Consequences of a strong phase transition in the dense matter equation of state for the rotational evolution of neutron stars
Key Words.:
stars: neutron – pulsars – equation of stateAbstract
Context:
Aims:We explore the implications of a strong firstorder phase transition region in the dense matter equation of state in the interiors of rotating neutron stars, and the resulting creation of two disjoint families of neutronstar configurations (the socalled highmass twins).
Methods:We numerically obtained rotating, axisymmetric, and stationary stellar configurations in the framework of general relativity, and studied their global parameters and stability.
Results:The instability induced by the equation of state divides stable neutron star configurations into two disjoint families: neutron stars (second family) and hybrid stars (third family), with an overlapping region in mass, the highmass twinstar region. These two regions are divided by an instability strip. Its existence has interesting astrophysical consequences for rotating neutron stars. We note that it provides a natural explanation for the rotational frequency cutoff in the observed distribution of neutron star spins, and for the apparent lack of backbending in pulsar timing. It also straightforwardly enables a substantial energy release in a minicollapse to another neutronstar configuration (core quake), or to a black hole.
Conclusions:
1 Introduction
Recent observations of the highmass pulsars PSR J16142230 (Demorest et al., 2010; Fonseca et al., 2016) and PSR J0348+0432 (Antoniadis et al., 2013) with masses has motivated the nuclear and particle physics communities to deepen their understanding of the equation of state (EOS) of highdensity matter and of the possible role of exotic states of matter in neutron star (NS) interiors. For further constraints on the highdensity EOS from NS and heavyion collision experiments see Klähn et al. (2006) and Klähn et al. (2012).
Measurements of high masses of NSs do not immediately imply that very high densities prevail in their cores, meaning that a transition to exotic forms of matter (hypernuclear matter, quark matter) has to be invoked. Of two models for the highdensity EOS with the same stellar mass, the stiffer model will lead to a lower central density but to a larger radius. Therefore, radius measurements for highmass pulsars are of the utmost importance.
Current radius measurements are controversial. Determinations of the radius range from about 9 km (Guillot et al., 2013) to 15 km (Bogdanov, 2013), but one must be aware of possible systematic flaws (see, e.g., Heinke et al. 2014; Elshamouty et al. 2016); for a recent critical assessment see, for example, Fortin et al. (2015); Miller & Lamb (2016); Haensel et al. (2016). At a gravitational mass of the range of radius values mentioned above would correspond to a range of central densities of the compact star between , where fm is the nuclear saturation density.
Several observational programs for simultaneous measuring of pulsar masses and radii are currently in preparation: the Neutron star Interior Composition ExploreR (NICER, Arzoumanian et al. 2014), the Square Kilometer Array (SKA, Watts et al. 2015), Athena (Motch et al., 2013), and possibly, a LOFTsize mission (Feroci et al., 2012). Thus there is hope that in the near future it will be possible to reconstruct the cold NS matter EOS (here is pressure and is energy density) within the measurement errors from the measured relation by means of inverting the Tolman–Oppenheimer–Volkoff (TOV, Tolman 1939; Oppenheimer & Volkoff 1939) equations. In addition, specific proposals to use gravitational waves (GWs) for measuring the NS radius from either the observations of the inspiral (e.g., Bejger et al. 2005; Damour et al. 2012) or postmerger waveforms (e.g., Bauswein et al. 2015) were presented.
The systematic investigation of a wide class of hybrid stars with varying stiffness of hadronic matter at high densities and the possibility of quark matter with varying highdensity stiffness has revealed interesting findings (Alvarez et al., 2016). Possible future measurements of radii of recently discovered highmass stars (Demorest et al., 2010; Antoniadis et al., 2013) would select a hybrid EOS with a strong firstorder phase transition if the outcome of their radius measurement were to show a difference of about 2 km with significance. Such a possibility has been suggested earlier on the basis of a new class of hybrid star EOS that fulfill the generic condition that the baryonic EOS is strongly stiffened at high densities, for instance, by effects of the Pauli exclusion principle (quark exchange interaction between baryons), and a strong firstorder deconfinement phase transition requiring sufficiently soft quark matter at the transition between baryonic and quark phases, . The quark matter EOS has, however, to stiffen quickly with increasing density so that immediate gravitational collapse that is due to the transition does not occur, allowing stable hybrid stars to exist.
