bridging the gap by squeezing superfluid matter

5
Bridging the Gap by Squeezing Superfluid Matter Mark G. Alford, 1 Sanjay Reddy, 2 and Kai Schwenzer 1 1 Department of Physics, Washington University, St. Louis, Missouri 63130, USA 2 Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA (Received 27 October 2011; published 14 March 2012) Cooper pairing between fermions in dense matter leads to the formation of a gap in the fermionic excitation spectrum and typically exponentially suppresses transport properties. However, we show here that reactions involving conversion between different fermion species, such as Urca reactions in nuclear matter, become strongly enhanced and approach their ungapped level when the matter undergoes density oscillations of sufficiently large amplitude. We study both the neutrino emissivity and the bulk viscosity due to direct Urca processes in hadronic, hyperonic, and quark matter and discuss different superfluid and superconducting pairing patterns. DOI: 10.1103/PhysRevLett.108.111102 PACS numbers: 26.60.c Ultradense matter, such as might be found in compact stars, has certain transport properties, such as bulk viscosity and neutrino emissivity, that are dominated by flavor- changing beta (weak interaction) processes. However, the -equilibration rate is exponentially suppressed when the matter cools below the critical temperature T c for super- conductivity or superfluidity, because Cooper pairing pro- duces a gap in the fermion excitation spectrum [1,2]. This suppression is relevant both in hadronic matter, where proton superconductivity and neutron superfluidity have T c ¼ 10 8 10 10 K [3], and in exotic phases such as quark matter, where critical temperatures are as high as T c ð35Þ 10 11 K if flavor antisymmetric pairing channels are favored, and as low as T c 10 7 K for singlet pairing [4]. In this Letter we show that compression oscillations can overcome the exponential suppression of flavor-changing processes if their amplitude is sufficiently high (but within the range for realistic oscillations in neutron stars). It is already known that -equilibration rates in unpaired mat- ter are enhanced by ‘‘suprathermal’’ contributions at high amplitude. These arise from higher order terms in the chemical potential imbalance [57]. Here we report on a similar but more dramatic phenomenon in superfluid or superconducting phases of matter, arising from a thresh- oldlike behavior with a rapid increase in available phase space when the typical energy in the equilibration pro- cesses approaches the gap. The mechanism we discuss is generic and can be ex- pected to operate in all situations where large perturbations drive the system out of equilibrium. One possible appli- cation is to unstable oscillations of rotating compact stars, like f- or r-mode instabilities [8]. The amplitude of these modes grows exponentially due to gravitational wave emission until they are saturated by nonlinear coupling to damped daughter modes [9] or damped by the suprather- mal bulk viscosity [10]. Note, that even at amplitudes well below the level where bulk viscosity alone would saturate such modes, large amplitude effects can noticeably affect the thermal evolution of the star, see also [11], via viscous heating and enhanced neutrino emissivity [6]. Another class of scenarios stems from oscillations caused by sin- gular events, like star quakes [12] or tidal forces preceding neutron star mergers [13]. Such events will inevitably cause large amplitude compressions locally and due to the enormous energies involved they may even trigger sizable global modes, which are only slowly damped by shear viscosity. The suprathermal bulk viscosity should then control the amplitude of the oscillation and corre- spondingly the emitted radiation. Here we will study the effect of large amplitude oscil- lations for various gapped phases of dense matter. When, as in nuclear matter, Cooper pairs contain two particles of the same flavor, density oscillations cannot lead to pair breaking. Nevertheless, oscillations displace the gapped Fermi seas, and at sufficiently large amplitude can bridge the gap, opening phase space for -equilibration pro- cesses. This is illustrated for the direct Urca process in nuclear matter in Fig. 1, but we emphasize that gap bridg- ing is relevant to any reaction where nucleons or quarks change flavor. The mechanism is similar to electron µ µ µ µ µ µ µ Beta equilibrium Compression Gap-bridging compression FIG. 1 (color online). Schematic illustration of the opening of phase space for direct Urca interactions in gapped nuclear matter at T ¼ 0 when the deviation from equilibrium " ¼ " n " p " e is sufficiently large relative to the pairing gaps, " > n þ p . PRL 108, 111102 (2012) PHYSICAL REVIEW LETTERS week ending 16 MARCH 2012 0031-9007= 12=108(11)=111102(5) 111102-1 Ó 2012 American Physical Society

