r-modes in low temperature colour-flavour-locked superconducting quark star
aa r X i v : . [ a s t r o - ph . S R ] M a y r-modes in low temperature colour-flavour-locked superconducting quark stars N. Andersson , B. Haskell and G.L. Comer School of Mathematics, University of Southampton, Southampton, UK Department of Physics and Center for Fluids at All Scales, Saint Louis University, St Louis, USA (Dated: November 20, 2018)We present the first multi-fluid analysis of a dense neutron star core with a deconfined colour-flavour-locked superconducting quark component. Accounting only for the condensate and (finitetemperature) phonons, we make progress by taking over results for superfluid He. The resultanttwo-fluid model accounts for a number of additional viscosity coefficients (compared to the Navier-Stokes equations) and we show how they enter the dissipation analysis for an oscillating star. Weprovide simple estimates for the gravitational-wave driven r-mode instability, demonstrating thatthe various phonon processes that we consider are not effective damping agents. Even though theresults are likely of little direct astrophysical importance (since we consider an overly simplisticstellar model) our analysis represents significant technical progress, laying the foundation for moredetailed numerical studies and preparing the ground for the inclusion of additional aspects (inparticular associated with kaons) of the problem.
I. INTRODUCTION
The state of matter at extreme densities continues to be an issue of vigorous investigation. The problem is compli-cated, not only from the theoretical point of view, but also by the fact that laboratory experiments are restricted. Forexample, while colliders like RHIC at Brookhaven, GSI in Darmstadt and the LHC at CERN probe hot quark-gluonplasmas they will never be able to explore the cold, extreme high density, region of the QCD phase diagram. In orderto test our understanding of the relevant physics we need to turn to astrophysics, and the dynamics of compact stars.In fact, “neutron stars” represent unique laboratories of such extreme physics. With core densities reaching aboutone order of magnitude beyond nuclear saturation, they are likely to contain exotic states of matter like hyperonphases with net strangeness and/or deconfined quarks. It is well-established that these states of matter should exhibitsuperfluidity/superconductivity at the relevant temperatures (neutron stars are born with temperatures ∼ Kand rapidly cool below ∼ K). Moreover, observed radio pulsar glitches provide strong evidence for the presenceof a partially decoupled superfluid component in these systems. The modelling of the dynamics of these, potentiallyvery complex, objects presents a serious challenge.A key aspect of the problem concerns the fact that a superfluid system has additional dynamical degrees of freedom.This is well-known from experiments on laboratory systems like He, which exhibit a second sound associated withthermal waves [1, 2]. Analogous “superfluid” modes have been studied in detail for superfluid neutron-proton-electronmixtures relevant for the outer core of a neutron star [3–5]. For simplicity, these studies have almost exclusively ignoredthermal effects (the work in [6] is a notable exception). While this is a useful first approximation it is clear that thezero temperature assumption must be relaxed in a realistic model. Basically, due to the density dependence of thevarious superfluid pairing gaps (see [7] for a guide to the literature), there will always be regions in a neutron starwhere thermal effects are important (in the vicinity of the critical density at which the phase transition occurs).Understanding the nature of these transition regions, and their effect on various aspects of neutron star dynamics, isone of the main challenges for research in this area.In this paper, we describe a first attempt at modelling thermal dynamics in a superfluid neutron star. We focus onstars with a colour-flavour-locked (CFL) superconducting quark core [8] at finite temperatures. This is an interestingproblem for several reasons. First of all, the simplest possible model for this system considers a quark condensatecoupled to a gas of phonons. This problem is analogous to He at low temperatures, and hence we can bring ourrecent dissipative two-fluid model [9] to bear on it (more or less directly). The lessons we learn from this exerciseshould inform the development of finite temperature models for the superfluids in the outer core and the neutron starcrust. Secondly, even though there have been discussions of the dynamics of the different CFL phases, in particularin the context of the gravitational-wave driven r-mode instability (see [10] for references), the superfluid aspects have(as far as we are aware) not previously been accounted for. Our discussion begins to address the relevance of theadditional degrees of freedom in these systems, and provides some insight into the nature of the different “fluids”involved.As an application with immediate astrophysical relevance, we will work out the inertial r-modes and estimate therelevant viscous damping rates for the “simplest” model of CFL matter. We consider the “cool” regime where thetemperature is significantly below all the quasiparticle energy gaps. In this regime, dissipation may mainly occurdue to phonon interactions. One reason for considering this model is that there are results in the literature forboth bulk- and shear viscosity [11, 12] as well as the mutual friction associated with superfluid vortices [13]. Ofcourse, the simple “condensate plus phonon” model that we consider is not the whole story. It should apply atasymptotically high densities, but may not be the true ground state at lower densities. The discussion in [15–17]adds extra dimensions to the problem by considering the bulk viscosity due to kaons (allowing for flavour-changingprocesses), which will be present at higher temperatures. At first sight, this mechanism will only be relevant for veryhot stars (
T > ∼ K) since it assumes that there is a thermal population of kaons. Below the criticalenergy where the kaons appear the contribution to the bulk viscosity is exponentially suppressed and may not bethat important. However, in more recent work on the so-called CFL-K phase [16], it is argued that the main lowtemperature mechanism involves condensed kaons. This is important since the kaon condensate will remain presentas T →
0. The upshot of this is that the problem requires a “multi-fluid” analysis at all temperatures. The simplestmodel would have three components; the quark condensate, the kaon condensate and finite temperature excitations(phonons and thermal kaons). This problem is more involved than the case that we focus on here. Nevertheless, ourresults provide an essential starting point for investigations into the dynamical role of the kaons. Most importantly,by providing the “hydrodynamics” view of the problem we illustrate the input needed to study various dynamicalscenarios. This should stimulate further discussion between experts on different aspects of this multi-faceted problem,as required to make progress in the future.
