revised version IPMU14-0005
On the Higgs-like Quintessence for Dark Energy
Sergei V. Ketov and Natsuki Watanabe
Department of Physics, Tokyo Metropolitan University
Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan
Kavli Institute for the Physics and Mathematics of the Universe (IPMU)
The University of Tokyo, Chiba 277-8568, Japan
Institute of Physics and Technology, Tomsk Polytechnic University
30 Lenin Ave.,Tomsk 634050, Russian Federation
We propose a dynamical (quintessence) model of dark energy in the current universe with a renormalizable (Higgs-like) scalar potential. We prove the viability of our model (after fine tuning) for the certain range of the average scalar curvature values, and study the cosmological signatures distinguishing our model from the standard description of dark energy in terms of a cosmological constant.
The dynamical dark energy models of modified gravity are usually constructed in the framework of gravity — see e.g., Refs. [1, 2, 3, 4, 5] for some reviews with many references therein, and Refs. [6, 7, 8] for some viable proposals to the -function, that are directly related to this paper. The current status of the gravity theories is phenomenological (or macroscopic) and is truly non-perturbative (or non-linear). The -functions are chosen ad hoc, in order to satisfy the existing phenomenological constraints coming from the Newtonian limit, classical and quantum stability, the Solar System tests and the cosmological tests. The usual treatment includes rewriting an -gravity model into the classically equivalent scalar-tensor gravity (or quintessence) by the Legendre-Weyl transform from the Jordan frame to the Einstein frame , and then applying the standard cosmology in terms of the dual (quintessence) scalar potential (see also Sec. 2). For instance, the 5th force (due to the quintessence scalar) is screened at high matter density (like that of the Solar system) by the Chameleon effect .
The scalar potentials, originating from the gravity functions of Refs. [6, 7, 8], are very complicated and non-renormalizable. In this Letter we physically motivate a shape of the quintessence scalar potential, and then use the inverse Legendre-Weyl transformation, in order to determine yet another function that is suitable for describing the present dark energy in the Universe and is related to a renormalizable quintessence scalar potential.
The paper is organized as follows. In Sec. 2 we review the Legendre-Weyl transformation and give its inverse form. In Sec. 3 we propose the scalar potential of the Higgs-type with the Uplifted-Double-Well (UDW) shape, and find the corresponding function. Further, we demonstrate that our model is viable, being close to the standard Einstein gravity with a (positive) cosmological constant . In Sec. 4 we study the modified gravity corrections (beyond the cosmological constant), in order to distinguish our model from the standard description of the present dark energy by the , by using MATHEMATICA. Sec. 5 is our conclusion.
Throughout this paper we use the natural units, , and the space-time signature . The Einstein-Hilbert action with a cosmological constant reads
where is the scalar curvature, , and is the (reduced) Planck mass in terms of the Newton constant . In our notation here, the cosmological constant is positive in a de-Sitter (dS) space-time (like the present Universe).
A generic -gravity action reads
with a function of the scalar curvature . So that, in our notation, a de-Sitter space-time has a negative scalar curvature. The gravity can be considered as the modified gravity theory extending the Einstein gravity theory with a cosmological constant (= static dark energy) defined by Eq. (1) to the gravitational theory with a dynamical dark energy.
An expansion of the function in power series of near a de-Sitter vacuum gives rise to the modified gravity corrections to the static dark energy. For example, the simplest model with
is known in cosmology as the Starobinsky model . It is well suitable for describing the early universe inflation, with the inflaton mass (see e.g., Ref. ). Hence, inflation can also be considered as a (primordial) dark energy.
2 The Legendre-Weyl transform and its inverse
The gravity action (2) is classically equivalent to
with the real scalar field , provided that that we always assume. Here the primes denote the differentiation. The equivalence is easy to verify because the -field equation implies .
The factor in front of the in eq. (4) can be eliminated by a Weyl transformation of metric , so that one can transform the action (4) into the action of the scalar field miminally coupled to the Einstein gravity and having the scalar potential 
The kinetic term of becomes canonically normalized after the field refedinition
in terms of the new scalar field . As a result, the action takes the standard quintessence form .
respectively. The first condition ensures the existence of a solution to Eq. (6).
The mass dimensions of various quantities are given by
Differentiating the scalar potential in Eq. (5) with respect to yields
where we have
It implies that
These two equations define the function in the parametric form in terms of a given scalar potential . It is the inverse transformation against defining the scalar potential in terms of a given function, according to Eqs. (5) and (6).
3 The UDW Scalar Potential
A choice of the quintessence scalar potential is usually dictated by desired phenomenology without imposing formal constraints. It is in the striking difference with the Standard Model of elementary particles whose Higgs scalar potential of the Double-Well (DW) shape is severely constrained by renormalizability to a quartic function of the Higgs field. In fact, it is one of the basic reasons for high predictability of the Higgs-based physics of elementary particles!
Since we are interested in the existence of a de Sitter vacuum for describing dark energy in gravity, we uplift one of the two DW-vacua of the Higgs scalar potential, while keeping another one to be a Minkowski (flat) space-time, as in Fig. 1. The potential barrier between those two vacua has to be high enough, in order to be consistent with the long lifetime of our Universe. It implies that the de Sitter vacuum is meta-stable and may be considered as a “false” vacuum against the “true” Minkowski vacuum.
