Non-singular string cosmology from all-order α′ corrections: first-principles foundations and physical viability

String theory's low-energy effective action receives an infinite tower of higher-derivative α′ corrections. For purely time-dependent (cosmological) backgrounds, Hohm and Zwiebach classified all α′ corrections compatible with O(d,d) duality, and Wang, Wu, Yang and Ying used this classification to build phenomenological, non-perturbative, non-singular bouncing solutions of the bosonic gravi-dilaton system, showing in principle that the big-bang singularity can be smoothed by α′ corrections. The open problem is to place this singularity resolution on a first-principles footing and to establish its physical viability: determine the higher-order coefficients of the tower from the underlying string theory rather than by ansatz, extend the construction beyond the pure gravi-dilaton system to the general class of backgrounds with a time-dependent Kalb–Ramond field, matter, and dilaton stabilization (all three), and beyond the bosonic string to both the heterotic and type II theories, construct solutions including α′ and quantum string-loop corrections simultaneously, and analyze the stability and primordial perturbation spectra of these backgrounds.

Background

In general relativity, spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) cosmologies — homogeneous, isotropic solutions with scale factor $a(t)$ and Hubble parameter $H = \dot{a}/a$ — generically begin at a big-bang singularity where the curvature diverges. String theory modifies the low-energy gravitational action by an infinite tower of higher-derivative corrections organized in powers of α′, the Regge slope parameter that sets the square of the fundamental string length; these terms become important precisely when the curvature approaches the string scale, i.e. near the putative singularity. The massless fields of the bosonic string's neutral sector are the metric $g_{ij}$, the dilaton $\phi$ (a scalar whose exponential sets the string coupling $g_s$, the expansion parameter for quantum string loops), and the Kalb–Ramond two-form $B_{ij}$; the simplest cosmological truncation keeps only the metric and a time-dependent dilaton (the gravi-dilaton system) with $B_{ij} = 0$. A classic result of Sen, building on Veneziano's scale-factor duality, states that for backgrounds depending only on time and with $d$ spatial translations, the classical string equations retain, to all orders in α′, an O(d,d) duality symmetry — O(d,d) being the group of orthogonal rotations of the 2d-dimensional space spanned by the momenta and winding numbers of string states; scale-factor duality, which sends $H \to -H$, maps an expanding branch to a contracting one and underlies the pre-big-bang scenario of Gasperini and Veneziano, in which a duality-related phase of accelerated contraction precedes the big bang. That scenario, however, is perturbative and the branches meet at a singularity where the α′ expansion breaks down. In 2019 Hohm and Zwiebach classified, for purely time-dependent backgrounds, the complete set of α′ corrections to the gravi-dilaton action compatible with O(d,d), and showed the corrected equations of motion can be kept second-order in time derivatives with an infinite sequence of numerical coefficients, only the lowest of which were fixed by first-principles string computations. Wang, Wu, Yang and Ying then chose the undetermined higher coefficients $c_{k \ge 3}$ phenomenologically by ansatz (their solution family is labeled by a positive integer $n$ tied to the known second-order coefficient $c_2$ of the string theory in question) and obtained closed-form, non-perturbative, non-singular bouncing solutions $H(t)$, $\phi(t)$ that exactly reproduce the perturbative results at early and late times and are regular at $t = 0$. This established that α′ corrections can resolve the big-bang singularity, but only at the level of a proof of principle: the coefficient choices were not derived from the underlying string theory, the Kalb–Ramond field and matter were set to zero, quantum loop corrections were treated separately at best, the applicability of the all-order action to heterotic and type II strings was unproven (their footnote notes that for the heterotic string $c_2 = 1/128$ and $n = 1$ already works, while for type II strings $c_2 = 0$ requires $n \ge 2$, and that in both cases the Hohm–Zwiebach action must first be proven for those theories), and neither the stability of these backgrounds nor the cosmological perturbations propagating on them was analyzed.

