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We explore a reduced phase space quantization of loop quantum cosmology (LQC)for a spatially flat FLRW universe filled with reference fields and an inflatonfield in a Starobinsky inflationary potential. We consider three separate casesin which the reference fields are taken to be the Gaussian dust, theBrown-Kucha\v{r} dust, and massless Klein-Gordon scalar reference fieldsrespectively. This is a "two-fluid" model in which reference fields act asglobal clocks providing a physical time in an inflationary spacetime, and allowbypassing various technical hurdles in conventional quantum cosmologicalmodels. The reduced phase space is obtained in terms of the Dirac observablesof the gravitational as well as the inflaton degrees of freedom. The physicalHamiltonians of the two dust models take the same form but turn out to be quitedifferent from that of the Klein-Gordon reference field which reflects anaspect of the multiple choice problem of time. Loop quantization is implementedusing the so-called $\bar \mu$ scheme and the Schr\"{o}dinger equationsinvolving the physical Hamiltonian operators generating the evolution in thephysical time in the dust and massless Klein-Gordon models are obtained. Theseturn out to be quantum difference equations with same non-singular structure asfor other models in LQC. We study some phenomenological implications of thequantization using the effective dynamics resulting from the reduced phasespace quantization including the resolution of the big bang singularity via aquantum bounce, and effects of the different reference fields on e-foldings inboth the pre-inflationary and the slow-roll inflationary phases. We find thatdifferent clocks, even when starting with a small but same energy density, canleave tiny but different imprints on the inflationary dynamics. (abridged)

Towards a reduced phase space quantization in loop quantum cosmology with an inflationary potential