On the Maximum Mass and Oblateness of Rotating Neutron Stars with Generic Equations of State

A considerable effort has been dedicated recently to the construction of generic equations of state (EOSs) for matter in neutron stars. The advantage of these approaches is that they can provide model-independent information on the interior structure and global properties of neutron stars. Making use of more than 106 generic EOSs, we assess the validity of quasi-universal relations of neutron-star properties for a broad range of rotation rates, from slow rotation up to the mass-shedding limit. In this way, we are able to determine with unprecedented accuracy the quasi-universal maximum-mass ratio between rotating and nonrotating stars and reveal the existence of a new relation for the surface oblateness, i.e., the ratio between the polar and equatorial proper radii. We discuss the impact that our findings have on the imminent detection of new binary neutron-star mergers and how they can be used to set new and more stringent limits on the maximum mass of nonrotating neutron stars, as well as to improve the modeling of the X-ray emission from the surface of rotating stars.


INTRODUCTION
Matter inside neutron stars is compressed by gravity to densities a few times larger than the saturation density of atomic nuclei, n s = 0.16 fm −3 , making them the most compact material objects known in our present universe.In principle, the properties of neutron stars are fully determined by the equation of state (EOS) of dense and cold, neutron-rich baryonic and possibly quark matter.In practice, however, because the EOS is known only with large uncertainties, our knowledge of even the most basic properties suffers from serious limitations that are only mildly mediated by astronomical observations and laboratory experiments.Among the various properties of neutron stars for any given EOS, the maximum mass of rotating M max and nonrotating configurations (TOV), M TOV , bear particular significance both in gravity and in nuclear physics.
Direct mass and radius measurements set strict lower limits on the maximum mass of neutron stars M TOV ≳ 2 M ⊙ (Fonseca et al. 2021) and constrain their radii to ≃ 11-14 km (Miller et al. 2021;Riley et al. 2021).Additional upper bounds on neutron-star radii and tidal deformabilities have been deduced from the first direct gravitational-wave detection of a binary neutron-star merger GW170817 by the LIGO and Virgo collaborations (The LIGO Scientific Collaboration et al. 2019).From this event, upper bounds on the maximum mass M TOV ≲ 2.33 M ⊙ have been derived from the associated gamma-ray burst GRB170817A (Margalit & Metzger 2017;Rezzolla et al. 2018a;Ruiz et al. 2018;Shibata et al. 2019).
One of the reasons why accurate theoretical predictions of neutron-star properties are difficult is that reliable calculations of the EOS are currently only available from Chiral Effective Field Theory at densities n b ≲ n s (see, e.g., Hebeler et al. 2013;Gandolfi et al. 2019;Keller et al. 2021;Drischler et al. 2020) and in the opposite limit from pQCD (see, e.g., Freedman & McLerran 1977;Vuorinen 2003;Gorda et al. 2021a,b) at densities n b ≳ 40 n s , that are much larger than those reached inside neutron stars (see also Komoltsev & Kurkela 2022).Between these limits, the only available options are to either build models that reproduce the expected behaviour of QCD (see, e.g., Beloin et al. 2019;Bastian 2021;Traversi et al. 2020;Jie Li et al. 2021;Demircik et al. 2022;Ivanytskyi & Blaschke 2022, for some recent attempts), or to construct agnostic (model independent) parametrizations for the EOS and constrain them with astronomical or multi-messenger measurements.
