Spherical Multiverse: A Theoretical Framework for a Perpetual Cosmological Model

Zenodo (2025) / DOI 10.5281/zenodo.17648071 1 Abstract The nature of the universe and the mechanisms driving its evolution remain subjects of intense research and debate. This work presents a theoretical model of a spherical multiverse in which the collapse of one universe (Big Crunch) initiates the expansion of an adjacent universe (Big Bang) through energy transfer processes. This mechanism ensures continuity in cosmological evolution and prevents thermal stagnation. The proposed mathematical framework extends the Friedmann equations to incorporate energy transfer between connected spherical universes. Additionally, the model introduces the concept of positive and negative dimensions of time and defines a habitable zone at the equator of each sphere, where physical conditions remain stable for life over long periods. This framework aims to deepen our understanding of cosmic dynamics and open new avenues for research in cosmology and theoretical physics. Key aspects include the role of entropy reconfiguration in avoiding thermal stagnation and the potential observational signatures of inter-universal energy transfers in the cosmic microwave background. Keywords: big bang; big crunch; cosmology; habitable zone; spherical model; negative time; multiverse 1. INTRODUCTION Modern cosmology has experienced significant advances in our understanding of the universe, from the Big Bang to the accelerated expansion observed today [1]. The Friedmann-Lemaître-Robertson-Walker (FLRW) equations have been fundamental in describing the dynamics of the universe in terms of its expansion and contraction. Nonetheless, the nature of the universe and its ultimate fate remain subjects of intense research and debate [2]. One of the theories proposed to explain the evolution of the universe is the cyclic model, where the universe undergoes successive expansions and contractions [3]. This model suggests that after a Big Bang, the universe expands to a maximum size, followed by a contraction towards a Big Crunch, which could give rise to a new Big Bang. However, conventional cyclic models do not specify a mechanism for how this transition occurs. Additionally, previous models do not explicitly describe how entropy behaves during such transitions or how time reversal could be physically realized. This work proposes the hypothesis that the universe may evolve through a perpetual sequence of expansions and contractions, where the collapse of one universe (Big Crunch) triggers the expansion of an adjacent universe (Big Bang) via energy transfer mechanisms. These transitions are assumed to preserve cosmological continuity and prevent thermal stagnation. The model also considers the possibility of time having both positive and negative dimensions and introduces the concept of a habitable zone located at the equator of each spherical universe, where physical conditions remain stable over extended periods. Accordingly, this work aims to accomplish the following goals: i) Develop a mathematical model that describes the dynamics of a closed universe using the Friedmann equations; ii) Generalize the model to include multiple connected spherical universes, describing the interactions between them; iii) Introduce the Open access | Version of record: November 19, 2025 TOMÁS DE J. MATEO SANGUINO Technology, Energy and Sustainability Research Center, University of Huelva, Spain [email protected] Spherical Multiverse: A Theoretical Framework for a Perpetual Cosmological Model Zenodo (2025) / DOI 10.5281/zenodo.17648071 2 concept of time with positive and negative dimensions, exploring how this affects the dynamics of the multiverse; and iv) Define a habitable zone in terms of the scale factor, describing the conditions necessary for life in the context of the spherical multiverse. The paper is structured as follows. Section 2 presents the mathematical foundations of the model, including the modified Friedmann equations and the incorporation of inter-universal energy transfer. Section 3 discusses the results of the simulations, focusing on the behavior of the scale factor, entropy evolution, and habitable zone stability. Section 4 provides a theoretical discussion of the implications of the model, particularly regarding time reversal, entropy regulation, and observational constraints. Finally, Section 5 summarizes the main conclusions and outlines directions for future research. 