The analytical approach in testing the Kaniadakis cosmology

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Abstract We know that a quantum-corrected cosmological scenario can emerge based on its corrected Friedmann equations corresponding to the corrected entropy of cosmic-horizon using the gravity-thermodynamics conjecture in deriving those equations. In this right, the Kaniadakis entropy associated with the apparent horizon of Friedmann-Robertson-Walker(FRW) Universe leads to a corrected Friedmann equation which contains a correction term as $\Omega_{Ka}=\frac{\alpha\left(H^2+\frac{k}{a^2}\right)^{-1}}{H^{2}}$ where $\alpha\equiv\frac{K^2 \pi^2}{2 G^2}$ ($K$ is the Kaniadakis parameter). Here, we derive the analytical relations between the energy density parameters $\Omega_{m},\Omega_{\Lambda}$ (and the ratio density $\Omega_{Ka}$ of correction term) and the geometrical cosmological parameters $\{q, j\}$. This leads to getting $\Omega_{Ka}=\frac{j-1}{4(q+1)^{2}-(j-1)}$ which enables us to put constrains on $\Omega_{Ka_{0}}$ using the measurable parameters $\{q_{0},j_{0}\}$ and $H_{0}$. It also reveals some interesting aspects of the Kaniadakis cosmology by explaining that the correction term plays different roles both in the presence and in the absence of the cosmological constant $\Lambda$.
This term plays the role of dark energy in the absence of the cosmological constant $\Lambda$, while in the presence of this component, it plays the role of a small correction to dark energy. The value of $\Omega_{Ka}$ then determines the amount of the deviation of Kaniadakis model from the concordance $\Lambda CDM$ model. As a result, the value of $j_{0}$ is found to be close to unit for vanishing cosmological constant $\Lambda=0$ and the maximum value of $\Omega_{Ka0}$; thereby we get $(j_{0}-1)\simeq 0.137$ and $(j_{0}-1)\simeq< 10^{-3}$ for the case of $\Lambda=0$ and $\Lambda\neq0$, respectively