Analyzing quantum gravity spillover in the semiclassical regime

One of the standard approaches of incorporating the quantum gravity (QG) effects into the semiclassical analysis is to adopt the notion of a quantum-corrected spacetime arising from the QG model. This procedure assumes that the expectation value of the metric variable effectively captures the relevant QG subtleties in the semiclassical regime. We investigate the viability of this effective geometry approach for the case of dust dominated and a dark energy dominated universe. We write the phase space expressions for the geometric observables and construct corresponding Hermitian operators. A general class of operator ordering of these observables is considered, and their expectation values are calculated for a unitarily evolving wave packet. In the case of dust dominated universe, the expectation value of the Hubble parameter matches the "semiclassical" expression, the expression computed from the scale factor expectation value. In the case of Ricci scalar, the relative difference between the semiclassical expression and quantum expectation is maximum at singularity and decays for late time. For a cosmological constant driven universe, the difference between the semiclassical expressions and the expectation value is most pronounced far away from the bounce point, hinting at the persistent quantum effect at the late time. The parameter related to the shape of the distribution appears as a control parameter in these models. In the limit of a sharply peaked distribution, the expectation value of the observables matches with their semiclassical counterpart, and the usage of effective geometry approach is justified.