Our incomplete knowledge of the EOS is partially compensated by a number of quasi-universal relations, i.e., essentially EOS-independent, that have been found among certain neutron-star quantities over the years, both in terms of iso-lated rotating and nonrotating stars (see, e.g., Yagi & Yunes 2013;Doneva et al. 2014;Haskell et al. 2014;Chakrabarti et al. 2014;Pappas & Apostolatos 2014;Breu & Rezzolla 2016;Weih et al. 2018;Konstantinou & Morsink 2022;Nath et al. 2023) and of the gravitational-wave signal from binary systems (see, e.g., Bauswein & Janka 2012;Read et al. 2013;Bernuzzi et al. 2014;Takami et al. 2015;Rezzolla & Takami 2016;Most et al. 2019;Bauswein et al. 2019;Weih et al. 2020;Gonzalez et al. 2022) and Yagi & Yunes (2017) for a review.Clearly, the robustness of these quasiuniversal relations depends on the number of EOSs that are employed in determining the relations.So far, the identification and study of quasi-universal relations was often limited by the relatively small number of available models.At the same time, a growing number of recent works have explored the possibility of building very large sets of modelindependent EOSs that satisfy all known theoretical and observational constraints and cover the physically allowed space of EOSs (Annala et al. 2020;Landry & Essick 2019) and of related quantities, such as the sound speed (Altiparmak et al. 2022;Ecker & Rezzolla 2023;Gorda et al. 2022), or the conformal anomaly (Marczenko et al. 2023;Annala et al. 2023;Brandes et al. 2023).These EOSs are built either using generic piecewise polytropes (see, e.g., Kurkela et al. 2014;Lattimer & Steiner 2014;Steiner et al. 2018;Most et al. 2018;Tews et al. 2018a,b), a parameterization of the sound speed (see, e.g., Annala et al. 2020;Altiparmak et al. 2022;Ecker & Rezzolla 2023) or non-parametric Gaussian process regression (see, e.g., Gorda et al. 2022;Gorda et al. 2023).
The purpose of this work is to exploit such large ensembles of generic EOSs to revisit the validity of quasi-universal relations of isolated slowly and rapidly rotating stars, and to study how the results depend on input from pQCD and on recently measured bounds.
2. QUASI-UNIVERSAL RELATIONS Among the quasi-universal relations found for isolated stars, one is particularly relevant for its impact on the stability of stellar models and was first discussed by Breu & Rezzolla (2016) (BR16 hereafter).More specifically, using a relatively small number of 28 tabulated EOSs, BR16 pointed out that the critical mass M crit , that is, the mass of uniformly rotating neutron stars on the turning-point line, can be expressed in a quasi-universal relation through the specific angular momentum j crit and that at the mass-shedding limit j Kep [j crit and j Kep are also indicated as χ crit and χ Kep , respectively (Most et al. 2020)] where a 2 = 0.132, a 4 = 0.071 (Breu & Rezzolla 2016).
A few remarks are worth making.First, expression (1) provides the stellar mass along the so-called "turning-point" line (Friedman et al. 1988), that is, the line in the (M, n c ) plane along which ∂M/∂n c | J = 0, where n c is the central number density.Second, because the turning-point criterion is a sufficient criterion for dynamical instability, the importance of (1) is that it allows one to determine for any rotation rate the critical mass above which a dynamical instability would trigger the collapse to a rotating black hole.Third, when considering the maximum value allowed for the specific angular momentum, Eq. ( 1) marks the maximum mass of stable uniformly rotating configurations, namely, M max := M crit (j crit = j Kep ).Finally, because the turningpoint criterion is only a sufficient but not a necessary criterion for dynamical instability, the maximum mass at the upper end of the dynamical-instability line is actually set by the upper end of the so-called "neutral-stability" line.Such a mass is normally slightly larger than M max and is attained at somewhat smaller central densities (see Takami et al. 2011;Weih et al. 2018, for a discussion).
In practice, the quasi-universal relation ( 1) is most often used to relate the endpoints of the critical line via the ratio of the maximum masses of rotating and nonrotating configurations, namely, Using Eq. ( 1), BR16 estimated the mass ratio to be R = 1.203 ± 0.022, in rough agreement with cruder estimates obtained previously with a much smaller number of EOSs (Cook et al. 1994a,b;Lasota et al. 1996) 2) and refine the quasi-universal behaviour exploiting a set of more than 10 6 different EOSs1 that covers the entire physically allowed space of EOSs consistent with constraints from dense-matter theory and neutron-star observations.
