Brans-Dicke理论中的关系框架等价与精确宇宙学

中文摘要 AI 辅助

我们将参考文献[Tavakoli:2026coc]中发展的正则时钟扇区框架应用于Brans-Dicke理论中的空间平坦FLRW宇宙学。目标是获得框架适应关系动力学的显式宇宙学实现，并确定相应的闭式宇宙学可观测量。从先前发展的完整ADM理论出发，将其限制到均匀扇区，我们得到正则时钟动量为$P_{\Phi}=p_{\phi}-ap_a/2\phi$。这一识别使得Jordan和Einstein框架约化系统能够在共同的正则相空间上表示。然后我们考虑一个二次Jordan框架势，其中Einstein框架势为常数。由此产生的关系FLRW系统是精确可积的。完整轨迹可以用单一参数$u$表达，得到尺度因子、宇宙时Hubble率、Hubble慢滚参数、e-fold数以及标量状态方程参数的闭式表达式。特别地，精确解提供了从$u=\operatorname{arctanh}(1/\sqrt{3})$开始的显式暴胀阶段，并过渡到渐近de Sitter阶段。最后，我们将精确解与适应Jordan和Einstein描述中的直接数值积分，以及基于未移位标量动量的朴素约化Jordan系统进行比较。数值结果再现了精确轨迹，并说明了不同约化正则表示如何通过正则时钟扇区构造所隐含的时钟依赖变换相关联。该分析为标量-张量宇宙学中的关系框架等价提供了一个精确可解的实验室，并给出了可用于扩展到更一般势和标量-张量模型的显式宇宙学可观测量。

英文摘要

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 $u=\operatorname{arctanh}(1/\sqrt{3})$ 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.

发表机构

• University of Gdansk（格但斯克大学）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

31 pages, 7 figures