Do Neutron Star Gravitational Waves Carry Superfluid Imprints?

1903

0015-9018/02/1200-1903/0 2003 Plenum Publishing Corporation

Foundations of Physics, Vol. 32, No. 12, December 2002 ( 2003)

Do Neutron Star Gravitational Waves CarrySuperfluid Imprints?

G. L. Comer1

1Department of Physics, Saint Louis University, Saint Louis, Missouri 63156-0907; e-mail:[email protected]

Received July 29, 2002

Isolated neutron stars undergoing non-radial oscillations are expected to emit gra-vitational waves in the kilohertz frequency range. To date, radio astronomers havelocated about 1,300 pulsars, and can estimate that there are about 2108 neutronstars in the galaxy. Many of these are surely old and cold enough that theirinteriors will contain matter in the superfluid or superconducting state. In fact, theso-called glitch phenomenon in pulsars (a sudden spin-up of the pulsars crust) isbest described by assuming the presence of superfluid neutrons and superconduct-ing protons in the inner crusts and cores of the pulsars. Recently there has beenmuch progress on modelling the dynamics of superfluid neutron stars in both theNewtonian and general relativistic regimes. We will discuss some of the mainresults of this recent work, perhaps the most important being that superfluidityshould affect the gravitational waves from neutron stars (emitted, for instance,during a glitch) by modifying both the rotational properties of the background starand the modes of oscillation of the perturbed configuration. Finally, we present ananalysis of the so-called zero-frequency subspace (i.e., the space of time-indepen-dent perturbations) and determine that it is spanned by two sets of polar (orspheroidal) and two sets of axial (or toroidal) degenerate perturbations for thegeneral relativistic system. As in the Newtonian case, the polar perturbations arethe g-modes which are missing from the pulsation spectrum of a non-rotating con-figuration, and the axial perturbations should lead to two sets of r-modes when thedegeneracy of the frequencies is broken by having the background rotate.

KEY WORDS: neutron stars; superfluidity; gravitational waves.

1. INTRODUCTION

Jacob Bekenstein is one of those rare individuals who can make significant,original contributions to diverse areas of theoretical physics. He is also a

man of great integrity and, I believe, has a humility that serves him well inadvising and supporting students and young scientists. I am profoundlygrateful that fate allowed me to be one of those young scientists and nowlets me participate in this celebration of his career. One of the areas oftheoretical physics that Jacob has worked on is relativistic fluid dynamics.This is an important component of my current area of research, which is todevelop models of Newtonian and general relativistic superfluid neutronstars. My original interest in superfluids, appropriately enough, wassparked by Jacob, when he suggested that I look at superfluid analogs ofeffects predicted for quantum fields in curved spacetimes (the Hawkingand FullingDaviesUnruh effects). My current interest in superfluids is todetermine how the dynamics of superfluid neutron stars differ from theirordinary, or perfect, fluid counterparts and if the different dynamicscan lead to observable effects in gravitational waves. In the remainder ofthis article, I will give an overview of what my collaborators and I haveaccomplished so far, including some new results (from work with NilsAndersson) on the structure of the so-called zero-frequency subspace (i.e.,the space of time-independent perturbations). The main purpose is to showthat superfluidity in neutron stars should affect their gravitational waves intwo ways, by modifying the rotational properties of the background starand the modes of oscillation of the perturbed configuration.

While there are many mysteries about neutron stars that remain to beexplained, we do have some significant observational facts to work with. Forinstance, Lorimer(1) reports that nearly 1300 pulsars (i.e., rotating neutronstars) have now been observed. By extrapolating the data on the local popu-lation, he can estimate that there are about 1.6105 normal pulsars andaround 4104 millisecond pulsars in our galaxy. Of course, there are alsoneutron stars that are no longer active pulsars. To get a handle on theirnumber Lorimer takes the observed supernova rate, which is about 1 per 60years, and the age of the universe to find about 2108 neutron stars in thegalaxy. The overwhelming majority of these objects must be very cold in thesense that their (local) temperatures are much less than the (local) Fermitemperatures of the independent species of the matter. One can estimate theFermi temperature to be about 1012 K for neutrons at supra-nuclear densi-ties, and it is generally accepted that within the first year (and probablymuch sooner than that) nascent neutron stars should cool to temperaturesless than 109K. This is an interesting fact, in that nuclear physics calcula-tions of the transition temperature for neutrons and protons to becomesuperfluid and superconducting, respectively, consistently yield a value thatis 109K in order of magnitude (for recent reviews see Refs. 2 and 3). Thuswe can expect that a significant portion of the neutron stars in our galaxywill have at least two (and perhaps more) superfluids in their cores.

1904 Comer

In addition to nuclear physics theory and experiment, the well-estab-lished glitch phenomenon in pulsars (e.g., Vela and Crab)(4, 5) is perhaps thebest piece of evidence that supports the existence of superfluids in neutronstars. A glitch is a sudden spin-up of the observed rotation rate of aneutron star, and can have a relaxation time of weeks to months.(6) Baymet al.(7) have noted that a relaxation mechanism based on ordinary fluidviscosity would be much too short to explain a weeks to months timescaleand so they argue that this signals the presence of a neutron superfluid.Now, a mainstay idea for explaining glitches is that of superfluids and theirvortex dynamics, i.e., how the vortices get pinned, unpinned, and thenrepinned(810) to nuclei in the inner crusts of the glitching pulsars. Thisis known as the vortex creep model and in it glitches are a transfer ofmomentum via vortices from one angular momentum carrying componentof the star to another. The model has worked well to describe both thegiant glitches in Vela and the smaller ones of the Crab. The vortex creepmodel can also be used to infer the internal temperature of a glitchingpulsar, and for Vela it implies a temperature of 107 K. (10) It is also interest-ing to note the work of Tsakadze and Tsakadze(11) who have experimentedwith rotating superfluid Helium II and find behaviour very much likeglitches in pulsars.

The classic description of superconductivity in ordinary condensedmatter systems is based on the so-called BCS mechanism (see, forinstance, Ref. 12 or Ref. 13 for excellent presentations): the particles thatbecome superconducting must be fermions, and below a certain transitiontemperature there must be an (usually effective) attractive interactionbetween them (at the Fermi surface with zero total momentum). Theinteraction leads to so-called Cooper-pairing where a pair of fermions actlike a single boson and a collection of them can behave as a condensate.The mechanism is very robust, which is why it also forms the basis for dis-cussion of nucleon superfluidity and superconductivity;(14) i.e., nucleons arefermions and the effective interaction between them at nuclear and supra-nuclear densities can be attractive. For instance, it is known experimentallythat the lowest excited states in even-even nuclei are systematically higherthan other nuclei because of pairing between nucleons which must bebroken.(12)

After many years of development, beginning with the work of Migdal,(15)

a consistent picture has emerged (in part, from gap calculations):(2, 3) Atlong-range the nuclear force is attractive and leads to neutron Cooperpairing in 1S0 states in the inner crust, but because of short-range repulsionin the nuclear force and the spin-orbit interaction neutrons pair into 3P2states in the more dense regions of the core.(16) In the crust protons arelocked inside of neutron rich nuclei embedded in a degenerate normal fluid

Do Neutron Star Gravitational Waves Carry Superfluid Imprints? 1905

of electrons. In the inner crust the nuclei are also embedded in, and evenpenetrated by, the superfluid neutrons. In the core, however, the nucleihave dissolved and the protons remain dilute enough that they feel only thelong-range attractive part of the nuclear force and pair in 1S0 states. Thereis no pairing between neutrons and protons anywhere in the core since theirrespective Fermi energies are so different. The core superfluid neutrons andsuperconducting protons are also embedded in a highly degenerate normalfluid of electrons. Other possibilities, such as pion or hyperon condenstates,have been put forward but we will keep to the simplest scenario that con-siders only superfluid neutrons, superconducting protons, crust nuclei, andnormal fluid electrons.

There are several ways in which the dynamics of a superfluid differfrom its ordinary fluid counterpart, and each difference should have someimpact on the gravitational waves that a superfluid neutron star emits. Onekey difference is that a pure superfluid is locally irrotational. A superfluid,however, can mimic closely ordinary fluid rotation by forming a densearray of (quantized) vortices. In the core of each vortex the superfluidity isdestroyed and the particles are in an ordinary fluid state, and can carry non-zero vorticity. A second, very important difference is when there are severalspecies of matter in a superfluid, or superconducting, state. The superfluidsof all the species will interpenetrate and each superfluid will be dynamicallyindependent having its own unit four-vector and local particle numberdensity. Lastly, superfluids have zero viscosity, but when vortices and exci-tations are present, then dissipative mechanisms can exist. For instance, thescattering of excitations off of the normal fluid in the vortex cores can leadto dissipative momentum exchange between the excitations and the super-fluid, the net effect being that the superfluid motion becomes dissipative.This form of dissipative mechanism is known as mutual friction.

