Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line
bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(πLk)
in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds
for families of relative dimension one. As a corollary, we show that when X is smooth, the line bundle η associated to X → B,
which was introduced in Phong-Sturm [12], coincides with the CM bundle defined by Paul-Tian [10, 11]. In a second and third corollaries,
we establish asymptotics for the K-energy along Bergman rays generalizing the formulas obtained in [11].