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We will now compare our approach with a number of other different models, all of them sharing as a background the same as for holonomy corrected LQC. These models are: LQC in the [*deformed algebra approach*]{}, the teleparallel LQC approach, the intrinsic curvature LQC approach and the mimetic LQC approach. First, we shall review the dynamical equations for each of these theories:

1.  LQC in the [*deformed algebra approach*]{} [@grain; @caitelleau].

    In this case the equation for the potential $\Phi$ decouples. It is given by $$\begin{aligned}
    {\ddot{\Phi}}-\frac{\Omega}{a^2}\Delta\Phi+\left(H-2\frac{\ddot{\bar\phi}}{\dot{\bar\phi}}-\frac{\dot{\Omega}}{\Omega}\right){\dot{\Phi}}+\left(2\left(\dot{H}-H\frac{\ddot{\bar\phi}}{\dot{\bar\phi}}\right)-H\frac{\dot{\Omega}}{\Omega} \right){\Phi}=0.
    $$ The variable M-S variable $v$ adquires the usual form $v=a(\delta\phi+\frac{{\bar{\phi}'}}{{\mathcal H}}\Phi) $ and the M-S equations decouple in the simple form $$\begin{aligned}
    u''-\Omega\Delta u-\frac{\theta''}{\theta}u=0, \quad v''-\Omega\Delta v-\frac{z''}{z}v=0.
    $$ In particular the equation $v''-\Omega\Delta v-\frac{z''}{z}v=0$ can be solved when the pivot scale is well inside and outside of the Hubble radius, obtaining the whole evolution of the variable $v$, which encodes all the information about scalar perturbations and, thus, the knowledge of the power spectrum for the [*curvature fluctuation in co-moving coordinates*]{}.

    On the other hand, for tensor perturbation the corresponding M-S equation is [@vidotto] $$\begin{aligned}
    h''-\Omega\Delta h-\frac{z_T''}{z_T}h=0,\end{aligned}$$ where $z_T\equiv \frac{a}{\sqrt{\Omega}}$.

2.  Teleparallel LQC [@haro13; @cai].

    In teleparallel LQC the equation 