Recipes forthe statistical properties
of Fast Radio Bursts
Yuu NIINO (NAOJ)
based on arXiv:1801.06578, accepted by ApJ

Fast Radio Bursts•••

Transient events in radio bands (~1 GHz) with a few milli-sec
durations.
First discovered in 2007 byLorimer et al.
Currently ~ 30 events are known.
• ~ 20 discovered by the Parkes
radio telescopeThornton et al. (2013)

Dispersion measures of FRBs
The dispersion measures(DMs) exceed the expected
MW contribution.
• DM• = delay of pulse arrival
time as a function of ν• = free electron column
density along the line ofsight• If the DMs arise from the IGM,
FRBs reside at z ~ 0.3-1.5.
• FRBs may enable us toobserved the IGM.
•Spitler+ (2014)

DM/DMMW

Lorimer+ (2007)

DMLocation and flux of FRBs•••

The Parkes multi-beamreceiver has 13 beams.
• The only information
about the location is thebeam pointing.
• ~ 14 arcmin in FWHM
Efficiency of a radio receivervaries within it's beam.
• Position of an FRB within
the beam is not known.
• Lower limits only.
Exception• FRB 121102: therepeating FRB detectedby Arecibo

Keane+ (2016)

DM distributionof cosmological FRBs

Cosmic FRB history
and the DM distribution•

If FRBs originate at cosmological distances,
DM distribution ~ redshift distribution.
• important clue to the cosmic FRB ratehistory.
• In the cases:••

ρFRBSFR

•

ρFRB

M★

The DM distribution is similar below DM ~1000 (or z ~ 1).
• Dramatically different beyond that.

DM [cm-3pc]

FRB luminosity function
and beam efficiency pattern••••

The DM distribution depends also on theluminosity function (LF) of FRBs
Receiver efficiency follows a certain probability
distribution function (PDF).
FRB LF is effectively convolved with the receiver
efficiency PDF.
• Fν,app ≡ εbeamFν
• Lν,app ≡ 4πdL2 Fν,app = 4εbeamπdL2 Fν
detection limit: Fν,app > 0.4 [Jy]power-lawstandard candle

power-law+ exp cutoff

DMIGM = c0|dz 0|

1 + z0dz .

(2)

ρFRB, FRB LF, & DM distribution
where ne,IGM is the electron density in the IGM. Herewe assume that the IGM is uniform at each redshift
with the comoving density ⇢crit ⌦b , composed of 75% H
and 25% He, and fully ionized throughout the redshift
range we consider. Under these assumptions, the IGM
electron density can be written as:

Lν,0ne,IGM (z) =L

7 ⇢crit ⌦b(1 + z)3 .
ν,08 mp(3)

Lν,0The upper horizontal axis of figure 1 indicates DMIGM
that corresponds to z in the lower axis (naivelyDMIGM ⇠ 1000z cm 3 pc in this redshift range).
In the above expression, it is assumed that thedominant fraction of baryons in the universe is in
the IGM, which is true when we consider di↵use
ionized gas associated with dark matter halos aspart of the IGM (e.g., Fukugita & Peebles 2004).
If a significant part of the IGM is associated darkmatter halos, the IGM might be inhomogeneous inreality, and the inhomogeneity might a↵ect the DMIGM
distribution of FRBs. We discuss the e↵ect of the IGM
inhomogeneity on our results in §6.1
The predicted redshift distributions with the three⇢FRB (z) models are shown in the bottom panel of figure 1. The redshift
ρFRBdistributions
SFR with the di↵erent
ρFRB⇢FRB (z) models are similar with. 1
34 each other at z LF1L=10
where majority of ν,0the currently known FRBs reside,
while the redshift distributions are dramatically di↵erent at z > 1, as previously shown by Dolag et al. (2015)

Figure 1.
Top panel: ⇢FRB models (occurrence
FRBs per comoving volume) normalized at z = 0.
panel: observed comoving volume per redshift per swith the assumed cosmology (dV /dzd⌦). Bottom pacurrence rate of FRBs per redshift per steradian inserver frame, which is proportional to ⇢FRB (z)/(1dV /dzd⌦. We note that the FRB rates shown in th
represents all FRBs regardless of their detectability.
that corresponds to each redshift is indicated in thhorizontal axis (see equation 2).

SFRof the FRB within the receiver beam, because effi
of a radio receiver largely varies within its beam.
sume beam efficiency pattern of a radio receiveDM distributionof Cosmological FRBs
LF and ρFRB are largely
degenerated.
• LF2 (power-law) can not reproduce
the DM distribution if ρFRB SFR.
••

LF needs bright-end cutoff
around logLν,0 [erg s-1Hz-1] ~ 34.

CosmologicalorLocal

Are FRBs cosmologicalor local?
• DM from the IGM• ~ 1000 × z [cm-3pc]• DM do not necessarily arise

solely from the IGM• host galaxy ISM• supernova remnant• HII region• If the IGM is not the dominant
DM component, FRBs might
be at lower redshifts.

host galaxyISM

Progenitor
CSM

IGM
Milky WayISM•••The power-law slope = -1.5 in theEuclidean space (e.g., local universe).
Cosmological effects modify theslope.
The fluence distribution is flatter thanthe Euclidean (Vedantham et al.
2016).
• affected by the incompleteness(a)• steeper in the bright-end Figure 6. Left panel: The source count distribution of Parkes FRBs. The sky
F > 2 Jy ms (see §5.4). Right panel: The slope ↵ of the integral source c• completeness limit ~ 2 [Jyforms]
obtain a slope of ↵ = 2.2+0.6
for FRBs above a fluence completeness limit o1.2
• ~ 50% of the sample fluence completeness limit and the horizontal dashed line indicates ↵ = 3/2Universe.
The S/N distribution agrees with theEuclidean.
Macquart & Ekers (2018)•Bhandari et al. (2017)

The logN-logSdistribution

The logN-logS distribution•Sν distribution• When ρFRB
SFR:
consistent with the Euclidean & obs.
When ρFRB M★:••

flatter than observed• close to the limit by the CHIME pathfinder (Amiri et al. 2017)
Fν distribution: steeper than the Sν distribution & the Euclidean••

Discussion

Flux, Fluence, and S/N
The faint end of the Fν distribution is affected by acomplicated incompleteness.
• The Sν distribution is sharply cut.
• S/N of FRB detections is determined by Sν rather than Fν.
• Better constraint might be obtained by using Sν insteadof Fν.
•DM-flux correlation•The Sν distribution of cosmological FRBs issimilar to the Euclidean when ρFRB SFR.

•

The correlation between DM and Sν is a clue.
• potentially provides better constraint thanthe logN-logS distributions.

K-correction• Constant Lν as a function of ν is assumed in

the discussion above.
• SED of FRBs is largely degenerated withρFRB.
• Observations in different frequency band areessential.

LF1
ρFRBSFRLν,0 = 1034

Summary•

The large DMs of FRBs suggest they are atcosmological distances.

•Even the single messenger studies can constrain
the nature of FRBs.
••

Many of the current limitations are peculiar to(single dish) radio telescopes.
••

FRBs look really interesting.

Discovery of other messengers will
revolutionize our understanding of FRBs.

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