Multiscalar-metric gravity: merging gravity, dark energy and dark matter

The status of a modification of General Relativity (GR) – Spontaneously Broken Relativity (SBR) – for merging gravity, dark energy (DE) and dark matter (DM) is presented. The modification is principally grounded on a multiscalar-metric concept of spacetime endowed with two dynamical structures: a basic metric and a set of the reversible multiscalar fields. The latter ones serve geometrically as exceptional dynamical coordinates among arbitrary kinematical/observer’s ones and physically as a kind of gravitational Higgs fields producing possible spontaneous breaking of the symmetry (SSB) of relativity. The effective field theory (EFT) of the extended gravity based on the multiscalar-metric spacetime – the metagravity – is discussed and some of its particular realizations beyond GR as SBR are explicated. Physically, SBR results in appearance of massive tensor and scalar gravitons. Some of the emerging consequences, problems and prospects for SBR for future deeper unification of gravity with matter in the context of the relativity and internal symmetries breaking are shortly discussed.

💡 Research Summary

The paper proposes a novel extension of General Relativity (GR) called Spontaneously Broken Relativity (SBR), built on a “multiscalar‑metric” spacetime. In this construction the geometry is equipped with two dynamical structures: (i) the usual metric g_{μν}(x) and (ii) a set of reversible scalar fields Z^{a}(x) (a = 0,…,d‑1). The scalars act as exceptional dynamical coordinates z^{α}(x)=δ^{α}{a} Z^{a}(x) and generate an auxiliary metric ζ{μν}=∂{μ}Z^{a}∂{ν}Z^{b}η_{ab}. By forming the mixed tensor æ_{μν}=g_{μλ}ζ^{λ}{ν} and taking its matrix logarithm Æ{μν}=log æ_{μν}, the authors separate a trace part and a traceless part. The trace yields a scalar field
σ ≡ ½ log(g/ζ)
which they call the “scalar graviton”. The traceless component \tilde{Æ}{μν}=Æ{μν}−(1/d)δ_{μν}Tr Æ is identified with a new tensorial degree of freedom that will play the role of dark energy (DE).

The key idea is that the multiscalar fields behave like gravitational Higgs fields. Under a constant rescaling Z^{a}→e^{γ}Z^{a} the scalar graviton shifts as σ→σ−γ d, revealing an approximate shift symmetry. If this symmetry is only softly broken, σ appears in the Lagrangian predominantly through derivatives, while the traceless part can couple directly to the metric.

The effective field theory (EFT) built from g_{μν} and ζ_{μν} is called “metagravity”. An effective metric is introduced,
\bar{g}{μν}=e^{\bar{γ}(σ)} g{μν},
where the function \bar{γ}(σ) parametrises different realizations of the theory. Two special choices are examined:

1. Spontaneously Broken GR (SBGR) with \bar{γ}=0, so \bar{g}{μν}=g{μν}.
2. Spontaneously Broken Weyl‑Transverse Relativity (SBWTR) with \bar{γ}=−σ/2, giving \bar{g}{μν}=g{μν}/(−g)^{1/4}(−ζ)^{1/4}.

The total Lagrangian consists of:
– Einstein‑Hilbert term for \bar{g}{μν},
– kinetic term for σ with coupling constant Υ (Υ = M_S / M_Pl),
– a dark‑energy potential V_DE( \tilde{Æ}{μν}, σ ), split into a pure‑σ part v_σ(σ) and a pure‑tensor part v_æ( \tilde{Æ}{μν} ),
– conventional matter (CM) sector, and
– a dark‑matter (DM) sector that can source σ through a current J^{μ}{DM}.

Varying the action with respect to g_{μν} yields modified Einstein equations expressed in terms of \bar{g}_{μν} and σ, while variation with respect to Z^{a} leads to an integrated scalar‑graviton equation of motion. The effective potential for σ is
V_eff(σ)=M_Pl²