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arXiv:1102.4654v1 [gr-qc] 23 Feb 2011

Mon. Not. R. Astron. Soc. 000, 1–17 (2010) Printed 24 February 2011(MN LATEX style file v2.2)

Designing a cross-correlation search for continuous-wave

gravitational radiation from a neutron star in the

supernova remnant SNR 1987A

C. T. Y. Chung1, A. Melatos1, B. Krishnan2, J. T. Whelan3

1School of Physics, University of Melbourne, Parkville, VIC 3010, Australia

2Max Planck Institut f¨ ur Gravitationsphysik, Am M¨ uhlenberg 1, D-14476 Golm, Germany

3Center for Computational Relativity and Gravitation and School of Mathematical Sciences,

Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USA

ligo-p1000089-v3

ABSTRACT

A strategy is devised for a semi-coherent cross-correlation search for a young neutron

star in the supernova remnant SNR 1987A, using science data from the Initial LIGO

and/or Virgo detectors. An astrophysical model for the gravitational wave phase is

introduced which describes the star’s spin down in terms of its magnetic field strength

B and ellipticity ǫ, instead of its frequency derivatives. The model accurately tracks the

gravitational wave phase from a rapidly decelerating neutron star under the restrictive

but computationally unavoidable assumption of constant braking index, an issue which

has hindered previous searches for such young objects. The theoretical sensitivity is

calculated and compared to the indirect, age-based wave strain upper limit. The age-

based limit lies above the detection threshold in the frequency band 75Hz ? ν ?

450Hz. The semi-coherent phase metric is also calculated and used to estimate the

optimal search template spacing for the search. The range of search parameters that

can be covered given our computational resources (∼ 109templates) is also estimated.

For Initial LIGO sensitivity, in the frequency band between 50Hz and 500Hz, in

the absence of a detected signal, we should be able to set limits of B ? 1011G and

ǫ ? 10−4.

Key words: gravitational waves — pulsars: general — stars: neutron — supernovae:

individual (SNR 1987A)

1 INTRODUCTION

The Laser Interferometer Gravitational Wave Observa-

tory (LIGO) achieved its design sensitivity during its

fifth science run [S5; (Abbott et al. 2009c)]. Analysis

of S5 data is progressing well, with new upper lim-

its being placed on the strength of various classes of

burst sources (Abbott et al. 2009d; Abadie et al. 2010a;

Abbott et al. 2010a), stochastic backgrounds (Giampanis

2008; Abbott et al. 2009a),

(Abbott et al. 2009e; Abadie et al. 2010c,d) and continuous-

wave sources (Abbott et al. 2009f,b, 2010b; Abadie et al.

2010b). In some cases, the LIGO limits on astrophysical

parameters beat those inferred from electromagnetic astron-

omy, e.g. the maximum ellipticity and internal magnetic field

strength of the Crab pulsar (Abbott et al. 2008, 2010b). Re-

cently, an S5 search was completed which placed upper lim-

its on the amplitude of r-mode oscillations of the neutron

compactbinarysources

star in the supernova remnant Cassiopeia A (Abadie et al.

2010b).

Aspherical,

promising

(Ostriker & Gunn 1969). The origin of the semi-permanent

quadrupole in these objects can be thermoelastic (Melatos

2000;Ushomirsky et al.2000;

Haskell 2008) or hydromagnetic (Bonazzola & Gourgoulhon

1996; Cutler2002; Haskell et al.

Akg¨ un & Wasserman 2008; Mastrano 2010). Thermoelastic

deformations arise due to uneven electron capture rates in

the neutron star crust. A persistent 5% temperature gra-

dient at the base of the crust produces a mass quadrupole

moment of ∼ 1038g cm2(ǫ ∼ 10−7; Ushomirsky et al.

2000). Hydromagnetic deformations, on the other hand, are

produced by large internal magnetic fields, and misaligned

magnetic and spin axes. For example, a neutron star

with spin frequency 300Hz and internal toroidal field

Bt ? 3.4 × 1012G has an ellipticity ǫ ∼ 10−6??Bt?/1015G?

isolated

of

neutronstarsconstitute

candidates

one

classcontinuous-wave source

Nayyar & Owen2006;

2008; Haskell2008;

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2C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

(Cutler 2002). The deformation of an ideal fluid star with

an arbitrary magnetic field distribution and a barotropic

equation of state can be computed, with ellipticities as high

as 10−5predicted for some configurations (Haskell et al.

2008). Additionally, some exotic neutron star models (e.g.

solid strange quark stars) allow for ellipticities as large

as 10−4(Owen 2005). Accreting neutron stars in binary

systems can form magnetic mountains, with ǫ ? 10−5

(Melatos & Payne 2005; Vigelius & Melatos 2009a). Defor-

mations of all kinds relax viscoelastically and resistively

over time, so that young neutron stars are expected to

be generally stronger gravitational wave emitters. For

example, thermoelastic deformations relax on the thermal

conduction time-scale (∼ 104yr), after the temperature

gradient in the crust has switched off (Brown & Bildsten

1998; Vigelius & Melatos 2010). Magnetic mountains relax

as the accreted matter diffuses through the magnetic field

on the Ohmic time-scale 105–108yr (Vigelius & Melatos

2009b).

A coherent search for 78 known radio pulsars was per-

formed on S3 and S4 LIGO and GEO 600 data. Upper limits

on the ellipticities of these pulsars were obtained, the small-

est being ǫ ? 7.1×10−7for PSR J2124–3358 (Abbott et al.

2007c). More recently, a coherent search for 116 known pul-

sars was carried out using data from both the LIGO and

Virgo detectors, placing an upper limit of ǫ < 7.0×10−8for

PSR J2124−3358.

The youngest isolated neutron star accessible to LIGO

probably resides in the supernova remnant SNR 1987A.

The coincident detection of neutrino bursts from the su-

pernova by detectors all over the world confirmed the core-

collapse event, strongly indicating the formation of a neutron

star (Aglietta et al. 1987; Hirata et al. 1987; Bionta et al.

1987; Bahcall et al. 1987).1Constraints have been placed

on the magnetic field strength, spin period, and other

birth properties of the putative neutron star (Michel 1994;

¨ Ogelman & Alpar 2004); see Section 2 for details. However,

searches for a pulsar in SNR 1987A have yielded no con-

firmed sightings; upper limits on its luminosity have been

placed in the radio, optical and X-ray bands (Percival et al.

1995; Burrows et al. 2000; Manchester 2007). An uncon-

firmed detection of a transitory 467.5 Hz optical/infra-red

pulsation in SNR 1987A was reported by Middleditch et al.

(2000).

The likely existence of a young neutron star in SNR

1987A makes it a good target for gravitational wave searches

(Piran & Nakamura 1988; Nakamura 1989). A coherent

matched filtering search was carried out in 2003 with the

TAMA 300 detector, searching 1.2×103hours of data from

its first science run over a 1-Hz band centered on 934.9Hz,

assuming a spin-down range of (2–3)×10−10Hz s−1. The

search yielded an upper limit on the wave strain of 5×10−23

1An unconfirmed correlation was also reported between data

taken by the Mont Blanc and Kamioka neutrino detectors

and gravitationalwave detectors

(Amaldi et al. 1989). Taken at face value, these observations are

consistent with a weak neutrino pulsar operating briefly during

the core-collapse event. However, a serious flaw in the original

analysis was found by Dickson & Schutz (1995), whose reanalysis

led them to conclude that the correlations were not physically

significant.

in Maryland and Rome

(Soida et al. 2003). An earlier matched filtering search was

conducted using 102hours of data taken in 1989 by the

Garching prototype laser interferometer. The latter search

was carried out over 4-Hz bands near 2kHz and 4kHz, did

not include any spin-down parameters, and yielded an upper

limit of 9×10−21on the wave strain (Niebauer et al. 1993).

There are two main types of continuous-wave LIGO

searches: coherent and semi-coherent. The former demand

phase coherence between the signal and search template over

the entire time series. Although sensitive, they are restricted

to small observation times and parameter ranges as they

are computationally intensive. Semi-coherent searches break

the full time series into many small chunks, analyse each

chunk coherently, then sum the results incoherently, trad-

ing off sensitivity for computational load. Santostasi et al.

(2003) discussed the detectability of gravitational waves

from SNR 1987A, estimating that a coherent search based

on the Middleditch et al. (2000) spin parameters requires 30

days of integration time and at least 1019search templates

covering just the frequency and its first derivative. In reality,

the task is even more daunting, because such a young object

spins down so rapidly, that five or six higher-order frequency

derivatives must be searched in order to accurately track

the gravitational wave phase. A Bayesian Markov Chain

Monte Carlo method was proposed as an alternative to

cover the parameter space efficiently (Umst¨ atter et al. 2004;

Umst¨ aetter et al. 2008). As yet, though, SNR 1987A has not

been considered a feasible search target, because even Monte

Carlo methods are too arduous. In this paper, we show how

to reduce the search space dramatically by assuming an as-

trophysically motivated phase model.

