Issue 
A&A
Volume 622, February 2019



Article Number  A174  
Number of page(s)  11  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201833969  
Published online  18 February 2019 
Tidal deformability and other global parameters of compact stars with strong phase transitions
^{1}
Nicolaus Copernicus Astronomical Center, PAS, ul. Bartycka 18, 00716 Warsaw, Poland
email: msieniawska@camk.edu.pl
^{2}
Astronomical Observatory, Adam Mickiewicz University, Poznań, Poland
^{3}
APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cité, 75205 Paris Cedex 13, France
Received:
27
July
2018
Accepted:
24
December
2018
Context. Using parametric equations of state (relativistic polytropes and a simple quark bag model) to model densematter phase transitions, we study global, measurable astrophysical parameters of compact stars such as their allowed radii and tidal deformabilities. We also investigate the influence of stiffness of matter before the onset of the phase transitions on the parameters of the possible exotic dense phase.
Aims. The aim of our study is to compare the parameter space of the dense matter equation of state permitting phase transitions to a subspace compatible with current observational constraints such as the maximum observable mass, tidal deformabilities of neutron star mergers, radii of configurations before the onset of the phase transition, and to give predictions for future observations.
Methods. We studied solutions of the TolmanOppenheimerVolkoff equations for a flexible set of parametric equations of state, constructed using a realistic description of neutronstar crust (up to the nuclear saturation density), and relativistic polytropes connected by a densityjump phase transition to a simple bag model description of deconfined quark matter.
Results. In order to be consistent with recent observations of massive neutron stars, a compact star with a strong highmass phase transition cannot have a radius smaller than 12 km in the range of masses 1.2 − 1.6 M_{⊙}. We also compare tidal deformabilities of stars with weak and strong phase transitions with the results of the GW170817 neutron star merger. Specifically, we study characteristic phase transition features in the Λ_{1} − Λ_{2} relation, and estimate the deviations of our results from the approximate formulæ for Λ∼ − R (M_{1}) and Λcompactness proposed in the literature. We find constraints on the hybrid equations of state to produce stable neutron stars on the twin branch. For the exemplary equations of state most of the highmass twins occur for the minimum values of the density jump λ = 1.33 − 1.54; corresponding values of the square of the speed of sound are α = 0.7 − 0.37. We compare results with observations of gravitational waves and with the theoretical causal limit and find that the minimum radius of a twin branch is between 9.5 and 10.5 km, and depends on the phase transition baryon density. For these solutions the phase transition occurs below 0.56 fm^{−3}.
Key words: stars: neutron / equation of state / dense matter
© ESO 2019
1. Introduction
The interior composition of neutron stars (NSs) is still not fully known today. Neutron stars are so compact that the density in their central regions exceeds by far the density of the atomic nuclei. At such extreme conditions matter may exist in a form impossible to obtain and study in terrestrial laboratories. One of the possibilities of very dense matter is the deconfinement of quarks.
Astrophysically, we study the dense matter in the interiors of NSs by measuring their parameters (masses, radii, etc.) and compare them to the theoretical models of a structure and equation of state (EOS). The multiple EOS models, which include a deconfined quarkmatter segment, were recently proposed in RaneaSandoval et al. (2016), Alford & Sedrakian (2017), Kaltenborn et al. (2017), Mellinger et al. (2017), Christian et al. (2018) and Typel & Blaschke (2018; see e.g. Buballa et al. 2014; Alford & Han 2016, for a review). A defining characteristic of these EOS is the phase transition between normal matter and quark matter. To be consistent with the observational constraints, all the EOS have to satisfy the maximum mass M_{max} criterion, which is now established to be approximately 2 M_{⊙} (see Antoniadis et al. 2013, for PSR J0348+0432 and Demorest et al. 2010, for PSR J1614–2230, as well as Fonseca et al. 2016, for the reevaluation of the mass of the latter).
Creation of a new quark phase–the appearance of a new phase core in the centre of a NS–may lead to a destabilisation of a part of the sequence of NS configurations (where the sequence is labelled by a central EOS parameter, such as central pressure, P_{c}). If the stability is regained further up the sequence, it is considered a detached (twin) branch of the compact stars (see right panel of Fig. 1 for a schematic depiction of this situation in the M(R) plane). In principle, both mass maxima (twin solutions) may exist around the maximum mass M_{max} ≈ 2 M_{⊙}, and is called the highmass twins solutions.
Fig. 1. Left panel: schematic EOS with a density jump phase transition on the pressure P–baryon density n_{b} plane. Right panel: gravitational mass M–radius R sequence of solutions in the case of a strong (destabilising) phase transition. We show the characteristic masses in this situation: the mass at the phase transition density M_{ph}, the maximum mass M_{max}, and the minimum mass M_{min} at the end of the instability (decreasing part of M(R) between M_{min} and M_{ph} indicated with a dotted line). Stable configurations between the M_{max} and M_{min} are sometimes called the twin branch. 
In this article we study the properties related to the twin branch of the NS using a simplified parametric EOS that allows us to analyse the space of NS solutions. Specifically, we study the influence of a strong (destabilising) phase transition on the tidal deformability of the NS, and apply these results to the recent measurements obtained by the LIGOVirgo Collaborations from the GW170817 event (Abbott et al. 2017, 2018, 2019). For the recent studies concerning the tidal deformability see Annala et al. (2018), Most et al. (2018), Nandi & Char (2018), Paschalidis et al. (2018) and Raithel et al. (2018). We also aim to constrain how the part of the EOS before the phase transition influences the parameters of the EOS after the phase transition. In other words, we aim to put constraints on the observable parameters of the NS (such as the radius that is potentially a measurable parameter) for the masses below that at which the phase transition occurs, and investigate how the shape of the M(R) is related to the properties of the twin branch.
The article is composed as follows. In Sect. 2 we describe the parametric EOS and the methods of obtaining the solutions of the hydrostatic equilibrium equations for the NSs. In Sect. 3 we present the properties of NS sequences compatible with existing observational constraints, such as the M_{max} requirement and the tidal constraints of GW170817. Section 4 contains discussion and summary.
2. Equations of state and methods
In order to survey the space of solutions corresponding to M(R) sequences with highdensity phase transitions, we employ a conservative approach and use the following simplified, parametric EOS. We assume the knowledge of the lowdensity part of the EOS and adopt the SLy4 EOS description of Haensel & Pichon (1994) and Douchin & Haensel (2001) up to a baryon density n_{0}, comparable to and typically larger than the nuclear saturation density (n_{s} ≡ 0.16 fm^{−3}). At n_{0} a relativistic polytrope (Tooper 1965) replaces the tabulated SLy4 EOS. The definition of the pressure P and the energydensity ρc^{2} are standard,
where κ is the pressure coefficient, γ is the index of the polytrope, and m_{b} is the mass of the baryon in this phase. The index γ is a parameter of choice; consequently, by demanding the chemical and mechanical equilibrium at n_{0}, κ and m_{b} are fixed. The polytrope ends at a density n_{1} > n_{0}, and is connected to a simple bag EOS (Chodos et al. 1974; Farhi & Jaffe 1984), characterising the quark matter. We use a linear pressuredensity relation of Zdunik (2000),
with α denoting square of the speed of sound in a quark matter, and ρ_{*} and n_{*} the energy density and baryon density of this matter at zero pressure, respectively. We assume that at the polytrope/bag boundary matter is softened by the firstorder phase transition defined by a density jump λ = n_{2}/n_{1}. Maxwell construction at this point results in a corresponding massenergy density jump ρ_{2}/ρ_{1} = λ + (λ − 1)P_{1}/ρ_{1}c^{2}. The values of n_{2}, ρ_{2} determined by the definition of the transition point (n_{1}, ρ_{1}, P_{1}), together with a given λ yield, from Eq. (2), the values of ρ_{*} = ρ_{2} − P_{1}/αc^{2} and n_{*}. The schematic pressure–density relation for such an EOS is presented in the left panel of Fig. 1. The initial parameter ranges are shown in Table 1.
Ranges of the polyquark EOS parameters used in the study.
Given the EOS, we solve the equations of hydrostatic equilibrium for a spherically symmetric distribution of mass (Tolman 1939; Oppenheimer & Volkoff 1939),
supplied with the equation for one of the metric functions,
for a spherically symmetric metric of the form
and the equation for the total gravitational mass inside the radius r:
We solve an additional equation for the tidal deformability of the star, defined as
It represents the reaction of the star on the external tidal field (such as that in a tight binary system; e.g. Abbott et al. 2017). Influence of the tidal field is obtained in the lowest order approximation, by calculating the second (quadrupole) tidal Love number k_{2} (Love 1911)
with the star’s compactness x = GM/Rc^{2}, and y the solution of
evaluated at the stellar surface (Flanagan & Hinderer 2008; Van Oeveren & Friedman 2017). In the following we use the normalised value of the λ_{td} parameter,
Concerning the current sensitivity of the detectors (Abbott et al. 2017), what is actually measured is the effective tidal deformability , defined as
with M_{1} and M_{2} denoting the component masses. The equations are solved using a RungeKutta fourthorder numerical scheme with a variable integration step (see e.g. Press et al. 1992 for details) for a range of central parameters of the EOS (central pressure P_{c}) to obtain global parameters of NSs: their masses and radii in the form of the massradius M(R) sequence.
3. Features of the twin branch
From all the possible M(R) sequences resulting from the parameter set presented in Table 1, we select those consistent with the following criteria:

