Accessibility of the stochastic gravitational wave background from magnetarsto the interferometric gravitational wave detectors

Cheng-Jian Wu (吳澄劍) and Vuk Mandic

School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

Tania Regimbau

Departement Artemis, Observatoire de la Cote d’Azur, CNRS, F-06304 Nice, France(Received 14 August 2012; published 26 February 2013)

Magnetars have been proposed as sources of gravitational waves, potentially observable by current and

future terrestrial gravitational-wave detectors. In this paper, we calculate the stochastic gravitational wave

background generated by summing the contributions from all magnetars in the Universe, and we study its

accessibility to the second- and third-generation gravitational-wave detector networks. We perform

systematic scans of the parameter space in this model, allowing the magnetic field, the ellipticity, the

initial period, and the rate of magnetars to vary over the currently believed range of values. We also

consider different proposed configurations of the magnetic field (poloidal, toroidal, and twisted torus) and

different proposed star-formation histories. We identify regions in the parameter space of poloidal and

toroidal models that will be accessible to the second- and third-generation gravitational-wave detectors

and conclude that the twisted-torus models are likely out of reach of these detectors. The poloidal field

configuration with a type II superconductor equation of state in the interior, or with a highly disordered

magnetic field, and the toroidal configuration with a very strong toroidal magnetic field in the interior

(> 1016 G) are the most promising in terms of gravitational-wave detection.

DOI: 10.1103/PhysRevD.87.042002 PACS numbers: 04.30.Db

I. INTRODUCTION

Stochastic gravitational-wave background (SGWB) isexpected to arise from the superposition of contributionsfrom many independent and unresolved gravitational-wave(GW) sources. The SGWB could be cosmological—forexample, arising in various inflationary models [1–4], orin models of cosmic (super)strings [5–9]. It could also beastrophysical, because of the superposition of wavesgenerated by many astrophysical sources, such as compactbinary coalescences [10–17] and neutron stars (quadrupoleemission [18,19] or initial instabilities [20–26], includingmagnetars [24,27–30]). Magnetars, neutron stars with verystrong magnetic fields, were first proposed [31] to explainthe observed features of soft gamma repeaters (SGRs) andanomalous x-ray pulsars (AXPs). Besides driving the power-ful electromagnetic radiation that enabled observation ofthese objects, the strong magnetic field is expected to inducea quadrupolar deformation in themagnetar structure, therebygenerating GWs during its rapid spinning (if its symmetryaxis is not aligned with the spin axis). Superposition ofGWs generated by all magnetars in the Universe results ina SGWB, which is the subject of this study.

Several interferometric GW detectors around the worldhave operated over the past decade. This includes the threeLIGO detectors operating at two USA sites (Hanford, WA,and Livingston Parish, LA) [32,33], Virgo detector in Italy[34,35], and GEO600 in Germany [36,37]. These detectorsreached their initial design sensitivities and acquired sev-eral years of data that are currently being analyzed in the

search for different types of GW signals. Among others,several searches for the SGWB have been completed[38–41], placing competitive upper limits on the amplitudeof the SGWB.Currently under construction is the second generation

of GW detectors, including Advanced LIGO [42,43],Advanced Virgo [44], GEO-HF [45], and KAGRA (alsoknown as LCGT) [46]. These detectors are expected toacquire their first data in 2014, with strain sensitivitiesabout 10 times better than the initial generation of detec-tors. Furthermore, efforts are under way to design the third-generation GW detectors, with yet another factor of 10 insensitivity improvement. Specifically, a design study ofthe Einstein Telescope project [47,48] was completed in2011 in Europe. While the second-generation detectors areexpected to make the first detections of GW signals, thethird-generation detectors are expected to explore the fullscientific potential of gravitational-wave astrophysics, en-abling systematic studies of various GW sources, as well asprobing the cosmology of the very early universe. Figure 1shows the strain sensitivity curves for the first, the second,and the third generations of GW detectors that will be usedin this study.It has been argued that the SGWB due to magnetars may

be observable by the second-generation GW detectors[24,27–30]. In this paper we aim to precisely identifyhow the SGWB measurements can be used to learn aboutthe physics that governs the behavior of magnetars, forexample, by constraining the equation of state in theinterior of magnetars. More specifically, we perform a

