Constraining the propagation speed of gravitational waves with compact binaries at cosmological distances

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by  Atsushi Nishizawa

Constraining the propagation speed of gravitational waves with compact binaries at
cosmological distances
Atsushi Nishizawa1,∗
1Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA
(Dated: February 5, 2016)
In testing gravity a model-independent way, one of crucial tests is measuring the propagation
speedofagravitationalwave(GW).Ingeneralrelativity,aGWpropagateswiththespeedoflight,
while in the alternative theories of gravity the propagation speed could deviate from the speed of
light due to the modification of gravity or spacetime structure at a quantum level. Previously we
proposed the method measuring the GW speed by directly comparing the arrival times between a
GWandaphotonfromthebinarymergerofneutronstarsorneutronstarandblackhole,assuming
that it is associated with a short gamma-ray burst. The sensitivity is limited by the intrinsic time
6
delaybetweenaGWandaphotonatthesource. Inthispaper,weextendthemethodtodistinguish
1
theintrinsictimedelayfromthetruesignalcausedbyanomalousGWspeedwithmultipleeventsat
0
cosmological distances,alsoconsideringtheredshiftdistributionofGWsources,redshift-dependent
2
GW propagation speed, and the statistics of intrinsic time delays. Weshow that an advanced GW

b detector such as Einstein Telescope will constrain the GW propagation speed at the precision of
e ∼ 10−16. We also discuss the optimal statistic to measure the GW speed, performing numerical
F simulations.

4

I. INTRODUCTION such cosmic rays on the Earth lead to the limit on GW
]
c speed, c υ < 2 10−15c [29]. The constraints on
g
q The second-generation laser-interferometric gravita- anisotrop−ic GW spee×d from the gravitationalCherenkov
-
r tional wave (GW) detectors would accomplish the first radiationhavebeen extensivelystudied in the contextof
g detectionofaGWinthecomingafewyearsandopenup gravitational standard-model extension via Lorentz vio-
[
GW astronomy [1]. After that, the detections of multi- lation [30]. However, the above constraint on isotropic
2 pleeventsatcosmologicaldistancewouldberealizedwith GW speed can be applied only to subluminal case. On
v the third-generation ground-based GW detector such as theotherhand,fromtheobservationaldataoftheorbital
2 Einstein telescope (ET) [2] and 40-km LIGO [3]. The decay of a binary pulsar, the constraint on superluminal
7 GW observations enable us not only to gain information GW speed has been obtained, c υg .10−2c [31]. Al-
0 | − |
aboutastronomicalobjectsandcosmology[4]butalsoto though this constaint can be applied to both super- and
1
0 test gravity theories in strong and dynamical regimes of subluminal propagations, there is still large parameter
. gravity (for reviews, see [5–8]). spaceallowedformodificationofgravity. Inaddition,all
1
To test gravity with GWs, it is crucial to search for the constraints above are indirect measurements of the
0
6 anomalous deviation from general relativity (GR) in a GW velocity. Therefore, the direct measurement of GW
1 model-independent way. There have been many sugges- propagationspeed is crucial in testing gravity theories.
: tions of such methods: seeking for the deviation from Sofartherehavebeenafewproposalstodirectlymea-
v
GR in GW phase evolution of compact-binary inspiral- sure the GW propagation speed. One is comparing the
i
X ing [9–12] and in GW waveforms of black-hole ringdown phasesofaGWanditselectromagneticcounterpartfrom
r [13, 14], and non-GR GW polarizations [15–18]. One of a periodic binary source [32, 33]. However, to eliminate
a
other tests is measuringthe propagationspeed ofa GW. unknown intrinsic phase lag between the GW and the
In GR, a GW propagates with the speed of light [19], electromagneticwaveatthe source,twosignalsatdiffer-
while in the alternative theories of gravity the propaga- ent times (e.g. a half year) on the Earth’s orbit around
tion speed could deviate from the speed of light due to the Sun have to be differentiated. Then the gain of the
the modification of gravity (see [20–23] for general for- differential signal is suppressed by the propagation dis-
mulations, and for more specific cases, nonzero graviton tance of the order of 1AU. A similar method us-
∼
mass [24, 25] and extra dimensions [26]). Also the mod- ing the Rømer time delay has been suggested recently
ification of spacetime structure at a quantum level may [34]. A GW signal from a periodic GW source is mod-
affect the propagationof a GW [27, 28]. ulated in phase due to the Earth revolution. Although
GWpropagationspeedhasbeenconstrainedindirectly this method does not require any electromagnetic obser-
fromultra-highenergycosmicrays. Assumingthecosmic vation, the measurement precision is again determined
raysoriginateinourGalaxy,theabsenceofgravitational basically by the baseline of the solar system.
