Designing a cross‐correlation search for continuous‐wave gravitational radiation from a neutron star in the supernova remnant SNR 1987A★
ABSTRACT ABSTRACTA strategy is devised for a semicoherent cross-correlation search for a young neutron star in the supernova remnant SNR 1987A, using science data from the Initial Laser Interferometer Gravitational Wave Observatory (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 ε, 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 75 ≲ν≲ 450 Hz. The semicoherent 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 (∼109 templates) is also estimated. For Initial LIGO sensitivity, in the frequency band between 50 and 500 Hz, in the absence of a detected signal, we should be able to set limits of B≳ 1011 G and ε≲ 10−4.
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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;
Page 2
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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