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systematic scan of the parameter space of the magnetarSGWB model, and we identify which parts of it are acces-sible to the second and third generations of GW detectors.We also explore three different types of the magnetic fieldconfiguration in magnetars (following Ref. [30]), as well asthe importance of the uncertainty in the star formationhistory. In Sec. II we present the model of SGWB due tomagnetars, following Ref. [29]. In Sec. III we present theresults of the parameter scan, and in Sec. IV we discuss theimplications for the three different proposed configurationsof magnetic field in magnetars. We conclude in Sec. V.

II. MAGNETAR SGWB MODEL

The SGWB is usually represented in terms of the nor-malized GW energy spectrum [49]

�GWðfÞ ¼ f

�c

d�GWðfÞdf

; (1)

where d�GW=df is the energy density in the frequencyband [f, fþ df] and �c is the critical energy densityrequired to close the universe,

�c ¼ 3H20c

2

8�G; (2)

whereH0 is the Hubble constant, c is the light speed, andGis the Newton gravitational constant. Equation (1) can berewritten in terms of the integrated flux density F [29],

�GWðfÞ ¼ 1

�ccfFðfÞ; (3)

which is given by

FðfÞ ¼Z

dP0pðP0ÞZ zsup ðf;P0Þ

0

RðzÞ4�r2ðzÞ

dEGW

dfeðfe;P0Þdz;

(4)

whereRðzÞ is the rate ofmagnetars as a function of redshift z,rðzÞ is the proper distance related to the luminosity distanceby dLðzÞ ¼ ð1þ zÞrðzÞ, and dEGW=dfe is the gravitationalwave energy spectrum emitted by a single source at thefrequency fe ¼ fð1þ zÞ (in the source frame), up to somemaximal frequency fmax . P0 denotes the initial period ofthe magnetar, which determines fmax . Since not all mag-netars are born with the same initial period, one in princi-ple must average over the probability distribution pðP0Þ ofinitial periods. However, assuming that magnetars areformed by the dynamo process, the range of initial periodsis rather limited, P0 2 ½1; 5� ms [31,50]. We can thereforeapproximate the integrated flux by replacing the dP0

integral with the average value of P0 to get

Fðf;P0Þ ¼Z zsup ðf;P0Þ

0

RðzÞ4�r2ðzÞ

dEGW

dfeðfe;P0Þdz: (5)

We will investigate below the importance of the choice ofthe P0 value. RðzÞ can be written in terms of the magnetarrate per unit comoving volume RVðzÞ,

RðzÞ ¼ RVðzÞdVðzÞdz¼ 4�c

H0

RVðzÞr2ðzÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Mð1þ zÞ3 þ��

p : (6)

We use the standard �CDM cosmology, with the matterenergy density �M ¼ 0:3, dark energy density �� ¼ 0:7,and Hubble constant H0 ¼ 70 km s�1 Mpc�1. Finally wewrite the magnetar rate in terms of the star formation rateR�ðzÞ,

RVðzÞ ¼ �R�ðzÞ1þ z

; (7)

where � is the mass fraction per M� that is converted intomagnetars. The (1þ z) term in the denominator correctsthe cosmic star formation rate by the time dilatation owingto the cosmic expansion.The parameter � captures the mass fraction �NS of

neutron star progenitors (in units of M�1� ) and the fractionfm of neutron stars that are born as magnetars, � ¼ �NSfm.We will treat � as a free parameter of this model, but weestimate its typical value here following [29]: First, assum-ing the Salpeter initial mass function �ðmÞ, and assumingthat the neutron star progenitors have masses larger than8M� and that stars with masses larger than 40M� give riseto black holes, we have

