Thanks! That was way more obvious than I thought. Summer break always breaks me up.vanhees71 said:
The General Friedmann Equation is a set of equations derived from Einstein's field equations of General Relativity. They describe the expansion of the universe in the context of a homogeneous and isotropic cosmological model. The most common form includes terms for the Hubble parameter, the density of different components (matter, radiation, dark energy), and the curvature of the universe.
The primary variables and parameters in the Friedmann Equation include the scale factor \(a(t)\), which describes how distances in the universe expand or contract over time; the Hubble parameter \(H(t)\), which is the rate of expansion of the universe; the density parameters for matter \(\Omega_m\), radiation \(\Omega_r\), and dark energy \(\Omega_\Lambda\); and the curvature parameter \(k\), which can be \(-1\), \(0\), or \(1\) corresponding to open, flat, or closed universes, respectively.
For a flat universe (\(k = 0\)) with only matter and dark energy, the Friedmann Equation simplifies to \(H^2 = H_0^2 \left( \Omega_m a^{-3} + \Omega_\Lambda \right)\). To solve this differential equation, you can separate variables and integrate. This typically involves expressing \(H\) as \(\dot{a}/a\) and integrating with respect to \(a\) to find \(a(t)\), the scale factor as a function of time.
To solve the Friedmann Equation, you need initial conditions such as the value of the scale factor \(a(t)\) at a particular time \(t\). Commonly, the present time \(t_0\) is chosen with \(a(t_0) = 1\). Additionally, the current values of the Hubble parameter \(H_0\) and the density parameters \(\Omega_m\), \(\Omega_r\), and \(\Omega_\Lambda\) are required.
Analytical solutions to the Friedmann Equation exist for some specific cases, such as a universe dominated by a single component (e.g., matter-only or radiation-only). For more complex scenarios involving multiple components (matter, radiation, dark energy) or non-zero curvature, numerical methods are typically employed to solve the equation. These involve discretizing the equation and using computational algorithms to approximate the solution.