Cosmology of a higher derivative scalar theory with non-minimal Maxwell coupling

Higher derivative scalar field theory in curved space-time belongs to the GLPV theory coupled non-minimally to the Maxwell field is considered. We will show that the theory admits two independent exact de Sitter solutions in the FRW background, one driven by the cosmological constant and the other by the GLPV scalar field. The dynamical system analysis of the theory shows that these two exact solutions are stable fixed points. Also, cosmological perturbations over these solutions shows that the cosmological constant based solution is healthy at linear level but the GLPV based solution suffers from a gradient instability in the scalar sector. This proves that the cosmological constant is needed in the GLPV-Maxwell system in order to have a healthy de Sitter solution.