Decisive multi-band gravitational-wave test of pre-big-bang string cosmology: breaking the (beta, z_s, z_d, z_sigma) degeneracy and bounding the dilaton-growth exponent

The pre-big-bang (PBB) scenario of string cosmology predicts a blue-tilted stochastic gravitational-wave background (SGWB) parameterized by the dilaton-growth exponent beta and three phase-transition ratios z_s, z_d, z_sigma. A Bayesian search in Advanced LIGO and Virgo O1-O3 cross-correlation data found no evidence for this background (Bayes factor 0.03 against it) and set only beta > -0.19, leaving the theoretically required region 0 <= beta < 3 intact and the four-parameter space large and degenerate. The open problem is to confront the PBB background decisively with multi-band gravitational-wave data - combining pulsar-timing, CMB/BBN, and interferometer constraints within one common PBB framework to break the parameter degeneracy and adjudicate the live minimal-versus-non-minimal tension over the nanohertz pulsar-timing signal.

Background

String cosmology replaces the standard inflationary Big Bang model with a bounce built on the low-energy effective action of string theory. Besides the metric, the theory contains the dilaton, a scalar field whose value controls the string coupling (and hence the strength of string interactions), and it obeys a scale-factor duality: a spatially flat, decelerated expanding solution is mapped, under $a\to 1/a$ combined with a dilaton shift, into an accelerated expanding one. Gasperini and Veneziano proposed in 1993 that this duality be taken seriously as cosmology: our post-Big-Bang decelerated era would then have a dual 'pre-big-bang' (PBB) era of accelerated expansion starting from a weak-coupling, low-curvature regime, with the Big Bang replaced by a high-curvature bounce joining the two branches.

A robust prediction of this scenario is a relic stochastic gravitational-wave background (SGWB) - a persistent, incoherent gravitational-wave 'noise', characterized by its present energy-density fraction per logarithmic frequency interval, $\Omega_{\rm GW}(f)$. Quantum tensor fluctuations are amplified during the PBB phases by the pump field, the effective scale factor (together with the dilaton) appearing in the tensor-mode equation $v_k''+(k^2-\xi''/\xi)v_k=0$; because the pump field grows toward the bounce, the graviton spectrum is blue-tilted (rising with frequency) and peaks at a frequency set by the curvature scale at the bounce, typically far above the LIGO band. The standard phenomenology approximates the background as five consecutive phases with piecewise power-law pump fields - the initial low-energy PBB era, a high-curvature string phase, and three post-bounce phases (radiation-like, an axion-oscillation-dominated stage, and the standard post-Big-Bang evolution) - joining the tensor modes and their derivatives continuously at four transition times $\tau_s$, $\tau_1$, $\tau_\sigma$, $\tau_d$, conventionally encoded as frequency ratios $z_s=f_1/f_s$, $z_\sigma=f_1/f_\sigma$, $z_d=f_1/f_d$. The spectrum is then fixed by four parameters: the exponent $\beta$, describing the high-energy growth of the dilaton and the dynamics of the internal (extra) dimensions during the string phase, and the three ratios $z_s$, $z_d$, $z_\sigma$. Theoretical consistency requires $0\le\beta<3$: $\beta\ge0$ so that the string coupling grows toward a smooth bounce, and $\beta<3$ to avoid background instabilities.

The predicted background must respect cosmological caps on gravitational-wave energy at ultra-low frequency (from anisotropies of the cosmic microwave background, CMB, and from big-bang nucleosynthesis, BBN), while its rising branch passes through the bands of pulsar-timing arrays (nanohertz), the space interferometer LISA (millihertz), and ground-based interferometers (hertz to kilohertz). Pulsar-timing arrays search for SGWBs via correlated timing residuals of millisecond pulsars; the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) reported a common nanohertz process in its 15-year dataset. The planned third-generation ground detectors Einstein Telescope and Cosmic Explorer would improve strain sensitivity by roughly an order of magnitude over Advanced LIGO/Virgo.

A Bayesian search for the PBB background in Advanced LIGO and Virgo O1-O3 cross-correlation data found no evidence for it: the Bayes factor between the PBB and noise-only models is 0.03, and the analysis establishes only $\beta\gtrsim-0.19$ (quoted at 95% confidence in the abstract), leaving the theoretically motivated region $0\le\beta<3$ untouched, while the posteriors on the other parameters span several orders of magnitude ($\log_{10}z_s=11.3^{+8.0}_{-6.6}$, $\log_{10}z_d=6.7^{+7.8}_{-4.9}$, $\log_{10}z_\sigma=3.7^{+7.2}_{-3.3}$). An earlier O1-O3 search for the string-cosmology background by a different group had likewise constrained mainly the overall amplitude. Two contemporaneous studies reached contrasting conclusions about whether the nanohertz pulsar-timing signal can be explained by PBB models: one finds the minimal PBB model excluded (its $\beta$ posterior lying outside $0\le\beta<3$ at more than $5\sigma$), while another, by the scenario's originators, finds the minimal version incompatible but a generalized 'non-minimal' PBB evolution - with a modified high-curvature string phase including string-source effects - able to fit the same data. What has not been done is a joint, multi-band confrontation of the PBB prediction with pulsar-timing, CMB, and interferometer data within one common PBB framework.

Problem Statement

The problem is to bring the pre-big-bang SGWB prediction to a decisive observational test, in the situation left by the O1-O3 non-detection: the allowed PBB parameter space ($\beta$, $z_s$, $z_d$, $z_\sigma$) is large and partially degenerate, and the only existing bound ($\beta\gtrsim-0.19$) does not reach the theoretically required region $0\le\beta<3$. The overarching objective is to determine whether the PBB graviton background can be detected or excluded with current and near-term gravitational-wave data through a joint multi-band confrontation. The problem is jointly posed: a constraint map of the PBB parameter space within one explicitly stated common PBB modeling framework, obtained by combining pulsar-timing, CMB/BBN, and interferometer (ground- and space-based) likelihoods over their respective frequency bands, giving joint constraints on ($\beta$, $z_s$, $z_d$, $z_\sigma$) and stating explicitly which bands enter as data and which as forecasts; an identification of which degeneracies of the O1-O3-only analysis are broken; and an adjudication of whether a PBB interpretation of the nanohertz pulsar-timing signal is viable in light of the contrasting conclusions of the minimal and non-minimal PBB formulations, carried out within that same common framework and stating the criterion used for 'viable'.

Partial progress exists separately in related directions (single-band LISA forecasts in a different parametrization; minimal-versus-non-minimal comparisons at nanohertz frequencies alone; an earlier amplitude-only O1-O3 search; ongoing non-minimal bounce-modeling studies), but no work has delivered the joint multi-band confrontation, and the problem remains open.

References

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