# Multiscalar-metric gravity: cosmological constant screening and emergence of
massive-graviton dark components of the Universe

Yury F. Pirogov111E-mail<EMAIL_ADDRESS>
Theory Division, Logunov Institute for High Energy Physics of NRC “Kurchatov
Institute”,
Protvino, 142281 Moscow Region, Russia

###### Abstract

In the multiscalar-metric frameworks, the issues of the vacuum
energy/cosmological constant (CC) screening due to the Weyl-scale enhancement
of the Diff gauge symmetry, along with emergence of the massive dark gravity
components through the gravitational Higgs mechanism are considered. A generic
dark gravity model is developed, with the two extreme versions of the model of
particular interest based on General Relativity (GR) and its classically
equivalent Weyl transverse alternative being compared and argued to be,
generally, inequivalent. The so constructed spontaneously broken Weyl
Transverse Relativity (WTR) is proposed as a viable beyond-GR effective field
theory of gravity, with screening of the Lagrangian CC, superseded by the
induced one, and emergence of the massive tensor and scalar gravitons as the
dark gravity components. A basic concept with the spontaneously broken Diff
gauge symmetry/relativity – in particular, WTR vs. GR – as a principle source
of the emergent dark gravity components of the Universe is put forward.

Keywords: modified gravity, Weyl transverse relativity, Higgs mechanism for
gravity, vacuum energy, cosmological constant, dark energy, dark matter.

##  1 Introduction

The issue of the vacuum energy/cosmological constant (CC) in General
Relativity (GR) due to the so-called zero-point quantum fluctuations is well-
known already for a long time.222For an original treatment of the vacuum
energy/CC problem, see, [1]. But only recently, in the wake of the appearance
of the convincing observational evidences for (an effective) CC responsible
for the late-time accelerated expansion of the Universe, as expressed by the
cosmological Standard Model (SM), or, otherwise, the LCDM model, the problem
of CC (or, more generally, the vacuum energy) seems to become the most crucial
one in the realm of the contemporary fundamental physics.333For the
observational status of CC, see, e.g., [2]. For the modern view of the vacuum
energy/CC problem, see, e.g., [3, 4]. Though not meaning any explicit
discrepancy with observations, this problem signifies nevertheless at least a
tension between the two present-day basic physical theories: the theory of
gravity – GR and the theory of matter and its interactions — the quantum field
theory (QFT) — expressed more particularly by the particle SM. Namely, in the
effective field theory (EFT) framework the CC problem is, in fact, at least
threefold:444For the importance of the EFT approach to the CC problem, cf,
e.g., [5].

(I) The (un)naturalness problem: What forbids the huge Lagrangian CC, expected
in the EFT framework, from the explicit manifestations, with a non-zero tiny
observational CC being (technically) “unnatural”?

(II) The coincidence problem: What defines such a tiny value of the
observational CC, so that the latter manifests itself in the accelerated
expansion of the Universe just “lately” on the cosmological time scale?

(III) The quantum stability problem: What prevents the tiny observational CC
from being drastically renormalized by the radiative corrections, a priori
expected in the EFT framework?

Solving these and related problems presents a challenge to the modern
fundamental physics and could ultimately imply either a revision of QFT, or a
GR modification, with a variety of the latter ones proposed to this end up to
now.555For the modifications of gravity, see, e.g., [6]–[8] and, in
particular, [9, 10] for the massive gravity. More particularly, to solve the
CC problem the modified gravity should, conceivably, be treated as EFT valid
at the relatively low energy scales (compared to the fundamental one given by
the Planck mass). At that, GR as EFT is well-known, first, to be constructed
from a symmetric second-rank tensor field – the metric $g_{\mu\nu}$, with its
determinant $g$ defining a four-volume element $\sqrt{-g}d^{4}x$ (in the
spacetime dimension $n=4$), with the spacetime measure $\sqrt{-g}$, and,
second, to be based on the gauge symmetry of the (full) diffeomorphisms
(Diff’s) incorporating, in particular, the longitudinal ones. The fact that
the spacetime measure $\sqrt{-g}$ transforms under the longitudinal
diffeomorphisms proves to be in GR (and its siblings) an ultimate reason of
the emergent CC problem. By this token, we take the issue of $g$ as a guide
for choosing a proper route towards solving the CC problems, with some
principle steps leading to this goal shortly indicated below.

(i) Unimodular relativity. First of all, long before the CC problem had become
so acute, for solving such a problem there was proposed in [11] (with the
numerous subsequent elaborations) to substitute GR based on the full Diff
symmetry by Unimodular Relativity (UR), known also as Unimodular Gravity,
based on the transverse diffeomorphism (TDiff $\subset$ Diff) symmetry.666For
the relevance of the TDiff gauge symmetry as a substitute of the full Diff one
for the consistent description of the massless tensor graviton, see [12]. By
construction, instead of the conventional ten-component (at $n=4$) metric
$g_{\mu\nu}$, UR/TDiff uses a reduced nine-component one $\hat{g}_{\mu\nu}$,
with the determinant $\hat{g}={{\kappa}}$ given by a fixed absolute/non-
dynamical scalar density ${{{\kappa}}}$ (for simplicity, say,
${{\kappa}}=-1$).777The notation in Introduction is in accord with what
follows. The latter defines simultaneously the spacetime volume element
$\sqrt{-{\kappa}}\,d^{4}x$. The Lagrangian CC within UR proves to be
irrelevant being superseded by an integration constant having yet no clear-cut
physical meaning. Due to this, UR remains, in a sense, dynamically incomplete.
Nevertheless, solving the first part of the CC problem, UR definitely presents
a step in the right direction, though hardly sufficient to completely solve
the problem, and implies further elaboration.

(ii) Weyl transverse relativity. In the latter respect, in [13] there was
proposed (cf. also [14]) a ten-component modification of UR constructed from a
relative tensor $\hat{g}_{\mu\nu}\equiv g_{\mu\nu}/(-g)^{1/4}$ (at $n=4$),
with a fixed determinant (put for definiteness $\hat{g}=-1$), the former being
liable to be converted into the true tensor through multiplication by
$(-{{\kappa}})^{1/4}$. By construction, the respective EFT of gravity — to be
called Weyl Transverse Relativity (WTR) — satisfies a gauge symmetry, WTDiff,
consisting of TDiff enhanced by the Weyl scale transformations, and provides
ultimately an improved, compared to UR, solution of the CC problem.888For a
development of WTR, known also as Weyl Transverse Gravity, cf., e.g., [15,
16]. Likewise UR/TDiff, its modification WTR/WTDiff may naturally give a
justification of the disappearance of the (huge) Lagrangian CC. At the
classical level (under the covariant conservation of the energy-momentum
tensor) WTR/WTDiff proves to be equivalent to GR/Diff, with an arbitrary
induced CC to be treated as a global degree of freedom. Moreover at the
quantum level, WTR/WTDiff could ensure the quantum stability of the residual
classical CC against the radiative corrections [17, 18]. Besides, on the
field-theoretic side, WTR may be obtained as a viable alternative to GR
proceeding in a self-consistent manner, in the complete analogy to GR, from a
theory of the precisely two-d.o.f. self-interacting massless graviton
[19].999This fact alone could, conceivably, justify WTR (and its siblings) to
be further explored at least on par with GR (and its siblings). Altogether,
solving (at least partially) the first and the third parts of the CC problem
and possessing a solid field-theoretic status WTR/WTDiff could thus constitute
a viable alternative to GR/Diff, realizing one more step in the right
direction, with the next one(s) still needed. More particularly, within WTR
per se (likewise UR) there appears no indication on a infrared mass scale for
the tiny residual CC, with the second part of the CC problem remaining to be
resolved, too.

(iii) Higgs mechanism for gravity. The looked-for infrared mass scale could
naturally be provided by a massive graviton, the latter serving as the
putative dark energy (DE) mimicking the observed CC. At that, the original
tensor-graviton mass term due to Fierz and Pauli [20] explicitly violates the
Diff gauge symmetry, and was later shown to result in the massive-massless
perturbative discontinuity [21, 22], to remedy which there was proposed [23] a
nonperturbative mechanism for recovering the transition. It was also found
[24] that such an explicit Diff gauge symmetry violation results in the
appearance of a ghost, with a violation of unitarity. Thus, with the massless
GR treated as a Diff gauge theory, the consistent ghost-free theory of massive
gravity would, conceivably, imply not the explicit violation but the
spontaneous breaking of such a gauge symmetry by a counterpart of the Higgs
mechanism for gravity. An attempt to approach such a goal, modifying GR
through the Higgs mechanism in terms of a quartet of the scalar fields, was
undertaken in [25] (see also [26, 27]), in the wake of which there was
proposed in [28]–[30] a simple solution to the problem elaborated further in
[31, 32]. In this respect, GR augmented by the Higgs mechanism for gravity
could naturally provide a required scale for the late exponential expansion of
the Universe, associating such a scale with the tiny tensor-graviton mass. The
latter, on the other hand, should naturally be treated as a parameter of the
theory (given still ad hoc) likewise any other fundamental parameter of
Nature. Out of the four (at $n=4$) scalar fields used in such a Higgs
mechanism within GR, the three components can, at a proper choice of the
(quasi-)Higgs potential, be absorbed as the additional components of the
massive tensor graviton. At that, one more component may be stated to become a
ghost independently of the potential, though in a sufficiently high order of
the perturbation theory [32]. Besides, in the massive-modified GR there still
remains unclear the reason of absence of the (huge) Lagrangian CC, as well as
the radiative stability of the Higgs mechanism (in the absence of a protecting
symmetry). So, the Higgs mechanism in GR is on its own hardly sufficient to
exhaust the CC problem, too.