2. METHOD The following mathematical model provides a theoretical framework for exploring how interactions between spheres, Big Crunch to Big Bang transitions, time dimensions and habitable zones might influence the dynamics of the multiverse. Let us describe the universe as a giant sphere that expands and contracts via the FLRW metric [4], whose formula describes how distance is measured in this spherical universe: 𝑑𝑠2=−𝑐2𝑑𝑡2+𝑎2(𝑡)( 𝑑𝑟2 1−𝑘𝑟2+𝑟2𝑑𝛺2) (1) where ds is a measure of distance in space-time, c is the speed of light, dt is a small interval of time, a(t) is the scale factor that indicates how the size of the universe changes over time, dr, dθ, dϕ represent small changes in spatial coordinates (i.e., longitude, latitude, and height on a sphere), and k defines whether the universe is flat (k = 0), closed (k = 1) or open (k = -1). On the other hand, Einstein's field equation [5] expresses how matter and energy affect the curvature of space-time: 𝐺𝜇𝜈 +Λ𝑔𝜇𝜈 =8𝜋𝐺 𝑐4𝑇𝜇𝜈 (2) where Gμν describes the curvature of space-time, Λ is the cosmological constant, related to dark energy, gμν is the metric tensor that describes the geometry of space-time, Tμν describes the distribution of matter and energy, and μy indicate the components of the tensors in the equation (i.e., one temporal dimension and three spatial dimensions). For a homogeneous and isotropic universe, the energy-momentum tensor can be written as follows: 𝑇𝜇𝜈 =(𝜌+𝑝)𝑢𝜇𝑢𝜈+𝑝𝑔𝜇𝜈 (3) where ρ is the energy density, p is the pressure, and uμ is the four-velocity and describes how matter moves in space-time. The Einstein tensor Gμν is obtained from the Ricci tensor Rμν and the Ricci scalar R as follows: 𝐺𝜇𝜈 =𝑅𝜇𝜈 −1 2𝑅𝑔𝜇𝜈 (4) For the FLRW metric, the non-zero components of the Ricci tensor are that associated with the curvature in the time direction and those associated with the curvature in the spatial directions: 𝑅00 =−3𝑎󰇘 𝑎 (5) 𝑅𝑖𝑗 =(𝑎󰇘 𝑎+2𝑎󰇗2 𝑎2+2 𝑘 𝑎2)𝑔𝑖𝑗 (6) Zenodo (2025) / DOI 10.5281/zenodo.17648071 3 where ä is is the second derivative of the scale factor a(t) with respect to time, which tells us how the expansion rate of the universe is changing, ȧ is the first derivative of the scale factor with respect to time, and gij is the spatial metric tensor, which describes the geometry of space. Furthermore, the Ricci scalar is defined as follows: 𝑅=6(𝑎󰇘 𝑎+𝑎󰇗2 𝑎2+𝑘 𝑎2) (7) Substituting these terms into the Einstein field formula, the first and second Friedmann equations are then obtained: (𝑎󰇗 𝑎)2=8𝜋𝐺 3𝜌−𝑘𝑐2 𝑎2+𝛬𝑐2 3 (8) 𝑎󰇘 𝑎=−4𝜋𝐺 3(𝜌+3𝑝 𝑐2)+𝛬𝑐2 3 (9) To model interactions between spheres and time dimensions in the multiverse, additional terms are introduced in the first and second Friedmann equations. The first equation is modified as follows: (𝑎󰇗 𝑎)2=8𝜋𝐺 3𝜌−𝑘𝑐2 𝑎2+𝛬𝑐2 3+𝜖(𝑡) 𝑄 𝑎𝑛 (10) where 𝑄 represents an interaction coefficient that quantifies energy transfer between neighboring spheres. The added term 𝑄/𝑎𝑛 accounts for inter-universal energy transfer. Its exponent 𝑛 depends on the underlying physical mechanism. If energy transfer behaves like non-relativistic matter, it should scale as 𝑄/𝑎3, analogous to the dilution of matter in an expanding universe. If the interaction is mediated by pressure effects, it could follow 𝑄/𝑎2, while radiative processes suggest 𝑄/𝑎4 [6]. Additionally, quantum tunneling models and vacuum fluctuations propose that interactions between universes could behave differently, affecting the choice of 𝑛. Further theoretical and observational constraints are needed to determine the most appropriate scaling. FIGURE 1. Model of spherical universe in a perpetual multiverse. Zenodo (2025) / DOI 10.5281/zenodo.17648071 4 2.1 Dimension of Time The proposed model explores a time-reversal mechanism at the transition from Big Crunch to Big Bang, suggesting that the arrow of time reverses direction due to entropy redistribution, in accordance with ChargeParity-Time (CPT) invariance [7]. This idea aligns with theoretical frameworks in thermodynamics and quantum gravity (such as the Wheeler-DeWitt equation), which explores time symmetry and its implications for the structure of the universe [8]. In this model, the transition of time is represented by the sign of 𝜖 (t) in the modified Friedmann equations: 𝜖(𝑡)={+1 for 𝑡 ≥from Big Bang to Big Crunch −1 𝑓𝑜𝑟 𝑡 <from Big Crunch to Big Bang (11) Additionally, observational constraints place restrictions on 𝑄. The term 𝑄/𝑎𝑛 must remain below the level of temperature anisotropies in the cosmic microwave background (CMB), observed at Δ𝑇/𝑇𝐶𝑀𝐵 ≈ 10−5. For 𝑛 = 3, this condition places 𝑄 within the range 10−6 to 10−4, ensuring that inter-universal interactions do not introduce inconsistencies with current cosmological data. Furthermore, a small 𝑄/𝑎3 contribution in the expansion phase could play a role in structure formation, influencing the evolution of density perturbations and avoiding excessive damping, while in contraction, it aids in entropy redistribution, ensuring smooth transitions between cycles. This also allows for the possibility that 𝑄/𝑎3 plays a subtle but non-negligible role in the evolution of large-scale structure, without disrupting the standard model of cosmology. However, if n ≠3, the scaling behavior changes. For n = 2, the effect is stronger in the late universe, whereas for n = 4, the dominant contribution occurs in the early universe. Any deviation from n = 3 would require further justification based on observational signatures and theoretical developments in quantum gravity or modified gravity models. Similarly, a term to represent the influence of the Big Crunch to Big Bang transitions is added into the second Friedmann equation: 𝑎󰇘 𝑎=−4𝜋𝐺 3(𝜌+3𝑝 𝑐2)+𝛬𝑐2 3+𝜖(𝑡) 𝑃 𝑎2 (12) where 𝑃 represents the pressure in the contracting phase, which influences the acceleration of expansion as the universe transitions from the Big Crunch to the Big Bang. The term 𝑃/𝑎2 accounts for the effects of pressure on the evolution of the universe’s expansion rate, which is tied to the geometry of spacetime and the energy density in the contracting phase. The function 𝜖(𝑡) modulates these terms depending on whether the universe is moving towards the Big Crunch or the Big Bang, with the sign of 𝜖(𝑡) indicating the direction of time. This reflects the idea that entropy increases in the expanding phase as structures grow, while the transition to the contracting phase 'resets' the local entropy conditions in the new universe, preserving total entropy constraints while allowing for the cyclic nature of the multiverse. Such a model extends the standard cosmological model without violating known physical principles, offering a broader understanding of the evolution of the universe. 2.2 Habitable Zone In a spherical multiverse model, this concept is introduced to define regions where physical conditions remain stable for extended periods. This is modeled mathematically as a density and pressure fluctuation around the equatorial region: 𝜌ℎ𝑎𝑏𝑖𝑡𝑎𝑏𝑙𝑒 =𝜌0(1+𝛿cos(𝜃)) (13) 𝑝ℎ𝑎𝑏𝑖𝑡𝑎𝑏𝑙𝑒 =𝑝0(1+𝛿cos(𝜃)) (14) Zenodo (2025) / DOI 10.5281/zenodo.17648071 5 where δ represents the amplitude of density and pressure variations, θ is the polar angle, and ρ₀, p₀ are reference values from cosmological data (e.g., cosmic microwave background energy density at recombination, ρ₀ ≈ 4.2 × 10⁻³¹ kg/m³ [9]). These equations describe how density and pressure fluctuate in a spherical multiverse model. In this context, the physical stability of galaxies or planetary systems in the habitable zone can be understood by considering the mechanisms of gravitational equilibrium and rotational stability. In a self-contained sphere, the equatorial regions experience lower density variations, which reduces the gravitational perturbations that could disrupt planetary formation. The equatorial zone's relatively uniform conditions act as a stabilizing factor, preventing drastic fluctuations in gravitational forces, thereby facilitating the long-term stability of planetary orbits. This stabilization can be compared to astrophysical models where rotational equilibrium stabilizes mass distributions, similar to how planetary disks in protoplanetary systems form and remain stable over time. In this context, the stability of planetary systems or galaxies within