Hubble without the Hubble: cosmology using advanced gravitational-wave detectors alone

0.47 MB·English

by  Stephen R. Taylor

Hubble without the Hubble:
Cosmology using advanced gravitational-wave detectors alone
Stephen R. Taylor∗
Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK
Jonathan R. Gair†
Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK
Ilya Mandel‡
NSF Astronomy and Astrophysics Postdoctoral Fellow,
MIT Kavli Institute, Cambridge, MA 02139; and
School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT
(Dated: January 31, 2012)
2
1 WeinvestigateanovelapproachtomeasuringtheHubbleconstantusinggravitational-wave(GW)
0
signals from compact binaries by exploiting the narrowness of the distribution of masses of the
2
underlying neutron-star population. Gravitational-wave observations with a network of detectors

n willpermitadirect,independentmeasurementofthedistancetothesourcesystems. Iftheredshift
a ofthesourceisknown,theseinspiralingdouble-neutron-starbinarysystemscanbeusedasstandard
J sirens to extract cosmological information. Unfortunately, the redshift and the system chirp mass

0 aredegenerateinGWobservations. Thus,mostpreviousworkhasassumedthatthesourceredshift
3 is obtained from electromagnetic counterparts. However, we investigate a novel method of using
these systems as standard sirens with GW observations alone. In this paper, we explore what we

] can learn about the background cosmology and the mass distribution of neutron stars from the
c
set of neutron-star (NS) mergers detected by such a network. We use a Bayesian formalism to
q
analyze catalogs of NS-NS inspiral detections. We find that it is possible to constrain the Hubble
-
r constant, H0, and the parameters of the NS mass function using gravitational-wave data alone,
g
withoutrelyingon electromagnetic counterparts. Underreasonable assumptions, wewill beableto
[ determineH0to±10%using∼100observations,providedtheGaussianhalf-widthoftheunderlying
2 double NS mass distribution is less than 0.04M⊙. The expected precision depends linearly on the
v intrinsic width of the NS mass function, but has only a weak dependence on H0 near the default
1 parametervalues. Finally,weconsiderwhathappensif,forsomefractionofourdatacatalog,wehave
6 an electromagnetically measured redshift. The detection, and cataloging, of these compact-object
1 mergers will allow precision astronomy, andprovidea determination of H0 which is independentof
5 thelocal distance scale.
.
8
PACSnumbers: 98.80.Es,04.30.Tv,04.80.Nn,95.85.Sz
0
1
1
I. INTRODUCTION GEO-600beganscience runs in2002,andLIGO reached
:
v its initial design sensitivity in 2005. The 3 km Virgo
i interferometer [6], located at Cascina, Italy, began com-
X
The previous decade has seen several ground-based missioning runs in 2005, and has participated in joint
r
a gravitational-wave (GW) interferometers built, and searches with LIGO and GEO-600 since 2007. The 300
brought to their design sensitivity. The construction m arm-length TAMA-300 detector [7], located in Tokyo,
of Initial LIGO, the Laser Interferometer Gravitational- Japan had undertaken nine observation runs by 2004
waveObservatory[1,2],wasakeystepinthequestfora to develop technologies for the proposed underground,
direct detection of gravitational waves, which are a fun- cryogenically-cooled,3kmarm-lengthLCGTproject[8].
damentalpredictionofEinstein’stheoryofgravity[3,4].
Gravitationalwavesfromthecoalescencesofcompact-
The three LIGO detectors are located in the USA, with
object binaries [9] consisting of neutron stars (NSs) and
twosited inHanford, Washingtonwithin a commonvac-
blackholes(BHs)areamongthe mostpromisingsources
uumenvelope(H1,H2 ofarm-lengths4kmand2 kmre-
for LIGO [10]. The first joint search for compact binary
spectively) and one in Livingston,Louisiana (L1 of arm-
coalescence signals using the LIGO S5 science run and
length 4 km) [1, 2]. The 600 m arm-lengthGEO-600de-
the Virgo VSR1 data has not resulted in direct detec-
tector[5]islocatednearHannover,Germany. LIGOand
tions, and the upper limits placed on the local NS-NS
merger rate are higher than existing astrophysicalupper
limits [2]. However, construction has already begun on
the Advanced LIGO detectors [11], which are expected
∗email: [email protected]
†email: [email protected] toincreasethehorizondistanceforNS-NSinspiralsfrom
‡email: [email protected] 33 to 445 Mpc. This thousandfold increase in de-
∼ ∼
2
tection volume is expected to yield detections of NS-NS rections to the phase evolution of DNS inspiral signals,
coalescences at a rate between once per few years and which break the degeneracy between mass parameters
several hundred per year, with a likely value of 40 de- and redshift to probe the distance-redshift relation [19],
∼
tections per year [12]. but which only enter at the fifth post-Newtonian order
The Advanced Virgo detector [13] is expected to be- andwilllikelybeverydifficulttomeasurewithAdvanced
come operational on a similar timescale as Advanced LIGO.Rather,werelyonmeasurementsoftheredshifted
LIGO ( 2015) and with similar sensitivity. We denote chirp mass, which is expected to be the best-determined
the netw∼ork of three AdLIGO detectors and AdVirgo as parameter, and the luminosity distance. This approach
HHLV in the following. These may later be joined by was introduced by Markovi´c in [20], where the author
additional detectors, such as LIGO Australia, IndIGO extracted candidate source redshifts from the redshifted
or LCGT, creatinga world-wide detector networkwhose chirp mass using a constant intrinsic chirp mass (this
sensitivitywillenablegravitational-waveastronomy. The is later extended to include some spread around the as-
network comprising LIGO-Australia, H1 and L1 will be sumed intrinsic value). Chernoff and Finn explored this
denoted as AHL. Regardless of the precise network con- technique in [21], which was elaborated upon by Finn in
figuration, the hope is that when AdLIGO (and either [22], where he suggested using the distribution of signal-
LIGO Australia or AdVirgo) achieve their design sensi- to-noise ratios and chirp masses to probe cosmological
tivitiesitwilltransformGWastronomyfromasearchfor parameters. In this paper, we use up-to-date cosmol-
the first detection, into a tool to explore many different ogy, mass-distribution models, expectations for detector
astrophysicaland cosmologicalphenomena. sensitivity and parametermeasurement accuraciesto in-
vestigate the precision with which the Hubble constant,
Gravitationalwaves directly encode the parametersof
andNSmassdistributionparameters,couldbemeasured
the emitting system, including the luminosity distance
D and the redshifted masses. Simultaneous measure- by the advanced GW detector network.