In neutron stars there is a very efficient form of mutual friction(14, 17, 18)

that depends on the entrainment effect,(19, 20) which Sauls(14) describes asfollows: even though the neutrons are superfluid and the protons aresuperconducting both will still feel the long-range attractive component ofthe nuclear force. In such a system of interacting fermions the resultingexcitations are quasiparticles. This means that the bare neutrons (orprotons) are dressed by a polarization cloud of nucleons comprised ofboth neutrons and protons. Since both types of nucleon contribute to thecloud the momentum of the neutrons, say, is modified so that it is a linearcombination of the neutron and proton particle number density currents.The same is true of the proton momentum. Thus when one of the nucleonfluids starts to flow it will, through entrainment, induce a momentum inthe other fluid. Alpar et al.(17) have shown that the electrons track veryclosely the superconducting protons (because of electromagnetic attraction).

1906 Comer

Around each vortex is a flow of the superfluid neutrons. Because ofentrainment, a portion of the protons, and thus electrons too, will be pulledalong with the superfluid neutrons. The motion of the plasma leads tomagnetic fields being attached to the vortices. The mutual friction in thiscase is the dissipative scattering of the normal fluid electrons off of themagnetic fields attached to the vortices.

There has been much effort put forward to develop Newtonian(2125)

and general relativistic formalisms(2634) for describing superfluid neutronstars. In the simplest, but still physically interesting, formalism one has asystem that consists of two interpenetrating fluidsthe superfluid neutronsin the inner crust and core and the remaining charged constituents (i.e.,crust nuclei, core protons, and crust and core electrons) that will be looselyreferred to as protonsand the entrainment effect that acts betweenthem. In principle the model can be expanded to have more than twointerpenetrating fluids (see, for instance, Ref. 35). As well a given super-fluid can be confined to a distinct region in the star.(36) In this way theproton fluid, say, can be made to extend out farther than the superfluidneutrons. This is a first approximation at incorporating the fact that thesuperfluid neutrons do not extend all the way to the surface of the star.

Our primary goal is to show that superfluidity will affect gravitationalwave emission from neutron stars. We will see that a suitably advanced,but plausible, detector will have enough sensitivity at high frequency to seemodes excited during a glitch. With such detections we will be able to placeconstraints, say, on the parameters that describe entrainment. But in addi-tion to studying glitches, we also need to analyze further the recently dis-covered instability in the r-modes of neutron stars.(3738) The instability isdriven by gravitational wave emission (the CFS mechanism)(3941) and thewaves are potentially detectable by LIGO II. (4244) The conventionalwisdom early on stated that mutual friction would act against the instabil-ity in a superfluid neutron star and thus effectively suppress the gravita-tional radiation. But Lindblom and Mendell(45) have found that mutualfriction is largely ineffective at suppressing the r-mode instability. However,there are many questions about the spectrum of oscillation modes allowedby a rotating superfluid neutron star and the analysis of instabilities is verylikely to be much richer than the ordinary fluid case. Thus, another goalhere is to lay some groundwork for a future detailed study of the CFSmechanism in superfluid neutron stars. Specifically, we will demonstratethat the zero frequency subspace is spanned by two sets of polar (orspheroidal) and two sets of axial (or toroidal) degenerate perturbations forthe general relativistic system. Like the Newtonian case,(46) the polar per-turbations are the g-modes which are missing from the pulsation spectrumof a non-rotating configuration, and the axial perturbations should lead to


two sets of r-modes when the degeneracy of the frequencies is broken byhaving the background rotate.

Below we will alternate between discussions based on Newtoniangravity and those using general relativity. Accuracy demands that a fullyrelativistic formalism be employed, however there are some questions ofprinciple for which a Newtonian formalism can suffice. For instance, indetermining the number of different modes of oscillation that a superfluidneutron star can undergo it is much more tractable to use the Newtonianequations. But, ultimately there is the need for a general relativistic for-malism. Newtonian gravity does not include gravitational waves and soone needs a fully relativistic formalism to get an accurate damping time ofa mode of oscillation due to gravitational wave emission. Also, there is thewell-known problem that Newtonian models do not produce reliable valuesfor the mass and radius, in that, for a given central density, the predictedmass and radius in Newtonian models may differ considerably from thoseof general relativity. This is a crucial point since the superfluid phase tran-sition in a neutron star is sensitive to density, as are the parameters relevantto entrainment, and the oscillation frequencies can depend sensitively onmass and radius.

2. GENERAL RELATIVISTIC AND NEWTONIAN SUPERFLUIDFORMALISMS

Based on the preceeding discussion, our formalism for modellingsuperfluid neutron stars must allow for two interpenetrating fluids (i.e., theneutrons and protons) and the entrainment effect. This must be the casewhether we are working in the general relativistic regime or the Newtonianlimit. A general relativistic superfluid formalism has been developed byCarter and Langlois(26, 27, 3032) and their collaborators.(28, 29, 33, 34) Here anaction principle will be outlined that yields the equations of motion and thestress-energy tensor for this system. Although a variational principle alsoexists for the Newtonian regime,(25) we will briefly indicate how to get theNewtonian equations by taking the appropriate limit of the general relativ-istic equations (see Ref. 46 for full details). In the same spirit, we will use aslow velocity approximation to motivate an expansion that can be usedto incorporate existing models of the entrainment effect.(36)

2.1. The General Relativistic Formalism

In this subsection the equations of motion and stress-energy tensor fora two-component general relativistic superfluid are obtained from an action

1908 Comer

principle. Specifically a so-called pull-back approach (see, for instance,Refs. 28 and 29) is used to construct Lagrangian displacements of thenumber density four-currents that form the basis of the variations of thefluid variables in the action principle (they will also be used later in theanalysis of the zerofrequency subspace). It will generalize to the superfluidcase some of the techniques used to analyze the CFS mechanism in ordi-nary fluid neutron stars.(40, 41) Finally, it can serve as a starting point forgeneralizing the Hamiltonian formalism developed by Comer and Langlois(29)

who limited their discussion to a superfluid in the Landau state(12) (i.e.,purely irrotational).

For both the rotation and mode calculations effects such as trans-fusion(33) of one component into the other (because of the weak interac-tion, for instance) will be ignored. The neutron and proton number densityfour-currents, to be denoted nm and pm respectively, are thus taken to beseparately conserved, meaning

Nmnm=0, Nm pm=0. (1)

Such an approximation is reasonable for the mode calculations since thetime-scale of the oscillations (which is milliseconds) is much less than theweak interaction time-scale in neutron stars, as long as the amplitudes ofthe oscillations remain small enough.(47) In the case of slowly rotatingneutron stars, it has been shown(48, 49) that when the neutrons and protonsrotate rigidly at different rates then chemical equilibrium cannot existbetween them. Of course, the energy associated with the relative rotationcould be dissipated through a process like transfusion. But Haensel(50) hasdemonstrated that such a process would take months to years in matureneutron stars, and so again it will be neglected (since ultimately we areinterested in gravitational waves emitted during a glitch, say, the timescaleof which would be much shorter than the transfusion timescale).

By introducing the duals to nm and pm, i.e.,

nnly=Enlymnm, nm=13!Emnlynnly, (2)

and

pnly=Enlym pm, pm=13!Emnlypnly, (3)

respectively, then the conservation rules are equivalent to having the twothree-forms be closed, i.e.,

N[mnnly]=0, N[m pnly]=0. (4)


The reason for introducing the duals is that it is straightforward to con-struct particle number density three-forms that are automatically closed.The point is that the conservation of the particle number density currentsshould notspeaking from a strict field theoretic point of view be a partof the system of equations of motion, rather they should be automaticallysatisfied when the true system of equations is imposed.

This can be made to happen by introducing two three-dimensionalabstract spaces which can be labelled by coordinates XA and YA, respec-tively, where A, B, C etc=1, 2, 3. By pulling-back each three-form ontoits respective abstract space we can construct threeforms that are automat-ically closed on spacetime, i.e., let

nnly=NABC(XD) NnXA NlXB NyXC,

pnly=PABC(YD) NnYA NlYB NyYC(5)

where NABC and PABC are completely antisymmetric in their indices. Becausethe abstract space indices are three-dimensional and the closure conditioninvolves four spacetime indices, and also that the XA and YA are scalars onspacetime (and thus two covariant differentiations commute), the pull-backconstruction does indeed produce a closed three-form

N[mnnly]=N[m(NABC(XD) NnXA NlXB Ny]XC) 0, (6)

and similarly for the protons. In terms of the scalar fields XA and YA, wenow have particle number density currents that are automatically con-served, and so another way of viewing the pull-back construction is thatthe fundamental fluid field variables are the spacetime functions XA andYA. (51) The variations of the three-forms can now be derived by varyingthem with respect to XA and YA.

