 occurring at the beginning of the EoR and during the cosmic dawn, an epoch preceding the EoR when first stars formed. As seen from the bottom panels of Figure 2, because of the width of the spectral profile the FRB rates do not drop abruptly at the cutoff redshifts specified above.

Sensitivity of a telescope to transients depends on the width of the pulse, W, which for FRBs depends on the redshift, intrinsic properties of the source, and broadening due to the finite sampling interval of the survey (Rajwade & Lorimer 2017). The limiting flux at the 10σ detection threshold is given by

$$\begin{eqnarray}&&{S}_{\mathrm{lim}}=10\displaystyle \frac{\mathrm{SEFD}}{\sqrt{2W{\rm{\Delta }}\nu }},\end{eqnarray} \tag{ 3 }$$

where SEFD is the system equivalent flux density, ${\rm{\Delta }}\nu$ is the bandwidth, and factor 2 accounts for the two polarization channels. Assuming that the observed duration of the pulse is dominated by the redshifted intrinsic width, we estimate 10σ sensitivity at the lowest redshift detectable in each one of the SKA bands (vertical lines in Figure 2). Adopting characteristic sensitivities for the final SKA configuration (phase 2, 0.8 mJy in the $0.95\mbox{--}1.76\,\mathrm{GHz}$ band, 0.8 mJy in the $0.35\mbox{--}0.95\,\mathrm{GHz}$ band, and 2 mJy in the 50–350 MHz band) we see that SKA-MID2 will detect FRBs out to $z\sim 3.2$ at a rate of ∼1000 sky⁻¹ day⁻¹ or higher. SKA-MID1 will detect FRBs out to $z\sim 9.1$ at a rate above 100 sky⁻¹ day⁻¹, with $\sim 2\times {10}^{4}$ sky⁻¹ day⁻¹ FRBs expected from the EoR ($z\gt 6$) for low-mass galactic hosts and ∼6000 sky⁻¹ day⁻¹ if the hosts are massive. Finally, SKA-LOW will only be sensitive to FRBs out to $z\sim 11$ (and only if the LF has a Schechter form) when the rate drops below 1 sky⁻¹ day⁻¹.

Low-frequency sensitive surveys such as SKA-LOW can be used to identify the nature of the host galaxies if the spectral shape is not flat. In Models A and C, we expect to find fewer FRBs at high redshifts in the model with massive hosts compared to the case of low-mass hosts. This is simply because massive galaxies are rare at higher redshifts and the population builds up at low redshifts. In the SKA-MID bands, where the signal is dominated by the events at low redshifts close to the peak of star formation, it is difficult to distinguish between the predictions for low-mass hosts and massive hosts.