It is well-known that some minimal (or maximal) hypersurfaces are stable. However, there is a growing recognition on unstable minimal (or maximal) hypersurfaces by introducing the concept of index of stability. Firstly, the index of stability for minimal hypersurfaces in the Euclidean n-sphere has been introduced by J. Simons [20], which is followed recently by many people (see for instance [3, 9, 18, 21]). Also, Barros and Sousa in [10] have payed attention to the concept of index of r-stability (as the r-extension of index of stability) on r-minimal hypersurfaces in the Euclidean n-sphere. They gave low bounds for r-stability index of r-minimal n-dimensional closed hypersurfaces in Sn+1. In this paper we give low bounds for the r-stability index of r-maximal closed spacelike hypersurfaces in the de Sitter space Sn+1 1 .