Let us introduce two Lagrangian displacements on spacetime for theneutrons and protons, to be denoted tmn and t

mp , respectively. These are

related to the variations dXA and dYA via a push-forward construction:

dXA=(NmXA) tmn , dY

A=(NmYA) tmp . (7)

Using the fact that

Nn dXA=Nn([NmXA] tmn )

=(NmXA) Nntmn (Nm NnX

A) tmn , (8)

1910 Comer

and similarly for the proton variation, we find(33)

dnnly=(tsn Nsnnly+nsly Nnt

sn+nnsy Nlt

sn+nnls Nyt

sn)=Ltnnnly,

dpnly=(tsp Ns pnly+psly Nnt

sp+pnsy Nlt

sp+pnls Nyt

sp)=Ltp pnly,

(9)

whereL is the Lie derivative. We can thus infer that

dnm=ns Nstmn t

sn Nsn

mnm(Nstsn+

12 gsr dgsr),

dpm=ps Nstmp t

sp Ns p

mpm(Nstsp+

12 gsr dgsr).

(10)

By introducing the two decompositions

nm=num, umum=1,

pm=pvm, vmvm=1,(11)

we can show furthermore that

dn=Ns(ntsn)n(unu

s Nstnn+

12 [g

sr+usur] dgsr),

dp=Ns(ptsp)p(vnv

s Nstnp+

12 [g

sr+vsvr] dgsr),(12)

and

dum=(dmr+umur)(us Nst

rn t

sn Nsu

r)+12 umusur dgsr,

dvm=(dmr+vmvr)(vs Nst

rp t

sp Nsv

r)+12 vmvsvr dgsr.

(13)

We will take one more step ahead and associate a notion of Lagrangianvariation with each Lagrangian displacement. These are defined to be

Dn d+Ltn , Dp d+Ltp , (14)

so that it now follows that

Dnnmly=0, Dp pmly=0, (15)

which is entirely consistent with the pull-back construction. We also findthat

Dnum=12 umusur Dn gsr, Dpvm=

12 vmvsvr Dp gsr, (16)

DnEnlys=12 Enlys g

mr Dn gmr, DpEnlys=12 Enlys g

mr Dp gmr, (17)


and

Dnn=n2(gsr+usur) Dn gsr, Dp p=

p2(gsr+vsvr) Dp gsr. (18)

However, in contrast to the ordinary fluid case,(40, 41) there are many moreoptions to consider. For instance, we could also look at the Lagrangianvariation of the neutron number density with respect to the proton flow,i.e., Dpn, or the Lagrangian variation of the proton number density withrespect to the neutron flow, i.e., Dn p. It is not clear at this point how theexistence of two preferred rest framesone that is attached to the neutronsand the other that is attached to the protonswill affect an analysis of theCFS mechanism in superfluid neutron stars.

Nevertheless, with a general variation of the conserved four-currents inhand, we can now use an action principle to derive the equations of motionand the stress-energy tensor. The central quantity is the so-called masterfunction L, which is a function of all the different scalars that can beformed from nm and pm, i.e., the three scalars n2=nmnm, p2=pm pm, andx2=pmnm. In the limit where the two currents are parallel, i.e., the twofluids are comoving, then the master function is such that L correspondsto the local thermodynamic energy density. In the action principle, themaster function is the Lagrangian density for the two fluids.

An unconstrained variation of L(n2, p2, x2) with respect to the inde-pendent vectors nm and pm and the metric gmn takes the form

dL=mm dnm+qm dpm+12 (n

mmn+pmqn) dgmn, (19)

where

mm=Bnm+Apm, qm=Cpm+Anm, (20)

and

A=Lx2 , B=2

Ln2 , C=2

Lp2 . (21)

The momentum covectors mm and qm are dynamically, and thermodynami-cally, conjugate to nm and pm, and their magnitudes are, respectively, thechemical potentials of the neutrons and the protons. The two momentumcovectors also show the entrainment effect since it is seen explicitly that themomentum of one constituent carries along some mass current of the other

1912 Comer

constituent (for example, mm is a linear combination of nm and pm). Wealso see that entrainment vanishes if L is independent of x2 (because thenA=0).

If the variations of the four-currents were left unconstrained, theequations of motion for the fluid implied from the above variation of Lwould require, incorrectly, that the momentum covectors should vanish inall cases. This reflects the fact that the variations of the conserved four-currents must be constrained. In terms of the constrained Lagrangiandisplacements, a variation of L now yields

d(`g L)=12`g(Ygmn+pmqn+nmmn) dgmn

2`g(nm N[mmn]tnn+pm N[mqn]tnp)+T.D. (22)

where the T.D. is a total divergence and thus does not contribute to thefield equations or stress-energy tensor. At this point we can return to theview that nm and pm are the fundamental variables for the fluids. Thus theequations of motion consist of the two original conservation conditions ofEq. (1) plus two Euler type equations

nm N[mmn]=0, pm N[mqn]=0, (23)

since the two Lagrangian displacements are independent. We see that thestress-energy tensor is

Tmn=Ydmn+p

mqn+nmmn, (24)

where the generalized pressure Y is defined to be

Y=Lnmmmpmqm. (25)

When the complete set of field equations is satisfied then it is automaticallytrue that NmT

mn=0.

In a later section we will be interested in the linearized version of thecombined Einstein and superfluid equations. It will thus be convenient towrite down the variations of the momentum covectors in terms of thevariations of the particle number density currents. Following the scheme ofCarter,(27) and Comer et al., (52) the variations of mm and qm due to a genericvariation of nm, pm, and gmn take the form

dmr=Asr dps+B

sr dns+(dgA) pr+(dgB) nr, (26)

dqr=Csr dps+A

sr dns+(dgC) pr+(dgA) nr, (27)


with

Amn=Agmn2Bp2 nm pn2

An2 nmnn2

Ap2 pm pn

Ax2 pmnn,

Bmn=Bgmn2Bn2 nmnn4

An2 p(mnn)

Ax2 pm pn,

Cmn=Cgmn2Cp2 pm pn4

Ap2 p(mnn)

Ax2 nmnn,

(28)

and the terms dgA, dgB, and dgC are given by

dgA=5An2 nmnn+Ap2 pmpn+Ax2 nmpn6 dgmn (29)(dgB and dgC being given by analogous formulas, with A replaced by Band C respectively).

2.2. The Newtonian Limit

The Newtonian superfluid equations can be obtained by writing thegeneral relativistic field equations to order c0, where c is the speed of light,and then taking the limit that c becomes infinite. To order c0 the metric canbe written as

ds2=c2 11+2Fc22 dt2+dij dx i dx j, (30)

where the x i (i=1, 2, 3) are Cartesian-like coordinates, and the gravitatio-nal potential F is assumed to be small in the sense that 1 F/c2 [ 0. Tothe same order the unit four-velocity components defined earlier are givenby

u t=1F

c2+v2n2c2, u i=v in, (31)

and

v t=1F

c2+v2p2c2, v i=v ip, (32)

where v2n, p=dijvin, pv

jn, p, and v

in and v

ip are the Newtonian three-velocities

of the neutron and proton fluids, respectively. The three-velocities are

1914 Comer

assumed to be small with respect to the speed of light. The entrainmentvariable x2 takes the limiting form

x2=nnnp 11+w22c22 , (33)where

w2=dij(vinv

ip)(v

jnv

jp) (34)

and we have introduced the new notation nn=n and np=p (to distinguishbetween Newtonian quantities and their general relativity counterparts).Finally, we separate the master function L into its mass part and a muchsmaller internal energy part E, i.e., we write it as

L=(mnnn+mpnp) c2E(n2n, n

2p, x

2), (35)

where mn (mp) is the neutron (proton) mass.To more closely agree with the Newtonian superfluid equations

derived by other means,(25) a different choice for the independent variablesis used which is the triplet of variables (n2n, n

2p, w

2). In this case E=E(n2n, n

2p, w

2) and the analog of the combined First and Second Law ofThermodynamics for the system takes the form

dE=mn dnn+mp dnp+a dw2, (36)

where

mn=Enn, mp=

Enp, a=

Ew2 . (37)

The generalized pressure Y, to be renamed P, takes the limiting form

P=E+mnnn+mpnp. (38)

Formally letting the speed of light become infinite in the combined Einsteinand general relativistic superfluid field equations, results in the followingset of 9 equations:

0=nnt +i(nnv

in),

0=npt +i(npv

ip),

(39)


and

0=t1v in+ 2amnnn [v ipv in]2+v jnj 1v in+ 2amnnn [v ipv in]2

+d ijj 1F+mnmn 2+ 2amnnn d ijdkl(v lpv ln) jvkn ,0=t1v ip+ 2ampnp [v inv ip]2+v jpj 1v ip+ 2ampnp [v inv ip]2

+d ijj 1F+mpmp 2+ 2ampnp d ijdkl(v lnv lp) jvkp.

