We study equilibrium configurations of hybrid stars with inhomogeneous Color SuperConducting (CSC) phases in the inner core and a mantle of nuclear matter

Ippolito, N. D.

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DOI 10.1393/ncc/i2009-10514-5Colloquia: IFAE 2009IL NUOVO CIMENTO Vol. 32 C, N. 3-4 Maggio-Agosto 2009Equilibrium sequences of hybrid stars with LOFF matter coreN. D. IppolitoUniversita` and INFN, Sezione di Bari - Bari, Italy(ricevuto il 19 Settembre 2009; pubblicato online il 21 Dicembre 2009)Summary. — We study equilibrium configurations of hybrid stars with inhomo-geneous Color SuperConducting (CSC) phases in the inner core and a mantle ofnuclear matter.PACS 12.38.Mh – Quark-gluon plasma.PACS 24.85.+p – Quarks, gluons, and QCD in nuclear reactions.Since the publication of the first articles about Color SuperConductivity, it was clearthat the primary place where these phases could be searched is in the core of verycompact stars, generically named Neutron Stars. Matter in the interiors of neutronstars is compressed by gravity to densities much larger than the density of an ordinarynucleus (by factors up to 10). At such densities baryons are likely to lose their identityand dissolve into deconfined quarks. If compact (hybrid) stars featuring quark coressurrounded by a nuclear mantle exist in nature, they could provide a unique window onthe properties of QCD at high baryon densities.Following the analysis given in [1], we study the possibility that the equation of stateof matter at high densities admits stable configurations of self-gravitating objects inGeneral Relativity featuring deconfined quark matter, and if so, we check if the grossparameters of these objects, like their mass and radius, are compatible with the knownastronomical bounds.Because of β-equilibrium in the light quark sector the chemical potentials of u andd quarks obey the constraint μd = μu + μe. Furthermore, the Fermi surface of strangequarks is mismatched with the Fermi surfaces of light u and d quarks because of thelarge strange quark mass. We have already incorporated these features in our previouslydescribed studies about the three flavor LOFF phases [2, 3].The nuclear equation of state can be constructed starting from a number of differ-ent principles [4]. We tested a large number of equations of state to construct hybridstar configurations. These belong to the classes of i) non-relativistic variational andBruckner-Hartree-Fock theories which use as an input a non-relativistic potential fittedto the elastic nucleon-nucleon scattering data; ii) relativistic mean-field models which arefitted to the bulk properties of nuclear matter, and iii) relativistic models which includec© Societa` Italiana di Fisica 329330 N. D. IPPOLITOcorrelations at the level of the covariant scattering amplitude (Dirac-Bruckner-Hartree-Fock theories). In the analysis presented here, it was sufficient to characterize theseequations of states by their stiffness; as we shall see, only the stiffest equations of stateare admissible for phase equilibrium between nuclear and quark matter.A straightforward normalization of the quark pressure in the NJL model requires thatthe pressure vanishes at zero density and temperature. In the terminology of the MITbag model, this is equivalent to subtraction of a bag constant from the thermodynamicpotential. Since the value of the bag constant is related to confinement which is absent inthe NJL model, it appears reasonable to consider changes in its value, and hence in thenormalization of the pressure. We shall consider the simple case of a constant shift in theasymptotic value of the pressure. In ref. [3] the calculation of masses and condensateshas been carried out for η = GD/GS = 0.75. This regime is usually referred to as“intermediate coupling”. We shall adopt a “strong coupling” regime with η = 1, sinceonly in the latter case the matching to the nuclear equations of state can be performedwithout variations in the bag constant (we will discuss this point in more detail in thefollowing subsection).Physically, the true nuclear equation of state must go over to some sort of quarkequation of state at some density if deconfinement takes place in nature. Since we haveonly models of deconfined matter and nuclear matter, this transition is modeled byrequiring that there