Such a solution for hybrid stars, which form a third family of compact stars that are disconnected from the baryonic branch of compact stars, is very interesting for the possible observational verification of specific features of phase transition to quark matter in NS cores. A sharp firstorder phase transition between pure B and Q phases, occurring at constant pressure , is associated with an energy density jump from to ( is the energy density including the rest energy of particles). Then, a general necessary condition for the existence of a disconnected family of NSs with Qphase cores is , with (Seidov, 1971). The second term in the brackets comes from general relativity. The condition implies that hybrid stars with small quark cores are unstable to radial perturbations and collapse into black holes (BH), and therefore implies the existence of a separate hybrid stars branch (a detailed review of the structure and stability of NSs with phase transitions in their cores is given in Sect. 7 of Haensel et al. 2007). This corresponds to a separate and small segment of the relation. High mass and small radii imply high spacetime curvature and strong gravitational pull, resulting in a relatively flat (nearly horizontal) segment. This hybrid star branch is characterized by a narrow range of , a broad range of , with weakly increasing with decreasing , up to a very flat maximum. These features of the hybridstar branch result in specific observational signatures of a strong BQ firstorder phase transition in NS cores. Moreover, these generic properties of the hybridstar branch indicate a relative ”softness” of their configurations with respect to perturbations that are due to rotation and oscillations, for example. All these generic features are studied in the present paper, using an illustrative example of an advanced EOS composed of a stiff baryonic segment and a strong firstorder phase transition to the quark phase (Sect. 3).
The phase transition might occur smoothed within a finite pressure interval through a mixed BQ phase layer. However, the interplay of the surface tension at the BQ interface and charge screening of Coulomb interaction (in a mixed state, B and Q phases are electrically charged) make the mixedstate layer very thin (Endo et al., 2006) so that the key features of the hybridstar branch remain intact (AlvarezCastillo & Blaschke, 2015).
Stable branches of static configurations in the plane have very specific generic features (see, e.g., Benic et al. 2015). The highmass baryon branch is very steep, not only nearly vertical, but even with increasing with , a feature characteristic of very stiff NS cores. The hybrid stable twinbranch is predicted to be flat, nearly horizontal with very broad maximum. Measuring radii of stars, which span a wide range of values from about to km, will clearly indicate a hybrid branch, and if moreover the radii for standard NS masses, , are roughly constant at approximately km, then the evidence for two distinct families that are separated as a result of a strong firstorder phase transition would be quite convincing. The range of central pressures for NSs of different radii at , which may bear a connection to the universal hadronization pressure found in heavyion collisions, is discussed in AlvarezCastillo et al. (2016).
This possibility of finding observational evidence for a firstorder phase transition in NS cores from the phenomenology of characteristics of NS populations offers the chance to answer to the currently controversial question of whether a critical endpoint of firstorder phase transitions in the QCD phase diagram exists (AlvarezCastillo & Blaschke, 2013; Blaschke et al., 2013).
Since the nature of the QCD transition at vanishing baryon density is beyond doubt identified as a crossover in two independent lattice QCD simulations at the physical point (Bazavov et al., 2014; Borsanyi et al., 2014), the evidence for a firstorder phase transition at zero temperature and finite baryon density necessarily implies the existence of a critical endpoint (CEP) of firstorder phase transitions in the QCD phase diagram (we note that there are theoretical conjectures about a continuity between hadronic and quark matter phases at low temperatures and finite densities in the QCD phase diagram (Schaefer & Wilczek, 1999; Hatsuda et al., 2006; Abuki et al., 2010); if, however, there were observational evidence for a horizontal branch in the diagram of compact stars, these considerations would become obsolete). The very existence of such a CEP is a landmark for identifying the universality class of QCD, and due to its importance for model building and phenomenology, a major target of experimental research programs with ultrarelativistic heavyion collisions at BNL RHIC (STAR beam energy scan, Stephans 2006) and CERN SPS (NA49 SHINE, Gazdzicki et al. 2006), in future at NICA in Dubna (Sissakian et al., 2006) and at FAIR in Darmstadt (Friman et al., 2011).
