#### Small dark energy and stable vacuum from Dilaton–Gauss–Bonnet coupling in TMT

Eur. Phys. J. C
Small dark energy and stable vacuum from Dilaton-Gauss-Bonnet coupling in TMT
Eduardo I. Guendelman 1
Hitoshi Nishino 0
Subhash Rajpoot 0
0 California State University at Long Beach , Long Beach, CA , USA
1 Department of Physics, Ben-Gurion University of the Negev , Beer-Sheva , Israel
In two measures theories (TMT), in addition to the Riemannian measure of integration, being the square root of the determinant of the metric, we introduce a metricindependent density in four dimensions defined in terms of scalars ϕa by = εμνρσ εabcd (∂μϕa )(∂ν ϕb)(∂ρ ϕc)(∂σ ϕd ). With the help of a dilaton field φ we construct theories that are globally scale invariant. In particular, by introducing couplings of the dilaton φ to the Gauss-Bonnet (GB) topological density √−g φ Rμ2νρσ − 4Rμ2ν + R2 we obtain a theory that is scale invariant up to a total divergence. Integration of the ϕa field equation leads to an integration constant that breaks the global scale symmetry. We discuss the stabilizing effects of the coupling of the dilaton to the GB-topological density on the vacua with a very small cosmological constant and the resolution of the 'TMT Vacuum-Manifold Problem' which exists in the zero cosmological-constant vacuum limit. This problem generically arises from an effective potential that is a perfect square, and it gives rise to a vacuum manifold instead of a unique vacuum solution in the presence of many different scalars, like the dilaton, the Higgs, etc. In the non-zero cosmological-constant case this problem disappears. Furthermore, the GB coupling to the dilaton eliminates flat directions in the effective potential, and it totally lifts the vacuum-manifold degeneracy.
1 Introduction
There are two basic formulations of the
CosmologicalConstant Problem (CCP), the first one is referred to as the
‘Old Cosmological-Constant Problem’ [
1,2
], where it was
believed that the vacuum energy density of the universe was
exactly zero and physicists invented mechanisms to obtain
exactly zero vacuum energy density for the universe today.
Since more recently, a different type of CCP is being
considered due to the evidence for the accelerating expansion of
the universe [3]. We therefore have a ‘New
CosmologicalConstant Problem’ [
4
]. The problem is now not to explain
the zero, but to explain a very small vacuum energy density.
The evidence of a very small vacuum energy density of the
universe means that getting a zero vacuum energy density for
the present universe is definitely not the full solution of the
problem, although it may be a first step toward its solution.
Many possibilities appear. It could be that the true vacuum of
the theory still has zero vacuum energy density, but we have
not reached this point yet, and that is why we see a small
vacuum energy density now. Alternatively, it could be that the
true vacuum state of the theory has a non-zero vacuum energy
density. In this case, there could as well be many possibilities.
It could be that, although there is a basic mechanism to drive
the vacuum energy density to zero, some ‘residual’
interaction exists that is responsible for slightly shifting the vacuum
energy density toward a small but non-zero value. Together
with identifying a certain mechanism that drives the vacuum
energy density to zero, we then consider a ‘residual’
interaction that provides a small vacuum energy density. Another
possibility is the existence of ‘cosmological see-saw
mechanism’, where the vacuum energy density arises as the ratio of
a certain coupling constant squared divided by another
coupling constant. In this scenario, if the first coupling is of the
order of electroweak scale to the fourth power and the
second is of the order of Planck-scale to the fourth power, the
presently observed vacuum energy of the universe is obtained
as a ‘see-saw’ effect, i.e., at O(M W8/MP4l ).
Interestingly enough, all the basic mechanisms described
above can be studied in the context of the TMT. It is a
mechanism able to drive the vacuum energy density of the universe
to zero [
5
], and can also be used in a spontaneously broken
scale-invariant model to obtain the ‘cosmological see-saw
mechanism’ [
6
]. Still in the TMT, by introducing couplings of
the dilaton to a topological density constructed out of gauge
fields, a small ‘residual’ interaction can provide a small
vacuum energy density (which is activated through instantons)
as explored in [
7
].
Concerning the basic mechanism to drive the vacuum
energy density to zero, we have studied a new class of theories
[
5
], based on the idea that the action-integral may contain the
new metric-independent measure of integration. For
example, in four dimensions the new measure can be built from a
three index field [
8
] as
(1.1)
(1.2)
= εμνρσ ∂μ Aνρσ ,
or out of four scalar fields ϕa (a = 1, 2, 3, 4) as
= εμνρσ εabcd (∂μϕa )(∂ν ϕb)(∂ρ ϕc)(∂σ ϕd ).
To provide parity conservation, one can choose one of the ϕa
to be a pseudo-scalar.
These two representations give the same results. There
is another inequivalent choice for constructing an
alternative measure using four Lorentz vectors and preserving local
Lorentz invariance [
9
].1
The is a scalar density under general coordinate
transformations and the action can be chosen in the form S =
L d4x . This has been applied to three different situations.
(1) Investigation of 4D gravity and matter fields models
containing the new measure of integration that appears to
be promising for resolving the dark-energy and dark-matter
problems, the fifth-force problem, etc.
(2) Studying new type of string and brane models based on
the use of a modified world-sheet/world-volume integration
measure [
10–14
]. It allows new types of objects and effects
like a spontaneously induced string tension; a classical
mechanism for a charge confinement; and a Weyl-conformally
invariant light-like (WILL) brane [
13
], obtaining promising
results for black hole physics.
(3) Studying higher-dimensional realization of the idea
of the modified measure in the context of the Kaluza–Klein
[
13
] and brane [
15
] scenarios with the aim to solve the CCP.
Finally a mechanism for supersymmetry breaking has also
been found using a modified measure formulation of
supergravity [
16
].