�NS ¼Z 40M�

8M��ðmÞdm ¼ 9� 10�3M�1� : (8)

The value of fm is rather uncertain. Some estimates [51]and population synthesis simulations [18] suggest fm �0:1, but other studies suggest that all neutron stars are

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Frequency (Hz)

Str

ain

(1/√

Hz)

LIGOaLIGO standardaLIGO 650 HzaLIGO 1000 HzET−D

FIG. 1. Strain sensitivity curves for the initial LIGO,Advanced LIGO, and ET-D (the currently most accurate sensi-tivity model for the third-generation GW detector, The EinsteinTelescope, developed by Hild et al. [68]). In addition to thestandard strain sensitivity, we also show the strain sensitivitiesfor Advanced LIGO detector configurations focusing on narrowfrequency bands around 650 Hz and 1 kHz. Advanced Virgo isexpected to have similar strain sensitivity to that of AdvancedLIGO.

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born as magnetars [52] (however, in this case most of themagnetars will dissipate their magnetic field in a very shorttime after formation). It is therefore unlikely that � >0:01M�1� , with typical values of order �typical ¼ 10�3M�1� .

The cosmic star formation rate R�ðzÞ has been studiedextensively in the literature. In our calculation, we adopt theone proposed by Hopkins and Beacom [53] and set themaximum redshift zmax ¼ 6. We also repeat our calculationusing four other star formation rates and compare these fiveresults: Fardal et al. [54],Wilkins et al. [55], Nagamine et al.[56], and Hernquist and Springel [57]. These models of starformation history are compared in Fig. 2.

We therefore rewrite Eq. (3) as

�GWðfÞ¼ 8�G

3c2H30

�fZ zsup ðf;P0Þ

0

R�ðzÞð1þ zÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�Mð1þ zÞ3þ��

p�dEGW

dfeðfe;P0Þdz: (9)

The upper limit of the integral zsup is constrained by both

zmax and fmax and is thus given by

zsup ðf; P0Þ ¼8<:zmax if f < fmax ðP0Þ

1þzmax

fmax ðP0Þf � 1 otherwise.

(10)

In other words, we exclude magnetars that produce gravi-tational waves with frequencies smaller than f (after red-shift), as well as magnetars beyond the maximal redshiftzmax of the models of star formation history we choose. Wecomplete the model following Regimbau and Mandic [29],

dEGW

dfe¼ dEGW

dt

��������dfedt

���������1

with fe 2 ½0� 2=P0�

¼ I�2f3e

�5c2R6

192�2GI2

��BBp

��2 þ f2e

��1; (11)

assuming that the spin axis of the neutron star is perpen-dicular to the magnetic axis so that gravitational waves areemitted only at twice the spinning frequency 2=P (with P0

being the initial period of the star).

The spin-down rate dfedt is calculated from the rate of the

rotational energy loss due to electromagnetic and GWradiation [30]. The first term in the brackets of Eq. (11)comes from the electromagnetic dipole radiation, which isproportional to the magnetic field strength at the poles B2

p.

The second term is due to the GW emission. Since bothdEGW=dt and the spin-down rate owing to GW emissionare proportional to �2B [30], this term of Eq. (11) does notdepend on �2B. Overall, however, the energy spectrumdEGW=dfe increases with ð�B=BpÞ2, as further discussed inSec. IV.As shown by Marassi et al. [30], if the spin and magnetic

axes are not perpendicular, GWs will also be emitted at the

frequency 1=P—in particular, the component of dEGW

dfewith

fe¼2=P (fe¼1=P) is proportional to sin 2� (cos 2�) wherethe wobble angle � is the angle between the two axes. Theprecession motion will drive� toward�=2 (0) if the neutronstar is prolate (oblate), and the time scale of this process isunknown. Since the value of � and its time evolution do notsignificantly affect the accessibility of the GW background[30], we simplify our model to use � ¼ �=2.We assume the neutron star radius R and the moment of

