Friedmann Equations - The Equations

The Equations

There are two independent Friedmann equations for modeling a homogeneous, isotropic universe. They are:

which is derived from the 00 component of Einstein's field equations, and

which is derived from the trace of Einstein's field equations. is the Hubble parameter, G, Λ, and c are universal constants (G is Newton's gravitational constant, Λ is the cosmological constant, c is the speed of light in vacuum). k is constant throughout a particular solution, but may vary from one solution to another. a, H, ρ, and p are functions of time. is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since in the Friedmann model. There are two commonly used choices for a and k which describe the same physics:

• k = +1, 0 or −1 depending on whether the shape of the universe is a closed 3-sphere, flat (i.e. Euclidean space) or an open 3-hyperboloid, respectively. If k = +1, then is the radius of curvature of the universe. If k = 0, then a may be fixed to any arbitrary positive number at one particular time. If k = −1, then (loosely speaking) one can say that i·a is the radius of curvature of the universe.
• a is the scale factor which is taken to be 1 at the present time. is the spatial curvature when (i.e. today). If the shape of the universe is hyperspherical and is the radius of curvature ( in the present-day), then . If is positive, then the universe is hyperspherical. If is zero, then the universe is flat. If is negative, then the universe is hyperbolic.

Using the first equation, the second equation can be re-expressed as

which eliminates and expresses the conservation of mass-energy.

These equations are sometimes simplified by replacing

to give:

And the simplified form of the second equation is invariant under this transformation.

The Hubble parameter can change over time if other parts of the equation are time dependent (in particular the mass density, the vacuum energy, or the spatial curvature). Evaluating the Hubble parameter at the present time yields Hubble's constant which is the proportionality constant of Hubble's law. Applied to a fluid with a given equation of state, the Friedmann equations yield the time evolution and geometry of the universe as a function of the fluid density.

Some cosmologists call the second of these two equations the Friedmann acceleration equation and reserve the term Friedmann equation for only the first equation.