Nonlinear Spontaneous Scalarization

• Nonlinear spontaneous scalarization is a strong-field mechanism where compact objects develop stable scalar hair once a critical parameter, such as curvature or spin, exceeds a threshold.
• It involves a transition from a scalar-free state to a scalarized state, driven by finite-amplitude perturbations that activate higher-order nonlinear couplings in the underlying theory.
• This phenomenon is studied in models including scalar-Gauss-Bonnet, Einstein-Maxwell-scalar, and rotating Kerr systems, revealing diverse branch structures and unique stability properties.

Searching arXiv for recent and foundational papers on nonlinear spontaneous scalarization. arxiv_search(query="nonlinear spontaneous scalarization black holes Gauss-Bonnet Einstein-Maxwell-scalar", max_results=10, sort_by="relevance") arxiv_search(query="nonlinear spontaneous scalarization black holes Gauss-Bonnet Einstein-Maxwell-scalar", max_results=10, sort_by="relevance") Nonlinear spontaneous scalarization is a strong-field, nonperturbative mechanism by which a compact object that admits a scalar-free solution develops a finite scalar configuration once a control parameter—such as compactness, charge, curvature, or spin—crosses a threshold. In the broad sense used in the scalarization literature, spontaneous scalarization is already nonlinear because a linear tachyonic instability is only the trigger, while nonlinear terms quench the growth and determine the final scalarized state. In the narrower and now common sense, “nonlinear scalarization” denotes models in which the bald solution is linearly stable and scalar hair forms only after a finite-amplitude perturbation activates higher-order couplings. The two usages are closely related but not identical, and both are established in the literature on compact objects and scalar-Gauss-Bonnet models (Doneva et al., 2022, Doneva et al., 2021).

1. Conceptual structure and terminology

The standard scalarization paradigm is organized around a linearized scalar equation of the form

$$\Box \,\delta\varphi-\mu_{\rm eff}^2\,\delta\varphi=0,$$

with a background-dependent effective mass. When $\mu_{\rm eff}^2<0$ in a sufficiently extended region, the trivial branch becomes tachyonically unstable; nonlinear interactions then quench the growth and produce a new equilibrium branch with finite scalar hair. The review literature describes this as a threshold, phase-transition-like phenomenon and emphasizes that the onset is linear but the end state is intrinsically nonlinear (Doneva et al., 2022).

A more restrictive definition emerged in later black-hole studies. There, one imposes conditions such as $f''(0)=0$ or $\zeta''(0)=0$, so that the scalar-free background is linearly stable and the linearized scalar equation reduces to a massless wave equation. Scalarization then requires a finite-amplitude seed and is therefore genuinely nonlinear rather than linearly tachyonic. This distinction is explicit in scalar-Gauss-Bonnet models where Schwarzschild or Kerr remains linearly stable but becomes nonlinearly unstable under sufficiently large perturbations (Doneva et al., 2021, Lai et al., 30 Apr 2026).

This terminological bifurcation is important because several papers use “nonlinear spontaneous scalarization” in the broad review sense, whereas others use it specifically for scalarization beyond linear tachyonic bifurcation. A precise reading therefore depends on whether “nonlinear” refers to saturation of an ordinary tachyonic instability or to the absence of any linear instability at all. The latter usage is the one most closely associated with “beyond spontaneous scalarization” in recent black-hole work (Staykov et al., 2022).

2. Nonlinear triggers in scalar-Gauss-Bonnet theories

In scalar-Gauss-Bonnet gravity, the scalar couples to the Gauss-Bonnet invariant

$$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$

and the coupling function determines whether the scalar-free solution is linearly unstable or only nonlinearly unstable. A representative sGB action is

$$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$

If $f'(0)=0$, the scalar-free GR solution is allowed; if $f''(0)\neq 0$, a tachyonic mass can appear at linear order; if $f''(0)=0$, standard spontaneous scalarization is removed and any scalarization must arise from higher-order terms (Doneva et al., 2021, Zhang et al., 2024).

