A theoretical cosmology analysis says that, during a continuous non-attractor phase, a scalar field can cover only a finite field-space distance while the nonconstant curvature mode is amplified outside the horizon. The upper distance depends on the field’s kinetic energy when the interval begins, and the formal high-amplification limit remains finite rather than becoming an unlimited excursion. The study frames this as an upper counterpart to the usual Lyth lower bound.
The work examines Einstein gravity with one canonically normalized scalar, accelerated expansion and continuous anti-damping of the nonconstant superhorizon mode. It presents the derivation as independent of the scalar potential and valid without the slow-roll approximation. The result comes from exact analytical relations for the background field distance and the superhorizon curvature mode, rather than from measured data.
The condition behind the ceiling
The anti-damped regime is defined by ε2 < −3. In that range, the paper reports a negative effective-friction term and a growing time derivative of the curvature perturbation, written R-dot. The calculation treats this growing mode and the field distance as linked quantities over the same continuous interval.
For a duration of ΔN e-folds, the bound is Δφ_NA/M_P < (2√(2ε_in)/3)(1−e^(−3ΔN/2)) < 2√(2ε_in)/3. Δφ_NA is the distance accumulated during the non-attractor phase, M_P is the Planck-mass unit, and ε_in is the entry parameter. The first part records the finite duration; the second gives the limiting ceiling as the interval is extended.
With the inflationary entry condition ε_in < 1, the weaker universal version becomes Δφ_NA/M_P < 2√2/3, or about 0.94. That is a sub-Planckian ceiling within the model’s assumptions and for the continuous interval covered by the calculation, rather than a statement about every inflationary construction.
Constant-roll evolution gives a stronger parameter-dependent form. When ε2 = −p remains fixed with p > 3, the finite-duration relation is Δφ/M_P = (2√(2ε_in)/p)(1−e^(−pΔN/2)) < 2√(2ε_in)/p. Ultra-slow roll is the p = 6 special case, with an asymptotic bound Δφ_USR < √(2ε_in) M_P/3.
Amplification does not buy unlimited distance
For constant ε2 = −p, the exact relation can be written in terms of the velocity-amplification factor G: G = e^((p−3)ΔN), and Δφ/M_P = (2√(2ε_in)/p)[1−G^(−p/[2(p−3)])]. As G rises without bound, the expression approaches a finite field distance. Larger amplification is therefore paired with progressively less additional field travel in the formal limit.
The paper makes a separate, conditional connection to scalar power. In the quasi-de Sitter limit, when the growing contribution dominates, the power-amplification factor A = P_R,f/P_R,i is approximately e^(2(p−3)ΔN), and the corresponding field-range expression is Δφ/M_P ≈ (2√(2ε_in)/p)[1−A^(−p/[4(p−3)])]. This is not a universal exact relation to total late-time power: the relative constant and nonconstant mode amplitudes at entry must also be specified.
That qualification also marks the result’s limits. The main inequality applies to distance accumulated during a continuous canonical non-attractor interval. Noncanonical kinetic structure, additional fields, or alternating attractor and non-attractor intervals can evade a bound on total inflationary field distance. At sufficiently small ε, stochastic quantum diffusion can compete with classical drift, so the formal limits G → ∞ and A → ∞ should not be read as arbitrarily long deterministic evolution.
The document is an arXiv version-one preprint. Its authorship disclosure says the central result and initial manuscript were generated by OpenAI’s GPT-5.6 Sol; William H. Kinney did not propose the result and later checked and edited the manuscript. The repository header gives Aug. 26, 2026, while the title page says Aug. 27, 2026.
Paper data and sources
Original title: An Inverse Lyth Bound for Non-Attractor Inflation
Authors: William H. Kinney
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text