  In standard general relativity the universe cannot be started with arbitrary
initial conditions, because four of the ten components of the Einstein's field
equations (EFE) are constraints on initial conditions. In the previous work it
was proposed to extend the gravity theory to allow free initial conditions,
with a motivation to solve the cosmological constant problem. This was done by
setting four constraints on metric variations in the action principle, which is
reasonable because the gravity's physical degrees of freedom are at most six.
However, there are two problems about this theory; the three constraints in
addition to the unimodular condition were introduced without clear physical
meanings, and the flat Minkowski spacetime is unstable against perturbations.
Here a new set of gravitational field equations is derived by replacing the
three constraints with new ones requiring that geodesic paths remain geodesic
against metric variations. The instability problem is then naturally solved.
Implications for the cosmological constant $\Lambda$ are unchanged; the theory
converges into EFE with nonzero $\Lambda$ by inflation, but $\Lambda$ varies on
scales much larger than the present Hubble horizon. Then galaxies are formed
only in small $\Lambda$ regions, and the cosmological constant problem is
solved by the anthropic argument. Because of the increased degrees of freedom
in metric dynamics, the theory predicts new non-oscillatory modes of metric
anisotropy generated by quantum fluctuation during inflation, and CMB B-mode
polarization would be observed differently from the standard predictions by
general relativity.