the maximum mass M_{max} is equal to or larger than the largest currently measured NS mass: M_{max} ≥ 2 M_{⊙} (Demorest et al. 2010; Antoniadis et al. 2013; Fonseca et al. 2016);

strong (destabilising) phase transition in the EOS, i.e. an instability near the phase transition point that results in the local minimum in the M(R) curve and the presence of the twin branch;

unless stated otherwise, the maximum mass is located on the twin branch, i.e. M_{max} ≥ M_{ph} (e.g. Sect. 3.3 for twin sequences with M_{max} ≤ M_{ph});

the EOS is causal (the speed of sound is smaller than the speed of light in vacuum).
3.1. Massradius diagram: minimum radius for the second branch and the tidal deformability
We are interested in establishing a lower limit on the radius R at densities below the phase transition point for the EOS compatible with the twin branch scenario. Since it is likely that a large fraction of the NS population has masses below M_{max} = 2 M_{⊙}, the future measurements of their radii will be indicative of whether the EOS is able to support a phase transition at higher densities (highmass twins scenario) in a way compatible with the observations of the massive NSs. In order to obtain the lower bound on the radius, we adopt the α parameter in the quark phase equal to its extreme value, α = 1. Characteristic shapes of the M(R) relations are presented in Fig. 2, where selected regions of the Λ parameters in the M(R) plane are also shown. The values of Λ decrease with M and increase with R thanks to the strong scaling with these values. A limit of 900 corresponds to the lower edge of the orange region. In principle, we can construct a specific EOS corresponding to limiting R_{ph} or M_{ph} still consistent with this bound. We conclude that the minimum radius R on the polytropic branch of the M(R) sequence that still allows for the twin branch scenario cannot be smaller than 12 km with the adopted SLy4 crust, for the astrophysically interesting range of masses (≃1.0 − 2.0 M_{⊙}). Softer prephasetransition polytropes which give R < 12 km result either in M_{max} < 2 M_{⊙}, or in M_{ph} > M_{max}. In addition to the polytropic index γ, factors that determine the radius for stars below the highdensity twin branch segment are the densities n_{0} at which the polytrope connects to the SLy4 EOS (lowdensity SLy4polytrope connection results in larger radii) and its ending point n_{1}. This simple exercise shows that the measurements of radii in this range of masses disclose information pertinent to our understanding of the EOS at densities above the nuclear saturation density.
Fig. 2. Massradius diagram for the polyquark EOS (α = 1) with the ranges of Λ(M, R) values colourcoded (see inset). The configurations on the M(R) plane may have significantly different Λ values. This is especially visible for the blackonwhite line denoting the SLy4 EOS M(R), which is placed on the top of its Λ ranges (see Fig. 3 for a closeup). The dotted horizontal line denotes 1.4 M_{⊙}. The dashed inclined black line denotes an approximate division between the purely polytropic M(R) sequences and sequences containing quark core (see Eq. (12)). The pale green and blue bands correspond to the component mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). For reference, in the upper left corner of the plot we indicate the regions excluded by the requirement of the EOS causality (speed of sound in the dense matter less than the speed of light in vacuum, Haensel et al. 1999), and the photon orbit (3GM/Rc^{2}) in the Schwarzschild case. 
Fig. 3. Zoomedin image of Fig. 2 around the SLy4 EOS M(R) sequence showing the differences in Λ for the configurations of the same M and R. 
Fig. 4. Massradius diagram for the polyquark EOS (α = 1) with the curves colourcoded according to their Λ(M_{1.4}) values. The blackonwhite solid line denotes the SLy4 EOS M(R). The regions with different Λ(M_{1.4}) overlap each other, also because one sequence may have several values of Λ(M_{1.4}) due to the nature of the strong phase transition around M = 1.4 M_{⊙}. The nontrivial behaviour of M(R) hinders the separation of the EOS on the M(R) plane with respect to the values of Λ(M_{1.4}) they yield (as proposed in Annala et al. 2018). 
The thick dashed black line in Fig. 2 approximates the minimum radius of the configurations before the phase transition,
with R_{SLy4}(1 M_{⊙}) = 11.85 km. For comparison, we also plot the SLy4 EOS curve, which resides entirely in the quarkcore part of the M(R) diagram. Figure 5 shows the relation between the central pressure P_{c} for configurations with M_{ph} and R(M_{ph}) at the end of the polytropic branch; the leftside boundaries of the scatter plots correspond to the minimum radius line in Fig. 2; for example, stars at the polytropic branch with radii smaller than 12 km and corresponding masses smaller than 1.2 M_{⊙} cannot exist with central pressures larger than approximately 7.5 × 10^{34} dyne cm^{−2}. The relation between M_{max} and R(M_{ph}) is shown in Fig. 6. Small radii at the transition point support lower maximum masses on the twin branch.
Fig. 5. Central pressure P_{c} vs. mass and radius M_{ph} and R(M_{ph}) at the end of the polytropic branch (corresponding to the phase transition point). 
Fig. 6. Maximum mass on the twin branch M_{max} vs. radii at the phase transition point R(M_{ph}). 
3.2. Tidal deformability of stars with weak and strong phase transitions
We now study the tidal deformability Λ parameters (Eq. (10)) using a subset of the parametric EOS. The ranges of astrophysically interesting Λ parameters are indicated in Fig. 2. We note that it is virtually impossible to define a clear boundary between different Λ ranges on the M(R) plane. As the value of Λ depends sensitively on the EOS of dense matter, it cannot be treated as a simple function of M and R only. This is particularly visible for the comparison of the SLy4 EOS in Fig. 3. Here, depending on the EOS, configurations with the same compactness GM/Rc^{2} but visibly different Λ occupy the same position on the M(R) diagram, e.g. for M = 1.32 M_{⊙} and R = 11.75 km, the end of the magenta segment yielding Λ = 450 is located in the middle of the Λ = 300 − 450 range for the parametric EOS.
Figure 4 shows a M(R) diagram with the EOS colourcoded according to their values of Λ(M_{1.4}). Here the situation is even less clear in terms of defining the Λ on the M(R) plane, also because of the nonmonotonic behaviour of M due to the strong phase transition: a given EOS may yield several values of Λ(M_{1.4}) corresponding to different R. We conclude that representing Λ(M_{1.4}) on the M(R) diagram (as proposed by e.g. Annala et al. 2018) is not the best way of uncovering the relation between the Λ and other EOS functionals. Our results are based on a relatively simple model of the EOS; adding more degrees of freedom will only further complicate this picture.
Using the selected EOS from Fig. 2 as an example, we discuss the tidal deformability Λ for the strongphase transition EOS by applying a measurement of the chirp mass,