We start with the action
S =
L1
d4x +
L2√−g d4x ,
(1.3)
including two Lagrangians L1 and L2 and two volume
elements: d4x and √−g d4x , respectively. For constructing
the field theory with the action (1.3), we make only the basic
additional assumption that L1 and L2 are independent of the
measure fields ϕa . The action (1.3) is invariant under volume
1 This third approach opens new possibilities, not fully explored in
detail as yet.
preserving diffeomorphisms. Besides, it is invariant (up to an
integral of a total divergence) under the infinite-dimensional
group of shifts of the measure fields ϕa : ϕa → ϕa + fa (L1),
where fa (L1) are arbitrary differentiable functions of the
Lagrangian L1. We can proceed in the first-order
formalism where all fields, including the metric gμν (or vierbeins
eaμ), connection coefficients (or spin-connection ωμab) and
the measure fields ϕa are treated as independent dynamical
variables. All the relations between them follow from
equations of motion. The field theory based on the above-listed
assumptions, we call TMT.
It turns out that the measure fields ϕa affect the theory
only via the ratio of the two measures
(1.4)
ζ ≡ √−g = e−1 ,
where e ≡ det (eμm ) = √−g, and ζ is a scalar field with zero
scaling-weight that is determined by a constraint in the form
of an algebraic equation which is a consistency condition of
the equations of motion.
TMT models naturally avoid the fifth-force problem [
17
]
since the coupling of the dilaton to ‘normal matter’, which
is more dense than the vacuum energy density, is negligibly
small and naturally provides a ground state with zero
vacuum energy [
4
], since the effective potential is generically a
perfect square. So when the function to be squared is zero
and we take a linear correction about this point, the resulting
effective potential, close to the point where it vanishes, will
be proportional to the square of the deviation of the field from
the point where the effective potential vanishes. Therefore,
the point where the resulting effective potential vanishes is
generically also a minimum. One should also notice that a
structure similar to the TMT has been found in the
Hodgedual formulation of supergravity theories [
18
]. TMTs also
have many points of similarity with the ‘Lagrange
Multiplier Gravity (LMG)’ [
19,20
]. The Lagrange-multiplier field
in LMG enforces the condition that a certain function be
zero. In the TMT this is equivalent to the constraint that
requires some Lagrangian to be constant. The two measure
models presented here, are different from the LMG models
of [
19,20
], and provide us with an arbitrary constant of
integration for the value of a given Lagrangian, this constant of
integration, if non-zero, can generate spontaneous symmetry
breaking of scale invariance, which is present in the theory
for example. To achieve a similar result in the
Lagrangemultiplier method, we would find that it cannot respect the
full scale symmetry of the theory, this symmetry would have
to be broken at the level of the action, but the TMT method
does not have the scale symmetry of the theory respected by
the action and it can be broken only after integration of the
equations of motion.
The introduction of constraints can cause Dirac fields to
contribute to dark energy [
17
] or scalar fields to behave like
dust in [
13
] and thus a dust behavior accompanied by a
floating dark-energy component can be obtained in TMT [
21–24
].
TMTs can also be used to construct non-singular
‘emergent’ scenarios [
25–27
] for the early universe that existed
since arbitrarily large early times in the form of a stable
Einstein universe. This phase then gets transformed into an
inflationary phase and subsequently into a slowly
accelerating one. The requirement that the early phase might exist can
impose restrictions on the possible values of the
cosmological constant at the end [
28
].
The structure of TMT has been generalized in the
dynamical space time theory [
29,30
], where instead of demanding
that a certain Lagrangian be constant, we require that a certain
energy momentum be covariantly conserved. In the case this
energy momentum is proportional to the metric, we recover
the TMT in the second-order formulation. To implement this
covariant conservation of a certain energy momentum
tensor, we must introduce a four vector. When this four vector
in restricted to be a gradient, we get a non conservation of the
energy momentum tensor, which gives rise a conserved total
energy momentum of the system but not to the separate
conservation of DE and DM energy momentum tensors, which
now interact in a diffusive fashion [31].
Recently a lot of interest has been attracted by the
socalled mimetic dark matter model proposed in [
32,33
]. The
latter employs a special covariant isolation of the
conformal degree of freedom in Einstein gravity, whose
dynamics mimics cold dark matter as a pressure-less dust.
Important questions concerning the stability of mimetic gravity
are studied in Refs. [
34,35
] also a formulates a generalized
mimetic tensor–vector–scalar mimetic gravity which avoids
those problems is studied. In [
36
] the idea is applied to
inflationary scenarios.
Most versions of the mimetic gravity, except for [
34
]
appears equivalent to a special kind of Lagrange-multiplier
theory or TMT models that were known before, where the
simple constraint that the kinetic term of a scalar field be
constant. This of course gives identical results to a very special
TMT, where the Lagrangian that couples to the new measure
is the kinetic term of this scalar field.
In this paper we will show how global scale invariance
can be maintained by a linear coupling of the dilaton to the
Gauss–Bonnet (GB) scalar density and how this can provide
stabilization of the vacuum in TMT for the case of a small
cosmological constant. Due to the scale invariance, the GB
coupling of the dilaton is restricted to a linear form. This
situation of non-zero vacuum energy does not have the problem
faced in the case of the zero cosmological constant, where
there is a huge degeneracy of the vacuum, and the vacuum
state is a manifold rather than a unique state. Furthermore the
linear coupling of the dilaton to the GB-scalar further
eliminates flat directions even in the multi-field case containing
fields, such as a dilaton, the Higgs fields, etc.
Some authors [
37
] have used a scalar field coupled to a
GB-density in cosmology. However, their methodology is
quite different from ours. For example, in our case scale
invariance restricts the coupling to the GB-density to be only
linear in the dilaton field, while the authors in [
37
] feel free to
choose any function that will eventually provide the desired
solutions. In fact, the full-structure of the theory is rigorously
derived, with no room for ‘string-inspired’ argument.
