Parameter choices and ranges for continuous gravitational wave searches for steadily spinning neutron stars

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by  D. I. Jones

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed16July2015 (MNLATEXstylefilev2.2)
Parameter choices and ranges for continuous gravitational
wave searches for steadily spinning neutron stars
D. I. Jones1
1 Mathematical Sciences and STAG Research Centre, University of Southampton, Southampton SO17 1BJ, UK
16July2015
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ABSTRACT
2
We consider the issue of selecting parameters and their associated ranges for carrying

l out searches for continuous gravitational waves from steadily rotating neutron stars.
u
J We consider three different cases (i) the ‘classic’ case of a star spinning about a prin-
cipal axis; (ii) a biaxial star, not spinning about a principal axis; (iii) a triaxial star
5
spinning steady, but not about a principal axis (as described in Jones, MNRAS 402,
1
2503 (2010)). The first of these emits only at one frequency; the other two at a pair

] of harmonically related frequencies. We show that in all three cases, when written in
c termsoftheoriginal‘sourceparameters’,thereexistanumberofdiscretedegeneracies,
q
with different parameter values giving rise to the same gravitational wave signal. We
-
r showhowthesecanberemovedbysuitablyrestrictingthesourceparameterranges.In
g
thecaseofthemodelaswrittendownbyJones,thereisalsoacontinuousdegeneracy.
[
Weshowhowtoremovethisthroughasuitablerewritingintermsof‘waveformparam-

2 eters’,chosensoastomakethespecialisationstotheotherstellarmodelsparticularly
v simple. We briefly consider the (non-trivial) relation between the assignment of prior
2 probabilities on one set of parameters verses the other. The results of this paper will
3 be of use when designing strategies for carrying out searches for such multi-harmonic
8 gravitational wave signals, and when performing parameter estimation in the event of
5
a detection.
0
. Key words: gravitational waves – methods: data analysis – stars: neutron – stars:
1
rotation
0
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1
:
v
1 OVERVIEW sar, where a small band around 2f was searched (Abbott
i
X et al. (2008), and Aasi et al. (2015)).
r However, as shown in Jones (2010), the presence of a
a Rotatingneutronstarsarepotentiallydetectablesourcesof
pinnedsuperfluidwithinthestarcanchangethispicture.A
continuousgravitationalradiation.Theymayemitasteady
starcanthenrotatesteadily,andstillproducegravitational
gravitational wave signal because of a non-axisymmetry in
radiation at both f and 2f, providingthe axis aboutwhich
their mass distributions, caused either by elastic strains in
the pinning takes place does not coincide with a principal
theirsolidphase(s),orbymagneticstrainssourcedbytheir
axis of the star’s moment of inertia tensor. This motivates
global magnetic field (see e.g. Andersson et al. (2011) for a
thecarryingoutofsearchesforsuchmulti-harmonicsignals
review). A star with such a ‘mountain’ will typically emit
from known pulsars, despite their lack of precession.
atafrequency2f,wheref isthespinfrequency.Ifthestar
does not rotate steadily but instead undergoes free preces- Giventheseconsiderations,wecanidentifythreetypes
sion, the gravitational wave signal is then emitted at fre- ofcontinuousgravitationalwaveemission.Thereisthegen-
quencies equal (or close to) both f and 2f (Zimmermann eral case (as per Jones (2010)), which we term the triaxial
&Szedenits1979;Jones&Andersson2002).However,such non-aligned case, with emission at both f and 2f. There
precessionwouldnormallyleaveanimprintontheobserved is also the simplest case, of a triaxial star spinning about
radiopulsationreceivedfromapulsar(seee.g.Jones&An- a principal axis. This is the sort of signal assumed in most
dersson (2001)). Such modulations are not typically seen targeted gravitational wave searches to-date, and produces
in the known pulsars. For this reason, most gravitational emission at only 2f. We term this the triaxial aligned case.
wavesearchestodatethathavetargetedknownpulsarshave There is also an intermediate biaxial case, where the star
searchedonlyatthefrequency2f;seeAasietal.(2014)and isassumedbiaxial.Thisproducesgravitationalradiationat
references therein. (The exceptions have been two ‘narrow bothf and2f,withawaveformofintermediatecomplexity.
band’searchesfortheCrabpulsarandonefortheVelapul- The waveform in this case is identical to that of a biaxial
(cid:13)c 0000RAS
2 Jones
precessingstar,asconsideredintheliterature(Zimmermann source parameters and priors written in terms of the wave-
&Szedenits1979;Jones&Andersson2002),soisanimpor- form parameters. We summarise our findings in Section 5.
tant case to include, but note that in the model of Jones Fortheconvenienceofthosecarryingoutgravitationalwave
(2010), the gravitational wave emission is not accompanied searches,intheAppendixweprovidesummarytablesshow-
by precession. ing(onepossiblechoice)ofrangesappropriatetothesource
A number of parameters appear in these models; we parameters(TableC1)andthewaveformparameters(Table
term these conventional parameter choices the source pa- C2).
rameters. It turns out that there are two issues with these
parameters, which we address in this paper. Both issues re-
late to the existence of degeneracies, i.e. the existence of 2 CHOOSING RANGES IN THE SOURCE
different values of the source parameters that produce the PARAMETERS
samedetectedgravitationalwavesignalh(t).Firstly,forall
We will consider gravitational wave emission from a star
ofthemodels,thereexistanumberofdiscrete degeneracies.
spinningsteadily,i.e.withanangularvelocityΩ,fixedinthe
These are related to discrete symmetries of the star’s mass
inertialframe.Thestarhasamomentofinertiatensorthat
distribution. Secondly, in the case of the model and param-
isconstantintherotatingframe,withprincipalcomponents
eterisation of Jones (2010), there also exists a continuous
(I ,I ,I ). In the most general case, the rotation axis need
symmetry, i.e. there exists a 1-parameter family of source 1 2 3
not coincide with one of these principal axes; as argued in
parameters that all produce the same h(t).
Jones (2010), the presence of an internal pinned superfluid,
We do two main things in this paper. We first identify
with pinning axis misaligned with a principal axes, will be
the discrete symmetries of the models written in terms of
of this class. As described above, and as will be elaborated
source parameters. This allows us to give minimal ranges
uponbelow,therearetwospecialcases,thetriaxialaligned
that are needed in these parameters, removing redundancy
case and the biaxial case. There are a number of features
in this form of the parameterisation, such that there is a
common to all three cases, which we will now describe.
unique set of parameters corresponding to any given star.