Upload: kai

Post on 09-Apr-2017

216 views

Category:

Documents


4 download

TRANSCRIPT

Page 1: Bridging the Gap by Squeezing Superfluid Matter

Bridging the Gap by Squeezing Superfluid Matter

Mark G. Alford,1 Sanjay Reddy,2 and Kai Schwenzer1

1Department of Physics, Washington University, St. Louis, Missouri 63130, USA2Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA

(Received 27 October 2011; published 14 March 2012)

Cooper pairing between fermions in dense matter leads to the formation of a gap in the fermionic

excitation spectrum and typically exponentially suppresses transport properties. However, we show here

that reactions involving conversion between different fermion species, such as Urca reactions in nuclear

matter, become strongly enhanced and approach their ungapped level when the matter undergoes density

oscillations of sufficiently large amplitude. We study both the neutrino emissivity and the bulk viscosity

due to direct Urca processes in hadronic, hyperonic, and quark matter and discuss different superfluid and

superconducting pairing patterns.

DOI: 10.1103/PhysRevLett.108.111102 PACS numbers: 26.60.�c

Ultradense matter, such as might be found in compactstars, has certain transport properties, such as bulk viscosityand neutrino emissivity, that are dominated by flavor-changing beta (weak interaction) processes. However, the�-equilibration rate is exponentially suppressed when thematter cools below the critical temperature Tc for super-conductivity or superfluidity, because Cooper pairing pro-duces a gap in the fermion excitation spectrum [1,2]. Thissuppression is relevant both in hadronic matter, whereproton superconductivity and neutron superfluidity haveTc ¼ 108–1010 K [3], and in exotic phases such as quarkmatter, where critical temperatures are as high as Tc ’ð3–5Þ � 1011 K if flavor antisymmetric pairing channelsare favored, and as low asTc ’ 107 K for singlet pairing [4].

In this Letter we show that compression oscillations canovercome the exponential suppression of flavor-changingprocesses if their amplitude is sufficiently high (but withinthe range for realistic oscillations in neutron stars). It isalready known that �-equilibration rates in unpaired mat-ter are enhanced by ‘‘suprathermal’’ contributions at highamplitude. These arise from higher order terms in thechemical potential imbalance [5–7]. Here we report on asimilar but more dramatic phenomenon in superfluid orsuperconducting phases of matter, arising from a thresh-oldlike behavior with a rapid increase in available phasespace when the typical energy in the equilibration pro-cesses approaches the gap.

The mechanism we discuss is generic and can be ex-pected to operate in all situations where large perturbationsdrive the system out of � equilibrium. One possible appli-cation is to unstable oscillations of rotating compact stars,like f- or r-mode instabilities [8]. The amplitude of thesemodes grows exponentially due to gravitational waveemission until they are saturated by nonlinear coupling todamped daughter modes [9] or damped by the suprather-mal bulk viscosity [10]. Note, that even at amplitudes wellbelow the level where bulk viscosity alone would saturatesuch modes, large amplitude effects can noticeably affect

the thermal evolution of the star, see also [11], via viscousheating and enhanced neutrino emissivity [6]. Anotherclass of scenarios stems from oscillations caused by sin-gular events, like star quakes [12] or tidal forces precedingneutron star mergers [13]. Such events will inevitablycause large amplitude compressions locally and due tothe enormous energies involved they may even triggersizable global modes, which are only slowly damped byshear viscosity. The suprathermal bulk viscosity shouldthen control the amplitude of the oscillation and corre-spondingly the emitted radiation.Here we will study the effect of large amplitude oscil-

