Friedmann equations

Friedmann equations. They are a set of dynamical equations central to the standard model of cosmology, describing the expansion of the universe within the framework of Albert Einstein's general relativity. Derived by applying the Einstein field equations to the Friedmann–Lemaître–Robertson–Walker metric, they govern the evolution of the scale factor in time, connecting it to the energy density and pressure of the universe's contents. Their solutions form the basis for all modern cosmological models, including the prevailing Lambda-CDM model, and provide the mathematical description of phenomena like the Big Bang and cosmic inflation.

Derivation from general relativity

The derivation begins with the Einstein field equations, which relate the Einstein tensor to the stress–energy tensor via the cosmological constant and gravitational constant. Assuming the cosmological principle, which posits homogeneity and isotropy on large scales, the geometry of the universe is described by the Friedmann–Lemaître–Robertson–Walker metric. Substituting this metric into the field equations, and modeling the cosmic fluid as a perfect fluid with a given equation of state, yields the two independent equations. Key steps involve calculating the Ricci curvature for the FLRW metric and applying the conservation of energy expressed through the continuity equation for the stress–energy tensor. This process was first rigorously completed by Alexander Friedmann following the publication of general relativity by Albert Einstein.

Solutions and cosmological models

Solutions to these equations depend critically on the values of cosmological parameters like the density parameter, curvature, and the cosmological constant. For a universe with only non-relativistic matter, known as the Einstein–de Sitter universe, the scale factor grows with a specific power law. Introducing a positive cosmological constant leads to solutions describing an accelerating expansion, as in the de Sitter universe or the modern Lambda-CDM model. Different curvature scenarios—flat, closed, or open—yield distinct evolutionary fates, from eternal expansion to eventual Big Crunch. Other important solutions include the Milne model and those incorporating radiation or dark energy with a phantom equation of state.

Physical interpretation

The first equation, often called the Friedmann equation, is essentially an energy equation relating the expansion rate (the Hubble parameter) to the total energy density and spatial curvature of the universe. It shows that the expansion dynamics are dictated by the competition between the attractive force of gravity from matter and the repulsive effect of dark energy or a cosmological constant. The second equation, an acceleration equation, describes how the scale factor's second derivative is driven by the combined pressure and energy density of all cosmic components. This directly leads to the prediction of cosmic acceleration when the universe is dominated by a component with strong negative pressure, such as in the quintessence or Lambda-CDM model.

Relation to cosmological parameters

These equations define the fundamental relationships between directly measurable cosmological parameters. The Hubble constant appears as the present-day value of the Hubble parameter in the first equation. The critical density is derived by setting the curvature term to zero. The density parameter for any component (Ω for matter, radiation, or dark energy) is its density relative to this critical density. Observations of the cosmic microwave background by missions like Planck (spacecraft) and Wilkinson Microwave Anisotropy Probe, combined with supernova data from the Supernova Legacy Survey, are used to constrain these parameters within the Lambda-CDM model framework, determining the composition and ultimate fate of the universe.

Historical context and development

The equations were first derived by Alexander Friedmann in 1922 from Einstein field equations, at a time when the static universe model was prevalent. Albert Einstein initially resisted this dynamic solution, having introduced the cosmological constant to achieve a static model. Following the observational work of Edwin Hubble on the recession of galaxies, which suggested an expanding universe, the theoretical importance of Friedmann's work was recognized. Key developments included the independent derivation by Georges Lemaître, who connected expansion to a primordial "atom", and the later contributions of Howard P. Robertson and Arthur Geoffrey Walker in rigorously establishing the metric. The modern era was catalyzed by the discovery of the cosmic microwave background by Arno Penzias and Robert Wilson, and later the observation of accelerating expansion by teams like the Supernova Cosmology Project and the High-Z Supernova Search Team.

Category:Physical cosmology Category:Equations Category:General relativity