For the construction of our EOSs we follow the procedure presented in a number of previous works where the interested reader can find additional details (Altiparmak et al. 2022;Ecker & Rezzolla 2023, 2022).What is relevant to recall here is that we construct the EOSs with a 10-segment parameterization of the sound speed in which we can either impose or not the pQCD constraints at ∼ 40 n s .From the astrophysics side, we impose constraints coming from radius measurements by the NICER collaboration on J0740+6620 (Miller  (Miller et al. 2019;Riley et al. 2019) by rejecting EOSs for which R 2.0 < 10.75 km or R 1.1 < 10.8 km, where the subscript indicates the corresponding gravitational mass of nonrotating stars.In addition, we impose an upper bound on the binary tidal deformability Λ as deduced from the LIGO/Virgo detection of GW170817 by rejecting all EOSs with Λ > 720 (low-spin prior) (The LIGO Scientific Collaboration et al. 2019) at a chirp mass M chirp = 1.186M ⊙ for mass ratios q > 0.73.The constraints from these astrophysical observations are imposed as sharp cutoffs on the EOS ensemble, which, as shown by Jiang et al. (2022), leads to good agreement of the important central part of the PDF obtained from a more expensive Bayesian analysis.Finally, we impose a lower bound M (−)  TOV on the maximum mass of nonrotating neutron stars (M TOV ) by rejecting all models with M TOV below a prescribed cutoff.Since this constraint has been shown to have a significant impact on the space of allowed EOSs (Ecker & Rezzolla 2023), we have explored different values for M (−) TOV to account for the uncertainties on this bound.
Once the EOS ensemble has been generated, the corresponding models for rotating stars are constructed with the RNS code (Stergioulas & Friedman 1995).In particular, we solve for stationary and axisymmetric equilibrium solutions of uniformly rotating perfect fluids with varying central densities along constant angular-momentum sequences between J = 0 and J = J Kep , where J Kep is the Keplerian (or "massshedding") angular momentum and is a function of n c (see also Konstantinou & Morsink 2022;Jie Li et al. 2023b, for similar analyses with a much smaller set of EOSs).The set of stellar models having J = J Kep is also referred to as the Keplerian limit because along this sequence the angular velocity is the largest possible; any increase of the angular momentum at constant central density would lead to a shedding of mass at the equator.The endpoint of the Keplerian sequence marks the maximum mass at a Keplerian frequency, M max,Kep and this is close to, but systematically larger than M max .The relative difference between the two masses is rather small, i.e., M max,Kep /M max − 1 ≲ 10 −2 , and therefore often ignored (e.g., Annala et al. 2022), but it is conceptually important that the two masses are kept distinct, as it is done here.
The left panel of Fig. 1 reports the 95% (dark lines) and 100% (light lines) confidence-level contours for our ensemble of 10 6 EOSs with M (−) TOV = 2.2 M ⊙ when the pQCD constraints are imposed (red) or not (blue).The right panel, on the other hand, shows the corresponding 95% confidencelevel contours for the mass-radius relations of nonrotating stars (j = 0) and stars rotating at the mass-shedding limit (j = j Kep ); in the rotating case, the equatorial radii are employed (see Appendix).

RESULTS
Making use of our ensemble of 10 6 EOSs, we have constructed more than 10 8 nonrotating and rotating stellar models up to the mass-shedding limit.In this way, we have reconsidered several quasi-universal relations governing the properties of neutron stars and first present the outcome of this analysis for the mass-ratio relation (1).In Fig. 2 we show results for two different ensembles where the pQCD constraints are either imposed (red colors) or not (blue colors).In both cases, we use a lower bound of M TOV > 2.2 M ⊙ to ensure consistency with the 1-σ confidence interval for the direct mass measurement of the black-widow binary pulsar PSR J0952-0607 (2.35 ± 0.17 M ⊙ ) (Romani et al. 2022).More specifically, the colored areas in the (large) left panel of Fig. 2 show the (95%) confidence intervals of the PDF with (red) and without (blue) imposing the pQCD constraint, while the solid lines of corresponding color mark the medians of the distributions.The median values and the corresponding confidence intervals of the distributions are still well approximated by expression (1), where we use a 2 and R as fitting parameters, thus fixing a 4 = R − (a 2 + 1); the best-fit coefficients are listed in Tab. 1 of the Appendix.Interestingly, the quadratic term in (1) provides a very good approximation up to j ∼ 0.7 j Kep and the quartic contribution becomes essential only for larger values of j.
