R-Modes of Accreting Hyperon Stars As Persistent Sources of Gravitational

Home , Thermal runaway

arXiv:astro-ph/0512041v2 10 Apr 2006 mltd sntapolm[,4 n h diinlvis- additional the and strange low 4] or scenario, [3, 21] LMXB problem 20, a the not 19, for is However, 18, amplitude hot 17, [22]. young, 16, stars 15, very quark [14, in stars altogether may neutron instability which mechanisms the quark—there down damping strange eliminate or a viscous by up additional replaced an is are neutrons nucleon where a than in hyperons, exotic quark neu- as more if protons—such Also, particles and [2]. contain supernova nearby stars ampli- very tron undetectable a mode signal is low wave there a gravitational unless spins, the of ef- renders regulation interesting tude the astrophysically as to such lead fects can still stars tron e e.[]frarcn eiwo h aypyia and physical many the of the to review related recent issues a astrophysical for stars. neutron [1] of waves Ref. spins See gravitational the regulating of for sources mechanisms attracted and possible have as stars interest neutron rotating much rapidly of force) lis eidcgaiainlwvs[,4.(eethe (Here of 4]. sources [3, scenario, persistent waves other be gravitational the could periodic In (LMXBs) binaries [2]. x-ray more a on or year frequency r a in changing of waves momen- timescale angular substan- gravitational and a as energy radiate rotational tum could its of star fraction sce- neutron tial one newborn In a scenarios. two nario, in emission wave gravitational ngaiainlwv msin rvttoa radiation Gravitational the drives emission. wave gravitational on o eeto.Teei o vdnefo eea ap- several an from of evidence amplitude now prospect the is brighter that There a proaches like detection. looks for LMXBs) (from emission torque general more 6]). a [5, for argument mechanism balance specific a provide c oarltvl ml au 7 ,9 0 1 2 13]. 12, 11, 10, 9, 8, dynam- [7, value fluid small nonlinear long-lived low-amplitude relatively by While a limited to is ics instability the to mdsi ail crtn eto tr nlow-mass in stars neutron accreting rapidly in -modes The urnl h atrseai o rvttoa wave gravitational for scenario latter the Currently R mdso crtn yeo tr spritn ore fg of sources persistent as stars hyperon accreting of -modes r mds(udoclain oendb h Corio- the by governed oscillations (fluid -modes r mdsusal n ol edt detectable to lead could and unstable -modes msini ut out lhuhi sdsaoe nstars in disfavored ro is of it effect although the robust, time quite first gravitat is persistent the of emission viability for the r on include the parameters various combine microphysics, We the theory. field of mean relativistic of framework qaino tt.I oecsspritn msini comp is emission persistent cases some In 10 state. of equation ASnmes 43.b 44.g 66.c 97.10.Sj 26.60.+c, 04.40.Dg, 04.30.Db, numbers: PACS feoi atce uha yeosaepeeti h oeo core for the in frequencies prevent present critical can are and matter hyperons stellar rates as the such damping of particles viscosity exotic bulk if the if waves nttt o rvttoa hsc n emtyadCente and Geometry and Physics Gravitational for Institute eateto hsc,TePnslai tt University, State Pennsylvania The Physics, of Department The 7 .INTRODUCTION I. ,osre nsm crtn eto tr nquiescence. in stars neutron accreting some in observed K, r mdso crtn eto tr ol eadtcal sour detectable a be could stars neutron accreting of -modes Dtd d ae.e, .220/32 44:1nya Ex nayyar 04:46:31 2006/03/22 1.92 paper.tex,v Id: (Dated: r mdsi ebr neu- newborn in -modes r mds eew focus we here -modes; r md rwn due growing -mode oi ayradBnai .Owen J. Benjamin and Nayyar Mohit r -modes r mdso eto tr otiighprn nthe in hyperons containing stars neutron of -modes ybl icst,ete rmteUc rcs (pertur- process Urca the dominated from probably either viscosity, At is bulk 27].) damping by 4, ex- the [3, indicated to temperatures region easier ob- the high in are the lies curve binaries but the x-ray if entirely, plain low-mass spins of of spins range served lie observed curves the low-temperature decreas- above (Some of temperature. but property with qualitative [26], ing the effects share magnetoviscous generally superfluid they or corresponding turbu- [25] plot, as such lence the mechanisms damping of complicated part more to curves temperature possible low other many the are for There dis- range [24]). augmented the Ref. [23], in in velocity cussed Ref. crust-core (from relative core constant a fluid by and be crust between to layer solid taken boundary the a is in 1 viscosity Fig. shear below.) by in see dominated damping LMXBs runaway; the thermal of temperatures a low temperatures At of observed because range quiescence most needed in than (The higher scenario. emission is LMXB the for ate h eprtr ag 10 range temperature the stable. marginally are it on stars and stable rqec,sasaoeteciia rqec uv have unstable curve more frequency or critical with one the of decreases above function timescale stars a driving frequency, as the Since frequency are of timescales critical temperature. locus damping a the graphically and is defines driving be that equal—this the can plane where viscosity that points in of all shown curve strength a as The by plane, represented temperature-frequency 1. the Fig. nearly in in be location star the to a consider to of assumed useful is (usually it Therefore star isothermal). the and frequency of rotation the temperature on expressed depend be which can timescales mechanisms as damping and driving the of o h atri ute n er explanation. bears and subtle, reason is The latter the better detection. for wave particles gravitational exotic for with candidates stars renders actually cosity icu adohr apn ehnsscmeewith the of compete driving mechanisms wave gravitational damping other) (and viscous iue1poseape fciia rqec uvsin curves frequency critical of examples plots 1 Figure elnurnsasaentpretflis n thus and fluids, perfect not are stars neutron Real oa aeeiso.W n htpersistent that find We emission. wave ional eo 1.3–1.5 below h tr ecmuebl viscous bulk compute We star. the f slso eea rvoscalculations previous several of esults hra uaa.Ti spossible is This runaway. thermal a o rvttoa aePhysics, Wave Gravitational for r tbewt eprtrsa o as low as temperatures with atible nvriyPr,P 16802-6300 PA Park, University ain n xlr h ffcsof effects the explore and tation, eo essetgravitational persistent of ce r M mds hl tr eo tare it below stars while -modes, ) p ⊙ 8 eedn nthe on depending aiainlwaves ravitational K ≪ r T mds h strengths The -modes. ≪ 10 10 appropri- K 2 bation of β-equilibrium) or from nonleptonic processes 700 involving strange particles such as hyperons. The Urca 600 process, which requires no exotic particles, does not affect the critical frequency curve for T < 1010 K and thus 500 does not show up in Fig. 1. Thus the top plot in Fig. 1 400 shows a critical frequency curve for a neutron star, and the bottom plot shows such a curve for a star with hyper- 300 ons (with the high-temperature part of the curve derived 200 from Ref. [19]). This type of plot looks encouraging for gravitational wave detection because there is plenty of Spin Frequency (Hz) 100 room above the curve for stars to be unstable and thus 0 emitting gravitational waves. 8 8.5 9 9.5 10 log Temperature (K) However, the gravitational wave emission duty cycle 10 700 could be much smaller than 100% due to a thermal runaway [28, 29]. This happens generically when the criti- 600 cal frequency decreases with temperature. In that case, 500 plotted at the top of Fig. 1, a star will execute a loop as shown and radiate only during the time it spends above 400 the curve. A stable star which begins at the bottom left 300 of the loop is spun up by accretion until it moves above the critical frequency and the instability is triggered. The 200 shear from the growing r-modes then causes the star to

