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

Loop Quantum Cosmology 405This happens by the occurrence of bounces where a turns around fromcontracting to expanding behavior. Thus, ȧ = 0 and ä > 0. The first conditionis not always realizable, as follows from the Friedmann equation (1).In particular, when the scalar potential is non-negative there is no bounce,which is not changed by the effective density. There are then two possibilitiesfor bounces in isotropic models, the first one if space has positivecurvature rather than being flat as assumed here, 60,61 the second one witha scalar potential which can become negative. 62,63 Both cases allow ȧ = 0even in the classical case, but this always corresponds to a maximum ratherthan minimum. This can easily be seen for the case of negative potentialfrom the Raychaudhuri equation (19) which in the classical case impliesnegative ä. With the modification, however, the additional term in theequation provides a positive contribution which can become large enoughfor ä to become positive at a value of ȧ = 0 such that there is a bounce.This provides intuitive explanations for the absence of singularities fromquantum gravity, but not a general one. The generic presence of bounces dependson details of the model such as its matter content or which correctionterms are being used, 64,65 and even with the effective modifications thereare always models which classically remain singular. Thus, the only generalargument for absence of singularities remains the quantum one based onthe difference equation (11), where the conclusion is model independent 41and which also confirms bounce pictures qualitatively. 59,464.4.2. InflationA repulsive contribution to the gravitational force can not only explain theabsence of singularities, but also enhances the expansion of the universeon scales close to the classical singularity. Thus, as seen also in Fig. 3 theuniverse accelerates just from quantum effects, providing a mechanism forinflation without choosing special matter.Via the generation of structure, inflationary phases of the universe canhave an imprint on the observable cosmic microwave background. Observationsimply that the predicted power spectrum of anisotropies must benearly independent of the scale on which the anisotropies are probed, whichimplies that the inflationary phase responsible for structure formation mustbe close to exponential acceleration. This is true for slow-roll inflation, butalso for the inflationary phase obtained from the effective density once anon-zero scalar potential is taken into account. 66 For more detailed comparisonsbetween theory and observations one needs to consider how inhomoge-