On modelling the Fast Radio Burst (FRB) population and event rate predictions

0.3 MB·

by  Apurba Bera

MNRAS000,1–11(2016) Preprint22January2016 CompiledusingMNRASLaTEXstylefilev3.0
On modelling the Fast Radio Burst (FRB) population and
event rate predictions
Apurba Bera1⋆, Siddhartha Bhattacharyya1†, Somnath Bharadwaj1,2,
N. D. Ramesh Bhat3,4 and Jayaram N. Chengalur5
1Department of Physics, Indian Institute of Technology, Kharagpur, India
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2Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur, India
1
3Australian Research Council Centre of Excellence for All-Sky Astrophysics (CAASTRO)
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4International Centre for Radio Astronomy Research, Curtin University,Bentley, WA 6102, Australia
2
5National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Pune 411007, India

n
a
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Accepted 19January2016

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ABSTRACT
]
Assuming that Fast Radio Bursts (FRBs) are of extragalactic origin, we have devel-
E
oped a formalism to predict the FRB detection rate and the redshift distribution of
H
the detected events for a telescope with given parameters. We have adopted FRB
h. 110220, for which the emitted pulse energy is estimated to be E0 = 5.4 × 1033J, as
p the reference event. The formalism requires us to assume models for (1) pulse broad-
- eningduetoscatteringintheionizedinter-galacticmedium-weconsidertwodifferent
o
modelsforthis,(2)thefrequencyspectrumofthe emittedpulse-weconsiderapower
tr law model Eν ∝ ν−α with −5 6 α 6 5, and (3) the comoving number density of
s
a the FRB occurrencerate n(E,wi,z)- we ignorethe z dependence andassume a fixed
[ intrinsic pulse width wi = 1ms for all the FRBs. The distribution of the emitted
pulse energy E is modelled through (a) a delta-function where all the FRBs have the
1
sameenergyE =E0,and(b)aSchechterluminosityfunctionwheretheenergieshave
v
a spread around E0. The models are all normalized using the 4 FRBs detected by
0
Thornton et al. (2013). Our model predictions for the Parkes telescope are all con-
1
sistent with the inferred redshift distribution of the fourteen FRBs detected there to
4
5 date. We also find that scattering places an upper limit on the redshift of the FRBs
0 detectableby agiventelescope;forthe Parkestelescopethis isz ∼2.Consideringthe
. upcomingOotyWideFieldArray,wepredictaFRBdetectionrateof∼0.01to∼103
1
per day.
0
6 Key words: Cosmology:observations
1
:
v
i
X
1 INTRODUCTION by a factor of 10 20 in most of the cases. This indi-
r ∼ −
a cates an extragalactic origin of the FRB sources. Notethat
Intherecentpastanewclassofradioburstsofmillisecond-
Loeb et al. (2014) have suggested an alternate interpreta-
duration, called Fast Radio Bursts (FRBs), have been de-
tionwheretheFRBsmaybeofGalacticorigin,howeverwe
tected at the Parkes and Arecibo telescopes (Lorimer et al.
donotconsiderthispossibilityhere.Theobservedfluxden-
2007; Thornton et al. 2013; Spitler et al. 2014). The ob-
sitytogetherwiththeredshift inferred from theextragalac-
served pulses show a dispersion index of 2.000 0.006
− ± ticcontributiontotheDMimplythatanenormousamount
and a scattering index of 4.0 0.4 both of which are
− ± ofenergy( 1031 1033J)isreleasedineachburst.Further,
the signatures of propagation through cold plasma. All of ∼ −
the source size of 100km inferred from the pulse widths
the FRBs barring FRB 010621 have been detected at high ∼
◦ of 1msimplyextremeenvironmentsinthesesources.Un-
Galactic latitudes ( b >5 ) and the large dispersion mea- ∼
sure (DM 400 | 11|00pccm−3) of these pulses exceed fortunately, no counterpart has yet been detected in any
∼ − other part of the electromagnetic spectrum, which makes
theexpectedGalacticcontributionpredictedbytheNE2001
it difficult to determine the physical origin of the FRBs
model (Cordes & Lazio 2002) in the direction of the bursts
(Petroff et al. 2015).
Several models have been proposed for the source
⋆ [email protected] of the FRBs. These include super-massive neutron
† [email protected] stars (Falcke& Rezzolla 2014), binary neutron star
(cid:13)c 2016TheAuthors
2 Bera et al.
FRB
Peakflux Measured
Pulse DM Spectral
FRB density width(ms) Fluence (pccm−3) z index Reference
(Jy) (Jyms)
FRB010621⋆ 0.4 7.8 3.12 746 0.1 − Keaneetal.(2012)
FRB010724 30 4.6 140 375 0.1 −4±1 Lorimeretal.(2007)
FRB011025 0.3 9.4 2.8 790 0.6 − Burke-Spolaor&Bannister(2014)
FRB090625 0.5 4.4 >2.2 900 0.9 − Championetal.(2015)
FRB110220 1.3 5.6 8.0 944 0.8 − Thorntonetal.(2013)
FRB110523 0.6 <6.3 3.8 623 0.5 −7.8 Masuietal.(2015)
FRB110627 0.4 <1.4 0.7 723 0.6 − Thorntonetal.(2013)
FRB110703 0.5 <4.3 1.8 1104 1.0 − Thorntonetal.(2013)
FRB120127 0.5 <1.1 0.6 553 0.5 − Thorntonetal.(2013)
FRB121002 0.4 2.1,3.7 1.5 1629 1.5 − Championetal.(2015)
FRB121102 0.4 3.0 1.2 557 0.3 7to11 Spitleretal.(2014)
FRB130626 0.5 3.2 >1.5 952 0.9 − Championetal.(2015)
FRB130628 0.9 1.4 >1.2 470 0.4 − Championetal.(2015)
FRB130729 0.1 23.4 >3.5 861 0.8 − Championetal.(2015)
FRB131104 1.1 <0.6 0.6 779 0.6 0.3±0.9 Ravietal.(2015)
FRB140514 0.5 2.8 1.3 563 0.4 − Petroffetal.(2015)
Table 1. Reported FRBs to date: This list contains a total of sixteen events among which FRB121102 and FRB110523 were
detectedatAreciboandGreenBankTelescoperespectivelywhiletheremainingfourteenwerealldetectedatParkes.TheFRBparameters,
includingtheredshiftsaretakenfromthepublishedreferenceslistedintheTable.Championetal.(2015)havenotmentionedtheredshift
but provide DMMW fromwhich we have estimated z using eq. (10). Note that none of the FRBs have adirect redshiftmeasurement,
and the z values inthis Table have all been inferredfromthe observed dispersionmeasure. Bannister&Madsen (2014)have proposed
thatFRB010621⋆ probablyhasagalacticorigininwhichcasetheestimatedredshiftisnotmeaningful.However,inourworkweassume
anextragalactic originforthisFRBandtaketheestimatedredshifttobecorrect.