Spin Frequency (Hz) 100 heat up, moving it rightwards on the loop. As the temperature rises, so does the rate of neutrino cooling, caus- 0 8 8.5 9 9.5 10 ing the star to drop down toward the critical frequency log Temperature (K) again at a temperature of a few times 109 K. After falling 10 below the critical frequency, the r-mode heating is re- moved and the star drifts leftward along the bottom of FIG. 1: Critical frequency curves are given (qualitatively) the loop until returning to its initial position. Although by the dashed lines. The top plot includes no bulk viscosity the time it takes for a star to complete the loop is dom- due to hyperons or other strange particles. In this case an accreting neutron star traversing the loop indicated undergoes inated by the accretion rate, the timescale for gravita- a thermal runaway and has a low gravitational radiation duty tional wave emission depends mainly on the saturation cycle. The bottom plot includes hyperon bulk viscosity. In amplitude of r-mode oscillations. For a saturation am- this case the thermal runaway is blocked, and an accreting plitude (α in the notation of Ref. [30]) of order unity, star is a source of persistent gravitational waves as it remains the duty cycle for gravitational wave emission is of the in equilibrium at the last arrowhead. order 10−6 [28]. Whereas the duty cycle can be as high as about 30% for the lowest predicted values of the satu- −5 ration amplitude (α ≃ 10 ), for typical estimates of the high bulk viscosity processes involving hyperons [35, 36] saturation amplitude (α ≃ 10−4) the duty cycle is only of −1 or strange quarks [37] are at work. Thus the case for the order 10 [31]. Advanced LIGO will be able to detect at gravitational wave emission scenario is in fact strength- most one LMXB (Sco X-1) without narrowbanding (and ened by high bulk viscosity from processes which are fun- hurting its ability to see other sources), or 6–7 LMXBs damentally quark-quark interactions. One might expect with narrowbanding around a series of different frequen- similar critical frequency curves to arise for stars contain- cies [32]. The small number of detectable systems and ing other forms of strange matter such as a kaon conden- the fact that the timescale for a star to complete a loop sate or mixed quark-baryon phase, but this has yet to is much longer than a human lifetime mean that a duty −1 be investigated. The thermal runaway is blocked if the cycle of order 10 or less is pessimistic for gravitational increase in temperature (width of the loop in Fig. 1) is wave searches for the r-modes. enough to take the star from the negatively sloped part of If the critical frequency increases with temperature as the instability curve to the positively sloped part. The in the bottom plot of Fig. 1, the thermal runaway can larger the saturation amplitude of r-modes the greater be blocked. A rapidly accreting star in an LMXB can is the increase in the temperature. Whether the star have a duty cycle of order unity for emission of gravi- makes the jump to the positively sloped curve also de- tational radiation as it sits on the curve or makes small pends on which cooling mechanisms are operative, and on peregrinations about it [33]. (It could also keep emitting, the shape of the low-temperature instability curve, which although less detectably, for some time after the rapid ac- in turn depends on which damping mechanism dominates cretion shuts off [34].) This could happen for stars which at low temperatures. Notwithstanding the wide range of exhibit a rise in the critical frequency curve at a low estimates for α, the shape of the negatively sloped curve, enough temperature. The rise is typical for stars where and the cooling mechanisms, the question of the r-modes 3 in LMXBs as a persistent source of gravitational radia- cal numbers such as hyperon coupling constants and the tion comes down to asking whether there is a rise in the superfluid bandgap, and even which nonleptonic reaction instability curve around 109 K or lower. is most important (taking into account the results of van Our purpose in this paper is to revisit the question Dalen and Dieperink [21]). We also investigate the de- of whether there is a rise in the critical frequency curve pendence on the mass of the star, since cooling observa- around 109 K or lower, as in the bottom of Fig. 1, lead- tions [40] and timing of radio pulsars in binaries [41, 42] ing to persistent gravitational wave emission. We focus indicate a wider mass range than the traditionally as- (for now) on neutron stars containing hyperons (hyperon sumed clustering around 1.4 M⊙. (Cooling observations stars), because hyperons are in some sense the most con- also might be interpreted to favor the existence of strange servative and robust of the many proposals for exotic particles such as hyperons, although the data still can be matter in the cores of neutron stars: Some properties fit by exotic cooling from purely nucleonic matter.) of hyperons can be measured in the laboratory, both in The organization of the rest of the paper is as fol- vacuum and in the environment of a light nucleus, which lows. In Sec. II we describe the microphysical model combined with astronomical observations allows one to which leads to the equation of state and ultimately the constrain some of the many uncertainties in building an macroscopic coefficient of bulk viscosity. In Sec. III we equation of state [38, 39]. Equations of state which allow plot the critical frequency curves for a range of neutron for hyperons generally produce them at densities relevant star masses and microphysical parameters. In Sec. IV for neutron stars, about twice nuclear density and up. we summarize our findings and discuss possible improve- We synthesize and extend results of previous work on ments. this topic. In arguing the case for persistent gravitational wave emission, Wagoner [35, 36] and Reiseneg- ger and Bonacic [34] base their critical frequency curves II. MICROPHYSICS on bulk viscosity coefficients obtained by combining the results of Lindblom and Owen [19] (hereafter LO) and We model the composition of a neutron star within Haensel, Levenfish, and Yakovlev [20] (hereafter HLY). the framework of relativistic mean field theory industri- The LO and HLY viscosities were obtained by differ- alized by Glendenning and described in Ref. [38]. Here we ent calculations and produced somewhat different re- do not consider meson condensates or exotica such as a sults. LO used a detailed self-consistent model of a multi- mixed quark-baryon phase, but do allow for the presence component fluid described by relativistic mean field the- of hyperons which are somewhat constrained by labora- ory [38], but since they were primarily concerned with the tory and astronomical data. high-temperature regime appropriate to newborn neutron stars they treated the effect of superfluidity with a rough approximation. LO also made some errors which A. Equation of state resulted in a bulk viscosity coefficient a factor of 20 or more too high. HLY were more careful with superfluidity, There are several free parameters in this framework, using consistent damping factors in the collision integrals, some of which are better known than others. We use a but instead of evaluating reaction rates and thermody- range of parameters set by combining the old laboratory namic derivatives within relativistic mean field theory constraints with recent astronomical observations [39]. they used “order of magnitude estimates” of some quan- At nuclear density (2–3×1015 g/cm3) the matter is, as in tities which resulted in more than an order of magnitude all models, composed of neutrons, protons, electrons, and disagreement with the microscopic results of LO (and muons. As the density rises hyperons generically appear, with Jones [18]). We correct some mistakes in LO and their order of appearance (and density thresholds) chang- combine their self-consistent microphysical model with a ing somewhat with the precise set of parameter values more careful treatment of superfluidity similar to that of but roughly corresponding to Σ−, Λ,Σ0, Σ+, Ξ−, start- HLY. We also treat the macroscopic physics more care- ing at roughly twice nuclear density. Other baryons typ- fully than LO, including the effect of rotation on stel- ically appear in this framework only at densities higher lar structure which in some cases can significantly affect than those found in the cores of the most massive neu- damping timescales and critical frequency curves. (HLY tron stars, nearly 10 times nuclear density. At such high made only order of magnitude estimates of mode damp- densities the asymptotic freedom of the strong nuclear ing timescales and did not plot critical frequency curves.) force is likely to lead to quark matter anyway, and any We also address the question “Does the viability of baryonic model must be considered somewhat suspect. persistent gravitational wave emission require fine tun- As we shall see, most of the astrophysically important ing of parameters?” Many of the parameters that go dissipation comes from lower densities (2–3 times nu- into building the equation of state have significant un- clear density) where this deficiency of the model is not certainties, and those that go into computing reaction important, i.e. low-energy effective field theory is good rates and bulk viscosities are even more uncertain. We enough. The framework of relativistic mean field theory investigate the viability of persistent gravitational wave has some advantages over others, such as an inherently emission with respect to variation of several microphysi- causal equation of state even at high densities, but has 4 some disadvantages including the neglect of correlations average mass of a nucleon and is used to make b and c between particles which should become more important dimensionless. The constraints of charge neutrality and at high densities. conservation of baryon number (if a particular baryon In this framework, the strong interaction between number density n is assumed) provide two more equa- baryons is modeled as a tree-level exchange of isoscalar tions which can be written, respectively, as mesons σ, vector mesons ω, and isovector mesons ρ. The effective Lagrangian includes kinetic and tree-level inter- niqi = 0, (5) action terms for the baryons, leptons, and mesons, as well i X as an effective potential for the σ expanded up to O(σ4). nB = n. (6) The variables of the theory are the Fermi momenta ki B of the baryons and leptons, supplemented by the field X strengths of the σ, ω, and ρ mesons. Expansion coeffi- Here ni denotes the number density of fermion species cients and coupling constants can be fit to numbers ex- i and qi is its electric charge in units of e. Now we are tracted from measurements of nuclei and hypernuclei at down to N unknowns, which can be eliminated with the saturation density. The fitting process (and Lagrangian) N equations of generalized β-equilibrium is described in detail in Ref. [38]. We use an updated set of fit parameters from Ref. [39]. µi = biµn − qiµe. (7) For N species of baryons, two of leptons, and three (The neutrinos are assumed to have zero chemical poten- of mesons, the composition of the matter consists of tial.) Here µi is the chemical potential of fermion species N + 5 unknowns (the baryon, lepton and meson fields) i and bi is its baryon charge. (Actually there are N +2 which must be determined by N + 5 equations. The first equations of β-equilibrium including the leptons, but the three equations are the Euler-Lagrange equations for the equations for the electron and neutron are identities and mesons determined by varying the effective Lagrangian thus do not count toward the elimination of unknowns.) of Ref. [38]. The equations are rendered tractable by as- The chemical potential of each baryon species is given by suming that the fields are given by their mean values in a uniform static ground state, and take the form 2 ∗2 µB = gωBω0 + gρBρ03I3B + kB + mB , (8) gωB q ω0 = 2 nB, (1) while that of each lepton species is given by mω XB gρB 2 2 ρ03 = 2 I3BnB, (2) µL = kL + mL. (9) mρ B q X 2 3 m2 σ = g ρ − bm g (g σ) − cg (g σ) .(3) Here kL and mL are, respectively, the Fermi momentum σ σB s,B N σ σ σ σ and mass of the Lth leptonic species. B X For a given baryon number density n, the baryon, me- Here the index B labels baryon species, ω0 is the timelike son, and lepton fields are found by simultaneously solv- component of the vector meson field (the spatial compo- ing Eqs. (1)–(7). The numerical technique is based on a nents vanish), nB is the number density of baryon B, and multi-dimensional root finder and thus requires an initial ρ03 is the isospin 3-component of the timelike component guess to ensure finding the correct root. We first solve for of the isovector meson field. Throughout this Section we a low value n below saturation density where the system use units such that ¯h = c = 1. The scalar density is given can be approximated as a weakly interacting Fermi gas by and analytic approximations can be found as in Ref. [38]. The code then steps up in baryon density n, at each step kB ⋆2 2JB +1 m (σ) using the previous step’s values for the unknowns as the ρ = B k2dk. (4) s,B 2π2 2 ⋆2 initial guess for the root finder. Once the fields are found 0 k + mB (σ) XB Z for a given n, the mass-energy density ǫ and pressure p The coupling constants betweenp the baryon B and the can be determined: mesons are given by gσB, gωB, and gρB. We as- 1 3 1 4 1 ǫ = bm (g σ) + c (g σ) + m2 σ2 sume, as in Ref. [38], that they are given by one 3 N σ 4 σ 2 σ set of values (gσ,gω,gρ) for the nucleons and another 1 2 2 1 2 2 (x g , x g , x g ) for the hyperons, so that the x’s mea- + m ω + m ρ σ σ ω ω ρ ρ 2 ω 0 2 ρ 03 sure relative coupling strengths for hyperons. The cou- kB pling constants for the self-interaction terms of the scalar 1 2 ∗2 2 + 2 k + mB k dk field are b and c. The masses of the mesons are given by π 0 XB Z q mσ, mω, and mρ. The quantities for the Bth baryon kL 1 2 I3 and J are, respectively, its 3-component of isospin, 2 2 B B + 2 k + mLk dk, (10) and its spin. The Fermi momentum of the baryon π 0 XL Z q species is given by kB while its effective mass is given 1 3 1 4 1 2 2 ∗ p = − bmN (gσσ) − c (gσσ) − m σ by mB = mB − gσBσ. The quantity mN represents the 3 4 2 σ 5

