The equations of state used in this work are constructed from two ingredients, namely a part that comes from a nuclear matter model and a part that comes from a holographic model. Nuclear matter models are based on well-understood physics, where the approximations used are well under control in the regime of small densities. These models can be extrapolated to dense nuclear matter, but the underlying assumptions, specifically that the interactions between the nucleons are weak enough to allow a perturbative treatment, eventually break down. In fact, there is a recent body of work [26–29] showing that nonperturbative effects can lead to drastic differences for the EOS for both hadronic and quark matter degrees of freedom.

Holographic models, instead, incorporate the assumption that the coupling is strong, and this property can be used in a way such that a holographic description can complement a nuclear matter model. Notice that the breakdown of weakly coupled nuclear matter models takes place at relatively low densities: These models are reliable at best up to about 2 times the nuclear saturation density [30,31], a density much smaller than the potential transition density from nuclear to quark matter. For realistic holographic modeling of NSs it is therefore not enough to just include quark matter, but the dense regime of the nuclear matter phase should also be modeled.

Holographic baryons and nuclear matter have been studied extensively in the literature. It is well established that a single baryon is dual to a soliton configuration of bulk gauge fields, similar to the Belavin-Polyakov-Schwarz-Tyupkin instanton in Yang-Mills theory [32,33]. This description has been analyzed in detail in the Witten-Sakai-Sugimoto model [34–37]; the nuclear matter phase turns out to be an instanton crystal [38,39]. In this article, however, we will employ a simpler approach which avoids the need to consider inhomogeneous configurations and therefore solving partial differential equations. Specifically, we use a homogeneous approximation originally developed in the context of the Witten-Sakai-Sugimoto model [40–43]. This approximation is expected to work best in the regime of large density, and therefore the holographic homogeneous approach and traditional models of nuclear matter complement each other’s regime of validity. By combining the two approaches in their respective regimes into one EOS with a matching procedure, we can then obtain a hybrid EOS.

Below, we will describe in more detail the model input and the precise matching procedure, and we will explore how large the set of equations of state is that one obtains in this way.

To determine how generic predictions from such a hybrid construction are, we perform this construction for a wide range of different nuclear matter models and holographic models. For the nuclear matter models, we take care to include both hard and soft choices, as well as several in between, so as to properly parametrize the uncertainty in these models. The models which we use, roughly from soft to stiff, are the following: Hebeler-Lattimer-Pethick-Schwenk (HLPS) soft variation [44], Akmal-Pandharipande-Ravenhall (APR) [45], Skyrme Lyon (SLy) [46,47], HLPS intermediate, IUF [48,49], and DD2 [50]. There are even stiffer EOSs available than DD2 in the range of densities where we will use them, but as we shall see, DD2 is already too stiff for our purposes: The hybrid constructions using it will fail to meet all constraints.²

Notice that the DD2 EOS as such, without matching with holography, is actually too stiff to pass the astrophysical bounds. It could happen though that matching would make the EOS softer at high densities, so that the hybrid DD2+holography EOS would pass the bounds. We will see below that this is not the case.

For the holographic model, we use V-QCD. This is a bottom-up holographic model, which is strongly inspired by string theory and top-down holographic models, but has been phenomenologically tuned to reproduce qualitative features of QCD. The model is in turn based on earlier models: improved holographic QCD (IHQCD) for the gluon sector [18,19] and tachyonic brane constructions for the flavor sector [53–58]. We work at zero quark mass. For a detailed exposition of the model, see Appendix A.

For a successful modeling of properties of cold QCD matter it is crucial to carry out a careful comparison to QCD data, in particular to the available lattice data for the thermodynamics of QCD at small densities. For the full V-QCD model this was done in [14] following similar work for IHQCD in [59,60]. A similar approach has been used to study, among other things, the critical point on the QCD phase diagram at finite temperature and density [61–65]. In V-QCD, the data fit was seen to produce a tightly constrained and viable prediction for the EOS of dense quark matter both at zero [14] and at finite [16] temperature. In the current article we will use V-QCD with three sets of potentials arising from the lattice fits of [14]; see Appendix A for details.