mergers (Totani 2013), binary white dwarf mergers placed nearly end-to-end along the focal line of the cylin-
(Kashiyama et al. 2013), flaring stars (Loeb et al. 2014), drical reflector. Since the dipoles are all oriented along the
pulsarcompanions (Mottez & Zarka2014) andmany more. same direction, the telescope is sensitive to only a single
However,wearestillfarfromhavingenoughinformationto linearpolarization component.TheORTcurrentlyoperates
validate any of these models. as a single antenna which combines the signal from all the
Fourteen outof thesixteenFRBsso far (Table1) were 1056 dipoles. We refer to this as the ORT Legacy system
detected with the Parkes telescope, nine of which were dis- (LS). Work is currently in progress to upgrade the ORT
coveredbytheHighTimeResolutionUniverse(HTRU)sur- so that it is possible to combine different numbers (N ) of
d
vey. The Parkes radio telescope is a fully steerable single successivedipolestoformmany(N )smallerindividualan-
A
dishtelescope of64mdiameter. TheParkesmulti-beam re- tennaewhichcanfunctionsasalinearinterferometricarray,
ceiver has 13 independent beams each with a narrow field theOotyWideFieldArray(OWFA;Prasad & Subrahmnya
of view (FoV) of 14.4′ (HPBW). The combination of low (2011a,b); Subrahmnya,Manoharan & Chengalur (2015)).
systemnoise(27K)andalargegain(G=0.74K/Jy)ofthe At completion we expect to have OWFA Phase I (PI),
primarybeammakesitoneofthemostsensitivesingledish OWFAPhaseII(PII)andtheLS,allofwhichcanfunction
telescopes currently in operation. The HTRU survey uses inparallel,andforwhichafewrelevantparametersaresum-
L-band receivers operating at 1.4GHz with a bandwidth marized in Table 2. The large field of view and reasonably
∼
of 400MHz. high sensitivity makesall threeversions of theORT(LS,PI
All the FRBs (except FRB 110523), to date (Table 1), and PII) very promising instruments for detecting FRBs.
have been detected in the L-band ( 1 2GHz) and their The PII, in particular, has a FoV that is 880 times larger
∼ −
emissionspectrumisnotconstrained,thoughitappearsthat thanthatofParkeswhilethesystemnoiseisonlyfivetimes
they may be consistent with a flat spectrum. Detections larger.Whilethetwoinstrumentsworkatdifferentfrequen-
with telescopes operating at lower frequencies will place cies, this comparison gives an idea of the tremendous po-
strong constraints on the spectrum which in turn will yield tentialof detecting alarge numberof FRBs. Thetwo other
very important constraints on models for the origin of the versions (LS and PI) will probe smaller FoVs with deeper
FRBs.Currentlyallthatisavailableareconstraintsonspec- sensitivity. We expect the three versions together provide
tral indices and event rates of FRBs from non-detections veryinterestingandusefulinputsastotheFRBpopulation
in observations at lower frequencies (Coenen et al. 2014; and theorigin of theFRBs.
Karastergiou et al. 2015).
In this paper we assume the FRBs to be of cosmolog-
TheOotyRadioTelescope(ORT1)isaparaboliccylin-
ical origin, and set up a general framework for predicting
dricalreflectorofdimensions530m 30mwhichoperatesat
× thedetectionrateforatelescopewithgivenparameters.As
anominalfrequencyofν =326.5MHzwithabandwidthof
o mentionedearlier,verylittleisknownabouttheFRBsandit
4MHz.Ithasalineararrayof1056 half-wavelengthdipoles
is necessary to make several assumptions to make progress.
To this end, we introduce a power law model for the spec-
tral energy density of a FRB and calculate the fluence and
1 http://rac.ncra.tifr.res.in/ pulse width that will be observed accounting for the vari-
MNRAS000,1–11(2016)
Modelling FRB Population & event rate predictions 3
Parameter ORTLegacy OWFAPhaseI OWFAPhaseII
Numberofdipoles(Nd) 1056 24 4
Numberofantennae(NA) 1 40 264
Aperturedimensions(b×d) 530m× 30m 11.5m× 30m 1.9m× 30m
Fieldofview 0.1◦× 1.75◦ 4.6◦×1.75◦ 27.4◦× 1.75◦
AngularResolution 0.1◦× 1.75◦ 7′×1.75◦ 6.3′× 1.75◦
BandwidthB inMHz 4 18 30
Spectral Resolution(∆νc)inkHz 125 24 48
[∆S]1ms inJy 0.343 1.179 2.151
Table2.ThisshowsthesystemparametersfortheORTLegacysystemandtheupcomingPhaseIandPhaseIIofOWFA.Theaperture
efficiency (η) is approximately 0.6, the system temperature (Tsys) is 150K for all the three systems and ∆S(1ms) is the 1 σ noise
for incoherent addition of the antenna signals with integration time of 1ms. For reference, the Parkes telescope has a FoV (HPBW)
of 0.23◦×0.23◦ and [∆S]1ms =0.05Jy in the L-band. We note that it is necessary to do offline beam forming to achieve the angular
resolutionquotedhereforOWFAPhasesIandII.Forthispaperwehaveonlyconsideredincoherentadditionwheretheangularresolution
isthesameasthefieldofviewlistedhere.
