### 3. FRIEDMANN OSCILLATIONS: THE RISE AND FALL OF DOMINANT COMPONENTS

Friedmann's Equation can be derived from Einstein's 4×4 matrix equation of general relativity (see for example Landau & Lifshitz 1975, Kolb and Turner 1992 or Liddle & Lyth 2000):

 (10)

where Rµ is the Ricci tensor, is the Ricci scalar, gµ is the metric tensor describing the local curvature of space (intervals of spacetime are described by ds2 = gµ dxµ dx), Tµ is the stress-energy tensor and is the cosmological constant. Taking the (µ, ) = (0, 0) terms of Eq. 10 and making the identifications of the metric tensor with the terms in the FRW metric of Eq. 1, yields the Friedmann Equation:

 (11)

where R is the scale factor of the Universe, H = / R is Hubble's constant, is the density of the Universe in relativistic or non-relativistic matter, k is the constant from Eq. 1 and is the cosmological constant. In words: the expansion (H) is controlled by the density (), the geometry (k) and the cosmological constant (). Dividing through by H2 yields

 (12)

where the critical density c = 3 H2 / 8 G. Defining = / c and = / 3H2 and using = + we get,

 (13)

or equivalently,

 (14)

If we are interested in only post-inflationary expansion in the radiation- or matter-dominated epochs we can ignore the term and multiply Eq. 11 by 3 / 8 G to get

 (15)

which can be rearranged to give

 (16)

A more heuristic Newtonian analysis can also be used to derive Eqs. 14 & 16 (e.g. Wright 2003). Consider a spherical shell of radius R expanding at a velocity v = HR, in a universe of density . Energy conservation requires,

 (17)

By setting the total energy equal to zero we obtain a critical density at which v = HR is the escape velocity,

 (18)

However, by requiring only energy conservation (2E = constant not necessarily E = 0) in Eq. 17, we find,

 (19)

Dividing Eq. 19 by H2 R2 we get

 (20)

which is the same as Eq. 14. Multiplying Eq. 19 by 3 / (8 G R2) we get

 (21)

which is the same as Eq. 16.