(iv) Quartet-metric gravity. Altogether, merging the above mentioned
directions of the gravity modification to reconcile the various parts of the
CC problem, with the two principle ingredients of the properly modified
gravity — an effective metric $\hat{g}_{\mu\nu}$, $\hat{g}={{\kappa}}$, and a
(quasi-)Higgs field — to stem from the same source — the scalar quartet — may
be obligatory.101010This, in turn, ensures one more important ingredient for
solving the CC problem – the dynamical non-gravitating spacetime measure
$\sqrt{-{{\kappa}}}$. For the relevance of such a measure built of the so-
called “measure fields” within the modified GR still with the conventional
metric $g_{\mu\nu}$, cf., e.g., [33, 34]. A new concept should, conceivably,
be invoked to this end, with a change of the spacetime paradigm [35, 36].
According to the latter, among the arbitrary kinematic observer’s coordinates
there exist some distinct dynamical ones, associated ultimately with the
vacuum and given by a quartet (at $n=4$) of the peculiar (invertible) scalar
fields. Such a scalar quartet is to be considered on par with the metric as a
basic field variable to construct EFT of the so-called quartet-metric gravity.
Introducing in such a framework, on the one hand, a proper effective metric,
without adding the (quasi-)Higgs potential for the massive tensor graviton,
one can eliminate the (huge) Lagrangian CC as in WTR. On the other hand,
introducing the proper (quasi-)Higgs potential without modifying metric, one
can produce, as in GR, the massive tensor graviton serving, conceivably, as
DE. Superimposing these two routes of the gravity modification within the
quartet-modified WTR may, conceivably, solve both the problem of screening of
the (huge) Lagrangian CC and emergence of the (tiny) observable one
supplemented by the associated dark gravity components – the massive tensor
and scalar gravitons. This is a goal of the present paper.

In section 2, the basics of the quartet-metric (dark) gravity are exposed. In
section 3, a sufficiently general prototype dark gravity model is constructed
to realize the concept of the spontaneously broken Diff gauge
symmetry/relativity as a source of the emergent dark gravity components. The
two extreme versions of such a generic model of the particular interest
corresponding, respectively, to the spontaneous breaking of GR vs. its
classically equivalent alternative — WTR — are worked out in sections 4 and 5.
These versions are compared in the context of the CC screening and emergence
of the dark gravity components through the gravitational Higgs mechanism, and
are argued to be, generally, non-equivalent. The consistency of the so
constructed spontaneously broken WTR as a theory of the massive tensor and
scalar gravitons is demonstrated in the weak-field limit. In Conclusion, the
viability of the spontaneously broken WTR as a beyond-GR EFT of gravity
solving the CC problem through screening the Lagrangian CC, superseding it by
the induced one and supplementing by the massive tensor and scalar gravitons
as the dark gravity components is argued. The importance of further developing
the basic concept of the spontaneously broken Diff gauge symmetry/relativity —
in particular, WTR vs. GR — as a principle reason of emergence of the dark
gravity components of the Universe is stressed.111111Though adhering primarily
to the spacetime dimension $n=4$, not to be formally confined by this value,
the term “scalar quartet” valid at $n=4$ is substituted in what follows by a
more general one — the “multiscalar” — and, accordingly, “quartet-metric” by
“multiscalar-metric”, etc.

##  2 Multiscalar-metric dark gravity: basics

###  2.1 Multiscalar-extended field set

Under the term “dark gravity” there will be understood a merger of the
(conventional) gravity with the emergent dark gravitational components. The
EFT of the multiscalar-metric dark gravity may basically be defined through
the generally covariant action functional121212Here we start directly from the
EFT level. For the justification of EFT of the multiscalar-metric gravity as
an affine-Goldstone nonlinear model, see [37].

$S[g_{\mu\nu},{{X}}^{a}]=\int{\cal L}(g_{\mu\nu},{{X}}^{a})\,d^{n}x$ (1)

in terms of a Lagrangian density $\cal L$ dependent on a tensor field — the
basic metric — $g_{\mu\nu}$ (possessing by an inverse $g^{-1\mu\nu}$) extended
through a set of the scalar fields $X^{a}$, with $a,b,\dots=0,1,\dots,n-1$
being the global Lorentz indices and $n$ a spacetime dimension. The scalar
multiplet $X^{a}=X^{a}(x)$ — the multiscalar — is peculiar by the fact of
being (in a patch-wise manner) invertible, $x^{\mu}=x^{\mu}(X)$. This allows
to use it as some distinct dynamical spacetime coordinates
$X^{\alpha}\equiv\delta^{\alpha}_{a}{X}^{a}(x)$ among the arbitrary kinematic
ones $x^{\mu}$, as well as to introduce the respective dynamical frame
$X^{\alpha}_{\mu}\equiv\partial_{\mu}X^{\alpha}$, $\det(X^{\alpha}_{\mu})\neq
0$, possessing by an inverse $X^{-1}{}_{\alpha}^{\mu}\equiv X_{\alpha}^{\mu}$.
Stress that such a multiscalar is not an auxiliary (non-dynamical) variable
introduced just to restore the general covariance, but a faithful dynamical
variable on par with the basic metric. Ultimately, the dynamical coordinates
$X^{\alpha}$ may be treated as those associated with the (dynamical) vacuum.
The action $S$ is moreover postulated to be invariant under the
reparametrizations of ${X}^{a}$ given by the Lorentz transformations and the
finite constant shifts, $X^{a}\to X^{a}+C^{a}$. By this token, ${X}^{a}$
should, in fact, enter through a derivative term ${{\kappa}}_{\mu\nu}$
constructed as

${{\kappa}}_{\mu\nu}\equiv\partial_{\mu}{X}^{a}\partial_{\nu}{X}^{b}\eta_{ab},$
(2)

with $\eta_{ab}$ being the Lorentz-invariant Minkowski symbol. By default,
${{\kappa}}_{\mu\nu}$, serving as a (quasi-)metric, possesses by an inverse
${{\kappa}}^{-1\mu\nu}$ (at least patch-wise with the proper matching), with

${{\kappa}}\equiv\det({{\kappa}}_{\mu\nu})=(\det(\partial_{\mu}{X}^{a}))^{2}\det(\eta_{ab})<0$
(3)

to ensure the non-degeneracy and invertibility of such a
(quasi-)metric.131313To this end, we assume the (patch-wise) Lorentz-
preservation in the field space of $X^{a}$, up to the physically equivalent
affine redefinitions of the fields $X^{a}$ and the metric $\eta_{ab}$. In
terms of the independent field variables, the multiscalar-metric action, in
fact, looks like

$S[g_{\mu\nu},{{X}}^{a}]=\int{\cal L}(g_{\mu\nu},{\kappa}_{\mu\nu})\,d^{n}x.$
(4)

Further, in terms of ${{\kappa}}$ and $g=\det(g_{\mu\nu})<0$ one can define an
effective scalar field

$\sigma=\ln(\sqrt{-g}/\sqrt{-{{\kappa}}}),$ (5)

to be called the scalar graviton. By this token, in terms of the fields
$g_{\mu\nu}$ and ${{\kappa}}_{\mu\nu}$ one can introduce the effective ones as
a conformal metric $\bar{g}_{\mu\nu}$ and a tensor field
$\bar{{\ae}}^{\mu}_{\nu}$ given by a correlator (partial contraction) of the
two metrics as follows:

$\bar{g}_{\mu\nu}\equiv
e^{{\bar{w}}}g_{\mu\nu},\quad\bar{{\ae}}^{\mu}_{\nu}\equiv\bar{g}^{\mu\lambda}{{\kappa}}_{\lambda\nu},$
(6)

where ${\bar{w}}={\bar{w}}(\sigma)$ is a gravitational scale factor,141414The
dependence $\bar{w}(\sigma)$ presents the simplest case. More generally, one
could envisage within EFT the dependence of the effective metric scale
$\bar{w}$ on the matter fields, environment, etc, with the ultraviolet and
infrared behavior of the effective metric being a priori quite different.
$\bar{g}^{\mu\nu}\equiv\bar{g}^{-1\mu\nu}=e^{-{\bar{w}}}g^{-1\mu\nu}$ is an
inverse of $\bar{g}_{\mu\nu}$ and
$\det(\bar{{\ae}}^{\mu}_{\nu})={{\kappa}}/\bar{g}=e^{-2(2{\bar{w}}+\sigma)}$,
$\bar{g}\equiv\det(\bar{g}_{\mu\nu})<0$.151515By default, the spacetime
indices are now lowered (raised) through the effective metric
$\bar{g}_{\mu\nu}$ ($\bar{g}^{\mu\nu}$) if not stated explicitly otherwise.
The tensor field $\bar{{\ae}}^{\mu}_{\nu}$ may be treated as a realization of
the (a priori loose) DE. By means of these effective fields, merging gravity
and its dark components, the effective action of the purely multiscalar-metric
dark gravity may be presented quite generally as