The first explicit black-hole realization of this fully nonlinear mechanism imposed

$f(0)=0,\qquad f'(0)=0,\qquad f''(0)=0,$

and studied couplings

$\mu_{\rm eff}^2<0$0

In these models Schwarzschild is linearly stable, yet finite-amplitude perturbations can drive the system toward scalarized black holes. The resulting equilibrium structure is richer than in standard scalarization: depending on the coupling parameters, two or three scalarized branches can exist, the stable branch is not continuously connected to Schwarzschild, and transitions occur with a jump rather than through infinitesimal hair (Doneva et al., 2021).

A later EsGB study sharpened the model-building criteria by testing polynomial couplings

$\mu_{\rm eff}^2<0$1

The numerical time evolution showed decay for $\mu_{\rm eff}^2<0$2, but nonlinear instability for sufficiently large perturbations in the $\mu_{\rm eff}^2<0$3 and $\mu_{\rm eff}^2<0$4 cases. The paper summarized this as requiring a coupling with sufficiently strong nonlinear self-interaction and stated that the coupling must include a term beyond $\mu_{\rm eff}^2<0$5 for the mechanism studied there. For the concrete model

$\mu_{\rm eff}^2<0$6

$\mu_{\rm eff}^2<0$7 plays a major role in making different nonlinear scalarized black holes, while $\mu_{\rm eff}^2<0$8 plays a supplementary role (Zhang et al., 2024).

The same logic extends to multi-scalar Gauss-Bonnet gravity. With target spaces $\mu_{\rm eff}^2<0$9, $f''(0)=0$0, and $f''(0)=0$1, and couplings

$f''(0)=0$2

scalarized black holes exist for all maximally symmetric scalar target spaces even though no tachyonic instability is present. A notable feature is the asymptotic falloff $f''(0)=0$3, which implies zero scalar charge and hence suppressed scalar dipole radiation. Thermodynamic analysis suggests that portions of upper branches are favored and likely stable, while lower branches are disfavored; this supports a discontinuous transition rather than a continuous bifurcation (Staykov et al., 2022).

3. Rotation and the emergence of spin-induced nonlinear scalarization

Rotation introduces a distinct control parameter. In the spin-induced spontaneous scalarization of Kerr black holes, the scalar is coupled quadratically to the Gauss-Bonnet invariant through

$f''(0)=0$4

with

$f''(0)=0$5

For Schwarzschild, $f''(0)=0$6, so the sign $f''(0)=0$7 gives $f''(0)=0$8 and no low-spin instability. For Kerr, however,

$f''(0)=0$9

and the sign of $\zeta''(0)=0$0 can change near the horizon. In the large-coupling limit the threshold approaches

$\zeta''(0)=0$1

to within percent-level numerical accuracy, yielding the characteristic onset $\zeta''(0)=0$2 (Dima et al., 2020).

This instability is not superradiant in the usual sense. The dominant unstable sector is $\zeta''(0)=0$3, including the spherical $\zeta''(0)=0$4 mode, so the usual superradiance condition $\zeta''(0)=0$5 cannot apply. The reported timescale can be as short as $\zeta''(0)=0$6, consistent with a tachyonic process rather than superradiant bound-state growth. The linear instability is therefore the seed of scalarization, while the final hairy Kerr solution is expected to arise only after nonlinear self-interactions quench the growth (Dima et al., 2020).

A decoupling-limit study then separated mixed and purely nonlinear Kerr scalarization. For

$\zeta''(0)=0$7

Kerr can scalarize through both tachyonic and nonlinear effects. For

$\zeta''(0)=0$8

one has $\zeta''(0)=0$9, no tachyonic instability, and therefore exclusively nonlinear scalarization. In that purely nonlinear case there is no probe limit at finite mass: bald and hairy Kerr black holes do not connect through an infinitesimal-hair branch, and the only zero-charge point is $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$0 (Doneva et al., 2022).

