Dual Cosmic Horizon Radius of Spacetime Curvature of a Multi-Path Connected Cosmic Topology

The necessity of the dark energy and dark matter in the present universe could be a consequence of the antimatter elimination assumption in the early universe. In this research, I derive a new model to obtain the cosmic horizon radius Rh(η) and the potential cosmic topology utilising a new construal of space geometry inspired by large-angle correlations of the cosmic microwave background (CMB). A version of the Big Bounce theory is utilised to avoid the Big Bang singularity and inflationary constraints, and to tune the initial conditions of the curvature density. The mathematical derivation of a positively curved universe governed by only gravity revealed ∓ cosmic horizon solutions. Although the positive horizon is conventionally associated with the evolution of the matter universe, the negative horizon solution could imply additional evolution in the opposite direction. This possibly suggests that the matter and antimatter could be evolving in opposite directions as distinct sides of the universe, as in the visualised Sloan Digital Sky Survey. The span of the cosmic horizon radius is Rh = ∓√(ct + a) and is found to be accountable for the universal space curvature. By implementing this model, we find a decelerated stage of expansion during the first ~10 Gyr, which is followed by a second stage of an accelerated expansion; potentially matching the tension in Hubble parameter measurements. In addition, the model predicts a final timereversal stage of spatial contraction leading to the Big Crunch of a cyclic universe. The predicted density is Ω0 = ~1.14 > 1. Other predictions are (1) a calculable flow rate of the matter side towards the antimatter side at the accelerated stage; conceivably explaining the dark flow observation, (2) a time-dependent spacetime curvature over horizon evolution, which could influence the galactic rotational speed; possibly explaining the high speed of stars, and (3) evolvable spacetime internal voids at the accelerated stage, which could contribute in continuously increasing the matter and antimatter densities elsewhere in both sides respectively. These findings may indicate the existence of the antimatter as a distinct side, which influences the evolution of the universe instead of the dark energy or dark matter.


INTRODUCTION
The fundamental CPT symmetry states that the matter and antimatter would have been created in the same quantities at the Big Bang [1]. In contrast, the matterantimatter asymmetry, by the violation of the CPT in the universe, the fast movement of stars and the contradictory Hubble parameter measurements from the early and present universe [12,15].
As an alternative, the non-singular Big Bounce theory assumes the primordial substance was concentrated from a previous collapsed universe, and the universe experiences continuous expansions and contractions [7,8]. However, a version of the Big Bounce theory is recommended that a Bang of a primordial substance at thermal equilibrium produced a hot and dense early universe where the matter and antimatter could have been separated by electromagnetic fields [9,31]. In addition, according to the theory, the initial scale factor of the universe must be greater than zero [7,8] where a constant 0 2 can represent the initial space curvature. Besides, I embrace a closed finite universe because it could aid a large-scale cut-off in primaeval density fluctuations and may provide an agreement with low CMB anisotropy quadrupole observations [10].
Recently, the ΛCDM model has faced inconsistency with the advancement of new astronomical observations [12,18]. The recent Planck Legacy 2018 (PL18) release indicated the existence of an enhanced lensing amplitude in the CMB that is higher than what is expected in the ΛCDM model [13,14].
This endorses the existence of a positive curvature of the universe with a level of confidence greater than 99% [15]. In addition, the precise Hubble parameter measurements from the early universe using the Planck datasets based on the CMB show a lower value of expansion rate in comparison with the value of Hubble parameter in the present universe using the type Ia supernovae distance-redshift method [11,19], where the variation is three standard deviations [15,16]. Riess [19] found that the expansion of the universe is faster than what ΛCDM estimates where the disagreement between several independent measurements taken from the early and present universe is at four to six-sigma. Accordingly, a profound adjustment of the ΛCDM model or new physics are now growing due to this new evidence underlying the model assumptions [18].
In this paper, I derive a model to obtain the cosmic horizon radius while utilising a metaphorical representation based on the large-angle correlations of the CMB to help to understand the cosmic topology. I also utilise the non-singular Big Bounce theory to avoid the Big Bang singularity and to provide further tuning of the curvature density. The paper is organised as follows: In Section 2, I modify the metric tensor to account for the initial space curvature according to the Big Bounce theory and derive the model. In Section 3, I discuss the model implementation, matter and antimatter growth, and the model outcomes. In Section 4, I present star speed of simulation over horizon evolution, while in Section 5, I introduce the predictions of the model. In Section 6, I discuss and summarise the conclusions. Finally, in Section 7, I suggest future work.