Additionally, we focus for the moment only on the vacuum state
of the universe, while the subject of [
37
] is a late-time
universe transition from a scaling-matter era to a dark-energy
universe. Those issues concerning the full history of the
universe in our model will be the subject of future studies.
This paper is organized as follows: in the next section, we
talk about the stability of scale-invariant TMT without any
dilaton–GB coupling. In Sect. 3, we introduce a dilaton–GB
coupling and study how the GB coupling eliminates vacuum
flat directions. Discussion and conclusion will be given in
Sect. 4.
2 Stability issues in scale-invariant TMT without
dilaton couplings to GB-scalar
We will study first the dynamics of a scalar field φ interacting
with gravity, without considering couplings to a GB-density.
We start with the following action:
In the variational principle, μν ρ , gμν , the measure-field
scalars ϕa and the ‘matter’-scalar field φ are all to be treated
as independent variables. It results in equations that allow us
to solve some of these variables in terms of others. Treating
the connection independent of the metric is what is referred
to as ‘first-order formalism’. If the connection is given by the
Christoffel symbol, this is referred as the ‘second-order
formulation’. It should be pointed out that the characterization
of these two procedures as merely two different formalisms,
leading to the same result is not correct. Indeed, except for
the special (although very important) case of general
relativity, these two procedures originate inequivalent theories, and
care must be exercised if equivalence is sought.
We can implement global scale invariance in this model
for very special exponential form for the U (φ) and V (φ)
potentials. Indeed, if we perform the global scale
transformation gμν → eθ gμν (θ = const.), then there is invariance
provided V (φ) and U (φ) are of the form
V (φ) = f1 eαφ , U (φ) = f2 e2αφ ,
and ϕa is transformed according to ϕa → λabϕb, which
implies that → det (λab) ≡ λ , such that λ = eθ and
φ → φ − θ /α.
Let us begin by considering the equations which are
obtained from the variation of the fields that appear in the
μ
measure, i.e. the ϕa -fields. We obtain Aa ∂μ L1 = 0, where
Aaμ = εμνρσ εabcd (∂ν ϕb)(∂ρ ϕc)(∂σ ϕd ), since det ( Aaμ) =
(4!/44) 3. If = 0, we obtain ∂μ L1 = 0, or
L1 = − κ1 R( , g) + 21 gμν (∂μφ)(∂ν φ) − V =. M,
.
where M is a constant, and the symbol = stands for a field
equation. Notice that this equation breaks spontaneously the
global scale invariance of the theory, since the left hand side
has a non-trivial transformation under scale transformations,
while the right hand side is equal to M , a constant that after
we integrate the equations, is fixed and cannot be changed.
Therefore, for any M = 0 we have obtained spontaneous
breaking of scale invariance.
By the field equations of gμν , φ and μν ρ out of the total
action S we get
μρ ν − gμν ρσ σ − δρ μ σ σ ν + ∂ρ gμν
where Tμν ρ ≡ μν ρ − νμρ is the torsion tensor. In our
present formulation, we look only for torsion-free solutions,
so that the last three T -linear terms in (2.4c) are ignored, and
we restrict μν ρ = (μν)ρ .
Solving (2.4a) for R( ) = gμν Rμν ( ), and substituting
it into (2.3), we obtain a constraint that allows us to solve for
the ratio ζ of the two measures:
. 2U (φ)
ζ ≡ √−g = M + V (φ)
.
As for the -field equation, we substitute
μν ρ =
ρ
μν
into (2.4c) to get
μνρ +
−4gμρ ∂ν (ln ζ ) =. 0.
ρμν − gνρ μσ σ + 2gμρ νσ σ + gνρ ∂μ(ln ζ )
(2.2)
This is solved for as
. 1
μνρ = + 2 ζ −1 gμρ ∂ν ζ + gνρ ∂μζ − gμν ∂ρ ζ .
To get the physical content of the theory, it is best to
consider variables that have a well-defined dynamical
interpretation. The original metric does not have a non-zero canonical
momenta in the first-order formalism as no derivatives of the
metric appear in the Lagrangian. All derivatives appear in the
connections, which are the fundamental dynamical variables
of the theory. The canonical momenta of these connections
are functions of gμν , where
where ζ is defined by (2.5). Interestingly enough, working
with gμν is the same as going to the ‘Einstein conformal
frame’. It turns out that in terms of gμν the non-Riemannian
contribution μν ρ disappears from the equations, that is,
¯ μν ρ = ρμ ν g¯στ . This is because the connection can be
recast as the Christoffel symbol of the metric gμν .2 In terms
of gμν the equations of motion for the metric can then be
written in the Einstein form,
1
Rμν (g¯) − 2 gμν R(g¯) =. κ2 Tμeνff (φ),
. 1
Tμeνff (φ) = (∂μφ)(∂ν φ) − 2 g¯μν g¯ρσ (∂ρ φ)(∂σ φ) + gμν V eff (φ),
Veff (φ) =. 4U1(φ) (V + M)2,
ρ .
μν = ¯ μν ρ (g¯) =
ρ
μ ν
g¯στ
(2.8)
(2.9)
(2.10a)
(2.10b)
(2.10c)
(2.10d)
(2.11)
(2.3)
gμν = ζ gμν ,
(2.4a)
(2.4b)
(2.4c)
(2.5)
(2.6)
(2.7)
where we define R¯μν (g¯) ≡ Rμν ( ¯ (g¯)), R¯ (g¯) ≡ R( ¯ (g¯)) ≡
gμν R¯μν (g¯).
¯
Using the metric g¯ αβ , the φ-field Eq. (2.4b) becomes
√1−g ∂μ(gμν √−g ∂ν φ) + Veff (φ) =. 0.