The fully backreacted rotating EsGB realization made this distinction sharper. With

$$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$1

the scalar-free Kerr solution has no linear tachyonic instability. For $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$2, sufficiently rapid rotation creates a negative near-horizon region in $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$3 near the poles, and this region acts as a geometric trapping region for finite-amplitude scalar growth. The threshold is the geometric value

$$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$4

The fully backreacted scalarized solutions occupy a finite low-mass high-spin wedge in the $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$5 plane rather than the narrow band typical of linearly bifurcating spin-induced spontaneous scalarization. Toward the high-spin end the hair strengthens and the solutions approach a near-extremal regime, whereas toward the low-spin boundary the scalar is strongly suppressed and approaches a weak-hair limit as $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$6 (Lai et al., 30 Apr 2026).

4. Electromagnetic, nonlinear-electrodynamic, and regular-black-hole realizations

In Einstein-Maxwell-scalar theory, the scalar couples directly to the Maxwell invariant,

$$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$7

and the Reissner-Nordström solution becomes unstable when the effective mass

$$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$8

is negative, as in the exponential coupling $$\mathcal{R}_{\rm GB}^2=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^2,$$9 with $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$0. For the representative choice $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$1, instability becomes possible for roughly $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$2, with the first unstable mode appearing around $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$3. The fully nonlinear evolution proceeds through wave-packet propagation, exponential growth, and damped relaxation, and the late-time state is interpreted as a perturbation around a scalarized charged black hole (Xiong et al., 2022).

The EMS model also accommodates coexistence of spontaneous and nonlinear channels through

$$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$4

Here the quadratic term controls the linear tachyonic onset and the quartic term controls nonlinear triggering or quenching. The paper labels the two mechanisms as Class II.A, with $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$5, and Class II.B, with $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$6. Mixed scalarization occurs when both are present. The analysis found an overall dominance of spontaneous scalarization over nonlinear scalarization and an entropical preference for mixed over the standard scalarization limits. Opposite-sign couplings act as counter-scalarization, mimicking a scalar mass or positive self-interaction term (Belkhadria et al., 2023).

Nonlinear electrodynamics enriches this structure further. In the Einstein-Born-Infeld-scalar model,

$$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$7

with $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$8, the scalar-free sector splits into RN-like BI black holes for $$S=\frac{1}{16\pi}\int d^4x \sqrt{-g}\, \Big[R - 2\nabla_\mu \varphi \nabla^\mu \varphi + \lambda^2 f(\varphi)\mathcal{R}_{GB}^2\Big].$$9 and Schwarzschild-like BI black holes for $f'(0)=0$0. The RN-like regime yields a single connected scalarized branch, but the Schwarzschild-like regime admits existence lines in addition to bifurcation lines, disconnected scalarized solutions, and one-, two-, or three-branch domains depending on $f'(0)=0$1. The large branch is often the entropically preferred one, whereas small and tiny branches are typically disfavored, especially near criticality (Wang et al., 2020).

Regular black holes supported by nonlinear electrodynamics provide another arena. In Einstein-nonlinear electromagnetic-scalar gravity, Hayward black holes scalarize because the nonlinear electromagnetic Lagrangian produces

$f'(0)=0$2

For $f'(0)=0$3 and $f'(0)=0$4, the first threshold is

$f'(0)=0$5

The scalarized charged black holes are labeled by a node number $f'(0)=0$6, and the $f'(0)=0$7 branch is stable against radial perturbations for both quadratic and exponential couplings. This identifies the fundamental branch as a dynamically viable endpoint (Cai et al., 21 Oct 2025).

A general inverse-problem framework later extended this construction to arbitrary static, spherically symmetric regular black holes supported by nonlinear electrodynamics. Using the P-dual formalism, the NLED sector is reconstructed from a seed metric and then coupled to a real scalar through

$f'(0)=0$8

with scalarization triggered when

$f'(0)=0$9

For the Balart-Vagenas example, scalarized and scalar-free branches coexist in a region where the scalarized configurations are entropically preferred. The same study found percent-level deviations in the shadow size and fundamental scalar quasi-normal modes, with $f''(0)\neq 0$0 shifts for small charge-to-mass ratios (Contreras et al., 3 Nov 2025).

