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The gravitational equations were derived in general relativity (GR) using theassumption of their covariance relative to arbitrary transformations ofcoordinates. It has been repeatedly expressed an opinion over the past centurythat such equality of all coordinate systems may not correspond to reality.Nevertheless, no actual verification of the necessity of this assumption hasbeen made to date. The paper proposes a theory of gravity with a constraint,the degenerate variants of which are general relativity (GR) and the unimodulartheory of gravity. This constraint is interpreted from a physical point of viewas a sufficient condition for the adiabaticity of the process of the evolutionof the space-time metric. The original equations of the theory of gravity withthe constraint are formulated. On this basis, a unified model of the evolutionof the modern, early, and very early Universe is constructed that is consistentwith the observational astronomical data but does not require the hypotheses ofthe existence of dark energy, dark matter or inflatons. It is claimed that:physical time is anisotropic, the gravitational field is the main source ofenergy of the Universe, the maximum global energy density in the Universe was64 orders of magnitude smaller the Planckian one, and the entropy density is 18orders of magnitude higher the value predicted by GR. The value of the relativedensity of neutrinos at the present time and the maximum temperature of matterin the early Universe are calculated. The wave equation of the gravitationalfield is formulated, its solution is found, and the nonstationary wave functionof the very early Universe is constructed. It is shown that the birth of theUniverse was random.

Foundations of a Theory of Gravity with a Constraint and its Canonical Quantization