As the final ingredient for the current setup, the implementation of nuclear matter in the homogeneous approach was studied for V-QCD in [15]. It was shown that this approach produces a physically reasonable phase diagram and a stiff EOS for dense nuclear matter, i.e., an EOS with a high speed of sound, reaching values well above the value for conformal field theories (cs2=1/3). Notice that the conformal value was long thought to set a ceiling, an obstruction for realistic holographic modeling of dense matter [66,67], but works [11,12] revived the interest in this context. High speed of sound makes it easier to alleviate the bounds to the EOS from the mass measurements of NSs [30]. In the current article we will demonstrate that the nuclear matter EOS is stiff independently of the values of those parameters of the holographic model which are left free by the fit to the lattice data.

We stress that, by choosing EOSs coming from specific models both for the low- and high-density regime and matching them in the middle, rather than using for example polytropic interpolations, we effectively restrict our approach to EOSs which are relatively smooth and regular (apart from the strong nuclear to quark matter phase transition predicted in the model). That is, we avoid EOSs where the speed of sound as a function of density, for example, varies quickly without obvious physical reason for the variation. In this sense our approach is similar to that of [23], which used families of analytic interpolating EOSs (including polytropes and piecewise continuous interpolations of the speed of sound) but regulated the EOSs by setting limits to the speed of sound as well as the parameters of the EOSs in consecutive intervals. Notice, however, that, as we shall see below, the use of the rather soft HLPS model (together with the stiff V-QCD predictions) is in slight tension with this idea: There will be a sizable jump in the speed of sound at the matching point when using this EOS at low density.

In summary, the constructed hybrid EOSs then include three regions. (i)

For nuclear matter at low densities, up to densities equal to roughly 2 times the saturation density, we use the various weakly coupled models of nuclear matter.

V-QCD contains several potential functions, which parametrize the remaining freedom in the model. In [14], these potentials have been quantitatively matched to available lattice data, resulting in several possible choices of potentials which are compatible with lattice data. Of these, we use potentials 5b, 7a, and 8b (see Appendix A). In this way, we also parametrize part of the uncertainty in the holographic model.

For each of these choices, the components need to be glued together to obtain a hybrid EOS. Since the two parts are supposed to be two descriptions of the same matter, we would ideally want the density at which the matching is performed to be a crossover. This is not possible, however, so instead we demand that the transition is as smooth as possible, which is a second-order phase transition. For a given density at which we want to perform the matching, we therefore require both the pressure and its first derivative with respect to the chemical potential to be identical for both the nuclear matter part and the holographic part. As these are two parameters, we need two parameters to tune to make this possible. These two parameters are (i)

the normalization of the pressure cb in the baryonic phase of the holographic model (see Appendix A).—This maps to the normalization of the nuclear matter action of V-QCD, and letting it be a free parameter amounts to taking the speed of sound as the input from the holographic model instead of the pressure itself.

In summary the hybrid EOSs depend on the choices of the models and parameters as follows: (i)

The low-density region up to the nuclear saturation density ns depends only on the choice of the nuclear matter model, HLPSs (HLPS soft), APR, SLy, HLPSi (HLPS intermediate), IUF, or DD2, from soft to stiff.

In this subsection we will discuss which constraints from the astrophysical observations will be relevant in our analysis.

The most precise bounds on the EOS come from the mass measurements of NSs. Over the years, convincing evidence for massive NSs with M∼2  M⊙ has been accumulated from various astronomical observations. A compact star may exist in the low-mass x-ray binary 4U 1636-536 M=2.0±0.1  M⊙ [72] and the pulsar B1516+02B in the globular cluster M5  M=2.08±0.19  M⊙ [73], as well as the millisecond pulsars J1614-2230 M=1.97±0.04  M⊙ [74], J0348+0432 M=2.01±0.04  M⊙ [75], and J0740+6620 M=2.14-0.09+0.10  M⊙ [76] accurately measured by using Shapiro delay. The existence of these massive stars sets a stringent lower bound to the maximum mass of NSs.