exists a baryo-chemical potential at which the pressures of thesephases are equal. This is equivalent to the condition that the pressure vs. the chemicalpotential curves P (μ) for these phases cross (matching). If the P (μ) curves for the chosenequations of state of nuclear and quark matter do not cross, the models are incompatiblein the sense that they cannot describe the desired transition between nuclear and quarkmatter. The low-density equation of state of nuclear matter and the high-density equationof state of CCS matter are matched at an interface via the Maxwell construction. Thephase with largest pressure is the one that is realized at a given chemical potential.The high-density regime contains two equations of state for crystalline color supercon-ductivity which differ for the normalization of pressure at zero density (or, equivalently,the value of the bag constant). The model A1 is normalized such that the pressure van-ishes at zero density. For the models A and B the zero-density pressure is shifted by anamount δp = 10MeV/fm3. We are aware of the arbitrariness of the latter procedure, thesole practical purpose of which is to produce an equation of state which can be matchedto a particular nuclear equation of state. Yet another possibility is to set δp = 0, butvary the value of the constituent masses of the light quarks in the fit of the parametersof the NJL model.A set of nuclear equations of state were tested for matching with the models above;it included about dozen equations of state, listed in refs. [4]. Only two equations ofstate based on the Dirac-Bruckner-Hartree-Fock approach are suitable to match withthe quark equations of state presented above. These are shown in both panels of fig. 1.We consider equilibrium and stability of cold hybrid stars with LOFF cores. Eachequation of state defines a sequence of equilibrium, non-rotating stellar configurations inGeneral Relativity, which can be parameterized in terms of the central density ρc of theconfiguration. It is assumed that the configurations are cold (T  0). The sphericallysymmetric solutions of Einstein’s equations for self-gravitating fluids are given by thewell-known Tolman-Opennheimer-Volkoff equations. A generic feature of these solutionsis the existence of a maximum mass for any equation of state; as the central densityis increased beyond the value corresponding to the maximum mass, the stars becomeunstable towards collapse to a black hole. One criterion for the stability of a sequenceEQUILIBRIUM SEQUENCES OF HYBRID STARS WITH LOFF MATTER CORE 3315e+14 1e+15 1.5e+15 2e+15 2.5e+15ρ  [g cm-3]0.40.81.21.62M/MOmodel Amodel A1model B10 12 14 16 18R [km]0.40.81.21.62M/MOEXO 0748-676J0737-3039BPSR B1516+02BFig. 1. – Left panel: dependence of masses of hybrid, non-rotating compact stars on their centraldensity for the models A, A1, and B. The dashed lines show the same for the associated nuclearequations of state. M0 stays for the mass of the Sun. We use indifferently the notations M0 andM. Right panel: mass-radius diagram for non-rotating configurations including the boundsfrom EXO 0748-676 and the bounds on the upper and lower pulsar masses. The sequences ofhybrid configurations for model A (heavy, black online), A1 (medium-light, red online), and B(light, blue online) are shown by solid lines. Models A and A1 share the same nuclear (low-density) equation of state, while the models A and B share the same quark (high-density)equation of state. The dashed lines are the sequences of purely nuclear stars with underlyingnuclear equations of state of model A (equivalently A1) and model B.of configurations is the requirement that the derivative dM/dρc should be positive (themass of the star should be an increasing function of the central density). At the point ofinstability the fundamental (pulsation) modes become unstable. If stability is regainedat higher central densities, the modes by which the stars become unstable towards theeventual collapse belong to higher-order harmonics.For configurations constructed from a purely nuclear equation of state the stablesequence extends up to a maximum mass of the order 2 M (right panel of fig. 1, dashedlines); the value of the maximum mass is large, since our chosen equations of state arerather hard. The hybrid configurations branch off from the nuclear configurations