Evolution of the f-mode instability in neutron stars and gravitational wave detectability

A. Passamonti,1,2 E. Gaertig,1 K.D. Kokkotas,1 and D. Doneva1,3

1Theoretical Astrophysics, IAAT, Eberhard Karls University of Tubingen, 72076 Tubingen, Germany2INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00044 Rome, Italy

3Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria(Received 24 September 2012; published 4 April 2013)

We study the dynamical evolution of the gravitational-wave driven instability of the f mode in rapidly

rotating relativistic stars. With an approach based on linear perturbation theory we describe the evolution

of the mode amplitude and follow the trajectory of a newborn neutron star through its instability window.

The influence on the f-mode instability of the magnetic field and the presence of an unstable r mode is

also considered. Two different configurations are studied in more detail, an N ¼ 1 polytrope with a typical

mass and radius and a more massive polytropic N ¼ 0:62 model with gravitational mass M ¼ 1:98M.We study several evolutions with different initial rotation rates and temperature and determine the

gravitational waves radiated during the instability. In more massive models, an unstable f mode with a

saturation energy of about 106Mc2 may generate a gravitational wave signal which can be detected by

the Advanced LIGO/Virgo detector from the Virgo cluster. The magnetic field affects the evolution and

then the detectability of the gravitational radiation when its strength is higher than 1012 G, while the

effects of an unstable r mode become dominant when this mode reaches the maximum saturation value

allowed by nonlinear mode couplings. However, the relative saturation amplitude of the f and r modes

must be known more accurately in order to provide a definitive answer to this issue. From the thermal

evolution we find also that the heat generated by shear viscosity during the saturation phase completely

balances the neutrinos’ cooling and prevents the star from entering the regime of mutual friction. The

evolution time of the instability is therefore longer and the star loses significantly larger amounts of

angular momentum via gravitational waves.

DOI: 10.1103/PhysRevD.87.084010 PACS numbers: 04.30.Db, 04.40.Dg, 95.30.Sf, 97.10.Sj

I. INTRODUCTION

After a core-collapse supernova explosion, a nascentneutron star, if rapidly rotating, may develop nonaxisym-metric instabilities and radiate a significant amount ofgravitational waves [1,2]. These instabilities may originatefrom different physical processes and grow on dynamicaland secular time scales.

The dynamical instabilities in uniformly rotating starsoccur at high rotation rates, the typical case of the bar modeinstability, i.e. driven by the l ¼ m ¼ 2 f mode, sets inwhen the kinetic to gravitational potential ratio is T=jWj *0:255 [3,4]. In stars with a high degree of differentialrotation, a dynamical instability may develop even at con-siderably lower rotation rates, T=jWj * 0:01, (see Ref. [1]and reference therein). Secular instabilities are driven in-stead by dissipative processes and thus develop on longertime scales but at smaller rotation rates. For instance, theviscosity driven bar-mode instability typically sets in whenT=jWj * 0:14 [5], while the gravitational wave driven l ¼m ¼ 2 f-mode instability requires roughly T=jWj * 0:13[6], and in more compact stars T=jWj * 0:07 [7]. Theconditions are much better for higher multipole f modeswhich can be gravitationally unstable at even lower rota-tion rates [6].

In rotating neutron stars, a nonaxisymmetric modemay be driven unstable by gravitational radiation via

the well-known Chandrasekhar-Friedman-Schutz (CFS)mechanism [8–10]. This instability occurs when a modewhich is counterrotating with respect to the star is seen ascorotating by an inertial observer. The angular momentumradiated by gravitational waves induces an increasinglynegative angular momentum of the mode which thus be-comes unstable [8–10]. The mode’s growth ends whensome dissipative process suppresses the instability.Nonaxisymmetric modes may be excited during the

protoneutron star formation, after the core bounce, andbecome CFS unstable as the star cools down below acritical temperature which depends on the star’s modeland rotation. The mode’s amplification expected duringthe instability may generate a significant gravitationalwave signal which can be potentially observed by thecurrent and next generation of Earth-based laser interfer-ometers. When the gravitational waves will be observed,the identification of the oscillation modes from the spec-trum will help us to unveil the properties of the densematter at super-nuclear densities and clarify the neutronstar physics. The analysis of the observed quasiperiodicoscillations in magnetars already provided important in-sights [11–15], and even more relevant results are expectedwhen asteroseismology will be applied to the gravitationalwave signal [16,17].Among the several modes which can be driven unstable,

the f and r modes are the most important due to their

PHYSICAL REVIEW D 87, 084010 (2013)

1550-7998=2013=87(8)=084010(15) 084010-1 2013 American Physical Society

relatively short growth time scale and an efficient emissionof gravitational waves. The r-mode instability has attractedmore attention so far as it is CFS unstable at any rotationrate and may grow in about a few tens of seconds in rapidlyrotating stars. There is in fact an extensive literature whichstudied the r-mode instability for various stellar modelsand considered it as a possible candidate for limiting thestar’s rotation below the Kepler velocity (see, for instance,Refs. [18,19] and references therein). The growth of the rmode can be, however, strongly limited by nonlinear modecoupling with other inertial modes [20,21].

The f-mode instability occurs instead in rapidly rotatingstars and is preferable in more compact objects ðM=R>0:2Þ [22]. In Newtonian stellar models with stiff polytropicequation of state (EoS), i.e. with polytropic index N < 1,the l ¼ m ¼ 2 f mode is unstable close to the maximumrotation rate, while in softer equations of state (N 1) thismode is always stable [23–25]. The multipoles for whichthe f mode becomes unstable for smaller rotation rates arethe l ¼ m ¼ 3 and 4, which have a large instability win-dow and still a reasonable short growth time [6,23–25].The conditions are more promising in relativistic stellarmodels as the f mode is driven unstable at smaller rotationrates, and also the quadrupole f mode might have a sig-nificant instability window [7,22,26].

To date, the evolution of the f-mode instability has beenstudied only for the l ¼ m ¼ 2 casewith nonlinear dynami-cal simulations in Newtonian gravity. These works usedboth ellipsoidal configurations [27,28] and compressiblestellar models with uniform [29] and differential [30] rota-tion. In these nonlinear simulations, the effects of viscosityhave been neglected and the bar-mode instability is drivenby a gravitational radiation reaction term which has beenincorporated in the Newtonian dynamical equations. Inparticular, the strength of this post-Newtonian term hasbeen artificially increased to shorten the instability evolu-tion and make feasible its analysis within the simulationtime. For a typical neutron star’s parameter, these nonlinearstudies find that the gravitational wave signal emitted dur-ing the bar-mode instability may be detected by AdvancedLIGO from a source located in the Virgo cluster. Moreimportantly, while the star spins down themode’s frequencydecreases toward the more sensitive frequency band(100 Hz) of the gravitational wave detectors.

In the present work, we study the evolution of thegravitational wave driven f-mode instability with a differ-ent approach. It is based on the linear perturbation frame-work developed for the r-mode instability in Refs. [31,32].From the evolution equations for the mode energy, totalangular momentum, and temperature we derive a system ofordinary differential equations to describe the mode’s am-plitude, the star’s rotation, and the thermal evolution. Inthis analysis we include the effects of viscosity, magneticfield, and consider also the impact of an unstable rmode onthe f-mode instability. The coefficients of these evolution

equations depend on the stellar model and mode properties.More specifically, for a given neutron star model we haveto determine the f-mode frequency and eigenfunctions,and with this information calculate the relevant viscousand gravitational radiation time scales.We consider rapidly rotating and relativistic models

with uniform rotation and extract the f-mode properties(frequency and eigenfunctions) from the time evolutions ofthe linearized dynamical equations. In particular, we sim-plify the problem by using the Cowling approximation, i.e.we neglect the space-time perturbations in our linearizedproblem. The accuracy of this approximation is to betterthan 20% for the quadrupole f mode, but the error de-creases considerably for higher multipoles [33]. In a sec-ond step, the mode’s frequency and eigenfunctions areinserted in appropriate volume integrals which determinethe viscous damping times and the gravitational radiationgrowth time [24].We study in more detail two polytropic neutron star

models with a different compactness and maximum rota-tion limit. The first is a star with polytropic index N ¼ 1and gravitational massM ¼ 1:4M, while the second is anN ¼ 0:62 polytrope with gravitational massM ¼ 1:98M.Considering several configurations we find that thegravitational-wave signal generated by an unstable fmode may be potentially detected with the Einstein tele-scope (ET) from a source located in the Virgo cluster. Forthe more massive stellar model the signal may be evendetected by the Advanced LIGO/Virgo detectors. For in-stance, the gravitational characteristic strain generatedby the l¼m¼3 and 4 f mode is shown in Fig. 1 for theN ¼ 1 and N ¼ 0:62 polytropic models with a relativelyweak magnetic field at the magnetic pole, Bp ¼ 1011 G.