(40)

The gravitational potential F is obtained from

i iF=4pG(mnnn+mpnp). (41)

For this system having no entrainment means setting the coefficient a tozero.

These equations are equivalent to those derived independently byPrix,(25) using a Newtonian variational principle. We also note that they areformally equivalent to the two-fluid set developed by Landau(53) for super-fluid He II. The two fluids in the Landau case are traditionally taken to bethe normal fluid (i.e., the phonons, rotons, and other excitations) thatcarries the entropy and the rest of the fluid (i.e., the superfluid) that carriesno entropy.

2.3. An Analytical Equation of State with Entrainment

The item that connects the microphysics to the global structure anddynamics of superfluid neutron stars is the master function, since it incor-porates all of the information about the local thermodynamic state of thematter. Ultimately, realistic models of superfluid neutron stars must bebuilt using realistic master functions (i.e., equations of state). Only in thisway can gravitational wave data be used to greatest effect to constrain themicrophysics, such as parameters that are important for entrainment.Unfortunately, there is not yet a fully relativistic determination ofentrainment (although Comer and Joynt are currently working on this),and the best that can be done is to adapt models that have been used in theNewtonian limit. Here we will describe an analytic expansion of the masterfunction that facilitates the process (see Ref. 36 for all the details).

1916 Comer

In the limit where the fluid velocities are small with respect to c we seefrom Eq. (33) that the combination x2np is small. Thus, it makes sense toconsider an expansion of the master function of the form(36)

L(n2, p2, x2)=C.

i=0l i(n2, p2)(x2np) i. (42)

The A, B, and C coefficients that appear in the definitions of the momen-tum covectors become

A= C.

i=1il i(n2, p2)(x2np) i1,

B=1nl0n

pnA

1nC.

i=1

l in (x

2np) i,

C=1pl0p

npA

1pC.

i=1

l ip (x

2np) i.

(43)

The expansion is especially useful for both the rotation and mode calcula-tions, since x2=np for any zeroth-order, or background, quantity. Inpractice this means that only the first few l i contribute, and for the modecalculations we need retain only l0 and l1. The first coefficient l0 isdirectly related to equations of state that describe a mixture of neutronsand protons that are locally at rest with respect to each other. The othercoefficient l1 contains the information concerning the entrainment effect.

We will use a sum of two polytropes for l0, i.e.,

l0(n2, p2)=mnnsnnbnmp psp pbp. (44)

In Table I are given various parameter values that have been used in speci-fic applications.(36, 48, 52)Model I corresponds to the values used in the initialstudy of modes on a nonrotating background by Comer et al., (52) and ismeant to describe only a neutron star core without an outer envelope ofordinary fluid. Thus, the neutron and proton number densities vanish atthe same radius. Model II is used in the mode study of Andersson et al. (36)

and is the most realistic. Its parameter values given in Table I have beenchosen specifically to yield a canonical static and spherically symmetricbackground model having the following characteristics (see Ref. 36 forcomplete details): (i) A mass of about 1.4M , (ii) a total radius of about10 km, (iii) an outer envelope of ordinary fluid of roughly one kilometerthickness, and (iv) a central proton fraction of about 10%. These are valuesthat are considered to be representative of a typical neutron star. The


Table I. Parameters Describing our Stellar Models I, II and III. Model I Is Identical toModel 2 of Ref. 52, and Has Only a Core with No Envelope. Model II Has An Envelope ofRoughly 1 km and Could Be Seen as a Slightly More Realistic Neutron Star Model. Model IIIHas No Distinct Envelope but Does Have a More Realistic Mass and Radius than Model I

model I model II model III

sn/mn 0.2 0.22 0.2sp/mn 0.5 1.95 2bn 2.5 2.01 2.3bp 2.0 2.38 1.95

nc (fm3) 1.3 1.21 0.93pc (fm3) 0.741 0.22 0.095M/M 1.355 1.37 1.409R (km) 7.92 10.19 10.076Rc (km) 8.90

Fig. 1. The radial profiles of the neutron and proton background particlenumber densities, n and p, respectively, for model II. The model has beenconstructed such that it accords well with a 1.4M neutron star deter-mined using the modern equation of state calculated by Akmal,Pandharipande and Ravenhall.(54) For reference, we show as horizontallines the number densities at which Akmal et al. suggest that (i) neutrondrip occurs; (ii) there is an equal number of nuclei and neutron gas; and(iii) the crust/core interface is located. It should be noted that the lattershould not coincide with our core/envelope interface since one wouldexpect there to be a region where crust nuclei are penetrated by a neutronsuperfluid.

1918 Comer

background distribution of particles for this model is determined inAndersson et al. (36) and their graph and caption is reproduced here inFig. 1. Finally, model III is used by Andersson and Comer(48) to studyslowly rotating configurations and is meant to be more realistic thanmodel I by extending the radius out further and having a better total massvalue, but with no distinct envelope so that the neutron and proton numberdensities vanish at the same radius.

Our strategy for incorporating entrainment is to adapt models thathave been used in Newtonian calculations. The particular model we willuse is that of Lindblom and Mendell.(45) It is a parametrized approximationof the more detailed model based on Fermi liquid theory(55) developed byBorumand, Joynt, and Kluzniak.(56) As demonstrated in Andersson et al. (36)

the Lindblom and Mendell model translates into the relation

l1=mnmp

r2nprnnrpprnp, (45)

where

mnn=rnn+rnp, mp p=rpp+rnp, (46)

and

rnp=Emnn. (47)

Here E is taken to be a constant, which is the approximation introduced byLindblom and Mendell.(45) Prix, Comer, and Andersson(49) argue that

E=mp pmnn1mpmgp12 , (48)

where mgp is the proton effective mass. But, Sjberg(57) has determined that

0.3 [ mgp/mp [ 0.8. Assuming a proton fraction of about 10% we can takeas physical for a neutron star core those values that lie in the range0.04 [ E [ 0.2.

3. SLOWLY ROTATING SUPERFLUID NEUTRON STARS

The first application to consider is to the problem of axisymmetric,stationary, and asymptotically flat configurations, i.e., the standardassumptions made for rotating neutron stars.(5860) Although accurate codes


exist for looking at rapidly rotating neutron stars in the ordinary fluidcase,(6062) and can in principle be adapted to the superfluid case (Prix,Novak, and Comer, work in progress), we will limit our discussion tosituations where rapid rotation accuracy is not needed. That is, we willdescribe models of rotating superfluid neutron stars that have beendeveloped(48, 49, 63) using a slow-rotation approximation. In the general rela-tivistic regime, we discuss results of Andersson and Comer(48) who haveadapted the one-fluid formalism of Hartle(64) and Hartle and Thorne(65) tothe superfluid case, whereas in the Newtonian limit it is the work Prixet al. (49) (who have built on the ChandrasekharMilne approach(66, 67)) thatwill be reviewed. The most important aspect of the superfluid case in boththe general relativistic regime and Newtonian limit is that the neutrons canrotate at a rate different from that of the protons, and this obviously hasno analog in the ordinary fluid case.

At the heart of the slow-rotation approximation is the assumption thatthe star is rotating slowly enough that the fractional changes in pressure,energy density, and gravitational field induced by the rotation are all rela-tively small.(64) If M and R represent the mass and radius, respectively, ofthe non-rotating configuration, and Wn and Wp the respective constantangular speeds of the neutrons and protons, then slow can be defined tomean rotation rates that satisfy the inequality

W2n or W2p or WnWp 1 cR22 GMRc2 . (49)

Since GM/Rc2 < 1, the inequality also implies Wn, pR c, i.e., that thelinear speed of the matter must be much less than the speed of light.

