Stealth de Sitter Solutions

• Stealth de Sitter solutions are gravitational configurations where the de Sitter metric is exactly maintained despite nontrivial, hidden contributions from matter fields or modified gravity sectors.
• They are realized in diverse frameworks such as nonlocal gravity, scalar-tensor, and vector-tensor theories, with specific functional forms ensuring the field’s influence is absorbed into the effective cosmological constant.
• Stability analyses and perturbation studies reveal that these solutions provide new insights into cosmic acceleration, black hole hair, and the cosmological constant problem while addressing quantum corrections and strong coupling issues.

A stealth de Sitter solution refers to a gravitational configuration in which the spacetime metric is precisely de Sitter (i.e., supports exponential expansion as in standard General Relativity), yet is sourced by nontrivial matter content or modified gravity sector that does not leave any direct, “active” signature in the field equations at the background level. Such solutions arise in a variety of contexts—including nonlocal gravity, scalar–tensor and vector–tensor theories, higher-order curvature corrections, and effective field theory frameworks—and often feature additional integration constants or free functions such that the underlying modifications or fields are hidden, or “stealthy,” with respect to classical observations of cosmic expansion. The stealth de Sitter mechanism plays a significant role in the theoretical landscape of cosmology, providing new perspectives on cosmic acceleration, hair for black holes, strong coupling issues, and the cosmological constant problem.

1. Core Definitions and Model Frameworks

Stealth de Sitter solutions are realized when a covariant classical or effective field theory admits configurations with the following attributes:

• The spacetime is (locally or globally) exactly de Sitter, with metric

$ds^2 = -dt^2 + e^{2 H_0 t} d\vec{x}^2$

where $H_0$ is the (constant) Hubble parameter characterizing exponential expansion.

• There exists a nontrivial matter field or modification—such as a scalar field $\phi$, a U(1) gauge field $A_\mu$, or a function of nonlocal operators—whose energy–momentum tensor, or more generally, its contribution to the modified gravitational field equations, vanishes identically or is absorbed into the effective cosmological constant, so that the field does not backreact on the metric at the background level.
• In generalized frameworks (e.g., nonlocal gravity, higher-order scalar–tensor, or vector–tensor actions), exact de Sitter or black hole geometries remain solutions even after the additional fields are switched on, provided certain functional or algebraic constraints are satisfied.

Notable classes include:

• Nonlocal Gravity with $f(\Box^{-1} R)$: Action is extended as $$S = \int d^4x \sqrt{-g}[ \frac{1}{2\kappa^2}( R (1+f(\Box^{-1} R)) - 2\Lambda ) + \mathcal{L}_\text{matter} ]$$, admitting de Sitter solutions when $f$ satisfies a linear differential equation (Elizalde et al., 2011).
• Scalar-Tensor Stealths: Nonminimally coupled scalar field configurations (e.g., with $\xi R \phi^2$ coupling) such that $T_{\mu\nu}^{\,\phi}=0$ on de Sitter (Maeda et al., 2012, Ayón-Beato et al., 2015).
• Vector Galileon/Generalized Proca Stealths: Vector field $A_\mu$ with certain divergence-free and null constraints leading to undisturbed (A)dS or Kerr backgrounds (Cisterna et al., 2016).
• Stealth Magnetic Fields: Homogeneous, non-vanishing magnetic fields with energy–momentum tensor proportional to $g_{\mu\nu}$, implemented via specific nonminimal couplings in Horndeski-type theories (Mukohyama, 2016, Mukohyama, 2018).
• Scalar Hair in Quadratic Higher-Order Scalar-Tensor Theories: Solutions with constant kinetic term for the scalar field, e.g., $X = \nabla^\mu \phi \nabla_\mu \phi = \text{const}$, allowing de Sitter or Kerr–de Sitter backgrounds with hidden scalar profiles (Takahashi et al., 2020, Babichev et al., 2023).
• Stealth Black Holes in Higher-Order Maxwell-Einstein Theories: Nontrivial electromagnetic fields (electric or dyonic) present but not sourcing the metric, which remains Schwarzschild–(A)dS (Minamitsuji, 12 Jun 2025).

