<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> Lorentzian quantum cosmology from spin-foams: opportunities an

Abstract We construct an effective cosmological spin-foam model for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> dimensional spatially flat Universe, discretized on a cubical lattice, containing both space- and time-like regions. Our starting point is the recently proposed coherent state spin-foam model for (2+1) Lorentzian quantum gravity. The full amplitude is assumed to factorize into single vertex amplitudes with boundary data corresponding to Lorentzian 3-frusta. A stationary phase approximation is performed at each vertex individually, where the inverse square root of the Hessian determinant serves as a measure for the effective path integral. Additionally, a massive scalar field is coupled to the geometry, and we show that its mass renders the partition function convergent. For a single 3-frustum with time-like struts, we compute the expectation value of the bulk strut length and show that it generically agrees with the classical solutions and that it is a discontinuous function of the scalar field mass. Allowing the struts to be space-like introduces causality violations, which drive the expectation values away from the classical solutions due to the lack of an exponential suppression of these configurations. This is a direct consequence of the semi-classical amplitude only containing the real part of deficit angles, in contrast with the Lorentzian Regge action used in effective spin-foams. We give an outlook on how to evaluate the partition function on an extended discretization including a bulk spatial slice. This serves as a foundation for future investigations of physically interesting scenarios such as a quantum bounce or the viability of massive scalar field clocks. Our results demonstrate that the effective path integral in the causally regular sector serves as a viable quantum cosmology model, but that the agreement of expectation values with classical solutions is tightly bound to the path integral measure.