and the component mass ranges from the one binary NS merger observed so far in gravitational waves, the GW170817 event (Abbott et al. 2017). We adopt the central value of the chirp mass, ℳ = 1.188 M_{⊙}, and component masses in the lowspin prior estimation case: M_{1} ∈ (1.36, 1.60), M_{2} ∈ (1.17, 1.36) (Abbott et al. 2017). An improved analysis that gives consistent results is described in Abbott et al. (2018, 2019), where revised values for the lowspin prior of the chirp mass and component masses are ℳ = 1.186 M_{⊙}, M_{1} ∈ (1.36, 1.60), and M_{2} ∈ (1.16, 1.36). A recent correction of the misprint reveal that (Table IV in Abbott et al. 2019), instead of (Abbott et al. 2017). Nevertheless, due to the very small differences between initial and reevaluated estimations of the mentioned values, and the fact that their estimation errors overlap (in the case of tidal deformability values, errors of the estimations are considerable and depend on the waveform model, e.g. for the symmetric TaylorF2 waveform model and for HPD PhenomDNRT , according to Abbott et al. 2019), we decided to use the results from the detection paper.
To visualise the influence of a phase transition softening on Λ, we manufacture a selection of the EOS that differ in the size of the density jump λ = n_{2}/n_{1}, and otherwise share their parameters (M(R) diagrams shown in Fig. 7). The phase transition from the polytropic segment of γ = 4.5 to the linear EOS of α = 1 occurs at n_{1} = 0.335 fm^{−3} (crustcore transition from the SLy4 EOS to a polytropic segment occurs at n_{0} = 0.21 fm^{−3}). The parameters are selected such that the onset of the softening occurs within the range of masses estimated for the GW170817 event (Abbott et al. 2017).
Fig. 7. Massradius relations for the polytropicquark EOS with realistic SLy4 crust. Selected M(R) relations correspond to a polytropic segment of γ = 4.5, connected to the SLy4 crust at n_{0} = 0.21 fm^{−3} and to a linear EOS of α = 1 (Eq. (2)) at n_{1} = 0.335 fm^{−3}. The curves differ by the density jump λ = n_{2}/n_{1}, as indicated in the plot. Regions destabilised by the phase transition are indicated by dotted segments. The black line indicates the SLy4 EOS. The pale green and blue bands correspond to the mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). 
In Fig. 8 we plot the Λ_{1} − Λ_{2} relations for the component masses of the GW170817 event by assuming the estimated central value of the chirp mass, ℳ = 1.188 M_{⊙}. Depending on the strength of the phase transition (measured by the size of the density jump λ), the Λ_{1} − Λ_{2} relation may exhibit a nonmonotonic behaviour, as shown in the left panel of Fig. 8 for the blue curve. This feature is a direct consequence of a nonmonotonic behaviour of the gravitational mass M as a function of the tidal deformability Λ (right panel of Fig. 8), which is connected to the instability caused by a sufficiently strong phase transition. Detecting this feature in the incoming observations of NS mergers will clearly signal a densematter softening in the range of central parameters corresponding to NS merger component masses.
Fig. 8. Λ_{2}(Λ_{1}) (left panel) and M(Λ) (right panel) relations for the M(R) sequences from Fig. 7. The values of the Λ parameter are based on the measurements of the chirp mass and the lowspin prior estimates of the component masses in the binary NS merger GW170817 (Abbott et al. 2017). Shaded areas in the left panel denote the estimated 50% and 90% confidence regions corresponding to the measurement (Abbott et al. 2017). 
Additionally, we produced a set of four strong phasetransition M(R) curves, displayed in Fig. 9: their common parameters are γ = 4.5, n_{1} = 0.335 fm^{−3}, λ = 1.7, α = 1. The EOS differ only in the values of the SLy4polytrope density matching point, n_{0} = 0.16, 0.185, 0.21, 0.235 fm^{−3}, respectively. These n_{0} values were selected so that the phase transition mass M_{ph} is placed below, above, or within the ranges of the component masses for the lowspin prior estimation of the GW170817 event (Abbott et al. 2017). The Λ_{1} − Λ_{2} relations are displayed in Fig. 10. Specifically, the green and blue curves are mirror rotated versions of themselves, reflecting the fact that the phase transition occurs in the range of M_{1} or M_{2}, respectively.
Fig. 9. Massradius relations for the polytropicquark EOS with realistic SLy4 crust. Selected M(R) relations correspond to γ = 4.5, connected to the SLy4 crust at the baryon densities n_{0} indicated on the plot (magenta EOS at n_{0} = 0.235 fm^{−3}, blue EOS at n_{0} = 0.21 fm^{−3}, green EOS at n_{0} = 0.185 fm^{−3}, red EOS at n_{0} = 0.16 fm^{−3}). In all the selected cases, the quark EOS starts at n_{1} = 0.335 fm^{−3}, the density jump is λ = 1.7, and α = 1 (Eq. (2)). Regions destabilised by the phase transition are indicated by dotted segments. The black line denotes the SLy4 EOS. The green and blue bands correspond to the mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). 
Fig. 10. Λ_{2}(Λ_{1}) (left panel) and M(Λ) (right panel) relations for the selected twin branch sequences, with colourcoding and symbols as in Fig. 9. 
When both configurations (twin case) are present in the relevant range of masses (e.g. blue and green curves in Fig. 9 and in the right panel of Fig. 10), the twin branch gives more compact M(R) configurations in comparison to those on the polytropic branch, their Λ values are in general smaller, and thus preferable from the point of view of the GW170817 observations (Abbott et al. 2017). Likewise, larger radii before the phase transition onset (e.g. red curve in Fig. 9) are generally less favoured by the observation of GW170817. This may put limits on the proposed highmass twin branch solutions in which the onset of a phase transition occurs near 2 M_{⊙} (Benić et al. 2015; Kaltenborn et al. 2017; Typel & Blaschke 2018). Figure 2 visualises the requirement for a highmass twin solution and the radius: the higher the M_{ph}, the larger the radius at M_{ph}. For M(R) dependencies presented in Fig. 2 in the very large range of masses radius increases with mass; this property is typical for a very stiff EOS. As a consequence, highmass twin configurations in the range of GW170817 masses (before the phase transition) have larger radii and larger Λ than lower mass twins.
The twin branch segments lie in the region compatible with the SLy4 EOS. Even though the magenta EOS (n_{0} = 0.235 fm^{−3}) and the blue EOS (n_{0} = 0.21 fm^{−3}) massradius relations cross the SLy4 EOS M(R) sequence, their Λ values differ in these points, as shown in the right panel of Fig. 10. For the SLy4magenta EOS crossing at M ≈ 1.17 M_{⊙}, the ΔΛ ≈ 120 (to be compared with the value of Λ_{Sly4} ≈ 970 at this point) and ΔΛ ≈ 35 for the SLy4blue EOS crossing at M ≈ 1.51 M_{⊙} (Λ_{Sly4} ≈ 190 at this point). Generally, configurations with the same M and R (same compactness M/R) have notably different tidal deformabilities depending on their EOS. We plot the set of configurations from Fig. 2 on the Λcompactness x = GM/Rc^{2} plane (Fig. 11), and compare the results with the fitting formula proposed in Maselli et al. (2013) and recently updated in Yagi & Yunes (2017), x_{fit} = ∑_{i}a_{i}(log(Λ))^{i} (see Eq. (78) and Fig. 15 in Yagi & Yunes 2017). The deviation from the Yagi & Yunes (2017) fit is, for the EOS used here, typically larger than 5% and increases above 10% for Λ > 100. The best fit to our set of EOS has the following parameters: a_{0} = 0.353, a_{1} = −0.0359, and a_{2} = 0.000790 (indicated by a green line in Fig. 11).
