$ll$-Boson stars in anti-de Sitter spacetime

In previous work, we introduced the $\ell$-boson stars, a generalization of standard boson stars, which are parameterized by an angular momentum number $\ell$, while still preserving the spacetime’s spherical symmetry. In this article, we present and study the properties of $\ell$-boson stars in spacetimes with a negative cosmological constant, such that they are asymptotically anti-de Sitter.

💡 Research Summary

In this paper the authors extend the recently introduced ℓ‑boson star model to spacetimes with a negative cosmological constant, i.e. asymptotically anti‑de Sitter (AdS) backgrounds. An ℓ‑boson star consists of 2ℓ + 1 complex scalar fields of equal mass μ, each carrying the same harmonic time dependence e^{iωℓt} and the same radial profile ψℓ(r), but multiplied by the spherical harmonics Yℓm(θ,φ) with m = −ℓ,…,ℓ. Because the angular dependence is summed over all m, the total stress‑energy tensor is spherically symmetric even though the individual fields are not. This allows the use of a static, spherically symmetric metric
ds² = −α²(r)dt² + a²(r)dr² + r²dΩ².

The Einstein–Klein‑Gordon equations are derived, and a convenient field redefinition ψℓ(r)=r^{ℓ}uℓ(r) is introduced to enforce regularity at the origin (uℓ(0)=u₀, u′ℓ(0)=0). Boundary conditions at r → ∞ require the scalar field to decay rapidly while the metric approaches the Schwarzschild‑AdS form with mass parameter M and AdS radius L = √(−3/Λ). The authors also present the low‑mass (linear) limit, obtaining an analytic spectrum for the frequency:
L ω_{ℓ,n}=2n+ℓ+3/2+½√{4(Lμ)²+9}, n=0,1,2,…
which reproduces known results for ℓ=0 and μ=0.

Numerically, the coupled ordinary differential equations are solved using a shooting method with high‑accuracy adaptive step integrators. Both a non‑compact radial coordinate and a compactified coordinate are employed to explore complementary regions of parameter space. The study focuses on ground‑state solutions (no radial nodes, n=0) for ℓ ranging from 0 to 15, with μ=0 and Lμ=1, while varying the central amplitude u₀ to span the “first region” (mass increasing up to a maximum) and part of the “second region” (mass decreasing beyond the maximum). The gravitational constant is set to G=1 for the numerical runs.

Key findings:

1. Mass–frequency curves – For each ℓ a characteristic M(ω) curve exhibits a single maximum. The branch up to the maximum (first region) is expected to be linearly stable, whereas the branch beyond the maximum (second region) shows the usual instability of boson stars.

2. Density profiles – As ℓ increases, the stars develop a hollow core and a thin, high‑density shell. For ℓ≥2 the central density is exactly zero, and the peak density occurs at a finite radius. By contrast, ℓ=0 (the standard boson star) has its maximum density at the centre, and ℓ=1 shows an intermediate behaviour with a non‑zero central density but an off‑centre peak.

3. Compactness – The compactness C = M/r_max decreases with ℓ, reflecting the more extended shell‑like structure. Nevertheless, for sufficiently large ℓ the compactness remains comparable to that of ordinary boson stars, indicating that ℓ‑boson stars can still be relatively compact despite their shell geometry.

4. Light rings – A striking result is the appearance of light‑ring pairs (stable inner and unstable outer circular null geodesics) already in the first region for ℓ ≳ 6. Earlier work on boson stars in AdS reported light rings only in the unstable second region. The present analysis shows that the angular momentum carried by the scalar multiplet can generate sufficient spacetime curvature to support photon orbits even while the configuration is still on the stable branch.

5. Role of the negative cosmological constant – The AdS boundary acts as a confining “mirror”, allowing boson‑star solutions even when the scalar field is massless (μ=0). In asymptotically flat space such configurations do not exist, highlighting the essential influence of Λ < 0.

Overall, the paper demonstrates that ℓ‑boson stars in AdS retain many qualitative features of their flat‑space counterparts (mass‑frequency turning point, stability regions) while exhibiting novel phenomena: shell‑like density distributions for high ℓ and the possibility of light‑ring pairs on the stable branch. These results enrich the landscape of self‑gravitating scalar configurations in AdS, with potential implications for holographic models (AdS/CFT correspondence), the study of nonlinear scalar dynamics in confined geometries, and the construction of exotic compact objects that could mimic black‑hole signatures. Future work is suggested on full linear stability analysis, inclusion of rotation or charge, and exploration of the dual CFT operators associated with the ℓ‑multiplet.