the habitable zone can be attributed to two main mechanisms: rotational equilibrium and reduced gravitational perturbations. For instance, in a protoplanetary disk, angular momentum conservation causes the material to accumulate in the equatorial plane, where gravitational forces and pressure gradients are balanced. This stability allows for the formation of planets with minimal disruption from external gravitational forces, much like the stable conditions expected in the habitable zone of a rotating multiverse. Additionally, the concept of a habitable zone extends to the stability of molecular structures necessary for life. The lower density gradients in the equatorial regions minimize abrupt energy shifts that might inhibit complex chemistry. This is analogous to the concept of the circumstellar habitable zone, where planets experience a stable climate, ideal for sustaining life. Similar to exoplanetary models, where equatorial regions of tidally locked planets are considered favorable for habitability due to stable heat distribution [10], the equatorial region of the multiversal model exhibits reduced density and pressure fluctuations. This stability could favor the formation of long-lived cosmic structures and provide conditions conducive to complex chemical processes necessary for life. Furthermore, analogies can be drawn with neutron star accretion disks, where matter in the equatorial bands remains preferentially stabilized by rotational forces, providing a stable environment for particle interactions, analogous to the stable conditions within the multiversal habitable zone. For a spherical multiverse, the terms to consider interactions between multiple connected spheres are the following: (𝑎𝑖 󰇗 𝑎𝑖)2=8𝜋𝐺 3𝜌𝑖−𝑘𝑖𝑐2 𝑎𝑖2+𝛬𝑐2 3+∑𝜖𝑖𝑗(𝑡)𝑄𝑖𝑗 𝑎𝑖3 𝑗≠𝑖 (15) 𝑎𝑖 󰇘 𝑎𝑖=−4𝜋𝐺 3(𝜌𝑖+3𝑝𝑖 𝑐2)+𝛬𝑐2 3+∑𝜖𝑖𝑗(𝑡)𝑃𝑖𝑗 𝑎𝑖2 𝑗≠𝑖 (16) where i and j denote the i-th and j-th adjacent universes, correspondingly. To help visualize the proposed model, Figure 1 illustrates the perpetual spherical multiverse model, where each sphere represents an individual universe. Arrows indicate the space-time flow and the transitions from the Big Crunch to the Big Bang. Habitable zones are marked at the equator of each sphere, where physical conditions allow for the existence of life. 3 RESULTS The function a(t) in Equation (17) describes a closed, cyclic universe model, where the scale factor oscillates between 0 (representing the Big Bang), 1 (the maximum expansion), and back to 0 (the Big Crunch). This behavior aligns with the Friedmann equations for closed cosmologies, fitting within the theory of cyclic universes [3]. To ensure the model's viability, cosmological parameters are adjusted such that the matter density is close to the critical density ρ ≈ ρc = 8.53 × 10−27 kg/m3 and the cosmological constant Λ ≈ 1.1056 × 10−52 m−2. These adjustments ensure that the cyclic expansion and contraction of the universe are Zenodo (2025) / DOI 10.5281/zenodo.17648071 6 consistent with current observational data and the model’s conditions. The period T defines the time required for the universe to undergo a complete cycle of expansion and contraction, with the scale factor oscillating accordingly. The cyclic nature of the universe is driven by gravitational dynamics, where the closed curvature naturally leads to expansion and contraction. The entropy evolution follows a self-regulating mechanism, with energy redistribution preventing a thermodynamic singularity at each cycle, consistent with the conformal cyclic cosmology (CCC). Gravitationally, the closed curvature of the universe naturally leads to a cycle of expansion and contraction, with the Friedmann equations ensuring that the scale factor follows a periodic behavior. By adjusting these parameters, the model adheres to the fundamental principles of cosmology, offering a consistent and plausible framework for understanding a cyclic, closed universe. Figure 2 illustrates the cyclic expansion and contraction of the universe, emphasizing its critical phases and aiding in understanding the temporal evolution of the model. 