values from the 2002 Particle Data Group review [45]). TABLE I: Parameters for five relativistic mean field equations As a result, the newer equations of state we use have of state used in this paper. higher hyperon populations and thus higher viscosities. ∗ Name K (MeV) m /mN xσ H3 300 0.70 0.60 H4 300 0.70 0.72 B. Relaxation timescale H5 300 0.80 0.66 H6 240 0.70 0.67 The reactions that contribute most to bulk viscosity H7 240 0.80 0.68 are the weak interaction processes, as their relaxation timescales are within a few orders of magnitude of the r-mode period (milliseconds). Among these, the most significant that can be calcu- 1 2 2 1 2 2 + mωω0 + mρρ03 lated from first principles are the non-leptonic weak in- 2 2 − teractions involving the lightest hyperons, Σ and Λ, as 1 kB k4dk + they occur at lower densities than the more massive par- 2 2 ∗2 π 0 k + m ticles and thus have higher populations in a given star. B Z B X Following LO, we calculate matrix elements for the reac- 1 kL p k4dk + . (11) tions 2 2 2 π 0 k + m L Z L X n + n ↔ p +Σ− (12) p Here the summations are over the baryon species (B) n + p ↔ p + Λ (13) and the lepton species (L). These expressions can be combined to produce tabulations of p(ǫ), the equation as tree-level Feynman diagrams involving the exchange of of state. a W boson. The latter reaction was also considered (with There remains the problem of choosing the values of a phenomenological free parameters) by HLY. These re- 2 2 2 the constants (gσ/mσ) , (gω/mω) , (gρ/mρ) , b, c, xσ, actions are combined to calculate an overall microscopic xω, and xρ. Each choice of constants produces a dif- relaxation timescale. (This is the timescale on which a ferent equation of state with different hyperon thresh- small perturbation of the neutron fraction returns to its old densities, neutron star maximum mass, etc. We use equilibrium value.) five equations of state from [39] chosen to be compatible In addition to these reactions, Jones [18] includes with recent measurements of neutron star masses [41, 42] and a gravitational redshift [43] as well as hypernuclear n + n ↔ n + Λ (14) data [38]. The constants in Eqs. (1)–(7) are fit in the manner of Glendenning [38] to a Λ binding of -28 MeV, which is the dominant nonleptonic process observed in Λ −3 saturation density 0.153 fm , binding energy per nu- hypernuclei in the laboratory. We do not consider this cleon -16.3 MeV at saturation, and isospin asymmetry reaction as it has no contribution based on a W-boson coefficient 32.5 MeV, as well as a range of incompressibil- exchange, even though the bare-mass interaction rate for ∗ ities K, nucleon effective masses m , and hyperon cou- this process is known. Using Eq. (15) from Jones [18] plings xσ. Since it is easiest to qualitatively understand we estimate that reaction (14) has a relaxation timescale the equations of state in terms of the latter three parame- that is longer than the timescale of the Λ reaction (13) ∗ ters, we give them in Table I. Low K and high m lead to by a factor of about 2 and thus does not change the ∗ a soft equation of state, while high K and low m lead to final viscosity much. (When accounting for the macro- a stiff equation of state. Since most of the matter in the scopic structure of the star, the Σ− process dominates the star is at high density, the hard core repulsion dominates overall bulk viscosity since the Σ− population extends to the strong interaction. Thus high xσ postpones hyperon lower densities and therefore a greater volume fraction of formation to higher densities, reducing hyperon popula- the star than the Λ population.) There are various other tions (and thus viscosity) and stiffening the equation of concurrently occuring processes which contribute to the state. net reaction rate but their rates are not easy to predict. All of these equations of state are somewhat different Even these rates are subject to substantial uncertain- from that used by LO and HLY. Though both HLY and ties: Van Dalen and Dieperink [21] use a meson-exchange LO use the “case 2” equation of state from an older paper model for all three hyperon reactions and find reaction by Glendenning [44], there were misprints in that paper rates of order 10–100 times greater than LO and thus which were corrected by HLY but not by LO. The scalar bulk viscosity coefficients 10–100 times lower than LO. self interaction coupling constants b and c in Ref. [44] (As we shall see in the next section, such a change has should be 3 and 4 times larger, respectively, than the less of an effect on the r-mode critical frequency than one values that are quoted for them. Also, case 2 of Ref. [44] might think.) Our result for the net rate is then a lower used K = 285 MeV and xσ = xω = xρ = 2/3, and limit, and consequently an upper limit on the bulk vis- had several minor differences in other constants (we use cosity, in the typical range of temperatures for LMXBs. p 6

