On Spacetime Quantum Duality and Bounce Cosmology of a Dual Universe

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 April 2021 doi:10.20944/preprints202005.0250.v9

Mohammed. B. Al-Fadhli1*

1College of Science, University of Lincoln, Lincoln, LN6 7TS, UK. *Correspondence: [email protected]; [email protected]

Abstract: The recent Planck Legacy release confirmed the existence of an enhanced lensing amplitude in the cosmic microwave background (CMB) power spectra, which endorses the positive curvature of the early Universe with a confidence level exceeding 99%. In this study, the pre-existing curvature is incorporated to extend the field equations where the derived wave function of the Universe is utilized to model Universe evolution with reference to the scale factor of the early Universe and its radius of curvature upon the emission of the CMB. The wave function reveals both positive and negative solutions, implying that matter and antimatter of early Universe plasma evolve in opposite directions as distinct Universe sides. The wave function indicates that a nascent hyperbolic expansion is followed by a first phase of decelerating expansion away from early plasma during the first ~10 Gyr, and then, a second phase of accelerating expansion in reverse directions, whereby both Universe sides free-fall towards each other under gravitational acceleration. Simulations of the predicted conformal curvature evolution demonstrate the fast orbital speed of outer stars owing to the external fields exerted on galaxies as they travel through conformally curved space-time. Finally, the wave function predicts an eventual time-reversal phase comprising rapid spatial contraction that culminates in a Big Crunch, signalling a cyclic Universe. These findings reveal that early plasma could have separated and evolved into distinct sides that collectively and geometrically influencing the Universe evolution, physically explanting the effects attributed to dark matter and energy.

Keywords: Accelerated Expansion; Fast orbital speed of outer stars; Duality; Antimatter.

1. INTRODUCTION

Bounce cosmology provides an alternative view The Planck Legacy PL18 release confirmed the of the Universe, in which our Universe expanded existence of an enhanced lensing amplitude in the from a hot and very dense state of a previous collap- power spectra of the CMB [5]. Notably, it is more than sed Universe [1]. This cosmology is free from the that estimated by the lambda cold dark matter model singularity problem and offers a clearer view of the (ΛCDM), which endorses the positive curvature of early Universe. However, the null-energy condition the early Universe with a confidence level more than is generally violated by the conjectured bounce of re- 99% [5,6]. Besides, the observed gravitational lensing expansion in several modified gravity theories [2]. by substructures of several galaxy clusters is an order Alternatively, this bounce could be realized through other scenarios, such as the phenomenon of plasma of magnitude higher than that estimated by the drift in the presence of electromagnetic fields. In this ΛCDM [7,8]. This endorses a curved Universe despite sense, matter and antimatter in early Universe of the present/local space-time flatness. plasma could be separated by primordial electro- The presumed concepts of dark energy, dark magnetic fields upon the emission of the cosmic matter, and antimatter early elimination are being microwave background (CMB), thereby evolving in challenged by new observations and measurements. opposite directions owing to their opposite spin and Riess in 2020 found the Universe expansion is faster charge. The plasma separation until and under the than the ΛCDM estimates [9] while Ryskin showed right conditions would allow the collapsed state to that vacuum energy cannot be causing the acceler- achieve thermal equilibrium while the gravitational ated expansion [10]. In addition, external field effect contributions of early Universe plasma boundary that influence galactic rotation curves were detected could realize a singularity-free paradigm. Signs of the at up to 8σ to 11σ, challenging traditional dark matter plasma separation or a significant predominant event concepts [11]. Additionally, advanced measurements might be observed as associated noise surrounding of the fine structure of hydrogen and antihydrogen the measured gravitational waves [3]. Additionally, atoms were found to be consistent with Quantum the magnetic fields that presently pervade the Electrodynamics Theory predictions at a margin of Universe at all probed scales favour a primordial 2%, including the Lamb-shift feature, undermining origin of this kind [4]. elimination assumptions [12].

Accordingly, a closed early Universe model is is consistent with Mach’s principle, the reliance of the considered in this study to approach the problems of small structure on the larger structure. Schrödinger in

accelerated expansion and fast orbital speed of stars 1925 pointed to the reliance of 퐺푡 on the distribution from a new perspective. The closed finite Universe of Universe’s masses and the Universe’s radius while can provide an agreement with the CMB anisotropy Dirac in 1938 proposed its correlation to the age of the

observations [13] and could explain the quantum Universe [16]. The evolution in 퐺푡 can be endorsed entanglement, where the total Universe energy is by the measured gradual evolution in the fine- finite; thus, cosmic conservation preserves the total structure ‘constant’ [17–19], which can reveal that the spin of a pair of particles regardless of their locations presumed fundamental constants may rely on other [14]. The pre-existing curvature is considered to properties of the Universe.

extend the field equations and the evolution of the On the other hand, 퐸퐷 is in terms of energy Universe from early plasma is modelled by utilising density and represents the continuum’s resistance to

the Universe wave function. This paper is organised deformation [20]. Eq. (1) shows 퐸퐷 is proportional to as follows. Section 2 presents the extended field the fourth-power of the speed of light, which, in turn equations. Sections 3 and 4 discuss the Universe directly proportional to the frequency; in accordance model, its evolution, and its minimal radius while with the frequency cut-off predictions of the vacuum Section 5 presents the spiral galaxy simulation under energy density in the Quantum Field Theory [21];

external fields. Finally, section 6 concludes this work thus, 퐸퐷 can characterize space-time continuum’s and suggests future works. resistance to curvature and could represent vacuum energy density. Further, the inverse proportionality 2. Extended Field Equations for the Curved in Eq. (1) could explain galaxy formation without Early Universe involving dark matter where 퐺푡 was larger in value The recent PL18 release revealed a positive early at the early Universe. Galaxy formation has been Universe curvature, that is, is evidence of a pre- simulated using modified Newtonian gravity [22]. existing or background curvature. To consider this By incorporating the pre-existing/background pre-existing curvature and its evolution over cosmic curvature as a function of cosmic time and complying with the law of energy conservation, the Einstein– time, 푡, a modulus of spacetime deformation, 퐸퐷, is utilized. Spacetime is regarded as a continuum with Hilbert action can be extended to 퐸 푅 a dual quantum nature, that it curves as waves 퐷 4 푆 = ∫ [ + ℒ푀] √−푔 푑 푥 (2) according to General Relativity while fluxing as 2 ℛ quantum energy particles; where the latter is justified where 푅 and ℛ are the Ricci and the pre-existing