Cherenkov radiation and the consequent observation of To extend the baseline and improve the sensitivity, in
ourpreviouswork[35],wehavereportedasimplemethod
directly comparing the arrival times between GWs, and
neutrinos or photons from supernovae (SN) and short
∗ [email protected]
gamma-ray burst (SGRB), assuming that the SGRB is
2
associated with a NS-NS or NS-BH binary merger [36], GW propagates with the speed of light. The sign of ∆T
whereNSandBHrepresentneutronstarandblackhole, can be both positive or negative, depending on whether
respectively. One might concern about unknown intrin- thepropagationspeedoftheGWissuperluminalorsub-
sic time delay at the source, which depends on the emis- luminal, respectively.
sion mechanisms of GWs, neutrinos, and photons. How- In order that the finite time lag due to the anomalous
ever, numerical simulations have been well developed in GW speed is detectable, ∆T has to exceed uncertainties
these days and start to allow us to predict the intrinsic intheintrinsictimelagoftheemissions,τ τ
int,min int
≤ ≤
time delays. Thanks to the developments of numerical τ , and satisfy one of the following two conditions:
int,max
simulations, the future multimessenger observations of a τ <∆T +τ for ∆T >0 and ∆T +τ <
int,max int,min int,max
GW, neutrinos, and photons can test the GW propaga- τ for ∆T <0, equivalently,
int,min
tion speed at the precision of 10−15, improving the
∼
previous suggestions by 8-10 orders of magnitude. In ∆τint < ∆T , (2)
| |
thispaper,weextendthepreviousmethodtoamultiple-
event case at cosmological distance, and show that the with ∆τint τint,max τint,min.
≡ −
intrinsic time delay canbe distinguished froma true sig- NoteinthederivationofEq.(2)thatwehavenottaken
nal due to anomalous GW speed by considering their into account the detection timing errors of a GW and a
redshift dependences. We also show that some combina- photonwhen they are detected onthe Earth. The phase
tions of signals cancel out the intrinsic time delay and errorofaGWsignificantlydependsonthesignal-to-noise
give nearly optimal sensitivity. ratio (SNR) and is given roughly by ∆φgw (SNR)−1
∼O
This paper is organized as follows. In Sec. II, we [37]. For a NS binary merger detected by aLIGO, SNR
briefly review the method comparing the arrival times is typically 10 at 200Mpc. Then the detection timing
of a GW and a high energy photon from a SGRB in or- error of a G∼W is at most 10−3sec. This is also true
∼
der to constrain GW propagation speed, extending the for ET because of a similar SNR for a NS binary even
previous formalism to compact binaries at cosmologi- at a high redshift. Because the intrinsic uncertainty of
cal distance. In Sec. III, we introduce the framework emission time, e.g. 10sec or more for SGRB photons,
∼
of Bayesian inference for parameter estimation of GW is much larger than the detection timing error, we can
propagationmodels. The method is numerically demon- neglect it when we consider the constraint on the GW
strated in Sec. IV, showing the expected constraints in speed.
the future. In Sec. V, several details of the method are Next we derive the explicit expression of ∆T, tak-
discussed, taking into account more practical situations: ing into account the redshift effect due to the cosmo-
optimality of the statistic, scaling of sensitivity, and the logical expansion, because the third-generation ground-
presence of high-z cutoff for SGRB detection and its ef- based GW detector such as ET enables us to observe
fect on sensitivity. Finally, Sec. VI is devoted to a sum- NS-NS binaries at cosmological distances up to z 2,
∼
mary. In this paper, we use the unit c=1. whileforNS-BHbinariesuptoz 4[38]. Letusassume
∼
a flat Lambda Cold Dark Matter (ΛCDM) universe for
simplicity. Strictlyspeaking,thisassumptionisnotvalid
II. ARRIVAL TIME DELAYS whenwedealwith modifiedgravitybecausedynamicsof
the cosmic expansion is also modified. However, to be
consistentwithobservationaldata,the cosmicexpansion
Let us start with a brief review of the method com-
has to be close to that in ΛCDM universe and is well
paringthe arrivaltimes ofa GWanda high-energypho-
approximated by ΛCDM model for our purpose here.