ous propagation effects including dispersion and scattering asthereisnoestimate forthehost contribution.Itispossi-
in the inter-galactic medium (IGM). We use this to deter- ble to avoid this last uncertainty to some extent by setting
mineaFRBdetectioncriteria.Itisalsonecessarytospecify thehost contribution tozerowherebywe mayinterpret the
the comoving number density of the FRB occurrence rate inferred redshift as an upper limit to the actual redshift of
n(E,w ,z) as a function of thepulse energy E, its intrinsic theFRB(Keane & Petroff 2015).Thevariousuncertainties
i
widthw ,andtheredshiftz.Wehaveconsideredtwosimple intheFRBmodelsadoptedlaterinthispaperfaroutweigh
i
models,bothofwhichassumen(E,w ,z)tobeindependent theuncertaintiesintheinferredredshifts,andforthepresent
i
ofz overtherelevantredshiftrange.Themodelsareallnor- work we haveadopted thevalues quoted in Table 1
malized to the FRB detection rate observed at the Parkes We have used (Ω ,Ω ,Ω ,h) = (0.32,0.68,0.04,0.7)
m Λ b
telescope.Finally,weusetheentireframeworktomakepre- for thecosmological parameters (Spergel et al. 2003).
dictions for the FRB detection rate for the three different
versions of the ORT (LS, PI and PII) which, in principle,
can work commensally.
2 BASIC FORMALISM
Abrief outlineof thepaperfollows. Theframework for
calculatingthedetectionrateispresentedinSection2.Sec- We assume that the spectral energy density Eν emitted by
tion 3 presents themodels for theFRB population, and we an FRB can beexpressed as
present the detection rates predicted for the three versions
E =Eφ(ν) (1)
ofORTinSection4.Thesummaryandconclusionsarepre- ν
sented in Section 5. whereφ(ν)istheemission profile.Asmentionedearlier, we
FRB 110220 detected by Thornton et al. (2013) is the haveusedFRB110220asthefiducialeventforouranalysis.
second brightest event observed so far (after the so-called ThisFRBwasinferred tohavearedshift z=0.8, forwhich
Lorimerburst;Lorimer et al.(2007)),anditisthebestchar- theParkes observational frequency band from 1182MHz to
acterized FRB at present. We have adopted FRB 110220 1582MHz corresponds to the frequencies ν = 2128MHz
a
as the reference event for our entire analysis. FRB 110220 and ν = 2848MHz respectively in the rest frame of the
b
wasdetectedinbeam 03oftheParkesmulti-beamreceiver, source.
howeveritsexactpositionrelativetothebeamcenterisnot We have used the frequency interval from ν =
a
known. For our analysis we have made the conservative as- 2128MHz to ν =2848MHz to normalize the emission line
b
sumptionthatitislocatedclosetothebeamcentre,i.e.the profile of all theFRBs such that
intrinsicfluenceisalmost equaltotheobservedfluence.For
νb
our calculations we assume that the Parkes has a Gaussian φ(ν)dν =1. (2)
Z
beam shape. νa
It is improtant to note that currently no FRB has Here E (eq. 1) is the energy emitted by the FRB in the
an independent redshift measurement, and all the redshifts frequency interval ν to ν , and we henceforth refer to E
a b
quoted in Table 1 have been inferred from the measured simply as the “energy” emitted by the FRB. For refer-
DM which is assumed to be a sum of three components. ence, the energy emitted by FRB 110220 is estimated to
TheNE2001model(Cordes & Lazio2002)givesanestimate be E = 5.4 1033J which we use as the fiducial value of
0
×
of theMilky Way contribution in thedirection of theFRB. E throughout this paper.
The contribution from the FRB host galaxy is unknown, WenowconsiderobservationsofanFRBofenergyEat
anddifferentauthorshaveuseddifferentvaluesforthis.The redshiftz.Thenumberofphotonsemittedinthefrequency
residualDM,afteraccountingforthesetwocomponents,is interval dν centred around ν in the rest frame of the
src src
attributed to an uniform, completely ionized IGM and this source is given by
isusedtoinfertheFRB’sredshift.BoththeMilkyWayISM
Eφ(ν )dν
and the IGM are clumpy and turbulent, and the respective dN = src src (3)
photon h ν
DM contributionsalongtheactuallineofsighttotheFRB p src
will differ from the model prediction used to infer the red- where h is the Planck constant. The same numberof pho-
p
shift. There is further uncertainity in the inferred redshift tons will be received in the frequency interval dν =(1+
obs
MNRAS000,1–11(2016)
4 Bera et al.
z)−1dν centredaroundthefrequencyν =(1+z)−1ν thefrequencyinthedenominatorofeq.(9)fixedatthevalue
src obs src
attheobserver.ThefluenceF observedinthisfrequency ν insteadvaryingitfromchanneltochannelwillintroduce
νobs 0
intervalis afewpercent errorinthecase ofbroadbandobservations.
dN h ν As mentioned earlier, the total line of sight DM has
Fνobs = 4pπhort2odnνp obs. (4) roughlythreecontributionsrespectivelyoriginatingfromthe
obs
MilkyWay(DM ),theInterGalacticMedium(DM )
MW IGM
where r is the comoving distance corresponding to redshift
and the host galaxy (DM ) of the source, and we can
Host
z. The usual unit of comoving distance is Mpc. Using eqs.