First examples for microscopically founded hybrid star EOS that simultaneously fulfill the above constraints for the existence of a disconnected hybrid star branch and for a high gravitational mass of about (which implies the existence of socalled highmass twin stars) have been given in Blaschke et al. (2013). The recent systematic investigation of highmass twin stars in AlvarezCastillo & Blaschke (2015) is based on the EOS developed in Benic et al. (2015), which joins a relativistic density functional for nuclear matter with nucleonic excluded volume stiffening with a Nambu–JonaLasinio (NJL) type model for quark matter that provides a highdensity stiffening as a result of higher order quark interactions. These examples belong to a new class of EOS that can be considered as a realization of the recently introduced threewindow picture for dense QCD matter (Kojo, 2016) as a microscopic foundation for the hybrid EOS that was conjectured by Masuda et al. (2013), suggesting a crossover construction between hadronic and quark matter. These three windows depicted in Fig. 1 of Kojo (2016) are characterized by the following:

at densities : occasional quark exchange between separated and still welldefined nucleons, leading to quark Pauli blocking effects in dense hadronic matter (Röpke et al., 1986) that can be modeled by a hadronic excluded volume or by repulsive shortrange interactions,

densities : multiple quark exchanges that lead to a partial delocalization of the hadron wavefunctions and to the formation of multiquark clusters and a softening of the EOS by an attractive mean field,
A still open and controversial question in this context is whether chiral symmetry restoration and deconfinement transition (which both coincide on the temperature axis according to lattice QCD simulations) would also occur simultaneously in the dense matter at zero temperature. If this were not the case, then there would be room for a more complex structure of the QCD phase diagram, for instance, with a triple point that is either due to a quarkyonic matter phase (light quarks confined in baryons that form parity doublets, McLerran & Pisarski 2007) or a massive quark matter phase (Schulz & Röpke, 1987) for which there is circumstantial evidence from particle production in ultrarelativistic heavyion collision experiments (Andronic et al., 2010). More circumstantial evidence for a region of strong firstorder phase transitions in the QCD phase diagram comes from the baryon stopping signal in the energy dependence of the curvature of the net proton rapidity distribution at midrapidity for energies in the intermediate range between former AGS experiments and the NA49 experiment at the CERN SPS (Ivanov, 2013). This signal has been proven to be quite robust under different experimental constraints (Ivanov & Blaschke, 2015) and against hadronic final state interactions (Batyuk et al., 2016).
On the basis of this discussion, the new class of EOS with a threewindow structure provides the theoretical background for a strong firstorder phase transition, the phenomena in the energy scan of heavyion collision experiments, and the creation of two disjoint families of NS configurations. Astrophysical observations of the NS twins with drastically different core compositions may be regarded as a manifestation of these dense matter EOS features in different regions of the QCD phase diagram and under different physical conditions.
In the present work we add to the discussion of highmass twin stars a detailed investigation of their properties under rigid rotation. This is because we expect a strong response of both branches to rotation. First, the high stiffness of the highdensity EOS of the baryonic branch results in large radii: km at , so that the effect of the centrifugal force will be large. Second, the stable static hybrid branch is flat, which makes it particularly susceptible to the effects of rotation: the margin of stability along this branch is narrow.
The neutron star instability induced by a strong phase transition in the EOS was studied in detail by Zdunik et al. (2006), who conjectured that the character of stability is not changed by the rotation rate of the star (disjoint families remain separated at any rotation rate). This question will be of particular interest for discussing compact star phenomena tied to the evolution of their rotational state, which eventually ends by collapse into a black hole (e.g., Falcke & Rezzolla 2014).
The general idea of the present work is to illustrate generic features of a class of highdensity quark phase transition EOSs using an exemplary EOS, in order to discuss the regions of stable and unstable configurations related to a strong firstorder phase transition. For illustration, we use one of the EOS that was recently developed by Benic et al. (2015).
The article is structured as follows. In Sect. 2 we describe the methods we used to calculate the rotating configurations, stressing particularly the need for high precision of numerical simulations. Precision is particularly important for testing stability criteria of stationary rotating configurations, which are formulated there. Section 3 starts with a brief presentation of the EOS that we use to illustrate generic properties of the rotating highmass twins. Then we construct families of rotating configurations, assess their stability, and classify the regions of instability and their generic features. Discussion of the results in Sect. 4 involves evolutionary considerations, potential scenarios leading to observational manifestations of massivetwin case, including dynamical phenomena triggered by the instabilities and their possible astrophysical appearances. The final part of Sect. 4 presents the conclusions.