$S[g_{\mu\nu},{{X}}^{a}]=\int\bar{L}(\bar{g}_{\mu\nu},\bar{{\ae}}^{\mu}_{\nu},\sigma)\sqrt{-\bar{g}}\,d^{n}x,$
(7)

with $\bar{L}$ being an effective Lagrangian defining, in the line with
${\bar{w}}$, a particular dark gravity model. Due to not all of the variables
in such a form of the effective action being independent, the action is
ultimately still a function of the independent basic variables $g_{\mu\nu}$
and $X^{a}$. At last, by means of the frame
$X^{a}_{\mu}\equiv\partial_{\mu}X^{a}$ and its inverse $X_{a}^{\mu}$, the DE
tensor $\bar{{\ae}}^{\mu}_{\nu}$ may equivalently be converted into the scalar
form as $\bar{{H}}^{a}_{b}\equiv
X_{\mu}^{a}\bar{{\ae}}^{\mu}_{\nu}X_{b}^{\nu}$, with
$\det(\bar{{\ae}}^{\mu}_{\nu})=\det(\bar{{H}}^{a}_{b})$, where

$\bar{{H}}^{a}_{c}\eta^{cb}\equiv\bar{{H}}^{ab}=\bar{g}^{{\kappa}\lambda}\partial_{\kappa}{X}^{a}\partial_{\lambda}{X}^{b}$
(8)

is a modification of the original (quasi-)Higgs field for gravity [28]–[30].
Ultimately, the effective action in the form (7), with the dependence on the
DE field $\bar{{\ae}}^{\mu}_{\nu}$ expressed through a scalar potential
$\bar{V}_{{\ae}}(\bar{{\ae}}^{\mu}_{\nu})=\bar{V}_{{H}}(\bar{{H}}^{ab})$, is
aimed at the spontaneous breaking of the Diff gauge symmetry/relativity, with
producing the dark gravity components ultimately due to the multiscalar.

###  2.2 Weyl-scale enhanced Diff gauge symmetry

Besides a set of fields, any EFT is basically characterized by a pattern of
its symmetries. For the generic multiscalar-metric dark gravity, the effective
action (7) is assumed to be generally covariant and gauge invariant under the
(full) Diff symmetry given by a Lie derivative $D_{\xi}$. The latter depends
on a (infinitesimal) vector parameter $\xi^{\mu}(x)$ and may by default be
constructed in terms of a covariant derivative $\nabla_{\lambda}(g_{\mu\nu})$
defined by the basic metric $g_{\mu\nu}$, subsequently expressed through the
effective fields as $g_{\mu\nu}=e^{-\bar{w}(\sigma)}\bar{g}_{\mu\nu}$. Under
$D_{\xi}$, all the basic fields transform conventionally in accord with their
covariant tensor structure still irrespective of their nature. To account for
the different origin of the basic fields, consider the (local) Weyl scale
transformations distinguishing such fields as follows:161616A priori, the Weyl
scale symmetry is assumed to be the specifically metric one, with the
conventional matter fields, likewise $X^{a}$, to be inert under the Weyl
rescalings, unless stated otherwise.

${\Delta}_{\zeta}g_{\mu\nu}=\zeta g_{\mu\nu},\quad{\Delta}_{\zeta}X^{a}=0,$
(9)

where $\zeta(x)$ is an arbitrary (infinitesimal) scalar parameter, so
that171717For definiteness, we restrict consideration here and in what follows
by the spacetime dimension $n=4$.

$\displaystyle{\Delta}_{\zeta}g$ $\displaystyle=$ $\displaystyle 4\zeta g,$
$\displaystyle{\Delta}_{\zeta}\Big{(}g_{\mu\nu}/(-g)^{1/4}\Big{)}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\Delta}_{\zeta}{{\kappa}}_{\mu\nu}={\Delta}_{\zeta}{{\kappa}}$
$\displaystyle=$ $\displaystyle 0.$ (10)

For the effective fields, this implies in turn

$\displaystyle{\Delta}_{\zeta}\sigma$ $\displaystyle=$ $\displaystyle 2\zeta,$
$\displaystyle{\Delta}_{\zeta}\bar{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\zeta(1+2{\bar{w}}^{\prime})\bar{g}_{\mu\nu},$
$\displaystyle{\Delta}_{\zeta}\bar{g}$ $\displaystyle=$ $\displaystyle
4\zeta(1+2{\bar{w}}^{\prime})\bar{g},$
$\displaystyle{\Delta}_{\zeta}\bar{{\ae}}^{\mu}_{\nu}$ $\displaystyle=$
$\displaystyle-\zeta(1+2{\bar{w}}^{\prime})\bar{{\ae}}^{\mu}_{\nu},$ (11)

where ${\bar{w}}^{\prime}=d{\bar{w}}/d\sigma$. Generally, this reflects the
explicit Weyl scale violation in the multiscalar-metric dark
gravity.181818Note though, that the exclusively derivative couplings of
$\sigma$ still admit the residual global Weyl scale symmetry $g_{\mu\nu}\to
e^{\zeta_{0}}g_{\mu\nu}$, with the constant shifts
$\sigma\to\sigma+2\zeta_{0}$. In an exceptional case
${\bar{w}}^{\prime}=-1/2$, one independent component of $g_{\mu\nu}$, namely,
the determinant $g$ is missing from $\bar{g}_{\mu\nu}$ and
$\bar{{\ae}}^{\mu}_{\nu}$. At that, the residual dependence of the effective
action (7) on $g$ is mediated only by $\sigma$. Under $g$ being a principle
source of the CC problem, such a Weyl scale symmetry, signifying otherwise the
$g$-independence, is of the particular importance for the CC screening in an
exceptional case of a generic dark gravity model to be exposed in what
follows.

##  3 Spontaneous Diff gauge symmetry/relativity breaking

###  3.1 Generic dark gravity model

To be more particular with the (multiscalar) spontaneous breaking of the Diff
gauge symmetry (or, otherwise, relativity) consider a prototype dark gravity
model merging tensor gravity and the associated dark gravity components, with
the effective Lagrangian in a restricted but still rather general partitioned
form

$\bar{L}=\bar{L}_{G}(\bar{g}_{\mu\nu},\bar{{\ae}}^{\mu}_{\nu})+\bar{L}_{M}(\bar{g}_{\mu\nu},\sigma,\phi_{I})+\Delta\bar{L},$
(12)

consisting of an extended gravity Lagrangian $\bar{L}_{G}$, an extended matter
Lagrangian $\bar{L}_{M}$ and a rest, $\Delta\bar{L}$. By default,
$\bar{L}_{G}$ may depend a priori arbitrarily on the tensor fields
$\bar{g}_{\mu\nu}$ and $\bar{\ae}^{\mu}_{\nu}$ (and thus on
${{\kappa}}_{\mu\nu}$), while $\bar{L}_{M}$ is assumed to be independent of
$\bar{\ae}^{\mu}_{\nu}$ and to depend only minimally on $\bar{g}_{\mu\nu}$,
but a priori arbitrarily on the scalar graviton $\sigma$ and the generic
matter fields $\phi_{I}$, $I=1,\dots$. The mixing part $\Delta\bar{L}$,
dependent, generally, on all the variables, is assumed to be negligible in the
main approximation.191919In the spirit of EFT, $\bar{L}_{G}$ is assumed to be
of the order of the Plank mass squared, $M_{\rm Pl}^{2}$, while $\bar{L}_{M}$
and $\bar{\Delta}L$ to be relatively suppressed in the scale of $M_{\rm Pl}$.
More particularly, adopt for $\bar{L}_{G}$ the partitioned form:

$\bar{L}_{G}=\bar{L}_{\Lambda}+\bar{L}_{g}(\bar{g}_{\mu\nu})-\bar{V}_{{\ae}}(\bar{{\ae}}^{\mu}_{\nu}),$
(13)

where the conventional gravity part, $\bar{L}_{g}$, is taken for simplicity in
the GR-like form for $\bar{g}_{\mu\nu}$:

$\bar{L}_{g}=-\frac{1}{2}M_{\rm Pl}^{2}R(\bar{g}_{\mu\nu}),$ (14)

with $M_{\rm Pl}=1/(8\pi G_{N})^{1/2}$ being the Planck mass and
$\bar{R}=R(\bar{g}_{\mu\nu})$ the Ricci scalar curvature.202020In principle,
any modification of $\bar{L}_{g}$, like $f(\bar{R})$, etc, is conceivable,
too. This term is supplemented by a constant one

$\bar{L}_{\Lambda}\equiv-M_{\rm Pl}^{2}\bar{\Lambda},$ (15)

with $\bar{\Lambda}$ being an effective Lagrangian CC. At last,
$\bar{V}_{{\ae}}$ is a scalar potential dependent on the traces of the
products of the DE tensor field $\bar{{\ae}}^{\mu}_{\nu}$:212121At that,
$\det(\bar{{\ae}}^{\mu}_{\nu})={{\kappa}}/\bar{g}=e^{-2(2{\bar{w}}(\sigma)+\sigma)}$
is explicitly missed being expressed, in principle, through $\sigma$ accounted
in $\bar{L}_{M}$.

