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We generalize the recently introduced extended scalar-tensor-Gauss-Bonnet(ESTGB) theories and their close relatives to include spontaneous growth ofnonscalar fields such as vectors. This is analogous to the program thatdeveloped spontaneous tensorization from the original spontaneous scalarizationtheory of Damour and Esposito-Far\`ese (DEF). The new larger family of theoriesconserves the appeal of the DEF theory in terms of conforming to weak-fieldtests and also providing large signals in strong gravity. Moreover, theyprovide a much richer phenomenology including spontaneous tensorization ofblack holes as in ESTGB. These theories, together with other possible futureextensions that we discuss, testify to the ubiquity of spontaneous growth ingravity. We also note that theories with derivative coupling require specialattention since they can lead to potentially problematic higher derivatives inthe equations of motion.

Spontaneous tensorization from curvature coupling and beyond