lations for various gapped phases of dense matter. When,as in nuclear matter, Cooper pairs contain two particles ofthe same flavor, density oscillations cannot lead to pairbreaking. Nevertheless, oscillations displace the gappedFermi seas, and at sufficiently large amplitude can bridgethe gap, opening phase space for �-equilibration pro-cesses. This is illustrated for the direct Urca process innuclear matter in Fig. 1, but we emphasize that gap bridg-ing is relevant to any reaction where nucleons or quarkschange flavor. The mechanism is similar to electron

µ∆

µ µµ∆µ∆

µ∆ ∆ ∆ ∆ ∆µ∆

Beta equilibrium Compression Gap−bridging compression

FIG. 1 (color online). Schematic illustration of the opening ofphase space for direct Urca interactions in gapped nuclear matterat T ¼ 0 when the deviation from � equilibrium �� ¼ �n ��p ��e is sufficiently large relative to the pairing gaps,

�� >�n þ �p.

PRL 108, 111102 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 MARCH 2012

0031-9007=12=108(11)=111102(5) 111102-1 � 2012 American Physical Society

Page 2: Bridging the Gap by Squeezing Superfluid Matter

tunneling between different superconductors separated bya large barrier when an electric potential is imposed [14].As we will show, the effect begins at amplitudes far belowthe level where nonlinear hydrodynamic effects, of higherorder in the fluid velocity, would arise [5,7,15] and causes ahuge increase in the corresponding rates, effectively bring-ing them up to the ungapped level.

�-equilibration processes have a finite time scale, so aharmonic local density fluctuation with amplitude �naround the equilibrium value �n induces an oscillatingdisplacement �� of the chemical potentials of the degen-erate particles from their equilibrium. This oscillationleads to both heat generation (via dissipation due to bulkviscosity) and heat loss (from enhanced neutrino emis-sion). Which of the two effects dominates depends amongother factors on the amplitude of the oscillation [6]. In[7,15] we found that over the entire range of physicallyreasonable density amplitudes the chemical potential os-cillation is given by the linear, harmonic relation

��ðtÞ¼ �̂�sinð!tÞ; �̂�¼C�n

�n; C� �n

@��

@n

��������x; (1)

where C characterizes the particular form of matter.Gap bridging is a generic phenomenon based on

straightforward energetics, so, to illustrate the large en-hancements that it produces, we will perform calculationsfor the simple case of direct Urca processes d ! uþ lþ��l and uþ l ! dþ �l. In reality, direct Urca occurs onlyin very dense or exotic forms of matter; elsewhere,modified-Urca processes occur instead. We defer themore complex modified-Urca calculation to future work.However, the energetics that give rise to gap bridging arethe same in both cases, so it seems inevitable that gap-bridging effects will be as dramatic for modified-Urcaprocesses (see, for example, [1,6]). To study all relevantcases within a unified framework we use a notation whered represents a down or strange quark or a hadron thatcontains it, u represents an up quark or a hadron thatcontains it; l represents a charged lepton (an electron, orin dense neutron matter a muon) and �l is the correspond-ing neutrino. We specialize to Fermi liquid theory, wherethe general dispersion relation of the matter fields is

ðEi ��iÞ2 ¼ v2Fiðpi � pFiÞ2 þ �2

i ; (2)

where i labels the species and �i represents a gap in theparticle spectrum arising from superfluidity or (color)-superconductivity. Non-Fermi liquid effects should notplay a role in a gapped system [16]. We neglect depen-dence of the gap on energy and momentum, but we includetemperature and density dependence �i ¼ �iðT;�Þ. Theleptons can to a very good approximation be described by afree dispersion relation E2

l ¼ p2l þm2

l , E� ¼ p�. Urca

processes enforce � equilibrium, �� � �d ��u ��l ¼0. Since in compact stars T � � we can restrict theanalysis to leading order in T=� where we find the generalresults for the net rate and the emissivity