The signal h(t) received by a gravitational wave detec-
We present such ranges in tabular form. Secondly, we iden-
torisgivenbyequation(4)ofJaranowskietal.(1998),here-
tify a convenient set of waveform parameters, in which the
after JKS:
continuous degeneracy present in the parameterisation of
Jones (2010) is removed by effectively reducing the number h(t)=F (t)h (t)+F (t)h (t), (1)
+ + × ×
of parameters by one. Any future gravitational wave search
wheretheantennafunctionsF ,F aregivenbyequations
could be first carried out using this waveform parameter- + ×
(10) and (11) of JKS:
isation. The conversion to the corresponding 1-parameter
(but possibly more insightful) family of source parameters F (t) = sinζ[a(t)cos2ψ +b(t)sin2ψ ], (2)
+ pol pol
could then be carried out by making use of formulae given
F (t) = sinζ[b(t)cos2ψ −a(t)sin2ψ ], (3)
here,withtherangesinsourceparametersrestrictedappro- × pol pol
priately. We also (very briefly) discuss the relationship be- where ζ is the angle between the interferometer arms and
tween prior probabilities assigned to the source parameters ψ is the polarisation angle; the subscript ‘pol’ has been
pol
and the corresponding prior probabilities on the waveform added to avoid confusion with the Euler angle ψ that ap-
parameters. pears below. The functions a(t) and b(t) are complicated
Theexistenceofthecontinuousdegeneracyofthemodel functions of source location in the sky (specified by right
of Jones (2010) was noted by Bejger & Królak (2014), who accession α and declination δ) and detector location on the
wrote down a reduced parameter set that removed this de- Earth, as given by equations (12) and (13) of JKS. The
generacy. The waveform parameters used here are differ- phasing of the signal, as give by JKS equation (14), also
ent from the parameter set introduced by Bejger & Królak depends upon the source location in a complicated way, so
(2014), and more closely tied to the fundamental scalar wewillalwaysneedtocoverthefullskyparameterspaceof
quantity, the mass quadrupole moment, that described the 0 ≤ α ≤ 2π, −π/2 ≤ δ ≤ π/2. Physically, the spin vector
gravitationalwaveemissionpropertiesofthestar.Ourcho- canpointinanydirection,overthefullrangeininclination
sen parameterisation makes the specialisation to simpler angle 0 < ι < π and the full range in polarisation angle
forms of gravitational wave emission particularly transpar- 0≤ψpol ≤2π.
ent. Collecting these results, we see that in all cases, the
The results of this paper will be of use to gravitational followingrangesingravitationwavefrequency,skylocation
wave observers when devising strategies for carrying out andspinvectororientationcorrespondtophysicallydistinct
searches for gravitational wave signals with such multiple sources:
frequency components. A study of the issues raised by such
0< Ω <∞, (4)
searches is currently underway, and will be presented else-
0≤ α ≤2π, (5)
where (Pitkin et al., in preparation).
The structure of this paper is as follows. In Section 2 −π/2≤ δ ≤π/2, (6)
welookatthesourceparameterdescription,carefullyiden- 0≤ ι ≤π, (7)
tifyingthediscretesymmetries,andtherebyfindingminimal
0≤ ψ <2π. (8)
rangestowhichtheparameterscanberestricted.InSection pol
3wegivethewaveformparameterisation,inwhichthecon- Strictly, Ω is a function of time, so Ω = Ω(t). In practice,
tinuous degeneracy is removed. In Section 4 we briefly dis- the full phase evolution is often parameterised as a Taylor
cuss the relationship between priors written in terms of the expansioninΩanditstimederivatives;bywritingthesingle
(cid:13)c 0000RAS,MNRAS000,000–000
Parameters for gravitational wave searches 3
where we have defined the constant
z
φ ≡φ −φ , (12)
(cid:54) gw,0 0 obs
z˜ where φ is the azimuthal location of the observer. It was
obs
(cid:64)(cid:73)
(cid:64) obvious that a constant of this form should appear in the
(cid:64) waveform, as it is only the t = 0 position of the source
(cid:64) (cid:8)(cid:8)(cid:42)y˜ relative to the observer that can affect the received signal.
(cid:64) (cid:8) Also,itisonlytheasymmetriesinthemomentofinertia
(cid:8)
(cid:64)θ (cid:8)(cid:8) tensor that appear in the waveforms, so we define:
(cid:64)(cid:8) (cid:45)
(cid:8) y
(cid:8)(cid:8) (cid:65) ∆I21 ≡ I2−I1, (13)
(cid:8) (cid:65)
(cid:8) φ ∆I31 ≡ I3−I1. (14)
(cid:8) (cid:65)
(cid:8) ψ
(cid:8) (cid:65) We therefore see that we have a set of ten parameters,
(cid:8)(cid:25)
x (cid:65) which we refer to as the source parameters:
(cid:65)
(cid:65)(cid:85) λ ={Ω,α,δ,ι,ψ ,∆I ,∆I ,θ,φ ,ψ}. (15)
source pol 21 31 gw,0
x˜
N
The ranges in the first five parameters that correspond to
physicallydistinctstellarconfigurationsweregiveninequa-
tions (4)–(8) above.
Therangesandvaluesoftheotherparametersdepends
uponwhichofthethreecasesweconsider,buttheirmaximal
Figure 1. The orientation of our body is specified by the three
ranges are:
standard Euler angles (θ,φ,ψ), as labelled above. The fixed
inertial-frameaxesaredenotedby(x,y,z),whilethebody-frame 0≤θ≤π, 0≤φ ≤2π, 0≤ψ≤2π, (16)
gw,0
axesaredenotedby(x˜,y˜,z˜)androtateabouttheinertialz-axis.
The so-called line of nodes, N, lies along the intersection of the fortheEuler-typeangles.Forthesakeofdefiniteness,wewill
xy andx˜y˜planes. alsoassumethereissomerestrictionontheallowedsizesof
the asymmetries in the moment of inertia tensor, although
this really depends upon the (poorly constrained) physical
parameter Ω we are subsuming all such phase information mechanismsthatproducethedeformationinthefirstplace:
intotheoneparameter,toavoidintroducingfurtherparam-
−∆I <∆I <∆I , −∆I <∆I <∆I .
eters that have no bearing on our considerations here. max 21 max max 31 max
(17)
Given the form of equations (2) and (3), if one only
Someoftheseparameterscanbesettozero,orsimplydon’t
caresaboutthereceivedwaveformh(t),itisclearlypossible
appear,in thetriaxial alignedand biaxial waveforms,while
to restrict the range in ψ further:
pol the appropriate ranges are also dependent upon the model.
0≤ψ <π. (9) In this Section, our purpose is two-fold:
pol
(i) To identify particular ranges in the parameters
In fact, as we argue below, if one only cares about the re-
{∆I ,∆I ,θ,φ ,ψ} such that all physically distinct
ceivedwaveformh(t),andhasno‘prior’informationonthe 21 31 gw,0
configurations of a star’s mass quadrupole (which is the
other parameters, it is possible to restrict the range in ψ
pol quantityresponsibleforgeneratingthegravitationalwaves)
further to 0 ≤ ψ ≤ π/2. However, the full range over a
pol canbedescribeduniquely.Itwillbepossibletoincorporate
complete circle given in equation (8) is the one required to
all conceivable additional prior information on the param-
represent all physically distinct stellar configurations, and
eters (possibly obtained by electromagnetic means) within
a value for ψ obtained by non-gravitational wave means
pol theseranges.Thiswilleliminatemostofthedegeneraciesin
could lie anywhere in this interval.
the waveform. We provide a summary of a possible choice
Wenowturntotheformofthepolarisationcomponents
of source parameter ranges in Table C1.