This is not as restrictive as one might guess, especially for theastrophysical scenarios we have in mind. For a solar mass neutron star ofradius 10 km, the square root of the combination on the right-hand-side ofEq. (49) works out to 11,500 s1. The Kepler limit (i.e., the rotation rate atwhich mass-shedding sets in at the equator) for the same star is roughly7700 s1. The fastest known pulsar rotates with a period of 1.56ms, whichtranslates into a rotation rate of 4000 s1 or nearly half the Kepler limit.This is why the slow-rotation approximation is still accurate to (say)1520% for stars rotating at the Kepler limit. This is seen in Fig. 2, whichis taken from Prix et al. (49) It shows a one-fluid rotating stars equatorial(Requ) and polar (Rpole) radii, determined using both the slow-rotationapproximation and the very accurate LORENE code developed by theMeudon Numerical Relativity group.(60, 62) Notice that the slow-rotationapproximation works quite well up to and including rotation rates equal tothe fastest known pulsar.

1920 Comer

Fig. 2. Comparison of one-fluid slow-rotation configurations withnumerical results obtained using the LORENE code. Requ and Rpoleare the stars equatorial and polar radius respectively. The stellarmodel is a N=1 polytrope with massM=1.4M and (static) radiusR=10 km.

The slow rotation scheme is based upon an expansion in terms of theconstant rotation rates of the fluids. The first rotationally induced effectthat one encounters in general relativity is the linear order frame-dragging(i.e., local inertial frames close to and inside the star are rotating withrespect to inertial frames at infinity). It is only at the second order thatrotationally induced changes in the total mass, shape, and distribution ofthe matter of the star are produced. Since the frame-dragging is a purelygeneral relativistic effect, the Newtonian slow rotation scheme yieldsnothing at linear order, but does have second order changes that parallelthose of general relativity.

Andersson and Comer(48) apply their slow rotation scheme using thetwo-polytrope equation of state with the model III parameters listed inTable I. Although their formalism is general enough to allow for entrain-ment, they do not consider it in their numerical solutions to the field equa-tions, since their main emphasis is to extract effects due to two rotationrates that can be independently specified. For instance, reproduced here inFig. 3 are some of their typical results for the frame-dragging for modestvalues of the relative rotation Wn/Wp between the neutrons and protons.The solutions exhibit a characteristic monotonic decrease of the frame-dragging from the center to the surface of the star. Although likely veryunrealistic, they also considered an extreme case of having the neutronsand protons counterrotate, with the result shown in Fig. 4 (reproduced,


Fig. 3. Typical results for the frame-dragging w for superfluid stars inwhich the neutrons and the protons rotate at slightly different rates.

Fig. 4. The frame dragging in an extreme (rather unphysical) situationwhere the neutrons and the protons counterrotate in such a way thatthe net frame dragging is backwards at the surface of the star butforwards in the central parts.

1922 Comer

Fig. 5. The Kepler limit WK is shown as a function of the relative rota-tion rate Wn/Wp for our model star. The filled squares show the maximumallowed rotation rate for the protons (Wp), while the open squares showthe corresponding neutron spin rate (Wn). The Kepler frequency simplycorresponds to the largest of the two.

again, from Ref. 48). Clearly the frame-dragging is no longer monotonic,and even changes sign. One can understand this behavior as follows: In theinner core of the star, the angular momentum in the protons dominates sothat the frame-dragging is positive. But in the outer layers of the star thefact that the neutrons contain roughly 90% of the mass means they beginto dominate so that the frame-dragging reverses. Finally, we reproducehere one other result of Andersson and Comer in Fig. 5, which is the effectof relative rotation on the Kepler limit. When the relative rotation is largerthan one there is little change in the Kepler limit, which is due simply tothe fact that the neutrons contain most of the mass. On the other hand,when the relative rotation is being decreased toward zero, the Kepler limitrises because the frequency of a particle orbiting at the equator isapproaching the non-rotating limit (again because the neutrons carry mostof the mass).

Prix et al. (49) apply their Newtonian slow rotation formalism in amanner similar to Andersson and Comer. There are important differences,however, even beyond the exclusion of general relativistic effects, and theseare (i) they use an equation of state that includes entrainment and termsrelated to symmetry energy (i.e., terms(68) which tend to force the system tohave as many neutrons as protons), and (ii) an exact solution to the slow


Fig. 6. Plots of the neutron (n) and proton (p) Kepler limits as functions of the relativerotation Wn/Wp, for s=0.5, 0, 0.5 and e=0, 0.4, 0.7.

rotation equations is used for the analysis. Thus, they are able to explorehow entrainment and the symmetry energy affects the rotational configu-ration of the star. We reproduce here in Fig. 6 their result for the Keplerlimit as the relative rotation is varied, for different values of entrainment(denoted e) and symmetry energy (denoted as s). The big surprise is thatthe symmetry energy has as much an impact as the entrainment. This facthas not been noticed before and should be explored in more depth, using amore realistic equation of state (like Ref. 68).

4. THE LINEARIZED NON-RADIAL OSCILLATIONS

The second application of our superfluid formalism is to the problemof non-radial oscillations. The ultimate goal is to calculate such oscilla-tions, and the gravitational waves that result from them, for rotatingneutron stars in the general relativistic regime. This is not an easy task, andthe problem has not been solved fully (for rapidly rotating backgrounds)even for the simpler case of the ordinary perfect fluid. That is, whilesome recent progress has been made to calculate the frequency of oscilla-tions,(69) there are as yet no complete determinations of the damping ratesof the modes because of gravitational wave emission. Even using the slowrotation approximation there are complications, due to questions about thebasic nature of the modes and if they can be separated into purely polarand axial parts, or if they are of the inertial hybrid mode class (as inRefs. 70 and 71). Nevertheless, we can gain valuable insight by consideringnon-radial oscillations on non-rotating backgrounds. We will use the New-tonian equations to reveal the nature of the various modes of oscillation,by given the highlights of the recent analysis of Andersson and Comer,(46)

1924 Comer

and we summarize the main results of general relativistic calculations(36, 52)

of mode frequencies and damping rates.

4.1. Linearized Oscillations in Newtonian Theory

The investigation of the nature of the modes of oscillation in both theNewtonian and general relativistic regimes has now a decade and a half ofhistory. Epsteins work(47) is the beginning, since he is the first to suggestthat there should be new oscillation modes because superfluidity allows theneutrons to move independently of the protons, and thereby increases thefluid degrees of freedom. Mendell(22) reaches the same conclusion andmoreover argues, using an analogy with coupled pendulums, that the newmodes should have the characteristic feature of a counter-motion betweenthe neutrons and protons, i.e., in the radial direction as the neutrons aremoving out, say, the protons will be moving in, which is to be contrastedwith the ordinary fluid modes that have the neutrons and protons movingin more or less lock-step. This basic picture has been confirmed by ana-lytical and numerical studies(23, 36, 46, 52, 7274) and the new modes of oscillationare known as superfluid modes. As we will see below they are predomina-tely acoustic in nature, and have a sensitive dependence on entrainmentparameters. It is worthwhile to mention again that our equations are for-mally equal to the two-fluid set of Landau for superfluid He II. With somehindsight, one recognizes that the superfluid modes could have perhapsbeen inferred to exist from a thermomechanical effect(12) in which the normalfluid and superfluid are forced into counter-motion by passing an alternat-ing current through a resistor placed in the container that holds the fluid.

The existence of the superfluid modes seems to confirm ones intuitionthat a doubling of the fluid degrees of freedom should lead to a doubling ofmodes. A review of the spectrum of modes for the ordinary fluid shouldthen give one an idea of what to expect in the superfluid. McDermottet al. (75) have given an excellent discussion of many of the possible modesin neutron stars, and these include the polar (or spheroidal) f-, p-, andg-modes and the axial (or toroidal) r-modes. But a puzzling aspect in allof this is that Lees(72) numerical analysis does not reveal a new set ofg-modes, and in fact does not find any g-modes of non-zero frequency. Ofcourse, the model he considers is that of a zero-temperature neutron star,and so one would not really expect g-modes like those of the sun (whichexist because of an entropy gradient) to be important in a mature neutronstar. However, Reisenegger and Goldreich(76) have shown conclusively thata composition gradient, such as the proton fraction in a neutron star, willalso lead to g-modes, and the model of Lee does have a composition gra-dient. Fortunately, this issue has been clarified by Andersson and Comer(46)


who use a local analysis of the Newtonian equations to confirm Leesnumerical result that there are no g-modes of non-zero frequency. Theiranalysis of the zero-frequency subspace reveals two sets of degeneratespheroidal modes, which they take to be the missing g-modes. They alsofind two sets of degenerate toroidal modes, which they interpret to ber-modes. And in fact when they add in rotation they find that the degener-acy is lifted and two sets of non-zero frequency r-modes exist.