2 Methods
In order to analyze the astrophysical consequences of the abovementioned specific type of EOS, we have obtained rigidly rotating, stationary, and axisymmetric NS configurations by means of the numerical library LORENE^{1}^{1}1http://www.lorene.obspm.fr nrotstar code, using the 3+1 formulation of general relativity of Bonazzola et al. (1993), and employing the multidomain pseudospectral decomposition (three domains inside the star). The accuracy is controlled by a 2D generalrelativistic virial theorem (Bonazzola & Gourgoulhon, 1994) and for the results presented here is typically on the order of . The high accuracy provided by the spectral method implementation of LORENE and a general low numerical viscosity of spectral methods is particularly suitable for studying the stability of NS models.
The evanescent error behavior of the solutions can be employed by expanding the number of coefficients to obtain the relevant values practically up to machine precision. In this context, we recall that a global parameter that is strictly conserved during the evolution of an isolated NS is its total baryon number (baryon charge) . Instead of it is convenient to use in relativistic astrophysics the baryon mass (or rest mass) of the NS defined by , where is a suitably defined baryon mass. is the total mass of noninteracting baryons; it is easy to extend these definitions to NS cores built of quarks (three quarks contribute +1 to the baryon number of NS). In our calculations we follow the LORENE unit convention and use a value of the mean baryon mass g. Other definitions of , suitable for specific applications in NS and supernova physics, are discussed in Sect. 6.2 of Haensel et al. (2007). We note that while is strictly constant in spinning down or cooling isolated NS, their gravitational mass is changing. We are in general interested in obtaining accurate values of the gravitational mass , the baryon mass and, the total angular momentum that are the properly defined functionals of the stellar structure and EOS suitable to study instabilities (for more details and a discussion comparing the 3+1 formulation with the slowrotation formulation see the recent review of the stability of rotating NSs with exotic cores, Haensel et al. 2016).
In the following we use the method described in Zdunik et al. (2004, 2006), who studied, among other things, the backbending phenomenon proposed for NSs in Glendenning et al. (1997). Backbending, a temporary spinup of an isolated NS that is decreasing its total angular momentum by the dipole radiation, for example, can be robustly quantified by analyzing the extrema of the baryon mass along the lines of constant spin frequency , that is, the rate of the baryon mass change with respect to a central EOS parameter (central pressure , e.g.,) changing along lines. The condition for backbending to occur is
(1) 
(see also Haensel et al. 2016, Sect. 7 for detailed description). Nevertheless, the susceptibility to backbending does not mean that the NS is indeed unstable. In order to study the regions of true instability, we use the turningpoint theorem formulated for rotating stellar models by Sorkin (1981, 1982); Friedman et al. (1988), which states that the sufficient condition for the change in stability corresponds to an extremum of the gravitational mass , or the baryon mass at fixed :
(2) 
or equivalently, to an extremum of at fixed either , or :
(3) 
In the following, we illustrate the generic features of the third family of compact stars with one of the EOSs developed by Benic et al. (2015). The baryon phase EOS is based on the relativistic meanfield (RMF) model DD2 of Typel et al. (2010). It fits the semiempirical parameters of nuclear matter at saturation well, and features densitydependent coupling constants. In order to obtain a highdensity stiffening of B phase, which is necessary to yield a strong BQ phase transition, while fulfilling for the hybrid Q branch, the DD2 model is modified using excluded volume (EV) effects that are included to account for the finite size of baryons, resulting in a DD2+EV model for the baryon phase. The EV effect was included without altering the good fit of DD2 to semiempirical nuclear matter parameter values. The quark phase description is based on the Nambu–JonaLasinio densitydependent model, with dimensionless coupling constants and . The complete EOS with a strong firstorder phase transition was obtained using Maxwell construction. We analyze the features of sequences of configurations along the lines of fixed baryon mass and total angular momentum .