write
(3) and (4), we have
DM =DM +DM +DM . (10)
Eφ(ν (1+z)) MW IGM Host
F = obs . (5)
νobs 4πr2 We can estimate the electron density along different
lines of sight in the Milky Way by from the NE2001 model
Wenowintroducetheassumptionthattheobservations
arebeingcarriedoutusingatelescopewithanobservational (Cordes & Lazio 2002) and use this to calculate DMMW
along the line of sight to the FRB. We use DM =
frequency band from ν to ν . In thiscontext it is useful to MW
introducethe average l1ine pr2ofile φ(z) defined as 60pccm−3asarepresentativevaluefordirectionsawayfrom
the Galactic plane (b> 5). For the host galaxy, we assume
1 ν2(1+z) that it is similar to the Milky Way with the difference that
φ(z)= φ(ν)dν. (6)
(1+z)(ν2−ν1)Zν1(1+z) we have no idea of the position of the FRB relative to the
disk of the host galaxy and we have to allow for the pos-
Further,wealsoassumethattheFRBislocatedatanangle
sibility that the FRB signal reaches us through the disk of
~θ relative to the telescope’s beam center, and use B(~θ) to
thehostgalaxy.Wethereforeexpectthatontheaveragethe
denote the normalized beam pattern. The fluence that will
FRB signal will traverse a larger distance through the ISM
beobserved by thistelescope is given by
of the host galaxy as compared to the distance it traverses
Eφ(z)B(~θ) through the Milky Way, and we use a slightly larger value
F = . (7) DM =100/(1+z)pccm−3forthehostgalaxycontribu-
4πr2 Host
tion. The (1+z) factor here arises due to the cosmological
expansion. We estimate the IGM contribution using (Ioka
2.1 Pulse Broadening 2003)
An electromagnetic pulse from cosmological distances gets 3cH Ω z (1+z′)dz′
broadened by three factors - cosmic expansion, dispersion DMIGM(z)= 8πG0mPb Z0 Ωm(1+z′)3+ΩΛ (11)
andscattering,thelatertwoduetopropagationthroughthe
p
ionized Inter Stellar Medium (ISM) and the Inter Galactic where mP is the proton mass and the other symbols have
Medium(IGM).Thecosmicexpansionsimplybroadensthe theusual interpretation.
pulsebyafactorof(1+z).Theobservedpulsewidthw for Multipath propagation through the ionized IGM and
an extragalactic event with and intrinsic pulse width of w the ISM of both the host galaxy and the Milky Way cause
i
is given by scatter broadening wsc of the pulse. We expect this to pre-
dominantly arise from the IGM due to a geometrical effect
w= w2 +w2 +w2 (8) known as thelever-arm effect (Vandenberg1976). The the-
cos DM sc
q
oryofscatterbroadeningintheionizedIGMisnotwellun-
where w =(1+z)w , w and w are respectively the
cos i DM sc derstood at present,and we consider two scattering models
contributions to the total pulse width from the cosmologi-
to calculate thepulse broadening.
callybroadenedintrinsicpulsewidth,theresidualdispersion
across a single frequency channel and scattering in the in- (i) Scattering Model I is based on the empirical fit
terveningmedium.
log w =C +0.15 log DM
The frequency dependent refractive index of the ion- sc 0 IGM (12)
izedcomponentsoftheISMandIGMcausesdispersionofa +1.1(log DMIGM)2 3.9 log ν0
−
pulse propagating through it. This dispersion, which has a
ν−2dependence,spreadstheobservedpulseoveralargetime with C0 = −6.46 given by Bhat et al. (2004) for pulsars in
the ISM of our Galaxy. We have used C = 3.2 to rescale
intervalacrosstheentireobservationalfrequencybandwidth 0
thisfor scattering in theIGM.This valueof C is based on
B. The signal is dedispersed by applying appropriate time 0
theassumptionthatthereferenceeventFRB110220hasan
delays to synchronize the pulse at all the frequency chan-
intrinsic pulse width of w =1ms. Eq. (9) predicts w =
nels across the band. However, it is not possible to correct i DM
0.17msfor z=0.8, andwe haveset thevalueof C sothat
for the dispersion within a single frequency channel width 0
eq. (12) gives w =5.3ms required to match the observed
∆ν .Thisintroducesaresidualdispersionbroadeningw sc
c DM pulsewidth w=5.6ms (Table1).Note that in eq.( 12), we
wushinicgh,undertheassumption∆νc/ν0 ≪1,canbecalculated use ν0 in MHz, wsc in ms, DM in pccm−3 and log denotes
log .
10
w 8.3 106 DM∆νc ms (9) (ii) Scattering Model II is based on the tempo-
DM ≈ × ν3 ral smearing equation for IGM turbulence given by
0
Macquart & Koay (2013)
where ν is the central frequency of the observation ex-
0
pprreesssseedd iinnpMcHcmz−a3n.dNtohtee tdhisaptetrhsieonfacmtetahsautrewe(DarMe)hoilsdienxg- wsc(z)= νk4ZscL Z0zDH(z′)dz′Z0z(1+z′)3DH(z′)dz′ (13)
MNRAS000,1–11(2016)
Modelling FRB Population & event rate predictions 5
that both the scattering models are normalized using FRB
102
110220 assuming w = 1ms for this event, and therefore
i
both the scattering models predict w = 5.6ms at z = 0.8.
PulseBroadening
We see that the residual dispersion measure w makes a
DM
very insignificant contribution to w at all redshifts. Scatter
broadeningis not veryimportant at low redshifts where we
havew w .Thetotalpulsewidthisdominatedbyscat-
101 ≈ cos
) terbroadeningatlargeredshifts.ForScatteringModelIthe
s
m
dth( itnoctraelapsueslseshwairdptlhy fisordzom>in0a.5te.dFobrySwcsactIteartinzg≈M0o.d5e,laInIdwwscsIcII
wi starts making a significant contribution to w at z 0.2,
≈
e however it dominates the total w only for z > 0.4. Unlike
s
ul ScatteringModelI,w increasesrelativelygraduallywith
P100 wDM z. For both the scattescrIinIg models, the total pulse width is
wcos considerably in excess of w for FRBs with z > 0.5. We
cos
wscI haverepeated the entire exercise for w =0.5 and 2ms, i.e.
i
wscII the intrinsic pulse width of the observed FRB and the nor-
wI malization of the scattering model are both changed. The
10−1 wII results, which are very similar to those shown in Figure (1)
for w = 1ms, are not shown here. We find that the qual-
0.1 0.2 0.5 1 2 4 i
itative features of the total pulse width as a function of z
Redshift(z)
are not very different if we change the value of w . Scat-
i
ter broadening starts to dominate at z 0.5, and the total
Figure 1.Thepredicted total pulsewidthassuminganFRB of ≈
pulsewidth is considerably larger than w for z >0.5.
intrinsic pulse width wi = 1ms located at redshift z observed cos
bytheParkestelescope. ThesubscriptsI andII denotethetwo
different scattering models referred to in the text. The different
2.2 Detection criteria and detection rate
components which contribute to the total pulse width are also
shownindividually. Considering an FRB of energy E and intrinsic pulse width
w located at redshift z, we have till now discussed how to
i
calculatethefluenceF (eq.7)andthepulsewidthw(eq.8)
where
that will be observed by a given telescope. We now discuss
DH(z)=(Ωm(1+z)3+ΩΛ)−1/2, (14) the criteria for this particular FRB to be detected by the
−1 given telescope.