L
Thepaperisorganizedasfollows. InSec.II,wepresent
ments of the redshift and the luminosity distance would
asimplifiedanalyticalcalculationandderivescalinglaws.
allow gravitational waves to be used as standard can-
SectionIII describes the assumptionsmade increating a
dles, probing the cosmic distance ladder and allowing
catalogofsources,includingadiscussionoftheDNSsys-
for measurements of cosmological parameters [14, 15].
tem properties we can deduce from a gravitational wave
However,the redshift and the intrinsic masses for point-
signal, as well as neutron-star mass distributions and
mass objects can not be individually determined from
merger rates. Section IV details the theoretical machin-
gravitational-wave observations alone. Therefore, pre-
eryfor analyzinga catalogofdetected DNS systems and
vious attempts to use gravitational waves as standard
the details of our implementation of this analysis. We
sirens have generally relied on the existence of electro-
describe our results in Sec. V, in which we illustrate the
magnetic counterparts which can be used to unambigu-
possibility of probing the Hubble constant and neutron-
ously measure the redshift and break the degeneracy
starmassdistributionviaGWdata,andconcludeinSec.
[16, 17], or at least do so statistically [18]. In this paper,
VI with a summary and discussion of future work.
wedemonstratethatsuchcounterpartsarenotnecessary
iftheintrinsicmassdistributionissufficientlynarrow,as
may be the case for double NS (DNS) binaries,although
II. ANALYTICAL MODEL
onecandoevenbetterbycombiningthetwoapproaches.
We show that it is possible to use the statistics from a
Here,we presenta simplified analyticalmodel that we
catalog of gravitational-wave observations of inspiraling
use to show the feasibility of our idea and to derive the
DNSsystemstosimultaneouslydeterminetheunderlying
main scaling relationships. We provide additional justi-
cosmological parameters and the NS mass distribution.
fication for the various assumptions made in this model
A given cosmological model determines the redshift as
later in the paper.
a function of luminosity distance, making it possible to
The network of advanced detectors will be sensitive
extract the intrinsic mass of a system from a measure-
to gravitational waves from NS-NS binaries only at rel-
ment of D and the redshifted mass. This permits us
L
atively low redshifts, z . 0.15 (see Sec. IIIA). At such
to statistically constrain the Hubble constant and the
low redshifts, the Hubble law is nearly linear, so that to
NS mass distribution via a Bayesian formalism, using
lowest order, we can write the Hubble constant as (see
only GW data. A narrower intrinsic NS mass distribu-
Section IIIA)
tion will more effectively penalize any model parameters
whichareoffsetfromthetruevalues. Weinvestigatehow z
H c . (1)
the precision with which we can recover the underlying 0 ≈ D
L
parametersscaleswith the number ofdetections andthe
Therefore, we expect that the uncertainty in the extrap-
values of the intrinsic parameters themselves.
olation of H from redshift and distance measurements
For the majority of our analysis, we do not consider 0
will scale as
difficult-to-detect electromagnetic (EM) counterparts to
theGWdetections,whichwerereliedoninpreviousanal- δH δz δD
| 0| . | | + | L|. (2)
yses, e.g., [17]. Nor do we consider tidal coupling cor- H z D
0 L
3
The detected neutron-starbinarieswill yield a catalog out a rigorous analysis below, and find that the results
of sources with measured parameters. These parame- ofour simplistic modelare accurateto within a factorof
ters will include estimates of the redshifted chirp mass 2 (see Sec. V).
∼
= (1+z) and luminosity distance D . The red-
z L
M M
shiftedchirpmasswillbemeasuredveryaccurately,sowe
can ignore measurementerrorsfor this parameter. How- III. SOURCE CATALOG
ever, our ability to extract the redshift of an individual
sourcefromthe redshiftedchirpmasswilldepend onthe
A. System properties from the gravitational
narrowness of the intrinsic chirp mass distribution, waveform
δz σ 1+z (σ / )
M M
M , (3) In the following analysis we consider an advanced
z ∼ z ∼ z
M globalnetworkdetectingthegravitationalradiationfrom
where the last approximation follows from the fact that an inspiraling double-neutron-star system. The wave-
z 1. On the other hand, the luminosity distance is formforsuchaninspiralhasadistinctivesignature. Such
est≪imateddirectlyfromthegravitational-wavesignal,but systemsaredenotedchirping binaries due tothe charac-
with a significant error that is inversely proportional to teristicfrequencyincreaseuptothepointofcoalescence.
the signal-to-noise-ratio(SNR) of the detection. We use the formalism of [22] to describe the response
Existing binary pulsar measurements suggest that the of an interferometric detector to the gravitationalradia-
chirp-mass distribution may be fairly narrow, σ tionfromaninspiralingbinary. The detectorresponseis
M
0.06M (see Sec. IIID). Meanwhile, for the most di≈s- a function of the system’s redshifted mass, the luminos-
⊙
tant sources at the threshold of detectability, z 0.15 ity distance to the source and the relative orientation of
and δD /D 0.3 (see Sec. IIIA). Therefore,th≈e first the source and detector. This relative orientation is de-
L L
term|in E|q. (2)≈is generally larger than the second term scribed by four angles. Two of them (θ, φ) are spherical
(though they become comparable for the most distant polar coordinates describing the angular position of the
sources),andthe intrinsicspreadinthe chirpmassdom- sourcerelativetothe detector. Theremainingtwo(ι, ψ)
inates as the source of error. describe the orientation of the binary orbit with respect
The errors described above were for a single detec- to the observer’s line of sight [22].
