# Analogies between Logistic Equation and Relativistic Cosmology

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## Abstract

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## 1. Introduction

## 2. Basics of FLRW Cosmology

#### 2.1. Einstein–Friedmann Equations

- Differentiating the Friedmann Equation (4) with respect to time yields$$2H\left(\frac{\ddot{a}}{a}-{H}^{2}\right)=\frac{8\pi G}{3}\phantom{\rule{0.166667em}{0ex}}\dot{\rho}+\frac{2KH}{{a}^{2}}\phantom{\rule{0.166667em}{0ex}};$$$${R}_{ab}-\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{g}_{ab}R+\Lambda {g}_{ab}=8\pi G{T}_{ab}$$
- The Friedmann Equation (4) follows from the acceleration Equation (5), the Einstein Equation (8), and the expression of the Ricci scalar in the FLRW geometry (3)$$R=6\left(\dot{H}+2{H}^{2}+\frac{K}{{a}^{2}}\right)\phantom{\rule{0.166667em}{0ex}}.$$In fact, a perfect fluid stress–energy tensor$${T}_{ab}=\left(P+\rho \right){u}_{a}{u}_{b}+P{g}_{ab}$$(where ${u}^{c}$ is the fluid four-velocity normalized to ${u}^{c}{u}_{c}=-1$) has trace $T=-\rho +3P$ and the contraction of the Einstein Equations (8) then gives$$R=8\pi G\left(\rho -3P\right)+4\Lambda \phantom{\rule{0.166667em}{0ex}}.$$
- The acceleration Equation (5) can be derived from the Friedmann Equation (4) and the energy conservation Equation (6). In fact, differentiating (4) with respect to time leads to$$2H\left(\frac{\ddot{a}}{a}-{H}^{2}\right)=\frac{8\pi G}{3}\phantom{\rule{0.166667em}{0ex}}\dot{\rho}+\frac{2KH}{{a}^{2}}\phantom{\rule{0.166667em}{0ex}};$$

#### 2.2. FLRW Lagrangian and Hamiltonian

#### 2.3. Symmetries of the Einstein–Friedmann Equations for Spatially Flat Universes

## 3. Cosmological Analogies

#### 3.1. First Analogy Using Comoving Time

#### 3.2. Second Analogy with Comoving Time

#### 3.3. First Analogy with Conformal Time

#### 3.4. Second Analogy with Conformal Time

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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Dussault, S.; Faraoni, V.; Giusti, A.
Analogies between Logistic Equation and Relativistic Cosmology. *Symmetry* **2021**, *13*, 704.
https://doi.org/10.3390/sym13040704

**AMA Style**

Dussault S, Faraoni V, Giusti A.
Analogies between Logistic Equation and Relativistic Cosmology. *Symmetry*. 2021; 13(4):704.
https://doi.org/10.3390/sym13040704

**Chicago/Turabian Style**

Dussault, Steve, Valerio Faraoni, and Andrea Giusti.
2021. "Analogies between Logistic Equation and Relativistic Cosmology" *Symmetry* 13, no. 4: 704.
https://doi.org/10.3390/sym13040704