Testing cosmic anisotropy with Pade approximation and Pantheon+ sample

Cosmography can be used to constrain the kinematics of universe in a model-independent way. In this work, we attempted to combine the Pad$\rm \acute{e}$ approximations with the latest Pantheon+ sample for testing cosmological principle. Based on the Pad$\rm \acute{e}$ approximations, we first gave the cosmographic constraints on the different order polynomials including third-order (Pad$\rm \acute{e}$$_{(2,1)}$), fourth-order (Pad$\rm \acute{e}$$_{(2,2)}$) and fifth-order (Pad$\rm \acute{e}$$_{(3,2)}$). Based on the Pad$\rm \acute{e}$$_{(2,1)}$ ($j_{0}$ = 1) polynomial and hemisphere comparison (HC) method, we tested the cosmological principle and found the preferred directions of cosmic anisotropy, such as (l, b) = (304.6$^{\circ}$$_{-37.4}^{+51.4}$, $-$18.7$^{\circ}$$_{-20.3}^{+14.7}$) and (311.1$^{\circ}$$_{-8.4}^{+17.4}$, $-$17.53$^{\circ}$$_{-7.7}^{+7.8}$) for $q_{0}$ and $H_{0}$, respectively. These two directions are consistent with each other in $1\sigma$ confidence level, but the corresponding results of statistical isotropy analyses including Isotropy and Isotropy with real positions (RP) are quite different. The statistical significance of $H_{0}$ are stronger than that of $q_{0}$, i.e., 4.75$\sigma$ and 4.39$\sigma$ for the Isotropy and Isotropy with RP respectively. Reanalysis with fixed $q_{0} = -0.55$ (corresponds to $\Omega_{m}$ = 0.30) gives similar results. Overall, our model-independent results provide clear indications for a possible cosmic anisotropy, which must be taken seriously. Further test is needed to better understand this signal.