Bounces and Nonsingular Cosmologies

A bounce replaces monotonic contraction by expansion at a nonzero scale factor. That kinematic achievement is only the first gate: a viable nonsingular cosmology must also be geodesically complete in its claimed domain, suppress ghosts and gradient instabilities, remain below its strong-coupling cutoff, control anisotropy and propagate scalar and tensor perturbations through the transition without imposed matching rules.

Helpful background. Einstein Equations fix the general-relativistic bounce condition, Relativistic Matter and Compact Objects supplies energy conditions, Causality and Global Structure supplies completeness tests, Relativistic Cosmology supplies background and perturbation variables, and Mathematical Relativity clarifies stability and extension claims.

The local bounce condition

At a smooth, nondegenerate bounce time $t_b$,

$$a_b\gt0,\qquad H_b=0,\qquad \dot H_b\gt0.$$

In general relativity the FLRW equations give

$$H^2=\frac{8\pi G}{3}\rho-\frac{K}{a^2}, \qquad \dot H=-4\pi G(\rho+p)+\frac{K}{a^2},$$

with any cosmological constant included in $\rho$ and $p$. For $K=0$, the second condition requires $\rho+p\lt0$: the null energy condition must be violated near the bounce. Positive spatial curvature can contribute instead, but then its magnitude, matter content and stability must be checked rather than hidden in the word “bounce.”

The flat-space constraint also gives $\rho_b=0$. Thus the ratio $w=p/\rho$ is generally undefined exactly at the bounce; the regular diagnostic is $\rho+p$, not a value assigned to $w_b$. With positive curvature the two equations instead give

$$\rho_b=\frac{3K}{8\pi G a_b^2}, \qquad \dot H_b=-\frac{4\pi G}{3}(\rho_b+3p_b).$$

A nondegenerate closed GR bounce therefore requires $\rho_b+3p_b\lt0$, but need not violate $\rho_b+p_b\ge0$. Global de Sitter is a simple example: with $a(t)=\cosh(H_\Lambda t)$, $K=H_\Lambda^2$, and $\rho=-p=3H_\Lambda^2/(8\pi G)$, it bounces at $t=0$ while saturating the null energy condition. Its earlier contracting phase is part of the same regular spacetime. Geodesic Completeness in Cosmology explains why this global solution and its expanding flat patch have different completeness properties.

A degenerate smooth minimum, such as $a=a_b+b(t-t_b)^4$ with $b\gt0$, has $\dot H_b=0$ and requires a higher-order test of contraction changing to expansion. It is not covered by the strict inequality used above.

A background solution with finite $a$, $H$ and curvature invariants is not automatically geodesically complete. Nor does it establish that perturbations are healthy. For the single-field stability test below, let $\zeta_u$ denote the spatial curvature perturbation on uniform-field slices, $\delta\phi=0$. This is the variable called $\zeta$ in the Horndeski source; it need not equal the uniform-density curvature perturbation used in the perturbation chapter. A useful quadratic diagnostic is

$$S_{\zeta_u}^{(2)} =\frac12\int dt\,d^3x\,a^3 Q_s \left[\dot\zeta_u^2-\frac{c_s^2}{a^2}(\nabla\zeta_u)^2\right].$$

One requires $Q_s\gt0$ to avoid a ghost and $c_s^2\gt0$ to avoid a short-wavelength gradient instability, with analogous positive kinetic and gradient coefficients for tensor modes. These linear conditions remain insufficient if the strong-coupling scale $\Lambda_{\mathrm{sc}}$ approaches the background frequency or the physical wavenumbers being evolved.

This test assumes that $\zeta_u$ is a regular physical variable after the constraints have been solved. A gauge choice or constraint elimination that divides by a vanishing background quantity can fail at a bounce. A pole in that variable’s action then needs examination in regular variables before it is interpreted as a physical instability. Conversely, finite ratios such as $c_s^2$ do not establish a healthy kinetic term if both its numerator and denominator vanish.