II. FLUX-CONSERVATIVE TWO-FLUID MODEL
We consider a deconfined quark system system that contains a single CFL superconducting condensate and aphonon gas. Formally, this problem is identical to that for He at low temperatures. Hence, we can take as ourstarting point the recent discussion of superfluid Helium [9], which is based on the convective variational approachto multi-fluid dynamics [18–20] and which we know is in one-to-one correspondence with the orthodox formulationdeveloped by, in particular, Khalatnikov [1].The variational model takes as its starting point an energy functional E , representing the equation of state. Thisenergy determines the relation between the various fluxes and the associated canonical momenta. In the presentcase, we distinguish between the massive particles in the system, with number density n and flowing with nv n i froma massless entropy component with number density s and flux sv s i . The latter represents the phonons (which aretreated in the fluid approximation, i.e., we assume that there exists a suitable defined “average” transport velocityfor each of the two components in the system) The two momentum densities then follow from [51] π n i = mnv n i − αw ns i , (1)and π s i = 2 αw ns i , (2)where w ns i = v n i − v s i and α represents the entropy entrainment [21]. It has been assumed that E = E ( n, s, w ), asexpected in an isotropic system, and we have defined α = ∂E∂w (cid:12)(cid:12)(cid:12)(cid:12) n,s , (3)omitting the indices on w for clarity.As discussed in [9, 19], the associated momentum equations can be written f n i = ∂ t π n i + ∇ j ( v j n π n i + D n ji ) + n ∇ i (cid:18) µ n − mv (cid:19) + π n j ∇ i v j n , (4)and f s i = ∂ t π s i + ∇ j ( v j s π s i + D s ji ) + s ∇ i T + π s j ∇ i v j s , (5)where µ n is the matter (quark) chemical potential; µ n = (cid:18) ∂E∂n (cid:19) s,w , (6)and we have used the fact that the temperature corresponds to the entropy chemical potential [21]; T = µ s = (cid:18) ∂E∂s (cid:19) n,w . (7)In these expressions, D x ij represent the viscous stresses while the “forces” f x i allow for momentum transfer betweenthe two components. In the following we will assume that the system is isolated, which means that f n i + f s i = 0. Aswe will discuss later, the force terms can also be used to account for “external” forces like gravity.In the present context, when we are dealing with a single matter quantity, we will have [19] ∂ t n + ∇ j ( nv j n ) = Γ n = 0 , (8)as there is no particle creation/destruction. At the same time the entropy can increase, so we have ∂ t s + ∇ j ( sv j s ) = Γ s ≥ . (9)We also have [19] T Γ s = − f n i w i ns − D ji ∇ j v i s − D n ji ∇ j w i ns , (10)where D ij = D n ij + D s ij . (11)For an isolated system we know that, if we impose the superfluid constraint of irrotationality the number ofdissipation coefficients reduces significantly. As discussed in [9], we have − D ij = g ij (ˆ ζ n ∇ l j l + ζ Θ s ) + 2 η Θ s ij . (12)where we have defined j i = nw i ns . We also have;1 n (cid:0) f n i − ∇ l D n li (cid:1) = ∇ i Ψ (13)with Ψ = ˆ ζ nn ∇ l j l + ˆ ζ n Θ s . (14)At this point, only four dissipation coefficients remain in the problem.In the above expressions, we have used the standard decomposition; ∇ i v s j = Θ s ij + 13 g ij Θ s + ǫ ijk W k s (15)in terms of, the expansion Θ s = ∇ j v j s , (16)the trace-free shear Θ s ij = 12 (cid:18) ∇ i v s j + ∇ j v s i − g ij Θ s (cid:19) , (17)and the “vorticity” W i s = 14 ǫ ijk ( ∇ j v s k − ∇ k v s j ) . (18)We use analogous expressions for gradients of the relative velocity. The definition of the various quantities should beobvious from the constituent indices.However, since the system that we consider is not irrotational we need to consider relaxing the assumptions on thedissipation coefficients. This involves making some subtle decisions. In the case of an irrotational flow it is natural toassume that f i n = 0. When the superfluid rotates, and vortices are present, the force will not vanish. It is necessary toaccount for dissipation due to, for example, the scattering of phonons off of the vortex cores. The standard approach tothe rotating problem is to add in this “mutual friction” force, keeping the other dissipative terms as in the irrotationalcase. This strategy ignores a number of dissipative terms that would, at least in principle, be allowed in the equationsof motion [9, 19]. The role of these additional terms has not yet, as far as we are aware, been investigated.Ignoring the potential relevance of most of the additional dissipation channels in the irrotational case, we simplyaccount for the presence of vortices by i) assuming that (4) represents fluid elements with both a condensate and asmooth-averaged vorticity arising from the vortices, and ii) accounting for the vortex mediated mutual friction byallowing a force [22] f mf i = B ′ ρ n n v ǫ ijk κ j w k ns + B ρ n n v ǫ ijk ǫ klm ˆ κ j κ l w ns m , (19)to act on the particles (with a balancing force affecting the excitations). Here n v is the vortex area density and thevector κ i = κ ˆ κ i (the hat represents a unit vector) is aligned with the rotation axis and has magnitude κ = h/ m .Since we will only consider the effect of phonon scattering off of vortices, we expect to be in the weak mutual frictionregime where B ′ ≪ B . This means that the first term in the force (19) can be ignored, leaving only the second,dissipative, contribution. III. ROTATING EQUILIBRIUM MODELS
In order to set the stage for the discussion of r-modes, we need to provide a suitable rotating equilibrium configura-tion. To do this, we note that the two components flow together (there is not heat flux) when the system is in thermalequilibrium. This means that w ns i = 0, which implies that π s i = 0. Thus, it follows from the entropy momentumequation (5) that we must have s ∇ i T = 0 −→ T = constant . (20)This is trivial; if the system is isothermal then there will be no heat flux.Rewriting the remaining momentum equation (4), making use of the continuity equation, we find (after accountingfor the gravitational force as the gradient of the gravitational potential Φ) ∂ t v n i + v j n ∇ j v n i + ∇ i (˜ µ n + Φ) = 0 , (21)where ˜ µ n = µ n /m . Using the Lie-derivative along the flow, L v n , we can rewrite this equation as ∂ t v n i + L v n v n i + ∇ i (cid:18) ˜ µ n + Φ − v (cid:19) = 0 . (22)Restricting ourselves to stationary models we have ∂ t v n i = 0 . (23)Next, uniform rotation implies v i n = Ω n e iϕ , (24)and, for axisymmetric models, it then follows that L ϕ v n i = 0 . (25)Hence, the required equilibrium models are determined from the usual Bernoulli-type equation˜ µ n + Φ − v = constant . (26)This analysis shows that a rotating configuration can be obtained in the usual way. Once we assume thermalequilibrium, we are dealing with a single-fluid problem. IV. LINEAR PERTURBATIONS
Our main aim is to develop the tools required to make quantitative estimates for the r-mode instability in neutronstars with a CFL core. In doing this, we want to account for the fact that CFL matter requires a multi-fluiddescription. The analysis proceeds in three steps. First we need to formulate the linear perturbation problem for thesystem. This step is interesting because, as far as we are aware, this is the first time that perturbations of a matterplus massless entropy system have been considered in an astrophysical context. The results provide useful insights intofinite temperature superfluid dynamics, and should be relevant in a broader context. The second step corresponds todetermining the pulsation modes of the system, in this case the r-modes, and finally we need to estimate the dampingtimescales associated with the different dissipation channels. In the discussion below, we only work out the first ofthese steps in detail. In order to obtain useful estimates for the r-mode instability we simplify last two steps byconsidering a uniform density model. This has the advantage that the r-mode solution is simple, and we can evaluatethe dissipation rates analytically. In a sense, we do not expect this approximation to be too bad because it is wellknown that the density profile for a canonical 1 . M ⊙ /10 km strange star (described by the MIT bag model) is almostflat (see [23]).Of course, we really need to develop a consistent model for a realistic equation of state. This is essential if we wantto study hybrid stars, where the CFL phase is only present in the core. However, the problem is complicated by thefact that the bulk viscosity damping of the r-modes requires the perturbations to be worked out to second order inthe slow-rotation approximation [24]. This necessitates a numerical solution, which makes some of the qualitativebehaviour of the results less clear. Another complicating factor is the well-known fact that it only makes senseto use a realistic equation of state in a fully general relativistic analysis (see [25] for discussion). The r-modes insingle component relativistic stars have been studied [26–29], but there has not yet been any serious analysis of thecorresponding multi-fluid problem (although see [30]). A focussed effort in this direction should be encouraged. A. The non-dissipative problem
