  The gravitational equations were derived in general relativity (GR) using the
assumption of their covariance relative to arbitrary transformations of
coordinates. It has been repeatedly expressed an opinion over the past century
that such equality of all coordinate systems may not correspond to reality.
Nevertheless, no actual verification of the necessity of this assumption has
been made to date. The paper proposes a theory of gravity with a constraint,
the degenerate variants of which are general relativity (GR) and the unimodular
theory of gravity. This constraint is interpreted from a physical point of view
as a sufficient condition for the adiabaticity of the process of the evolution
of the space-time metric. The original equations of the theory of gravity with
the constraint are formulated. On this basis, a unified model of the evolution
of the modern, early, and very early Universe is constructed that is consistent
with the observational astronomical data but does not require the hypotheses of
the existence of dark energy, dark matter or inflatons. It is claimed that:
physical time is anisotropic, the gravitational field is the main source of
energy of the Universe, the maximum global energy density in the Universe was
64 orders of magnitude smaller the Planckian one, and the entropy density is 18
orders of magnitude higher the value predicted by GR. The value of the relative
density of neutrinos at the present time and the maximum temperature of matter
in the early Universe are calculated. The wave equation of the gravitational
field is formulated, its solution is found, and the nonstationary wave function
of the very early Universe is constructed. It is shown that the birth of the
Universe was random.