tion, but, as usual, both sources of uncertainty are re- In the quadrupolar approximation, the dependence
duced with more detections as 1/√N, where N is the of the detector response, h(t), on these four angles is
total number ofdetected binaries. In principle, we could completely encapsulated in one variable, Θ, through the
worry whether a few very precise measurements domi- equation
nate overthe remaining N, affecting the overall1/√N
sscoatlihneg.bTeshtemteeramsur(eσmMe/n∼Mts )w/izllisbelartgheorsethwahne|rδeDtLh|e/DfoLr-, h(t)= MD5zL/3Θ(πf)2/3cos[χ+Φ(t)], for t<T, (4)
(0, for t>T,
mer term is minimized. The spreadin the intrinsic chirp
mass σ / is independent of the SNR. Thus, we will
M where f is the GW frequency, χ is a constant phase,
M
learnthemostfrommeasurementsathighz,eventhough
Φ(t) is the signal’s phase, and T is taken as the time of
these will have a worse uncertainty in D (the SNR
L coalescence. =(1+z) istheredshiftedchirpmass,
z
scalesinverselywithD ). Therefore,somewhatcounter- M M
L while encodes an accurately measurable combination
intuitively, the low SNR observations will be most infor- M
of the neutron star masses,
mative. However, since the detections are roughly dis-
tributed uniformly in the volume in which the detector 3/5
m m
is sensitive, we expect half of all detections to be within 1 2
= (m +m ). (5)
20%ofthe mostdistantdetection; therefore,wedo ex- M (m1+m2)2! 1 2
∼pect a 1/√N scaling in δH /H .1
0 0
∝
Usingthevaluesquotedabove,forN 100detections, Analysisofthe gravitational-wavephaseevolutionyields
∼
we may expect that it will be possible to extract the errors on the deduced redshifted chirp mass which vary
Hubble constant with an uncertainty of 5%. We carry accordingtothewaveformfamilybeingused. Regardless,
∼
the precision is expected to be extremely high, with a
characteristic error of 0.04%2 [23].
∼
Θ is defined as
1 This scaling holds whenever the number of detections is in-
creased,eitherbecausethemergerrateishigherorbecausedata Θ 2[F2(1+cos2ι)2+4F2cos2ι]1/2, (6)
aretakenforlonger. Ontheotherhand,ifthenumberofdetec- ≡ + ×
tionsincreasesbecausethedetectorsbecomemoresensitive,the
distance or redshift to the furthest detection will also increase,
scaling with N1/3. In that case, as long as the first term in
Eq. (2) is still dominant, the overall improvement in δH0/H0 2 z canbedeterminedfromthestrainsignalinoneinterferom-
scalesas1/N5/6. eMter throughthephaseevolution.
4
where 0<Θ<4, and LIGO 4-km detectors at Hanford and one at Livingston
(HHL), probably sharing data with a 3-km European
1
F (1+cos2θ)cos2φcos2ψ cosθsin2φsin2ψ, Virgodetector(HHLV).Alternatively,oneoftheHanford
+
≡ 2 − detectors may be moved to Australia (AHL or AHLV),
1
F (1+cos2θ)cos2φsin2ψ+cosθsin2φcos2ψ. improving the network’s parameter-estimation accuracy
×
≡ 2 [26, 28], while the Japanese detector LCGT and/or an
(7)
Indian detector IndIGO may join the network at a later
date.
The luminosity distance D is encoded directly in the
L
In the HHL configuration all of the sites are located
gravitational waveform; however a single interferometer
in the United States, such that we may use the ap-
cannot deduce this. The degeneracy in the detector re-
proximation of assuming the AdLIGO interferometers
sponse between Θ and D must be broken and this re-
L
can be used in triple coincidence to constitute a super-
quires a network of three or more separated interferom-
interferometer. This assumptionismotivatedbythe ori-
eters to triangulate the sky location [14]. A network of
entation of the interferometer arms being approximately
separateddetectors will be sensitive to the different GW
parallel [29], and also has precedents in the literature
polarization states, the amplitudes of which have differ-
[e.g., 22, 30]. However, source localization and D de-
ent dependences on the binary orbital inclination angle, L
termination is very poor in HHL, and would be greatly
ι. Thus, the degree of elliptical polarization measured
improvedby the inclusionof data fromVirgo or anAus-
by a network can constrain ι [9]. The interferometers
tralian detector.
comprising a network will be misaligned such that their
varying responses to an incoming GW can constrain the The single-interferometer approximation is less obvi-
polarization angle, ψ. ously valid for networks with distant, nonaligned detec-
Once Θ is constrained, D can then be deduced from tors,suchasAHL(V)orHHLV.In[31],theauthorscom-
L
thedetectorresponse,givingatypicalmeasurementerror ment that the proposed LIGO-Australia site was chosen
of (300/ρ)%, where ρ is the signal-to-noise ratio of the to be nearly antipodal to the LIGO sites such that all
det∼ection (e.g., [23–25]). The accuracy with which the three interferometers in the AHL configuration would
distance can be measured will depend on the exact net- have similar antenna patterns. Furthermore, since the
work configuration (for example, the introduction of an same hardware configuration would be used for LIGO-
Australiandetectionwill partiallybreakthe inclination– Australia and AdLIGO, the noise spectra are expected
distance degeneracy[26]), but we will use the above as a to be similar [32]. Meanwhile, Virgo does not have the
representative value. same antenna pattern as the LIGO detectors, and the
Wedon’tincludetheimpactofdetectoramplitudecal- Advanced Virgo noise spectrum [33] will be somewhat
ibration errors, which could lead to systematic biases in different from the Advanced LIGO spectrum.
distance estimates. Unlike statistical measurement er- In any case, precise comparisons of the sensitivity of
rors,thesebiaseswouldnotbeamelioratedbyincreasing different networks depend on assumptions about search
thenumberofdetections. Forexample,calibrationerrors strategies(e.g.,coincidentvsfullycoherentsearches)and
of order 10%, as estimated for the LIGO S5 search [27], sourcedistributions (see, e.g., [26,31, 34]). We therefore
would translate directly into 10% systematic biases in penalize our super-interferometer assumption in two dif-
H estimates. Thus, systematic calibration errors could ferent ways. Firstly, we set the network SNR threshold
0
become the limiting factor on the accuracy of measuring to correspond to the expected SNR from three identi-
H if they exceed the statistical errors estimated in this cal interferometers, as described below, rather than the
0
paper. fourinterferometerscomprisingtheAHLVorHHLVnet-
works. WefurtherpenalizetheHHLVnetworkrelativeto
thenetworkincludingthe moreoptimallylocatedLIGO-
B. Detector characteristics Australia by raising the SNR threshold from 8 to 10.
∼
These increases in SNR thresholds have the effect of re-
Forthepurposesofcreatingacatalogofsourcesforour stricting the network’s reach in luminosity distance or
study, we are only interested in determining which bina- redshift; however, similar numbers of detections can be
ries are detectable, and how accurately the parameters achieved by longer observation times.
