# Accelerating Universe and the Scalar-Tensor Theory

## Abstract

**:**

## 1. Introduction and Summary

## 2. Scalar-Tensor Theory Due to Jordan

- The first term with ${\varphi}^{2}$, multiplied with R, is called a nonminimal coupling term, designed to give an effective gravitational “constant” represented by $\xi {\varphi}^{2}={\left(8\pi {G}_{\mathrm{eff}}\right)}^{-1}$ allowing it to be spacetime-dependent, obviously intended to implement Dirac’s idea of the time-dependent gravitational constant [20]. Note the continued use of the Planckian unit system.
- In the sign factor $\u03f5=\pm 1$, the positive choice corresponds to a positive kinetic energy of ϕ, though the negative choice is not excluded immediately because ϕ at the level of (7) is not a fully diagonalized field. See Section 2.6 of [11] for details on the mixing interaction between ϕ and the spinless part of the metric, taking place in the nonminimal coupling term.
- We also use the parameter $\xi >0$, instead of the original $\omega =\u03f5{\left(4\xi \right)}^{-1}$, excluding repulsive gravity.
- We have the ordinary type of the matter Lagrangian ${L}_{\mathrm{matter}}$, specifically representing the fundamental fields in the microscopic world.
- Also to be added beyond Jordan is the cosmological constant $\Lambda >0$ introduced to expect the theory now to provide us with a new way of understanding the accelerating universe. Respecting the arguments on the Unification program, we assume Λ of the order unity in the Planckian unit system, or $\sim {M}_{\mathrm{P}}^{4}$.

## 3. Conformal Transformation/Frames

**Figure 1.**${\zeta}^{2}$ given by (13) restricted to be positive is plotted against $\xi >0$, taken from Figure 1 of [25]. There are two branches for $\u03f5=\pm 1$ bounded by the two (dotted) straight lines $\xi =1/6$ and ${\zeta}^{2}=1/6$. The results from the solar-system experiments correspond to the points, like the one marked by +, converging toward the origin $\xi ={\zeta}^{2}=0$ with $\u03f5=+1$. The symbol ) also marked with r at the point $(\xi =0.5,{\zeta}^{2}=0.25)$ naturally with $\u03f5=-1$ provides with the boundary of selecting the portion of the curve to the upper-left, arising from the positive radiation-dominated matter energy both in J frame, (26) and (27), and E frame, (38). The same $(\xi =1.5,{\zeta}^{2}=3/16=0.1875)$ marked by d is for the dust-dominated universe. The symbol × shows a prediction of string theory in higher-dimensional spacetime, $\u03f5=-1,\xi =1/4$ thus $\omega =-1$, and ${\zeta}^{2}=1/2$, based on (4.1) of [25], taken originally from (3.4.58) of [26].

## 4. Simple Cosmology

#### 4.1. Radiation Dominance in J Frame

#### 4.2. Radiation Dominance in E Frame

**Figure 3.**An example of the phase-diagrams in E frame taken from Figure 3 of [28]. The evolution variable is chosen to be ${\tau}_{*}=\sqrt{V\left(\sigma \right)}d{t}_{*}$, while the coordinates are defined by $x\left({\tau}_{*}\right)=d\sigma /d{\tau}_{*}$ and $y\left({\tau}_{*}\right)={\zeta}^{-1}(d{a}_{*}/d{\tau}_{*})/{a}_{*}$, which satisfy the self-autonomous equations (3.15) and (3.16) of [28]. The solid and dashed curves in (

**a**) are for the null curves of $dx/d{\tau}_{*}=0$ and $dy/d{\tau}_{*}=0$, respectively, bounding the area of $dx/d{\tau}_{*}>0$ marked by ${+}_{x}$, for example. The fixed points are $x=y=1$ and $x=y=-1$ for an attractor and repeller, respectively, as shown in the close-up views in (

**b**) and (

**c**). The trajectory shown by a dotted curve enters the frame of (

**b**), with $\xi =1/4$ thus ${\zeta}^{2}=1/2$, near the lower-left corner, going out across the right edge, re-entering again at the top, spiraling finally into the attractor at $x=1=1$. No such trajectory is shown naturally in (