because the energy flux from early Universe plasma scalar curvatures respectively, ℒ푀 is the Lagrangian into space at the speed of light creating a ‘spacetime density and 푔 is the determinant of the metric 푔푢푣. continuum’ or ‘vacuum energy’. This concept of The indistinctive variation in the action yields spacetime quantum duality can be corroborated by 퐸 훿푅√−푔 훿ℛ√−푔푅 훿√−푔푅 [ 퐷 ( − + )] 푑4푥 recent findings regarding the polarization of light 2 ℛ ℛ2 ℛ 훿푆 = ∫ from the CMB, which indicated the possible existence 푅 (3) 4 of an exotic substance throughout space that causes + [훿ℒ푀√−푔 + 훿√−푔ℒ푀] 푑 푥 ( ) these measured polarizations [15]. By using the trace- reversed Einstein field equations, the modulus of the By differentiating the determinant according to the 휇휈 Jacobi's formula as 훿√−푔 = −√−푔 푔휇휈훿푔 /2 where continuum, 퐸퐷 = (stress/strain), is expressed as the variation in the metric times the inverse metric, 4 푇 − 푇푔 /2 c 휇 휇휈 휇휈 푔휇휈 푔 = 훿 , yields 푔휇휈 훿푔 + 푔 훿푔휇휈 = 0, while the 퐸퐷 = = 2 (1) 휇휈 휈 휇휈 휇휈 푅휇휈/ℛ 8π퐺푡푟푡 휇휈 variation in the scalar curvature, 푅 = 푅휇휈푔 , is 훿푅 = 휇휈 푢푣 where the stress is signified by the stress-energy 푅휇휈훿푔 + 푔 훿푅휇휈 [23]. Thus, Eq. (3) is expressed as

tensor, 푇휇휈, of trace, 푇, while the strain is signified by 퐸퐷 [ (푅 훿푔휇휈 + 푔휇휈훿푅 )] √−푔 푑4푥 2ℛ 휇휈 휇휈 the Ricci curvature tensor, 푅휇휈 , as the change in 2 퐸 푅 curvature divided by ℛ = 1/푟푡 , the scalar of the 퐷 휇휈 휇휈 4 − [ 2 (ℛ휇휈훿푔 + 푔 훿ℛ휇휈)] √−푔 푑 푥 2ℛ pre-existing curvature, 푟푡 is the Universe’s radius of 훿푆 = ∫ 퐸 푅 (4) 퐷 휇휈 4 curvature and 푔휇휈 is the metric tensor. According to − [ 푔휇휈훿푔 ] √−푔 푑 푥 4ℛ

the theory of elasticity, 퐸퐷 is a constant; therefore, ℒ푀 Eq. (1) shows an inverse proportionality between the + [훿ℒ − 푔 훿푔휇휈] √−푔 푑4푥 푀 2 휇휈 ( ) gravitational ‘constant‘, 퐺푡, and 푟푡, where 퐺푡 follows the inverse square law with respect to the Universe’s The lambda/pressure is not considered, which could radius as a function of cosmic time. This relationship be implicitly incorporated into stress-energy tensors.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 April 2021 doi:10.20944/preprints202005.0250.v9

By considering the first boundary term in Eq. (4): can be considered as a scalar. Otherwise, its variational terms can be incorporated in the conformal 퐸퐷 휇휈 4 ∫ 푔 훿푅휇휈√−푔 푑 푥 2 2ℛ (5) transformation function Ω as follows. By utilizing Jacobi's formula for the determinant differentiation as The variation in the Ricci curvature tensor 훿푅푢푣 can 훿√|푞| = −√|푞| 푞 훿푞휇휈/2; and the variation in 푞휇휈푞 = be written in terms of the covariant derivative of the 휇휈 휇휈 훿휇 as 푞휇휈 = −푞 훿푞휇휈/훿푞 ; Therefore, Eq. (10) can be difference between two Levi-Civita connections, the 휈 휇휈 휇휈 expressed as Palatini identity: (11) 훿퐾 휌 휌 퐸퐷휖 퐾 휇휈 3 휇휈 훿푅휇휈 = ∇휌(훿Γ 휇휈) − ∇휈(훿Γ 휇휌) (6) ∫ (퐾휇휈 − (푞휇휈 + 2푞휇휈 )) √|푞| 푑 푥 훿푞 2ℛ 2 훿푞휇휈퐾 The variation in the Ricci curvature tensor with here 훿퐾 /훿푞 퐾 = (훿퐾 /퐾 )(푞 /훿푞 ) = 훿ln퐾 /훿ln푞 respect to the inverse metric 푔휇휈 can be obtained by 휇휈 휇휈 휇휈 휇휈 휇휈 휇휈 휇휈 휇휈 can resemble the Ricci flow in a normalised form utilising the metric compatibility of the covariant 휇휈 reflecting the conformal distortion in the boundary, derivative, ∇휌푔 = 0 [23] as which can be expressed as a positive function Ω2 휇휈 휇휈 휌 휇휈 휌 푔 훿푅휇휈 = ∇휌(푔 훿Γ 휇휈) − ∇휈(푔 훿Γ 휇휌) (7) based on Weyl’s conformal transformation [25] as 푞̅ = 푞 Ω2. Therefore, Eq. (11) is written as By substituting Eq.(7) to Eq. (5), the boundary term as 휇휈 휇휈 퐸퐷휖 1 a total derivative for any tensor density can be ∫ (퐾 − 퐾푞̂ ) √|푞| 푑3푥 훿푞휇휈 2ℛ 휇휈 2 휇휈 (12) transformed according to the Stokes' theorem with

renaming the dummy indices as where 푞̂휇휈 = 푞휇휈 + 2푞̅휇휈 is the conformal transformation of the induced metric where Einstein spaces are 퐸퐷 휎휈 휇 휎휇 휇 4 퐸퐷 휇 ∫ ∇휇(푔 훿Γ휈휎 − 푔 훿Γ 휇휎)√−푔푑 푥 ≡ ∭ ∇휇퐴 √−푔푑푉 2ℛ 2ℛ 푉 a subclass of the conformal space [26]. The same is applied for the second boundary term. Thus, the 퐸퐷 휇 퐸퐷 3 = ∯ 퐴 . 푛̂푢√|q| 푑푆 = ∮ 퐾휖√|q| 푑 푥 2ℛ 푆 2ℛ 휕푉 (8) action in Eq. (9) is rewritten as

