  For any $s \in [-\infty, 0] $ and oriented homology 3-sphere $Y$, we
introduce a homology cobordism invariant $r_s(Y)\in (0,\infty]$. The values
$\{r_s(Y)\}$ are included in the critical values of the $SU(2)$-Chern-Simons
functional of $Y$, and we show a negative definite cobordism inequality and a
connected sum formula for $r_s$. As applications, we obtain several new results
on the homology cobordism group. First, we give infinitely many homology
3-spheres which cannot bound any definite 4-manifold. Next, we show that if the
1-surgery of $S^3$ along a knot has the Fr{\o}yshov invariant negative, then
all positive $1/n$-surgeries along the knot are linearly independent in the
homology cobordism group. In another direction, we use $\{r_s\}$ to define a
filtration on the homology cobordism group which is parametrized by
$[0,\infty]$. Moreover, we compute an approximate value of $r_s$ for the
hyperbolic 3-manifold obtained by $1/2$-surgery along the mirror of the knot
$5_2$.
