Experimental test of pre-big bang cosmology through its relic gravitational-wave background: multi-band confirmation or exclusion of the string parameter space

The pre-big bang (PBB) scenario of string cosmology replaces standard slow-roll inflation with a dilaton-driven phase of growing curvature before a high-curvature bounce, and predicts a blue-tilted stochastic background of relic gravitational waves (GWs) that may be detectable by ground-based interferometers and LISA, together with a very low CMB tensor-to-scalar ratio ($r_{0.05}\lesssim 0.01$), while the observed scalar perturbations are produced by the axion/curvaton mechanism. Gasperini (2016) showed that a wide region of the model's parameter space (the string redshift $z_s$ and the coupling ratio $g_s/g_1=z_s^{-\beta}$) is compatible with Planck data and yields a potentially observable background, so that cross-correlated CMB and GW observations could confirm or disprove the scenario. This problem asks for the multi-band observational confrontation (CMB B-modes; pulsar-timing, LIGO/Virgo/KAGRA and LISA bands) that establishes which regions of PBB parameter space are excluded or confirmed, in a setting where the theoretical prediction currently rests on simplified benchmark assumptions (axion parameters $\sigma_i=M_P$ and $m=H_1$, simplified post-bounce evolution) and where non-minimal string-phase dynamics broaden the model class to be confronted. A decade of partial progress — LIGO/Virgo O1–O3 non-detections, $r_{0.05}<0.036$ from BICEP/Keck, and exclusion of the minimal PBB model as the source of the NANOGrav signal — has not yet decided the question; the decisive test awaits design-sensitivity and LISA-class data together with generalized-model predictions.

Background

According to the conventional cosmological picture, the large-scale homogeneity of the Universe and the nearly scale-invariant spectrum of cosmic microwave background (CMB) anisotropies are explained by a phase of slow-roll inflation at decreasing curvature, after the big bang. String cosmology suggests an alternative: the pre-big bang (PBB) scenario, motivated by the T-duality symmetry of string theory (the equivalence of string physics under exchange of large and small length scales), in which the Universe starts near the string perturbative vacuum — a state of low curvature and weak coupling — and evolves through a long dilaton-driven phase of accelerated evolution with growing curvature ('superinflation'), followed by a high-curvature 'string phase' and a non-singular bouncing transition that plays the role of the big bang and leads into the standard radiation era. The dilaton is the string-theory scalar field whose value fixes the string coupling $g_s$ (and hence the gauge coupling); its growth makes the end of the pre-bounce era a strong-coupling regime.

Quantum fluctuations of the geometry amplified by the evolving background produce a stochastic background of relic gravitational waves, characterized by its energy density per logarithmic frequency interval, $\Omega_{GW}(f)$. The amplification is controlled by a 'pump field' — the background quantity whose time variation stretches perturbation modes beyond the horizon. Because the pump field grows with curvature before the bounce, the PBB tensor spectrum is blue-tilted: its energy density grows with frequency, in sharp contrast with the nearly flat or red spectrum of slow-roll inflation. The background is therefore weak at CMB scales but potentially strong in the bands of ground-based interferometers (Advanced LIGO/Virgo/KAGRA, roughly 10–$10^3$ Hz), of the space antenna LISA (millihertz; 'eLISA' in Gasperini (2016)), and in the nanohertz band probed by pulsar timing arrays, which search for correlated timing deviations of millisecond pulsars following the Hellings–Downs angular pattern expected for a stochastic background.

The obstacle to viability is the scalar sector: the same pump field amplifies scalar metric perturbations with a growing spectrum, in conflict with the observed slightly red CMB spectrum. The standard remedy is the axion/curvaton mechanism (a curvaton being a light field whose late oscillations or decay generate the observed curvature perturbations): the CMB anisotropies are sourced by quantum fluctuations of a Kalb–Ramond axion — the pseudoscalar partner of the dilaton in the string gravitational multiplet — with initial amplitude $\sigma_i$ and mass $m$. Fixing the scalar amplitude then fixes a combination of parameters, and the tensor background is controlled by the string redshift parameter $z_s$ (the total redshift of physical frequencies across the high-curvature string phase), by the ratio $g_s/g_1=z_s^{-\beta}$ between the string coupling at the end of the string phase and at the bounce, where the exponent $\beta\geq 0$ governs the high-energy growth of the dilaton, and by the Hubble parameter at the bounce, $H_1$.

Gasperini (2016) asked whether, within this framework, one can simultaneously reproduce the Planck scalar-perturbation data and produce a GW background strong enough for Advanced LIGO/Virgo and/or eLISA at design sensitivity. Working with the illustrative benchmark $\sigma_i=M_P$ (the Planck mass) and $m=H_1$, so that the oscillating axion dominates immediately at the start of the post-bounce era, he found that a wide region of the $(z_s, g_s/g_1)$ plane works, and that the model predicts very low values of the tensor-to-scalar ratio $r$ (the ratio of tensor to scalar primordial perturbation amplitudes) at the conventional CMB pivot scale $k=0.05\ \mathrm{Mpc}^{-1}$: $r_{0.05}\lesssim 0.01$. This benchmark — two-phase dilaton-driven/string-phase evolution with an instantaneous bounce, axion parameters $\sigma_i=M_P$ and $m=H_1$, and a simplified post-bounce evolution — defines the minimal model; the non-minimal class denotes extensions in which high-energy string sources modify the high-curvature string phase, an explicit example being the fluid-dynamic source model of reference 12. The paper emphasized the discriminating power of cross-correlated observations: a direct detection of the stochastic background in the interferometer bands, together with a low or undetected $r$, would indicate a growing tensor spectrum and hence inflation localized before the bounce; conversely, a non-detection at design sensitivity combined with a measured $r$ not much below that ceiling would favor standard post-big-bang inflation. What remained unknown were the actual values of the string parameters, the detailed physics of the high-curvature string phase and of the bouncing transition, and the robustness of the quantitative boundaries to the simplified post-bounce evolution and to the particular axion-parameter choice.

Problem Statement

The problem is to decide whether the pre-big bang scenario describes the very early Universe, by confronting its predicted relic gravitational-wave background with observations — turning the pre-big bang scenario into a decided (confirmed or excluded) hypothesis.

Multi-band observational confrontation. Determine what CMB B-mode polarization measurements of the tensor-to-scalar ratio at the pivot scale $k=0.05\ \mathrm{Mpc}^{-1}$ (where the minimal model predicts $r_{0.05}\lesssim 0.01$), together with searches for the stochastic gravitational-wave background in the nanohertz (pulsar-timing), 10–$10^3$ Hz (Advanced LIGO/Virgo/KAGRA) and millihertz (LISA) bands, imply for the pre-big bang parameter space — namely, which regions of the $(z_s, g_s/g_1)$ plane (equivalently the $(z_s, \beta)$ plane, since $g_s/g_1=z_s^{-\beta}$), together with which ranges of the remaining model parameters $H_1$, $\sigma_i$ and $m$, are excluded at what confidence, and for which version of the model (minimal, or non-minimal with high-energy string sources).

The discriminating test is the following: a direct detection of the blue-tilted stochastic background in the interferometer bands together with a low CMB tensor-to-scalar ratio would support pre-bounce (pre-big bang) inflation; a non-detection at design sensitivity combined with $r_{0.05}$ significantly above the pre-big-bang ceiling of order $0.01$ would disfavour it. Determine the parameter-space exclusion or confirmation, at a stated confidence level, for both the minimal model and the non-minimal class with high-energy string sources, together with the overall support-or-disfavour verdict.

References

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