Fig. 11. Compactness GM/Rc^{2} as a function of Λ for the α = 1 EOS from Figs. 2–4 (grey lines). The black line denotes the SLy4 EOS result. The dashed red line is the fitting formula of Yagi & Yunes (2017), whereas the green line is the best fit to the values described in the text. The lower panel shows the relative difference between the Yagi & Yunes (2017) formula and the compactness values resulting from the M(R) curves. 
Figure 12 shows the resulting (Eq. (11)) in a full range of the GW170817 event lowspin component masses (1.36 − 1.60 M_{⊙} for M_{1}) as a function of the radius R(M_{1}) of the more massive component for the selected strong phase transition EOS. These results are compared with the approximate formula for proposed in Raithel et al. (2018), based on a set of the “standard” realistic EOS used in Abbott et al. (2017). The twin branch spans a considerably wider range in R for the allowed component masses of GW170817 than the standard EOS, e.g. the SLy4 EOS, for which the radii are relatively constant. This feature results in a much broader range of allowed values in comparison to the standard EOS: for comparison see Fig. 1 of Raithel et al. (2018), where varies to a much smaller extent than here. The behaviour of the red EOS (but also the violet EOS, in which the phase transition occurs below the lower bound on the M_{2} mass) is qualitatively similar to the SLy4 EOS, but since the radii of the components change more than in the case of the SLy4 EOS in the appropriate mass ranges, the response of the star to the tidal deformation is larger, as the Λ paramteter is a sensitive function of the radius. We expect a similar behaviour for any other EOS with a similar M(R) sequence. The deviation from the fit is of the order of 100 in and of several hundred metres in R(M_{1}). The character of the curves depends on whether the phase transition occurs within the range of M_{2} or M_{1} (blue and green lines). In the first case, we note a discontinuity in related to the phasetransition induced instability; in the second case, however, the R(M_{1}) also behaves noncontinuously because a range of M_{1} for which two solutions with different values of R is possible.
Fig. 12. as a function of the radius of the more massive component, R(M_{1}) for the lowspin estimation of parameters of the GW170817 event. Squares denote the lowest estimated mass of the primary component, M_{1} = 1.36 M_{⊙}, while triangles denote the highest mass, M_{1} = 1.6 M_{⊙}. The dashed line corresponds to the fit by Raithel et al. (2018). The thick dotted line indicates the 90% confidence limit of 900 for (corrected value from Abbott et al. 2019, Table IV). Information about varying with the mass ratio q = M_{2}/M_{1} for fixed chirp mass is encoded in the plot: curves for selected EOS follow the fixed chirp mass of GW170817. The square denotes the M_{1} = 1.36 M_{⊙} (i.e. M_{1} = M_{2}, q = 1), while the triangle denotes M_{1} = 1.60 M_{⊙} (M_{2} = 1.17 M_{⊙}, q = 0.725). 
3.3. Features of the quark twin branch for given polytropic outer layer
In the following we present our study of the polytropequark matter EOS in order to connect the properties of the quark part of the M(R) diagram to its lessdense, outer part i.e. the polytrope. The last M(R) configuration of the polytropic EOS (corresponding to the transition point) is denoted by C_{ph}. For the sake of presentation we use a specific C_{ph} configuration with M_{ph} = 2 M_{⊙}. It was selected because of its relation to the current maximum observed mass, which we assume to be equal M_{max} = 2 M_{⊙}.
Figure 13 shows a complimentary set of features. Here we want to study the influence of the C_{ph} configuration on the twin branch. We fix the α parameter to the maximum (α = 1). The stiffness of the polytropic EOS (related to the value of γ) is directly connected with the derivatives dM/dρ_{c} or dM/dR at C_{ph}, and has a direct impact on the presence of the twin branch. Stiffer EOS at C_{ph} (higher γ resulting in higher value of dR/dM) favours the presence and size of the twin branch. Additionally, we investigate the point before the onset of phase transition. Figure 14 shows the α versus P relation at the end of the polytope (related to the n_{1} parameter from Table 2 for each EOS). The colours correspond to the M(R) relations in Fig. 13. For illustrative purposes to investigate the conditions at which the stable twin branch may be present, we select several massradius relations with M_{max} > M_{ph} (configurations denoted dark red, DR, and dark blue, DB), M_{max} = M_{ph} (red, R, and blue, B), M_{max} < M_{ph} (orange, O, and cyan, C), and one sequence without a twin branch (yellow, Y). The parameters of these configurations are given in Table 2.
Fig. 13. Example of the relation between the various EOS parameters and properties of the twin branch. The state of matter at the phase transition point (here quantified by the dM/dR slope on the M(R) plane) is related to the parameters of the interior EOS. For the DB and DR EOS M_{max} > M_{ph}; for the R and B EOS M_{max} = M_{ph}; for the O and C EOS M_{max} < M_{ph}; for the Y EOS there are no stable twin solutions. 
Fig. 14. Relation between the square of the speed of sound and pressure P in the polytropic part of EOS (before the phase transition point) for the exemplary EOS introduced in Table 2. M(R) relations produced for these EOS are shown in Fig. 13. 
Figure 15 demonstrates the behaviour of the M(R) relations for the selected values of the speed of sound parameter α (right panel) and the density jump parameter λ (left panel). Only critical values of α and λ parameters for which stable twin branches can be formed are shown. As expected, the value of the sound velocity of a quark matter influences the stiffness of the core and ability to stop gravitational collapse due to instability resulting from softening by the phase transition. On the other hand, the stiffness λ influences the size of the instability region between polytropic onephase stars and hybrid stars with a quark core. For the EOS with smaller radii, at the point of the phase transition, it is harder to produce a stable twin branch. This means that all twin solutions for the R and Y EOS occupy a similar area on the massradius plane, while twin branches for B and DB span a much wider space. From the observational point of view, according to Figs. 15 and 17, observations of the NS with smaller radii would put stronger constraints on the parameters of the possible phase transition to quark matter and the quark matter EOS, e.g. values of α and λ, as well as the macroscopic properties of the NSs, e.g. the minimum radius R_{min}(M_{max}), corresponding to the maximum mass on the twin branch.
