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

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.