h (t) and h (t). In the model of Jones (2010), the orienta-
+ × (ii) To identify and exploit one further discrete symme-
tionofthebody,andthereforethegravitationalwaveemis-
try in the waveform connected with the polarisation angle
sion, depend upon three angles θ,φ,ψ, basically just Euler
ψ that, in the absence of prior information, would allow
angles giving the orientation of the body with respect to pol
a search to be carried out over a slightly smaller parameter
the inertial frame; see Figure 1. (This triaxial non-aligned
range(leadingtoasimplersearch),togetherwitharulefor
casehasthetriaxialalignedandbiaxialsolutionsasspecial
generating the other, equally acceptable parameter values,
cases). Inthe pinned superfluid caseθ and ψ are constants,
in the event of a successful detection.
while φ is the angle that generates the rotation, so that
In terms of our first aim, we will exploit the fact that,
φ=Ωt+φ , (10)
0 as we are considering gravitational waves generated by the
mass quadruple of the star, we only care about the actual
with φ a constant, giving the orientation of the body at
0
orientation of the star up to a π rotation about any one of
time t=0. However, as we show below, the phase function
thebodyaxes(x˜,y˜,z˜),renderingthefullrangesofequation
thatactuallyappearsinthewaveformsisφ (ortwicethis),
gw
(16) redundantly large. For instance, suppose some glowing
given by
hotspot is observed in the rotational equator of a triaxial
φ =Ωt+φ , (11) aligned star, and some astronomer’s theory said that this
gw gw,0
(cid:13)c 0000RAS,MNRAS000,000–000
4 Jones
mustcorrespondtothepositionoftheaxisofleastmoment (iv) Probably most sensibly of all, one could search over
of inertia. Suppose this hotspot is such that at time t = 0 the reduced range 0 < ψ < π/2. This eliminates the
pol
it lies on the far side of the star, relative to the observer. discrete degeneracy, and a detection would be expected to
Wewouldthensetφ =π.However,theπ rotationsym- manifest itself as a single set of parameters. In the event of
gw,0
metry of the mass quadrupole means that there must also a detection, three other sets of parameters, corresponding
be an axis of least moment of inertia on the nearside of the toidenticalwaveforms,canthenbegeneratedbysuccessive
star at t = 0, so we could equally well set φ = 0, i.e. useofthetransformψ →ψ +π/2,whilesimultaneously
gw,0 pol pol
we have the freedom to map all values of φ into the carrying out a transform on (all or some) of the parameter
gw,0
range 0 ≤ φ < π. Other rotational symmetries can be set {∆I ,∆I ,θ,φ ,ψ}, in a way that depends upon
gw,0 21 31 gw,0
exploited for the biaxial and triaxial non-aligned cases, al- the model under consideration, as will be described in this
though their description in terms of Euler angles will be Section.
more complicated, as will be described below.
Wewillnowturntoaconsiderationofeachofourthree
Intermsofoursecondaim,notethatitisobviousfrom
models.Whenwewriteoutthepolarisationcomponentsh
the equations of JKS (i.e. equations (1)–(3) above), a ro- +
andh below,werefertoagravitationalwavedetectorwith
tation in orientation angle ψ → ψ +π/2 produces a ×
pol pol its1-armalonge andits2-armalonge ,relativetothein-
change in sign of h(t), i.e. h(t)→−h(t). But there is a sec- θ φ
ertialframeaxes(x,y,z)describedabove.SeeJones(2012)
ondwayofproducingan(alsophysicallydistinct)starwith
for the generalisation to an arbitrary detector orientation.
waveform−h(t).Consideremissionfromastardescribedby
a given set of parameters, producing a waveform h(t). Now
consideremissionforastarthathasanidenticalspinvector
2.1 Triaxial star, not spinning about a principal
and sky location, but whose density perturbation δρ away
axis
from sphericity is reversed in sign, i.e. δρ → −δρ. Such a
star will produce a waveform −h(t), and will be described In the general case the wave field can be written as (see
by a different set of parameters. If follows that if both op- Jones (2010) and Jones (2012) for details):
ei.rea.ttiohnerseairseacadrergieedneoruatcyatinontcheetwhaevwefaovremfo.rEmxipsliucnitclhy,anggiveedn, h2+Ω = 2Ωr2(1+cos2ι)(cid:8)[∆I21(sin2ψ−cos2ψcos2θ)
asignalwithaparticularsetofparameters,therewillexist −∆I sin2θ]cos2φ +∆I sin2ψcosθsin2φ(1(cid:9)8,)
three other sets of parameters corresponding to the same 31 gw 21 gw
signal, obtained, by transforming ψpol → ψpol +π/2 and h2Ω = −2Ω22cosι(cid:8)∆I sin2ψcosθcos2φ
simultaneously transforming some or all of the parameters × r 21 gw
{θ,φgw,0,ψ,∆I31,∆I21}, in a way that depends upon the −[∆I21(sin2ψ−cos2ψcos2θ)
choicesalreadymadefortheallowedrangesoftheseparam- −∆I sin2θ]sin2φ (cid:9), (19)
31 gw
eters. Note that solutions that differ by ψ → ψ +π/2
pol pol
wraimllebteersd,ewschriilbeesdolbuytiodniffsetrheanttd{iffθ,eφrgbwy,0ψ,ψ,∆→I3ψ1,∆I+21π}wpail-l hΩ+ = Ωr2 sinιcosι(cid:8)∆I21sin2ψsinθcosφgw
pol pol
bedescribedbythesamevaluesof{θ,φgw,0,ψ,∆I31,∆I21}, +(∆I21cos2ψ−∆I31)sin2θsinφgw(cid:9), (20)
seoxisdtieffnecreoannldy einffetchtes oofritehnitsastoiornt doefgtehneeirracspyinwevreectdoerssc.riTbhede hΩ× = −Ωr2 sinι(cid:8)(∆I21cos2ψ−∆I31)sin2θcosφgw
long ago in the context of biaxial precessing stars by Zim- (cid:9)
−∆I sin2ψsinθsinφ . (21)
mermann & Szedenits (1979). 21 gw
Thismeansthatthereseveralwaysinwhichthepolar- (ThisdiffersonlytriviallyfromthewaveformgiveninJones
isationanglecanbehandledinagravitationalwavesearch: (2010),weretheobserverlocationwasfixedtoφobs =−π/2,
thedetectorwithrespecttowhichh andh werereferred
+ ×
(i) If electromagnetic observations have provided a value hadits1-armalonge1 =eφ,andits2-armalonge2 =−eθ.
