Generalizing the Cosmic Energy Equation
  We generalize the cosmic energy equation to the case when massive particles
interact via a modified gravitational potential of the form phi(a, r), which is
allowed to explicitly depend upon the cosmological time through the expansion
factor a(t). Using the nonrelativistic approximation for particle dynamics, we
derive the equation for the cosmological expansion which has the form of the
Friedmann equation with a renormalized gravitational constant. The generalized
Layzer-Irvine cosmic energy equation and the associated cosmic virial theorem
are applied to some recently proposed modifications of the Newtonian
gravitational interaction between dark-matter particles. We also draw attention
to the possibility that the cosmic energy equation may be used to probe the
expansion history of the universe thereby throwing light on the nature of dark
matter and dark energy.