$\bar{V}_{{\ae}}=M_{\rm
Pl}^{2}\sum\bar{v}_{n_{1}n_{2},\dots}\Big{(}\mbox{tr}\,({\bar{{\ae}}}^{\mu}_{\nu})^{n_{1}}\mbox{tr}\,({\bar{{\ae}}}^{\mu}_{\nu})^{n_{2}}\dots\Big{)},$
(16)

with a priori arbitrary coefficients $\bar{v}_{n_{1}n_{2}\dots}$. The powers
$n_{i}=0,1,2,\dots$ are to be positive finite for the potential to be analytic
near $\bar{{\ae}}^{\mu}_{\nu}=0$, allowing in such a case a smooth limit to
GR. The potential may be normalized as $\bar{V}_{{\ae}}|_{\star}=0$ in a
priori arbitrary background point in the unbroken or spontaneously broken
phases, say, at $\bar{{\ae}}_{\star}{}^{\mu}_{\nu}=0$ or
$\bar{{\ae}}_{\star}{}^{\mu}_{\nu}=\delta^{\mu}_{\nu}$, respectively,
defining, in turn, the proper normalization of $\bar{\Lambda}$. In what
follows, we restrict consideration by the simplest quartic potential which
proves to be sufficient in the leading approximation (see, section 5). The
consistency of the theory in the higher-order approximations may require a
further elaboration of the potential including, in particular, the fractional
powers of $\bar{{\ae}}^{\mu}_{\nu}$, as well as the negative ones
corresponding to
$(\bar{{\ae}}^{\mu}_{\nu})^{-1}\equiv\bar{{\ae}}^{-1}{}^{\mu}_{\nu}={{\kappa}}^{-1}{}^{\mu\lambda}\bar{g}_{\lambda\nu}$,
etc.222222A priori, one may admit another mode of the spontaneous braking,
with the potential $\bar{V}_{\ae}$ dependent instead of
$\bar{\ae}^{\mu}_{\nu}$ on its inverse $\bar{\ae}^{-1}{}^{\mu}_{\nu}$, subject
to the analyticity requirement near $\bar{\ae}^{-1}{}^{\mu}_{\nu}=0$ and
possessing by the flat background at the symmetric point
$\bar{\ae}^{-1}|_{\star}{}^{\mu}_{\nu}=\bar{\ae}|_{\star}{}^{\mu}_{\nu}=\delta^{\mu}_{\nu}$.
Finally, the extended matter Lagrangian $\bar{L}_{M}$ (incorporating,
conceivably, some kinds of the particle DM in addition to the scalar-graviton
dark component) remains a priori arbitrary, elaborating further the purely-
tensor dark gravity due to $\bar{L}_{G}$.

###  3.2 Generic field equations

A straightforward way of deriving the dynamical field equations (FEs) for EFT
of the multiscalar-metric dark gravity is to extremize its effective action
with respect to the independent variations of the primary metric $g_{\mu\nu}$
(or, rather, its inverse $g^{-1\mu\nu}$) and the multiscalar ${X}^{a}$,
expressing afterwards the coefficients at the independent variations $\delta
g^{-1\mu\nu}$ and $\delta{X}^{a}$ through the effective fields
$\bar{g}_{\mu\nu}$ (and its inverse $\bar{g}^{\mu\nu}$), as well as $\sigma$
(in particular, ${\bar{w}}(\sigma)$). By this token, the effective variations
look like

$\displaystyle\delta\bar{g}^{\mu\nu}$ $\displaystyle=$ $\displaystyle
e^{-{\bar{w}}}\delta
g^{-1\mu\nu}-{\bar{w}}^{\prime}\bar{g}^{\mu\nu}\delta\sigma,$
$\displaystyle\frac{\delta\sqrt{-\bar{g}}}{\sqrt{-\bar{g}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{g}_{{\kappa}\lambda}\delta\bar{g}^{{\kappa}\lambda}=-\frac{1}{2}e^{-{\bar{w}}}\bar{g}_{{\kappa}\lambda}\delta
g^{-1{\kappa}\lambda}+2{\bar{w}}^{\prime}\delta\sigma,$ (17)

with a prime meaning the derivative with respect to $\sigma$, as well as

$\delta\bar{{\ae}}^{\mu}_{\nu}=\bar{g}^{\mu\lambda}\delta{{{\kappa}}}_{\lambda\nu}+{{\kappa}}_{\nu\lambda}\delta\bar{g}^{\lambda\mu}.$
(18)

In the above, one should also account for the constraints (5) and (2)
(implying $\sigma$ and ${{\kappa}}_{\mu\nu}$ to be the effective composite
fields), resulting due to

$\frac{\delta\sqrt{-g}}{\sqrt{-g}}=-\frac{1}{2}e^{-{\bar{w}}}\bar{g}_{{\kappa}\lambda}\delta
g^{-1{\kappa}\lambda},\quad\frac{\delta\sqrt{-{{{\kappa}}}}}{\sqrt{-{{{\kappa}}}}}=\frac{1}{2}{{\kappa}}^{-1{\kappa}\lambda}\delta{{\kappa}}_{{\kappa}\lambda},$
(19)

in the respective variation of $\sigma$

$\delta\sigma=\frac{\delta\sqrt{-g}}{\sqrt{-g}}-\frac{\delta\sqrt{-{{\kappa}}}}{\sqrt{-{{\kappa}}}}=-\frac{1}{2}(e^{-{\bar{w}}}\bar{g}_{{\kappa}\lambda}\delta{g}^{-1{\kappa}\lambda}+{{{\kappa}}}^{-1{\kappa}\lambda}\delta{{{\kappa}}}_{{\kappa}\lambda}),$
(20)

where

$\delta{{{\kappa}}}_{\mu\nu}=({{X}}^{a}_{\mu}\delta{{X}}^{b}_{\nu}+{{X}}^{a}_{\nu}\delta{{X}}^{b}_{\mu})\eta_{ab},\
\ \delta{X}^{a}_{\mu}\equiv\partial_{\mu}\delta{X}^{a}.$ (21)

Extremizing then the effective action (7), given by the effective Lagrangian
(12)–(16), with respect to the independent variations $\delta g^{-1\mu\nu}$
and $\delta{{\kappa}}_{\mu\nu}$ (the latter expressed finally through
$\delta{X}^{a}$), we get the generic (under an arbitrary ${\bar{w}}$) FEs:

$\displaystyle\bar{R}_{\mu\nu}-\frac{1}{4}\bar{R}\bar{g}_{\mu\nu}-\frac{1}{M_{\rm
Pl}^{2}}(\bar{T}_{\mu\nu}-\frac{1}{4}\bar{T}\bar{g}_{\mu\nu})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle-\frac{1}{4}(1+2{\bar{w}}^{\prime})(\bar{R}+4\bar{\Lambda}+\frac{1}{M_{\rm
Pl}^{2}}\bar{T})+\frac{1}{M_{\rm
Pl}^{2}}\frac{\delta\bar{L}_{M}}{\delta\sigma}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\bar{\nabla}_{\lambda}\Big{(}\frac{1}{M_{\rm
Pl}^{2}}\bar{g}^{\lambda\rho}\frac{\partial\bar{V}_{{\ae}}}{\partial\bar{{\ae}}^{\rho}_{\kappa}}{X}_{{\kappa}a}+\frac{1}{8}(\bar{R}+4\bar{\Lambda}+\frac{1}{M_{\rm
Pl}^{2}}\bar{T}){X}^{\lambda}_{a}\Big{)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{\delta\bar{L}_{M}}{\delta\phi_{I}}$ $\displaystyle=$
$\displaystyle 0$ (22)

for the three sectors of the multiscalar-metric dark gravity — respectively,
tensor-traceless, tensor-trace and multiscalar — supplemented by FE for
matter. Here and in what follows, $\partial/\partial$ and $\delta/\delta$
mean, respectively, the partial and total (incorporating the derivatives with
respect to the derivatives of the proper fields) variational
derivatives.232323At that, the scalar-graviton FE as such is absent, with the
composite $\sigma$ being, generally, off-mass-shell,
$\delta\bar{L}_{M}/\delta\sigma\neq 0$. Besides, $X_{{\kappa}a}\equiv
X^{b}_{\kappa}\eta_{ba}$ and

$\bar{T}_{\mu\nu}\equiv\bar{T}_{{\ae}\mu\nu}+\bar{T}_{M\mu\nu},\quad\bar{T}\equiv\bar{g}^{{\kappa}\lambda}\bar{T}_{{\kappa}\lambda}=\bar{T}_{{\ae}}+\bar{T}_{M},$
(23)

is the total energy-momentum tensor for the DE field $\bar{\ae}^{\mu}_{\nu}$
and the extended matter $M=(\sigma,\phi_{I})$, with the partial contributions,
respectively, as follows:

$\displaystyle\bar{T}_{{\ae}\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\Big{(}\frac{\partial\bar{V}_{{\ae}}}{\partial\bar{\ae}^{\mu}_{\lambda}}\bar{g}_{\rho\nu}+\frac{\partial\bar{V}_{{\ae}}}{\partial\bar{\ae}^{\nu}_{\lambda}}\bar{g}_{\rho\mu}\Big{)}\bar{\ae}^{\rho}_{\lambda}+\bar{V}_{{\ae}}\bar{g}_{\mu\nu},$
$\displaystyle\bar{T}_{M\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{\sqrt{-\bar{g}}}\frac{\partial(\sqrt{-\bar{g}}\bar{L}_{M})}{\partial\bar{g}^{\mu\nu}}=2\frac{\partial\bar{L}_{M}}{\partial\bar{g}^{\mu\nu}}-\bar{L}_{M}\bar{g}_{\mu\nu}.$
(24)

A priori, the derived generic FEs do not automatically imply the covariant
conservation of the total energy-momentum tensor, to be discussed below.

