A Class of Exact Single-Field Inflationary Solutions beyond Slow Roll
Abstract
We construct exact solutions for single-field inflaton dynamics without invoking the slow-roll approximation. A suitable change of variables reduces the background equation to an Abel equation of the first kind. Although a generic Abel equation is not analytically solvable, we identify a class of inflaton potentials for which the transformed equation admits exact solutions. The resulting framework contains constant-roll inflation as a special case and also accommodates solutions with a constant second Hubble-flow parameter. We analyze the linear local attractor behavior and superhorizon evolution of these rolling backgrounds. Using the public joint CMB likelihood contours in the (ns,r) plane, we identify compatible parameter regions and show that one rolling branch can also yield 50≤ N*<60. Direct numerical evolution of the scalar and tensor modes at representative points validates the local-index predictions to better than 7×10−4 in ns and 2×10−6 in r. The exact family extends beyond slow roll, although the observationally selected regions displayed here lie close to the slow-roll regime.
Related
- Nonreciprocal black holes with primary hair in beyond Horndeski gravity: Exact solutions and stability
- Algebraic classification of single-field inflationary dynamics
- Exact Solutions of Maxwell Vacuum Equations in Null Homogeneous Petrov Spaces with Non-Solvable Motion Groups
- Exact black hole solutions with semi-symmetric vectorial torsion
- Pushing the Primordial Frontier: Exact Linear Solutions in Multifield Inflation
BibTeX
@misc{zhang2026class,
title = {{A Class of Exact Single-Field Inflationary Solutions beyond Slow Roll}},
author = {Jia-Wei Zhang and Bai-Cian Ke and Yao Yu and Dong-Ze He and Shou-Jia Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01003},
eprint = {2609.01003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01003},
}
RIS
TY - UNPB
TI - A Class of Exact Single-Field Inflationary Solutions beyond Slow Roll
AU - Jia-Wei Zhang
AU - Bai-Cian Ke
AU - Yao Yu
AU - Dong-Ze He
AU - Shou-Jia Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01003
UR - https://arxiv.org/abs/2609.01003
AB - We construct exact solutions for single-field inflaton dynamics without invoking the slow-roll approximation. A suitable change of variables reduces the background equation to an Abel equation of the first kind. Although a generic Abel equation is not analytically solvable, we identify a class of inflaton potentials for which the transformed equation admits exact solutions. The resulting framework contains constant-roll inflation as a special case and also accommodates solutions with a constant second Hubble-flow parameter. We analyze the linear local attractor behavior and superhorizon evolution of these rolling backgrounds. Using the public joint CMB likelihood contours in the (n_s,r) plane, we identify compatible parameter regions and show that one rolling branch can also yield 50≤ N_*<60. Direct numerical evolution of the scalar and tensor modes at representative points validates the local-index predictions to better than 7×10^−4 in n_s and 2×10^−6 in r. The exact family extends beyond slow roll, although the observationally selected regions displayed here lie close to the slow-roll regime.
ER -