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Simple $φΛ$ CDM dynamics

Pierros Ntelis, Jackson Levi Said

Abstract

In this paper, we introduce the $φΛ$CDM model and compare it to the concordance model. We present their respective epoch evolutions, perform a detailed dynamical analysis for each model, and conduct a comparative analysis between the two. This study revitalizes these models by considering systems with a higher number of variables. Additionally, the $φΛ$CDM model we present is both more comprehensive and simpler than those found in the literature, as it accounts for all known epochs, including the radiation, matter, and dark energy phases. Notably, existing studies on this model often omit the radiation epoch and focus on simpler dynamics. We find that both models, the $φΛ$CDM and the $Λ$CDM, can describe the generally accepted scenario of cosmic evolution, and current observations. Both models, describe qualitative and quantitative current observations about the epoch behaviour of the species of the universe. We find the $φΛ$CDM model has the following exotic transition from a \textit{dominant radiation energy density ratio} epoch, or a \textit{low scalar kinetic term energy density ratio} epoch, towards a \textit{dominant cosmological constant energy density ratio} epoch. This renders the $φΛ$CDM a richer phenomenologically model than the concordance cosmological model. The software of the study is publicly available online.

Simple $φΛ$ CDM dynamics

Abstract

In this paper, we introduce the CDM model and compare it to the concordance model. We present their respective epoch evolutions, perform a detailed dynamical analysis for each model, and conduct a comparative analysis between the two. This study revitalizes these models by considering systems with a higher number of variables. Additionally, the CDM model we present is both more comprehensive and simpler than those found in the literature, as it accounts for all known epochs, including the radiation, matter, and dark energy phases. Notably, existing studies on this model often omit the radiation epoch and focus on simpler dynamics. We find that both models, the CDM and the CDM, can describe the generally accepted scenario of cosmic evolution, and current observations. Both models, describe qualitative and quantitative current observations about the epoch behaviour of the species of the universe. We find the CDM model has the following exotic transition from a \textit{dominant radiation energy density ratio} epoch, or a \textit{low scalar kinetic term energy density ratio} epoch, towards a \textit{dominant cosmological constant energy density ratio} epoch. This renders the CDM a richer phenomenologically model than the concordance cosmological model. The software of the study is publicly available online.

Paper Structure

This paper contains 19 sections, 14 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overall methodology and physical preliminaries
  3. Dynamical analysis on $\phi\Lambda$CDM
  4. Dimensionless variables and the representative 3D system
  5. Critical points characterisation summary
  6. Phase portraits of the projections of 3D system
  7. Numerical solution of $\Lambda$CDM and $\phi\Lambda$CDM model comparison
  8. Phase portraits of $\Lambda$CDM and $\phi\Lambda$CDM model comparison
  9. Conclusions and discussion
  10. Triples in different epochs
  11. Dynamical system analysis on $\Lambda$CDM
  12. The $\Lambda$CDM model
  13. Defining the dimensionless variables for $\Lambda$CDM
  14. $\Lambda$CDM model: system in 2D, 3D projected to 2D and 3D approaches
  15. Dynamical analysis of the 3D set of differential equations of $\Lambda$CDM model
  16. ...and 4 more sections

Figures (4)

  • Figure 1: We illustrate the numerical solutions of the model of $\phi\Lambda$CDM cosmology, in the case which $\texttt{var-zero-is}=3 \times 10^{-17}$, and we provide more details. [See section ]
  • Figure 2: We illustrate the phase portrait of the 3D model. Left Panel: We illustrate the phase portrait of the $(m,x)$. We cannot observe the $(m,x)=(0,0)$ as a unstable initial point, but we can observe it as a saddle point. This is due to that this phase portrait is a projection of the 3D case, where both points appear. Center Panel: We illustrate the phase portrait of the $(m,v)$ Right Panel: We illustrate the phase portrait of the $(x,v)$. [See section ].
  • Figure 3: Comparison of numerical solutions of the energy density ratio epochs between $\Lambda$CDM and $\phi\Lambda$CDM. [See section ]
  • Figure 4: We illustrate the phase portrait of $\Lambda$CDM cosmology. [See appendix ]