In this paper, we discuss how to use a cross-correlation

algorithm to search for periodic gravitational waves from a

neutron star in SNR 1987A. The search is semi-coherent

(Dhurandhar et al. 2008). The signal-to-noise ratio is en-

hanced by cross-correlating two data sets separated by an

adjustable time lag, or two simultaneous data sets from dif-

ferent interferometers, thereby nullifying short-term timing

noise (e.g. from rotational glitches). This is a modification

of the method used in searches for a cosmological stochastic

background (Abbott et al. 2007a, 2009a) and for the low-

mass X-ray binary Sco X-1 (Abbott et al. 2007b). In Section

2, we review the properties of SNR 1987A and its putative

neutron star. Section 3 briefly describes the cross-correlation

algorithm and the data format. We estimate the theoretical

sensitivity of the search in Section 4. Section 5 describes

an astrophysical model, which expresses the gravitational

wave phase in terms of the initial spin, ellipticity, magnetic

field, and electromagnetic braking index of the neutron star.

We calculate the semi-coherent phase metric and the num-

ber of templates required for the search in the context of

the astrophysical phase model. Given the computational re-

sources available to us, we derive upper limits on the grav-

itational wave strain, ellipticity and magnetic field which

can be placed on a neutron star in SNR 1987A with a cross-

correlation search. Finally, Section 7 summarises the results.

2A YOUNG NEUTRON STAR IN SNR 1987A

SNR 1987A is the remnant of a Type II core-collapse su-

pernova which occurred in February 1987, 51.4kpc away

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Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A3

in the Large Magellanic Cloud (α = 5h 35m 28.03s, δ

= −69◦16′11.79′′) (see reviews by Panagia 2008 and in

Immler et al. 2007). Its progenitor was the blue supergiant

Sk 1 (Panagia 1987; Gilmozzi et al. 1987; Barkat & Wheeler

1988; Woosley et al. 2002). The color of the progeni-

tor, as well as the origin of the complex three-ring

nebula in the remnant, are still unexplained. Detailed

simulations of the evolutionary history of Sk 1, per-

formed by Podsiadlowski et al. (2007), support the the-

ory that two massive stars merged to form an oversized

20M⊙ red supergiant 2 × 105years before the supernova,

which eventually shrank as its envelope evaporated (e.g.

Podsiadlowski & Joss 1989; Podsiadlowski et al. 1990). An

alternative theory suggests that Sk 1 was instead a single 18–

20M⊙ red supergiant which evolved into a blue supergiant

via wind-driven mass loss (e.g. Woosley 1988; Saio et al.

1988; Sugerman et al. 2005).

There is strong evidence for the existence of a neu-

tron star in SNR 1987A. The progenitor mass range re-

quired to produce Type II supernovae, 10–25M⊙, which in-

cludes the above evolutionary scenarios, is the same range

required to produce neutron star remnants (Woosley et al.

2002; Heger et al. 2003). The secure neutrino detections

mentioned in Section 1 support this conclusion. Although

there have been no confirmed pulsar detections, numerous

searches have placed upper limits on the flux and luminos-

ity at radio (< 115 µJy at 1390 MHz, Manchester 2007),

optical/near-UV (< 8×1033ergs s−1, Graves & et al. 2005),

and soft X-ray (< 2.3 × 1034erg s−1, Burrows et al. 2000)

wavelengths. Middleditch et al. (2000) reported finding an

optical pulsar in SNR 1987A with a frequency of 467.5Hz,

modulated sinusoidally with a ∼ 1-ks period, consistent

with precession for an ellipticity of ǫ ∼ 10−6. However,

the pulsations were reported to have disappeared after 1996

(Middleditch et al. 2000) and were never confirmed indepen-

dently.

There are several possible reasons why a pulsar in SNR

1987A has not yet been detected. If its spin period is greater

than 0.1s, it would not be bright enough to be detectable in

the optical band (Pacini & Salvati 1987; Manchester 2007).

If the radio emission is incoherent or the emission region is

patchy, the pulses may have been missed, even if the beam

width is as wide as is typical for young pulsars (Manchester

2007). Shternin & Yakovlev (2008) argued that, although

the neutron star’s theoretical X-ray luminosity exceeds the

observational upper limits by a factor of 20–100, the current

upper limits still allow for concealment behind an opaque

shell formed by fallback (Woosley & Weaver 1995). How-

ever, simulations by Fryer et al. (1999) suggest that, once

fallback ceases, the accreted material cools, leaving no ob-

scuring atmosphere.

Another possible reason why a pulsar has not yet been

detected is that its magnetic field is too weak. The weak-

field theory is supported by theoretical models, in which the

field grows only after the neutron star is formed and can

take up to 103years to develop (e.g. Blandford & Romani

1988; Reisenegger 2003). A growth model for SNR 1987A

was proposed by Michel (1994), in which the magnetic

field of a millisecond pulsar intensifies from 1010G at

birth to ∼ 1012G after several hundred years (exponen-

tial and linear growth were considered, yielding growth

times of ∼ 0.3–0.7 kyr), before the pulsar has time to spin

down significantly. In an alternative model, the neutron

star is born with a strong magnetic field, which is ampli-

fied during the first few seconds of its life by dynamo ac-

tion (e.g. Duncan & Thompson 1992; Bonanno et al. 2005).

Assuming this model, measurements of the known spin

periods of isolated radio pulsars imply a distribution of

birth magnetic field strengths between 1012G and 1013G

(Arzoumanian et al. 2002; Faucher-Gigu` ere & Kaspi 2006).

Several birth scenarios for the pulsar in SNR 1987A were

considered by¨ Ogelman & Alpar (2004) in this context, who

concluded that the maximum magnetic dipole moment is

< 1.1×1026G cm3, 2.5×1028G cm3, and 2.5×1030G cm3

for birth periods of 2ms, 30ms, and 0.3s respectively. How-

ever, the dynamo model also accommodates a magnetar in

SNR 1987A, with magnetic dipole moment > 2.4 × 1034G

cm3, regardless of the initial spin period (¨ Ogelman & Alpar

2004).

Estimates of the birth spin of the pulsar in SNR

1987A are more uncertain. Simulations of the bounce and

post-bounce phases of core collapse were performed by

Ott et al. (2006) to determine the correlation between pro-

genitor properties and birth spin. These authors found

proto-neutron star spin periods of between 4.7–140 ms,

proportional to the progenitor’s spin period. A Monte

Carlo population synthesis study using known velocity

distributions (Arzoumanian et al. 2002) favoured shorter

millisecond periods, but a similar population study by

Faucher-Gigu` ere & Kaspi (2006) argued that the birth spin

periods could be as high as several hundred milliseconds.

Faint, non-pulsed X-ray emission from SNR 1987A was first

observed four months after the supernova and decreased

steadily in 1989 (Dotani et al. 1987; Inoue et al. 1991), lead-

ing to the suggestion that a neutron star could be powering

a plerion that is partially obscured by a fragmented super-

nova envelope. Bandiera et al. (1988) modelled the X-ray

spectrum from a nebula containing a central pulsar, with a

magnetic field of 1012G and an expansion rate of 5×108cm

s−1. The authors found a fit to the SNR 1987A data for a

pulsar spin period of 18ms.

3THE CROSS-CORRELATION ALGORITHM

In this section, we briefly summarise the cross-correlation

method described in Dhurandhar et al. (2008), a semi-

coherent search algorithm designed specifically to search

for continuous-wave gravitational radiation. It operates on

Short Fourier Transforms (SFTs) of data segments of length

∆T = 30 min, whose duration is chosen to minimise the

Doppler effects due to Earth’s rotation. In each SFT, the

kth frequency bin corresponds to the frequency νk= k/∆T

for 0 ? k ? N/2 and νk= (k−N)/∆T for N/2 ? k ? N−1,

where N is the total number of frequency bins in the SFT.

The output x(t) of a detector is the sum of the instan-

taneous noise, n(t), and the gravitational wave signal, h(t).

The noise is assumed to be zero mean, stationary, and Gaus-

sian. Its power is characterised by Sn(ν), the single-sided

power spectral density (i.e. the frequency-dependent noise

floor) in the following way:

?˜ n(ν)∗˜ n(ν′)? =1

2Sn(ν)δ(ν − ν′),(1)

where∗denotes complex conjugation. Therefore, in the low

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4C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

signal limit (|h(t)| ≪ |n(t)|), the power in the k-th frequency

bin of SFT I can be approximated by

?|˜ xk,I|2? ≈∆T

2

Sn(νk), (2)

where we apply the finite time approximation to the delta

function in (1), i.e. δ∆T(ν) = sin(πν∆T)/(πν) ≈ ∆T.