The non-boundary term 퐸 /2ℛ is left outside the 퐸퐷 4 휇휈 퐷 + [ 푅 ] √−푔 푑 푥 훿푔 2ℛ 휇휈 integral transformation as it only acts as a scalar to 퐸퐷휖 1 3 휇휈 the integral that called 푆퐺퐻푌 [24]. 퐾 is the trace of the + [ (퐾 − 퐾푞̂ )] √|푞| 푑 푥 훿푞 2ℛ 휇휈 2 휇휈 extrinsic curvature tensor, 푞 is the determinant of 퐸 푅 퐷 4 휇휈 the induced metric and 휖 equals 1 when the normal − [ ℛ휇휈] √−푔 푑 푥 훿푔 2ℛ2 훿푆 = ∫ 푛̂푢 is a spacelike entity and equals -1 when it is a 퐸 푅휖 1 (13) − [ 퐷 (퐾 − 퐾푞̂ )] √|푞| 푑3푥 훿푞휇휈 timelike entity. By substituting Eq. (8) to Eq. (4): 2ℛ2 휇휈 2 휇휈

퐸퐷푅 퐸퐷 4 휇휈 4 휇휈 + [ 푅휇휈] √−푔 푑 푥 훿푔 − [ 푔휇휈] √−푔 푑 푥 훿푔 2ℛ 4ℛ

훿ℒ ℒ 퐸퐷휖 3 푀 푀 4 휇휈 + [ 퐾] √|q| 푑 푥 + [ − 푔휇휈] √−푔 푑 푥 훿푔 2ℛ ( 훿푔휇휈 2 )

퐸 푅 퐷 4 휇휈 − [ ℛ휇휈] √−푔 푑 푥 훿푔 The stress-energy tensor is proportional to the 2ℛ2 훿푆 = ∫ Lagrangian term by definition [23] as 퐸 푅휖 (9) − [ 퐷 퐾] √|q| 푑3푥 2ℛ2 훿ℒ푀 푇 : = ℒ 푔 − 2 (14) 휇휈 푀 휇휈 훿푔휇휈 퐸퐷푅 − [ 푔 ] √−푔 푑4푥 훿푔휇휈 4ℛ 휇휈 By substituting Eq.(14) in Eq. (13), choosing 휖 as a 훿ℒ푀 ℒ푀 + [ − 푔 ] √−푔 푑4푥 훿푔휇휈 timelike entity and then applying the principle of ( 훿푔휇휈 2 휇휈 ) stationary action yield It is worth noting that some terms satisfy the criteria 퐾푞̂ 퐾푞̂ 퐾 − 휇휈 퐾 − 휇휈 훿푆 푅휇휈 휇휈 2 푅ℛ휇휈 휇휈 2 푅푔휇휈 푇휇휈 that the variation in the action is with respect to − − + 푅 − = (15) ℛ ℛ ℛ2 ℛ2 2ℛ 퐸 the variation in their inverse metric 훿푔휇휈 excluding 퐷 the boundary terms that still lack this feature. To Eqs. (15) can be rearranged as achieve the consistency of the action, the variation in ℛ푅휇휈 − 푅ℛ휇휈 푅 푅 − ℛ 1 푇휇휈 the boundary action has to be determined. Thus, the − 푔 + (퐾 − 퐾푞̂ ) = (16 ) ℛ2 2ℛ 휇휈 ℛ2 휇휈 2 휇휈 퐸 indistinctive variation in the first boundary term is 퐷 휇휈 since ℛ /ℛ = ℛ /ℛ 푔̅ = 푔̅ corresponds to Weyl’s | | 휇휈 휇휈 휇휈 휇휈 퐸퐷휖 휇휈 휇휈 훿√ 푞 3 ∫ (퐾휇휈훿푞 + 푞 훿퐾휇휈 + 퐾 ) √|푞| 푑 푥 (10) conformal transformation of the metric [25,26], where 2ℛ √|푞| the conformal transformation can describe the tidal 휇휈 where 퐾 = 퐾휇휈푞 . 퐾휇휈 is the extrinsic curvature tensor distortion and the gravitational waves in the absence and 푞휇휈 is the induced metric on the boundary. The of matter [27], and by using a conformally transform- 퐸 휖/2ℛ non-boundary term 퐷 is left outside, where it ed metric, 푔̂휇휈 = 푔휇휈 + 2푔휇휈̅ , and rearranging Eqs. (16),

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 April 2021 doi:10.20944/preprints202005.0250.v9

the extended field equations are Figure 1 shows the referenced metric model of the early Universe plasma expansion. 푅휇휈 1 푅 푅 − ℛ 1 푇휇휈 (17) − 푔̂휇휈 + 2 (퐾휇휈 − 퐾푞̂휇휈) = ℛ 2 ℛ ℛ 2 퐸퐷

Eqs. (17) can be interpreted as indicating that the induced curvature over the pre-existing curvature 푛̂ 푡⃗⃗⃗⃗ 푢 푣 equals the ratio of the imposed energy density and its ⃗푡⃗⃗푤⃗⃗ flux to the vacuum energy density and its flux thro- 푎 ugh an expanding/contracting Universe. They can be 푟푝 푝 푟 a simplified by substituting Eq. (1) to Eqs. (17) as 휃 1 푅 − ℛ 1 8휋퐺푡 푅 − 푅푔̂ + (퐾 − 퐾푞̂ ) = 푇 (18) 휇휈 2 휇휈 ℛ 휇휈 2 휇휈 푐4 휇휈 휙

