General relativity extended through a dynamical scalar quartet is proposed as a theory of the scalar-vectortensor gravity, generically describing the unified gravitational dark matter (DM) and dark energy (DE). The implementation in the weak-field limit of the Higgs mechanism for the extended gravity, with a redefinition of metric field, is exposed in a generally covariant form. Under a natural restriction on the parameters, the redefined theory possesses in the linearized approximation a residual transversediffeomorphism invariance, and consistently comprises the massless tensor graviton and a massive scalar one as a DM particle. The number of adjustable parameters in the full nonlinear theory and a partial decoupling of the latter from its weak-field limit noticeably extend the perspectives for the unified description of the gravity DM and DE in the various phenomena at the different scales.

In the general frameworks of an earlier introduced quartet-metric/multi-component gravity, a theory of a massive scalar graviton supplementing the massless tensor one is consistently deduced. The peculiarities of the scalar-graviton field compared to the canonical scalar one are demonstrated. The (ultra-)light scalar graviton is treated as an emergent dark substance of the Universe: dark matter and/or dark energy depending on the solution. The case with scalar graviton as dark energy responsible for the late-time accelerated expansion of the Universe is studied in more detail. In particular, it is shown that due to an attractor solution for the light scalar graviton there naturally emerges at the classical level a tiny nonzero effective cosmological constant, even in the absence of the Lagrangian one. The prospects of going beyond LCDM model per scalar graviton are shortly indicated.

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