𝑎(𝑡)=sin𝜋𝑡 𝑇 (17) Figure 3 shows the evolution of the modified Friedmann formula defined in Equation (10) as a function of the scale factor, using different values of the interaction coefficient Q. The values are significantly larger in the modified model compared to the original Friedmann model, especially in the early stages of the universe (small values of a). This is due to the additional term Q/a3, which introduces a variation in the expansion rate of the universe. For comparison purposes, the original Friedmann model uses the scalar function described in Equation (18), which is derived under the assumption that the energy density is dominated by nonrelativistic matter. It is widely used in cosmology to describe the expansion of a matter-dominated universe because of its ability to simplify calculations and provide a good approximation to the actual evolution of the universe, validated by cosmological observations [11]. The findings obtained with the modified model in Figure 3 suggest that interactions between adjacent universes can have a significant impact on the expansion rate of the universe, especially in its early stages. This opens new avenues of research in cosmology, as the inclusion of additional terms in the Friedmann equations could provide a better understanding of the dynamics of the universe and its possible interactions with other universes in a multiversal context. 𝑎(𝑡)=(𝑡 𝑡0)2 3 (18) Figure 4 illustrates the variation of habitable density defined in Equation (13) as a function of the angle θ, from the pole (θ = 0) to the opposite pole (θ = π). In this relationship, ρ0 represents the density of the cosmic background at recombination (4.2 × 10-31 kg/m³) and δ is the relative fluctuation in density (0.1). The graph FIGURE 2. Evolution of the scale factor over time. FIGURE 3. Evolution of the modified Friedmann equation as a function of the scale factor. Zenodo (2025) / DOI 10.5281/zenodo.17648071 7 reveals a non-uniform distribution of habitable density, with a maximum of 4.62 × 10-31 kg/m³ at the pole (θ = 0), gradually decreasing to 4.2 × 10-31 kg/m³ at the equator (θ = π/2, marked by a red dashed line), and reaching a minimum of 3.78 × 10-31 kg/m³ at the opposite pole (θ = π). This cosine variation in habitable density suggests a dipolar structure in the distribution of matter, with potential implications for the formation of cosmic structures and habitability in different regions of the early universe, as well as its extinction. The equatorial region, where density and pressure fluctuations are minimized, may play a key role in the formation and long-term stability of galaxies and planetary systems, making it a relevant subject for future cosmological studies. Figure 5 illustrates the evolution of the energy transfer fluctuation (Q/a³) as a function of time (t) in billions of years, spanning from the Big Bang to the hypothetical Big Crunch in a cyclic universe model. Let us assume Q small enough not to completely dominate the energy density, but large enough to have an observable impact. A reasonable value of Q consistent with cosmological observations and theoretical models might be in the range 10−6 to 10−4 [11]. At the Big Bang, Q/a³ exhibits extremely high values, attributed to the very small scale factor a(t) at the universe's inception. As the universe expands, a(t) increases, leading to a substantial decrease in Q/a³. This trend continues until the point of maximum expansion, where Q/a³ reaches its minimum. Subsequently, as the universe potentially begins to contract towards the Big Crunch, Q/a³ shows a marked increase, mirroring the reduction in a(t).This pattern of fluctuation in energy transfer is consistent with the proposed cyclic model, where physical conditions are expected to vary significantly during expansion and contraction phases. The graph captures the non-constant nature of energy transfer fluctuations, aligning with the model's predictions of oscillating energy density and pressure throughout the cosmic cycle. The extreme values observed at the Big Bang and Big Crunch, contrasted with lower values mid-cycle, provide crucial insights into the dynamic energy distribution and transfer mechanisms in a potentially cyclic universe. Figure 6 shows the fluctuation of entropy as a function of a(t) and ϵ(t), that can be expressed as follows: 𝑆(𝑡)=𝑆0(1−cos2𝜋𝑡 𝑇)·𝜖(𝑡) (19) where S0 is the maximum