The new boundary tensor/term is only significant at high-energy limits such as within black holes [24] and the early Universe, and could remove the singular- ities and satisfy a conformal invariance theory. The evolution in 퐺푡 can accommodate the pre-existing curvature, ℛ, evolution over cosmic time, 푡, against

the constant 퐺 for the special flat space-time case. Figure 1. The hypersphere of a positively curved early Universe plasma expansion upon the CMB emissions. 푟 is 3. Bounce from a Closed Early Universe 푝 the reference radius of the intrinsic curvature and 푎푝 is the 3.1 Early Universe Plasma Model corresponding reference scale factor of the early Universe at ⃗ The Friedmann–Lemaître–Robertson–Walker reference cosmic time 푡푝 while 푛̂푢 and 푡푣 are the normal (FLRW) metric is the standard cosmological metric and tangential vectors on the manifold boundary respect- model, which assumes an isotropic and homogenous ively regarding the extrinsic curvature. Universe [28,29], where the isotropy and homog- eneity of the early Universe plasma based on the 3.2 Universe Evolution Model CMB are consistent with this metric model. The PL18 By solving the field equations for a perfect fluid release revealed a closed early Universe. Thus, the 2 given by 푇휇휈 = (휌 + 푃/푐 ) 푢휇푢휈 + 푃푔휇휈 [30] and plasma reference radius of curvature 푟푃 upon the substituting Eqs. (19-21), the Friedmann equations emission of the CMB and the corresponding early that count for the plasma reference radius and Universe scale factor 푎푃 at the reference cosmic time reference scale factor are 푡푝 are incorporated in order to reference the metric as 2 2 2 shown in Figure 1. Hence, the referenced metric is 2 푎̇ 8휋퐺푡 휌 푐 푎푝 퐻 ≡ 2 = − 2 2 , (22) 푎 3 푎 푟푝 푎2 푑푟2 푑푠2 = 푐2 푑푡2 − ( + 푟2 푑휃2 + 푟2푠푖푛2휃 푑휙2) 2 2 (19 ) 푎̈ 4휋퐺 푃 푎푝 푟 ̇ 푡 1 − 2 퐻 ≡ = − (휌 + 3 2). (23) 푟푝 푎 3 푐

where 푎/푎푝 is a new dimensionless scale factor . No where 퐻, 푃, and 휌 are Hubble parameter, pressure, conformal distortion is included in this metric, thus, and density respectively. its outcomes are comparable with the literature. The By utilizing the imaginary cosmic time, 휏 = 푖푡, Ricci curvature tensor and Ricci scalar curvature are the referenced Friedman equations in Eqs. (22, 23) are solved using Christoffel symbols of the second kind solved over conformal time by rewriting Eq. (22) in

for 푔푢푣 in Eqs. (19) (derivations in Appendix A) as terms of the conformal time in its parametric form, 푎푝 푑푎 푎푝 2 2 2 푑휂 = −푖 푑휏 (where ȧ = 푖 ); thus, 푑휂 = 푑푎 as 푎̈ 푟 푎푎̈ 2푎̇ 2푐 푎 푑휏 aȧ 푅푡푡 = −3 , 푅휃휃 = 2 ( 2 + 2 + 2 ), 푎 푐 푎푝 푎푝 푟푝 휂 2휋 3 2 2 −1/2 8휋퐺푝휌푝푎푝 푐 푎푝 2 2 2 2 ∫ 푑휂 = ∫ 푎푝 ( 푎 − 2 푎 ) 푑푎 (24) 1 푎푎̈ 2푎̇ 2푐 푟 0 0 3 푟푝 푅푟푟 = 2 ( 2 + 2 + 2 ) / (1 − 2), (20) 푐 푎푝 푎푝 푟푝 푟푝 3 휌푝푎푝 where 휌 = 3 [31]. By integrating, the evolution of 푟2푠푖푛2휃 푎푎̈ 2푎̇ 2 2푐2 푎 푅휙휙 = 2 ( 2 + 2 + 2 ). the scale factor is 푐 푎푝 푎푝 푟푝 푀푝퐺푝 푐 푎(휂)/푎 = (1 − 푐표푠 휂) 2 2 2 푝 2 (25) 푐 푟푝 푟푝 휇휈 −6 푎̈ 푎̇ 푐 푎푝 푅 = 푅휇휈푔 = 2 ( + 2 + 2 2 ) . (21) 푐 푎 푎 푎 푟 4 푝 푀 = 휋휌 푟 3 퐺 where 푝 3 푝 푝 is the plasma mass and 푝 is the

where the dot represents a cosmic time derivative. gravitational parameter value at 휏푝.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 29 April 2021 doi:10.20944/preprints202005.0250.v9

The constant in Eq. (25) can be rewritten in terms of 3.3 Plasma Boundary Contribution

the modulus 퐸퐷 representing the vacuum energy For high energy limits, the gravitational contribution density and the Universe energy density 퐸 by using of the plasma boundary can be obtained using the Eq. (1) as 퐸/6퐸퐷 . Additionally, the evolution of the boundary term, 푅−ℛ (퐾 − 1 퐾푞 ) = 8휋퐺푡 푇 , where at imaginary time 휏(휂) can be obtained by integrating ℛ 휇휈 2 휇휈 푐4 휇휈 the length of the spatial factor contour over the 휏푝, there is no conformal transformation. Therefore,

expansion speed 퐻휂 while initiating at the reference the induced metric tensor on the plasma hypersphere

