Title: "Inflation, reheating and present acceleration of the Universe in f(R) gravity "

Since 1980, the simplest non-trivial variant of f(R) fourth-order theory of gravity (with small one-loop corrections) was known to provide an internally self-consistent scenario of the early Universe with an initial quasi-de Sitter (inflationary) stage followed by a graceful exit to the radiation-dominated FRW stage via the period of reheating in which all matter in the Universe arises as a result of gravitational particle creation [1]. Its predictions regarding spectra of primordial density perturbations and gravitational waves remain in agreement with the most recent observational data. Several years ago it was proposed to use this class of models for description of dark energy in the present Universe. However, this problem appeared to be more complicated, so many attempts in this direction failed. Still recently some f(R) models were found which can satisfy laboratory, Solar system and cosmological tests, in particular [2]. These viable models typically exhibit phantom behaviour of dark energy during the matter-dominated stage and recent crossing of the phantom boundary ωDE = -1. As a consequence of the anomalous growth of density perturbations in the cold dark matter + baryon component at recent redshifts, their growth index evolves non- monotonically with time and may even become negative temporarily [3]. However, all these models generically could not reproduce the correct evolution of the Universe in the past due to formation of additional weak singularities and other problems. Now it is shown at last that it is possible to cure all these problems and to construct viable f(R) models of present dark energy which do not destroy any of previous successes of the early Universe cosmology. Such models have the same asymptotic behaviour for large R as f(R) models of inflation. Combined description of inflation and present dark energy using one f(R) function is possible, but leads to completely different reheating after inflation during which strongly non-linear oscillations of R occur [4].

[1] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
[2] A. A. Starobinsky, JETP Lett. 86, 157 (2007) [arXiv:0706.2041].
[3] H. Motohashi, A. A. Starobinsky and J. Yokoyama, Progr. Theor. Phys. 123, 887 (2010) [arXiv:1002.1141].
[4] S. A. Appleby, R. A. Battye and A. A. Starobinsky, JCAP 1006, 005 (2010) [arXiv:0909.1737].