Inflation and transition to a slowly accelerating phase from SSB of scale invariance

We consider the effects of adding a scale-invariant R² term to the action of the scale-invariant model (SIM) studied previously by one of us (Guendelman E I 1999 Mod. Phys. Lett. A 14 1043). The SIM belongs to the general class of theories, where an integration measure independent of the metric is introduced. To implement scale invariance (SI), a dilaton field is introduced. The integration of the equations of motion associated with the new measure gives rise to the spontaneous symmetry breaking (SSB) of SI. After SSB of SI in the model with the R² term, it is found that a non-trivial potential for the dilaton is generated. This potential contains two flat regions: one associated with the Planck scale and with an inflationary phase, while the other flat region is associated with a very small vacuum energy (VE) and is associated with the present slowly accelerated phase of the universe (SAPH). The smallness of the VE in the SAPH is understood through the see-saw mechanism introduced in SIM.