Analogies between Logistic Equation and Relativistic Cosmology
Abstract
:1. Introduction
2. Basics of FLRW Cosmology
2.1. Einstein–Friedmann Equations
- Differentiating the Friedmann Equation (4) with respect to time yields
- 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)In fact, a perfect fluid stress–energy tensor(where is the fluid four-velocity normalized to ) has trace and the contraction of the Einstein Equations (8) then gives
- 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
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
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 StyleDussault, 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
APA StyleDussault, S., Faraoni, V., & Giusti, A. (2021). Analogies between Logistic Equation and Relativistic Cosmology. Symmetry, 13(4), 704. https://doi.org/10.3390/sym13040704