inertia I to be 10 km and 1045 g cm2, respectively. Themagnetic field strength Bp is evaluated at the magnetic

poles and is expected to be between about 1014 and 1015 G[58] (in this study we consider newborn magnetars andneglect a possible dissipation of the magnetic field).The star quadrupole ellipticity �B is determined by the

magnetic field configuration and strength. However, wewill first study the model assuming no relationship betweenthe ellipticity and the magnetic field, and we will thenconsider three specific magnetic field configurations (poloi-dal, toroidal, and twisted torus), which will imply differentrelations between the magnetic field strength and ellipticity.The continuity of the GW signal produced by the mag-

netars is characterized by its duty cycle D. Assuming �ðzÞis the average signal duration of magnetars at redshift z, theduty cycle is defined by

D ¼Z zsup

0RðzÞ�ðzÞð1þ zÞdz; (12)

where 1þ z rescales �ðzÞ to account for the time-dilationeffect. By summing up the time duration of all magnetars,Marassi et al. [30] showed that the duty cycle of the SGWBfrom magnetar population D � 1, confirming that theproduced background is continuous in time.

III. ACCESSIBILITY TO THE SECOND- ANDTHIRD-GENERATION DETECTORS

We scan the parameter space of the magnetar SGWBmodel described in Sec. II. In particular, since the spectrum

0 2 4 6 8 1010

−2

10−1

z

R* (

Mso

lar M

pc−

3 yr−

1 )

Hopkins & Beacom 2006Fardal et al. 2007Wilkins et al. 2008Nagamine et al. 2006Herquist & Springel 2003

FIG. 2. Comparison of different models of star formation his-tory: Hopkins and Beacom [53], Fardal et al. [54], Wilkins et al.[55], Nagamine et al. [56], and Hernquist and Springel [57].

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depends on the ratio of ellipticity and the polar magneticfield, we will treat �B=Bp as one free parameter, ranging

between 10�20 and 10�14 G�1. We will also treat � as afree parameter, allowing it to take values in the range10�4–1M�1� , although the likely value of this parameteris expected to be � < 10�2M�1� . We therefore scan the�B=Bp � � plane, and for each point in this plane we

compute �GWðfÞ. We then compare the spectrum to theprojected sensitivities for the second- and third-generationGW detectors. For the second generation, we assume thestandard Advanced LIGO expected strain sensitivity[42,43], and we also explore the detector configurationswhere the strain sensitivity is amplified at 650 Hz or 1 kHz.For the third generation, we assume the ET-D strain sensi-tivity curve of the Einstein Telescope [47,48]. In all cases,the projected sensitivity is defined as the signal-to-noiseratio of 2, assuming two colocated detectors and one yearof exposure.

The results of the scan are shown in Fig. 3 using the starformation rate from Ref. [53] and P0 ¼ 1 ms. The second-generation detectors will be able to probe large parts of thisparameter space, reaching even the typical values of themagnetar rate, �� 10�3M�1� . Some parts of the parameterspace, corresponding to � * 10�2M�1� and �B=Bp *

10�17 G�1, may be accessible to multiple frequency bandsof the second-generation detectors, allowing a more de-tailed study of the SGWB spectrum. The vertical part of theLIGO and Advanced LIGO curves, where the spectrumis not sensitive to �B=Bp, corresponds to the regime where

GW emission dominates over the electromagneticemission—in this case, the second term in the brackets ofEq. (11) dominates over the first one, so the whole brackets

can be approximated at f�2e , independent of �B=Bp. The

third-generation detectors will be able to probe a substan-tially larger part of this parameter space, reaching muchlower values of the magnetar rate and �B=Bp.

To estimate the importance of the choice of the starformation rate model and of the initial period P0, wecomputed the sensitivity curves for the second- and third-generation detectors using five different models of the starformation rate [53–57] and five different values of P0. Theresults are shown in Fig. 4—the uncertainties in the starformation rate and in the initial period lead to a factor of3–4 uncertainty in the �B=Bp parameter.