ton from the same source to constrain GW propagation
The comoving distance from the observer at z to a
speed. As a source, in this paper we concentrate on a o
SGRB, assuming that the SGRB is associated with a source at redshift z is
NS-NS or NS-BH binary merger. z υ
g
AGW is emitted atthe time t=t andis detected on χ(z ,z)= dz , (3)
e o H(z)
the Earthatt=te+Tg,wherethe arrivaltime refersto, Zzo
for instance, the merger time of a NS binary and T is
g andiswrittenχ (z ,z)whentheGWpropagationspeed
the propagation time of the GW from the source to the 0 o
is υ =c. Here H(z) is the Hubble parameter given by
Earth. Ontheotherhand,aγ-rayphotonaccompanying g
tothepromptemissionofSGRBisemittedatt=t +τ
e int H(z)=H Ω (1+z)3+Ω . (4)
with some intrinsic time delay τ and is detected at 0 m Λ
int
t = te +τint +Tγ, where Tγ is the propagation time of where Ω and Ω = 1pΩ are the energy densities of
thephotonfromthesourcetotheEarth. Theobservable m Λ − m
matter and a cosmologicalconstant, and H is the Hub-
isthedifferenceofthearrivaltimesbetweentheGWand 0
ble constant at present. In this paper, we use the cos-
the photon and is given by mological parameters, H = 100h kmMpc−1s−1 with
0 0
h = 0.68, h2Ω = 0.14 [39]. It is convenient to define
τobs =∆T +τint . (1) 0 0 m
δ (c υ )/c. The GW propagation speed υ is in
g g g
≡ −
Here wedefined ∆T T T ,whichvanishes whenthe general time-dependent [20–22] and should deviate from
γ g
≡ −
3
We define the difference of arrival times in the source
frame by
∆T(z) τ (z)
∆T˜(z) , τ˜ (z) obs . (8)
obs
≡ 1+z ≡ 1+z
Then Eq. (7) converted to in a source frame is
τ˜ (z)=∆T˜(z)+τ˜ . (9)
obs int
Thisexpressionisusefulbecauseonlythesignaldepends
onredshift,notthenoise. Furthermore,forthelateruse,
we write τ˜ as the sum of the expectation value τ˜
int int
h i
andafluctuatingpartaroundtheexpectationvalueδτ˜ .
int
Then Eq. (9) can be separated into the systematic and
Figure 1. Arrival time lags due to GW speed δg and the
intrinsictimedelayasafunctionofredshift. Forillustration, statistical terms:
theparameters are chosen as δg =10−15 and τ˜int=10sec. τ˜ (z)= τ˜ (z) +δτ˜ , (10)
obs obs obs
h i
τ˜ (z) ∆T˜(z)+ τ˜ , (11)
obs int
h i≡ h i
δτ˜ δτ˜ . (12)
obs int
c in the current epoch of the Universe if modification ≡
of gravity allows the GW speed to change and simulta-
neously explains the self-acceleration of the cosmic ex- III. BAYESIAN INFERENCE
pansion [40]. Motivated by these facts, we parameterize
the functional form as δ = δ (1+z)−n, where δ is δ
g 0 0 g The detections of multiple events at cosmological
at present and n = 0 corresponds to the constant case
distance would be realized with the third-generation
δ = δ . The index n is different in each gravity model
g 0 ground-based GW detector such as ET. From the con-
and has no preferred value from the observational point
siderationofthebeamingangleofSGRB[41],morethan
of view, but as pointed out in [40] it might be increasing
several tens of GW-SGRB coincidence events would be
faster than the decrease of the matter energy density to
observed with ET and gamma-ray detectors in a real-
affect the cosmic expansion of the current Universe. n
istic observation time, e.g. 1yr. With these coincidence
shouldnotbe largenegativenumberso asnotto diverge
events,onecandistinguishthetruesignalduetofiniteδ
g
at high redshifts. Therefore, we consider in this paper
from the intrinsic time delay of the emission at a source
the range 1 n 4. From χ( ∆z,z) = χ (0,z), the
− ≤ ≤ − 0 by utilizing their redshift dependences. To utilize multi-
time delay (or advance) induced by δ is
g plecoincidenceeventsofNS-NSbinariesorNS-BHbina-
riesandSGRB formeasuringthe propagationspeedofa
∆z z dz
∆T = =δ . (5) GW,weintroducetheframeworkofBayesianinferenceto
H 0 (1+z)nH(z)