Contraction amplifies what expansion dilutes

In a Bianchi I contraction the shear energy density scales as $\rho_\sigma\propto a^{-6}$. A perfect-fluid component scales as $\rho_w\propto a^{-3(1+w)}$, hence

$$\frac{\rho_\sigma}{\rho_w}\propto a^{3(w-1)}.$$

As $a$ decreases, shear overtakes matter-like contraction with $w\lt1$. An ekpyrotic component with $w\gt1$ instead suppresses the ratio and can smooth a homogeneous patch. Brandenberger and Peter derive the shear scaling and explain why matter bounces need an additional smoothing mechanism Brandenberger and Peter 2017, § V.D, pp. 19–20 (PDF).

This does not dispose of inhomogeneous collapse, mixmaster behavior, relics or initial-condition sensitivity. A complete calculation evolves the background, constraints, scalar, vector and tensor modes from a declared contracting state through the same bounce equations. Junction conditions imposed at an artificial surface are not evidence that a nonsingular field theory produced the matching.

Stability claims and their loopholes

For spatially flat Horndeski cosmology, Kobayashi writes the scalar and tensor quadratic actions with coefficients $\mathcal F_S,\mathcal G_S,\mathcal F_T,\mathcal G_T$. His scalar normalization corresponds to $Q_s=2\mathcal G_S$ and $c_s^2=\mathcal F_S/\mathcal G_S$ above. The key identity is

$$\mathcal F_S=\frac1a\frac{d\xi}{dt}-\mathcal F_T, \qquad \xi=\frac{a\mathcal G_T^2}{\Theta},$$

where $\Theta$ is a background coefficient from the scalar constraint. Strict positivity implies $\dot\xi\gt a\mathcal F_T$. Consider an all-time history, $-\infty\lt t\lt\infty$, with $a\ge a_{\min}\gt0$, finite $\Theta$, continuous $\xi$, and smooth, strictly positive kinetic and gradient coefficients. The endpoint assumptions are, for any fixed finite $t_0$,

$$\int_{-\infty}^{t_0}a\mathcal F_T\,dt=\infty, \qquad \int_{t_0}^{\infty}a\mathcal F_T\,dt=\infty.$$

Integrating the inequality forces $\xi$ to be negative sufficiently far in the past and positive sufficiently far in the future. Continuity then requires a zero, but $a\mathcal G_T^2/\Theta$ cannot vanish with a positive numerator and finite denominator. Thus all these conditions cannot hold together. The continuity proviso matters: a zero of $\Theta$ can make $\xi$ discontinuous and invalidates this particular integral argument Kobayashi 2016, §II, pp. 2–3, Eqs. (8)–(15) and footnote 1 (v2 PDF). This is an obstruction in a specified theory and asymptotic class, not a no-go result for every possible bounce.

Proposed evasions change one of those assumptions. Beyond-Horndeski interactions can change the constraint structure; an effective Planck mass can tend to zero; higher-spatial-derivative theories modify the high-wavenumber dispersion relation; quantum-geometry proposals change the background equation. Each route needs its own stability and cutoff analysis. Ijjas and Steinhardt construct a geodesically complete $L_4$ Horndeski background whose scalar and tensor sound speeds remain finite. In their explicit solution, however, both tensor coefficients and the scalar kinetic and gradient coefficients vanish quadratically at a finite $\gamma$ crossing; strict kinetic positivity therefore fails there, and the quadratic calculation alone does not establish weak coupling or nonlinear stability. This crossing is not generally the $H=0$ bounce: it can occur earlier, while the coefficients can remain regular throughout the bounce itself Ijjas–Steinhardt 2017, §§4–6, especially Eqs. (29)–(31). Ageeva, Petrov and Rubakov construct Horndeski examples with effective Planck masses tending to zero but an estimated strong-coupling scale above the classical background scale; they also stress the large functional freedom used in the construction Ageeva et al. 2021, §5.

Results at the stated cutoff

The evidence and literature status below is fixed at 2026-08-19. “Stable” always names the perturbative order, wavenumber range and model class actually tested.