Overall, we find that the new and much larger set of EOSs shows again a quasi-universal behaviour similar to that proposed by BR16, reported with a black solid line in Fig. 2. The variance of the new universal relations is slightly larger than what was found by BR16, but this is expected due to the much larger set of EOSs considered here.Note also that the difference between the PDFs obtained when imposing and not imposing the pQCD constraints is small but significant and has a clear physical origin, which we discuss below.More importantly, the new quasi-universal relations are slightly but systematically larger than those reported in BR16 and, indeed, the median of the latter falls outside the 95% confidence interval for both the ensembles with and without pQCD constraints.This is best seen in the right panel of Fig. 2, which reports the one-dimensional PDFs of M crit /M TOV at j = j Kep , i.e., the PDFs of R. The medians of these PDFs are marked with red/blue dashed lines for EOSs built with/without the pQCD constraints, and while they differ by only ≃ 1%, they are clearly separated.The reason for this is that the pQCD constraint leads to a softening of the EOS at large densities and, as a result, to smaller values of M TOV .Because the central density in maximally rotating stars is smaller than in TOV stars, the impact on M max is also smaller; the combination of the two trends drives R towards slightly larger values.Finally, and more importantly, when considering the PDF obtained with the pQCD constraint being imposed, we can extract a new estimate for the mass ratio R = 1.255 +0.047 −0.040 , which is therefore ≃ 4% larger than the BR16 estimate.
While the quasi-universality of the mass ratio R is robust, it is clear that the constraints imposed on the EOS ensemble will influence this ratio.To quantify this dependence, we also compute R for a smaller (M (−) TOV = 2.0 M ⊙ ) and a larger (M (−) TOV = 2.35 M ⊙ ) lower bound on M TOV .The results are reported in Tab. 1 2 and show that the median of the PDFs of R increases monotonically with the imposed lower bound on M TOV .Stated differently, imposing a larger cutoff on 2 Given the rapid convergence of the PDFs with the number of EOSs (see Appendix), we have considered for these different bounds 10 4 EOSs.the TOV mass in the EOS ensemble drives the posterior for M max upwards faster than that of M TOV .While this might seem counter-intuitive at first, it is simple to understand.We recall that imposing larger bounds on M TOV leads to a significant stiffening of the EOSs, i.e., in the energy-density range 0.5 < e/(GeV fm −3 ) < 0.8 [see Fig. 1 of Ecker & Rezzolla (2023)] and that are relevant for the cores of rapidly rotating stars.At the same time, such bounds have a negligible impact on energies e ≈ 1 GeV/fm 3 , that are those relevant for the cores of nonrotating stars (Altiparmak et al. 2022).This increase of M max not balanced by an equal increase in M TOV leads to the measured growth of R with M (−) TOV .The new and more accurate estimate of R provides two important tools to be employed when a binary neutron-star merger is detected together with its electromagnetic counterpart.First, as discussed by Rezzolla et al. (2018b) in the case of GW170817, the detection of a gamma-ray burst counterpart following a binary neutron-star merger can be taken as an indirect evidence for the formation of a black hole from the remnant and the emergence of a relativistic jet from the ejected matter (Gill et al. 2019).This implies that the mass of the merger remnant, properly reduced by the ejected restmass and the mass lost in gravitational waves, is very close to M max , setting an upper limit for it.In turn, making use of R, this can be used to set an upper limit on M TOV .Following this logic and employing the iterative procedure to account for an adaptive adjustment of the mass ratio (see Appendix for details), we obtain the following upper limit for the maximum mass of a nonrotating neutron star M (+) TOV = 2.24 +0.07 −0.11 .Clearly, the same method can be employed to further refine the estimate of M (+) TOV as new detections of merging binaryneutron star systems will be made.