imaginary time 휏푝 with the corresponding scale boundary 푞휇휈 [32] is factor 푎푝. Therefore, by rewriting Eq. (25) in terms of 푎2(푡) 푎2(푡) 푞 = 푑푖푎푔 −푐2, 푅2, 푅2푠푖푛2휃 (33) the Hubble parameter by its definition and initiating [ 휇휈] ( 2 2 ) 푎푝 푎푝 휏 푑휏 = 푖 푑푎(휂) at 푝 as 퐻 푎 : 휂 푝 The extrinsic curvature tensor is solved utilizing the 휏 휂 퐸 푐 ⃗⃗⃗⃗ ∫ 푑휏 = 푖 ∫ (1 − cos 휂) 푑휂 formula 퐾푢푣 = −푡휇 . ∇휈푛̂푣 . Due to the hypersphere 6퐻 퐸 푟 (26) 휏푃 0 휂 퐷 푝 smoothness, the covariant derivative reduces to ⃗⃗⃗휈⃗ By integration, the evolution of imaginary time is partial derivative as 퐾푢푣 = −푡⃗⃗⃗휇⃗ 휕푛̂ /휕푡 [32]; thus: 2( ) 2 퐸 푐 푎 푡 푎 (푡) 2 (34) 휏(휂) = 푖 (휂 − sin 휂) + 휏푝 (27) [퐾휇휈] = 푑푖푎푔 (0, − 2 푅, − 2 푅푠푖푛 휃) 6퐻휂퐸퐷 푟0 푎푝 푎푝 휇휈 According to the law of energy conservation, The trace of the extrinsic curvature is 퐾 = 퐾휇휈푞 = 2 the divergence of the stress-energy tensor vanishes, −2/푅 = −2/푟 푝 . The pre-existing curvature of the 푎̇ 푎̇ ∂휌 푃 푎̇ ∂휌 2 ∆ 푇푢푣 푇푢 + 3 휌 − 푖 = 0 3 (휌 + ) − 푖 = 0 plasma boundary at 휏푝 is ℛ푝 = 1/푟 푝 (Gaussian 푣 , thus, a 푢 a ∂휏 , 푐2 a ∂휏 . By combining these outcomes, integrating, and curvature). On the other hand, the Ricci scalar substituting the spatial scale factor rate in Eq. (25) to curvature can be written as the difference between their outcome, the density evolution of matter is the kinetic and potential energy densities whereby

−3 substituting Friedmann equations in Eqs. (22, 23) to 푐 휌휂,푚 = 퐷푝 (1 − cos 휂 ) (28) the Ricci scalar curvature in Eqs. (21) at 휏푝 as 푟푝 24휋푃푝 8휋휌푝 where 퐷 is constant. According to Eq. (23), the rate 푅푝 = 퐺푝 ( − ) (35) 푝 푐4 푐2 of Hubble parameter, 퐻̇ , relies on the density; thus, by substituting Eqs. (28) to Eq. (23): By solving the boundary for a perfect fluid given by 푇 = (휌 + 푃/푐2) 푢 푢 + 푃푔 −3 휇휈 휇 휈 휇휈 and substituting Eqs. (35- 푎̈ 4휋퐺푡 푐 퐻̇ ≡ = − 퐷푝 (1 − cos 휂) (29) 33) into the boundary term as 푎 3 푟푝 24휋푃푝 8휋휌푝 1 퐺푝 ( 4 − 2 ) − 2 2 By substituting Eq. (25) to Eq. (29) and rewriting them 푐 푐 푟 푝 −c (36) 2 ( 2 ) = 8휋퐺푝휌푝 in terms of Hubble parameter rate 퐻̇ 휂 by its definition 1/푟 푝 푟 푝

and initiating the integration at 휏푝 as ∫ 퐻̇ = ∫ 푎̈/푎 as 푝 By multiplying by both sides by plasma volume 푉푝: 1 푐 푐 3 4퐺푝푃푝푉푝 퐻휂,푚 = 퐻푚 ( cot 휂 + cot 휂) + 퐻푝 (30) 푟 = (37) 3 2푟푝 2푟푝 푝 푐4 The reference radius 푟 > 0 because any reduction Similarly, by applying the same procedure for the 푝 radiation-only density evolution, Hubble parameter in plasma volume causes an increase in its pressure. evolution for radiation-only is 4. Universe Evolution 1 5 푐 2 3 푐 푐 퐻휂,푟 = 퐻푟 ( cot 휂 + cot 휂 + cot 휂) + 퐻푝 (31) 5 2푟푝 3 2푟푝 2푟푝 The positive and negative solutions of the wave 휓 where 퐻푚, 퐻푟 and 퐻푝 are constants. The quantized function 퐿 imply that matter and antimatter in the

wave function 휓퐿 with respect to its reference value plasma evolved in opposite directions. The evolution

휓푝 at 휏푝 is obtained by using Eqs. (25, 27-31) as a of the Universe according to the wave function for potential third quantization as both matter and radiation-only in addition to the

2 light cone are shown in Figure 2a; where only the 퐸 푐 휓퐿(휂)/휓푝 = ∓ ((1 − cos 휂 ) ) positive solution of one Universe side is shown due 6퐸퐷 푟푝 (32) 푐 to their symmetry. A chosen mean evolution value of |퐻휂|푎푝(1−푐표푠 푟 휂 ) 1/2 −1 푝 1 -1 2 2 푖 cot 푐 the Hubble parameter of ~70 km∙s- ∙Mpc and a 푐 푐 푐 (휂−푠푖푛 휂 ) 푟푝 + 2 2 ((휂 − sin 휂) ) 푒 phase transition of expansion at age of ~10 Gyr were 퐻휂 푎푝 푟푝 applied to tune the integration constants of the

here 퐸 /퐸퐷 is a new dimensionless energy density model; accordingly, the predicted energy density parameter as the ratio of the Universe energy density parameter is ~1.16. Further, the Hubble parameter to the vacuum energy density. evolution and its rate are shown in Figure 2b.

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According to the wave function (Figure 2a, orange the expansion speed shown in Figure 2b (blue curve, curve), the cosmic evolution can be interpreted as Hubble parameter) starts with a hyperbolic rate at the comprising three distinct phases. Firstly, matter and nascent stage, then, the rate decreases due to gravity antimatter sides of the Universe expanded in oppo- between the two sides, until it reached its minimal site directions away from the early plasma during the value at the phase transition at age of ~ 10 Gyr. The first phase perhaps due to the phenomenon of plasma evolution in Hubble parameter is evidence from three

drift in the presence of electromagnetic fields where time-delay cosmography observations [33].

Time (Gyr) Time Hubble Parameter Parameter Hubble

Hyperspherical Radius Time (Gyr)

Figure 2. A) Evolution of the wave function of matter of one side of the Universe, radiation only wave function in addition to the straight line of light cone (diagram is not to scale). B) The evolution of the Hubble parameter 퐻 and its rate 퐻̇ in addition to a rectified Hubble parameter reflecting the reverse expansion direction.

