MNRAS 460, L25–L29 (2016) doi:10.

1093/mnrasl/slw068
Advance Access publication 2016 April 8

Inferring the distances of fast radio bursts through associated

21-cm absorption

Ben Margalit1,2‹ and Abraham Loeb2

1 Physics Department, Columbia University, 538 West 120th St, New York, NY 10027, USA
2 Institute for Theory and Computation, Harvard University, 60 Garden St, Cambridge, MA 02138, USA

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Accepted 2016 April 6. Received 2016 April 6; in original form 2015 November 17

ABSTRACT
The distances of fast radio burst (FRB) sources are currently unknown. We show that the 21-cm
absorption line of hydrogen can be used to infer the redshifts of FRB sources, and determine
whether they are Galactic or extragalactic. We calculate a probability of ∼10 per cent for the
host galaxy of an FRB to exhibit a 21-cm absorption feature of equivalent width 10 km s−1 .
Arecibo, along with several future radio observatories, should be capable of detecting such
associated 21-cm absorption signals for strong bursts of several Jy peak flux densities.
Key words: galaxies: distances and redshifts – quasars: absorption lines – radio lines: general.

ers (Kashiyama, Ioka & Mészáros 2013), collapse of supramas-

1 I N T RO D U C T I O N
sive neutron stars (Falcke & Rezzolla 2014; Zhang 2014), orbiting
Fast radio bursts (FRBs), a recently discovered class of transient bodies immersed in pulsar winds (Mottez & Zarka 2014), magne-
events in which a ∼1 Jy signal of duration ∼1 ms is observed at ra- tar pulse–wind interactions (Lyubarsky 2014), giant pulses from
dio frequencies of ∼1–10 GHz, have become the source of great de- magnetars near galactic centres (Pen & Connor 2015), collisions
bate in the astrophysical community. The large dispersion measures between neutron stars and asteroids (Geng & Huang 2015), and gi-
(DM) of these events have led to the suggestion of their extragalac- ant pulses from young pulsars within supernova remnants (Connor,
tic origin, implying high isotropic luminosities of ∼1042 erg s−1 . Sievers & Pen 2016; Katz 2016). Alternatively, Loeb, Shvartzvald
Despite an estimated rate of order ∼103 d−1 (4π sr)−1 (Rane et al. & Maoz (2014) suggested that FRBs may originate from nearby
2016), only a handful of events have been reported to date. Of these, flaring stars in our own Galaxy (see also Maoz et al. 2015), and
12 were found in analysis of archival data from the Parkes radio tele- Kulkarni et al. (2014), motivated by discoveries of terrestrial FRB-
scope (Lorimer et al. 2007; Keane et al. 2012; Thornton et al. 2013; like impostors (Burke-Spolaor et al. 2011), suggested a terrestrial
Burke-Spolaor & Bannister 2014; Champion et al. 2015), one in origin for FRBs. Note that a terrestrial origin has largely been ruled
archival data from the Arecibo radio telescope (FRB121102; Spitler out due to FRB121102’s repetitions, which have been detected at
et al. 2014), and another in archival data of the Green Bank Tele- both Arecibo and GBT with consistent localization and DM. These
scope (GBT; Masui et al. 2015). Three FRBs have been discovered repetitions are also in strong tension with catastrophic event models
in real-time at the Parkes radio telescope (Petroff et al. 2015; Ravi, (usually extragalactic), unless they form a sub-population of FRBs.
Shannon & Jameson 2015; Keane et al. 2016). Keane et al. (2016) In this Letter, we propose a novel method for inferring the redshift
claimed to have identified an associated FRB ‘afterglow’ which of FRBs. This method could allow precise distance measurements
would have provided the first FRB redshift measurement, but this independent of the DM, and may therefore break the degeneracy
claim has largely been refuted by Williams & Berger (2016) and between terrestrial, Galactic, and extragalactic interpretations of the
later Vedantham et al. 2016 (although see Li & Zhang 2016). In large electron column densities. Specifically, we suggest measuring
an important discovery, Spitler et al. (2016) have recently reported the absorption signature of the 21-cm line of neutral hydrogen in
repetitions following FRB121102, later confirmed with additional the FRB’s host galaxy. Detection of an absorption line at longer
repetitions found by Scholz et al. (2016). wavelengths λobs yields an estimated measurement of the FRB pro-
Although many theoretical models for these events had been pro- genitor’s redshift, z = (λobs /21.106 cm) − 1. The precision of this
posed, the physical mechanism producing FRBs as well as their ori- method is not limited by the unknown electron column density along
gin remains illusive. Most models assume an extragalactic origin in line of sight, in sharp contrast to DM estimates of the distance scale.
the possible contexts of black hole evaporation (Keane et al. 2012), Recently, Fender & Oosterloo (2015) have also considered the
magnetar hyperflares (Popov & Postnov 2013; Katz 2014), neutron signature of the 21-cm line, but focused on absorption by interven-
star mergers (Totani 2013; Ravi & Lasky 2014), white dwarf merg- ing intergalactic clouds (as well as absorption in the Milky Way),
whereas we discuss absorption in the FRB’s host galaxy. These
studies are complementary in this sense. Our estimates indicate
 E-mail: btm2134@columbia.edu that associated absorption is more likely to yield high H I column