of these binaries can be estimated. We use the crite- Withtheaforementionedcaveats,weproceedwithour
rionthat the networksignal-to-noiseratio, ρ , must be assumptionthataglobalnetworkcanbeapproximatedas
net
greater than 8 for detection. Actual searches use signifi- a single super-interferometer. This is to provide a proof
cantlymorecomplicateddetectionstatisticsthatdepend ofprincipleforthe abilityofsuchanetworktoprobethe
on the network configuration, data quality, and search background cosmology and aspects of the source distri-
techniques, whichmight make ourassumeddetectability bution. We do not anchor our analysis to precise knowl-
threshold optimistic. Here, we are interested only in a edge of the individual interferometer site locations and
sensible approximation of the detectability criterion. orientations,but will attempt to correctfor any possible
The network configuration for the advanced detector bias.
era is uncertain at present. Possibilities include two Following [22] (and correcting for a missing square
5
root), we can write the matched filtering signal-to-noise be 4.2M .3 TheAdLIGOhorizondistancefor1.4M –
⊙ ⊙
∼
ratio in a single detector as 1.4M inspirals is 445 Mpc, which corresponds to
⊙
∼
z 0.1 in the ΛCDM cosmology. Given that we are
∼
r 5/6 evaluating different cosmological parameters, we adopt
0 z
ρ=8Θ M ζ(fmax), (8) z 1 as a generous upper redshift limit to a second-
DL(cid:18)1.2M⊙(cid:19) gen∼eration network’s reach. This redshift exceeds the
p
reachofAdLIGOinallconsideredcosmologies4andchirp
where
masses. With these extreme choices for the variables,
the orbital frequency at the ISCO, f , could be as
5 3 5/3 max
r2 x M2, low as 262 Hz. For the latest zero-detuning-high-power
0 ≡ 192π 20 7/3 ⊙ AdLIG∼O noise curve [32], ζ(f = 262Hz) & 0.98.
(cid:18) (cid:19) max
∞ df(πM )2 Thus, we feel justified in adopting ζ(f ) 1 for the
x ⊙ , max ≃
7/3 ≡ (πfM )7/3S (f) ensuing analysis.
Z0 ⊙ h
1 2fmax df(πM⊙)2 Thus matched filtering, with an SNR thresholdof 8, a
ζ(f ) . (9)
max ≡ x7/3Z0 (πfM⊙)7/3Sh(f) characteristicdistancereachof∼176Mpcandζ(fmax)≃
1,providesacriteriontodetermine the detectabilityofa
S (f) denotes the detector’s noise power spectral den- source by our network.5
h
sity (the Fourier transform of the noise auto-correlation
function), and 2f is the wave frequency at which the
max
inspiral detection template ends [35]. The SNR of a de-
tected system will vary between the individual network
sites, as a result of the different S (f)’s and angular de-
h
pendencies. The network SNR of a detected system is
C. Orientation function, Θ
givenby the quadraturesummation of the individual in-
terferometer SNRs,
The angular dependence of the SNR is encapsulated
ρ2 = ρ2. (10)
net k withinthevariableΘ,whichvariesbetween0and4,and
Xk has a functional form given by Eq. (6). From our cat-
alog of coincident DNS inspiral detections we will use
We approximate the sensitivity of the super-
only and D for each system. The sky location and
interferometer by assuming 3 identical interferometers Mz L
binaryorientationcanbededucedfromthenetworkanal-
in the network with the sensitivity of Advanced LIGO,
ysis, however we will not explicitly consider them here.
such that r √3r . Different target noise curves
0,net ≈ 0 Without specific vales for the angles (θ, φ, ι, ψ) we can
for AdLIGO produce different values for r , which vary
0 still write down the probability density function for Θ
between 80 120 Mpc [32]. We adopt the median
∼ − [22]. Taking cosθ, φ/π, cosι and ψ/π to be uncorre-
value of 100 Mpc for a single interferometer, yielding
lated and distributed uniformly over the range [ 1,1],
r 176 Mpc for the network. −
0,net ∼ the cumulative probability distribution for Θ was calcu-
The SNR also depends on ζ(f ), which increases
max lated numerically in [37]. The probability distribution
monotonicallyasafunctionoff . Thisfactordescribes
max can be accurately approximated [22] by,
the overlap of the signal power with the detector band-
width [22], which will depend on the wave frequency at
which the post-Newtonian approximation breaks down,
and the inspiral ends. It is usual to assume that the in- 5 Θ(4 Θ)3, if 0<Θ<4,
(Θ)= 256 − (12)
spiral phase terminates when the evolution reaches the PΘ (0, otherwise.
innermost stable circular orbit (ISCO), whereupon the
neutron stars merge in less than an orbital period. This
gives
3 Both neutron stars in the binary system would need to have
fGW =2f =2 fISCO = 1570 Hz 2.8M⊙ , masses2σabovethedistributionmeanatthemaximumconsid-
max max 1+z 1+z M eredµandσ,whereµNS [1.0,1.5]M⊙,σNS [0,0.3]M⊙.
(cid:18) (cid:19) (cid:18) (cid:19)(11) 54 THh0e∈re[0w.0il,l20b0e.0s]okmmesb−i1aMs∈pinc−t1h;isΩak,p0p=rox0i;mΩamt∈i,o0n∈, [s0in.0c,e0.w5]e are
where M is the total mass of the binary system [12].
assuming each interferometer records the same SNR for each
fISCO also depends directly on the mass ratio µ/M (µ event. The fact that the different interferometers are not colo-
is the system’s reduced mass); however this mass asym- catedmeansthatthismayoverestimatethenumberofcoincident
metry term has a negligible effect on f for the mass detections. Wecarryouttheanalysishereawareof,butchoosing
max
toignore,thisbias,andinSec.VCconsiderraisingthenetwork
range of neutron stars considered here [35, 36]. SNRthreshold, whichhasthesameeffect asreducingthechar-
The maximum binary system mass could conceivably acteristic distance reachofthenetwork.
