Field Ranges and the Lyth Bound in Inflationary Cosmology
The Lyth bound provides a simple yet profound connection between observable primordial gravitational waves and the displacement of the inflaton field during cosmic inflation. Derived by David Lyth in 1997, the bound shows that a measurable tensor-to-scalar ratio r implies that the inflaton traversed a significant distance in field space, potentially exceeding the Planck scale. Such large excursions raise theoretical questions about the validity of effective field theory and the need for ultraviolet completions. This calculator implements the canonical form of the bound to relate r and the number of e-folds N to the field excursion Δφ.
In slow-roll inflation, the tensor-to-scalar ratio is related to the first slow-roll parameter ε via r = 16ε. The inflaton's field variation over an interval of N e-folds is approximately
Combining these relations yields the Lyth bound, often written in reduced Planck units (MPl = 1):
The calculator employs the equality version Δφ = (MPl/√8) √r N, providing the minimal field excursion compatible with a given r and N. Users can explore how even modest tensor signals rapidly push Δφ into super-Planckian territory when N ≈ 50–60, the range associated with observable scales exiting the horizon during inflation.
The long explanation accompanying the tool delves into the derivation of the bound from the dynamics of the inflaton and discusses its implications. For instance, models with r ≈ 0.01 and N = 50 require Δφ ≈ 1.1 MPl. This suggests that simple small-field models struggle to produce detectable gravitational waves without invoking mechanisms like monodromy or multi-field dynamics. The narrative also covers alternative formulations, including cases where the inflaton trajectory meanders in field space, reducing the net displacement even with large tensor modes, as well as scenarios in which r varies with scale.
Observationally, the search for primordial B-mode polarization in the cosmic microwave background (CMB) serves as the prime avenue for constraining r. Experiments like BICEP/Keck, the Simons Observatory, and satellite missions aspire to reach sensitivities of r ~ 10-3 or lower. The calculator contextualizes such efforts by translating prospective r values into field excursions, illustrating the challenge of constructing theoretically consistent models that accommodate them. For example, if future observations detect r = 0.005, the Lyth bound implies Δφ ≥ 0.79 MPl for N = 50.
Beyond its immediate application, the bound intersects with broader theoretical frameworks. In string theory, large field excursions may clash with the so-called swampland conjectures, which posit limits on scalar field ranges in quantum gravity. The explanatory text reviews these conjectures and their relevance to inflation, highlighting the tension between observable tensor modes and the swampland distance conjecture. It also examines proposals like axion alignment, clockwork mechanisms, and trapped inflation that attempt to reconcile large r with controlled effective theories.
The table below provides example calculations for representative parameter choices, showcasing the threshold between sub-Planckian and super-Planckian excursions:
| r | N | Δφ/MPl | Classification |
|---|---|---|---|
| 0.001 | 50 | 0.56 | Sub-Planckian |
| 0.01 | 60 | 2.12 | Super-Planckian |
| 0.05 | 50 | 3.95 | Super-Planckian |
These examples demonstrate how quickly Δφ grows with r, underscoring why the Lyth bound plays a central role in assessing the feasibility of inflationary models. The calculator's classification output mirrors the table by reporting whether the computed field excursion exceeds one reduced Planck mass.
The explanatory text continues by addressing nuances such as the dependence on the reheating history, which can alter the number of e-folds corresponding to observable scales, and by extension the inferred Δφ. It also notes that the bound assumes monotonic evolution of the inflaton; models with features or non-canonical kinetic terms can evade simple interpretations. Nevertheless, the core message remains: detectable primordial tensors generally point toward large field ranges.
In theoretical explorations, the Lyth bound informs discussions about the ultraviolet sensitivity of inflationary potentials. Super-Planckian excursions require knowledge of the potential over a wide field range, inviting concerns about Planck-suppressed operators and the need for symmetry protection. The narrative surveys approaches like shift symmetries, supersymmetric flat directions, and the use of higher-dimensional brane constructions, linking back to the implications of the bound for model builders.
To conclude, the calculator equips users with a quantitative handle on the Lyth bound while the extensive exposition provides the necessary context to appreciate its significance. Whether one is a student grappling with the basics of inflationary cosmology or a researcher evaluating model viability against upcoming CMB data, this tool aims to demystify the relationship between tensor modes and field excursions in a self-contained, client-side package.