Non-Polynomial Gravities in 4D

• Non-polynomial gravities are theories that extend the Einstein-Hilbert action with infinite curvature invariants, preserving second-order field equations and avoiding ghost instabilities.
• They yield regular black holes and frozen neutron star solutions by modifying metric functions through characteristic functions like rational and square-root forms.
• Observational implications include altered mass–radius relations and cosmological self-tuning, with constraints emerging from NICER measurements and gravitational wave data.

Non-Polynomial Gravities

Non-polynomial gravities refer to classes of metric theories of gravity in four dimensions where the action is supplemented by infinite (or sufficiently high order) series of curvature scalars, generally constructed as non-polynomial functions of the Riemann tensor, Ricci tensor, and related invariants. This produces modifications to Einstein gravity that are neither purely local, nor truncations at finite curvature order, but involve infinite towers of higher-derivative contributions resummed in ways that maintain desirable physical properties, such as the absence of ghosts and preservation of second-order field equations for certain backgrounds. Non-polynomial gravity actions have attracted interest for resolving spacetime singularities, enabling novel compact object structure, and providing new models for cosmological self-tuning.

1. Action and Structure of Four-Dimensional Non-Polynomial Gravities

Four-dimensional non-polynomial gravities are constructed by augmenting the Einstein-Hilbert action with an infinite sum of “quasi-topological” curvature invariants $Z_{(n)}$ of the spacetime metric %%%%1%%%%, each with a coupling parameter $\alpha_n$. The generic action takes the form

$$S=\frac{c^3}{16\pi G}\int d^4x\,\sqrt{-g}\,\left[ R+\sum_{n=2}^{\infty} \alpha_n\, Z_{(n)}[g] \right]$$

where $R$ is the Ricci scalar, and $Z_{(n)}$ are non-polynomial, quasi-topological densities of order $n$ in curvature (Tan et al., 29 Dec 2025, Bueno et al., 23 Sep 2025).

All couplings $\{\alpha_n\}_{n=2}^{\infty}$ are encoded in a “characteristic function” $h(\psi)$,

$$h(\psi) = \psi + \sum_{n=2}^{\infty} (2-n)\alpha_n\,\psi^n,$$

with $\psi \equiv (1-X)/\varphi^2$, $X = \nabla_a \varphi \nabla^a\varphi$, and $\varphi$ an auxiliary scalar gauge-fixed to $\varphi = r$ under spherical symmetry.

Two characteristic families of $h(\psi)$ used to generate specific non-polynomial theories are:

• Rational: $h(\psi) = \psi/(1 - \alpha^2\psi^2)$,
• Square-root: $h(\psi)=\psi/\sqrt{1-\alpha^2\psi^2}$, where $\alpha$ is the representative modification parameter. Both forms encode infinite towers of curvature couplings with definite coefficients (Tan et al., 29 Dec 2025).

2. Field Equations, Spherical Reduction, and Second-Order Properties

Varying the non-polynomial action with respect to $g^{ab}$ yields field equations,

$$\mathcal{E}_{ab} \equiv G_{ab} + \sum_{n=2}^{\infty} \alpha_n E^{(n)}_{ab} = \frac{8\pi G}{c^4}T_{ab},$$

where $G_{ab}$ is the Einstein tensor and $E^{(n)}_{ab}$ are the variational derivatives of the curvature densities (Tan et al., 29 Dec 2025). On general spherically symmetric spacetimes, the theory undergoes a spherical reduction where all higher than second derivatives vanish, mapping into a two-dimensional subclass of Horndeski scalar-tensor models: $$S_{2d}=(2G_N)^{-1}\int d^2x \sqrt{|\gamma|}\,\mathcal{L}_{2d}(\gamma_{\mu\nu},\varphi),$$ with $\mathcal{L}_{2d}$ specified by the characteristic $h(\psi)$ and Horndeski functions $G_2, G_3, G_4$ as explicit functions of $(\varphi, X)$ (Bueno et al., 23 Sep 2025).

This structure ensures second-order equations of motion for generic spherical symmetry, precluding the propagation of Ostrogradsky ghosts and maintaining a unique, static solution per spherical mass—effectively enforcing a Birkhoff theorem akin to GR (Bueno et al., 23 Sep 2025).

3. Static Solutions: Regular Black Holes and “Frozen” Neutron Stars

Static, spherically symmetric solutions are governed by an algebraic “master equation” for the metric function $f(r)$: $$h(\psi) = \frac{2G_N M}{r^3}, \qquad \psi = \frac{1-f(r)}{r^2},$$ where $M$ is the ADM mass parameter (Bueno et al., 23 Sep 2025).

In vacuum, this yields regular black-hole metrics, e.g., the “modified Hayward” solution for $h(\psi)=\psi/\sqrt{1-\alpha^2\psi^2}$: $$f(r) = 1 - \frac{2G_N M r^2}{\sqrt{r^6 + (2G_N M \alpha)^2}}.$$ These solutions possess no central curvature singularity, with all curvature invariants remaining finite at $r=0$, ensuring geodesic completeness (Bueno et al., 23 Sep 2025). Dynamically, the collapse of dust stars or thin shells approaches the regular black-hole geometry at late times, with bounces at finite radius reflecting singularity resolution.