Apart from questions about the g-modes, Andersson and Comer(46)

also illuminate the character of the superfluid modes. Although they do notsolve the linearized equations for global mode frequencies, they are able todemonstrate that the equation that describes the radial behaviour of thesuperfluid modes is of the SturmLiouville form for large frequencies w.Thus, one can expect there to be a set of modes for which w2n Q. as theindex nQ.. The equation that describes the ordinary fluid modes is alsoof the SturmLiouville form for large frequencies and so it also has a set ofmodes with the same mode frequency behavior. That is both sets of modeswill be interlaced in the pulsation spectrum of the neutron star.

Finally, Andersson and Comer(46) also use a local analysis of theNewtonian equations to find a (local) dispersion relation for the modefrequencies. Letting

c2n nnmn

mnnn, c2p

npmp

mpnp, (50)

and assuming that the proton fraction (i.e., the ratio of the proton numberdensity over the total number density) is small, then they find that onesolution to the dispersion relation is

w2o % L2l , (51)

where

L2l %l(l+1) c2nr2

. (52)

Here cn is essentially the speed of sound in the neutron fluid, l is the indexof the associated spherical harmonic Yml of the mode, and r is the radialdistance from the center of the star, and so this is the classic ordinary fluidsolution in terms of the Lamb frequency Ll. (77) Likewise, they find anothersolution of the form

w2s %mpmgp

l(l+1)r2

c2p, (53)

1926 Comer

where cp is roughly the speed of sound in the proton fluid. Thus both solu-tions are of predominately acoustic nature, but we see that the secondsolution, which corresponds to the superfluid mode, has a sensitive depen-dence on entrainment (through the appearance of the proton effectivemass). An observational determination of the superfluid mode frequencyvia gravitational waves, say, could be used to constrain the proton effectivemass.(36, 78) This would translate into a deeper understanding of superfluid-ity at supra-nuclear densities since the effective proton mass is part of theinput information for BCS gap calculations.(79)

4.2. Quasinormal Modes in General Relativity

We have just seen that the spectrum of mode pulsations in a superfluidneutron star is significantly different from its ordinary fluid counterpart.Moreover, we have also seen that the superfluid modes have a sensitivedependence on entrainment. We will now confirm, and build on, this basicpicture in a quantitative way by looking at numerical results for the modesobtained using the general relativistic formalism. The key results to bedescribed come from the work of Comer et al.(52) and Andersson et al. (36) Itis worth noting that the set of linearized equations that describe the modeoscillations of superfluid neutron stars has much in common with theordinary fluid set, and so many of the computational and numerical tech-niques that have been developed for the ordinary fluid(8088) can be adaptedto the superfluid case. That being said, Andersson et al. (36) have developeda new technique for calculating the damping rates of the modes due togravitational wave emission.

Before we get to the ordinary fluid and superfluid modes, there isanother set of modes, called w-modes, that we will discuss that exist only ina general relativistic setting. They were first discovered by Kokkotas andSchutz(89) and are due mostly to oscillations of spacetime, coupling onlyvery weakly to the matter. For instance, Andersson et al. (90) use an InverseCowling Approximation where all the fluid degrees of freedom are frozenout and were able to find the w-modes. Comer et al. (52) have obtainedw-modes in the superfluid neutron star case and find that they look verymuch like those of ordinary fluid neutron stars. As well they do not find asecond set of w-modes because of superfluidity. Both results are due to thefact that the w-modes are primarily oscillations of spacetime.

From the previous subsection we expect that there will be no g-modesin the pulsation spectrum (cf. Sec. 5 where we find two sets of polar per-turbations in the zero frequency subspace), but there should be interlacedin it the ordinary and superfluid modes. In general relativity one obtainsquasinormal modes because each frequency will have an imaginary part


Fig. 7. This figure shows the asymptotic amplitude Ain as a functionof the (real) frequency wM for our model I. The slowly-dampedQNMs of the star show up as zeros of Ain, i.e., deep minima in thefigure. The first few ordinary and superfluid modes are identifiedin the figure.

due to dissipation via gravitational wave emission. Such modes correspondto those particular solutions that have no incoming gravitational waves atinfinity. In Fig. 7, taken from Ref. 36, is graphed the asymptotic amplitudeof the incoming wave versus the real part of the mode frequency formodel I of Table I. The zeroes of the asymptotic incoming wave amplitudecorrespond with the deep minima of the figure. As expected, the ordinaryand superfluid modes are interlaced in the spectrum and the lowest fewhave been identified in the figure.

We recall that the ordinary fluid modes should be characterized by theneutrons and protons flowing in lock-step whereas the superfluid modesshould have the particles in counter-motion. As well, the superfluid modesshould have a sensitive dependence on entrainment. Both are confirmed bythe next two figures (Figs. 8 and 9 both taken from Ref. 36), which alsoreveal a phenomenon known as avoided crossings. Figure 8 graphs the realpart of the first few ordinary and superfluid mode frequencies as a functionof the entrainment parameter E (for the physical range discussed earlierin Sec. 2). The solid lines in the figure are for the ordinary fluid modes, andwe see that the first few are essentially flat as the entrainment parameter isvaried, but the superfluid modes (the dashed lines) are clearly dependent onthe entrainment parameter. Figure 9 is a graph of the Lagrangian varia-tions in the neutron and proton number densities. The two plots on the far

1928 Comer

Fig. 8. This figure shows how the frequen-cies of the fluid pulsation modes for ourmodel II vary with the entrainment param-eter E. The modes shown as solid lines aresuch that the two fluids are essentiallycomoving in the EQ 0 limit, while the modesshown as dashed lines are countermoving. Asis apparent from the data, the higher ordermodes exhibit avoided crossings as E varies.Recall that the range often taken as physi-cally relevant is 0.04 [ E [ 0.2. We indicateby aE and bE the particular modes for whichthe eigenfunctions are shown in Fig. 9.

Fig. 9. An illustration of the fact that the modes exchange properties during anavoided crossing. We consider two modes, labelled by aE and bE (cf. Fig. 8). Themode eigenfunctions are represented by the two Lagrangian number density varia-tions, Dn and Dp (solid and dashed lines, respectively). For mode a the two fluids areessentially countermoving in the EQ 0 limit (it is a superfluid mode), while the twofluids comove for mode b (it is similar to a standard p-mode). After the avoidedcrossing (which takes place roughly at E=0.1) the two modes have exchangedproperties.


left of the figure are for the modes whose frequencies are labelled by a0 (forthe superfluid mode) and b0 (for the ordinary fluid mode) in Fig. 8. The a0graph in Fig. 9 clearly indicates a counter-motion of the neutrons withrespect to the protons, whereas the b0 graph shows the opposite behavior.

Another obvious feature of both figures is the avoided crossings phe-nomenon. Near the top of Fig. 8 we see that there are points in the(E, Re wM) plane where the solid and dashed lines approach each other,but just before crossing they diverge away from each other. The mostinteresting aspect of an avoided crossing is how the mode functions behavebefore, during, and after the avoided crossing. This is shown in Fig. 9,where the middle and right-hand-side graphs are for the modes of Fig. 8labelled by {a0.1, b0.1} and {a0.2, b0.2}, respectively. In the middle graphs wesee that the modes no longer have a clear distinction as to whether or notthe neutrons and protons are flowing together or in counter-motion. But inthe {a0.2, b0.2} graphs of Fig. 8 we see that now it is the superfluid modesthat have the particles flowing together and the ordinary fluid modes showthe counter-motion. Although we will not go into details here, Anderssonand Comer(46) and Andersson et al.(36) have suggested that this may explainone of the interesting results of Lindblom and Mendell(45) on the effect ofmutual friction damping on the r-modes in superfluid neutron stars, andthat is that mutual friction damping is negligible for the r-modes except fora small subset of the values of the entrainment parameter E. Mutual frictionshould be most effective when the neutrons and protons are in counter-motion, as in the superfluid modes, and what may be happening is that asthe entrainment parameter is varied it is possible that they find modesbeyond an avoided crossing where the ordinary fluid modes take on thecharacteristic of counter-motion.

4.3. Detectable Gravitational Wave Signals?

While the study of modes in superfluid neutron stars is a fascinatingand complex mathematical and theoretical physics problem, the real hopeis that one can use the results to make scientific progress. This is why itis important to understand superfluid neutron star dynamics for realisticastrophysical scenarios, because one wants to know if there are detectablegravitational waves that carry imprints of superfluidity. In other words,one wants to develop a gravitational wave asteroseismology as a probe ofneutron star interiors, in much the same way that helioseismology is alreadya probe of the sun and asteroseismology is a probe of distant stars.(91) Thisexciting possibility is being discussed(78, 9295) in the literature and alreadysome quantitative statements are in hand. Unfortunately, estimates forLIGO II suggest that one needs modes of unrealistic amplitudes to

1930 Comer

ensure detection. The best possibility is for detection of modes followingneutron star formation from gravitational collapse, but even this has to bequalified(92, 93) because of low event rates and uncertainties in the energythat will get deposited in the oscillations. However, there is no reason whyone should only be pessimistic, since clearly gravitational wave detectionwill improve as more experience is gained, and new technology will alsolead to improved sensitivities. For instance, there is already the so-calledEURO(96) detector being discussed, which is a configuration of severalnarrow-banded (cryogenic) detectors operating as a xylophone thatshould allow high sensitivity at high frequencies.

