The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain Ω B d 0 (µ) endowed with the canonical metric g(µ), we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space H α of square integrable holomorphic functions on (Ω B d 0 (µ), g(µ)) with the weight exp{−αϕ} (where ϕ is a globally defined Kähler potential for g(µ)) for α > 0, and, furthermore, we give an explicit expression of the Rawnsley's ε-function expansion for (Ω B d 0 (µ), g(µ)). Secondly, using the explicit expression of the Rawnsley's ε-function expansion, we show that the coefficient a 2 of the Rawnsley's ε-function expansion for the Cartan-Hartogs domain (Ω B d 0 (µ), g(µ)) is constant on Ω B d 0 (µ) if and only if (Ω B d 0 (µ), g(µ)) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.