C 2016 The Authors

Published by Oxford University Press on behalf of the Royal Astronomical Society

L26 B. Margalit and A. Loeb
densities. This is consistent with the fact that only a small fraction
of quasars show evidence for damped Ly α absorption which flags
galactic discs (Fumagalli et al. 2014, and references therein). Even
so, a redshift inferred from 21-cm absorption should conservatively
be interpreted as only a lower-bound on the true FRB redshift due
to the possibility of intervening absorption.
Successful application of our method relies on H I absorption fea-
tures being strong and common among FRBs, which in turn depend
on the H I column density traversed by a typical FRB. In the fol-
lowing sections, we quantify the associated statistics. Throughout Figure 1. Schematic diagram of a galactic disc (cross-section) illustrating
our analysis, we assume that FRBs originate from common gas- the geometry of our model. The lightly shaded purple region demarcates the

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rich galaxies, and not from rare galaxies which are deficient in cold neutral hydrogen whereas the hatched light blue area shows the FRB source
gas and would likely not exhibit any H I absorption. Our results for distribution. Our exponential disc model assumes that both components
are characterized by the same radial scalelength, Rd . The FRB population’s
21-cm absorption are presented in Section 2. We continue by esti-
vertical scaleheight can vary by a factor of 1/γ relative to the H I scaleheight,
mating the probability for significant H I absorption in FRBs (Sec- zd . The brown circles are two example FRB progenitors, and the long green
tion 3). Finally, Section 4 summarizes our results and their applica- arrows indicate the line of sight towards the observer.
tion to current and future radio telescope surveys.
edge on 2R_d?
2 21-CM ABSORPTION though within a single CNM cloud at T = 100 K, the sound speed is
only ∼1 km s−1 . Additional broadening from large-scale motions of
The optical depth to 21-cm absorption of H I gas is given by (e.g.
the gas, such as galactic rotation, depends on the inclination angle
Loeb 2008),
v of the host galaxy relative to the line of sight. Since the typical
3A10 hc2 NH I distance propagated through the host galaxy disc is of the order of
τ (ν) = φ(ν), (1)
32πν10 kB Ts the vertical scaleheight zd , this effect results in v ∼ (zd /Rd )v rot
≈ σ t as well. Here, Rd refers to the disc radial scalelength, and
where A10 ≈ 2.8689 × 10−15 s−1 is the Einstein coefficient for the
we have made use of the well-known thin disc relation (obtained
hyperfine transition, ν 10 ≈ 1.4204 GHz is the frequency associated
from vertical hydrostatic balance) zd /r ≈ σ /v rot (e.g. Frank, King
 energy, φ(ν) is the line profile function (normal-
with the transition
& Raine 2002). Note that the mean distance travelled through the
ized such that φ dν = 1), NH I is the intervening neutral hydrogen
galaxy is 2zd and that in rare cases if the disc is viewed nearly edge
column density, and Ts is the spin temperature.
on, v can be substantially larger due to galactic rotation.
It is commonly assumed that the spin temperature is in equilib-
The value of τ v in equation (2) is characteristic of a Milky Way
rium with the kinetic temperature of the H I gas, since collisions
like galaxy, since a reasonable estimate for the H I column density is
dominate the level population of the hyperfine transition at the gas