An analytically solvable flat-spacetime toy model makes the role of nonlinear completion unusually transparent. For a conducting charged sphere in Maxwell-scalar theory, the exactly linearizing choice

$f''(0)\neq 0$1

produces a runaway regime in which neither the Coulomb branch nor the scalarized branch is stable. By contrast, nonlinear continuations such as

$f''(0)\neq 0$2

heal the instability and can render the 0-node scalarized branch dynamically preferred. This parallels the role of nonlinear completion in black-hole scalarization (Herdeiro et al., 2020).

5. Branch structure, stability, and dynamical selection

A recurring structural distinction is whether the scalarized family bifurcates continuously from the bald branch or is separated by a gap. Standard spontaneous scalarization generally yields a zero-mode bifurcation and infinitesimal hair at onset. Fully nonlinear scalarization can instead produce disconnected or only finitely connected branches. In sGB black holes, the stable scalarized branch is not continuously connected to Schwarzschild, so the transition is a jump; in purely nonlinear Kerr scalarization there is likewise no finite-mass probe limit, and the bald and hairy families meet only as both mass and scalar charge go to zero (Doneva et al., 2021, Doneva et al., 2022).

Multiplicity of branches is also common. In nonlinear sGB and multi-scalar Gauss-Bonnet gravity, lower and upper branches, inspirals, and loop-like structures can occur, and only one branch is typically dynamically preferred. The stable one is identified either by direct time evolution, by having the largest entropy, or by radial stability. In the Hayward regular-black-hole case, the $f''(0)\neq 0$3 branch is stable against radial perturbations, while excited $f''(0)\neq 0$4 branches are expected to be unstable. In BI-supported black holes, the large branch is often both more entropic and radially stable when sufficiently far from criticality (Staykov et al., 2022, Cai et al., 21 Oct 2025, Wang et al., 2020).

Thermodynamic preference and dynamical stability are related but not identical. Some studies use entropy or reduced horizon area as a proxy for selection. In multi-scalar Gauss-Bonnet gravity, lower branches generally have smaller entropy than Schwarzschild, whereas portions of upper branches can have larger entropy and are therefore likely stable. In mixed EMS scalarization, spontaneously dominated solutions are entropically preferred over the corresponding RN backgrounds, while nonlinear-dominated solutions split into cold and hot branches, only the hot branch being potentially favored. These results suggest, but do not by themselves prove, the dynamically realized endpoint (Staykov et al., 2022, Belkhadria et al., 2023).

Fully nonlinear time evolutions clarify how selection occurs. In EMS, the early-time growth is dominated by the fundamental unstable mode from linear analysis, and the late-time oscillations are interpreted as perturbations around a scalarized black hole (Xiong et al., 2022). In sGB models without any linear instability, decoupling-limit evolutions show threshold behavior: sub-threshold perturbations decay, whereas super-threshold perturbations saturate into a stationary scalar configuration (Doneva et al., 2021). In rotating EsGB with $f''(0)\neq 0$5, the same dichotomy appears on Kerr: low spin or insufficient amplitude leads to decay, whereas high spin and sufficient amplitude lead to nonlinear trapping and saturation (Lai et al., 30 Apr 2026).

The AdS Einstein-Born-Infeld-scalar model exhibits an even more elaborate dynamical picture. There the scalarized evolution can undergo a single flip under changes in the initial amplitude or scalar-electromagnetic coupling, and a double flip under changes in the black-hole charge. Near the critical point, the relaxation time obeys

$f''(0)\neq 0$6

and the separatrix is an unstable AdS-Born-Infeld black hole. The Born-Infeld parameter is decisive: in the strong nonlinearity limit, the scalar hair vanishes (Wu et al., 2024).

