Stealth de Sitter Solutions in Modified Gravity

• Stealth de Sitter solutions are exact de Sitter geometries where extra fields, despite having nontrivial profiles, do not affect the background metric.
• Key realizations include nonminimally coupled scalar fields, homogeneous magnetic configurations, scordatura-regulated DHOST models, and massive-gravity formulations.
• These configurations enable applications in inflationary magnetogenesis, dark energy modeling, and stealth black-hole solutions while maintaining de Sitter expansion.

Stealth de Sitter solutions are configurations in which the spacetime geometry is exactly de Sitter, or de Sitter-like, while nontrivial scalar, vector, gauge, or Stückelberg sectors remain present without generating the anisotropic or time-dependent backreaction that would ordinarily deform the background. In the literature, “stealth” has two closely related meanings: either the extra field has a vanishing energy–momentum tensor on the chosen background, or its total stress–energy is of pure cosmological-constant form, $T_{\mu\nu}\propto g_{\mu\nu}$, despite a nontrivial field profile and, in some realizations, a preferred spatial direction (Ayón-Beato et al., 2015, Mukohyama, 2016, Aguilar-Pérez et al., 30 Jun 2026). The subject spans nonminimally coupled scalar fields in FRW and de Sitter cosmology, fully backreacted homogeneous magnetic fields in scalar–vector–tensor Horndeski-type theories, scordatura-regulated DHOST dark-energy models, de Sitter vacua in ghost-free massive gravity, and stealth black holes with de Sitter asymptotics (Motohashi et al., 2019, Mazuet et al., 2015, Minamitsuji, 12 Jun 2025).

1. Core notion and principal realizations

The unifying feature is geometrical invisibility at the background level. A stealth field is dynamically nontrivial, but the metric remains the same as a GR de Sitter solution, or as a de Sitter black-hole geometry, because the extra sector either cancels within its own effective stress tensor or contributes only an isotropic vacuum term. In scalar-tensor cosmology this is implemented through $T^{(S)}_{\mu\nu}=0$ for a nonminimally coupled scalar. In magnetic stealth constructions, the gauge field is nonzero but the total stress tensor is exactly of cosmological-constant form. In dRGT massive gravity, the physical metric is de Sitter while the nontrivial Stückelberg field sits entirely in the reference metric sector (Ayón-Beato et al., 2015, Mukohyama, 2016, Mazuet et al., 2015).

Realization	Background geometry	Stealth content
Nonminimally coupled scalar	FRW or de Sitter	$T^{(S)}_{\mu\nu}=0$
Scalar–vector–tensor magnetic solution	Exact de Sitter	Homogeneous magnetic field with isotropic total stress tensor
Scordatura DHOST dark energy	Exact $\Lambda$CDM / de Sitter	Linearly time-dependent scalar
dRGT massive gravity	De Sitter hyperboloid	Nontrivial Stückelberg field $T(t,r)$
Higher-order Maxwell–Einstein black holes	Schwarzschild-(A)dS or RN-(A)dS	Electric or dyonic field invisible to the metric

A recurrent distinction is between exact stealth and generalized de Sitter. Exact stealth usually means vanishing effective stress–energy of the extra field on the chosen metric. Generalized de Sitter denotes constant-curvature solutions with a rolling scalar or nontrivial auxiliary sector that leaves the geometry de Sitter but not necessarily through a strictly vanishing stress tensor. This distinction is explicit in scalar-tensor thermodynamic analyses and in massive-gravity constructions (Giardino et al., 2023, Kakushadze, 2014).

2. Scalar stealths in FRW and exact de Sitter

A standard cosmological realization begins with a nonminimally coupled scalar field $\Psi$ on an FRW background,

$$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$

with stealth condition

$T^{\mathrm{s}}_{\mu\nu}=0.$

For FRW metrics in conformal time, the key result is that for nonconformal couplings $\xi\neq 1/6$, inhomogeneous stealths exist only when

$\mathcal{H}'-\mathcal{H}^2-k=0,$

and this condition is equivalent to the background being de Sitter. Consequently, nonconformal inhomogeneous stealths are restricted to de Sitter FRW backgrounds, whereas homogeneous stealths can coexist with the various power-law phases of $T^{(S)}_{\mu\nu}=0$0CDM cosmology (Ayón-Beato et al., 2015).