In this case the vacuum is obtained for V + M = 0, where
Veff = 0 and Veff = 0 also, provided V is finite and U = 0
there. This means that the vacuum with zero
cosmologicalconstant state is achieved without any fine-tuning. That is,
independent of whether we add to V a constant piece, or
whether we change the value of M , as long as there is still a
point where V + M = 0, then still Veff = 0 and Veff = 0. This
is the basic feature that characterizes the TMT and allows it
to solve the ‘old’ CCP.
If V (φ) = f1eαφ and U (φ) = f2e2αφ as required by
global scale invariance, we obtain from (2.10c)
Veff = 41f2 ( f1 + M e−αφ )2.
(2.12)
2 This is also understandable from the viewpoint of Weyl rescaling: the
factor ≡ ζ √−g in front of the scalar curvature can be completely
absorbed into the Weyl rescaling of the metric.
Since we can always perform the transformation φ → −φ
we can choose by convention α > 0. We then see that as
φ → ∞, Veff → f12/(4 f2) = const., providing an infinite
flat region for the effective potential (see Fig. 1 below). Also
a minimum is achieved at zero cosmological constant for the
case f1/M < 0 at the point
(2.13)
φmin = − α1 ln Mf1 .
For the case when f1/M > 0, the effective potential never
reaches zero as in Fig. 1.
In both cases the asymptotic value of Veff as φ → ∞ is,
according to Eq. (2.12), Veff (∞) = f12/(4 f2), which can be
made very small by means of a see-saw mechanism. By
considering f1 ≈ O(ME4W ) and f2 ≈ O(MP4lanck ), the observed
vacuum energy is indeed of O(ME8W/M 4
Planck ). So this flat
region is indeed a good candidate for representing the
vacuum of the present universe if this see-saw mechanism is
invoked. There is the issue, however, that there is no
particular point at which the dilaton is stabilized, leading to the
appearance of a massless field. Notice that in spite of the
existence of a massless field, TMT models naturally avoid
the fifth-force problem [
38
], since the coupling of the dilaton
to ‘normal matter’, which is more dense than the vacuum
energy density, is negligibly small, but still the vacuum is
still undetermined, since all values in the infinite flat region
of the potential (2.12) appear equivalent
If we have several scalars, however, like a dilaton and a
Higgs field, the sum of the potentials of each of the fields that
couple to the measure squared will appear in the effective
potential and the vacuum with zero cosmological constant
will be realized by a manifold of infinite possibilities, where
the sum of the potentials of each of the fields vanishes. So it
is of interest to see how this degeneracy can be lifted.
As will be demonstrated in the next section, enriching the
scale-invariant TMT with a linear coupling of the dilaton
with the GB-density, does indeed lift both of the
degeneracies described above. In the case of a single dilaton field, this
coupling can determine where in the flat region of the
effective potential the dilaton must reside in the vacuum state, and
in the case of the vacuum with zero cosmological constant,
the manifold degeneracy will be lifted because in addition
to the condition that the sum of the potentials of each of the
fields that couple to the measure be zero, another equation
from the dilaton-field equation will appear that will assist in
the process. All the details are explicitly presented in what
follows.
3 Introducing the dilaton–GB coupling and stabilizing
the vacuum-degeneracy problem
Coupling the dilaton to the GB-density, which is a total
derivative, provide us with an interesting stabilization
property in several situations in TMT in the cases of zero or small
cosmological constant.
In this paper we consider a linear coupling of the dilaton
to the GB-scalar:
G ≡ Rμνρσ ( )Rμνρσ ( ) − 4Rμν ( )Rμν ( ) + R2( ),
(3.1)
Only for the GB-term, we restrict later the affinity to
the Christoffel connection νμρ . For the other parts of the
Lagrangian we leave the connection to be defined by the
equations of motion of the connection. In other words, we
adopt the first-order formulation for the Lagrangians L1 and
L2, while we have the second-order formulation for our
GBterm.3
There are good reasons why the GB-density is considered
in the second-order formulation. Because when we consider
a torsion [
40
], or other non-Riemannian structures in the
firstorder formulation, the GB-density becomes non-topological.
Since the invariance of the action under scale transformations
requires the GB-term to be topological, we must consider
only the GB-term in the second-order formalism.
In our work, the GB-term in the second-order formalism
is coupled linearly to the dilaton and the rest of the action
formulated in the first-order formalism. They give a rise to
second-order differential equations, and the ratio between the
new measure and the Riemannian measure is determined by
3 In passing, we note that terms of the type √−g G ln G were
considered in [
39
].
an algebraic constraint. This way, the new measure does not
introduce new degrees of freedom. In this case, we achieve
invariance under global scale-invariance transformations up
to a total derivative. This total derivative can give a
nontrivial contribution in the presence of gravitational instantons
[
41,42
].
We will now study the dynamics of a scalar field φ
interacting with gravity as given by the following action:
S =
R( , g) ≡ gμν Rμν ( ), Rμν ( ) ≡ Rμρν ρ ( ),
Rμνρ σ ( ) ≡ +∂μ νρ σ − ∂ν μρ σ − μρ τ ντ σ + νρ τ μτ σ ,
where N is a non-zero real constant. In the variational
principle, except for the GB-term, μν ρ , gμν , the measure-field
scalars ϕa , and the ‘matter’-scalar field φ are all to be treated
as independent variables. In the GB-term, only gμν and φ are
involved, because Rμνρ σ (g) ≡ Rμνρ σ ({}). In other words,
the Lagrangians L1 and L2 are in the first-order formulation,
while the GB-term is in the second-order formulation.