The gravitational wave observations by the LIGO/Virgo Collaborations have also set constraints on compact star properties. The (non)observation of the squishiness of the star constrains the tidal deformability for the compact star of mass 1.4  M⊙ in the range [2] 580≥Λ(1.4  M⊙)≥70(GW170817).(1)This study assumed that both stars obey the same EOS, therefore effectively assuming that neither of the objects in the merger event GW170817 were black holes for which Λ would be strictly vanishing (see, however, [77]). The upper bound essentially requires stars of mass 1.4  M⊙ to be small, so that the tidal forces have very little effect: The matter is kept very tightly close to the center of the star. However, this is a merely a rule of thumb. The presence of a strong phase transition in the outer crust can lead to nonmonotonic behavior of the dependence of the radius R with Λ1.4  M⊙ [10].

Of the astrophysical observables relevant for the study of dense EOSs, reliable constraints on NS radii are more difficult to obtain than are the mass and tidal deformability estimates. There are, however, some interesting recent results for the isolated millisecond pulsar PSR J0030+0451 by the NICER Collaboration. Both Riley et al. [69] and Miller et al. [68] have independently done Bayesian parameter estimation on the energy-dependent x-ray pulse waveform data with independently developed codes to determine the pulse waveforms and different modeling of the emitting spots on the stellar surface and the instrumental response. Both of these works find consistent estimates for M and R, with 1σ results for both studies shown in Fig. 1, with the data by Riley et al. in black and Miller et al. in blue. There is an ongoing discussion on the atmospheric model used in the analysis to model the emitting hot spots [78], leading perhaps to slightly updated constraints in the future. The additional contours in Fig. 1 are Bayesian parameter fits applying state-of-the-art atmospheric models directly to observed time-evolving thermonuclear x-ray burst cooling tail spectra, obtained by the Rossi X-Ray Timing Explorer for low-mass x-ray binaries 4U 1702-429 (in dark cyan) from Ref. [70] and 4U 1724-307 (in magenta) and SAX J1810.8-2609 (in light green), both from Ref. [71].

We now discuss how we take into account these astrophysical bounds in our setup. As we showed above, the most stringent bounds on the maximum mass of the NS come from the measurements of J0348+0432 and J0740+6620 [75,76]: The former has significantly smaller error bars, whereas the latter sets a higher lower bound at 1σ level. In the following we will not strictly follow either of these results but we adopt the following rule: We exclude any hybrid EOS for which maximum NS mass Mmax falls below two solar masses; i.e., we require that Mmax≥2  M⊙.(2)We will also exclude all EOSs which do not satisfy the LIGO/Virgo bound (1). We will not impose any of the radius measurements directly but will rather compare our results to them. As we will demonstrate below, the results constrained by maximum mass and tidal deformability are already consistent with the NICER analyses. The measurement of 4U 1702-429 appears more constraining, but one should recall that all the x-ray measurements may contain sizable systematic uncertainties due to the modeling of the NS atmosphere.

The effect of imposing the bounds (1) and (2) is shown in Fig. 2, where we plot the tidal deformability Λ(1.4  M⊙) and the maximum mass Mmax as a function of the matching density ntr. The excluded zones are indicated in the plots by the gray regions, and the thick curves show the EOSs which pass both bounds. We also show the 1σ lower bounds from the mass measurements of J0348+0432 and J0740+6620, in addition to our mass cutoff, for comparison. Both figures exhibit a downward trend, as a higher matching density corresponds to a softer EOS, which supports smaller maximum masses and smaller tidal deformabilities. Thus, because we have an upper limit for Λ(1.4  M⊙) and a lower limit for Mmax and the curves in question are monotonic, we can bracket the values of ntr/ns from these astrophysical constraints. The LIGO/Virgo bound (left plot) excludes some of the stiffest EOSs, including all EOSs with ntr/ns<1.4 and all hybrids with the DD2 nuclear EOSs. The effect of the maximum mass bound is less severe because it has the tendency of ruling out softer equations of state, and the V-QCD nuclear matter EOS is quite stiff. Some of the hybrids with HLPSs and APR nuclear matter models are, however, excluded for large values of ntr. From Fig. 2 we can also note that the nonexcluded transition densities differ greatly between the different nuclear matter and potential combinations used. For example, HLPSs with potential 5b has a window of 1.4<ntr/ns<1.6, whereas the same nuclear matter EOS combined with potential 8b allows for 1.5<ntr/ns<2.1.