Z =(1+z)2 (1+z) z(1+z) (15)
L h −p i Thedetectioncriteria isdecidedbythetelescope’s sys-
tem noise. The r.m.s. flux density fluctuation ∆S is given
andweusek =8.5 1013msMHz4 forthenormalization
sc × by
constant. As with Scattering Model I, the value of the nor-
malization constant is set to reproduce the observed pulse T 1ms
∆S = sys [∆S] (16)
widthofFRB110220assumingthatithasanintrinsicpulse G ∆tBN ≡r w × 1ms
pol
width w =1ms.
i p
For both the scattering models that we have considered where G is the antenna gain of the primary beam, Tsys is
here,thevalueofthenormalizationconstantwouldchangeif the telescope’s system temperature, Npol is the number of
weassumeadifferentintrinsicpulsewidthforFRB110220. polarizationsthetelescopedetectsand∆tistheintegration
Wehavetried out w =0.5 and 2msin orderto assess how time. We assume an integration time equal to the observed
i
this would affect theresults of ouranalysis. pulse width w. Since the observed FRB pulse widths are
Note that Scattering Model I is based on an empirical of the order of a few milliseconds, it is then convenient to
fit which is observationally well established within the ISM express∆S (eq.16)intermsofw and[∆S]1ms whichisthe
of our Galaxy. Given the high-DM part of the model is r.m.s. noise for ∆t=1ms.
largelyconstrainedbymeasurementsofpulsarsintheGalac- An FRB with average observed flux density S = F/w
ticplane,itiseffectivelyarepresentation ofturbulenceand will result in a detection if
clumpiness in the ISM. The nature of turbulence may be S F
= >n, (17)
differentforIGMandhenceitisnotclearwhetherthesame ∆S w∆S
fit can be rescaled to correctly quantify IGM scattering. In
where n is theminimum signal to noise ratio required for a
contrast,ScatteringModelIIisbasedentirelyonatheoreti-
detection. The same criteria can be expressed in terms of a
calmodelfortheIGMscattering.Thismodel,however,has
limiting fluenceF =n (1ms) [∆S] as
not been observationally verified. Given our present lack of l × × 1ms
knowledge,wehaveusedthetwodifferentscatteringmodels 1ms
F >F (18)
to estimate the possible impact on the pulsewidth. ×r w l
Figure (1) shows the total pulse width w (eq. 8) cor- Thornton et al. (2013) have only considered events with a
responding to an FRB of intrinsic pulse width w = 1ms signal to noise ratio greater than nine as a detection, fol-
i
located at a redshift z observed by the Parkes telescope lowing these authors we have used n = 9 for the Parkes
for which ν = 1382MHz and ∆ν = 390kHz. Recollect telescope.
0 c
MNRAS000,1–11(2016)
6 Bera et al.
102
ScatteringModelI ScatteringModelII
101
α=1.4 α=1.4
0 100
E
/
n
mi
E10−1
10−2 wi=0.5ms
wi=1.0ms
wi=2.0ms
10−3
102
ScatteringModelI ScatteringModelII
101
wi=1ms wi=1ms
0 100
E
/
n
Emi10−1
α=−5
α=−2
10−2 α=0
α=1.4
α=5.0
10−3
0.1 0.2 0.5 1 2 4 0.1 0.2 0.5 1 2 4
Redshift(z) Redshift(z)
Figure 2. The minimum energy Emin for an FRB at redshift z to be detected at the Parkes telescope assuming that the source is
locatedatthebeamcentre.TheleftandrightpanelsareforScatteringModelsIandIIrespectively.Theupperpanelsconsiderdifferent
values of the intrinsic pulse widths wi with spectral index α = 1.4 fixed, while the lower panels consider different values of α with
wi=1msfixed.
The detection criteria (eq. 17) combined with (eq. 7) numberof eventsdetected (N ) is expected to be
det
implies a minimum energy
dr r2
N (T)=T dz
E = 4πr2Fl w (19) det Z dz (cid:18)1+∞z(cid:19) (21)
min φ(z)B(~θ)r1ms dΩ dw dE n(E,w ,z).
i i
Z Z Z
Emin(z)
for a telescope to detect an FRB at a redshift z and a sky We use eq. (21) to predict the FRB detection rate for any
position (~θ) relative to the telescope’s beam centre. A tele- given telescope. Here it is assumed that thetelescope has a
scope’s primary beam pattern B(~θ), the antenna gain G of sampling time 6 1ms so as to be able to resolve the FRB.
theprimarybeamandsystemtemperatureTsyswill,ingen- The factor of (1+z) in the denominator arises from the
eral,varyasthetelescopeispointedtodifferentpartsofthe fact that a time interval of T in the observer’s frame corre-
sky. To simplify the analysis, we have assumed these tele- sponds to a time interval of T/(1+z) in the source frame.
scope parameters to beconstant. The quantity within the square brackets gives the redshift
The number of FRB events at redshift z per unit time distribution of the detected events. We may interpret the
(inthesourceframe)perunitcomovingvolumewithenergy latterasthedetectionratewiththeFRBsourceoriginating
in the range E to E+dE and intrinsic pulse width in the in the redshift intervalz to z+dz.
range w and w +dw can be expressed as
i i i
dN =n(E,w ,z)dEdw . (20)
i i
3 MODELLING THE FRB POPULATION
where n(E,wi,z) is the comoving number density of the The basic formalism introduced in the previous section re-
FRB occurrence rate. quirestheFRBemission lineprofileφ(ν)andthecomoving
For an observation time T with a given telescope, the number density of the FRB occurrence rate n(E,w ,z) as
i
MNRAS000,1–11(2016)
Modelling FRB Population & event rate predictions 7
inputs in order to predict the FRB detection rate for any redshifts,apredictionthatisinconsistent withtheobserva-
given telescope. With only sixteen FRB events detected to tions summarized in Table 1. A similar problem also arises
date, we do not as yet have any established models and for forα= 2,howeveritisnotassevereasforα= 5.Based
− −
our work we assume very simple models for for these two onthesefindings,wehaverestrictedthesubsequentanalysis
quantities. Wediscuss these models below. to αvalues in therange 26α65.
