the Friedmann equations
/ FREED-mahn /
If the scale factor tells you how big the universe is, the Friedmann equations tell you how fast it grows and whether it is speeding up or slowing down. They are Newton's second law for the whole cosmos: pour in what the universe is made of, matter, radiation, dark energy, and out comes the fate of space itself, whether it expands forever, coasts, or one day recollapses.
They follow from Einstein's field equations under the assumption that the universe is homogeneous and isotropic, the same everywhere and in every direction, which fixes the geometry to the FLRW form. The first Friedmann equation is (a_dot / a)^2 = (8 pi G / 3) rho - k c^2 / a^2 + Lambda c^2 / 3, relating the expansion rate H = a_dot / a to the total energy density rho, the spatial curvature k, and the cosmological constant Lambda. The second, or acceleration, equation is a_ddot / a = -(4 pi G / 3)(rho + 3 p / c^2) + Lambda c^2 / 3, showing that ordinary pressure p and density both decelerate the expansion, while Lambda pushes it to accelerate. Together they govern a(t).
The lesson hidden in the acceleration equation is profound: because gravity in general relativity responds to pressure as well as density, a component with sufficiently negative pressure (p more negative than minus rho c^2 / 3) causes the expansion to accelerate rather than slow. This is exactly how dark energy or a cosmological constant drives the observed cosmic acceleration. The equations assume perfect homogeneity, an idealization that holds only on the largest scales; on the scale of galaxies the real universe is very lumpy.
For a flat, matter-dominated universe (k = 0, Lambda = 0), the first equation gives a(t) proportional to t^(2/3), a steadily decelerating expansion. Switch on a cosmological constant and at late times a(t) instead grows like e^(Ht), an accelerating expansion, which is the regime our universe entered a few billion years ago.
Contents in, expansion history out: the Friedmann equations are cosmology's engine.
The first Friedmann equation is a constraint (first order in a_dot), the second a genuine equation of motion; together with an equation of state relating p to rho they close the system. Their homogeneity assumption is an idealization valid only on scales above a few hundred million light-years.