2. Mathematical Structure and Conditions

The precise realization of a stealth de Sitter solution depends on the field content and structure of the action. Common features:

• Existence Conditions: For models with nonlocal gravity, the functional form $f(n)$ (where $n = -\Box^{-1}R$) must solve a linear second-order ODE,

$f''(n) + C_1 f'(n) = \mathcal{F}(n)$

with exponential solutions $f(n) = f_0 e^{n/B}$ often emerging (Elizalde et al., 2011).

• Scalar-Tensor Theories: Nonminimal couplings (e.g., $\xi \neq 1/6$) restrict inhomogeneous stealth solutions to de Sitter backgrounds, as other FLRW cosmologies only support homogeneous stealths, enforced by the constraint $\mathscr{H}' - \mathscr{H}^2 - k = 0$ (with $\mathscr{H}$ the conformal Hubble parameter) (Ayón-Beato et al., 2015).
• Generalized Proca Theories: Stealth requires

$\nabla_\mu A^\mu = 0,\quad A_\mu A^\mu = 0,$

ensuring cancellation of the vector field’s effect on the Einstein equations (Cisterna et al., 2016).

• Stealth Hair in Scalar-Tensor Models: Imposing constant kinetic term $X = X_0$ leads to a system of algebraic and differential equations on the action’s functional coefficients, guaranteeing that all contributions from $\phi$ vanish in the background Einstein equations (Takahashi et al., 2020).
• Stealth Electromagnetic Fields: In higher-order Maxwell–Einstein setups, algebraic relations among the higher-derivative couplings ensure that the field equations for the metric coincide with those of Schwarzschild–(A)dS despite a nonzero electric field (Minamitsuji, 12 Jun 2025).

3. Stability and Perturbation Analysis

Stability properties of stealth de Sitter solutions are highly model dependent:

• Nonlocal Gravity: De Sitter solutions are linearly stable in both FLRW and Bianchi I cases for a positive Hubble parameter and positive functional parameter $B$ in the exponential ansatz for $f(n)$. Stability holds even when the cosmological constant is negative, and for $\Lambda=0$ provided $4/3 < B < 2$ (Elizalde et al., 2011).
• Stealth Scalar Fields: Stability is not guaranteed. In the context of DHOST or quadratic higher-order scalar–tensor theories, stealth backgrounds can have perturbations that are either infinitely strongly coupled (zero sound speed) or gradient-unstable (negative sound speed squared). The “scordatura” mechanism (controlled detuning of degeneracy conditions, adding $k^4$ operators) resolves this issue and ensures weak coupling of fluctuations, preserving the stealth background at macroscopic scales (Motohashi et al., 2019, Gorji et al., 2020).
• Stealth Magnetic Fields: Full dynamical stability proven for the de Sitter solution with magnetic field, provided Horndeski-type nonminimal couplings and scalar sector are properly chosen. Linear perturbations are ghost-free and free from gradient instabilities in both subhorizon and superhorizon regimes, with explicit algebraic conditions on model parameters (Mukohyama, 2018).
• Thermal Stability in Scalar-Tensor Gravity: Analysis using a gravitational “temperature” $𝒦𝒯$ shows that generic stealth and nonconstant scalar de Sitter solutions are either thermally unstable ($m_\text{eff}^2<0$) or at best marginally stable. Only GR ($𝒦𝒯=0$) is a true thermal equilibrium (Giardino et al., 2023).
• Starobinsky–Bel–Robinson Gravity: The de Sitter inflationary solution is unstable due to a positive eigenvalue in the perturbation equations, ensuring a graceful exit from inflation in the early universe while precluding this phase as a late-time attractor (Do, 8 Jul 2025).

4. Physical Interpretation and Cosmological Implications

Stealth de Sitter solutions provide theoretical frameworks for a variety of phenomenological scenarios:

