Following an introduction to the observed properties of FRBs, we discuss some basic physical inferences that can be made from the most readily observable parameters. A selection of the current sample of FRBs is shown in Fig. 3, which displays all those found with the Parkes telescope to date.

The FRB search process is described in detail in
Section 4. In brief, it consists of looking
for dispersed pulses like the one shown in
Fig. 1 in radio astronomical data
that is sampled in frequency and time. Searches are most commonly done
by forming a large number of time series corresponding to
different amounts of dispersion over a wide range. The amount of
dispersion is quantified by the time delay of the pulse between the
highest and lowest radio frequencies of the observation,
ν_{hi} and ν_{lo} are the high,
respectively, as

(1) |

where *m*_{e} is the mass of the electron, and *c* is
the speed of light. The second approximate equality holds when
ν_{lo} and ν_{hi} are in units of GHz. The
dispersion measure is given as

(2) |

In this expression, *n*_{e} is the electron number density,
*l* is a path length
and *d* is the distance to the FRB, which we will estimate
below. Note that, as in pulsar astronomy, DM is typically quoted in
units of cm^{−3} pc. This makes the numerical value of DM
more easy to quote compared to using column density units of, e.g.,
cm^{−2}. In practice, depending on the observational
setup and signal-to-noise ratio (S/N), the DM can be
measured with a precision of about 0.1 cm^{−3} pc.

The process for finding the optimum DM of a pulse is described in
Section 4.1. Once the DM value has been
optimised, a de-dispersed time series can be formed in which the
pulse S/N is maximized. If this time series
can be calibrated such that intensity can be converted to flux density
as a function of time, *S*(*t*), the pulse
can be characterized in terms of its width and peak
flux density, *S*_{peak}. In practice, the
calibration process is approximated from a measurement of the
root-mean-square (rms) fluctuations in the dedispersed time series,
σ_{S}. From radiometer noise considerations (see,
e.g.,
Lorimer
and Kramer, 2012),

(3) |

where *T*_{sys} is the system temperature,
*G* is the antenna gain, Δν is the receiver
bandwidth and *t*_{samp} is the data sampling interval.

For each FRB, the observed pulse width, *W*, is typically thought
of as a combination of an intrinsic pulse of width
*W*_{int} and instrumental broadening contributions.
In general, for a top-hat pulse,

(4) |

where *t*_{samp} is the sampling time as above,
Δ*t*_{DM} is the dispersive delay across an individual
frequency channel and Δ*t*_{DMerr} represents
the dispersive delay due to de-dispersion at
a slightly incorrect DM. FRB pulses can also be temporally broadened
by multi-path propagation through a turbulent medium. The
so called ‘scattering time-scale’
τ_{s} due to this effect is discussed in detail in
Section 3.3.

Pulse width is often measured at 50% and 10% of the peak
(Lorimer
and Kramer, 2012);
however, for a pulse of arbitrary shape, it is also common
to quote the equivalent width *W*_{eq} of
a top-hat pulse with the same *S*_{peak}. Such a pulse has an
energy or fluence

(5) |

A complicating factor with quoting flux density or fluence values is the
fact that, for many FRBs, the true sky position is not known well enough
to uniquely pinpoint the source to a position in the beam. Here,
‘beam’ is defined as the field of view of the radio
telescope, which is typically diffraction limited, as discussed more in
Section 4.3.1. The sensitivity across
this beam is not uniform, with the response as a function of angular
distance from the center being approximately Gaussian, in most cases. As
a result, with the exception of the ASKAP FRBs
(Bannister
*et al.*, 2017,
Shannon
*et al.*, 2018)
most one-off FRB fluxes and fluences determined so far are lower
limits. In addition, the limited angular resolution of most FRB searches
so far leads to typical positional uncertainties that are on the order
of a few arcminutes.

As is commonly done for other radio sources, measurements of the flux
density spectrum of FRBs as described by *S*_{ν}
∝ ν^{α}, where
α is the spectral index, are typically complicated by the
small available observing bandwidth. As a result, α is
usually rather poorly constrained. An additional complication also
arises from the poor localization of FRBs within the telescope beam,
where the uncertain positional offset and variable beam response with
radio frequency can lead to significant variations in measured
α values. We also note that a simple power-law spectral model may
not be an optimal model of the intrinsic FRB emission process (e.g.,
Hessels
*et al.*, 2018).
As discussed in Section 3, the spectrum can
also be modified by propagation effects.

One exception to these positional uncertainty limitations
is the repeating source FRB 121102, which is discussed further below
(Section 5.4). We
note here that flux density *S*(*t*)
defined above is the integral of the flux
per unit frequency interval over some observing band from
ν_{lo} to ν_{hi}. For the purposes of
the discussion below, and in the absence
of any spectral information, we assume α = 0 so that

(6) |

For a few FRBs, measurements of polarized flux are also available (see,
e.g.,
Petroff
*et al.*, 2015a,
Masui
*et al.*, 2015,
Ravi
*et al.*, 2016,
Michilli
*et al.*, 2018a).
In these cases it is often possible to measure the change in the
position angle of linear polarization, which scales with wavelength
squared. As discussed in Section 3.4,
the constant of proportionality for this scaling is the rotation measure
(RM), which probes the magnetic field component along the line of sight,
weighted by electron density.