0 Z0 estimateerrorsinmodelparametersofGWpropagation.
According to the Bayes theorem, the posterior proba-
Also the intrinsic time delay is redshifted. Denoting
bility distribution is given by
the intrinsic time delay at the source as τ˜ , the time
int
delay we observe on the Earth is p(D ~θ, )p(~θ )
p(~θ D, )= | H |H , (13)
| H p(D )
τ (z)=(1+z)τ˜ . (6) |H
int int
where ~θ is a set of model parameters, is a hypothesis,
H
The difference of arrival times observed on the Earth is and D is observational data. On the right-hand side of
Eq.(13), p(D ~θ, ) is the likelihood, p(~θ )) is the prior
τobs(z)=∆T(z)+τint(z). (7) distribution,a|ndHp(D )istheevidence.|HTheevidenceis
|H
merelyanormalizationfactoroftheposteriorprobability
In Fig. 1, the GW time delay due to δg and intrinsic distribution and does not affect physical consequences.
time delay are illustrated for the case of δg =10−15 and We assume that the statistical fluctuation of the in-
τ˜int =10sec. TheGWtimedelayincreasesatlowz,pro- trinsictime delayobeys the Gaussiandistributionwhose
portional to the distance to the source. At high z, how- variance is given by σ2 = (δτ˜ )2 . This assumption is
ever, the cosmic expansion modifies the dependence of τ int
the time delay on the distance (redshift) and the growth equivalentto writing the uDnnormalEizedlikelihoodproba-
bility of a single event using Eq. (10) as
of the time delay slows down. As the index n increases
from 1 to 4, the contribution of the time delay at high
2 2
− δτ˜ τ˜ (z ) τ˜ (z )
z is more suppressed. On the other hand, the intrinsic exp { obs,i} =exp { obs i −h obs i i} ,
time delay is constant at low z but linearly increases at "− 2στ2 # "− 2στ2 #
high z. (14)
4
where the index i discriminates each event. Since each
event is independent one another, the total likelihood is
τ˜ (z ) τ˜ (z ) 2
p(D ~θ, ) exp { obs i −h obs i i}
| H ∝ i "− 2στ2 #
Y
2
τ˜ (z ) τ˜ (z )
obs i obs i
=exp { −h i} . (15)
"− i 2στ2 #
X
In our case, the hypothesis is that the Universe is
H
described by flat ΛCDM model. However,the cosmolog-
ical parameters in the flat ΛCDM model, H and Ω ,
0 m
arewelldeterminedwithin5%precisionfromthecosmo-
logical observations [39] and their uncertainties do not
much affect the errors in the measurement of GW prop-
agation speed. Thus, we exclude H and Ω from free
0 m
parameters in our analysis and take ~θ = δ ,n, τ˜ as Figure 2. Number of NS-NS binaries (in the unit of 104)
0 int
{ h i} in each redshift bin of ∆z = 0.1 at a redshift z during 1yr
free parameters. In other words, the priors on H and
0
observation.
Ω are regarded as the delta functions. On the other
m
hand, we apply flat priorsfor δ , n, and τ˜ . Our fidu-
0 int
h i
cial values for the model parameters are δ = 0, n = 0,
0
τ˜int = 150sec. The choice of τ˜int = 150sec might where~θ′ =δ ,nandN isthetotalnumberofsources.
h i h i 0 total
seemtobeintentional. However,asdiscussedinSec.VA, Particularly,whenN ,Qˆ /N approachesthe
total 1 total
it is irrelevantto constrainthe GW speed because it can →∞
expectation value q¯. Therefore,
always be canceled out by pairing the signals.