−
The spectrum of the FRB emission is very poorly con- We now shift our attention to the comoving number
strained at present, the detection so far being all (except density of the FRB occurrence rate n(E,w ,z). Since the
i
FRB 110523) in the L-band. The high observed flux sug- cut-offredshiftz forFRBdetection,atleastfortheParkes
c
geststhat theemission mechanism iscoherent forwhich we telescope, does not depend on the intrinsic pulse width w
i
expect a negative spectral index. Here we have assumed a (Figure 2) we assume that all the FRBs have the same in-
simple power law spectrum S ν−α with a spectral index trinsic pulse width of w = 1ms. Further, since we expect
ν i
∝
α vary from 5 to +5 for which we have the normalized all the detected FRBs to be within z 6 2, as assume that
− −
(eq.2) emission profile n(E,w ,z)isconstantoverthelimitedredshiftrangeofour
i
interest. The function n(E,w ,z) is now just a function of
φ(ν)= 1−α ν−α (22) E, and we have considered tiwo simple models for the E
(cid:20)ν1−α ν1−α(cid:21)
b − a dependence.
and the average line profile has theform
(i) Thedelta-functionmodelwherealltheFRBsemit
φ(z)=K(α)(1+z)−α (23) thesame energy E and
0
where K(α) is n(E,w ,z)=n δ(E E ). (26)
i 0 0
−
1 ν1−α ν1−α (ii) TheSchechterluminosityfunctionmodelwhere
K(α)= 2 − 1 . (24)
ν2−ν1 (cid:20)νb1−α−νa1−α(cid:21) theFRBenergieshaveaspread,theenergydistributionbe-
ing given by theSchechterfunction
Althoughnon-detectioninsearchesatdifferentwavelengths
giveconstraintsonthespectralindex,weconsidertherange n E γ E
n(E,w ,z)= 0 exp . (27)
−56α65 for completeness. i E0 (cid:18)E0(cid:19) (cid:18)−E0(cid:19)
The minimum energy E (eq. 19) can now be ex-
min We consider both positive and negative values of the expo-
pressed as
nent γ. The negative values of γ require a lower cut-off to
E = 4πr2Fl (1+z)α w . (25) make the distribution normalizable. We have considered a
min B(~θ) (cid:20) K(α) (cid:21) r1ms cut-off energy of E0/100 for ouranalysis.
Figure 2 shows E as a function of z assuming the We have used the FRBs observed by the Parkes tele-
min
FRB to be located at the center of one of the beams of the scope to determine the value of the normalization con-
Parkes telescope (B(θ) = 1). We expect the FRBs to typi- stant n which is a free parameter in both the models for
0
cally have an energy E E , and we have shown E in n(E,w ,z). Though there are fourteen FRBs detected at
0 min i
∼
unitsofthereferenceenergyE .Theupperpanelsshowthe the Parkes telescope, it is not possible to use all of them
0
results for threedifferent valuesof theintrinsic pulsewidth to calculate an event rate because the exact duration of
w with α = 1.4 fixed, while the bottom panels shows the the observation is not known. The four FRBs detected by
i
results for three different values of α with w = 1ms fixed. Thornton et al. (2013) correspond to an effective observa-
i
We do not find a very big difference if the value of w is tion time of 298 days with a single beam of the Parkes
i
changed, however the results vary considerably if the value telescope, and we have used the inferred detection rate
of α is changed. We first discuss only the positive values of along with eq. (21) to determine the valueof n . Note that
0
α(> 0). For Scattering Model I, we find that the value of Champion et al.(2015)haveestimatedaslightlylowerFRB
E increases sharply for z > 1 due to the steep increase occuranceratethanThornton et al.(2013)buttheyarecon-
min
in the pulse width (Figure 1). The value of E increases sistent with each other within 1-σ uncertainities.
min
moregraduallyinScatteringModelII.Inallcasesthemain As noted earlier, the FRB distribution predicted for
feature is that E increases with redshift and exceedsE Parkesextendstoarbitrarilylargeredshiftsfornegativeval-
min 0
intherange16z 62.AssumingthattheFRBshaveenergy uesofα. Wesee thisin thetopmost right panelofFigure 3
E E , this imposes a cut-off redshift z such that obser- (α = 2 and Scattering Model II) where the FRB predic-
0 c
∼ −
vationwith theParkestelescopeareonly sensitivetoFRBs tions do not fall of even at z >4. This is even more severe
withz6z .Weseethatthevalueofz islargelyinsensitive for α = 5 which has not been shown here. This poses a
c c
−
to w , however it shifts to smaller z if α is increased from problem for thez integral in eq.(21), andit isnecessary to
I
0 to 5. In all cases we find 1 6 z 6 2 for the Parkes tele- assumeanupperlimittoobtainafiniteprediction.Wehave
c
scope. Forα>0, ourmodels predict that we donot expect assumed an upper limit of z = 5 for calculating n . While
0
the Parkes telescope to detect FRBs with z >2, consistent the upper limit z = 5 has been introduced here for math-
with the observations summarized in Table 1. Next, con- ematical convenience, we may interpret this as the redshift
sidering the negative values of α we find that the results beyond which the FRB population ends abruptly. Finally,
are quite different from those for α > 0. The difference is we note that the non-detection of any FRBs in the search
particularly pronounced for Scattering Model II where we byRaneet al.(2015)seem toindicatethattheFRBrateis
see that the value of E decreases with z for α= 5. In possibly a factor of 3 to 5 times smaller than that inferred
min
−
this case we do not have a cut-off redshift and we expect from these four Parkes FRBs.
