Towards a Warm Holographic Equation of State by an Einstein–Maxwell-Dilaton Model

Abstract

The holographic Einstein–Maxwell-dilaton model is employed to map state-of-the-art lattice QCD thermodynamics data from the temperature (T) axis towards the baryon–chemical potential (${\mu }_B$) axis and aims to gain a warm equation of state (EoS) of deconfined QCD matter which can be supplemented with a cool and confined part suitable for subsequent compact (neutron) star (merger) investigations. The model exhibits a critical end point (CEP) at $T_{\mathrm{CEP}}=\mathcal{O}\left(100\right)$ MeV and ${\mu }_{B\mathrm{CEP}}=500\dots 700$ MeV with an emerging first-order phase transition (FOPT) curve which extends to large values of ${\mu }_B$ without approaching the ${\mu }_B$ axis. We consider the impact and peculiarities of the related phase structure on the EoS for the employed dilaton potential and dynamical coupling parameterizations. These seem to prevent the design of an overall trustable EoS without recourse to hybrid constructions.

1. Introduction

The advent of detecting gravitational waves of merging neutron stars [1] and improved determinations of mass–radius relations of neutron stars, e.g., by NICER [2,3,4], has triggered a cascade of related investigations, most notably focused on accessing the cool equation of state (EoS) of dense strong-interaction matter. Among the various approaches to compact (neutron) star EoS is the application of the famous AdS/CFT correspondence, which mimics dense matter by a suitable gravity dual of QCD [5,6,7]. Here, we employ such a holographic approach based on the Einstein–Maxwell-dilaton (EMd) model pioneered in [8,9] and used further in [10,11,12,13,14,15,16,17,18,19,20,21]. References [22,23] provide a survey of and a valuable comparison between of Dirac–Born–Infeld AdS/CFT models.

Our motivation is as follows: Given lattice QCD thermodynamics results, e.g., the scaled pressure, $p/T^4$, on the temperature (T) axis [24,25] supplemented with the susceptibility ${\chi }_2={\partial }^{2}p/\partial {\mu }_B^2{|}_{{\mu }_B=0}$ [26], the EMd model primarily delivers the entropy density $s(T,{\mu }_B)$ and the baryon density $n_B(T,{\mu }_B)$, which can be integrated to arrive at the potential $p(T,{\mu }_B)$ (even at ${\mu }_B=0$, $p(T,{\mu }_B=0)=p\left(T_{min}\right)+{\int }_{T_{min}}^T\mathrm{d}\overline{T}s\left(\overline{T}\right)$ is a nonlocal quantity which needs $p\left(T_{min}\right)$ as input parameter, supposing that $s(T\ge T_{min},{\mu }_B=0)$ is accurately given. More favorable is to directly use $p(T,{\mu }_B=0)$ from lattice QCD results and map it without further assumptions into the T-${\mu }_B$ plane, as advocated below). These quantities additionally depend on the baryon–chemical potential ${\mu }_B$. The lattice input [24] is provided, at ${\mu }_B=0$, for $T\in [T_1,T_2]=[125,240]$ MeV and allows the adjustment of the dilaton potential and the dynamical coupling of the EMd model. The lattice data [24] for ${\mu }_B/T=0.5,1,\dots ,3.5$ serve as a control, again with $T\in [125,240]$ MeV for 2+1 flavors (for other relevant lattice datasets, cf. [27,28,29]). Curves of $p(T,{\mu }_B)=const$, i.e., isobars, can then be utilized to continue the lattice-given EoS $p(T,{\mu }_B=0)$ into the T-${\mu }_B$ plane, and finally—if no obstacle is met—to $p(T=0,{\mu }_B)$. That is, the EoS—here, the pressure—is directly mapped from the T axis on the ${\mu }_B$ axis. Curves of $p(T,{\mu }_B)=const$ are determined by solving

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{\mathrm{d}T\left({\mu }_B\right)}{\mathrm{d}{\mu }_B}}\right|}_{p=const}=-{\displaystyle \frac{n_B(T,{\mu }_B)}{s(T,{\mu }_B)}},\end{array}$$

(1)

where the inverse entropy-per-baryon determines the slope field. Solutions are $T\left({\mu }_B\right){|}_{p=p_0}$ with $T({\mu }_B=0){|}_{p=p_0}=T_0$ and $p(T_0,{\mu }_B=0)=p_0$. Via Gibbs–Duhem equation, the energy density (e) follows from $e=-p+sT+n_B{\mu }_B$.

