Controlled string-theoretic bounce completion

Choose a time-oriented smooth Lorentzian spacetime $(\mathcal{M},g_F)$, a named field frame $F$, and a smooth future-directed unit timelike vector field $u$ with $g_F(u,u)=-1$. Its integral curves are the chosen observer congruence: each curve $c(s)$ is parametrized by proper time $s$ and satisfies $dc\mathrm{/}ds=u(c(s))$. Its expansion scalar is ${\theta }_F:={\mathrm{\nabla }}_{\mu }^{(F)}{u}^{\mu }$, where ${\mathrm{\nabla }}^{(F)}$ is the Levi-Civita connection of $g_F$. A bounce in this item means that for a specified integral curve there are $s_{-}<s_{+}$ across the transition with ${\theta }_F(c(s_{-}))<0<{\theta }_F(c(s_{+}))$; for a homogeneous cosmology these signs must hold on the entire stated pre- and post-transition hypersurfaces. In the unit-lapse FLRW convention $d{s}_F^2=-d{t}^2+a(t{)}^2{\gamma }_{ij}d{x}^{i}d{x}^j$ with $c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}=1$, $u={\mathrm{\partial }}_t$, $H=\dot{a}\mathrm{/}a$, and ${\theta }_F=dH$ for $d$ spatial dimensions. If $g_{tt}=-c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^2$, then $u=c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{-1}{\mathrm{\partial }}_t$ and ${\theta }_F=dH\mathrm{/}{c}_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$.

For this question, a causal geodesic is a $C^2$ curve $\gamma$ with ${\mathrm{\nabla }}_{\dot{\gamma }}^{(F)}\dot{\gamma }=0$ and $g_F(\dot{\gamma },\dot{\gamma })\le 0$; use an affine parameter for a null geodesic and proper time for a timelike geodesic. A background is nonsingular through the bounce if it is defined on a smooth time-oriented manifold containing an open neighborhood of the transition, its Lorentzian metric is nondegenerate and $C^2$, its background fields are at least $C^2$, every such causal geodesic reaching the transition at finite parameter extends as a causal geodesic through it, and the specified curvature diagnostics and fields remain finite there. A globally consistent compactification consists of a specified compact internal space $Y$ (or a separately specified allowed orbifold), its metric and all flux, brane, gauge, boundary and patching data, and a stated string theory whose field equations, Bianchi identities, flux-quantization, tadpole, anomaly-cancellation and other global consistency conditions are satisfied on all of $Y$. Let $\mathfrak{M}(Y)$ be the complete declared set of lower-dimensional scalar fields parametrizing continuous deformations of those internal data that are not fixed by the stated constraints. “All moduli stabilized” means every $\chi \in \mathfrak{M}(Y)$ satisfies the candidate's explicit stabilization criterion throughout the bounce and the interval needed for perturbation propagation; omitting a member of $\mathfrak{M}(Y)$ does not count as stabilization.

A perturbation state is specified on an initial hypersurface as an element of a stated classical perturbation phase space (or a specified quantum state space), with its variables, gauge or invariant formulation, regularity, constraints and domain given. The transfer rule is a stated map $\mathcal{T}:{\mathcal{P}}_{-}\to {\mathcal{P}}_{+}$ across the bounce, derived from the candidate's linearized or nonlinear field equations; late-time observables are a separate specified map from ${\mathcal{P}}_{+}$ to the quantities compared with data. “Controlled” is model-specific data: the candidate must define dimensionless functions $q_j$ measuring ${\alpha }^{\mathrm{\prime }}$ corrections, string loops, omitted-mode effects and backreaction, and dimensionless tolerances ${ϵ}_j>0$ such that $q_j\le {ϵ}_j$ throughout the claimed interval for every declared control channel $j$. The coupling/frame convention, curvature diagnostic, cutoff, omitted-mode conditions and all ${ϵ}_j$ must be stated; no universal control threshold is assumed here.

Does there exist such a globally consistent compactification and nonsingular, controlled time-dependent solution connecting a contracting cosmological branch to an expanding branch, stabilizing every member of $\mathfrak{M}(Y)$ by the stated criteria, and propagating a specified perturbation state through a specified transfer map to late-time observables while satisfying all declared control bounds?

Alternative string-cosmology model data

M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo and I. Zavala, String Cosmology: from the Early Universe to Today, arXiv:2303.04819v2

O. Hohm and B. Zwiebach, Duality Invariant Cosmology to all Orders in ${\alpha }^{\mathrm{\prime }}$, arXiv:1905.06963v2

M. Gasperini and G. Veneziano, Non-singular pre-big bang scenarios from all-order ${\alpha }^{\mathrm{\prime }}$ corrections, arXiv:2305.00222v4

I. Ben-Dayan et al., Gravitational-wave background in bouncing models from semi-classical, quantum and string gravity, arXiv:2406.13521v2

Status date: 2026-10-30. The full relevant text of Cicoli et al. (2023, §§7.3–7.4, PDF pp. 135–141) identifies the two-derivative pre-big-bang graceful exit and the ekpyrotic bounce as unresolved high-curvature/strong-coupling problems, and treats moduli stabilization and a global string realization as additional requirements. Later work supplies partial candidates. Hohm and Zwiebach (2019, arXiv:1905.06963v2, §§1, 4–6) classify homogeneous, time-dependent, duality-invariant metric, antisymmetric-tensor and dilaton actions to all orders in ${\alpha }^{\mathrm{\prime }}$ and exhibit de Sitter solutions for some functions in that theory space; their conclusion leaves determination of the string-theory-specific coefficients, realistic matter, and dilaton/loop control open. Gasperini and Veneziano (2023, revised 2025, arXiv:2305.00222v4, §§1–4 and Appendix A) construct regular pre-to-post-big-bang backgrounds conditional on assumed properties of an unspecified all-orders duality-invariant function. Their §4 cautions that string theory may not realize those properties, compatibility with a general-covariant action is unclear, and the post-bounce evolution reaches strong coupling, where string-loop and nonperturbative effects become important. It describes dilaton and other-moduli stabilization by the resulting potentials as a possibility, and leaves observational consistency of the later history open. Ben-Dayan et al. (2024, arXiv:2406.13521v2, Abstract, §1 and relevant parts of §§3–5) calculate model-dependent gravitational-wave spectra for selected bouncing scenarios, including a parameterized pre-big-bang history; their §1 explicitly treats the strong-coupling dilaton dynamics as phenomenological. These results do not establish the full conjunction formulated here. The source review is targeted, not an exhaustive search of all literature through 2026-10-30; no claim is made that no later construction exists. This status is unresolved rather than a proof of nonexistence, and the open problem is not a premise for another result.