In order to work out the r-mode solutions, it is advantageous to work in a rotating frame. Focussing, for themoment, on the non-dissipative equations we find that, in a frame rotating uniformly with Ω i we have ∂ t π n i + ∇ j ( v j n π n i ) + n ∇ i (cid:18) µ n + m Φ − mv (cid:19) + π n j ∇ i v j n + 2 ρǫ ijk Ω j v k n = 0 , (27)and ∂ t π s i + ∇ j ( v j s π s i ) + s ∇ i T + π s j ∇ i v j s = 0 , (28)It is notable that the Coriolis force does not affect the entropy equation (28). This is, however, not surprising. It iswell-known that inertial forces are proportional to the mass, and since our entropy component is taken to be masslessit should not be affected by the Coriolis force. From a technical point of view, it means that the problem we consideris subtly different from the two-fluid r-mode problem discussed in [31, 32].Let us now consider perturbations of the rotating equilibrium models discussed in the previous section. ConsideringEulerian perturbations (denoted by δ ) we have, in the rotating frame (using the Cowling approximation δ Φ = 0, andnot explicity denoting the velocities as perturbations since they vanish in the background anyway); ρ∂ t v n i − α∂ t w ns i + ρ ∇ i δ ˜ µ n + 2 ρǫ ijk Ω j v k n = 0 , (29)and 2 α∂ t w ns i + s ∇ i δT = 0 . (30)Adding these we get an equation for the total perturbed momentum ρ∂ t v n i + ρ ∇ i δ ˜ µ n + s ∇ i δT + 2 ρǫ ijk Ω j v k n = 0 . (31)Noting that the pressure p is defined by (for a co-rotating equilibrium model) ∇ i p = ρ ∇ i ˜ µ n + s ∇ i T , (32)we see that ∇ i δp = δρ ∇ i ˜ µ n + ρ ∇ i δ ˜ µ n + δs ∇ i T + s ∇ i δT = δρρ ∇ i p + ρ ∇ i δ ˜ µ n + s ∇ i δT , (33)since ∇ i T = 0 in the background. Hence, we have the usual Euler equation ρ∂ t v n i + ∇ i δp − δρρ ∇ i p + 2 ρǫ ijk Ω j v k n = 0 . (34)We also have the perturbed continuity equation for the particles ∂ t δρ + ∇ i ( ρv i n ) = 0 , (35)and a conservation law for the entropy ∂ t δs + ∇ i ( sv i s ) = ∂ t δs + ∇ i ( sv i n ) − ∇ i ( sw i ns ) = 0 . (36)The entropy is conserved since there is no heat flux in the background (Γ s is quadratic in the heat flux [21]).Once we provide an equation of state for matter, we have all relations we need to study the linear dynamics of anon-dissipative finite temperature CFL quark core. B. Energy integrals and dissipation
Our main aim is to work out the r-modes and establish the parameter range in which gravitational-wave emissiontriggers a secular instability. In order to assess the relevance of this instability we need to consider the variousdissipative mechanisms that counteract the growth of an unstable mode. The damping due to shear- and bulkviscosity in a two-component system can be estimated using the strategy set out in [31, 32]. That is, we use energyintegrals to estimate the various timescales. In essence, the damping timescale τ associated with any given process isobtained from τ ≈ E (cid:20) dEdt (cid:21) − , (37)where E is the energy associated with the flow and dE/dt is the rate of energy loss due to dissipation. This estimateshould be accurate as long as the dissipation is weak enough that it does not affect the flow on a dynamical timescale.The energy associated with a given perturbation can be obtained from the equations of motion (29) and (30).Multiplying the first equation by ¯ v i n and the second by ¯ v i s (where the bars denote complex conjugates) and adding,we find that the kinetic energy is given by E = 12 Z ρ (cid:18) | v n | − αρ | w ns | (cid:19) dV . (38)In the case of the r-modes, this is the leading order contribution to the energy. In fact, it turns out that the secondterm in the bracket is of higher order in the slow-rotation approximation (for the same reasons as in [32]). Hence, ther-mode energy is well approximated by E ≈ Z ρ | v n | dV . (39)In order to evaluate the energy loss due to viscous damping we add the dissipative part of the stress tensor D ij tothe equations of motion. The total energy dissipation then follows from the body integral of − ¯ v i n ∇ j D n ij − ¯ v i s ∇ j D s ij . After some algebra, this leads to dEdt = − Z h η ¯Θ ij s Θ s ij + ˆ ζ nn | Θ | + 2ˆ ζ n Re (cid:0) ¯ΘΘ s (cid:1) + ζ | Θ s | i dV , (40)where we have defined Θ = ∇ i j i . (41)Recall the relations (16) and (17) and the definition j i = nw i ns .In order to combine this result with existing results for the viscosity coefficients we need to translate our variablesinto those of the “orthodox” two-fluid model due to, for example, Khalatnikov [1]. The required translation has beendiscussed at length in [9], and the key relations that we need are given in the Appendix. Carrying out the comparison,we find that the shear viscosity terms are exactly the same in the two descriptions. Hence, we can use η from [12]without change. In the case of the bulk viscosity, we find that the two sets of coefficients have different mass scalings,and we identify ζ = ζ , ˆ ζ n = mζ = mζ , ˆ ζ nn = m ζ . (42)This means that, when expressed in terms of the standard coefficients, the bulk viscosity dissipation rate becomes dEdt = − Z h η ¯Θ ij s Θ s ij + m ζ | Θ | + 2 mζ Re (cid:0) ¯ΘΘ s (cid:1) + ζ | Θ s | i dV . (43)To conclude, there are three bulk viscosity coefficients rather than the usual single one, and both dynamical degreesof freedom are needed if we want to evaluate the dissipation integrals. Note that the dissipation is expressed in termsof j i and v is , rather than v i n and w i ns , the variables that we will solve for when we determine the r-modes. It is, ofcourse, straightforward to express (43) in terms of these variables. Unfortunately, the dissipation integrand is thenrather messy. We find dEdt = − Z n η (cid:2) ¯Θ ij n Θ n ij − (cid:0) ¯Θ ij n Θ ns ij (cid:1) + ¯Θ ij ns Θ ns ij (cid:3) + ζ | Θ n | + (cid:2) ζ − ρζ + ρ ζ (cid:3) | Θ ns | − ζ − ρζ )Re (cid:0) ¯Θ n Θ ns (cid:1) + 2 ζ Re (cid:2) ¯Θ n (cid:0) w j ns ∇ j ρ (cid:1)(cid:3) − ζ − ρζ ) (cid:2) ¯Θ ns (cid:0) w j ns ∇ j ρ (cid:1)(cid:3) + ζ (cid:12)(cid:12) w j ns ∇ j ρ (cid:12)(cid:12) o dV . (44)where Θ n = ∇ j v j n and Θ ns = ∇ j w j ns . (45)However, as we will see later, this expression simplifies considerably in the r-mode problem.Before moving on, it is worth making a general point. It might be tempting to suggest that the terms involving w i ns in (44) should be less relevant than those involving only v i n . However, without solving for an actual oscillationmode one cannot make this argument precise. In general, the bulk viscosity contributions from terms involving w i ns cannot be neglected. Since a mode-oscillation typically involves both degrees of freedom (unless the star is unstratified[33, 34]), we need to determine the nature of the oscillations before we make further simplifications. V. A SIMPLE MODEL EQUATION OF STATE