We now discuss the predictions of our construction to the EOS and the nuclear to quark matter transition. First, in Fig. 1 (left), the light red band is spanned by all hybrid equations of state which satisfy the constraints of Eqs. (1) and (2). The light blue band is spanned by the quadrutropic interpolations [79] between low-density nuclear matter and perturbative QCD results which satisfy the same constraints,⁴

We use quadrutropes with continuous pressure and number density, i.e., with second-order phase transitions at the joints. In principle one should allow for a first-order deconfinement transition. This, however, would change the results very little because first-order transitions can be mimicked to arbitrary precision by nearby second-order transitions.

In the leftmost section of the curves and the band, up to the matching density which is where the curves for the examples of EOSs have kinks (i.e., for ε≲300  MeV/fm3), the EOSs are determined by the nuclear matter models. In this regime our construction excludes the stiffest EOSs (having higher pressures). This happens because the V-QCD EOS at higher densities is stiff, which makes the effect of the LIGO/Virgo bound in (1) for the low-density part of the EOSs more severe.

In the regime of dense nuclear matter up to the phase transition (which appears as a horizontal line on the curves) the EOS is given by the nuclear matter phase in V-QCD. Because this EOS is stiff and in part due to the maximum mass bound in (2), the softest EOSs are excluded in this region. In addition, the light red band of the hybrid EOSs is narrower than the light blue band of the quadrutrope interpolations, because our construction restricts to “regular” EOSs, excluding combinations of very soft and very stiff sections. As we pointed out above, our approach is therefore similar in spirit to that of [23], which used constraints on the speed of sound to regulate families of analytic interpolating EOSs. Our results in the nuclear matter regime are also similar to this reference.⁵

The width of the light red band in Fig. 1 (left) roughly corresponds to requiring cs2<0.5 or cs2<0.6 in [23]. Notice, however, that due to differences in assumptions one should be cautious when comparing the results; see the discussion in Sec. IV.

In order to analyze the hybrid EOSs in more detail, we show the speed of sound squared cs2=dp/dε as a function of the baryon number density in Fig. 3. In the top, bottom left, and bottom right plots we vary the nuclear matter model, the matching density ntr, and the potentials of the V-QCD model, respectively. In these plots, the EOS arises from nuclear theory models for n≲2ns, from nuclear matter in V-QCD for 2ns≲n≲5ns, and from quark matter in V-QCD for n≳10ns, so that the gap at 5ns≲n≲10ns appears due to the first-order nuclear to quark matter transition. As discussed in Sec. II B, varying the different parameters mostly affects the speed of sound in distinct regions: The nuclear matter model affects the low-density region, ntr affects the intermediate-density region, and the potential choice of V-QCD affects the high-density region. The nuclear to quark matter transition density, however, depends on a combination of the parameters. We also note that for all hybrid equations of state, the speed of sound rises well above the conformal value cs2=1/3 in the dense nuclear matter region, and even values cs2≈0.6 can be reached with potentials 8b.

