Figure 1

The free energy density for various models compares to the free gas case. For the reason that all gravity dual theories are strongly coupled, we scale $(•)/(•{)}_{q,\mathrm{SB}}$ to $3/4$ as $T\to \infty$, based on the $\mathcal{N}=4$ SYM result in Eq. (1), where $(•)$ can be replaced by any thermodynamical quantities, such as entropy, free energy, energy, or pressure. The MIT bag model [11, 12] and the fuzzy bag model [36] are also scaled to $3/4$ by some comparison reasons; they can be easily transform back to their original form if needed. For clarity, we classify and tag our models by numbers. From the arrow direction marked in this figure, the thick lines are for the models ($1\to 4\to 5\to 7\to 2\to 3\to 6$), respectively. Line (Model) (1) denotes the hard-wall model with the Ricci flat horizon calculated in Eq. (4), models considered in [42, 43], and the MIT bag model itself. We neglect their divergent when $T<2^{-1/4}{T}_{\mathrm{dec}}$. Line (2) denotes the hard-wall model with the spherical horizon in Eq. (5). Line (3) indicate the soft-wall model case, as Eq. (6) shows. Line (4) indicate the ten-dimensional “AdS/QCD cousin” model in Eq. (7). Line (5) denotes the fuzzy bag model result for comparison with Line (4). Line (6) is for the Gürsoy et al. model given in 51. Line (7) is calculated by the phenomenological model of 22, with a scalar potential $V(\varphi )=[-12\cosh (\sqrt{7/12}\varphi )+2{\varphi }^2]/L^2$. The thin gray lines are the p4-action result [117], in which the solid line indicates the pure glue case, the dashed line for the ($2+1$) flavor case, and the dashed-dotted line for the 3 flavor case. The points are calculated by the lattice methods with almost physical quark masses [118], where small solid bullets for $N_{\tau }=4$ case and solid squares for $N_{\tau }=6$ case. The small dark region near the critical temperature is enlarged and shown in the top-right corner, where the trianglelike shape formed by some line segments shows clearly the multivalued nature of Line (6) and (7).Reuse & Permissions

Figure 2

The entropy density $s=-df/dT$ for various models compares to the free gas case. In fact, in the large-$N_c$ limit, we have $s_q\propto N_c^2$ and $s_h\propto N_c^0$, hence $s_h=0$. The notations are as in Fig. 1. The thick lines are models ($1\to 4\to 5\to 7\to 2\to 3\to 6$), respectively, seeing from the arrow direction.Reuse & Permissions

Figure 3

The square of sound speed $c_s^2=d\log T/d\log s$ for various models compares to the free gas case. For the ideal gas case, or the strongly coupled $\mathcal{N}=4$ SYM theory indicated in Eq. (1), we have $c_s^2=1/3$. The notations are as in Fig. 1. The thick lines are models ($1\to 4\to 5\to 7\to 6\to 2\to 3$), respectively, seeing from the arrow direction.Reuse & Permissions

Figure 4

The normalization of the latent heat for pure glue fields of various models. In this case, we have $({\pi }^2/15)(\Delta s/s_{\mathrm{SB}})=(L_h/T_{\mathrm{dec}}^4)/(N_c^2-1)$. The left part is calculated for various models as tagged in Fig. 1. The right part shows that lattice results for $N_c=3$, 4, 6, and 8 [37], which suggest that the phase transition is second order for $N_c=2$, weakly first order for $N_c=3$, and robustly first order for $N_c\ge 4$. The black error bars are for $L_t=5$, and the gray ones for $L_t=6$ and 8. The fitting line for $L_t=5$ case is informal, but the extend to $N_c\to \infty$ case can be guess in this fitting. Notice that the original MIT bag model has the value ${\pi }^2/15\simeq 0.658$ in this figure.Reuse & Permissions

Figure 5

The prefactor $(\kappa /2\pi ){\Omega }_0$ in the nucleation rate formula. The thin gray dashed and dashed-dotted lines on the top are for dimensional values $T_c^4$ and $T^4$ respectively. The thin gray solid line using the same parameters as in [103], is shown for comparison reasons. Its value seems much larger than all other cases, mainly because it uses a rather large $\sigma =50\mathrm{MeV}/{\mathrm{fm}}^2$ (although other parameters also affect the curve); however, we choose a rather small value of $\sigma =0.02T_{\mathrm{dec}}^3\simeq 3.64\mathrm{MeV}/{\mathrm{fm}}^2$ for $T_{\mathrm{dec}}=192\mathrm{MeV}$ [104] in all other estimations. The gray solid bullet and square lines are for the pure gluon $SU(3)$ lattice result $L_h=1.4T_{\mathrm{dec}}^4$ and $\sigma =0.02T_{\mathrm{dec}}^3$ [37]. Nevertheless, when calculating the effectively massless degrees of freedom, we also count the fermionic contributions. The difference is the former case uses the perturbative result ${\eta }_q\simeq 1.12T^3/{\alpha }_s^2\log (1/{\alpha }_s)$ and ${\alpha }_s\sim 0.23$, but the latter case uses the AdS/CFT result ${\eta }_q=s_q/4\pi$. The thick color lines are for models discussed above. Seeing from the arrow direction, they are models ($7\to 2\to 3\to 5\to 4\to 1$), respectively. The process of scaling those large-$N_c$ theories to real QCD, and the rationality of that scaling, are discussed in the main text. The shear viscosity of models (1) and (4) are evaluated by 76; while for all other cases, we choose ${\eta }_q=s_q/4\pi$. The bulk viscosities are chosen by the relation ${\zeta }_q/{\eta }_q=2(1/3-c_s^2)$ of Eq. (15), and the shadow regions show the differences between them and the ${\zeta }_q=0$ cases. ${\zeta }_q$ of model (7) can be calculated from more sophistical numerical results given by 23, 24 if needed; however, we deal with it similarly with others for simplification. The black dotted line near the bottom is for the original MIT bag model with ${\eta }_q=(s_q-s_h)/4\pi$. We choose the correlation length ${\xi }_q=0.48(T_{\mathrm{dec}}/T)\mathrm{fm}$ [119] from lattice result for all our estimations, except the thin gray solid comparison line; in the gravity side, a lower limit of ${\xi }_q$ is given by 66.Reuse & Permissions

