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A New Equation for a Scalar Field from Thermodynamics First Law and Its Cosmological Implications

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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.

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Notes

  1. The convention used here is that Greek indices runs from 0 to 3 and Latin indices run from 1 to 3.

  2. The use of these cosmological probes in a joint analysis is useful to brake the degenerescence between the model parameters.

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Funding

E.M.C.A. is the corresponding author. This research was financially supported in part by the Coordenaзão de Aperfeiзoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. E.M.C.A. and J.A.N. thank CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, for partial financial support, Grants number 406894/2018-3 (E.M.C.A.) and 303140/2017-8 (J.A.N.). CAPES and CNPq are Brazilian scientific support federal agencies.

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Authors and Affiliations

  1. Grupo de Física Teórica e Física Matemática, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971, Seropédica, RJ, Brazil

    Everton M. C. Abreu

  2. Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil

    Everton M. C. Abreu & Jorge Ananias Neto

  3. Programa de Pós-Graduaзão Interdisciplinar em Física Aplicada, Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil

    Everton M. C. Abreu

  4. Departamento de Física, Universidade do Estado do Rio Grande do Norte, 59610-210, Mossoró, RN, Brazil

    Edésio M. Barboza Jr.

Authors
  1. Everton M. C. Abreu
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  2. Edésio M. Barboza Jr.
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  3. Jorge Ananias Neto
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Correspondence to Everton M. C. Abreu, Edésio M. Barboza Jr. or Jorge Ananias Neto.

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Abreu, E.M., Barboza Jr., E.M. & Neto, J.A. A New Equation for a Scalar Field from Thermodynamics First Law and Its Cosmological Implications. Gravit. Cosmol. 27, 1–10 (2021). https://doi.org/10.1134/S0202289321010023

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  • DOI: https://doi.org/10.1134/S0202289321010023

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