The Friedmann equations are the set of differential equations governing the expansion dynamics of a homogeneous and isotropic universe.Imagine that the Universe is a giant balloon with galaxies living on its surface. The Friedmann equations are instructions for how this balloon inflates or deflates over time. They say: 'If you know how much matter and energy is inside, you can predict how fast the balloon expands, and even find out whether it will expand forever or collapse back.' These equations are the foundation of modern ideas about the Big Bang.
Alexander Friedmann in 1922 first found non-stationary solutions of general relativity, predicting the expansion of the Universe. Einstein initially rejected them but later acknowledged their correctness. Georges Lemaître independently derived the same equations in 1927 and was the first to link them with astronomical observations, pointing out a linear relationship between speed and distance. In 2011, Saul Perlmutter, Brian Schmidt, and Adam Riess proved accelerated expansion, confirming the role of the Λ term.
The Friedmann equations serve as the basis for the standard ΛCDM cosmological model. They allow, knowing the current parameters (Hubble constant, fractions of matter, radiation, and dark energy), to reconstruct the history of the Universe from the Big Bang to the future. The limit of applicability is scales greater than ~100 Mpc, where the Universe is indeed homogeneous; on smaller scales, the structure is inhomogeneous, and perturbations must be taken into account.
Alexander Friedmann himself was a mathematician and meteorologist. He loved solving complex equations and even participated in launching stratospheric balloons. His boldness in applying general relativity to the entire Universe initially displeased Einstein, who at the time believed in an eternal and unchanging Universe.
| H | variable | Hubble parameter — hubble parameter per second |
| z | variable | redshift — redshift |
| H0 | variable | Hubble constant — hubble constant per second |
| Ωm | variable | matter density parameter — density parameter |
| Ωr | variable | radiation density parameter — density parameter |
| Ωk | variable | curvature density parameter — density parameter |
| ΩΛ | variable | cosmological constant density parameter — density parameter |
| Ωx | variable | dark energy density parameter (general) — density parameter |
| f(z) | variable | dark energy equation-of-state function — equation of state function |
| sqrt | operator | square root — square root |
| Ω_M | variable | Matter density parameter (including baryonic and dark matter) — density parameter |
| Ω_Λ | variable | Cosmological constant (dark energy) density parameter — density parameter |
| Ω_k | variable | Curvature density parameter — density parameter |
| ρ | variable | density matrix of the open quantum system — density matrix |
| t | variable | time — time second |
| H | variable | Hamiltonian of the system — hamiltonian joule |
| L_n | variable | Lindblad operators (jump operators) for each dissipation channel — lindblad operator |
| i | constant | imaginary unit, square root of -1 — imaginary unit = 1 |
| ħ | constant | Planck constant divided by 2π — reduced planck constant = 1.054571817e-34 joule second |
| d/dt | operator | derivative with respect to time — time derivative |
| [H, ρ] | operator | commutator of H and ρ — commutator |
| {L_n† L_n, ρ} | operator | anticommutator of L_n† L_n and ρ — anticommutator |
| ∑_n | operator | sum over all dissipation channels — sum over channels |