IV. IMPLICATIONS

We now investigate the implications of the results ofour scans for different magnetic field configurations ofmagnetars. Since different field configurations imply dif-ferent relations between the magnetic field and the ellip-ticity of the magnetar, the SGWB searches can be used toprobe these different field configurations and the physicsthat gives rise to them. In general, a poloidal magnetic fieldextends throughout both the interior and the exterior ofthe magnetar with the expected field strength of Bp �1014–1015 G [58]. In addition, a toroidal magnetic fieldcomponent has been suggested to exist within a torus-shaped region inside the magnetar [27,59,60] in order toaccount for the observed features of the SGRs and theAXPs. Following Marassi et al. [30] we consider threedifferent cases: poloidal-dominated, toroidal-dominated,and twisted-torus configurations. For each case we assumethe star formation rate from Hopkins and Beacom [53] andP0 ¼ 1 ms.

A. Poloidal-dominated field configuration

In the poloidal magnetic field configuration, the elliptic-ity is given by

�B ¼ �R8B2

p

4GI2; (13)

following Bonazzola and Gourgoulhon [61]. The distortionparameter � is dimensionless, and it accounts for the mag-netic field geometry and the equation of state. Numericalsimulations [61] indicate that if the interior of the neutronstar is a normal conductor, the distortion parameter istypically �< 10, even if the current flow is limited to thecore of the star. If the star’s interior is a type I superconduc-tor, such that the magnetic field is expelled from parts of thestar, the distortion parameter could be significantly larger,reaching values of several hundred. Even larger distortionsare expected in the case of the highly disordered (stochastic)magnetic field in the interior—in such cases the overall(average) magnetic dipole moment could be very small,but the distortion parameter can reach values �> 1000.Similarly, in the case of the type II superconducting interior,

10−4

10−3

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100

10−20

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10−14

ε B /B

p (G

−1 )

λ (M−1solar

)

λ typi

cal

λ NS

ET−DAdvLIGOAdvLIGO−n650AdvLIGO−n1000

FIG. 3. We show regions of the parameter space of the mag-netar SGWB model accessible to the future second-generationdetectors (Advanced LIGO [42,43]) and third-generation detec-tors (Einstein Telescope [47,48]). The accessible regions areabove the corresponding curves. We assume star formationrate from Ref. [53] and the initial period P0 ¼ 1 ms.

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very large stresses in the crust of the star could also lead tovalues of �> 1000. It is therefore possible to turn theSGWB constraints on �B into constraints on �, hence ex-tracting information about the equation of state in magne-tars, in the framework of poloidal-dominatedmagnetic fieldconfiguration.

In Fig. 5 we show the sensitivity curves for the second-and third-generation detectors in the �-� plane, for severalvalues of Bp in the range 10

14–1016 G. As noted above, the

second-generation detectors will be able to probe onlymodels where the magnetar rate is relatively large, largerthan the ‘‘typical’’ value of 10�3M�1� . Depending on theaverage strength of the poloidal magnetic field, these de-tectors will probe models with the largest values of �,corresponding to the type II superconductor or to highlydisordered (stochastic) magnetic field in the interior of thestar. They may also be able to probe some of the type-I

superconductor models. The nonsuperconductor modelswill be out of reach of the second-generation detectors.The third-generation detectors will be able to explore asignificantly larger part of this parameter space, potentiallyexploring the entire type-I and type-II superconductorregions down to rather low magnetar rates of ��10�4M�1� (for the largest values of Bp). These detectors

may even be able to probe the nonsuperconductor equationof state for the largest values of Bp and �.