whenthe central density reaches that of the deconfinement phase transition. The jump inthe density at constant pressure causes a plateau of marginal stability beyond the pointwhere the hybrid stars bifurcate. This is followed by an unstable branch (dM/dρc < 0).Most importantly, the stability is regained at larger central densities: a stable branchof hybrid stars emerges in the range of central densities 1.3 ≤ ρc ≤ 2.5 × 1015 g cm−3.The models A and B feature the same high-density quark matter, whereas the models Aand A1 the same nuclear equation of state. It is seen that the effect of having differentnuclear equations of state (the models A and B) at intermediate densities is substantial(at densities below 1013 g cm3 all models are matched to the same equation of state). Atthe same time, the small shift δp by which the models A and A1 differ does not influencethe masses of stable hybrid stars, although it is necessary for matching of nuclear andquark EOS in the models A and B. It is evident that there will exist purely nuclearand hybrid configurations with different central densities but the same masses. Thisis reminiscent of the situation encountered in non-superconducting hybrid stars [5]; the332 N. D. IPPOLITOsecond branch of hybrid stars was called twin, since for each hybrid star there alwaysexists a counterpart with the same mass composed entirely of nuclear matter.The right panel of fig. 1 displays the astronomical bounds on the masses and radiiof compact stars along with the “tracks” for our models on the mass-radius diagram.All bounds are quoted at the 1σ level. The bound inferred from the star EXO0748-676,which combines information from redshifted O and F lines, the emitting area of X-rayradiation, and the Eddington luminosity, constrain the mass and the radius of a compactstar to lie on a straight line shown in the right panel of fig. 1 [6]. Both the hybrid starsand their nuclear counterparts have masses and radii within these bounds. For sequencesconstructed from models A and A1 there is a range of masses and radii that correspondto the new family of twin stars discussed above. The stars belonging to the new familylie to the left from the sharp kink at the point R  13.5 km and M  1.9M. Thestable branch of this new family of stars is separated from the stable nuclear sequenceby an instability region. The sequences corresponding to model B do not show such aninstability region.Some evidence for massive neutron stars with M ∼ 2M has been inferred fromrecent measurements on the pulsar PSR B1516+02B in the Globular Cluster M5 gaveM = 1.96± 0.1M [7]. The models A and A1 are consistent with these bounds. For themodel B these bounds correspond to the stable configuration with the largest mass. Forcompleteness, the lower bound on the neutron star mass 1.249±0.001 M is shown inthe right panel of fig. 1, which is inferred from the millisecond binary J0737-3039 [8].We stress that our models of color superconducting hybrid stars with LOFF mattercore can be used to obtain quantitative limits on the strain of gravitational wave emissionthat can be emitted. This topic will be studied in a future work.REFERENCES[1] Ippolito N., Ruggieri M., Rischke D., Sedrakian A. and Weber F., Phys. Rev. D,77 (2008) 023004 [arXiv:0710.3874 [astro-ph]].[2] Casalbuoni R., Gatto R., Ippolito N., Nardulli G. and Ruggieri M., Phys. Lett. B,627 (2005) 89; 634 (2006) 565 (Erratum) [arXiv:hep-ph/0507247].[3] Ippolito N. D., Nardulli G. and Ruggieri M., JHEP, 0704 (2007) 036 [arXiv:hep-ph/0701113]; Anglani R., Gatto R., Ippolito N. D., Nardulli G. and Ruggieri M.,Phys. Rev. D, 76 (2007) 054007 [arXiv:0706.1781 [hep-ph]].[4] Weber F., Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics (Instituteof Physics, Bristol, U.K.) 1999; Sedrakian A., Prog. Part. Nucl. Phys., 58 (2007) 168[arXiv:nucl-th/0601086].[5] Glendenning N. K. and Kettner C., Astron. Astrophys., 353 (2000) L9 [arXiv:astro-ph/9807155];[6] Ozel F., Nature, 441 (2006) 1115.[7] Freire P. C. C., Wolczczan A., van den Berg M. and Hessel W. T., arXiv:0712.3826[astro-ph].[8] Lyne A. G. et al., Science, 303 (2004) 1153.

Equilibrium sequences of hybrid stars with LOFF matter core

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Equilibrium sequences of hybrid stars with LOFF matter core

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