On Approximation Methods in the Study of Boson Stars

TL;DR

This paper evaluates various variational wavefunctions for boson stars, identifying the most accurate ansätze for different interaction regimes and particle numbers, and critically assesses the Thomas-Fermi approximation in the relativistic context.

Contribution

It systematically compares multiple variational ansätze against exact solutions, proposing optimized forms like the double exponential for improved accuracy and computational efficiency.

Findings

The 'linear+exponential' wavefunction best fits stable attractive interactions.

The Gaussian ansatz performs poorly across most parameter space.

The double exponential ansatz can be tuned to match the sech ansatz.

Abstract

We analyze the accuracy of the variational method in computing physical quantities relevant for gravitationally bound Bose-Einstein condensates. Using a variety of variational ans\"atze found in existing literature, we determine physical quantities and compare them to exact numerical solutions. We conclude that a "linear+exponential" wavefunction proportional to $(1 + \xi)\exp(-\xi)$ (where $\xi$ is a dimensionless radial variable) is the best fit for attractive self-interactions along the stable branch of solutions, while for small particle number $N$ it is also the best fit for repulsive self-interactions. For attractive self-interactions along the unstable branch, a single exponential is the best fit for small $N$, while a sech wavefunction fits better for large $N$. The Gaussian wavefunction ansatz, which is used often in the literature, is exceedingly poor across most of the parameter space, with the exception of repulsive interactions for large $N$. We investigate a "double exponential" ansatz with a free constant parameter, which is computationally efficient and can be optimized to fit the exact solutions in different limits. We show that the double exponential can be tuned to fit the sech ansatz, which is computationally slow. We also show how to generalize the addition of free parameters in order to create more computationally efficient ans\"atze using the double exponential. Determining the best ansatz, according to several comparison parameters, will be important for analytic descriptions of dynamical systems. Finally, we examine the underlying relativistic theory, and critically analyze the Thomas-Fermi approximation often used in the literature.