###  3.3 Geometry consistency condition

To explicitly account for the Riemannian structure of the spacetime in terms
of the effective metric $\bar{g}_{\mu\nu}$, apply the proper covariant
derivative $\bar{\nabla}_{\mu}$ to the sum of the trace-free and trace (times
$\bar{g}_{\mu\nu}$) parts of the tensor-gravity FEs. Due to the reduced
Bianchi identity,
$\bar{\nabla}^{\lambda}(\bar{R}_{\lambda\nu}-1/2\,\bar{R}\bar{g}_{\lambda\nu})=0$,
this gives a geometrical consistency condition

$-\frac{1}{2}\partial_{\mu}\Big{(}{\bar{w}}^{\prime}(\bar{R}+4\bar{\Lambda}+\frac{1}{M_{\rm
Pl}^{2}}\bar{T})\Big{)}=\frac{1}{M_{\rm
Pl}^{2}}\bar{\nabla}^{\lambda}\bar{\Theta}_{\lambda\mu},$ (25)

to be satisfied for all the solutions to the multiscalar-metric FEs. In the
above, the tensor

$\bar{\Theta}_{\mu\nu}\equiv\bar{T}_{\mu\nu}+\bar{\theta}_{s\mu\nu}=\bar{T}_{{\ae}\mu\nu}+\bar{T}_{M\mu\nu}+\bar{\theta}_{s\mu\nu}$
(26)

is an extension of the total energy-momentum tensor accounting also for the
off-mass-shell contribution

$\bar{\theta}_{s\mu\nu}\equiv-\delta\bar{L}_{M}/\delta\sigma\,\bar{g}_{\mu\nu}$
(27)

of the (composite) scalar graviton $\sigma$.242424Note that the value
$\bar{\theta}_{s\mu\nu}\neq 0$ additionally closes or opens a spacetime
region, i.e., acts in this region as an effective dark matter (DM) or DE,
depending on $\delta\bar{L}_{M}/\delta\sigma$ being positive or negative,
respectively. A priori, $\bar{\Theta}_{\mu\nu}$ is not bound to covariantly
conserve, likewise any its part (26) separately, in particular,
$\bar{\theta}_{s\mu\nu}$. The covariant conservation of
$\bar{\Theta}_{\mu\nu}$ can still be imposed at an arbitrary $\bar{w}$ as an
additional restriction on the solutions. The two extreme versions of the so
constructed generic dark gravity model, with $\bar{w}^{\prime}=0$ and $-1/2$,
presenting the special interest — respectively, as the spontaneously broken GR
vs. WTR — are worked out in more detail below.

###  3.4 Linear approximation

To illustrate the spontaneous breaking of the Diff gauge symmetry/relativity
expand the basic fields $g_{\mu\nu}$ and ${X}^{a}$ on the background ones
$g_{\star\mu\nu}$ and ${X}_{\star}^{a}$ supplemented by some perturbations:

$g_{\mu\nu}\equiv
g_{\star\mu\nu}(x)+h_{\mu\nu}(x),\quad{X}^{a}\equiv{X}_{\star}^{a}(x)+{\chi}^{a}(x).$
(28)

Choose then the proper background coordinates
$x_{\star}^{\alpha}\equiv\delta^{\alpha}_{a}X^{a}_{\star}(x)$, with the flat
background looking in such coordinates as follows:

$g_{\star\alpha\beta}=\bar{g}_{\star\alpha\beta}={{\kappa}}_{\star\alpha\beta}=\eta_{\alpha\beta},\quad\bar{{\ae}}^{\pm
1}_{\star}{}^{\alpha}_{\beta}=\delta^{\alpha}_{\beta},\quad\sigma_{\star}=0,$
(29)

where $\eta_{\alpha\beta}$ is the Minkowski symbol. Identifying the global
Lorentz indices $a,b,\dots$ with the flat background ones $\alpha,\beta,\dots$
results then for the basic fields in the linear approximation (LA) in the
relations:

$\displaystyle g^{\pm 1}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta}\pm h_{\alpha\beta}({x}_{\star}),$
$\displaystyle{X}^{\pm 1}{}_{\alpha}^{\beta}$ $\displaystyle=$
$\displaystyle\delta_{\alpha}^{\beta}\pm\partial_{\alpha}{\chi}^{\beta}({x}_{\star}),$
$\displaystyle{{\kappa}}^{\pm 1}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta}\pm k_{\alpha\beta}({x}_{\star}),$
$\displaystyle k_{\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle\partial_{\alpha}{\chi}_{\beta}({x}_{\star})+\partial_{\beta}{\chi}_{\alpha}({x}_{\star}),$
(30)

with $\partial_{\alpha}\equiv\partial/\partial x_{\star}^{\alpha}$ and
$\chi_{\alpha}\equiv\eta_{\alpha\beta}\chi^{\beta}$, etc. Here and in what
follows, the tensors are explicitly marked vs. their inverse values by the
superscripts $\pm 1$. For the effective fields, this in turn implies in LA:

$\displaystyle\bar{g}^{\pm 1}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\eta_{\alpha\beta}\pm\bar{h}_{\alpha\beta},$
$\displaystyle\bar{{\ae}}^{\pm 1}{}^{\alpha}_{\beta}$ $\displaystyle=$
$\displaystyle\delta^{\alpha}_{\beta}\mp\bar{f}^{\alpha}_{\beta}$ (31)

and $\sigma={s}$. Designating in LA $\bar{w}=\bar{\alpha}\sigma$, with
$\bar{\alpha}$ a constant interpolating between $\bar{\alpha}=0$ for the
modified GR and $\bar{\alpha}=-1/2$ for the modified WTR, we get in the same
approximation:

$\displaystyle\bar{h}_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle
h_{\alpha\beta}+\bar{\alpha}/2(h-2\partial{\chi})\eta_{\alpha\beta},$
$\displaystyle\bar{f}_{\alpha\beta}\equiv\eta_{\alpha\gamma}\bar{f}^{\gamma}_{\beta}$
$\displaystyle=$
$\displaystyle\bar{h}_{\alpha\beta}-k_{\alpha\beta},\quad\bar{f}=(1+\bar{2}\bar{\alpha})(h-2\partial\chi),$
$\displaystyle{s}$ $\displaystyle=$ $\displaystyle 1/2(h-2\partial\chi),$ (32)

with $\partial\chi\equiv\partial_{\gamma}\chi^{\gamma}$,
$h=h_{\alpha\beta}\eta^{\alpha\beta}$,
$\bar{f}=\bar{f}_{\alpha\beta}\eta^{\alpha\beta}$, etc. By construction, under
the Diff gauge symmetry enhanced by the Weyl rescalings, for the basic fields
there takes place:

$\displaystyle{\chi}^{\alpha}$ $\displaystyle\to$
$\displaystyle{\chi}^{\alpha}+\xi^{\alpha},$ $\displaystyle{k}_{\alpha\beta}$
$\displaystyle\to$
$\displaystyle{k}_{\alpha\beta}+(\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha})=\partial_{\alpha}(\chi_{\beta}+\xi_{\beta})+\partial_{\beta}(\chi_{\alpha}+\xi_{\alpha}),$
$\displaystyle h_{\alpha\beta}$ $\displaystyle\to$ $\displaystyle
h_{\alpha\beta}+(\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha})+\zeta\eta_{\alpha\beta},$
$\displaystyle h$ $\displaystyle\to$ $\displaystyle h+2\partial\xi+4\zeta,$
(33)

with the arbitrary functions $\xi^{\alpha}(x_{\star})$ and $\zeta(x_{\star})$
as the gauge parameters. For the effective fields, this results in LA in
transformations:

$\displaystyle\bar{h}_{\alpha\beta}$ $\displaystyle\to$
$\displaystyle\bar{h}_{\alpha\beta}+(\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha})+(1+2\bar{\alpha})\zeta\eta_{\alpha\beta},$
$\displaystyle\bar{f}_{\alpha\beta}$ $\displaystyle\to$
$\displaystyle\bar{f}_{\alpha\beta}+(1+2\bar{\alpha})\zeta\eta_{\alpha\beta},$
$\displaystyle{s}$ $\displaystyle\to$ $\displaystyle{s}+2\zeta.$ (34)

Otherwise, $\bar{h}_{\alpha\beta}$ is a tensor under Diff, coinciding in the
“unitary” gauge $\chi^{\alpha}=0$ (and thus $k_{\alpha\beta}=0$) according to
(3.4) with the Diff gauge-invariant $\bar{f}_{\alpha\beta}$, whereas $s$ is a
scalar under Diff. Hence, the fields $\bar{f}_{\alpha\beta}$ and ${s}$ may be
used to describe the spontaneous Diff gauge symmetry/relativity breaking in
the dark gravity in the explicitly Diff gauge-invariant terms. At that,
$\bar{f}_{\alpha\beta}$ at $\bar{\alpha}\neq-1/2$ and $s$ explicitly violate
the Weyl-scale gauge symmetry, with $s$ serving, in effect, as a Weyl scale in
disguise.252525To be more concise, the scalar graviton being a measure of the
(Weyl) scale transformations may be called the systolon, with the tensor
graviton being conventionally the graviton.