For simplicity of computation, it is advantageous to use the
Lorentz connection ωμrs , corresponding to the Christoffel
connection:
Cμνr ≡ ∂μeνr − ∂ν eμr ,
1
ωmrs (e) ≡ − 2 (Cmrs − Cmsr − Crsm ) ,
so the Lorentz-covariant derivative acts like Dμ(ω)Xm ≡
∂μ Xm − ωμ m n(e) Xn for an arbitrary vector Xm with a local
Lorentz index. Therefore, the Lorentz curvature is
Rμνrs (ω(e)) ≡ +∂μωνrs (e) − ∂ν ωμrs (e)
− ωμrt (e) ωνt s (e) + ωνrt (e) ωμt s (e). (3.4)
Accordingly, there is a direct relationship between Rμνρ σ ({})
≡ Rμνρ σ (g) and Rμνρ σ (ω(e)):
Rμνρ σ (g) = eρr es σ Rμνrs (ω(e)),
(3.2a)
(3.6a)
G = Rμνρσ (g)Rμνρσ (g) − 4Rμν (g)Rμν (g) + R2(g)
1
= − 4 e−1 μνρσ emnrs Rμν mn(ω(e))Rρσ rs (ω(e)). (3.6b)
Equation (3.6b) simplifies the computation, because of there
being no involvement of the metric or vierbein other than
ω(e).
As alluded to before, we can have global scale invariance
in this model for the very special exponential form for the
U (φ) and V (φ) potentials (2.2). Indeed, if we perform the
global scale transformation gμν → eθ gμν (θ = const.),
then there is invariance provided V (φ) and U (φ) are of the
form (2.2).4
Let us begin by considering the equations that are obtained
from the variation of the fields that appear in the
measure, i.e. the ϕa fields. We obtain Aa μ ∂μ L1 = 0 where
Aa μ = εμνρσ εabcd (∂ν ϕb)(∂ρ ϕc)(∂σ ϕd ). Now det ( Aaμ) =
(4!/44) 3. If = 0, we obtain the same result as (2.3).
The field equations of gμν , φ and μν ρ are
where Gmn ≡ Rmn − (1/2) ηmn R is the Einstein tensor, and
∗Rm∗nrs ≡ (1/4) mn tu rs vw Rtuvw.
Note that the last two terms in (3.7a) are from the variation
of the GB-term with at least one derivative on φ. Under our
principle of choosing the second-order formulation for the
GB-term, the GB-term has no -field, so that the -field
equation (3.7c) is exactly the same as (2.4c). As in the last
section, we can drop the T -linear terms if we ignore torsion.
For a slow-varying or constant φ, the last two terms in
(3.7a) can be ignored. Solving (3.3) for R( ) = gμν Rμν ( ),
and substituting it into (2.3), we obtain a constraint that
allows us to solve for the ratio ζ of the two measures as
which is confirmed under the ‘metricity’ or
‘vierbeinpostulate’: Dμeνr ≡ ∂μeνr − ωμrs eνs − σμ ν eσ r ≡
Dμ(ω) eνr − σμ ν eσ r = 0 in the second-order
formulation. Relevantly, there are two ways to express our GB-term
in (3.2a) in the second-order formulation:
Here the last term of O(∂ν φ ) is from the last two terms in
(3.7a) from the GB-term with at least one derivative on φ,
which we ignore, since we assume that φ is slowly varying
or constant.
4 The case without coupling to the GB-topological density was studied
in Ref. [
43
].
To get the physical content of the theory, it is best to
consider variables that have a well-defined dynamical
interpretation. The original metric does not have a non-zero canonical
momenta in the first-order formalism as no derivatives of such
metric appear in the Lagrangian. All derivatives appear in the
connections, which are the fundamental dynamical variables
of the theory. The canonical momenta of those connections
are functions of gμν given by (2.9) where ζ is defined by
(3.8). As in the last section, we can go to the ‘Einstein
Conformal Frame’, and use the metric g¯μν instead of gμν . As
in the previous case without the GB-term, all ζ -dependent
terms in μν ρ are absorbed into the Weyl rescaling (2.9), and
therefore the affinity is entirely expressed by the Christoffel
symbol: μν ρ =. ¯ μν ρ =. ρμ ν g¯στ .
Eventually, the forms of (2.10a) and (2.10c) are intact,
while (2.10b) is now modified to
1
Tμeνff (φ) = (∂μφ)(∂ν φ) − 2 g¯μν g¯ρσ (∂ρ φ)(∂σ φ)
+ gμν Veff (φ) + O(∂ν φ ).
Using the metric gρσ , the φ-field equation (3.7b) becomes
1
∂μ(gμν
√−g
+N O(∂ν φ ) =. 0.
−g ∂ν φ) + Veff (φ) + N G(g)
The case N → 0 coincides with the result in Sect. 2. Also,
notice that in the above equation everything must be
calculated using the metric in the Einstein frame, the O(∂ν φ )
terms originate from the fact that the GB has a simple scaling
transformation for a rescaling of the metric, but for a space
time dependent scaling (through the space time dependence
of φ) these terms will appear, but are irrelevant for the case of
constant φ. However, the non-zero GB-term changes this
picture (cf. Fig. 1). Because of the additional N -term, the scalar
field may be determined to assume a constant value at a value
different from the extremum of Veff , Furthermore, even if the
effective potential Veff has no extremum, the equation of the
φ-field can be stabilized at a certain value
Veff ( φ ) + N G =. 0.
(3.10)
(3.11)
This equation follows from (3.10), since we take φ to be
slowly varying or constant.
As for the GB-contribution, two remarks are in order:
First, the -field equation (3.7c) is the same as the N = 0
case, so that all ζ -dependent terms are absorbed into the
rescaling (2.9). Second, all ∂ζ -terms arising after the
rescaling (2.9) in (3.7a) and (3.7b) contain the factor ∂φ, which is
negligibly small for constant or slowly varying φ-field.