�ð$ÞdU � 17G2

120�D�T4��R�

���

T;�d

T;�u

T

�; (3)

�dU � 457�G2

5040D�T6R"

���

T;�d

T;�u

T

�: (4)

In these expressionsG is the effective coupling of the fieldsto the weak current, D is a function depending on thedensity and the equation of state

D ¼ pFdpFu

�uvFdvFu

ð�2d � p2

Fd ��2u þ p2

Fu �m2l Þ (5)

in which the parenthesis vanishes for a free masslessdispersion relation. Any modification thereof due to inter-actions or masses opens phase space and yields a nonzeroresult. Equations (3) and (4) also contain a thresholdfunction � � �ðpFu þ pFl � pFdÞ, a characteristic tem-perature and amplitude dependence, and a modificationfunction R given for singlet gaps by the dimensionlessintegrals

Ri

���

T;�d

T;�u

T

�¼ Ni

Z 1

0dx�x

2þ�i�

�Z ��d=T

�1þ

Z 1

�d=T

� dxdjxdjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2d � �2

d

T2

q�Z ��u=T

�1þZ 1

�u=T

� dxujxujffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2u � �2

u

T2

q ~nðxdÞ~nð�xuÞ

� ½~nðX�Þ � ð�1Þ�i ~nðXþÞ�; (6)

where N� ¼ 6017�4 , N� ¼ 2520

457�6 , �� ¼ 0, and �� ¼ 1, X� �x� þ xu � xd ���=T, and ~nðxÞ � 1=ðexpðxÞ þ 1Þ. Thesefunctions reflect the phase space available to Urca reac-tions. They are normalized so that Rð0; 0; 0Þ ¼ 1 and aresymmetric in the gap parameters Rðx; y; zÞ ¼ Rðx; z; yÞ. Inequilibrium they are pure reduction factors Rð0; � � �Þ 1that can become extremely small [1]. However, for high-amplitude density oscillations (�� > T) the modificationfunctions gain a suprathermal enhancement and can thengreatly exceed one [5–7]. They depend on the equation ofstate only via dimensionless ratios and are the same for alldirect Urca processes [1]. In ungapped matter they have ananalytic polynomial form [6,7]. The point of this Letter isthat they show an even more dramatic enhancement inmatter with gapped fermions.The oscillation period of a compact star is much smaller

than its evolution time scale, so the relevant quantity is theaveraged emissivity ��dU � 1

2�

R2�0 d’�dUð��ð’ÞÞ, which

takes the same form Eq. (4) with the averaged modificationfunction R �� depending on �̂�. The bulk viscosity of densematter is [7]

� ¼ � C

�!2

�n�n

Z 2�

0d’ cosð’Þ

Z ’

0d’0�ð$Þ½��ð’0Þ�:

(7)

PRL 108, 111102 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 MARCH 2012

111102-2

Page 3: Bridging the Gap by Squeezing Superfluid Matter

Inserting the rate Eq. (3) and the chemical potential oscil-lation Eq. (1) gives at low T valid for superfluid matter

�dU ¼ 17G2

120�C2D�

T4

!2R�

��̂�

T;�d

T;�u

T

�(8)

with an analogous modification function

R�

��̂�

T;�d

T;�u

T

�¼ � 1

Z 2�

0d’ cosð’Þ

Z ’

0d’0 sinð’0Þ

� R�

��̂�

Tsinð’0Þ;�d

T;�u

T

�: (9)

In hyperon and quark matter the viscosity is dominated bynonleptonic rather than Urca processes, but for those wealso expect a high-amplitude enhancement [7].

