Quartet-metric gravity and dark components of the Universe

In the report there are presented the general frameworks for the quartet-metric gravity based upon the two physical concepts. First, there exist in space-time the distinct dynamical coordinates, given by a scalar quartet, playing the role of the Higgs fields for gravity. Second, the physical gravity fields arising due to the spontaneous symmetry breaking serve as the dark components of the Universe. It is argued that the mere admixture to metric of the scalar quartet may give rise to an extremely wide spectrum of the emergent gravity phenomena beyond General Relativity (GR). Developing the proposed frameworks further to find out the next-to-GR theory of gravity is a challenge.


Introduction: dark components and quartet-metric gravity
The current state-of-the-art in cosmology may be summarized by the so-called Standard Cosmological Model (or, otherwise, the ΛCDM model). 1 According to the model, the Universe expands with acceleration, is spatially flat (K = 0) with high precision, and possesses at the present epoch by the following • Energy content: Ω K = 0, Ω r + Ω b + Ω DM + Ω Λ = 1, with the partial contributions: -Radiation negligibly small Ω r 1; -Baryonic matter Ω b 0.04; -Cold dark matter (CDM) Ω DM 0.26 (the total matter Ω b+DM 0.3); -Vacuum energy/Λ-term Ω Λ 0.7.
The ΛCDM model proves to be highly successful phenomenologically revealing nevertheless some (mainly theoretical) • Problems: -Nature and properties of DM; -Cosmological constant (CC)/Λ-term (un)naturalness: why Λ is extremely small compared to the Planck mass M P l (or even the electroweak scale M W )?
The solution to the problems may imply, among other things, going beyond General Relativity (GR), upon which ΛCDM is based. In the absence of the convincing observational and/or theoretical arguments in favor of a preferred choice for the beyond GR theory of gravity (if any), it it is natural to try and modify GR restricting himself exclusively by the gravity components already present in metric. 2 In GR, the metric components beyond the tensor graviton get non-physical due to the general relativity, or, technically, the general diffeomorphism (GDiff) invariance. To lift these gauge components from the gauge "cellar" to the observational "attic", converting them into the physical dark components (DCs), the mechanism of the spontaneous symmetry breaking (SSB) for gravity is to be implemented. To this end, in Ref. [2] there were developed the frameworks of the spontaneously broken GR, with the distinct dynamical coordinates, given by a scalar quartet, playing the role of the Higgs-like fields for gravity. The so-constructed quartet-metric gravity (QMG) describes, in general, the scalar-vector-tensor gravity (or some of its reductions). At that, the physical gravity components arising due to SSB are to be treated as the gravitational DM and/or DE, depending on the context [3,4]. The respective frameworks, with some elaboration, are exposed in what follows.

Basic ingredients
Whatever the underlying theory of gravity might be, to be eventually confronted with Particle Standard Model, as well as the observations, it should be realized at the present-day level (or even at the level of a foreseeable future) as the effective field theory (EFT). The latter is generically characterized by the three basic ingredients: the set of fields, the pattern of symmetries and the type of interactions among the given fields satisfying the given symmetries. QMG may be characterized as follows.
• Symmetry pattern: -General covariance (GC): all the (smooth enough) coordinates x µ (including, in particular, z α ) on the space-time manifold are a priori admissible on par . 3

-General relativity (GR)/general diffeomorphism (GDiff) invariance under
GDiff : followed by the associated transformation of the matter fields through their Lie derivative. At that, GDiff serves as the gauge group for gravity.
-Poincare invariance: with the (chart-wise) global translations c a and Lorentz transformations Λ living invariant the Minkowski symbol η ab .
• Type of interactions: defined generically by the gauge and Lorentz-invariant GC scalar action with L the Lagrangian density in terms of the tetrad Z a µ ≡ ∂ µ Z a and a generic matter field Ψ. 4