For $T^{(S)}_{\mu\nu}=0$1, the scalar can be written as

$T^{(S)}_{\mu\nu}=0$2

with a separable auxiliary function $T^{(S)}_{\mu\nu}=0$3. On de Sitter, the compatible potential is fixed to a specific three-term power-law form involving $T^{(S)}_{\mu\nu}=0$4, $T^{(S)}_{\mu\nu}=0$5, and the de Sitter radius $T^{(S)}_{\mu\nu}=0$6; for the special case $T^{(S)}_{\mu\nu}=0$7, the potential becomes logarithmic,

$T^{(S)}_{\mu\nu}=0$8

This gives a large family of exact de Sitter stealths rather than a unique solution (Ayón-Beato et al., 2015).

The Euclidean de Sitter instanton furnishes an additional selection principle. With metric

$T^{(S)}_{\mu\nu}=0$9

and $T^{(S)}_{\mu\nu}=0$0-symmetric scalar $T^{(S)}_{\mu\nu}=0$1, one finds, for $T^{(S)}_{\mu\nu}=0$2,

$T^{(S)}_{\mu\nu}=0$3

together with a tuned potential $T^{(S)}_{\mu\nu}=0$4. Regularity of the instanton and reality of the Lorentzian continuation restrict the allowed $T^{(S)}_{\mu\nu}=0$5, hence the allowed $T^{(S)}_{\mu\nu}=0$6, to discrete branches. In the Hartle–Hawking semiclassical treatment, creation with a stealth field is possible for a discrete value of the coupling constant, and the creation probability is always less than that with a trivial scalar field, although the rates can be almost the same depending on the parameters of the theory (Maeda et al., 2012).

3. Fully backreacted magnetic stealth de Sitter

A distinct realization replaces the scalar stealth by a homogeneous magnetic field in a scalar–vector–tensor Horndeski-type theory. The action is built from a metric $T^{(S)}_{\mu\nu}=0$7, a scalar $T^{(S)}_{\mu\nu}=0$8, and a $T^{(S)}_{\mu\nu}=0$9 gauge field $\Lambda$0, with a scaling-type global symmetry

$\Lambda$1

and with gauge invariants constructed from $\Lambda$2. Besides the general $\Lambda$3, the model includes the shift-symmetric Horndeski terms $\Lambda$4 and a Horndeski-type vector–curvature coupling

$\Lambda$5

The background ansatz is initially axisymmetric Bianchi I,

$\Lambda$6

with a homogeneous magnetic field generated by

$\Lambda$7

The fixed-point solution is parameterized by constants $\Lambda$8, where $\Lambda$9, $T(t,r)$0, and $T(t,r)$1. Exact de Sitter corresponds to $T(t,r)$2, and the stealth magnetic configuration further imposes $T(t,r)$3 with $T(t,r)$4 (Mukohyama, 2016).

For $T(t,r)$5 and $T(t,r)$6, the algebraic background conditions reduce to

$T(t,r)$7

$T(t,r)$8

$T(t,r)$9

$\Psi$0

Because these are four equations for two background unknowns $\Psi$1, two relations among couplings are required. One is automatically satisfied if $\Psi$2 is even in $\Psi$3, so that $\Psi$4. The remaining relation is a genuine fine-tuning among $\Psi$5. When it is imposed, the metric is exactly de Sitter, the magnetic field is homogeneous and nonzero, the electric field vanishes, and the scalar rolls linearly so that $\Psi$6 is constant. The magnetic field is then stealth in the precise sense that the matter sector is nontrivial while the background curvature invariants are those of pure de Sitter (Mukohyama, 2016).

If the fine-tuning is relaxed, the solution deforms continuously to an axisymmetric Bianchi type-I universe with constant curvature invariants, nonzero shear, homogeneous magnetic field, and homogeneous electric field. Exact stealth de Sitter is therefore a tuned point inside a broader family of constant-curvature, constant-field cosmological fixed points (Mukohyama, 2016).

The perturbative status of this magnetic solution was worked out in detail later. The linear stability analysis yields five propagating degrees of freedom and separates UV ghost conditions, UV gradient conditions, and IR attractor conditions. Ghost freedom in the subhorizon regime requires

$\Psi$7

while gradient stability is encoded by the positivity of a set of coefficients $\Psi$8, together with additional inequalities ensuring positive squared sound speeds. Superhorizon stability requires the attractor condition $\Psi$9 or, equivalently in the notation of the paper, $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$0, plus positivity of three coefficients $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$1. The paper gives explicit parameter choices satisfying all conditions and concludes that a stable de Sitter solution with a homogeneous magnetic field opens up a new possibility for inflationary magnetogenesis (Mukohyama, 2018).