To see the gap-bridging enhancement of �-equilibrationrates for oscillations of suprathermal amplitude, withoutspecifying a particular form of matter, we show in Fig. 2the modification functions R �� and R� . We vary the maxi-

mum gap � and the ratio of the two gaps. For stronglygapped particles �i=T � 1, the larger gap tends todominate, e.g., ð�d;�uÞ ¼ ð2�; 0Þ is more suppressedthan (�, �). Once the amplitude �̂� becomes comparableto �, the Urca reactions can bridge the gap(s) so that themodification functions rise steeply, and quickly reach theirungapped levels. Interestingly, for matter where the twogaps are the same (dotted lines) this rise starts already for�̂� � �i and eventually merges into the solid curve for a

single gap �d þ �u. In contrast, in the asymmetric casewhere one gap is much larger than the other, the effects ofthe smaller gap are only visible at large amplitudes andbecome negligible when it is more than an order of mag-nitude smaller. In the large amplitude limit �̂� � �i, T

the modification functions approach R �� � 1057312�6

�̂6�

T6 and

R� � 5136�4

�̂4�

T4 .

Figure 2 shows that suprathermal enhancement ingapped matter is considerably bigger than in ungappedmatter (the �=T ¼ 0 line) [7]. This enhancement couldbe realistically achieved in compact star oscillations: �̂� isrelated to the density amplitude �n= �n by the factor CEq. (1). For hadronic matter with an APR equation of state[17], C increases from Cðn0=4Þ � 20 MeV to Cð5n0Þ �150 MeV [2]; for quark matter it decreases from Cðn0Þ �30 MeV to Cð5n0Þ � 20 MeV. Thus for amplitudes�n= �n� 0:01 the amplitude �̂� can indeed become largeenough to bridge gaps �� 1 MeV [15].To facilitate calculations of gap-bridging in various

forms of matter, we show in Table I the values of therelevant parameters for direct Urca processes in hadronic[2,18,20], hyperonic [21,22] and quark matter [23,24].Inserted in the general results Eqs. (4) and (8) they repro-duce previous expressions in the subthermal case�� � T.In the ideal gas approximation our results simplify to

��¼ ~�

��n

n0

���

T69R ��; � ¼ ~�

��n

n0

����1kHz

!

�2T49R�; (10)

T 50

T 25

T 10

T 5

T 0

T 100

1 2 5 10 20 50 100

10 20

10 15

10 10

10 5

1

105

T

R

T 50

T 25

T 10

T 5

T 0

T 100

1 2 5 10 20 50 100

10 20

10 15

10 10

10 5

1

105

T

R

FIG. 2 (color online). The modification functions Ri Eqs. (4) and (8) as a function of oscillation amplitude �̂�, for different pairingpatterns with maximum gap �: asymmetric case where only one particle is gapped, e.g. �d ¼ �, �u ¼ 0 (solid lines); intermediatecases �d ¼ 5�u ¼ � (long dashed lines); �d ¼ 2�u ¼ � (short dashed lines); symmetric case �d ¼ �u ¼ � (dotted lines). Gapranges from �=T ¼ 0 to �=T ¼ 100. Left panel: R �� for the averaged neutrino emissivity. Right panel: R� Eq. (9) for the bulk

viscosity.

PRL 108, 111102 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 MARCH 2012

111102-3

Page 4: Bridging the Gap by Squeezing Superfluid Matter

where T9 is the temperature in units of 109 K and the

values of ~�, ~� , �� and �� are given in Table I.

Finally, to illustrate the likely magnitude and scope ofgap-bridging effects in the neutron star context, we per-form an illustrative calculation of the rate for direct Urcaprocesses in a star, using a ‘‘toy model’’ of nuclear matterin which there is gapping of protons and neutrons, anddirect Urca processes can occur at all densities. The modeluses the APR equation of state [17] with the in-mediumhadron masses given in [25]. For the density dependence ofthe 1S0 proton gap we use a Gaussian fit to data from [3],

with maximum �p0 ¼ 1 MeV centered at n � 1:3n0. For

illustrative purposes we assume the neutrons also have a

1S0 gap (in reality it is expected to be3P2) with the density

dependence proposed to explain the Cas A cooling data in[26], i.e., with maximum �n0 ¼ 0:12 MeV at n � 3:7n0.The gaps have a BCS temperature dependence, but this hasonly a modest effect in small regions where T � �p, �n.