Basic constructions
The looked-for GC scalar action may straightforwardly be constructed in terms of the GC elements as follows.
• Higgs-like scalar : or, equivalently, the Higgs-like tensor with Z µ a ≡ ∂x µ /∂z α | z=Z(x) the tetrad inverse to Z a µ . This implies the relation and defines another scalar composed of the two basic metrics: as well as similar constructions of the higher degrees, like s 2 = Σ µ ν Σ ν µ = s 2 1 , etc.
• Space-time measure M for the volume element dV ≡ Md 4 x: 4 Due to the assumed (chart-wise) global Poincare invariance, reflecting the homogeneity of the vacuum, there is no derivativeless input of Z a . This allows to treat the second derivative of Z a just as the first derivative of the tetrad Z a µ . 5 It follows that in the quasi-affine coordinates z α one has ζ αβ (z) = η αβ , ζ = −1 and γ γ αβ (z) = 0, what, in fact, distinguishes such coordinates. 6 Such a scalar to implement the Higgs mechanism for graviton was proposed originally in Refs. [5,6,7].
-Gravitational: M g = − det(g µν ) ≡ √ −g; -Quasi-affine/non-gravitating: All the constructions are understood ultimately as the functions of the basic fields g µν and Z a µ = ∂ µ Z a . In building the action, one should, in particular, account for • Measure/Lagrangian ambiguity: without loss of generality one can always choose either the gravitational or quasi-affine measure (or a combination of both) supplemented by a proper redefinition of the Lagrangian, which one gets simpler.
Imposing on space-time the geometrical structure one gets more elaborate elements.
• Quartet-modified metric:g as a linear combination of the two basic metrics, with the Σ-dependent coefficients. This defines, in turn, the quartet-modified Higgs-like tensor the connection-like tensorB and the measureM = − det(g µν ) ≡ −g.
A priori , it is conceivable using in the action S the different metrics. To reduce such an ambiguity one may additionally impose • Geometry universality: all the geometrical constructions on the space-time, such as the Christoffel connectionΓ λ µν , the covariant derivative∇ µ , the measureM = √ −g, etc, are to be attributed to the single metricg µν . Atg µν = g µν the geometry/gravity equivalence is violated due to influence of the vacuum on metric.
Under the chosen geometry, the residual freedom in the action S is to be attributed to the choice of the Lagrangian L specifying the dynamics. By this token one gets Admitting in L conventionally not higher then the first degree of derivative of g µν and Z a µ one may, following the simplicity spirit, impose on S additionally the reduction of the two following principle types.
• Quadratic-derivative: leaving not higher than the second power of the first derivative of g µν and Z a µ , but a priori any input of ζ µν (including those through ζ); • Scalar-graviton: leaving the input of ζ µν only through its determinant ζ, but a priori any power of the first derivative of σ.
Imposing both types of reduction for the scalar graviton results in the maximal simplification (see, below).

Non-linear theory
To illustrate the general frameworks let us consider in more detail the quartet-modified GR [2], which still possesses by the geometry/gravity equivalence as GR,g µν = g µν and M = √ −g, though with the modified quadratic-derivative action: In the above, is the GR Lagrangian, κ g the Planck mass, and is the additional gravity Lagrangian consisting of the potential V , containing the mass and self-interaction terms, and the quadratic kinetic term ∆K. The latter is given by the sum, ∆K = κ i K i , of the partial contributions as follows: with the coefficients κ i , generally, dependent on Σ.

Linear approximation: physics content
Choosing the flat backgrounds, putting x µ = δ µ α z α , designating Z α = δ α a Z a , and (with indices operated by η αβ ) decomposing the fields as one arrives in the linear approximation (LA) at the substitutions This is nothing but the realization in LA of the SSB mechanizm for gravity. In the general case, GDiff gets completely broken/"hidden", with all ten gravity components χ αβ becoming physical and describing the scalar, vector and tensor gravitons. Nevertheless, depending on the particular choice for the Lagrangian parameters, there may be left some residual Diff symmetry to eliminate a part of the components (especially those corresponding to the putatively "dangerous" vector graviton) [2]. To illustrate, let us note the two following physically typical cases.
• Massive scalar graviton [2] σ = χ/2 at with κ 1 , generally, dependent on σ, and V = V (σ) m 2 s /2 σ 2 in addition to L g (χ αβ ) of GR. Due to the residual (three-parameter) transverse diffeomorphism (TDiff) symmetry transforming the trace and traceless part of χ αβ separately as the proper irreducible representations, there is left the three physical components: one for the massive scalar graviton plus two for the massless tensor one, as in GR. 8 • Massive tensor graviton [5,6,7] γ αβ = χ αβ /2 at ∆K = 0 and V = V (Σ) chosen properly to reduce in LA to the conventional Fiertz-Pauli potential V FP (χ αβ ). Under such a choice, there is left for the massive tensor graviton the five physical components, without ghosts.
A wide variety of the more sophisticated cases for SSB of GDiff may a priori be envisaged in QMG, with DCs beyond the two simplest realizations discussed above. All such cases can hardly be rejected just ab initio. To explore them is a challenge.

Resume
• QMG presents a bold and new type of the GR modification built on the mechanism of SSB for gravity. This is assured by a set of the (chart-wise reversible) quasi-affine dynamical coordinates, given by the scalar quartet, serving as the gravity counterpart of the Higgs fields.
• The mere admixture to metric of the scalar quartet is able to give rise to the extremely wide spectrum of the emergent gravity phenomena beyond GR, such as the massive tensor and scalar gravitons, the gravitational DM and/or DE, etc.
• Further theoretical and phenomenological study of QMG is needed to validate it (if any) as a candidate on the beyond GR theory of gravity. Accounting for the influence of vacuum, QMG as EFT may ultimately pave the way towards a more fundamental theory of gravity and space-time.