Formulation or claim	Evidence or result	Scope and limitation
Flat GR bounce with ordinary matter	Friedmann equations require NEC violation at $H=0$	Canonical matter does not supply it; positive curvature or changed dynamics alters the premise
All-time flat Horndeski with $a\ge a_{\min}\gt0$	The integral inequality is incompatible with simultaneous strict positivity and a continuous $\xi$	Requires smooth coefficients, finite $\Theta$, and divergence of both displayed endpoint integrals; a zero or pole requires a separate regular-variable and coupling analysis
$L_4$ Horndeski inverse construction	Explicit background is complete and linear sound speeds stay finite	Kinetic and gradient coefficients touch zero at the finite crossing; weak coupling, nonlinear stability and UV completion are not established
Horndeski with strong gravity in the past	Explicit models estimate $\Lambda_{\mathrm{sc}}$ above the classical scale and avoid linear superluminality	Effective Planck masses vanish asymptotically; realism and radiative stability remain model-dependent
Primordial perturbations from contraction	Some models reproduce a red scalar tilt and acceptable amplitudes	Tensor ratio, non-Gaussianity, strong coupling and bounce transfer can be correlated; no unique observed bounce signature is established

The 2023 perturbation analysis by Ageeva, Petrov and Rubakov illustrates the last tension: making the tensor-to-scalar ratio very small through a small scalar sound speed moves scattering amplitudes toward strong coupling Ageeva et al. 2023, §§ 2.5–2.6 and § 4. Meanwhile, the broad scalar spectrum is not a discriminant by itself; the Brandenberger–Peter review concludes that then-current observations did not uniquely favor inflation or a bounce, while simple matter-bounce realizations faced data tension Brandenberger and Peter 2017, § VI, pp. 22–23 (PDF).

The evidence does not establish a universal, UV-complete nonsingular origin. The results distinguish conditional examples from conditional obstructions in different formulations. Simulation plots of one background are not an independent nonlinear stability test, and fitting primordial spectra does not verify the transition that generated them.

What a complete demonstration must contain

A persuasive calculation declares the action and EFT cutoff; solves the constraints and background through the entire transition; shows geodesic completeness in the intended extension; reports scalar and tensor kinetic matrices and dispersion relations; compares every physical mode with $\Lambda_{\mathrm{sc}}$; controls anisotropy and inhomogeneity; derives rather than imposes matching; and propagates observables to a likelihood with stated degeneracies. Failure at any stage narrows the claim even if the scale factor remains finite.

The interpretation could change after a UV-complete or demonstrably controlled model evolves stable perturbations across the full transition; after a nonlinear simulation establishes an attractor for a broad initial-data class; or after a distinctive observation survives degeneracy with inflation, foregrounds and late-time transfer.

Exercises

During contraction, compare shear with (i) matter, $w=0$, and (ii) an ekpyrotic component, $w=3$. Which component controls the approach to the bounce as $a\to0$?

Solution

The ratio is $\rho_\sigma/\rho_w\propto a^{3(w-1)}$. For matter it scales as $a^{-3}$ and grows without bound, so shear threatens the FLRW bounce. For $w=3$ it scales as $a^6$ and tends to zero, so the ekpyrotic component suppresses homogeneous shear. This check does not prove control of nonlinear inhomogeneities or the bounce itself.

References

• Ageeva, Y., P. Petrov, and V. Rubakov. “Generating Cosmological Perturbations in Non-singular Horndeski Cosmologies.” Journal of Cosmology and Astroparticle Physics 2023, no. 01 (2023): 019. DOI. arXiv:2207.04071.
• Ageeva, Y., P. Petrov, and V. Rubakov. “Nonsingular Cosmological Models with Strong Gravity in the Past.” Journal of High Energy Physics 2021, no. 12 (2021): 107. DOI. arXiv:2104.13412.
• Brandenberger, Robert, and Patrick Peter. “Bouncing Cosmologies: Progress and Problems.” Foundations of Physics 47 (2017): 797–850. DOI. Open PDF.
• Ijjas, Anna, and Paul J. Steinhardt. “Fully Stable Cosmological Solutions with a Non-singular Classical Bounce.” Physics Letters B 764 (2017): 289–294. DOI. arXiv:1609.01253.
• Kobayashi, Tsutomu. “Generic Instabilities of Nonsingular Cosmologies in Horndeski Theory: A No-Go Theorem.” Physical Review D 94 (2016): 043511. DOI. Open PDF, v2.