Instability of de-Sitter black hole with massive scalar field coupled to Gauss-Bonnet invariant and the scalarized black holes

The black hole scalarization in a special Einstein-scalar-Gauss-Bonnet (EsGB) gravity has been widely investigated in recent years. Especially, the spontaneous scalarization of scalar-free black hole in de-Sitter (dS) spacetime possesses interesting features due to the existence of cosmological horizon. In this work, firstly, we focus on the massive scalar field perturbation on Schwarzschild dS (SdS) black hole in a special EsGB theory. By analyzing the fundamental QNM frequency and time evolution of the scalar field perturbation, we figure out the unstable/stable regions in $(\Lambda,\alpha)$-plane as well as in $(m,\alpha)$-plane for various perturbation modes, where $\Lambda$, $\alpha$ and $m$ denote the cosmological constant, the GB coupling strength and the mass of scalar field, respectively. Then by solving the static perturbation equation, we analyze the bifurcation point at which the SdS black hole supports spherical scalar clouds, and we find that the bifurcation points match well with $\alpha_c$ on the border of unstable/stable region. Finally, after addressing that the scalarised solutions could only emerge from the scalar could with node $k\geq 1$. we explicitly construct the scalarized hairy solutions for different scalar masses and compare the profile of scalar field to the corresponding scalar clouds.