The gained cool EoS $p\left(e\right)$ can be then used as input for compact (neutron) star calculations. This vision is illustrated in Figure A1 in Appendix A with a toy model. The abovementioned state-of-the-art lattice data uncovers, in fact, a relevant pressure interval: $p(T_1,{\mu }_B=0)=\mathcal{O}\left({10}^1\right){\mathrm{MeV}/\mathrm{fm}}^3$ and $p(T_2,{\mu }_B=0)=\mathcal{O}\left({10}^3\right){\mathrm{MeV}/\mathrm{fm}}^3$. For the toy model, these values translate into $p(T=0,{\mu }_{B1})=\mathcal{O}\left({10}^1\right){\mathrm{MeV}/\mathrm{fm}}^3$ and $p(T=0,{\mu }_{B2})=\mathcal{O}\left({10}^3\right){\mathrm{MeV}/\mathrm{fm}}^3$ localized just above the reliable pressure interval accessible by nuclear-physics many-body methods (cf. Figure 1 in [30,31] and Figure 12 in [32] which suggest the matching point to be at about 3 MeV/fm³ (pressure) and 200 MeV/fm³ (energy density)). The corresponding energy densities $e(T=0,{\mu }_{B1})$ and $e(T=0,{\mu }_{B2})$ depend, of course, on the details of the mapping along $p=const$ curves from the T axis to the ${\mu }_B$ axis, as do the actual values of ${\mu }_{B1,2}$.

However, such a vision meets potential obstacles. These are (i) already for ${\mu }_B=0$, an unwanted first-order or Hawking–Page phase transition may occur outside the controlled region $T\in [T_1,T_2]$, where $T_{1,2}$ are again the lower and upper limits of the safe EoS, which are used to adjust dilaton potential and dynamical coupling of the EMd model; (ii) a critical end point (CEP) and related first-order phase transition (FOPT) curve may disturb the expected pattern of curves $T\left({\mu }_B\right){|}_{p=const}$, displayed in Appendix A for the toy model; and (iii) the expected pattern of curves $T\left({\mu }_B\right){|}_{p=const}$ does not continue smoothly to the ${\mu }_B$ axis. In fact, item (i) is met in [13] ($T_{\mathrm{FOPT}}({\mu }_B=0)\approx 45$ MeV), which focuses on the neighborhood of a CEP at ${\mu }_B>0$, and in [11] ($T_{\mathrm{FOPT}}({\mu }_B=0)\approx 1$ MeV). The occurrence of a CEP at ${\mu }_B>0$ is known [8,9] and a welcome effect to study. Within such a model class is its impact on the EoS and related dynamics, thus explicating the conjecture posed in [33,34,35,36] and motivating a continuation of the ongoing energy scan at RHIC [37,38]. (For the tight connection of heavy ion–neutron star physics, cf. [39,40].)

To avoid irritations by item (i), we optionally impose here novel side conditions for the dilaton potential. It happens that this enforces parameterizations which facilitate a change of the FOPT curve below the CEP from convex to concave, i.e., the FOPT curve levels off and seems to asymptotically approach to the ${\mu }_B$ axis, thus hindering a concise cool EoS prediction. Nevertheless, the warm EoS, characterized by ${p\left(e\right)|}_T$ or the scaled trace anomaly/conformality measure ${\Delta \left(e\right)|}_T{:=(e-3p)/\left(3e\right)|}_T$ [41,42], is accessible and may be useful for numerical studies.