This characteristic strain is determined by integrating intime the signal generated by an f mode during the insta-bility and setting a maximum saturation energy of E106Mc2. However in order to not overestimate the signalwhen the instability evolves on long time scales, we haveconsidered a maximum detector integration time of 1 yr.For more details on these results and the effects of highermagnetic fields and r modes, see Sec. V.A general expectation is that superfluidity should further

restrict the parameter space of the instability. In fact, if astar cools down below the transition temperature where theneutrons of the core become superfluid, the f-mode insta-bility should be efficiently suppressed by the mutual fric-tion force [34]. However, for typical stellar parameters wefind that the heat generated by shear viscosity during thesaturation phase of an unstable f mode may counterbal-ance the neutrino cooling. In fact, our results show that anunstable star may follow a quasi-isothermal trajectorywithin the instability window without cooling below thecritical temperature of the core’s neutrons. As a result, theevolution time is longer and the star loses more angularmomentum via the emission of gravitational waves.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-2

We present the formalism for studying the f-mode in-stability in Secs. II and III, while in the Appendix weprovide the equations for the gravitational wave instabilitydriven simultaneously by two modes. The results forthe instability evolution are described in Sec. IV, and thegravitational wave signal is determined in Sec. V. Theconcluding remarks can be found in Sec. VI.

II. THE f-MODE INSTABILITY

The standard approach for studying secular instabilities

is based on linear perturbation theory and consists of two

distinct steps. First, we determine the main properties of a

mode (frequencies and eigenfunctions), which will be in-

serted in a second step into appropriate volume integrals

for the estimation of the relevant time scales due to vis-

cosity and gravitational radiation. Since these time scales

are generally much longer than the oscillation period of the

mode, their properties can be determined by solving the

inviscid problem [24].In this work, we consider relativistic models of rapidly

rotating neutron stars and determine the f-mode frequencyand eigenfunctions by postprocessing time evolutiondata of the relativistic perturbation equations. All thedetails of the formalism and numerical methods for study-ing the oscillations of rapidly rotating stars are given inRefs. [35,36].

In a realistic star, an oscillation mode approximately

evolves as ei!tt=, where ! is the mode’s frequency and the damping/growth time scale. Away to determine is tocalculate the variation in time of the mode’s energy given inthe rotating frame by

E ¼ 1

2

ZdV

uiui þ

1

2ðh þ hÞ

; (1)

where the asterisk denotes a complex conjugate and theintegral extends over the volume of the star. The quantity represents the background mass density, while , h,and ui are perturbations of the mass density, the enthalpy,and the velocity, respectively. The Latin indices denote thespatial components of a generic four-vector. Equation (1)is also defined as the canonical energy in the rotatingframe [37].As long as the energy of a mode depends quadratically

on its perturbations, one can show that [24]

dE

dt¼ 2E

; (2)

where the global time scale of Eq. (2) results from theindividual dissipative mechanisms that act on the mode.For stars in which only gravitational waves, shear, and bulkviscosity dissipate energy, we have [24]

0.5 1.0 1.5 2.0 2.5

ν [ kHz ]

10-26

10-25

10-24

10-23

10-22

10-21

10-20

h c44

d = 20 Mpc

1.0

1.0

ET

Adv. LIGOαsat

= 10-4

N = 1

N = 0.62

0.95

0.98

l = m = 4 f-mode

0.88

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

ν [ kHz ]

10-26

10-25

10-24

10-23

10-22

10-21

10-20

h c33

d = 20 Mpc

1.0

ET

Adv. LIGOαsat

= 10-4

N = 0.62

0.95

l = m = 3 f-mode

0.91

FIG. 1 (color online). Characteristic strain generated by the f-mode instability for a polytrope with Bp ¼ 1011 G. The sourceis located at 20 Mpc and the saturation amplitude of the f mode is set to sat ¼ 104 (see Secs. III A and IVA). Left panel:The gravitational wave signal emitted by the l ¼ m ¼ 4 f mode for the two stellar models with N ¼ 1 and N ¼ 0:62. Thenumber near the vertical lines denotes the initial rotation rate =K, where K represents the mass shedding limit. Right panel: Thesignal emitted by the l ¼ m ¼ 3 f mode for the more massive model with N ¼ 0:62. The sensitivity curves of Advanced LIGO andthe ET [60,61] are shown in both panels. The gravitational wave signal generated during the f-mode instability may be detected byET for the most part of the instability window of the N ¼ 0:62 model. In particular, the l ¼ m ¼ 3 f mode may be detectable even byAdvanced LIGO for 0:95K.

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-3

1

¼ 1

gwþ 1

sþ 1

b: (3)

The gravitational radiation time scale can be calculatedby using the standard multipole formula1 [39],

1

gw¼ !

2E

Xl2

Nlð!mÞ2lþ1ðjDlmj2 þ jJlmj2Þ; (4)

where ! is the mode frequency as measured in the rotatingframe, Dlm, Jlm are the mass and current multipolemoments, and Nl is given by

Nl ¼ 4G

c2lþ1

ðlþ 1Þðlþ 2Þlðl 1Þ½ð2lþ 1Þ!!2 ; (5)

where !! denotes a double factorial.Since the f mode mainly radiates via the mass multipole

moments, which are determined by

Dlm ¼Z

dVrlYlm; (6)

we neglect the current multipole moments for computingits damping times.

The dissipative time scales due to bulk and shear viscos-ity can be determined by the following volume integrals:

1

b¼ 1

2E

ZdV; (7)

1

s¼ 1

E

ZdVij

ij; (8)

where and are the bulk and shear viscosity coefficients,while the stress tensor ij is given in terms of velocityperturbations,

ij ¼ 1

2

riuj þrjui 2

3gijr

; (9)

¼ rjuj; (10)

and gij is the spatial metric tensor.

The bulk and shear viscosity coefficients depend on thestate and composition of the neutron star matter. For neu-tron, proton, electron matter in a normal state, i.e. with nosuperfluid/superconducting components, these coefficientscan be written in a simple analytical form [24,40]. Morespecifically, the bulk viscosity coefficient is given by [41]

¼ 6 10592!2T6 g cm1 s1; (11)

where T is the star’s temperature. This expression is strictlyvalid for ‘‘small’’ oscillation amplitudes, within the

so-called subthermal regime. If the mode amplitude issufficiently large, nonlinear terms may become importantand will increase the strength of the bulk viscosity [42,43].However, the impact of the nonlinear bulk viscosity on thef-mode instability is moderate and appears at rather largemode amplitudes [44]. In the present work we limit thef-mode growth to comparatively small amplitudes andthen neglect the effects of nonlinear bulk viscosity.In normal neutron, proton, electron matter, shear viscos-

ity is dominated by neutron collisions and can be parame-trized with the following coefficient [24,40]:

¼ 3479=4T2 g cm1 s1: (12)

For an inviscid star, there is now the following situation:if the star is oscillating at a sufficiently high rotationrate, the condition for the secular CFS instability,!ð!mÞ 0, might be fulfilled. In this case, the stel-lar rotation drags a counterrotating mode into corotation asseen from an inertial observer, leading to a gravitationalwave driven exponential growth of the initial perturbation[it is gw 0 then; see Eq. (4)]. However, the other dis-

sipative mechanisms described above also operate in real-istic stars and tend to stabilize the mode evolution. Theinstability onset and the mode growth is therefore con-trolled by the global time scale which has to be 0. Thisglobal time scale is generally a function of the star’srotation rate and temperature T so that the instabilityhas a natural representation in a T- plane, where theregion above the critical curve for ¼ 0 denotes theso-called instability window (e.g., see Fig. 2).This description of the instability is clearly static as no

information about the evolution of the star and modeamplitude is provided. The evolution problem will beaddressed in the next section.

