# Transition to Zero Cosmological Constant and Phantom Dark Energy
as Solutions Involving Change of Orientation of Space-Time Manifold

E. I. Guendelman and A. B. Kaganovich guendel<EMAIL_ADDRESS>Physics
Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

###### Abstract

The main conclusion of long-standing discussions concerning the role of
solutions with degenerate metric ($g\equiv det(g_{\mu\nu})=0$ and even with
$g_{\mu\nu}=0$) was that in the first order formalism they are physically
acceptable and must be included in the path integral. In particular, they may
describe topology changes and reduction of ”metrical dimension” of space-time.
The latter implies disappearance of the volume element $\sqrt{-g}d^{4}x$ of a
4-D space-time in a neighborhood of the point with $g=0$. We pay attention
that besides $\sqrt{-g}$, the 4-D space-time differentiable manifold possesses
also a ”manifold volume measure” (MVM) described by a 4-form which is sign
indefinite and generically independent of the metric. The first order
formalism proceeds with originally independent connection and metric
structures of the space-time manifold. In this paper we bring up the question
whether the first order formalism should be supplemented with degrees of
freedom of the space-time differentiable manifold itself, e.g. by means of the
MVM. It turns out that adding the MVM degrees of freedom to the action
principle in the first order formalism one can realize very interesting
dynamics. Such Two Measures Field Theory enables radically new approaches to
resolution of the cosmological constant problem. We show that fine tuning free
solutions describing a transition to $\Lambda=0$ state involve oscillations of
$g_{\mu\nu}$ and MVM around zero. The latter can be treated as a dynamics
involving changes of orientation of the space-time manifold. As we have shown
earlier, in realistic scale invariant models (SIM), solutions formulated in
the Einstein frame satisfy all existing tests of General Relativity (GR). Here
we reveal surprisingly that in SIM, all ground state solutions with
$\Lambda\neq 0$ appear to be degenerate either in $g_{00}$ or in MVM. Sign
indefiniteness of MVM in a natural way yields a dynamical realization of a
phantom cosmology ($w<-1$). It is very important that for all solutions, the
metric tensor rewritten in the Einstein frame has regularity properties
exactly as in GR. We discuss new physical effects which arise from this theory
and in particular strong gravity effect in high energy physics experiments.

###### pacs:

04.50.Kd, 02.40.Sf, 95.36.+x, 98.80.-k, 25.75.Dw

##  I Introduction: Degenerate Metric, Manifold Volume Measure and Orientation
of the Space-Time Manifold

Solutions with degenerate metric were a subject of a long-standing discussions
starting probably with the paper by Einstein and RosenEinstein . In spite of
some difficulty interpreting solutions with degenerate metric in classical
theory of gravitation, the prevailing view was that they have physical meaning
and must be included in the path integralHawking1979 ,D'Auria-Regge
,Tseytlin1982 . As it was shown in Refs.Hawking1979 ,Horowitz , in the first
order formulation of an appropriately extended general relativity, solutions
with $g(x)\equiv\det(g_{\mu\nu})=0$ can describe changes of the space-time
topology. Similar idea is realized also in the Ashtekar’s
variablesAshtekar-3-in-Jacobson ,Jacobson-2-in-Jacobson . There are known also
classical solutionsDray-0 -Senovilla  with change of the signature of the
metric tensor.

The space-time regions with $g(x)=0$ can be treated as having ’metrical
dimension’ $D<4$ (using terminology by TseytlinTseytlin1982 ).

The simplest solution with $g(x)=0$ is $g_{\mu\nu}=0$ while the affine
connection is arbitrary (or, in the Einstein-Cartan formulation, the vierbein
$e_{\mu}^{a}=0$ and $\omega_{\mu}^{ab}$ is arbitrary). Such solutions have
been studied by D’Auria and ReggeD'Auria-Regge , TseytlinTseytlin1982 ),
WittenWitten-15-and-16-in-Horowitz , HorowitzHorowitz , GiddingsGiddings ,
BañadosBanados ; it has been suggested that $g_{\mu\nu}=0$ should be
interpreted as essentially non-classical phase in which diffeomorphism
invariance is unbroken and it is realized at high temperature and curvature.

Now we would like to bring up a question: whether the equality $g(x)=0$ really
with a necessity means that the dimension of the space-time manifold in a
small neighborhood of the point $x$ may become $D<4$? At first sight it should
be so because the volume element is

$dV_{(metrical)}=\sqrt{-g}d^{4}x.$ (1)

Note that the latter is the ”metrical” volume element, and the possibility to
describe the volume of the space-time manifold in this way appears after the
4-dimensional differentiable manifold $M_{4}$ is equipped with the metric
structure. For a solution with $g_{\mu\nu}=0$, the situation with description
of the space-time becomes even worse . However, in spite of lack of the
metric, the manifold $M_{4}$ may still possess a nonzero volume element and
have the dimension $D=4$. The well known way to realize it consists in the
construction of a differential 4-form build for example by means of four
differential 1-forms $d\varphi_{a}$, ($a=1,2,3,4$): $d\varphi_{1}\wedge
d\varphi_{2}\wedge d\varphi_{3}\wedge d\varphi_{4}$. Each of the 1-forms
$d\varphi_{a}$ may be defined by a scalar field $\varphi_{a}(x)$. The
appropriate volume element of the 4-dimensional differentiable manifold
$M_{4}$ can be represented in the following way

$dV_{(manifold)}={4!}d\varphi_{1}\wedge d\varphi_{2}\wedge d\varphi_{3}\wedge
d\varphi_{4}\equiv\Phi d^{4}x$ (2)

where

$\Phi\equiv\varepsilon_{abcd}\varepsilon^{\mu\nu\lambda\sigma}(\partial_{\mu}\varphi_{a})(\partial_{\nu}\varphi_{b})(\partial_{\lambda}\varphi_{c})(\partial_{\sigma}\varphi_{d}).$
(3)

is the volume measure independent of $g_{\mu\nu}$ as opposed to the case of
the metrical volume measure $\sqrt{-g}$. In order to emphasize the fact that
the volume element (2) is metric independent we will call it a manifold volume
element and the measure $\Phi$ \- a manifold volume measure.

If $\Phi(x)\neq 0$ one can think of four scalar fields $\varphi_{a}(x)$ as
describing a homeomorphism of an open neighborhood of the point $x$ on the
4-dimensional Euclidean space $R^{4}$. However if one allows a dynamical
mechanism of metrical dimensional reduction of the space-time by means of
degeneracy of the metrical volume measure $\sqrt{-g}$, there is no reason to
ignore a possibility of a similar effect permitting degenerate manifold volume
measure $\Phi$. The possibility of such (or even stronger, with a sign change
of $\Phi$) dynamical effect seems to be here more natural since the manifold
volume measure $\Phi$ is sign indefinite (in Measure Theory, sign indefinite
measures are known as signed measuressigned ) . Note that the metrical and
manifold volume measures are not obliged generically to be simultaneously
nonzero.

The original idea to use differential forms as describing dynamical degrees of
freedom of the space-time differentiable manifold has been developed by Taylor
in his attemptTaylor  to quantize the gravity. Taylor argued that quantum
mechanics is not compatible with a Riemannian metric space-time; moreover, in
the quantum regime space-time is not even an affine manifold. Only in the
classical limit the metric and connection emerge, that one allows then to
construct a traditional space-time description. Of course, the transition to
the classical limit is described in Ref.Taylor  rather in the form of a
general prescription. Thereupon we would like to pay attention to the
additional possibility which was ignored in Ref.Taylor . Namely, in the
classical limit not only the metric and connection emerge but also some of the
differential forms could keep (or restore) certain dynamical effect in the
classical limit. In such a case, the traditional space-time description may
occur to be incomplete. Our key idea is that one of such lost differential
forms, the 4-form (2) survives in the classical limit as describing dynamical
degrees of freedom of the volume measure of the space-time manifold, and hence
can affect the gravity theory on the classical level too111An opposite view on
the role of the volume element has been studied by WilczekWilczek . Another
idea of modified volume element was studied in Ref.Mosna ..

If we add four scalar fields $\varphi_{a}(x)$ as new variables to a set of
usual variables (like metric, connection and matter degrees of freedom) which
undergo variations in the action principle222Fore a more detailed discussion
of the role of scalars $\varphi_{a}(x)$ in the TMT dynamics, see the end of
Sec.II. then one can expect an effect of gravity and matter on the manifold
volume measure $\Phi$ and vice versa. We will see later in this paper that in
fact such effects exist and in particular classical cosmology solutions of a
significant interest exist where $\Phi$ vanishes and changes sign.

As is well known, the 4-dimensional differentiable manifold is orientable if
it possesses a differential form of degree 4 which is nonzero at every point
on the manifold. Therefore two possible signs of the manifold volume measure
(3) are associated with two possible orientations of the space-time manifold.
The latter means that besides a dimensional reduction and topology changes on
the level of the differentiable manifold, the incorporation of the manifold
volume measure $\Phi$ allows to realize solutions describing dynamical change
of the orientation of the space-time manifold.

In the light of existence of two volume measures, the simplest way to take
into account this fact in the action principle consists in the modification of
the action which should now consist of two terms, one with the usual measure
$\sqrt{-g}$ and another - with the measure $\Phi$,

$S_{mod}=\int\left(\Phi L_{1}+\sqrt{-g}L_{2}\right)d^{4}x,$ (4)

where two Lagrangians $L_{1}$ and $L_{2}$ coupled with manifold and metrical
volume measures appear respectively. According to our previous experienceGK1
-GK10  in Two Measures Field Theory (TMT) we will proceed with an additional
basic assumption that, at least on the classical level, the Lagrangians
$L_{1}$ and $L_{2}$ are independent of the scalar fields $\varphi_{a}(x)$,
i.e. the manifold volume measure degrees of freedom enter into TMT only
through the manifold volume measure $\Phi$. In such a case, the action (4)
possesses an infinite dimensional symmetry

$\varphi_{a}\rightarrow\varphi_{a}+f_{a}(L_{1}),$ (5)

where $f_{a}(L_{1})$ are arbitrary functions of $L_{1}$ (see details in
Ref.GK3 ). One can hope that this symmetry should prevent emergence of the
scalar fields $\varphi_{a}(x)$ dependence in $L_{1}$ and $L_{2}$ after quantum
effects are taken into account.

Note that Eq.(4) is just a convenient way for presentation of the theory in a
general form. In concrete models studied in the present paper, we will see
that the action (4) can be always rewritten in an equivalent form where each
term in the action has its own total volume measure and the latter is a linear
combination of $\Phi$ and $\sqrt{-g}$.

In the next section it will be shown that the space-time geometry described in
terms of the original metric and connection of the underlying action (4) is
not generically Riemannian. However by making use a change of variables to the
Einstein frame one can represent the resulting equations of motion in the
Riemannian (or Einstein-Cartan) space-time.

In our previous investigations we have shown that TMT enables radically new
approaches to resolution of the cosmological constant (CC) problemGK3 ,G1 ,GK9
(for an alternative approach see Ref.Comelli ). Intrinsic features of TMT
allow to realize a scalar field dark energy model where all dependence of the
scalar field appears as a result of spontaneous breakdown of the dilatation
symmetry. Solutions of this model formulated in the Einstein frame satisfy all
existing tests of General Relativity (GR)GK6 ,GK7 ,GK10 . A new sort of
dynamical protection from the initial singularity of the curvature becomes
possibleGK9 . It also allows us to realize a phantom dark energy in the late
time universe without explicit introducing phantom scalar fieldGK9 .