A second consequence of the new estimate of R follows from the discussion made in Most et al. (2020).In particular, once a gravitational-wave merger event with a significant mass difference such as GW190814 is detected, the knowledge of the total mass of the system and of the universal mass ratio R allows one to set constraints on the spin of the secondary [see Fig. 1 of Most et al. (2020)].More specifically, assuming that the secondary of GW190814 was a neutron star at the merger (see Nathanail et al. 2021, for a different conclusion), we can employ the new value of R to improve the constraints on the dimensionless spin of the secondary in GW190814 to be 0.52 ≲ χ 2 ≲ 0.72.Also in this case, the methodology discussed above can be applied to new detections of binaries with small mass ratios.
We next employ our large set of rotating-star models to assess the validity of other quasi-universal relations and we start by reporting a novel quasi-universal relation found for the surface oblateness, i.e., the ratio between the polar and equatorial proper radii, R p /R e .This ratio is obviously unity in the case of nonrotating stars and decreases as the angular momentum is increased, since the equatorial radius becomes larger and the quadrupolar deformation of the star increases (see also Konstantinou & Morsink 2022, for different but equally interesting relations).The result of our analysis in this case is presented in Fig. 3, which is logically similar to Fig. 2, but now for the ratio R p /R e .Interestingly, the quasiuniversal relation has an extremely small variance and this is most probably due to the fact that R p /R e depends most sensitively on the low-density part of the EOSs which is rather well constrained.This hypothesis is also corroborated by the fact that the result for this relation seems to be essentially independent on whether or not the perturbative QCD constraint is imposed, implying that the high-density portion of the EOS, where this constraint has an impact, is not involved in the deformation seen in Fig. 3. Given the simple behavior of the relation, the corresponding medians can be very accurately described by a second-order polynomial of the form R p /R e = 1 − b 2 (j/j Kep ) 2 , and the corresponding bestfit coefficients are reported in Tab. 1 of the Appendix.The relevance of this result is that it allows for a much more accurate modelling of the emission from rotating neutron stars, such as those observed by NICER, where a precise knowledge of the surface deformation is essential for the modelling of the X-ray emission from the hot spots of rotating stars (Morsink et al. 2007;Bogdanov et al. 2019).Furthermore, it can be employed when studying black-widow systems, where rapid rotation is expected and indeed observed [with a spin frequency of f = 706 Hz, PSR J0952-0607 is the second-fastest-spinning pulsar known (Romani et al. 2022) and has j/j Kep ≲ 0.46].Finally, we use our EOS ensemble to assess the validity of the so-called "I-Q" quasi-universal relations, where I is the stellar moment of inertia and Q the mass-quadrupole moment.The IQ relations were first investigated by Yagi & Yunes (2013) in the context of slowly rotating neutron stars and subsequently extended to stars in rapid rotation by Doneva et al. (2014); Chakrabarti et al. (2014), andPappas &Apostolatos (2014).The latter, in particular, have found quasi-universal relations between I and Q also for rapidly rotating stars and expressed them in terms of j.Since these functions were validated using a dozen of nuclear-physics EOSs only, it is interesting and important to assess the validity of such relations when employing a much larger set of model-independent EOSs (see also Nath et al. 2023, for similar recent work).Overall, we find that the fit proposed by Pappas & Apostolatos (2014) performs very well also with the much larger ensemble of EOSs and the differences in the newly estimated best-fit coefficients are of the order of 1%, with the largest differences of ∼ 2.5% being attained in the low-Q and high-j region of the parameter space (see Appendix for details).

