Bouncing cosmologies from Born-Infeld-type gravity

Abstract: We construct a Born-Infeld-type $f(R,{\cal G})$ modification of gravity, where ${\cal G}$ is the Gauss-Bonnet term, by embedding Born-Infeld electrodynamics in a five-dimensional pure modified gravity. This method leads to the correspondence between curvature scalars and electromagnetic field strength scalars -- $R\leftrightarrow F_{μν}F<sup>{μν}$ and $${\cal G}\leftrightarrow (ε_{μνρσ}F<sup>{μν}F<sup>{ρσ})<sup>2$$ -- allowing us to replicate the structure of Born-Infeld electrodynamics in the gravitational sector. The resulting Born-Infeld-type gravity is a ghost-free $f(R,{\cal G})$ theory which reduces to Einstein gravity in the low energy limit. In this work we focus on bouncing cosmological solutions of such a theory, which require positive spatial curvature. By using both the Jordan and Einstein frame analyses, we find a vast space of bouncing solutions with different asymptotic behaviors, including solutions with multiple bounces grouped together. Observational consequences of such solutions will be investigated in the future.

• The paper constructs a one-parameter, ghost-free f(R,𝒢) theory from five-dimensional Kaluza–Klein reduction that reproduces Born–Infeld electrodynamics and approaches Einstein gravity at low energies.
• Numerical analysis finds non-singular FLRW bounces only for positive spatial curvature, separating runaway solutions from oscillatory and multi-bounce regimes near a critical initial scale factor of 3.674 in rescaled units.
• The stability analysis identifies the runaway bounce as a late-time attractor, while oscillatory solutions can develop singularities unless a positive cosmological constant prevents recollapse; perturbation generation remains unresolved.

This paper constructs a Born–Infeld (BI)-type modification of gravity by embedding four-dimensional BI electrodynamics into a five-dimensional modified gravitational theory via the Kaluza–Klein (KK) mechanism, and then analyzes non-singular bouncing cosmological solutions of the resulting theory (2604.24860). The construction yields a ghost-free $f(R,\mathcal{G})$ gravity — where $\mathcal{G}$ is the Gauss–Bonnet invariant — that reduces to Einstein gravity in the low-energy limit, and the paper provides a systematic classification of bouncing FLRW solutions, including multi-bounce scenarios, together with a linear stability analysis.

Construction from Kaluza–Klein reduction

The central idea exploits the KK correspondence between curvature scalars and electromagnetic field-strength scalars: dimensional reduction of the 5D scalar curvature gives $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$, while reduction of the 5D Gauss–Bonnet term produces terms proportional to $(F_{\mu\nu}\tilde F^{\mu\nu})^2$ plus derivative corrections. The authors start from a 5D action $\int d^5x\sqrt{-\hat g}\, f(\hat L)$ with $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$ and the BI-type function

$$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$

where $b$ is the single BI parameter of inverse mass dimension. After reduction on the KK ansatz with a constant radion ($\Phi=1$) and elimination of the auxiliary scalar $\Sigma$, fixing the metric to Minkowski reproduces the BI Lagrangian up to field-strength derivatives; notably these derivatives do not introduce higher derivatives of $\mathcal{G}$0 in the equations of motion, so no extra gauge-field degrees of freedom arise. Setting $\mathcal{G}$1 yields the gravitational sector

$\mathcal{G}$2

a one-parameter ghost-free extension of GR whose equations of motion are second order. In the Einstein frame the theory becomes a Horndeski-type scalar-vector-tensor system with canonical scalaron $\mathcal{G}$3 and potential $\mathcal{G}$4, which possesses a stable Minkowski minimum at $\mathcal{G}$5 and a runaway minimum at large $\mathcal{G}$6. The paper adopts the pragmatic view of the construction as a formalism for building BI-type gravity rather than as a genuine higher-dimensional theory; consequently, radion stabilization is deferred.

Bouncing solutions in the simplified $\mathcal{G}$7 model

Most of the bounce phenomenology is captured by dropping the Gauss–Bonnet term, leaving $\mathcal{G}$8. In an FLRW background with curvature $\mathcal{G}$9, the nondimensionalized field equation for the scale factor is third order but parameter-free after rescaling $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$0, $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$1. The bounce condition $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$2 constrains the initial acceleration through an algebraic relation whose solvability immediately excludes flat and open universes: positive spatial curvature $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$3 is a necessary condition for a successful bounce, with consistent initial data requiring $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$4 and a fixed $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$5.

Numerical integration reveals two classes of symmetric bouncing solutions separated by a sharp critical value:

Class	Range of $$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$6	Asymptotic behavior	Curvature / EOS
Runaway	$$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$7	$$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$8	$$\hat R \leftrightarrow -\tfrac12 F_{\mu\nu}F^{\mu\nu}$$9; $(F_{\mu\nu}\tilde F^{\mu\nu})^2$0
Oscillatory	$(F_{\mu\nu}\tilde F^{\mu\nu})^2$1	damped oscillations around $(F_{\mu\nu}\tilde F^{\mu\nu})^2$2	oscillations around $(F_{\mu\nu}\tilde F^{\mu\nu})^2$3

The exact critical value is $(F_{\mu\nu}\tilde F^{\mu\nu})^2$4, tied to the local maximum of the scalaron potential. Runaway solutions admit an analytic asymptotic approximation $(F_{\mu\nu}\tilde F^{\mu\nu})^2$5. Beyond single bounces, the scan uncovers multi-bounce solutions: odd-bounce runaway solutions confined to narrow "islands" of $(F_{\mu\nu}\tilde F^{\mu\nu})^2$6 (e.g., triple bounces near $(F_{\mu\nu}\tilde F^{\mu\nu})^2$7 and $(F_{\mu\nu}\tilde F^{\mu\nu})^2$8, five bounces near $(F_{\mu\nu}\tilde F^{\mu\nu})^2$9, seven near $\int d^5x\sqrt{-\hat g}\, f(\hat L)$0), and even-bounce solutions obtained by imposing an anti-bounce (local maximum of the scale factor) at the origin with negative initial acceleration. Between islands, solutions generically develop singularities away from the main bounce.