(or at least small range) in ψ (whose value may lie any- See Jones (2012) for the equations for the general metric
pol
whereintherange0<ψpol <2π),thenthisshouldbeused perturbationhab,andforprojectingthisontoanarbitrarily
inthesearch,andadetectionwillcorrespondtoasingleset orientated detector).
of parameters {∆I ,∆I ,θ,φ ,ψ}. The physical and therefore default ranges in the Euler
21 31 gw,0
(ii) Ifelectromagneticobservationsdonotconstrainψ , angles are given by equation (16) above. One could allow
pol
one could search over the interval 0 < ψpol < 2π, but the parameters ∆I21 and ∆I31 to take either sign (positive
thiswouldbehighlyredundant,withadetectionpotentially or negative). However, we are free to follow the common
manifesting itself at four different points over the searched conventionofrigidbodydynamicsandchooseouraxessuch
parameter range, with each inferred parameter set differing that I3 >I2 >I1, so that
by π/2 in ψ .
pol ∆I >∆I >∆I >0. (22)
(iii) More sensibly, one could search over the reduced max 31 21
range 0 < ψ < π, but again mindful of a degeneracy, There exist discrete symmetries that can be exploited
pol
withadetectionmanifestingitselfattwodifferentvaluesof to reduce the range in parameters further. The quadrupole
ψ within this range, differing by π/2, and corresponding moment tensor, and therefore the gravitational wave field,
pol
totwodifferentparametersets{∆I ,∆I ,θ,φ ,ψ}.The is invariant under rotation of π about any one of the three
21 31 gw,0
remaining parameter sets, corresponding to identical wave- bodyaxes,Ox˜,Oy˜orOz˜.ArotationaboutOz˜corresponds
forms, can then be generated by the simple transformation to ψ → ψ +π (this is obvious from the definition of ψ,
ψ →ψ +π, keeping the other parameters fixed. but a proof is given in Appendix A1). The waveform above
pol pol
(cid:13)c 0000RAS,MNRAS000,000–000
Parameters for gravitational wave searches 5
clearlyonlydependsuponsin2ψandcos2ψ(orasfunctions may be needed, e.g. the π rotation about Oz˜ of Appendix
that we be re-written in terms of these), so indeed has this A1).
symmetry. This means we can halve the range in this angle Note that if we weren’t enforcing the inequalities of
to 0<ψ<π, leaving equation(22)andwereinsteadallowingthequantities∆I
21
and∆I totakeeithersign,theoperationδρ→−δρcould
0≤θ≤π, 0≤φ ≤2π, 0≤ψ≤π. (23) 31
gw,0 beachievedmuchmoresimply,bymakingthereplacements
A rotation of π about the body’s Oy˜ axis corresponds ∆I21 → −∆I21 and ∆I31 → −∆I31. This would, however,
to the mapping (θ,φ ,ψ) → (π−θ,φ +π,−ψ); see leadtodifferentandlesseasilyderivablechoicesinminimal
gw,0 gw,0
AppendixA2.Thewaveformabovecanindeedbeshownto ranges of the Euler-type angles.
posses this symmetry. This means we can make a further See the first column of Table C1 for a summary of a
reduction,halvingtheparameterrangeofanyone(butonly possiblechoiceofparameterranges,where,fordefiniteness,
one)oftheparameters(θ,φ ,ψ).(Notethatthewaveform weusetherangesofequation(24)fortheEuler-likeangles.
gw,0
mustalsobeinvariantunderaπrotationaboutOx˜,butthis
isequivalenttothecompositionoftheabovetworotations,
2.2 Intermediate case: A biaxial star, not
so cannot generate any further reduction in the parameter
spinning about a principal axis
space.) This means there are three options for the Euler
angle parameters: Thesimplest(andconventionalway)ofdescribingabiaxial
bodyistosingleoutthez˜-axisasspecial,i.e.tosetI =I ,
0 ≤θ≤π/2, 0≤φ ≤2π, 0≤ψ≤π, or (24) 1 2
gw,0
so that ∆I = 0. The waveform can then be shown to be
21
0 ≤θ≤π, 0≤φ ≤π, 0≤ψ≤π, or (25)
gw,0 (Jones 2012)
0 ≤θ≤π, 0≤φ ≤2π, 0≤ψ≤π/2. (26)
gw,0 2Ω2
h2Ω = − (1+cos2ι)∆I sin2θcos2φ , (29)
Any one of these choices, together with the ranges of equa- + r 31 gw
tion(22)andtheparameterrangesofequations(4)–(8),will 2Ω2
always be able to accommodate any additional prior infor- h2×Ω = − r 2cosι∆I31sin2θsin2φgw, (30)
mation on the star.
Ω2
However, this parameter space is redundantly large hΩ+ = − r sinιcosι∆I31sin2θsinφgw, (31)
from the point of view of carrying out a gravitational
Ω2
wave search without additional prior information. As ar- hΩ = sinι∆I sin2θcosφ . (32)
× r 31 gw
guedabove,ifthereexistsastarproducingawavefieldwith
components hΩ,2Ω there must exist another, physically dis- Thisisthewavefieldofabiaxialstarspinningaboutanaxis
+,×
tinct star, whose wavefield has the sign of all these compo- other than a principal axis. In the model of Jones (2010) it
nents reversed, i.e. hΩ,2Ω → −hΩ,2Ω. Physically, this cor- corresponds to a non-precessing star with a pinned super-
+,× +,×
responds to reversing the sign of the density perturbation fluid. It is also identical to the GW field of a precessing
δρ that deforms the star away from spherical symmetry. biaxialstarwithoutpinning,ofthesortconsideredbyZim-
The transformation that produces the mapping hΩ,2Ω → mermann&Szedenits(1979)andJKS.So,itisanimportant
+,×
−hΩ,2Ω is rather complicated. The details are given in Ap- case to cover.
+,× Bysinglingoutthez˜-axisasthesymmetryaxis,wehave
pendixB,andinvolveatransformationmixingtheEuleran-
∆I =0. Having made this choice, we can no longer insist
gles (θ,φ ,ψ) and also mixing the amplitude parameters 21
gw,0
that I is the axis of greatest moment of inertia. Instead,
(∆I ,∆I ). If this transformation is carried out together 3
21 31
wemustallowforthestarbeingeitheroblate(∆I >0)or
with the operation ψ → ψ +π/2, the full waveform 31
pol pol
prolate(∆I <0),dependinguponthesignof∆I ,sowe
h(t) is invariant. The significance of this is that a further 31 31
have:
reduction in the parameter space is possible. One option is
to reduce the range in ψpol to the range: ∆I21 =0, −∆Imax <∆I31 <∆Imax. (33)
0<ψ <π/2. (27)
pol Now consider the Euler angles, whose ‘default’ ranges
were given in equation (16). We can again exploit symme-
There is presumably another option, involving some reduc-
tries. The waveform no longer depends upon the angle ψ,
tion in the parameter set (θ,φ ,ψ,∆I ,∆I ), but it is
gw,0 21 31
a consequence of the axisymmetry of the body about the
notclearhowtoimplementthissecondoption.Forinstance,
Oz˜body axis, so this angle is removed from our considera-
the Euler angles aren’t simply increased by π or multiplied
tions. As described in Section A2 the operation of perform-
by −1, so the reduction presumably isn’t a simple halving
ingarotationofπrotationabouttheOy˜axistakestheform
their ranges (see equations (B13)–(B15) in Section B).