Andersson and Comer(78) and Andersson et al.(36) have taken theestimated spectral noise density for such a configuration and have used itto determine the signal-to-noise for a detection of oscillation modes fromsuperfluid neutron stars that have been excited during a glitch. These esti-mates have been for two of the best studied glitching pulsars, which arethe Crab and Vela pulsars. One assumes that a typical gravitational wavesignal from a mode takes the form of a damped sinusoidal, where thedamping time is that of the mode itself.(36, 78) The amplitude of the signalcan be expressed in terms of the total energy radiated through the mode.Andersson and Comer and Andersson et al. assume that this total energy iscomparable to the amount of energy released during a glitch, which for theCrab and Vela pulsars can be of order 10121013Mc2. (97, 98) Unfortuna-tely, an error in the signal-to-noise calculation of Andersson and Comer, tobe corrected in Andersson et al., has incorrectly estimated the predictedsignal-to-noise for the EURO configuration. Fortunately, the revised datastill indicate sufficient sensitivity to expect detection for Crab- and Vela-like glitches.

5. THE GENERAL RELATIVISTIC ZERO-FREQUENCY SUBSPACE

Andersson and Comer(46) have determined that the zero-frequencysubspace in Newtonian theory is spanned by two sets of polar and two setsof axial degenerate perturbations. These solutions are time-independentconvective currents. They were stated to be the two missing sets of g-modesand the two sets of r-modes that become non-degenerate when rotation isadded. We will extend this analysis to the general relativistic case and showthat there are two sets of polar perturbations, thus supporting our earlierclaim that there will be no non-zero frequency g-modes in the pulsationspectrum of a general relativistic superfluid neutron star. We will also findtwo sets of axial perturbations, which could presumably lead to two sets ofr-modes (or more general hybrid modes) when the background rotates.


5.1. The Background Spacetime and Fluid Configuration

The background is treated exactly as in Comer et al., (52) i.e., it isspherically symmetric and static, and thus the metric can be written in theSchwarzschild form

ds2=en(r) dt2+el(r) dr2+r2(dh2+sin2 h df2). (54)

The two conserved currents nn and pn are parallel with the timelike Killingvector tn=(1, 0, 0, 0), and are thus of the form

nn=n(r)Un, pn=(r)Un, (55)

whereUn=tn/|t|. Likewise, the chemical potential covectors mn and qn become

qn=q(r) Un, mn=m(r) Un, (56)

where m=Bn+Ap and q=Cp+An. Explicit solutions for the back-ground configurations can be constructed following the procedure ofComer et al. (52)

5.2. The Linearized Fluid and Metric Variables

Making no assumptions yet on the metric and matter variations, wewill first insert the background metric and matter variables into Eqs. (12),(13), and (27). Without too much effort, one finds

du0=12e3n/2 dg00, du i=en/2

t t

in,

dv0=12e3n/2 dg00, dv i=en/2

t t

ip,

(57)

for the fluid velocity perturbations,

dn=n21el dgrr+1r2 5dghh+ 1sin2 h dgff62 1r2el/2 r (nr2el/2t rn)

n 1 h thn+ f tfn+cot hthn 2 ,dp=

p21el dgrr+1r2 5dghh+ 1sin2 h dgff62 1r2el/2 r (pr2el/2t rp)

p 1 h thp+ f tfp+cot hthp 2 ,

(58)

1932 Comer

for the density variations and

dm0=12men/2 dg00en/2(B

00 dn+A

00 dp),

dmi=men/2 dg0i+en/2gij 1Bn t t jn+Ap t t jp 2 ,dq0=

12qen/2 dg00en/2(C

00 dp+A

00 dn),

dqi=qen/2 dg0i+en/2gij 1Cp t t jp+An t t jn 2 ,

(59)

for the momentum covector variations, where A00, B00, and C

00 can be cal-

culated from Eq. (28). One can also show that

dL=m dnq dp, dY=(B00n+A00p) dn+(C

00p+A

00n) dp. (60)

Since the four velocity perturbations must be time-independent we see thatthe Lagrangian displacements must take the form

t in=en/2t du i+z in, t

ip=e

n/2t dv i+z ip, (61)

where z in and zip are integration constants (i.e., they are independent of

time but depend on the spatial coordinates). We will see below that theironly role is that once they have been determined they specify dn and dp viaEq. (58).

Although the Einstein equations must be analyzed using a decomposi-tion in terms of spherical harmonics for the perturbations, we can actuallycompletely solve the linearized Euler equations, which take the form

tdmi=idmt, tdqi=idqt. (62)

One can easily verify that the left-hand-sides of each equation is zero (bytaking a time derivative of dmi and dqi using Eq. (59)) and it thus followsthat

dmt=0, dqt=0. (63)

From Eq. (59) we see that these last two equations can be used to find dnand dp in terms of dg00. Given that the conservation equations are satisfiedautomatically by the Lagrangian displacements, then all of the fluid equa-tions have been solved.


We have seen earlier that the question of chemical equilibrium is animportant aspect of the types of perturbations that can be induced on asuperfluid neutron star. Langlois et al.(33) have argued that the generalcondition for chemical equilibrium to exist between the two fluids is that

b vn(mnqn)=0. (64)

For perturbations that do not maintain chemical equilibrium then it is thecase that db ] 0. Using the variations above we find

db=(A00B00) dn(A

00C

00) dp. (65)

We note here that axial perturbations are such that dn=dp=0 and thusit follows that db=0, that is axial perturbations on spherically symmetricand static backgrounds must necessarily maintain chemical equilibriumbetween the two fluids if the background fluids were in chemical equilibrium.

5.3. The Zero-Frequency Subspace

We have exhausted the information that can be extracted from justusing the background configuration in the formulas for the variations. Tofinish mapping out the zero-frequency subspace we must examine theEinstein equations. In doing this it will be very convenient to consider twotypes of perturbations and those are the polar (or spheroidal) and the axial(or toroidal) perturbations. For the metric perturbations we will use theReggeWheeler gauge,(80) in which the polar components of the metricperturbations can be written as

dPgmn=ren(r)H0(r) H1(r) 0 0H1(r) el(r)H2(r) 0 00 0 r2K(r) 00 0 0 r2 sin2 hK(r)

s Yml (h, f), (66)and the axial components as

dAgmn=| 0 0 h0(r) ( 1sin h) f h0(r) sin h h0 0 h1(r) ( 1sin h) f h1(r) sin h hh0(r) (

1sin h)

f h1(r) (

1sin h)

f 0 0

h0(r) sin hh h1(r) sin h

h 0 0

} Yml (h, f),(67)

1934 Comer

where the Yml are the spherical harmonics. We write the polar unit four-velocity perturbations as

dPum=en/2 | 12H01rWn(r)1r2Vn(r)

h

1r2 sin2 h

Vn(r)f

} Yml (h, f),dPvm=en/2 | 12H01rWp(r)

1r2Vp(r)

h

1r2 sin2 h

Vp(r)f

} Yml (h, f), (68)and the axial perturbations as

dAum=en/2

r2 sin h| 00Un(r)

f

Un(r)h

} Yml (h, f),dAvm=

en/2

r2 sin h| 00Up(r)

f

Up(r)h

} Yml (h, f). (69)Although we have already seen that the particle number density perturba-tions can be solved in terms of dg00, it is worthwhile to comment just a littlemore on their form. The easy case is that of axial perturbations since forthem dAg00=0 and so dAn and dAp must both vanish for a generic masterfunction. The polar perturbations in the particle number densities are a bitmore complicated to determine, but in the end take a simple form. SincedPn and dPp must both be time independent, a setting of time derivatives ofEq. (58) to zero yields constraints on the velocity perturbation functions,i.e.,

l(l+1) Vn=el/2

n(rnel/2Wn), l(l+1) Vp=

el/2

p(rpel/2Wp). (70)


But recall that the Lagrangian displacements still have the integrationconstant terms. However, using the same type of decomposition for z in, pas used for dPu i and dPv i above, and in place of the Wn, p(r) and Vn, p(r)coefficients we substitute some new coefficients An, p(r) and Bn, p(r), say,then it follows that

dPn=dn(r) Yml , dPp=dp(r) Yml , (71)

where the dn(r) and dp(r) are linear combinations of An, p(r) and Bn, p(r)and their derivatives. These new coefficients An, p(r) and Bn, p(r) appearnowhere else but in dPn and dPp and thus this is why we stated earlier thatthe only role of the integration constants is to determine the particlenumber density perturbations. Of course at this point we can forgo usingAn, p(r) and Bn, p(r) and just solve for the dn(r) and dp(r) instead.

