Friedmann Equations
What It Means
The Friedmann Equations describe how the universe expands or contracts over time given its contents: matter, radiation, and dark energy. These equations emerge from Einstein's General Relativity when applied to a homogeneous and isotropic universe—one that looks approximately the same in all directions and at all locations on large scales. The left side of the equation (ȧ/a)² represents the square of the expansion rate: a is the scale factor describing the size of the universe, and ȧ is how fast it's changing. The right side contains three terms: the matter and radiation density (ρ), the spatial curvature (k), and the cosmological constant (Λ representing dark energy).
The beauty of the Friedmann equations is that they show how the composition of the universe determines its expansion history. The matter and radiation term (8πGρ/3) acts to slow expansion through gravity—the attractive pull of matter decelerates cosmic expansion. The cosmological constant term (Λ/3) accelerates expansion, representing the repulsive effect of dark energy. The curvature term (−k/a²) depends on the overall geometry of space: whether the universe is flat, closed (spherical), or open (hyperbolic). Modern observations show the universe is spatially flat and dominated by dark energy, meaning it will expand forever with accelerating speed.
From these equations, cosmologists can trace the entire history of the universe. In the distant past, when the universe was small and dense, radiation dominated, and the universe expanded very rapidly. As it cooled and expanded, matter became dominant, and expansion slowed. Eventually, as the universe continued expanding and matter dispersed, dark energy came to dominate the expansion, and the expansion began accelerating again. These equations predict the Big Bang (where a=0 in the infinite past), the expansion Hubble observed, and the current acceleration observed in distant supernovae. Understanding the Friedmann equations is essential to understanding cosmology.
The Variables
| Symbol | Meaning | Unit |
|---|---|---|
| a(t) | Scale factor (relative size of the universe) | Dimensionless |
| ȧ | Time derivative of scale factor (expansion rate) | 1/second |
| ρ | Total density of matter and radiation | Kilograms per cubic meter (kg/m³) |
| G | Gravitational constant | 6.674 × 10⁻¹¹ N·m²/kg² |
| k | Spatial curvature parameter | Dimensionless (0=flat, +1=closed, −1=open) |
| Λ | Cosmological constant (dark energy) | 1/meter² |
Historical Context
Alexander Friedmann, a Russian mathematician and physicist, derived his equations in 1922 as solutions to Einstein's field equations applied to a homogeneous, isotropic universe. Remarkably, Friedmann discovered this almost immediately after Einstein published General Relativity, demonstrating mathematical insight and physical intuition. Friedmann's work showed that a dynamic, expanding or contracting universe was mathematically consistent with General Relativity—a conclusion that surprised many scientists, including Einstein himself. Initially, Einstein dismissed Friedmann's work, but eventually acknowledged its correctness.
The Friedmann equations remained somewhat abstract theoretical constructs until Hubble's 1929 observational discovery that the universe is indeed expanding. This observation provided direct confirmation of Friedmann's theoretical prediction. Friedmann himself died in 1925 and never saw his equations vindicated observationally. In the following decades, as astronomical observations improved, the Friedmann framework became the standard tool of cosmology. The discovery in 1998 that the universe's expansion is accelerating—later attributed to dark energy—was unexpected but explained naturally by the Friedmann equations with a non-zero cosmological constant. These equations remain the foundation of modern cosmological models and our understanding of the universe's past, present, and future.
Why It Matters
The Friedmann Equations are the master equations of cosmology, describing the evolution of the entire universe from the Big Bang to the infinite future. They enable calculation of the universe's age, the density of matter required for gravitational bound structures, and the ultimate fate of cosmic expansion. When combined with observations of cosmic microwave background radiation, the distribution of galaxies, and supernova distances, the Friedmann equations allow determination of the universe's composition: 5% ordinary matter, 27% dark matter, and 68% dark energy. These equations unite General Relativity with observational cosmology and provide the mathematical framework for understanding the cosmos at the largest scales.
Applications
- Age of the Universe: Solving the Friedmann equations and measuring the expansion rate allows calculation of the universe's age, determined to be approximately 13.8 billion years with remarkable precision.
- Composition of the Universe: Comparing Friedmann equation predictions with observations of cosmic microwave background radiation, supernovae, and large-scale structure determines the universe's composition of dark energy, dark matter, and ordinary matter.
- Inflation Theory: During the first fraction of a second after the Big Bang, Friedmann equations with specific matter properties predict a period of exponential expansion (inflation) that explains the universe's uniformity and large-scale structure.
- Big Bang Nucleosynthesis: The Friedmann equations, combined with nuclear physics, predict the abundance of light elements (hydrogen, helium, lithium) created in the early universe, matching observed cosmic abundances.
- Dark Energy and Fate of the Universe: Friedmann equations show that if dark energy dominates (as observations indicate), the universe will expand forever at accelerating speeds, eventually reaching a cold, empty state billions of years in the future.