4. Perturbative viability, strong coupling, and scordatura

A separate line of work concerns stealth de Sitter solutions in degenerate scalar-tensor theories. The simplest background is ghost-condensation-like: $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$2 so that $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$3 is constant. In the decoupling limit, the effective field theory of perturbations around stealth Minkowski or stealth de Sitter admits the universal dispersion relation

$$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$4

If $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$5 and of order unity, a scaling argument shows that the EFT is weakly coupled up to $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$6. In exactly degenerate DHOST-like theories, however, the structure of the underlying theory forces $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$7, so the dispersion relation loses its dependence on the spatial momentum. This is the origin of the strong-coupling problem of stealth backgrounds in degenerate theories. The proposed remedy is a controlled detuning of the degeneracy condition, called scordatura, which reintroduces the $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$8 term and pushes the Ostrogradsky mode above the EFT cutoff without changing the background stealth solution at astrophysical scales (Motohashi et al., 2019).

This logic was then applied directly to cosmology in stealth dark-energy models built on scordatura DHOST. The background metric is exactly that of standard $$S_{\mathrm s}[g,\Psi] = -\int d^4x\,\sqrt{-g}\left( \frac{1}{2}\partial_\mu\Psi\,\partial^\mu\Psi + \frac{\xi}{2}R\,\Psi^2 + U(\Psi) \right),$$9CDM, written in dimensionless variables as

$T^{\mathrm{s}}_{\mu\nu}=0.$0

and the scalar profile is linearly time-dependent,

$T^{\mathrm{s}}_{\mu\nu}=0.$1

At the background level, the scalar merely reproduces an effective cosmological constant, so the metric is the same as in GR. At the perturbative level, exact DHOST stealth solutions suffer from either vanishing or negative scalar sound speed squared. Scordatura resolves both the infinite strong coupling and the gradient instability, and it also makes the quasi-static limit well-defined. In this framework the subhorizon matter contrast obeys

$T^{\mathrm{s}}_{\mu\nu}=0.$2

with modified effective gravitational coupling, friction term, Weyl potential, and gravitational slip parameter. A central conclusion is that, in the absence of scordatura, the quasi-static approximation would break down at all scales around stealth cosmological solutions, so previous subhorizon estimates in modified gravity need to be revisited when stealth backgrounds are involved (Gorji et al., 2020).

5. Non-equilibrium and thermodynamic reinterpretations

Cosmological stealths have also been recast in explicitly thermodynamic language. In a spatially flat FLRW background filled by a dissipative fluid with bulk viscous pressure $T^{\mathrm{s}}_{\mu\nu}=0.$3, a nonminimally coupled scalar can be required to satisfy

$T^{\mathrm{s}}_{\mu\nu}=0.$4

identically on the evolving background. Defining

$T^{\mathrm{s}}_{\mu\nu}=0.$5

the stealth condition leads to a generalized Riccati equation,

$T^{\mathrm{s}}_{\mu\nu}=0.$6

with $T^{\mathrm{s}}_{\mu\nu}=0.$7 and $T^{\mathrm{s}}_{\mu\nu}=0.$8. The dissipative pressure enters as a thermodynamic driving term, and entropy production selects the stable branch $T^{\mathrm{s}}_{\mu\nu}=0.$9 over the unstable repeller $\xi\neq 1/6$0. In the asymptotic de Sitter limit $\xi\neq 1/6$1, the scalar tracks the background exponentially,

$\xi\neq 1/6$2

while remaining gravitationally invisible. The paper identifies a critical coupling $\xi\neq 1/6$3, where the Riccati nonlinearity disappears and the unstable branch is pushed to infinity in phase space (Aguilar-Pérez et al., 30 Jun 2026).

A different thermodynamic approach studies stealth and generalized de Sitter solutions in Brans–Dicke–type scalar-tensor gravity using the “temperature of gravity”

$\xi\neq 1/6$4

and an effective thermal mass

$\xi\neq 1/6$5

For generalized de Sitter solutions with

$\xi\neq 1/6$6

one finds

$\xi\neq 1/6$7

so thermal stability would require $\xi\neq 1/6$8. The detailed survey of stealth Minkowski and generalized de Sitter examples nevertheless concludes that no robust non-GR equilibrium states are found: solutions are unstable, marginal only in pathological limits, or thermodynamically singular. This further validates the special role of GR, and especially GR de Sitter, as an equilibrium state in the landscape of gravity theories (Giardino et al., 2023).