NH I ∼ nH I zd ∼ 1020 cm−2 . Even so, significantly smaller or larger
densities of interest. The interstellar medium (ISM) is known to ex-
value are possible depending on the FRB source’s location within
hibit multiple phases of neutral hydrogen in pressure equilibrium:
its host galaxy, the distribution of neutral hydrogen in the galaxy,
the cold neutral medium (CNM) whose characteristic temperature
the inclination angle, and the host disc properties. In the following
is ∼100 K, and the warm neutral medium (WNM) which is typ-
section, we calculate the probability of measuring various values of
ically at ∼5000 K. In order to maintain pressure equilibrium, the
τ v, taking these factors into account.
CNM must be significantly denser than the WNM, however, it is
also observed to have a lower volume-filling factor. Coincidentally,
these opposing factors conspire to give very similar average den-
sities fCNM nCNM ∼ fWNM nWNM ∼ 0.1 cm−3 (e.g. Draine 2011). It 3 P RO BA B I L I T Y D I S T R I B U T I O N
therefore follows that the main contribution to H I absorption origi- OF H I ABSORPTION SIGNALS
nates from
 the CNM. This is because the average column densities
NH I = f nH I ds through either the CNM or the WNM are of the In estimating the probability distribution for finding various τ v
same order of magnitude, whereas the T−1 falloff in equation (1) values, we assume a fixed CNM spin temperature of Ts =
favours the lower temperature CNM. 100 K. Calculating τ v then reduces to calculating the H I col-
 umn density, which we split into two separate components, NH I =
Defining the integrated optical depth as τ v ≡ τ (v) dv, we
find N0 (MH I )η(x 0 , i, ϕ). Here,
   −1 mH
NH I Ts MH I /mH
τ v ≈ 0.55 km s−1 . (2) N0 (MH I ) ≡ ∼ nH I zd , (3)
1020 cm−2 100 K 4πRd (MH I )2
This quantity provides a measure of the total attenuation over all is a characteristic column density for a host galaxy containing a
frequency bands, and is commonly referred to as the ‘equivalent mass MH I in neutral hydrogen. The second term, η(x 0 , i, ϕ), is a
width’ of the line. Once the characteristic line width v is pre- dimensionless geometrical factor which depends on the position of
scribed, it is straightforward to calculate the typical optical depth the FRB source inside its host galaxy, x 0 , and on the inclination and
τ from equation (2). Viewed this way,τ v can be thought of as a azimuthal angles i, ϕ relative to the line of sight.
top-hat approximation for the integral τ (v) dv. We consider the neutral hydrogen gas and FRB source distribu-
The intrinsic hyperfine line width is extremely narrow, so that in tions to be exponential discs. A schematic diagram illustrating our
any practical application the minimal observed line width is set by model geometry and key parameters is given in Fig. 1. We take the
the thermal, turbulent, or bulk velocities of the gas. For a thin Milky radial scalelength of the discs, Rd , to be the same for both compo-
Way-like disc, the turbulent velocity is typically σ t ∼ 10 km s−1 even nents, yet allow the FRB source population’s vertical scaleheight to

MNRASL 460, L25–L29 (2016)

 Associated 21-cm FRB absorption L27
vary by a factor of 1/γ relative to the H I vertical scaleheight zd ,
namely
  
r exp (−|z|/zd ) , HI
n ∝ exp − × (4)
Rd exp (−γ |z|/zd ) , FRB .
Under this prescription, the geometrical factor η can be expressed
as

η(r0 , z0 , i, ϕ) ≡ nH I ds/N0
 ∞  

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=α exp − r̃02 + s̃ 2 sin2 i + 2r̃0 s̃ sin i cos ϕ
0