The magnitude of a measurement noise in the time-
1
delay signal is determined by σ , which depends on the p(~θ′ D, ) exp qˆ (qˆ q¯)
emission mechanism of SGRB.τIn this paper, we con- | H ∝ "−2στ2 i i i− #
X
siderthreecases: σ =10,25,50sec. Thereasonofthese
τ 1
2
choice is because the duration of SGRB is typically less =exp (qˆ q¯)
than∼2secandthefluctuationsofτ˜int isexpectedtobe "−2στ2 (Xi i−
the same order of magnitude or less from consideration +q¯ Qˆ N q¯ . (17)
of the emission mechanisms [42]. However, to be conser- 1− total
vative, we consider not only 10sec but also larger noises (cid:16) (cid:17)oi
By the definition of the expectation value, the second
25sec and 50sec.
terminthebracketvanishes. Thus,themarginalizedpos-
When δ is a time-varying function and contains two
g
terior distribution obeys the Gaussian distribution with
free parameters, it is convenient to show the posterior
respecttoqˆ. Namely,thelogarithmicposteriordistribu-
distribution by marginalizing over τ˜ . The marginal- i
h inti tion marginalized over τ˜ obeys χ2 distribution. The
ized distribution can be derived as follows. We write int
h i
q τ˜ andqˆ τ˜ (z ) ∆T˜(z )forsimplicityofno- above result is derived for infinite Ntotal. However, it is
int i obs i i
≡h i ≡ − expectedthatEq.(17)alsoholdsforthelargenumberof
tation. From Eqs. (11), (13), and (15), the marginalized
sources.
posterior distribution is
2
qˆ q
p(~θ′ D, ) dqexp { i− } IV. NUMERICAL IMPLEMENTATION
| H ∝Z "− i 2στ2 #
X
1 Qˆ2 In this section, we numerically generate mock data of
=exp Qˆ 1
"−2στ2 2− Ntotal!# eventsandinvestigateexpectedconstraintsonmodelpa-
rameters, δ , n, and τ˜ , based on the Bayesian ap-
0 int
2 h i
1 Qˆ proach.
1
dqexp N q
×Z −2στ2 total − Ntotal! 
1 Qˆ2  A. Procedures
exp Qˆ 1 , (16)
∝ "−2στ2 2− Ntotal!#
Theproceduresofdataanalysisarecomposedofthree
stages.
Qˆ qˆ , Qˆ qˆ2 ,
1 ≡ i 2 ≡ i (i). Redshift distribution of NS binary merger events
i i
X X
5
n˙(z) is the NS merger rate per unit comoving vol-
6000
ume per unit proper time at a redshift z. The fit-
ting formula based on the observation of star for-
] 5000
mation history is given in [43] by c
e
s 4000
1+2z (z 1)
n˙(z)=n˙ 3(5 z) (1<≤z 5) , (18) y [
0× 4 − ≤ a 3000
 0 (5<z) el
d
where the quantity n˙0 represents the merger rate e 2000
m
atpresent. Although the normalizationof n˙ is still
largely uncertain, we adopt the intermediate value ti 1000
of recent estimates, n˙ = 10−6Mpc−3yr−1, as a
0
reliable estimate based on extrapolations from the 0
observed binary pulsars in our Galaxy [44]. The 0.5 z 1 1.5 2
numberofNSbinarymergerintheredshiftinterval
[z,z+dz]observedduringtheobservationtimeT
obs
is given by [43] Figure3. Onerealizationoftimedelaysignalsasafunctionof
redshift when δg =10−14 (n=0), στ =50sec,and ǫ=10−3.
dN(z) 4πr2(z) n˙(z) The red points are mock time-delay signals and the dashed
=T , (19)
dz obs H(z) 1+z curveis thetime delay dueto finiteδg.
where r(z) is the comoving radial distance and is
related to the luminosity distance d (z) by r(z)=
L
d (z)/(1+z) in the flat universe. In Fig. 2, using be canceled by taking the difference of two signals
L
Eq. (19), the redshift distribution of NS binaries at different redshifts.
per year is shown.
(iii). Computation of the posterior distribution
To generate NS binary merger events that obeys
Since we apply flat priors for δ , n, and τ˜ , the
theredshiftdistributioninEq.(19)fromahomoge- 0 int
h i
posterior distribution is obtained from the likeli-
neous random distribution, we use the Box-Muller
hood distribution in Eq. (15) except for its nor-
method [45]. In our numerical simulation, we take
malization. The posterior distribution marginal-
into account NS binary merger events only at the
ized over τ˜ is given by Eq. (16). From these
redshift range z < 2, because the electromagnetic int
h i
posterior distributions, we compute parameter es-
identification of SGRB at higher redshifts would
timationerrorsat68%CL.Tosuppressasampling
be difficult and it seems to be realistic to assume
error, we average the parameter estimation errors
that sources at z < 2 can be identified as coin-
over 100 realizations of the event list. As a result,
cident events between electromagnetic waves and
the averaged constraints are less fluctuating, but
GWs. Wedenotethefractionofcoincidenceevents
still fluctuate by 5%, at most 10%.