6
We canuse Eq.(12) toevaluate the cumulative distribu- distributions,
tion of Θ,
X N(µ ,σ2) ; X N(µ ,σ2)
1 ∼ 1 1 2 ∼ 2 2
CΘ(x)≡ ∞PΘ(Θ)dΘ≃1(1,+x2)5(46−x)4, iiff 0x≤≤x0≤4 SinceatXh1e+nbeXut2ro∼n-Nst(aarµ1m+asbsµ2d,iast2rσib12u+tibo2nσ22i)s. symm(1e4t)-
Zx 0, if x>4. ric around the mean (and all neutron star masses are
(13) O(1M⊙) with the values spread over a relatively nar-
 ∼row range), then we can assume a characteristic value
forthepre-factorinEq.(5)isthevaluetakenwhenboth
masses are equal i.e. (0.25)3/5. The chirp mass distri-
D. Mass distribution bution should then be∼approximately normal
N(µ ,σ2),
Inrecentyears,the number ofcatalogedpulsarbinary M∼ c c
systems has increased to the level that the underlying with mean and standard deviation
neutron-star mass distribution can be probed. There is
now a concordance across the literature that the neu- µc 2(0.25)3/5µNS, σc √2(0.25)3/5σNS. (15)
≈ ≈
tron star mass distribution is multimodal, which reflects
whereµ andσ arethemeanandstandarddeviation
the different evolutionarypathsofpulsar binarysystems NS NS
ofthe underlyingneutron-starmassdistribution, respec-
[38, 39]. However, we are only concerned with neu-
tively.
tron stars in NS-NS systems for this analysis, and their
The accuracyof such anansatz depends upon the size
distribution appears to be quite narrow. In particular
of mass asymmetries which could arise in a DNS binary
[39] found that the neutron stars in DNS systems pop-
system. We investigated the percentage offset between
ulate a lower mass peak at m 1.37M 0.042M .
∼ ⊙ ± ⊙ the actual distribution parameters (deduced from least-
Meanwhile, [38] restricted their sample of neutron stars
squares fitting to the sample number-density distribu-
to those with secure mass measurements and predicted
tion) and the ansatz parameters, for a few values of µ
that the posterior density for the DNS systems peaked NS
and σ . The largest offset of the ansatz parameters
at µ 1.34M , σ 0.06M . NS
NS ⊙ NS ⊙
∼ ∼ from the true chirp mass distribution was on the or-
Population synthesis studies of binary evolution pre-
der of a few percent ( 2.5% for µ , and 3.5% for σ
dictsimilarlynarrowmassdistributionsforneutronstars ∼ c ∼ c
when µ = 1.0M , σ = 0.3M ), and the agreement
in NS-NS binaries (see, e.g., [10, 30, 40] and references NS ⊙ NS ⊙
improved with a narrower underlying neutron star mass
therein). Some models predict that the mass of neutron
distribution. For the case of σ 0.05M , the agree-
starsatformationisbimodal,withpeaksaround1.3and NS ∼ ⊙
mentwas 0.1%forµ and<0.1%forσ . Inthecaseof
1.8 solar masses, and any post-formation mass transfer ∼ c c
σ 0.15M , the agreement was within a percent for
in DNS systems is not expected to change that distribu- NS ∼ ⊙
both parameters. The signofthese offsets indicates that
tion significantly, but the 1.8M mode is anticipated to
⊙ µtrue <µmodel and σtrue >σmodel.
be very rare for DNS systems, with the vast majority of c c c c
Given that the literature indicates an underlying
mergingneutronstarsbelongingtothe 1.3M peak[41].
⊙ neutron-star mass distribution in DNS systems with
Thus, population synthesis results support the anticipa-
σ . 0.15M , we anticipate that Eq. (15) will be ap-
tionthatNSbinariesmayhaveanarrowrangeofmasses NS ⊙
propriate for generating data sets and we use this in the
that could be modeled by a Gaussian distribution.
ensuing analysis. The assumption throughoutis that for
To lowest order, the GW signal depends on the two
thevolumeoftheUniverseprobedbyourglobalnetwork,
neutron star masses through the chirp mass, . We
the neutron-star mass distribution does not change.
M
assume that the distribution of individual neutron-star
The observed data will tell us how wide the intrinsic
masses is normal, as suggested above. For σ µ ,
NS NS chirpmassdistributionisinreality. Ifitiswiderthanan-
≪
this should yield an approximately normal distribution
ticipated,wemaynotbeabletomeasureH asprecisely
0
for the chirp mass as well.
as we find here, but we will know this from the observa-
We carried out O(105) iterations, drawing two ran- tions. Inprinciple,wecouldstillbesystematicallybiased
∼
dom variates from a normal distribution (representing if the mass distribution turned out to be significantly
theindividualneutron-starmasses),andthencomputing non-Gaussian, since we are assuming a Gaussian model.
. Wevariedthemeanandwidthoftheunderlyingdis- However,itwillbefairlyobviousifthemassdistribution
M
tribution within the allowed ranges (Sec. IIIB). Binning is significantly non-Gaussian (e.g., has a non-negligible
the values,the resulting distributionwasfoundto secondarypeakaround1.8M ),sinceredshiftcouldonly
M M ⊙
be normal, as expected. introduce a 10% spread in the very precise redshifted
∼
We postulate a simple ansatz for the relationship be- chirpmass measurementsfor detectorsthat aresensitive
tween the chirp mass distribution parameters and the to z 0.1. In such a case, we would not attempt to fit
∼
underlyingneutronstarmassdistribution. If X andX thedatatoaGaussianmodelfortheintrinsicchirpmass
1 2
aretwoindependentrandomvariatesdrawnfromnormal distribution.
7
To convert from merger-rate densities to detection
TABLE I: AcompilationofNS-NSmergerratedensitiesin rates, [12] take the product of the merger-rate density
variousformsfromTablesII,IIIandIVin[12]. Thefirstcolumn
withthe volumeofaspherewithradiusequalto the vol-
givestheunits. Thesecond,thirdandfourthcolumnsdenote the
plausiblepessimistic,likely,andplausibleoptimisticmergerrates ume averaged horizon distance. The horizon distance is
extrapolatedfromtheobservedsampleofGalacticbinaryneutron thedistanceatwhichanoptimallyoriented,optimallylo-
stars[44]. Thefifthcolumndenotes theupperratelimitdeduced catedbinarysystemofinspiraling1.4M neutronstarsis
⊙
fromtherateofTypeIb/Icsupernovae[45].
detected with the threshold SNR. This is then averaged
over all sky locations and binary orientations.
Source Rlow Rre Rhigh Rmax
NS-NS(MWEG−1Myr−1) 1 100 1000 4000 3
4π D
NS-NS(L−1Myr−1) 0.6 60 600 2000 ND =n˙0 horizon (2.26)−3, (18)
10 × 3 Mpc
NS-NS(Mpc−3Myr−1) 0.01 1 10 50 (cid:18) (cid:19)
where the (1/2.26) factor represents the average over all
sky locations and binary orientations.
