Simplicial minisuperspace models in the presence of a massive scalar field with arbitrary scalar coupling ηRφ

We consider a simplicial minisuperspace model based on a cone over the α4 triangulation of the 3−sphere, in the presence of a massive scalar field, φ, with arbitrary scalar coupling term ηRφ. By restricting all the interior edge lengths and all the boundary edge lengths to be equivalent and the scalar field to be homogenous on the 3−space, we obtain a two dimensional minisuperspace model {si, φi} for what is one of the most relevant triangulations of the spatial universe. We solve the two classical equations and find that there are both real Euclidean and Lorentzian classical solutions for any size of the boundary 3−space, α4. After studying the analytic properties of the action in the space of complex edge lengths we then obtain steepest descents contours of constant imaginary action passing through Lorentzian classical geometries giving the dominant contribution and yielding a convergent wavefunction of the universe. We also show that the semiclassical wavefunctions for the Euclidean solutions associated with large boundary 3−spaces are exponentially suppressed, Consequently we can be confident that by using the SD contour associated with classical Lorentzian solutions the semiclassical approximation based on those classical solutions is well justified, clearly predicting classical spacetime in the late universe. This wavefunction is then evaluated numerically. e-mail address : [email protected] e-mail address : [email protected]