Solitonic Hayward-Boson Stars

• Solitonic Hayward–Boson Stars are horizonless, self-gravitating solitonic configurations emerging from a massive complex scalar field and a nonlinear magnetic monopole core.
• Their construction employs a spherically symmetric ansatz and adaptive shooting methods to solve the coupled Einstein, Klein–Gordon, and modified Maxwell equations with strict regularity and asymptotic flatness.
• The solutions display mass-frequency spirals and shell-like energy density profiles, offering potential astrophysical signatures as horizonless black hole mimickers.

Solitonic Hayward–Boson Stars (SHBSs) are horizonless, self-gravitating solitonic configurations arising from the interplay of a massive complex scalar field and a nonlinear magnetic monopole core within the Fan–Wang nonlinear electrodynamics (NLED) model. These objects interpolate between ordinary boson stars and regular Hayward monopole cores, existing only when a critical threshold involving the NLED coupling and magnetic charge is satisfied. SHBSs exhibit exponential localization of energy density and characteristic shell-like profiles, distinguishing them as astrophysically relevant candidates for horizonless black hole mimickers and fundamental studies of gravitating solitons (Chicaiza-Medina et al., 16 Aug 2025).

1. Theoretical Framework and Field Equations

SHBSs are constructed from four-dimensional, asymptotically flat General Relativity, minimally coupled to both a complex scalar field and a NLED sector engineered to asymptotically produce the Hayward regular metric. The model includes:

• A complex scalar field $\Psi$ with mass $\mu$ and quadratic potential $U(|\Psi|)=\mu^2 |\Psi|^2$.
• A Fan–Wang type NLED Lagrangian,

$$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$

where $F = F_{\mu\nu}F^{\mu\nu}$ and $\beta>0$ is the nonlinear electrodynamics coupling.

The action governing SHBSs is:

$$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$

The resulting dynamical equations are:

• Einstein equations: $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$,
• Klein-Gordon equation: $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$,
• Modified Maxwell equations: $\nabla_\mu[ \mathcal{L}_F F^{\mu\nu} ] = 0$, with $\mu$0.

The energy-momentum tensors encode the contributions from both the scalar ($\mu$1) and electromagnetic ($\mu$2) sectors. The specific NLED is engineered so that the theory admits static, regular magnetically charged Hayward cores in the absence of scalar hair.

2. Ansatz, Regularity, and Boundary Conditions

A spherically symmetric, time-independent ansatz is imposed:

• Metric:

$\mu$3

• Scalar field: $\mu$4.
• Magnetic potential: $\mu$5 with monopole profile $\mu$6; $\mu$7 yields $\mu$8.

Regularity at the origin requires a Taylor series:

$\mu$9

with

$U(|\Psi|)=\mu^2 |\Psi|^2$0

Asymptotic flatness enforces $U(|\Psi|)=\mu^2 |\Psi|^2$1, $U(|\Psi|)=\mu^2 |\Psi|^2$2, $U(|\Psi|)=\mu^2 |\Psi|^2$3 (ADM mass), and the scalar field decays exponentially if $U(|\Psi|)=\mu^2 |\Psi|^2$4.

A necessary and sufficient condition for a horizonless Hayward background is $U(|\Psi|)=\mu^2 |\Psi|^2$5, derived from the relation $U(|\Psi|)=\mu^2 |\Psi|^2$6 with $U(|\Psi|)=\mu^2 |\Psi|^2$7, $U(|\Psi|)=\mu^2 |\Psi|^2$8. If this inequality is not met, the spacetime develops an event horizon and supports no regular scalar field.

3. Numerical Construction and Solution Structure

SHBS solutions are computed via adaptive shooting methods in dimensionless ($U(|\Psi|)=\mu^2 |\Psi|^2$9) units:

• The coupled ODEs for $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$0, $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$1, and $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$2 are integrated from the origin to large $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$3, imposing regularity and decay boundary conditions.
• For fixed $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$4 and a trial central scalar amplitude $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$5, the eigenvalue $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$6 is tuned such that $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$7 decays at infinity and $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$8.
• Only ground-state (nodeless) solutions are constructed.

The primary solution families are labeled by the magnetic charge $$\mathcal{L}(F) = \frac{3}{\pi\beta} \frac{(\beta F)^{3/2}}{\left[1+(\beta F)^{3/4}\right]^2},$$9 and central amplitude $F = F_{\mu\nu}F^{\mu\nu}$0. The admissible parameter space is restricted by the horizonless condition, and only those with $F = F_{\mu\nu}F^{\mu\nu}$1 correspond to physically regular SHBSs.