In the cross-correlation algorithm, SFTs are paired ac-

cording to some criterion (e.g. time lag or interferome-

ter combination) and multiplied to form the raw cross-

correlation variable

Yk,IJ =˜ x∗

kI,I˜ xkJ,J

(∆T)2

, (3)

where I and J index the SFTs in the pair. The gravitational

wave signal is assumed to be concentrated in a single fre-

quency bin in each SFT (because ∆T ≪ ν/˙ ν due to sidereal

or intrinsic effects), whose index is denoted by kI or kJ.

The frequency bins in the two SFTs are not necessarily the

same; they are related by the time lag between the pair and

between interferometers, as well as spin-down and Doppler

effects. For an isolated source, the instantaneous frequency

at time t is given by

ν(t) = ˆ ν(t) + ˆ ν(t)v · n

c

,(4)

where ˆ ν(t) is the instantaneous frequency in the rest frame of

the source, v is the detector velocity relative to the source,

n is the position vector pointing from the detector to the

source, and c is the speed of light. The instantaneous signal

frequencies in SFTs I and J, νI and νJ, are calculated at

the times corresponding to the midpoints of the SFTs, TI

and TJ. The frequency bin kJ is therefore shifted from kI by

an amount ∆TδνIJ, with δνIJ = νJ−νI (Dhurandhar et al.

2008). For convenience, we now drop the subscripts kI and

kJ.

In the low signal limit, YIJ is a random, complex vari-

able. The cross-correlation statistic comprises a weighted

sum of YIJ over all pairs IJ. YIJ has variance σ2

S(I)

tral density of SFT I at frequency νI, and S(J)

power spectral density of SFT J at frequency νJ.

The parameters describing the amplitude and the phase

of the signal are contained within the signal cross-correlation

function˜GIJ, defined as

˜GIJ

+FI×FJ×A2

with ∆ΦIJ = ΦI(TI) − ΦJ(TJ). ΦI(TI) and νI(TI) are the

phase and frequency at time TI, whereas ΦJ(TJ) and νJ(TJ)

are evaluated at time TJ. Note that there is an error in equa-

tion (3.10) of Dhurandhar et al. (2008), which omits the fac-

tor of e−iπ∆T[νI(TI)−νJ(TJ)]arising from the choice of time

origin of the Fourier transforms. The phase factors are deter-

mined by the astrophysical phase model described in Section

5.

The terms in square brackets in (5) depend on the po-

larization angle ψ, and the inclination angle ι between n and

the rotation axis of the pulsar, in the following way:

IJ =

n (νI)S(J)

n (νJ)/(4∆T2), where S(I)

n (νI) is the power spec-

n (νJ) is the

=

1

4e−i∆ΦIJe−iπ∆T[νI(TI)−νJ(TJ)]?FI+FJ+A2

×− i(FI+FJ×− FI×FJ+)A+A×

+

?,(5)

A+

A×

=

1 + cos2ι

2

cosι,

,(6)

=(7)

F+(t;n,ψ)=a(t;n)cos2ψ + b(t;n)sin2ψ,(8)

F×(t;n,ψ)=b(t;n)cos2ψ − a(t;n)sin2ψ, (9)

where a(t;n) and b(t;n) are the detector response functions

for a given sky position, and are defined in equations (12)

and (13) of Jaranowski et al. (1998). A geometrical defini-

tion is also given in Prix & Whelan (2007). The gravitational

wave strain tensor is

← →

h (t) = h0A+cosΦ(t)← →

where h0 is the gravitational wave strain, and← →

basis tensors for the + and × polarizations in the transverse-

traceless gauge.

In principle, one should search over the unknowns cosι

and ψ, but this adds to the already sizeable computational

burden. Accordingly, it is customary to average over cosι

and ψ when computing˜GIJ, with

?˜GIJ?cos ι,ψ=

e ++ h0A×sinΦ(t)← →

e ×

(10)

e +,× are the

1

10exp−i∆ΦIJe−iπ∆T[νI(TI)−νJ(TJ)](aIaJ+bIbJ),

(11)

where aI,J = a(TI,J;n) and bI,J = b(TI,J;n). Once the first-

pass search is complete, a follow-up search on any promising

candidates can then be performed, which searches explicitly

over cosι and ψ. Preliminary Monte Carlo tests indicate that

the detection statistic resulting from (11) is approximately

10−15% smaller than if the exact cosι and ψ values are used.

The cross-correlation detection statistic is a weighted

sum of YIJ over SFT pairs. The number of pairs which can

be summed over are limited by the available computational

power. We discuss the computational costs of running the

search in Section 6.2. The cross-correlation detection statis-

tic is given by

ρ = ΣIJ(uIJYIJ + u∗

IJY∗

IJ),(12)

where the weights are defined by

uIJ =

˜G∗

σ2

IJ

IJ

.(13)

For each frequency and sky position that is searched, we

obtain one real value of ρ, which is a sum of the Fourier

power from all the pairs. Ignoring self-correlations (i.e. no

SFT is paired with itself), the mean of ρ is given by µρ =

h2

0

?

and if self-correlations are included, µρ and σ2

(Dhurandhar et al. 2008).

IJ|˜GIJ|2/σ2

ρ= 2ΣIJ|˜GIJ|2/σ2

IJ. In the low signal limit, the variance of ρ

IJ. In the presence of a strong signal,is σ2

ρscale as h2

0

4SENSITIVITY

4.1Detection threshold

Detection candidates are selected if they exceed a threshold

value, ρth. For a given false alarm rate Fa, this threshold is

given by (Dhurandhar et al. 2008)

ρth= 21/2σρerfc−1(2Fa/N),(14)

where erfc is the complementary error function, and N is

the number of search templates used. In the presence of a

signal, the detection rate for events with ρ > ρthis given by

γ =1

2erfc

?ρth− µρ

√2σρ

?

.(15)

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Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A5

As µρ

tional wave strain that is detectable by the search to be

(Dhurandhar et al. 2008)

∝h2

0, one can calculate the lowest gravita-

hth(ν) =

S1/2

√2?|˜GIJ|2?1/4N1/4

pairs

?Sn(ν)

∆T

?1/2

. (16)

In (16), we define S = erfc−1(2Fa)+erfc−1(2Fd), Fd is the

false dismissal rate, ?|˜GIJ|2? is the mean-square of the signal

cross-correlation function defined in (5), Npairsis the number

of SFT pairs, and Sn(ν) is the single-sided power spectral

density of the interferometers (assumed to be identical).

One can estimate ?|GIJ|2?1/4theoretically for the spe-

cial case where TI = TJ and˜GIJ is averaged over cosι, ψ,

and sidereal time. In this case, the primary contribution to

∆ΦIJ is the term [r(TI) − r(TJ)] · n/c, where r(t) is the

position of the detector at time t in the frame of the solar

system barycentre. Under these assumptions, equation (11)

can be expressed in terms of the overlap reduction func-

tion (Whelan 2006), which depends only on ν,α, and δ. For

SNR 1987A, we have (α,δ) = (1.46375 rad, −1.20899 rad),

and hence ?|˜GIJ|2?−1/4= 4.6882. Assuming Fa = Fd = 0.1,

∆T = 1800s, and Npairs = 105(approximately 1 year of

SFTs), equation (16) gives

hth(ν) = 5.92 × 10−3

?Sn(ν)

Hz−1

?1/2

. (17)

Figure 1 is a graph of hth as a function of ν. The val-

ues of Sn(ν) are based on LIGO’s S5 noise characteristics.2

The S5 run began in November 2005 and accumulated a

year’s worth of triple coincidence data. For a signal from

SNR 1987A to be detectable, we must have hth? h0.

4.2Minimum ellipticity and indirect, age-based

limit

The deformation of a neutron star is parameterised by its

ellipticity ǫ. The gravitational wave strain at Earth emitted

by a biaxial neutron star is

h0 =4π2G

c4

Iǫν2

D

(18)

where G is Newton’s gravitational constant, c is the speed

of light, I is the moment of inertia, D is the distance to the

source, and ν is the gravitational wave frequency, assumed

to be twice the spin frequency (Jaranowski et al. 1998).

An upper limit on h0 can be derived from existing elec-

tromagnetic data by assuming all the observed spin down

comes from the gravitational wave torque, i.e. the observed

frequency derivative ˙ ν satisfies ˙ ν = −(32π4Gǫ2Iν5)/(5c5)

(Wette et al. 2008). Combining this with (18) to eliminate ǫ

gives

h0 ?1

D

?5GI|˙ ν|

2c3ν

?1/2

.(19)

Hence, for SNR 1987A to be detectable (i.e. hth ? h0), we

require

hth(ν)

?