While many preceding EMD studies focus on the CEP localization and/or transport coefficients near the CEP and across the FOPT curve, here, we place emphasis on the EoS. Among the hitherto less discussed issues is the pattern of isentropes near the CEP and FOPT. In [13], it has been pointed out that the isentropes, with respect to “incoming” and “outgoing” under adiabatic expansion, in various EMd models can be fairly different. A general classification of such isentropic patterns was proposed recently in [43]. Such a study and holographic perspective is of relevance for heavy-ion collisions experiments and speculations on sourcing stochastic gravitation waves in the cosmic confinement transition [44,45].

Our paper is organized as follows. We recap the EMd model and its data adjustment in Section 2. Numerical results are presented in Section 3, where we discuss the CEP position, the shape of the FOPT curve up to fairly large values of ${\mu }_B$, the shape of the $p=const$ curves, and a closer inspection of the resulting EoS by means of various contour plots and related cross-sections of thermodynamic quantities. We conclude and summarize in Section 4. A brief series of appendices complements our analysis. Appendix A sketches our vision of mapping QCD data from the T axis into the T-${\mu }_B$ plane, and Appendix B supplements details of the EMd model used. Appendix C considers the density and pressure near and across the FOPT curve. Appendix D presents a numerical study of a new dilaton potential parameterization adjusted to data, which allows for the construction of a ${\chi }^2$ landscape.

2. Holographic Einstein–Maxwell-Dilaton Model

In line with [8,11,12] we employ the EMd action (reference [6] relates the EMd model (2) to a D3-D7 action and Taylor-expanded versions of string theory-anchored approaches. Hadron, nucleon, and quark degrees of freedom are added separately in [21]) in a fiducial five-dimensional pseudo-Riemann spacetime with asymptotic AdS symmetry:

$$S_{\mathrm{EMd}}={\displaystyle \frac{1}{2{\kappa }_5^2}}\int {\mathrm{d}}^{4}x\mathrm{d}r\sqrt{-g_5}\left(R-{\displaystyle \frac{1}{2}}{\partial }^M\varphi {\partial }_M\varphi -V\left(\varphi \right)-{\displaystyle \frac{1}{4}}\mathcal{G}\left(\varphi \right)F_{\mathcal{B}}^2\right),$$

(2)

where R denotes the Einstein–Hilbert gravity part, and $F_{\mathcal{B}}^{MN}={\partial }^M{\mathcal{B}}^N-{\partial }^N{\mathcal{B}}^M$ stands for the field strength tensor of an Abelian gauge field $\mathcal{B}$ à la Maxwell with ${\mathcal{B}}_M\mathrm{d}{x}^M=\Phi \left(r\right)\mathrm{d}t$ defining the electrostatic potential. An embedded black hole facilitates the description of a hot and dense medium (due to its Hawking surface temperature) and sources an electric field, thus holographically encoding a temperature and an entropy density of the system. Dynamical objects are the scalar dilaton field $\varphi$ and a Maxwell-type field $\Phi$ which are governed by a dilaton potential $V\left(\varphi \right)$, a dynamical coupling $\mathcal{G}\left(\varphi \right)$, and the spacetime which is described by the line element squared

$$\begin{array}{c}\hfill \mathrm{d}{s}^2=g_{MN}\mathrm{d}{x}^M\mathrm{d}{x}^N=e^{2A\left(r\right)}\left(-f\left(r\right)\mathrm{d}{t}^2+\mathrm{d}{\overrightarrow{x}}^2\right)+{\displaystyle \frac{\mathrm{d}{r}^2}{f\left(r\right)}},\end{array}$$

(3)

where $r=0$ is the horizon position, $r\in [0,\infty ]$ the radial coordinate, A the warp factor, and f the blackness function. The resulting Einstein equations are a set of coupled second-order ODEs to be solved with appropriate boundary conditions; see Appendix B. Within the present bottom-up approach, the quantities V and $\mathcal{G}$ are tuned (“tuning” within the bottom-up approach faces three issues: choices of the functional forms of V and $\mathcal{G}$ and parameter adjustments at data. For Bayesian analyses cf. [46]) to reproduce the lattice QCD data [24,47] for 2+1 flavor strong-interaction matter (quark–gluon plasma).