III. EVOLUTION EQUATIONS FORTHE f-MODE INSTABILITY

To study the evolution of the f-mode instability wederive a system of equations by using and extending theformalism already developed for the r-mode instability[31]. Our equations consider also the effects of thermalheating, magnetic torque, and the impact of an unstable rmode on the evolution of the f-mode instability and itsgravitational wave signal. The formalism we develop hereis general, it will be used in Sec. IV to study the gravita-tional wave instability of the l ¼ m ¼ 3 and 4 f modes.In the nonlinear study of the secular bar-mode instability

[27–30], it is reasonable to discern two dynamical phasesof the f-mode instability, namely, the mode’s exponentialgrowth and its nonlinear saturation [31]. As the star entersthe instability window, the mode grows exponentiallywhile the star slowly spins down on viscous time scales.This initial growth phase ends either when nonlinear dy-namics or dissipative processes saturate the mode ampli-tude at a finite value. After this point, the mode amplitude

1In our calculations we correct the gw determined via Eq. (4)with a constant factor equal to 3. This procedure provides abetter agreement with the relativistic results (no Cowling ap-proximation) of the gravitational radiation time scale; seeRef. [38] for more details.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-4

is nearly constant while the star loses angular momentumvia gravitational waves [29,30]. Eventually, the star leavesthe instability window and the f mode is damped bygravitational radiation.

The dynamical equations that describe these two re-gimes of the instability can be derived using the relationsfor the canonical mode energy (in the rotating frame) andthe stellar angular momentum. In this section we focus on astar which is driven unstable by a single f mode. In theAppendix we generalize the equations for studying the casein which two modes can be simultaneously driven unstableby gravitational radiation, and we apply later this formal-ism for the case of a simultaneous f- and r-modeinstability.

A. Mode evolution and star’s spin-down

The mode energy can be considered as a function of themode amplitude and the stellar rotation rate. Hence, wemay assume that E ¼ ~EðÞ, where the parameter denotes the mode amplitude and ~EðÞ describes the angu-lar velocity dependence. It is worth noticing that in contrastto Refs. [31,44] we define from the mode energy, hence

the fluid perturbations scale as 1=2. After taking thetime derivative, Eq. (2) can be written as

d

dtþ

d ln ~E

d

d

dt¼ 2

: (13)

A second equation can be derived from the evolution ofthe angular momentum,

dJ

dt¼ dJgw

dtþ dJmag

dt; (14)

where J ¼ Js þ Jc is the total angular momentum, whichconsists of the star’s angular momentum Js and thecanonical angular momentum of the mode Jc. The latterone can be related to the mode energy by the well-knownequation [37]:

Jc ¼ m

!E: (15)

The dissipative terms in Eq. (14) are the gravitationalradiation torque,

dJgwdt

¼ 2Jcgw

; (16)

and the magnetic torque dJmag=dt.

We consider a standard oblique rotator model in vacuumwith a dipolar magnetic field, which has [45]

dJmag

dt¼ sin 2

6c3B2pR

63; (17)

where R is the stellar radius, Bp is the magnetic field at the

magnetic pole and is the inclination angle between therotation and magnetic axes. For simplicity we assume anorthogonal rotator ( ¼ =2). This configuration providesalso a magnetic spin-down comparable to that of a standardpulsar model with magnetosphere [45,46].From Eq. (15) and the functional form of E we can write

the canonical angular momentum as Jc ¼ ~JcðÞ. In this

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

t [yr]

10-6

10-5

10-4

α

l=m=4 f-mode

N = 1

Ω = 1.0 ΩK

Ω = 0.99 ΩK

Ω = 0.98 ΩK

Ω = 0.97 ΩK

0.95

0.96

0.97

0.98

0.99

1

1.01

Ω /

ΩK

108

109

1010

T [K]

0.95

0.96

0.97

0.98

0.99

1

Ω /

ΩK

Tcn

.

N = 1

αsat

= 10-4

M = 1.4 M

FIG. 2 (color online). Evolution of the l ¼ m ¼ 4 f-mode instability for a relativistic polytrope with N ¼ 1, gravitational massM ¼ 1:4M and Bp ¼ 1011 G. The star enters the instability window at different spins (see right panel) and correspondingly the left

panel depicts the time evolution of the stellar rotation rate (top left panel) and mode amplitude (bottom left panel), where the initialamplitude ¼ 106 saturates at sat ¼ 104. The shaded region in the right panel represents the temperature range where theneutrons and protons of the core are expected to be in a superfluid/superconducting state in accordance with recent models for theobserved cooling of Cassiopeia A [55,56].

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-5

way we can simplify the calculation and determine fromEq. (14) the following expression:

d

dtþ 1

~Jc

dJsd

þ dJcd

d

dt¼ 2

gwþ 1

mag

; (18)

where we have introduced a time scale for the magnetictorque,

1

mag¼ 1

2~Jc

dJmag

dt: (19)

From Eqs. (13) and (18) we can now derive a system ofordinary differential equations:

d

dt¼ 2

gw 2

v

1þ Q

Dþ 2P

D

mag

; (20)

d

dt¼ 2F

D

v 1

mag

; (21)

where the total viscous damping time is defined as

1

v¼ 1

sþ 1

b; (22)

and the coefficients are given by the following expressions:

Q ¼ d~Jcd

dJsd

1; F ¼ ~Jc

dJsd

1; (23)

P ¼ d ln ~E

d F; D ¼ 1þ ðQ PÞ: (24)

The form of Eqs. (20) and (21) agrees with the evolutionexpected during the initial phase of the instability, in which

the f mode grows exponentially as et=jgwj and the starslows down on viscous time scales.

As we have already pointed out, we expect that duringthe nonlinear saturation regime the amplitude of the moderemains nearly constant [29,30]. We can therefore approxi-mate this evolution phase by assuming that d=dt ¼ 0,which allows us to recast Eqs. (20) and (21) into a singlerelation,

d

dt¼ 2F

1þ Q

gwþ 1

mag

: (25)

This expression shows that for neutron stars with a weakmagnetic field, e.g., Bp 1011 G (see Sec. IVB), the spin-

down during the nonlinear saturation phase is dominated,as expected, by gravitational radiation.

Equations (20), (21), and (25) describe the time evolu-tion of the mode amplitude and the star’s angular velocity,but do not provide any information about the thermalevolution. In the next section, we address the coolingproblem and derive the equations for the thermal balanceof the star.

B. Thermal evolution

Few minutes after a core collapse, a neutron star be-comes transparent to neutrinos which dominate the coolingfor at least 1000 yr [45]. However, an isolated neutronstar may be reheated by viscosity and magnetic fielddecay in later stages of its life. The evolution of themagnetic field in the core is not well known in stars youngerthan 104 yr [47], and it is typically described by approxi-mate relations [47–49]. For instance, an equation used inRef. [47] reads

B ¼ B0

1þ t

dec

1; (26)

where B0 is the initial magnetic field and dec is the fielddecay time scale. This is approximately dec 104 yr andit is much longer than the typical evolution of the f-modeinstability. Hence, the magnetic field decay can be reason-ably neglected during the unstable phase of the f mode.The two viscous processes that operate in our models

have an opposite effect on the thermal evolution. Bulkviscosity dissipates the mode energy by neutrino emission,as a result of the Urca reactions that attempt to restore thebeta equilibrium in a displaced fluid element. These neu-trinos induced by mode oscillations escape at infinity andcool down the star. However, this effect is very smallcompared to the dominant cooling rate of the backgroundstar and can be therefore neglected. More important is theshear viscosity which dissipates energy through particlecollisions. The heat generated by shear viscosity is propor-tional to the mode amplitude and it can be relevant during amode instability. In fact, as we will show in Sec. IV, shearviscosity may reduce and even completely balance theneutrino cooling of an unstable star.The thermal evolution of a neutron star can be studied

with a global energy balance between the relevant radiativeand viscous processes [45]:

Cv

dT

dt¼ L þHs; (27)

where Cv is the total heat capacity at constant volume, L

is the neutrino’s luminosity produced by the modified Urcaprocesses that operate in the background star, and Hs isthe heating rate generated by shear viscosity. AlthoughEq. (27) is strictly valid for isothermal neutron stars weconsider it as a reasonable approximation for this work.The heat capacity depends in general on the neutron star

EoS and temperature. An expression frequently used inliterature is given by [45]

Cv ¼ 1:2 1039M

M

0

2=3 T

109 KergK1; (28)

which has been strictly derived for an ideal Fermi gas ofneutrons in which & 20, where 0 is the nuclear satu-ration density.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-6

The neutrinos’ luminosity can be determined by thefollowing equation [45]:

L ¼ 5:3 1039M

M

0

1=3

T

109 K

8erg s1; (29)

which has already been used for the r-mode instability [31].The last term we need to specify in Eq. (27) is the

heating rate induced by shear viscosity, which can bedescribed by the following equation [31]:

Hs ¼ 2E

s: (30)

We now have all the ingredients to study the evolution ofthe f-mode instability with the system of Eqs. (20), (21),and (27).

