Ultracompact Spinning Boson Stars

• Ultracompact spinning boson stars are horizonless, compact objects formed from complex scalar fields with quantized angular momentum and toroidal energy distribution.
• They are constructed via the Einstein–Klein–Gordon framework using specialized numerical techniques to resolve steep scalar field gradients and capture thin-shell behavior.
• Their long-term stability, presence of light rings, and near-Kerr multipolar structure make them promising black hole mimickers with distinct astrophysical signatures.

Ultracompact spinning boson stars are horizonless, self-gravitating, stationary or dynamically formed objects composed of complex scalar (or, more generally, bosonic) fields with nonzero angular momentum and a compactness approaching that of black holes. These models are uniquely characterized by their scalar field structure, quantized angular momentum, toroidal energy distribution, and the possibility of supporting features such as light rings typically associated with event horizons, while remaining nonsingular and regular everywhere. In recent years, an extensive body of work has established their construction, internal structure, geodesic properties, observable signatures, and, critically, their dynamical stability on long timescales in the nonlinear regime.

1. Construction and Mathematical Framework

The prototypical ultracompact spinning boson star is realized in the Einstein–Klein–Gordon system with a complex scalar field $\Phi$ and a self-interaction potential $V(|\Phi|^2)$. The line element is usually assumed to be stationary and axisymmetric, and the field ansatz incorporates both harmonic time dependence and azimuthal winding: $$\Phi(r, \theta, t, \phi) = \phi(r, \theta)\, e^{i(\omega t - m\phi)}$$ where $\omega$ is the field frequency and $m \in \mathbb{Z}$ is the winding number (rotational quantum number), directly tied to quantized angular momentum $J = m \mathcal{N}$.

Several classes of self-interaction potentials are employed:

• Free field: $V(|\Phi|^2) = \mu^2 |\Phi|^2$
• Quartic: $V(|\Phi|^2) = \mu^2 |\Phi|^2 + \lambda |\Phi|^4$
• Sextic/solitonic: $$V(|\Phi|^2) = \mu^2 |\Phi|^2 \left[1 - 2|\Phi|^2/\sigma_0^2\right]^2$$
• Axion-inspired (periodic): $V$ periodic, controlled by axion mass $m_a$ and decay constant $f_a$

The construction proceeds by solving the coupled Einstein–Klein–Gordon equations in a multidomain pseudo-spectral scheme, specialized to resolve steep gradients in the scalar field and to capture thin-shell behavior in the ultracompact regime (Grandclement et al., 2014, Sukhov, 5 Apr 2024). The boundary conditions enforce regularity at the rotation axis and origin, with asymptotic flatness at spatial infinity. Physical parameters of interest—the ADM mass ($M$), angular momentum ($J$), and scalar Noether charge ($\mathcal{N}$)—are computed from asymptotic expansions and Komar integrals.

2. Interior Structure and Compactness

The internal structure of ultracompact spinning boson stars is governed by the scalar field profile and potential. In the ultracompact regime, the scalar field often forms a shell-like configuration: a nearly constant interior scalar field sharply bounded by a thin shell where the field transitions rapidly to zero. For rotating solutions, the energy density assumes a toroidal topology, with the field vanishing along the rotation axis as $\phi \sim (r\sin\theta)^m$.

Compactness, defined as $C = M/R$ (with $R$ an effective radius, e.g., enclosing 99% of the mass), approaches and may slightly exceed $C = 1/3$, the threshold for the appearance of a light ring outside the object's surface. For sextic or stiff polynomial potentials ($V \propto \phi^n$ with $n \gg 4$), compactness can approach the causal limit $C_\text{max} \approx 0.354$ (Pitz et al., 2023). Table 1 summarizes key compactness benchmarks:

Model Class	Max Compactness $C_\text{max}$	Light Ring Exists?
Mini-boson star ($\phi^4$)	$\sim 0.16$	No
Solitonic/sextic	$\sim 0.33$	Yes (stable/unstable pairs)
$V\propto \phi^n$, $n\to\infty$	$0.354$	Yes

As potential stiffness increases, the mass–radius relation transitions from neutron-star-like (soft EOS) to self-bound star behavior, ultimately mimicking the characteristics of an incompressible fluid.

3. Geodesics, Light Rings, and Astrophysical Signatures

Ultracompact spinning boson stars, due to their compactness, admit light rings (closed null geodesics) analogous to those around black holes. Numerical integration of the geodesic equations in these spacetimes reveals several distinctive features:

• Stable and unstable light rings: Typically, both a stable inner and an unstable outer light ring are present for high compactness configurations (Sukhov, 5 Apr 2024, Marks et al., 24 Apr 2025).
• Stable circular orbits: In contrast to Kerr, all equatorial circular orbits around boson stars remain stable; there is no innermost stable circular orbit (ISCO) below which instability sets in (Grandclement et al., 2014).
• "Pointy petal" orbits: Zero-angular-momentum timelike geodesics in rotating spacetimes exhibit nonclosed, spiked trajectories near the center, reflecting strong frame-dragging and transparent interior structure—qualitatively distinct from Kerr black hole geodesics (Grandclement et al., 2014).

Signal characteristics:

• The resulting shadow/lensing "annulus" is thinner than the BH shadow, due to the more compact light-ring, and features a cascade of Einstein rings accumulating at the photon ring (Cunha et al., 2017).
• These features potentially enable electromagnetic discrimination between ultracompact boson stars and black holes, as well as constraints from observed star orbits at the Galactic center.

