Propagation of statistical uncertainties of Skyrme mass models to simulations of $r$-process nucleosynthesis

Uncertainties in nuclear models have a major impact on simulations that aim at understanding the origin of heavy elements in the universe through the rapid neutron capture process ($r$ process) of nucleosynthesis. Within the framework of the nuclear density functional theory, we use results of Bayesian statistical analysis to propagate uncertainties in the parameters of energy density functionals to the predicted $r$-process abundance pattern, by way not only of the nuclear masses but also through the influence of the masses on $\beta$-decay and neutron capture rates. We point out the importance of the nonequilibrium end stage of the $r$ process in determining the width of the resulting abundance pattern uncertainty bands. We additionally make the first identifications of specific parameters of Skyrme-like energy density functionals which show tentative correlations with particular aspects of the $r$-process abundance pattern. While previous studies have explored the reduction in the abundance pattern uncertainties due to anticipated new measurements of neutron-rich nuclei, here we point out that an even larger reduction will occur when these new measurements are used to reduce the uncertainty of model predictions of masses, which are then propagated through to the abundance pattern. We make a quantitative prediction for how large this reduction will be.

Figure 1

Abundance patterns $Y\left(A\right)$ versus $A$ for 50 $r$-process simulations with astrophysical conditions corresponding to high-entropy (a), low-entropy (b), and fission-recycling (c) outflows, as described in the text. The shaded region shows the full range of abundance patterns produced, and the black line shows their mean. All patterns are scaled to solar abundances from [13].

Figure 2

Average standard deviation of calculated nuclear abundances as a function of cumulative number of unedf1 mass tables considered in this work. Abundance pattern variations begin to converge once the first $\approx 35$ mass tables are considered for each of the high-entropy (green diamonds), low-entropy (red circles), and fission recycling (blue triangles) conditions from Fig. 1. Convergence holds as the number of unedf1 mass tables is extended to 50.

Figure 3

Total abundance patterns $Y\left(A\right)$ vs $A$ for 50 $r$-process simulations based on the simulated disk ejecta conditions of [34]. The shaded region corresponds to the total range of abundance patterns and the black line to their mean, as in Fig. 1.

Figure 4

Relationship between $r$-process abundance pattern and unedf1 functional parameters for 50 unedf1 mass tables. (a) shows the relationship between the weighted average mass number $A$ of the rare earth peak and the isovector surface coupling constant $C_1^{\rho \mathrm{\Delta }\rho }$ for the low-entropy wind conditions of Fig. 1. A linear fit to the data set is given by the solid line with correlation coefficient $r=0.68$. (b) shows the relationship between the proton pairing strength $V_0^p$ and the ratio of summed abundances in the rare earth region to the $A=195$ region for the high-entropy (green diamonds), low-entropy (red circles), and fission recycling (blue triangles) conditions from Fig. 1, with linear fits given for the high-entropy data set by the green dashed line ($r=0.66$), the low-entropy data set by the red solid line ($r=0.76$), and the fission recycling data set by the blue dot-dashed line ($r=0.75$). The gray shaded region in each figure indicates the range of values in each metric admitted by the solar abundances of [13, 35].

Figure 5

Variations $\sigma$ among our set of 50 unedf1 mass tables (light green shaded region) and our set of simulated mass tables (dark green shaded region), each with respect to the nominal unedf1 masses, for the tin ($Z=50$) isotopic chain. The AME2016 range of known masses [28] and anticipated FRIB reach are indicated, respectively, by black and gray solid lines. The vertical darkened band ($N=105-121$) indicates the range in location for the one-neutron dripline.

Figure 6

Ratios of the abundances $Y\left(A\right)$ to the mean abundance $Y_{\text{mean}}\left(A\right)$ for the set of 50 simulations with the example high entropy wind (a), low entropy wind (b), and fission recycling outflow (c) astrophysical conditions, as in Fig. 1. The light shaded band shows theory-only calculations, the medium shaded band implements AME2016 masses and NUBASE2016 decay properties where available, and the dark shaded band additionally includes the simulated mass tables described in the text. The lightest-shaded grey region shows the analogous range for a set of 15 distinct, non-unedf1 mass tables (FRDM1995 [50], FRDM2012 [51], Duflo-Zuker [52], ETFSI [53], ETFSI-q [54],Thomas-Fermi [55], Weizsäcker-Skyrme (WS) [56, 57], KTUY05 [58], HFB21 [59], HFB22 [60], SkM* [61], SkP [62], SLy4 [63], SV-min [64], and UNEDF0 [65]).