The matching between the nuclear matter and holographic model in effect leads to a second-order transition, where the speed of sound jumps. Such a jump may be understood as an approximation for a region of rapid change of the speed of sound as we move from weakly coupled low-density nuclear matter to the strongly coupled high-density region. The jump is moderate for most of the hybrid models and even practically absent for some hybrids with the HLPS intermediate EOS but very drastic for hybrids with the HLPS soft EOS. An example of such an EOS is shown as the thin blue curve in the top row plot. Even though some of the EOSs with most extreme discontinuities (such as the one shown in the plot) are excluded by the maximum mass bound, we expect that the hybrids with the HLPS soft EOS are less likely to be good models of the nuclear matter EOS in this region than the other hybrids, and results based on them should be interpreted with caution.

Our results for the speed of sound have very similar features to the results obtained by using the functional renormalization group approach (see, e.g., [26,80,81], and references therein) in both the nuclear and quark matter phases [27–29]. This is particularly interesting as these studies use a somewhat similar approach to the current article: the use of a nonperturbative method in the regime where ab initio calculations cannot be trusted.

We also study the closely related quantity, the adiabatic index γ=dlogp/dlogε=εcs2/p in Appendix B. We note that it shows much weaker dependence on the parameters of V-QCD than the speed of sound. Near the phase transition we obtain that γ≈1.5 universally for all EOSs, which is a very low value compared to typical predictions (γ≳2) of nuclear matter models [23].

The plots in Fig. 4 show the effect of the constraints imposed on the nuclear matter–quark matter first-order transition. In toto, the astrophysical constraints do not limit the quark matter transition parameters as strongly as they do the astrophysical observables. However, in specific cases, like for potential 7a with HLPSs, the constraint is quite stringent, limiting 4.7<nb/ns<6.8 for all allowed ntr (top left plot). None of the constrained hybrid EOSs support a quark matter transition below four saturation densities, and for ntr/ns<1.9, all constrained hybrid EOSs have nb/ns<8.

The latent heat of the quark matter transition for each hybrid EOS is presented in the top right plot in Fig. 4. There we see that all hybrid EOSs produce a Δε>700  MeV/fm3, implying that the transition is strongly first order for all transition densities and EOSs. For ntr/ns<2.2 the values of Δε are bounded both from below and from above, so that 700  MeV/fm3<Δε<2000  MeV/fm3. This, combined with the solutions for the TOV equations, tells us that cold NSs cannot contain a quark matter core for these transition densities because of the energy barrier. This finding is consistent with the analysis of [14] which used V-QCD for quark matter and a model-independent approach employing quadrutropes for nuclear matter. It is furthermore expected that the latent heats will decrease with a moderate rate with increasing temperature [16], leaving ample room for quasistable quark matter core generation in mergers.

Finally, the critical value of the chemical potential is shown in Fig. 4 (bottom). We observe that the critical value depends even more distinctly on the holographic model than, say, nb/ns. The dependence on ntr and on the nuclear matter model, which appears due to the matching procedure, is mostly a small correction. The results fall within the range 470  MeV≲μc≲680  MeV. The relatively large spread reflects a similar spread of the dependence of the thermodynamic potentials on the chemical potential in the quark matter phase [14] as the parameters of the holographic model are varied. Notice, however, that this dependence mostly cancels in the EOS, which is expressed as a relation between the potentials, ε=ε(p).

As a final remark, we notice that the upper bounds of all parameters shown in Fig. 4 are set by the hybrids using the soft variant of the HLPS EOSs for low-density nuclear matter. As we have pointed out above, this means that there is a jump from a very soft to a somewhat stiff EOS at the matching density ntr, rendering the hybrid EOS potentially unreliable. The problem is enhanced by the fact that these EOSs (thick dashed blue curves) use the stiffest version of V-QCD with potentials 8b. Therefore the other curves should be regarded as more realistic predictions of our construction.