IV. RESULTS

In this section, we discuss the dynamical evolution of thef mode with and without the impact of an additionalmagnetic field. Furthermore, we study also the effect ofan unstable r mode on the evolution of the f-mode insta-bility. For this, we model the compact objects as relativisticneutron stars which obey a polytropic EoS,

p ¼ K1þ1=N; (31)

where p is the fluid pressure, K is the polytropic constant,and N is the polytropic index. The rest-mass density is related to the fluid energy density " via the relation" ¼ þ Np.

To study the dependence of the f-mode instability on theneutron star model, we consider more in detail two sequen-ces of uniformly rotating stars up to the mass sheddinglimit K. The first model represents a neutron star withpolytropic index N ¼ 1. The nonrotating member of thissequence has a gravitational mass of M ¼ 1:4M andcircumferential radius of R ¼ 14:15 km, while the maxi-mum rotating configuration has K ¼ K=2 ¼ 673:1 Hzand T=jWj ¼ 0:095. The second model is described by apolytrope withN ¼ 0:62, which has a nonrotating configu-ration with M ¼ 1:98M and R ¼ 11:95 km. This star atthe mass shedding limit rotates with K ¼ K=2 ¼1086 Hz and has T=jWj ¼ 0:139.

In this work we consider for simplicity only uniformlyrotating stars. This is however a reasonable assumption, asit is expected that magnetohydrodynamical processesshould redistribute the angular momentum inside a proto-neutron star on few tens of rotation periods [50].

In order to extract physical results from our simulationswe must normalize the mode amplitude . In this work, weuse the following definition:

E ¼ Erot; (32)

where Erot is the stellar rotational energy. For our poly-tropic models an ¼ 1 corresponds to E 102Mc2.

The definition (32) implies that the fluid variables and the

gravitational wave strain scale as 1=2.The numerical scheme used to study the evolution of the

f-mode instability is based on a fourth order Runge-Kuttaalgorithm. The numerical code evolves separately the twophases of the instability. After having specified the initialconditions for , , and T, we evolve the initial growthphase of the f mode according to Eqs. (20) and (21). Whenthe mode amplitude reaches a predetermined saturationvalue sat, the nonlinear saturation phase is studied withrelation (25) until the star leaves the instability window andthe numerical simulation ends. In both phases of the in-stability we use Eq. (27) for the thermal evolution.

A. The evolution of the f-mode instability

We first consider the evolution of the f-mode instabilityin a low magnetized star (Bp 1011 G), and study the

effect of the magnetic torque and r mode in the followingtwo sections.The ideal conditions for the f-mode instability to work

are high temperatures and rotation rates. At birth, thetemperature of a neutron star is around T ’ 1011 K anddrops down rapidly in the following few days. We thereforeexpect that as a rapidly rotating star cools down, it entersthe f-mode instability window from the high temperatureside (see Figs. 2 and 3). In our numerical scheme, weprescribe this behavior by choosing appropriate initialconditions for and T. In particular, for a given rotationrate we specify the highest temperature allowed by thef-mode instability (e.g., see again Figs. 2 and 3). Theinitial mode amplitude is set to ¼ 106; this correspondsto an energy of E 108Mc2 which is a typical value forthe total energy loss in gravitational waves due to quadru-pole deformations shortly after a gravitational core col-lapse (see for instance Refs. [51–53]). Although the initialpulsation energy of the l ¼ m ¼ 3 and 4 f modes might besmaller, our results do not change significantly for differentinitial values of , as during the initial phase of theinstability the mode’s amplitude grows exponentially.Another parameter one needs to specify in the evolution

scheme is the saturation amplitude of the f mode. Fromnonlinear dynamical simulations one can determine upperlimits on the quadrupole f mode [54], but the problem isstill open for higher multipole f modes. We then considerthe saturation amplitude as a free parameter. In our simu-lations we choose sat ¼ 104, which is equivalent to E106Mc2. These values are ‘‘reasonable’’ and muchsmaller than what is frequently used in literature [28,31],where sat ¼ 1.For the N ¼ 1 model we consider four cases for the l ¼

m ¼ 4 f-mode instability with different initial rotationrates and temperatures. Figure 2 shows the results for thestar’s spin, the mode’s amplitude, and the star’s trajectorythrough the instability window. The entire evolution lastsfor about 500 yr, while the duration of the initial growth

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-7

phase depends on the initial rotation rate. For a star thatbecomes unstable at ¼ K ( ¼ 0:97K) the growthphase lasts for about 1 day (6 yr). This phase is dominatedby the growth time gw which increases gradually for

slower rotating stars, from gw ’ 104 s at the mass shed-

ding limit to gw ’ 108–109 s at the bottom of the insta-

bility window, 0:95K [22]. After the mode saturates,Fig. 2 (right panel) shows that the star spins down at nearlyconstant temperature (T ’ 109 K) as a result of the heatgenerated by shear viscosity which completely balancesthe neutrino cooling. This effect prevents the star fromentering the superfluid transition zone and increases theduration of the instability. In fact, for T Tcn the neutronsof the core become superfluid and mutual friction dampsthe f mode very efficiently on short time scales [34]. Forthe superfluid critical temperature we chose a value ofTcn ’ 5 108 K which has recently been determinedfrom the cooling curves of Cassiopeia [55,56].

In Fig. 2 (right panel) it is noticeable that the trajectoryof a star which becomes unstable at ¼ 0:97K isslightly different from the other cases. This is due to theslower growth phase of the mode which delays the effect ofthe shear viscosity heating on the thermal evolution.Therefore, the star first cools down to a minimum tempera-ture, then is reheated by shear viscosity and finally evolvesquasi-isothermally in the last part of the trajectory.

The evolution of the l ¼ m ¼ 4 f-mode instability forthe N ¼ 0:62 model is qualitatively similar to the N ¼ 1case, but it develops faster and within a significantly largerinstability window (see right panel of Fig. 3). At the massshedding limit the gravitational growth time is only gw ’700 s in contrast to the gw ’ 104 s for the N ¼ 1 model,

and the total evolution now lasts for about 200 yr. The

exponential growth of the f mode takes about 26 min(5 days) for a star with an initial ¼ K ( ¼ 0:95K).For the N ¼ 0:62 model we have a relevant instability

window also for the l ¼ m ¼ 3 f mode. In fact, at theKepler limit the gravitational growth time, gw ’ 300 s, is

even shorter than the l ¼ m ¼ 4 f mode but increasesmore rapidly with decreasing rotation. The resulting insta-bility region is shown in Fig. 4 for a star that becomesunstable at different rotation rates. The total instabilityevolution time is about 200 yr and the duration of thegrowth phase is comparable to the l ¼ m ¼ 4 f-modecase. Furthermore, Fig. 4 shows that when a star becomesunstable at lower rotation rates near the minimum of theinstability window, e.g., ’ 0:90K, the mode amplituderemains small and the shear viscosity heating does notbalance the neutron star’s cooling.The properties of the instability window and gravitational

growth time make the N ¼ 0:62 massive star a better gravi-tational wave source than the N ¼ 1 model (see Sec. V).

B. Magnetic torque

The main mechanism that gradually slows down a neu-tron star is electromagnetic radiation, which is powered bythe star’s rotational energy. Since the magnetic torqueincreases considerably with rotation, as shown by thedipole formula (17), the magnetic braking may affectthe evolution of the f-mode instability by acceleratingthe spin-down. This means that an unstable f mode maynot have the time to grow significantly and generate adetectable gravitational wave signal.We study the impact of the magnetic field on the l ¼

m ¼ 4 f-mode instability for the N ¼ 1 and N ¼ 0:62models. For each model we consider two dipolar magnetic

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1Ω

/ Ω

K

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

t [yr]

10-6

10-5

10-4

α

l=m=4 f-mode

N = 0.62

Ω/ΩK

=1

Ω/ΩK

=0.88

108

109

1010

T [K]

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Ω /

ΩK

.