Fig. 15. Presence of a twin branch as a function the speed of sound in quark matter (blue and red curves, left panel) and the density jump λ (dark blue and yellow, right panel) for the exemplary EOS. Only configurations with the critical values of α and λ for which a stable twin branch can be produced are shown. The highmass twin branch segments have α in the range from 0.78 (bottom red curve) to 1.00 (top red curve) and 0.60 (bottom blue curve) to 1.00 (top blue curve). Values of λ are in the range from 1.58 (top dark blue curve) to 2.5 (bottom dark blue curve) and 1.34 (top yellow curve) to 1.48 (bottom yellow curve). The remaining EOS parameters (γ, n_{1}, λ, n_{0} for the right panel and γ, n_{1}, n_{0}, α = 1 for the left panel) correspond to the EOS in Table 2. 
Fig. 16. Exemplary B EOS case where the phase transition occurs at M_{ph} = 2 M_{⊙}. The relation between the values of maximum mass M_{max} and the difference between M_{max} and M_{min}. For a given λ this dependence is approximately linear for a broad range of λ parameter and α parameters. Along each curve the α parameter decreases from top to bottom if the figure, with a step of 0.1. The topmost dots correspond to α = 1.0. 
Fig. 17. Ranges of λ and α compatible with the conditions of the presence of a stable second twin branch (above the solid curves) and M_{max} > 2 M_{⊙} (above the longdashed lines). The region between the dotted and solid vertical lines and the solid curve shows the region in which the solution denoted “Both” in Alford et al. (2013) are present. For the Y, O, R, and DR EOS the region is limited by the similar value of λ ≈ 1.7, while for the C, B, and DB EOS λ ≈ 1.67. 
The onset and parameters of the polytrope have a crucial influence on the radius for configurations below 2 M_{⊙} and on the behaviour of the twin branch. The smaller the radius (the higher the value of n_{1}, the smaller the value of γ), the higher the value of α needed to yield a stable twin branch (compare the lowest red and blue sequences in the left panel of Fig. 15, where lower limit for the R EOS equals α = 0.78 for λ = 1.60, and α = 0.60 for λ = 1.90 for the B EOS). A similar situation, but for the λ parameter, is presented in the right panel of the same figure: stable twin branches for the DB EOS can be formed for values of λ between 1.58 and 2.5, while for the Y EOS the allowed λ range is much smaller, and lies between 1.34 and 1.48 (both for α = 1).
Figure 16 illustrates the conditions for a stable twin branch. For this, local mass minimum and maximum, M_{min} and M_{max}, is required. A marginal case corresponds to M_{min} = M_{max}. These configurations correspond to the zeros of the approximately linear relation between M_{max} and the difference M_{max} − M_{min} in Fig. 16 for the values of n_{0}, n_{1}, and γ as for the B EOS. We note that for smaller values of λ: i) a smaller value of α is required to produce a stable twin branch, ii) higher values of M_{max} are reached, iii) (M_{max} − M_{min}) obtains higher values.
For a given C_{ph} configuration we can plot a λ(α) relation, defining the onset of a twin branch, from the crossing point of the curve plotted in Fig. 16 with a horizontal line (y = 0). For a given λ this condition defines the critical value of α. For the presence of a new, stable twin branch, matter in a dense phase should be sufficiently stiff: α > α_{crit}.
Another astrophysically important condition corresponds to the presence of stellar configurations on the twin branch with baryon masses higher than the maximum mass of onephase star M_{ph}. In this case there is a natural evolutionary track which allows for the creation of twin stars. The condition M_{B, max} = M_{B, ph} defines for each λ the limiting value of α – α_{Mmax}. This condition is stronger than just the presence of stable twin branch and α_{Mmax} > α_{crit}. Strictly speaking this condition corresponds to the comparison of baryon masses. However, the relative difference in the gravitational mass of normal and twin sequence for the same M_{B} is of the order of ∼10^{−3}, and within this accuracy we can consider the condition M_{max} = M_{ph} = 2 M_{⊙}, which is equivalent to the crossing points of curves presented in Fig. 16 with vertical line M_{max} = 2 M_{⊙}. Conditions for the α and λ parameters for which a stable twin branch is possible are shown in Fig. 17. A schematic explanation of this plot is presented in Fig. 18. Stable twin solutions are not possible to the left of the dotted vertical lines, nor below the solid curves. Between the solid curves and the dashed lines, a stable hybrid branch can be formed with M_{max} < 2 M_{⊙}. Dashed lines correspond to the situation in which M_{ph} = M_{max} = 2 M_{⊙}, while the regions between the dashed and dotted curves correspond to M_{max} > M_{ph}.
Fig. 18. Schematic explanation of Fig. 17. Above a critical λ (above the solid line) twin solutions are possible. With increasing α, the mass on the twin branch may be higher than the mass at the phase transition point M_{ph} (regions above the longdashed line). The region between the vertical dotted and solid lines corresponds to the “Both” class, described in Alford et al. (2013), which is characterised by a delayed onset of instability with respect to configurations with larger λ. 
We also indicate a region of the “Both” class proposed by Alford et al. (2013): it lies between the vertical dotted and vertical solid lines. These configurations are not immediately destabilised after the phase transition; both maxima in the massradius relation occur for a finite size quarkphase core. For the Y EOS, all twins are of the “Both” class. Limiting λ for the DB, B, and C EOS at which “Both” configurations can still be formed ≈1.67, while for the Y, O, R, and DR EOS it equals ≈1.70. A clear separation between these two types of EOS is, in this context, the effect of the separation of the radii at the phasetransition point.
Parameter n_{1} decides where the transition point C_{ph} occurs. Here we investigate the properties of the twin branch with respect to the density at which the polytrope ends.
For the set of EOS from Table 2, we changed parameters n_{1} and chose only the configurations with stable twin branches, as shown in Fig. 19. The configurations with larger radii at the polytropic branch (the C, B, DB EOS) span a wider range of n_{1} for which our assumptions are fulfilled (see also Fig. 20). This kind of EOS support situation, where the phase transition point occurs at low n_{1} and where very massive, stable NSs (with M_{max} even higher than 3 M_{⊙}) can exist on the quark branch. We note that all possible R_{min}(M_{max}) for the DR EOS are in the range ≈9.8 − 10.5 km, while for the C EOS this range is much wider: ≈9.7 − 14.8 km (shown in Fig. 19 as thick lines and in Fig. 20 as a function of n_{1}). We compare these results with the causal limit (see e.g. Appendix D in Haensel et al. 2007):
