(where $\gamma$ is a Barbero-Immirzi parameter, which value has been set following [@gamma]) and are global degrees of freedom of the geometry. In particular $p=a^2$.

Within selected gauge all the constraints except the Hamiltonian one are automatically satisfied. The remaining constraint takes the form $C = N(C_{\rm gr}+C_{\phi})$ where $$\label{eq:Cgr-def}
  C_{\rm gr} = -\frac{1}{\gamma^2}\int_{{\mathcal{V}}}\rd^3x\, \epsilon_{ijk} e^{-1} 
  E^{ai} E^{bj} F^k_{ab} ,$$ with $e:=\sqrt{|\det(E)|}$ and $F$ being a curvature of a connection $A$. To deal with the noncompactness of $\Sigma$ the integration of a Hamiltonian density was performed only over a chosen cubic cell ${\mathcal{V}}$ constant in comoving coordinates, which is an equivalent to the infrared cut-off. Despite this, the physical predictions are invariant with respect to the choice of the cell [@aps-imp].

The remaining matter part of the constraint equals $$\label{eq:Cphi-def}
   C_{\phi} = 8\pi G p_{\phi}^2 / p^{3/2} ,$$ where $\phi$ and $p_{\phi}$ are, respectively, the value of the scalar field and its conjugate momentum with $\{\phi,p_{\phi}\}=1$.

### Loop quantization {#sec:frame-FRW-q}

The classical system specified in Sec. \[sec:frame-FRW-class\] is next quantized via methods of Loop Quantum Gravity. In particular, the Dirac program is employed to construct the physical Hilbert space. It consists of the following steps:

$\bullet$ *Quantization on the kinematical 