3 Results
The static configurations of NSs for the sample EOS have been studied in Benic et al. (2015). The basic stellar parameters were the gravitational mass and the circumferential radius . To analyze the stability of rotating stars, it is more convenient to consider the baryon mass (also called the rest mass) of the star and the equatorial circular radius (see, e.g., Haensel et al. 2016). Generally, the study of rapidly rotating star configurations offers advantages for the investigation of phase transitions in NS interiors because as a function of the angular momentum (rotation frequency), which is an additional parameter, the profile of the density distribution and therefore also the interior composition would change, which is expected to allow for observational signatures in the course of the rotational evolution of the star (for early examples, see Glendenning et al. 1997; Chubarian et al. 2000). The spin frequency range we studied spans the astrophysically relevant range from Hz (static configurations, corresponding to the solutions of the TolmanOppenheimerVolkoff equations), to Hz (i.e., much higher than the frequency of Hz of the most rapid pulsar known to date, PSR J17482446ad of Hessels et al. 2006).
Stationary uniformly rotating configurations are labeled (determined) by two parameters. In Fig. 1 we show various continuous twoparameter curves in the plane. Here, the parameter can for instance be the central pressure , or central baryon chemical potential (both behaving continuously and monotonously along the curve). The quantity characterizes the uniform rotation of the star; we chose it to be equal to the frequency of rotation or the total stellar angular momentum . In Fig. 1 we study the regions of backbending and stability on the baryon mass equatorial radius plane. The region in which the backbending phenomenon is present is marked with the dashed lines. An isolated NS () that decreases its angular momentum (by electromagnetic dipole radiation, e.g.) crosses the lines of while moving from the right to the left side of Fig. 1. According to Eq. (1), in the region of dashed lines, the line crosses the curves such that it results in a spinup while the angular momentum is monotonically decreasing, that is, the backbending. The wavy pattern region denotes the opposite, usually observed behavior, that is, spindown with angular momentum loss.
Figure 1 also shows two instability regions. The first is the familiar instability with respect to axisymmetric perturbations related to the existence of the maximum mass; the blue line corresponding to maxima of and denotes its boundary. The second instability is induced by the strong phase transition (red strip between the local minimum and maximum of ). The latter divides the space of stable solutions into two disjoint families at any rotation rate (Zdunik et al., 2006). Assuming that at some moment in its evolution an isolated NS enters the instability strip, it becomes then unstable and collapses to another, more compact configuration along the lines of and (Dimmelmeier et al., 2009).
Configurations that survive such a minicollapse are located in the green region. We note that in this idealized picture no mass and angular momentum loss is assumed. Realistically, some mass and angular momentum loss may occur, and so the green region will decrease toward lower spin rates and toward the region that is stable against the backbending, which is marked with the wavy pattern in Fig. 1. Careful analysis of Fig. 1 reveals the existence of a critical angular momentum , such that for exceeding the maximum mass on the B branch (resulting from accretion) implies in a direct collapse into a rotating BH. This situation is depicted in detail in Fig. 2.
The existence of and its potential observational signatures are discussed in more detail in Sect. 4. Here we restrict ourselves to a few comments on how the existence of is due to generic features of the rotatingtwin curves. Consider first the static () B and Q branches. They have very different dependence. The maximum mass at (with ) is due is to the strong phase transition, and is somewhat lower than the flat maximum on the Q branch, . We proceed to the case of an increasing , when an NS on the B branch acquires mass and angular momentum from an accretion disk. At some moment it reaches a maximum on the B branch; it has then , . Under further accretion it collapses into a Q configuration of the same and , provided . The equality is reached exactly at , and for higher the B star collapses directly into a BH. A critical is equivalent to maximum however, which can be reached by the baryon stars. This situation is depicted in detail in Fig. 2 and is discussed further in Sect. 4. A separate question is related to the way in which the NS reaches the instability line. As shown in Zdunik et al. (2005) and Bejger et al. (2011), the efficiency of transfer of the angular momentum in the process of disk accretion governs the evolution of a spinningup NS. To illustrate how an NS can reach the unstable region, we present in Fig. 3 the mass–equatorial radius dependence for evolutionary tracks of accreting NS. A realistic evolutionary path depends on many physical details, such as the configuration and strength of the magnetic field, the accretion rate, and the way the magnetic field interacts with the disk. We assume thindisk accretion in the presence of the magnetic field by employing a model used previously in Bejger et al. (2011) and Fortin et al. (2016). Two different efficiencies of angular momentum transfer, and , are considered for initially nonrotating configurations of mass and . The initial magnetic field is G. If the accretion stops before the instability is reached, the star evolves by moving horizontally from right to left (secular spindown). Figure 3 is intended to show that it is possible to reach the instability region at low or high angular momentum corresponding to the two cases considered in Fig. 2.