theParkes telescope to detect FRBs out to arbitrarily high Redshift estimates are available for all the fourteen
MNRAS000,1–11(2016)
8 Bera et al.
0.6 ScatteringModelI Scattering ModelII
α=−2 α=−2
N 0.4
/
N
∆
0.2
0
0.6 ScatteringModelI Scattering ModelII
α=0 α=0
N 0.4
/
N
∆
0.2
0
0.6 ScatteringModelI Scattering ModelII
α=1.4 α=1.4
N 0.4
/
N
∆
0.2
0
0.6 ScatteringModelI Scattering ModelII
α=5 α=5
N 0.4
/
N
∆
0.2
0
0 0.6 1.2 1.8 2.4 3 3.6 4.2 0 0.6 1.2 1.8 2.4 3 3.6 4.2
Redshift(z) Redshift(z)
Populationmodels
Deltafunction Schecter functionwithγ=0
Schecter functionwithγ=−2 Schecter functionwithγ=1
Schecter functionwithγ=−1 Schecter functionwithγ=2
Figure 3. The data points show the redshiftdistribution ∆N/N of the fourteen FRBs detected at the Parkes telescope, the data has
been binned with ∆z = 0.3, ∆N is the number of FRBs in each bin and N is the total number of FRBs. The error bars show the
1−σPoissonerrorsforthedata.Thetheoreticalpredictionsforthedifferentmodelsareshownascontinuouscurves.Thesecurvesshow
N−1(dN/dz)normalizedsothatthetotal areaissameforall.Unfortunately, thenumberofFRBswhichhavebeendetected todateis
toosmalltoplacemeaningfulconstraints onthemodelswhichwehaveconsidered.
MNRAS000,1–11(2016)
Modelling FRB Population & event rate predictions 9
FRBs detected by the Parkes telescope (Table 1). The red- wehaveusedtheflat-skyapproximation,and(θ ,θ )arethe
x y
shiftdistributionofthedetectedFRBsprovidesanindepen- componentsofthevector~θontheplaneofthesky.Wenote
dent constraint on any model for the FRB population. We that the flat sky approximation does not hold for Phase II
nowcomputetheredshiftdistributionpredictedbyourmod- whichhasaverylargefieldofview,howeverthisisjustified
els ( eq. 21) and compare these with the observed redshift by the fact that the error introduced by this assumption is
distributionofthefourteenParkesFRBs.Sixofthefourteen small compared to the other uncertainties in our modelling
ParkesFRBsareintheredshiftrange0.46z 60.6.Wesee of thescattering and theFRBpopulation.
that most of the models also predict redshift distributions Our predictions for the FRB detection rate are shown
which peak around the same z range. Considering first the in Figure 4. We have taken minimum signal to noise ratio
delta-function modelwherealltheFRBshavethesameen- n=10 for these predictions. We see that the predicted de-
ergyE0,weseethattheredshiftdistributionextendsoutto tection rate is highest for PII which has the largest field of
largerredshiftsinScatteringModelIIascomparedtoScat- viewandthelargestfrequencybandwidth.Weexpecttode-
teringModelI.Theredshiftdistributionalsoextendsoutto tectsomewherebetween 0.01to1,000FRBsperdaywith
∼
larger redshifts if the value of α is decreased. Both of these PII, depending on the spectral index of the FRB emission.
effects can be understood in terms of the cut-off redshift Wehaveahigherdetectionrateforlargerpositivevaluesof
zc introduced while discussing Figure 2. The Schechter lu- α,thedetectionratesarealsohigherforScatteringModelI
minosity function introduces a spread in the FRB energies. ascompared toModel II.Thepredicted detection rates fall
Therelativeabundanceoflow energy FRBsincreases if the by roughly an order of magnitude for PI, and roughly two
value of γ is reduced, and we see that the entire redshift ordersofmagnitudeforLSascomparedtoPII.Considering
distributionshiftstolowerredshiftsfornegativevaluesofγ. α=1.4whichhasbeenproposedtobethemostlikelyvalue
We see that most of the models considered here are forthecoherentFRBemission(Lorimer et al.2013),weex-
roughly consistent with the redshift distribution of the ob- pecttodetect 10to 100FRBsperdaywithPII.Thisis
∼ ∼
servedFRBs(Figure3).Asmentionedearlier,thepredicted quiteencouragingevenifwetakeintoaccountthefactthat
FRB distribution extends beyond z = 4 for α = 2 and Raneet al. (2015) havesuggested thattheFRBoccurrence
−
Scattering Model II whereas theobserved redshift distribu- ratesestimated from theFRBsdetected byThornton et al.
tionfalls offwellwithinz=1.5.Unfortunately,thenumber (2013)maybeafactorof3to5largerthantheactualFRB
of FRBs which have been detected to date is too small to occurrence rate.
place meaningful constraints on the models which we have
considered. However, we anticipate that larger numbers of
FRBs will be detected in future and it will be possible to
5 SUMMARY & CONCLUSION
constrain both the scattering models as well as the models
fortheFRBpopulationusingtheredshiftdistributionofthe The source of the FRBs is still largely unknown. Assuming
detected events. theFRBstobeofextragalactic origin,wehavedevelopeda
formalismtopredicttheFRBdetectionrate(eq.21)andthe
redshift distribution of the detected events for a telescope
with given parameters. We have adopted FRB 110220 (Ta-
4 PREDICTIONS FOR OWFA
ble 1) as the reference event for our entire analysis. None
Wenowusetheformalismandthevariousmodelspresented of the FRBs detected to date have a direct redshift mea-
in the earlier parts of this paper to study the prospects of surement,andtheredshiftsofallthedetectedFRBs(Table
detectingFRBswiththeORTLegacy System(LS)andthe 1) have been inferred from the observed DMs. The value
two different phases of OFWA (PI and PII). In Phase I, oftheinferred redshift dependson theMilkyWayandhost
the signal from Nd =24 successive dipoles are combined to galaxyDM contributionassumedintheanalysis,anddiffer-
formanindividualantenna,andthereareatotalofNA =40 ent authors have assumed different values. For our analysis
suchantennas.WehaveNd =4andNA =264forPhaseII. we have adopted the inferred z values from the references
The aperture dimensions and other details are tabulated in listed in Table 1. In contrast, the redshift z in our analytic
Table 2. calculations and in Figures 1, 2 and 3 refer to the actual
EachphaseofOWFAhasNAantennaswhichcanbeop- cosmological redshiftoftheFRBwhichisunaffectedbyour
eratedtogetherasalinearradio-interferometricarray.How- assumptionsfor DM and DM .Theassumed values
MW Host
ever,forthepresentanalysisweconsiderasimplersituation onlyaffecttheobservedpulsewidthw througheqs.(8),(9)
where the signals from the NA antennas are incoherently and(10).Further,theDM makesasubdominantcontribu-
added. A more detailed analysis using the full beam form- tion to the total pulse width w for the entire range consid-
ing capability of OWFA will be presented in a later paper. ered here (Figure 1), and consequently our predictions are
Consequently,the field of view is thesame as that of a sin- largely unaffected byDM and DM .