Our ansätze for the dilaton potential V and dynamical coupling $\mathcal{G}$ are

$$\begin{array}{cc}\hfill \mathcal{W}\equiv {\partial }_{\varphi }lnV\left(\varphi \right)& =(p_1\varphi +p_2{\varphi }^2+p_3{\varphi }^3)exp\{-\gamma \varphi \},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathcal{G}\left(\varphi \right)& ={\displaystyle \frac{1}{1+c_3}}\left({\displaystyle \frac{1}{cosh(c_1\varphi +c_2{\varphi }^2)}}+{\displaystyle \frac{c_3}{cosh{c}_4\varphi }}\right)\hfill \end{array}$$

(5)

with parameters $\left\{p_{1,2,3}\right\}=\{0.165919,$$0.269459,$$-0.017133\}$, $\gamma =0.471384$, and $\left\{c_{1,2,3,4}\right\}=\{-0.276851,$$0.394100,$$0.651725,$$101.6378\}$; the prefactor $1/(1+c_3)$ ensures $\mathcal{G}(\varphi =0)=1$. The scale is set by $L^{-1}=216$ MeV, in $L^{2}V\left(\varphi \right)=-12exp\left\{{\int }_0^{\varphi }\mathrm{d}\tilde{\phi }\mathcal{W}\left(\tilde{\phi }\right)\right\}$ and ${\kappa }_5=1.87L^{3/2}$ is used. The conditions (i) ${lim}_{\varphi \to \infty }\mathcal{W}(p_3>0)=0$ and (ii) monotony of V towards the boundary exclude a purely thermal phase transition at ${\mu }_B=0$, cf. [48,49]. The parameters chosen here facilitate a smooth shape of $\mathcal{W}$ with maximum of $0.605$ at $\varphi =3.373$ and a zero at $\varphi =16.321$, which corresponds to an exceedingly small temperature far below 1 MeV. Appendix D presents more details on the impact of the dilaton potential.

In contrast to former work, here, we place emphasis on the optional side condition $p_3>0$ which ensures that, at ${\mu }_B=0$, no phase transition is facilitated outside the temperature range uncovered by the lattice data. Details of our handling of the EMd model are relegated to the Appendix B.

Despite the richness of the dataset [24], only in scarce cases is a direct comparison with our EoS results exhibited in various figures below possible. We therefore present in Figure 1 a more detailed comparison focused on $T\in [100,250]$ MeV and for ${\mu }_B/T=0.5,\dots ,3.0$. While the scaled pressure (left column) seems to perfectly agree, the scaled baryon density (middle column) points to some tension, in particular for larger values of ${\mu }_B/T$, as already noticed in [11] and stressed in [10]. This calls for an improved dynamical coupling ansätze.

However, we could also argue for the failure of the model when extending it into the confinement region, in particular towards lower temperatures being of relevance for compact star physics. This is in line with statements in [11]. Thus, one could scrutinize the implications of the critical end point and related first-order phase transition which emerge in the EMd model for parameterizations employed hitherto in the literature. Reference [19] sharpens the above warning, further enhanced by statements in [23] (“…, the fact that the present EMD model is in good quantitative agreement with the latest lattice QCD data at finite baryon density does not automatically guarantee that the predictions made for regions of the QCD phase diagram well beyond the reach of current lattice simulations are phenomenologically reliable. Indeed, the fact that the EMD model of Ref. [10] is also able to obtain a good quantitative agreement with lattice QCD thermodynamics at zero and finite baryon density, while still predicting the QCD CEP at a significantly different location than in our model, shows that the available lattice data is not enough to strongly constraint such a prediction in the EMD class of holographic models.” Further critical remarks on specific limitations and drawbacks of the holographic EMd model, see [22]).