However, the wave function reverses its direction in A congruence of space-time worldlines has been the second phase with both sides of matter and anti- simulated to virtualize the matter wave function matter entering a state of free-fall towards each other (Figure 3a). In the first phase, the simulation of the

at gravitational acceleration possibly causing current worldlines produced a curved geometry in harmony accelerated expansion; the Hubble parameter starts to with the PL18 [5,6]. Conversely, for the accelerated increase in the reverse direction. present phase in reverse directions, the simulation of According to mechanics, the minus sign of the the worldlines produced a flat end or flat space-time, speed indicates an opposite direction while the oppo- explaining current space-time flatness. The apparent site signs of the acceleration (Figure 2b, green curve) Universe’s topology due to the gravitational lensing and the speed in the first phase indicate a slowing effects is shown in Figure 3b, possibility matching the down while the matching signs in the second phase large-angle correlations of the CMB [34] and the indicate the speed of expansion is increasing. SLOAN Digital Sky Survey data virtualization [35].

Local observer at age of ~14 Gyr – Local group

Light

1st Phase

Matter side Matter

Antimatter side Antimatter

(a) (b)

Figure 3. A) The predicted cosmic topology of both sides. Firstly, a decelerated phase of both sides away from early Universe plasma;

secondly, an accelerated phase in reverse directions where both sides free fall toward each other under gravitational acceleration. Finally, a rapid contraction phase leading to a Big Crunch. B) The apparent topology due to the gravitational lensing effects. 6

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In addition, this reasoning for current accelerated Furthermore, the measured gradual evolution in the expansion, due to the free-fall of both sides towards fine structure ‘constant’, where its observed variation each other under gravitational acceleration, can also occurs both with time and across a specific axis [17], explain the observed dark flow towards the great could evolve along with the evolution of the matter attractor that cannot be observed [36]. Further, the wave function that effects the average density. observed strong “dipole repeller” [37] can endorse Interestingly, the matter wave function predicts this model; the authors argued their observation is a final phase of spatial contraction that appears after incompatible with an attractive gravitational force ~18 Gyr, which could be because of the future high while the under-dense and over-dense regions expla- concentration of matter/antimatter at both sides, nation for this phenomenon seems incomplete as leading to a Big Crunch. On the other hand, the accordingly numerous regions that differently dense radiation-only according to its wave function, which create multiple weak attractors-repellers throughout propagates faster than matter, is predicted to pass the Universe. On the other hand, since our local from a side to other side (Figure 2a, blue curve), group (Figure 3a, blue dot at age of ~14 Gyr) flows in which could explain why CMB light can be observed a direction towards the antimatter side while the even though matter moves much slowly than light. corresponding galaxy group in the antimatter side Figure 4 shows 3D spatial and 1D temporal flows in a direction towards our group. Therefore, the dimensions schematic of the space-time evolutions of flow of the matter side including our group can both sides, which can be in correspondence with the signify the great attractor while the flow of the Kaluza–Klein unified theory. Due to the anisotropy antimatter side towards our group can signify the in the expansion of the early Universe hypersphere corresponding repeller. into two sides, the expansion in time and hence the Further, since at the second phase both sides time dimension can be analyses and considered as move closer to each other; this would increase the two dimensions, the imaginary (future-past) and the average density, which can explain the reason of the real (forward-backward) time components. Thus, the current increase in the average temperature of the fifth dimension would be the real time component Universe [38] in contrast to the state of cooling down while the cylindrical condition belongs to the scale from hot plasma during the first phase. factor; antimatter travels backwards in real time.

3D slice of the

universe at an Plasma Separation

instant of time Decelerated spatial expansion

Big Crunch Spatial contraction

(a) (b)

Figure 4. A) Schematic in 3D spatial and 1D temporal dimensions of both sides, according to the space-time worldline

evolution. (a) In the first phase, both sides expand away from the early plasma. B) In the second phase, both sides expand in

reverse directions and free-fall towards each other at gravitational acceleration. In the third phase, both sides contract, due to

high concentrations of matter/antimatter, leading to the Big Crunch. Blue circles represent a 3D Universe slice, which is not

necessarily a simply path connected.

The inverse square law of 퐺푡 with respect to the lete cycle of the wave function, the minimal radius is Universe’s radius could explain the galaxy formation equal the distance that light travels in one second in

without involving dark matter where 퐺푡 was larger free space, which may hint the origin of the speed in value at the early Universe. This could explain the value. Moreover, the model can be consistence with gravity hierarchy problem while for a proper comp- galaxy rotation curve as discussed in the next section.

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5. Simulation of a Spiral Galaxy

The consistent patterns of galactic rotation Alternatively, the curvature of the space-time curves observed using precise and independent continuum is predicted to change over the age of the galactic redshift data have confirmed that hydrogen Universe, that is, a conformal curvature evolution is clouds and outermost stars are orbiting galaxies at expected, with the highest degree of curvature occur- speeds faster than those calculated using Newtonian ing at the phase transition as determined by the laws. Accordingly, the dark matter hypothesis was derived wave function (see Figure 3a). Accordingly, introduced to account for the apparently missing it can be inferred that the fast orbital speeds observed galactic mass and explain these fast-orbital velocities for outer stars in galaxies are a result of their travel [39,40]. However, no evidence for the existence of through conformally curved space-time. dark matter, which is hypothesized to account for the To evaluate this inference, a fluid dynamics majority of galactic mass, has been observed since its simulation was performed based on Newtonian introduction. The failure to find dark matter led to the dynamics using the Fluid Pressure and Flow software introduction of new theories such as modified gravity [47]. In this simulation, the fluid was deemed to and MOND [41–45]. Several recent studies have represent the space-time continuum that flows reported that many galaxies do not contain dark throughout an incrementally flattening curvature matter [46]. This scenario was used to inform galaxy paths. Using these conditions, a fluid model was built formation simulations using modified Newtonian to analyse the external momenta exerted on objects dynamics without considering the effects of dark flowing along an incrementally flattening curvature, matter [22]. In addition, external fields that influence which represents the conformal curvature evolution galactic rotation speeds were detected at up to 8σ to as shown in Figure 5a. The momenta yielded by the 11σ, which challenge traditional dark matter concepts fluid modelling were used to inform a simulation of [11]. Therefore, there is no evidence for the existence a spiral galaxy as a forced vortex (under external or nature of dark matter. fields), as shown in Figure 5b.