As mentioned above, given a hybrid EOS constructed in Sec. II, we can solve the full Einstein equations, which for a spherical configuration of a self-gravitating perfect fluid are called the TOV equations. As a solution of these equations one obtains the mass-radius (M-R) relation for nonrotating NSs, examples of which are shown in Fig. 1 (right), along with a band spanned by all of the hybrid EOSs, NICER results for PSR J0030+0451, and a selection of results from measurements of cooling of x-ray bursts. We note that the hybrid EOSs favor relatively large NS radii, above 11 km for typical NS masses. Moreover, we note that the stars with lowest radii are obtained by using the HLPS soft model at low density, and as noted above, this means that such EOSs have a significant jump in the speed of sound at the matching density. The more regular EOSs (examples of which are shown in the plot) give NS radii around 12 or even close to 13 km. We will discuss this in more detail below. The radii being large is a consequence of the V-QCD nuclear matter EOS being stiff and contrasts with results obtained through the systematic use of effective field theory only [82] which gives R(1.4  M⊙)<11.9  km at 90% confidence level. Notice that the regular EOSs are however in good agreement with the radius measurement of 4U 1702-429 (dashed dark cyan curve in Fig. 1), which has the smallest errors from the x-ray results we have included, and are also consistent with the other direct measurements of radii.

In Fig. 5 we show the radii of the hybrid EOSs at NS mass M=1.4  M⊙ and M=2.0  M⊙. From the left plot in Fig. 5 we notice that the radius of a 1.4  M⊙ NS for all hybrid EOSs is limited to 10.9  km≲R(1.4  M⊙)≲12.8  km.(3)The lower limit on the radius of just under 11 km is achieved by HLPSs combined with the stiffest set of potentials 8b in the V-QCD model, meaning that the jump in the speed of sound at the matching density is highest among the hybrid EOSs we have constructed. Other hybrid EOSs produce minimal radii of around 11.5 to 12.5 km; see⁶

Notice that in Table I we restricted ntr<2.2ns whereas the plots of Fig. 6 use the whole range of constructed hybrids up to ntr=2.6ns. Therefore some numbers in the table differ from those obtained from the plots.

For a given EOS and a star of a given mass and radius one can compute the second Love number k2 and, from that, the tidal deformability Λ=(2/3)k2[(c2/G)R/M]5, which quantifies the effect of an external tidal field on the induced quadrupole moment of the star [85,86]. The tidal deformability Λ is one of the most important parameters describing the gravitational wave signal from NS mergers. Apart from characterizing the deviation from the point mass limit for the inspiral part of the signal, it can also be used to roughly predict some properties of the merger and postmerger parts (as we will discuss below).

We show the tidal deformability of a 1.4  M⊙ NS and the maximum mass supported by the EOSs for various hybrid EOSs as a function of the matching density ntr/ns from nuclear models to V-QCD in Fig. 2 (left). We also show the relation of the Λ parameter to the NS radius at three different NS masses, M=1.4  M⊙, 1.8  M⊙, and 2.0  M⊙, in the top left, top right, and bottom plots of Fig. 6, respectively. We notice that the requirement of Eq. (2) also constrains the values of Λ. As an example of a constraint following from our construction, we obtain a lower bound for Λ at M=1.4  M⊙: Λ(1.4  M⊙)≳230.(4)Here one should note that the lowest values are obtained (again) by the hybrid EOS using the HLPS soft model together with the stiffest (8b) version of V-QCD, so that there is a sizable jump in the stiffness of the EOS at the matching point. Using only more regular hybrid EOSs would push the lower bound close to 300 (see Table I). Interestingly, these lower bounds are close to the value (∼300) obtained by analyzing the electromagnetic signal from the GW170817 event in [87]. Notice that for transition densities between 1.4≤ntr/ns≤2.2, even without constraints from the maximum mass, Λ(1.4  M⊙)≳150 for all hybrid EOSs. As is well known the tidal deformability Λ (at fixed mass) depends on the EOS to a rough approximation only through the radius. We can see this from Fig. 6: The spread of the curves in all panels is relatively small. Also, tidal deformability decreases fast with increasing mass.

