VOLUME 86, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 JUNE 2001Evolution of Proto-Neutron Stars with Quarks
Jos A. Pons, Andrew W. Steiner, Madappa Prakash, and James M. LattimerDepartment of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, New York 11794-3800
(Received 1 February 2001)
Neutrino fluxes from proto-neutron stars with and without quarks are studied. Observable differencesbecome apparent after 1020 s of evolution. Sufficiently massive stars containing negatively charged,strongly interacting, particles collapse to black holes during the first minute of evolution. Since theneutrino flux vanishes when a black hole forms, this is the most obvious signal that quarks (or other typesof strange matter) have appeared. The metastability time scales for stars with quarks are intermediatebetween those containing hyperons and kaon condensates.
DOI: 10.1103/PhysRevLett.86.5223 PACS numbers: 97.60.Jd, 12.38.Mh, 21.65.+f, 26.60.+cA proto-neutron star (PNS) is born following the gravi-tational collapse of the core of a massive star, in conjunc-tion with a successful supernova explosion. During thefirst tens of seconds of evolution, nearly all (99%) of theremnants binding energy is radiated away in neutrinos ofall flavors . The n luminosities and the evolutionarytime scale are controlled by several factors, such as thetotal mass of the PNS and the n opacity at supranucleardensity, which depends on the composition and equationof state (EOS). One of the chief objectives in modelingPNSs is to infer their internal compositions from n sig-nals detected from future supernovae by detectors such asSuper-K, SNO, and others under consideration, includingUNO .
In their landmark paper, Collins and Perry  noted thatthe superdense matter in neutron star cores might consistof weakly interacting quarks rather than of hadrons, dueto the asymptotic freedom of QCD. The appearance ofquarks causes a softening of the EOS which leads to a re-duction of the maximum mass and radius . In addition,quarks would alter n emissivities and thereby influence thesurface temperature of a neutron star  during the hun-dreds of thousands or millions of years that they might re-main observable with such instruments as HST, Chandra,and XMM. Quarks would also alter the spin-down ratesof neutron stars .
Many calculations of dense matter predict the appear-ance of other kinds of exotic matter in addition to quarks:for example, hyperons or a Bose (pion, kaon) conden-sate (cf.  and references therein). An important issue iswhether or not n observations from a supernova could re-veal the presence of such exotic matter and, further, couldunambiguously point to the appearance of quarks. The de-tection of quarks in neutron stars would go a long way to-ward the delineation of QCD at finite baryon density whichwould be complementary to current relativistic heavy ioncollider experiments, which largely address the finite tem-perature, but baryon-poor regime.
An important consequence of the existence of exoticmatter in neutron stars (in whatever form, as long as itcontains a negatively charged component) is that a suffi-ciently massive PNS becomes metastable . After a delay0031-90070186(23)5223(4)$15.00of up to 100 s, depending upon which component appears,a metastable PNS collapses into a black hole . Thecollapse to a black hole proceeds on a free-fall time scaleof less than a millisecond , much shorter than n dif-fusion times, and the neutrinos still trapped in the innerregions cannot escape. Such an event should be straight-forward to observe as an abrupt cessation of n flux whenthe instability is triggered .
The evolution of the PNS in the so-called Kelvin-Helmholtz phase, during which the remnant changesfrom a hot, n-trapped, and lepton-rich object to a coldand n-free star, occurs in near-hydrostatic equilibrium.The n-matter interaction time scales are much smallerthan the dynamical time scale of PNS evolution, whichis of the order of seconds. Thus, until neutrinos enterthe semitransparent region at the edge of the star, theyremain close to thermal equilibrium with matter and maybe treated in the diffusion approximation.
In this Letter we provide a benchmark calculation withquarks by solving the general relativistic n transport andhydrostatic equations (as in ) and then compare ourresults with those of our previous work [10,11] in whichother compositions were studied. In addition, we assessthe prospects of observing PNS metastability and its sub-sequent collapse to a black hole through observations incurrent and planned detectors.