This gives 40 detection events per year in AdLIGO
E. DNS binary merger rate density, n˙(z)
∼
(using R ), assuming that D = 445 Mpc and all
re horizon
neutron stars have a mass of 1.4M .
We assume that merging DNS systems are distributed ⊙
homogeneously and isotropically. The total number of
thesesystemsthatwillbedetectedbytheglobalnetwork
F. Cosmological model assumptions
dependsontheintrinsicrateofcoalescingbinarysystems
per comoving volume. Werequiresomeknowledgeofthis
We assume aflat cosmology,Ω =0,throughout,for
in order to generate our mock data sets. Any sort of k,0
which the luminosity distance as a function of the radial
redshift evolution of this quantity (as a result of star-
comoving distance is given by
formation rate evolution etc.) can be factorised out [22],
such that z dz′
D (z)=(1+z)D (z)=(1+z)D , (19)
d2N L c H E(z′)
n˙(t) n˙(z)=n˙ ξ(z), (16) Z0
0
≡ dt dV ≡
e c where D =c/H (the “Hubble length scale”) and
H 0
whereN isthenumberofcoalescingsystems,t isproper
e
time, Vc is comovingvolume,andn˙0 representsthe local E(z)= Ωm,0(1+z)3+ΩΛ,0. (20)
merger-rate density. q
We willconsider anevolvingmerger-ratedensity, such Insuchacosmology,theredshiftderivativeofthecomov-
that, ing volume is given by
ξ(z)=1+αz =1+2z, for z 1, (17) dVc 4πDc(z)2DH
≤ = . (21)
dz E(z)
which is motivated by a piecewise linear fit [42] to the
merger-rate evolution deduced from the UV-luminosity- At low redshifts, we can use an approximate simplified
inferred star-formation-rate history [43]. form for the relationship between redshift and luminos-
The appropriate value for n˙ is discussed in detail in ity distance. Using a Taylor expansion of the comoving
0
[12]. In that paper, the authors review the range of distance around z = 0 up to O(z2), and taking the ap-
values quoted in the literature for compact binary co- propriatepositiveroot,wefindD (z)=D (z)/(1+z)is
c L
alescence rates, i.e. not only NS-NS mergers but NS-BH given by
and BH-BH. The binary coalescence rates are quoted
∂D z2 ∂2D
(p1e0r1M0tiilmkyesWtahyeSEoqluairvballeunet-liGghaltalxuym(iMnoWsitEyG,L)Ba,n⊙d),paesrwLe1l0l Dc(z)≈Dc(z =0)+z ∂zc(cid:12)z=0+ 2! ∂z2c(cid:12)z=0+...
achsaprearctuenriitzecdombyovfoinugrvvoalluumese,.aI“nloewac,”h“caresae,listthiec,r”a“tehsigahr”e ≈DH z− 43Ωm,0z2 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (22)
and “maximum” rate, which coverthe full range of pub- (cid:20) (cid:21)
lished estimates. Hence,
The values for the NS-NS merger rate given by [12]
3
are listed in Table I. The second row of Table I is de- D D z+ 1 Ω z2
L H m,0
≈ − 4
rived assuming that coalescence rates are proportional (cid:20) (cid:18) (cid:19) (cid:21)
tothe star-formationrateinnearbyspiralgalaxies. This Therefore,
star-formationrate is crudely estimated from their blue-
ltuhmeicnoonsviteyr,siaonndfatchteormoefr1g.e7r-Lrate/MdeWnsEitGy i[s46d].edTuhceeddavtiaa z 1 1+ 4 1− 43Ωm,0 DL 1 .
in the third row is obtained u1s0ing the conversion factor ≈ 2 1− 34Ωm,0 s (cid:0) DH (cid:1) − 
of 0.0198 L10/Mpc3 [47]. (cid:0) (cid:1) (23)
8
This approximation is very accurate for the range of which gives,
parameters investigated (H [0,200] km s−1Mpc−1,
0
∈
Ωm,0 [0,0.5]), and for DL . 1 Gpc (which is comfort- ∂Θ Θ
∈ (ρ ,D )= (Θ) = (Θ)
ably beyond the reach of AdLIGO for NS-NS binaries). ρ z L Θ Θ
P |M P ∂ρ P ρ
In this parameter range, the largest offset of this ap- (cid:12)Mz,DL
(cid:12)
proximationfromafullredshiftroot-findingalgorithmis ρD (cid:12) 1.2M 5/6
= L(cid:12) ⊙
∼4.6%, at a luminosity distance of 1 Gpc. PΘ"8 r0 (cid:18) Mz (cid:19) #
5/6
D 1.2M
L ⊙
, (27)
× 8r
G. Distribution of detectable DNS systems 0 (cid:18) Mz (cid:19)
such that we finally obtain,
The two system properties we will use in our analysis
are the redshifted chirp mass, , and the luminosity
Mz d4N 1 ∂z dV n˙(z)
distance, DL. Only systems with an SNR greater than = c
threshold will be detected. Thus, we must include SNR dtdρdDLd z (1+z)∂DL dz (1+z)
M
selection effects in the calculation for the number of de- Θ
( z) (Θ)
tections. Wecanwritedownthedistributionofthenum- ×P M| × PΘ ρ
ber of events per year with , z and Θ [22, 30],
M Pρ(ρ|Mz,DL)
d4N = dVc n˙(z) ( ) (Θ), (24) = 4πDc(z)2|DH{z } n˙(z)
dtdΘdzd dz (1+z)P M PΘ Dc(z)E(z)+DH(1+z)(1+z)2
M
z
wheretisthetimemeasuredintheobserver’sframe,such M DL ρ(ρ z,DL).