CONCLUSION
Motivated by recent advancements in model-agnostic sampling methods for the EOS of neutron-star matter (Kurkela et al. 2014;Lattimer & Steiner 2014;Steiner et al. 2018;Most et al. 2018;Tews et al. 2018a;Annala et al. 2020;Altiparmak et al. 2022), we have revisited the quasi-universal behaviour of isolated rotating and nonrotating neutron stars by employing a very large sample of generic EOSs constructed with a 10-segment parameterization of the sound speed (Annala et al. 2020;Altiparmak et al. 2022).By exploring the whole physically allowed space of solutions, our approach has the advantage of being able to test rigorously whether a quasi-universality is present or not, and to study the dependence of the quasi-universal behaviour on the imposed constraints.
Our attention here has been focussed on the quasi-universal relation of the mass along the critical line and, more in particular, on the mass ratio R between the maximum masses of rotating and nonrotating stars (Breu & Rezzolla 2016).Our analysis has revealed that such a relation is still valid when considering an ensemble of up to 10 6 EOSs obtained either imposing or not the constraints from pQCD at high densities, and with different lower bounds the maximum mass of nonrotating stars M TOV .While the variance in the new estimates is comparable but larger than in BR16, the median values are a few percent larger than the original estimate of BR16.This is not surprising given the comparatively small number of EOSs considered by BR16.Furthermore, we have found that increasing the lower bound on M TOV also leads to a larger value of R, while neglecting the pQCD constraints at high densities leads to a decrease of this ratio.
Finally, we have employed our large set of EOSs to assess the validity of known and novel quasi-universal relations.In particular, we have revealed the existence of a new and tight quasi-universal relation for the surface oblateness, which can be used to model more accurately the emission from the hot spots on the surface of rotating neutron stars, such as those observed by NICER.At the same time, we were able to confirm the validity of the I-Q relations over a broad range of rotations, from slow rotation up to the mass-shedding limit, and to improve the best-fit coefficients of the functional behaviour proposed by Pappas & Apostolatos (2014) in the low-Q and high-j region of the parameter space.
Besides improving previously known results, the more accurate estimate of R offers two important and direct applications whenever a binary neutron-star merger is detected together with its electromagnetic counterpart.In particular, it can be used to set new and tighter constraints on the maximum mass of nonrotating stars and hence on the EOS.For example, when considering GW170817, this implies a new upper limit of M (+) TOV = 2.24 +0.07 −0.11 M ⊙ .Furthermore, in binaries with small mass ratio it can be employed to set constraints on the dimensionless spin of the secondary object when this is represented by a neutron star.For example, when considering the case of GW190814, the new value of R implies a new bracketing interval of 0.52 ≲ χ 2 ≲ 0.72.The present data-taking runs of the LIGO-Virgo-Kagra collaboration will hopefully provide us with a number of potential applications of these findings.Launching, propagation and emission of relativistic jets from binary mergers and across mass scales" (Grant No. 884631).CE acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 "Strong-interaction matter under extreme conditions"-project number 315477589 -TRR 211.LR acknowledges the Walter Greiner Gesellschaft zur Förderung der physikalischen Grundlagenforschung e.V. through the Carl W. Fueck Laureatus Chair.The calculations were performed on the local ITP Supercomputing Clusters Iboga and Calea.

OVERVIEW OF THE STELLAR PROPERTIES
In Fig. 4 we show the PDFs for the EOSs (left panel) and the corresponding mass-radius relation of static (middle panel) and maximally rotating stars (right panel) when imposing (top row) or not (bottom row) the pQCD constraint and assuming M (−) TOV = 2.2 M ⊙ .As clearly shown by the figure, the allowed pressure band is influenced by imposing the pQCD constraint down to energy densities e ∼ 800 MeV, which are relevant for the cores of heavies nonrotating stars TOV and even for rapidly rotating stars.In particular, the EOSs are allowed to be significantly stiffer and with higher sound speeds [i.e., with steeper curves in the (p, e) space] in the range 0.8 − 1.2 GeV since the EOSs do not need to be causally connected to the pQCD band at high densities.As can be observed in the two rightmost columns of Fig. 4, and in Fig. 1, the effect of this stiffening in the intermediate to high-density regime is to allow for higher masses and to increase the radii of stars close to the stability limit.In particular, the lower panels in these figures show that the distributions of radii for 1.4 M ⊙ models varies only modestly whether or not the EOS ensemble is restricted to be compatible with results of pQCD.On the other hand, the contours of the distribution close to the critical limit clearly show that the constraint has non-negligible impact on TOVs (which was already shown in Gorda et al. 2022), as well as on rapidly rotating stars, which we show here for the first time.
