Friedmann Equations in Cosmology

The Friedmann equations are a pair of differential equations in physical cosmology that describe how the geometry of the universe evolves over cosmic time. Derived from Einstein’s field equations under the assumption of a homogeneous and isotropic spacetime, they connect the universe’s expansion rate to its matter and energy content, spatial curvature, and (in many applications) a cosmological constant. The equations are fundamental to the standard model of cosmology and are used to model eras such as radiation domination, matter domination, and late-time acceleration.

Mathematical formulation

In the Friedmann–Lemaître–Robertson–Walker (FLRW) framework, the spacetime metric is written in terms of a scale factor (a(t)), which encodes how distances between comoving observers change with time. The Friedmann equations provide the dynamics of (a(t)) and are typically written using the Hubble parameter (H=\dot a/a).

The first Friedmann equation relates the expansion rate to the total energy density (\rho) and spatial curvature (k): [ H^2=\left(\frac{\dot a}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}+\frac{\Lambda}{3}. ] Here (G) is Newton’s gravitational constant, (k) is the (normalized) curvature constant associated with the geometry of spatial slices, and (\Lambda) is the cosmological constant. The second Friedmann equation (often expressed via the scale-factor acceleration) can be written as [ \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+3p\right)+\frac{\Lambda}{3}, ] where (p) is pressure. Together, these equations determine whether the expansion accelerates or decelerates given an assumed equation of state, such as those used for radiation, matter, or dark-energy-like components.

Derivation and relation to general relativity

The Friedmann equations are obtained by applying Einstein’s field equations to an FLRW metric, an approach that assumes the universe is homogeneous and isotropic on large scales. This assumption leads to the cosmological principle and reduces the field equations to ordinary differential equations in time for (a(t)). The derivation is typically presented in terms of the Robertson–Walker metric and the stress–energy tensor of cosmic fluids.

In practice, one also uses local energy-momentum conservation, often written as the continuity equation, [ \dot\rho+3H(\rho+p)=0, ] which determines how each component’s density evolves with the scale factor. For example, radiation density scales approximately as (a^{-4}), while nonrelativistic matter scales as (a^{-3}), assumptions that underpin many standard cosmological calculations involving Big Bang cosmology. The equations are therefore closely tied to the general-relativistic description of gravity and to the fluid approximation for cosmic contents.

Cosmological parameters and observational interpretation

A convenient re-expression of the Friedmann equations uses density parameters normalized to the critical density (\rho_\text{c}=3H^2/(8\pi G)). In this form, the curvature contribution is parameterized in terms of (\Omega_k), while matter, radiation, and the cosmological constant are encoded in (\Omega_m), (\Omega_r), and (\Omega_\Lambda). These parameters are used to connect theory with observations, such as expansion history and distance-redshift relationships.

The Friedmann framework underlies the calculation of key observable quantities, including the comoving distance and the age of the universe, which depend on the integral form of (H(z)) as a function of redshift (z). In many modern analyses, the time evolution derived from the Friedmann equations is compared with measurements from cosmic microwave background observations and from large-scale structure. Parameter constraints are often reported using a specific cosmological model, such as Lambda-CDM, where a cosmological constant plus cold dark matter is assumed.

Solutions and cosmic expansion histories

Because the Friedmann equations reduce to a finite set of ordinary differential equations, they admit solutions corresponding to different cosmic contents and curvature choices. For a spatially flat universe ((k=0)), and for a single fluid with equation of state parameter (w=p/\rho), the scale factor often follows a power-law behavior in time. For instance, (w=0) (pressureless matter) yields (a(t)\propto t^{2/3}), while (w=1/3) (radiation) gives (a(t)\propto t^{1/2}), illustrating how the dominant component shapes expansion.

When a cosmological constant dominates ((\Lambda>0) with (w\approx -1)), solutions approach exponential expansion, with (a(t)) growing approximately as (e^{Ht}). This late-time behavior is consistent with observed cosmic acceleration in the standard cosmological picture. More broadly, by combining multiple components—radiation, matter, and dark-energy-like terms—the Friedmann equations generate the full expansion history used to interpret observational evidence for the early universe and its subsequent evolution.

Historical context and development

The equations are named after Alexander Friedmann, who first obtained expanding-universe solutions in the 1920s by applying Einstein’s theory to homogeneous cosmologies. The framework was later refined and popularized within a broader relativistic cosmology program, including contributions from Georges Lemaître and others. Their work helped establish that the universe need not be static and that dynamical solutions naturally arise in general relativity under reasonable symmetry assumptions.

The Friedmann equations also set the stage for later developments in cosmology, including the use of inflation-motivated models and the development of Friedmann–Lemaître–Robertson–Walker cosmological modeling. Even when more complex scenarios are considered—such as modified gravity or additional relativistic components—the Friedmann equations often remain the baseline starting point for describing how expansion proceeds in time.

See also

• Friedmann–Lemaître–Robertson–Walker metric
• Lambda-CDM model
• Cosmic acceleration
• Big Bang cosmology
• Alexander Friedmann

Categories: Cosmological models, General relativity, Physical cosmology, Differential equations, Friedmann equations