Galaxy

curvature curvature

Conformal

Universe age (Gyr) Spiral galaxy simulation

(a) (b)

Figure 5. A) External momenta exerted on a galaxy due to its travel through a conformally curved space-time. Green curves represent an incrementally flattening conformal curvature of space-time worldlines. Blue curves show the resultant external fields exerted on galaxies. B) Simulation of spiral galaxy rotation. Blue represents the slowest tangential speeds, and red represents the fastest speeds. 6. Conclusions and Future Works The simulation shows that the tangential speeds of outer parts of the spiral galaxy are rotating faster In this study, the pre-existing curvature is in comparison with the rotational speeds of inner incorporated to extend the field equations where a parts. Additionally, the galaxies of the same mass in closed early Universe model is considered by the present Universe would rotate faster than they utilising a referenced FLRW metric model. The were in the past because of the increase in the external evolution of the Universe from early plasma is momenta due to the highest conformal curvature at modelled utilising the Universe wave function. The the phase transition. Based on these results, it can be wave function revealed both positive and negative concluded the conformal curvature is responsible for solutions implying that the Universe evolved into the fast orbital speed of outer stars in galaxies. two opposite sides: matter and antimatter.

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The derived wave function predicted that a Acknowledgements: I am grateful to the Preprints nascent hyperbolic expansion is followed by a phase Editor Ms Mila Marinkovic for her rapid and of decelerating spatial expansion during the first ~ 10 excellent attention in processing the submissions. Gyr, and then, a second phase of accelerating expa- Conflicts of Interest: The author declares no conflict nsion. The plasma drift in the presence of electro- of interest magnetic fields could provide a physical explanation for the expansion of the early Universe against the Appendix A

gravity of its early hot and very dense state; where The Ricci curvature tensor 푅푢푣 is solved using both sides of the Universe expand away from the the Christoffel symbols of the second kind given by 1 훤휌 = 푔휌휆(휕 푔 + 휕 푔 − 휕 푔 ) early plasma during the first phase. Then, during the 휇휈 2 휇 휆휈 휈 휆휇 휆 휇휈 for the referenced

second phase, they reverse their directions and free metric tensor 푔휇휈 in Eq. (19): fall towards each other. It is conceivable that, the 푎푎̇ r matter and antimatter are free-falling towards each 0 Γ1 = , (A.1) Γ 11 = 2 , 11 푟2 2 2 푟 푟 2 (1 − ) other, causing the present accelerating expansion of 푐 푎푝 (1 − 2) 푝 푟 2 푟푝 푝 the Universe. This could explain the effects attributed 2 2 to dark energy as well as the observed dark flow and 0 푟 푎푎̇ 1 푟 (A.2) Γ 22 = 2 2, Γ 22 = −r (1 − 2), dipole repeller. 푐 푎푝 푟푝

The radiation only worldlines which predicted 2 2 2 0 푟 푎푎̇ 푠푖푛 휃 1 2 푟 (A.3) to pass from one side to another could explain why Γ 33 = 2 2 , Γ 33 = −r푠푖푛 휃 (1 − 2), 푐 푎푝 푟푝 CMB light can be observed even though matter moves much more slowly than light while the 푎̇ Γ1 = Γ2 = Γ3 = Γ1 = Γ2 = Γ3 = , (A.4) apparent topology is possibly in accordance with the 01 02 03 10 20 30 푎

large-angle correlations of the CMB and the SLOAN 1 Γ2 = Γ2 = Γ3 = Γ3 = , (A.5) Digital Sky Survey data. 12 21 13 31 푟 Further, the simulated spacetime worldlines 2 3 3 during the decelerating phase were found to be Γ 33 = −푠푖푛 휃 푐표푠휃, Γ 23 = Γ 32 = 푐표푡 휃 (A.6) flattened during the accelerating phase due to the 휆 The Ricci curvature tensor given by 푅휇휈 = 휕휆훤 휇휈 − reverse direction of the continuum worldlines, 휆 휌 휆 휌 휆 휕휈훤 휇휆 + 훤 휇휈훤 휌휆 − 훤 휇휆훤 휌휈. The 푡 − 푡 component of the explaining the current space flatness while at the Ricci tensor is early Universe the simulation showed a positive 1 2 3 1 1 curvature in accordance with PL18. In addition, the 푅 푡푡 = 푅 00 = −휕0Γ 01 − 휕0Γ 02 − 휕0Γ 03 − Γ 01Γ 10 (A.7) 2 2 3 3 predicted topology could fit well the observed super- − Γ 02Γ 20 − Γ 03Γ 30

voids, galaxy filaments and walls. 2 2 2 푎̇ 푎̇ 푎̈푎 − 푎̇ 푎̇ 푎̈ Regarding the fast-orbital speed of outer stars, 푅 푡푡 = −3휕푡 − 3 ( ) = −3 − 3 = −3 (A.8) 푎 푎 푎2 푎2 푎 the simulation could provide a physical explanation by which the fast orbital speed of outer stars owing to the external fields exerted on galaxies as they travel The 푟 − 푟 component 0 2 3 0 2 through conformally curved space-time rather than 푅푟푟 = 푅11 = 휕0Γ 11 − 휕1Γ 12 − 휕1Γ 13 + Γ 11Γ 02 0 3 1 0 1 2 1 3 (A.9) the existence of dark matter. The inverse square law +Γ 11Γ 03 − Γ 10Γ 11 + Γ 11Γ 12 + Γ 11Γ 13 of 퐺푡 with respect to the Universe’s radius could 푎푎̇ 1 푎푎̇ 푎̇ explain the galaxy formation without involving dark 푅푟푟 = 휕푡 2 − 2휕푟 + 2 2 2 푟 푟 2 2 푟 푎 matter where 퐺 was larger at the early Universe, 푐 푎푝 (1 − 2) 푐 푎푝 (1 − 2) 푡 푟푝 푟푝 (A.10) which could explain the gravity hierarchy problem. r 1 1 + 2 2 − 2 2 The model predicted a final phase of time- 2 푟 푟 푟 푟푝 (1 − 2) reversal of spatial contraction leading to a Big 푟푝 Crunch, signifying a cyclic Universe. The derived 푎푎̈ 푎̇ 2 푅 = + 푟푟 푟2 푟2 smallest possible reference radius of the early plasma 푐2푎 2 (1 − ) 푐2푎 2 (1 − ) (A.11) 푝 푟 2 푝 푟 2 due to its boundary gravitational contributions can 푝 푝 reveal that the early Universe expansion upon 푎̇ 2 2 + + 푟2 푟2 emission of the CMB might mark the beginning of the 푐2푎 2 (1 − ) 푟 2 (1 − ) 푝 푟 2 푝 푟 2 Universe from a previous collapse one. 푝 푝 푎푎̈ 2푎̇ 2 2푐2 (A.12) Finally, this theoretical work will be tested ( + + ) 푎 2 푎 2 푟 2 푅 = 푝 푝 푝 against observational data in future works. 푟푟 푟2 푐2 (1 − ) 푟 2 Funding: This research received no funding. 푝