**c**).

- According to (35), the universe not only expands but also does so in conformity with the assumed radiation-dominance.
- Equation (37) can be interpreted as implementing the scenario of a decaying cosmological constant, represented by (6) supposed to apply to quite wide a time span, as a first step toward understanding the numerical relation (4), which we have focused upon as the simplest yet probably the most important achievement arising from the discovery of the acceleration of the universe. As we also add, the behavior of the inverse-square of time is simply a standard result deriving a critical density. As we recall, this nearly desired behavior is a consequence of the exponential potential (15), also traced back to the simplest imaginable choice, Λ, added to (7). We notice, however, that multiplying Λ by ${\varphi}^{q}$ results only in replacing ζ in $V\left(\sigma \right)$ by $\zeta -q/4$.

#### 4.3. The Brans–Dicke Requirement

## 5. Scale-Invariance Model

#### 5.1. Leaving the Brans–Dicke Model

#### 5.2. Spontaneously Broken Scale Invariance

#### 5.3. Quantum Loop Effects

**Figure 4.**The simple Yukawa interaction with the coefficient $2-d$ as in (

**a**), but now with a non-gravitational radiative correction included, like in (

**b**), where the dashed curve is for a non-gravitational field with the associated coupling constant ${g}_{c}$. Heavy dotted lines drawn vertically are for σ.

**Figure 5.**A loop diagram generating a mass of the field σ (heavy dotted lines), while solid lines inside a loop represent quarks or leptons, with the coupling strength proportional to their own masses divided by ${M}_{\mathrm{P}}$. We also assume the integral to be cut off roughly around ${M}_{\mathrm{ssb}}$, the mass scale of supersymmetry breaking.

**Figure 6.**(

**a**) 1-loop photon self-energy part with σ (heavy dotted line) attached to two of the vertices. (

**b**) The same but one of the photon lines (thin dotted lines) attached to another charged field (vertical solid line), with σ attached to three of the vertices.

#### 5.4. A Slight Deviation from E Frame

## 6. Trapping Mechanism

#### 6.1. Hesitation Behavior

**Figure 7.**An example of hesitation behavior, taken from Figure 5.6 of [11]. The solid curve in the upper-half of the plot shows $2ln{a}_{*}$, while the dashed curve represents $2\sigma $. In the lower-half of the plot, the dashed and the solid curves are for ${log}_{10}{\rho}_{*}$ and ${log}_{10}{\rho}_{\sigma}$, respectively. We chose $\zeta =1.5823$, the same as will be used in the next subsection. The initial values at $log{t}_{*1}=10$ is given by ${\sigma}_{1}=6.75442,{\dot{\sigma}}_{1}=0$, while the matter density assumed to be radiation-dominated is $3.7352\times {10}^{-23}$.

#### 6.2. Mini-Inflation(s)

**Figure 8.**The potential $V(\sigma ,\chi )$ given by (98), taken from Figure 5.7 of [11]. Along the central valley with $\chi =0$, the potential reduces back to the simpler behavior $\Lambda {e}^{-4\zeta \sigma}$, but with $\chi \ne 0$, it shows an oscillation in the σ direction. The configuration of σ and χ is represented by a point, which is trapped in one of the valleys in the χ direction stays there, hence contributing a lasting ${\rho}_{\sigma \chi}$ that acts like a cosmological “constant”. As the time elapses, however, the force in the χ direction towards the central valley becomes strong, because of the increase of ${t}_{*}^{2}$ in the last term on LHS of (101), eventually releasing the point in the positive σ direction, the end of the mini-inflation. For more details, see also Figure 5.14 of [11].

