100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

404 M. BojowaldThe effective Friedmann equation then takes the formaȧ 2 = 8π 3 G ( 12 (a−3 ) eff p 2 φ + a 3 V (φ) ) (17)with (a −3 ) eff as in (15) with a choice of ambiguity parameters. Since thematter Hamiltonian does not just act as a source for the gravitational fieldon the right hand side of the Friedmann equation, but also generates Hamiltonianequations of motion, the modification entails further changes in thematter equations of motion. The Klein–Gordon equation (3) then takes theeffective form¨φ = ˙φ ȧ d log(a−3 ) effdaand finally there is the Raychaudhuri equationäa = −8πG 3((a −3 d(a) −1 ˙φ 2 eff1 − 1 4 ad log(a3 d(a) eff )da− a 3 (a −3 ) eff V ′ (φ) (18)) )− V (φ)(19)which follows from the above equation and the continuity equation ofmatter.4.4.1. BouncesThe resulting equations can be studied numerically or with qualitativeanalytic techniques. We first note that the right hand side of (17) behavesdifferently at small scales since it increases with a at fixed φ andp φ . Viewing this equation as analogous to a constant energy equationin classical ( mechanics with kinetic ) term ȧ 2 and potential term V(a) :=− 8π 3 Ga−1 12 (a−3 ) eff p 2 φ + a3 V (φ) illustrates the classically attractive natureof gravity: The dominant part of this potential behaves like −a −4which is increasing. Treating the scale factor analogously to the position ofa classical particle shows that a will be driven toward smaller values, implyingattraction of matter and energy in the universe. This changes whenwe approach smaller scales and take into account the quantum modification.Below the peak of the effective density the classical potential V(a)will now decrease, −V(a) behaving like a positive power of a. This impliesthat the scale factor will be repelled away from a = 0 such that there isnow a small-scale repulsive component to the gravitational force if we allowfor quantum effects. The collapse of matter can then be prevented if repulsionis taken into account, which indeed can be observed in some modelswhere the effective classical equations alone are sufficient to demonstratesingularity-free evolution.