Constrained Brownian-bridge prior for neutron-star equation-of-state inference

Abstract

We set forth a new method for generating model-agnostic, nonparametric priors for neutron star equation-of-state inference that are stable, causal and thermodynamically consistent by construction. This generalizes Gaussian processes to include global thermodynamic constraints, specifically allowing the inclusion of any number of training points in the form $(μ, n, p)$ while retaining thermodynamic consistency between them. The method is based on constructing constrained Brownian bridges, whose correlation properties can be tuned at will allowing flexibility between a conservative prior and a theory-informed prior. The method does not require any shooting to obey multiple constraints and provides an efficient and informed way to include both chiral effective field theory and perturbative quantum chromodynamics constraints within the same framework.

Figures (9)

• Figure 1: Example of a two-step self-similar refinement (fractal). Knowledge of the EoS at low and high densities ($\beta_\mathrm{L} = (\mu_\mathrm{L}, n_\mathrm{L}, p_\mathrm{L})$ and $\beta_\mathrm{H} = (\mu_\mathrm{H}, n_\mathrm{H}, p_\mathrm{H})$) implies constraints on points $\beta_0$ lying between them. Once a point $\beta_0$ is chosen, similar constraints are imposed on the intervals $[\beta_\mathrm{L}, \beta_0]$ and $[\beta_0, \beta_\mathrm{H}]$. Iteratively refining each interval by introducing new points eventually leads to a complete EoS illustrated by the blue dotted line obtained after multiple iterations passing through the specified points.
• Figure 2: Three-dimensional rendering of the volume in the ($\mu, n, p$) that can be reached by a stable, causal and consistent EoS interpolating between known low- and high-density limits arising from chiral EFT and perturbative QCD. The limits are depicted as the thick lines emerging from the lower-right and upper-left corners of the volume. (Left) Sample of fractal EoSs constructed using 10 iterations of the self-similar refinement process discussed in Sec. \ref{['sec:fractal']}. The coloring of the EoSs is arbitrary and used solely to improve visual clarity. (Right) Sample of Brownian-bridge EoSs constructed from the self-similar EoSs by diffusive smoothing as described in Sec. \ref{['sec:implementation']}.
• Figure 3: The two-point correlation function of the sound speed, $\langle \delta c_s^2(n_0)\, \delta c_s^2(n) \rangle$, showing correlations between values at $n' = 0.8\ \mathrm{fm}^{-3}$ and different $n$ for varying levels of diffusion (left panel), and for a fixed amount of diffusion with varying $n'$ (right panel). Thin dashed lines show the corresponding Gaussian correlations, with correlation lengths set by the amount of diffusion, $\sigma = \sqrt{4\,D(n_0)\,t}$. The local correlations closely follow the Gaussian form, but the global structure imposed by physical constraints produces a long-range anti-correlation.
• Figure 4: The procedure for incorporating the low- and high-density inputs into the construction of the constrained Brownian bridge. The blue dashed lines correspond to the samples of the fractal EoSs together with their diffused counterparts represented by purple solid lines. The two colored bands correspond to the low- and high-density limits from the chiral EFT and pQCD EoSs. The fractal EoS is constructed from two endpoints, $\beta_\mathrm{L}$ and $\beta_\mathrm{H}$, obtained from the sampled chiral-EFT and pQCD EoSs at the corresponding densities $n_\mathrm{L} = 2\,n_{\rm s}$ and $n_\mathrm{H} = 30\,n_{\rm s}$. The diffusion is then performed over the broader density range $[0.34,\,40]\,n_{\rm s}$ to ensure a smooth transition of $c_s^2$ to both limits.
• Figure 5: Progression of the posterior equation of state ($\varepsilon$ vs $p$), speed of sound ($c_s^2$ vs $n$) and the mass-radius relation ($M$ vs $R$) as a function of the relative correlation length $\sigma/n = 1\%, 10\%, 20\%$, and $40\%$. Each EoS drawn from the prior is colored according to the combined likelihood function. The likelihood function is normalized separately to the maximum likelihood for each fixed correlation length. As the correlation length increases, the global structure of the EoS with significant softening at high densities becomes clearer. The bar marked with $\sigma$ indicates the correlation length associated with each column. The relative correlation length corresponds to a fixed length on a logarithmic axis.