Problem Statement

Establish, from first principles and with physically realistic content, whether the infinite tower of α′ corrections of string theory resolves the big-bang singularity in the duality-invariant cosmological sector, going beyond the phenomenological proof of principle of Wang, Wu, Yang and Ying (2019). Concretely: (i) determine the higher-order coefficients of the O(d,d)-invariant cosmological effective action from the underlying string theory rather than by ansatz, at orders beyond the now-known α′³ term, the target being the complete tower to all orders (equivalently, a generating prescription that reproduces the full coefficient sequence) rather than only the next order; (ii) extend the all-order construction beyond the pure gravi-dilaton system to include all three of a time-dependent Kalb–Ramond field, matter sources with an energy density, pressure, and dilatonic charge coupled in a duality-covariant way, and stabilization of the string coupling, and extend the all-order duality-invariant framework itself to both heterotic and type II string theory, in every case to all orders in α′; (iii) construct non-singular solutions in which α′ corrections and quantum string-loop corrections are included simultaneously, and identify explicitly which features of the combined regime, if any, are qualitatively new relative to the α′-only and loop-only solutions; (iv) determine the stability of these non-perturbative bouncing backgrounds — including at minimum the linear stability of cosmological perturbations about them — and the primordial scalar and tensor perturbation spectra they predict, so that singularity resolution can be assessed against early-universe phenomenology. These four components are complementary routes toward one objective — a first-principles, stable, observationally meaningful non-singular string cosmology — and progress on each feeds the others (the coefficient determination pins the equations the solutions must satisfy, the field content and loop corrections determine their realism, and stability plus spectra determine their viability). Each component is an independently well-posed sub-question with the resolution criterion stated above. Results already reported in the cited literature — in particular the all-order O(d,d)-invariant constructions with matter and Kalb–Ramond field (refs 6, 8), the order-α′³ reduction to heterotic and type II (ref 11), and the non-linear stability of the α′-corrected Friedmann equations (ref 15) — address parts of components (ii) and (iv), but they do not by themselves resolve the task: the targets stated above require work beyond them.