To summarize the results of LO, the relaxation when both Λ and Σ− are present. Here kT is Boltz- timescale can be computed as a function of the equi- mann’s constant times the temperature. The matrix ele- librium matter fields. Neglecting superfluidity, the result ments M are computed as tree-level Feynman diagrams, is squared, summed over initial spinors, and averaged over 2 the angular part of the collision integral (indicated by 1 (kT ) 2 δµ hi). (Note that van Dalen and Dieperink [21] do not an- = 3 kΣh|MΣ|i (15) τ 192π n xn gle average.) They are functions of the Fermi momenta and meson fields given by Eqs. (4.28) and (4.29) of LO, when the Λ hyperons have not yet appeared and

2 1 (kT ) 2 2 δµ = 3 kΣh|MΣ|i + kΛh|MΛ|i (16) τ 192π n xn

2 2 2 GF sin 2θC 2 2 ∗ ∗2 ∗ 2 2 ∗ ∗ 2 h|MΛ| i = 120 1 − g 1 − g m m m − 20 1 − g 1+ g m m 3µ µΛ − k 15 np pΛ n p Λ np pΛ n p p Λ 2 2 ∗ ∗ 2 2 2 2 −10 1+ gnp 1 − gpΛ mpm Λ 6µnµp − 3kn + kΛ +2 1+ gnp 1+ gpΛ +4gnpgpΛ 2 2 2 2 2 2 2 2 × 5µpµΛ 6µn µp +3kn − kΛ + kΛ 10µnµp +5kn+ 10kp − kΛ + 1+ gnp 1+ gpΛ − 4gnpgpΛ ∗2 2 2 2 2 2 2 2 2 2 2 2 × 10µnµΛ 6mp +3kn + kΛ + kΛ −20µp + 15kp − 3kΛ + 5(kn − kp) /(kp − kΛ) , (17) 2 2 2 2 2 2 ∗2 ∗ ∗ 2 2 ∗ ∗ 2 h|MΣ| i = G sin 2θ 180 1 − g 1 − g m m m − 40 1 − g 1+ g m m 3µ µΣ − k − 20 15 F C np nΣ n p Σ np nΣ n p n Σ 2 2 ∗ ∗ 2 2 2 2 ∗ ∗ 2 2 2 2 × 1+ gnp 1 − gnΣ mnmΣ 6µnµp −3kp + kΣ − 5 1 − gnp 1 − gnΣ mpmΣ 6µ n +6kn − 3kp − kΣ 2 2 2 2 2 2 2 2 2 2 +4 1+ gnp 1+ gnΣ +4gnp gnΣ 10µn 3µpµΣ + kΣ +5µpµ Σ 6kn − 3kp − kΣ + kΣ 10kn +5kp 2 2 2 ∗2 2 2 2 −kΣ + 1+ gnp 1+ gnΣ − 4gnpgnΣ −10mn 3µpµΣ + kΣ + 10µn 6µnµpµΣ − 2µpkΣ − 3µΣkp 2 2 2 4 +µΣkΣ + 15kpkΣ− 3kΣ . (18)

We reproduce them here since the latter equation in LO is of the leading-order terms and produces a running value missing some terms (although the code used to generate of the term which can increase by more than an order of the LO results is not missing them). Also, LO actually magnitude above 0.1. used kinetic energies instead of chemical potentials which should be used to be consistent with the quasiparticle picture of Fermi liquid theory. Correcting this reduces C. Bulk viscosity the relaxation time by a factor of 10 or more, making up much of the difference with the results of van Dalen We use for ζ, the macroscopic coefficient of bulk vis- and Dieperink [21] who use a drastically different model cosity, the special relativistic expression derived by Lind- of the interaction. In the matrix elements, GF is the blom and Owen [19]: Fermi constant and θC is the Cabibbo angle. The g’s are axial-vector coupling constants, which we take to have p(γ∞ − γ0)τ their values measured from vacuum β-decay of particles ζ = 2 . (20) at rest [45]. The remaining factor on the far right of 1+(ˆωτ) Eqs. (15) and (16) is explained in the next subsection. where p is the pressure,ω ˆ is the angular frequency of the HLY, in contrast, used the nonrelativistic limit r-mode in a frame corotating with the star, and τ is the 2 2 2 2 net microscopic relaxation time we just computed. The h|MΣ| i =4GF sin 2θC (1+3gnpgnΣ) (19) fast adiabatic index γ∞ (for infinite-frequency perturba- of Eq. (18) and left the term in parentheses as a phe- tions) can be written as nomenological parameter, which they simply set to 0.1. ni ∂p This is somewhat justified because the Particle Data γ∞ = , (21) n ∂n group values for the couplings gnp and gnΣ result in a i i near-perfect cancellation of the term. Any in-medium X 3 2 change of these values would have a disproportionately where ni = ki /(3π ) is the number density of species large effect as a result. We (and LO) find that including i and the partial derivatives can be evaluated explicitly the full Eq. (18) erases the effect of the near-cancellation from Eq. (11). The slow adiabatic index γ0 (for zero- 7 frequency perturbations) can be written as D. Superfluidity

n dp 10 γ0 = , (22) In the temperature range of interest (below 10 K), p dn nucleons and hyperons are expected to form Cooper pairs near their Fermi surfaces and act as superfluids. This and is straightforward to evaluate for example by dif- greatly slows reaction rates and has an important effect ferentiating the equation of state. However, while LO on transport coefficients such as bulk viscosity. derived the correct relativistic expression and noted that The energy associated with the Cooper pairing is 1 it reduces the factor γ∞ − γ0 by a factor of 2–2.5 from given by the bandgap. Assuming S0 pairing, the zero- the nonrelativistic expression, their Fig. 2 and code for temperature bandgap ∆0 is related to the superfluid crit- the rest of the paper incorrectly used a data file with the ical temperature TC by [46] nonrelativistic expression and thus are a factor 2–2.5 too high. kTC =0.57∆0. (29) Now we address the factor δµ/δxn in Eqs. (15) and (16) We use the LO fit to the finite-temperature bandgap in the same manner as LO. This factor is determined 3 4 0.53 by the constraints of charge and baryon conservation (5) T . and (6), plus the constraint that the reaction ∆(T ) = ∆0 1 − . (30) T " C # + − n +Λ ↔ p +Σ (23) For the Λ hyperons we use the empirical fit made by LO to the zero-temperature gap function ∆0Λ of the Λ is in equilibrium since its reaction rate is many orders hyperon as computed by Balberg and Barnea [47]. The of magnitude greater than the weak interaction rates. calculation by Balberg and Barnea [47] is constrained by This implies that both the non-leptonic reactions have experiments on double Λ hypernuclei, though like other the same chemical potential imbalance: calculations of ∆Λ it is likely only good to within a factor of 2 or 3. The zero-temperature gap depends on the total δµ ≡ δµn − δµΛ =2δµn − δµp − δµΣ. (24) baryon number density n and on the Fermi momentum kΛ in a way that LO found was well fit by Assuming small perturbations and using the neutron 3 3 fraction xn, this yields the relation ∆0Λ (kΛ,n) = 5.1kΛ (1.52 − kΛ) 2 × 0.77+0.043 (6.2n − 0.88) . (31) δµ (βn − βΛ)(αnp − αΛp + αnΣ − αΛΣ) = αnn + nBδxn 2βΛ − βp − βΣ h −1 −3 i Here ∆0Λ is in MeV, kΛ in fm , and n in fm . Thus the (2βn − βp − βΣ)(αnΛ − αΛΛ) critical temperature for Λ superfluidity peaks somewhat −αΛn − , (25) 10 2βΛ − βp − βΣ below 10 K. The Σ− superfluid bandgaps are not as well known as where βi is given by ∆Λ due to the absence of similar experiments on hypernuclei containing the Σ− hyperon. Determination of the − βi = αni + αΛi − αpi − αΣi. (26) hypernuclei energy levels for the Σ hyperon are constrained by the very short decay time of the hyperon. and αi is given by Takatsuka et al [48] have however calculated, using several models of the nuclear interaction, that the bandgap ∂µ i lies in the range ∆Λ ≤ ∆ − ≤ 10∆Λ. To account for αij = . (27) Σ ∂nj = the uncertainty in the bandgap we perform our calcula- nk ,k6 j tion for two cases as in LO: when ∆Σ− = ∆Λ and when − These expressions apply for densities where the Λ and Σ ∆Σ− = 10∆Λ. hyperons are both present. In the regime where only the For the superfluidity of protons we use model 2p of Σ− hyperon is present the strong interaction constraint Ref. [49], which is one of the bandgap models used by is no longer a factor and one obtains a simpler form HLY. (LO did not model proton superfluidity, since it was not relevant at the temperatures greater than 1010 K 2δµ that they considered.) This bandgap model is not based = 4αnn − 2(αpn + αΣn + αnp + αnΣ) nBδxn on a specific calculation of the bandgap but preserves the +αpp + αΣp + αpΣ + αΣΣ. (28) general features of bandgaps predicted by various microscopic theories. In this model the zero temperature gap LO used numerical differencing to calculate the deriva- depends only on the Fermi momentum kp of the proton. tives αij , but due to a coding error the σ meson field The relation for TC is given by was not differenced properly. The maximum error this 2 2 kp (kp − k2) induces in the value of δµ/nBδxn is less than 0.3%, well TC = T0 2 2 2 2 , (32) k + k (k − k2) + k below the uncertainties of the problem. " p 1 # p 3 8