**Figure 9.**An example of the solution, taken from Figure 5.8 of [11]. In accordance with the argument following (102), we chose ${\zeta}_{\mathrm{dm}}=0$, without much affecting the result around today. Upper diagram: $b=ln{a}_{*}$ (dotted), σ (solid) and $2\chi $ (dashed) are plotted against ${\tau}_{10}=log{t}_{*}$. The present epoch corresponds to ${\tau}_{10}=60.1-60.2$, while the primordial nucleosynthesis must have taken place at ${\tau}_{10}\sim 45$. The parameters are $\Lambda =1,\zeta =1.5823,m=4.75,\gamma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.8,\kappa =10$. The initial values at ${\tau}_{10}={10}^{10}$ are ${\sigma}_{1}=6.7544,{\sigma}_{1}^{\prime}=0$ (a prime implies differentiation with respect to $\tau =ln{t}_{*}$), ${\chi}_{1}=0.21,{\chi}_{1}^{\prime}=-0.005,{\rho}_{1\mathrm{rad}}=3.7352\times {10}^{-23},{\rho}_{1\mathrm{dust}}=4.0\times {10}^{-45}$. The dashed-dotted straight line represents the asymptote of σ given by $\tau /\left(2\zeta \right)$. Notice long plateaus of σ and χ, and their rapid changes during relatively “short” periods. Middle diagram: ${p}_{*}={b}^{\prime}={t}_{*}{H}_{*}$ for an effective exponent in the local power-law expansion ${a}_{*}\sim {t}_{*}^{{p}_{*}}$ of the universe. Notable leveling-offs can be seen at 0.333, 0.5 and 0.667 corresponding to the epochs dominated by the kinetic terms of the scalar fields, the radiation matter and the dust matter, respectively. Lower diagram: $log{\rho}_{\sigma \chi}$ (solid), the total energy density of the σ-χ system, and $log{\rho}_{*}$ (dashed), the matter energy density. Notice an “interlacing" pattern of ${\rho}_{\sigma \chi}$ and ${\rho}_{*}$, still obeying $\sim {t}_{*}^{-2}$ as an overall behavior. Nearly flat plateaus of ${\rho}_{\sigma \chi}$ precede before it overtakes ${\rho}_{*}$, hence with ${\Omega}_{\Lambda}$ passing through 0.5.

**Figure 10.**An example of the solution, taken from Figure 5.11 of [11], showing no mini-inflation around the present epoch, though another mini-inflation at ${\tau}_{10}\sim 27$ is still present. Symbols and initial values are the same as explained in Figure 9, except for ${\sigma}_{1}=6.761$, which is different form 6.7544 in Figure 9 only slightly. This indicates how sensitively the result might depend on the choice of some of the parameters.