A frequent misconception is that all threshold phenomena in rotating black holes are superradiant or that entropic preference alone settles the endpoint. The Kerr Gauss-Bonnet studies explicitly exclude a superradiant explanation for the dominant $f''(0)\neq 0$7 instability, while the branch analyses across sGB, EMS, and BI models show that entropy, existence, and linear stability can diverge in nontrivial ways (Dima et al., 2020, Wang et al., 2020).

6. Extensions, astrophysical implications, and unresolved issues

Nonlinear scalarization is not restricted to vacuum black holes. In scalar-tensor neutron-star models, the linear onset can occur even for $f''(0)\neq 0$8 provided sufficiently compact stars satisfy $f''(0)\neq 0$9, but the nonlinear endpoint is highly sensitive to the full coupling function. One bounded coupling permits spontaneous scalarization, whereas the standard quadratic truncation drives collapse instead. This dependence on higher-order structure is the stellar analogue of nonlinear completion in black-hole scalarization (Mendes et al., 2016).

The mechanism also generalizes beyond scalars in the sense of spontaneous tensorization. The vectorization program emphasized that spontaneous scalarization is a specific case of spontaneous growth induced by nonminimal coupling to matter rather than by any uniquely scalar property. The same logic yields spontaneous vectorization and, more generally, spontaneous tensorization, though later reviews stress that most nonscalar generalizations encounter ghost or gradient problems, with spinorization being a notable exception in the survey literature (Ramazanoğlu, 2017, Doneva et al., 2022).

Core-collapse simulations show that scalarized compact objects can form dynamically from nonscalarized progenitors. In scalar-Gauss-Bonnet gravity, fully nonlinear stellar collapse can produce scalarized neutron stars and scalarized black holes. The channels depend on the sign $f''(0)=0$0: for $f''(0)=0$1, both neutron stars and black holes can scalarize; for $f''(0)=0$2, black-hole final states descalarize even if a proto-neutron-star stage was temporarily scalarized. The reported scalar-radiation energy is $f''(0)=0$3, exceeding the typical tensor gravitational-wave energy from core collapse, $f''(0)=0$4, in the scenarios studied (Kuan et al., 2021).

Rotation, charge, and branch multiplicity create observationally selective deviations from general relativity. Spin-induced Kerr scalarization implies theories in which low-spin black holes look effectively like GR solutions while high-spin black holes carry scalar hair, with possible consequences for extra dipole gravitational radiation in binaries, inspiral waveforms, ringdown spectra, accretion signatures, and black-hole shadows (Dima et al., 2020). Regular-black-hole scalarization frameworks presently find only percent-level shadow deviations and $f''(0)=0$5 changes in scalar quasi-normal modes for small charge-to-mass ratios, suggesting that current electromagnetic and gravitational-wave observations do not yet exclude such solutions (Contreras et al., 3 Nov 2025).

A further extension reaches cosmology. In quadratic Einstein-scalar-Gauss-Bonnet cosmology with $f''(0)=0$6, rigorous analysis has established global existence of singularity-free FLRW solutions and proved nonlinear spontaneous scalarization triggered by a tachyonic instability induced by the Gauss-Bonnet term. The proof is based on decoupled differential inequalities for the Hubble parameter derived from a structural relation termed the power identity. This suggests that nonlinear scalarization is not solely a compact-object phenomenon but can also be formulated as a global dynamical effect in cosmological settings (He et al., 21 Jul 2025).

The main unresolved issues are therefore not the existence of nonlinear scalarization mechanisms but their domain of dynamical well-posedness, the relation between thermodynamic and dynamical selection, and the construction of realistic waveform and multimessenger templates. Review work emphasizes that the original massless Damour-Esposito-Farèse model is essentially ruled out by binary pulsars, whereas massive scalars, mixed curvature couplings, multiscalar models, and spin-induced black-hole scalarization remain viable directions. In this broader landscape, nonlinear spontaneous scalarization is best understood as a family of threshold-driven strong-field instabilities whose final state is controlled by nonlinear saturation rather than by perturbative corrections to general relativity (Doneva et al., 2022).