Einstein–Aether primordial universe with radiation and dark energy

Abstract We investigate a quantum cosmological model within the framework of Einstein–Aether gravity. The model consists of a positively curved FLRW Universe whose matter content is described by a radiation fluid, in the presence of a positive cosmological constant $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> . We first perform a classical Hamiltonian analysis, constructing the phase space and deriving the dynamics of the scale factor. Notably, we identify specific initial conditions that lead to an inflationary expansion. Subsequently, we quantize the model following Dirac’s formalism, obtaining the Wheeler-DeWitt equation for the wave function of the Universe. We solve this equation numerically and compare our results to those of the WKB approximation, computing quantum tunneling probabilities and the expectation value of the scale factor, along with its standard deviation. Finally, we analyze how these tunneling probabilities are influenced by variations in the fundamental parameters of the theory ( $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> , $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> , $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> ) and the energy of the fluid E . Our results indicate a higher probability for the quantum birth of the Universe in a parameter regime characterized by larger values of $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> , $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> , and fluid energy E , together with a smaller value of $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> .