Fig. 19. Massradius relations for the marginal values of n_{1}, for which stable twin branches can be formed, in the case of the DR and C EOS (colours correspond to the same parameters, except n_{1} values, as shown in Fig. 13 and in Table 2). Thick lines correspond to the M_{max} (or R_{min} on the twins branches) for all ranges of n_{1}. The causal limit is shown as a dark shaded area, when the light shaded space corresponds to the photon orbit in the Schwarzschild case. Some stable configurations can be formed within the photon orbit. For the C EOS, when M_{max} is high, the quark phase occurs for very low masses. 
Fig. 20. Minimum radii R_{min} (thick lines in Fig. 19) as a function of baryon density where the phase transition occurs n_{1}. 
The maximum mass M_{max} is then related to the density ρ_{*} as
where ρ_{*, 15} = ρ_{*}/10^{15} g cm^{−3} (see Eq. (2)). The M(R) relations for all the EOS are limited by Eq. (14) and asymptotically converge to the causal limit with decreasing n_{1}.
Minimum radii corresponding to M_{max} are indicated by thick lines in Fig. 19, with R_{min} between 9.5 and 10.5 km, depending on the EOS. For low values of n_{1} the radii decrease approximately along the causal limit, reach a value of R_{min}, and become bigger again for high values of n_{1}.
We also investigate our results in the context of Eq. (15), as shown in Fig. 21. For the configurations with dominant contribution of the quark phase (like the C EOS), the results converge to the causal limit. If the phase transition starts at higher densities (corresponding to ρ_{*} ≈ 0.9 × 10^{15} g cm^{−3}), calculated curves diverge from the causal limit. The divergence is especially strong for the O, R, and DR EOS, for which twin branches can be formed for relatively small radii at the transition point. According to the recent analysis of the GW170817 event in Abbott et al. (2018), these EOS are more favourable than C, B, and DB, due to the smaller radii for configurations without phase transition in the centre and with masses relevant to GW170817 (M ∼ 1.2 − 1.6 M_{⊙}).
Fig. 21. Maximum mass M_{max} on the twin branch vs. ρ_{*} (see Eq. (2)). The causal limit (Eq. (15)) is shown by a black line. 
On the twin branch the minimum radius of the hybrid star depends on the density at phase transition n_{1} (see Fig. 20). For the adopted EOS models the maximum n_{1} for which a stable, twin branch exists is about 0.56 fm^{−3}.
4. Conclusions
In our work we consider stable NSs configurations, described by the EOS constructed using the SLy4 prescription of the crust, joined with the relativistic polytrope, which ends in the transition point where the quark matter phase occurs. This internal structure may reflect as a twin branch in the massradius relation, which means that objects with different R might have the same M.
Recent detection and improved analysis of the gravitational wave event, originated as a NSNS merger and named GW170817, delivered information about tidal deformabilities of the compact objects. By using the multiple hybrid EOS, we show that the behaviour of the Λ(M, R) is not straightforward, as regions with different Λ values occupy the same space in the M(R) plane. We point out that representing Λ(M_{1.4}) on the M(R) plane, as was recently used in the literature, is not optimal for examining the M(R) dependency of the tidal deformability on the EOS. We also show that the nonmonotonic behaviour of Λ_{1} − Λ_{2} relation depends on the size of the density jump λ, and directly reflects the nonmonotonic behaviour of M(Λ) function. An even more complicated situation occurs when the dependency on the SLy4 crustpolytrope connection point is included.
We find that objects on the twin branch are more compact (in comparison to the polytropic branch) and give smaller Λ values. Such solutions are more favoured from the point of view of the GW170817 analysis. In the case of a highmass twin branch (where the transition point occurs around 2 M_{⊙}), stars in the range of GW170817 masses have larger R and larger Λ than lowermass twins. Because the NS with the same compactness can have different tidal deformability (due to the different EOS), we show that the deviation from the Λ − GM/Rc^{2} fit proposed in the literature, for the broad range of the hybrid EOS, is around 5% for Λ < 100 and increases to above 10% for the higher values of Λ. We find a similar inaccuracy in the fit on the versus R(M_{1}) plane, where discontinuous relations for the hybrid EOS span a wider range in R than the standard ones (e.g. the SLy4 EOS).
Finally, we focus on the influence of the transition point on the twin branch. Approximately linear relations for M_{max} − M_{ph} versus M_{max} (for a fixed λ and changeable α) are used to establish conditions for a density jump and speed of sound, that allow for the creation of different types of twin configurations (i.e. “Both” NSs). For the exemplary EOS, most of the twins with M_{max} > M_{ph} can be formed for at least λ = 1.33 − 1.54 and corresponding values of α = 0.7 − 0.37, while “Both” stars occupy the λ ≲ 1.7 range. Generally, the smaller the R on the polytropic branch, the more finetuned the conditions need to be to produce a stable twin branch. Our whole set of solutions asymptotically converges to the limit of the causal EOS with decreasing values of n_{1}, and it can be approximated by the causal limit when the central density at the beginning of the twin branch ρ_{*} ≲ 0.9 × 10^{15} g cm^{−3} (when contribution to the EOS of the quark phase is dominant). We also show that the minimum radius on the twin branch has a size between 9.5 and 10.5 km and depends on the n_{1} value. We note that analyses were performed for exemplary EOS, and at most two varying parameters at once from the parameter set (n_{0}, n_{1}, γ, α, λ), when other parameters were constant. This strategy simplified investigation and allowed us to determine the constraints mentioned above. Nevertheless, a full analysis is needed and is dedicated to future studies.
Acknowledgments
We acknowledge support from the Polish National Science Centre via grant nos. 2014/13/B/ST9/02621, 2014/14/M/ST9/00707, and 2016/22/E/ST9/00037, and from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 653477. WT acknowledges the support of the Nicolaus Copernicus Astronomical Center 2015 summer studies programme.
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All Tables
All Figures
Fig. 1. Left panel: schematic EOS with a density jump phase transition on the pressure P–baryon density n_{b} plane. Right panel: gravitational mass M–radius R sequence of solutions in the case of a strong (destabilising) phase transition. We show the characteristic masses in this situation: the mass at the phase transition density M_{ph}, the maximum mass M_{max}, and the minimum mass M_{min} at the end of the instability (decreasing part of M(R) between M_{min} and M_{ph} indicated with a dotted line). Stable configurations between the M_{max} and M_{min} are sometimes called the twin branch. 