Our results for neutrino emissivity and bulk viscosity atT ¼ 108 K are shown in Fig. 3. At infinitesimal amplitudes(thick �n= �n ¼ 0 lines) there is enormous suppression bythe proton gap, and moderate suppression by the smallerneutron gap. As the density oscillation amplitude in-creases, it first, at �n= �n * 10�3, overcomes the suppres-sion of emissivity and viscosity due to the neutron gap,increasing them at densities potentially relevant for directUrca processes by many orders of magnitude. Then, at�n= �n * 10�2, even the large proton gap at lower densityis bridged, boosting the results by up to 1050. Our calcu-lation is for the direct Urca process, but the enhancementarises from pure energetics, so we expect that for the caseof modified-Urca reactions it will also provide enhance-ment that is capable of canceling the suppression due toCooper pairing.Figure 3 shows that, if the star is cold enough that Urca

processes are blocked by pairing gaps throughout the star,density oscillations could provide the dominant contribu-tion to those processes, by bridging the pairing gaps, start-ing in the regions where the rate-controlling gap is thesmallest. The smaller this minimum gap, the lower theamplitude required to bridge it. In hyperonic [21,22] orquark matter [27,28], as well as for a smaller neutron gap[29], there are processes which are only suppressed by� &0:01 MeV, that could be bridged entirely by oscillationswith amplitude as small as �n= �n & 10�4.We are grateful to C. Horowitz and J. Read for helpful

discussions. This work was initiated during the workshopINT-11-2b: ‘‘Astrophysical Transients: MultimessengerProbes of Nuclear Physics.’’ We thank the Institute forNuclear Theory at the University of Washington for itshospitality and the U.S. Department of Energy for supportunder Contracts No. DE-FG02-91ER40628, No. DE-FG02-05ER41375, No. DE-FG02-00ER41132 and by theTopical Collaboration ‘‘Neutrinos and nucleosynthesis inhot dense matter.’’

TABLE I. Parameters for direct Urca processes in dense matter, with Fermi constant GF, Cabbibo angle �C, vector and axialcouplings fV and gA [1], proton fraction x, symmetry energy Sð �nÞ [18], in-medium hadron masses mð �nÞ and quark interactionparameter c [7,19]. The quark expression is to leading order in s and all results to leading order in T=�, ml=�, and ��=�e.

Process G fV gA D �e C ~�ð ergscm3 s

Þ ~�ð gcm sÞ �� ��

n ! pl ��l12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2V þ 3g2A

qcos�CGF 1 1.23 2m

nmp�e 4ð1–2xÞS 4ð1–2xÞðn @S

@n � S3Þ 6:82� 1026 7:86� 1022 2

3 2

� ! pl ��l12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2V þ 3g2A

qsin�CGF �1:23 0.89 2m

�mp�e or ð3�2nÞ2=3

2mnor ð3�2nÞ2=3

6mn3:05� 1025 3:52� 1021

�� ! nl ��l12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2V þ 3g2A

qsin�CGF �1 0.28 2m

�mn�e for a free gas for a free gas 1:04� 1025 1:20� 1021

d ! ue ��e

ffiffiffi3

pcos�CGF � � � � � � 8s

3� �2q�e

m2s

4�q� m2

d�m2

u

3ð1�cÞ�q(gas: c ¼ 0) 1:57� 1025s 5:09� 1021s

13 � 1

3

s ! ue ��e

ffiffiffi3

psin�CGF � � � � � � m2

s�q � m2s

3ð1�cÞ�q(gas: c ¼ 0) 3:98� 1024 1:29� 1021

FIG. 3 (color online). Illustration of the enhancement viasuprathermal oscillations of the neutrino emissivity and thebulk viscosity of hadronic matter with a 1S0 proton gap �p0 ¼1 MeV at n � 1:3n0 and a neutron gap �n0 � 0:1 MeV atn � 4n0.