B. Twisted-torus magnetic field configuration

The twisted-torus magnetic field configuration was in-troduced by Braithwaite and Spruit [59]. It is argued to be auniversal equilibrium structure of the magnetic field [62],while the pure poloidal field configuration is potentiallyunstable [63]. The twisted-torus field configuration takesa toroidal magnetic field closed in the interior of the

10−4

10−3

10−2

10−19

10−18

10−17

λ (M−1solar

)

ε B /B

p (G

−1 )

ET−D

Hopkins & Beacom 2006Fardal et al. 2007Wilkins et al. 2008Nagamine et al. 2006Hernquist & Springel 2003

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)

ε B/B

p (G

−1 )

ET−D

P0 = 1 ms

P0 = 2 ms

P0 = 3 ms

P0 = 4 ms

P0 = 5 ms

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10−2

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λ (M−1solar

)

ε B/B

p (G

−1 )

Advanced LIGO

Hopkins & Beacom 2006Fardal et al. 2007Wilkins et al. 2008Nagamine et al. 2006Hernquist & Springel 2003

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10−2

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10−17

10−16

10−15

λ (M−1solar

)

ε B/B

p (G

−1 )

Advanced LIGO

P0 = 1 ms

P0 = 2 ms

P0 = 3 ms

P0 = 4 ms

P0 = 5 ms

FIG. 4. Top left: Comparison of the third-generation sensitivity curves (ET-D) for P0 ¼ 1 ms and for five different choices of starformation rate [53–57]. Top right: Comparison of the third-generation sensitivity curves (ET-D) for the Hopkins and Beacom starformation rate model [53] and for five different values of P0. Bottom left: Same as top left but for the second-generation detectors(standard Advanced LIGO strain sensitivity). Bottom right: Same as top right but for the second-generation detectors (standardAdvanced LIGO strain sensitivity).

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magnetar into account. The toroidal field is twisted andstabilizes the poloidal field. The resulting ellipticity can bemodeled by

�B ¼ k

�Bp

1015 G

�2 � 10�6; (14)

where k is a dimensionless parameter dependent on thefield geometry and the equation of state. Ciolfi et al. [64]argue that for the twisted-torus field configuration k ¼ 4–9spans a realistic range of the star compactness at Bp ¼1� 1016 G in the framework of general relativity. Morespecifically, k ¼ 4 corresponds to a small compactnessgiven by the Akmal-Pandharipande-Ravenhall equation

of state [65], and k ¼ 9 corresponds to a large compactnessgiven by the Glendenning equation of state [66].Figure 6 shows the sensitivity curves of the second- and

third-generation detectors in the k-Bp plane for several

values of �. The realistic range of k values is far belowthe sensitivities of both detector generations, implying thatthe twisted-torus field configuration models will not bereachable by these detectors.

C. Toroidal-dominated field configuration

A very strong toroidal magnetic field component, Bt *1016 G, has been proposed for the interior of the magnetarto explain the enormous energy emission in the December27, 2004, giant flare from the SGR 1806-20, as argued by

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105

106

Advanced LIGO

β

λ (M−1solar

)

Non−superconducting Interior

Type I Superconductor

Type II Superconductor or Stochastic Magnetic Field

Bp = 1014 G

Bp = 6 × 1014 G

Bp = 3 × 1015 G

Bp = 1016 G

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10−2

10−1

100

101

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103

104

ET−D

β

λ (M−1solar

)

Non−superconducting Interior

Type I Superconductor

Type II Superconductor or Stochastic Magnetic Field

Bp = 1014 G

Bp = 6 × 1014 G

Bp = 3 × 1015 G

Bp = 1016 G

FIG. 5. Sensitivity curves in the �-� plane for the poloidal magnetic field configuration with different values of Bp are shown for thesecond (left) and third (right) generations of GW detectors. We use the star formation rate from Hopkins and Beacom [53] andP0 ¼ 1 ms. The gray horizontal dashed lines denote different types of the equation of state in the interior of magnetars, in theframework of a pure poloidal field configuration.