The effect of the EOSinduced instability strip may also be studied on the angular momentum–spin frequency plane, assuming a fixed baryon mass sequence, see Fig. 4. The evolution of an isolated NS that decreases its angular momentum corresponds to a downward movement in this figure. This also means that the central density (and pressure) of such a star increases. A spinningdown NS on the B branch decreases its until it becomes unstable (point B), which forces it to collapse (dynamically migrate) to another stable branch (point Q) with the same and . We note here that in principle by adopting a certain EOS we may place constraints on the parameters of massive pulsars with known spin frequency. In the example from Fig. 4, an NS with and the spin frequency of PSR 16142230 (317 Hz) is located on either the upper baryonic branch denoted by B, or on the lower branch (hybrid stars  Q), the two configurations differing greatly in the total angular momentum because of very different moments of inertia. A configuration with a slightly lower (blue line) may, for this particular EOS, exist at 317 Hz only in the B phase, since the instability transfers it to frequency Hz (significantly lower than 317 Hz). A configuration with a slightly higher (red line) is excluded as a model of PSR 16142230; during its evolution, it never spins down to reach 317 Hz: after the dynamical migration and a period of spindown, it will collapse to a BH.
The time evolution of the rotation period of an isolated NS loosing its energy and angular momentum due to the dipole radiation is presented in Fig. 5. The assumption of dipole radiation from the pulsar leads to the formula
(4) 
where is the energy of rotating pulsar and the dipole moment of a star. In the framework of general relativity, we should use the total massenergy of the star in place of . For the evolution of an isolated NS with a fixed total number of baryons, the relation holds, resulting in the equation for the time evolution,
(5) 
where is the stellar angular momentum in the units of and is time in kyr (the same relation in a Newtonian case could be obtained for and , with being the NS moment of inertia). The timescales of slowing down of a solitary pulsar from milliseconds (close to maximum massshedding frequency) down to ms before and after the minicollapse are clearly comparable  we assumed that the magnetic moment does either not change during the evolution, or that the magnetic field decreases.
A minicollapse is a dynamical process, provided the BQ conversion has detonation character (Haensel et al., 2016). It involves considerable spinup and a substantial reorganization of the interior of the star (Dimmelmeier et al., 2009). It is also exoenergetic: a substantial amount of energy, quantified as the difference between the initial and the final gravitational mass, , is released in the process. The left panel of Fig. 6 shows the relation between the angular momentum of the star and the energy difference . The relation is approximately quadratic in : . For the particular EOS used in this study, and , for in and in units. The line ends at a critical where the dynamical collapse forces an NS to the instability region, where it collapses to a BH (see also the right panel of Fig. 2). The right panel of Fig. 6 shows the amount of spinup (spin frequency difference) that is acquired in the minicollapse as a function of the initial spin frequency . The situation presented here corresponds to a specific case studied in detail in Zdunik et al. (2008), where the overpressure of the new metastable phase is set to zero, .
From the same plot one may estimate the ‘Newtonian’ change of kinetic energy and the luminosity of the process. Assuming that the total angular momentum does not change during the process, we obtain
(6) 
For the value of the change of the spin frequency is approximately Hz ( rad/s). For these figures we obtain erg. This value overestimates the difference in total gravitational mass by one order of magnitude ( erg). The process occurs on a dynamical timescale of a millisecond.
4 Discussion
The goal of this article is to bona fide consider a specific class of EOS featuring a substantial phase transition motivated by the theory of dense matter physics. The EOSinduced instability region divides stable NS configurations into two disjoint families (twin families). Its existence has interesting astrophysical consequences for rotating NSs. We note that it facilitates a natural (i.e., not finetuned) way for various astrophysical phenomena that we list below.