Completing the discussion of Figure 1, we mention that $s/T^3$ (not displayed) is in similarly good agreement with data to $p/T^4$. The resulting energy density, in particular, when displayed as the ratio $e/p$ (right column), again displays some deviations: the EMd model does not reproduce the apparent structures of the data when ignoring the error bars; including them (not displayed) makes such structures much less pronounced. Note the slight increase of the peak of $e/p$ and its left-shift with increasing values of ${\mu }_B/T$. Recap also the relations of $e/p$ to the conformality measure $e/p={({\displaystyle \frac{1}{3}}-\Delta )}^{-1}$ and the scaled trace anomaly $(e-3p)/T^4=(p/T^4)(e/p-3)$.

The dimensionless susceptibility ${\chi }_2\left(T\right)$ is compared successfully to precision lattice data in Figure 7 (right) in [50].

3. Numerical Results: EoS

3.1. CEP Location and FOPT

The employed parameterizations of V and $\mathcal{G}$ facilitate a CEP with coordinates $T_{\mathrm{CEP}}=97.2$ MeV and ${\mu }_{B\mathrm{CEP}}=694.7$ MeV (fat bullet) (similar CEP locations are obtained in [8,11,12,13]. Interestingly, QCD-functional methods deliver fairly consistent values, cf. [51], and for finite-volume effects [52]. CEP and FOPT loci are inherent in the present model, steered by the parameterizations of V and $\mathcal{G}$, in contrast, e.g., to the approach in [53,54] which allows for a free choice) and an FOPT curve $T_{\mathrm{FOPT}}\left({\mu }_B\right)$ as exhibited in Figure 2 and Figure 3. Remarkably, the FOPT curve displays a concave shape near the CEP (as in [8,11,12,13]; note the agreement of $T_{\mathrm{FOPT}}({\mu }_B=1000\mathrm{MeV})\approx 55\mathrm{MeV}$ with Figure 11 in [11]) which, however, turns, for larger values of ${\mu }_B$, into a convex shape, obviously asymptotic to the ${\mu }_B$ axis with dramatic consequences for the cool EoS. Our numerically accessible domain is by far larger than that of [11].

3.2. Scaled Entropy, Density, Pressure, and Specific Entropy

The contour plots of $s/T^3$, $n_B/T^3$, $s/n_B$ and $p/T^4$ exhibited in Figure 2 point to a mysteriously weak dependence of $s/T^3$ and $p/T^4$ on ${\mu }_B$ to the left and below the FOPT (the contour plot of $n_B/T^3$ (right top panel) can be used to construct the unstable region in a T vs. $n_B$ diagram: A given point $(T,{\mu }_B)$ on the FOPT curve $T_{\mathrm{FOPT}}\left({\mu }_B\right)$ leads to two values of the density, $n_B^{\pm }\left(T\right):=n_B(T,{\mu }_B){|}_{T=T_{\mathrm{FOPT}}\left({\mu }_B\right)\pm ϵ}$. The unstable region has a maximum at $T_{\mathrm{CEP}}$, where $n_B^{\pm }\left(T_{\mathrm{CEP}}\right)$ merge. The left-hand side flank, i.e., for the smaller values of $n_B\left(T\right)$, may be more or less steep, depending on details of the underlying model setup. Figure 4 in [13] exhibits that the slopes of isentropes are either to bypass the unstable region or enter and exit it for not too small values of $s/n_B$. As noted in [13], the examples in [8,9] are such to enter also the unstable region on the left-hand side without graceful exit. Subtle deformations of the isentropic trajectories and/or the left-hand side of the unstable region (e.g., by a bumpy structure) may cause a more involved picture with bypassing and entering-only and entering-and-exiting trajectories). Additionally, the quantities $s/T^3$, $n_B/T^3$, and $s/n_B$ jump across the FOPT, while $p/T^4$ is continuous. The right-hand side continuations of the left-hand side iso-lines, e.g., $p/T^4={10}^{-0.5}$ and ${10}^{-1}$, are not visible on the displayed scale; they are squeezed into a narrow corridor between the FOPT curve and the hardly visible curve $p/T^4=10$ (see Appendix C for some details). The agreement of the EMd model results and lattice data [24] (crosses, error bars are ignored here and below), whenever possible, looks near-perfect (see also Figure 8 in [50]).