4. Multipole Structure and Universal Relations

The external spacetime of ultracompact spinning boson stars is characterized by a set of relativistic multipole moments ($M_\ell$, $S_\ell$), extracted as Geroch–Hansen invariants. For spinning solutions with moderate shell thickness (parameter range $\sigma_0 \in [0.3, 0.5]$ in solitonic models), the reduced multipole moments ($m_2$, $s_3$ etc.) approach those of Kerr black holes (unity), making their gravitational field nearly indistinguishable from Kerr outside the star (Sukhov, 5 Apr 2024).

Nevertheless, universal relations among the dimensionless moment of inertia $\bar{I} = I/M_{99}^3$, dimensionless spin $\chi = J/M_{99}^2$, and reduced quadrupole $\bar{Q} = Q/(M_{99}^3\chi^2)$ differ substantially from both neutron stars and Kerr: $$\log_{10} \bar{I} = A_0 + \sum_{m,n} A_n^m\,\chi^m \left(\log_{10} \bar{Q} - B\right)^n$$ These relations are robust under changes in the scalar potential, even when extending the kinetic sector to nonlinear sigma models (Adam et al., 28 Feb 2025). This effective no-hair property allows for unique identification of boson stars through gravitational wave analysis, as their multipolar structure deviates from the Kerr "no-hair" sequence (Vaglio et al., 2022, Adam et al., 2023, Adam et al., 2022).

5. Dynamical and Nonlinear Stability

A central question is whether ultracompact spinning boson stars with light rings—often assumed to be generically unstable—can be dynamically stable in the nonlinear regime. Recent works resolve this:

• Eigenmode analysis shows that radial stability (real fundamental oscillation frequency $\chi_0^2 > 0$) correlates with the "rising" (lower-amplitude) side of the mass–amplitude curve, with a critical transition at the maximal-mass point or other extremal configurations (Marks et al., 24 Apr 2025). Both perturbative calculations and numerical-relativity results are in precise agreement.
• Long-term nonlinear evolutions in full 3+1D and 2+1D general-relativistic codes (using CCZ4, generalized harmonic, and BSSN formulations) demonstrate that spinning boson stars with a stable light ring remain regular and close to stationary for evolution times $t\mu \sim 10^4$ without developing an instability—even under non-axisymmetric perturbations or perturbations localized at the light ring (Evstafyeva et al., 15 Aug 2025).
• Perturbation details: Non-axisymmetric perturbations of the form

$\phi \to \phi \cdot e^{-2\epsilon \cos(n\phi)}$

primarily excite the corresponding $m$-modes in the azimuthal decomposition, while Gaussian perturbations centered at the light ring induce persistent radial oscillations without triggering instability unless the amplitude exceeds a sharp collapse threshold. In all subcritical cases, perturbations exhibit only slow decay.

These findings demonstrate that ultracompact spinning boson stars—provided they are on the stable branch of the solution space—can function as long-lived black hole mimickers, maintaining their light-ring structure and multipole moments over astrophysically relevant timescales (Marks et al., 24 Apr 2025, Evstafyeva et al., 15 Aug 2025).

6. Formation and Astrophysical Relevance

The formation of ultracompact spinning boson stars is tightly constrained by dynamical processes:

• Binary mergers: Simulations show that binary boson star mergers overwhelmingly produce nonspinning remnants due to rapid shedding of angular momentum unless specially tuned initial conditions induce a nonzero net vortex; only then can a spinning remnant with quantized angular momentum form (Siemonsen et al., 2023). The quantization $J = m \mathcal{N}$ restricts possible angular momenta after merger, thus making spinning remnant formation a rare outcome for generic initial data (Palenzuela et al., 2017, Bezares et al., 2022).
• Gravitational collapse: Direct collapse of a rotating bosonic cloud via gravitational cooling is found to be unstable for scalar fields (toroidal energy density, subject to non-axisymmetric instabilities), while vector (Proca) stars with spheroidal profiles may remain robust (Sanchis-Gual et al., 2019).
• Stability with light ring: The existence of a stable branch for ultracompact configurations with light rings implies that horizonless black hole mimickers, indistinguishable via gravitational-wave inspiral and electromagnetic shadow signatures from Kerr in certain parameter regimes, exist robustly (Sukhov, 5 Apr 2024, Marks et al., 24 Apr 2025).

Astrophysically, these objects provide testable alternatives to black holes, as only a few external multipole moments need to be measured to distinguish them, given their universal multipolar structure. Their existence also underpins interpretations of unusual compact object observations that deviate from canonical neutron star or black hole models.

7. Summary and Outlook

Ultracompact spinning boson stars with and without light rings represent a theoretically robust class of horizonless, quantized, regular objects whose external gravitational field can closely match that of Kerr black holes for a wide range of equations of state and self-interaction potentials. They possess genuine internal structure (e.g., thin toroidal shells, violations of the strong energy condition in some models), admit unique geodesic phenomenology (“pointy petal” orbits), and support a universal set of (reduced) multipole moment relations distinct from either neutron stars or the Kerr sequence. Most importantly, the combination of perturbative and long-term nonlinear evolution studies show that a substantial fraction of these stars are dynamically stable on timescales relevant for observational astrophysics, even in the presence of a stable light ring and under generic perturbations (Marks et al., 24 Apr 2025, Evstafyeva et al., 15 Aug 2025). As such, they constitute a predictive, falsifiable family of black-hole mimickers with direct connections to fundamental scalar field physics, self-interaction models, and beyond-Standard-Model particle phenomenology.