Slow-Roll Inflation at Arbitrary Order
Abstract
We derive explicit expressions for the primordial scalar and tensor spectra at arbitrary finite order p in the slow-roll expansion, valid both for canonical single-field inflation and k-inflation. At any order, the spectra are determined by known combinations of slow-roll parameters weighted by universal coefficients entirely fixed by the free de Sitter solutions, without requiring perturbatively corrected mode functions. Using these spectra, we obtain closed-form consistency relations that connect the tensor tilt nt and its runnings αnt≡ dn nt/d(ln k)n to the tensor-to-scalar ratio r, the scalar spectral index ns, and its runnings αns≡ dn(ns−1)/d(ln k)n. The consistency relations are valid for arbitrary n≥ 0 and any finite p≥ 0, without recursion over lower-order relations. For k-inflation, they additionally depend on the scalar sound speed and its scale derivatives. We develop an algorithm that generates both the spectra and consistency relations for any pair (n,p). We reproduce known results through next-to-next-to-next-to-leading order (N3LO) and extend the explicit calculations up to N10LO.
Related
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BibTeX
@misc{giare2026slowroll,
title = {{Slow-Roll Inflation at Arbitrary Order}},
author = {William Giarè},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28608},
eprint = {2610.28608},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28608},
}
RIS
TY - UNPB
TI - Slow-Roll Inflation at Arbitrary Order
AU - William Giarè
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28608
UR - https://arxiv.org/abs/2610.28608
AB - We derive explicit expressions for the primordial scalar and tensor spectra at arbitrary finite order p in the slow-roll expansion, valid both for canonical single-field inflation and k-inflation. At any order, the spectra are determined by known combinations of slow-roll parameters weighted by universal coefficients entirely fixed by the free de Sitter solutions, without requiring perturbatively corrected mode functions. Using these spectra, we obtain closed-form consistency relations that connect the tensor tilt n_t and its runnings α_n^t≡ d^n n_t/d(ln k)^n to the tensor-to-scalar ratio r, the scalar spectral index n_s, and its runnings α_n^s≡ d^n(n_s−1)/d(ln k)^n. The consistency relations are valid for arbitrary n≥ 0 and any finite p≥ 0, without recursion over lower-order relations. For k-inflation, they additionally depend on the scalar sound speed and its scale derivatives. We develop an algorithm that generates both the spectra and consistency relations for any pair (n,p). We reproduce known results through next-to-next-to-next-to-leading order (N3LO) and extend the explicit calculations up to N10LO.
ER -