×P 1+z P |M
that the 1/(1+z) factor accounts for the redshifting of (cid:18) (cid:12) (cid:19)
(cid:12) (28)
the merger rate [30]. (cid:12)
(cid:12)
We convert this to a distribution in , D and ρ
Mz L We may not necessarily care about the specific SNR
using,
of a detection; rather only that a system with
z
M
and D has SNR above threshold (and is thus de-
∂M ∂M ∂M L
d4N ∂Mz ∂DL ∂ρ d4N tectable). Fortunately the SNR only enters Eq. (28)
= ∂z ∂z ∂z . (25) through (ρ ,D ), such that we can simply inte-
dtdρdDLdMz (cid:12)(cid:12)(cid:12)∂∂MΘz ∂∂DΘL ∂∂Θρ (cid:12)(cid:12)(cid:12)× dtdΘdzdM grate ovePrρthis|Mtezrm aLnd apply Eq. (13),
(cid:12)∂Mz ∂DL ∂ρ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) ∞ ∞
We use the definiti(cid:12)ons of the variab(cid:12)les in Sec. IIIA and (ρ ,D )dρ= (Θ)dΘ C (x),
ρ z L Θ Θ
IIIB to evaluate the Jacobian matrix determinant. The Zρ0 P |M Zx P ≡
redshift is only a function of DL (in a given cosmology); ρ D 1.2M 5/6
0 L ⊙
the intrinsic chirp mass, , is the redshifted chirp mass where, x= .
M 8 r
divided by (1 + z) (again the redshift is a function of 0 (cid:18) Mz (cid:19)
(29)
D ); Θ is a function of , D and ρ according to Eq.
L z L
M
(8). The(1,3)component ∂ /∂ρ (∂ /∂ρ)
M ≡ M Mz,DL In this case, Eq. (28) is modified to give,
iszerobecausewearediffe(cid:16)rentiatingintrinsicch(cid:12)irpmas(cid:17)s
(a function of redshifted chirp mass and distan(cid:12)ce) with
d3N
respecttoSNR,butkeepingdistanceandredshiftedchirp
dtdD d
mass constant. If these variables are held constant then L Mz(cid:12)ρ>ρ0
(cid:12)
the derivative must be zero. Similar considerations of 4πD (z(cid:12))2D n˙(z)
which variables are held constant in the partial deriva- = c (cid:12) H
D (z)E(z)+D (1+z)(1+z)2
c H
tives are used to evaluate the remaining elements of the
matrix. Hence, z ρ0DL 1.2M⊙ 5/6
M D C . (30)
L Θ
×P(cid:18)1+z(cid:12) (cid:19) " 8 r0 (cid:18) Mz (cid:19) #
1 M ∂z 0 (cid:12)
(cid:12)(cid:12)(cid:12) (15+0zΘ) −(1+∂∂DΘzz)L∂DL Θ0(cid:12)(cid:12)(cid:12)= (1+1 z)∂∂DzLΘρ. (26) Tofoccoasmlcuollaotgeictahlea(cid:12)(cid:12)nndumNbSemr oafssddetisetcrtiebdutsiyosntepmarsa(mgievteenrsa,−→sµet)
(cid:12)−6Mz DL ρ(cid:12) weintegrateoverthisdistribution,whichisequivalentto
(cid:12) (cid:12)
(cid:12) (cid:12) integrating over the distribution of events with redshift
We n(cid:12)ote that, (cid:12)
and chirp mass, i.e. N = T ∞ ∞ d3N dzd ,
µ × 0 0 dtdzdM M
Pρ(ρ)δρ=PΘ(Θ)δΘ, where T is the duration of the oRbseRrvat(cid:16)ion run.(cid:17)
9
year,which compareswellto the 40 eventsfound in [12].
TABLE II: AsummaryoftheWMAP7-yearobservations.
ThedatafromColumn1isfromTable3of[48],containing
parametersderivedfromfittingmodelstoWMAPdataonly.
Column2contains thederivedparametersfromTable8of[49], IV. ANALYSIS METHODOLOGY
wherethevaluesresultfromasix-parameterflatΛCDMmodelfit
toWMAP+BAO+SNedata.
We will use Bayesian analysis techniques to simulta-
Parameter WMAP only WMAP+BAO+SNe neously compute posterior distribution functions on the
H0 / (km s−1Mpc−1) 71.0±2.5 70.4+−11..34 mean and standard deviation of the intrinsic NS mass
Ω 0.0449±0.0028 0.0456±0.0016 distribution (in DNS systems) and the cosmological pa-
b,0 rameters given a catalog of simulated sources with mea-
Ωc,0 0.222±0.026 0.227±0.014 sured redshifted chirp masses and luminosity distances.
ΩΛ,0 0.734±0.029 0.728+−00..001156
A. Bayesian analysis using Markov Chain Monte
Carlo techniques
H. Creating mock catalogs of DNS binary
inspiraling systems
Bayes’theoremstatesthattheinferredposterior prob-
The model parameter space we investigate is the 5D ability distribution of the parameters −→µ based on a hy-
pothesis model , and given data D is given by
space of[H ,µ ,σ ,Ω ,α] with a flatcosmologyas-
0 NS NS m,0 H
sumed. To generate a catalog of events, we choose a set
of reference parameters, motivated by previous analysis p(−→µ D, )= L(D|−→µ,H)π(−→µ|H), (31)
in the literature. The seven-year WMAP observations | H p(D )
|H
gavethecosmologicalparametersinTableII.Forourref-
erence cosmology, we adopt H = 70.4 km s−1Mpc−1, where (D −→µ, ) is the likelihood (the probability of
0 L | H
Ω = 0.27 and Ω = 0.73. The parameters of the measuring the data, given a model with parameters −→µ),
m,0 Λ,0
neutron-starmassdistributionwerediscussedearlier,but π(−→µ ) is the prior (any constraints already existing
|H
on the model parameters) and finally p(D ) is the ev-
as reference we use µ = 1.35M and σ = 0.06M .
NS ⊙ NS ⊙ |H
idence (this is important in model selection, but in the
The merger-rate density was also discussed earlier, and
we take α = 2.0 and n˙ = 10−6 Mpc−3yr−1 as the ref- subsequent analysis in this paper can be ignored as a
0
normalization constant).
erence. Later, we will investigate how the results change
if the width of the NS mass distribution is as large as In this analysis, the data in Eq. (31) is not from a
0.13M , as indicated by the predictive density estimate single source, but rather from a set of sources, and we
⊙
of [38]. want to use it to constrain certain aspects of the source
distribution, as well as the background cosmology. The
These reference parameters are used to calculate an
uncertainty arises from the fact that any model cannot
expectednumberofevents,6 andthenumberofobserved
predict the exact events we will see, but rather an as-
events is drawn from a Poisson distribution (assuming
trophysical rate of events that gives rise to the observed
each binary system is independent of all others) with
events. The probability distribution for the set of events
that mean. Monte-Carlo acceptance/rejection sampling
will be discussed in Sec. IVB.
is used to draw random redshifts and chirp masses from
To compute the posterior on the model parameters,
the distribution in Eq. (24) for each of the N events.
o
we use Markov Chain Monte Carlo (MCMC) techniques
The D and are then computed from the sampled
L z
M sincethey provideanefficientwayto explorethe full pa-
and z.
