  In the path integral formulation of the reduced phase space Loop Quantum
Gravity (LQG), we propose a new approach to allow the spatial cubic lattice
(graph) to change dynamically in the physical time evolution. The equations of
motion of the path integral derive the effective dynamics of cosmology from the
full LQG, when we focus on solutions with homogeneous and isotropic symmetry.
The resulting cosmological effective dynamics with the dynamical lattice
improves the effective dynamics obtained earlier from the path integral with
fixed spatial lattice: The improved effective dynamics recovers the FLRW
cosmology at low energy density and resolves the big-bang singularity with a
bounce. The critical density $\rho_c$ at the bounce is Planckian $\rho_c\sim
\Delta^{-1}$ where $\Delta$ is a Planckian area serving as certain UV cut-off
of the effective theory. The effective dynamics gives the unsymmetric bounce
and has the de-Sitter (dS) spacetime in the past of the bounce. The
cosmological constant $\Lambda_{eff}$ of the dS spacetime is emergent from the
quantum effect $\Lambda_{eff}\sim\Delta^{-1}$. These results are qualitatively
similar to the properties of $\bar{\mu}$-scheme Loop Quantum Cosmology (LQC).
Moreover, we generalize the earlier path integral formulation of the full LQG
by taking into account the coupling with an additional real scalar field, which
drives the slow-roll inflation of the effective cosmological dynamics. In
addition, we discuss the cosmological perturbation theory on the dynamical
lattice, and the relation to the Mukhanov-Sasaki equation.