##  4 Spontaneously broken General Relativity

Let first ${\bar{w}}^{\prime}=0$, so that without loss of generality
${\bar{w}}={{w}}=0$ and $\bar{g}_{\mu\nu}=g_{\mu\nu}$, with the bar-sign to be
dropped-off everywhere. In the Lagrangian $L=L_{G}+L_{M}$, both $L_{G}$ and
$L_{M}$ are Diff gauge invariant, with the explicitly violated Weyl scale
symmetry. The FEs (3.2) can now be combined as follows:

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}-\Lambda
g_{\mu\nu}-\frac{1}{M_{\rm Pl}^{2}}{\Theta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\nabla_{\lambda}\Big{(}g^{\lambda\rho}\frac{\partial
V_{{\ae}}}{\partial{\ae}^{\rho}_{\kappa}}{X}_{{\kappa}a}+\frac{1}{2}\frac{\delta
L_{M}}{\delta\sigma}{X}^{\lambda}_{a}\Big{)}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle\frac{\delta L_{M}}{\delta\phi_{I}}$ $\displaystyle=$
$\displaystyle 0,$ (35)

with the scalar-graviton FE $\delta L_{M}/\delta\sigma=0$ as such being,
generally, missing. Due to the reduced Bianchi identity, the extended total
energy-momentum tensor ${\Theta}_{\mu\nu}$ should for the geometrical
consistency be covariantly conserved for all the solutions to FEs,
$\nabla^{\lambda}{\Theta}_{\lambda\nu}\equiv 0$, though separately each of its
three contributions to the total sum (26) may, generally, not satisfy this
requirement. In general, FEs (4) correspond to the multiscalar-
modified/spontaneously broken GR, with the Lagrangian CC $\Lambda$
supplemented by the emergent tensor- and scalar-graviton dark components
contributing through ${\Theta}_{\mu\nu}$. Under the conventional FE for the
scalar graviton $\sigma$, $\delta L_{M}/\delta\sigma=0$, and the absence of
the tensor DE, $V_{\ae}\equiv 0$, the multiscalar part of FEs (4) is clearly
trivial, with the tensor FE reducing due to $T_{\ae\mu\nu}=0$ to that of the
unbroken GR supplemented by a scalar field $\sigma$.

In LA at $\bar{\alpha}=\alpha=0$, one has
$\bar{h}_{\alpha\beta}=h_{\alpha\beta}$, with the tensor and scalar dark
gravity fields

$\displaystyle{f}_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
h_{\alpha\beta}-k_{\alpha\beta},\quad{f}=h-2\partial\chi,$ $\displaystyle{s}$
$\displaystyle=$ $\displaystyle 1/2\,(h-2\partial\chi)$ (36)

transforming under the enhanced Diff gauge symmetry as follows:

$\displaystyle h_{\alpha\beta}$ $\displaystyle\to$ $\displaystyle
h_{\alpha\beta}+(\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha})+\zeta\eta_{\alpha\beta},$
$\displaystyle{f}_{\alpha\beta}$ $\displaystyle\to$
$\displaystyle{f}_{\alpha\beta}+\zeta\eta_{\alpha\beta},\quad{s}\to{s}+2\zeta,$
(37)

resulting in the Weyl scale symmetry to be explicitly violated. At that, the
trace metric part $h$ enters both the Diff gauge-invariant tensor field
$f_{\alpha\beta}$ and scalar $s$, and, generally, results in a ghost. In the
case with the missing $\sigma$, $\delta L_{M}/\delta\sigma\equiv 0$, under the
proper choice of the potential $V_{\ae}$, there appears in the broken phase
the massive tensor graviton, with $h$ as a ghost missing. But the ghost
inevitably reappears in a sufficiently high order of the perturbation theory
[28]–[32]. This may, conceivably, be traced back to the absence of a
protecting symmetry. On the other hand, in the reduced case $V_{\ae}=0$ the
tensor graviton remains massless, with $h$ in the transverse gauge
$\partial\chi=0$ becoming the physical massive scalar graviton.262626For the
reduced case $V_{\ae}\equiv 0$, with the (ultralight) physical $\sigma$
serving as a scalar DE superseding the Lagrangian CC, see [38]. For the
similar case, with $\sigma$ as a scalar DM, cf. [39]. The general tensor-
scalar mixed case beyond LA, with both $V_{\ae}\neq 0$ and $\delta
L_{M}/\delta\sigma\neq 0$, is to be investigated.

Altogether, though being able to produce the dark gravity components, the
spontaneously broken GR at its face value can hardly help in solving the
entire CC problem, with the latter still to be resolved by some other means.
Nevertheless, an alternative to GR classically equivalent to the latter up to
CC — WTR — proves a priori to be well suited to this end, as is discussed in
more detail below, with the case of GR reserved as the canonical reference
one.

##  5 Spontaneously broken Weyl Transverse Relativity

###  5.1 Full nonlinear theory

####  5.1.1 General field equations

Let then ${\bar{w}}^{\prime}=-1/2$, with the effective gravity scale factor
${\bar{w}}={\hat{w}}\equiv-\sigma/2$. Replacing in this case everywhere the
bar-sign by the hat, one has for the effective metric

$\hat{g}_{\mu\nu}=e^{-\sigma/2}g_{\mu\nu}=({{\kappa}}/g)^{1/4}g_{\mu\nu},\quad\hat{g}={\kappa},$
(38)

with $\det(\hat{\ae}^{\mu}_{\nu})={{\kappa}}/\hat{g}=1$. In the effective
Lagrangian $\hat{L}=\hat{L}_{G}+\hat{L}_{M}$, the purely tensor-gravity part
$\hat{L}_{G}$ possesses by the Diff gauge symmetry enhanced by the “hidden”
local Weyl scale symmetry. Due to this, one independent component in
$\hat{L}_{G}$ is, in fact, missing. With account for $\hat{L}_{M}$, the
missing component reappears again as $\sigma$ signifying the explicit
violation of the local Weyl scale symmetry. The FEs (3.2) now read as follows:

$\displaystyle\hat{R}_{\mu\nu}-\frac{1}{4}\hat{R}\hat{g}_{\mu\nu}-\frac{1}{M_{\rm
Pl}^{2}}(\hat{T}_{\mu\nu}-\frac{1}{4}\hat{T}\hat{g}_{\mu\nu})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\hat{\nabla}_{\lambda}\Big{(}\frac{1}{M_{\rm
Pl}^{2}}\hat{g}^{\lambda\rho}\frac{\partial\hat{V}_{{\ae}}}{\partial\hat{{\ae}}^{\rho}_{\kappa}}{X}_{{\kappa}a}+\frac{1}{8}(\hat{R}+4\hat{\Lambda}+\frac{1}{M_{\rm
Pl}^{2}}\hat{T}){X}^{\lambda}_{a}\Big{)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{\delta\hat{L}_{M}}{\delta\sigma}=\frac{\delta\hat{L}_{M}}{\delta\phi_{I}}$
$\displaystyle=$ $\displaystyle 0,$ (39)

with $\hat{T}_{\mu\nu}=\hat{T}_{{\ae}\mu\nu}+\hat{T}_{M\mu\nu}$ given by eq.
(3.2). The tensor-gravity FE (5.1.1) is trace-free, being principle in the
context of the CC problem.272727For the viability of the trace-free tensor-
gravity FE in cosmology, cf., e.g., [40, 41]. In the multiscalar-metric
framework, such a FE alone is, in fact, incomplete, to be supplemented by the
multiscalar one. At that. the Lagrangian CC $\hat{\Lambda}$ clearly influences
the tensor gravity just indirectly through the multiscalar FE. A crucial point
here is that the dynamical measure $\sqrt{-{\kappa}}$, becoming, in a sense,
non-gravitating, depends entirely on the multiscalar $X^{a}$, but not on the
basic metric $g_{\mu\nu}$. The scalar graviton behaves now likewise a
conventional scalar particle, with
${\hat{\theta}}_{s\mu\nu}\equiv-\delta\hat{L}_{M}/\delta\sigma\,\hat{g}_{\mu\nu}=0$.
Due to the general covariance and the conventional FEs for $\sigma$ and
$\phi_{I}$, the extended matter energy-momentum tensor $\hat{T}_{M\mu\nu}$
should be covariantly conserved,
$\hat{\nabla}^{\lambda}\hat{T}_{M\lambda\nu}=0$, whereas the DE
$\hat{T}_{{\ae}\mu\nu}$ and thus the total $\hat{T}_{\mu\nu}$ may, generally,
not satisfy this requirement.282828In addition to the scalar graviton
$\sigma$, the extended matter Lagrangian
$\hat{L}_{M}=\hat{L}_{M}(\hat{g}_{\mu\nu},\sigma,\phi_{I})$ could, in
principle, account for the (more conventional) DM components. In the absence
of matter, the purely scalar-graviton Lagrangian
$\hat{L}_{M}=\hat{L}_{s}(\hat{g}_{\mu\nu},\sigma)$ could be looked-for, e.g.,
in a most general form for the scalar field $\sigma$ satisfying the second-
order FE to avoid the explicit Ostrogradsky instabilities [42, 43].