MW Host
gle antenna (given in Table 2) but the r.m.s. flux density The FRB pulse width plays an important role in de-
fluctuation is reduced to [∆S]1ms/√NA. The ORT LS and termining the detection rates. At present we lack adequate
OWFAPIandPIIallhaveanisotropicbeampatternswhich understanding of pulse broadening due to scattering in the
we haveparametrized as ionized IGM, and we consider two different alternatives to
modelthis.ScatteringModel Iis based onan observational
πdθ πbθ
B(~θ)=sinc2 x sinc2 y (28) fitgiven byBhat et al. (2004) for pulsars in theISMof our
(cid:18) λ (cid:19) (cid:18) λ (cid:19)
Galaxy, we have extrapolated this for FRB pulse broaden-
(Ali & Bharadwaj 2014) where we have the antenna aper- ingintheIGM.Incontrast,ScatteringModelIIisbasedon
tureb d(Table2)andλistheobservingwavelength.Here atheoreticalcalculationgivenbyMacquart & Koay(2013),
×
MNRAS000,1–11(2016)
10 Bera et al.
104
103 ORTLS OWFAPI OWFAPII
y
da 102
r
pe 101
s
n
o 100
ti
c
te10−1
e
D
10−2
10−3
-2 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5
α α α
Figure 4.TheexpectedFRBdetectionratesasafunctionofαforORTLS,andOWFAPIandPIIassuming1024frequencychannels
spanningthebandwidthBgiveninTable2.ThegrayshadedandhatchedregionscorrespondtoScatteringModelIandIIrespectively,
thesolidcurvespassingthroughtheseregionsshowpredictionsforthedelta-functionFRBpopulationmodelwhiletheboundariesofthe
regionsenclosethecurvescorrespondingtoalltheothermodelsconsideredinFigure3.
and it has no observational confirmation at present. Both detect FRBs with redshifts z>2 with theParkes telescope
the scattering models are normalized to reproduce the ob- (Figure2).Theredshiftdistribution(Figure3)peaksinthe
served pulsewidth of FRB110220, assuming that it hasan range 0.4 6 z 6 0.6 for most of the models which we have
intrinsicpulsewidthofw =1ms.FortheParkestelescope, considered. We find that most of the models that we have
i
we find that in both the models scatter broadening starts considered are consistent with the redshift distribution of
to dominate the total pulse width at z 0.5 (Figure 1). In the fourteen FRBs observed by the Parkes telescope. How-
≈
Model I, the total pulse width increases steeply for z >0.5 ever, some of the models with α 6 2 predict a redshift
−
whereas a more gradual increases is predicted by Model II. distribution that extends beyond z > 4 while the observed
We also find that the z >0.5 behaviour is not significantly FRBdistributionisrestrictedwithinz61.5.Thenumberof
modified if we assume an intrinsic pulse width of w = 0.5 FRBsobserved to dateis tosmall toconclusively constrain
i
or2msforFRB110220.Thecosmologicalbroadeningofthe the models which we have considered here. Our prediction
intrinsic pulse width dominates at lower z. however indicate that it will be possible to distinguish be-
The total energy in the FRB pulse and its spectrum tween the different models when more FRB data becomes
also play an important role in determining the FRB detec- available in future.
tion rate. We have introduced the FRB energy E and the Finally, we have used the formalism and the different
FRB emission profile φ(ν) (eq. 1) to model the FRB en- models presented here to predict the FRB detection rate
ergy spectrum. For our work we have assumed a power law expected at the ORT LS and the OWFA PI and PII. The
φ(ν) ν−α(eq.22)whereαisthe(negative)spectralindex. main point to note here is that OWFA PII has a field of
In th∝is paper we have presented results for α values in the view which is 880 times larger than the individual beam of
range 56α65,howevermostoftheanalysisisrestricted theParkestelescopewheremostoftheFRBshavebeende-
to α>− 2. tected.Further,theexistingFRBshaveallbeendetectedin
− the L-band ( 1 2GHz) whereas ORT and OWFA oper-
ItisnecessarytomodeltheFRBpopulationinorderto ∼ −
atearound 326.5MHz. Wepredict thatwe expecttodetect
makepredictionsfor thedetection rate. Wehavequantified
somewhere between 0.01 to 1,000 FRBs per day with
theFRBpopulationthroughn(E,wi,z)whichgivestheco- PII,dependingonthe∼valueofα.TheupcomingOWFAPII
movingnumberdensityoftheFRBoccurrencerate.Forour
holds the potential of dramatically increasing the popula-
work we have assumed that n(E,w ,z) does not vary with
i tion of detected FRBs, thereby opening a new window to
z over the limited redshift range of our interest. Further,
unravel the source of these mysterious events. We plan to
all the FRBs are assumed to have the same intrinsic pulse
presentadetailedtreatmentofthepredictionsforORTand
width w = 1ms. For the E dependence we have adopted
i OWFA in a subsequentwork.
the simplest delta-function model where all the FRBs have
the same energy E = 5.4 1033J which is the estimated
0
×
energy for FRB 110220. We have also considered a set of
modelswheretheE valueshaveaspreadaroundE .Inthis
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