In order to obtain quantitative results for the r-mode instability we need to provide an equation of state. As ourmain focus is on the two-fluid aspects of the problem (ignored in previous studies), the equation of state must accountfor the thermal excitations, i.e. the phonons. Moreover, if we want to investigate the relative importance of thedifferent bulk viscosity terms we need to allow for a relative flow at the perturbative level. This is also importantif we want to consider the superfluid mutual friction [13]. However, if we insist on the model being truly “realistic”then the problem becomes much more challenging. Hence, we will make a number of simplifying assumptions. Thisis not only practical, it is also quite reasonable since this is our first attempt at a quantitative analysis. Future workshould aim to relax some of our assumptions.As discussed in Section II, the equation of state that we require takes the form of an energy functional E = E ( n, s, w ). Once this functional is provided, we can work out all the quantities that we need to construct a rotatingmodel and calculate the r-modes. Unfortunately, we do not have an equation of state of the required form. In fact,some of the parameters that we need are usually not considered, essentially since they are not required if one is onlyinterested in equilibrium configurations. In order to make progress we have to build a suitably simple model equationof state. To do this, we assume that the thermal contribution due to the phonons adds to a zero-temperature equationof state for the condensate. This strategy has previously been developed for a relativistic superfluid [35], and buildson the classic expressions for a non-relativistic phonon gas, see [1]. A. The phonon gas
Let us assume that, at zero temperature matter is described by a barotropic model such that E ( n, s = 0) = E ( n ).Then the chemical potential follows from µ = dE dn , (46)the pressure is obtained from dp = ndµ , (47)and it is easy to work out the speed of (first) sound; c = dp dρ , (48)Now consider thermal excitations represented by a phonon gas with a linear dispersion relation, with slope c .Working out the thermodynamics of such a gas one can show that its contribution to the pressure, ψ , is given by [35] ψ = 4 π
45 ( kT ) (2 π ~ c ) (cid:18) − w c (cid:19) − . (49)Here, and in the following, it is to be understood that w corresponds to w . From this we obtain the entropy density(in the matter frame) via s = ∂ψ∂T . (50)This leads to the explicit result; s = 4 ψkT = 16 π (cid:18) kT π ~ c (cid:19) (cid:18) − w c (cid:19) − , (51)which means that the heat capacity is given by c v = T ∂s∂T = 2 π (cid:18) kT ~ c (cid:19) , (52)in agreement with the phonon result in [16].Finally, one can show that [35] the “normal fluid” density, ρ N , required in the orthodox superfluid formalism (seeAppendix) is given by ρ N = 16 π
45 1(2 π ~ ) (cid:18) kTc (cid:19) (cid:18) − w c (cid:19) − c . (53)This result is consistent with ρ N = 43 E ph c (cid:18) − w c (cid:19) − . (54)where the phonon energy is E ph = 4 π (cid:18) kT π ~ c (cid:19) kT . (55)In our formulation of the problem, we need the entrainment between particles and entropy. As discussed in [9] therelevant entrainment parameter is related to ρ N according to α = − ρ N (cid:18) − ρ N ρ (cid:19) − ≈ − ρ N , (56)where the last approximation is accurate in the low temperature limit, when ρ N ≪ ρ . This is the regime that we areconsidering here.It is worth noting that the above results lead to ρ N ≈ (cid:18) T K (cid:19) g / cm . (57)In order to be consistent we need ρ N ≪ ρ ≈ × g / cm (the average density for a canonical neutron star), whichtranslates into T ≪ × K. In other words, the model should be valid for all astrophysical neutron stars.Let us now use the phonon gas results to construct a “complete” model equation of state. That is, we want todeduce a consistent energy functional E . This functional should be such that the temperature is obtained from T = ∂E∂s (cid:12)(cid:12)(cid:12)(cid:12) n,w . (58)Now, combining (49) with (51) we find that ∂E∂s (cid:12)(cid:12)(cid:12)(cid:12) n,w = 1 B s / , (59)where B = (cid:18) π (cid:19) / π ~ c (cid:18) − w c (cid:19) − / . (60)Since c = c ( n ) (by definition) we can integrate to get E = E ( n, w ) + 34 B s / = E + 3 ψ = E + E ph (cid:18) − w c (cid:19) − . (61)We learn that in the limit of a low relative velocity ( w ≪ c ) the equation of state is simply given by E = E + E ph . (62)This result is quite intuitive. Using the fundamental relation of thermodynamics p + E = nµ + sT , (63)it is also straightforward to show that the total pressure is given by p = p + ψ , (64)as one might have expected.Now that we have the required energy functional, we can determine the entrainment parameter from α = ∂E∂w (cid:12)(cid:12)(cid:12)(cid:12) n,s . (65)Working this out, we find that (after some algebra) α = ∂E ∂w − ρ N . (66)Comparing to (56) we see that the model is consistent with ∂E ∂w = 0 . (67)Hence, it is natural to take E = E ( n ), i.e. simply add the thermal phonon energy to the zero-temperature equationof state. The final result is then E = E + 3 ψ = E + E ph (cid:18) − w c (cid:19) − . (68)In order to proceed, we need to provide the zero-temperature equation of state. A “realistic” model should obviouslybe based on QCD. It is natural [10] to use the MIT bag model. In the case of CFL matter, this leads to (using unitswhere the speed of light is unity, c = 1) p ≈ µ π − B + 3 µ π (4∆ − m s ) , (69)0and ρ ≈ µ π + B − µ m s π , (70)where B is the bag constant and ∆ is the pairing gap associated with the CFL condensate.For simplicity, we will use p ≈
13 ( ρ − B ) , (71)in which case the speed of sound is constant; c = dp dρ ≈ . (72)As discussed in [10, 17], this approximation may be quite good, essentially because the two contributions to the lastterm in (69) almost exactly cancel each other. Later, when we work out the various r-mode dissipation integrals, wewill simplify the problem further by assuming that the density is uniform, with a constant speed of sound. B. Viscosity
In order to determine the damping timescale for the unstable r-modes, we need to supplement the non-dissipativemodel with viscous terms. From the discussion in section IIIB we know that we need one shear viscosity coefficient andthree bulk viscosity coefficients. In order to account for dissipation associated with the superfluid vortices, we shouldalso consider the mutual friction. Some of the required viscosity coefficients have been discussed in the literature. Inparticular, the dissipation due to various phonon interactions has been investigated [11–13]. In fact, the availabilityof these results is one of the key reasons for us focussing on the condensate plus phonon model. Having said that, theavailable results are incomplete. To what extent this is the case, and how we have dealt with this problem, will bediscussed below.Before we proceed it is worth iterating the point that we are focusing on the phonons because they represent thesimplest in a hierarchy of relevant multi-fluid problems. The presence of kaons, either as thermal excitations or acondensate [15, 16], would require us to extend the model to account for additional degrees of freedom. The issuesinvolved are similar to those for a superfluid hyperon core [25], and we expect to consider them in the future.Let us first consider the phonon shear viscosity. The relevant viscosity coefficient is determined in [12]. We shoulduse (following the discussion above, we are using c = 1 / η = 2 . × (cid:16) µ q
300 MeV (cid:17) (cid:18) K T (cid:19) g / cm s , (73)where µ q is the quark chemical potential. We arrive at this result by taking both µ n and the quark “mass” m in thefluid model to be equal to the quark chemical potential µ q . The temperature scaling of η is the same as in the case of He, which makes sense given that the involved phonon processes are the same. However, in the context of neutronstars, this model is severely limited. Basically, the “hydrodynamic” treatment of the phonons is no longer valid iftheir mean-free path is larger than (or comparable to) the size of the system. As in kinetic theory, the mean-free pathassociated with the shear viscosity follows from (up to a factor of order unity) η ≃ ρ N c λ . (74)For the phonon gas model this leads to λ ≃ × (cid:16) µ q