In contrast to all our previous investigations of TMT, the purpose of the
present paper is to study the dynamics of the metric $g_{\mu\nu}$ and the
manifold volume measure $\Phi$ (used in the underlying action (4)) in a number
of TMT models. The main attention is concentrated on the analysis of the
amazing features of ”irregularity” of $g_{\mu\nu}$ and $\Phi$ (involving
change of orientation of the space-time manifold) in the course of transitions
to a ground state and in the phantom dark energy. It is very important to note
immediately that in the Einstein frame the metric tensor has regularity
properties exactly as in GR. The organization of the paper is the following.
In Sec. II we discuss general features of classical dynamics in TMT. In Sec.
III we consider the pure gravity model. In Secs. IV and V, in the framework of
a simple scalar field model I, we analyze in detail the behavior of two volume
measures in the course of transition to the ground state with zero CC. In Sec.
VI we study a (generically broken) intrinsic TMT symmetry which is restored in
the ground states; the relation of this symmetry restoration to the old CC
problemWeinberg1  is also analyzed; a discussion of this effect is continued
in Sec.VIII. In Sec.VII we shortly present the scalar field model II with
spontaneously broken global scale invarianceG1  studied in detail in Ref.GK9 .
In the framework of such class of models, an interesting dynamics of the
metric and the manifold volume measure in the course of transition to ground
states is analyzed in Sec.VIII. In section IX we reveal that a possibility to
realize a phantom dark energy without explicit introducing a phantom scalar
field (demonstrated in GK9 ) has the origin in a sign indefiniteness of the
manifold volume measure (3). Finally, since one cannot check directly whether
a tiny/zero cosmological constant is fine-tuned or not, in Sec.X we discuss
the new physical effects which arise from this theory and in particular a
strong gravity effect in high energy physics experiments.

##  II Classical equations of motion

Varying the measure fields $\varphi_{a}$, we get
$B^{\mu}_{a}\partial_{\mu}L_{1}=0$ where
$B^{\mu}_{a}=\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}\partial_{\nu}\varphi_{b}\partial_{\alpha}\varphi_{c}\partial_{\beta}\varphi_{d}$.
Since $Det(B^{\mu}_{a})=\frac{4^{-4}}{4!}\Phi^{3}$ it follows that if
$\Phi\neq 0$ the constraint

$L_{1}=sM^{4}=const.$ (6)

must be satisfied, where $s=\pm 1$ and $M$ is a constant of integration with
the dimension of mass. Variation of the metric $g^{\mu\nu}$ gives

$\zeta\frac{\partial L_{1}}{\partial g^{\mu\nu}}+\frac{\partial
L_{2}}{\partial g^{\mu\nu}}-\frac{1}{2}g_{\mu\nu}L_{2}=0,$ (7)

where

$\zeta\equiv\frac{\Phi}{\sqrt{-g}}$ (8)

is the scalar field build of the scalar densities $\Phi$ and $\sqrt{-g}$.

We study models with the Lagrangians of the form

$L_{1}=-\frac{1}{b_{g}\kappa}R(\Gamma,g)+L_{1}^{m},\quad
L_{2}=-\frac{1}{\kappa}R(\Gamma,g)+L_{2}^{m}$ (9)

where $\Gamma$ stands for affine connection,
$R(\Gamma,g)=g^{\mu\nu}R_{\mu\nu}(\Gamma)$,
$R_{\mu\nu}(\Gamma)=R^{\lambda}_{\mu\nu\lambda}(\Gamma)$ and
$R^{\lambda}_{\mu\nu\sigma}(\Gamma)\equiv\Gamma^{\lambda}_{\mu\nu,\sigma}+\Gamma^{\lambda}_{\alpha\sigma}\Gamma^{\alpha}_{\mu\nu}-(\nu\leftrightarrow\sigma)$.
Dimensionless factor $b_{g}^{-1}$ in front of $R(\Gamma,g)$ in $L_{1}$ appears
because there is no reason for couplings of the scalar curvature to the
measures $\Phi$ and $\sqrt{-g}$ to be equal. We choose $b_{g}>0$ and
$\kappa=16\pi G$, $G$ is the Newton constant. $L_{1}^{m}$ and $L_{2}^{m}$ are
the matter Lagrangians which can include all possible terms used in regular
(with only volume measure $\sqrt{-g}$) field theory models.

Since the measure $\Phi$ is sign indefinite, the total volume measure
$(\Phi/b_{g}+\sqrt{-g})$ in the gravitational term $-\kappa^{-1}\int
R(\Gamma,g)(\Phi/b_{g}+\sqrt{-g})d^{4}x$ is generically also sign indefinite.

Variation of the connection yields the equations we have solved earlierGK3 .
The result is

$\Gamma^{\lambda}_{\mu\nu}=\\{^{\lambda}_{\mu\nu}\\}+\frac{1}{2}(\delta^{\alpha}_{\mu}\sigma,_{\nu}+\delta^{\alpha}_{\nu}\sigma,_{\mu}-\sigma,_{\beta}g_{\mu\nu}g^{\alpha\beta})$
(10)

where $\\{^{\lambda}_{\mu\nu}\\}$ are the Christoffel’s connection
coefficients of the metric $g_{\mu\nu}$ and

$\sigma,_{\mu}\equiv\frac{1}{\zeta+b_{g}}\zeta,_{\mu},$ (11)

If $\zeta\neq const.$ the covariant derivative of $g_{\mu\nu}$ with this
connection is nonzero (nonmetricity) and consequently geometry of the space-
time with the metric $g_{\mu\nu}$ is generically non-Riemannian. The gravity
and matter field equations obtained by means of the first order formalism
contain both $\zeta$ and its gradient as well. It turns out that at least at
the classical level, the measure fields $\varphi_{a}$ affect the theory only
through the scalar field $\zeta$.

For the class of models (9), the consistency of the constraint (6) and the
gravitational equations (7) has the form of the following constraint

$(\zeta-b_{g})(sM^{4}-L_{1}^{m})+g^{\mu\nu}\left(\zeta\frac{\partial
L_{1m}}{\partial g^{\mu\nu}}+\frac{\partial L_{2}^{m}}{\partial
g^{\mu\nu}}\right)-2L_{2}^{m}=0,$ (12)

which determines $\zeta(x)$ (up to the chosen value of the integration
constant $sM^{4}$) as a local function of matter fields and metric. Note that
the geometrical object $\zeta(x)$ does not have its own dynamical equation of
motion and its space-time behavior is totally determined by the metric and
matter fields dynamics via the constraint (12). Together with this, since
$\zeta$ enters into all equations of motion, it generically has
straightforward effects on dynamics of the matter and gravity through the
forms of potentials, variable fermion masses and selfinteractionsGK1 -GK10 .

For understanding the structure of TMT it is important to note that TMT
(where, as we suppose, the scalar fields $\varphi_{a}$ enter only via the
measure $\Phi$) is a constrained dynamical system. In fact, the volume measure
$\Phi$ depends only upon the first derivatives of fields $\varphi_{a}$ and
this dependence is linear. The fields $\varphi_{a}$ do not have their own
dynamical equations: they are auxiliary fields. All their dynamical effect is
displayed only in the following two ways: a) in generating the constraint (6)
(or (12)); b) in the appearance of the scalar field $\zeta$ and its gradient
in all equations of motion.

##  III Pure Gravity TMT model

Let us start from the simplest TMT model with action (4) where

$L_{1}=-\frac{1}{b_{g}\kappa}R(\Gamma,g)-\Lambda_{1},\quad
L_{2}=-\frac{1}{\kappa}R(\Gamma,g)-\Lambda_{2}$ (13)

and $\Lambda_{1}$, $\Lambda_{2}$ are constants. Note that $\Lambda_{1}=const.$
cannot have a physical contribution to the field equations (in this model -
only gravitational) because $\Phi\Lambda_{1}$ is a total derivative.
Nevertheless we keep $\Lambda_{1}$ to see explicitly how $\Lambda_{1}$ appears
in the result. $\Lambda_{2}/2$ would have a sense of the cosmological constant
in the regular, non TMT, theory (i.e. with the only measure $\sqrt{-g}$).

Following the procedure described in Sec.II we obtain the gravitational
equations (7) and the constraint (12) in the following form:

$R_{\mu\nu}(\Gamma)=\frac{\kappa}{2}\frac{b_{g}\Lambda_{2}}{\zeta-
b_{g}}g_{\mu\nu}$ (14)
$\zeta=b_{g}-\frac{2\Lambda_{2}}{sM^{4}+\Lambda_{1}}=const.,$ (15)

where $sM^{4}$ is the constant of integration that appears in Eq.(6) and we
have assumed that the total volume measure in the gravitational term of the
action is nonzero, that is $\Phi/b_{g}+\sqrt{-g}\neq 0$.

Since $\zeta=const.$ the connection $\Gamma^{\lambda}_{\mu\nu}$, Eq.(10),
coincides with the Christoffel’s connection coefficients of the metric
$g_{\mu\nu}$. Therefore in the model under consideration, the space-time with
the metric $g_{\mu\nu}$ is (pseudo) Riemannian. It follows from Eqs.(14) and
(15) the resulting Einstein equations

$G_{\mu\nu}(g)=\frac{\kappa}{2}\Lambda
g_{\mu\nu};\qquad\Lambda=\frac{b_{g}}{2}(sM^{4}+\Lambda_{1})$ (16)

Constancy of $\zeta(x)$ on the mass shell, Eq.(15), means that for the
described solution the manifold and metrical volume measures coincide up to a
normalization factor. However, this is true only on the mass shell; if we were
try to start from this assumption in the underlying action the resulting
solution would be different completely.

The model possesses a few interesting features in what it concerns the CC:

(1) The effective CC $\Lambda$ appears as a constant of integration (as we
noticed above, the parameter $\Lambda_{1}$ has no a physical meaning and it
can be absorbed by the constant of integration). The effective regular, non
TMT, gravity theory provides the same equations if the cosmological constant
is added explicitly.

(2) The effective cosmological constant $\Lambda$ does not depend at all on
the CC-like parameter $\Lambda_{2}$ (which should describe a total vacuum
energy density including vacuum fluctuations of all matter fields). The latter
resembles the situation in the unimodular theoryunimodular-1 ,unimodular-2 .

(3) Note that $\Lambda$ becomes very small if the integration constant is
chosen such that $sM^{4}+\Lambda_{1}$ is very small. The latter is equivalent
to a solution with $\Phi/b_{g}\gg\sqrt{-g}$. In the limit where the metrical
volume measure $\sqrt{-g}\to 0$ while the manifold volume measure $\Phi$
remains nonzero, we get $\Lambda\to 0$. Thus a $\Lambda=0$ state is realized
for a solution which involves a reduction of the metrical dimension to
$D^{(g)}<4$ and at the same time the dimension of the space-time as a
differentiable manifold remains $D^{(m)}=4$.

(4) In the limit where the free parameter $b_{g}\to\infty$, the gravitational
term in the underlying action (Eqs.(4) and (13)) with coupling to the manifold
measure $\Phi$ approaches zero; then TMT takes the form of a regular (non TMT)
field theory, but the effective cosmological constant $\Lambda$ becomes
infinite. If we wish to reach a very small value of $\Lambda$ keeping the
integration constant arbitrary, one should take the opposite limit where
$b_{g}\ll 1$. Then in the underlying action, the weight of the gravitational
term with coupling to the manifold volume measure $\Phi$ increases with
respect to the regular one with coupling to the metrical measure $\sqrt{-g}$.

The above speculations can be regarded as a strong indication that TMT
possesses a potential for resolution of the CC problem. In the next sections
we will study this issue in more realistic models.

##  IV Scalar Field Model I

Let us now study a model including gravity as in Eqs.(9) and a scalar field
$\phi$. The action has the same structure as in Eq.(4) but it is more
convenient to write down it in the following form

$S_{mod}^{(1)}=\frac{1}{b_{g}}\int
d^{4}x\left[-\frac{1}{\kappa}(\Phi+b_{g}\sqrt{-g})R(\Gamma,g)+(\Phi+b_{\phi}\sqrt{-g})\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-\Phi
V_{1}(\phi)-\sqrt{-g}\,V_{2}(\phi)\right]$ (17)

The appearance of the dimensionless factor $b_{\phi}$ is explained by the fact
that without fine tuning it is impossible in general to provide the same
coupling of the $\phi$ kinetic term to the measures $\Phi$ and $\sqrt{-g}$.
$V_{1}(\phi)$ and $V_{2}(\phi)$ are potential-like functions; we will see
below that the physical potential of the scalar $\phi$ is a complicated
function of $V_{1}(\phi)$ and $V_{2}(\phi)$.