(θ,φ )→(π−θ,φ +π),allowingustohalvetherange
If a gravitational wave search is carried out over this gw,0 gw,0
in θ or φ . We can therefore have:
restrictedrange0<ψ <π/2,threeotherequallyaccept- gw,0
pol
able solutions can be obtained by successive applications of
0≤θ≤π/2, 0≤φ ≤2π, or (34)
gw,0
the transformation
0≤θ≤π, 0≤φ ≤π. (35)
gw,0
ψ →ψ +π/2, (28)
pol pol Eitheroneofthesetwooptions,togetherwiththeparameter
with a corresponding transformation of ranges of equation (33) and equations (4)–(8), will always
(θ,φ ,ψ,∆I ,∆I )givenbyequations(B13)–(B15).(If beabletoaccommodateanyadditionalpriorinformationon
gw,0 21 31
one wishes the parameters to remain confined to one of the the star.
minimal ranges identified above, further transformations However, this parameter space is again redundantly
(cid:13)c 0000RAS,MNRAS000,000–000
6 Jones
large from the point of view of carrying out a gravitational (i) ψ →ψ +π/2 ,
pol pol
wavesearchwithoutadditionpriorinformation.Toseethis, (ii) (φ +ψ)→(φ +ψ)+π/2 .
gw,0 gw,0
note that the transformation ∆I →−∆I (equivalent to
31 31
Thesecondofthesetransformationisequivalenttoswapping
thetransformationδρ→−δρ)changesthesignoftheabove
polarisation components, i.e. hΩ,2Ω →−hΩ,2Ω, thereby flip- overtheaxesoflargestandsmallestmomentofinertiathat
+,× +,× lie in the rotational equatorial plane; as such it is not the
pingthesignofh(t).Thetransformationψ →ψ +π/2
pol pol
same as the transformation δρ→−δρ discussed above, but
also flips the sign of h(t), so the two transformations to-
ratherflipsthesignofh(t)inawaythatpreservesourchoice
gether leave h(t) unchanged. It follows we can reduce the
of fixing the Oz˜axis as the rotation axis.
range in ∆I or the range in ψ , i.e. we have the choice
31 pol
It follows that, in carrying out a gravitational wave
0<ψpol <π/2, −Imax <∆I31 <Imax, or (36) search, we can reduce the range in any one (but only one)
0<ψ <π, 0<∆I <I . (37) of these parameters, so the options are:
pol 31 max
This degeneracy was noted by Zimmermann & Szedenits 0≤ψpol ≤π/2, 0≤(φgw,0+ψ)≤π, or (42)
(1979). A gravitational wave search for a biaxial star could 0≤ψ ≤π, 0≤(φ +ψ)≤π/2. (43)
pol gw,0
thenbecarriedoutwitheitherofthetwochoiceshardwired
The first of these three options has traditionally been used
in, with the understanding that in the event of a detection,
in gravitational wave searches (see e.g. Aasi et al. (2014)),
three other equally valid solution can be obtained via suc-
although it should be noted that the phase angle that ap-
cessive uses of the transformation:
pears in the literature is actually
ψ →ψ +π/2, ∆I →−∆I . (38)
pol pol 31 31
Φ =2(φ +ψ), (44)
GW,0 gw,0
See the middle column of Table C1 for a summary of
a possible choice in parameter ranges, where for definite- so that searches have traditionally searched over the range
nesswechoosetherangesofequation(34)fortheEuler-like 0<ΦGW,0 <2π.
angles. Ifoneoftheserestrictedparameterspacesisusedanda
signaldetectedwithparameters(ψ ,(φ +ψ)),threead-
pol gw,0
ditionalsolutionscanbeobtainedbysuccessiveapplications
2.3 Simplest case: A triaxial star, spinning about of the transform
a principal axis
ψ →ψ +π/2, (φ +ψ)→(φ +ψ)+π/2. (45)
pol pol gw,0 gw,0
Thisisthestandardcaseofatriaxialstarspinningabouta
Withoutadditional(non-gravitationalwaveinformation)all
principal axis, emitting only at 2Ω, i.e. the sort of emission
such solutions are equally valid. (If one wishes the parame-
normallyassumedincontinuousgravitationalwavesearches.
terstoremainconfinedto,say,therange0≤(φ +ψ)≤
The convention is to choose the rotation axis to be the z- gw,0
π,thenafurthertransformation(φ +ψ)→(φ +ψ)+π
axis. This is accomplished by setting θ = 0 in equations gw,0 gw,0
can be applied when necessary).
(18–21) to give
SeethefinalcolumnofTableC1forasummaryofthese
h = −2Ω2∆I (1+cos2ι)cos2[Ωt+(φ +ψ)],(39) possible choices in parameter ranges.
+ r 21 gw,0
2Ω2
h = − ∆I 2cosιsin2[Ωt+(φ +ψ)]. (40)
× r 21 gw,0
3 REFORMULATING IN TERMS OF
Notethattheparametersanglesφ andψaredegenerate,
gw,0 WAVEFORM PARAMETERS
i.e. only their sum appears in the waveform.
We are free to lay down the (Ox˜,Oy˜) axes such that The 10-parameter triaxial non-aligned waveform of Section
∆I > 0. The symmetry of rotating by π about Oz˜ then 2.1 contains a degeneracy, and in fact only depends upon 9
21
correspondsto(φ +ψ)→(φ +ψ)+π.Thewaveform parameters.Aneasywayofseeingthisistonotethatifthe
gw,0 gw,0
clearly has this symmetry, suggesting we need to cover the cosine and sine terms in each of the four equations giving
range0<(φ +ψ)<π.Aswehavefixedθ=0,thisrota- thepolarisationcomponents(equations(18)–(21))arecom-
gw,0
tionaboutOz˜istheonlyangulardegreeoffreedom,sothere binedintosingletrigonometricterms(essentiallywritingthe
are no further symmetries we can exploit, corresponding to equations in ‘amplitude-phase’ form), the five parameters
rotationsaboutOx˜orOy˜.So,foratriaxialalignedrotator, (θ,φgw,0,ψ,∆I21,∆I31) appear in only four different com-
the set of physically distinct configurations is spanned by: binations. Another, possibly more insightful, way of under-
standing this is to return to first principles, making use of
∆I >0, 0≤(φ +ψ)<π. (41)
21 gw,0 the multipole formalism for gravitational wave emission, as
These choices, together with the parameter ranges of equa- described in Thorne (1980).
tions (4)–(8), will always be able to accommodate any ad- The fundamental quantities that appears in the wave
ditional prior information on the star. generation equations Thorne are the mass quadrupole mo-
However,theseparameterrangesareredundantlylarge, mentscalars,relatedtothesource’sdensityfieldρbyequa-
in the sense that, in the absence of such prior information, tion (5.27a) of Thorne (1980):
smaller parameter ranges can be used to carry out gravi- 16π√3(cid:90)
tational wave searches. To see this, note that the waveform I2m = ρY∗ r2dV. (46)
15 2m
changessign(h(t)→−h(t))wheneitheroneofthefollowing
transformations is performed: Thetransversetraceless(TT)descriptionoftheGWfieldis
(cid:13)c 0000RAS,MNRAS000,000–000
Parameters for gravitational wave searches 7
given by equation (4.3) of Thorne (1980): equaltothenumberofsourceparameters,sothereisnode-
generacyforbiaxialstars,onlyfortriaxialnon-alignedones.
hTabT(t)= 1r (cid:88)I¨2mTaEb2,2m, (47) Wecancollecttherelevantparameterstogethertogive
m the nine waveform parameters:
where TaEb2,2m is a tensor spherical harmonic. λwaveform ={Ω,α,δ,ι,ψpol,C˜21,ΦC21,C˜22,ΦC22}. (57)
For rigid rotation about the z-axes at rate Ω, the mass
quadrupole scalars can be shown to take the from (Jones Comparingwiththetensourceparametersofequation(15),
2012) weseethatthefirstfiveparameters{Ω,α,δ,ι,ψpol}arecom-
monbetweenthetwoparameterisations,whilethesetoffive
I2m =C2cmomplexe−im(Ωt+φ0), (48) source parameters {∆I21,∆I31,θ,φgw,0,ψ} are replaced by
whereCcomplex isacomplexnumberthatencodesdetailsof the set of four waveform parameters {C˜21,ΦC21,C˜22,ΦC22}.