1.66 × 10−20

2Available at http://www.ligo.caltech.edu/˜jzweizig/distribution/LSC Data

×

?

I

1038kgm2/s

?1/2?|˙ ν|

ν

?1/2?

D

51.4kpc

?−1

(20)

Unfortunately, without having observed any pulsations

from SNR 1987A, it is impossible to determine ν or |˙ ν| a

priori. Instead, we note that ˙ ν can be re-expressed in terms of

the characteristic age of the source, τc = ν/(4|˙ ν|), assuming

that ν today is much less than ν at birth. The factor 4

arises if one assumes the gravitational radiation dominates

electromagnetic spin down, in order to remain consistent

with (19); in reality, electromagnetic spin down is expected

to dominate, with τc = ν/(2|˙ ν|). Equation (20) then reduces

to

hth(ν) ? 3.39 × 10−25

?

τc

19yr

?−1/2?

D

51.4kpc

?−1

.(21)

The right-hand side of (21) is graphed as a horizontal red

line in Figure 1. The detectability condition (21) is then

satisfied for spins in the range 75Hz ? ν ? 450Hz. Note

that we have chosen τc = 19yr, the age of SNR 1987A in

2006 when the S5 search began.

It is important to note here that the assumption that ν

is currently much less than at birth is likely untrue for the

object in SNR 1987A, as it is so young. Hence, the indirect,

age-based limit in equation (21) and the horizontal line in

Figure 1 are only indicative of the expected gravitational-

wave emission strength (in fact, they are upper limits). Ex-

act calculations of ν and ˙ ν are performed in Section 5.1.

5 AN ASTROPHYSICAL MODEL FOR THE

GRAVITATIONAL WAVE PHASE

All continuous wave searches to date have used the stan-

dard model for the gravitational wave phase, described in

terms of a Taylor expansion involving spin frequency deriva-

tives (Jaranowski et al. 1998). For a young object like SNR

1987A, which spins down rapidly, it is not computationally

feasible to search over the six or more frequency derivatives

typically needed to track the phase accurately. In this sec-

tion, we present an alternative model for the gravitational

wave phase, stated in terms of astrophysical parameters (i.e.

the magnetic field strength and the neutron star ellipticity)

instead of spin frequency derivatives. It tracks the phase ex-

actly using four parameters, under the restrictive assump-

tion (justified further below) that the braking index is con-

stant.

The phase of a slowly evolving gravitational wave signal,

Φ(t) = Φ(t0) + 2π

?t

the

t0

dtν(t),(22)

can

(Jaranowski et al. 1998)

beapproximated byTaylor expansion

Φ(t) = Φ(t0) + 2π

s

?

k=0

ν(k)

tk+1

(k + 1)!+2πn · r(t0)

c

s

?

k=0

ν(k)tk

k!

(23)

where ν(k)is the k-th derivative of the gravitational wave

frequency at time t0, and s is the number of spin-down pa-

rameters required to achieve a given accuracy. The compu-

tational cost of using (23) is substantial for rapidly deceler-

ating objects. For a maximum allowable phase error of one

cycle, the maximum bin size in the k-th derivative is ν(k)is

Page 6

6C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

Figure 1. Theoretical sensitivity of the cross-correlation search for SNR 1987A as a function of gravitational wave frequency (blue curve),

assuming the initial LIGO detector power spectral density, a false alarm rate of 0.1, and a false dismissal rate of 0.1. The blue curve

shows the theoretical sensitivity for the special case where the search uses 105pairs of time-coincident 30-minute SFTs, and averages

over inclination angle, polarization angle, and sidereal time (see discussion in Section 4.1). The horizontal red line shows the indirect,

age-based limit assuming ν ≪ νb(see discussion in Section 4.2).

∆ν(k)= (k+1)!/Tk+1

in that derivative and Ntotal =?s

To improve on the above situation, we recognize that ˙ ν

for an isolated neutron star is the sum of gravitational-wave

and electromagnetic torque contributions:

lag, implying Nk≈ ν(k)/∆ν(k)templates

k=0Nk templates overall.

We discuss this matter further in Section 6.2.

˙ ν=−32π4Gǫ2Iν5

5c5

−Q′

−2π3R6

⋆B2νn

3µ0Ic3

?πR⋆

c

?n−3

(24)

=

1ν5− Q′

2νn,(25)

where R⋆ is the neutron star radius, B is the polar magnetic

field, n is the electromagnetic braking index (theoretically

equal to 3, but could be as low as 1.8; Melatos 1997; Palomba

2005). Assuming that the electromagnetic torque is propor-

tional to a power of ν, then ν must enter the torque in the

combination R⋆ν/c, (i.e. the ratio of R⋆to the characteristic

lever arm, the light cylinder distance, c/2πν) on dimensional

grounds. In terms of an arbitrary reference frequency, νref,

we write ˙ ν = −Q1(ν/νref)5−Q2(ν/νref)n, with Q1 = Q′

and Q2 = Q′

for simplicity.

There may, of course, be other torques acting on a newly

born neutron star. For example, nonlinear r-mode instabil-

ities can emit a significant amount of gravitational radia-

tion under certain conditions (Owen et al. 1998). If there is

a rapidly rotating pulsar with B ? 1011G in SNR 1987A,

1ν5

ref

2νn

ref. Throughout this paper, we set νref= 1Hz

its instability time scale (27 years) would exceed its age,

and the gravitational radiation from the instabilities alone

should be detectable by Advanced LIGO (Brink et al. 2004;

Bondarescu et al. 2009). However, for the purposes of our

search, we assume that the spin down is described by (25).

An equally serious issue is that n may change over the 1yr

integration period, although in (25), we assume that n is

constant. Young pulsars have n < 3, and it can be argued

that n approaches 3 over the spin-down time-scale (Melatos

1997). In this search, we maintain the assumption of con-

stant n. However, it is possible to extend (25) to include

time-dependent n in future searches. We aim, in the first

instance, to exclude the simplest astrophysical model while

recognizing that it covers only a small fraction of the total

parameter space.

When implementing the search, instead of stepping

through a grid of frequency derivatives, we search instead

over ν,Q1,Q2, and n. This reduces the number of parame-

ters and allows one to track the phase more accurately for

a given computational cost, as errors stemming from incor-

rect choices of (ν,Q1,Q2,n) grow more slowly with observa-

tion time than errors stemming from higher-order frequency

derivatives. The improvement is quantified in Section 6.2.

We note that the search targets a source with a known po-

sition, hence in our estimates we consider only a single sky

position.

Page 7

Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A7

5.1Historical spin down

We can use the possible spin histories of a source like SNR

1987A with a known age to constrain the invisible values

of (ν,Q1,Q2,n) today and hence the maximum amount of

phase evolution to be expected during a LIGO integration.

There are two ways of estimating ν and ˙ ν for a source

whose age is known. In the simplest situation, where the

current spin frequency ν is much smaller than the birth

frequency νb, the characteristic age τc ≈ −ν/[(?n? − 1) ˙ ν]

closely approximates the true age irrespective of νb, where

?n? is the mean braking index, averaged over the time since

birth. Under these conditions, a source with unknown ν and

˙ ν lies on a line of slope −τc(?n? − 1) in the ν-˙ ν plane. How-

ever, as discussed in Section 4.2, this is not necessarily true

for SNR 1987A, which was only 19 years old at the start of

the S5 search. In order to calculate ν and ˙ ν exactly with-

out using the characteristic age approximation, one must

integrate (25) over the lifetime of the source. Accordingly,

we adopt this approach and map out the regions in the ν-˙ ν

plane which can be reached from νbby electromagnetic-plus-

gravitational-wave spin down and physically sensible choices

of ǫ,B and n.

Figure 2 shows the range of possible ν and ˙ ν values at

t = 19yr obtained by solving (25) for 10−6? ǫ ? 10−3,

1011.5? B ? 1013G and 0.1kHz ? νb? 1.2kHz. For refer-

ence, we plot the search sensitivity (black curve in the ν-˙ ν

plane) obtained from (17). According to (17), the search is

only sensitive to combinations of ν and ˙ ν above the black

line. The conservative limits set by the characteristic age ap-

proximation are plotted as cyan lines. The lines correspond

to ?n? = 1.8 (top), ?n? = 3 (middle) and ?n? = 5 (bottom).

For a given value of ?n?, an object lies on the line for ν ≪ νb,

and below the lines for ν ? νb, but never above the line.