PRL 108, 111102 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 MARCH 2012

111102-4

Page 5: Bridging the Gap by Squeezing Superfluid Matter

[1] D. Yakovlev, A. Kaminker, O. Y. Gnedin, and P. Haensel,Phys. Rep. 354, 1 (2001).

[2] P. Haensel, K. P. Levenfish, and D.G. Yakovlev, Astron.Astrophys. 357, 1157 (2000).

[3] D. J. Dean and M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607(2003).

[4] M.G. Alford, A. Schmitt, K. Rajagopal, and T. Schafer,Rev. Mod. Phys. 80, 1455 (2008).

[5] J. Madsen, Phys. Rev. D 46, 3290 (1992).[6] A. Reisenegger, Astrophys. J. 442, 749 (1995).[7] M.G. Alford, S. Mahmoodifar, and K. Schwenzer, J. Phys.

G 37, 125202 (2010).[8] N. Stergioulas, Living Rev. Relativity 6, 3 (2003).[9] P. Arras et al., Astrophys. J. 591, 1129 (2003).[10] M.G. Alford, S. Mahmoodifar, and K. Schwenzer, Phys.

Rev. D 85, 044051 (2012).[11] E.M. Kantor and M. E. Gusakov, Phys. Rev. D 83, 103008

(2011).[12] L.M. Franco, B. Link, and R. I. Epstein, Astrophys. J. 543,

987 (2000).[13] D. Tsang, J. S. Read, T. Hinderer, A. L. Piro, and R.

Bondarescu, Phys. Rev. Lett. 108, 011102 (2012)..[14] D. Tilley and J. Tilley (IOP Publishing Ltd., Bristol,

1990).[15] M.G. Alford, S. Mahmoodifar, and K. Schwenzer, AIP

Conf. Proc. 1343, 580 (2011).

[16] T. Schafer and K. Schwenzer, Phys. Rev. D 70, 114037(2004).

[17] A. Akmal, V. R. Pandharipande, and D.G. Ravenhall,Phys. Rev. C 58, 1804 (1998).

[18] J.M. Lattimer, M. Prakash, C. J. Pethick, and P. Haensel,Phys. Rev. Lett. 66, 2701 (1991).

[19] M. Alford, M. Braby, M.W. Paris, and S. Reddy,Astrophys. J. 629, 969 (2005).

[20] P. Haensel and R. Schaeffer, Phys. Rev. D 45, 4708 (1992).[21] M. Prakash, M. Prakash, J.M. Lattimer, and C. J. Pethick,

Astrophys. J. Lett. 390, L77 (1992).[22] P. Haensel, K. P. Levenfish, and D.G. Yakovlev, Astron.

Astrophys. 381, 1080 (2002).[23] N. Iwamoto, Phys. Rev. Lett. 44, 1637 (1980).[24] B. A. Sa’d, I. A. Shovkovy, and D.H. Rischke, Phys. Rev.

D 75, 125004 (2007).[25] D. Page, J.M. Lattimer, M. Prakash, and A.W. Steiner,

Astrophys. J. Suppl. Ser. 155, 623 (2004).[26] P. S. Shternin et al., Mon. Not. R. Astron. Soc. 412, L108

(2011).[27] A. Schmitt, I. A. Shovkovy, and Q. Wang, Phys. Rev. D 73,

034012 (2006).[28] X. Wang and I. A. Shovkovy, Phys. Rev. D 82, 085007

(2010).[29] A. Schwenk and B. Friman, Phys. Rev. Lett. 92, 082501

(2004).

PRL 108, 111102 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

16 MARCH 2012

111102-5