• Dark Energy and Cosmic Acceleration: Stealth sectors can underpin late-time acceleration while preserving a standard $\Lambda$CDM metric, with all novel effects relegated to the dynamics of perturbations (e.g., modified growth rates, effective gravitational couplings), and possibly impacting redshift-space distortions or lensing (Gorji et al., 2020).
• Initial Conditions and Baryogenesis: In models of decaying vacuum energy connecting two de Sitter phases, the initial de Sitter state solves the horizon, singularity, and even baryogenesis problems via entropy production through vacuum decay, within a dynamical “stealth” framework (Zilioti et al., 2015).
• Inflationary Magnetogenesis: With solutions allowing stable, homogeneous magnetic fields during inflation, the problem of backreaction and anisotropy in early-universe magnetogenesis is circumvented. The stealth nature of the field ensures its energy–momentum tensor is isotropic and compatible with the observed CMB (Mukohyama, 2016, Mukohyama, 2018).
• Quantum Creation Probability: In semiclassical gravity, the presence of stealth fields (e.g., nonminimally coupled scalars with $T_{\mu\nu}=0$) modifies the instanton (Euclidean) action controlling universe creation probabilities. Creation with a stealth scalar is possible for discrete choices of coupling constants, with the universe without stealth field generally favored, but the difference in creation rates can be made arbitrarily small for appropriate parameter choices (Maeda et al., 2012).
• Uniqueness and Selection Principles: Thermal stability arguments position stealth solutions as typically non-equilibrium, reinforcing the privileged status of classical GR solutions as stable attractors in the landscape of gravitational theories (Giardino et al., 2023).

5. Black Holes, Hair, and Rotating Stealth Solutions

Stealth structures are not restricted to cosmologies but also occur in black hole and rotating spacetimes:

• Stealth Black Holes: Schwarzschild–(A)dS black holes with nontrivial electric fields (“hair”) that do not source the metric, plus generalized Proca sectors yielding Schwarzschild–(A)dS or Kerr geometries with divergence-free and null vector profiles. Such configurations bypass traditional no-hair restrictions and remain exact GR geometries at the background level (Cisterna et al., 2016, Minamitsuji, 12 Jun 2025).
• Rotating Kerr–de Sitter Stealths and Cosmological Embedding: By conformally transforming a stealth Kerr–(de Sitter) solution (with scalar field linearly dependent on time) with a time-dependent conformal factor $C(\phi)=A^2(\tau)$, one obtains metrics describing rotating black holes embedded in evolving FLRW-like universes. Cosmological asymptotics and trapping horizon structures can be explicitly analyzed, and the construction is fully covariant within the DHOST class (Babichev et al., 2023).

6. Model-Building, Effective Field Theory, and Quantum Considerations

Stealth de Sitter solutions are also central to modern approaches in field theory and quantum gravity:

• Effective Field Theory and Strong Coupling: Generic stealth backgrounds in scalar-tensor or DHOST theories may render perturbations strongly coupled. Introducing higher-derivative operators (“scordatura”) cures this, producing a healthy dispersion relation $\omega^2 = \alpha k^4/M^2$ and weak coupling up to the cutoff scale $M$, with no observable deviation at astrophysical/macroscopic distances (Motohashi et al., 2019, Gorji et al., 2020).
• S-Matrix and Quantum Break-Time: Arguments from S-matrix theory (string theory or quantum field theory in curved spacetime) exclude strictly eternal de Sitter vacua due to quantum break-time effects, which are inversely related to the coupling strength. In this view, de Sitter is only possible as a transient or "stealth" state, with corpuscular quantum corrections (scaling as $1/N$) leading to departures from eternality—impacting cosmological constant interpretations (Dvali, 2020).
• Supergravity and Swampland Constraints: Embedding supergravity models with nonpositive F-term potentials into SU(2,1|1) invariant frameworks supports stealth de Sitter solutions along the inflationary trajectory. Stabilization of heavy fields and slow-roll (concave) inflaton potentials ensures these models simultaneously satisfy observational slow-roll and the softened de Sitter swampland conjecture bounds (Atli et al., 2020).

7. Broader Significance and Outlook

Stealth de Sitter solutions, by allowing nontrivial matter sectors or gravitational modifications that evade detection at the level of the background geometry, have wide-ranging implications:

• They furnish theoretically robust mechanisms for realizing cosmic acceleration, inflation, and dark energy without the need for strict cosmological constant tuning.
• Provide platforms to probe scalar, vector, and higher-spin degrees of freedom that evade traditional no-hair theorems.
• Serve as testbeds for constraining modified gravity through the behavior of cosmological and black hole perturbations, especially in view of gravitational wave observations.
• Motivate novel model-building directions where new sectors or higher-derivative corrections remain stealthy at the background level, but may induce observable corrections in the dynamics of perturbations or non-equilibrium cosmological phases.

Stealth de Sitter frameworks thus continue to inform the interplay between modified gravity, cosmology, black hole physics, and quantum field theory, providing both foundational and phenomenological insights into the architecture of gravitational theories.