The essential microphysical ingredients in our study arethe EOS of dense matter and its associated n opacity. Webegin by considering two generic compositions: charge-neutral, beta equilibrated matter containing (i) nucleonsonly (np) and (ii) nucleons with quark matter (npQ). Inthe npQ case, a mixed phase of baryons and quarks (purequark matter exists only for very large baryon densities,except for extreme choices of model parameters) is con-structed by satisfying Gibbs phase rules for mechanical,chemical, and thermal equilibrium . The EOS of bary-onic matter is calculated using a field-theoretic model atthe mean field level . The results reported with thisEOS are quite general, as we verified by alternatively us-ing a potential model approach . The quark matter EOSis calculated using a MIT baglike model (similar resultsare obtained with the NambuJona-Lasinio quark model). 2001 The American Physical Society 5223
VOLUME 86, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 JUNE 2001The details of the EOS may be found in Ref. . We usen opacities [17,18] consistent with the EOS. When quarksappear, the n absorption and scattering cross sections dra-matically decrease, the precise reduction being sensitive tothe thermodynamic conditions in the mixed phase .
Figure 1 shows the evolutions of some thermodynamicquantites at the center of npQ stars of various, fixed, bary-onic masses (MB). In the absence of accretion, MB re-mains constant during the evolution, while the gravitationalmass MG decreases. With the EOS utilized, stars withMB & 1.1M do not contain quarks and those with MB *1.7M are metastable. The latter value is 0.05M largerthan the maximum mass for cold, catalyzed npQ matter,because the maximum mass of hot n-free npQ matter isthis much less than that of hot n-trapped matter . Gen-erally, due to the high lepton (n) content initially presentin the PNS, the electron chemical potential at the center istoo large for quarks to exist. For sufficiently massive stars,quarks eventually appear after a certain amount of n lossoccurs. For the MB 1.6M star, for example, quarks ap-pear after about 15 s (indicated by a diamond). Thereafter,the stars central density increases for a further 1520 s,until a new stationary state with a quark-hadron mixedphase core is reached (for stable stars) or for MB * 1.7M,instability occurs (indicated by asterisks). It is interestingthat for this EOS the lifetimes for all masses are restrictedto the range 1030 s, and slowly decrease with increasingmass. The appearance of quarks is accompanied by an in-
FIG. 1. Evolutions of the central baryon density nB, n con-centration Yn , quark volume fraction x, and temperature T fordifferent baryonic masses MB. Solid lines correspond to stablestars; stars with larger masses are metastable (dashed lines).Diamonds indicate when quarks appear at the stars center, andasterisks denote when metastable stars become gravitationallyunstable.5224crease in Yn because of the depletion of electrons; for thelargest masses, the increase is very large.
To point out the major differences one might observebetween the np and npQ cases, we have estimated the necount rate in the Super-K detector in Fig. 2. For this es-timate, we assumed the total n luminosity from a PNSat 8.5 kpc distance, corresponding to a Galactic super-nova, was equally divided among the six n species. Then-energy spectra were taken to be Fermi-Dirac with zerochemical potential and a temperature corresponding to thematter temperature in the PNS where the n-optical depthwas approximately unity. Only the dominant reaction, neabsorption on protons, was included. (For details, see.) It is difficult to discern much difference in the early(t , 10 s) count rates from np and npQ stars. For starswith MB , 1.7M, this is because quarks have not yet ap-peared. For more massive stars, the fact that neutrinos arestrongly trapped inhibits any discriminatory signal fromreaching the surface before this time. The signals at latertimes (t * 25 s), however, are substantially larger for thenpQ case, due to the decrease in n opacity of npQ mat-ter and the increased binding energy of npQ stars. Mostimportantly, the n signal from metastable npQ stars haltsabruptly when the instability occurs. Qualitatively, thesefeatures are also found for npH and npK stars [10,11].
We compare n signals observable with different de-tectors in Fig. 3, which displays the n-light curves as afunction of MB for npQ stars. The two upper shadedbands correspond to estimated SN 1987A (50 kpc dis-tance) detection limits with KII and IMB, and the lowerbands correspond to estimated detection limits in SNO,Super-K, and UNO, for a Galactic supernova (8.5 kpc dis-tance). The detection limits have been set to a ne countrate dNdt 0.2 Hz  with the same assumptions asin Fig. 2. The general rise with time of the detector lim-its is chiefly due to the steady decrease in the ne averageenergy. It is possible that these limits are too conservative
FIG. 2. A comparison of ne count rates expected in Super-Kfrom a PNS containing either np or npQ matter. The left panelshows times less than 10 s, while the right panel shows timesgreater than 10 s.
