SSCL-Preprint-117 June 1992 The idea [1] that bulk matter can be bound by purely strong forces has im• BU-HEP-92-08, SU-1TP 92/09 plications for cosmology (2, 3, 4, 5, 6, 7] and astrophysics [5, 8, 9, 10, 11]; various species include strange quark matter [6, 7] and baryonic matter [8, 9, 10]. Bary- An Upper Bound on Q-star Masses onic matter is particularly interesting because it offers a consistent description of DAVID A. HOOIIRON the known properties of nuclei on the one hand, and exotic hadronic bulk matter with very different properties on the other hand. The bulk properties of nuclei Department of Physics, Stanford University, Stanford, CM 94305 (binding energies, charge densities, bulk compressibility, optical potential, spin BRYAN W. LYNN observables in polarized proton scattering, etc.) have been simply and success• Department of Physics, Stanford University, Stanford, CA 94305 and fully modeled with a field theory of a baryon field ty, a phenomenological classical Superconducting Super Collider Laboratory, Dallas, TX 75237-3946 scalar field a and a vector field Vu [12]. Such models of ordinary nuclei can also be made compatible with the hadronic successes of chiral SU(3)i, X SU(3)R symme• STEPHEN B.SEUPSKY try [8] (baryon and pseudoscalar meson octet mass spectrum, low energy hadronic Department of Physics, Boston University, Boston, MA 02215-2507 scattering, Adler-Weisberger sum rule, Goldberger-Trieman relation etc.) Exotic ABSTRACT bulk matter can appear because, for certain forms of the tr-self interaction po• tential U(tr), there exist large "Q-balP solutions with hundreds of MeV binding Q-stars (the gravitational generalization of Q-balls, strongly bound bulk mat• energy per baryon. This possibility stems directly from our ignorance of the ef• ter that an appear in field theories of strongly interacting hadrons) are the only fective potential U(a). A crucial fact is that the bulk properties of the Q-ball known mpact objects consistent with the known bulk structure of nuclei and solutions are virtually independent of the properties of ordinary nuclei. chiral s umctry that evade the Rhoades-Ruffini upper bound of 3.2M0. Generic bounds are quite weak: Mq-,M < 4OOA/0. If, however, we assume that the 1.558 This scenario has important implications for neutron stars. The genetalixa- ms pulsar is a Q-star, equilibrium , 1 stability criteria of rotating fluids place a tion of Q-balJs to include gravity are called Q-stars [8, 9, 10, 13, 14], Here, much stronger upper bound of Afy „ < 5.3M,.-, on such models under certain Q stands for the conserved charge that stabilizes the solutions; in field theories special assumptions. This has important implications for heavy compact objects of strongly interacting hadrons it is the baryon number. They can have large such as f.ygnus X-l. masses or high rotation rates, and should differ from conventional neutron stars in many important ways; for a large range of parameters they give counterex• Submitted to Classical and Quantum Cravity. amples to the li.'IMp upper bound on neutron star masses proposed by Rhoades MASTER and Ruffini [15]. In fact, prior to this work, the only upper bound on Q-star We consider the Lagrangian masses, Afq_.t», < 400A/©, was given by the observation that white dwarfs are 2 not Q-stars [14], thereby setting a lower limit on Q-ball densities. In this paper C = *R.r - m(a) - gvV]* + i(eV) - U(a) + \m\V,V, (1) we bound a certain Q-star model with astrophysical data. in which * is a baryon field, and a and V are phenomenological scalar and vector There is a wealth of data on neutron stars that confront any model of bulk fields. Serr and Walerka [21] have shown neglect of the vector field dynamics to nuclear matter, particularly if, as is widely accepted, pulsars are interpreted as be a good approximation for ordinary nuclei such as 40Ca and ^Pb in their rotating neutron stars. These include observed masses, population statistics, mean-field model. This Lagrangian then gives a good mean field account of bulk magnetic field strengths, cooling curves, rotation rates, pulsar glitches, and high nuclear properties and we examine its consequences for neutron stars. For certain space velocities* For example, the conventional interpretation of pulsar glitches choices of U(a) compatible with bulk nuclear properties, the stable neutron star as resulting from plate motion in