Curing singularities in cosmological evolution of F(R) gravity
  We study $F(R)$ modified gravity models which are capable of driving the
accelerating epoch of the Universe at the present time whilst not destroying
the standard Big Bang and inflationary cosmology. Recent studies have shown
that a weak curvature singularity with $|R|\to\infty$ can arise generically in
viable $F(R)$ models of present dark energy (DE) signaling an internal
incompleteness of these models. In this work we study how this problem is cured
by adding a quadratic correction with a sufficiently small coefficient to the
$F(R)$ function at large curvatures. At the same time, this correction
eliminates two more serious problems of previously constructed viable $F(R)$ DE
models: unboundedness of the mass of a scalar particle (scalaron) arising in
$F(R)$ gravity and the scalaron overabundance problem. Such carefully
constructed models can also yield both an early time inflationary epoch and a
late time de Sitter phase with vastly different values of $R$. The reheating
epoch in these combined models of primordial and present dark energy is
completely different from that of the old $R + R^{2}/6M^{2}$ inflationary
model, mainly due to the fact that values of the effective gravitational
constant at low and intermediate curvatures are different for positive and
negative $R$. This changes the number of e-folds during the observable part of
inflation that results in a different value of the primordial power spectrum
index.
