Relational Frame Equivalence and Exact Cosmology in Brans--Dicke Theory
Abstract
We apply the canonical clock-sector framework developed in Ref.~\cite{Tavakoli:2026coc} to spatially flat FLRW cosmology in Brans--Dicke theory. The aim is to obtain an explicit cosmological realization of the frame-adapted relational dynamics and to determine the corresponding cosmological observables in closed form. Starting from the previously developed full ADM theory and restricting it to the homogeneous sector, we obtain the canonical clock momentum as PΦ=pϕ−apa/2ϕ. This identification allows the Jordan and Einstein-frame reduced systems to be represented on a common canonical phase space. We then consider a quadratic Jordan-frame potential for which the Einstein-frame potential is constant. The resulting relational FLRW system is exactly integrable. The complete trajectory can be expressed in terms of a single parameter u, yielding closed-form expressions for the scale factor, the cosmic-time Hubble rate, the Hubble slow-roll parameter, the number of e-folds, and the scalar equation-of-state parameter. In particular, the exact solution provides an explicit inflationary regime that begins at u=arctanh(1/√3) and transitions toward an asymptotic de Sitter regime. We finally compare the exact solution with direct numerical integration in the adapted Jordan and Einstein descriptions, and with the naively reduced Jordan system based on the unshifted scalar momentum. The numerical results reproduce the exact trajectory and illustrate how the different reduced canonical representations are related by the clock-dependent transformation implied by the canonical clock-sector construction. The analysis provides an exactly solvable laboratory for relational frame equivalence in scalar--tensor cosmology and gives explicit cosmological observables that can be used in extensions to more general potentials and scalar--tensor models.
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BibTeX
@misc{tavakoli2026relational,
title = {{Relational Frame Equivalence and Exact Cosmology in Brans--Dicke Theory}},
author = {Yaser Tavakoli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32047},
eprint = {2609.32047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32047},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Relational Frame Equivalence and Exact Cosmology in Brans--Dicke Theory
AU - Yaser Tavakoli
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32047
UR - https://arxiv.org/abs/2609.32047
AB - We apply the canonical clock-sector framework developed in Ref.~\cite{Tavakoli:2026coc} to spatially flat FLRW cosmology in Brans--Dicke theory. The aim is to obtain an explicit cosmological realization of the frame-adapted relational dynamics and to determine the corresponding cosmological observables in closed form. Starting from the previously developed full ADM theory and restricting it to the homogeneous sector, we obtain the canonical clock momentum as P_Φ=p_ϕ−ap_a/2ϕ. This identification allows the Jordan and Einstein-frame reduced systems to be represented on a common canonical phase space. We then consider a quadratic Jordan-frame potential for which the Einstein-frame potential is constant. The resulting relational FLRW system is exactly integrable. The complete trajectory can be expressed in terms of a single parameter u, yielding closed-form expressions for the scale factor, the cosmic-time Hubble rate, the Hubble slow-roll parameter, the number of e-folds, and the scalar equation-of-state parameter. In particular, the exact solution provides an explicit inflationary regime that begins at u=arctanh(1/√3) and transitions toward an asymptotic de Sitter regime. We finally compare the exact solution with direct numerical integration in the adapted Jordan and Einstein descriptions, and with the naively reduced Jordan system based on the unshifted scalar momentum. The numerical results reproduce the exact trajectory and illustrate how the different reduced canonical representations are related by the clock-dependent transformation implied by the canonical clock-sector construction. The analysis provides an exactly solvable laboratory for relational frame equivalence in scalar--tensor cosmology and gives explicit cosmological observables that can be used in extensions to more general potentials and scalar--tensor models.
ER -