entropy reached during the cycle and T is the complete period of the cycle, spanning from Big Bang to Big Crunch or vice versa. This equation models cyclic entropy, with the maximum value S0 reached at the midpoint of the cycle. The function, ϵ(t), modulates the direction of the transition, reflecting changes in entropy depending on whether the universe is transitioning from the Big Bang to the Big Crunch or vice versa, while maintaining its cyclic structure. To estimate S0 based on modern cosmological theories, the concept of universal entropy is considered, which is related to the number of microstates available in the system. In modern cosmology, the maximum entropy of the universe is associated with the entropy of the largest possible black hole, known as Bekenstein-Hawking entropy. This value is FIGURE 4. Habitable density over angle. FIGURE 5. Fluctuation of energy transfer over time. Zenodo (2025) / DOI 10.5281/zenodo.17648071 8 significant because black hole entropy serves as a good approximation for the total entropy of the universe on a large scale [12]. The entropy of the universe, in terms of Boltzmann's constant (kB) and the total mass of the universe (Mtotal) can be approximated as: 𝑆0~𝐾𝐵𝑀𝑡𝑜𝑡𝑎𝑙𝑐2 ℏ𝐺 (20) where Mtotal is the total mass of the universe (~ 1053 kg), G is the gravitational constant, c is the speed of light, and ℏ is the reduced Planck constant. This value is extraordinarily large, and for simulation purposes, a value close to S0∼10120 is assumed, which is an approximation based on known magnitudes of observable universe entropy in modern cosmology. 4 DISCUSSION The proposed spherical multiverse model introduces a novel perspective on cyclic universes by incorporating inter-universal interactions, time reversal dynamics and the concept of a habitable zone within each universe. A key component of this model is the mechanism by which the Big Crunch of one universe leads to the Big Bang of another. Energy transfer between adjacent universes in the multiverse model can be understood through a combination of thermodynamic principles, gravitational effects and quantum fluctuations. The term Q/an introduced in the Friedmann equations models this interaction, but its exact form depends on the dominant mechanism. If the transfer occurs through classical gravitational interactions, a term Q/a2 might be justified, reflecting an influence that scales with spatial curvature. If energy is carried by particles behaving like non-relativistic matter, then Q/a3 is appropriate, as it follows the dilution of energy density. Alternatively, if quantum tunneling or vacuum fluctuations play a major role, a term Q/a4 could emerge, mimicking the scaling behavior of radiation energy density. Current observational constraints on the cosmic microwave background impose limits on Q, ensuring that its contribution does not exceed the measured anisotropies. Further investigations, including potential observational signatures in large-scale structure formation or relic radiation, could help determine the most suitable value of n. Furthermore, observational constraints from the CMB impose strict limits on Q/an. To remain consistent with current measurements, the term must satisfy that Q/an << 10-5 x ρCMB. Given that the radiation density at recombination (z ≈ 1100) was 4.2×10−31 kg/m³, any deviation in Q/an beyond this limit could introduce inconsistencies in the CMB power spectrum, affecting the formation of large-scale structures [13]. This process is analogous to vacuum decay in false vacuum states and could leave observable signatures in the cosmic microwave background. Based on black hole thermodynamics and Hawking radiation models, an approximate energy transfer efficiency of 10⁻² to 10⁻¹ could be assumed for interactions between universes FIGURE 6. Fluctuation of entropy over time. Zenodo (2025) / DOI 10.5281/zenodo.17648071 9 [14]. Additionally, quantum tunneling effects similar to those observed in vacuum decay processes suggest that fluctuations in the energy density near the transition point might facilitate the transfer of a fraction of the collapsing universe’s energy to its adjacent expanding counterpart. This process may leave observable imprints, such as anisotropies