300 MeV (cid:17) (cid:18) K T (cid:19) cm . (75)In other words, for a 10 km star the model will not be valid below ∼ K . This is obviously a problem, sincemature neutron stars are expected to be significantly colder than this. In essence, we ought to model the phonons asballistic in the low temperature regime. However, there is some evidence from laboratory experiments on He thatthe fluid model remains relatively accurate also at lower temperatures. Given this, we will use the two-fluid modelalso in the regime where it is no longer formally valid. Still, in order to describe such systems we need a different1model for the shear viscosity. In order to make progress we will simply assume that the mean free path is limited bythe size of the system. That is, we will use an effective shear viscosity given by η eff ≃ ρ N c R , (76)where R is the radius of the star. This may seem like a rather drastic assumption, but it has recently been shown [36]that in the case of Helium it leads to results that agree quite well with the observed sound attenuation. In particular,we would have η eff ∼ ρ N ∼ T at low temperatures, which means that the viscosity weakens as the phonon densitydecreases. This makes (at least qualitative) sense. Explicitly, we get η eff ≈ × (cid:18) T K (cid:19) g / cms . (77)This model produces a maximum in the viscosity as a function of temperature, a feature that seems quite natural.Having said that, more detailed work on the low-temperature phonon problem is obviously needed in order to improveon our results. This is, in fact, a very interesting problem; when the mean-free path is large, the phonons interactwith the “surface” more frequently than with each other. In principle, the hydrodynamics approach should not bevalid in this regime. Yet, there is some evidence from studies of heat conduction in nano-systems (see [21] for a recentdiscussion) that a judicious choice of surface boundary condition for the phonons leads to a useful “fluid” model. Weplan to explore this idea further in the future.We now turn to the bulk viscosity. Only one of the three required coefficients has, so far, been calculated in detail.In the static limit, the analysis in [11] leads to the result ζ ( ω = 0) = 1 . × (cid:16) m s (cid:17) (cid:18) K T (cid:19) g / cms , (78)where m s is the strange quark mass. However, in order to use this result to study oscillation modes, we need considerthe frequency dependence in more detail. This is essential since the bulk viscosity is a “resonant” mechanism that isparticularly effective when the involved dynamics has a timescale similar to the relevant relaxation time. The staticlimit only provides partial information. Following [14] we will use ζ i = τ ω τ α i ( T ) , i = 1 − , (79)where the “amplitudes” α i , in general, depend on the different physical scales of the system, while τ is the relaxationtime for the processes that give rise to bulk viscosity. According to [14], the relaxation time scales according to τ ∼ c µ q T . (80)The scaling of all three viscosity coefficients, in the static limit, has been determined in [14]. We should have ζ ≈ ˜ α m s T µ q , (81) ζ ≈ ˜ α m s T , (82) ζ ≈ ˜ α T µ q , (83)where ˜ α i are constants. This, together with the relations in (79) and (80) allows us to determine the frequencydependent bulk viscosity coefficients, up to the unknown amplitudes ˜ α i and β . These can not be determined with theapproximations used in [14], although we can obviously infer ˜ α from (78). From the analogous problem for He, weknow that the three bulk viscosity coefficients may be of the same order of magnitude. In absence of detailed resultsfor the quark case, it thus makes sense to assume that ζ and ζ have a similar amplitude to ζ . In particular, thiswould suggest that we parametrise ζ , in the static limit, according to ζ ( ω = 0) ρ g / cm ! = ¯ α × (cid:18) K T (cid:19) (cid:18)
300 MeV µ q (cid:19) g / cms , (84)2where ¯ α is an unspecified parameter. The frequency dependent result then takes the form ζ = ζ ( ω = 0)1 + ω τ , (85)with τ = β × (cid:18) K T (cid:19) (cid:16) µ q
300 MeV (cid:17) s . (86)Here, β is another undetermined parameter. In principle, one would expect both ¯ α and β to be of order unity, butgiven the lack of precise information we can vary them and assess how this affects the bulk viscosity damping ofthe r-modes. Note that the coefficient in (85) has a maximum at the resonance frequency ω = 1 /τ . We know fromprevious work that it is essential to understand how this resonance frequency relates to the r-mode frequency. A precisestatement to this effect is not possible, given the free parameters in our model, but the strong temperature dependencein (86) means that even a change of several orders of magnitude in τ would produce a relatively small change in thetemperature at which the resonance occurs for a given mode frequency. This suggests that the uncertainty associatedwith β may not affect our analysis too severely (assuming that the temperature scaling is correct, of course). If we, forexample, consider a mode frequency ω = 10 s − and an amplitude such that τ = 10 s, we would have a resonancearound T ≈ . × K. If, on the other hand, we take a drastically shorter relaxation timescale of τ = 1 s, thenthe resonance appears near T ≈ . × K. That is, the resonance temperature would shift by less than an orderof magnitude. This model is obviously phenomenological, and allows us to proceed, but an actual derivation of thefrequency dependent viscosity coefficients is needed if we want more detailed results.It is obviously important to compare our results to other relevant estimates for the r-modes. The natural, and mostimmediate, comparison would be to unpaired quark matter. In that case, which corresponds to a single fluid problem,we have [10, 37] η ≈ . × (cid:18) . α s (cid:19) ρ g/cm ! / (cid:18) T K (cid:19) − g / cms , (87)where α s is the strong interaction coupling constant, and ζ = A ( T ) ω + B ( T ) , (88)where A ( T ) = 1 . × (cid:16) µ q (cid:17) (cid:16) m s (cid:17) (cid:18) T K (cid:19) g / cms , (89)and B ( T ) = 2 . × (cid:16) µ q (cid:17) (cid:18) m s µ q (cid:19) (cid:18) T K (cid:19) s − . (90)We also want to quantify the relevance of the vortex mutual friction for the r-mode instability in CFL matter. Inorder to do this we need a representation of the counter-moving degree of freedom associated with an r-mode (whichis of order Ω in the slow-rotation approximation) [32]. Before we consider this problem, we need to translate themutual friction results from [13] to our formalism. In order to relate the parameter B to the results in [13], let usconsider equation (30) where we now include a mutual friction force of the form (19). This leads to an equation forthe evolution of the relative velocity of form ∂w ns i ∂t + s α ∇ i δT = − ρ α (cid:2) B ′ n v ǫ ijk κ j w k ns + B n v ǫ ijk ǫ klm ˆ κ j κ l w ns m (cid:3) . (91)Given this equation, and the results in the Appendix, we can compare our formalism to the standard mutual frictiondescription, see e.g. equations (2.2)-(2.3) in [38]. This allows us to determine B from the parameter ˜ α that is calculatedin [13]. This leads to B = 2 π (cid:18) − αρ (cid:19) (cid:0) − c (cid:1) c (cid:18) Tµ q (cid:19) , (92)3or B ≈ . × − (cid:18) − αρ (cid:19) (cid:18)
300 MeV µ q (cid:19) (cid:18) T K (cid:19) . (93)We see that the mutual friction vanishes as the temperature decreases. This obviously makes sense since there will beno phonon-vortex scattering when the phonon gas becomes dilute. Moreover, according to this result, we are safelyin the extreme “weak drag” regime (c.f. the discussion in [39]) where B ′ ≈ B ≪ . (94)This implies that the phonon mutual friction will not damp the r-modes efficiently (probably as expected) [32]. Thisresult is confirmed by the detailed analysis in section VII. However, these results come with an important caveat.Strictly speaking, our analysis is only valid as long as the phonons can be described as a fluid. As we have alreadyexplained, this is not the case for mature neutron stars. In fact, the results of [13] are supposedly derived for ballisticphonons. This means that our analysis is somewhat inconsistent. At temperatures above (say) 10 K the mutualfriction parameter from [13] may not apply, and below this temperature our two-fluid model may not be appropriate.However, the latter issue may not be that important. After all, one would expect the phonon mutual friction toweaken dramatically at lower temperature, c.f. (93), meaning the any quantitative errors in the analysis will be of noreal importance.