For most FRBs, the only observables are position, flux density, pulse width, and DM. We now provide the simplest set of derived expressions that can be used to estimate relevant physical parameters for FRBs.

Starting with the observed DM, we follow what is now tending towards standard practice (see, e.g., Deng and Zhang, 2014) and define the dispersion measure excess

(7) |

where DM_{MW} is the Galactic (i.e. Milky Way) contribution from
this line of sight, typically obtained from electron density models such
as NE2001
(Cordes
and Lazio, 2002)
or YMW16
(Yao
*et al.*, 2017),
DM_{IGM} is the contribution from the intergalactic medium (IGM) and
DM_{Host} is the contribution from the host galaxy. The (1 +
*z*) factor accounts for cosmological time dilation for a source at
redshift *z*. The last term on the right-hand side of Eq. 7
could be further broken down into host galaxy free electrons and local
source terms, as needed. In any case, DM_{e} provides an upper
limit for DM_{IGM}, and most conservatively
DM_{IGM} < DM_{e}. We note that
DM_{MW} is likely uncertain at least at the tens of percent
level, but could in rare cases be quite far off if there are unmodelled
Hii regions along the line
of sight
(Bannister
and Madsen, 2014).

To find a relationship between DM and *z*, following, e.g.,
Deng and
Zhang (2014),
one can assume all baryons are homogeneously distributed and ionized
with an ionization fraction *x*(*z*). In this case, the mean
contribution from the IGM,

(8) |

where the constant *K*_{IGM} = 933 cm^{−3}
pc assumes standard Planck cosmological parameters
^{3}
and a baryonic mass fraction of 83%
(Yang
and Zhang, 2016) and
Ω_{m} and Ω_{Λ} are,
respectively, the energy densities of matter and dark energy.
At low redshifts, the ionization fraction
*x*(*z*) ≃ 7/8, and we find (see Fig. 1a of
Yang and
Zhang, 2016)
DM_{IGM} ≃ *z* 1000 cm^{−3}
pc. For a given FRB with a particular observed DM, a very crude but
commonly used rule of thumb is to estimate redshift as *z* <
DM / 1000 cm^{−3} pc.

Finally, to convert this redshift estimate to a luminosity distance,
*d*_{L}, we can make use of the approximation
^{4}
*d*_{L} ≃ 2*z*(*z* + 2.4) Gpc, which is
valid for *z* < 1. In this case, for the most conservative
assumption, we find that

(9) |

For the repeating FRB 121102, where *d*_{L} can be
inferred directly from the measured redshift of the host galaxy,
and constraints on dispersion in the host galaxy can be made, these
expressions can be used instead to place constraints on
DM_{IGM}, as discussed in
Section 5.4.

Having obtained a distance limit, for an FRB observed over some bandwidth Δν, we can place constraints on the isotropic equivalent source luminosity

(10) |

In arriving at this expression, we have started from the differential flux
per unit logarithmic frequency interval,
*S*_{ν} Δ ν (see, e.g. Eq. 24 of
Hogg,
1999)
in the simplest case of a flat spectrum source (i.e. constant
*S*_{ν}, see Eq. 6). The (1 + *z*) factor accounts
for the redshifting of the frequencies between
the source and observer frames. We also note that replacing
*S*_{ν} with
fluence in the above
expression yields the equivalent isotropic
energy release for a flat spectrum source.

As an example, we apply Eq. 9 to a typical FRB
(FRB 140514) with a
DM of 563 cm^{−3} pc and a
peak flux density of 0.5 Jy. The limiting luminosity distance
*d*_{L} < 3.3 Gpc, i.e. *z* < 0.56. The limiting
luminosity *L* < 44 Jy Gpc^{2} per unit
bandwidth. Assuming a 300-MHz bandwidth, this translates to a luminosity
release of approximately 10^{17} W (10^{24} erg
s^{−1}).

As shown by
Yang
*et al.* (2017),
for *z* < 1, the luminosity distance can
be directly related to the IGM DM as follows:

(11) |

Yang
*et al.* (2017)
find the following useful approximate relationship:

(12) |

where the constant *K* can be computed in
terms of the assumed values of the constants in Eq. 11 at a particular
observing frequency (for details, see
Yang
*et al.*, 2017).
Such a trend is apparent in the observed sample, albeit with a
considerable amount of scatter. Applying this model to the FRBs found
with the Parkes telescope, the authors constrain host galaxy DMs to have
a broad distribution with a mean value
⟨ DM_{Host} ⟩ =
270_{−110}^{+170} cm^{−3} pc and
*L* ∼ 10^{36} W (∼ 10^{43} erg
s^{−1}).

As in the case of other radio sources, where the emission mechanism is
likely to be non-thermal in origin, it is often useful to quote the
brightness temperature inferred from the source, *T*_{B},
which is defined as the thermodynamic temperature of a black body of
equivalent luminosity. Making similar arguments as is commonly done for
pulsars (see, e.g., Section 3.4 of
Lorimer
and Kramer, 2012),
we find

(13) |

Again evaluating this for our example FRB 140514 from the
previous section, where the pulse width *W* = 2.8 ms, we find
*T*_{B} < 3.5 × 10^{35} K.

^{3} For details, see Eq. 6 of
Yang & Zhang (2016).
Back.

^{4} This result is not widely used, but can
be easily verified by numerical integration.
Back.