The constraint (12) reads now

$(\zeta-
b_{g})[sM^{4}+V_{1}(\phi)]+2V_{2}(\phi)+b_{g}\frac{\delta}{2}g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}=0,$
(18)

where

$\delta=\frac{b_{g}-b_{\phi}}{b_{g}}$ (19)

Since $\zeta\neq const.$ the connection (10) differs from the connection of
the metric $g_{\mu\nu}$. Therefore the space-time with the metric $g_{\mu\nu}$
is non-Riemannian. To see the physical meaning of the model we perform a
transition to a new metric

$\tilde{g}_{\mu\nu}=(\zeta+b_{g})g_{\mu\nu},$ (20)

where the connection $\Gamma^{\lambda}_{\mu\nu}$ becomes equal to the
Christoffel connection coefficients of the metric $\tilde{g}_{\mu\nu}$ and the
space-time turns into (pseudo) Riemannian. This is why the set of dynamical
variables using the metric $\tilde{g}_{\mu\nu}$ we call the Einstein frame.
One should point out that the transformation (20) is not a conformal one since
$(\zeta+b_{g})$ is sign indefinite. But $\tilde{g}_{\mu\nu}$ is a regular
pseudo-Riemannian metric. For the action (17), gravitational equations (7) in
the Einstein frame take canonical GR form with the same $\kappa=16\pi G$

$G_{\mu\nu}(\tilde{g}_{\alpha\beta})=\frac{\kappa}{2}T_{\mu\nu}^{eff}$ (21)

Here $G_{\mu\nu}(\tilde{g}_{\alpha\beta})$ is the Einstein tensor in the
Riemannian space-time with the metric $\tilde{g}_{\mu\nu}$ and the energy-
momentum tensor reads

$\displaystyle T_{\mu\nu}^{eff}$ $\displaystyle=$
$\displaystyle\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\left(\phi_{,\mu}\phi_{,\nu}-\tilde{g}_{\mu\nu}X\right)-\tilde{g}_{\mu\nu}\frac{b_{g}-b_{\phi}}{(\zeta+b_{g})}X+\tilde{g}_{\mu\nu}V_{eff}(\phi;\zeta,M)$
(22)

where

$X\equiv\frac{1}{2}\tilde{g}^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}$ (23)

and the function $V_{eff}(\phi;\zeta,M)$ is defined as following:

$V_{eff}(\phi;\zeta,M)=\frac{b_{g}\left[sM^{4}+V_{1}(\phi)\right]-V_{2}(\phi)}{(\zeta+b_{g})^{2}}.$
(24)

The scalar $\phi$ field equation following from Eq.(17) and rewritten in the
Einstein frame reads

$\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left[\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi\right]+\frac{\zeta
V_{1}^{\prime}+V_{2}^{\prime}}{(\zeta+b_{g})^{2}}=0$ (25)

The scalar field $\zeta$ in Eqs.(22)-(25) is determined by means of the
consistency equation (12) which in the Einstein frame (20) takes the form

$(\zeta-b_{g})[sM^{4}+V_{1}(\phi)]+2V_{2}(\phi)+\delta\cdot
b_{g}(\zeta+b_{g})X=0.$ (26)

##  V Manifold Measure and Old Cosmological Constant Problem: Cosmological
Dynamics with $|\Phi|/\sqrt{-g}\to\infty$

It is interesting to see the role of the manifold volume measure in the
resolution of the CC problem. We accomplish this now in the framework of the
scalar field model I of previous section. The $\zeta$-dependence of
$V_{eff}(\phi;\zeta,M)$, Eq.(24), in the form of inverse square like
$(\zeta+b_{g})^{-2}$ has a key role in the resolution of the old CC problem in
TMT. One can show that if quantum corrections to the underlying action
generate nonminimal coupling like $\propto R(\Gamma,g)\phi^{2}$ in both
$L_{1}$ and $L_{2}$, the general form of the $\zeta$-dependence of $V_{eff}$
remains similar: $V_{eff}\propto(\zeta+f(\phi))^{-2}$, where $f(\phi)$ is a
function. The fact that only such type of $\zeta$-dependence emerges in
$V_{eff}(\phi;\zeta,M)$, and a $\zeta$-dependence is absent for example in the
numerator of $V_{eff}(\phi;\zeta,M)$, is a direct result of our basic
assumption that $L_{1}$ and $L_{2}$ in the action (4) are independent of the
manifold measure fields $\varphi_{a}$.

Generically, in the action (17), $b_{\phi}\neq b_{g}$ that yields a nonlinear
kinetic term (i.e. the $k$-essence type dynamics) in the Einstein frame333See
also Ref.GK9  where we study in detail a model with dilatation symmetry which
also results in the $k$-essence type dynamics. But for purposes of this
section it is enough to take a simplified model with $b_{\phi}=b_{g}$ (which
is in fact a fine tuning) since the nonlinear kinetic term has no qualitative
effect on the zero CC problem. In such a case solving the constraint (26) for
$\zeta$ and substituting into Eqs.(22)-(25) we obtain equations for scalar-
gravity system which can be described by the regular GR effective action with
the scalar field potential

$V_{eff}(\phi)=\frac{(sM^{4}+V_{1})^{2}}{4[b_{g}(sM^{4}+V_{1}(\phi))-V_{2}(\phi)]}$
(27)

For an arbitrary nonconstant function $V_{1}(\phi)$ there exist infinitely
many values of the integration constant $sM^{4}$ such that $V_{eff}(\phi)$ has
the absolute minimum at some $\phi=\phi_{0}$ with $V_{eff}(\phi_{0})=0$
(provided $b_{g}[sM^{4}+V_{1}(\phi)]-V_{2}(\phi)>0$). This effect takes place
as $sM^{4}+V_{1}(\phi_{0})=0$ without fine tuning of the parameters and
initial conditions. Note that the choice of the scalar field potential in the
GR effective action in a form proportional to a perfect square like emerging
in Eq.(27) would mean a fine tuning.

For illustrative purpose let us consider the model with

$V_{1}(\phi)=\frac{1}{2}\mu_{1}^{2}\phi^{2},\qquad
V_{2}(\phi)=V_{2}^{(0)}+\frac{1}{2}\mu_{2}^{2}\phi^{2}.$ (28)

Recall that adding a constant to $V_{1}$ does not effect equations of motion,
while $V_{2}^{(0)}$ absorbs the bare CC and all possible vacuum contributions.
We take negative integration constant, i.e. $s=-1$, and the only restriction
on the values of the integration constant $M$ and the parameters is that
denominator in (27) is positive444For the case $s=+1$ and the ground state
with nonzero cosmological constant see Appendix A.

Consider spatially flat FRW universe with the metric in the Einstein frame

$\tilde{g}_{\mu\nu}=diag(1,-a^{2},-a^{2},-a^{2}),$ (29)

where $a=a(t)$ is the scale factor. Each cosmological solution ends with the
transition to a $\Lambda=0$ state via damping oscillations of the scalar field
$\phi$ towards its absolute minimum $\phi_{0}$. The appropriate oscillatory
regime in the phase plane is presented in Fig. 1.

Figure 1: Typical phase curve (in the phase plane ($\phi$,$\dot{\phi})$) of
the scalar field $\phi$ during the transition to $\Lambda=0$ state. For
illustrative purposes the parameters are chosen such that
$V_{eff}=(M^{2}/2b_{g})(\phi^{2}-M^{2})^{2}/(\phi^{2}+4M^{2})$ and
$\phi_{0}=\pm M$ and $\delta=0$. In the case without fine tuning of the
parameters $b_{g}\neq b_{\phi}$, i.e. $\delta\neq 0$, the phase portrait is
qualitatively the same.

It follows from the constraint (26) (where we took $\delta=0$) that
$|\zeta|\to\infty$ as $\phi\to\phi_{0}$. More exactly, oscillations of
$sM^{4}+V_{1}$ around zero are accompanied with a singular behavior of $\zeta$
each time when $\phi$ crosses $\phi_{0}$

$\frac{1}{\zeta}\sim sM^{4}+V_{1}(\phi)\to 0\qquad as\qquad\phi\to\phi_{0}$
(30)

and $\zeta^{-1}$ oscillates around zero together with $sM^{4}+V_{1}(\phi)$.
Taking into account that the metric in the Einstein frame
$\tilde{g}_{\mu\nu}$, Eq.(29), is regular we deduce from Eq.(20) that the
metric $g_{\mu\nu}$ used in the underlying action (17) becomes degenerate each
time when $\phi$ crosses $\phi_{0}$

$g_{00}=\frac{\tilde{g}_{00}}{\zeta+b_{g}}\sim\frac{1}{\zeta}\to 0;\qquad
g_{ii}=\frac{\tilde{g}_{ii}}{\zeta+b_{g}}\sim-\frac{1}{\zeta}\to 0\qquad
as\qquad\phi\to\phi_{0},$ (31)

where we have taken into account that the energy density approaches zero and
therefore for this cosmological solution the scale factor $a(t)$ remains
finite in all times $t$. Therefore

$\sqrt{-g}\sim\frac{1}{\zeta^{2}}\to 0\qquad
and\qquad\Phi=\zeta\sqrt{-g}\sim\frac{1}{\zeta}\to 0\qquad
as\qquad\phi\to\phi_{0}$ (32)

The detailed behavior of $\zeta$, the manifold measure $\Phi$ and $g_{\mu\nu}$
\- components555Since the metric in the Einstein frame $\tilde{g}_{\mu\nu}$ is
diagonal, Eq.(29), it is clear from the transformation (20) that $g_{\mu\nu}$
is also diagonal. are shown in Fig. 2.

Figure 2: Oscillations of the measure $\Phi$, the original metric $g_{\mu\nu}$
and the r.h.s. of Eq.(41) during the transition to $\Lambda=0$ state.

Recall that the manifold volume measure $\Phi$ is a signed measuresigned  and
therefore it is not a surprise that it can change sign. But TMT shows that
including the manifold degrees of freedom into the dynamics of the scalar-
gravity system we discover an interesting dynamical effect: a transition to
zero vacuum energy is accompanied by oscillations of $\Phi$ around zero.
Similar oscillations666Note however that these oscillations do not effect the
sign of the metrical volume measure $g=det(g_{\mu\nu})$ used in the underlying
action (17). This notion is useful when comparing our model with an approach
developed in Refs.Erdem -reflection2  simultaneously occur with all components
of the metric $g_{\mu\nu}$ used in the underlying action (17).

The measure $\Phi$ and the metric $g_{\mu\nu}$ pass zero only in a discrete
set of moments in the course of transition to the $\Lambda=0$ state. Therefore
there is no problem with the condition $\Phi\neq 0$ used for the solution (6).
Also there is no problem with singularity of $g^{\mu\nu}$ in the underlying
action since

$\lim_{\phi\to\phi_{0}}\Phi g^{\mu\nu}=finite\qquad
and\qquad\sqrt{-g}g^{\mu\nu}\sim\frac{1}{\zeta}\to 0\qquad
as\qquad\phi\to\phi_{0}$ (33)

The metric in the Einstein frame $\tilde{g}_{\mu\nu}$ is always regular
because degeneracy of $g_{\mu\nu}$ is compensated in Eq.(20) by singularity of
the ratio of two measures $\zeta\equiv\Phi/\sqrt{-g}$.

