Variable conventions for boson stars

General Theory

For boson stars, we assume the following Lagrangian

$$ \begin{equation} S = \int \frac{\sqrt{-g}}{2} \left( \frac{R}{8 \pi G}- \left[g^{\mu \nu} \nabla_{\mu} \bar{\varphi} \nabla_{\nu} \varphi + V(\varphi) \right] \right) d ^4x, \end{equation} $$

where for a spherically symmetric complex solution we have

$$ \varphi = A(r) e^{i(\epsilon \omega t + \delta \phi)} $$

Here $A(r)$ is the amplitude, $\omega$ is the constant BS frequency, $\epsilon = \pm 1$ determines whether the frequency is positive or negative and $\delta \phi$ introduces a constant phase-offset.

In our code we have 3 possible potentials you can choose from:

• Mini potential, with $\mu$ denoting the mass of the scalar field and in numerics always set to 1 for simplicity:

  $$V=\mu^2|\varphi|^2$$

• Repulsive potential, with $\lambda$ denoting the self-interaction term:

  $$V=\mu^2 |\varphi|^2 + \frac{\lambda}{2} |\varphi|^4$$

• Solitonic potential, with $\sigma_0$ denoting the self-interaction term:

  $$V=\mu^2|\varphi|^2 \left(1-2\frac{|\varphi|^2}{\sigma_0^2} \right)^2$$

Implementation of other BS potentials should be straight-forward.

Computation of 1D boson star solutions

The 1D solver has been developed by Robin Croft and the detailed implementation can be found in his thesis. In brief, in isotropic gauge we assume:

$$ \begin{equation} \rm{d}s^2 = -\Omega^2(r) \rm{d}t^2 + \Psi^2(r) \rm{d}\underline{x}^2, \end{equation} $$

where $\Omega$ is the lapse function and $\Psi$ is the conformal factor. When solving for the BS solution, one needs to provide the central amplitude of the boson star, i.e. $A(0)$, and find the rest of the functions $\Omega$, $\Psi$ and constant $\omega$ that satisfy the Einstein-Klein-Gordon system of equations. In order to obtain the right solution, we use a shooting method, where first the initial guesses for the central values of $\Psi(0)$ and $\Omega(0)$ are provided and then refined to obtain the correct asymptotics of the BS solution. A bisection algorithm is applied to find the correct frequency $\omega$, yielding the required ground state.