What is the critical density?

The critical density is the energy density for which the universe is spatially flat; a density above it implies positive curvature and a density below it negative curvature, so comparing the actual density with the critical value determines the geometry of space.

Do the Friedmann equations predict a Big Bang?

Run backward, the equations for a universe dominated by matter and radiation reach a moment of zero scale factor and infinite density, a singular origin; this Big Bang is where classical general relativity breaks down and new physics is expected to be needed.

Friedmann Equations and Expanding Universe

The Friedmann equations are the cosmological form of the Einstein equations, linking the expansion rate of the universe to its energy density, pressure, and spatial curvature.

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Definition

The Friedmann equations are two ordinary differential equations for the cosmic scale factor, derived from the Einstein equations under the assumption of homogeneity and isotropy, that determine how the size of the universe evolves given its energy content and curvature.

Scope

This topic covers the derivation of the Friedmann and acceleration equations from the Robertson-Walker metric, the scale factor and the Hubble parameter, the critical density and the density parameters of matter, radiation, curvature, and dark energy, the different expansion histories, radiation-, matter-, and dark-energy-dominated, and the conditions for accelerated expansion and a Big Bang origin.

Core questions

How do the Einstein equations reduce to equations for a single scale factor?
What determines whether the universe expands forever, recollapses, or accelerates?
How do matter, radiation, and dark energy each drive the expansion differently?

Key concepts

Scale factor
Hubble parameter
Critical density
Density parameters
Deceleration and acceleration
Big Bang singularity

Key theories

Friedmann equation: The square of the Hubble expansion rate equals a sum of the energy density and a curvature term, so the present densities of matter, radiation, curvature, and dark energy fully determine the expansion rate and geometry of the universe.
Acceleration equation: The second Friedmann equation shows that the expansion decelerates under ordinary matter and radiation but accelerates when a component with sufficiently negative pressure, such as a cosmological constant, dominates the energy budget.

Clinical relevance

The Friedmann equations are the quantitative core of the standard cosmological model, used to fit the measured expansion history, infer the densities of dark matter and dark energy, compute the age of the universe, and trace the thermal history back toward the Big Bang.

History

Friedmann derived the expanding and contracting solutions in 1922, and Lemaitre independently rediscovered them in 1927 while linking them to the observed recession of galaxies; Hubble's 1929 measurement of the redshift-distance relation confirmed cosmic expansion, vindicating the dynamical Friedmann models over Einstein's static universe.

Key figures

Aleksandr Friedmann
Georges Lemaitre
Edwin Hubble

Seminal works

friedmann1922
weinberg2008

Frequently asked questions

What is the critical density?: The critical density is the energy density for which the universe is spatially flat; a density above it implies positive curvature and a density below it negative curvature, so comparing the actual density with the critical value determines the geometry of space.
Do the Friedmann equations predict a Big Bang?: Run backward, the equations for a universe dominated by matter and radiation reach a moment of zero scale factor and infinite density, a singular origin; this Big Bang is where classical general relativity breaks down and new physics is expected to be needed.