The blue, red and purple boxes contain combinations of

(ν, ˙ ν) that can be reached for various choices of ǫ, B, n, and

νb. The blue box covers the region in which B ? 1011.5G and

n = 3, and the gravitational wave torque (Q1) dominates,

i.e. ˙ νGW ≫ ˙ νEM, where the subscripts EM and GW denote

the electromagnetic and gravitational wave components of

the spin down respectively. The red box covers the region

in which the electromagnetic torque (Q2) dominates, with

B = 1013G and n = 3. The purple box also shows a region

in which the Q2 term dominates, where we have chosen B =

1013G and n = 2.3. As a rule of thumb, ǫ determines the

size of the box along the ν-axis, and νb determines the size

of the box along the ˙ ν-axis.

Let us first investigate what happens to the blue box

when we vary the minimum and maximum ellipticity, ǫmin

and ǫmax. The Q1 term dominates in the region bounded by

the blue box. The absolute value of the RQ slope increases

as ǫmin decreases, shrinking the range of ˙ ν. The curve PQ

shifts to the left as ǫmax increases, increasing |˙ ν|, and hence

lowering ν.

Let us now see what happens when we vary the min-

imum and maximum magnetic field, Bmin and Bmax. The

absolute value of the RS slope decreases as Bmin in-

creases, stretching the box sideways as we retreat from the

gravitational-wave dominated limit. The blue box is always

bounded above by the ?n? = 5 age line. It shrinks, and flat-

tens as the role of Q1 diminishes.

We now discuss the purple and red boxes in which Q2

dominates. The region bounded by the purple box has B =

1013G, and n = 2.3, whereas the red box has the same

B, but n = 3. Reducing n increases the spin-down rate by

a factor of (πR⋆/c)n−3. Hence, for the same ǫ and B, the

purple box covers a smaller range of ν than the red box.

Both are considerably smaller than the blue box for the same

range of ǫ and νb. Again, if ǫmax increases, the purple and

red boxes expand downwards. In Figure 2, we choose to plot

the purple box with n = 2.3 because it lies partially within

the sensitivity range of the search. Importantly, ν and ˙ ν end

up outside the search sensitivity range for n < 2.3 or B >

1013G, restricting the range of astrophysical birth scenarios

that our search is sensitive to.

The range of ν covered in the Q2-dominated limit is

sensitive to B. In Figure 3, we show explicitly how varying B

affects ν, ˙ ν. We plot eight red boxes, for 1011G (largest box)

? B ? 1014.5G (smallest box), and n = 3. As B increases,

the red boxes shift to the left. For B ? 5 × 1013G, the

box falls out of the sensitivity range of the search. Also, the

boxes shrink as Bmax increases. This happens because as B

increases, ˙ νEM increases. For B ? 1014G, we find ν ≪ νb

after 19 years, and the boxes end up on the ?n? = 3 line. All

the red boxes are bounded above by the ?n? = 3 age line.

Figures 2 and 3 provide constraints on the detectable

range of ǫ,B,n, over a broad range of νb. We conclude that,

in preparing to select the search templates, it is sensible

to consider the parameter range 10−5? ǫ, B ? 1013G,

2.3 ? n ? 5. A more detailed breakdown of the detectable

and computationally feasible parameter ranges is presented

in Section 6.2. Note that even though the particular boxes

drawn as examples in Figures 2 and 3 do not cover the entire

region between the sensitivity curve and the ?n? = 1.8 line,

one can potentially reach any point in that region with some

combination of n, ǫ and B. Also, each (ν, ˙ ν) pair in the

figures can be reached by an infinite set of combinations (ǫ,

B, n and νb). However, there are combinations of ν and ˙ ν

which are allowed in principle by age-based indirect limits

but which cannot be reached from νb with realistic choices

of ǫ,B, and n.

6 TEMPLATE SPACING

The cross-correlation search for SNR 1987A is computa-

tionally limited rather than sensitivity limited over much

of the parameter space in Figures 2 and 3. Therefore, the

placement of templates is crucial. If the template grid is

too coarse, the risk of missing the signal increases; if it

is too fine, time is wasted searching redundant templates.

In order to compute the optimal spacing, we construct

a phase metric (Balasubramanian et al. 1996; Owen 1996)

which computes the signal-to-noise ratio as a function of

template spacing along each axis of the four-dimensional

parameter space (ν,ǫ,B,n). The coherent phase metric for

the conventional Taylor-expansion phase model is widely

used in LIGO in both coherent and semi-coherent searches

(Brady & Creighton 2000; Prix 2007; Wette et al. 2008), al-

though its semi-coherent form has not been fully investi-

gated. In this section, we derive the semi-coherent phase

metric for the astrophysical phase model defined by integrat-

ing (25). We also estimate the range of detectable spin-down

Page 8

8C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

Figure 2. Final states (ν, ˙ ν) calculated from equation (25) on the ν-|˙ ν| plane for a range of ellipticities (10−6? ǫ ? 10−3), and birth

spin frequencies (0.10kHz ? νb? 1.2kHz), and for a 19 yr old pulsar. The blue lines surround the region where the Q1term dominates

(B ? 1011.5G, all n), the red lines surround the region where the Q2term dominates (B = 1013G,n = 3), and the purple lines surround

the region where the Q2 term dominates (B = 1013G,n = 2.3). The black curve shows the theoretical search sensitivity from solving

equation (17). The ν ≪ νbage limits are shown in cyan for ?n? = 1.8 (top), ?n? = 3 (middle) and ?n? = 5 (bottom).

Figure 3. Final states (ν, ˙ ν) calculated from equation (25) on the ν-|˙ ν| plane, for a range of magnetic field strengths. The eight red

boxes surround regions which have n = 3 and cover the same range of ǫ and νbas Figure 2. Their magnetic fields range from B = 1011G

(largest box) to B = 1014.5G (smallest box).

Page 9

Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A9

values as well as magnetic field, ellipticity and braking index

values given a computationally feasible number of templates.

6.1 Semi-coherent phase metric

When searching a template grid, it is extremely unlikely

that one particular set of parameters will match the true

signal exactly. What we have instead is a set of guessed

parameters θ +∆θ, describing the closest match, which are

offset from the true values by a small amount, ∆θ. For a

given set of guessed parameters, the power spectrum of a

time-coincident SFT pair is

P(θ,∆θ) =

2A

√∆T

????

?Tstart+∆T

Tstart

dtei∆Φ(t)

????

2

,(26)

where ∆Φ(t) = Φ(t,θ + ∆θ) − Φ(t,θ) is the mismatch be-

tween the actual and guessed phases, Tstart is the time at

the midpoint of the SFT, and A is the gravitational wave

amplitude.

The mismatch between (26) and the power spectrum of

the SFT pair if ∆θ = 0 is defined to be

m(θ,∆θ) = 1 −P(θ,∆θ)

P(θ,0)

(27)

and is related to the semi-coherent phase metric sij by

m(θ,∆θ) = sij(θ)∆θi∆θj, (28)

where 1 ? i,j ? 4 label the various search parameters.

Forthecross-correlation

(ν,Q1,Q2,n). Hence, for a given mismatch m, the mini-

mum (i.e. most conservative) template spacings are given

by ∆ν(θ) =

?m/s00(θ),∆Q1(θ) =

sible to do better (i.e. expand the spacing) by taking advan-

tage of the covariances between parameters embodied in the

metric through (28); this issue deserves further study.

In order to calculate sij, we must first calculate the

coherent phase metric, defined to be

search,wehave

θ

=

?m/s11(θ),∆Q2(θ) =

?m/s22(θ),∆n(θ) =

?m/s33(θ). Note that it may be pos-

gij = ?∂i∆Φ∂j∆Φ? − ?∂i∆Φ??∂j∆Φ?,

with ?...? =

evaluated at ∆θ = 0. Calculating gij analytically by inte-

grating (25) is non-trivial. However, a good approximation

results if we integrate (25) separately for the gravitational-

wave and electromagnetic torques, and combine the answers

in quadrature. Details of the calculation are shown in Ap-

pendix A. In brief, tracking the gravitational-wave and elec-

tromagnetic spin down separately yields two “sub-metrics”,

one comprising ν and Q1 (gravitational) and the other com-

prising ν,Q2, and n (electromagnetic). Diagonal elements of

sij can be obtained by summing the two sub-metrics.

The semi-coherent metric sij is the average of the co-

herent metric from Tstart = 0 to Tstart = Tobs, where Tobs

is the entire observation time spanned by all SFT pairs. It

is defined to be sij = (Tobs)−1?Tobs

are:

?5

1

360K2

(29)

1

Tlag

?Tstart+Tlag

Tstart

dt... and ∂i∆Φ = ∂∆Φ/∂∆θi

0

dTstartgij. From Ap-

pendix B, the diagonal elements of the semi-coherent metric

s00

≈10T2

lagT2

obs

72K2

1ν8Q2

1−

1

36K1K2nνn+3Q1Q2

+

2n2ν2n−2Q2

2

?