−1 −1 where k1 = 1.117 fm , k2 = 1.329 fm , k3 = −1 9 1 0.1179 fm and T0 = 17×10 K. The Fermi momentum −1 kp is in fm units and TC is non-zero only if kp < k1. -2 10 This leads to TC for the protons peaking somewhat below 10 8

10 K, similar to the Λ critical temperature. ( s ) -4 T=10 K

τ 10 8.5 Since neutrons are, after all, more abundant than other 10 K 9 particles in neutron stars, they have higher energies and -6 10 K 1 10 9.5 at supernuclear densities are unlikely to form S0 pairs. 10 K 3 10 They can have a P2 pairing as considered in Ref. [49], -8 10 K but it is much weaker and corresponds to a maximum 10 8 critical temperature TC ≃ 3×10 K. Below this tempera- 0.4 0.8 1.2 1.6 2.0 ture the contribution to the critical frequency curve from 15 3 hyperon bulk viscosity is quite low, most of the contri- Density (10 g/cm ) bution coming from viscosity at the crust-core interface. Therefore neutron superfluidity does not much affect the FIG. 2: Microscopic relaxation timescale as a function of density for various temperatures. The equation of state is H3 viability of persistent gravitational wave emission from − and the Σ and Λ bandgaps are assumed equal. Thus at the the r-modes, and we neglect it here. highest temperature plotted, no superfluid effects are present. The effect of superfluidity on the microscopic relaxation timescale (15) or (16) appears, after doing the 34 lengthy collision integrals (see HLY and LO), as mul- 10 tiplicative factors in the average matrix elements: 32 8 2 2 10 T=10 K h|MΣ| i → RnnpΣh|MΣ| i, (33) 8.5 2 2 10 K h|MΛ| i → RnppΛh|MΛ| i. (34) 30 9 10 10 K If none of the participating particle species is superfluid, 9.5 (g/cm s) 10 K an R factor is 1. Each R factor is reduced as more ζ 28 10 species become superfluid. Our treatment of these fac- 10 10 K tors is based on that used by HLY, who calculate R for 26 the cases when either one or two of the particles partic- 10 ipating in the reaction are superfluid. HLY do not how- 0.4 0.8 1.2 1.6 2.0 15 3 ever provide a 3-particle R which is required for calculat- Density (10 g/cm ) ing the timescale of the Λ reaction (13). Our numerical investigations found (III C) that expressing RnnpΣ as a FIG. 3: Bulk viscosity coefficient as a function of density for product of two single-particle factors RpRΣ− instead of various temperatures, for the same model as Fig. 2. The star HLY’s two-particle factor does not affect the r-mode crit- is rotating with angular frequency 2300 radians per second. ical frequency much. Therefore throughout this paper we express all R’s as products of the appropriate single- particle factors. The expression for the single-particle R consistent with the quasiparticle picture, and most of the from Eq. (30) of HLY is rest comes from changes to the equation of state (including accounting for the typos in Ref. [44] which propagated a5/4 + b1/2 R = exp 0.5068 − 0.50682 + y2 , (35) to LO). Fig. 3 is reduced from LO by a further factor of 2 2 due to the relativistic factor γ∞ − γ0. Including super- p 2 fluidity, the shapes of the curves are different from LO where y = ∆(T )/kT , a = 1+0.3118y and b = 1+ 2 − but the peak timescales are comparable. It is difficult 2.566y . LO used a simple factor e y for the hyperons to compare to HLY because at these temperatures the only, since they were focusing on the ordinary fluid case. high-frequency approximation they use (ˆωτ ≫ 1) does The exponential is indeed the dominant behavior of the not hold. HLY expression for y ≫ 1, but the HLY expression is significantly more accurate for y less than about 5–10, which turns out to be the astrophysically interesting region. We plot the most important results of our micro- III. MACROPHYSICS physics calculations. Figure 2 shows a typical relaxation timescale as a function of density, and Fig. 3 shows the corresponding bulk viscosity coefficient. Among the With the bulk viscosity coefficient ζ in hand, we can equations of state we use, H3 has a relaxation timescale in proceed to compute r-mode damping times and critical the middle of the distribution. Neglecting superfluidity, frequencies. For this we also need the hydrodynamic the timescales are about a factor of 40 shorter than in LO. structure of the r-mode as well as equilibrium models of About 20 of this comes from making the matrix elements the structure of a neutron star (i.e. its density profile). 9

