This document describes the cosmological constants and equations implemented in the MPP system.
| Constant | Symbol | Description | Dimensions |
|---|---|---|---|
| Cosmological Constant | Λ (Lambda) | Driving accelerated expansion | [T⁻²] |
| Hubble Constant | H | Rate of universe expansion | [T⁻¹] |
| Critical Density | ρₖ | Density threshold for universe geometry | [M/L³] |
| Density Parameter | Ω | Ratio of actual to critical density | Dimensionless |
| Dark Energy Density | Ωᴧ | Dark energy component | Dimensionless |
| Matter Density | Ωₘ | Total matter component | Dimensionless |
| Dark Matter Density | Ωₘ | Dark matter component | Dimensionless |
| Baryonic Matter Density | Ωᵦ | Regular matter component | Dimensionless |
| Radiation Density | Ωᵣ | Radiation component | Dimensionless |
| Curvature Density | Ωₖ | Spatial curvature component | Dimensionless |
| Scale Factor | a(t) | Universe expansion factor | Dimensionless |
| Redshift | z | Cosmological redshift | Dimensionless |
The dynamics of an expanding universe in general relativity are described by the Friedmann equations.
The first Friedmann equation relates the expansion rate to the energy density:
\[H^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} + \frac{\Lambda}{3}\]Where:
The second Friedmann equation describes the acceleration of the expansion:
\[\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda}{3}\]Where p is the pressure.
The Lambda Cold Dark Matter (ΛCDM) model is the standard model of cosmology, including:
The Hubble parameter evolves with redshift according to:
\[H(z) = H_0\sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}\]Different cosmological components have different equations of state:
Where the equation of state parameter w relates pressure to density:
\[p = w\rho c^2\]The MPP system implements these concepts with full symbolic closure and dimensional consistency. All cosmological equations are represented as symbolic expressions that can be manipulated, simplified, and analyzed.
The implementation includes:
All quantities maintain proper dimensional analysis throughout calculations.