in the cosmic microwave background, which could serve as indirect evidence of inter-universal interactions. Further analysis is required to refine the exact nature of these transfers and determine potential observational signatures. At the transition from Big Crunch to Big Bang, entropy dissipation may occur through inter-universal interactions, preventing thermal stagnation. Previous studies have explored similar cyclical models, providing a theoretical basis for this hypothesis [15]. This work proposes that near maximal contraction, the extreme curvature of space-time induces vacuum fluctuations analogous to Hawking radiation, allowing part of the entropy to be transferred to neighbouring regions of the multiverse. This mechanism ensures that entropic growth within each individual cycle does not lead to progressive thermal decay, maintaining compliance with the second law of thermodynamics in a multiverse context [16]. This mechanism ensures that high-entropy states do not accumulate indefinitely, allowing for a smoother transition between cycles while maintaining the second law of thermodynamics at a multiversal scale. During the Big Bang and the Big Crunch, the entropy of the universe decreases as structures collapse. The reversal of time implies that the entropy evolution is mirrored in the new expanding universe, effectively resetting local entropy conditions while preserving total entropy constraints [17]. Some approaches, such as the CCC proposed [18], suggest that the transition from Big Crunch to Big Bang could involve a loss of classical time, leading to a reemergence of time in a new form. This framework allows for a cyclic multiverse where each transition between universes does not violate known physical principles but instead extends the standard cosmological model, ensuring consistency with thermodynamic and relativistic constraints. Another essential aspect of this model is the role of time reversal at the transition point between universes. The sign change of 𝜖(t) in the modified Friedmann equations suggests that the arrow of time may invert at the moment of maximum contraction. This could be interpreted in terms of CPT invariance, where the transition from a collapsing universe to a new expanding universe is modeled as a transformation that preserves the fundamental laws of physics. In this scenario, time in the new universe would advance in the same direction locally, but with global symmetry across the multiverse. However, experimental verification of these effects remains an open challenge in modern cosmology [19]. Additionally, the concept of a habitable zone within each universe offers a new way to consider cosmic conditions necessary for life. While the present model suggests that the equatorial region of each sphere may provide long-term stability due to minimized density fluctuations, further studies are needed to determine whether this stability can support complex chemistry and planetary formation processes. Future work may focus on simulating such conditions to assess their viability within known physical constraints. The implications of this model extend beyond theoretical cosmology, potentially influencing how we interpret the structure of the multiverse and its observable effects. If inter-universal energy transfers produce detectable signatures in the cosmic microwave background or large-scale structure formation, new observational strategies could be developed to test these predictions. Future research should explore how alternative formulations of quantum gravity and entropy evolution might refine or challenge the assumptions presented in this work. 5 CONCLUSIONS This paper presents a theoretical framework for a spherical multiverse, where each universe is a closed sphere connected to others. The model explains how the Big Crunch of one sphere can lead to the Big Bang of another, creating a perpetual cycle of expansion and contraction. The introduction of positive and negative dimensions of time provides a novel approach to understanding time symmetry and its implications for the multiverse. In addition, the concept of a habitable zone at the equator of each sphere offers insights into the stability of physical conditions necessary for life.