2m Itwouldthereforeseemthattheremaybeanadvantage
the source, and φ is as defined above, i.e. a phase angle
0
in using the waveform parameters, rather than the source
giving the rotational phase of the body at time t = 0. We
can write Ccomplex in amplitude-angle form: parameters that naturally come out of rigid body calcula-
2m tions.Letuslookatthewaveformparameterdescriptionof
C2cmomplex =C2meiΦ2m, (49) the gravitational wave signal for the three particular cases
where C ≡|Ccomplex|≥0 and 0<Φ <2π. of interest. We have two goals: (i) to relate the source pa-
2m 2m 2m rameters to the waveform parameters, and (ii) to identify
The waveform for an arbitrary rigidly rotating source
sensible ranges to search over in the waveform parameters.
isthengivenbyequation(47).Inwritingitdown,itiscon-
A summary of the identified parameter ranges is given in
venient to include some additional factors in our amplitude
the Appendix; see table C2.
parameters; we define
(cid:114)
Ω2 5
C˜2m = r 2πC2m, (50) 3.1 Triaxial star, not spinning about a principal
axis
so that the waveform can then be shown to take the very
simple form (Jones 2012): Starting with equation (46), the motion of a triaxial non-
aligned star leads to (Jones 2012)
h+(2Ω) = −C˜ cos[2Ωt+ΦC](1+cos2ι), (51)
22 22
(cid:114)
h×(2Ω) = −C1˜22sin[2Ωt+ΦC22]2cosι, (52) I22 = −e−2i(Ωt+φ0) 85π[∆I21(sin2ψ−cos2ψcos2θ)−
h+(Ω) = − C˜ cos[Ωt+ΦC]sinιcosι, (53)
2 21 21 ∆I31sin2θ+i∆I21sin2ψcosθ], (58)
1 (cid:114)
h×(Ω) = −2C˜21sin[Ωt+ΦC21]sinι, (54) I21 = −e−i(Ωt+φ0) 85π[∆I21sin2ψsinθ+
where the phases ΦC2m are related to previously introduced i(∆I21cos2ψ−∆I31)sin2θ], (59)
quantities by
so that
ΦC = 2φ −Φ , (55)
22 gw,0 22 (cid:114)
8π
ΦC = φ −Φ . (56) Ccomplex = − [∆I (sin2ψ−cos2ψcos2θ)
21 gw,0 21 22 5 21
These equations can then be specialised to the three cases −∆I sin2θ+i∆I sin2ψcosθ], (60)
31 21
considered above. They are clearly rather simple in form, (cid:114)
8π
with all the (potentially) complicated details of the source Ccomplex = − [∆I sin2ψsinθ+
parametersbeingburiedwithintheamplitudesC˜ andC˜ , 21 5 21
22 21
and the phases ΦC22 and ΦC21. i(∆I21cos2ψ−∆I31)sin2θ], (61)
This approach also has the advantage of making the
from which we see
countingofthenumberofparametersmorestraightforward.
We can count as follows. We need the set of five parame- C˜ = Ω22{[∆I (sin2ψ−cos2ψcos2θ)−
ters{Ω,α,δ,ι,ψ }givingthespinfrequency,skylocation, 22 r 21
pol
andspinorientationofthesource,asbefore.(Themaximal ∆I sin2θ]2+(∆I sin2ψcosθ)2}1/2, (62)
31 21
ranges in these parameters were given in equations (4)–(8) Ω2
earlier).Forasteadilyrotatingsourceemittinggravitational C˜21 = r 2{(∆I21sin2ψsinθ)2+
waves only at 2Ω, we then have the amplitude-phase pair
C˜ ,ΦC also, giving seven parameters, consistent with the (∆I21cos2ψ−∆I31)2sin22θ}1/2, (63)
22 22
number of source parameters in this case. However, in the ΦC22 = 2φgw,0
triaxial non-aligned case, where the Ω-harmonic is present ∆I sin2ψcosθ
− tan−1 21 (64,)
too, we also have the amplitude-phase pair C˜21,ΦC21. This ∆I21(sin2ψ−cos2ψcos2θ)−∆I31sin2θ
givesatotalofnineparameters,notthetenthatonewould (∆I cos2ψ−∆I )sin2θ
arrive at by examining the waveform as written previously, ΦC21 = φgw,0−tan−1 21∆I sin2ψsi3n1θ . (65)
confirming the existence of a continuous degeneracy in the 21
source parameters in this triaxial non-aligned case. For the If values are given for the quantities {C˜ ,ΦC,C˜ ,ΦC},
22 22 21 21
biaxial case, we will find that there is a particular relation as would be the case in the event of a detec-
between the phases ΦC and ΦC, giving eight parameters, tion, the above four equations in the five unknowns
21 22
(cid:13)c 0000RAS,MNRAS000,000–000
8 Jones
{Ω2∆I /r,Ω2∆I /r,θ,ψ,φ } would then generate a 1- As discussed in Section 2.2, we are always free to insist
21 31 gw,0
parameter family of solutions. Note that, when evaluating 0 ≤ θ ≤ π/2, as made explicit in equation (34). With this
theinversetangentfunctionsofequations(64)and(65),care choice, both sin2θ and sin2θ will always be non-negative.
mustbetakentoselectthecorrectrootsoastocorrectlyre- In contrast, ∆I can be either positive or negative, so we
31
construct the complex mass numbers of equations (60) and should treat the ∆I >0 and ∆I <0 cases separately.
31 31
(61). For ∆I >0 case, we can read-off
31
Having found the algebraic relationship between the (cid:114)
8π
source and waveform parameters we can now turn to the C = ∆I sin2θ, Φ =0, (76)
22 5 31 22
issue of selecting ranges in the waveform parameters. Care-
(cid:114)
fulstudyofequations(62)and(63)showsthatifoneselects 8π
C = ∆I sin2θ, Φ =π/2. (77)
∆I21 and ∆I31 according to equation (22) then the corre- 21 5 31 21
sponding bounds on the amplitude parameters are:
Converting to the parameters C˜ and Φc that actually
2m 2m
2Ω2 appear in the waveform:
0 ≤ C˜ ≤ ∆I , (66)
22 r max
2Ω2
0 ≤ C˜ ≤ 2Ω2∆I . (67) C˜22 = r ∆I31sin2θ, Φc22 =2φgw,0, (78)
21 r max
2Ω2 π
C˜ = ∆I sin2θ, Φc =φ − . (79)
For the two phase parameters ΦC , the default range is: 21 r 31 21 gw,0 2
2m
0<ΦC <2π. (68) These equations can be inverted to give:
2m
Ω2∆I 1 (cid:34) (cid:18) C˜ (cid:19)2(cid:35)
Together with the ranges given in equations (4)–(8), these 31 = C˜ 1+ 21 , (80)
ranges are sufficiently wide to cover all physically distinct r 2 22 2C˜22
stellar configurations, and accommodate all possible ad- π
φ = ΦC + , (81)
ditional information obtained by non-gravitational wave gw,0 21 2
means. 2C˜
tanθ = 22. (82)
However, from the point of view of carrying out a C˜
21
gravitational wave search without such extra information,
Note that in this case
these ranges are redundantly large. The polarisation com-
ponents change sign under the operation ΦC2m → ΦC2m+π. Φc22 =2Φc21+π, (83)
The waveform h(t) also changes sign under the operation
ψ →ψ +π/2,sowecanhalvetherangeinoneorother arelationthatcouldbehardwiredintoanysearchforoblate
pol pol
of the polarisation angle or the phases. We therefore have biaxial stars using the waveform parameterisation.