6. Massive-gravity de Sitter vacua and stealth-like branches

Ghost-free massive gravity provides a structurally different class of de Sitter stealth-like solutions. In dRGT theory, with physical metric $\xi\neq 1/6$9 and flat reference metric $\mathcal{H}'-\mathcal{H}^2-k=0,$0, the Gordon ansatz reduces the field equations to the algebraic condition $\mathcal{H}'-\mathcal{H}^2-k=0,$1, a de Sitter Einstein equation for $\mathcal{H}'-\mathcal{H}^2-k=0,$2, and a first-order PDE for the Stückelberg field $\mathcal{H}'-\mathcal{H}^2-k=0,$3,

$\mathcal{H}'-\mathcal{H}^2-k=0,$4

This equation has infinitely many solutions. Consequently there are infinitely many de Sitter vacua with the same physical metric but different reference metrics. The simplest solution,

$\mathcal{H}'-\mathcal{H}^2-k=0,$5

is manifestly homogeneous and isotropic and had been studied previously, but it is unstable. The paper then singles out solutions that share a timelike isometry between $\mathcal{H}'-\mathcal{H}^2-k=0,$6 and $\mathcal{H}'-\mathcal{H}^2-k=0,$7; only for these is the canonical energy time-independent, and they are conjectured to minimize the energy and therefore to be stable. Some of these vacua are homogeneous and isotropic in a non-manifest way, so their symmetries are obscured in standard coordinate systems (Mazuet et al., 2015).

A more radical massive-gravity construction arises in the gravitational Higgs mechanism. There, besides exact de Sitter, the full non-perturbative equations admit additional vacuum solutions in which spacetime is asymptotically de Sitter both in the past and in the future, first contracts, then the contraction slows down and reverses into expansion, and there is an epoch where the space appears to be nearly flat even though the vacuum energy density is non-vanishing. The existence of these solutions is controlled by a critical graviton mass

$\mathcal{H}'-\mathcal{H}^2-k=0,$8

For the Fierz–Pauli mass term this critical mass coincides with the perturbative Higuchi bound, which is thereby reinterpreted non-perturbatively. The synthesis given in the paper characterizes these as stealth de Sitter in the sense that large vacuum energy is dynamically hidden during an almost flat epoch (Kakushadze, 2014).

7. Black-hole realizations and broader applications

Stealth de Sitter ideas also appear in black-hole spacetimes. In a truncation of generalized Proca theory with action

$\mathcal{H}'-\mathcal{H}^2-k=0,$9

the conditions

$T^{(S)}_{\mu\nu}=0$00

make the vector equation automatic when $T^{(S)}_{\mu\nu}=0$01 and cause the vector stress tensor to vanish. For the static ansatz one obtains

$T^{(S)}_{\mu\nu}=0$02

while the metric is exactly Schwarzschild-(A)dS,

$T^{(S)}_{\mu\nu}=0$03

The same mechanism extends to rotating stealth solutions with Kerr or Kerr-(A)dS metric and nontrivial vector hair. When a nonlinear electrodynamic term $T^{(S)}_{\mu\nu}=0$04 is turned on, the vector Galileon sector remains stealth while the nonlinear Maxwell sector backreacts, giving asymptotically (A)dS black holes with a modified falloff (Cisterna et al., 2016).

Gauge-invariant higher-order Maxwell–Einstein theories supply a closely related but technically distinct class. In generic higher-order Maxwell–Einstein theories with ordinary Maxwell term $T^{(S)}_{\mu\nu}=0$05, the standard RN-(A)dS solution remains an exact solution if the higher-derivative couplings satisfy specific algebraic relations. When the ordinary Maxwell kinetic term is absent, $T^{(S)}_{\mu\nu}=0$06, the same theories can admit

$T^{(S)}_{\mu\nu}=0$07

so that the metric is exactly Schwarzschild-(A)dS even though the electric field is nonzero. In the degenerate Class $T^{(S)}_{\mu\nu}=0$08II, dyonic RN-(A)dS is also a solution, and with $T^{(S)}_{\mu\nu}=0$09 one obtains dyonic stealth Schwarzschild-(A)dS black holes in which both electric and magnetic fields are present but invisible to the metric. The paper does not perform a stability analysis, leaving perturbations and dynamical viability open (Minamitsuji, 12 Jun 2025).

Across these realizations, the main applications are inflationary magnetogenesis, dark-energy model building, hidden gravitational sectors with GR-like background expansion, and nontrivial de Sitter vacua in massive gravity. The main unresolved issues are equally consistent across the literature: full inhomogeneous perturbations beyond special limits, confrontation with CMB anisotropy and large-scale-structure data, causal non-equilibrium generalizations beyond Eckart theory, ultraviolet completion of scordatura-regulated higher-derivative models, and the dynamical stability of black-hole stealth sectors (Mukohyama, 2018, Aguilar-Pérez et al., 30 Jun 2026, Mazuet et al., 2015).