##  VI Restoration of Intrinsic TMT Symmetry in the Course of Transition to
Zero Cosmological Constant State

Let us now turn to intrinsic symmetry of TMT which can reveal itself in a
model with only the manifold volume measure $\Phi$. Indeed, if in Eq.(9) we
take the limit777for the model of Sec.IV it means that in Eq.(17) we take the
limit $b_{g}\to 0$, $b_{\phi}\to 0$ and $V_{2}\to 0$) $b_{g}\to 0$ and
$L_{2}^{m}\to 0$ then Eq.(12) reads

$L_{1}^{m}-g^{\mu\nu}\frac{\partial L_{1}^{m}}{\partial
g^{\mu\nu}}=sM^{4},\quad{\text{i}f}\quad\zeta\neq 0.$ (34)

If in addition $L_{1}^{m}$ is homogeneous of degree 1 in $g^{\mu\nu}$ then the
integration constant $M$ must be zero. The simplest example of a model for
$L_{1}^{m}$ satisfying this property is the massless scalar field. In such a
case the theory is invariant under transformations in the space of the scalar
fields $\varphi_{a}$

$\varphi_{a}\rightarrow\varphi^{\prime}_{a}=\varphi^{\prime}_{a}(\varphi_{b})$
(35)

resulting in the transformation of the manifold volume measure $\Phi$

$\Phi(x)\rightarrow\Phi^{\prime}(x)=J(x)\Phi(x),\qquad
J(x)=Det(\frac{\partial\varphi^{\prime}_{a}}{\partial\varphi_{b}})$ (36)

simultaneously with the local transformation of the metric

$g_{\mu\nu}(x)\rightarrow g^{\prime}_{\mu\nu}(x)=J(x)g_{\mu\nu}(x).$ (37)

This symmetry was studied in earlier pulicationsGK1  where we called it the
local Einstein symmetry (LES).

Consider now linear transformations in the space of the scalar fields
$\varphi_{a}$

$\varphi_{a}\rightarrow\varphi^{\prime}_{a}=A_{a}^{b}\varphi_{b}+C_{b},\quad
a,b=1,2,3,4$ (38)

where $A_{a}^{b}=constants$, $C_{b}=constants$. Then LES (35)-(37) is reduced
to transformations of the global Einstein symmetry (GES) with
$J=det(A_{a}^{b})=const$. Notice that the Einstein symmetry contains a
$\mathbb{Z}_{2}$ subgroup of the sign inversions when $J=-1$:

$\Phi\rightarrow-\Phi,\quad g_{\mu\nu}\rightarrow-g_{\mu\nu}$ (39)

LES as well as GES appear to be explicitly broken if $L_{1}^{m}$ is not a
homogeneous function of degree 1 in $g^{\mu\nu}$, for example as in the model
where $L_{1}^{m}$ describes a scalar field with a nontrivial potential888Note
that the pure gravity model of Sec.III is invariant both under the LES and the
GES.. The Lagrangian $L_{2}^{m}$ generically breaks the Einstein symmetry too.
The transformation of GES originated by the infinitesimal linear
transformations
$\varphi_{a}(x)\rightarrow\varphi^{\prime}_{a}(x)=(1+\epsilon/4)\varphi_{a}(x)$,
$\epsilon=const.$, yields the following variation of the action
$(\ref{sec-2-S-modif})$ written in the form $S=\int{\cal L}d^{4}x$ where
${\cal L}=\Phi L_{1}+\sqrt{-g}L_{2}$:

$\delta S=\int\left[-\frac{\partial{\cal L}}{\partial
g^{\mu\nu}}g^{\mu\nu}+L_{1}\frac{\partial\Phi}{\partial\varphi_{a,\mu}}\varphi_{a,\mu}\right]\epsilon
d^{4}x.$ (40)

The first term in (40) equals zero on the mass shell giving the gravitational
equation (7); recall that we proceed in the first order formalism. Integrating
the second term by part, using Eq.(6) and the definition (3) of the measure
$\Phi$, we reduce the variation (40) to $\delta
S=\epsilon\int\partial_{\mu}j^{\mu}d^{4}x$ where
$\partial_{\mu}j^{\mu}=sM^{4}\Phi$ and $j^{\mu}=sM^{4}B^{\mu}_{a}\varphi_{a}$.
In the presence of topological defects with $\Phi=0$, Eq.(6) does not hold
anymore all over space-time, and one should keep $L_{1}$ in the definition of
the current: $j^{\mu}=L_{1}B^{\mu}_{a}\varphi_{a}$. In Subs.VIII.D we will see
how such a situation may be realized.

To present the current conservation in the generally coordinate invariant form
one has to use the covariant divergence. However when doing this using the
original metric $g_{\mu\nu}$ we encounter the non-metricity. It is much more
transparent to use the Einstein frame (20) where the space-time becomes
pseudo-Riemannian and the covariant derivative of the metric
$\tilde{g}_{\mu\nu}$ equals zero identically. Thus with the definition
$j^{\mu}=\sqrt{-\tilde{g}}J^{\mu}$, using the definition of $\zeta$ in Eq.(7)
and the transformation to the Einstein frame (20) we obtain

$\tilde{\nabla}_{\mu}J^{\mu}\equiv\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left(\sqrt{-\tilde{g}}J^{\mu}\right)=sM^{4}\frac{\zeta}{(\zeta+b_{g})^{2}}$
(41)

As one should expect, when $L_{2}\equiv 0$ and $L_{1}^{m}$ is homogeneous of
degree 1 in $g^{\mu\nu}$, i.e. in the case of unbroken GES, the current is
conserved because in this case the integration constant $M=0$.

As we have seen in the framework of the scalar field model of Sec.V, the
dynamical evolution pushes $|\zeta|\equiv|\Phi|/\sqrt{-g}\to\infty$ as the
gravity$+$scalar field $\phi$ -system approaches (without fine tuning) the
$\Lambda=0$ ground state $\phi=\phi_{0}$. Therefore according to Eq.(41),

$\tilde{\nabla}_{\mu}J^{\mu}\to 0\qquad as\qquad\phi\to\phi_{0}.$ (42)

For the model of Sec.V, the damping oscillations of the r.h.s. of Eq.(41)
around zero are shown in Fig. 2. Thus, the GES explicitly broken in the
underlying action, emerges in the vacuum which, as it turns out, has zero
energy density. And vice versa, emergence of GES due to $|\zeta|\to\infty$
implies, according to Eq.(24), a transition to a $\Lambda=0$ ground state.

Other way to reach the same conclusion is to look at the underlying action
(17). In virtue of Eq.(33), it is evident that in the course of transition to
the ground state, the terms in (17) coupled to the metric volume measure
$\sqrt{-g}$ become negligible in comparison with the corresponding terms
coupled to the manifold volume measure $\Phi$; besides the term $-\int
V_{1}(\phi)\Phi d^{4}x$ (which also breaks the GES) disappears as
$\phi\to\phi_{0}$. The only terms surviving in the transition to the
$\Lambda=0$ ground state are the following

$\frac{1}{b_{g}}\int\Phi
d^{4}x\left[-\frac{1}{\kappa}R(\Gamma,g)+\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right]$
(43)

and they respect the GES.

One should notice however that one can regard the GES as the symmetry
responsible for a zero CC only if TMT is taken in the strict framework
formulated in Sec.I. In fact, let us consider for example a modified TMT model
where the manifold volume measure degrees of freedom enter in the Lagrangian
$L_{1}$ in contrast to our additional basic assumption made Sec.I ( after
Eq.(4)). Namely let us assume that the Lagrangian $L_{1}$ in Eq.(4) involves a
term proportional to $\Phi/\sqrt{-g}$ that explicitly breaks the infinite
dimensional symmetry (5). To be more concrete we consider a model with the
action

$S=S_{mod}^{(1)}-\lambda\int\frac{\Phi^{2}}{\sqrt{-g}}d^{4}x$ (44)

where $S_{mod}^{(1)}$ is the action defined in Eq.(17). Such an addition to
the action (17) respects the GES but it is easy to see that it affects the
theory in such a way that without fine tuning it is impossible generically to
reach a zero CC (see Appendix B).

##  VII Scalar Field Model II. Global Scale Invariance

Let us now turn to the analyze of the results of the TMT model possessing a
global scale invariance studied early in detailG1 -GKatz ,GK5 -GK9 . The
scalar field $\phi$ playing the role of a model of dark energy appears here as
a dilaton, and a spontaneous breakdown of the scale symmetry results directly
from the presence of the manifold volume measure $\Phi$. In other words, this
SSB is an intrinsic feature of TMT.

In the context of the present paper this model is of significant interest
because cosmological solutions of the FRW universe exhibit two unexpected
results: (a) the ground state as well as the asymptotic of quintessence-like
evolution (in co-moving frame) possess certain degeneracies in $\Phi$ or
$g_{\mu\nu}$; (b) superaccelerating expansion of the universe (phantom
cosmology) appears as the direct dynamical effect when $\Phi<0$, i.e. as the
orientation of the space-time manifold is opposite to the regular one. In this
section we present the model and some of its relevant results. Regimes (a) and
(b) will be analyzed in the next two sections.

The action of the model reads

$\displaystyle S=$ $\displaystyle\frac{1}{b_{g}}\int
d^{4}xe^{\alpha\phi/M_{p}}\left[-\frac{1}{\kappa}R(\Gamma,g)(\Phi+b_{g}\sqrt{-g})+(\Phi+b_{\phi}\sqrt{-g})\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}-e^{\alpha\phi/M_{p}}\left(\Phi
V_{1}+\sqrt{-g}V_{2}\right)\right]$ (45)

and it is invariant under the global scale transformations ($\theta=const.$):

$g_{\mu\nu}\rightarrow
e^{\theta}g_{\mu\nu},\quad\Gamma^{\mu}_{\alpha\beta}\rightarrow\Gamma^{\mu}_{\alpha\beta},\quad\varphi_{a}\rightarrow\lambda_{ab}\varphi_{b}\quad\text{where}\quad\det(\lambda_{ab})=e^{2\theta},\quad\phi\rightarrow\phi-\frac{M_{p}}{\alpha}\theta.$
(46)

The appearance of the dimensionless parameters $b_{g}$ and $b_{\phi}$ is
explained by the same reasons we mentioned after Eqs.(9) and (17). In contrast
to the model of Sec.IV, now we deal with exponential (pre-) potentials where
$V_{1}$ and $V_{2}$ are constant dimensionfull parameters. The remarkable
feature of this TMT model is that Eq.(6), being the solution of the equation
of motion resulting from variation of the manifold volume measure degrees of
freedom, breaks spontaneously the scale symmetry (46): this happens due to the
appearance of a dimensionfull integration constant $sM^{4}$ in Eq.(6). One can
showGK9  that in the case of the negative integration constant ($s=-1$) and
$V_{1}>0$, the ground state appears to be again (as it was in the scalar field
model I of Sec.IV) a zero CC state without fine tuning of the parameters and
initial conditions. The behavior of $\Phi$ and $g_{\mu\nu}$ in the course of
transition to the $\Lambda=0$ state is qualitatively the same as we observed
in Sec.V for the scalar field model I. Therefore in the present paper studying
the model (45) we restrict ourself with the choice $s=+1$ and $V_{1}>0$.

Similar to the model of Sec.IV, equations of motion resulting from the action
(45) are noncanonical and the space-time is non Riemannian when using the
original set of variables. This is because all the equations of motion and the
solution for the connection coefficients include terms proportional to
$\partial_{\mu}\zeta$. However, when working with the new metric ($\phi$
remains the same)

$\tilde{g}_{\mu\nu}=e^{\alpha\phi/M_{p}}(\zeta+b_{g})g_{\mu\nu},$ (47)

which we call the Einstein frame, the connection becomes Riemannian and
general form of all equations becomes canonical. Since $\tilde{g}_{\mu\nu}$ is
invariant under the scale transformations (46), spontaneous breaking of the
scale symmetry is reduced in the Einstein frame to the spontaneous breakdown
of the shift symmetry

$\phi\rightarrow\phi+const.$ (48)

After the change of variables to the Einstein frame (47) the gravitational
equation takes the standard GR form with the same Newton constant as in the
action (45)

$G_{\mu\nu}(\tilde{g}_{\alpha\beta})=\frac{\kappa}{2}T_{\mu\nu}^{eff}$ (49)

where $G_{\mu\nu}(\tilde{g}_{\alpha\beta})$ is the Einstein tensor in the
Riemannian space-time with the metric $\tilde{g}_{\mu\nu}$. The energy-
momentum tensor $T_{\mu\nu}^{eff}$ reads

$\displaystyle T_{\mu\nu}^{eff}$ $\displaystyle=$
$\displaystyle\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\left(\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\tilde{g}_{\mu\nu}\tilde{g}^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\right)-\tilde{g}_{\mu\nu}\frac{b_{g}-b_{\phi}}{2(\zeta+b_{g})}\tilde{g}^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}+\tilde{g}_{\mu\nu}V_{eff}(\phi,\zeta;M)$
(50)

where the function $V_{eff}(\phi,\zeta;M)$ is defined as following:

$V_{eff}(\phi,\zeta;M)=\frac{b_{g}\left[M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right]-V_{2}}{(\zeta+b_{g})^{2}}.$
(51)

Note that the $\zeta$-dependence of $V_{eff}(\phi,\zeta;M)$ is the same as in
Eq.(24) of the model of Sec.IV.

The scalar field $\zeta$ is determined by means of the constraint similar to
Eq.(26) of Sec.IV

$\displaystyle(b_{g}-\zeta)\left[M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right]-2V_{2}-\delta\cdot
b_{g}(\zeta+b_{g})X=0$ (52)

where

$X\equiv\frac{1}{2}\tilde{g}^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}\qquad\text{and}\qquad\delta=\frac{b_{g}-b_{\phi}}{b_{g}}$
(53)

The dilaton $\phi$ field equation in the Einstein frame is reduced to the
following

$\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left[\frac{\zeta+b_{\phi}}{\zeta+b_{g}}\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi\right]-\frac{2\alpha\zeta}{(\zeta+b_{g})^{2}M_{p}}M^{4}e^{-2\alpha\phi/M_{p}}=0.$
(54)

where again $\zeta$ is a solution of the constraint (52). Note that the
dilaton $\phi$ dependence in all equations of motion in the Einstein frame
appears only in the form $M^{4}e^{-2\alpha\phi/M_{p}}$, i.e. it results only
from the spontaneous breakdown of the scale symmetry (46).

The effective energy-momentum tensor (50) can be represented in a form of that
of a perfect fluid

$T_{\mu\nu}^{eff}=(\rho+p)u_{\mu}u_{\nu}-p\tilde{g}_{\mu\nu},\qquad\text{where}\qquad
u_{\mu}=\frac{\phi_{,\mu}}{(2X)^{1/2}}$ (55)

with the following energy and pressure densities resulting from Eqs.(50) and
(51) after inserting the solution $\zeta=\zeta(\phi,X;M)$ of Eq.(52):

$\rho(\phi,X;M)=X+\frac{(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})^{2}-2\delta
b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})X-3\delta^{2}b_{g}^{2}X^{2}}{4[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]},$
(56) $p(\phi,X;M)=X-\frac{\left(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}+\delta
b_{g}X\right)^{2}}{4[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]}.$ (57)

In a spatially flat FRW universe with the metric
$\tilde{g}_{\mu\nu}=diag(1,-a^{2},-a^{2},-a^{2})$ filled with the homogeneous
scalar field $\phi(t)$, the $\phi$ field equation of motion takes the form

$Q_{1}\ddot{\phi}+3HQ_{2}\dot{\phi}-\frac{\alpha}{M_{p}}Q_{3}M^{4}e^{-2\alpha\phi/M_{p}}=0$
(58)

where $H$ is the Hubble parameter and we have used the following notations

$\dot{\phi}\equiv\frac{d\phi}{dt}$ (59)
$Q_{1}=2[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]\rho_{,X}=(b_{g}+b_{\phi})(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-2V_{2}-3\delta^{2}b_{g}^{2}X$
(60)
$Q_{2}=2[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]p_{,X}=(b_{g}+b_{\phi})(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-2V_{2}-\delta^{2}b_{g}^{2}X$
(61)
$Q_{3}=\frac{1}{[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]}\left[(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-2V_{2}]+2\delta
b_{g}V_{2}X+3\delta^{2}b_{g}^{3}X^{2}\right]$ (62)

The non-linear $X$-dependence appears here in the framework of the fundamental
theory without exotic terms in the Lagrangians $L_{1}$ and $L_{2}$. This
effect follows just from the fact that there are no reasons to choose the
parameters $b_{g}$ and $b_{\phi}$ in the action (45) to be equal in general;
on the contrary, the choice $b_{g}=b_{\phi}$ would be a fine tuning. Thus the
above equations represent an explicit example of $k$-essencek-essence
resulting from first principles. The system of equations (21), (56)-(58)
accompanied with the functions (60)-(62) and written in the metric
$\tilde{g}_{\mu\nu}=diag(1,-a^{2},-a^{2},-a^{2})$ can be obtained from the
k-essence type effective action

$S_{eff}=\int\sqrt{-\tilde{g}}d^{4}x\left[-\frac{1}{\kappa}R(\tilde{g})+p\left(\phi,X;M\right)\right],$
(63)

where $p(\phi,X;M)$ is given by Eq.(57). In contrast to the simplified models
studied in literaturek-essence , it is impossible here to represent
$p\left(\phi,X;M\right)$ in a factorizable form like
$\tilde{K}(\phi)\tilde{p}(X)$. The scalar field effective Lagrangian, Eq.(57),
can be represented in the form

$p\left(\phi,X;M\right)=K(\phi)X+L(\phi)X^{2}-U(\phi)$ (64)

where the potential

$U(\phi)=\frac{[V_{1}+M^{4}e^{-2\alpha\phi/M_{p}}]^{2}}{4[b_{g}\left(V_{1}+M^{4}e^{-2\alpha\phi/M_{p}}\right)-V_{2}]}$
(65)

and $K(\phi)$ and $L(\phi)$ depend on $\phi$ only via
$M^{4}e^{-2\alpha\phi/M_{p}}$. Notice that $U(\phi)>0$ for any $\phi$ provided

$b_{g}>0,\qquad V_{1}>0\qquad and\qquad b_{g}V_{1}\geq V_{2},$ (66)

that we will assume in what follows. Note that besides the presence of the
effective potential $U(\phi)$, the Lagrangian $p\left(\phi,X;M\right)$ differs
from that of Ref.k-inflation-Mukhanov  by the sign of $L(\phi)$: in our case
$L(\phi)<0$ provided the conditions (66). This result cannot be removed by a
choice of the parameters of the underlying action (45) while in
Ref.k-inflation-Mukhanov  the positivity of $L(\phi)$ was an essential
assumption. This difference plays a crucial role for a possibility of a
dynamical protection from the initial singularity of the curvature studied in
detail in RefGK9 .

The model allows a power law inflation (where the dilaton $\phi$ plays the
role of the inflaton) with a graceful exit to a zero or tiny cosmological
constant state. In what it concerns to primordial perturbations of $\phi$ and
their evolution, there are no difference with the usual (i.e. one-measure)
model with the action (63)-(65).

In the model under consideration, the conservation law corresponding to the
GES (38) has the form

$\tilde{\nabla}_{\mu}J^{\mu}\equiv\frac{1}{\sqrt{-\tilde{g}}}\partial_{\mu}\left(\sqrt{-\tilde{g}}J^{\mu}\right)=sM^{4}\frac{\zeta}{(\zeta+b_{g})^{2}}e^{-2\alpha\phi/M_{p}}$
(67)

with the same definition of the current $J^{\mu}$ as in Sec.VI.

##  VIII Degeneracies of $g_{00}$ and $\Phi$ in $\Lambda\neq 0$ Ground States

###  VIII.1 Fine-Tuned $\delta=0$ Models

We are going now to analyze some of the cosmological solutions for the late
universe in the framework of the scale invariant model of the previous
section. These solutions surprisingly exhibit that asymptotically, as
$t\to\infty$, either $g_{00}\to 0$ or $\Phi\to 0$.

In the late universe, the kinetic energy $X\to 0$. Therefore in many cases the
role of the nonlinear $X$ dependence becomes qualitatively unessential. This
is why, for simplicity, in this section we can restrict ourself with the fine
tuned model with $\delta=0$. In such a case the constraint (52) yields

$\zeta=\frac{b_{g}V_{1}-2V_{2}+b_{g}M^{4}e^{-2\alpha\phi/M_{p}}}{V_{1}+M^{4}e^{-2\alpha\phi/M_{p}}},$
(68)

The energy density and pressure take then the form

$\rho^{(0)}=\rho|_{\delta=0}=\frac{1}{2}\dot{\phi}^{2}+U(\phi);\qquad
p^{(0)}=p|_{\delta=0}=\frac{1}{2}\dot{\phi}^{2}-U(\phi),$ (69)

where $U(\phi)$ is determined by Eq.(65). The $\phi$-equation (58) is reduced
to

$\ddot{\phi}+3H\dot{\phi}+\frac{dU(\phi)}{d\phi}=0.$ (70)

Applying this model to of the late time cosmology of the spatially flat
universe and assuming that the scalar field $\phi\rightarrow\infty$ as
$t\rightarrow\infty$, it is convenient to rewrite the potential $U(\phi)$ in
the form

$U(\phi)=\Lambda+V(\phi),$ (71)

where

$\Lambda=\frac{V_{1}^{2}}{4(b_{g}V_{1}-V_{2})}.$ (72)

is the positive cosmological constant and

$V(\phi)=\frac{(b_{g}V_{1}-2V_{2})V_{1}M^{4}e^{-2\alpha\phi/M_{p}}+(b_{g}V_{1}-V_{2})M^{8}e^{-4\alpha\phi/M_{p}}}{4(b_{g}V_{1}-V_{2})[b_{g}(V_{1}+M^{4}e^{-2\alpha\phi/M_{p}})-V_{2}]}.$
(73)

It is evident that if $b_{g}V_{1}>2V_{2}$ or $b_{g}V_{1}=2V_{2}$ then
$V(\phi)$ is a sort of a quintessence-like potential and therefore
quintessence-like scenarios can be realized. This means that the dynamics of
the late time universe is governed by the dark energy which consists of both
the cosmological constant and the potential slow decaying to zero as
$\phi\to\infty$. In the opposite case, $b_{g}V_{1}<2V_{2}$, the potential
$V(\phi)$, and also $U(\phi)$, has an absolute minimum at some finite value of
$\phi$, and therefore the cosmological scenario is different from the
quintessence-like. Details of the cosmological evolution starting from the
early inflation and up to the late time universe governed by the potential
$U(\phi)$ have been studied in Ref.GK9  for each of these three cases. Here we
want to analyze what kind of degeneracy appears in ground state depending on
the region in the parameter space.