−α |z̃0 + s̃ cos i| ds̃ , (5)

where r˜0 (z˜0 ) is defined as r0 /Rd (z0 /Rd ) and α ≡ Rd /zd is the ratio
of radial to vertical scalelengths and is roughly ∼15 for the Milky
Way.
The cumulative probability distribution for measuring a value of η
Figure 2. Cumulative probability distribution for measuring an H I ab-
larger than η , P(η > η ), can then be calculated by populating a grid
sorption feature with equivalent width larger than τ v, P(>τ v). The
in (r0 , z0 , i, ϕ). Using equation (5), each gridpoint can be assigned
solid blue curve represents the resulting probability distribution for our
a well-defined value of η, as well as a probability density for FRB canonical values: α = 15, γ = 1, Ts = 100 K. These values are char-
sources to populate its region, f (x 0 , i, ϕ) ∝ nFRB (r0 , z0 )r0 sin i. acteristic of a CNM and a thin disc FRB population. The lightly shaded
Generating a sufficiently extended and finely coarsed grid, we sort blue region illustrates the variability due to the ∼40 per cent uncertainty in
the gridpoints in decreasing order of η and cumulatively sum the equation (6). This should not be strictly interpreted as the total uncertainty in
probabilities dP ≈ f(r0 , z0 , i) dr0 dz0 di dϕ. This results in a nu- our results, as the systematics of our model assumptions will likely contribute
merical evaluation of P(η > η ) which is equivalent to solving the significantly. The dashed red curve shows the result when the ratio of H I
integral dr0 dz0 di dϕ f(r0 , z0 , i) [η > η (r0 , z0 , i)], where is the to FRB vertical scaleheight is changed to γ = 1/4, characteristic of a thick
Heaviside function. disc FRB population. As expected, the larger vertical spread in the FRB
source distribution reduces the average H I column density and decreases
Once P(>η) is obtained, all that remains in the process of calcu-
the typical τ v. The horizontal axis trivially scales as (Ts /100 K)−1 and/or
lating P(>τ v) is an estimation of N0 . Several studies have shown
(N0 /9.1 × 1020 cm−2 ), if different spin temperatures and/or characteristic
that the H I mass component of spiral galaxies scales as the disc column densities are assumed (see equation 2).
scale-radius squared, namely MH I ∝ Rd2 (e.g. Broeils & Rhee 1997;
Verheijen & Sancisi 2001), so that the surface density N0 is essen- thick blue curve shows the cumulative probability distribution for
tially constant and does not vary appreciably with galactic mass. a nominal value of γ = 1, whereas the dashed red curve represents
We calculate the characteristic (mean) column density N0 using the results for γ = 1/4. For our canonical values (γ = 1), we obtain
the tabulated H I masses and radii of Broeils & Rhee (1997). We a median τ v of ≈1.7 km s−1 , and a 10 per cent probability of find-
relate our exponential disc scalelength, Rd , to the radius Reff which ing values larger than ≈11 km s−1 . Since the parameter γ is defined
encloses half the H I mass and is quoted by these authors. For our as the ratio of the neutral hydrogen vertical scaleheight to that of
exponential disc model (4), we find that the radius encompassing the FRB source population, the canonical γ = 1 scenario describes
50 per cent of the total H I mass is Reff ≈ 1.68Rd . Substituting this an FRB population that is localized to the thin disc component of
relation in equation (3) and using values of MH I and Reff from table the galaxy, similarly to the H I distribution.
1 of Broeils & Rhee (1997), we find that the resulting mean column Fig. 2 also shows the resulting cumulative probability distribu-
density and its standard deviation are tion for γ = 1/4. This value of γ is expected if the FRB source
N0 = (9.1 ± 3.6) × 1020 cm−2 . (6) distribution is predominantly associated with the thick disc com-
ponent of the galaxy, so that the FRB vertical scaleheight is larger
Note that a crude estimate of this result may be obtained without than the scaleheight of the H I disc. As expected, the probability
the full tabulated data of Broeils & Rhee (1997), by using the mean distribution curve in this case is shifted to lower τ v values, since
‘hybrid’ density found by these authors, σH∗ I = 11.2 M pc−2 . there is a larger chance for an FRB to originate further away from
Here, the ‘hybrid’ density is defined as σH∗ I ≡ MH I /πR25 2
, and R25 the galactic plane where the H I column density along the line of
−2
is the optical radius at an isophote of 25 mag arcsec . Broeils & path would (if the line of sight is pointed away from the galactic
Rhee (1997) find that the optical radius strongly correlates with plane) decrease. The median value of τ v, in this case, drops to
the H I half-mass radius Reff , R25 /Reff ≈ 1, which combined with ≈0.8 km s−1 , although there is still a ≈10 per cent chance of mea-
the relation Reff ≈ 1.68Rd and the factor 4 difference in definition suring values larger than 9 km s−1 .
between σH∗ I and N0 , yields an estimate of N0 ≈ 0.7 σH∗ I . For high values of τ v, the γ = 1/4 curve converges to the
The cumulative probability distribution for measuring H I column canonical γ = 1 curve, since the largest values of τ v are obtained
density larger than N is in this case simply related to the probability when the FRB progenitor is positioned ‘behind’ its host galaxy.
distribution of η, When viewed from the FRB source, the signal propagates towards
the galactic plane, passes through the mid-plane, and continues
P (N > N  ) = P [η > N  /N0 ]. (7)
outwards towards the observer (see Fig. 1). These large column
Combining equation (2), and (7), we numerically evaluate the prob- density events are nearly unaffected by the vertical scaleheight of
ability distribution for τ v. The results are illustrated in Fig. 2. The FRB progenitors, hence our predictions for P(10 km s−1 ) are