As we commented above, one of the clearest outcomes of the our analysis for NSs is that quark matter cores are always unstable, i.e., dM/dR>0 by a wide margin for solutions containing quark matter. Moreover, we do not find EOSs supporting additional stable branches, i.e., twin stars. The instability of the quark matter cores also limits the maximum mass of the NSs, shown in Fig. 2 (right), which marks roughly the point on the M-R curve where the star becomes unstable. For hybrid EOSs with potentials 5b and 7a, Mmax is set by the star where the central density reaches the phase transition density, so that the instability of the star around M=Mmax is driven by the phase transition. For potentials 8b, however, the maximum mass is reached within the nuclear matter phase, at densities slightly lower than the transition density. We also notice that the upper limit for Λ(1.4  M⊙) limits the highest masses in the model. For example, for HLPSs, we notice that for potential 5b the highest maximum mass of around 2.1 solar masses is achieved by ntr/ns≈1.4 and for potential 8b the corresponding values are Mmax≈2.4 and ntr/ns≈1.5.

Features of the electromagnetic signal related to GW170817 suggest that the remnant collapsed into a black hole soon after the merger. It has been estimated that this sets an upper bound of around 2.2  M⊙ to the maximal mass Mmax of nonrotating NS [88–91]. Since Mmax is sensitive to the parameters of the holographic model, such a bound would severely constrain these parameters. It would exclude most (but not all) hybrid EOSs using the stiffest version of V-QCD considered in the article defined using potentials 8b; see Fig. 2 (right). Almost all the hybrids with potentials 5b and 7a would however pass this bound.

For slowly rotating compact stars there exist approximately universal relations between the moment of inertia I, the quadrupole moment Q, and the Love numbers, called the I-Love-Q relations [92], which arise due to all of these three parameters being most sensitive to the star structure far from the core, where all the realistic EOSs are relatively similar. To specify the relations one first defines the dimensionless combinations I¯=c4G2M3I,Q¯=-MI2QΩ2/c2,(5)where Ω is the angular velocity of the star. The dimensionless combination corresponding to the Love number is the tidal deformability Λ. The three relations between the possible pairs from the set {I¯,Q¯,Λ} are then obtained as polynomial fits in log-log scale. We use the fit parameters from [93].

In Fig. 7 we show the relative deviation of the constrained hybrid EOSs at ntr=1.9ns from the universal I-Love-Q relations. We note that the deviations from universality are small, with the largest deviation in the I-Q relation being around 0.6%, again with the combination of potential 8b with HLPSs. For more regular hybrid EOSs the deviation from universality is even smaller. Therefore, the hybrid EOSs do not lead to larger deviations than, e.g., various traditional nuclear matter models for which the relations were checked in [92]. Even though we present the results here at ntr=1.9ns, the agreement with the universal relations is generic for all the constrained hybrid EOSs over all the transition densities ntr.

Finally we also show in Table I how large a fraction of the NS is described by holographic nuclear matter, i.e., has density above the matching density. This is given in terms of the minimal and maximal values for the ratio of the radius of the “holographic core” to the radius of the whole star. We see that typical numbers for this ratio are around 0.7 at M=1.4  M⊙ and around 0.8 or a bit above at M=2.0  M⊙.

In this subsection we discuss some of the key properties of the signal from mergers of NSs described by using the hybrid EOSs: the characteristic frequencies of the merger and postmerger parts of the signal. Such frequencies include the peak frequencies f1, f2, and f3 of the power spectral density of the postmerger signal, as well as the value of the instantaneous frequency at the time of the merger fmrg [94,95]. Here we will restrict to the frequency of the most prominent peak f2, which is linked the rotation frequency of the hypermassive NS formed in the merger, and the merger frequency fmrg. Notice that f2 is, however, absent if the binary promptly collapses into a black hole after the merger. Based on numerical simulations, it has been found that whether or not prompt collapse happens depends, to a good approximation, only on the parameter [96] κ2T≡3MBMA4ΛA+MAMB4ΛB(MA+MB)5,(6)where the subscripts A and B refer to the two components of the merger. We follow here [97] and set the following threshold to single out the cases of prompt collapse to a BH: κ2T>70.(7)This limit excludes the mergers with the heaviest masses, including all mergers with average NS mass above ∼1.5  M⊙. For equal NS masses, MA=MB, we have κ2T=3Λ/16, and the limit becomes Λ≳370, i.e., a value well below the LIGO/Virgo bound of (1).