Let us consider the vacuum as a de Sitter space
represented as a spatially flat FRW space: ds2 = +dt 2 −a2(dx 2 +
dy2 + dz2). Then G = 24H 4 and the gravitational equation
(3.9)
determine that H 2 = (constant) Veff .5 In the presence of the
GB-term, the Hubble constant is determined only by Veff .
The GB-term effect only enters into the φ-field equation as
Veff ( φ ) − AVe2ff ( φ ) =. 0.
where A is a constant. Using the form for Veff ( φ ) in
(2.12), we find that the above equation can be written as
the extremum of a new effective potential
VeGff ( φ ) = + 1A6Mφf2 − 64MAα f2 ( f1 + Me−α φ )4
+
1 A f22
4 f2 − 32Mα f2
f1 + Me−α φ 2
A
− 16Mα f2
f13( f1 + Me−α φ )+ f31 ( f1 + Me−α φ )3 .
(3.12)
(3.13)
We see that the new potential stabilizes the effective potential
for positive M , which for zero N does not have an extremum;
see Fig. 2.
The non-zero cosmological-constant vacuum in the
presence of the dilaton–GB coupling does not have the
vacuummanifold degeneracy problem in the multi-field situation.
In the case the dilaton couples linearly to the GB-density,
the GB–dilaton coupling eliminates flat directions and
stabilizes the dilaton. Thus there will be no vacuum-manifold
problem in the non-zero cosmological-constant case.
5 The exact equation is G = 24(H 2 + k/a2)(H˙ + H 2), which becomes
G = 24H 4 for k = 0 and H˙ = 0.
4 Discussion and conclusions
The enrichment of the TMT by the addition of a linear
coupling of the dilaton to the GB-topological density has the
effect of stabilizing the dilaton for small cosmological
constant at a certain well-defined value of the dilaton field.
We have studied in particular the solution of an effective
potential of the form Veff = ( f1 + M e−αφ )2/(4 f2), where
f1/M > 0, before the GB–dilaton coupling is introduced.
After the GB–dilaton coupling is introduced, this same Veff
determines the value of the cosmological constant for a
constant φ, but the equilibrium value of φ is determined by
another potential VeGff , which does have an extremum (which
is a minimum).
A related question concerns the lifting of the
‘vacuummanifold degeneracy’ in the TMT in the multi-field case,
when, e.g., we have in addition to the dilaton a Higgs field,
etc. In this case given that Veff is a perfect square, the
vacuum for the Higgs and dilaton, even for the case there is
dilaton–GB coupling is found to be defined by Veff = 0,
which defines a manifold in the multi-field case. In the
nonzero cosmological-constant case, this degeneracy is absent,
furthermore, in the presence of the GB–dilaton couplings, as
we have seen, we obtain further stabilization of the dilaton
in the single-field case. In the multi-field case with dilatons
and Higgs fields, we expect full stabilization and unique
vacuum after spontaneous breaking of gauge symmetries. Some
aspects of the theory could be further analyzed, for example
the effects of matter on the coupling of the dilaton. In [
38
]
for example we have found that, for a high matter density, the
dilaton decouples from matter, one could study if such
properties remain when the dilaton–GB coupling is added. In Ref.
[
7
], we also obtained another way to give the dilaton a mass
generated from instantons, which is another way of
stabilizing the vacuum independent of this one, which could also
solve the vacuum-manifold degeneracy problem, although
this one breaks discrete symmetries like P and CP
symmetries, since it relies on the coupling of the dilaton to a gauge
field topological density that changes sign under parity, but
since the dilaton is a scalar, such a term breaks the parity
symmetry. It would be interesting to study the full effects of
the couplings of the dilaton to both topological densities.
We have so far shown that the cosmological constant is
controlled to a desirable amount based on TMT. The reader
may wonder about the quantum corrections to the
classicallevel cosmological constant. Even though we do not complete
our answer to this question, we mention the important points
not to be overlooked.
First, no theory except for superstring theory [
44
], has ever
formulated consistent quantum corrections in gravitational
theories. Except for superstring with zero cosmological
constant [
44
], quantum gravity theory is not renormalizable, so
that there are no commonly accepted prescriptions for
quantum corrections. Therefore, at the present time, the problem
of quantum corrections is a common problem to any gravity
theory dealing with a non-zero cosmological constant.
Second, the quantum corrections may well be controlled
by scale invariance. In fact, two of the authors (HN and SR)
have proposed a scale-invariant theory [
45–50
], in which the
underlying scale invariance excludes considerable number of
counter-terms [51]. In other words, compared with the
conventional general relativity, scale-invariant theory [
45–51
]
drastically reduces possible counter-terms. From this
viewpoint, it is not far-fetched to expect that the quantum
behavior of scale-invariant theory [
45–51
] is much more
controllable than general relativity. Therefore, combining TMT [
5
]
with scale-invariant theory [
45–51
], the quantum corrections
to the cosmological constant are expected to be more
suppressed than conventional gravity theories.
Third, as we have mentioned, the Two Measures Theories
enjoy the infinite-dimensional group of shifts of the
measure fields ϕa : ϕa → ϕa + fa (L1), where fa (L1) are
arbitrary differentiable functions of the Lagrangian L1, which is
the Lagrangian that couples to the metric-independent
measure. This symmetry is valid only if the metric-independent
measure remains linear in the effective Lagrangian. So if the
quantum corrections were to respect this infinite-dimensional
symmetry, the structure of the two measures theories will be
preserved even after quantum corrections, and also the
general consequences will remain, like the fact that the effective
potential is a perfect square will remain and therefore the
natural zero vacuum energy density will still be obtained.
Intrinsically related to the structure of the TMT is an
additional symmetry, not of the full theory, but of the vacuum with
zero cosmological constant [
52,53
]. Symmetries of this type
for the zero cosmological-constant state have been searched
by many authors, but in the two measures theories they arise
without constructing the theory specially for this purpose;
for further references see [
52,53
].