the options: For ∆I31 <0 case, we can read-off
(cid:114)
00<<ψψpol <<ππ/,2, 00<<ΦΦC2Cm <<2ππ., or ((6790)) C22 = − 85π∆I31sin2θ, Φ22 =π, (84)
pol 2m (cid:114)
8π
If one or other of these restricted ranges are employed in a C21 = − 5 ∆I31sin2θ, Φ21 =−π/2. (85)
search,andadetectionismadewithparameters(ψ ,ΦC ),
three other equally acceptable solutions can bepoobltai2nmed Converting to the parameters C˜2m and Φc2m that actually
through successive applications of the transformation appear in the waveform:
ψpol →ψpol+π/2, ΦC2m →ΦC2m+π. (71) C˜22 = −2Ωr2∆I31sin2θ, Φc22 =2φgw,0+π, (86)
SeethefirstcolumnofTableC2forasummaryofthese 2Ω2 π
C˜ = − ∆I sin2θ, Φc =φ + . (87)
possible choices in parameter ranges. 21 r 31 21 gw,0 2
These equations can be inverted to give:
3.2 Intermediate case: A biaxial star, not Ω2∆I 1 (cid:34) (cid:18) C˜ (cid:19)2(cid:35)
31 = − C˜ 1+ 21 , (88)
spinning about a principal axis r 2 22 2C˜
22
Setting ∆I21 =0 in equations (58) and (59) leads to φ = ΦC − π, (89)
gw,0 21 2
(cid:114)
8π
I22 = e−2i(Ωt+φ0) ∆I sin2θ, (72) with θ given by equation (82). Note that in this case
5 31
(cid:114) Φc =2Φc , (90)
8π 22 21
I21 = ie−i(Ωt+φ0) ∆I sin2θ, (73)
5 31 a relation that could be hardwired into any search for pro-
late biaxial stars using the waveform parameterisation. In
so that
the event of a successful detection, the measured values for
(cid:114)
Ccomplex = 8π∆I sin2θ, (74) the waveform parameters could then be inserted into the
22 5 31 equationsabove,todeducethecorrespondingsourceparam-
(cid:114)8π eters.
Ccomplex = ∆I sin2θeiπ/2. (75)
21 5 31 Toidentifytherangesinthesewaveformparametersto
(cid:13)c 0000RAS,MNRAS000,000–000
Parameters for gravitational wave searches 9
searchover,wecanconverttherangesinthesourceparam- If using such a reduced parameter ranges, other equally ac-
eters of equations (33) and (34) using equations (78)–(79) ceptablesolutionscanbegeneratedthroughsuccessiveuses
above (or, equivalently, equations (86)–(87)), to give: of the transformation
0<C˜ < 2Ω2∆I , (91) ψpol →ψpol+π/2, C˜2m →−C˜2m. (98)
2m r max
We mention this possibility as, while not fitting into the
0≤ΦC <2π, (92) scheme of C2m being the modulus of a complex number,
21
it has the advantage of possibly being easier to implement,
being mindful to use both equations (83) (for oblate stars) as there is only one relation connecting the phases ΦC and
22
and equation (90) (for prolate stars) to calculate ΦC22 as a ΦC21, regardless of whether the body is oblate or prolate, so
functionofΦC21.Togetherwiththeparameterrangesofequa- some users may find it easier to integrate into their search
tions(4)–(8),theserangesarewideenoughtoaccommodate method.
all physically distinct stellar configurations.
However,fromthepointofviewofcarryingoutagrav-
itational wave search, these ranges are redundantly large. 3.3 Simplest case: A triaxial star, spinning about
The waveform changes sign under the operation ψpol → a principal axis
ψ +π/2. It also changes sign under the operation ΦC →
pol 21
ΦC +π and simultaneously, swapping the relationship be- In this case we can set θ=0 in equations (58) and (59), so
tw2e1enΦC andΦC fromequation(83)to(90),orviceversa, that I21 =0, while
21 22
whichsimplyamounttothetransformationΦC →ΦC +π. (cid:114)
22 22 8π
Itfollowsthat,intheabsenceofadditionalnon-gravitational I22 = 5 ∆I21e−2i(Ωt+φ0+ψ), (99)
wave information, we can carry out a gravitational wave
search over the reduced ranges of either so that
(cid:114)
0≤ψpol <π/2, 0≤ΦC21 <2π, (93) Ccomplex = 8π∆I e−2iψ. (100)
22 5 21
allowing for both the oblate and prolate relations of equa-
tions (83) (for oblate stars) and (90) (for prolate stars) in If we follow the convention used in Section 2.3 and choose
calculating ΦC(ΦC), or to insist that ∆I21 >0, we can then immediately read-off
22 21
(cid:114)
0≤ψ <π, 0≤ΦC <2π, (94) 8π
pol 21 C = ∆I , (101)
22 5 21
withonlyoneorother(butnotboth)oftheoblateandpro-
Φ = −2ψ. (102)
laterelationsofequations(83)(foroblatestars)and(90)in 22
calculatingΦC22(ΦC21).Ifeitheroftheserestrictedparameter Using the relation of equation (55) these equations can be
spaces is used in a search, and a detection is made, with inverted to give:
parameters (ψ ,ΦC,ΦC), three other equally acceptable
pol 21 22 Ω2∆I 1
solutionscanbeobtainedthroughsuccessiveapplicationsof 21 = C˜ , (103)
the transformation r 2 22
2(φ +ψ) = ΦC. (104)
ψ →ψ +π/2, ΦC →ΦC +π. (95) gw,0 22
pol pol 2m 2m
Note that the parameters φ and ψ are degenerate, as
Alternatively,inasearchforageneraltriaxialbody(as gw,0
expectedforthiscase.Intheeventofadetection,themea-
describedinSection3.1),findingarelationshipbetweenΦC
21 sured values of (C˜ ,ΦC) could be inserted into the above
andΦC oftheformofeitherofequations(83)or(90)would 22 22
22 equationstocomputethecorrespondingsourceparameters,
be a sign that the detected signal is coming from a biaxial
whichinthiscasearerelatedinaverystraight-forwardway.
star.