among all NS binary merger events by ǫ and use
∼
ǫ = 10−3, which is estimated from the simple con-
sideration of SGRB jet opening angle [41]. Thus,
B. Expected errors of model parameters
thecumulativenumberofcoincidenceeventsoutto
a redshift z is ǫN(z) and the total number of co-
incidence events is N =ǫN(z ), where N(z) In Fig. 3, the generated time-delay signals of events
total max
is the cumulative number of GW events out to a areplottedasafunctionofredshift. Justfortheillustra-
redshift z and z is the maximum redshift that tive purpose, the parameters are chosen as δ = 10−14,
max g
an electromagnetic counterpart of a GW source is στ = 50sec, and ǫ = 10−3 (Ntotal = 63). It is seen that
detected. the intrinsic time delay in the observer’s frame is red-
shifted and larger at high z and that the more sources
(ii). Generating time delay signals
aredistributedatredshiftsfrom1to1.5asexpectedfrom
Time delay signals are generated using Eq. (7) for the redshift distribution in Fig. 2.
fixed parameters δ , n, and τ˜ . The error of The posterior distribution of δ and τ˜ in the case
0 int 0 int
h i h i
the intrinsic time delay is added to each signal by of constant δ (n = 0) is shown in Fig. 4. The er-
g
generating a Gaussian error with the standard de- rors in δ and τ˜ are strongly correlated. This is
0 int
h i
viation σ , for which we choose σ =10,25,50sec. because the larger τ˜ is equal to negative δ (su-
τ τ int 0
h i
We fix the expectation value of an intrinsic time perluminal propagation) in the observational signal in
delay to τ˜ = 150sec. However, this does not Eq. (7). However, they do not completely degenerate
int
h i
loose generality because as discussed in Sec. VA because of different redshift dependence. The expected
theexpectationvalueofanintrinsictimedelaycan constraints on δ (68% CL) are 0.6 < δ /10−16 < 0.8,
g g
−
6
Figure 5. Constraint on δ0 and n when hτ˜inti distribution
is marginalized and ǫ = 10−3 (Ntotal = 63). The fiducial
parameters are chosen δ0 = 0, n = 0, and hτ˜inti = 150sec,
Figure4. Constraintonconstantcaseδg =δ0andhτ˜intiwhen represented by a black point at the center of the figure. The
ǫ = 10−3 (Ntotal = 63). The fiducial parameters are chosen fluctuations of intrinsic time delays are στ = 10 (red, solid),
δg =δ0 =0andhτ˜inti=150sec,representedbyablackpoint 25 (green, dotted), and 50sec (blue, dashed).
at the center of the figure. From the smaller ellipses to the
larger, the fluctuations of intrinsic time delays are στ = 10,
25, and 50sec.
στ =10sec στ =25sec στ =50sec
n=−1 −0.3<δ0 <0.4 −1.0<δ0 <0.9 −1.8<δ0 <2.0
n=0 −0.8<δ0 <1.0 −2.2<δ0 <2.0 −4.8<δ0 <4.7
n=1 −2.1<δ0 <2.1 −4.9<δ0 <5.3 −10.2<δ0 <10.0
n=2 −2.9<δ0 <2.6 −6.1<δ0 <7.5 −13.8<δ0 <12.8
n=4 −3.1<δ0 <2.9 −6.8<δ0 <8.0 −15.0<δ0 <14.9
Table I.Expected constraint on δ0 (68% CL) for different στ
and n in the redshift-dependent δg case with fiducial param-
eters δ0 = 0 and n = 0. The values of δ0 is in the unit of
10−16.
Figure 6. A differential signal of arrival time delays
|hs(z1,z2)i| in the unit of sec when δg =10−15 (n=0). The
diagonal line is z1 =z2.
2.0 < δ /10−16 < 1.7, and 3.6 < δ /10−16 < 3.5 for
g g
− −
σ =10, 25, and 50sec.