, (30)

s11

≈

K2

1ν10

36

K2

36

K2

T2

lagT2

obs,(31)

s22

≈

2ν2n

T2

lagT2

obs,(32)

s33

≈

2log(ν)2ν2nQ2

36

2

T2

lagT2

obs,(33)

with K1 = K1(ν,Q1) and K2 = K2(ν,Q2,n). For pure

gravitational-wave and electromagnetic spin down, we have

(K1,K2) = (1,0) and (0,1) respectively. The full expres-

sions for (30)–(33) are presented in Appendix B. Note that

in (30)–(33), all frequency terms are normalised by νref. For

clarity, we have set νref= 1Hz and do not display it.

In Appendix C, we estimate the phase error which ac-

cumulates after a time Tlag from mismatches in ν,Q1,Q2,

and n. We find that it scales with Tlag similarly to (30)–

(33) for Q1 and Q2. For ν, the phase error scales instead as

Tlag, and for n, it scales as Tlaglog?1 + (n − 1)Q2Tlagνn−1?.

within π/4 over the interval Tlag, not Tobs, unlike in fully

coherent searches. Across the entire observation time Tobs,

we require only that the frequency of the signal be tracked

to within 1/∆T. This adds an overall T2

(30)–(33).

In a semi-coherent search, the phase needs to be tracked to

obsdependence to

6.2Computational cost of the search

The run-time of the search code is proportional to

NpairsNtotal, where Ntotal is the total number of templates

required to search the parameter space. Trials with Npairs =

105comprising 1 year’s worth of SFTs (from the two in-

terferometers H1 and L1), and Tlag = 1hour take ∼ 1s per

template on a single, 1-gigaflop computational node. We can

therefore search ∼ 109templates in a realistic run using 103

nodes over two weeks.

We now compare the computational cost of the as-

trophysical phase model (25) against the Taylor-expansion

model (23). The semi-coherent metric for the latter model is

not well studied, however recent work has yielded analytic

expressions for the metric (Pletsch & Allen 2009; Pletsch

2010). Based on these expressions, we can estimate the num-

ber of templates in the following way.

Firstly, we consider the number of templates required

to track the phase coherently over a time Tlag. For the

k-th frequency derivative in the Taylor expansion model,

the corresponding diagonal term of the coherent metric

scales as (gij)(k)

lag

(Whitbeck 2006). The number

of templates required to track the k-th frequency deriva-

coh∝ T2k+2

tive coherently is then Nk ∝

tal number of templates required for each coherent chunk

of length Tlag is therefore given by Ncoh =

Ncoh ∝

derivatives required to track the gravitational wave phase

(see Section 5). Now, assume that over a time Tobs, we sum a

number of chunks incoherently, approximately proportional

to Nchunks∝ Tobs/Tlag.3Now, using the semi-coherent met-

ric (Pletsch & Allen 2009; Pletsch 2010), the number of tem-

?

(gij)(k)

coh∝ Tk+1

lag. The to-

?s

k=0Nk, i.e.

?s

k=0Tk+1

lag, where s is the number of frequency

3We emphasize that this is only an approximate estimate, as the

cross-correlation method sums SFT pairs separated by a time up

to and including Tlag. Strictly speaking, Nchunks> Tobs/Tlag.

Page 10

10C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

plates required for s frequency derivatives is proportional to

γsNcoh, where γs is a ‘refinement factor’ which scales as

Ns(s+1)/2

chunks

. The total number of templates is then approxi-

mately

Ntotal

∝Ns(s+1)/2

chunks

s?

k=0

Tk+1

lag

(34)

∝

?Tobs

Tlag

?s(s+1)/2

s?

k=0

Tk+1

lag.(35)

For the range of (νb,ǫ,B,n) considered in Section 5.1, for

Tlag = 1hr, we must track terms up to and including ν(4)

in (23) in order to keep the phase error overall below π/4.

This gives Ntotal∝ T10

Under the astrophysical phase model, we estimate

Ntotal = NνNQ1NQ2Nn from (30)–(33), where the sub-

scripts denote the number of templates required for

each individual parameter, e.g. Nν

yields different results for Nν in the gravitational and

electromagnetic limits, we bound Nν by taking it to

be the sum of squares of the two limits, i.e. Nν

?N2

Ntotal∝ m−2nν10+3nlog(ν)Q1Q2

obsT5

lag.

≈ ν/∆ν. As (30)

=

ν,(K1,K2)=(1,0)+ N2

match m, we obtain

ν,(K1,K2)=(0,1)

?1/2. For a given mis-

?Q2

21+ n2Q2

2

?1/2T4

lagT4

obs

(36)

Equation (36) is an approximate result, achieved by

combining the two sub-metrics used in equations (30)–(33).

It should be regarded as a rule of thumb. If gravitational-

wave spin down dominates, we have s22 = s33 = 0, τc =

(4Q1ν4)−1, and hence

Ntotal∝ m−1ν2τ−2

c T2

lagT2

obs. (37)

If electromagnetic spin down dominates, we have s11 = 0,

τc =?(n − 1)Q2νn−1?−1, and hence

Ntotal∝ m−3/2n2ν3log(ν)[(n − 1)τc]−3T3

The required template spacing therefore varies dramati-

cally across the astrophysical parameter range. To illustrate,

let us consider 0.1kHz ? ν ? 1kHz, 10−22s3? Q1 ?

10−18s3, 10−21s2? Q2 ? 10−13s2, and 2.3 ? n ? 3.0, and

hence 8×10−6? ǫ ? 8×10−4, 4×109G ? B ? 4×1013G.

We assume a mismatch m of 0.2. The required resolutions

in the four search parameters range across

lagT3

obs. (38)

2.935 × 10−4?

2.973 × 10−26?

3.448 × 10−20?

8.674 × 10−8?

in this search volume. The number of templates required for

each parameter is its range divided by its bin resolution. If

the bin resolution is larger than its range, we require only

one template. Equations (39)–(42) imply a total number of

templates between 2.958 × 105? Ntotal ? 4.347 × 1026to

cover the entire parameter space. Smaller values of ν,Q1,Q2

and n require fewer templates to cover their neighbourhood.

Unfortunately, given the computational restrictions

that we face, we cannot search the entire region of astro-

physical parameters in Figure 2. In the following analysis,

we therefore divide each axis in parameter space into (say)

ten bins, i.e. a 10 ×10 × 10 ×10 hypercubic grid containing

∆ν/Hz

∆Q1/s3

∆Q2/s2

? 9.632 × 10−4,

? 3.685 × 10−22,

? 2.120 × 10−16,

? 1.166 × 104

(39)

(40)

(41)

∆n(42)

νb(kHz)ǫB (1011G)

0.19–0.28

0.28–0.55

0.55–1.00

? 1.6 × 10−4

? 1.0 × 10−4

? 7.9 × 10−5

? 2.0

? 1.3

? 0.8

Table 1. Table of νb, ǫ, and B ranges (approximate) which are

detectable by the cross-correlation search for SNR 1987A using

LIGO S5 data, for 2.3 ? n ? 3.0. The numbers in the table are

based on the regions in which the computationally feasible (dark

blue dots) and search-sensitive (cyan shaded) regions overlap in

Figure 5. We assume standard values for the neutron star mass

and radius, i.e. M⋆= 1.4M⊙,R⋆= 10km, and n = 3.

104“boxes”, and calculate the localised resolution at the cen-

troid of each box. The grid is spaced logarithmically along ǫ

and B to cover Q1 and Q2 in a representative fashion. Only

those boxes requiring N ? 109are practical to search.

6.3Astrophysical upper limits

In this section, we combine the estimates of sensitivity and

computational cost in Sections 4 and 6.2 respectively to iden-

tify the ranges of the astrophysical parameters B and ǫ that

can be probed by a realistic search. In the event of a non-

detection, upper limits on B and ǫ can be placed.

We solve (25) for a range of νb,ǫ, and B, and calculate

the characteristic wave strain h0from (18). Figure 4 displays

contours of h0 versus B and ǫ for n = 3 at two frequencies

corresponding to νb = 300Hz and νb = 1200Hz. The cyan

shaded areas indicate where h0 ? hth. The search is sensitive

to a larger range of ǫ and B as νb rises. This occurs because

the search sensitivity peaks at ν ≈ 150Hz. For small νb and

large ǫ and B, the pulsar spins down after τc = 19yr to

give ν < 150Hz. In the best case scenario, for νb= 1200Hz,

upper and lower limits on the magnetic field and ellipticity

of B ? 2.5 × 1013G and ǫ ? 8 × 10−5can be achieved.