A. Stellar structure and the boundary conditions that at the center of the star ω¯(0) =ω ¯i, an arbitrary constant, and (dω¯(r)/dr)r=0 = 0. We are motivated by the sensitivity of the hyperon From Eq. (40) one determines Ω and the angular momen- population to the central density to include the effects tum J corresponding toω ¯i as follows: of general relativity and rotation in our stellar models. 1 dω¯(r) Relativistic stellar models have higher densities than non- J = R4 , (42) 6 dr relativistic ones, and rotating models have lower densi- r=R ties than nonrotating ones. LO used relativistic, non- 2J Ω=ω ¯(R)+ . (43) rotating stellar models; while other work such as HLY R3 and Ref. [21] did not treat the stellar or mode structure in much detail. Since we are interested in exploring a Here R is the radius of the star. To obtain the desired broader parameter space, we do not restrict ourselves to value of Ω one can scaleω ¯ thus: nonrotating stars. ω¯(r) =ω ¯(r) (Ω /Ω ). (44) We take into account the effect of rotation on the neu- new old new old tron star structure using Hartle’s slow-rotation approx- The angular momentum J corresponding to Ωnew can imation [50]. Hartle’s formalism is based on treating a then be calculated using Eq. (42). rotating star as a perturbation on a non-rotating star. A measure of the change in the star structure caused We start by solving the Oppenheimer-Volkoff equations by rotation is ξ(r, θ) which represents the displacement for a static spherically symmetric star in a form due to of constant energy density surfaces between the rotating Lindblom [51] star and the corresponding non-rotating star. This dis- dm 4πǫ(h)r(h)3[r(h) − 2m(h)] placement’s angular dependence can be separated out: = − , (36) dh m(h)+4πr(h)3p(h) ξ(r, θ)= ξ0(r)+ ξ2(r)P2. (45) dr r(h)[r(h) − 2m(h)] = − , (37) dh m(h)+4πr(h)3p(h) The l = 0 part of this deformation, ξ0, can be obtained by solving the following equations: using as independent variable the specific enthalpy p ′ ′ ′ dm0(r) 2 dǫ(r) ∗ h(p)= dp /[p + ǫ(p )]. (38) = + 4πr [ǫ(r)+ p(r)] p0(r) 0 dr dp(r) Z 2 From r(h), the radius at which the enthalpy is h, and 1 2 4 ω¯(r) + j (r)r m(h), the mass contained within a sphere of that radius, 12 dr we find m(r). We also find ν(r), the logarithm of the 1 dj2(r) time-time metric component which is based on h(r). We − r3 ω¯2(r), (46) 3 dr then use this solution of the static star to solve for a ∗ 2 2 rotating star with the same central pressure. dp0(r) m0 1+8πr p(r) = − The metric of a slowly rotating star to second order in dr r − 2M(r) Ω, the spin frequency, can be written as 2 4π [ǫ(r)+ p(r)] r ∗ 2 ν 2 − p0(r) ds = − e [1 + 2(h0 + h2P2)] dt r − 2M(r) −1 4 2 2 2(m0 + m2P2) 2M(r) r j (r) dω¯(r) + 1+ 1 − dr2 + (r − 2M(r)) r 12 [r − 2M(r)] dr 2 2 2 2 3 2 2 + r [1 + 2(v2 − h2)P2] dθ + sin θ(dφ − ωdt) 1 d r j (r)¯ω (r) + , (47) 3 3 dr r − 2M(r) + O(Ω ). (39) ∗ Here P2 = P2(cos θ) is the 2nd order Lagrangian poly- where p0 is a pressure perturbation, the density profile nomial. The quantity ω is the frame dragging frequency ǫ(r) is the energy density of the non-rotating star as func- that is proportional to Ω, while h0, h2, m0, m2, and ν2 tion of r, and p(r) is the pressure of the star as a function are all functions of r that are proportional to Ω2. Before of r. The boundary conditions for the above equations ∗ we solve the Einstein equations for all the subscripted are that both m0 and p0 vanish at the origin. quantities in Eq. (39) we need to solve forω ¯ = Ω − ω. The gravitational mass of the rotating star is given by This is done by solving 2 ′ J 1 d dω¯(r) 4 dj(r) M (R)= M(R)+ m0(R)+ , (48) r4j(r) + ω¯(r)=0, (40) R3 r4 dr dr r dr which is greater than the mass of the nonrotating star with the definition with the same central pressure. We wish to build se- 1/2 2M(r) quences of stars rotating at various rates all with the j(r) = exp−ν/2 1 − , (41) r same gravitational mass. To do this we first solve the 10

Oppenheimer-Volkoff equations; then, solve Eq. (40), Here α is a dimensionless amplitude coefficient which (46), and (47); then calculate the new mass of the ro- cancels out of the driving and damping timescales. tating star using Eq. (48). This procedure is repeated The viscous damping time scale τV is given by with lower values of the central pressure until the same mass as the nonrotating star is obtained. 1 1 dE˜ To perform the viscosity calculations we need informa- = − , (54) τV 2E˜ dt ! tion on the structure of the rotating star contained in its V density profile. Once the correct mass is obtained the where (dE/dt˜ ) is the rate at which energy is being density profile ǫ(r) is calculated by using V drained from the mode by viscosity. Here we consider dǫstat(r) only hyperon bulk viscosity from the nonleptonic pro- ǫ (r)= ǫ (r) − ξ0(r), (49) rot stat dr cesses discussed in the previous section. Leptonic (Urca) where processes also contribute, but in LO they were shown to be overwhelmed by the viscosity due to nonleptonic pro- ∗ ǫ(r)+ p(r) ξ0(r)= −p0(r) . (50) cesses. Although the crust-core viscosity is important at dp(r)/dr low temperatures—it determines the shape of the nega- Here, ǫrot(r) is the density profile of the rotating star, tively sloped part of the instability curve which in turn while ǫstat(r) is the density profile of the non-rotating puts a constraint on how much farther along the tempera- one. This yields only the spherically symmetric part of ture axis the rising part of the instability curve should be the density perturbation, which is the dominant one for located at to prevent a thermal runaway—we neglect it. our purposes. For a reasonable estimate of the shape of the crust-core viscosity curve and of the value of the saturation amplitude the stars temperatures hould increase to around B. Driving and damping timescales 109 K [31]. To explore the effect of the crust-core viscosity and the saturation amplitude in more details is The stability of an r-mode is determined by calculat- beyond the scope of this paper. We can then compute ing the damping and driving timescales. The mode is the viscous damping timescale as in LO: To lowest order unstable if the driving timescale is shorter than the vis- in Ω, we can write cous damping timescale. This can be expressed in terms ˜ R of an overall r-mode timescale τ such that dE 2 2 r = −4π ζ(ǫ(r))h|∇~ · δ~v| ir dr, (55) 1 1 1 dt ! 0 = + , (51) V Z τr(Ω,T ) τGR(Ω) τV (Ω,T ) where h|∇~ · δ~v|2i is the angle average of the square of the where τ < 0 is the damping timescale of the mode GR hydrodynamic expansion. LO found that the expansion due to gravitational radiation and τ > 0 is the damp- V found numerically in Ref. [52] could be fit well by ing timescale resulting from the sum of all viscous processes in the star. The star’s spin angular frequency and 2 α2Ω2 r 6 r 2 Ω2 temperature are represented by Ω and T , respectively. h|∇~ · δ~v|2i = 1+0.86 , A mode unstable to gravitational radiation corresponds 690 R R πGǫ¯ to τr < 0 in Eq. (51), and the critical frequency Ωc of (56) the star as a function of temperature is found by solving whereǫ ¯ is the mean density of the (nonrotating) star. 1/τr(Ωc,T ) = 0. [In Eq. (6.6) of LO the last factor is missing due to a The gravitational radiation timescale is given by misprint, but is properly included in the calculations.] This approximation breaks down for r/R ≈ 1, but since 1 1 dE˜ the bulk viscosity is coming from the hyperon core that is = − , (52) τGR 2E˜ dt ! enough. In Table II we give typical radii of nonrotating GR stars and their hyperon cores. where E˜ is the mode energy in a frame rotating with the star and (dE/dt˜ )GR is the rate at which energy is emitted in gravitational waves in the same frame. For the cur- C. Critical frequency curves rent quadrupole r-mode, there exist simple expressions (to lowest order in mode amplitude and angular veloc- We are now in a position to solve for the critical fre- ity of the star) for the energy and gravitational radiation quency as a function of temperature for various neutron timescale: [30] star models. This must be done numerically. R Because of the inclusion of rotation the process is some- 1 E˜ = α2Ω2R−2 ǫ(r)r6dr, what more involved than in LO. The critical frequency 2 Z0 curve is calculated in two stages to ensure that the correct 1 131 072π R gravitational mass is retained after including the correc- = − ǫ(r)r6 dr. (53) τ 164 025 tions to it from rotation. In the first stage we calculate GR Z0 11