###  VIII.2 The case $b_{g}V_{1}>2V_{2}$

Let us consider the case when the relation between the parameters $V_{1}$ and
$V_{2}$ satisfies the condition $b_{g}V_{1}>2V_{2}$. It follows from Eq.(68)
that

$\zeta\to\frac{b_{g}V_{1}-2V_{2}}{V_{1}}=const>0\quad as\quad\phi\to\infty$
(74)

By making use the $(00)$ component of Eq.(47), we see that

$g_{00}=\frac{e^{-\alpha\phi/M_{p}}}{\zeta+b_{g}}\to 0$ (75)

In order to get the asymptotic time dependence of $g_{00}$ and the spatial
components of the metric

$g_{ii}=-\frac{e^{-\alpha\phi/M_{p}}}{\zeta+b_{g}}\,a(t)^{2},\qquad i=1,2,3$
(76)

as $t\to\infty$, we have to know a solution $a=a(t)$, $\phi=\phi(t)$. We can
find analytically the asymptotic (as $\phi\to\infty$) behavior of a
cosmological solution for a particular value of the parameter
$\alpha=\sqrt{3/8}$. In such a case, keeping only the leading contribution of
the $\phi$-exponent in Eq.(73), we deal with the following system of equations

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3M_{p}^{2}}\rho^{(0)}$ (77)
$\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}-\frac{2\alpha}{M_{p}}\frac{V_{1}(b_{g}V_{1}-2V_{2})}{4(b_{g}V_{1}-V_{2})^{2}}M^{4}e^{-2\alpha\phi/M_{p}}=0,$
(78)

where $\rho^{(0)}$ is determined by Eq.(69). The exact analytic solution for
these equations is as followsGK3 :

$\phi(t)=const.+\frac{M_{p}}{2\alpha}\ln(M_{p}t),\qquad a(t)\propto
t^{1/3}e^{\lambda t},\qquad\lambda=\frac{1}{M_{p}}\sqrt{\frac{\Lambda}{3}}$
(79)

where $\Lambda$ is determined by Eq.(72). Therefore we obtain for the
asymptotic cosmic time behavior of the components of the metric $g_{\mu\nu}$

$g_{00}\sim\frac{1}{t^{1/2}}\to 0;\qquad g_{ii}\sim-t^{1/6}e^{2\lambda
t}\qquad as\qquad t\to\infty$ (80)

So in the course of the expansion of the very late universe, only $g_{00}$
asymptotically vanishes while the space components $g_{ii}$ behave
qualitatively in the same manner as the space components of the metric in the
Einstein frame $\tilde{g}_{ii}$. Respectively, the asymptotic behavior of the
volume measures is as follows:

$\Phi\approx\frac{b_{g}V_{1}-2V_{2}}{V_{1}}\sqrt{-g}\sim e^{3\lambda t}\qquad
as\qquad t\to\infty$ (81)

The GES is asymptotically restored that can be seen from the asymptotic time
behavior of the conservation law (67)

$\tilde{\nabla}_{\mu}J^{\mu}\sim const\cdot
e^{-2\alpha\phi/M_{p}}\sim\frac{1}{t}.$ (82)

###  VIII.3 The case $b_{g}V_{1}=2V_{2}$

In this case the asymptotic form of $V(\phi)$ is

$V(\phi)\approx\frac{M^{8}}{2b_{g}V_{1}}e^{-4\alpha\phi/M_{p}}$ (83)

and $\zeta$ asymptotically approaches zero according to

$\zeta=\frac{b_{g}M^{4}e^{-2\alpha\phi/M_{p}}}{V_{1}+M^{4}e^{-2\alpha\phi/M_{p}}}\to
0\qquad as\qquad\phi\to\infty.$ (84)

Similar to the previous subsection, the analytic form of the asymptotic (as
$\phi\gg M_{p}$) cosmological solution exists for a particular value of the
parameter $\alpha=\sqrt{3/32}$:

$\phi(t)=const.+\frac{M_{p}}{4\alpha}\ln(M_{p}t),\qquad a(t)\propto
t^{1/3}e^{\lambda t},\qquad\lambda=\frac{1}{M_{p}}\sqrt{\frac{\Lambda}{3}},$
(85)

where now

$\Lambda=\frac{V_{1}}{2b_{g}}$ (86)

For this solution we obtain the following asymptotic cosmic time behavior for
the components of the metric $g_{\mu\nu}$ and volume measures:

$g_{00}\sim\frac{1}{t^{1/4}}\to 0;\qquad g_{ii}\sim-t^{5/12}e^{2\lambda
t}\qquad as\qquad t\to\infty$ (87) $\sqrt{-g}\sim\sqrt{t}e^{3\lambda
t},\qquad\Phi\sim e^{3\lambda t}\qquad as\qquad t\to\infty$ (88)

The asymptotic time behavior of the conservation law describing the asymptotic
restoration of the GES is the same as in Eq.(82).

###  VIII.4 The case $0<b_{g}V_{1}<2V_{2}$

In this case the potential $U(\phi)$, Eq.(65), has an absolute minimum

$\Lambda=U(\phi_{min})=\frac{V_{2}}{b_{g}^{2}}\qquad\text{at}\qquad\phi=\phi_{min}=-\frac{M_{p}}{2\alpha}\ln\left(\frac{2V_{2}-b_{g}V_{1}}{b_{g}M^{4}}\right).$
(89)

The spatially flat universe described in the Einstein frame with the metric
$\tilde{g}_{\mu\nu}=diag(1,-a^{2},-a^{2},-a^{2})$, in a finite timeGK2
reaches this ground state where it expands exponentially

$a\propto e^{\lambda t},\qquad\lambda=M_{p}^{-1}\sqrt{\Lambda/3}$ (90)

and $\Lambda$ is given by Eq.(89). A surprising feature of this case is that
$\zeta$, Eq.(68), disappears in the minimum:

$\zeta(\phi_{min})=0$ (91)

The components of the metric $g_{\mu\nu}$ in the ground state are as follows

$g_{00}|_{(ground\,state)}=\left(\frac{2V_{2}-b_{g}V_{1}}{b_{g}^{3}M^{4}}\right)^{1/2}=const,\qquad
g_{ii}|_{(ground\,state)}=-g_{00}|_{(ground\,state)}\cdot e^{2\lambda t}$ (92)

with the respective behavior of the metrical volume measure $\sqrt{-g}\propto
exp(3\lambda t)$. Hence the manifold volume measure in the ground state
disappears

$\Phi|_{(ground\,state)}=0$ (93)

in view of Eq.(91).

Disappearance of the manifold volume measure $\Phi$ in the ground state may
not allow to get the equation (6) by varying the $\varphi_{a}$ fields in the
action (45). Therefore in the conservation law (67) one should use the current
in the form $j^{\mu}=L_{1}B^{\mu}_{a}\varphi_{a}$ as we have noticed after
Eq.(40). Recall that $L_{1}$ is constituted by the terms of the Lagrangian in
(45) coupled to the measure $\Phi$. However, after using the gravitational
equation obtained by varying $g^{\mu\nu}$ in (45) and substituting the ground
state value $\phi=\phi_{min}$ into $L_{1}$, we obtain $L_{1}=M^{4}$. Hence,
the conservation law (67) in the ground state reads just

$\tilde{\nabla}_{\mu}J^{\mu}|_{(ground\,state)}=0.$ (94)

Figure 3: The phase portrait (in the phase plane ($\phi$,$\dot{\phi})$) for
the model with $\alpha=0.2$, $\delta=0.1$, $V_{1}=10M^{4}$ and
$V_{2}=9.9b_{g}M^{4}$. The region with $\rho>0$ is divided into two
dynamically disconnected regions by the line $Q_{1}(\phi,\dot{\phi})=0$. To
the left of this line $Q_{1}>0$ (the appropriate zone we call zone 1) and to
the right $Q_{1}<0$. The $\rho>0$ region to the right of the line
$Q_{1}(\phi,\dot{\phi})=0$ is divided into two zones (zone 2 and zone 3) by
the line $Q_{2}=0$ (the latter coincides with the line where $w=-1$). In zone
2 $w>-1$ but $c_{s}^{2}<0$. In zone 3 $w<-1$ and $c_{s}^{2}>0$. Phase curves
started in zone 2 cross the line $w=-1$. All phase curves in zone 3 exhibit
processes with super-accelerating expansion of the universe. Besides all the
phase curves in zone 3 demonstrate dynamical attractor behavior to the line
which asymptotically, as $\phi\to\infty$, approaches the straight line
$\dot{\phi}=0$. Figure 4: For the model with $\alpha=0.2$, $\delta=0.1$,
$V_{1}=10M^{4}$ and $V_{2}=9.9b_{g}M^{4}$: typical scalar factor dependence of
$\phi$ (Fig.(a)) and of the energy density $\rho$, defined by Eq.(56),
(Fig(b)) in the regime corresponding to the phase curves started in zone 2.
Both graphs correspond to the initial conditions $\phi_{in}=M_{p}$,
$\dot{\phi}_{in}=5.7M^{2}/\sqrt{b_{g}}$; $\rho$ increases approaching
asymptotically the value $\frac{M^{4}}{b_{g}}e^{5.52}$ Figure 5: For the same
model and with the same initial conditions as in Fig. 4: crossing the phantom
divide $w=-1$ and changing sign of the total volume measure
$(\Phi+b_{\phi}\sqrt{-g})$ in the scalar field $\phi$ kinetic term (in the
underlying action (45)) occur simultaneously.

##  IX Sign Indefiniteness of the Manifold Volume Measure
as the Origin of a Phantom Dark Energy

We turn now to the non fine-tuned case of the model of Sec.VII applied to the
spatially flat universe. We start from a short review of our recent resultsGK9
concerning qualitative structure of the appropriate dynamical system which
consists of Eq.(58) and the equation

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3M_{p}^{2}}\rho$ (95)

where the energy density $\rho$ is defined by Eq.(56). The case of the
interest of this section is realized when the parameters of the model satisfy
the condition

$(b_{g}+b_{\phi})V_{1}-2V_{2}<0$ (96)

In this case the phase plane has a very interesting structure presented in
Fig.3. Recall that the functions $Q_{1}$, $Q_{2}$, $Q_{3}$ are defined by
Eqs.(60)-(62).

We are interested in the equation of state $w=p/\rho<-1$, where pressure $p$
and energy density $\rho$ are given by Eqs.(56) and (57). The line indicated
in Fig. 3 as ”line $w=-1$” coincides with the line $Q_{2}(\phi,X)=0$ because

$w+1=\frac{X}{\rho}\cdot\frac{Q_{2}}{\left[b_{g}\left(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right)-V_{2}\right]}$
(97)

Phase curves in zone 3 correspond to the cosmological solutions with the
equation of state $w<-1$. In zone 2, $w>-1$ but this zone has no physical
meaning since the squared sound speed of perturbations

$c_{s}^{2}=\frac{Q_{2}}{Q_{1}}$ (98)

is negative in zone 3. But in zone 2, $c_{s}^{2}>0$. Some details of numerical
solutions describing the cross of the phantom divide $w=-1$ and the super-
accelerating expansion of the universe are presented in Figs. 4 and 5.

Note that the superaccelerating cosmological expansion is obtained here
without introducing an explicit phantom scalar field into the underlying
action (45). In Ref.GK9  we have discussed this effect from the point of view
of the effective k-essence model realized in the Einstein frame when starting
from the action (45). A deeper analysis of the same effect yields the
conclusion that the true and profound origin of the appearance of an effective
phantom dynamics in our model is sign-indefiniteness of the manifold volume
measure $\Phi$. In fact, using the constraint (52), Eqs.(97) and (47) it is
easy to show that

$\Phi+b_{\phi}\sqrt{-g}=(w+1)\,\frac{\rho}{4X}\,\frac{[M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}+\delta\cdot
b_{g}X]}{[b_{g}(M^{4}e^{-2\alpha\phi/M_{p}}+V_{1})-V_{2}]}\,a^{3},$ (99)

where $a$ is the scale factor. The expression in the l.h.s of this equation is
the total volume measure of the $\phi$ kinetic term in the underlying action
(45):

$\int
d^{4}x(\Phi+b_{\phi}\sqrt{-g})\frac{1}{2}g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$
(100)

The sign of this volume measure coincides with the sign of $w+1$ as well as
with the sign of the function $Q_{2}$ (see Eq.(97)). In Fig. 5 we present the
result of numerical solution for the scale factor dependence of $w$ and
$(\Phi+b_{\phi}\sqrt{-g})/a^{3}$. Thus crossing the phantom divide occurs when
the total volume measure of the $\phi$ kinetic term in the underlying action
changes sign from positive to negative for dynamical reasons. This dynamical
effect appears here as a dynamically well-founded alternative to the usually
postulated phantom kinetic term of a scalar field LagrangianPhantom-usual .

##  X Summary and Discussion

Introducing the space-time manifold volume element (2) and adding the
appropriate degrees of freedom to a set of traditional variables (metric,
connection, matter fields) we reveal that such a two measures theory (TMT)
takes up a special position between alternative theories. First, the equations
of motion can be rewritten in the Einstein frame (where the space-time becomes
Riemannian) with the same Newtonian constant as in the underlying action
(where the space-time is generically non-Riemannian). Second, the theory
possesses remarkable features in what it concerns the CC problem. Third, the
TMT model with spontaneously broken dilatation symmetry satisfies all existing
tests of GR. There are other interesting results, for example a possibility of
a dynamical protection from the initial singularity of the curvature.