N = 0.62

αsat

= 10-4

M = 1.98 M

Tcn

FIG. 3 (color online). Evolution of the l ¼ m ¼ 4 f-mode instability for a relativistic polytrope with N ¼ 0:62, massM ¼ 1:98M,and Bp ¼ 1011 G. The various panels depict the same quantities as described in Fig. 2.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-8

field configurations with, respectively, Bp ¼ 1012 G and

Bp ¼ 1013 G, and focus on an orthogonal rotator (see

Sec. III A).We evolve various initial conditions with different tem-

peratures and rotation rates. The evolution of the modeamplitude and the star spin-down is shown in Fig. 5 for theN ¼ 0:62 model. For a magnetic field of Bp ¼ 1012 G the

total duration of the instability is shorter than a factor ofabout 20 with respect to the Bp ¼ 1011 G model. In fact,

the evolution lasts about 10 yr for the N ¼ 0:62 and N ¼ 1models. In slower rotating models, the magnetic torqueclearly dominates the evolution even during the initialphase of the instability and limits considerably the growthof the mode amplitude. This behavior is for instance evi-dent in Fig. 5 for a N ¼ 0:62 model with an initial rotationrate of ¼ 0:88K. The magnetic torque dominates com-pletely the evolution when Bp ¼ 1013 G. The unstable star

is quickly spun down and the mode amplitude is stronglylimited even in the more massive model (see Fig. 5).

C. The f-mode versus the r-mode instability

In rapidly rotating neutron stars several modes can bedriven unstable by gravitational radiation at the same time.The most important ones are definitely the f and r modes,as they have a comparatively short growth time and there-fore can generate a significant gravitational wave signal. Arelevant difference between these two classes of modes isthat the f mode only gets unstable in very rapidly rotatingstars while the r mode is CFS unstable at any rotation rate.In addition, the growth time is typically shorter for the rmode which consequently has a larger instability window.It is then reasonable to think that the r mode shoulddominate the evolution of the gravitational wave driveninstability. However before drawing any secure conclusion,it is necessary to know the maximum amplitude that eachmode can reach during the instability.Nonlinear perturbation calculations show that the r

mode may transfer energy to other inertial modes throughnonlinear mode coupling. As a result the maximum ampli-tude of the r mode may be limited to c ¼ 103–105,

108

109

1010

T [K]

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Ω /

ΩK

108

109

1010

T [K]

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Ω /

ΩK

10-6

10-5

10-4

10-3

10-2

10-1

100

101

t [yr]

10-6

10-5

10-4

α

10-6

10-5

10-4

10-3

10-2

10-1

100

101

t [yr]

10-6

10-5

10-4

α

Bp = 10

12G

N = 0.62

N = 0.62

Bp = 10

13G

FIG. 5 (color online). The impact of the magnetic torque onthe evolution of the l ¼ m ¼ 4 f-mode instability for the N ¼0:62 model. The magnetic field is dipolar and orthogonal to therotation axis. The two top panels display the mode amplitude(left panel) and the star’s evolution through the instability region(right panel) for Bp ¼ 1012 G. The same quantities are depicted

in the two lower panels for Bp ¼ 1013 G.

0.88

0.9

0.92

0.94

0.96

0.98

1Ω

/ Ω

K

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

t [yr]

10-6

10-5

10-4

α

l=m=3 f-mode

N = 0.62Ω/Ω

K=1

Ω/ΩK

=0.90

108

109

1010

T [K]

0.9

0.92

0.94

0.96

0.98

1

Ω /

ΩK

.

N = 0.62

αsat

= 10-4

M = 1.98 M

Tcn

FIG. 4 (color online). Evolution of the l ¼ m ¼ 3 f-mode instability for a relativistic polytropic star with N ¼ 0:62, mass ofM ¼ 1:98M, and Bp ¼ 1011 G. The various panels depict the same quantities as described in Figs. 2 and 3.

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-9

where E ¼ c2Erot [21,57]. In our notation, these valuescorrespond to a saturation amplitude of r ¼ c2 ¼106–1010, since we parametrize the mode energyaccording to E ¼ rErot.

As it was already pointed out, the maximum amplitudeof the f mode is still uncertain. This forces us to consider a‘‘reasonable’’ value for the f-mode saturation which we setto sat ¼ 104, the same value that was used in the pre-vious sections. We then study the combined evolution of fand r modes for different r-mode saturation amplitudes. Inorder to address this problem we derive an additional set ofequations which are given in the Appendix.

We focus on the l ¼ m ¼ 2 rmode and the l ¼ m ¼ 4 fmode, which are the most relevant unstable modes in theirrespective classes due to their relatively short instabilitygrowth time with respect to the dissipative processes [5].We consider the more massive stellar model withN ¼ 0:62as it is potentially a better source for gravitational wavedetection, and neglect for simplicity the effect of the rmode on the magnetic field. As shown in Ref. [58], thetoroidal magnetic field component may be amplified by anr mode and this can change the evolution of the instability.In order to assess this effect it would be necessary to studyalso the backreaction of the magnetic field on the mode anddetermine a maximum magnetic field amplification.

The properties of the f mode have been already deter-mined in Sec. IVA, while for the r mode one needs toconsider some approximations of its viscous damping andgravitational radiation growth times. More precisely, thevarious dissipative time scales are determined via theanalytical relations given in Ref. [59] which have been

derived for a uniform density star. For the model underconsideration here, the l ¼ m ¼ 2 r-mode growth time atthe mass shedding limit is gw ’ 2:9 s, which is more than

2 orders of magnitude smaller than the l ¼ m ¼ 4 f-modegrowth time (gw ¼ 700 s). The oscillation frequency in

the rotating frame of the l ¼ m ¼ 2 r mode is determinedat leading order by ! ¼ 2=3, and for simplicity it is alsoassumed that the canonical angular momenta of the r and fmodes are equal. These approximations prevent us fromdetermining the properties of the r mode from numericalevolutions. However, since the r mode grows much fasterthan the f mode it is not expected that the qualitativebehavior of the results will change considerably with amore accurate description of the r mode.As the star cools down, it is expected that the r mode

gets unstable before the f mode as its instability windowextends towards higher temperatures (see Fig. 6). The rmode then quickly reaches its nonlinear saturation valuesatr and spins down the star. However if the r-mode growth

is limited by nonlinear mode coupling, the star evolvestowards the critical curve of the fmode and eventually alsothis mode is driven unstable by gravitational radiation.In Fig. 6 we show a representative case in which the star

enters the critical curve of the rmode at ¼ 0:98K withan initial amplitude r ¼ 1010. We consider severalsimulations where the r mode grows up to a maximumamplitude of sat

r ¼ 109–105, while for the f mode weset sat ¼ 104. The impact of the r mode on the star’sspin-down and the total evolution of the f mode is evidentalready at low saturation r. The star leaves the instabilitywindow of the f mode after an evolution time given by

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

t [yr]

10-6

10-5

10-4

α l=m=4 f-mode

l=m=2 r-mode

N = 0.62

αr = 10

-9

αr = 10

-7

αr = 10

-5

0.86

0.88

0.9

0.92

0.94

0.96

0.98

Ω /

ΩK

108

109

1010

1011

T [K]

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Ω /

ΩK

N = 0.62

αsat

= 10-4

τr = 0

Tcn

αr = 10

-9

αr = 10

-7

αr = 10

-5

FIG. 6 (color online). The effects of an unstable l ¼ m ¼ 2 r mode on the evolution of the l ¼ m ¼ 4 f-mode instability for arelativistic polytrope with N ¼ 0:62. The saturation amplitude of the r mode is varied between r ¼ 109–105 while the maximumamplitude of the f mode is kept constant at sat ¼ 104. The left panel shows the evolution of the stellar rotation (top panel) and thef-mode amplitude (lower panel) for the various examples. The filled circle in the lower left panel denotes the end of the f-modeinstability for r ¼ 107. The right panel shows the evolution through the instability window, where the dashed line denotes the hightemperature part of the r-mode critical curve (r ¼ 0).