MWith a reference rate of n˙ = 10−6 Mpc−3yr−1 and rameter space. An initial point, −x→0, is drawn from the
0 prior distributionandthenateachsubsequentiteration,
a constant merger-rate density, we estimate that there
shouldbe 90yr−1detections,whilsttakingintoaccount i, a new point, −→y, is drawnfrom a proposal distribution,
merger-rat∼eevolutionusingEq.(17)boosts this to 100 q(−→y|−→x) (uniform in all cases, covering the range of pa-
yr−1. ThesenumbersareforanetworkSNRthresho∼ldof rameter investigation). The Metropolis-Hastings ratio is
then evaluated,
8. If we ignore merger-rate evolution and raise the SNR
threshold to 10 (to represent an AdVirgo-HHL network
π(−→y) (D −→y, )q(−→xi −→y)
for which the coincident detection rate is roughly halved R= L | H | . (32)
relativetotheHHL-onlynetwork)weget 45eventsin1 π(−→xi)L(D|−→xi,H)q(−→y|−→xi)
∼
Arandomsampleisdrawnfromauniformdistribution,
u U[0,1], and if u < R the move to the new point is
∈
6 The observation time, T, is assumed to be 1 year (but the ex- accepted, so that we set −x−i+→1 = −→y. If u > R, the move
pected number of detections simply scales linearly with time) is rejected and we set −x−i+→1 = −→xi. If R > 1 the move is
always accepted, however if R<1 there is still a chance
andanetworkactingasasuper-interferometerwithr0,net 176
Mpcisalsoassumed. ≃ that the move will be accepted.
10
TheMCMCsamplescanbeusedtocarryoutintegrals theproductoftheindividualPoissonprobabilitiesforde-
over the posterior, e.g. tecting n events in a bin, i, where the expected (model-
i
dependent) number of detections is ri(−→µ). For the ith
N bin,
1
f(−→x)p(−→x D, )d−→x = f(−→xi). (33)
| H N →−
Z Xi=1 (ri(−→µ))nie−ri(µ)
p(ni −→µ, )= , (34)
| H n !
The 1D marginalized posterior probability distributions i
in individual model parameters can be obtained by bin-
and so the likelihood of the cataloged detections is,
ning the chain samples in that parameter.
X (ri(−→µ))nie−ri(→−µ)
(−→n −→µ, )= . (35)
L | H n !
B. Modelling the likelihood i=1 i
Y
In this work, we take the continuum limit of Eq. (35).
1. Expressing the likelihood
In this case, the number of events in each infinitesimal
bin is either 0 or 1. Everyinfinitesimal bin contributes a
We use a theoreticalframeworksimilar to that of [50]. factor of e−ri(→−µ), whilst the remaining terms in Eq. (35)
Thedataareassumedtobeacatalogofeventsforwhich
evaluateto1foremptybins,andri(−→µ)forfullbins. The
redshifted chirp mass, Mz, and luminosity distance, DL productof the exponential factorsgives e−Nµ, where Nµ
have been estimated. These two parameters for the
isthenumberofDNSinspiraldetectionspredictedbythe
events can be used to probe the underlying cosmology
model, withparameters−→µ. The continuumlikelihoodof
and neutron-star mass distribution. In this analysis, we
a catalog of discrete events is therefore
focus on what we can learn about the Hubble constant,
H , the Gaussian mean of the (DNS system) neutron-
0 No
starmassdistribution,µNS,andtheGaussianhalf-width, (−→−→Λ −→µ, )=e−Nµ r(−→λi −→µ), (36)
σ . We could also include the present-day matter den- L | H |
NS i=1
Y
sity, Ω , however we expect that this will not be well
m,0
constrained due to the low luminosity distances of the
−→
where −→Λ = −λ→,−λ→,...,−λ−→ is the vector of measured
sources. We could also include the gradient parameter, { 1 2 No}
α, describing the redshift evolution of the merger-rate systemproperties,with−→λi =( z,DL)iforsystemi,and
M
density. Noisthenumberofdetectedsystems. Finally,r(−→λi −→µ)is
|
The measurement errors were discussed earlier (Sec. therateofeventswithproperties andD ,evaluated
z L
IIIA) and we will account for these later. For the first for the ith detection under modelMparameters −→µ, which
analysis we assume that the observable properties of in- is given by Eq. (30).
dividual binaries are measured exactly.
We consider first a binned analysis. We divide the
parameter space of Mz and DL into bins, such that the 2. Marginalizing over n˙0
data is the number of events measured in a particular
range of redshifted chirp mass and luminosity distance.
We may also modify the likelihood calculation to
Each binary system can be modeled as independent of
marginalizeoverthepoorlyconstrainedmerger-rateden-
all other systems, so that within a given galaxy we can
sity, n˙ .7 This quantityis sopoorlyknown(seeTable I),
modelthenumberofinspiralsthatoccurwithinacertain 0
that it is preferable to use a new statistic that does not
time asa Poissonprocess,with DNS binariesmerging at
rely on the local merger-rate density, by integrating the
aparticularrate(e.g.,[51,52]). ThemeanofthePoisson
likelihood given in Eq. (36) over this quantity,
process will be equal to the model-dependent rate times
the observation time, and the actual number of inspirals ∞
occurring in the galaxy is a random-variate drawn from ˜(−→−→Λ −→µ, )= (−→−→Λ −→µ, )dn˙0. (37)
L | H L | H
the Poisson distribution. Z0
A bin in the space of system properties may contain
TheexpectednumberofdetectionsdescribedinSec.IIIG
events from several galaxies, but these galaxies will be-
can be expressed as,
have independently and the number of recorded detec-
tions in a given bin will then be a Poisson process, with
a mean equal to the model-dependent expected number Nµ =n˙0 d zdDL,
× I M
of detections in that bin [50]. The data can be written Z Z
asavectorofnumbersinlabelledbinsinthe2Dspaceof
system properties, i.e. −→n = (n1,n2,...,nX), where X is
the number of bins. Therefore, the likelihood of record- 7 Asimilartechnique wasusedin[53],wherethetotal numberof
ingdataDundermodel (withmodelparameters−→µ)is events predictedbythemodelismarginalizedover.
H