The Possibility of Formation of Compact Boson Stars via Cosmological Evolution of a Background Scalar Field

Abstract

Boson stars, hypothetical astrophysical objects bound by the self-gravity of a scalar field, have been widely studied as a type of exotic compact object that is horizonless and provides a testing ground for physics beyond the Standard Model. In particular, many previous works have demonstrated methods for distinguishing compact boson stars from black holes in general relativity through gravitational wave observations. However, the formation scenario of compact boson stars within the age of the universe remains unclear. In this paper, we explore a possible scenario for the formation of compact boson stars. The model we consider requires two coupled scalar fields: a complex scalar field that forms a boson star and a spatially homogeneous background field, as formation of a compact boson star cannot be achieved in a single filed model. Using the adiabatic approximation, we show that non-relativistic boson clouds can evolve into compact boson stars through the cosmological time-evolution of the background field. In our model the background field evolves to increase the effective mass of the scalar field, and as a result compact boson stars can form within the cosmological timescale, if the variation of the background field is as large as the Planck scale. However, further investigation is required because the required initial states are not the configurations that can be described by the well-studied Schrödinger-Poisson system.

Figures (15)

• Figure 1: The total mass $M$ of stationary solutions for mini-boson stars where the Lagrangian is given by Eq. \ref{['eq_mini']}. The horizontal axis represents $1-\omega/m_\phi$, where $m_\phi$ is the mass of the $\phi$-field and $\omega$ is the eigenvalue defined by Eq. \ref{['eq_scalar']}. In mini-boson star cases, the stationary configurations form a one-parameter family, parameterized by either the central field value $\phi_0=\phi(0)$ or the eigenvalue $\omega$. The blue point at $\omega/m_\phi=0.85$ represents the maximum mass of mini-boson stars as well as the critical point for gravitational instability. The solid curve represents the stable branch, while the dashed curve represents the unstable branch.
• Figure 2: Profiles of $\hat{\phi}(\hat{r})$, $\hat{\chi}(\hat{r})$, $N^2(\hat{r})$, and $G^2(\hat{r})$ of the stationary solutions for $\hat{\phi}_0=0.02$. Here, we define $\hat{\phi}$, $\hat{\chi}$, $\hat{r}$ and $\hat{\omega}$ as given in Eqs. \ref{['eq_hat_def']}. The blue, red, and orange curves represent the profiles for $\tilde{\chi}_\mathrm{out} = 0.5$, $1$, and $10$, respectively. When $\hat{\phi}_0$ is fixed to $0.02$, the eigenvalue $\omega$ becomes $1-\hat{\omega}=1.4\times10^{-2},1.8\times10^{-2}$ and $1.2\times10^{-1}$ for $\tilde{\chi}_\mathrm{out} = 0.5$, $1$, and $10$, respectively.
• Figure 3: The normalized total mass $\hat{M}$, total particle number $\hat{J}$, compactness $C$, radius $\hat{R}$, central $\chi$-field value $\tilde{\chi}(0)$, and magnitude of the gradient of $\chi$-field $\delta\chi/\chi_\mathrm{out}$ of stationary solutions as functions of the eigenvalue $\hat{\omega}$ for various values of $\tilde{\chi}_\mathrm{out}$, where $\delta\chi=\chi_\mathrm{out}-\chi(0)$. Here, we define $\hat{M}$, $\hat{J}$, $C$ and $\hat{R}$ as given in Eqs. \ref{['eq_hat_def2']}. These are plotted up to the final states where $\hat{M}$ and $\hat{J}$ reach their maximum values for each $\chi_\mathrm{out}$. The horizontal axis represents $1-\hat{\omega}$, with smaller values corresponding to non-relativistic configurations and values near 1 indicating relativistic ones. In each panel, the dotted red curves represent the quantities for mini-boson stars. These quantities are defined using Eq. \ref{['eq_mani_dimensionless1']} and Eq. \ref{['eq_mani_dimensionless2']}. The solid, dashed, and dotted curves, labeled Adiabatic Evolution 1, 2, and 3, represent the adiabatic evolution paths from the initial states defined by Eq. \ref{['eq_start_n']} to the final states corresponding to $\tilde{\chi}_\mathrm{out,f} \simeq 0.7$, 2, and 10, respectively. The black asterisks represent the initial state of mini-boson stars. Note that non-relativistic regime of mini-boson stars can be described by Shrödinger-Poisson equationsPhysRevD.42.384.
• Figure 4: Profiles of $\tilde{\phi}(\tilde{r})$, $\tilde{\chi}(\tilde{r})$, $N^2(\tilde{r})$, and $G^2(\tilde{r})$ of the stationary solutions for a normalized total particle number $\tilde{J}=1.6\times10^{-1}$. These profiles are in the adiabatic evolution path 1 shown in Fig. \ref{['fig_nmphi0']}. The blue, and red curves represent the profiles for $\tilde{\chi}_\mathrm{out} = 1$, and 2, respectively. Each horizontal axis is $\tilde{r}\equiv rgm_\mathrm{pl}/\sqrt{4\pi}$. Note that $m_\phi$, the effective mass of $\phi$-field, varies in the adiabatic evolution.
• Figure 5: The ratio of the binding energy associated with the $\chi$-field gradient to the gradient energy, given by $2 C/\tilde{\chi}_{\mathrm{out}} \delta\tilde{\chi}$, as a function of $1 - \hat{\omega}$. Similarly to Fig. \ref{['fig_nmphi0']}, each colored curve represents a sequence of stationary solutions for each $\chi_{\mathrm{out}}$. The solid, dashed, and dotted curves correspond to the three different adiabatic evolution paths.