Initial Conditions of Inhomogeneous Universe and the Cosmological
  Constant Problem
  Deriving the Einstein field equations (EFE) with matter fluid from the action
principle is not straightforward, because mass conservation must be added as an
additional constraint to make rest-frame mass density variable in reaction to
metric variation. This can be avoided by introducing a constraint
$\delta(\sqrt{-g}) = 0$ to metric variations $\delta g^{\mu\nu}$, and then the
cosmological constant $\Lambda$ emerges as an integration constant. This is a
removal of one of the four constraints on initial conditions forced by EFE at
the birth of the universe, and it may imply that EFE are unnecessarily
restrictive about initial conditions. I then adopt a principle that the theory
of gravity should be able to solve time evolution starting from arbitrary
inhomogeneous initial conditions about spacetime and matter. The equations of
gravitational fields satisfying this principle are obtained, by setting four
auxiliary constraints on $\delta g^{\mu\nu}$ to extract six degrees of freedom
for gravity. The cost of achieving this is a loss of general covariance, but
these equations constitute a consistent theory if they hold in the special
coordinate systems that can be uniquely specified with respect to the initial
space-like hypersurface when the universe was born. This theory predicts that
gravity is described by EFE with non-zero $\Lambda$ in a homogeneous patch of
the universe created by inflation, but $\Lambda$ changes continuously across
different patches. Then both the smallness and coincidence problems of the
cosmological constant are solved by the anthropic argument. This is just a
result of inhomogeneous initial conditions, not requiring any change of the
fundamental physical laws in different patches.