14 3 Equation of state H6 is soft and has the highest hyperon EOS ǫc (10 g/cm ) R (km) RΣ (km) RΛ (km) population for a given mass, and thus its curve is higher H3 6.86 13.74 6.38 0.00 than for H3 which is intermediate. This plot shows the H4 6.12 13.85 4.50 0.00 general trend that the effect of rotation is more significant H5 7.70 13.33 5.37 0.00 for softer equations of state. For persistent gravitational H6 7.84 13.44 6.70 2.46 wave emission the key question is whether or not a neu- H7 9.00 13.01 6.03 0.00 tron star undergoing thermal runaway will be blocked by hitting the curve at a temperature lower than a few times 109 K. Even for the soft H6 equation of state, the TABLE II: Parameters of non-rotating 1.4M⊙ equilibrium effect of rotation on stellar structure changes the temper- stars for each equation of state (EOS). The central mass density is ǫc, R is the stellar radius, RΣ is the distance out to ature at which a star hits the curve by no more than a − which Σ hyperons appear, and RΛ is the corresponding dis- factor of 2 (still within the region that is good for gravi- tance for Λ hyperons. tational wave emission). More significant is the fact that the frequency at which the curve peaks can be considerably reduced from the value neglecting rotation. If the peak is low enough, thermal runaway can still occur— for example in this plot it can occur for H3 stars if they 1000 hit above 700 Hz—and the outlook is bad for persistent gravitational wave emission. In Fig. 5 we show the effect of the equation of state 500 on the critical frequency curve for 1.4 M⊙ stars. The curves generically reach their peak at temperatures of 1– 2×109 K, which is good for persistent gravitational wave Spin Frequency (Hz) emission. However, the heights (frequencies) of the peaks 0 8 9 10 change significantly. The peak is highest for H6, which 10 10 10 has the largest hyperon population. It is almost iden- Temperature (K) tical for H3 and H7, although those two equations of ∗ state are the furthest separated in the (K,m , xσ ) pa- FIG. 4: Critical frequency curves 1.4M⊙ neutron stars with − rameter space. This seems to be due to the similar radii equal bandgaps for Σ and Λ hyperons. The dashed curve is for equation of state H3 and the solid curve is for H6. The of their hyperon-containing cores as in Table II. All three thinner curves neglect the effect of rotation on stellar struc- of these equations of state then allow persistent gravita- ture while the thicker curves include it. The curves including tional wave emission in 1.4 M⊙ stars. The peak is lowest rotation are extended to their peaks, but no further. for H4, which is not surprising since that is the stiffest equation of state and has the lowest hyperon population. For H4 the peak is so low that, depending on the details the critical frequency as a function of temperature for a of the crust-related damping, the full curve including that nonrotating star using the Oppenheimer-Volkoff formal- damping might show no significant effect from hyperon ism. Then a set of points on this curve is taken. In the bulk viscosity—it might decrease throughout the tem- second stage, for each point we find the structure (density perature range and never allow persistent emission. In profile) of a star rotating at that frequency with the same any case, for H4 and H5 persistent emission is not pos- gravitational mass as the nonrotating star. Since Har- sible for the most rapidly rotating stars in LMXBs (up tle’s formalism produces stars with greater gravitational to 619 Hz) if they are 1.4 M⊙. This demonstrates in masses, this step requires several iterative trials of the principle how observation of r-mode gravitational waves central pressure to find the star with the same gravita- from an LMXB could rule out some equations of state tional mass. Using the new density profile with rotation, or, more broadly, constrain hyperon populations. we keep the frequency fixed and use the corresponding In Fig. 6 we plot critical frequency curves for stars with temperature on the nonrotating curve as an initial guess different masses for the H4 equation of state, which is the for a routine that finds the root of 1/τr, thereby obtaining stiffest of those we use. (For other equations of state, the the correct temperature for the marginally star rotating stellar mass affects the curve more but is constrained to at that frequency. Because of this it is cumbersome to a narrower range, so the overall variation is greatest for trace out parts of the curves where the critical frequency H4.) We also vary the Σ− bandgap from the Λ value decreases with temperature, and since those parts of the to 10 times that value. It can be seen that increasing bulk viscosity curves are not important for determining the mass pushes the curve to lower temperatures and the viability of persistent gravitational wave emission we raises its peak. This is due to the fact that higher mass neglect them. stars have higher central densities and hence, in abso- The importance of rotation is shown in the critical fre- lute terms, larger hyperon populations than their lower quency curves plotted in Fig. 4. (Kepler frequencies for mass counterparts. Persistent r-mode gravitational wave these equations of state are 750–850 Hz for 1.4 M⊙ stars.) emission is viable in all but the 1.4 M⊙ neutron stars. 12

750 750

H3 500 500 H5 H6

250 H7 H4 250 Spin Frequency (Hz) Spin Frequency (Hz)

0 0 8 8.5 9 9.5 8.6 9 9.4 10 10 10 10 10 10 10 Temperature (K) Temperature (K)

FIG. 5: Critical frequency as a function of temperature for FIG. 7: Variations on the critical frequency curve for a − 1.4 M⊙ stars with different equations of state. All curves 1.4 M⊙ H3 star with equal Λ and Σ bandgaps. The solid assume the superfluid bandgaps of the Λ and Σ− hyperons curve is our standard model. The dotted curve uses asymp- are equal. totic values for the axial couplings instead of vacuum at-rest values. The dot-dash curve uses the 2-particle superfluid re- − 750 duction factor for the Σ interaction instead of two 1-particle factors. The dashed curve simulates the fast timescales predicted by van Dalen and Dieperink.

500

and equation of state models. We thereby obtain a tem- 7 8 250 perature range (1.5×10 –4×10 K) which covers LMXBs whose thermal emission spectra have been observed. This result is probably statistically biased, since LMXBs with Spin Frequency (Hz) higher accretion rates are likely to have higher tempera- 0 7 7.5 8 8.5 9 9.5 tures but surface temperatures are harder to observe be- 10 10 10 10 10 10 cause of the accretion discs. At any rate, the most mas- Temperature (K) sive H4 stars have critical frequency curves which would allow gravitational wave emission scenarios [34, 35] to op- FIG. 6: Instability curves for H4 stars with different masses. erate in that temperature range. Less massive stars allow The thicker curves use the larger Σ− bandgap while the thinner curves use the smaller one. The solid lines represent it at higher temperatures, which may be consistent with the more rapidly accreting neutron stars such as Sco X-1 2.0M⊙ stars, the dot-dash lines repsents 1.8M⊙ stars, the dotted line represents 1.6M⊙ stars, and the dashed line rep- whose temperatures are poorly known. resents 1.4M⊙ stars. The effects of varying some additional microphysical parameters are shown in Fig. 7, which depicts curves for 1.4 M⊙ neutron stars with the H3 equation of state and The temperature at which this can happen is strongly the Σ− superfluidity bandgap equal to that of the Λ. dependent on the mass, and to a lesser but still signifi- The variations include using the asymptotic (quark) val- − cant extent (factor 3) on the Σ bandgap, with higher ues for the axial-vector couplings in the reaction rates, bandgaps leading to higher temperatures at a given fre- using HLY’s two-particle superfluid reduction factor in- quency. (The small horizontal sections of the 1.4M⊙ and stead of the product of two one-particle factors, and us- 1.6M⊙ large bandgap curves should actually dip slightly. ing relaxation times an order of magnitude faster from This behavior is missed due to our method of including van Dalen and Dieperink [21]. All of these curves peak rotation, but is unimportant for the thermal runaway and at almost identical frequencies, and the temperature at persistent gravitational wave emission.) a fixed frequency changes by less than a factor of 1.5 The curves for 1.8–2.0 M⊙ stars are particularly in- Therefore, the changes to these microphysical parametrs teresting in light of predicted interior temperatures of do not affect our earlier conclusions. From an astrophys- neutron stars in LMXBs. These can be determined from ical point of view, the most sensitive unknowns in this observations in quiescence of the surface temperatures of complicated calculation are the hyperon population (thus three slowly accreting neutron stars mentioned in Brown, the equation of state and stellar mass) and superfluid Bildsten, and Chang [53]. To calculate the interior tem- bandgaps. At a very simple level this makes sense be- peratures we use Fig. 2 from Yakovlev et al [54], which cause the hyperons appear only after a threshold density depicts the relationship between the core and the mea- and the bandgaps appear in exponential factors rather sured surface temperatures using different atmospheric than power laws. 13

