What does the critical density mean?

The critical density is the total energy density that makes the universe spatially flat in the Friedmann framework; densities above it imply positive curvature and densities below it imply negative curvature, so comparing the actual density to the critical value sets the geometry of space.

Why does the universe accelerate today?

In the Friedmann equations a component with sufficiently negative pressure, such as a cosmological constant, drives accelerated expansion; once dark energy dominates the energy budget at late times, the second Friedmann equation predicts the observed acceleration.

Friedmann Equations and Cosmological Models

The Friedmann equations govern how the scale factor of a homogeneous universe evolves with time, turning the contents of the cosmos into a prediction for its expansion history.

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Definition

The Friedmann equations are the two relations obtained from Einstein's field equations for an FLRW universe, expressing the square of the expansion rate and the acceleration of the scale factor in terms of the total energy density, pressure, spatial curvature, and cosmological constant.

Scope

This topic covers the derivation of the Friedmann equations from general relativity applied to the FLRW metric, the equation of state and continuity relation for each energy component, the succession of radiation-, matter-, and dark-energy-dominated eras, the density parameters and critical density that determine spatial geometry, and the assembly of these elements into the standard Lambda-CDM model.

Core questions

How does the energy content of the universe determine its expansion history?
Why does the universe pass through radiation-, matter-, and dark-energy-dominated eras?
How do the density parameters fix the spatial geometry of the cosmos?

Key concepts

Scale factor
Critical density
Density parameter
Equation of state
Cosmological constant
Deceleration parameter
Spatial curvature

Key theories

Friedmann equations: Two coupled equations derived from general relativity relate the expansion rate and its acceleration to the density, pressure, curvature, and cosmological constant, fully determining the evolution of the scale factor for a given energy budget.
Equation of state and cosmic eras: Each component scales with the scale factor according to its equation of state, so radiation dominates first, then matter, then the cosmological constant, producing the characteristic sequence of expansion regimes.
Lambda-CDM model: The standard cosmological model combines cold dark matter and a cosmological constant within the Friedmann framework, fitting a wide range of observations with a small set of parameters.

Mechanisms

Substituting the FLRW metric and a perfect-fluid stress-energy tensor into Einstein's equations yields the Friedmann equations; combining them with the continuity equation gives how each component's density dilutes with expansion, and integrating determines the scale factor and hence the full expansion history.

Clinical relevance

The Friedmann equations are the computational core of cosmology: they predict the age of the universe, the expansion history that calibrates distances and look-back times, and the era-by-era behaviour required to model nucleosynthesis, recombination, and the growth of structure.

History

Friedmann obtained expanding and contracting solutions of Einstein's equations in 1922, initially dismissed by Einstein; Lemaitre rediscovered them with physical interpretation, and over the twentieth century the equations were combined with measurements of the matter and dark-energy densities to yield the concordance Lambda-CDM model.

Debates

Naturalness of the cosmological constant: Including a cosmological constant in the Friedmann equations fits the data, but its tiny observed value compared with quantum-field-theory estimates makes its origin one of the deepest open problems in physics.

Key figures

Alexander Friedmann
Georges Lemaitre
Albert Einstein
Willem de Sitter

Seminal works

friedmann1922

Frequently asked questions

What does the critical density mean?: The critical density is the total energy density that makes the universe spatially flat in the Friedmann framework; densities above it imply positive curvature and densities below it imply negative curvature, so comparing the actual density to the critical value sets the geometry of space.
Why does the universe accelerate today?: In the Friedmann equations a component with sufficiently negative pressure, such as a cosmological constant, drives accelerated expansion; once dark energy dominates the energy budget at late times, the second Friedmann equation predicts the observed acceleration.