*4.2. New Version of CCC*

The main aim of this subsection is to explain a new factor we would like to add to the CCC which has more than a decade of research history. At the present moment, we are not sure whether our new factor will fit consistently into the basic schemes of the CCC so far investigated. However, we hope that our proposal presented here could be a somewhat useful contribution to the CCC which is related, for instance, to a particular study by Lübbe [35] who discussed the inclusion problem of a cosmological constant. As our dark energy model introduced in (47) is related to de Sitter space, we start from the run-through of the well-known characteristics of it by looking into the Einstein field equation

$$R^{\,\,\,\nu}\_{\mu} - \frac{R}{2} \mathbf{g}^{\,\,\,\nu}\_{\mu} - \Lambda\_{d\varepsilon} \mathbf{g}^{\,\,\,\nu}\_{\mu} = 0,\tag{56}$$

which yields a familiar solution given by

$$ds^2 = (cdt)^2 - (R\_0)^2 \exp\left[2\sqrt{\frac{\Lambda\_{dc}}{3}}ct\right] [dr^2 + r^2(d\theta^2 + \sin^2\theta)d\phi^2],\tag{57}$$

where the constant *R*0 serves as the coefficient of the time-dependent scale factor. In the use of (50), this solution can be simplified by taking *R*0 = *lp* into

$$ds^2 = \left(cdt\right)^2 - \left(l\_{\mathcal{P}}\right)^2 \exp\left[2\sqrt{\Lambda\_{dm}}ct\right] \left[dr^2 + r^2(d\theta^2 + \sin^2\theta)d\varphi^2\right].\tag{58}$$

At the end of Section 3, the simultaneous CSB in electromagnetic and gravitational fields was mentioned. We now explain what this exactly means. Recall that the energymomentum tensor *T*ˆ *νμ* of the spacelike (*UνU<sup>ν</sup>* < 0) CD field is given in Section 2.1 by (26),

which is isomorphic to the Einstein tensor *G νμ* . The same quantity *T* ˆ *ν μ* also emerges from the light-like case of *UνU<sup>ν</sup>* = 0 by replacing *∂ν∂νφ* = 0 with [*∂ν∂ν* − (*<sup>κ</sup>*0)<sup>2</sup>]*φ* = 0, which can be regarded as the breaking of both symmetries, i.e., conformal and gauge (cf. (10)). Therefore, this CSB from the light-like to the spacelike CD field can be seen as responsible simultaneously for the breaking from *ds*<sup>2</sup> = 0 to nonzero *ds*<sup>2</sup> in (58) through (53), which corresponds to the CSB of gravitational field with the scale parameter Λ*dm*.

A well-known remarkable characteristic of the solution (58) is that it is transformed into a stationary solution

$$ds^2 = \left(1 - \Lambda\_{dm}(r')^2\right) \left(cdt'\right)^2 - \frac{(dr')^2}{\left(1 - \Lambda\_{dm}(r')^2\right)} - \left(r'\right)^2 \left(d\theta^2 + \sin^2\theta d\phi^2\right) \tag{59}$$

by the following variable changes:

$$d\_p r = \frac{r'}{\sqrt{D}} \exp\left[-\sqrt{\Lambda\_{dm}} ct'\right], \quad t = t' + \frac{1}{2c} \sqrt{\frac{1}{\Lambda\_{dm}}} \ln D,\tag{60}$$

where *D* is defined either by 1 > *D* := 1 − <sup>Λ</sup>*dm*(*r*)<sup>2</sup> > 0 (case I) or by 1 > *D* := <sup>Λ</sup>*dm*(*r*)<sup>2</sup> − 1 > 0 (case II). Note that the metric (59) is similar in form to the Schwarzschild metric given below, for which an event horizon exists at *r* = *α*, while that in (59) exists at *r* = √1/Λ*dm*. (See Figure 1)

$$ds^2 = \left(1 - \frac{a}{r'}\right) (cdt')^2 - \frac{(dr')^2}{\left(1 - \frac{a}{r'}\right)} - (r')^2 (d\theta^2 + \sin^2\theta d\varphi^2). \tag{61}$$