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1015

1016

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Bp (G)

|k|

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λ = 0.002 Msolar−1

λ = 0.01 Msolar−1

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1015

1016

100

101

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105

Bp (G)

|k|

ET−D

λ = 0.0001 Msolar−1

λ = 0.001 Msolar−1

λ = 0.002 Msolar−1

λ = 0.01 Msolar−1

FIG. 6. Sensitivity curves in the k-Bp plane of the twisted-torus field configuration models are shown for several values of � for thesecond (left) and third (right) generations of gravitational-wave detectors. We assume the star formation rate model from Hopkins andBeacom [53] and P0 ¼ 1 ms. The two gray horizontal dashed lines denote the realistic range of values for k [64].

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Stella et al. [60]. Braithwaite further investigated thestability of the relative strengths of the toroidal and poloi-dal components, confirming that a magnetar can indeedhave such a toroidal-dominated field configuration [67]. Tomodel the ellipticity, we adopt the estimate by Cutler [27]:

�B ¼(�1:6� 10�6ðBt=10

15 GÞ; Bt < 1015 G

�1:6� 10�6ðBt=1015 GÞ2; Bt > 1015 G:

(15)

In the framework of the toroidal-dominated models, it istherefore natural to study the GW detector sensitivities inthe Bt-Bp plane, as shown in Fig. 7. While the second-

generation detectors will be able to probe only models witha relatively strong toroidal field component (reaching Bt �1016 G only for the largest magnetar rate values �, andBt � 1017 G for the more typical �� 10�3M�1� ), the third-generation detectors will be able to probe much of theinteresting parameter space in these models.

V. CONCLUSION

In this paper we conducted a systematic study of theparameter space in the model of stochastic gravitational-wave background due to magnetars, and we identified theregions of this parameter space that will be probed by theupcoming second- and third-generation gravitational wavedetectors. We first computed these regions for the generalcase, without assuming a specific magnetic field configu-ration in the magnetars, in the plane spanned by twocritical parameters: the ratio of ellipticity and the polarmagnetic field, �B=Bp, and the rate of magnetars �. We

found that different choices of the star formation history

and of the initial period of magnetars P0 lead to a factor of3–4 uncertainty in the computed accessible regions of thisparameter space. A similar level of uncertainty is alsoexpected because of the uncertainty in the assumed valuesof the average neutron star radius (R � 10 km) and themoment of inertia (I � 1045 g cm2).We then proceeded to apply our results to three different

magnetic field configurations proposed in the literature,which imply different relations between the magnetarellipticity and its magnetic field strength. We found thatthe twisted-torus field configuration models will not bereachable by either the second or the third generationof gravitational wave detectors. In the case of poloidal-dominated configuration, the second-generation detectorswill be able to probe models with the largest distortions inthe shape of the magnetar, corresponding to the type-IIsuperconductor equation of state or the highly disorderedmagnetic field in the interior, for magnetar rates larger thantypical, � > 10�3M�1� . The third-generation detectors willalso be able to probe models with a type-I superconductorequation of state, even for very low magnetar rates�� 10�4M�1� . The toroidal-dominated magnetic fieldconfiguration models will be probed by both second- andthird-generation detectors—the third-generation detectorswill be able to explore a large fraction of the interestingparameter space in these models, reaching relatively lowvalues of the toroidal magnetic field strength Bt � 1016 G.

ACKNOWLEDGMENTS

The work of C.-J.W. and V.M. was in part supported bythe NSF Grant No. PHY0758036.

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Bt (

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λ = 0.01 Msolar−1

1014

1015

1016

1015

1016

1017

Bp (G)

Bt (

G)

ET−D

λ = 0.0001 Msolar−1

λ = 0.001 Msolar−1

λ = 0.002 Msolar−1

λ = 0.01 Msolar−1

FIG. 7. Sensitivity curves in the Bt-Bp plane of toroidal-dominated field configuration models are shown for different values of �, forthe second (left) and third (right) generations of gravitational-wave detectors. We assume the star formation rate from Hopkins andBeacom [53] and P0 ¼ 1 ms.

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