VI. APPROXIMATE R-MODE RESULTS, SHEAR- AND BULK VISCOSITY
Our (simple) model for a two-component cool CFL core is now “complete”, and we may conduct the r-mode analysisas in [31, 32]. In order to quantify the damping due to bulk viscosity, we need to account for terms of order Ω inthe analysis. This complicates the problem, since the centrifugal deformation of the star’s shape enters at the samelevel. In view of this, and the fact that this is a first exploratory study, we will make a sequence of approximationsthat allow us to proceed analytically. This strategy also makes sense since we are using a simplified equation of state.The results of our analysis should help determine whether a full numerical study is worthwhile.First of all, we need to make a choice of primary variables. The linearised equations link the four scalar variables δp , δT , δs and δρ . In principle, we can use the equation of state to express any two of these in terms of the othertwo. Following the strategy set out in [32] we will work with δp and δT . For our model equation of state we have(since the background is co-rotating) δp = δp + (cid:18) ∂ψ∂T (cid:19) δT = δp + sδT , (95)which means that we can express the equations in terms of δp and δT . We then have δρ = (cid:18) ∂ρ∂p (cid:19) δp + (cid:18) ∂ρ∂T (cid:19) δT = 1 c δp , (96)since c is contant, and δs = (cid:18) ∂s∂p (cid:19) δp + (cid:18) ∂s∂T (cid:19) δT = c v T δT , (97)since p = p ( n ).In order to avoid numerics we will make use of a “trick” used by Lindblom, Owen and Morsink in one of theearly r-mode instability papers [40]. The basic idea is to neglect the rotational change in shape in different terms inthe perturbation equations. Once this is done one can “estimate” the bulk viscosity from the leading order r-modesolution. Although this simplification is not not based on a rigorous argument, one can show that the bulk viscositydamping timescale is of the right order of magnitude (within about a factor of 5 of the true second-order slow-rotationresult [41]). This is good enough for our present purposes.In our case, the calculation would proceed as follows: First consider the continuity equation iωδρ + ρ ∇ i v i n + v i n ∇ i ρ = 0 , (98)4where we have assumed that all perturbed quantities behave as e i ( ωt + mϕ ) . If we simply omit the last term (that wouldvanish for a spherical star since an r-mode is purely toroidal to leading order), we get ∇ i v i n ≈ − iωρ δρ = − iωρc δp . (99)This may not be a very accurate estimate, but it allows us to progress without having to determine the rotationalcorrections to the r-mode (which depend on the term that we have neglected). Now consider the second degree offreedom in a similar way. From the entropy equation we get iωδs + s (cid:0) ∇ i v in − ∇ i w i ns (cid:1) + ( v i n − w i ns ) ∇ i s = 0 . (100)In fact, for the model equation of state the last term vanishes identically since s = s ( T ) and ∇ i T = 0 for ourbackground model. It follows that ∇ i v i n − ∇ i w i ns ≈ − iω δss . (101)The approximation is completed by the ϕ -components of the two Euler equations. Assuming that the r-modevelocity field takes the form v i n = (cid:2) v r e ir + v θ e iθ + v ϕ e iϕ (cid:3) e i ( ωt + mϕ ) , (102)and that w i ns is similar, although of higher slow-rotation order than v i n [32], we get iωρr sin θv ϕ + imδp + 2 ρ Ω r sin θ cos θv θ = 0 , (103)and 2 iαωr sin θw ϕ + imsδT = 0 . (104)The second relation shows immediately that δT is of higher slow-rotation order for the classic r-mode (since w ϕ is ofhigher order and ω ∼ Ω for inertial modes). This means that we can justifiably neglect the temperature variation inthe pressure term in the first equation. That is, we have iωρr sin θv ϕ + imδp + 2 ρ Ω r sin θ cos θv θ = 0 . (105)This is exactly the result used in [40]. Moreover, we now see that the first term in (100) should also be of higherorder, which means that we are left with ∇ i w i ns ≈ ∇ i v i n . (106)The above relations allow us to estimate the terms needed to evaluate the bulk viscosity integrals once we have theleading order r-mode solution. Since the required contribution to the r-mode velocity field is purely toroidal, we have v θ = − U l r sin θ ∂ ϕ Y ml , v ϕ = U l r sin θ ∂ θ Y ml . (107)Moreover, the mode-solution is such that the only contribution comes from l = m in which case U m = ( r/R ) m +1 (noting that the normalisation is irrelevant at the linear perturbation level). Finally, to the accuracy needed, themode frequency is (in the rotating frame) ω = 2Ω m + 1 . (108)Using these results, and expanding the (scalar) pressure perturbation in spherical harmonics, i.e. using δp = P l δp l Y ml e i ( ωt + mϕ ) , we find that (105) leads to δp m +1 ρ = 1 √ m + 3 2 m Ω m + 1 U m , (109)and thus we have ∇ i v i n ≈ ∇ i w i ns ≈ − i √ m + 3 4 m Ω ( m + 1) c U m Y mm +1 . (110)This completes the approximation scheme, which provides all the expressions we need to make a rough estimate ofthe bulk viscosity damping timescale.5 VII. MUTUAL FRICTION
In order to estimate the mutual friction damping, we need to determine the detailed counter-moving degree offreedom associated with the r-modes. In principle, this calculation proceeds as in [32] and involves the rotationalcorrections to the shape of the star. However, in order to be consistent we will instead build on the approximationsintroduced in the previous section. First of all, it is easy to show that the countermoving degree of freedom will bepoloidal (in contrast to the toroidal leading order r-mode solution). This means that we have w r = 1 r W l Y ml , w θ = V l r ∂ θ Y ml , w ϕ = V l r sin θ ∂ ϕ Y ml . (111)If we also expand the temperature perturbation in such a way that δT = P l T l Y ml , then it follows from (104) that V l = is αω T l . (112)Similarly, we get from the radial component of (30); W l = is αω ∂ r T l . (113)Combining these results with (108) and (110), we find that sT m +1 = 8 mα Ω √ m + 3( m + 1) (2 m + 5) R c (cid:16) rR (cid:17) m +1 (cid:20) m + 3 m + 1 − (cid:16) rR (cid:17) (cid:21) . (114)From this result we easily obtain the required counter-moving components of the r-mode.We arrive at (114) by assuming that w r = 0 at the surface (in accordance with the discussion in [32]). This conditionis unlikely to be correct in the present problem, and a detailed analysis of the appropriate condition to impose onthe phonons needs to be carried out in the future (c.f. the discussion in [21]). At this point we simply note that ourassumption that the whole star is composed of CFL matter is artificial anyway. Moreover, the boundary conditiondoes not affect the parameter scaling of the result. This is the dominant factor in determining the r-mode dampingtimescale. VIII. THE R-MODE INSTABILITY WINDOW
To estimate the growth/damping timescales for the r-mode instability we need to evaluate various energy integrals,see [24, 40]. First of all, the mode energy is given by E = 12 m ( m + 1) R − (2 m +2) Z R ρr m +2 dr , (115)while the growth time-scale due to (current multipole) gravitational-wave emission is obtained from1 τ gw = − πG Ω m +2 c m +3 ( m − m [(2 m + 1)!!] (cid:18) m + 2 m + 1 (cid:19) m +2 Z R ρr m +2 dr . (116)Turning to the shear viscosity, we know that | w ns | ≪ | v n | for the superfluid r-mode. This means that the shearviscosity integral simplifies to the usual, single fluid result. Hence, the leading-order energy loss due to shear viscosityfollows from the expression˙ E sv = − m ( m + 1) (Z R ηr " r (cid:12)(cid:12)(cid:12)(cid:12) ∂ r (cid:18) U m r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + ( m − m + 2) | U m | dr = − m ( m − m + 1) R − (2 m +2) Z R ηr m dr , (117)leading to 1 τ sv = − ( m − m + 1) Z R ηr m dr "Z R ρr m +2 dr − . (118)6Next, we need to estimate the bulk viscosity damping. From (44) we know that the general bulk viscosity expressionis complicated, especially since we cannot rule out the possibility that the three different bulk viscosity contributionsare of similar magnitude. However, the approximations that we have made simplify the problem considerably. Ne-glecting the terms that arise due to the change in shape of the rotating star [the last line in (44)], and making use of(106) we find that ˙ E bv ≈ − Z R ζ eff (cid:12)(cid:12) ∇ i v i n (cid:12)(cid:12) dV ≈ − m Ω ( m + 1) (2 m + 3) R − (2 m +2) Z R ζ eff c r m +4 dr , (119)which leads to 1 τ bv = − m Ω ( m + 1) (2 m + 3) Z R ζ eff c r m +4 dr "Z R ρr m +2 dr − , (120)where ζ eff ≈ ρ ζ . (121)This result follows after some surprising cancellations, associated with (106). It is notable that only one of the threebulk viscosity coefficients plays a role in our simplified case. Moreover, it is not the one ( ζ ) that remains in theNavier-Stokes limit. Of course, one would not expect this drastic simplification for a more realistic stellar model.This fact provides strong motivation for a more detailed, numerical, r-mode analysis. We also learn that we can not,in general, ignore the additional bulk viscosities in the multi-fluid problem.Finally, we have the dissipation integral for the mutual friction. From the results discussed in [31, 32], we find that˙ E mf ≈ − Z ρ B Ω (cid:16) δ mi − ˆΩ m ˆΩ i (cid:17) w i ns ¯ w ns m dV . (122)Working out the angular integrals, this reduces to˙ E mf ≈ − Z ρ B Ω (cid:8)(cid:2) − Q m +2 − Q m +1 (cid:3) W m +1 + (cid:2) ( m + 1)( m + 2) − ( m + 1) Q m +2 − ( m + 2) Q m +1 (cid:3) V m +1 (cid:9) dr , (123)where we have used Q l = ( l + m )( l − m )(2 l − l + 1) . (124)The integral in (123) is easily evaluated using the eigenfunctions from the previous section.In order to evaluate the relevant integrals, and obtain estimated timescales with explicit parameter scaling, we nowassume that the density is uniform. Taking all parameters constant, we easily find that τ gw = − (cid:18) . M ⊙ M (cid:19) (cid:18)
10 km R (cid:19) (cid:18) P (cid:19) s , (125)for the l = m = 2 r-mode (the sign indicates that the mode is unstable). We also get, for the shear viscosity,1 τ sv ≈ . × − (cid:18) η / cms (cid:19) (cid:18) R
10 km (cid:19) (cid:18) . M ⊙ M (cid:19) s − , (126)and for the bulk viscosity we find1 τ bv ≈ . × − (cid:18) ζ eff / cms (cid:19) (cid:18) R
10 km (cid:19) (cid:18) . M ⊙ M (cid:19) (cid:18) P (cid:19) s − . (127)Finally, in the case of the mutual friction we find1 τ mf ≈ . B (cid:18) R
10 km (cid:19) (cid:18) P (cid:19) − s − . (128)7 T(K) -3 P c ( s ) mutual friction, eq (128)unpaired mattershear viscosity, eq (126)bulk viscosity, eq (127)mutual friction, ref [13] s bulkshear LMXB mutual friction
FIG. 1: The r-mode instability window for a dense core comprising a CFL superconducting quark condensate and (finitetemperature) phonons. The figure shows the critical rotational period vs core temperature for the instability. Gravitationalwaves drive the l = m = 2 r-mode unstable for systems located below the different curves. The effect of shear- and bulkviscosity associated with phonon processes are shown as blue (dashed) and red (solid/dashed) curves, respectively. We showthe result for both the canonical value β = 1 (solid red line) and the rather extreme value β = 10 − (dashed red line) in (86),leading to relaxation times of 10 s and 1 s (as indicated in the figure), respectively. We also show the result for the case when¯ α is increased by a factor of 10 (thin solid red line). Our result for the mutual friction damping (this dash-dot black line) iscompared to the result from [13] (thin dash-dot black line), demonstrating the dramatic effect of the different scaling with therotation. For comparison, we also provide the instability curve for the combined shear- and bulk viscosity, obtained from (87)and (88), in the case of unpaired quark matter (black line at bottom of figure). Finally, we show the observed spin period andinferred core temperature for a number of astrophysical systems (filled squares), and we also indicate the region of parameterspace where accreting neutron stars in low-mass X-ray binaries (LMXB) are thought to be located. The region below 10 K,where the phonon-fluid model may not be appropriate, is shown with grey background to emphasize the need for improvedmodelling in the astrophysically relevant part of parameter space.
This result differs significantly from the estimate in [13], in particular, in terms of the scaling with the rotation rate.However, our result is obtained from the actual counter-moving part of the r-mode solution and the final scalingaccords with the standard result (see, for example, [32]). The main difference between our result and that in [13] canbe explained by noting that, if we take the r-mode energy to be proportional to w rather than v then we arrive at amutual friction damping time scale very similar to that obtained in [13]. The large discrepancy between the estimatesthus follows from the fact that the velocites v i n and w i ns associated with the r-mode solution are of different orders ofΩ in the slow-rotation approximation.Combining these estimates with (73), (77), (84) and (93), i.e. balancing the four timescales (125)–(128), we arriveat the r-mode instability window in Figure 1. The figure shows the critical rotational period for the instability asfunction of the core temperature for a canonical star with mass 1 . M ⊙ and radius 10 km. Because of the less efficientdamping mechanisms, we show the period rather than the rotation rate. This makes sense because the long timescalesinvolved beome much clearer than when the critical rotation rate is expressed as a small fraction of the rotationalbreak-up limit. As a result, gravitational waves drive the l = m = 2 r-mode unstable for systems located below thedifferent curves in the figure. Our estimates show that the bulk viscosity is much weaker than the shear viscosity atall temperatures of interest. In order to show how the bulk viscosity result depends on the relaxation time, c.f. (86),we show the result for both the canonical value β = 1 and the rather extreme value β = 10 − , leading to relaxationtimes of 10 s and 1 s, respectively. The fact that the associated instability curves do not differ much shows thatthe results are not very sensitive to this parameter. This is due to the strong scaling with temperature. In order toexplore the importance of the overall strength of the bulk viscosity, we also show the result for the case when ¯ α isincreased by a factor of 10 compared to our canonical value ¯ α = 1. This result shows that the bulk viscosity has tobe vastly different from our assumed model in order to be relevant. The data illustrated in figure 1 also shows thatthe instability curve obtained from balancing the gravitational-wave driving and our result for the phonon mutualfriction damping is vastly different from the result in [13]. The different dependence on rotation is obvious, and it is8clear that the phonon mutual friction would be irrelevant for all astrophysical stars with CFL cores. For comparison,we show the instability curve for the combined shear- and bulk viscosity, obtained using (87) and (88), in the case ofunpaired quark matter. It is interesting to note that the phonon shear viscosity in the CFL case leads to the strongestr-mode damping at temperatures above 10 K. Of course, astrophysical neutron stars will cool to temperatures muchlower than this soon after birth so this region may not be that relevant. The results in the figure show, quite clearly,that we need to improve our understanding of the low-temperature regime. The instability curve below the 10 K isobtained using our phenomenological model (77). Future work needs to model this regime in detail.For comparison, we also show the observed spin and inferred core temperature for a number of astrophysical systemsin Figure 1 (the data is taken from [42]). This comparison is somewhat inconsistent because we have assumed thatthe core is shielded by a normal crust with a thickness of, at least, a few hundred meters (so that the standardheat-blanket argument [43] applies. The presence of a normal matter outer region should provide addition r-modedissipation channels that we have not considered. In particular, one would expect a viscous boundary layer at thecrust-core interface to be a more efficient damping agent (see [44, 45] for the most recent discussion) than the phononmechanisms that we have considered. We also indicate the region of parameter space where accreting neutron stars inlow-mass X-ray binaries would be located (given the usual arguments [46, 47]). These systems would be at variancewith the pure CFL plus phonon model in the sense that they would be located deep into the unstable region. If therewere an active r-mode instability in these systems, one would expect it to have observational effects on, for example,the spin evolution. Observations do not provide any evidence of this.
