Nuclear pasta in hot dense matter and its implications for neutrino scattering

The abundance of large clusters of nucleons in neutron-rich matter at subnuclear density is found to be greatly reduced by finite-temperature effects when matter is close to $\beta$ equilibrium, compared to the case where the electron fraction is fixed at $Y_e>0.1$, as often considered in the literature. Large nuclei and exotic nonspherical nuclear configurations called pasta, favored in the vicinity of the transition to uniform matter at $T=0$, dissolve at a relatively low temperature $T_{\mathrm{u}}$ as protons leak out of nuclei and pasta. For matter at $\beta$ equilibrium with a negligible neutrino chemical potential we find that $T_{\mathrm{u}}^{\beta }\simeq 4\pm 1$ MeV for realistic equations of state. This is lower than the maximum temperature $T_{\max }^{\beta }\simeq 9\pm 1$ MeV at which nuclei can coexist with a gas of nucleons and can be explained by a change in the nature of the transition to uniform matter called retrograde condensation. An important new finding is that coherent neutrino scattering from nuclei and pasta makes a modest contribution to the opacity under the conditions encountered in supernovas and neutron star mergers. This is because large nuclear clusters dissolve at most relevant temperatures, and at lower temperatures, when clusters are present, Coulomb correlations between them suppress coherent neutrino scattering off individual clusters. Implications for neutrino signals from galactic supernovas are briefly discussed.

Figure 1

$\beta$-equilibrium path (thick solid blue line) for SLy4 Skyrme interaction in the coexistence region (delimited by the thick solid black line) in the ($n_n$, $n_p$) plane and for $T=1$–9 MeV. A sample of the Gibbs construction paths is shown (thin solid purple lines). The global density, which intersects the $\beta$-equilibrium path line with the Gibbs construction one (see, for instance, the “+”symbol at $T=5$ MeV), represents the equilibrium state connecting the two phases at equilibrium which are located at the boundaries (see the points labeled “l” and “h,” which are the low- and high-density phases associated with the global density identified by the + symbol at $\beta$ equilibrium). The constant $Y_e=0.2$, 0.3, and 0.4 paths are represented by the thick dashed yellow lines. The spinodal instability region is also shown (thin solid green line) and the critical points are shown by filled red circles with error bars.

Figure 2

Volume fraction of the high-density phase in heterogeneous matter for SLy4 Skyrme interaction. (a) For the $\beta$-equilibrium path; (b) for the constant $Y_e=0.2$ path.

Figure 3

This schematic shows the occupied proton energy levels in the low- and high-density phases that coexist in the heterogeneous phase. The $T=0$ situation is shown at the left, and the finite temperature at which a significant thermal population exists in the low-density phase is shown at the right. See text for additional details.

Figure 4

(a) Energy differences $\mathrm{\Delta }{E}_p/T$ and (b) charge particle densities ($n_e$, $n_p^l$, $n_p^h$) as a function of the density at different temperatures $T=1$–6 MeV. The SLy4 EOS is considered here.

Figure 5

(a) Dissolution temperature $T_{\mathrm{u}}$ and maximal temperatures $T_{\max }$ at $\beta$ equilibrium, compared to $T_{\max }$ at fixed $x_e=0.2$ and 0.4 for the set of EOSs compatible with the chiral EFT EOS in neutron matter. (b) Highest density reached by the pasta phase for $\beta$ equilibrium or at fixed $Y_e$. These predictions are shown versus $L_{\mathrm{sym}}$.

Figure 6

Ratio $\mathcal{R}$ from Eq. (13) under two conditions: (a) fixed proton fraction $Y_e=0.4$ and (b) $\beta$ equilibrium. The gray central band indicates the density region where nonspherical pasta may be present, and the curves terminate in a circle at the density at which $u=1/8$. Dotted curves indicate the contribution of the external gas only. Neutrinos are considered to be at thermal equilibrium, except for the dashed black curves, where $E_{\nu }=30$ MeV.

Figure 7

Energy dependence of the total ratio of cross section $\mathcal{R}$ at various temperatures and for two densities: $n_B=0.01{\mathrm{fm}}^{-3}$ (solid lines) and the threshold density at which $u=1/8$ (dashed lines). As for Fig. 6, (a) shows results for a fixed proton fraction $Y_e=0.4$, and (b) for $\beta$-equilibrium conditions. In both cases the filled circles indicate the energies for thermal neutrinos ($E_{\nu }=3T$).