####  5.1.2 Covariant conservation condition

Impose additionally on a class of the solutions to FEs (5.1.1) the condition
of the covariant conservation of the total energy-momentum tensor
$\hat{\Theta}_{\mu\nu}=\hat{T}_{\mu\nu}$ in eq. (25), resulting upon
integration in the relation

$\hat{R}+4\hat{\Lambda}_{0}+\frac{1}{M_{\rm Pl}^{2}}\hat{T}=0,$ (40)

where $\hat{\Lambda}_{0}$ is an integration constant, in particular,
$\hat{\Lambda}_{0}=0$ or the Lagrangian CC $\hat{\Lambda}$. Incorporating this
relation refines, in turn, the tensor-gravity and multiscalar FEs as follow:

$\displaystyle\hat{R}_{\mu\nu}-\frac{1}{2}\hat{R}\hat{g}_{\mu\nu}-\hat{\Lambda}_{0}\hat{g}_{\mu\nu}-\frac{1}{M_{\rm
Pl}^{2}}\hat{T}_{\mu\nu}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\hat{\nabla}_{\lambda}\Big{(}\frac{1}{M_{\rm
Pl}^{2}}\hat{g}^{\lambda\rho}\frac{\partial\hat{V}_{{\ae}}}{\partial\hat{{\ae}}^{\rho}_{\kappa}}{X}_{{\kappa}a}+\frac{1}{2}(\hat{\Lambda}-\hat{\Lambda}_{0}){X}^{\lambda}_{a}\Big{)}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{\delta\hat{L}_{M}}{\delta\sigma}=\frac{\delta\hat{L}_{M}}{\delta\phi_{I}}$
$\displaystyle=$ $\displaystyle 0,$ (41)

with the Lagrangian CC $\hat{\Lambda}$ clearly screened from the tensor-
gravity FE, and $\hat{T}_{\mu\nu}$ explicitly covariantly conserved due to the
reduced Bianchi identity. Moreover, due to
$\hat{\nabla}^{\lambda}\hat{T}_{M\lambda\nu}=0$ there should now separately
fulfill $\hat{\nabla}^{\lambda}\hat{T}_{{\ae}\lambda\nu}=0$. Under an
arbitrary $\hat{V}_{\ae}$, FEs (5.1.2) describe the multiscalar-
modified/spontaneously broken WTR, with DE manifesting itself through the
covariantly conserved $\hat{T}_{{\ae}\mu\nu}$, as well as the induced and
Lagrangian CCs $\hat{\Lambda}_{0}$ and $\hat{\Lambda}$, respectively. In the
case $\hat{V}_{\ae}\equiv 0$, the multiscalar part of FEs (5.1.2) factorizes
due to $\hat{g}={{\kappa}}$ as
$\partial_{\lambda}(\sqrt{-{{\kappa}}}X^{\lambda}_{a})=0$. By this token,
${{\kappa}}$ can be chosen arbitrary, looking formally as non-dynamical
(similarly to UR). With account for $\hat{T}_{{\ae}\mu\nu}=0$, FEs in this
case reduce to those of UR, equivalent classically to GR up to CC, with the
matter set (containing, conceivably, DM) extended through the scalar graviton
$\sigma$ as a conventional scalar particle.292929Otherwise, the multiscalar-
modified UR could be obtained as a reduction of the multiscalar-modified WTR
under imposing ab initio the constraint $g={{\kappa}}$, consistent with
$\hat{g}={{\kappa}}$. This would imply, in particular, the explicit absence of
the scalar graviton $\sigma$, which is to be added ad hock as an additional
scalar particle.

###  5.2 Weak-field limit

####  5.2.1 Linear approximation

At $\bar{\alpha}=\hat{\alpha}=-1/2$, in the flat background one has in LA

$\displaystyle\hat{h}_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle
h_{\alpha\beta}-1/4\,(h-2\partial{\chi})\eta_{\alpha\beta},\quad\hat{h}=2\partial\chi,$
$\displaystyle\hat{f}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\hat{h}_{\alpha\beta}-k_{\alpha\beta},\quad\hat{f}=0,$
$\displaystyle{s}$ $\displaystyle=$ $\displaystyle 1/2\,(h-2\partial\chi),$
(42)

Under the Weyl-scale enhanced Diff gauge transformations one gets:

$\displaystyle\hat{h}_{\alpha\beta}$ $\displaystyle\to$
$\displaystyle\hat{h}_{\alpha\beta}+(\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}),$
$\displaystyle\hat{f}_{\alpha\beta}$ $\displaystyle\to$
$\displaystyle\hat{f}_{\alpha\beta},\quad{s}\to{s}+2\zeta.$ (43)

This means that $\hat{h}_{\alpha\beta}$, being a tensor under Diff, is Weyl-
scale invariant, $\hat{f}_{\alpha\beta}$ is completely invariant under the
Weyl-scale enhanced Diff gauge symmetry, while ${s}$, being a scalar under
Diff, transforms inhomogeneously under the Weyl rescalings. Hence, the fields
$\hat{f}_{\alpha\beta}$ and ${s}$ may be used to describe the spontaneously
broken Diff gauge symmetry in the explicitly Diff-invariant terms realizing
the unitary gauge ($\chi^{\alpha}=0$,
$\hat{h}_{\alpha\beta}=\hat{f}_{\alpha\beta}$) for the dark gravity. At that,
one independent DE component in the tensor $\hat{f}_{\alpha\beta}$ is missing
due to $\hat{f}=0$ as a manifestation of the Weyl scale symmetry. In the end,
such a component reappears again through the scalar graviton ${\sigma}$ as a
conventional particle, reflecting ultimately the explicit local Weyl scale
violation.303030Note that imposing ab initio the UR restriction $g={{\kappa}}$
eliminating, in fact, one independent component would result in LA in
$h=\hat{h}=2\partial\chi$ (still preserving $\hat{f}=0$) and
$\hat{h}_{\alpha\beta}=h_{\alpha\beta}$, as well as ${s}=0$, with the lost of
the Weyl scale symmetry. The account beyond LA for the higher orders in the
presence of the protecting Weyl scale symmetry, as well as the local Weyl-
scale violation due to $\sigma$, are to be investigated.

####  5.2.2 Massive tensor graviton

To clarify the viability of the multiscalar-modified/spontaneously broken WTR
as a theory of gravitons retain first only the Weyl-scale invariant purely
tensor-gravity Lagrangian $\hat{L}_{G}$ putting
$\hat{L}_{M}(\sigma,\phi_{I})=0$ in neglect by matter and the explicit scalar-
graviton contribution. Choosing then as independent variables for the DE
tensor field ${\hat{\ae}}^{\mu}_{\nu}$ its trace $\hat{\ae}$ and the traceless
part $\hat{\tilde{\ae}}^{\mu}_{\nu}$, respectively, as

$\hat{\ae}\equiv{\hat{\ae}}^{\kappa}_{\lambda}\delta_{\kappa}^{\lambda},\quad\hat{\tilde{\ae}}^{\mu}_{\nu}\equiv\hat{\ae}^{\mu}_{\nu}-\frac{1}{4}\hat{\ae}\delta^{\mu}_{\nu},$
(44)

consider the even quartic potential
$\hat{V}_{\ae}(\hat{\tilde{\ae}}^{\mu}_{\nu},\hat{\ae})$ as follows:

$\hat{V}_{\ae}=\frac{1}{2}M_{\rm
Pl}^{2}\hat{v}_{2}\bigg{(}\hat{\tilde{\ae}}^{\mu}_{\nu}\hat{\tilde{\ae}}^{\nu}_{\mu}+\hat{\beta}_{4}\Big{(}(\frac{1}{4}\hat{\ae})^{2}-1\Big{)}^{2}\bigg{)},$
(45)

where $\hat{v}_{2}>0$ and $\hat{\beta}_{4}$ are some constant
parameters.313131In particular, a similar potential $V_{\ae}$ at a proper
$\beta_{4}$ is used for the consistent description of the massive tensor
graviton within the spontaneously broken GR [28]. For definiteness,
$\hat{V}_{\ae}$ is normalized as $\hat{V}_{\ae}|_{\star}=0$ in the flat
background at $\hat{\ae}_{\star}{}^{\mu}_{\nu}=\delta^{\mu}_{\nu}$,
$\hat{\ae}_{\star}=4$, with an additional constant contribution assigned by
default to the Lagrangian CC $\hat{\Lambda}$. Such a potential is peculiar by
possessing by the two extremums under $\hat{\ae}$: at $\hat{\ae}=0$ and
$|\hat{\ae}|=4$, which may be ascribed at $\hat{\beta}_{4}>0$ to the unbroken
trivial and broken flat backgrounds, respectively. In the trivial background
at $\hat{\ae}_{\star}{}^{\mu}_{\nu}=0$, $\hat{\ae}_{\star}=0$, the potential
$\hat{V}_{\ae}|_{\star}$ at $\hat{\beta}_{4}>0$ lies higher, being positive
and unstable near $\hat{\ae}_{\star}=0$. In a vicinity of the flat background
at $\hat{\ae}_{\star}{}^{\alpha}_{\beta}=\delta^{\alpha}_{\beta}$, the
potential $\hat{V}_{\ae}$ in the quadratic approximation is as
follows:323232In fact, due to $\hat{f}=0$ in LA the second term in (46) in the
quadratic approximation is irrelevant, with the potential being (quasi-)stable
irrespective of $\hat{\beta}_{4}$, admitting, in particular,
$\hat{\beta}_{4}=0$ and the purely-quadratic
$\hat{V}_{\ae}(\hat{\tilde{\ae}}^{\mu}_{\nu})$.

$\hat{V}_{\ae}=\frac{1}{2}M_{\rm
Pl}^{2}\hat{v}_{2}\hat{f}^{\alpha}_{\beta}\hat{f}_{\alpha}^{\beta}+\frac{1}{4}(\hat{\beta}_{4}-1)\hat{f}^{2},$
(46)

being in LA due to $\hat{f}=0$, in fact, independent of $\hat{f}$ irrespective
of $\hat{\beta}_{4}$. This gives in turn:

$\hat{T}_{{\ae}\alpha\beta}=2M_{\rm
Pl}^{2}\hat{v}_{2}\hat{f}_{\alpha\beta},\quad\hat{T}_{{\ae}}\equiv\hat{T}_{{\ae}\alpha\beta}\eta^{\alpha\beta}=0.$
(47)

Due to the Diff gauge symmetry, accounting under the unitary gauge
$\chi^{\alpha}=0$ (and thus $k_{\alpha\beta}=0$) for (5.2.1), one can
substitute in $\hat{R}_{\alpha\beta}$ the tensor-graviton field
$\hat{h}_{\alpha\beta}$, with $\hat{h}=0$, by the DE one
$\hat{f}_{\alpha\beta}$, with $\hat{f}=0$, getting in LA

$\hat{R}_{\alpha\beta}=\frac{1}{2}(\partial_{\alpha}\partial_{\gamma}\hat{f}^{\gamma}_{\beta}+\partial_{\beta}\partial_{\gamma}\hat{f}^{\gamma}_{\alpha}-\partial^{2}\hat{f}_{\alpha\beta}),\quad\hat{R}=\partial_{\gamma}\partial_{\delta}\hat{f}^{\gamma\delta}.$
(48)

This results in LA in the tensor and multiscalar FEs (5.1.1) as follows:

$\displaystyle\partial_{\alpha}\partial_{\gamma}\hat{f}^{\gamma}_{\beta}+\partial_{\beta}\partial_{\gamma}\hat{f}^{\gamma}_{\alpha}-\partial^{2}\hat{f}_{\alpha\beta}-\frac{1}{2}\partial_{\gamma}\partial_{\delta}\hat{f}^{\gamma\delta}\eta_{\alpha\beta}$
$\displaystyle=$ $\displaystyle m_{g}^{2}\hat{f}_{\alpha\beta},$
$\displaystyle\partial_{\alpha}(m_{g}^{2}\hat{f}^{\alpha}_{\beta}-\frac{1}{2}\partial_{\gamma}\partial_{\delta}\hat{f}^{\gamma\delta}\delta^{\alpha}_{\beta})$
$\displaystyle=$ $\displaystyle 0,$ (49)

clearly independent of $\hat{\Lambda}$. The latter property can be explicitly
confirmed under an arbitrary $\chi^{\alpha}$ due to the Diff gauge symmetry.
In the above, we have put $\hat{v}_{2}\equiv m_{g}^{2}/4$, with $m_{g}$ to be
associated with the tensor-graviton mass. It proves that the multiscalar FE
follows, in fact, from the tensor one, with all the nine dark-gravity
components remaining thus independent. At that, one missing component
corresponds to the neglected scalar graviton, the inclusion of which would
explicitly violate the hidden Weyl scale symmetry. To reduce the number of the
independent components, impose additionally the covariant conservation
condition (25) resulting in LA in the transversality condition

$\partial_{\gamma}\hat{f}^{\gamma\alpha}=0$ (50)

followed by $\hat{\Lambda}_{0}=0$. This finally results in FE

$(\partial^{2}+m_{g}^{2})\hat{f}_{\alpha\beta}=0$ (51)

for the gauge-invariant $\hat{f}_{\alpha\beta}$ coinciding with the gauge-
variant $\hat{h}_{\alpha\beta}=h_{\alpha\beta}-1/4\,h\eta_{\alpha\beta}$ under
the unitary gauge $\chi_{\alpha}=0$. The FEs (50) and (51) describe precisely
the five independent DE degrees of freedom without ghosts, similarly to the
Fierz-Pauli case for the massive tensor graviton within the spontaneously
broken GR [28]. The covariant conservation condition in the spontaneously
broken WTR, likewise GR, proves thus to be of principle to ensure the
consistency of the massive tensor graviton description. Still, the conceivable
violation of the covariant conservation condition in the context of DE is of
particular interest.

####  5.2.3 Massless limit

At last, in the massless limit $m_{g}\to 0$, in FEs (5.2.2) there appears a
three-parameter residual gauge symmetry

$\hat{f}_{\alpha\beta}\to\hat{f}_{\alpha\beta}-(\partial_{\alpha}\hat{\varphi}_{\beta}+\partial_{\beta}\hat{\varphi}_{\alpha}),\quad\partial\hat{\varphi}=0,\quad\partial^{2}\hat{\varphi}_{\alpha}=0,$
(52)

satisfied also by the constraint (50). This reduces the number of the
independent tensor-graviton components on-mass-shell $m_{g}=0$ to two, making
thus the massless limit in LA consistent. In fact, in this limit
$\hat{f}_{\alpha\beta}$ reduces to the conventional massless unimodular
$\hat{h}_{\alpha\beta}$, with $\hat{h}=0$. The presence of the residual gauge
symmetry ensuring in the spontaneously broken WTR the smooth massless limit to
UR is thus crucial for the explicit consistency in LA of the massive tensor-
graviton description.

####  5.2.4 Scalar graviton

Finally, retaining the tensor-graviton description through the Weyl-scale
invariant Lagrangian $\hat{L}_{G}$, with the three (at $n=4$) gauge components
converted into the additional physical ones for the massive tensor graviton,
one could add the explicitly Weyl-scale violating $\hat{L}_{M}(\sigma)$ to
consistently realize the last component contained in the multiscalar $X^{a}$.
In the neglect by matter, this would result in LA in FE for the scalar
graviton $s$ as a conventional particle:

$(\partial^{2}+m_{s}^{2})s=0,$ (53)

with a scalar-graviton mass $m_{s}$. In the above, the gauge-invariant scalar
field $s$ coincides with the gauge-variant metric component $h/2$ under the
transverse gauge $\partial\chi=0$ (embodying, in particular, the unitary one
$\chi^{\alpha}=0$). Going beyond LA and accounting for the mixing of the
scalar graviton $\sigma$ with matter remain to be studied.

##  6 Conclusion

The generic multiscalar-metric framework provides a promising route for the
modification of gravity, with the spontaneous breaking of the Diff gauge
symmetry/relativity as a basic reason of the dark gravity – a merger of
gravity with the emergent dark components. The multiscalar-
modified/spontaneously broken WTR, built on this route, incorporating the Weyl
scale symmetry and the Higgs mechanism for gravity, may well serve as a viable
beyond-GR EFT of gravity. It is able to screen the Lagrangian CC
$\hat{\Lambda}$, with emergence of the induced one $\hat{\Lambda}_{0}$
supplemented by the massive tensor and scalar gravitons as the dark gravity
components. As EFT, the so constructed dark gravity model, being applied to
the (extensions of the) particle SM, allows, in principle, to account also for
the spontaneous breaking of the internal symmetries. This should be followed
by the step-wise modifications of the Lagrangian CC and matching of the
emergent descriptions at the phase transition scales. In the leading
approximation, the spontaneously broken WTR proves to be well-behaving as the
theory of the massive tensor and scalar gravitons without ghosts. Further
developing the basic concept of the spontaneously broken Diff gauge
symmetry/relativity — in particular, WTR vs. GR — as a principle source of the
emergent dark gravity components of the Universe is urgent.

##### Acknowledgment

The author is sincerely grateful to S. S. Gershtein for encouraging
discussions.

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