Global viability of extended pre-big-bang cosmology as the source of the nanohertz gravitational-wave background

Pulsar timing arrays have detected a nanohertz stochastic gravitational-wave background (SGWB), opening a window on the very early Universe. Tan, Wu and Liu (Phys. Rev. D 111, 123525, 2025) showed that the minimal pre-big-bang (PBB) scenario of string cosmology cannot explain the NANOGrav 15-year signal: Bayesian analysis of the model's dilaton/internal-dimension parameter $\beta$ gives $\beta=-0.12^{+0.06}_{-0.21}$ (90% credible interval), outside the theoretically allowed range $0\lesssim\beta<3$ at more than $5\sigma$, and a Bayes factor of about 468 favors a simple power-law spectrum over the PBB model. The open problem is whether a theoretically consistent extension of the PBB framework — a modified high-curvature string phase, broken $S$-duality of the perturbation spectra ('non-minimal' evolution), extra sources, or altered matching conditions — can simultaneously satisfy the theoretical consistency requirements, reproduce the observed nanohertz spectrum in a global fit of the complete pulsar-timing data, overcome the strong Bayesian preference for a simple power law, and remain compatible with all non-PTA bounds (cosmic microwave background, nucleosynthesis, ground-based interferometers). Later work has constructed phenomenologically viable non-minimal PBB spectra that match the signal and effectively relax the $\beta$ bound, but the global-fit, model-comparison, fine-tuning, and observational-discrimination questions remain unresolved.

Background

A pulsar timing array (PTA) monitors millisecond pulsars as ultra-stable clocks; a stochastic gravitational-wave background (SGWB) — a diffuse superposition of gravitational waves from unresolved sources — imprints correlated timing deviations with a characteristic angular pattern. In 2023 the NANOGrav collaboration reported, from its 15-year dataset, strong evidence for such a correlated nanohertz signal, consistent with a power-law spectrum and commonly attributed to inspiraling supermassive black hole binaries, though cosmological sources are not excluded. The pre-big-bang (PBB) scenario, proposed by Gasperini and Veneziano, is an inflationary alternative built on string theory's scale-factor duality (a symmetry of the low-energy string equations relating expanding and contracting branches): the Universe evolves from a weak-coupling, low-curvature initial state through a dilaton-driven accelerating phase (the dilaton being the scalar field that controls the string coupling) into a high-curvature 'string phase', then a bounce into the standard hot big bang. A distinctive prediction is a relic graviton background whose energy density per logarithmic frequency interval, $\Omega_{\mathrm{gw}}(f)$, is a steeply growing (blue) broken power law at low frequency. In the phenomenological parametrization of Ben-Dayan et al. (JCAP 09, 058, 2024), the string-phase dynamics is encoded in $\beta\equiv d\log g_4/d\log a$, the growth rate of the four-dimensional string coupling (a combined effect of the dilaton and the time-dependent volume of internal dimensions), together with three epoch ratios $z_s, z_\sigma, z_d$ fixing the frequencies of the transitions between the dilaton, string, and post-bounce phases. Theoretical consistency of this 'minimal' scenario (a stable bounce and perturbation spectra respecting the $S$-duality symmetry relating pre- and post-big-bang spectra) requires $0\lesssim\beta<3$. The same authors of the source paper first fitted a simplified, phenomenological PBB spectrum to the NANOGrav 15-year data (Eur. Phys. J. C 85, 327, 2025; arXiv:2409.17846), finding a mild Bayes factor of 2.2 in its favor — a Bayes factor being the ratio of Bayesian evidences of two models. When the theoretically consistent minimal parametrization is instead used, the conclusion reverses: the posterior for $\beta$ is negative at more than $5\sigma$, violating the consistency bound, and the Bayesian evidence strongly favors a simple power law (Bayes factor $\approx 468$, i.e. about $e^{6.1}$). The scenario as formulated is therefore excluded by the data, and the question becomes whether theoretically justified extensions of the PBB framework can evade this exclusion.

Problem Statement

Determine whether a theoretically consistent extension of the pre-big-bang scenario can explain the nanohertz stochastic gravitational-wave background observed by pulsar timing arrays, in the full sense in which the minimal scenario was excluded. The question has two possible resolutions. (1) Existence: construct an extended PBB model — modified high-curvature string-phase dynamics, corrected dilaton/internal-dimension evolution, broken $S$-duality ('non-minimal' evolution), additional string sources, or altered matching conditions between the string phase and the post-big-bang phase — that (i) satisfies the theoretical consistency requirements of string cosmology (a stable bounce with the required coupling behavior; in the minimal parametrization this enforces $0\lesssim\beta<3$, and any relaxation such as $\beta<0$ must be justified by the extended dynamics), (ii) predicts a relic graviton spectrum $\Omega_{\mathrm{gw}}(f)$ consistent with the PTA-inferred nanohertz signal — either as its dominant source or as a cosmological component resolved in a joint fit alongside the astrophysical binary background — and with all other existing bounds (the cosmic microwave background and big-bang nucleosynthesis integrated bounds and the LIGO/Virgo upper limits), and (iii) survives a global Bayesian analysis of the complete pulsar-timing data — parameter posteriors consistent with the theoretical constraints and a Bayesian evidence that, on the complete dataset, leaves the extended model no longer strongly disfavored relative to the simple power law (in the minimal case the simple power law is instead strongly favored, with a Bayes factor of $\approx 468$) — together with the observable signatures (spectral-break frequencies, high-frequency tail, and its detectability prospects at future interferometers such as LISA, ET and DECIGO) that would confirm or refute the PBB interpretation. (2) Exclusion: demonstrate that no theoretically consistent PBB extension of the kinds enumerated in (1) — modified high-curvature string-phase dynamics, corrected dilaton/internal-dimension evolution, broken $S$-duality, additional string sources, or altered matching conditions — can meet conditions (i)–(iii), so that the nanohertz signal must be attributed to supermassive black hole binaries or to other cosmological sources.

References

1. Qin Tan, You Wu, Lang Liu, "Pre-big-bang cosmology cannot explain the NANOGrav 15-year signal", Phys. Rev. D 111, 123525 (2025), DOI 10.1103/l7kb-r667, arXiv:2411.16505, https://arxiv.org/abs/2411.16505
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