Preserving the structure of TMT does not impose
restrictions on counter-terms, as long as they are introduced in a
way consistent with this structure in the original frame. When
going to the Einstein frame we obtain a reorganization of
the interactions, for example a cosmological constant in the
original frame enters in the Einstein frame in the
denominator of the effective potential of the scalar fields. So a big
cosmological constant in the original frame is not
necessarily an undesirable feature; we in fact use such a
possibility when discussing the cosmological see-saw mechanism
in TMT.
Fourth, one may observe that although the models we
have discussed have scale invariance, scale symmetry
breaking is obtained through the integration of the measure-field
equations of motion and the effects considered in this paper
survive this scale symmetry breaking. Further scale
symmetry breaking from quantum corrections, if they preserve the
structure of the TMT, are also expected to preserve the main
results of this paper.
Finally, we also would like to recall that the theorem by
Weinberg [
1, 2
] stating that it is impossible to solve the CCP
without invoking fine-tuning does not apply in the TMT [
5,
54
], because here the assumptions that Weinberg used do not
apply.
Acknowledgements One of the authors (EG) gratefully acknowledges
the hospitality at CSULB where this work was carried out.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
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Funded by SCOAP3.
1. S. Weinberg , The cosmological constant problem . Rev. Mod. Phys. 61 , 1 ( 1989 )
2. Y.J. Ng , The cosmological constant problem . Int. J. Mod. Phys. D 1 , 145 ( 1992 )
3. B.K. Gibson , T.S. Axelrod , M.E. Putman , N. Bahcall , J.P. Ostriker , S.J. Perlmutter , P.J. Steinhardt , For reviews of this subject, see for example, M.S. Turner in the third Stromlo Symposium: The Galactic Halo , ASP Conference Series, vol. 666 , 1999 , (eds). Science 284 , 1481 ( 1999 )
4. S. Weinberg , The cosmological constant problems . Talk given at Conference: C00-02-23 , pp. 18 - 26 Proceedings. arXiv:astro-ph/0005265
5. E.I. Guendelman , A.B. Kaganovich , Phys. Rev. D 60 , 065004 ( 1999 ). arXiv:gr-qc/9905029
6. E.I. Guendelman , Scale invariance, new inflation and decaying lambda terms . Mod. Phys. Lett. A 14 , 1043 ( 1999 ). arXiv:gr-qc/9901017
7. E.I. Guendelman , H. Nishino , S. Rajpoot, Phys. Lett. B 732 , 156 ( 2014 ). arXiv: 1403 .4199 [hep-th]
8. E. Guendelman , H. Nishino , S. Rajpoot , Two-measure theory with 3rd-rank antisymmetric tensor for local scale symmetry breaking . (To appear in Phys. Rev. D)
9. E. Guendelman , H. Nishino , S. Rajpoot, Phys. Rev. D 87 , 027702 ( 2013 )
10. E.I. Guendelman , Class. Quant. Gravit. 17 , 3673 ( 2000 )
11. E.I. Guendelman , Phys. Rev. D 63 , 046006 ( 2001 )
12. E.I. Guendelman , A.B. Kaganovich , E. Nissimov , S. Pacheva, Phys. Rev. D 66 , 046003 ( 2002 ). arXiv:hep-th/0203024
13. E.I. Guendelman , A. Kaganovich , E. Nissimov , S. Pacheva, Phys. Rev. D 72 , 086011 ( 2005 ). arXiv:hep-th/0507193
14. H. Nishino , S. Rajpoot, Phys. Lett. B 736 , 350 ( 2014 )
15. E.I. Guendelman , Phys. Lett. B 580 , 87 ( 2004 ). arXiv:gr-qc/0303048
16. E. Guendelman , E. Nissimov , S. Pacheva , M. Vasihoun , Dynamical volume element in scale-invariant and supergravity theories . Bulg. J. Phys . 40 , 121 ( 2013 ). arXiv: 1310 .2772 [hep-th]
17. E.I. Guendelman , A.B. Kaganovich , Neutrino generated dynamical dark energy with no dark energy field . Phys. Rev. D 87 , 044021 ( 2013 ). arXiv: 1208 .2132 [gr-qc]
18. H. Nishino , S. Rajpoot , Mod. Phys. Lett. A 21 , 127 ( 2006 )
19. E. Lim, I. Sawicki , A. Vikman , Dust of dark energy . JCAP 1005 , 012 ( 2010 )
20. S. Capozziello , J. Matsumoto , S. Nojiri , S.D. Odintsov , Phys. Lett. B 693 , 198 ( 2010 ). arXiv: 1004 .369 [hep-th]
21. E.I. Guendelman , D. Singleton , N. Yongram , A two measure model of dark energy and dark matter . JCAP 1211 , 044 ( 2012 ). doi: 10 . 1088/ 1475 - 7516 / 2012 /11/044. arXiv: 1205 .1056 [gr-qc]
22. Stefano Ansoldi (ICRA, Pescara and INFN , Udine and Udine U.), Eduardo I. Guendelman. Sep 2012 . Unified dark energy-dark matter model with inverse quintessence . JCAP 1305 , 036 ( 2013 ). doi: 10 .1088/ 1475 - 7516 / 2013 /05/036. arXiv: 1209 .4758 [gr-qc]
23. E. Guendelman , E. Nissimov , S. Pacheva , Unified dark energy and dust dark matter dual to quadratic purely kinetic k-essence . Eur. Phys. J. C 76 , 90 ( 2016 ). arXiv: 1511 .07071 [gr-qc]