MNRASL 460, L25–L29 (2016)

L28 B. Margalit and A. Loeb
robust with respect to this parameter. We have additionally cal- one FRB (Scholz et al. 2016; Spitler et al. 2016) could greatly help
culated the probability distributions obtained for various values of reduce such noise-confusion and increase a detected absorption fea-
α (the ratio of radial to vertical scalelengths) in the range 10–20, ture’s significance by implementing stacking techniques (although
yet find no appreciable change in the results. the large variability in spectral index between repeating bursts poses
a clear difficulty).
We also assessed the feasibility of applying the method with fu-
4 DISCUSSION ture radio observatories such as the Canadian Hydrogen Intensity
We have proposed a novel method for measuring the cosmological Mapping Experiment telescope (Bandura et al. 2014), the Five hun-
distances to FRBs, based on the 21-cm absorption signature of H I dred meter Aperture Spherical Telescope (FAST),2 and the Square
gas in the FRB progenitor’s host galaxy. The method is only useful if Kilometer Array (SKA) telescope.3 We repeated the calculation
the absorption feature is strong enough to be detected. We therefore for these observatories and found similar results for FAST and

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estimated the probability of obtaining strong H I absorption to assess SKA, since the planned sensitivities are roughly comparable with
the feasibility of using the method with current and near future Arecibo: Aeff /Tsys ≈ 1250, 1630 m2 K−1 for FAST and SKA-mid,
observational facilities. respectively.4 SKA can be an exception to this case, if it is ever
Using a simple exponential disc model for both FRB and H I realized with a filling factor significantly larger than a few per cent.
galactic distributions, we find a probability of ∼10 per cent for mea- Our analysis in Section 3 is somewhat simplified. To begin with,
suring values of τ v larger than 10 km s−1 , and a median value of an exponential disc model with radially fixed vertical scaleheight is
∼1 km s−1 . only a crude model for Milky Way-like spiral galaxies. Secondly, we
The observational significance of these values can be quanti- have assumed diffuse H I distributions as opposed to more clumpy
fied in terms of the signal-to-noise ratio (SNR) and the channel cloud features representative of the CNM. Furthermore, we ne-
bandwidth of the radio telescope. If the observational frequency glected any redshift evolution in N0 and based our results on studies
bandwidth (measured in velocity units), v obs , exceeds the absorp- of galaxies at z ≈ 0. This last assumption is conservative, since
tion feature’s inherent v, the absorption profile will be unresolved. MH I (z) > MH I (0) and Rd (z) < Rd (0), so that we have underesti-
This, however, does not mean that the H I absorption will be unde- mated the typical galactic column densities. We also ignored the
tectable, since the entire signal in the bandwidth bin will merely possibility that some FRBs originate in elliptical galaxies. As such,
be attenuated by a factor of ≈τ v/v obs . Smaller bandwidth is our analysis of P(>τ v) should be treated as a first crude estimate
therefore highly desirable in detecting H I absorption (and note that of the viability of our proposed method.
the previous expression is only correct for v obs > v). Scattering effects may also play a role in determining the
As a feasibility demonstration we focus on the Arecibo observed absorption signal and should generally be considered.
Observatory.1 The sensitivity of this telescope is estimated as Diffractive scintillations caused by the intervening ISM or inter-