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-10

tev 300 ð109=satr Þ yr. When r 105 the evolu-

tion is completely dominated by the r mode and thef-mode amplitude is strongly constrained to small values.However, for r 107 the f mode still has time to growand potentially generates a detectable gravitational wavesignal (see Sec. V). As expected, it is therefore crucial toknow more accurately the relative saturation amplitudebetween these two modes. This is an interesting aspectthat must be clarified in a future work.

V. GRAVITATIONALWAVES

The instability of the f mode is potentially a strongsource of gravitational radiation and it is then logical toestimate the detectability prospectives by the current andnext generation of Earth-based laser interferometers.

The two independent polarization states of the gravita-tional wave strain can be expressed in terms of the spin-weighted spherical harmonic 2Y

lm as

hlm ihlm ¼ hlm2Ylm; (33)

where hlm is defined as

hlm G

clþ2

l

rði!IÞlDlm; (34)

and !I ¼ !m is the mode frequency in the inertialframe. The coefficient l is given by

l 8

lðl 1Þð2lþ 1Þ!!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlþ 2Þ!ðl 2Þ!

s: (35)

A more suitable quantity to evaluate the gravitationalwave detection is the characteristic strain, which also takes

into account the statistical amplification of the signal dueto the number of oscillations accumulated in a given fre-quency bandwidth. The characteristic strain is defined bythe following equation:

hlmc hjhlm2Ylmji

ffiffiffiffiffiffiffiffiffiNcyc

q; (36)

where h i denotes an averaging over the angles ð;Þ.The number of oscillation cycles can be expressed asNcyc ¼ ev, which is written in terms of the mode fre-

quency ¼ !I=2 and the time spent near a givenfrequency ev.For magnetic fields with a magnitude of Bp 1011 G,

the impact of the magnetic torque on the f-mode instabilityis irrelevant (see Sec. IV), and the mode evolves like in anunmagnetized star. This means that during the initialexponential growth of the mode the star spins down onviscous time scales, i.e. is nearly constant, and theradiated gravitational wave signal is virtually monochro-matic. As a result, the quantity ev can be approximated bythe accumulated evolution time of the star during thisinitial phase with ’ const. After the f mode saturates,the amplitude remains approximately constant and gravi-tational radiation removes angular momentum from thestar. This leads to a frequency variation which can bewell approximated by the following relation:

ev ¼

dt

d

: (37)

When a star has higher magnetic field strengths, Bp >

1011 G, the magnetic torque affects also the initial phase ofthe instability and accelerates the spin-down. The f-mode

0.5 1.0 1.5 2.0 2.5

ν [ kHz ]

10-24

10-23

10-22

10-21

10-20

h c44

d = 20 Mpc

1.0

0.95

0.98 1.0

ET

Adv. LIGOα

sat = 10

-4

N = 1

Bp = 10

12G

0.90

l = m = 4 f-mode

N = 0.62

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

10-24

10-23

10-22

10-21

10-20

h c33

d = 20 Mpc 1.0

Bp = 10

12G

ET

Adv. LIGOαsat

= 10-4

N = 0.62

0.95

l = m = 3 f-mode

0.91

ν [ kHz ]

FIG. 7 (color online). The characteristic strain generated by the f-mode instability for a polytropic star with Bp ¼ 1012 G. The leftpanel shows the gravitational wave signal of the l ¼ m ¼ 4 f mode for the N ¼ 1 and N ¼ 0:62 models, while the right panel depictsthe signal of the l ¼ m ¼ 3 f mode for the N ¼ 0:62 model. The source is located at 20 Mpc and the saturation amplitude of the fmode is set to sat ¼ 104. The notation used in this figure is the same as in Fig. 1.

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-11

frequency therefore evolves as the star loses angular mo-mentum by magnetic braking and Eq. (37) has to be usedthroughout the entire instability evolution.

However, realistic calculations of the characteristicstrain also have to consider the maximum signal integra-tion time allowed by the detector technology. We assumethat a detector may integrate the signal at most for 1 yr, andconsequently calculate hlmc for ev 1 yr. This means thatin Ncyc we use the evolution time of our simulations

whenever ev < 1 yr, otherwise we set ev ¼ 1 yr.The results for the N ¼ 1 and N ¼ 0:62 models are

shown in Fig. 1 and in Figs. 7 and 8 together with thesensitivity curves of Advanced LIGO and ET [60,61]. Thedifferent noise curves of the detectors are determined by

using hrms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiShðÞ

p, where ShðÞ is the power spectrum

of the detector at hand.For each model we determine the characteristic strain of

the various evolutionary paths studied in Sec. IV in whichthe onset of the f-mode instability happens at differentrotation rates. As shown in Fig. 1, the signal is initiallymonochromatic during the growth phase of the mode andevolves successively to lower frequencies as the star spinsdown. We find that if an unstable f mode with multipolesl ¼ m ¼ 3, 4 saturates at sat ¼ 104, the gravitationalwave signal can be detected by ET for the most part ofthe instability evolution for the more massive stellar model(N ¼ 0:62) located in the Virgo Cluster at a distance of20 Mpc. Actually, the l ¼ m ¼ 3 f mode may be detect-able also by Advanced LIGO/Virgo. For the N ¼ 1 model,an unstable l ¼ m ¼ 4 f mode may generate a detectablesignal only if the star rotates near the Kepler limit * 0:98K.

A higher magnetic field may affect significantly thecharacteristic strain of the f mode, as the number ofaccumulated cycles Ncyc decreases. The results for the

Bp ¼ 1012 G case are shown in Fig. 7 for both the N ¼ 1

and N ¼ 0:62 models. Although the gravitational wavesignal is slightly weaker than in the previous Bp ¼1011 G case, it is still detectable by ET and the l ¼ m ¼3 f mode of the N ¼ 0:62 model is also above the sensi-tivity curve of Advanced LIGO. The effect of the magneticbreaking is particularly evident if the star gets unstable atlower rotation rates. For instance, the characteristic strainof the N ¼ 1 model with an initial rotation rate of ¼ 0:97K is significantly smaller than for the lowmagnetized model. We have also calculated the character-istic strain for models with a magnetic field strength ofBp ¼ 1013 G and found that the gravitational wave signal

of the N ¼ 1 model drops below the detectors’ sensitivitycurves. This happens also to the N ¼ 0:62 model when & 0:96K. The results presented in this section can beeasily rescaled with the f-mode saturation amplitude as

hc 1=2sat .

If in addition an rmode is present as well, the results forthe gravitational wave strain depend on the value of the

saturation amplitude for this auxiliary mode; see Fig. 8 forthe N ¼ 0:62 polytrope. Supposing a small r-mode satu-ration, the expected signal can still be detected with ETfrom a source in the Virgo cluster. However, as the satura-tion amplitude of the r mode is increased, its instabilitydominates the evolution of the f-mode instability. The starloses angular momentum too fast for the f mode to growsubstantially before it leaves its instability window (seealso Fig. 6), leading to a significant reduction in the gravi-tational wave strain for large r-mode saturation values. Inthese cases the signal of the f mode will drop even belowthe ET sensitivity curve. However, estimating the gravita-tional wave growth time of the r mode with a uniformdensity stellar model [59], we find that an unstable l ¼m ¼ 2 r mode can generate a gravitational wave signaldetectable by ET. This means that when both the f and rmodes are simultaneously excited and the r-mode satura-tion amplitude is sat

r & 107, ET should be able to detectthe gravitational wave signal emitted by both these twomodes. Instead, for larger sat

r the r mode will be stilldetectable but the f-mode signal should be below the ETsensitivity curve.

VI. CONCLUSIONS

We have presented in this work the first dynamical studyof the f-mode instability which considers relativistic rap-idly rotating stars and incorporates the effect of viscosity,magnetic fields, and unstable r modes. These are the mostdominant effects which may have a significant impact on

0.6 0.8 1.0 1.2 1.4 1.6ν [ kHz ]

10-24

10-23

10-22

10-21

10-20

h c44

d = 20 Mpc

Ω/ΩK

= 0.98

ET

Adv. LIGO

N = 0.62

αr = 10

-9

αr = 10

-7

αr = 10

-6

FIG. 8 (color online). The characteristic strain generated bythe l ¼ m ¼ 4 f-mode instability when an l ¼ m ¼ 2 r mode isexcited as well. The star is a relativistic polytrope with N ¼0:62, Bp ¼ 1011 G and is located at 20 Mpc. It enters the r-mode

instability window at ¼ 0:98K. The figure shows three caseswith different r-mode saturation amplitudes while the saturationamplitude of the f mode is fixed to sat ¼ 104.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-12

the evolution of this mode. Our neutron star’s models arerelativistic and described by a polytropic equation of state.We consider more in detail two sequences of uniformlyrotating stars which can rotate up to the Kepler limitand describe, respectively, neutron star models withM ¼ 1:4M and M ¼ 1:98M.

