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For any s∈[−∞,0] and oriented homology 3-sphere Y, we introduce a homology cobordism invariant r s ​ (Y)∈(0, ∞]. 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 Frøyshov 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, ∞]. 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 ​. [ABSTRACT FROM AUTHOR]