In this paper we have studied the behavior of the manifold volume measure
$\Phi$ and the metric tensor $g_{\mu\nu}$ (used in the underlying TMT action)
in cosmological solutions for a number of scalar field models of dark energy.
We have made a special accent on the sign indefiniteness of the manifold
volume measure $\Phi$ that may yield interesting physical effects. An example
of such type of effects we have seen in Sec.IX: the total volume measure of
the dilaton scalar field kinetic term in the underlying action can change sign
from positive to negative in the course of dynamical evolution of the late
time universe. In the Einstein frame, this transition corresponds to the
crossing of the phantom divide of the dark energy.

We have found out that in all studied models, the transition to the ground
state is always accompanied by a certain degeneracy either in the metric
(e.g., in $g_{00}$ or in all components) or in the manifold volume measure
$\Phi$, or even in both of them. This result differs sharply from what was
expected e.g. in Refs.Hawking1979 -Tseytlin1982  where degenerate metric
solutions have been associated with high curvature and temperature phases. One
should only take into account that degeneracy of $g_{\mu\nu}$ and/or $\Phi$ in
the (transition to) ground state takes place only when one works with the set
of variables of the underlying TMT action. In the Einstein frame, we deal with
the effective picture where the measure $\Phi$ does not present at all and the
metric tensor $\tilde{g}_{\mu\nu}$ (see Eqs.(20) or (47)) has the same
regularity properties as in GR. The regularity of $\tilde{g}_{\mu\nu}$ results
from the singularity of the transformations (20) or (47)): degeneracy of
$g_{\mu\nu}$ in a discrete set of moments is compensated by a singularity of
$\zeta$.

###  X.1 The CC problem

TMT provides two different possibilities for resolution of the CC problem: one
which guarantees zero CC without fine tuning (see however the end of Sec.VI
and Appendix B); another which allows an unexpected way to reach a tiny CC.
Which of these possibilities is realized depends on the sign of the
integration constant $sM^{4}$, $s=\pm 1$. We are going now to discuss these
two issues.

####  X.1.1 The case $\Lambda=0$ in TMT

This case is of a special interest for two reasons. First, as it was shown
earlierGK9 , the conditions of the Weinberg’s no-go theoremWeinberg1  fail and
a transition to a zero CC state in TMT can be realized without fine tuning.
This becomes possible for example if $V_{1}(\phi)>0$ and the integration
constant $sM^{4}<0$. Second, as we have shown in Sec.V, in the course of
transition to a zero CC state, $g_{\mu\nu}$ and $\Phi$ oscillate synchronously
around zero and they cross zero each time $t_{i}$ ($i=1,2,3,...$) when the
scalar field $\phi$ crosses the (zero) absolute minimum of the potential (27)
(or of the potential (65) for the model of Sec.VII with $V_{1}<0$, see GK9 ).

One should recall that $\zeta(x)$ does not have its own dynamics: its values
at the space-time point $x$ are determined directly and immediately by the
local configuration of the matter fields and gravity through the algebraic
constraint, which is nothing but a consistency condition of equations of
motion. $\zeta(x)$ does not possess inertia and therefore it changes together
and synchronously with changing matter and gravity fields. This notion is very
important when trying to answer the natural question: can oscillations of
$\zeta(x)$ be a source for particle creation? The answer is - no, it cannot.
In fact, there is a coupling of $\zeta$ with fermions. But the structure of
this coupling in the Einstein frame has very surprising features which we will
shortly review in the next subsection. Here we are only formulating the
conclusion: emergence of even a tiny amount of fermionic matter immediately
yields a rearrangement of the vacuum999A possibility of a vacuum deformation
in a different approach has been shown by MacKenzie, Wilczek and ZeeZee  in
such a way that $\zeta$ instantly ceases the regime of oscillations and
rapidly enters into a regime of monotonous approach to a nonzero constant. It
is interesting to note that the latter effect may explain why the present day
cosmological constant most likely is tiny but nonzero, in spite of the
existence of a fine tuning free classical solution described a transition to
the $\Lambda=0$ state.

An overall change of sign of $g_{\mu\nu}$ in the course of these oscillations
means a change of the signature from $(+---)$ to $(-+++)$ and vice versa,
while oscillations of the sign of $\Phi$ describe the change of orientation of
the space-time manifold. The latter means that the arena of the gravitational
dynamics should contain two space-time manifolds with opposite orientations.
The discrete set of changes of the orientations happens in the form of a
smooth dynamical process in the course of which the space-time passes the
”degenerate” phase where both the metrical structure and the total 4D-volume
measure disappear. The latter means also that the term ”orientation of the
space-time manifold” loses any sense at moments $t_{i}$ ($i=1,2,3,...$). We
conclude therefore that two 4D differentiable manifolds with opposite
orientations (described by means of a sign indefinite volume 4-form) equipped
with connection and metrical structure still are not enough to describe the
arena of the gravitational dynamics: the complete description of the space-
time dynamics requires also the mentioned degenerate phase. This situation is
somewhat similar to that discussed in Introduction: first-order formulation of
GR where the degenerate phase with $g_{\mu\nu}=0$ should be also
addedHawking1979 ,D'Auria-Regge ,Tseytlin1982 ; see e.g. the recent discussion
by BañadosBanados  where the limiting process $g_{\mu\nu}\to 0$ is analyzed.

A new interesting feature of ground states in TMT we have revealed in the
present paper concerns the so-called global Einstein symmetry (GES),
Eqs.(35)-(38), which turns out generically to be explicitly broken in all
models with non-trivial dynamics. The surprising result we have discovered
here on the basis of a number of models is that the GES is restored in the
course of transitions to the ground state in all models considered. Hence its
subgroup of the sign inversions of $g_{\mu\nu}$ and $\Phi$, Eq.(39), is also
restored. Therefore the oscillations of $g_{\mu\nu}$ and $\Phi$ around zero in
the course of transition to a $\Lambda=0$ ground state provoke a wish to
compare this dynamical effect with the attempts to solve the old CC problem
developed in Refs.Erdem -reflection2 . The main idea of these approaches is
that the field theory or at least the ground state't Hooft  should be
invariant under transformations of a discrete symmetry. According to
Refs.Erdem -reflection2  it might be either an invariance under the metric
reversal symmetry or under the space-time coordinate transformations with the
imaginary unit $i$: $x^{A}\rightarrow ix^{A}$. In contrast with these
approaches, in TMT there is no need to postulate such exotic enough
symmetries. Nevertheless we have seen that sign inversions of $g_{\mu\nu}$
emerge as a dynamical effect in the course of the cosmological evolution and
this effect has indeed a relation to the resolution of the old CC problem.

####  X.1.2 The case of a tiny CC

In the scalar field models of dark energy, an interesting feature of TMT
consists in a possibility to provide a small value of the CC. If in the model
of Sec.VIII.B, the parameter $V_{2}<0$ and $|V_{2}|\gg b_{g}V_{1}$ then the CC
can be very small without the need for $V_{1}$ and $V_{2}$ to be very small.
For example, if $V_{1}$ is determined by the energy scale of electroweak
symmetry breaking $V_{1}\sim(10^{3}GeV)^{4}$ and $V_{2}$ is determined by the
Planck scale $V_{2}\sim(10^{18}GeV)^{4}$ then
$\Lambda_{1}\sim(10^{-3}eV)^{4}$. Along with such a seesaw mechanismG1 ,
seesaw , there exists another way to explain the smallness of the CC
applicable in all types of scenarios discussed in Secs.VIII.B-VIII.D (see also
Appendix A). As one can see from Eqs.(72), (85) and (89), the value of
$\Lambda$ appears to be inverse proportional101010In the pure gravity model,
Sec.III, $\Lambda$ is proportional to $b_{g}$ to the dimensionless parameter
$b_{g}$ which characterizes the relative strength of the ’manifold’ and
’metrical’ parts of the gravitational action. If for example
$V_{1}\sim(10^{3}GeV)^{4}$ then for getting $\Lambda_{1}\sim(10^{-3}eV)^{4}$
one should assume that $b_{g}\sim 10^{60}$. Such a large value of $b_{g}$ (see
Eq.(9)) permits to formulate a correspondence principleGK9  between TMT and
regular (i.e. one-measure) field theories: when $\zeta/b_{g}\ll 1$ then one
can neglect the gravitational term in $L_{1}$ with respect to that in $L_{2}$
(see Eq.(9) or Eq.(17) or Eq.(45)). More detailed analysis shows that in such
a case the manifold volume measure $\Phi=\zeta\sqrt{-g}$ has no a dynamical
effect and TMT is reduced to GR. This happens e.g. in the model of Sec.VIII.C
where the late time evolution proceeds in a quintessence-like manner: the
energy density decreases to the cosmological constant, Eq.(86), and $\zeta\to
0$, Eq.(84). Another example is the model of Sec.VIII.D where $\Phi=0$ in the
ground state, Eq.(91), while $\sqrt{-g}$ is finite. However generically
$\zeta/b_{g}$ is not small, as it happens for example in the quintessence-like
scenario of the late time universe in the model of Sec.VIII.B (see Eq.(74)).

###  X.2 Possibilities for predictions of new physical effects

####  X.2.1 Short review of the TMT model with spontaneously broken dilatation
symmetry in the presence of matter

It would be interesting to find out other possible physical manifestations of
the sign indefiniteness of the manifold volume measure. In fact, the model
with spontaneously broken dilatation symmetry studied in Secs.VII-IX and in
Ref.GK9  allows extensions which include fermion and gauge fieldsGK5 -GK7  or,
alternatively, dust as a phenomenological matter modelGK10 . In the former
case, for example, the constraint (52) is modified to the following

$\frac{1}{(\zeta+b)^{2}}\left\\{(b-\zeta)\left[M^{4}e^{-2\alpha\phi/M_{p}}+V_{1}\right]-2V_{2})\right\\}=\frac{\mu(\zeta-\zeta_{1})(\zeta-\zeta_{2})}{2(\zeta+k)^{2}(\zeta+b)^{1/2}}\bar{\Psi}\Psi,$
(101)

where $\Psi$ is the fermion field in the Einstein frame and for simplicity we
have chosen $\delta=0$ (that is $b_{\phi}=b_{g}=b$); $\zeta_{1,2}$ are defined
by

$\zeta_{1,2}=\frac{1}{2}\left[k-3h\pm\sqrt{(k-3h)^{2}+8b(k-h)-4kh}\,\right].$
(102)

and the dimensionless parameters $k$ and $h$ appear in the underlying action
in the total volume measures of the fermion kinetic term

$\int
d^{4}xe^{\alpha\phi/M_{p}}(\Phi+k\sqrt{-g})\frac{i}{2}\overline{\Psi}\left(\gamma^{a}e_{a}^{\mu}\overrightarrow{\nabla}_{\mu}-\overleftarrow{\nabla}_{\mu}\gamma^{a}e_{a}^{\mu}\right)\Psi$
(103)

and the fermion mass term

$-\int
d^{4}xe^{\frac{3}{2}\alpha\phi/M_{p}}(\Phi+h\sqrt{-g})\mu\overline{\Psi}\Psi$
(104)

respectively. Note that the fermion equation in the Einstein frame has a
canonical form but the mass of the fermion turns out $\zeta$ dependent

$m(\zeta)=\frac{\mu(\zeta+h)}{(\zeta+k)(\zeta+b)^{1/2}}$ (105)

The constraint (101) describes the local balance between the fermion energy
density and the scalar field $\phi$ contribution to the dark energy density in
the space-time region where the wave function of the primordial fermion is not
equal to zero. By means of this balance the constraint determines the scalar
$\zeta(x)$.