Figure 6

The supercooling scale $\Delta =1-T_f/T_{\mathrm{dec}}$ depends on the surface tension $\sigma$ for various models, which is estimated by Eq. (23). The notations are as in Fig. 5, except the shadow regions around the lines show the difference between Eq. (23) and the rough criterion $\Gamma \simeq 1/d_H^4$. The two gray vertical dashed lines are marked for $\sigma =0.02T_{\mathrm{dec}}^3$ and $\sigma =0.2T_{\mathrm{dec}}^3$, which are chosen as typical parameters in Fig. 9, 10. It can be seen that the supercooling scale $\Delta$ is really unsensitive to the method we estimate it, even in the small $\sigma$ regions where $d_{\mathrm{nuc}}\ll d_H$. The thick lines are models ($7\to 2\to 3\to 5\to 4\to 1$), respectively, seeing from the arrow direction. Although $v_{\mathrm{sh}}$ can be calculated accurately by 89, we choose $v_{\mathrm{sh}}=c_s$ for simplification, where the differences between them are imperceptible.Reuse & Permissions

Figure 7

The mean nucleation distance $d_{\mathrm{nuc}}$ depends on various surface tension $\sigma$, estimated by Eq. (24). The notations are as in Fig. 5. The thick lines are models ($7\to 2\to 3\to 5\to 4\to 1$), respectively, seeing from the arrow direction. Although the terminal point “$\star$” marked for model (7) is factual, the terminal point “$\circ$” marked for model (2) is the numerical limit of our calculation. A maximum $\sigma$ exists for the maximum expected supercooling scale to be achieved; as $L_h=0$ for $\Delta ={\Delta }_{<}$ in model (2), $d_{\mathrm{nuc}}\to \infty$ when $\sigma$ tending towards this limit.Reuse & Permissions

Figure 8

A Hawking-Page phase transition [21] should always have a minimum temperature $T_{\min }$, below which the high temperature phase cannot exist. This minimum temperature is intrinsic, rather than caused by impurities or perturbations in the old supercooling scenarios. The long curved arrows show the behavior of the system from high temperature to low temperature phase, if no supercooling happens.Reuse & Permissions

Figure 9

The mean nucleation distance $d_{\mathrm{nuc}}$ depends on the latent heat $L_h$ for the Gubser et al. model [22]. The gray vertical dashed line is $L_h=2.67T_{\mathrm{dec}}^4$ deduced from the potential $V(\varphi )=[-12\cosh (\sqrt{7/12}\varphi )+2{\varphi }^2]/L^2$ in the formal estimations. The three thick lines are for $\sigma =0.2T_{\mathrm{dec}}^3$, $0.02T_{\mathrm{dec}}^3$ and $0.002T_{\mathrm{dec}}^3$ (from up down), respectively.Reuse & Permissions

Figure 10

The various supercooling scales depend on the latent heat $L_h$ for the Gubser et al. model [22]. The black dotted curve from the top-right corner to the bottom-left corner is the maximum supercooling scale ${\Delta }_{<}$; hence the shadow region above it, is forbidden by that model. The actual supercooling scales $\Delta =1-T_f/T_{\mathrm{dec}}$ calculated by Eq. (24) are denoted by the thick solid lines, which are for $\sigma =0.2T_{\mathrm{dec}}^3$, $0.02T_{\mathrm{dec}}^3$ and $0.002T_{\mathrm{dec}}^3$ (from up down), respectively. The dotted lines a little below them constrain the phase transition to be completed, which are roughly calculated by $\Gamma \simeq 1/d_H^4$; that is, $d_{\mathrm{nuc}}\simeq d_H$. For some particular $\sigma$, $d_{\mathrm{nuc}}$ can easily be much larger, providing that the latent heat $L_h$ is small enough that the maximum supercooling ${\Delta }_{<}$ is saturated. However, it is unlikely that the larger $d_{\mathrm{nuc}}$ can help us understanding the formation of quark nuggets or the inhomogeneous initial conditions of the big-bang nucleosynthesis, because the parameter $L_h$ should be fine-tuned.Reuse & Permissions