τ
V. DISCUSSIONS
In Fig. 5, the posterior distribution marginalized over
τ˜ is shown. The constraint is tighter at smaller n
int
jhustibecause of the redshift dependence of δg. When n In this section, we focus on the case of constant δg
is negative, the absolute value of δ increases at higher (n = 0) and investigate physical aspects of sensitivity
g
redshifts. On the other hand, when n is positive, δ is and a concrete statistic to interpret the results.
g
| |
suppressedathigherredshiftsandbecomesmoredifficult
to detect. In Table I, the projected constraints on δ for
0
differentσ andn arelisted. Itshouldbe notedthatthe A. Optimal statistic
τ
constraints on δ for n =0 are those obtained when the
0
6
fiducialparametersareδ =0andn=0. Inotherwords, If one has multiple SGRB events observed coinciden-
0
thosearewhatisderivedfromthedatawhennopositive tallybyGWandγ-raydetectors,onecandistinguishthe
detectionisachievedandtrueparametersareδ =0and true signal due to finite δ and the intrinsic time delay
0 g
n=0. at a source by looking at the redshift dependence. To
7
added. Indeed, for the event pairs with small redshift
separation, the signals are canceled out and only noises
are added. Then SNR is not improved at all. Thus, the
efficientwayofthe summationis pairingthe eventsfrom
thehighestandlowestredshiftsandaddingtheminturn.
Thisorderofsummationisalsocomputationallyefficient
to reach the maximum sensitivity on δ and would be
g
useful in a practical data analysis. The SNR for all pair
of events is
2
s
SNR2 = h mi , (23)
√2σ
m (cid:20)P τ (cid:21)
X
wheremrepresentseventpairsandrunsfromthelargest
Figure7. Constraintstoconstantδg =δ0 asafunctionofthe redshift-separation pair to the smallest one.
number of event pairs for στ = 50, 25, and 10sec from the Wenumericallygeneratemockdataofeventsthesame
toptothebottom,respectively. mrepresentseventpairsand wayasinSec.IVAtoshowexplicitlythatthenewstatis-
runs from thelargest redshift-separation pair to the smallest tic is efficient in computation and gives almost optimal
one. constraint on δ with the small number of signal pairs.
g
In Fig. 7, constraints on δ = δ (n = 0) as a function
g 0
of the number of event pairs for different σ are shown.
τ
It indicates that the SNR is dominated by only several
do so, we consider a new statistic that could be used in
event pairs m with large redshift separation and is not
a real data analysis. The observed quantity is the ar-
improvedbyaddingeventpairswithsmallerredshiftsep-
rival time delay τ˜ , from which we can construct the
obs
aration. These asymptotic values of the constraints on
following statistic:
δ = δ agree well with the error ellipses in Fig. 4. This
g 0
s(z ,z ) τ˜ (z ) τ˜ (z ) meansthatthestatisticintroducedinthissubsectionhas
i j obs i obs j
≡ − almost optimal sensitivity to δ .
=∆T˜(z ) ∆T˜(z )+δτ˜ δτ˜ . (20) g
i j int,i int,j
− −
where i and j denote i-th and j-th events. The second
B. Scaling of SNR
term is stochastic with zero mean, while there remains
finite contribution from GW. Therefore, we have
From some consideration about signal and noise, we
s(z ,z ) =∆T˜(z ) ∆T˜(z ). (21) canderivescalingrelationswithmodelparameters. Since
i j i j
h i −
the observable is given by Eq. (20), we do not have to
careabout τ˜ . Onlynoisescaleswithσ andtheSNR
int τ
Var[s(zi,zj)]= (s(zi,zj) s(zi,zj) )2 scales withhσ−1ifrom Eq. (23). Then the constraint on
h −h i i τ
=2hδτ˜i2nti δogr ǫli,ntehaerlsycaslcianlgesofwδithisσsτi.mAplsyfǫo−r1t/h2ebnecuamubseerǫodfoeevsennotst
=2σ2 (22) g
τ changetheredshiftdependenceofthesourcedistribution
but its normalization. Therefore, the scaling relation for
InFig.6,weshowtheredshiftdependenceof s(z ,z ) .