**Figure 1.** Dual configuration of twin universes.

In case I of the stationary metric (59), we have *r* = 0 by the synchronization *t* = *t* of *t* and *t* owing to (60). If *t* is adjusted as *t* = Θ*t*, (Θ > <sup>1</sup>), then we see that *r* moves from 0 to 1/ (<sup>Λ</sup>*dm*) as *t* moves from 0 to +<sup>∞</sup>. Similarly, in case II, we see that *r* moves from 2/(<sup>Λ</sup>*dm*) to 1/ (<sup>Λ</sup>*dm*) as *t* moves from 0 to +<sup>∞</sup>. This dual structure, illustrated in Figure 1, clearly shows that by taking *t* = 0 as the origin of time from which twin Big Bang universes evolve, they will meet at the event horizon in (59) an eon later (*t* = ∞). To the best of our knowledge, the concept of twin universes with matter vs. antimatter duality was first discussed by Petit [15]. We believe that his cosmological model fits exactly into the configuration illustrated in Figure 1, which tells us that (<sup>Λ</sup>*dm*)−<sup>1</sup> is a genuine characteristic length scale of our universe. This justifies the fact that Λ*dm* defined in (50) is the cosmological constant that appears in the form of (49). The forward and backward time evolutions of twin universes correspond, respectively, to positive and negative field operators of the 4-momentum, while the existence of twin universes naturally explains the reason why one-sided energy spectra at the level of state vector space works for many practical situations in each universe. If the birth of these twin universes was brought about by conformal symmetry breaking of certain light fields in which the duality between "matter (with positive energy) and antimatter (with negative energy)" works as the separation rule of the twin structure, then the twin pair will return to the original light

fields when they meet at the event horizon. The next Big Bangs of the twin pair will occur at certain locations on this event horizon distant from each other by √2/ Λ*dm*.

According to the arguments developed thus far, we can say that the original conformal light field is composed of light fields with the following duality structures:

$$\left[T\_{\mu}^{\;\;\;\nu} = -F\_{\mu\sigma}F^{\nu\sigma}, \; \; ^\*\hat{T}\_{\mu}^{\;\;\nu} = ^\*(S\_{\mu\sigma}S^{\nu\sigma}), \; T\_0^0 > 0, \; ^\*\hat{T}\_0^0 > 0\right],\tag{62}$$

$$\mathcal{T}^\* T^\nu\_\mu = -^\* (F\_{\mu\nu} F^{\nu\sigma}), \ \ \mathcal{T}^\nu\_\mu = \mathcal{S}\_{\mu\nu} S^{\nu\sigma}, \ \ ^\*T^0\_0 < 0, \ \mathcal{T}^0\_0 < 0 \ \ \_\downarrow \tag{63}$$

where the symbol ∗ denotes the Hodge duality explained in the derivation of (44). Although (62) and (63) can be considered as light and anti-light (light with positive energy moving backward in time) fields, respectively, they can coexist as free modes without interacting with each other, unlike the case of matter and antimatter interactions. As all of these fields are trace free, the associated Ricci scalar curvature is zero. Equation (26) tells us that the Riemann curvature associated with these light fields takes the form *<sup>R</sup>λρμν* = *<sup>F</sup>λρFμν*(= *<sup>S</sup>λρSμν*). In addition to *R ν ν* = 0, we can readily show *<sup>R</sup>μν Rμν* = 0 using (23). Under the former condition *R ν ν* = 0, the Weyl tensor *<sup>W</sup>λρμν* assumes the form

$$\mathcal{W}\_{\lambda\rho\mu\nu} = R\_{\lambda\rho\mu\nu} + \frac{1}{2} (R\_{\lambda\mu}\mathcal{g}\_{\rho\nu} - R\_{\lambda\nu}\mathcal{g}\_{\rho\mu} - R\_{\rho\mu}\mathcal{g}\_{\lambda\nu} + R\_{\rho\nu}\mathcal{g}\_{\lambda\mu});\tag{64}$$

thus, by direct calculations using the latter condition of *<sup>R</sup>μν Rμν* = 0, we obtain *W*<sup>2</sup> = 0. Therefore, for light fields (62) and (63), we have

$$R\_{\nu}^{\;\;\nu} = 0, \quad \mathcal{W}^2 = \mathcal{W}\_{\nu\alpha\beta\gamma} \mathcal{W}^{\nu\alpha\beta\gamma} = 0. \tag{65}$$

The second equation in (65) is related to Penrose's Weyl curvature hypothesis [14].