We emphasize the pattern of isentropic trajectories, $s/n_B=const$ (left bottom panel), which points to a nonmonotonic specific entropy at the low-temperature side in the proposed classification of [43]. The pattern is qualitatively analogous to the one of [8,9] but different to that found in [13], where “outgoing” isentropic trajectories are attributed to “incoming” trajectories across the FOPT upon adiabatic expansion. It seems that various “good fits” of the same lattice data can deliver quite different contours of $s/n_B=const$, even though we did not attempt here a dedicated ${\chi }^2$ optimization with our dilaton potential and dynamical coupling parameterizations. Curves $e(T,{\mu }_B)=const$ are determined by ${\displaystyle \frac{\mathrm{d}T\left({\mu }_B\right)}{\mathrm{d}{\mu }_B}}{|}_{e=const}=-(T\partial s/\partial {\mu }_B+{\mu }_B\partial n_B/\partial {\mu }_B)/(T\partial s/\partial T+{\mu }_B\partial n_B/\partial T)$.

3.3. Isobars and Iso-Energy Lines

The pattern of the isobars, see left panel in Figure 3, strongly deviates from the naive expectations shown in Figure A1 for the toy model. Neither left nor right of the FOPT, does one recognize a near-vertical dropping of the curves $p=const$. Instead, the mysterious low-temperature–low-${\mu }_B$ behavior of the scaled pressure exhibited in the right panel of Figure 2 is retained, and, most importantly for our goals, the FOPT curves seem to repel the isobars. Thus, a smooth continuation of curves $p=const$ to the ${\mu }_B$ axis is hindered in the explored range. A warm EoS at $T>50$ MeV is conceivable by our approach, but the envisaged cool EoS at small or zero temperature remains elusive. Also in the current case, the agreement with lattice data [24] (crosses) looks fine.

3.4. Warm EoS

To illustrate further features of the obtained EoS, we exhibit in Figure 4 (left panel) energy density e and pressure $p_0$ as functions of temperature T along the isobars $T\left({\mu }_B\right){|}_{p=p_0}$ (or ${\mu }_B{\left(T\right)|}_{p=p_0}$ by inversion) for various values of $T_0$, which generate $p_0=p(T_0,{\mu }_B=0)$. That is, the EoS in parametric form, $e(T,{\mu }_B\left(T\right){|}_{p=p_0})$ and $p(T,{\mu }_B\left(T\right){|}_{p=p_0})$, can be inferred from this information on $e/p$ due to the log scale. Note the relation to the trace anomaly measure [41] via $e/p=1/({\displaystyle \frac{1}{3}}-\Delta )$) or $p\left(e\right)$ (see right panel). As mentioned above, the envisaged access to the small-T region is hampered by the failed approach of isobars down to the baryon–chemical potential axis, even for the “safer” isobars emerging from or running through the T-${\mu }_B$ region uncovered by the lattice QCD results [24]. We stress the enormous stiffness of the EOS on the right-hand side of the FOPT. In line with the peculiar features of the EoS at $T<T_{\mathrm{CEP}}$ and matter properties for the left-hand side of the FOPT is the huge energy density jump.

The connection of the holographic EoS with the nuclear-physics-based EoS at a certain matching point requires a special treatment, beyond the goal of the present paper. Analogously, the transit to the perturbative QCD regime above about $4\times {10}^3$ MeV/fm³ (pressure) and $1.3\times {10}^4$ MeV/fm³ (energy density) also requires separate work. As a guideline, one could follow the construction of a hybrid EoS in the spirit of Figure 15 in [5] and plug in the present holographic EoS as one building block among others. However, we find that, at a given energy density, e.g., ${10}^3$ MeV/fm³, our pressure seems too low for $T<100$ MeV in comparison with currently advocated credible EoSs, cf. Figure 1 in [30,31].