After some algebra it can be shown that the perturbations for l \ 2result in three distinct groups of the linearized Einstein and superfluid fieldequations, and these are

(i) l \ 1. Group I:

0=elr2K+el 13rl22 rK1 l(l+1)

212 K

elrH 01 l(l+1)2 18pr2Y2H0+4pr2([3qnA00pC

00] dp+[3mpA

00nB

00] dn),

0=el 11+rn22 rK1 l(l+1)

212 K

elrH 0+1 l(l+1)2 18pr2Y2H0+4pr2([q3nA003pC

00] dp+[m3pA

003nB

00] dn),

0=elr2K+el 1 r(nl2

+22 rK16pr2([nA00+pC

00] dp+[pA

00+nB

00] dn)

elr2H'0 el 1 r(3nl)

2+22 rH 016pr2YH0,

H2=H0, K=en(enH0),m

2H0=A

00 dp+B

00 dn,

q

2H0=A

00 dn+C

00 dp.

(72)

1936 Comer

(ii) l \ 2. Group II:

0=H1+16prel

l(l+1)(mnWn+qpWp),

0=l(l+1) Vnel/2

n(rnel/2Wn),

0=l(l+1) Vpel/2

p(rpel/2Wp),

0=e(nl)/2(e (nl)/2H1)+16pel(mnVn+qpVp).

(73)

(iii) l \ 2. Group III:

0=h'0 n+l2h 0+12l2lr2 el n+lr 2r22 h016pel(qpUp+mnUn),

0=(l1)(l+2) h1,

0=e(nl)/2(e (nl)/2h1).

(74)

A quick counting of the number of independent functions, and the numberof equations, shows that Group I appears to have more equations thanunknowns. However, such a result is not unexpected because of the Bianchiidentities. For the present discussion, the important point about theGroup I equations is that their solutions represent a subset that map staticand spherically symmetric stars, with no mass currents, into other (nearby)static and spherically symmetric stars, also having no mass currents. Moreinteresting counting comes from the Group II and III equations. ForGroup II one can show that the last equation in the group is a consequenceof the other three. Thus, we can specify arbitrarilyWn andWp, for instance,and then the other variables (H1, Vn, and Vp) can be determined from thefield equations. Likewise, for Group III we can specify freely Un and Up,and then the remaining variable h0 is determined from its field equation(because it is clear that h1=0).

For various reasons, the cases of l=0, 1 must be distinguished fromthat of l \ 2. One reason is that there is more gauge freedom, which allowsus to set K(r)=h1(r)=0 for l=1 and in addition H1(r)=0 for l=0(which also has no axial perturbations). Without listing all the formulas, wefind that the counting of the number of equations and unknown functionsis similar to l \ 2. In particular, the l=1 analysis of the Groups II and IIIequations reveals that each have two arbitrary functions that must be spe-cified before a solution can be obtained. For l=0, the polar currents must


also vanish (otherwise they would diverge at the center of the star). Hence,there are only l=0 solutions that map static and spherically symmetricstars into other static and spherically symmetric stars.

5.4. Decomposition of the Zero-Frequency Subspace

Regardless of the form of the equation of state for the background, orfor the perturbations, we can make the following conclusions: Any solution

{H0, H1, H2, K, h0, Wn, Wp, Vn, Vp, Un, Up, dn, dp} (75)

to the equations governing the time-independent perturbations of a static,spherical superfluid neutron star is a superposition of (i) a solution

{H0, 0, H2, K, 0, 0, 0, 0, 0, 0, 0, dn, dp} (76)

and (ii) a solution

{0, H1, 0, 0, h0, Wn, Wp, Vn, Vp, Un, Up, 0, 0}. (77)

The solutions in (i) are those that satisfy the Group I equations that mapone static and spherically symmetric star to another (nearby) static andspherically symmetric star. The solutions in (ii) are those that satisfy theGroup II and III equations. It is not difficult to see that the solutions (ii)contain two sub-classes, the purely polar solutions that satisfy the Group IIequations, i.e.,

{0, H1, 0, 0, 0, Wn, Wp, Vn, Vp, 0, 0, 0, 0} (78)

and the purely axial solutions that satisfy the Group III equations, i.e.,

{0, 0, 0, 0, h0, 0, 0, 0, 0, Un, Up, 0, 0}. (79)

The first subclass is made of the g-modes because they are (1) purely polarand (2) the particle number densities (and likewise the energy density andpressure) vanish. The second subclass is made of the r-modes because theyare (1) purely axial and (2) the particle number densities (and likewise theenergy density and pressure) vanish. The final conclusion is that the zero-frequency subspace of superfluid neutron stars is spanned by two sets ofg-modes and two sets of r-modes. This is qualitatively the same conclusionas found for the Newtonian equations,(48) and we assert that there will beno non-zero frequency g-modes in the pulsation spectrum of non-rotatingsuperfluid neutron stars. As for the axial perturbations, presumably rotation

1938 Comer

will break the degeneracy and we will find two sets of r-modes (or moregeneral hybrid modes(70, 71)), as in the Newtonian superfluid case, but thismust be verified.

6. CONCLUDING REMARKS

We have reviewed recent work to model the rotation and oscillationdynamics of Newtonian and general relativistic superfluid neutron stars.We have seen that superfluidity affects both the background and the per-turbation spectrum of neutron stars and both should therefore cause animprint of superfluidity to be placed in the stars gravitational waves. Inparticular our local analysis of the Newtonian mode equations indicatesthat the superfluid modes should have a sensitive dependence on entrain-ment parameters, something that is supported by the quasinormal modecalculations. Given a suitably advanced detector, like the EURO configu-ration, we have seen that gravitational waves emitted during a Vela glitch,say, should be detectable. The key conclusion is that direct detection ofgravitational waves from glitching pulsars can be used to greatly improveour understanding of the local state of matter in superfluid neutron stars.This may become more important in the next few years because of indica-tions of free precession in neutron stars,(99) which if true means that thevortex creep model will have to be reconsidered.

We have also put in place some groundwork for a future analysis ofthe CFS mechanism in superfluid neutron stars. We have done this byusing the pull-back formalism to motivate fluid variations in terms ofconstrained Lagrangian displacements. We have also used them as pertur-bations to help map out the zero frequency subspace, and found that it isspanned by two sets of polar and two sets of axial perturbations. It remainsto be seen if adding rotation will lift the degeneracy and yield two sets ofpolar and two sets of axial oscillation modes, or if the more general inertialhybrid modes result.

Finally, I would like to elaborate a little more on why I continueto treasure my two years in Israel with Jacob Bekenstein. After I hadcompleted working on the superfluid analogs of quantum field theory incurved spacetime effects I tried in vain to publish the work. True to hischaracter, Jacob had some very kind words of advise, which were to neverworry that effort is lost when a project does not play out exactly asexpected, because his own experience was that one, or many, pieces of itwould eventually be of direct importance for something else. For a youngscientist, those were words of comfort and hope, and remain so for onethat now has a little more experience.


ACKNOWLEDGMENTS

The bulk of the work discussed here is very much the result of a teameffort, and would not have happened without my teammates Nils Andersson,David Langlois, Lap Ming Lin, and Reinhard Prix. Thanks guys! I alsothank Ian Jones for providing me information and references about theestimates for galactic neutron star populations, and Brandon Carter andJohn Friedman for discussions on fluid variational principles. Finally,I gratefully acknowledge partial support from a Saint Louis UniversitySLU2000 Faculty Research Leave award, and EPSRC visitors grantGR/R52169/01 in the United Kingdom.

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1. INTRODUCTION2. GENERAL RELATIVISTIC AND NEWTONIAN SUPERFLUID FORMALISMS3. SLOWLY ROTATING SUPERFLUID NEUTRON STARS4. THE LINEARIZED NON-RADIAL OSCILLATIONS5. THE GENERAL RELATIVISTIC ZERO-FREQUENCY SUBSPACE6. CONCLUDING REMARKSACKNOWLEDGMENTS

Do Neutron Star Gravitational Waves Carry Superfluid Imprints?

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