4. Physical Quantities and Characteristic Relations

Key observable and diagnostic quantities:

• ADM Mass: $F = F_{\mu\nu}F^{\mu\nu}$2.
• Noether Charge (Particle Number):

$F = F_{\mu\nu}F^{\mu\nu}$3

computed from the conserved current $F = F_{\mu\nu}F^{\mu\nu}$4.

• Family Label: Central scalar amplitude $F = F_{\mu\nu}F^{\mu\nu}$5.

SHBSs organize into one-parameter families in $F = F_{\mu\nu}F^{\mu\nu}$6 or $F = F_{\mu\nu}F^{\mu\nu}$7 space. As $F = F_{\mu\nu}F^{\mu\nu}$8 increases, $F = F_{\mu\nu}F^{\mu\nu}$9 grows from the Hayward vacuum mass $\beta>0$0 to a maximal value $\beta>0$1, then decreases along an unstable branch. The particle number $\beta>0$2 exhibits similar nonmonotonic dependence.

In the $\beta>0$3 diagram, for each $\beta>0$4:

• $\beta>0$5 rises monotonically from $\beta>0$6 (vanishing scalar field), peaks at $\beta>0$7, then turns back, marking the onset of instability.
• As $\beta>0$8 increases, $\beta>0$9 decreases, and the characteristic mass–frequency spiral of the $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$0 (mini-boson star) case unwinds and eventually disappears.

The effective radius $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$1 (enclosing 99% of $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$2) also shows a turning point in $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$3; unlike standard mini-boson stars, the curve bends back toward the Hayward core radius, indicating the dominance of the nonscalar NLED core at large $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$4.

5. Solitonic Nature and Morphological Properties

SHBSs are everywhere regular, with energy density,

$$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$5

where

$$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$6

which is exponentially localized at large $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$7. The metric function $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$8 for all $$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi} - \mathcal{L}(F) - \nabla_\alpha\Psi^* \nabla^\alpha\Psi - \mu^2 |\Psi|^2 \right].$$9, confirming horizonless nature, and $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$0 asymptotically.

A defining feature is the possible shell-like structure in $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$1: for certain parameter values, the total energy density peaks away from the origin due to competition between the electromagnetic core ($$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$2) and the scalar profile. The scalar charge density,

$$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$3

is similarly localized, reinforcing the solitonic interpretation.

6. Stability Analysis and Special Limits

By analogy with mini-boson stars, the branch of solutions with $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$4 and $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$5 (from $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$6 up to $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$7) is expected to be linearly stable. Beyond $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$8, both $$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{(\Psi)} + T_{\mu\nu}^{(EM)})$$9 and the sign of the binding energy $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$0 indicate instability. A full linear perturbation analysis remains a principal open problem, but turning-point and binding-energy criteria support this conclusion.

Several notable limits are:

• As $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$1, the NLED decouples, and the standard mini-boson star sequence (featuring the familiar spiral in $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$2–$\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$3) is recovered.
• As $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$4, the spacetime develops an extremal horizon, and no regular solitonic solution exists—a no-scalar-hair theorem for Hayward black holes.
• In the "frozen limit" $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$5 at large $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$6, the solution approaches "frozen stars" introduced by Yue & Wang (Yue et al., 2023).

7. Interpretation and Astrophysical Outlook

SHBSs provide a continuum between pure regular Hayward magnetic monopole cores and mini-boson stars, determined by the interplay of the scalar field and NLED charge. Their existence strictly requires the parameter combination $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$7 (equivalent to $\nabla_\mu \nabla^\mu \Psi = \mu^2 \Psi$8).

With maximal masses below the mini-boson star limit and morphologically distinct, potentially shell-like energy profiles, SHBSs represent robust, horizonless, gravitating solitons. Their regularity and localization suggest potential astrophysical signatures—most notably, in lensing and gravitational-wave scenarios—and motivate further numerical and analytical studies of their perturbative stability and phenomenology. Their viability as black-hole mimickers or exotic dark compact objects remains an active direction, contingent on further investigation of their dynamical response and observational distinguishability from standard compact objects (Chicaiza-Medina et al., 16 Aug 2025).