IX. CONCLUDING REMARKS
We have presented the first true multi-fluid analysis of a dense neutron star core with a deconfined, colour-flavour-locked superconducting, quark core. By focussing on a cool system, and accounting only for the condensate and(finite temperature) phonons, we made progress by taking over much of the formalism from the analogous problemfor superfluid He, the archetypal two-fluid laboratory system. The additional fluid degree of freedom, in the presentcase represented by the phonon gas, leads to the system not being well represented by the Navier-Stokes equations.In particular, a complete model requires a number of additional viscosity coefficients.Without an actual calculation it is not easy to establish whether the multi-fluid aspects are relevant or not. Forexample, in the case of the gravitational-wave driven instability of the f-modes it is known that the superfluid degreeof freedom is very important, since the vortex mutual friction may completely suppress that instability below a criticaltemperature (see [31, 48]). It is known that, because of the different nature of the associated velocity field, the mutualfriction does not affect the r-mode instability in the same drastic fashion [32, 49]. These examples provide clearevidence that different problems need to be considered on a case-by-case basis.We have provided a detailed dissipative formulation for a system comprising a quark (CFL) condensate and phonons.The model builds on recent improvements in our understanding of the analogous problems of superfluid He [9]and causal heat conductivity [21]. A key ingredient is the massless entropy component that represents the phononexcitations. We have discussed how the superfluid constraint of irrotationality reduces the number of required viscositycoefficients to four (one shear and three bulk), and provided a translation between results in the literature [11, 12]and the coefficients in our formalism. We also emphasised that many more dissipative channels may come into playin a rotating system where superfluid vortices are present [9, 19]. In particular, we showed how the vortex mediatedmutual friction is accounted for in the model, and translated the available results for the associated coefficients [13].In order to be able to make relevant estimates for the r-mode instability, we developed a simple two-componentequation of state based on the MIT bag model at zero temperatures with an additional phonon gas representing thethermal component. This example highlights the additional information that is needed in a multi-fluid analysis, inparticular, regarding the entrainment coupling. Future work needs to provide a consistent equation of state, includingall the key aspects. The model equation of state completed the formulation of the problem and we could, in principle,have carried out a numerical study of the r-modes. We opted not to do this, instead introducing a sequence ofsimplifying assumptions, because we felt that it would be useful to start by working out some less precise estimatesfor the relevant dissipation timescales. A numerical analysis of the problem should, of course, be encouraged. Theproblem has a number of interesting aspects, and may shed light on how one should deal with finite temperaturesuperfluid neutron stars in general.This work was motivated by a desire to understand the different phases of CFL matter from a hydrodynamicspoint of view. It is, obviously, an interesting problem and the notion that observations of gravitational waves fromrelativistic stars may help shed light on the extreme QCD phase diagram is exciting. The r-mode instability has beendiscussed in this context for some time, see for example [11–13, 15, 16, 37, 50], but there has not been any previousdiscussion of the multi-fluid aspects of the problem. We hope that this work will stimulate a more detailed discussionbetween experts in the relevant areas.9The fact that the various phonon processes that we have accounted for can be shown to have little effect on ther-mode instability should not discourage future efforts. In fact, the result could probably have been anticipated. Thereal challenge will be to account for additional degrees of freedom, especially associated with the kaons (either thermalor in a condensate) [15, 16], and the modelling of “hybrid” stars with quark cores and various phase transitions. Theseproblems have additional features that, while possibly understood in principle, have never been considered in practice.This makes the modelling more complex, but also intriguing since the richer dynamics may lead to surprises.
Appendix: Translation to the orthodox framework
The relationship between the flux conservative formulation and the orthodox framework for superfluid Helium hasbeen examined in detail elsewhere [9, 18]. Still, it is useful to summarise the main results that are needed to relatethe different dissipation coefficients. First of all, we identify the “normal” fluid in the standard description with thegas of excitations, which is directly associated with the entropy of the system. This leads to v i N = v i s . (129)Secondly, the so-called “superfluid” velocity is given by v i S = π i n ρ = (1 − ε ) v i n + εv i s , (130)where ε = αρ is the entropy entrainment parameter. This means that the total mass flux takes the form ρv i n = ρ N v iN + ρ S v i S , (131)from which we learn that ρ S = ρ − ε and ρ N = − ερ − ε , (132)where ρ S and ρ N are the superfluid and normal fluid densities. Acknowledgments
We are grateful to Mark Alford and Cristina Manuel for a number of useful discussions. NA and BH acknowledgesupport from STFC via grant number PP/E001025/1. BH also acknowledges support from the European ScienceFoundation (ESF) for the activity entitled “The New Physics of Compact Stars” , under exchange grant 2449, andthanks the Dipartimento di Fisica, Universit`a degli studi di Milano for kind hospitality during part of this work. [1] I.M. Khalatnikov,
An introduction to the theory of superfluidity (W. A. Benjamin, Inc., New York, 1965).[2] S.J. Putterman,
Superfluid hydrodynamics (North-Holland, Amsterdam, 1974).[3] R.I. Epstein, Ap. J. , 880 (1988).[4] L. Lindblom & G. Mendell, Ap. J.
689 (1994).[5] N. Andersson & G.L. Comer, MNRAS , 1776 (2006).[7] N. Andersson, G.L. Comer & K. Glampedakis, Nucl. Phys. A , 212 (2005).[8] M. Alford, A. Schmitt, K. Rajagopal & T. Sch¨afer, Rev. Mod. Phys. , 1455 (2008).[9] N. Andersson & G. L. Comer, Entropy entrainment in finite temperature superfluids, preprint arXiv:0811.1660[10] P. Jaikumar, G. Rupak & A.W. Steiner, Phys. Rev. D , 1 (2007).[12] C. Manuel, A. Dobado & F.J. Llanes-Estrada, JHEP , 76 (2005).[13] M. Mannarelli, C. Manuel & B.A. Sa’d, Phys. Rev. Lett. , 241101 (2008).[14] M. Mannarelli & C. Manuel, Phys. Rev. D , 043002 (2010).[15] M.G. Alford, M. Braby, S. Reddy & T Sch¨afer, Phys. Rev. C. , 115007 (2008).[17] M.G. Alford, M. Braby & S. Mahmoodifar, Phys. Rev. C [18] R. Prix, Phys. Rev. D no. 1 (2007).[21] N. Andersson & G.L. Comer, Proc. R. Soc. London A, doi 10.1098/rspa.2009.0423[22] N. Andersson, T. Sidery & G.L. Comer, MNRAS , 162 (2006).[23] C. Alcock. E. Farhi & A. Olinto, Ap. J. , 261 (1986).[24] N. Andersson & K.D. Kokkotas, Int. J. Mod. Phys. D , 381 (2001).[25] B. Haskell & N. Andersson, Superfluid hyperon bulk viscosity and the r-mode instability of rotating neutron stars, preprintarXiv:1003.584[26] K.H. Lockitch, N. Andersson & J.L. Friedman, Phys. Rev. D , 024019 (2001).[27] K.H. Lockitch, J.L. Friedman & N. Andersson, Phys. Rev. D , 124010 (2003).[28] J. Ruoff & K.D. Kokkotas, MNRAS , 678 (2001).[29] J. Ruoff & K.D. Kokkotas, MNRAS , 1027 (2002).[30] G.L. Comer, Found. Phys. , 103009 (2009).[32] B. Haskell, N. Andersson & A. Passamonti, MNRAS , 1464 (2009).[33] R. Prix & M. Rieutord, Astron. Astrophys. , 949 (2002).[34] A. Passamonti, B. Haskell & N. Andersson, MNRAS , 951 (2009).[35] B. Carter & D. Langlois, Phys. Rev. D , 100 (2009).[37] J. Madsen, Phys. Rev. D , 3290 (1992).[38] R.J. Donnelly, Quantized vortices in Helium II (Cambridge University Press, Cambridge, 1991).[39] K. Glampedakis, N. Andersson & D.I. Jones, MNRAS , 1908 (2009).[40] L. Lindblom, B.J. Owen & S.M. Morsink, Phys. Rev. Lett. , 4843 (1998)[41] L. Lindblom & G. Mendell, Phys. Rev. D , 064006 (1999).[42] D.N. Aguilera, J.A. Pons, J.A. Miralles, Ap. J. Lett. , 167 (2008).[43] E.H. Gudmundsson, C.J. Pethick & R.I. Epstein, Ap. J. , 286 (1983).[44] K. Glampedakis & N. Andersson, MNRAS , 1311 (2006).[45] K. Glampedakis & N. Andersson, Phys. Rev. D , 044040 (2006).[46] L. Bildsten, Ap. J. Lett. , 89 (1998).[47] N. Andersson, K.D. Kokkotas & N. Stergioulas, Ap. J. , 307 (1999).[48] L. Lindblom & G. Mendell, Ap. J. , 804 (1995).[49] L. Lindblom & G. Mendell, Phys. Rev. D , 104003 (2000).[50] N. Andersson, D.I. Jones & K.D. Kokkotas, MNRAS , 1224 (2002).[51] Throughout this paper we use a coordinate basis to represent tensorial relations. That is, we distinguish between co- andcontra-variant objects, v i and v i , respectively. Indices, which range from 1 to 3, can be raised and lowered with the (flatspace) metric g ij , i.e., v i = g ij v j . Derivatives are expressed in terms of the covariant derivative ∇ i which is consistentwith the metric in the sense that ∇ i g klkl