Chains of rotating boson stars with quartic or sextic self-interaction

TL;DR

This work analyzes chains of rotating boson stars in Einstein gravity with either quartic or sextic self-interactions in the scalar sector. Using axisymmetric Einstein–Klein–Gordon equations and high-precision finite-element numerics, it constructs multi-component configurations and tracks their ADM mass $M$, angular momentum $J$, and ergospheres as functions of the frequency $\omega$ and coupling $\lambda$. A key finding is a parity-dependent morphology: even-numbered chains produce spiral $M$-$\omega$ and $J$-$\omega$ curves, while odd-numbered chains produce loops, with ergospheres merging as $\omega$ grows; quartic interactions impose stronger existence bounds, especially for odd chains, whereas sextic interactions allow stronger couplings but do not exceed quartic maxima. Across both interactions, ergospheres arise for 1, 2, and 4 constituents and merge at higher $\omega$, highlighting a robust macroscopic signature potentially relevant for gravitational-wave or lensing observations.

Abstract

This paper investigates chains of rotating boson stars (BSs) within Einstein gravity coupled to a complex scalar field. The model incorporates quartic or sextic self-interactions in the scalar Lagrangian, which support the existence of stationary, solitonic, gravitationally bound solutions. We numerically construct these multi-component systems and investigate how the self-interactions alter their global properties -- specifically the Arnowitt-Deser-Misner (ADM) mass $M$, the angular momentum $J$ and the morphology of their ergospheres. A central result is the distinct dependence of the $(ω, M)$ and $(ω, J)$ relations on the parity of the chains. Specifically, systems with an even number of constituents display spiraling curves, while those with an odd number exhibit loop structures. Moreover, we observe that two initially distinct ergospheres merge into a single one as the frequency $ω$ increases. Our analysis also indicates that the quartic interaction imposes more restrictive existence bounds, particularly for odd-numbered chains, thereby restricting stable configurations to the weak-coupling regime. In contrast, the sextic interaction has a weaker effect and enables stable solutions at substantially stronger couplings.

Figures (21)

• Figure 1: The ADM mass $M$ as a function of the frequency $\omega$. Left: The rotating BSs with quartic self-interaction of one constituent. Middle: The rotating BSs with quartic self-interaction of two constituents. Right: The rotating BSs with quartic self-interaction of four constituents.
• Figure 2: The total angular momentum $J$ as a function of the frequency $\omega$. Left: The rotating BSs with quartic self-interaction of one constituent. Middle: The rotating BSs with quartic self-interaction of two constituents. Right: The rotating BSs with quartic self-interaction of four constituents.
• Figure 3: The curves of $(\omega, M)$ and $(\omega, J)$ for three-constituent case. Left: The ADM mass $M$ with the different frequency $\omega$ of the rotating BSs with quartic self-interaction of three constituents. Right: The total angular momentum $J$ with different frequency $\omega$ of the rotating BSs with quartic self-interaction of three constituents.
• Figure 4: Contours of $g_{tt}$ for chains of rotating BSs with one constituent. Left: When $\lambda = 0$, the ergosphere emerges at $\omega = 0.658$ on the first branch. Middle: When $\lambda = 400$, the ergosphere emerges at $\omega = 0.874$ on the second branch. Right: When $\lambda = 800$, the ergosphere emerges at $\omega = 0.851$ on the third branch. Warm and cold color schemes denote negative and positive $g_{tt}$ values, respectively. The red dashed line represents ergospheres ($g_{tt}$ = 0), and the black solid lines represent the contours of $g_{tt}$.
• Figure 5: Contours of $g_{tt}$ for chains of rotating BSs with two constituents. Left: The ergospheres emerge at $\omega=0.719$ on the second branch. Middle: With the increase of $\omega$, the ergospheres begin to merge at $\omega=0.730$ on the second branch. Right: The two ergospheres merge into one at $\omega=0.790$ on the second branch. Warm and cold color schemes denote negative and positive $g_{tt}$ values, respectively. The red dashed line represents ergospheres ($g_{tt}$ = 0), and the black solid lines represent the contours of $g_{tt}$.