VOLUME 86, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 JUNE 2001FIG. 3. The total n luminosity for npQ stars of various baryonmasses. Shaded bands illustrate the limiting luminosities corre-sponding to count rates of 0.2 Hz for the indicated supernovaein some detectors.
and could be lowered with identifiable backgrounds andknowledge of the direction of the signal. The width ofthe bands represents the uncertainty in the ne average en-ergy due to the flux-limited diffusion approximation. Weconclude that it should be possible to distinguish betweenstable and metastable stars, since the luminosities whenmetastability is reached are always above conservative de-tection limits.
The drop in n luminosity for stable stars is associatedwith the end of the Kelvin-Helmholtz epoch when the PNSis becoming optically thin. This portion of the n-lightcurve is approximate due to the breakdown of the diffu-sion approximation. It is an apparent coincidence that thisoccurs simultaneously with the collapse of the lower massmetastable stars.
Our choice of bag constant, B 150 MeV fm23, inconjunction with the baryonic EOS we used, was moti-vated to maximize the extent of the quark matter phasein a cold neutron star and was limited by the necessity ofproducing a maximum mass cold star in line with accu-rate observational constraints (MG 1.444M ). In-creasing B, or employing an alternative quark EOS thatotherwise produces a larger maximum mass, delays theappearance of quarks and raises the metastability windowto larger stellar masses. Necessarily, this results in an in-creased time scale for metastability for a given mass, andFIG. 4. Lifetimes of metastable stars versus the PNS MB forvarious assumed compositions. Thick lines denote cases inwhich the maximum masses of cold, catalyzed stars are nearMG 1.45M, which minimizes the metastability lifetimes.The thin lines for the npQ and npH cases are for EOSswith larger maximum masses (MG 1.85M and 1.55M,respectively.)
a lower n luminosity when metastability occurs. Figure 4shows the relation between time to instability and MB forthe original case (B 150 MeV fm23, thick solid curve)and a case with B 200 MeV fm23 (thin solid curve), inwhich the maximum gravitational mass of a cold neutronstar is about 1.85M. For the latter case, the metastabil-ity time scales lie in a narrow range 4045 s. These, andthe metastability masses, are both larger than obtained forB 150 MeV fm23 and have narrower ranges. Furtherincreases in B diminish the size of the instability window,because the appearance of quarks is shifted to progres-sively larger densities.
Figure 4 also compares the metastability time-MB rela-tion found for matter containing hyperons (npH, dashedlines ) or matter with kaons (npK , dotted line ) in-stead of quarks. All three types of strange matter are sup-pressed by trapped neutrinos [7,16], but hyperons alwaysexist in npH matter at finite temperatures and the transi-tion to quark matter can occur at lower densities than thatfor very optimistic kaon cases . Thus, the metastabil-ity time scales for npH matter can be very short, and thosefor npK matter are generally larger than for npQ matter.Note the relatively steep dependence of the metastabilitytime with MB for npH stars, which decreases to very smallvalues near the maximum mass limit of hot, lepton-rich,stars. The thick npH and npQ lines, as well as the npKline, represent minimum metastability times for a given5225
VOLUME 86, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 4 JUNE 2001MB as discussed above. The thin npQ and npH lines arefor EOSs with larger cold, catalyzed maximum mass.
Clearly, the observation of a single case of metastabil-ity, and the determination of the metastability time alone,will not necessarily permit one to distinguish among thevarious possibilities. Only if the metastability time is lessthan 1015 s could one decide on this basis that the starscomposition was that of npH matter. However, as in thecase of SN 1987A, independent estimates of MB mightbe available . In addition, the observation of two ormore metastable neutron stars might permit one to differ-entiate among these models, but given the estimated rateof Galactic supernova (one per 3050 yr ), this mayprove time-consuming.
Our study has focused on times longer than approxi-mately 1 s after core bounce, after which effects of dynam-ics and accretion become unimportant. Studies of the nsignal during the first second, during which approximately13 of the energy is emitted, and at late times, as thestar becomes optically thin to neutrinos, requires moreaccurate techniques for n transport. In addition, the earli-est time periods require the incorporation of hydrodynam-ics .
Our conclusions are that (1) the metastability and subse-quent collapse to a black hole of a PNS containing quarkmatter, or other types of matter including hyperons or aBose condensate, are observable in current and planned ndetectors and (2) discriminating among these compositionsmay require more than one such observation. This high-lights the need for breakthroughs in lattice simulations ofQCD...