We write down such Uplifted-Double-Well (UDW) quartic scalar potential in the form 
where we have introduced the field
and four real parameters , , and of (mass) dimension
Amongst those parameters, the sets the dimensional scale, the is physically irrelevant (it can be changed by a shift of the field ), so that the shape of the UDW potential is determined by only two dimensionless parameters .
The parametrization of the Higgs-type scalar potential used in Eq. (14) is convenient for several reasons. First, it is a generic parametrization compatible with the UDW shape. Second, it allows us to have the mass (of the quintessence scalar), the cosmological constant in the dS vacuum, and the height of the barrier between the dS and Minkowski vacua to be fully independent (see below). Third, the scalar potential (14) admits a supersymmetric extension of the quintessence theory, as was already demonstrated in Ref. , by using a particular (O’Raifertaigh-type) model of spontaneous supersymmetry breaking. Fourth, it appears to be easy to find the vacua of the scalar potental Eq. (14) analytically.
The UDW scalar potential (14) has three extrema at
where and are the positions of the de Sitter and Minkowski vacua, respectively, and is the position of the maximum of the potential barrier (Fig. 1).
The height of the barrier is given by
Differentiating the with respect to yields
where we have
Substituting these equations into the inverse transformation formulae of Sec. 2 yields
Equations (21) and (22) determine the exact function in the parametric form, which is suitable for numerical calculations (e.g., by using MATHEMATICA computing software). Moreover, those equations are greatly simplified when using the very small cosmological constant and the very high potential barrier, corresponding to the present dark energy and the current age of our Universe, respectvely. Both conditions necessarily imply
The required smallness of the cosmological constant is then easily achieved by fine-tuning the value of , whereas the high potential barrier related to (meta)stability of our Universe in the de Sitter vacuum can be achieved by raising the parameter to the desired value — see Eq. (18). In addition, we choose
Near the de Sitter vacuum, the values of are close to zero, so that we can expand both and to the order as
The first equation (26) can be easily inverted as
Substituting it into the second equation (26) for yields
It takes the form of Eq. (1) with the cosmological constant
The mass of the canonically normalized quintessence scalar is given by
in the dS vacuum at , so that it is independent upon the value of the cosmological constant in Eq. (29) indeed. Actually, we have the relation
As is clear from Eq. (28), the standard cosmological model of the current dark energy, described by the observed cosmological constant (29), is recovered by the extreme fine-tuning of the parameters in the UDW potential. Fine-tuning of the parameters of the effective (quintessence) scalar potential is the common property of all known models of the present dark energy. Usually, one assumes that the mass of the quintessence scalar is of the order of the current Hubble scale, ie. eV. In our context it means that both parameters are of the order one. However, in our UDW model, we can also assume that and , so that the values of and decouple from each other.
4 The gravity corrections in our model
The difference between the cosmological constant and an gravity model is described by the corrections beyond the leading term in Eq. (28), which follow from the gravity function. In our case, when using the parametric form of the -function given by Eq. (25) in the approximation (26) and keeping the subleading terms, a tedious calculation gives
where we have ignored the higher-order terms in and , and have introduced the background scalar curvature .
We also verifed that the stability conditions (7) are satisfied for the values of near and larger than the in the regions I and II (as, e.g., in the Solar System and beyond it), i.e. and , as long as is negative.
Let us consider the regions I and II in Fig. 1 away from the dS minimum, with much larger than , in more detail. The behavior of the function in the first line of Eq. (25) after its renormalization by is plotted in Fig. 2.
For numerical calculations by the use of MATHEMATICA, we choose the values of the shape parameters as and , in agreement with our basic assumptions that is small and is large.
The function has four extrema denoted by and with . The and are in the vicinity of , and they are strongly dependent upon the value of but are slightly dependent upon the value of . When , and reach the values and , respectively. When becomes large, , the values of and are essentially independent upon the values of and . For very large both and become imaginary and the extrema of the for those ’s vanish.
The extremal values of and are approximately given by
whereas . As increases, the reaches zero. Since the values of and strongly depend upon the value of , both and are close to zero in the large region. As is shown in Fig. 2, the has extrema only in the region where is positive. When is negative, the rapidly falls down to . We find the following approximations for :
Region I: it appears that already for we can take . Then the higher order terms in inside the square brackets of the can be ignored. Hence, we can approximate the as
Region II: when is larger than the , the reaches zero and, therefore, the condition is not valid in this region. However, both and are still significantly larger than the . Hence, an expansion of the around the can be valid too.
To the end of this Section, we calculate an expansion of the function in the vicinity of up to the second order in . It is the very extreme (non-physical) case far away from the dS vacuum, which helps us to make some qualitative conclusions in Sec. 5. Given and , we find
At we find a violation of the classical stability condition .
We conclude that the Higgs-like quintessence scalar potential (14) after fine-tuning of its parameters is a viable proposal for the present dark energy. According to the proposal, our Universe is in a meta-stable dS vacuum. This description is still valid for the average scalar curvatures (near an observer) but breaks down for approaching (in the case of the Solar system, its average scalar curvature obeys ).
In our simple model the cosmological constant and the mass of the quintessence scalar decouple, being independent upon each other.
Our results also imply that the effective “coupling constants” are very slowly dependent upon of the observer. Hence, our dynamical model of dark energy can be distinguished from the standard description (by via a possible time (and/or space) dependence of the observed values of on cosmological scales.
This work was supported by the Tokyo Metropolitan University and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The authors thank Muhammad Usman for correspondence.
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