Review

, Volume: 12( 6) DOI: 10.37532/2320-6756.2023.12(6).278

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Dimensional complement of the mathematical solution to the cosmological constant problem.

*Correspondence:

Stéphane Wojnow Independent Researcher, France,E-mail: wojnow.stephane@gmail.com

Abstract

We have proposed a mathematical solution to the cosmological constant problem with an attempted physical explanation. Here we propose a complement to this solution to validate the hypothetical energy density value of the cosmological constant in Quantum Field Theory (QFT), showing that the dimensional method used can be applied to find the critical energy density of the ΛCDM mode

Keywords

Cosmology; Quantum field theory; Cosmological constant problem; Vacuum catastrophe; Cosmological constant; Zero point energy; Critical energy density; ΛCDM model

Introduction

This document presents a complement to the proposed mathematical solution to the cosmological constant problem, with the aim of validating the hypothetical energy density value of the cosmological constant in quantum field theory. The proposed method utilizes dimensional analysis to find the critical energy density of the ΛCDM model. The document provides a reminder of the mathematical solution and defines relevant parameters, including Planck mass, Planck length, and Hubble constant. It then presents the proposed formula for the quantum critical energy density of the universe and demonstrates how it can be used to calculate the critical energy density of the ΛCDM model.

Reminder of the result of the mathematical solution to the problem of the cosmological constant

Here we define parameters with m_p as Planck mass, l_p as Planck length, ℏ reduced Planck constant, c speed of light in vacuum, G as Newton's constant, Λ as cosmological constant, A as zero-point energy density in quantum field theory [1], B as vacuum energy density assumed for the cosmological constant in the QFT, H₀ as Hubble contant, and ρ_c as critical energy density of the ΛCDM model.

The energy density of the quantum vacuum in Planck units, i.e. that of the zero point of the QFT is :

By dimensional analysis, we can propose this hypothetical quantum energy density of the cosmological constant in the QFT [2] :

To demonstrate that the cosmological constant C in J/m³ is [2] :

Dimensional complement of the mathematical solution to the cosmological constant problem

Let us consider H₀ the Hubble parameter (or Hubble constant) of dimension (T^-1). We want a dimension in (L^-2) to replace in m Eq(3),

As c² is used to convert to s by writing [2],

we will write

To write a formula B' as "quantum critical energy density of the universe for H₀" assumed in the QFT with Eq(7) of dimension (L^-2) :

Finally, consider the critical energy density of the ΛCDM model for H₀ :

We have:

This can be proved using Eq (2) and Eq (9) :

Eq (15) is the definition of the critical energy density of the ΛCDM model for a flat universe, i.e. Eq (10).

Conclusion

The same dimensional methodology, to assume on the one hand the hypothetical quantum energy density of the cosmological constant QFT, on the other hand the hypothetical quantum critical energy density of the QFT, allows to find their equations in the ΛCDM model via their geometric mean with the zero-point energy density. In addition to attempting to make physical sense of the square roots of the energy density as a Hildebrand solubility parameter, the reproducibility of the method reciprocally strengthens both results obtained. This result coud open a new approch of the cosmology.

References

1. Lombriser LJPLB., "On the cosmological constant problem," 2019;797:134804. [Google Scholar] [Crossref]
2. Wojnow S. A simple mathematical solution to the cosmological constant problem.: Cosmology. Hyperscience Int. J. 2022;2(3):57-9.[Google Scholar] [Crossref]