The structure of the paper is as follows. Einstein-aether scalar field theory are of special interest in the scientific society and there are various studies in the literature on the subject, some of these studies for homogeneous and inhomogeneous spacetimes can be found in. In this work we realize this idea for the case of a spatially flat FLRW universe and a scalar field with arbitrary potential and we present, for the first time, the quantization of the Einstein-aether scalar field cosmology we also derive the generic algebraic classical solution to the field equations. A classical slow-roll era and a Lorentz violating epoch. Such a theory has been proposed before as an alternative inflationary model which provides two periods of inflation. One can contemplate a non-trivial coupling of the scalar to the aether field by allowing the coefficients of its kinetic action to be functions of the scalar field. Moreover, scalar fields have also been proposed as dark energy models, while they can attribute the geometrodynamical degrees of freedom provided by higher-order theories which belong to the class of modified theories of gravity, for more details we refer the reader to for a complete review on the cosmological implications in modified theories of gravity see and references therein. The main mechanism for the description of the inflation is based on the domination of a scalar field potential, known as inflaton. On the other hand, scalar fields play a prominent role in modern cosmology. These new terms break the Lorentz symmetry, by selecting a preferred frame at each space time point, while keeping the field equations of second-order as in the case of General Relativity. Specifically, in Einstein-aether theory quadratic kinematic quantities of the unitary time-like vector fields are introduced in the gravitational Action Integral. In the Einstein-aether theory, a unit time-like rotationally invariant vector field, called the “aether” is also responsible for the same effect. The Hořava–Lifshitz gravity is certainly violating Lorentz invariance by construction, since arbitrary time re-definitions are not among its covariances. Fun fact the Greeks associated to aether the dodecahedron that they thought as the perfect shape, and those how have already read book 8 will know why this is important.The interest in Lorentz violating cosmological theories comes together with the idea of a variable speed of light. For a brief period was use to think permeate the universe and link to Gravity, of I recall correctly was use to explain interaction without field theory (don't quote me on that I am not really sure) The version use in TBATE is used as the concept of quintessence the fifth element the divine element of the 4 fire, water, earth and wind (this interpretation explain why in TBATE in the atmosphere exists only mana of the 4 fundamentaltype and noneof the deviant kind), the substance that greeks and ancient world thought made the divine (sun, moon, stars ecc.), but it retain the concept of medium when is explain its connection with mana as the idea of cup that holds it (even if is an incomple explanation according to Arthur). Aether was a real scientific term it was (as a vague explanation) the medium use by the light to propagate, the theory is of course abandoned (unless we prove with a quantum physic's experiment that our understanding of light is false) and was call luminos aether.
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