- Generally speaking, the accelerating universe we are now watching might be one of the repeated events, in the past or the future. Due to this feature, we may expect to lessen the weight of the coincidence problem, though by a little bit, not entirely.
- The mini-inflations do occur as step-like behavior superimposed on the smooth and overall behaviors $\sim {t}_{*}^{-2}$, as expected. The values of the height of each plateau, essentially ${\Lambda}_{\mathrm{obs}}^{\left(i\right)}$ for $i=1,2,\cdots $, are given basically by $\sim {t}_{*i}^{-2}$, with the times ${t}_{*i}$ at which the mini-inflation occurred. This is because ${\rho}_{*}$ and ${\rho}_{\sigma \chi}$ falling off interlacingly with a common overall behavior as $\sim {t}_{*}^{-2}$. The exact time ${t}_{*i}$ of a mini-inflation is obviously determined by the height of the plateau, which depends critically on the parameter values at preceding times. For this reason exact timing of mini-inflations are hard to be determined theoretically. But once determined, the relation ${\Lambda}^{\left(i\right)}\sim {t}_{*i}^{-2}$ always follows, no matter when it occurs. This is what is truly meant by (4).
- At this point we take up again the question we asked ourselves toward the end of Section 1: Can we include the presence of a finite-range force between matter objects in a manner consistent with the cosmological evolution in terms of σ, which rolls slowly down the smooth slope of an exponential potential? The key of the points lies in noticing that we found the solution, as in the upper panel of Figure 9 in which σ varies in a range of the order unity in the Planckian units. Even the initial value ${\sigma}_{1}=6.7544$ is overwhelmingly larger than the value corresponding to the resonance; ${\sigma}_{\mathrm{res}}\sim {m}_{\sigma}\sim {10}^{-9}\phantom{\rule{3.33333pt}{0ex}}\mathrm{eV}\sim {10}^{-36}$ in the Planckian units. This implies that ${\sigma}_{\mathrm{res}}$ is completely outside the range corresponding to the entire history of the universe. In this sense a massive pseudo dilaton exerts a force nearly independent of the accelerating universe, in conformity with the former phenomenon belonging to the local physics, as we remarked.
- There are so many different fine details as were discussed in Section 5.4.2. of [11]. We will make a comment particularly on Figure 11, in which the vertical scale is enlarged by more than 300 times of Figure 9, also with the horizontal scale by more than 10 times. Around the present time, what appears to be a little upward kink of σ (solid curve) in the upper panel of Figure 9 is magnified to reveal a small oscillatory structure unique to the trapping dynamics. This small variation of σ may be substituted to the RHS of the second equation of (93) to predict an oscillatory time-variation of the fine-structure constant, to be tested by near-future experiments [40,41,42,43,44]. In order to give an example, we are going to show a set of plots in Figure 12 calculated based on Figure 9 to be compared with the observation [46,47]. We expect that future experiments with improved accuracy will probe the proposed trapping mechanism probably with even better precision than in measuring the way of accelerating universe itself. It seems important to emphasize that we do provide with this way of observational verification, though our trapping mechanism implemented in terms of the potential (98) is not yet necessarily proved unique at present. For more recent observations on the ratio of the electron to the proton masses, see [49,50], and papers cited therein.
- The equations are highly nonlinear, as was discussed in [48] and also in 5.4.3 of [11]. As an example, an apparently repeated occurrence of mini-inflations, particularly its separations, or “frequencies”, are determined by the initial values or the parameters, in the presence of the cosmological friction, but not prepared in the starting Lagrangian like a harmonic oscillator, for example. Consequently, some of the final results may depend sharply on the choice of the initial values or the parameters. This might even be at odds with a traditional attitude that the cosmologically computed results on the present era should be as insensitive as possible to the initial states of the universe. But is anything wrong if certain aspect of the universe is as chaotic as what is happening daily around us? Some of the parameters have been fine-tuned, but not to the extreme extent of 120 orders of magnitude. We have obtained Figure 9, for example, after a few days of working on Mathematica. It might be still worth quoting, among others, that the occurrence of another mini-inflation around ${\tau}_{10}\sim 27.5$ was a consequence of requiring none of the significant presence of ${\rho}_{\sigma \chi}$ in the era of primordial nucleo-synthesis around ${\tau}_{10}\sim 45$.

**Figure 12.**Typical plots of the theoretical curves for $(\Delta \alpha /\alpha )\times {10}^{6}$ as a function of redshift z, translated from the same of ${t}_{*}$, taken from Figure 1 of [40]. See also [41,42,43,44]. The Oklo phenomenon having occurred $\approx 1.95\times {10}^{9}\mathrm{y}$ ago corresponds to $z\sim 0.15$[45], while two QSO data [46,47] are shown; $-0.12\pm 1.79$ and $5.66\pm 2.67$ for $z=1.15$ and $1.84$, also for the fractional look-back time 0.59 and 0.73, respectively. We commonly choose the initial values at ${t}_{1}={10}^{10}$ in the reduced Planckian unit system, as in Figure 5.8 of [11]; ${\sigma}_{1}=6.77341501,{\sigma}_{1}^{\prime}=0,{\chi}_{1}=0.21,{\chi}_{1}^{\prime}=0$, where the prime is for the derivative with respect to $\tau =lnt$.

## 7. Proposed Experimental Search for the Scalar Field

**Figure 13.**σ-dominated tree diagrams for the photon-photon scattering process, taken from Figure 3 of [18]. Solid lines are for the photons with the attached momenta p’s while the dashed lines for σ, in the s-, t-, and u-channels, respectively.

**Figure 14.**A single Gaussian laser beam focused by an ideal lens where a scalar field exchange entails a frequency-upshifted photon in the forward direction, taken from Figure 5 of [18]. The frequency of the incident laser beam is assumed to be within a narrow band, while the incident angle varies largely including the value $\sim {10}^{-9}$.