Unfortunately, the number of search templates required

to cover the shaded region in Figure 4 is prohibitively large,

as discussed in Section 6.2. Figure 5 shows both sensitivity

and computational cost. Regions in which the search is sen-

sitive (i.e. h0 ? hth) for given νb and n are shaded in cyan.

Overplotted as dark blue dots are the central coordinates

of our grid boxes with N ? 109. The panels correspond to

a range of birth frequencies, νb, and are grouped in pairs:

n = 2.335 (left panel in pair) and n = 2.965 (right panel in

pair). The top pair shows the sensitivity and computational

cost for νb = 235Hz, whereas the bottom pair corresponds

to νb= 955Hz (bottom right).

Figure 5 shows that the search sensitivity increases with

νband ǫ, and decreases with B. On the other hand, the com-

putational efficiency of the search decreases with νb and ǫ,

and increases with B. Even so, there is substantial over-

lap between the regions in which the search is sensitive and

the regions which are computationally permissible. We note

that as each individual dot in Figure 5 represents a region

in which N ? 109, it is not feasible to search over all the

dotted areas, as this would mean Ntotal ? 109. Therefore,

when implementing the search, we will choose an appropri-

ate range of parameters such that Ntotal? 109, using Figure

5 as a guide.

Table 1 summarises the approximate range of νb, ǫ, and

Page 11

Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A11

Figure 4. Contour plots of h0 as a function of ǫ and B(1011G) for SNR 1987A for values of νb= 250 (left panel) and 1200Hz (right

panel). We assume n = 3 and a pulsar age of 19 years, as the S5 run began in 2006. The cyan shaded areas correspond to h0 ? hth,

where hthis defined in (16).

Figure 5. Log-log contour plots of h0 as a function of ǫ and B (104G) for birth frequencies νb= 235Hz (top panel) and νb= 955Hz

(bottom panel), and a range braking indices, n. The frequency of the signal ν is obtained by solving (25), and integrating over τc= 19yr.

The cyan shaded areas indicate the regions in which h0? hth, where hthis defined in (16). The panels are arranged in pairs. Each pair

shows n = 2.335 (left) and n = 2.965 (right). The dark blue dots indicate parameter combinations for which one has N ? 109.

Page 12

12C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

B in which the two regions in Figure 5 overlap. If the pulsar

in SNR 1987A was born with a frequency between 0.19kHz

and 0.28kHz, the search is sensitive to ǫ ? 1.6 × 10−4and

B ? 2.0 × 1011G. This range narrows as νb increases; for

birth frequencies between 0.55kHz and 1.00kHz, the search

is sensitive to ǫ ? 7.9×10−5and B ? 0.8×1011G. We note

that these estimates, derived from the limits on Q1 and Q2,

assume the standard values for the neutron star mass and ra-

dius, M⋆ = 1.4M⊙ and R⋆ = 10km. It is possible that SNR

1987A contains a low-mass neutron star with M⋆ ≈ 0.13M⊙

(Imshenik 1992), in which case the limit on the ellipticity,

for νb= 1.00kHz, would be ǫ ? 2.5 × 10−4.

We now comment briefly on the relevance of these

limits. The range of B listed in Table 1 is within the

expected theoretical range discussed in Section 2 (Michel

1994;¨ Ogelman & Alpar 2004). The range of ǫ listed in Ta-

ble 1, however, is larger than the maximum ellipticity sus-

tainable by the unmagnetized neutron star crust for many

equations of state. For example, conventional neutron stars

are expected to support ǫ ? 10−6, while hybrid quark-

baryon or meson-condensate stars can support ǫ ? 10−5

(Ushomirsky et al. 2000; Owen 2005; Horowitz & Kadau

2009). However, some exotic models do allow for larger el-

lipticities. Solid strange quark stars are predicted to be able

to sustain ǫ ? 6 × 10−4(Owen 2005). For low-mass neutron

stars, the limit is ǫ ? 5 × 10−3(Imshenik 1992; Horowitz

2010). We note also that these limits apply only to elasti-

cally supported deformations; magnetically supported defor-

mations can be larger (Melatos 2007; Akg¨ un & Wasserman

2008; Haskell 2008). Therefore, even placing the relatively

large upper limit of ǫ ? 10−4on the putative neutron star

in SNR 1987A will be useful to some degree in constraining

its mass and/or equation of state.

7 CONCLUSION

In this paper, we describe the steps taken to quantify the

astrophysical significance of a cross-correlation search for the

supernova remnant SNR 1987A in LIGO S5 data.

• We estimate the theoretical sensitivity of the cross-

correlation search, and compare it to the conservative, age-

based, wave strain estimate. In the frequency band 75Hz

? ν ? 450Hz, the age-based estimate lies above the detec-

tion threshold.

• We introduce an alternative to the Taylor expansion

model of the gravitational wave phase based on a set of four

astrophysical search parameters (ν,ǫ,B,n). The new phase

model renders a search for a neutron star like SNR 1987A

with a high spin-down rate computationally feasible.

• To estimate the optimal template spacing for the search,

we calculate the semi-coherent phase metric corresponding

to this astrophysical model.

• We place detection limits on ǫ and B for a range of birth

spin frequencies, 0.1kHz ? νb? 1.2kHz.

With the required template spacing and current com-

putational capabilities discussed in Section 6.2, we will be

able to search up to approximately 109templates. In the

event of a non-detection, considering the parameter range

discussed in this paper and assuming the standard neutron

star mass and radius, we expect to place the following limits

on the pulsar’s ellipticity and magnetic field: ǫ ? 8 × 10−5,

B ? 2.0 × 1011G. The search is also expected to be sensi-

tive to electromagnetic braking indices 2.3 ? n ? 3.0. Its

greatest weakness remains that it assumes n to be constant

throughout the semi-coherent integration. Constant n is the

simplest possible astrophysical scenario, and it certainly de-

serves to be considered in its own right, in view of the over-

whelming computational cost of a variable-n search. Never-

theless, it is vital to recognize that the constant-n hypothesis

covers a small fraction of the astrophysical parameter space.

A search using gravitational wave data is anticipated to

begin soon and would be the first application of the cross-

correlation method to a continuous wave search.

ACKNOWLEDGEMENTS

CC acknowledges the support of an Australian Postgradu-

ate Award and the Albert Shimmins Memorial Fund. JTW

acknowledges the support of NSF grant PHY-0855494, the

College of Science at Rochester Institute of Technology, and

the German Aerospace Center (DLR). This paper has been

designated LIGO Document No. ligo-p1000089-v3.

Page 13

Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A13

APPENDIX A: CALCULATION OF THE COHERENT METRIC GIJ

This appendix details the calculation of the diagonal terms of the coherent metric, gij (29). We start by evaluating the

frequency ν(t) at time t, by assuming that ν(t) is a simple sum of separate, independent contributions from gravitational-wave

and electromagnetic spin down:

ν(t)=K1

?

−Q1ν(t)5dt + K2

K1ν

(1 + 4Q1ν4t)1/4+

?

−Q2ν(t)ndt

K2ν

(43)

=

[1 + (n − 1)Q2νn−1t]1/n−1.(44)

Here, K1 and K2 are constants which satisfy K1 + K2 = 1, and the search parameters ν,Q1,Q2, and n are defined at a

reference time t0. Recall that ν is normalised by νref, which we set to 1Hz and do not write down, for simplicity. The first

term in (44) follows from the first integral in (43) by assuming Q2 = 0. The second term in (44) follows from the second

integral in (43) by assuming Q1 = 0. Needless to say, the exact solution for ν(t) follows from solving (25) self-consistently

for Q1 ?= 0,Q2 ?= 0, but this is too difficult to solve analytically. As the phase metric calculation is useful only in an analytic

form, we adopt the approximation in (43).

The phase at time t is given by,

Φ(t,θ)=

?t+t0

K1

t0

dtν(t)(45)

=

?1 + 4Q1ν4?t +r.n

+K2

c

??3/4

3Q1ν3

−K1

?1 + 4Q1ν4t0

??2−n

?3/4

?1 + (n − 1)Q2νn−1t0

3Q1ν3

?1 + (n − 1)Q2νn−1?t +r.n

c

1−n

(n − 2)Q2νn−2

−K2

?2−n

1−n

(n − 2)Q2νn−2

. (46)

We can expand each term in the regimes (Q1ν4)−1≫ t and (Q2νn−1)−1≫ t, giving

?

+K2ν

?

Φ(t,θ)=K1νt +r.n

c

− t0

?

−K1

−K2

2Q1ν5

2Q2νn[(t +r.n

??

t +r.n

c

?2

)2− t2

− t2

0

?

t +r.n

c

− t0

?

c

0], (47)

and

Φ(t,θ + ∆θ)=K1(ν + ∆ν)

?

t +r.n

c

− t0

?