The Mathematical Model
I utilise Einstein's field equations to derive the model: where is Ricci curvature tensor, is the scalar curvature, is the metric tensor, is Newton's gravitational constant, is the speed of light in vacuum, and is the energy-momentum tensor. I omitted the cosmological constant while considering gravity as the only force governing the universe [20].
The metric tensor can be characterised using the Friedmann-Lemaître-Robertson-Walker (FLRW) model [23,24]. Based on equivalent Newtonian dynamics of this metric, the isotropic spherical coordinates of the FLRW can be enhanced to account for the initial space curvature as follows: where is the four-dimension spacetime interval in polar coordinates, is the scale factor, and & 0 2 are constants representing the space curvature [25,29] and the initial space curvature respectively. The tensor signature (−, +, +, +) is utilised throughout this research. Einstein's field equations can be solved for a perfect fluid. By using the notation = 1 and rising one index in eq (1), thus it can be expressed in term of mixed component tensors: By resolving these outputs, we get the Friedmann equation that counts the initial space curvature as follows: Eq (4) can be solved at the time 0 where 0 is normalised to 1, and = 1 by the definition of the metric for the positively curved universe as follows: where 0 0 3 = constant [26]. By rewriting eq (5) with regards to the conformal time ≡ in the parametric form, which is represented in the range of (0 < < 2 ), we get the following: curved universe and is matter-dominated [26].

The Friedmann equation of deceleration and
acceleration of the universe is given as follows [17]: Through inspiration by CMB large-angle correlations where the temperature correlation across the microwave sky vanishes on scalars wider than 60° The evolution of the horizon radius ℎ ⃡⃗⃗⃗ ( ) is in the direction of the cosmic evolution while the scale factor can be understood as a local radius.
Thus, we can interpret the negative horizon radius as an indication of the existence of the antimatter as a distinct side, which is evolving in the opposite direction with regards to the matter side evolution path. The span of the cosmic horizon radius is: According to this new interpretation, we can differentiate ℎ ( ) which represents the overall space curvature radius due to the overall age of the universe while ( ) represents the universe slice radius at an instant of the time.

Model Implementation
We can utilise the derived model to find the evolution  The slope of the evolution curves shows that the rate of expansion was slowing down, which could be due

Matter-Antimatter Density Growth
As we interpreted that the universe is predicted to pass through two spatial stages. In the first

Adjusted Model Outcomes
Both Hubble parameter measurements at the early  Table 1 were investigated: By implementing these parameters, we obtain the predicted evolution paths of the horizon radius as shown in Figure 3. The model predicts that cosmic evolution might experience three distinct stages.
Firstly, the matter side of the universe was expanding away from the Big Bounce during first ~10 Gyr with a decelerated spatial expansion, which could be due to gravity between the two sides, until it reaches its critical horizon radius.

N-body Simulations of Galaxy Rotation
The consistent patterns of galactic rotation curves using precise and independent galactic redshift data confirmed that the hydrogen clouds and outer stars are orbiting galaxies at speeds faster than that calculated using Newtonian laws. Accordingly, the dark matter hypothesis was introduced to account for the apparently missing galactic mass and to explain the fast-orbital velocity [32,33]. However, no evidence of the existence of the dark matter, which is supposed to account for the majority galactic mass, was observed since its introduction. The failure to find dark matter led to the introduction of new theories such as modified gravity and modified Newtonian dynamics [21, 22, 34 -36]. On the other hand, several recent studies found that many galaxies do not contain dark matter [37 -39]. This observation was considered in some studies where the galaxy formation was simulated using modified Newtonian dynamics without considering the dark matter [40].
Thus, it seems that there is no evidence or agreement on the existence or nature of the dark matter as well as it is not an essential element in some galaxies.
As an alternative, I introduce a new hypothesis based on the variation of the curvature of both sides along with the evolution of the horizon radius as was shown argue that the observed various star movements could occur as a result of the variation of the universe curvature, where the gravitational attraction between both sides of the universe can highly bend the spacetime over the evolution of the horizon radius.
To evaluate this hypothesis, I perform a fluid simulation study based on the Newtonian dynamics using the Fluid -Pressure and Flow software [41]. In this simulation, a perfect fluid of mass density and isotropic pressure was assumed to represent the matter of the universe while the fluid particles were assumed to represent the stars. The fluid was considered as a perfect fluid because it is frictionless with no heat conductivity [42].
Using these conditions, the fluid model was built to simulate the star movement speed between an incrementing horizon curvature and their distance from it to a lesser curved spacetime, as shown in Figure 6. Figure 6: The particles that passing away from the highly curved horizon are moving faster than their counterparts that are passing within lesser curved spacetime. As the bending of horizon increases, the speed difference increases.
Considering these outcomes, it could be concluded that the variation of the curvature of the spacetime over the horizon evolution can influence the speed of star movement as they are located through different curvatures of the universe.  The expansion rate was decreasing at the first stage, and this conforms with the lower value of the Hubble parameter obtained by the Planck datasets of the early universe [12,15]. However, at the second stage, the expansion rate is increasing, and this aligns with the higher value of the Hubble parameter obtained using the supernovae type Ia distance-redshift method at the present universe.

The Predictions of the Model
In addition, at the second stage of expansion, the model predicts a flow of the matter side of the universe due to the gravitational attraction by the antimatter side.
We can predict the flow rate using the derivative of the optimised matter model with regards to the conformal time as follows: that the antimatter travels backwards in time with regards to the travel direction of the matter.

Conclusions
In

Future Work
The values of the matter density, curvature density, energy growth factor, the Hubble