Abstract

We use the canonical formalism of classical mechanics together with the first law of thermodynamics to unveil a differential equation for a minimally coupled canonical scalar field which should be solved simultaneously with the Klein–Gordon equation. This equation was obtained by realizing that the thermodynamic work is equal to the work done by the generalized force to increase the scalar field by an amount \(d\phi\). Then, using the first law of thermodynamics, we obtain the equation \({\dot{\phi}}(dV/d\phi)={-}3H[{\dot{\phi}}^{2}/2-V(\phi)]\) which, together with the Klein–Gordon equation, rules the dynamics of this minimally coupled scalar field. We demonstrate precisely that the potential \(V\), that satisfies both equations simultaneously, evolves with the scale factor as \(V\propto a^{-3\sqrt{2}}\) and dominated the early stages of the Universe expansion history. We carry out an observational analysis together with some of the most updated data currently available, and we prove that the energy density of this field is comparable to the radiation density energy at redshifts of order \(10^{8}\), which is nearly 100 times smaller than the radiation energy density today. Although negligible today, we have shown that this scalar field can alleviate enormously the so-called \(H_{0}\) tension.

中文翻译：

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热力学第一定律的标量场新方程及其宇宙学意义

摘要

我们使用经典力学的规范形式学和热力学第一定律来揭示最小耦合规范标量场的微分方程，该方程必须与Klein-Gordon方程同时求解。通过认识到热力学功等于广义力使标量场增加量\（d \ phi \）所做的功，可以得出该方程。然后，使用热力学第一定律，获得方程\（{\ dot {\ phi}}（dV / d \ phi）= {-} 3H [{\ dot {\ phi}} ^ {2} / 2 -V（\ phi）] \），与Klein-Gordon方程一起，控制该最小耦合标量场的动力学。我们精确地证明了潜在的\（V \），同时满足两个方程，并随着比例因子\（V \ proto a ^ {-3 \ sqrt {2}} \）演化，并主导了宇宙膨胀历史的早期阶段。我们与目前可用的一些最新数据一起进行了观测分析，我们证明了该场的能量密度与阶数\（10 ^ {8} \）的红移下的辐射密度能量相当，这是比当今的辐射能量密度小近100倍。尽管今天可​​以忽略不计，但我们已经表明，该标量场可以极大地缓解所谓的\（H_ {0} \）张力。