The f-mode instability may develop in the aftermath ofa supernova explosion, when a new born neutron star mayspin very rapidly. The most unstable f mode is expected inthe l ¼ m ¼ 3 and 4 multipoles, which have a relativelylarge instability window with respect to the l ¼ m ¼ 2case. Considering various cases in which a neutron starenters the instability window at different rotation rates, wefind that the gravitational wave signal emitted during theinstability may be detected by ET from a source located inthe Virgo cluster, and as expected the gravitational signal isstronger for the more massive model with N ¼ 0:62.Actually for this model, the l ¼ m ¼ 3 f mode may bedetected also by Advanced LIGO/Virgo. This result is validfor an f mode which saturates at E ’ 106Mc2 and for asignal integration time 1 yr. From the Virgo cluster weshould expect about 30–60 supernova explosions per year,by assuming a rate of 2–3 events per century in our Galaxy.This means that if few of them leave behind a very rapidlyspinning protoneutron star, we might be able to detect theseevents from the gravitational radiation emitted during thef-mode instability. Stars with high mass and angular mo-mentum are certainly more promising gravitational wavesources. They may originate from roughly the 1% of corecollapses of progenitor stars with M> 10M [62]. Thenumber of these potential sources may be even morepromising if we note that in more massive stars the gravi-tational wave signal may remain detectable for many years.

Another important result of our simulations is that theheat generated by shear viscosity during the f-mode satu-ration phase prevents the star from entering the regime ofmutual friction. In fact, the shear viscosity reheating bal-ances the neutrino cooling and leads to a nearly isothermalevolution. The star therefore leaves the instability windowat lower rotation rates, and even a moderate change in thesuperfluid transition temperature does not change thisresult.

The magnetic torque as well as an unstable r mode mayaccelerate the transition of the star through the instabilitywindow of the f mode. This may limit the growth of themode’s amplitude and therefore the strain of the gravita-tional wave radiation. Our results show that the magneticfield affects the f-mode instability when Bp * 1012 G, and

its influence is more relevant in the N ¼ 1 neutron starmodel. For the r mode we find that it must reachthe maximum value expected from nonlinear mode cou-pling studies in order to affect considerably the f-modeevolution. However, a definitive answer to this issuecannot be given as the maximum f-mode amplitude is stillunknown.

In this study, we have not addressed the ‘‘classical’’ l ¼m ¼ 2 f-mode instability, which is potentially the lowestorder unstable mode driven by gravitational wave emis-sion. As it was shown previously in Ref. [22], the prospectsfor a dynamical long-term evolution of a quadrupolarinstability are rather pessimistic given the diminishedsize of the corresponding instability window. However,dropping the Cowling approximation which we used herewill certainly improve the situation for the l ¼ m ¼ 2 casesince in full general relativity the critical angular velocityfor the onset of the CFS instability is shifted towards lowervalues [7]. On the other hand, including relativistic effectsfor the r mode might have the opposite effect in the sensethat relativistic perturbation theory hints towards a weak-ening, i.e. much larger growth times, of the instabilitywhen compared to Newtonian results [63]. This also worksin favor of an enhanced detectability even if f and rmodesare unstable at the same time.So far we used polytropic equations of state for mod-

eling relativistic neutron stars. A natural extension ofthis work should also incorporate a wide range of realisticequations of state. This most probably will not change themain results of this work but for sure will lead to alter-ations in mode frequencies, growth times, and gravita-tional wave detectability. At the same time, in morecompact neutron star models one needs also to take intoaccount the effect of direct Urca processes. These -reaction processes are energetically favorable whenthe proton fraction is sufficiently large, xp > 1=9, [64],

and they lead to a stronger bulk viscosity which may limitconsiderably the instability window of the f mode [44].Another potentially significant damping mechanism ne-

glected here is the formation of a crust and its importanceon the f-mode instability. Additional dissipation will occurat a viscous boundary layer at the crust-core interface ormore directly, the crust could break by a large amplitude fmode. This would release a non-negligible amount ofenergy for dissipation at the fracture sites. Of course thismechanism depends on how fast a crust is formed after thecreation of a hot, rapidly rotating protoneutron star. Largeamplitude oscillations might even delay the formation ofthe crust in the nascent neutron star.

ACKNOWLEDGMENTS

A. P. acknowledge support from the German ScienceFoundation (DFG) via SFB/TR7, and from the EuropeanCommission Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 267251. E. G. ac-knowledges support from VESF and the GermanScience Foundation (DFG) via SFB/TR7. D. D. acknowl-edges support from the German Science Foundation(DFG) via SFB/TR7 and by the Bulgarian NationalScience Fund under Grant No. DMU-03/6. We wouldlike to thank K. Glampedakis and L. Rezzolla for stimu-lating discussions and comments.

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-13

APPENDIX: THE EVOLUTION EQUATIONSOF THE f- AND THE r-MODE INSTABILITY

In this section, we derive the evolution equations for

studying the evolution of the gravitational wave instability

driven by two modes. Although these equations have been

applied in Sec. IVC for studying the f and r modes the

formalism is general. As in the single mode case the main

equations are (2) and (14), which now have to be extended

to describe the instability of two modes.

The first two equations can be derived from the mode

energy variation (2)

dx

dtþ x

d ln ~Ex

d

d

dt¼ 2x

x; (A1)

where the index x ¼ f, r identifies the quantities related to

the f and r modes, respectively, and we have defined the

energy of each x mode as Ex ¼ x~ExðÞ.

A third equation can be determined from the angular

momentum evolution equation (14), with the canonical

angular momentum now given by

Jc ¼ f~JfcðÞ þ r

~JrcðÞ; (A2)

where ~JfcðÞ and ~JrcðÞ describe the dependence of the

canonical angular momentum of each mode on the angular

velocity of the star. The other term that changes in Eq. (14)

is the gravitational radiation torque, which now can be

decomposed as

dJgwdt

¼ 2Jfcfgw

2Jrcrgw

; (A3)

where fgw and rgw are the gravitational growth time of the

f and r modes. Taking the time derivatives, the angular

momentum conservation equation reads

Jfcdf

dtþ Jrc

dr

dtþ

dJsd

þ f

dJfcd

þ r

dJrcd

d

dt

¼ 2Jfcfgw

2Jrcrgw

þ dJmag

dt: (A4)

As final step, we can combine Eqs. (A1) and (A4) and

obtain the following system of evolution equations:

dx

dt¼ 2x

xgw 2x

xv

Dx

D 2xy

yv

AxFy

D

x

Ax

D

dJsd

1 dJmag

dt; (A5)

Dd

dt¼ X

x

2Fxx

xvþ

dJsd

1 dJmag

dt; (A6)

where y also denotes the f- and r-mode quantities with the

condition that x y.In Eqs. (A5) and (A6) we have defined the following

functions:

Ax ¼ d ln ~Ex

d; Fx ¼ ~Jxc

dJsd

1; (A7)

Px ¼ Ax Fx; Qx ¼ d~Jxcd

dJsd

1; (A8)

where

D ¼ 1þXx

xðQx PxÞ; (A9)

Dx ¼ Dþ xPx: (A10)

Similar to the single mode instability, we may determinethe evolution equations of a star in case one of the modessaturates.When the ymode saturates, its amplitude is nearlyconstant, i.e. dy=dt ¼ 0, and Eqs. (A5) and (A6) now read

dx

dt¼ 2x

xgw 2x

xv

Dq

Dy

þ 2xy

yv

AxFy

Dy

xAx

Dy

dJsd

1 dJmag

dt; (A11)

Dy

d

dt¼ 2Fyy

ygw

þ 2Fxx

xvþ

dJsd

1 dJmag

dt; (A12)

where Dq ¼ 1þPxxQx.

