Approximation methods in the study of boson stars

We analyze the accuracy of the variational method in computing physical quantities relevant for gravitationally bound Bose-Einstein condensates. Using a variety of spherically symmetric variational ansätze found in existing literature, we determine physical quantities and compare them to numerical solutions. We conclude that a “$\mathrm{linear}+\mathrm{exponential}$” wave function 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 wave function fits better for large $N$. The Gaussian wave function 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 numerical 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 ansätze 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.

Figure 1

The rescaled mass $M$ and radius $R_{99}$ for boson stars. The black lines denote the numerical solutions, whereas the ansätze are color-coded in the legend. The left panel is the attractive interaction case, where the maximum mass is calculable in a nonrelativistic method; the right panel is the repulsive interaction case.

Figure 2

Quantitative comparisons for attractive interactions, on the stable branch of solutions. In the upper two panels, we show the rescaled wave functions for two choices of particle number $N=0.12N_c$ (left) and $N=0.98N_c$ (right). In the lower two panels, we show the corresponding deviations ${\mathrm{\Delta }}_{\psi }$ (left) and ${\mathrm{\Delta }}_r$ (right) of the ansätze from the numerical solution, as defined in Eqs. (4.12) and (4.13).

Figure 3

Quantitative comparisons for attractive interactions, on the unstable branch of solutions. In the upper two panels, we show the rescaled wave functions for two choices of particle number $N=0.59N_c$ (left) and $N=0.98N_c$ (right). In the lower two panels, we show the corresponding deviations ${\mathrm{\Delta }}_{\psi }$ (left) and ${\mathrm{\Delta }}_r$ (right) of the ansätze from the numerical solution, as defined in Eqs. (4.12) and (4.13).

Figure 4

Quantitative comparisons for repulsive interactions. In the upper two panels, we show the rescaled wave functions for two choices of particle number $N=0.12N_{*}$ (left) and $N=40N_{*}$ (right). In the lower two panels, we show the corresponding deviations ${\mathrm{\Delta }}_{\psi }$ (left) and ${\mathrm{\Delta }}_r$ (right) of the ansätze from the numerical solution, as defined in Eqs. (4.12) and (4.13).