CompactObject-TOV package Tutorials

An open-source package for neutron star whole workflow Bayesian nference constraining Neutron star EOS package

CompactObject is designed to open-source a full-scope neutron star equation inference. Currently, this equation of state framework is mainly based on relativistic mean field (RMF) theory. The package integrates three independent modules:

1. A user-friendly Tolman–Oppenheimer–Volkoff (TOV) equation solver for determining neutron star structure based on a given equation of state (EOS).

2. A neutron star full density range equation of state generator. At present, it only contains the relativistic mean field theory equation of state solver. In the future, it will easily accommodate polytropes and more equation of states.

3. A complete Bayesian inference workflow package for constraining the equation of state of neutron stars. This includes defining the likelihood of inference from observations and nuclear experiments, as well as from simulated astrophysical observations. It also involves defining priors for the given parameters and running a nested sampling of the posterior space.

We should mention that these three parts are independent, which means users can embed these modules as part of their work. The usage of this package extends beyond conducting inference studies. Additionally, we welcome contributions of new features for our package.

The papers generated from this package are listed below: [1] (Huang, C., Raaijmakers, G., Watts, A. L., Tolos, L., and Providência, C., “Constraining fundamental nuclear physics parameters using neutron star mass-radius measurements I: Nucleonic models”, 2023. doi:10.48550/arXiv.2303.17518)

Concept

Bayesian inference studies of the neutron star equation of state have become a trending field nowadays, particularly due to significant advancements such as the Neutron Star Interior Composition Explorer (NICER) measuring the mass and radius of neutron stars through X-ray timing, as well as the detection of neutron star merger events through gravitational wave observations by the LIGO detector.

Here below is the overall pipline of this field:

As depicted in this plot, fundamental physics can provide the equation of state for neutron stars. By inputting the equation of state into the Tolman–Oppenheimer–Volkoff (TOV) equation, we can obtain parameters related to the neutron star structure, such as mass, radius, and tidal properties. Instruments like NICER and LIGO can be utilized to measure these properties. With the information obtained from observations, it becomes possible to perform Bayesian inference constraining, such as determining the region in the Mass-Radius space where neutron stars can exist. Subsequently, we can derive constraints on the equation of state space and, ideally, obtain insights into the fundamental composition of the neutron star’s interior.

Equation of State

The neutron star equation of state (EOS) plays a crucial role in reflecting the composition of a neutron star. It is closely connected to the microphysical properties of neutron stars. In this context, we are utilizing an equation of state derived from a model known as the Relativistic Mean Field theory (RMF). The Lagrangian of this model is represented as follows:

\[\mathcal{L}=\sum_N \mathcal{L}_N+\mathcal{L}_{\mathcal{M}}+\sum_l \mathcal{L}_l\]

Where the \(\mathcal{L}_N\) is the neucleonic Lagrangian, \(\mathcal{L}_M\) is the meson part Lagrangian, \(\mathcal{L}_l\) is the lepton Lagrangian. details of the Lagrangian are

where \(\Psi_{N}\) and \(\psi_{l}\) are the nucleon and lepton spinors, and \(\bar{I}_{N}\) is the nucleon isospin operator. The strong interaction coupling of a meson to a nucleon is denoted by \(g\), while the masses of the nucleons, mesons, and leptons are denoted by \(m\). The parameters \(\kappa\), \(\lambda_0\), \(\zeta\) and \(\Lambda_{\omega}\) plus the meson-nucleon coupling constants are coupling constants to be determined by the inference method.

These free parameters represent the degrees of freedom in the RMF model and can be determined through nuclear experiments. However, in addition to nuclear experiments, we can also explore the possibility of constraining these parameters through astrophysical observations. The complete list of parameters includes:

These are the parameters that you should input to generate the equation of state from our EOSgenerators module, different equation of state parameter will have different effect on mass radius like we showed here.More details about this physics can check (Glendenning, 1996)

Tolman–Oppenheimer–Volkoff(TOV) equation

TOV equation is a general relativity equation that constrain the structure of a spherical symmetrical body by gravity. This is the original equation:

\[\frac{d P}{d r}=-\frac{G m}{r^2} \rho\left(1+\frac{P}{\rho c^2}\right)\left(1+\frac{4 \pi r^3 P}{m c^2}\right)\left(1-\frac{2 G m}{r c^2}\right)^{-1}\]

To solve this problem, the essential ingredient is the equation of state (EOS). Once you have the EOS, the basic strategy for solving the equation is as follows: at a given central density, you input it into the neutron star EOS to obtain the pressure. Then, you integrate the density from the center to the boundary, repeating this process across the entire possible density range.

In our code, we provide two different functions that you can call. The default option allows you to solve the Tolman-Oppenheimer-Volkoff (TOV) equation within a predefined density range, which is log(14.3, 15.6) on a scale of 10. Alternatively, you can choose to solve the equation point by point, allowing you to select any central density range you prefer. More information about TOV you could check (wiki page)

The following image integrates (and thus averages) over waveband (a range of photon energies). We also decrease the mode frequency relative to the stellar spin frequency, such that the mode is not as equatorially trapped.

Bayesian Inference

Using Bayesian inference tools to explore the constraint of neutron star equation of state is common nowadays, the basic equation of it is Bayes theorem:

\[P(A \mid B)=\frac{P(B \mid A) P(A)}{P(B)}\]

That is, Posterior probablity is propotional to the prior probablity times likelihood. Posterior is — after correction of the new observations/experiment, the probablity of something is true. Prior is — before the new observations/experiment come in, my initial thought about the probablity of something is true. Likelihood is — the correction that we get from the new observation/experiments.

Here, the likelihood will be mostly come from three different families:

1. Mass Raius measurements from x-ray timing (like NICER).

2. Tidal measurements from gravitational wave detection

3. Mass measurements from radio timing.

4. Nuclear physics constraint comes from the nuclear experiments.

NICER Mass radius measurements are remarkable achievement of this centry of neutron star physics, same as the gravitational wave detection. many references out there for this topic. About the Nuclear physics connection between our equation of state and the nuclear quantities, please check (Chen & Piekarewicz 2014a)

Here the nuclear physics quantities we cared are K, J and L, that is the decompressibility of nuclear matter K, symmerty energy at saturation density J, and the slope of symmetry energy at saturation density L. These all can be computed out by posterior samples (will add the nuclear properties computation code in near future). Also they could be independent group of constraint on our equation of state of neutron star.

When you do a Real astrophysical sampling, the important thing is you should also sampling the neutron star central density of that measurement you are using, which means if you want to investigate what the constraining effect for neutron star EOS by two mass radius measurements, then you need define another two free parameters — the central densities of these measurements, other-wise, this could be proved to be a not full-scope equation of state inference, that is why our likelihood functions once you want to constraint from observation, always need a parameter d1, that is the density parameter of this observation.

Illustration Notebooks

• Equation of State Generators illustration notebook
  • Load crust EOS and prepare interface EOS
  • Defining the paramters
  • Connecting Core and crust EOS
• TOV solver sampled notebook
• Bayesian inference analysis notebook
  • Set up prior
  • Set up likelihood
  • Set up sampler
  • Posterior plot

EOS generators module

• Crust Equation of state interpolation
• RMF EOS solver
• RMF EOS solver (numba speed up version)

TOV solver module

• Import Equation of State
• Main functions
• Solve TOV Equation
• Calculate Speed of Sound

Bayesian Inference module

• Bayesian Inference smapler setup module
• Prior Definition Module
• Likelihood function definition module

Indices and tables

• Index

• Module Index

• Search Page