In the text 
Fig. 2. Massradius diagram for the polyquark EOS (α = 1) with the ranges of Λ(M, R) values colourcoded (see inset). The configurations on the M(R) plane may have significantly different Λ values. This is especially visible for the blackonwhite line denoting the SLy4 EOS M(R), which is placed on the top of its Λ ranges (see Fig. 3 for a closeup). The dotted horizontal line denotes 1.4 M_{⊙}. The dashed inclined black line denotes an approximate division between the purely polytropic M(R) sequences and sequences containing quark core (see Eq. (12)). The pale green and blue bands correspond to the component mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). For reference, in the upper left corner of the plot we indicate the regions excluded by the requirement of the EOS causality (speed of sound in the dense matter less than the speed of light in vacuum, Haensel et al. 1999), and the photon orbit (3GM/Rc^{2}) in the Schwarzschild case. 

In the text 
Fig. 3. Zoomedin image of Fig. 2 around the SLy4 EOS M(R) sequence showing the differences in Λ for the configurations of the same M and R. 

In the text 
Fig. 4. Massradius diagram for the polyquark EOS (α = 1) with the curves colourcoded according to their Λ(M_{1.4}) values. The blackonwhite solid line denotes the SLy4 EOS M(R). The regions with different Λ(M_{1.4}) overlap each other, also because one sequence may have several values of Λ(M_{1.4}) due to the nature of the strong phase transition around M = 1.4 M_{⊙}. The nontrivial behaviour of M(R) hinders the separation of the EOS on the M(R) plane with respect to the values of Λ(M_{1.4}) they yield (as proposed in Annala et al. 2018). 