For finite-density perfect fluids, namely neutron stars, the structure equations become a modified Tolman-Oppenheimer-Volkoff (TOV) system: $$\begin{aligned} m'(r) &= 4\pi r^2 \rho(r), \ p'(r) &= -\frac{r\,[\rho(r)c^2+p(r)]\, [3h(\psi)-2\psi h'(\psi)+\, 8\pi G p(r)/c^4]}{2f(r)h'(\psi)}, \end{aligned}$$ with mass function $m(r)=\frac{c^2}{2G}r^3 h(\psi(r))$ (Tan et al., 29 Dec 2025). As the modification parameter $\alpha$ increases, stable neutron-star solutions become larger in mass and radius; compactness and mean density also increase.

Crucially, for $\alpha\gtrsim 10^{7-8}$ m$^2$, a critical phenomenon appears: at the maximal central density, the metric develops a “double zero” at the surface, producing a critical horizon ($f(r_\text{min})\to 0$ at $r_\text{min}\lesssim R$). The external geometry is then almost indistinguishable from an extremal black hole, but an infinitesimal surface layer of matter still persists—these are referred to as “frozen neutron stars” and represent universal endpoints for the neutron-star sequence in this regime (Tan et al., 29 Dec 2025).

4. Observational Consequences and Astrophysical Implications

The flexibility in the deformation function $h(\psi)$, controlled by a single parameter $\alpha$, alters the predicted mass-radius curves for neutron stars. For $\alpha\gtrsim 10^8$ m$^2$, the allowed $M$–$R$ region is shifted to larger radii and masses; softer equations of state (EOS) such as BSk19, ruled out in GR, become viable for observed heavy pulsars when non-polynomial corrections are included (Tan et al., 29 Dec 2025).

Observational bounds can be set from joint NICER mass–radius measurements and tidal-deformability constraints from GW170817: $$0\lesssim \alpha \lesssim \mathcal{O}(10^8)\ \mathrm{m}^2.$$ Within this interval, the theory predicts that “frozen neutron stars” are allowed and should exist at the leftmost limit of the $M$–$R$ relation, with maximal masses exceeding $2.1\, M_\odot$ for the entire class of suitable EOS. A plausible implication is that observed neutron stars with extreme compactness may already be “black hole mimickers,” indistinguishable from true event horizons at astronomical distances, offering new possibilities for interpreting gravitational-wave and timing data (Tan et al., 29 Dec 2025).

5. Dynamical Spacetime Evolution and Singularity Resolution

Dynamical collapse scenarios, specifically pressureless dust stars (Oppenheimer–Snyder model) and thin shells, have been analyzed in these theories. The evolution is controlled by the non-polynomial deformation $h(\psi)$, leading to junction conditions matching the regular black-hole exterior with the interior (e.g., FLRW for dust). Near the bounce, the equations of motion generically enforce a minimum radius, with the trajectory reversing before a singularity can form.

For thin shell collapse, similar arguments show a bounce at finite radius $R>0$, demonstrating singularity avoidance. The absence of a central spacetime singularity is robust across families of $h(\psi)$ with the necessary non-polynomial structure, and such collapse dynamics are fully compatible with energy conservation and causality for generic initial data (Bueno et al., 23 Sep 2025).

6. Linearization, Spectrum, and Theoretical Consistency

Linear perturbations around maximally symmetric backgrounds (Minkowski, dS, AdS) in non-polynomial gravity remain well-posed and free of ghosts. The second-order nature of the reduced field equations ensures that only the standard two transverse graviton polarizations propagate, matching the expectations from polynomial quasi-topological gravity and avoiding the introduction of higher-derivative propagating degrees of freedom (Bueno et al., 23 Sep 2025). Explicitly, the linearized action to quadratic order is found to depend only on a combination $R^n$, $R^{n-2}W_{abcd}W^{abcd}$, $R^{n-2}Z_{ab}Z^{ab}$ (where $W$ is the Weyl tensor and $Z_{ab}$ the traceless Ricci), with coefficients ensuring no additional modes at linear level.

7. Extensions: Non-Polynomial Horndeski Models and Cosmological Applications

Analytic infinite series as encountered in the non-polynomial $h(\psi)$ exist in wider scalar-tensor frameworks. Well-tempered gravity provides a parallel context where a non-polynomial dependence in shift-symmetric Horndeski functions (kinetic and braiding terms) results from imposing a nonlinear “degeneracy” equation, guaranteeing the existence of a de Sitter attractor for any vacuum energy: $g(X) = \sum_{n=-1}^\infty a_n X^{n/2},$ with $K_X(X)$ and $G_3(X)$ coupled uniquely (Linder et al., 2020). Such “golden” theories collapse the enormous functional space of scalar-tensor gravity down to just a handful of constants, drastically enhancing predictivity for cosmological self-tuning, late-time cosmic acceleration, and large-scale structure signals.

The interplay between non-polynomial gravities and scalar-tensor degeneracy conditions hints at a broad landscape of technically natural, observationally viable, and structurally consistent models beyond the polynomial paradigm.

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References

• [Frozen Neutron Stars in Four-Dimensional Non-polynomial Gravities, (Tan et al., 29 Dec 2025)]
• [Regular black hole formation in four-dimensional non-polynomial gravities, (Bueno et al., 23 Sep 2025)]
• [An Expansion of Well Tempered Gravity, (Linder et al., 2020)]