**Figure 15.**Definitions of kinematical variables, taken from Figure 1 of [18], in the Quasi-Parallel-Frame.

## 8. Concluding Remarks

- The overall Scenario of a decaying cosmological constant $\Lambda \sim {t}_{*}^{-2}$ is shown to be implemented naturally in E frame in terms of the simplest version of the scalar-tensor theory with the term of $\Lambda \sim 1\sim {M}_{\mathrm{P}}^{4}$ added in J frame, a model much simpler than any of those discussed by [54,55], for example. With the help of a rather phenomenological trapping mechanism, we reproduce a mini-inflation, with ${\Lambda}_{\mathrm{obs}}\sim {t}_{0*}^{-2}$, undoubtedly a core of the message of the observed accelerating universe, leaving us free from the fine-tuning problem. This E frame is close to the physical conformal frame in which we have a constant unit of length/time provided by microscopic particles, with reference to which an expansion/acceleration of the universe is measured. We must be ready, however, to leave the long-held idea of the Brans–Dicke requirement, replacing it by another model of global scale invariance, thus allowing WEP violation expected to be somewhere below the directly available observational upper bounds.
- It is crucially important to recognize that the invariance just mentioned is broken spontaneously in which the gravitational scalar field plays a role of a dilaton, a pseudo NG boson, allowing us to understand two vastly different scales of size, cosmological and microscopic, in an entirely new perspective. Remarkably enough, this dilaton mediates a WEP violating force, likely with the finite range around the order of 100 m, or so. In order to search for it, we propose two types of the experiment; photon-photon scattering using strong laser beams, and time-dependent variation of the fine-structure constant and the ratio of the masses of electron and proton. As we also point out, the dilaton is responsible for providing us with the Higgs field in the Standard Model. It then follows that all the masses in the world have their common origin ultimately in the scalar field of the scalar-tensor theory, re-formulated to understand the observed accelerating universe.
- In this connection we recall a comic drawn by Sato [56] in 1983, when he symbolized a then new era of Unification inspired particularly by GUT (Grand Unified Theories), as reproduced in Figure 16. Decades later, we still appreciate the same comic, but this time reminding us of how strongly the cosmology is tied with the microscopic physics, specifically in terms of the choice of the physical conformal frame, a unique concept of the scalar-tensor theory, with the role of a dilaton.
- Also noteworthy is the importance of J frame, although it is not a physical frame. Above-mentioned scale-invariance is made visible in J frame, in which the radiation-dominated universe turns out to be static asymptotically. Obviously, J frame is an indispensable ingredient from a theoretical point of view. In this connection we point out that a string-theory model in higher-dimensional spacetime was shown in J frame, with $\u03f5=-1$, as indicated in Figure 1.
- We still know little about how various masses and other coupling constants vary with time beyond the lowest-order perturbation estimates. This might be a problem if we look deeper into the early universe. In the more practical side, on the other hand, we already have examples of “composite” units, like the Rydberg constant, the reduced mass multiplied with the electric charge, as pointed out in Subsection 5.1. Even different reduced masses might depend on time differently, if the leptons and quarks yield different variations. The presence by itself of a number of different units in this sense might be an issue from a more general point of view. The exercises attempted in Subsection 5.4 are expected to provide us with a first step toward possible complications. We should be prepared with other types of complications arising both from theoretical and phenomenological aspects.
- One of the aspects we failed to discuss from a truly more significant view is another possible origin of the cosmological constant; the vacuum energy expected from the relativistic quantum field theory, as was emphasized in [57], for example. In view of our success in understanding the scenario of a decaying cosmological constant as a theoretical goal, probably an entirely different approach appears to be called for. See, however, [58], for example.

**Figure 16.**Fumitaka Sato’s image of Unification in 1983. His original caption in Japanese goes like “Understanding microscopic world now provides us with a powerful tool to understand the hyper-macroscopic world”. In his own drawing, a guy is looking into a microscope instead of a telescope, yelling “Look, I got the universe!”.

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**MDPI and ACS Style**

Fujii, Y.
Accelerating Universe and the Scalar-Tensor Theory. *Entropy* **2012**, *14*, 1997-2035.
https://doi.org/10.3390/e14101997

**AMA Style**

Fujii Y.
Accelerating Universe and the Scalar-Tensor Theory. *Entropy*. 2012; 14(10):1997-2035.
https://doi.org/10.3390/e14101997

**Chicago/Turabian Style**

Fujii, Yasunori.
2012. "Accelerating Universe and the Scalar-Tensor Theory" *Entropy* 14, no. 10: 1997-2035.
https://doi.org/10.3390/e14101997