In the text 
Fig. 5. Central pressure P_{c} vs. mass and radius M_{ph} and R(M_{ph}) at the end of the polytropic branch (corresponding to the phase transition point). 

In the text 
Fig. 6. Maximum mass on the twin branch M_{max} vs. radii at the phase transition point R(M_{ph}). 

In the text 
Fig. 7. Massradius relations for the polytropicquark EOS with realistic SLy4 crust. Selected M(R) relations correspond to a polytropic segment of γ = 4.5, connected to the SLy4 crust at n_{0} = 0.21 fm^{−3} and to a linear EOS of α = 1 (Eq. (2)) at n_{1} = 0.335 fm^{−3}. The curves differ by the density jump λ = n_{2}/n_{1}, as indicated in the plot. Regions destabilised by the phase transition are indicated by dotted segments. The black line indicates the SLy4 EOS. The pale green and blue bands correspond to the mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). 

In the text 
Fig. 8. Λ_{2}(Λ_{1}) (left panel) and M(Λ) (right panel) relations for the M(R) sequences from Fig. 7. The values of the Λ parameter are based on the measurements of the chirp mass and the lowspin prior estimates of the component masses in the binary NS merger GW170817 (Abbott et al. 2017). Shaded areas in the left panel denote the estimated 50% and 90% confidence regions corresponding to the measurement (Abbott et al. 2017). 

In the text 
Fig. 9. Massradius relations for the polytropicquark EOS with realistic SLy4 crust. Selected M(R) relations correspond to γ = 4.5, connected to the SLy4 crust at the baryon densities n_{0} indicated on the plot (magenta EOS at n_{0} = 0.235 fm^{−3}, blue EOS at n_{0} = 0.21 fm^{−3}, green EOS at n_{0} = 0.185 fm^{−3}, red EOS at n_{0} = 0.16 fm^{−3}). In all the selected cases, the quark EOS starts at n_{1} = 0.335 fm^{−3}, the density jump is λ = 1.7, and α = 1 (Eq. (2)). Regions destabilised by the phase transition are indicated by dotted segments. The black line denotes the SLy4 EOS. The green and blue bands correspond to the mass ranges estimated in the lowspin prior case of GW170817 (Abbott et al. 2017). 

In the text 
Fig. 10. Λ_{2}(Λ_{1}) (left panel) and M(Λ) (right panel) relations for the selected twin branch sequences, with colourcoding and symbols as in Fig. 9. 

In the text 
Fig. 11. Compactness GM/Rc^{2} as a function of Λ for the α = 1 EOS from Figs. 2–4 (grey lines). The black line denotes the SLy4 EOS result. The dashed red line is the fitting formula of Yagi & Yunes (2017), whereas the green line is the best fit to the values described in the text. The lower panel shows the relative difference between the Yagi & Yunes (2017) formula and the compactness values resulting from the M(R) curves. 

In the text 
Fig. 12. as a function of the radius of the more massive component, R(M_{1}) for the lowspin estimation of parameters of the GW170817 event. Squares denote the lowest estimated mass of the primary component, M_{1} = 1.36 M_{⊙}, while triangles denote the highest mass, M_{1} = 1.6 M_{⊙}. The dashed line corresponds to the fit by Raithel et al. (2018). The thick dotted line indicates the 90% confidence limit of 900 for (corrected value from Abbott et al. 2019, Table IV). Information about varying with the mass ratio q = M_{2}/M_{1} for fixed chirp mass is encoded in the plot: curves for selected EOS follow the fixed chirp mass of GW170817. The square denotes the M_{1} = 1.36 M_{⊙} (i.e. M_{1} = M_{2}, q = 1), while the triangle denotes M_{1} = 1.60 M_{⊙} (M_{2} = 1.17 M_{⊙}, q = 0.725). 

In the text 
Fig. 13. Example of the relation between the various EOS parameters and properties of the twin branch. The state of matter at the phase transition point (here quantified by the dM/dR slope on the M(R) plane) is related to the parameters of the interior EOS. For the DB and DR EOS M_{max} > M_{ph}; for the R and B EOS M_{max} = M_{ph}; for the O and C EOS M_{max} < M_{ph}; for the Y EOS there are no stable twin solutions. 

In the text 
Fig. 14. Relation between the square of the speed of sound and pressure P in the polytropic part of EOS (before the phase transition point) for the exemplary EOS introduced in Table 2. M(R) relations produced for these EOS are shown in Fig. 13. 

In the text 
Fig. 15. Presence of a twin branch as a function the speed of sound in quark matter (blue and red curves, left panel) and the density jump λ (dark blue and yellow, right panel) for the exemplary EOS. Only configurations with the critical values of α and λ for which a stable twin branch can be produced are shown. The highmass twin branch segments have α in the range from 0.78 (bottom red curve) to 1.00 (top red curve) and 0.60 (bottom blue curve) to 1.00 (top blue curve). Values of λ are in the range from 1.58 (top dark blue curve) to 2.5 (bottom dark blue curve) and 1.34 (top yellow curve) to 1.48 (bottom yellow curve). The remaining EOS parameters (γ, n_{1}, λ, n_{0} for the right panel and γ, n_{1}, n_{0}, α = 1 for the left panel) correspond to the EOS in Table 2. 

In the text 
Fig. 16. Exemplary B EOS case where the phase transition occurs at M_{ph} = 2 M_{⊙}. The relation between the values of maximum mass M_{max} and the difference between M_{max} and M_{min}. For a given λ this dependence is approximately linear for a broad range of λ parameter and α parameters. Along each curve the α parameter decreases from top to bottom if the figure, with a step of 0.1. The topmost dots correspond to α = 1.0. 

In the text 
Fig. 17. Ranges of λ and α compatible with the conditions of the presence of a stable second twin branch (above the solid curves) and M_{max} > 2 M_{⊙} (above the longdashed lines). The region between the dotted and solid vertical lines and the solid curve shows the region in which the solution denoted “Both” in Alford et al. (2013) are present. For the Y, O, R, and DR EOS the region is limited by the similar value of λ ≈ 1.7, while for the C, B, and DB EOS λ ≈ 1.67. 

In the text 
Fig. 18. Schematic explanation of Fig. 17. Above a critical λ (above the solid line) twin solutions are possible. With increasing α, the mass on the twin branch may be higher than the mass at the phase transition point M_{ph} (regions above the longdashed line). The region between the vertical dotted and solid lines corresponds to the “Both” class, described in Alford et al. (2013), which is characterised by a delayed onset of instability with respect to configurations with larger λ. 

In the text 
Fig. 19. Massradius relations for the marginal values of n_{1}, for which stable twin branches can be formed, in the case of the DR and C EOS (colours correspond to the same parameters, except n_{1} values, as shown in Fig. 13 and in Table 2). Thick lines correspond to the M_{max} (or R_{min} on the twins branches) for all ranges of n_{1}. The causal limit is shown as a dark shaded area, when the light shaded space corresponds to the photon orbit in the Schwarzschild case. Some stable configurations can be formed within the photon orbit. For the C EOS, when M_{max} is high, the quark phase occurs for very low masses. 

In the text 
Fig. 20. Minimum radii R_{min} (thick lines in Fig. 19) as a function of baryon density where the phase transition occurs n_{1}. 

In the text 
Fig. 21. Maximum mass M_{max} on the twin branch vs. ρ_{*} (see Eq. (2)). The causal limit (Eq. (15)) is shown by a black line. 

In the text 
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