Oscillatory solutions suffer from past/future singularities in Minkowski vacuum — the universe collapses after the bounce and emerges from a singularity before it — because the scalaron eventually oscillates around the stable minimum and drives contraction. A sufficiently large positive cosmological constant removes these singularities by preventing the contracting phase, analogous to $\int d^5x\sqrt{-\hat g}\, f(\hat L)$1CDM with positive curvature. Asymmetric solutions are generated by small nonzero $\int d^5x\sqrt{-\hat g}\, f(\hat L)$2 at the origin and can interpolate between runaway behavior on one side of the bounce and oscillatory behavior on the other. Minimally coupled matter with sub-BI-scale initial density ($\int d^5x\sqrt{-\hat g}\, f(\hat L)$3) does not qualitatively alter the bounce structure, though dust shortens the post-bounce expansion of oscillatory solutions.

Einstein-frame interpretation

The classification has a transparent interpretation in the Einstein frame. The Weyl factor $\int d^5x\sqrt{-\hat g}\, f(\hat L)$4 relates the frames via $\int d^5x\sqrt{-\hat g}\, f(\hat L)$5, and the bounce condition in the Jordan frame translates to constraints on $\int d^5x\sqrt{-\hat g}\, f(\hat L)$6 that fix the critical value $\int d^5x\sqrt{-\hat g}\, f(\hat L)$7 — precisely the local maximum of the scalaron potential. Runaway solutions live entirely in the region $\int d^5x\sqrt{-\hat g}\, f(\hat L)$8 rolling toward the runaway vacuum; oscillatory solutions live around the stable Minkowski minimum, with the scalaron rolling off the hilltop post-bounce and oscillating as in reheating, ultimately triggering collapse. Multi-bounce solutions involve the scalaron overshooting the local maximum, producing "sub-bounces" on the Minkowski side before returning to the runaway region. This framing also explains why the Jordan-frame Hubble function plateaus for runaway solutions while decaying exponentially in the Einstein frame.

Effects of the Gauss–Bonnet term

Introducing a free coefficient $\int d^5x\sqrt{-\hat g}\, f(\hat L)$9 in $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$0 (with $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$1 recovering the KK-derived theory) modifies the bounce initial conditions only mildly for the "+" branch, but opens a genuinely new class: for sufficiently negative $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$2, the previously excluded "−" branch becomes viable, yielding GB-supported bounces with no lower bound on $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$3. The effective scalaron critical point shifts according to $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$4 (e.g., $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$5 for $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$6, shifting $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$7 from $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$8 to $\hat L = \hat R + (b^2/24)\hat{\mathcal G}$9), which explains why positive $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$0 converts some oscillatory solutions into runaway ones. The local maximum persists for $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$1; for $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$2 it is uplifted and only oscillatory solutions remain. The effective potential takes the closed form $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$3.

Stability analysis

Following Nojiri–Odintsov's method, the authors perturb $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$4 in e-fold time $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$5 and examine the characteristic exponent $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$6. Although $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$7 at the bounce itself, this does not signal instability: the coefficient of the highest-derivative term vanishes there, degenerating the perturbation equation to first order, and the local solution $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$8 shows perturbations cannot grow within the $$f(\Sigma) = \frac{2}{b^2}\left(1-\sqrt{1-b^2\Sigma}\right),$$9 e-folds over which $b$0 turns negative. Analytic late-time results confirm that the runaway attractor $b$1 is stable ($b$2, $b$3, $b$4), while oscillatory solutions enter a strongly damped regime as $b$5 ($b$6). With a cosmological constant, a de Sitter fixed point exists for $b$7 and is linearly stable; the $b$8 limit is continuous for backgrounds but singular for the perturbation coefficients. The same structure carries over to the GB case, with the caveat that for large $b$9 on the "−" branch, growing modes persist longer before stabilization. The method's applicability is limited to intervals where $\Phi=1$0 does not change sign, since $\Phi=1$1 becomes multivalued otherwise — a genuine restriction for multi-bounce and oscillatory phases.

Limitations and open questions

Several caveats are stated explicitly. The radion is held fixed throughout, so its stabilization in the full KK interpretation remains unresolved. Bounces require positive spatial curvature, which current observations permit but do not establish. The stability analysis cannot be extended globally across sign changes of $\Phi=1$2, leaving multi-bounce and late-time oscillatory dynamics only partially controlled. Most importantly, the generation of cosmological perturbations with observationally viable power spectra — whether during contraction or post-bounce inflation — has not yet been addressed, nor have regular black hole solutions with the non-minimally coupled BI electrodynamics been constructed in this framework.

Conclusion

The paper delivers two concrete results: a KK-based derivation of a ghost-free, one-parameter BI-type $\Phi=1$3 gravity whose vector sector reproduces BI electrodynamics, and a comprehensive classification of non-singular bouncing cosmologies within it, characterized entirely by the initial scale factor at the bounce and interpretable through scalaron potential dynamics. Combined with prior results on regular black holes in the same Lagrangian class, the work indicates that both cosmological and black hole singularities can be removed in this theory, though its viability as a cosmological model hinges on the still-open perturbation analysis.