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$\acute{\mathrm{e}}$ approximations with the latest Pantheon+ sample for testing cosmological principle. Based on the Pad$\acute{\mathrm{e}}$ approximations, we first gave the cosmographic constraints on the different order polynomials including third-order (Pad$\acute{\mathrm{e}}$${}_{(2,1)}$), fourth-order (Pad$\acute{\mathrm{e}}$${}_{(2,2)}$) and fifth-order (Pad$\acute{\mathrm{e}}$${}_{(3,2)}$). Based on the Pad$\acute{\mathrm{e}}$${}_{(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 ${\mathrm{\Omega }}_m$ = 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.