4. Conclusions and Summary

Einstein–Maxwell-dilaton (EMd) models have become quite popular in the last few years. They are aimed at obtaining a quantified hint of the location of a (hypothetical) critical end point (CEP) and emerging first-order phase transition (FOPT) curve in the phase diagram of QCD. The experimental search in relativistic heavy-ion collisions and the conjectured impact on neutron star merger dynamics and compact (neutron) star configurations up to sources of stochastic gravitational waves formed in the cosmological confinement epoch provide strong motivation to apply the class of EMd models in a regime where direct first-principle QCD calculations are not (yet) at our disposal.

EMd models are argued to deliver reliable information in the deconfinement regime. Once the dilaton potential and the dynamical coupling are adjusted, a specific EMd model parameterization delivers as primary quantities the entropy density s and the baryon density $n_B$, thus allowing the construction of isobars $p=const$. These curves $T\left({\mu }_B\right){|}_{p=const}$ map the pressure profile, given by lattice QCD data on the temperature axis, onto the temperature–baryon–chemical potential plane towards the baryon–chemical potential axis. When supplemented with the available information on s and $n_B$, the accompanying energy density e and, thus, the warm equation of state $p\left(e\right)$ become accessible.

The model facilitates a critical end point and a peculiar first-order phase transition curve, which levels off partially due to an imposed optionally novel side condition to suppress the appearance of an unwanted purely thermal phase transition. Thus the pressure interval directly accessible by lattice QCD on the temperature axis is transformed to a pressure profile at smaller temperatures: $p(T,{\mu }_B=0)\mapsto p(\mathrm{smaller}T\approx 0,{\mu }_B>0)$. We emphasize that the lattice QCD pressures in the temperature interval $T\in [125,240]$ MeV continue (numerically) the equation of state of strong-interaction matter towards a region relevant for compact (neutron) stars and their merging dynamics but are not accessible reliably by nuclear many-body theory. We also emphasize that pressures $p(T\notin [125,240]\mathrm{MeV},{\mu }_B=0)$ are largely unconstrained (and could be merely hampered by the employed ansätze of the dilaton potential and dynamical coupling of the EMd model), and, therefore, their mapping $p(T,{\mu }_B=0)\mapsto p(T\approx 0,{\mu }_B)$ is hazardous, in contrast to the valuable control of pressures $p(T\in [125,240]\mathrm{MeV},{\mu }_B=0)\mapsto p(T\in [125,240]\mathrm{MeV},{\mu }_B>0)$ by lattice data with $0<{\mu }_B/T\le 3.5$.

Since the EMd model with our presently employed parameterization points to a peculiar shape of the first-order phase transition curve, one should test other ansätze to elucidate whether suitable side conditions have a strong impact. In a larger context, a systematic approach to reliable parameterizations of the dilaton potential and dynamical coupling within EMd models is desirable. The low-temperature–low-density behavior of the model appears somewhat mysterious and could point to the need of including explicitly fermionic degrees of freedom, i.e., nucleons and their hard-core repulsion. First-principle input would be highly welcome to better constrain the model and provide confidence at larger densities. For practical applications, however, that region is uncovered by nuclear many-body approaches, and the present model should be constrained to the high-density continuation relevant to compact stars. The matching to perturbative QCD is a further constraint to the holographic EoS to be considered in follow-up work towards a hybrid EoS, where, eventually, also local charge neutrality and $\beta$ equilibrium should be imposed. In hybrid constructions [55], inhomogeneity effects [56] should also be accounted for. The role of the velocity of sound is to be emphasized [57,58,59] as well as the general finite-temperature behavior [60].