In the case of dust as a phenomenological matter model, the r.h.s. of the
constraint (101) looks

$\frac{\zeta-b_{m}+2b}{2\sqrt{\zeta+b}}\,m\,\tilde{n},$ (106)

where the dimensionless parameter $b_{m}$ appears in the total volume measure
of the dust contribution to the underlying action

$S_{m}=\int(\Phi+b_{m}\sqrt{-g})L_{m}d^{4}x$ (107) $L_{m}=-m\sum_{i}\int
e^{\frac{1}{2}\alpha\phi/M_{p}}\sqrt{g_{\alpha\beta}\frac{dx_{i}^{\alpha}}{d\lambda}\frac{dx_{i}^{\beta}}{d\lambda}}\,\frac{\delta^{4}(x-x_{i}(\lambda))}{\sqrt{-g}}d\lambda$
(108)

and $m$ is the mass parameter.

The wonderful feature of these models in the Einstein frame consists of the
exact coincidence of the following three quantities: a) the noncanonical (in
comparison with GR) terms in the energy-momentum tensor; b) the effective
coupling ”constant” of the dilaton $\phi$ to the matter (up to the factor
$\alpha/M_{p}$); c) the expressions in the r.h.s. of the above mentioned
constraints (101) and (106) for fermionic matter and dust respectively. For
matter in normal conditions, the local matter energy density (i.e. in the
space-time region occupied by the matter) is many orders of magnitude larger
than the vacuum energy density. Detailed analysisGK5 -GK7 ,GK10  shows that
when the matter is in the normal conditions, the balance dictated by the
constraint becomes possible if $\zeta$ with very high accuracy takes the
constant values: $\zeta\approx\zeta_{1}$ or $\zeta\approx\zeta_{2}$ for
fermions (and therefore the fermion masses become constant)) and $\zeta\approx
b_{m}-2b$ for the dust. Then the mentioned three quantities simultaneously
become extremely small. Besides for the matter in normal conditions the
gravitational equations are reduced to the canonical GR equations. The
practical disappearance of the dilaton-to-matter coupling ”constant” for the
matter in normal conditions which occurs without fine tuning of the parameters
allows us to assert that in such type of models the fifth force problem is
resolvedGK7 ,GK10 .

It does not mean however that matter does not interact with the dilaton at
all. When the matter is in states different from normal, the effect of
dilaton-to-matter coupling may yield new very interesting phenomena. One of
such effects appears when the neutrino energy density decreases to the order
of magnitude close to the vacuum energy density. The latter can happen due to
spreading of the neutrino wave packet. Then the cold gas of uniformly
distributed nonrelativistic neutrinos causes a reconstruction of the vacuum to
a state with $\zeta\to|k|$ and as a result the neutrino gas rapidly transmute
into an exotic state called neutrino dark energy(see e.g. Ref.Nelson ). This
effect was studied in details in Ref.GK7  where we have shown that
transmutation from the pure scalar field dark energy to the neutrino dark
energy regime is favorable from the energetic point of view.

####  X.2.2 Prediction of strong gravity effect in high energy physics
experiments

For the solutions $\zeta\approx\zeta_{1}$ or $\zeta\approx\zeta_{2}$ of the
constraint (101), the l.h.s. of the constraint has the order of magnitude
close to the vacuum energy density. There exists however another solution if
one allows a possibility that in the core of the support of the fermion wave
function the local dark energy density may be much bigger than the vacuum
energy density. Such a solution turns out to be possible as fermion density is
very big and $\zeta$ becomes negative and close enough to the value
$\zeta\approx-b$. Then the solution of the constraint (101) looksGK5

$\frac{1}{\sqrt{\zeta+b}}\approx\left[\frac{\mu(b-h)}{4M^{4}b(b-k)}\bar{\Psi}\Psi
e^{2\alpha\phi/M_{p}}\right]^{1/3}.$ (109)

In such a case, instead of constant masses, as it was for
$\zeta\approx\zeta_{1,2}$, Eq.(105) results in the following fermion self-
interaction term in the effective fermion Lagrangian

$L^{ferm}_{selfint}=3\left[\frac{1}{b}\left(\frac{\mu(b-h)}{4M(b-k)}\bar{\Psi}\Psi\right)^{4}e^{2\alpha\phi/M_{p}}\right]^{1/3}.$
(110)

It is very interesting that the described effect is the direct consequence of
the strong gravity. In fact, in the regime where $\zeta+b\ll 1$ the effective
Newton constant in the gravitational term of underlying action(45)

$S_{grav}=-\int d^{4}x\sqrt{-g}\,\frac{\zeta+b}{\kappa
b}R(\Gamma,g)e^{\alpha\phi/M_{p}}$ (111)

becomes anomalously large. Recall that for simplicity we have chosen here
$b_{\phi}=b_{g}=b$. But if one do not to imply this fine tuning then one can
immediately see from Eqs.(49)-(51) that in the Einstein frame the regime of
the strong gravity dictated by the dense fermion matter is manifested for the
dilaton too.

The coupling constant in Eq.(110) is dimensionless and depends exponentially
of the dilaton $\phi$ if one can regard $\phi$ as a background field
$\phi=\bar{\phi}$. But in a more general case Eq.(110) may be treated as
describing an anomalous dilaton-to-fermion interaction very much different
from the discussed above case of interaction of the dilaton to the fermion
matter in normal conditions where the coupling constant practically vanishes.
Such an anomalous dilaton-to-fermion interaction should result in creation of
quanta of the dilaton field in processes with very heavy fermions. The
probability of these processes is of course proportional to the Newton
constant $M_{p}^{-2}$. But the new effect consists of the fact that the
effective coupling constant of the anomalous dilaton-to-fermion interaction is
proportional to $e^{2\alpha\bar{\phi}/3M_{p}}$. If the dilaton is the scalar
field responsible for the quintessential inflation type of the cosmological
scenarioQuint-ess  then one should expect an exponential amplification of the
effective coupling of this interaction in the present day universe in
comparison with the early universe. One can hope that the described effect of
the strong gravity might be revealed in the LHC experiments in the form of
missing energy due to the multiple production of quanta of the dilaton field
(recall that coupling of the dilaton to fermions in normal conditions
practically vanishes and therefore the dilaton will not be observed after
being emitted).

####  X.2.3 Some other possible effects

1\. Dark matter as effect of gravitational enhancement. In the case of dust as
a phenomenological matter model, the constraint (101) with the r.h.s. (106) is
the fifth degree algebraic equation with respect to $\sqrt{\zeta+b}$. There
are some indications that in a certain region of the parameters a solution of
the constraint exists which could provide a very interesting effect of an
amplification of the gravitational field of visible diluted galactic and
intergalactic dust or/and neutrinos. Such an effect might imply that the dark
matter is not a new sort of matter but it is just a result of a so far unknown
enhancement of the gravitational field of low density states of usual matter.

2\. Dilaton to photon coupling. Astrophysical observations of few last years
indicate anomalously large transparency of the Universe to gamma raysNature1
,Nature2 . It is hard to explain this astrophysical puzzle in the framework of
extragalactic background light. Recently a natural mechanism was suggested by
De Angelis, Mansutti and RoncadelliAngelis  in order to resolve this puzzle.
The idea is to suppose that there exists a very light spin-zero boson coupled
to the photon:

$L_{\phi\gamma}=-\frac{1}{4\mu}F_{\mu\nu}\tilde{F}_{\mu\nu}\phi.$ (112)

where $\mu$ is a mass parameter. In the context of quintessential scenario
such a coupling was studied by Carrollcarroll . Then
$\gamma\longrightarrow\phi\longrightarrow\gamma$ oscillations emerge which
explainAngelis  the observed transparency of the Universe to gamma rays in a
natural way if mass of the spin-zero boson $m<10^{-10}eV$. The crucial feature
of this boson is that no other coupling of this scalar to matter exists. In
the standard quintessence models this feature seems to be a real problem. But
in TMT, as we already mentioned (see also Refs.GK7 ,GK10 ) the dilaton playing
the role of quintessence field decouples from matter in normal conditions. At
the same time its coupling to the photon in the form (112) is not suppressed.

3\. Creation of a universe in the laboratory. A theoretical attempt by Farhi,
Guth and Guven to describe a creation of a universe in the laboratoryGuth
runs across a need to allow vanishing and changing sign of $\sqrt{-g}$. In
Ref.Guth , this need is naturally regarded as a pathology. If similar approach
to the problem of creation of a universe in the laboratory could be formulated
in the framework of TMT then instead of $\sqrt{-g}$ there should appear a
linear combination of $\Phi$ and $\sqrt{-g}$ which, as we already know, is
able to vanish and change sign. In recent paperGS  by Guendelman and Sakai a
model of child universe production without initial singularities was studied.
To provide the desirable absence of initial singularity a crucial point is
that the energy momentum tensor of the domain wall should be dominated by a
sort of phantom energy. A possible way to realize this idea is to apply the
dynamical brane tensionGKNP  obtained when using the modified volume measure
similar to the signed measure $\Phi$ of the present paper. So it could be that
applying the notions explored in the present paper one can obtain also a
framework for formulating non singular child universe production.

4\. Unparticle physics. $\zeta$ dependence of the fermion mass, Eq.(105),
together with the constraint (101) can be treated as a $\bar{\Psi}\Psi$
dependence of the fermion mass. This means that in states different from the
normal one, the fermion mass spectrum may be continuous, that allows to think
of a possibility to establish relation with the idea of unparticle
physicsGeorgi .

Note finally that for the matter in normal conditions the model does not
impose essential constraints on the parameters of the model (such as $b_{g}$,
$b_{\phi}$, $b_{m}$, $k$, $h$). But the appropriate constraints should appear
when more progress in the study of the listed and another possible new effects
will be achieved.

##  XI Acknowledgements

We acknowledge V. Goldstein, M. Lin, E. Nissimov and S. Pacheva for useful
discussions of some mathematical subjects. We also thank M. Duff for
explaining us his approach and M. Bañados for useful conversations. We are
also grateful to the referee whose constructive remarks assisted us in the
improvement of this paper.

##  Appendix A The Ground State with Non Zero CC in Model I

Let us consider the scalar field model I (Eqs.(17) and (28)) with $\delta=0$
where we now choose a positive integration constant ($s=+1$) and the
parameters $V_{2}^{(0)}<0$, $b_{g}\mu_{1}^{2}>\mu_{2}^{2}$. Then the ground
state is realized for $\phi=0$ and the vacuum energy is

$\Lambda=V_{eff}(0)=\frac{M^{8}}{4b_{g}M^{4}-V_{2}^{(0)}}$ (113)

In this ground state, both the measure $\Phi>0$ and all components of the
metric tensor $g_{\mu\nu}$ are regular. Note that the presence of the free
dimensionless parameter $b_{g}$ in the denominator allows again to reach a
small vacuum energy by means of the correspondence principle discussed in item
2 of Sec.X.

##  Appendix B Global Einstein Symmetry does not Guarantee Resolution of the
CC Problem

In the model (44), the gravitational equations are modified to the following

$G_{\mu\nu}(\tilde{g})=\frac{\kappa}{2}\left[\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\tilde{g}_{\mu\nu}X+\frac{b_{g}[sM^{4}+V_{1}(\phi)+2\lambda\zeta]-V_{2}(\phi)-\lambda\zeta^{2}}{(\zeta+b_{g})^{2}}\right]$
(114)

while the form of the scalar field $\phi$ equation remains the same as in
Eq.(25). However the constraint is now very much differs from Eq.(26):

$4\lambda\zeta^{2}+[sM^{4}+V_{1}(\phi)-2b_{g}\lambda]\zeta+2V_{2}(\phi)-b_{g}[sM^{4}+V_{1}(\phi)]=0$
(115)

One can see from Eq.(114) that $\zeta$-dependence emerges now in the numerator
of the effective potential. Besides, it is evident that in contrast with what
was in Sec.V, the regime with $\zeta\to\infty$ cannot be a solution of the
constraint. It is evident that a zero minimum of the effective potential
cannot be now reached without fine tuning. Thus although the second term in
the action (44) is invariant under the GES, adding this term we loss the
ability to resolve the old CC problem.

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