Aeff /Tsys ≈ 1150 m2 K−1 , where Aeff is the effective collecting area galactic medium can, in principle, smear the spectral feature of the
of the instrument and Tsys is the system temperature. The L-Wide re- 21-cm absorption. Although H I absorption by galactic discs has
ceiver covers the frequency range 1.15–1.73 GHz, corresponding to been successfully observed for quasars (e.g. Borthakur et al. 2011),
a maximum H I line redshift of ∼0.24, whereas the Arecibo L-band the potentially smaller source size of FRBs warrants further analysis
Feed Array covers 1.225–1.525 GHz corresponding to z ≤ 0.16. of the effects of scintillations on their absorption signal.
These are consistent with the low end of DM inferred redshifts Finally, we note that while our method provides a means of mea-
for known FRBs. With these parameters, and using the radiome- suring the FRB’s precise redshift, observationally, this scenario is
ter equation (e.g. Loeb & Furlanetto 2013, ch. 12.7), we estimate mostly indistinguishable from a possible intervening absorption,
the SNR of an FRB with pulse duration t and peak-flux Fν in a which only provides a lower limit on the redshift. Fender & Ooster-
frequency bin of bandwidth v obs centred around ν obs , loo (2015) estimate a probability of significant intervening absorp-
tion (τ v > 2 km s−1 ) at z = 2 of ∼10 per cent, but scaling this to
Fν Aeff  typical DM inferred FRB redshifts of z ∼ 0.5 reduces the probabil-
SNR ≈ νobs t
2kB Tsys ity to 2 per cent. This is significantly smaller than the probabilities
    1/2
Fν t 1/2 vobs νobs 1/2 estimated in this work, and consistent with the fact that only a small
≈ 4.6 . fraction of all quasars show damped Ly α absorption in their spec-
1 Jy 3 ms 10 km s−1 1.2 GHz
tra, although caution is necessary in interpreting this result due to
(8)
the many uncertainties in both models. Further detailed analysis of
Only for a strong burst would the sensitivity implied by both associated and intervening absorption is therefore necessary to
equation (8) be sufficient to detect an H I absorption signature. Based more accurately assess the relative importance of the two effects.
on our analysis, and assuming a bandwidth of v obs = 10 km s−1 , It is important to stress that the limitations of our analysis affect
we find a ∼10 per cent probability of measuring effective H I opti- only the probability distributions for measuring τ v. This informa-
cal depths larger than ∼1. These would be detectable at an SNR tion is important in estimating the likelihood of a 21-cm absorption
of 8 for an FRB of Fν  2 Jy. This optimistic assessment does not detection, but bears no implications whatsoever to the application
take into account difficulties associated with noise-confusion, which of our method once such an absorption signal is detected. In partic-
may be substantial for a single-frequency channel ‘detection’. Even ular, precise redshift measurements to within ∼(v obs /c) accuracy
with such difficulties, a strong burst such as FRB010724 (Lorimer (neglecting peculiar motions of the host galaxies) are immediately
et al. 2007) which had an estimated flux density of 30 ± 10 Jy would
likely overcome this issue, and is an ideal candidate for detecting H I
absorption. We additionally note that the repeating nature of at least
2 http://fast.bao.ac.cn/en/
3 https://www.skatelescope.org
1 http://www.naic.edu/ 4 see e.g. table 1 of https://www.skatelescope.org/?attachment_id=5400.

MNRASL 460, L25–L29 (2016)

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MNRASL 460, L25–L29 (2016)