---
title: Solitonic Hayward-Boson Stars
url: https://www.emergentmind.com/topics/solitonic-hayward-boson-stars-shbss
type: topic
---

# Solitonic Hayward-Boson Stars

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 [2508.11906].

## 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 $\mathcal{L}_F = d\mathcal{L}/dF$.

The energy-momentum tensors encode the contributions from both the scalar ($T_{\mu\nu}^{(\Psi)}$) and electromagnetic ($T_{\mu\nu}^{(EM)}$) 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:
  $$
  ds^2 = -N(r)\sigma(r)^2 dt^2 + \frac{dr^2}{N(r)} + r^2(d\theta^2+\sin^2\theta\,d\phi^2), \quad N(r) = 1-\frac{2m(r)}{r}.
  $$
- Scalar field: $\Psi(r,t) = \psi(r) e^{-i\omega t}$.
- Magnetic potential: $A=a(\theta)\,d\phi$ with monopole profile $a(\theta)=Q\cos\theta$; $F_{θφ}=Q\sin\theta$ yields $F=2Q^2/r^4$.

Regularity at the origin requires a Taylor series:
$$
m(r) = m_3 r^3 + \ldots, \quad \psi(r) = \psi_0 + \mathcal{O}(r^2), \quad \sigma(r) = \sigma_0 + \mathcal{O}(r^2),
$$
with
$$
m_3 = \frac{4\pi}{3} \psi_0^2\left( \frac{\omega^2}{\sigma_0^2} + \mu^2 \right) + \frac{1}{\beta}.
$$
Asymptotic flatness enforces $\psi(r\to\infty)\to0$, $\sigma(r\to\infty)\to1$, $m(r\to\infty)\to M$ (ADM mass), and the scalar field decays exponentially if $\mu>\omega>0$.

A necessary and sufficient condition for a horizonless Hayward background is $\sqrt{\beta} Q > 1.49661$, derived from the relation $\ell/m_0 > 1.05827$ with $\ell^2 = Q\sqrt{2\beta}$, $m_0 = Q^{3/2}(8/\beta)^{1/4}$. 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 ($\mu=1$) units:
- The coupled ODEs for $m(r)$, $\sigma(r)$, and $\psi(r)$ are integrated from the origin to large $r$, imposing regularity and decay boundary conditions.
- For fixed $(\beta, Q)$ and a trial central scalar amplitude $\psi_0$, the eigenvalue $\omega$ is tuned such that $\psi(r)$ decays at infinity and $\sigma(\infty)=1$.
- Only ground-state (nodeless) solutions are constructed.

The primary solution families are labeled by the magnetic charge $Q$ and central amplitude $\psi_0$. The admissible parameter space is restricted by the horizonless condition, and only those with $\sqrt{\beta}Q > 1.49661$ correspond to physically regular SHBSs.

## 4. Physical Quantities and Characteristic Relations

Key observable and diagnostic quantities:
- **ADM Mass**: $M = \lim_{r\to\infty} m(r)$.
- **Noether Charge (Particle Number)**:
  $$
  N = 8\pi \omega \int_0^\infty \frac{r^2 \psi^2}{\sigma N} dr,
  $$
  computed from the conserved current $j^\mu = -i[\Psi^*\partial^\mu\Psi - \Psi\partial^\mu \Psi^*]$.
- **Family Label**: Central scalar amplitude $\psi_0 \equiv \psi(0)$.

SHBSs organize into one-parameter families in $(\psi_0,\omega)$ or $(M,\omega)$ space. As $\psi_0$ increases, $M$ grows from the Hayward vacuum mass $m_0$ to a maximal value $M_{max}$, then decreases along an unstable branch. The particle number $N$ exhibits similar nonmonotonic dependence.

In the $(M, \psi_0)$ diagram, for each $Q$:
- $M(\psi_0)$ rises monotonically from $m_0$ (vanishing scalar field), peaks at $\psi_0^{crit}$, then turns back, marking the onset of instability.
- As $Q$ increases, $M_{max}(Q)$ decreases, and the characteristic mass–frequency spiral of the $Q=0$ (mini-boson star) case unwinds and eventually disappears.

The effective radius $R_{99}$ (enclosing 99% of $M$) also shows a turning point in $M(R_{99})$; unlike standard mini-boson stars, the curve bends back toward the Hayward core radius, indicating the dominance of the nonscalar NLED core at large $Q$.

## 5. Solitonic Nature and Morphological Properties

SHBSs are everywhere regular, with energy density,
$$
\rho(r) = \rho_\Psi + \rho_{EM},
$$
where
$$
\rho_\Psi = N\psi'^2 + \mu^2\psi^2 + \frac{\omega^2\psi^2}{N\sigma^2}, \quad \rho_{EM} = \frac{3}{4\pi\beta} \frac{(\beta F)^{3/2}}{\left[1 + (\beta F)^{3/4}\right]^2},
$$
which is exponentially localized at large $r$. The metric function $N(r)>0$ for all $r$, confirming horizonless nature, and $N(r)\to1$ asymptotically.

A defining feature is the possible shell-like structure in $\rho(r)$: for certain parameter values, the total energy density peaks away from the origin due to competition between the electromagnetic core ($\propto r^{-6}$) and the scalar profile. The scalar charge density,
$$
j^t(r) \propto \frac{\omega \psi^2}{N\sigma^2},
$$
is similarly localized, reinforcing the solitonic interpretation.

## 6. Stability Analysis and Special Limits

By analogy with mini-boson stars, the branch of solutions with $dM/d\omega<0$ and $dM/d\psi_0>0$ (from $\psi_0=0$ up to $M_{max}$) is expected to be linearly stable. Beyond $M_{max}$, both $dM/d\omega$ and the sign of the binding energy $E_B=\mu N-M$ 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 $Q\to0$, the NLED decouples, and the standard mini-boson star sequence (featuring the familiar spiral in $M$–$\omega$) is recovered.
- As $\sqrt{\beta} Q\to1.49661$, 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" $\omega\to0$ at large $Q$, the solution approaches "frozen stars" introduced by Yue & Wang [2312.07224].

## 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 $\ell/m_0 = (1/Q)\sqrt{\beta/2} > 1.05827$ (equivalent to $\sqrt{\beta} Q > 1.49661$).

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 [2508.11906].

Source: https://www.emergentmind.com/topics/solitonic-hayward-boson-stars-shbss