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The 휃 − 휃 component is 2 2 2 2 푟 1 푎̇ 푟 푎푎̇ 푠푖푛 휃 − r푠푖푛 휃 (1 − 2) − 2 2 0 1 3 푟푝 푟 푎 푐 푎푝 푅휃휃 = 푅22 = 휕0Γ 22 + 휕1Γ 22 − 휕2Γ 23

0 1 0 3 2 + Γ 22Γ 01 + Γ 22Γ 03 2 푟 1 + r푠푖푛 휃 (1 − 2) 1 1 1 3 푟푝 푟 + Γ 22Γ 11 + Γ 22Γ 13 (A.13)

2 0 2 1 − Γ 20Γ 22 − Γ 21Γ 22 + 푠푖푛휃 푐표푠휃 co푡휃

3 3 2 2 2 2 2 − Γ 23Γ 32 푟 푎푎̈푠푖푛 휃 푟 푎̇ 푠푖푛 휃 푅 = 푅 = + 휙휙 33 푐2푎 2 푐2푎 2 푟2푎푎̇ 푟2 푝 푝 푅 = 휕 − 휕 r (1 − ) − 휕 cot(휃) 휃휃 푡 푐2푎 2 푟 푟 2 휃 푝 푝 푟2 (A.14) − 푠푖푛2휃 (1 + 3 ) 푟 2 푟2푎푎̇ 푎̇ 푟2 1 푝 (A.20) 2( ) + 2 2 − r (1 − 2) − cot 휃 푐 푎푝 푎 푟푝 푟 + 푠푖푛2휃 − 푐표푠2휃

푟2푎푎̈ 푟2푎̇ 2 푟2 푟2푎̇ 2푠푖푛2휃 푟2 2( ) 2 2 푅휃휃 = 2 2 + 2 2 + (3 2 − 1) + csc 휃 + 2 2 − 푠푖푛 휃 ( 2) + 푐표푠 휃 푐 푎푝 푐 푎푝 푟푝 푐 푎푝 푟푝

2 2 2 (A.15) 2 2 2 2 2 푟 푎̇ 푟 푟 푎푎̈푠푖푛 휃 푟 푎̇ 푠푖푛 휃 + 2 2 − (1 − 2) 푅휙휙 = 푅33 = 2 2 + 2 2 2 푐 푎푝 푟푝 푐 푎푝 푐 푎푝 (A.21) 2 − cot (휃) 푟2 + 2푠푖푛2휃 푟 2 푟2푎푎̈ 푟2푎̇ 2 푟2 푝 푅 = + 2 + (2 ) − 1 휃휃 푐2푎 2 푐2푎 2 푟 2 푝 푝 푝 (A.16) 푟2푠푖푛2휃 푎푎̈ 2푎̇ 2 2푐2 푅휙휙 = 2 ( 2 + 2 + 2 ) (A.22) + csc2(휃) − cot2(휃) 푐 푎푝 푎푝 푟푝

푟2 푎푎̈ 2푎̇ 2 2푐2 The inverse metric tensor 푔푢푣 is 푅휃휃 = 2 ( 2 + 2 + 2 ) (A.17) 푐 푎푝 푎푝 푟푝 푔푢푣 =

The 휙 − 휙 component is 1 2푎푝 2 0 0 (1 0 ) 0 푅 = 푅 = 휕 Γ0 + 휕 Γ1 + 휕 Γ2 푐 푎 휙휙 33 0 33 1 33 2 33 푟2 (1 − ) 0 1 0 2 푟 2 + Γ 33Γ 01 + Γ 33Γ 02 0 − 푝 0 0 푎2 1 1 1 2 ( 2) + Γ 33Γ 11 + Γ 33Γ 12 (A.18) 푎푝 (A.23) 3 0 3 1 −1 − Γ 30Γ 33 − Γ 31Γ 33 0 0 2 0 푎 2 3 2 ( 2) 푟 − Γ Γ 푎푝 32 33 −1 2 2 푟 푎푎̇ 푠푖푛 휃 0 0 0 2 푎 2 2 푅휙휙 = 푅33 = 휕푡 2 2 ( 2) 푟 푠푖푛 휃 푐 푎푝 ( 푎푝 ) 2 2 푟 − 휕푟 r푠푖푛 휃 (1 − 2) 푟푝 푢푣 푟2푎푎̇ 푠푖푛2휃 푎̇ Finally, the Ricci scalar curvature 푅 = 푅푢푣푔 which −휕 푠푖푛 휃 푐표푠휃 + 2 휃 푐2푎 2 푎 equals the Ricci curvature tensor time the inverse 푝 (A.19) 2 metric tensor is 2 푟 r − r푠푖푛 휃 (1 − 2) 2 푟푝 2 푟 2 2 2 푟푝 (1 − 2) 푢푣 −6 푎̈ 푎̇ 푐 푎푝 푟푝 푅 = 푅푢푣푔 = 2 ( + 2 + 2 2 ) (A.24) 푐 푎 푎 푎 푟푝

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