When both modes have reached their respective satura-tion amplitudes, we approximately have that df=dt ¼dr=dt ¼ 0. As a result the star rotation evolves accordingto the following equation:

Dq

d

dt¼ 2

Xx

Fxx

xgwþ

dJsd

1 dJmag

dt; (A13)

where in a lowmagnetized star both fgw and rgw govern the

spin-down rate of the star.

PASSAMONTI et al. PHYSICAL REVIEW D 87, 084010 (2013)

084010-14

[1] C. D. Ott, Classical Quantum Gravity 26, 063001(2009).

[2] N. Andersson, V. Ferrari, D. I. Jones, K. D. Kokkotas, B.Krishnan, J. S. Read, L. Rezzolla, and B. Zink, Gen.Relativ. Gravit. 43, 409 (2011).

[3] L. Baiotti, R. De Pietri, G.M. Manca, and L. Rezzolla,Phys. Rev. D 75, 044023 (2007).

[4] G.M. Manca, L. Baiotti, R. De Pietri, and L. Rezzolla,Classical Quantum Gravity 24, S171 (2007).

[5] N. Andersson, Classical Quantum Gravity 20, R105(2003).

[6] J. L. Friedman, Phys. Rev. Lett. 51, 11 (1983).[7] B. Zink, O. Korobkin, E. Schnetter, and N. Stergioulas,

Phys. Rev. D 81, 084055 (2010).[8] S. Chandrasekhar, Phys. Rev. Lett. 24, 611 (1970).[9] J. L. Friedman and B. F. Schutz, Astrophys. J. Lett. 199,

L157 (1975).[10] J. L. Friedman and B. F. Schutz, Astrophys. J. Lett. 221,

937 (1978).[11] L. Samuelsson and N. Andersson, Mon. Not. R. Astron.

Soc. 374, 256 (2007).[12] H. Sotani, A. Colaiuda, and K.D. Kokkotas, Mon. Not. R.

Astron. Soc. 385, 2161 (2008).[13] A.W. Steiner and A. L. Watts, Phys. Rev. Lett. 103,

181101 (2009).[14] A. Colaiuda and K.D. Kokkotas, Mon. Not. R. Astron.

Soc. 414, 3014 (2011).[15] M. Gabler, P. Cerda-Duran, J. A. Font, E. Muller, and N.

Stergioulas, arXiv:1208.6443.[16] N. Andersson and K.D. Kokkotas, Phys. Rev. Lett.77,

4134 (1996).[17] N. Andersson and K.D. Kokkotas, Mon. Not. R. Astron.

Soc. 299, 1059 (1998).[18] N. Andersson and K.D. Kokkotas, Int. J. Mod. Phys. D 10,

381 (2001).[19] A. Patruno, B. Haskell, and C. D’Angelo, Astrophys. J.

746, 9 (2012).[20] P. Arras, E. E. Flanagan, S.M. Morsink, A.K. Schenk,

S. A. Teukolsky, and I. Wasserman, Astrophys. J. 591,1129 (2003).

[21] R. Bondarescu, S. A. Teukolsky, and I. Wasserman, Phys.Rev. D 76, 064019 (2007).

[22] E. Gaertig, K. Glampedakis, K. D. Kokkotas, and B. Zink,Phys. Rev. Lett. 107, 101102 (2011).

[23] J. R. Ipser and L. Lindblom, Astrophys. J. 355, 226 (1990).[24] J. R. Ipser and L. Lindblom, Astrophys. J. 373, 213 (1991).[25] L. Lindblom, Astrophys. J. 438, 265 (1995).[26] N. Stergioulas and J. L. Friedman, Astrophys. J. 492, 301

(1998).[27] S. L. Detweiler and L. Lindblom, Astrophys. J. 213, 193

(1977).[28] D. Lai and S. L. Shapiro, Astrophys. J. 442, 259 (1995).[29] S. Ou, J. E. Tohline, and L. Lindblom, Astrophys. J. 617,

490 (2004).[30] M. Shibata and S. Karino, Phys. Rev. D 70, 084022

(2004).[31] B. J. Owen, L. Lindblom, C. Cutler, B. F. Schutz, A.

Vecchio, and N. Andersson, Phys. Rev. D 58, 084020(1998).

[32] N. Andersson, D. I. Jones, and K.D. Kokkotas, Mon. Not.R. Astron. Soc. 337, 1224 (2002).

[33] N. Stergioulas, Living Rev. Relativity 6, 3 (2003), http://www.livingreviews.org/lrr-2003-3.

[34] L. Lindblom and G. Mendell, Astrophys. J. 444, 804(1995).

[35] E. Gaertig and K.D. Kokkotas, Phys. Rev. D 78, 064063(2008).

[36] E. Gaertig and K.D. Kokkotas, Phys. Rev. D 83, 064031(2011).

[37] J. L. Friedman and B. F. Schutz, Astrophys. J. 221, 937(1978).

[38] D. Doneva, E. Gaertig, K. Kokkotas, and C. Krueger (inpreparation).

[39] K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980).[40] E. Flowers and N. Itoh, Astrophys. J. 230, 847 (1979).[41] R. F. Sawyer, Phys. Rev. D 39, 3804 (1989).[42] M.G. Alford, S. Mahmoodifar, and K. Schwenzer, Phys.

Rev. D 85, 024007 (2012).[43] M.G. Alford, S. Mahmoodifar, and K. Schwenzer, Phys.

Rev. D 85, 044051 (2012).[44] A. Passamonti and K. Glampedakis, Mon. Not. R. Astron.

Soc. 422, 3327 (2012).[45] S. L. Shapiro and S.A. Teukolsky, Black Holes,

White Dwarfs, and Neutron Stars (Wiley-Interscience,New York, 1983).

[46] A. Spitkovsky, Astrophys. J. Lett. 648, L51 (2006).[47] W.C.G. Ho, K. Glampedakis, and N. Andersson, Mon.

Not. R. Astron. Soc. 422, 2632 (2012).[48] S. Dall’Osso, S. N. Shore, and L. Stella, Mon. Not. R.

Astron. Soc. 398, 1869 (2009).[49] S. Dall’Osso, J. Granot, and T. Piran, Mon. Not. R. Astron.

Soc. 422, 2878 (2012).[50] M.D. Duez, Y. T. Liu, S. L. Shapiro, M. Shibata, and B. C.

Stephens, Phys. Rev. D 73, 104015 (2006).[51] H. Dimmelmeier, C. D. Ott, A. Marek, and H. Thomas

Janka, Phys. Rev. D 78, 064056 (2008).[52] S. Scheidegger, R. Kappeli, S. C. Whitehouse, T. Fischer,

and M. Liebendorfer, Astron. Astrophys. 514, A51 (2010).[53] B. Muller, H.-T. Janka, and A. Marek, Astrophys. J. 766,

43 (2013).[54] W. Kastaun, B. Willburger, and K.D. Kokkotas, Phys.

Rev. D 82, 104036 (2010).[55] D. Page, M. Prakash, J.M. Lattimer, and A.W. Steiner,

Phys. Rev. Lett. 106, 081101 (2011).[56] P. S. Shternin, D. G. Yakovlev, C.O. Heinke, W.C.G. Ho,

and D. J. Patnaude, Mon. Not. R. Astron. Soc. 412, L108(2011).

[57] A. K. Schenk, P. Arras, E. E. Flanagan, A. Teukolsky, andI. Wasserman, Phys. Rev. D 65, 024001 (2001).

[58] L. Rezzolla, F. K. Lamb, and S. L. Shapiro, Astrophys. J.Lett. 531, L139 (2000).

[59] K. D. Kokkotas and N. Stergioulas, Astron. Astrophys.341, 110 (1999).

[60] S. Hild, S. Chelkowski, and A. Freise, arXiv:0810.0604.[61] G.M. Harry, Classical Quantum Gravity 27, 084006

(2010).[62] S. E. Woosley and A. Heger, Astrophys. J. 637, 914

(2006).[63] J. Ruoff and K.D. Kokkotas, Mon. Not. R. Astron. Soc.

330, 1027 (2002).[64] J.M. Lattimer, C. J. Pethick, M. Prakash, and P. Haensel,

Phys. Rev. Lett. 66, 2701 (1991).

EVOLUTION OF THE f-MODE INSTABILITY IN . . . PHYSICAL REVIEW D 87, 084010 (2013)

084010-15