i j
|h i| the constraint on δ in the case of constant δ is
Since the noise δτ˜int does not depend on a redshift, the 0 g
redshift dependence of the SNR is identical to that of
a signal. This implies two crucial facts to construct an δ 6 10−17 10−3 1/2 στ . (24)
optimal statistic. Firstly, it hardly depends on the red- | 0|≤ × ǫ 10sec
shift for z > 1. In other words, high-z sources at z & 2, (cid:18) (cid:19) (cid:16) (cid:17)
for which it is more difficult to have an electromagnetic ThisformulaagreeswellwiththeerrorsinFig.7andthe
counterpart, do not play an important role in obtaining errors from Bayesian inference in Fig. 4 except for some
large SNR. Secondly, since ∆T˜(z) is monotonously in- statisticalfluctuations. Thescalingalsoholdsforthecase
creasing (decreasing) function for positive (negative) δ of nonzero n. In Table I, the scaling of the constraints
g
below z = 1, the largest SNR is obtained by taking the withστ agreewell. However,themagnitudesoftheerrors
difference of time delays at largely separating redshifts, deteriorate because of some parameter degeneracies.
z z & 1. Thus, when one has multiple events, the
i j
| − |
tightest constraint on δ would be imposed by a part of
g
event pairs whose redshift difference is large. C. Maximum redshift dependence
One possible way to combine all signals at different
redshifts is summing the signals over z > z . However, It may happen that SGRB events are seenonly at low
i j
this is suboptimal because the signals are redundantly redshifts, having low-z cutoff at z < 2. In this case,
8
VI. CONCLUSION
In this paper, we have extensively studied the method
measuring the GW propagation speed by directly com-
paring the arrival times between GWs and photons
from NS binary mergers associated with SGRB. Partic-
ularly we have considered multiple coincidence events at
cosmological distance, the redshift distribution of GW
sources, redshift-dependent GW propagation speed, and
the statistics of intrinsic time delays. Based on the
Bayesian parameter inference in the realistic observa-
tional situationwith ET, we have obtainedthe expected
constraints on δ (68% CL): 0.6 < δ /10−16 < 0.8,
g g
2.0 < δ /10−16 < 1.7, and− 3.6 < δ /10−16 < 3.5
Figure 8. The total number of sources up to z =zmax when − g − g
ǫ=10−3. when στ = 10, 25, and 50sec, respectively, for constant
δ (n = 0), and the similar values of the same order in
g
TableIfortime-varyingGWpropagationspeed(nonzero
n). Furthermore, we have proposed an optimal statistic
that would be useful in a real data analysis. From nu-
mericalinvestigationofthisstatistic,wehaveshownthat
a systematic part of the intrinsic time delay can be can-
celedoutfromsignals,distinguishingitfromatruesignal
due to finite δ , and that the statistic gives nearly opti-
g
mal sensitivity. We also have shown that by changing
the maximum redshift below which coincidence events
areavailable,high-z SGRBaroundz =2orhigheraffect
the sensitivity modestly, while those at 1 . z . 1.5 are
crucial in constraining δ .
g
FinallywecommentonconstraintonGWpropagation
at much higher redshifts. As a measurement method of
GW propagation speed other than the one using differ-
Figure 9. Constraint on constant δg = δ0 as a function of enceofarrivaltimes,thereisasuggestionthatGWspeed
zmax for στ =50, 25, and 10sec from thetop to the bottom, different from c at high redshifts, 103, affects the cos-
respectively. ǫ=10−3. mic microwavebackground(CMB)∼spectrumand canbe
measured indirectly [46, 47]. Since the detection of B-
mode polarization by BICEP 2 turned out to be caused
by dust emissions, GW speed has not been measured
the constraint on δg is degraded in two ways. Firstly, by the CMB observation. However, since the method
the number of sources decreases, as shown in Fig. 8 as a with CMB is complementary to the method with arrival
function of maximum redshift zmax. Secondly, the signal timesfromtheviewofredshift,bothmethodswouldgive
s(zi,zj) in Eq. (21) is likely to be small due to lack a tight constraint on the redshift evolution of the GW
h i
of high-z sources. By these two effects, the constraint is propagationspeed in the future observations.
degradedinanontrivialwayasz decreases. Asshown
max
intheFig.9,thesensitivitytoδ isdrasticallydegradedif
g
thereisacutoffattheredshiftlessthanz =1. However, ACKNOWLEDGMENTS
interestingly, the degradation is modest for the cutoff at
z > 1 because SNR is almost constant for sources at TheauthorwouldliketothankG.Ballesteros,Y.Fan,
z > 1, as shown in Fig. 6. Therefore, we conclude that J. B. Jimenez, E. Malec, and J. D. Tasson for valuable
we do not necessarily have to see high-z SGRBs around comments. A. N. was supported by JSPS Postdoctoral
z =2 or higher, but those at 1.z .1.5 are crucial. Fellowships for Research Abroad.
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