24. E. Guendelman , E. Nissimov , S. Pacheva , Dark energy and dark matter from hidden symmetry of gravity model with a nonriemannian volume form . Eur. Phys. J. C 75 , 10 - 472 ( 2015 ). doi: 10 . 1140/epjc/s10052-015-3699-8. arXiv: 1508 . 02008 [gr-qc]
25. S. del Campo , E.I. Guendelman , A.B. Kaganovich , R. Herrera , P. Labrana, Phys. Lett. B 699 , 211 ( 2011 ). arXiv: 1105 . 0651 [astroph .CO]
26. S. del Campo , E.I. Guendelman , R. Herrera , P. Labrana , Emerging universe from scale invariance . JCAP 1006 , 026 ( 2010 ). arXiv:1006.5734 [astro-ph.CO]
27. E. Guendelman, R. Herrera , P. Labrana , E. Nissimov , S. Pacheva , Gen. Relat. Gravit. 47 , 10 ( 2015 ). doi: 10 .1007/ s10714-015 -1852-1 . arXiv: 1408 .5344 [gr-qc]
28. E.I. Guendelman , Int. J. Mod. Phys. A 26 , 2951 ( 2011 ). arXiv: 1103 .1427 [gr-qc]
29. E.I. Guendelman , Gravitational theory with a dynamical time . Int. J. Mod. Phys. A 25 , 4081 - 4099 ( 2010 ). doi: 10 .1142/ S0217751X10050317. arXiv: 0911 . 0178
30. David Benisty , E.I. Guendelman , Radiation like scalar field and gauge fields in cosmology for a theory with dynamical time . Mod. Phys. Lett. A 31 , 33 ( 2016 ). doi: 10 .1142/ S0217732316501881. arXiv: 1609 .03189 [gr-qc]
31. D. Benisty , E.I. Guendelman , Interactive diffusive unified dark energy and dark matter from scalar fields . arXiv:1701 .08667 [grqc]
32. A. Chamseddine , V. Mukhanov , JHEP 1311 , 135 ( 2013 ). arXiv: 1308 . 5410
33. A. Chamseddine , V. Mukhanov , A. Vikman , JCAP 1406 , 017 ( 2014 ). arXiv: 1403 . 3961
34. M. Chaichian , J. Kluson , M. Oksanen , A. Tureanu , JHEP 1412 , 102 ( 2014 ). arXiv: 1404 . 4008
35. A.O. Barvinsky , Dark matter as a ghost free conformal extension of Einstein theory . JCAP 1401 , 014 ( 2014 ). doi: 10 .1088/ 1475 - 7516 / 2014 /01/014. arXiv: 1311 .3111 [hep-th]
36. R. Myrzakulov , L. Sebastiani , S. Vagnozzi , Eur. Phys. J. C 75 , 444 ( 2015 ). arXiv: 1504 . 07984
37. S. Tsujikawa , M. Sami , See, e.g., JCAP 0701 , 006 ( 2007 ). doi: 10 . 1088/ 1475 - 7516 / 2007 /01/006. arXiv: hep-th/0608178
38. E.I. Guendelman , A.B. Kaganovich , Ann. Phys. 323 , 866 ( 2008 ). arXiv:0704 . 1998 [gr-qc]
39. H.J. Schmit , D. Singleton , J. Math. Phys. 54 , 062502 ( 2013 ). arXiv: 1212 .1769 [gr-qc]
40. H. Niu , D.G. Pak , Non-topological Gauss-Bonnet type model of gravity with torsion . arXiv:0709 .2109 [gr-qc]
41. G.W. Gibbons , S.W. Hawking , Phys. Lett. B 76 , 430 ( 1978 )
42. G.W. Gibbons , S.W. Hawking , Commun. Math. Phys. 66 , 291 ( 1979 )
43. E.I. Guendelman , O. Katz , Class. Quant. Gravit. 20 , 1715 ( 2003 ). arXiv:gr-qc/0211095
44. J.H. Schwarz , M. Green , See, e.g., 'Superstring Theory', vols. I and II (Cambridge University Press, Cambridge, 1986 )
45. H. Nishino , S. Rajpoot , Broken scale invariance in the standard model . Phys. Rev. D 79 , 125025 ( 2009 ). arXiv:hep-th/0403039
46. H. Nishino , S. Rajpoot , Broken scale invariance in the standard model . AIP Conf. Proc. 881 , 82 ( 2007 ). (Melville, New York, 2006 )
47. H. Nishino , S. Rajpoot , Weyl's scale invariance: inflation, dark matter and dark energy connections . in Proceedings of Dark Side of Universe , eds. Shgabaan Khalil, Egypt ( 2008 )
48. H. Nishino , S. Rajpoot, Comment on shadow and non-shadow extensions of the standard model . arXiv:hep-th/0702080
49. H. Nishino , S. Rajpoot, Comment on electroweak higgs as a pseudo-goldstone Boson of broken scale invariance . arXiv:0704 . 1836 [hep-ph]
50. H. Nishino , S. Rajpoot, Comment on papers by Foot, Kobakhidze, McDonald and Volkas relating to scale invariance symmetry . arXiv:0712 .2487 [hep-ph]
51. H. Nishino , S. Rajpoot , Class. Quant. Gravit. 28 , 145014 ( 2011 )
52. E.I. Guendelman , A.B. Kaganovich , Transition to zero cosmological constant and phantom dark energy as solutions involving change of orientation of space-time manifold . Class. Quant. Gravit . 25 , 235015 ( 2008 ). doi: 10 .1088/ 0264 -9381/ 25/23/235015. arXiv: 0804 .1278 [gr-qc]
53. E.I. Guendelman , A.B. Kaganovich , Symmetry supporting a transition to zero cosmological constant state . arXiv:0707 .4286 [hep-th]
54. E.I. Guendelman , A.B. Kaganovich , Fine tuning free paradigm of two measures theory: k-essence, absence of initial singularity of the curvature and inflation with graceful exit to zero cosmological constant state . Phys. Rev. D 75 , 083505 ( 2007 ). doi: 10 .1103/ PhysRevD.75.083505. arXiv:gr-qc/ 0607111