  In this work, we study unified picture of inflation and dark matter in the
Higgs-Starobinsky (HS) model. As pointed out in the literature, Starobinsky
$R^2$ inflation is induced by quantum correction effect from the large
Higgs-curvature (graviton) coupling. We start with non-minimal coupling HS
action in Jordan frame. We then transform the Jordan frame action into the
Einstein one using the conformal transformation. The inflation potential is
derived from the gravitational action of non-minimal-Higgs coupling and
Starobinsky term in Einstein frame where the $R^2$ term is dominated in the
inflationary phase of the universe. For model of inflation, we compute the
inflationary parameters and confront them with Planck 2015 data. We discover
that the predictions of the model are in excellent agreement with the Planck
analysis. In addition, we study the Higgs field as a candidate for dark matter.
The renormalization group equations (RGEs) of Higgs-Starobinsky scenario with
standard model at one-loop is qualitatively analyzed. By using the solutions of
parameter spaces from RGE analysis, the remnants of the quantum effect ($R^2$
term) from Higgs-graviton coupling will be implied by dark matter relic
abundance.