IV. DISCUSSION sue requires further investigation. Stars with high masses and stiff equations of state could exist in thermal and 7 We have extended previous investigations of r-modes torque equilibrium at temperatures down to 10 K. in accreting hyperon stars in LMXBs as persistent One possible avenue for substantial improvement is the sources of gravitational waves, focusing on the bulk vis- hydrodynamics. We used a fluid expansion in the dis- cosity which is needed to prevent thermal runaway. We sipation integrals which is a reasonable approximation have used improved microphysics compared to previous for a Newtonian normal fluid, but not for a superfluid treatments, and have accounted for the most important which can be much more complicated due to multiple macroscopic correction due to rotation of the star. components and entrainment. Work is underway to deal We find that persistent gravitational wave emission is with the superfluid problem in general ([55] and refer- quite robust. Even the stiffest hyperonic equations of ences therein). state in relativistic mean field theory produce enough damping to stop the runaway and persistently radiate, although some of them require neutron stars somewhat Acknowledgments more massive than 1.4 M⊙. Stars below about 1.3 M⊙ are not likely to be persistent sources regardless of the This work was supported by the National Sci- equation of state. The mass thresholds are somewhat ence Foundation under grants PHY-0245649 and PHY- more favorable for lower superfluid bandgaps than for 0114375 (the Penn State Center for Gravitational Wave higher bandgaps. Other details of the microphysics are Physics). We are especially grateful to Lee Lindblom for found to be considerably less important. Our results many helpful discussions, including confirming errors in seem robust for typical values of the crust-core viscosity previous work. We also thank Lars Bildsten, Ian Jones and of saturation amplitude of r-modes, though this is- and Dimitri Yakovlev for helpful discussions.

[1] N. Stergioulas, Living Rev. Rel. 6, 3 (2003), and refer- Astrophys. 381, 1080 (2002), astro-ph/0110575. ences therein, gr-qc/0302034. [21] E. N. E. van Dalen and A. E. L. Dieperink, Phys. Rev. [2] B. J. Owen et al., Phys. Rev. D58, 084020 (1998), gr- C69, 025802 (2004), nucl-th/0311103. qc/9804044. [22] J. Madsen, Phys. Rev. Lett. 81, 3311 (1998), astro- [3] L. Bildsten, Astrophys. J. 501, L89 (1998), astro- ph/9806032. ph/9804325. [23] L. Lindblom, B. J. Owen, and G. Ushomirsky, Phys. Rev. [4] N. Andersson, K. D. Kokkotas, and N. Stergioulas, As- D62, 084030 (2000), astro-ph/0006242. trophys. J. 516, 307 (1999), astro-ph/9806089. [24] Y. Levin and G. Ushomirsky, Mon. Not. Roy. Astron. [5] J. Papaloizou and J. Pringle, Mon. Not. R. Astron. Soc. Soc. 324, 917 (2001), astro-ph/0006028. 184, 501 (1978). [25] Y. Wu, C. D. Matzner, and P. Arras, Astrophys. J. 549, [6] R. V. Wagoner, Astrophys. J. 278, 345 (1984). 1011 (2001), astro-ph/0006123. [7] A. K. Schenk, P. Arras, E. E. Flanagan, S. A. Teukolsky, [26] J. B. Kinney and G. Mendell, Phys. Rev. D67, 024032 and I. Wasserman, Phys. Rev. D65, 024001 (2002), gr- (2003), gr-qc/0206001. qc/0101092. [27] D. Chakrabarty et al., Nature 424, 42 (2003), astro- [8] S. M. Morsink, Astrophys. J. 571, 435 (2002), astro- ph/0307029. ph/0202051. [28] Y. Levin, Astrophys. J. 517, 328 (1999), astro- [9] P. Arras et al., Astrophys. J. 591, 1129 (2003), astro- ph/9810471. ph/0202345. [29] H. C. Spruit, Astron. Astrophys. 341, L1 (1999), astro- [10] J. Brink, S. A. Teukolsky, and I. Wasserman, Phys. Rev. ph/9811007. D70, 121501 (2004), gr-qc/0406085. [30] L. Lindblom, B. J. Owen, and S. M. Morsink, Phys. Rev. [11] J. Brink, S. A. Teukolsky, and I. Wasserman, Phys. Rev. Lett. 80, 4843 (1998), gr-qc/9803053. D70, 124017 (2004), gr-qc/0409048. [31] J. S. Heyl, Astrophys. J. 574, L57 (2002). [12] J. Brink, S. A. Teukolsky, and I. Wasserman, Phys. Rev. [32] C. Cutler and K. S. Thorne (2002), gr-qc/0204090. D71, 064029 (2005), gr-qc/0410072. [33] R. V. Wagoner, J. F. Hennawi, and J.-s. Liu (2001), [13] P. M. Sa and B. Tome, Phys. Rev. D71, 044007 (2005), astro-ph/0107229. gr-qc/0411072. [34] A. Reisenegger and A. A. Bonacic, Phys. Rev. Lett. 91, [14] W. D. Langer and A. G. W. Cameron, Astrophys. Space 201103 (2003), astro-ph/0303375. Sci. 5, 213 (1969). [35] R. V. Wagoner, Astrophys. J. 578, L63 (2002), astro- [15] P. B. Jones, Astrophys. Lett. 5, 33 (1970). ph/0207589. [16] P. B. Jones, Proc. Roy. Soc. (London) A323, 111 (1971). [36] R. V. Wagoner, AIP Conf. Proc. 714, 224 (2004), astro- [17] P. B. Jones, Phys. Rev. Lett. 86, 1384 (2001). ph/0312433. [18] P. B. Jones, Phys. Rev. D64, 084003 (2001). [37] N. Andersson, D. I. Jones, and K. D. Kokkotas, [19] L. Lindblom and B. J. Owen, Phys. Rev. D65, 063006 Mon. Not. Roy. Astron. Soc. 337, 1224 (2002), astro- (2002), astro-ph/0110558. ph/0111582. [20] P. Haensel, K. P. Levenﬁsh, and D. G. Yakovlev, Astron. [38] N. K. Glendenning, Compact Stars: Nuclear Physics, 14

Particle Physics, and General Relativity (Springer, New [47] S. Balberg and N. Barnea, Phys. Rev. C57, 409 (1998), York, 2000), 2nd ed. nucl-th/9709013. [39] B. D. Lackey, M. Nayyar, and B. J. Owen, Phys. Rev. [48] T. Takatsuka, S. Nishizaki, Y. Yamamoto, and R. Tam- D73, 024021 (2006), astro-ph/0507312. agaki, Prog. Theor. Phys. 105, 179 (2001). [40] D. G. Yakovlev and C. J. Pethick, Ann. Rev. Astron. [49] A. D. Kaminker, D. G. Yakovlev, and O. Y. Gnedin, Astrophys. 42, 169 (2004), astro-ph/0402143. Astron. Astrophys. 383, 1076 (2002). [41] D. J. Nice et al., Astrophys. J. 634, 1242 (2005), astro- [50] J. B. Hartle, Astrophys. J. 150, 1005 (1967). ph/0508050. [51] L. Lindblom, Astrophys. J. 398, 569 (1992). [42] S. M. Ransom et al., Science 307, 892 (2005), astro- [52] L. Lindblom, G. Mendell, and B. J. Owen, Phys. Rev. ph/0501230. D60, 064006 (1999), gr-qc/9902052. [43] J. Cottam, F. Paerels, and M. Mendez, Nature 420, 51 [53] E. F. Brown, L. Bildsten, and P. Chang, Astrophys. J. (2002), astro-ph/0211126. 574, 920 (2002), astro-ph/0204102. [44] N. K. Glendenning, Astrophys. J. 293, 470 (1985). [54] D. G. Yakovlev, K. P. Levenﬁsh, A. Y. Potekhin, O. Y. [45] K. Hagiwara et al. (Particle Data Group), Phys. Rev. Gnedin, and G. Chabrier, Astron. Astrophys. 417, 169 D66, 010001 (2002). (2004), astro-ph/0310259. [46] N. W. Ashcroft and N. D. Mermin, Solid State Physics [55] N. Andersson and G. L. Comer (2005), physics/0509241. (Brookes Cole, 1976), 1st ed.