Spin frequency cutoff. Even though theoretical models of NSs allow for spin rotation rates much above 1 kHz and although with current observational techniques such rapidly rotating pulsars could be detected (see, e.g., Patruno 2010; Davoust et al. 2011), so far, the most rapidly rotating NS observed is PSR J17482446ad (716 Hz, Hessels et al. 2006). It cannot be excluded a priori that some rapidly rotating and massive NSs were created close to their currently observed state, that is, in a specific type of corecollapse supernovæ. If this were the case, then they might appear practically everywhere on the right side of the thick blue line of Fig. 1, with the exception of the instability strip (red area), where no stationary axisymmetric solutions are possible. However, as the observations, evolutionary arguments, and numerical simulations tend to suggest, NS that become radio pulsars are not born with 1  3 ms periods, but with much longer periods of ms, see FaucherGiguère & Kaspi (2006), Kramer et al. (2003), Table 7.6 in Lyne & GrahamSmith (1998), and references therein (a specific class of NS with millisecond periods at birth in massive corecollapse supernovae are thought to be progenitors of magnetars, which are observed as softgamma ray repeaters or anomalous Xray pulsars, see e.g., Kargaltsev & Pavlov 2008). Then, after slowing down to a period of a few seconds and entering the pulsar graveyard, they gain their angular momentum, as well as mass, during longterm accretion processes in lowmass binary systems (in the socalled recycling of dead pulsars, see Alpar et al. 1982; Radhakrishnan & Srinivasan 1982; Wijnands & van der Klis 1998), and it is possible that some of them enter the strong phase transition instability strip sometime in their evolution. Sufficiently massive and sufficiently rapidly rotating NS will then migrate dynamically along the track in the direction of the twin branch (see, e.g., Dimmelmeier et al. 2009). Moreover, for some critical angular momentum (critical spin frequency) the value of on the right side of the instability strip (the peaks in Fig. 1) is higher than the corresponding maximum of on the twin branch  in that case, the star collapses to a BH. No observations of backbending in radiopulsar timing. One of the observational predictions related to substantial densematter phase transitions is the detection of the backbending phenomenon, which occurs at spin frequencies of known pulsars. As we showed in Sect. 3, Fig. 1, NSs exhibiting an instability that is caused by a strong phase transition avoid the vast majority of the backbending region for spin frequencies lower than some critical value. The most rapidly rotating currently known pulsar, PSR J17482446ad, has a spin period of 716 Hz. From Fig. 1 we note that the Hz line is the first dashed line above the no backbending (wavy pattern) region. When we assume that the EOS used for illustration is the true EOS of dense matter, this means that PSR J17482446ad, which does not show the features of backbending in the timing, still resides on the hadronic branch (does not contain the quark core). Additionally, the most massive stars that are in the backbending region may not be effective pulsars  they may be electromagnetically exhausted, with their magnetic field dissipated in the violent process of minicollapse, and therefore not easily detectable.
Radius gap. The existence of an instability strip creates in a mass and spinfrequencydependent radius region of avoidance between the allowed green region and stable baryonic branch on the right of Fig. 1, which is broadened with increasing mass (see also Fig. 3 of Benic et al. 2015). For nonrotating NS, the predicted gap is km. Smallradius Qbranch twins have within a very narrow mass range . The measurement of a radius km for a star indicates a Bbranch configuration. If for another NS the radius is determined to be within the range , then we obtain strong evidence in favor of distinct B and Q twin branches as a result of a strong BQ phase transition.