Similarly, in Fig. 5 we report the one-dimensional PDFs for the sound speed (left column), for the normalized trace anomaly ∆ = 1/3 − p/e (Fujimoto et al. 2022) (middle column) and for the measure of the non-conformality of the matter d c := ∆ 2 + (∆ ′ ) 2 (right column), where ∆ ′ is the logarithmic derivative of ∆ with respect to the energy density.All quantities are computed at the center of j = 0 (top row) and j = j Kep (bottom row) stellar models.The median values, together with the 95% error bars are reported in Tab. 1. Overall, our results compare well with the corresponding slices calculated by Annala et al. (2023), as well as with the results of Gorda et al. (2022), obtained with a Gaussian-process regression technique (Landry & Essick 2019).
Finally, in Tab. 1 we also report the median values with 95% error bars for R, the best-fit value for the coefficient a 2 , the central sound speed squared in either maximally rotating rotating (c 2 s,c,Kep ) or static stars (c 2 s,c,TOV ).The last four columns report

Figure 1 .
Figure1.Left: 95% (dark colors) and 100% (light colors) confidence-level contours for an ensemble of 10 6 EOSs with M (−) TOV = 2.2 M⊙ for which the pQCD constraints are imposed (red) or not (blue).Right: Corresponding 95% confidence-level contours for the mass-radius relations of nonrotating stars (j = 0) and stars rotating at the mass-shedding frequency (j = jKep; in this case the equatorial proper radii are used).Shaded areas indicate the imposed astrophysical constraints or the limits from the threshold mass to collapseKoeppel et al. (2019); the bottom panels report slices of the PDFs at M = 1.4 M⊙. et al. 2021;Riley et al. 2021) and J0030+0451(Miller et al. 2019;Riley et al. 2019) by rejecting EOSs for which R 2.0 < 10.75 km or R 1.1 < 10.8 km, where the subscript indicates the corresponding gravitational mass of nonrotating stars.In addition, we impose an upper bound on the binary tidal deformability Λ as deduced from the LIGO/Virgo detection of GW170817 by rejecting all EOSs with Λ > 720 (low-spin prior) (TheLIGO Scientific Collaboration et al. 2019) at a chirp mass M chirp = 1.186M ⊙ for mass ratios q > 0.73.The constraints from these astrophysical observations are imposed as sharp cutoffs on the EOS ensemble, which, as shown byJiang et al. (2022), leads to good agreement of the important central part of the PDF obtained from a more expensive Bayesian analysis.Finally, we impose a lower bound M(−)  TOV on the maximum mass of nonrotating neutron stars (M TOV ) by rejecting all models with M TOV below a prescribed cutoff.Since this constraint has been shown to have a significant impact on the space of allowed EOSs(Ecker &  Rezzolla 2023), we have explored different values for M(−)

Figure 2 .
Figure 2. Left: Quasi-universal relation of the gravitational mass along the critical line between.Red-shaded (blue-shaded) areas indicate the 95% confidence intervals with (without) the pQCD constraint, while the solid lines mark the corresponding median values.The black solid line reports the prediction by Breu & Rezzolla (2016) (BR16).Right: One-dimensional slice of the PDFs for j = jKep.

Figure 3 .
Figure3.The same as in Fig.2but for the novel quasi-universal relation for the ratio of the polar-to-equatorial radii.

Figure 4 .
Figure 4. PDFs of the EOSs (left panel), of the mass-radius relation of nonrotating stars (middle panel) and of stars rotating at the massshedding frequency (right panel).The top and bottom row refer respectively to ensembles where the pQCD constraints are imposed or not; in all cases M (−) TOV = 2.2 M⊙.