Comparison of static properties of NSs and the results of numerical merger simulations has shown that the frequencies can be estimated to a good accuracy by using universal relations involving the masses and tidal deformabilities of the individual stars [94,95,97–99]. We use here the relations for the frequencies f2 and fmrg from [97], where they are fitted as rational functions of the variable ξ=κ2T+c(1-4ν),(8)where ν=MAMB/(MA+MB)2 is the symmetric mass ratio and c is also a fit parameter. The results for our hybrid EOSs are shown in Figs. 8 and 9.

Figure 8 shows the predictions for the frequencies f2 and fmrg for equal mass binaries as a function of the mass of an individual star. Notation is as in Fig. 1: We show the bands spanned by all hybrid EOSs (striped band) and those passing the astrophysical constraints (1) and (2) (light red band) compared to the band for polytropic interpolations satisfying the same constraints, as well as examples of results for hybrid EOSs with ntr=1.9ns. We also required the bound of (7) for f2 which explains the sharp cutoff at large masses in the left plot. Notice that the bound contains a sizable uncertainty [96], which is not shown. We did not apply the cutoff for fmrg, which is also well defined in the case of prompt collapse. The fits of [97] however used data with κ2T≳70, so that the results for large masses and frequencies in the right plot are based on extrapolation. We did not include the uncertainty of the universal relations in the plots—this uncertainty is less than 10% at 90% confidence level [97]. The simulations carried out in [17] typically found frequencies that were a bit smaller than those predicted by the universal relations. These deviations were also less than 10% in all cases. However this suggests that the bands in Fig. 8 (and also below in Fig. 9) systematically slightly overestimate the frequencies for the hybrid EOSs.

We note that the hybrid EOSs in general favor smaller frequencies than generic polytropic interpolations, because the hybrid EOSs are stiffer. This is in agreement with the results of [17] where mergers were simulated by using the hybrid EOS with the SLy model and V-QCD with potentials 7a, and it was found that the characteristic frequencies for the hybrid EOS were significantly lower than for the pure SLy EOS. Notice however that, compared to the polytropic results, our construction also excludes a bunch of stiff EOSs producing lower characteristic frequencies than the hybrid models. The excluded polytropic EOSs are very stiff at low densities but can still meet the LIGO/Virgo bound due to their softness at high densities, well above the saturation density. The hybrid construction does not allow for such softening: All hybrid EOSs are relatively stiff at high densities due to the input from holography.

Notice that at the average mass M¯=1.4  M⊙, corresponding roughly to the gravitational wave event GW170817, only a restricted set of hybrid EOSs passes the constraint (7), producing a frequencies f2 in a narrow interval in Fig. 8 (left). That is, the binaries with softest hybrid EOSs (including typical EOSs using the soft variant of the HLPS models) are predicted to lead to prompt collapse, whereas stiffer hybrids lead to a (possibly short-lived) remnant NS. It has been estimated based on the observed electromagnetic signal from GW170817 that a hypermassive NS was formed, which collapsed into a black hole about 100 ms after the merger [100] (see also [101]). Consequently, this observation favors a hybrid EOS with one of the stiffer nuclear matter models, which (as we have argued above) are more regular than those obtained by using the soft HLPS model and are also favored by the direct radius measurements using the x-ray channel.

Figure 9 shows the results similarly at fixed average mass M¯=1.35  M⊙ varying the mass ratio q=MA/MB. The merger frequency fmrg shows much stronger dependence on q than f2. Notice that the upper edges of both the light red and light blue bands are determined by the bound (7) for q≲1.4 in the left plot and therefore they coincide.