−K1

−K2

2(Q1+ ∆Q1)(ν + ∆ν)5

??

t +r.n

c

?2

− t2

0

?

+K2(ν + ∆ν)

?

t +r.n

c

− t0

?

2(Q2+ ∆Q2)(ν + ∆ν)n+∆n

??

t +r.n

c

?2

− t2

0

?

. (48)

Subtracting (47) from (48) gives

∆Φ(t)=∆ν(K2+ K1)

?

t +r.n

c

− t0

??(ν + ∆ν)5(Q1+ ∆Q1) − ν5Q1

??

?

−K1

2

??

??

t +r.n

c

?2

?2

− t2

0

?

−K2

2

t +r.n

c

− t2

0

(ν + ∆ν)n+∆n(Q2+ ∆Q2) − νnQ2

?

(49)

We now take the derivative of (49) with respect to ∆ν, ∆Q1, ∆Q2, and ∆n. We have

∂∆ν∆Φ(t)|∆θ=0

=(K1+ K2)

?

t +r.n

c

− t0

?

−5

2ν4Q1

??

t +r.n

c

?2

− t2

0

?

−K2

2nνn−1Q2

??

t −r.n

c

?2

− t2

0

?

(50)

∂∆Q1∆Φ(t)|∆θ=0

=−K1

2ν5

??

??

t +r.n

c

?2

?2

t +r.n

− t2

0

?

(51)

∂∆Q2∆Φ(t)|∆θ=0

=−K2

2νn

t +r.n

c

− t2

0

?

(52)

∂∆n∆Φ(t)|∆θ=0

=−K2

2νnQ2ln(ν)

??

c

?2

− t2

0

?

(53)

We construct gij by substituting (50)–(53) into (29). In this paper, we only require the diagonal terms of the metric. The

relevant terms (g00,g11,g22,g33) are

g00

=T2

lag

?K2

1

12+K1K2

6

+K2

2

12

?

+ T2

lag

?

Tlag+ 2r.n

c

+ 2Tstart

?

Page 14

14C. T. Y. Chung, A. Melatos, B. Krishnan, J. T. Whelan

?

−1

+15r.n

−5

12K2

1ν4Q1−5

12K1K2ν4Q1−1

?

??5

12K1K2nνn−1Q2

12K2

2nνn−1Q2

+ T2

lag

?

36K2

4T2

lag+ 15

?r.n

1+1

c

?2

+ 15TlagTstart+ 15T2

start

c

(Tlag+ 2Tstart)

1ν8Q2

18K1K2nνn+3Q1Q2+

1

180n2ν2n−2Q2

2

?

(54)

g11

=

K2

180

1ν10

?

4T4

lag+ 15T2

lag

?r.n

c

?2

+ 15T3

lagTstart+ 15T2

lagT2

start+ 15T2

lagr.n

c

(Tlag+ 2Tstart)

?

(55)

g22

=

K2

2ν2n

180

?

4T4

lag+ 15T2

lag

?r.n

c

?2

+ 15T3

lagTstart+ 15T2

lagT2

start+ 15T2

lagr.n

c

(Tlag+ 2Tstart)

?

(56)

g33

=

K2

2log(ν)2ν2nQ2

180

2

?

4T4

lag+ 15T2

lag

?r.n

c

?2

+ 15T3

lagTstart+ 15T2

lagT2

start+ 15T2

lagr.n

c

(Tlag+ 2Tstart)

?

.(57)

APPENDIX B: SEMI-COHERENT METRIC

In this appendix, we list in full the diagonal terms of the semi-coherent metric presented in (30)–(33). The relevant terms

(s00,s11,s22,s33) are

s00

=T2

lag

?1

?

?

12K2

1+1

6K1K2+

1

12K2

??

+r.n

2

?

12ν4Q1K2

2+r.n

+T2

lag

Tlag+ 2r.n

c

+ Tobs

−5

1−5

12K1K2ν4Q1−1

12K1K2nνn−1Q2−1

12K2

2nνn−1Q2

?

+T2

lag

8T2

lag+ 30

?

Tlagr.n

ccc

Tobs

?

2n2ν2n−2Q2

+ 15TlagTobs+ 10T2

obs

?

?5

K2

360

K2

360

K2

72K2

1ν8Q2

1−

1

36K1K2nνn+3Q1Q2+

1

360K2

2

?

lagTobsr.n

(58)

s11

=

1ν10

?

?

8T4

lag+ 30T2

lagr.n

c

2+ 15T3

lag

?

?

2r.n

c

+ Tobs

?

?

+ 30T2

c

+ 10T2

lagT2

obs

?

?

(59)

s22

=

2ν2n

8T4

lag+ 30T2

lagr.n

c

2+ 15T3

lag

2r.n

c

+ Tobs

+ 30T2

lagTobsr.n

c

+ 10T2

lagT2

obs

(60)

s33

=

2log(ν)2ν2nQ2

360

2

?

8T4

lag+ 30T2

lag

?r.n

c

?2

+ 15T3

lag

?

2r.n

c

+ Tobs

?

+ 30T2

lagTobsr.n

c

+ 10T2

lagT2

obs

?

(61)

APPENDIX C: ANALYTIC ACCURACY ESTIMATES FOR THE ASTROPHYSICAL PHASE MODEL

In this appendix, we motivate (30)–(33) physically by calculating the phase error in two special cases: (i) pure gravitational-

wave spin down, and (ii) pure electromagnetic spin down. In the gravitational wave case, (25) reduces to

dν

dt

=−Q1ν5

(62)

ν(t)=

ν

(1 + 4Q1ν4t)1/4

(63)

where we take νref= 1Hz for simplicity and ν = ν(t = 0). The gravitational wave phase at t = Tlag is then

Φ(Tlag) − Φ(t0)=

?Tlag+t0

(1 + 4Q1ν4Tlag)3/4− 1

3Q1ν3

t0

dtν(t) (64)

=

.(65)

There are two regimes to be considered: (i) Tlag ≫ (4Q1ν4)−1, and ii) Tlag ≪ (4Q1ν4)−1. In terms of the characteristic

age

τc(t) =

ν(t)

4|˙

ν(t)|

,(66)

the two regimes correspond to (i) Tlag ≫ τc(t0), and (ii) Tlag ≪ τc(t0). In the case of SNR 1987A, we have τc ≈ 19years (in

2006, when the S5 run began) and Tlag≈ 1hr, i.e. regime (ii).

Page 15

Cross-correlation search for continuous-wave gravitational radiation from a neutron star in SNR 1987A15

Given small errors ∆Q1 and ∆ν in Q1 and ν, the phase error that accumulates between the template and the signal after

a time Tlag is

∆Φ=

dΦ

dQ1∆Q1+dΦ

−1

Overall, therefore, the number of templates required scales as T3

down. This scaling matches the conventional Taylor expansion if ν and ˙ ν suffice to track the signal (Ntotal∝ T3

more economical if ¨ ν is needed (Ntotal∝ T6

1987A search, we cover frequencies above 0.1kHz, so ¨ ν always contributes significantly. Hence the phase model (25) is always

preferable.

Now suppose the electromagnetic term dominates. Equation (25) with νref= 1Hz reduces to

dν∆ν(67)

=

2ν5T2

lag∆Q1+ Tlag∆ν(68)

lagregardless of how rapidly the neutron star is spinning

lag) but is much

lag), which happens for ν > 1.7×10−5Hz?τc/102yr?−2(Tlag/1hour)−3. In the SNR

dν

dt

=−Q2νn

(69)

ν(t)=

ν

[1 + (n − 1)Q2νn−1t]1/(n−1)

and the gravitational wave phase after a time Tlag is

(70)

Φ(Tlag)=

?Tlag+t0

?1 + (n − 1)Q2νn−1Tlag

For small errors ∆Q2, ∆ν and ∆n in Q2, ν and n, the phase error between the template at the signal after a time Tlag is

t0

dtν(t) (71)

=

?2−n

1−n− 1

(n − 2)Q2νn−2

.(72)

∆Φ=

dΦ

dQ2∆Q2+dΦ

?

2

Tlaglog?1 + (n − 1)Q2Tlagνn−1?∆n

Hence in the electromagnetic limit, the phase error due to ∆ν scales in the same way as in the gravitational wave limit. The

phase error due to ∆Q2 scales as T2

number of templates required scales as T4

which, as shown above, is true for the range of signal frequencies considered in this search.

dν∆ν +dΦ

?

dn∆n(73)

=−

νnT2

lag

∆Q2+ Tlag∆ν +

(74)

lag, and the phase error due to ∆n scales as Tlaglog?1 + (n − 1)Q2Tlagνn−1?. Overall, the

laglog(Tlag). This represents a saving if the second frequency derivative is important

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