In modern cosmology, cosmic inflation theory was introduced to explain the observed highly tuned initial condition of the Big Bang, in which the notion of "false vacua" plays a key role in explaining the tremendous exponential expansion of space. In the introduction, however, we pointed out that the notion of the vacuum state in conventional QFT is highly biased by the one in Fock space, which may be called "Fock vacuum prejudice" if adhering to the idea of creation from emptiness. One of the aims of our present paper is to overcome this prejudice in the spirit of Occam's razor as follows: in view of the present circumstances showing that inflation theory seems to be "lost in a maze" in achieving the above-mentioned original goal, the basic premise of our working hypothesis in cosmology can be shifted from the Fock vacuum to the phase transition of the extended light field arising from its CSB, according to which a simpler alternative view emerges such that the initial condition of the Big Bang and the dynamics of both dark energy and matter can be naturally explained.

For light fields, *ds*<sup>2</sup> = 0, the amplitude of the smallest perturbations of CSB in the length scale would be *lp* in (58), but its magnitude in the converted energy scale is tremendously large because energy is inversely proportional to length. By virtue of the Weyl curvature hypothesis of (65), and especially of the peculiar form of (49) through which the Weyl tensor contributes to part of the energy-momentum field, we see that the Weyl contribution to the energy field is a very low value of Λ*dm*. Therefore, the energy field with extremely high density thus created must have a distribution in spacetime very close to the FRW metric on which small amplitude perturbations of *W*<sup>2</sup> exist. The emergence of the FRW metric is the result of unfolding the "blueprint" (14) encoded in the lightlike CD field. Note that in the limit of *W*<sup>2</sup> → 0, the energy-momentum field (49) approaches the anti-de Sitter (AdS) space; thus, the weak gravitational field and high energy conformal field share a common AdS spacetime, which is an essential part of the Maldacena duality [36]. In our new revised version of the CCC of twin universes, the beginning and end of the cycle are, respectively, compared to the pair creation and annihilation of elementary particles through the intervention of conformal light fields. Within the cycle in each universe, a couple of

different classes of entities exist, i.e., both visible matter and invisible dark energy and dark matter exist. In S3O, we already discussed an extended thermodynamical viewpoint on the dynamics at cosmological scales.

When we take into account the remarkable abundance ratios of invisible dark energy and dark matter in comparison to the negligible one of ordinary visible matter, the time evolution of visible material subsystems in the universe, for instance, galaxy cluster formations, may be compared to the "heat engines" working between invisible "heat reservoirs" with higher and lower temperature, which, respectively, correspond to dark matter with positive energy and negative dark energy. If we denote the space averaged *W*<sup>2</sup> by *<sup>W</sup>*<sup>2</sup>|*ave*., then due to the property of universal gravitation, it will increase with the passage of time and thus may be related to the gravitational entropy of the visible subsystem in the universe. From this viewpoint, the effect of the gravitational field, including that of dark matter, modeled as <sup>Λ</sup>*dmgμν* in our theory, can be interpreted by a certain model of thermodynamics. Actually, attempts at this have already been made, for instance, in [37,38].

As the final remarks on CCC, first, we note that the conformal symmetry of sourcefree Maxwell's equation holds well only in four dimensions, which may explain why the dimensions of spacetime in which we live are four. Second, the first author would appreciate if his philosophical preference of helical evolution to cyclic motion is reflected in CCC. His speculative "Book of Genesis" on CCC is as follows:

In the beginning, God, as a mathematician, created the primordial light with conformal symmetry, and God said: "Let there be conformal symmetry breaking, and there were twin universes, beginning their long journey towards a brighter future of a light world one stage higher in eternal evolution."

**Author Contributions:** H.S. contributed to the basic structure of this article. I.O. provided the knowledge on fundamental quantum field theory and the new perspective on the involvement of the *ζ*-function singularity in the quantum walk models describing the behaviors of dressed photons. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This research was partially supported in the form of a collaboration with the Institute of Mathematics for Industry, Kyushu University. We thank anonymous reviewers for their helpful comments and suggestions to improve the quality of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