Wehavealreadyidentifiedrangesinthesourceparam-
See the second column of Table C2 for a summary of
eters that cover all possible physicality distinct stellar con-
these possible choices in parameter ranges.
figurations; see equation (41). These immediately translate
An alternative choice would have been to instead use
into the waveform parameter ranges:
equations(76)and(77)forboththe∆I >0and∆I <0
31 31
cases, with the understanding that C22 and C21 can now 0<C˜ < 2Ω2∆I , 0≤ΦC <2π. (105)
be either positive or negative, but both of the same sign 22 r max 22
(i.e. both positive, or both negative). Equations (78)–(83)
These parameter ranges, together with those of equations
then apply in both the oblate and prolate cases. The wave-
(4)–(8), will be wide enough to accommodate all possible
formcanthemademadetochangesignundertheoperation
non-gravitational wave priors.
C →−C (applied simultaneously to both the C and
2m 2m 22 However,fromthepointofviewofcarryingoutagrav-
C ). It follows that, in the absence of other information,
21 itational wave search without such prior information, these
onecansearchoverthereducedparameterrangesofeither
ranges are redundantly large. The waveform changes sign
2Ω2 2Ω2 under the operation ψ → ψ +π/2 and also under the
0<ψ <π/2, − ∆I <C˜ < ∆I , pol pol
pol r max 2m r max operation ΦC →ΦC +π, and so we can reduce the ranges
22 22
(96) in one or other (but not both) of those parameters:
or
0≤ψ ≤π/2, 0≤ΦC ≤2π,or (106)
2Ω2 pol 22
0<ψpol <π, 0<C˜2m < r ∆Imax, (97) 0≤ψpol ≤π, 0≤ΦC22 ≤π. (107)
(cid:13)c 0000RAS,MNRAS000,000–000
10 Jones
Thefirstchoiceistheonethatreflectsthechoicetradition- Tosimplifythings,wecanworkwithadimensionlessquan-
allymadeingravitationalwavesearches(seee.g.Aasietal. tity Iˆ:
(2014)).Intheeventofasuccessfuldetectionwithparame-
∆I
ters(ψ ,ΦC),threeotherequallyacceptablesolutionscan Iˆ≡ 31 . (111)
pol 22 ∆I
be generated by successive applications the of the transfor- max
mation The corresponding separately normalised priors are
ψ →ψ +π/2, ΦC →ΦC +π. (108) P(θ) = sinθ, 0≤θ≤π/2, (112)
pol pol 22 22
P(Iˆ) = 1, 0≤Iˆ≤1, (113)
SeethefinalcolumnofTableC2forasummaryofthese
possible choices in parameter ranges.
giving a joint prior
P(θ,Iˆ)=sinθ. (114)
4 RELATION BETWEEN THE PRIORS The parameter space is simply a rectangle in (θ,Iˆ) coordi-
As described above, in carrying out a gravitational wave nates, with a probability density that depends only upon
searchonehasachoiceastowhichsetofvariablesareused, θ.
the source parameters or the waveform parameters. If one To eliminate annoying factors, and made our ampli-
is using Bayesian methods to conduct the search, one also tudes dimensionless, define
needs to specify prior information on the range of each pa- C
Cˆ ≡ 21 (115)
rameter, and supply a function giving one’s initial belief as 21 (cid:113)
8π∆I
to its probability distribution. For the triaxial star rotating 5 max
aboutaprincipalaxis,thetwosetsareessentiallythesame, C
Cˆ ≡ 22 (116)
but for the biaxial star, and the triaxial star not rotating 22 (cid:113)
8π∆I
about a principal axis, there is a non-trivial conversion to 5 max
be made. so that our transformation equations become
Unfortunately, it is not obvious what a physically mo-
Cˆ = Iˆsin2θ, (117)
tivated choice of priors would be, in terms of either set 21
of parameters. This is particularly true for the amplitude- Cˆ = Iˆsin2θ. (118)
22
like source parameters ∆I ,∆I or the wave parame-
ters C˜ ,C˜ . The choice of21priors31would be related to the To see the shape of the parameter space in the
21 22 (Cˆ ,Cˆ ) variables we can look at the images of all four
strengthofthesolidcrust,theprecisemechanismproducing 21 22
sides of the rectangle formed by the (θ,Iˆ) variables:
crustaldeformation,and,forthemodelofJones(2010),the
strength and orientation of the superfluid pinning. We will (i) Thesideθ=0,0≤Iˆ≤1mapstoCˆ =Cˆ =0,i.e.
21 22
thereforecontentourselvesherewitharelativelysimplecon- collapses to the origin.
sideration: if we make some simple choice of priors for the (ii) The side Iˆ = 0, 0 ≤ θ ≤ π/2 also collapses to the
source parameters, we will evaluate the corresponding pri- origin.
orsforthewaveformparameters.Thiswillillustratethefact (iii) The side θ = π/2, 0 ≤ Iˆ ≤ 1 maps to Cˆ = 0,
21
thatasimplechoiceof,say,arelatively simplerectangular- Cˆ =Iˆ⇒0≤Cˆ ≤1.
22 22
type distribution in one set of parameters, does not corre- (iv) The side Iˆ= 1, 0 ≤ θ ≤ π/2 maps to Cˆ = sin2θ,
21
spond to such a simple distribution when expressed in the Cˆ = sin2θ. This can be shown to be equivalent to the
22
other set. curve Cˆ =+2(cid:113)Cˆ (1−Cˆ ).
Wewillonlyconsiderthebiaxialcase,asthereitiseasy 21 22 22
to carry out calculations analytically. The non-trivial con- To find the actual probability distribution within this
version is between the ‘wobble angle’ θ and the asymmetry closed region we can use the conversion formula
∆I forthesourceparameters,andtheamplitudesC and
31 21 P(Cˆ ,Cˆ )det(J)=P(θ)P(Iˆ), (119)
C in for the waveform parameters. The relevant formulae 21 22
22
are: where J is the Jacobian of the transformation:
C21 = (cid:114)85π∆I31sin2θ, (109) J =(cid:32) ∂∂CC∂ˆˆIˆ2212 ∂∂CC∂ˆˆθ2212 (cid:33)=(cid:18) ssiinn22θθ IˆIˆ2scions22θθ (cid:19). (120)
(cid:114)8π ∂Iˆ ∂θ
C22 = 5 ∆I31sin2θ. (110) Then
We will take as a simple example of a set of priors the fol- det(J)=2Iˆsin2θ=2Cˆ . (121)
22
lowing:
Eliminating ∆I between equations (109) and (110) we
31
(i) θ drawn by choosing a point randomly and uniformly have
fromtheupperhalfoftheunitsphere0≤θ≤π/2.(Thefull
range 0≤θ <π is not required, because of the degeneracy (cid:34) (cid:32) Cˆ (cid:33)2(cid:35)−1/2
sinθ= 1+ 21 , (122)
discussedinSection2.2;wecanalwaysinsistourθvaluelies 2Cˆ
22
in this upper hemisphere).
(ii) ∆I drawn uniformly over the interval (0,∆I ), and so we obtain
31 max
independently of the value of θ, corresponding to an oblate P(Cˆ ,Cˆ )= 1 . (123)
star. 21 22 [Cˆ2 +(2Cˆ )2]1/2
21 22
(cid:13)c 0000RAS,MNRAS000,000–000