Moment of inertia gap. The twins on the B and Q branches have different internal structure. The Bstar twin is more compact, and its mass is concentrated in the dense quark core. Consequently, at the same of twins, one has significantly smaller than , as shown on Fig. 7. Moreover, the strong firstorder phase transition BQ results in an gap between the B and Q twins (Fig. 7). The moment of inertia of NS can be measured through the spinorbit effect contribution to the timing parameters for a binary of two radio pulsars (Damour & Schaefer, 1988). The first binary of this type, PSR J07373039A,B was discovered more than a decade ago (Lyne et al., 2004). Pulsar B has become invisible to terrestrial observers in March 2008 because its beam wandered out of our line of sight as a consequence of the geodetic precession effect (Perera et al., 2010). It may reappear as late as in 2034 (or later), depending on the model of the pulsar magnetosphere (see, e.g., Lomiashvili & Lyutikov 2014). Since the mass of pulsar A has been accurately determined, a measurement of its moment of inertia through the spinorbit momentum coupling would allow us to constrain the radius and hence the EOS (Lattimer & Schutz, 2005). Given the present timing accuracy of the system’s postKeplerian parameters, that is, the periastron advance, the decrease in the orbital period and the Shapiro shape parameter, from which the spinorbit coupling contribution is derived, reasonable accuracy may be achieved around the time of pulsar B reappearance in years (Kramer & Wex, 2009). However, in the forthcoming era of large radio telescopes (e.g., FAST, SKA) the number of known pulsars will increase by orders of magnitude, including many thousands of millisecond pulsars, out of which we may hopefully expect tens of binary systems with two pulsars suitable for simultaneously measuring and of an NS. A sufficiently dense set of pairs resulting from these future measurements could then be used to confirm or reject the generic shape in Fig. 7.
EOSinduced dynamical collapse as an energy reservoir. We assume that the BQ phase transition, mediated by strong interaction, occurs in the detonation regime. A dynamical process involving NS, triggered by the loss of stability, is associated with a substantial energy release (), heating of dense matter, kinetic energy flow, and some emission of radiation, on both a short timescale (NS quake, minicollapse) and long timescale (surface glowing). The process is probably also associated with a substantial rearrangement of the NS magnetic field. For NS with total angular momenta larger than some critical value, it leads to a direct collapse to a BH. This event is related to the expulsion of the magnetic field and thus the dynamical migration to the highmass twin branch may be considered a natural extension of the Falcke & Rezzolla (2014) cataclysmic scenario. Alternatively, for NSs below the angularmomentum threshold, the minicollapse dynamics may influence the magnetic field to such an extent that it becomes a transient source of observable magnetospheric emission after the final configuration ends up on the twin branch.
To conclude, the assumption of the strong phase transition in the NS EOS leads to a number of falsifiable (at least in principle) astrophysical predictions. As we described in Sect. 3 and in Fig. 4, a transition to a new exotic phase (deconfined quarks in this example) constrains the range of available and . This reasoning can be extended to other EOS functionals, like the moment of inertia ; moreover, these constraints may be combined with the observations that are sensitive to the composition of the core, for example, NS cooling studies. The evolutionary scenario in which some of NS collapse to BHs produces a specific NSBH mass function without a mass gap, which should be possible to test with current and future searches for lowmass BHs with microlensing surveys (Wyrzykowski et al., 2016), for instance. Unless the NS magnetic field is amplified and/or reoriented during the minicollapse event, it is likely that it is dissipated and disordered during the process  in the latter case, we expect a population of massive ineffective pulsars with a low magnetic field. Moreover, a dynamical minicollapse creates a characteristic signature of the GW emission, strongly dependent on the EOS and mass of the NS; short transient GW radiation of this type should be detectable by the advanced era interferometric detectors (Dimmelmeier et al., 2009). If the collapse is not entirely axisymmetric, the final configuration may retain an asymmetry, thus creating a rotating NS that spins down while continuously emitting the almostmonochromatic GWs. Such objects are among the prime astrophysical targets of the Advanced LIGO and Advanced Virgo interferometric detectors (see the demonstration of search pipelines used on the initial LIGO and initial Virgo detector data, e.g., Aasi et al. 2014a, b, 2015).
There are several open questions related to the various aspects of the highmass twin scenario, such as what the conditions are for the NS to reach the instability region via the disk accretion spinup, what the influence of the possible metastability of the quark phase core is on the NS population on the twin branch, and how the electromagnetic and gravitationalwave emission depends on the parameters of the NS. These questions will be addressed in subsequent studies.
Acknowledgements.
This work has been supported in part by the Polish National Science Centre (NCN) under grant No. UMO2014/13/B/ST9/02621. D.B. has been supported in part by the MEPhI Academic Excellence Project under contract No. 02.a03.21.0005. The authors acknowledge support from the COST Action MP1304 ”NewCompStar” for their networking activities.References
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