Unifying Inflation with Early and Late-time Dark Energy in $F(R)$
  Gravity
  In this work we shall present models of $F(R)$ gravity which realize in a
unified way the inflationary era along with a post-inflationary early dark
energy era, with the late-time dark energy era. We shall use two approach
methods in order to realize the unified cosmological eras, firstly we specify a
Hubble rate which may describe the three distinct acceleration eras, and then
by using well known $F(R)$ gravity reconstruction techniques, we shall find the
differential equation which may yield the $F(R)$ gravity that realizes the
cosmologies. In our second approach, we shall present in a qualitative way,
several $F(R)$ gravities which unify the inflationary era with the early and
late-time dark energy eras, and we discuss several qualitative issues related
to the terms that realize the post-inflationary early dark energy era. We
quantify our analysis by numerically solving the Friedmann equation, using the
redshift as the main variable, and expressing all the physical quantities as
functions of the statefinder $y_\mathrm{H}(z)$, which depends on the redshift
and the Hubble rate $H(z)$. For the model studied numerically, we present the
behavior of some statefinder quantities, like the deceleration parameter, and
we calculate the dark energy density parameter and the dark energy equation of
state parameter at present time. After demonstrating that the dark energy era
is viable, we investigate when the early dark energy term does not affect the
late-time era, and this restricts the free parameters of the model.