References

1. Peng Wang, Houwen Wu, Haitang Yang, Shuxuan Ying, 'Non-singular string cosmology via α′ corrections', J. High Energ. Phys. 2019, 263 (2019). DOI: 10.1007/jhep10(2019)263, arXiv:1909.00830, https://doi.org/10.1007/jhep10(2019)263
2. Olaf Hohm and Barton Zwiebach, 'Duality invariant cosmology to all orders in α′', Phys. Rev. D 100, 126011 (2019). DOI: 10.1103/PhysRevD.100.126011, arXiv:1905.06963, https://doi.org/10.1103/PhysRevD.100.126011
3. Olaf Hohm and Barton Zwiebach, 'Nonperturbative de Sitter vacua via α′ corrections', Int. J. Mod. Phys. D 28, 1943002 (2019). DOI: 10.1142/S0218271819430028, arXiv:1905.06583, https://doi.org/10.1142/S0218271819430028
4. Ashoke Sen, 'O(d)×O(d) symmetry of the space of cosmological solutions in string theory, scale factor duality and two-dimensional black holes', Phys. Lett. B 271 (1991) 295. DOI: 10.1016/0370-2693(91)90090-D, https://doi.org/10.1016/0370-2693(91)90090-D
5. Peng Wang, Houwen Wu, Haitang Yang, Shuxuan Ying, 'Construct α′ corrected or loop corrected solutions without curvature singularities', J. High Energ. Phys. 2020, 164 (2020). DOI: 10.1007/JHEP01(2020)164, arXiv:1910.05808, https://doi.org/10.1007/JHEP01(2020)164
6. Heliudson Bernardo, Robert Brandenberger, Guilherme Franzmann, 'O(d,d) covariant string cosmology to all orders in α′', J. High Energ. Phys. 2020, 178 (2020). DOI: 10.1007/JHEP02(2020)178, arXiv:1911.00088, https://doi.org/10.1007/JHEP02(2020)178
7. Heliudson Bernardo, Guilherme Franzmann, 'α′-Cosmology: solutions and stability analysis', J. High Energ. Phys. 2020, 073 (2020). DOI: 10.1007/JHEP05(2020)073, arXiv:2002.09856, https://doi.org/10.1007/JHEP05(2020)073
8. Heliudson Bernardo, Paul-Robert Chouha, Guilherme Franzmann, 'Kalb-Ramond backgrounds in α′-complete cosmology', J. High Energ. Phys. 2021, 109 (2021). DOI: 10.1007/JHEP09(2021)109, arXiv:2104.15131, https://doi.org/10.1007/JHEP09(2021)109
9. Jerome Quintin, Heliudson Bernardo, Guilherme Franzmann, 'Cosmology at the top of the α′ tower', J. High Energ. Phys. 2021, 149 (2021). DOI: 10.1007/JHEP07(2021)149, arXiv:2105.01083, https://doi.org/10.1007/JHEP07(2021)149
10. Tomás Codina, Olaf Hohm, Diego Marqués, 'String Dualities at Order α′³', Phys. Rev. Lett. 126, 171602 (2021). DOI: 10.1103/PhysRevLett.126.171602, arXiv:2012.15677, https://doi.org/10.1103/PhysRevLett.126.171602
11. Tomás Codina, Olaf Hohm, Diego Marqués, 'General string cosmologies at order α′³', Phys. Rev. D 104, 106007 (2021). DOI: 10.1103/PhysRevD.104.106007, arXiv:2107.00053, https://doi.org/10.1103/PhysRevD.104.106007
12. Maurizio Gasperini and Gabriele Veneziano, 'Non-singular pre-big bang scenarios from all-order α′ corrections', J. High Energ. Phys. 2023, 144 (2023). DOI: 10.1007/JHEP07(2023)144, arXiv:2305.00222, https://doi.org/10.1007/JHEP07(2023)144
13. Pietro Conzinu, Giovanni Fanizza, Maurizio Gasperini, Eliseo Pavone, Luigi Tedesco, Gabriele Veneziano, 'From the string vacuum to FLRW or de Sitter via α′ corrections', J. Cosmol. Astropart. Phys. 2023, 019 (2023). DOI: 10.1088/1475-7516/2023/12/019, arXiv:2308.16076, https://doi.org/10.1088/1475-7516/2023/12/019
14. Song Li and Deyou Chen, 'Two non-perturbative α′ or loop corrected string cosmological solutions', Chin. Phys. C 47, 095102 (2023). DOI: 10.1088/1674-1137/acddd5, arXiv:2306.07031, https://doi.org/10.1088/1674-1137/acddd5
15. Heliudson Bernardo, Jan Chojnacki, Vincent Comeau, 'Non-linear stability of α′-corrected Friedmann equations', J. High Energ. Phys. 2023, 119 (2023). DOI: 10.1007/JHEP03(2023)119, arXiv:2212.11392, https://doi.org/10.1007/JHEP03(2023)119
16. Przemysław Bieniek, Jan Chojnacki, Jan H. Kwapisz, Krzysztof A. Meissner, 'Stability of the nonperturbative O(D,D) de Sitter spacetime: The isotropic case', Phys. Rev. D 106, 106009 (2022). DOI: 10.1103/PhysRevD.106.106009, arXiv:2208.06010, https://doi.org/10.1103/PhysRevD.106.106009
17. Maurizio Gasperini, 'Cosmological horizons in regular bouncing backgrounds' (2026), arXiv:2605.13636, https://arxiv.org/abs/2605.13636
18. Gabriele Veneziano, 'Recent progress in classical string cosmology', Gen. Relativ. Gravit. 57, 151 (2025). DOI: 10.1007/s10714-025-03482-2, https://doi.org/10.1007/s10714-025-03482-2