Implicit correlations within phenomenological parametric models of the neutron star equation of state

The rapid increase in the number and precision of astrophysical probes of neutron stars in recent years allows for the inference of their equation of state. Observations target different macroscopic properties of neutron stars which vary from star to star, such as mass and radius, but the equation of state allows for a common description of all neutron stars. To connect these observations and infer the properties of dense matter and neutron stars simultaneously, models for the equation of state are introduced. Parametric models rely on carefully engineered functional forms that reproduce a large array of realistic equations of state. Such models benefit from their simplicity but are limited because any finite-parameter model cannot accurately approximate all possible equations of state. Nonparametric methods overcome this by increasing model freedom at the cost of increased complexity. In this study, we compare common parametric and nonparametric models, quantify the limitations of the former, and study the impact of modeling on our current understanding of high-density physics. We show that parametric models impose strongly model-dependent, and sometimes opaque, correlations between density scales. Such interdensity correlations result in tighter constraints that are unsupported by data and can lead to biased inference of the equation of state and of individual neutron star properties.

Figure 1

Symmetric 90% credible region for the pressure $p$ at each density $\rho$ in units of the nuclear saturation density using the nonparametric and spectral prior processes. We show results including all astrophysical data (labeled “astro,” solid lines) and restricting to the heavy pulsars only (labeled “psr,” dashed lines). The latter choice ensures that prior choices on the $M_{\max }$ supported by each model are irrelevant. Other parametric models are shown in Fig. 10. In all cases, we find that the $p\text{−}\rho$ posterior depends on the EoS model even when identical data and inference schemes are employed.

Figure 2

Prior (dashed) and posterior (solid) for the radius of a $1.4M_{\odot }$ NS, $R_{1.4}$, and maximum mass, $M_{\max }$, of a NS for two choices of the marginal $M_{\max }$ prior: default (left) and flat (right). We show results with the nonparametric and spectral EoS models, and contours denote 90% credible regions. The black line in the two-dimensional plot represents a maximally stiff $M\text{−}R$ curve [48] stitched to a fiducial low density EoS [49]. Due to the low-density stitching, this “causality” line should be interpreted as a fuzzy boundary and not a sharp line. Both panels demonstrate that the spectral and nonparametric EoS models produce different $M_{\max }$ posteriors and that these differences cannot be attributed to the marginal priors. They are instead caused by correlations between low and high densities (equivalently, between $M_{\max }$ and $R_{1.4}$) imposed by the models. The correlations in the nonparametric case are due to causality, while the spectral case exhibits additional correlations and model dependence.

Figure 3

68% ($\pm 1\text{−}\sigma$) marginal credible regions for the pressure at each density under the point+slope (top, green), two-point (middle, red), and GP (bottom, blue) prior processes. Shaded regions correspond to marginal distributions induced by the prior process at each energy density (light colors) and conditioned distributions for $p(ϵ)$ given a precise observation of $p_c$ (dark colors). Only the GP process “fills the prior volume” rapidly as one moves away from the observation point ${ϵ}_c$. Compare to Fig. 5.

Figure 4

Joint and marginal distributions for the pressures at the three reference densities called out in Fig. 3 for the $\mathrm{point}+\mathrm{slope}$ (green), two-point (red), and GP (blue) prior processes. Contours in the joint distribution represent 90% credible regions. While certain choices for the parametrization and priors can uncorrelate pairs of variables, only the GP prior process induces minimal correlations between all variables simultaneously.

Figure 5

Similar to Fig. 1 but with a mock constraint injected directly into $p(\rho =2{\rho }_{\mathrm{nuc}})$ for each EoS prior process. The posterior after the simulated constraint is included (“$\mathrm{astro}+\mathrm{mock}$”) is overplotted on the posterior with all current data (“astro”) and in a darker color. The constraint at a single density affects the parametric posteriors over a much wider range of density scales than the nonparametric one. The inset focuses around $\rho =2{\rho }_{\mathrm{nuc}}$. The two black straight lines provide an estimate of the constraints imposed by causality ($c_s^2<1$) and thermodynamic stability $(c_s^2>0.1)$ around $\rho =2{\rho }_{\mathrm{nuc}}$, subject to the heavy pulsar measurements. The nonparametric posterior quickly “fills” more of the physically available region after satisfying the mock constraint, while the parametric posteriors do not.

Figure 6

Inferred posterior for $R_{1.4}$ and $M_{\max }$ using the nonparametric model and the spectral model and mock x-ray-radio (blue and orange dashed lines, respectively) and GW (blue and orange solid line, respectively) observations. All posteriors also include all current astrophysical data. The vertical and horizontal red lines show the injected value of $R_{1.4}$ and $M_{\max }$. The $M_{\max }$ posterior is only weakly informed by the GW data as they typically cannot lead to a definitive identification of a $>2M_{\odot }$ object as a NS, and it is thus similar to that of Fig. 2.

Figure 7

Marginal one- and two-dimensional prior and posterior distributions for the parameters of the spectral model, ${\gamma }_i$, as well as the maximum NS mass, $M_{\max }$, and the radius, $R_{1.4}$. We show the default prior as well as reweighted results that upweight more extreme values of ${\gamma }_i$. In both cases, $M_{\max }$ and $R_{1.4}$ have very similar posteriors, showing that the reweighting does not efficiently extend the coverage of the prior toward the causality threshold in the $M_{\max }\text{−}{R}_{1.4}$ plane. Additionally, extending the prior ranges for ${\gamma }_i$ is unlikely to change the results as the posteriors are not limited by the prior ranges.

Figure 8

Example EoS draws from the nonparametric prior plotted in terms of the pressure $p$ vs the density $\rho$ (top), the mass $M$ vs the radius $R$ (middle), and the speed of sound $c_s$ vs the density $\rho$ (bottom). We only draw EoS with non-negligible contribution to the posterior. For reference, we also plot the 90% symmetric credible intervals for the posterior using only heavy pulsar observations and all astrophysical data.

Figure 9

Comparison between using strictly causal ($c_s^2<c^2$) parametric EoS with each model (gray) and the headline results allowing some violation of the causal limit ($c_s^2<1.1c^2$). When restricting to only causal EoS, the issues of model dependence and insufficient coverage of the physically allowed $M_{\max }\text{−}{R}_{1.4}$ space are more severe, especially for the piecewise polytrope.

Figure 10

Symmetric 90% credible region for the pressure $p$ at each density $\rho$ in units of the nuclear saturation density (left) and the radius $R$ as a function of the mass $M$. From top to bottom, we show results with the spectral, piecewise-polytrope and speed-of-sound parametric models. At each panel, we overplot the corresponding nonparametric result for comparison. The spectral $p\text{−}\rho$ panel is identical to Fig. 1, but we show it for completeness. As in Fig. 1, we show results with all astrophysical data (labeled “astro,” solid lines) and restricting to the heavy pulsars only (labeled “psr,” dashed lines).

Figure 11

The speed of sound squared as a function of baryon density in the spectral (top), piecewise-polytrope (middle) models, and speed-of-sound (bottom) models, compared to our nonparametric model.

Figure 12

Similar to Fig. 2 but for the piecewise-polytrope (top panels) and the speed-of-sound (bottom panels) models. As with the spectral case, we find that the parametric model leads to tighter posteriors for $M_{\max }$ compared to the nonparametric model. The two-dimensional plots show that this is again due to model-dependent correlations in $M_{\max }\text{−}{R}_{1.4}$.