the neutron star crust is problematic for Q-stars, configuration will be a Q-star rather than the metastable conventional neutron for which crusts are not generic [16]. In this work, we assume that the fastest star configuration. pulsar PSR 1937+214 (which is not known to glitch) is a Q-star and restrict Spherical Q-stars in which m(<r) = 0 and {/ = Ua in the interior have the Q-star parameters from its (presumed) rotation rate period r = 1.558 ms. We following equation of state (EOS) [10] (£ and P are respectively the total energy consider here for simplicity only the case of Q-stars in which baryons are massless density and pressure): in the interior. e~3P-Ur0 + av{£-P -2</0)5 =0, (2) Shapiro, Teukolsky, and Wasserman [19] have outlined general criteria, in light of the pulsar 1937+214 [20], period 1.588 ms, for limiting the mass and 2 in which av = \/Z(gv/mv) /x- It is convenient to define ( = ctvU£-j- , with radius of the nonspinning progenitor. If the mass M of a neutron star is too low, which Q-star properties scale. For given parameters av and U0, there will be a instabilities set in for conventional neutron star models (secular and dynamical, range of possible masses and radii. For low masses the density is constant, so depending on the regime and the compressibility of the star). If M is too high, a description in terms of an EOS is possible. The density increases as gravity there are two restrictions: the formation of destabilizing ergotoroids and gravita• becomes important, and the maximum mass occurs when ^ vanishes (II is the tional collapse to a black hole. Here we examine the possible equations of state neutron star radius). R<;f. [14] summarizes many properties of Q-stars in terms of of Q-stars that are consistent with all the stability criteria. the scaling parameter C, and we will quote many of their results in what follows. For any fixed (, there is a locus of such maximum masses (^ = 0) obtained #1 See, for example, lef. [17]. A recent review of the properties of millisecond pulsars can be I'o ;., .el' [18]. when av is varied. An example is shown in figure 1. Note that there is a smallest 2 :t DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or use• fulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any spe• cific commercial product, process, or service by trade name, trademark, manufac• turer, or otherwise does not necessarily constitute or imply its endorsement, recom• mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. maximum, which occurs when the scalar energy density U0 is so large that the m = 3 earlier, etc. From (3) this means smaller critical CI for increasing m. For corresponding Q-ball is unbound. The unlabeled solid line of figure 2 shows the example, stability of the m = 2 mode requires locus of such minimum maxima for different values of (; the point of maximum mass of any Q-star must lie to the right of that line. im *108- (5) We now seek to constrain the space of possible Q-star masses and radii from Each higher mode has a longer growth time, but sets in at lower angular momen• considerations of rotational stability and equilibrium assuming that the 1.558 ms tum. For sufficiently high m the instability is either damped by viscous forces or pulsar is a Q-star. grows slowly on stellar timescales, so generally m > 5 need not be considered. A There is a long history of the analysis of the maximum rotation velocity of useful parameter for describing the onset of instability is a gravitationally bound fluid with a given EOS. The standard classical reference is by Chandrasekhar [22]. The simplest F.OS is the case of incompressibility (see _ rotational kinetic energy ~ gravitational binding energy [17]). We will later argue that this is a good approximation for Q-stars with EOS (2). In the Newtonian regime rotation produces the MacLaurin spheroids. The The insensitivity of critical t of the fastest-growing mode to the EOS is a remark• angular velocity 0 and eccentricity e are related by able and useful result; i.e., m = 2 sets in at t = .14, not only for incompressible stars but also for centrally condensed ones. For higher modes there is dependence -91 = ^p-?'3 - 2eVcsine - 1(1 - e2). (3) on the EOS [24, 25]. The maximum fi occurs at e = .93; beyond that point, an increase in angular Equilibrium and stability criteria have also been established for polytropes 1+ momentum decreases $1.