To properly describe the physics of a 5D membrane embedded in a 6D higher-dimensional bulk space, the induced metric on the membrane must be defined first. The 6D background bulk spacetime metric in higher dimensions is given by [13] :

$g_{MN}^{\left(6\right)}\text{d}{x}^M\text{d}{x}^N=g_{ab}^{\left(5\right)}\text{d}{x}^a\text{d}{x}^b+\text{d}{y}^2$(1)

where the induced 5D brane metric is derived from the bulk metric via [14] :

$g_{ab}^{\left(5\right)}=g_{MN}^{\left(6\right)}\frac{\partial x^M}{\partial {\xi }^a}\frac{\partial x^N}{\partial {\xi }^b}$(2)

The 4D metric of the observable universe is extracted as [13] :

$h_{\mu \nu }=g_{ab}^{\left(5\right)}\frac{\partial {\xi }^a}{\partial x^{\mu }}\frac{\partial {\xi }^b}{\partial x^{\nu }}.$(3)

These reductions establish the basis for translating higher-dimensional geometric effects into 4D observable phenomena [15] .

The deformation waves are undamped longitudinal waves; the deformation wave field equation is given by:

$\Phi =A\left(x^{\mu }\right)\cos \left(k_{wave}{x}^4-\omega t+\phi \right),\left(\mu =0,1,2,3,4\right)$(4)

where:

The deformation field Φ in our model is not only a classical wave but also exhibits microscopic quantum characteristics through its localized structures, termed deformation “knots.” These knots are quantized excitations of the field Φ at its extrema—namely the crests and troughs of the deformation wave—it forms localized structures known as deformation knots. These knots represent the geometric origin of matter particles, with their energy concentrated in specific regions rather than dispersing across multiple vibrational modes during wave propagation. This localization ensures that the vibrational energy of the knots does not dissipate as the deformation wave propagates, resulting in undamped propagation across the 5D brane, consistent with the conservation of mass and energy in the universe.

Physically, these knots arise at points where ${\partial }_{x^4}\Phi =0$, indicating local maxima or minima in the deformation field. Their stability results from the effective potential $V\left(\Phi \right)$, which reaches a local minimum at these points. We describe these structures using the following effective Lagrangian on 4D slices:

${ℒ}_{slice}=-\frac{1}{2}{h}^{\mu \nu }{\partial }_{\mu }A\left(x^{\mu }\right){\partial }_{\nu }A\left(x^{\mu }\right)-V\left(A\left(x^{\mu }\right)\right)$(5)

where $A\left(x\right)\equiv \Phi$ is the scalar field representing the amplitude of local microscopic deformation within the wave’s crests and troughs. In these regions, the gradient ${\partial }_{x^4}{V}_{\Phi }$ vanishes, leading to near-zero effective gauge field mass and enabling stable interactions.

These deformation knots exhibit localized energy, stability, and particle-like properties in the 4D perspective, effectively behaving as matter particles. Analogous mechanisms are seen in solitonic models such as Q-balls and Skyrmions [16] ; our model generalizes this concept to a higher-dimensional brane setting.

In our model, the deformation knots are not static field defects localized in the brane; instead, they are dynamically propagating maxima within the longitudinal wave packet along the fifth dimension $x^4$. Their stability is ensured not by topological protection, but by the robust form of the wave peak, which remains a local maximum under small perturbations during propagation. So the knot structures are stable in the sense that they ride on the crest and trough of the deformation wave, maintaining their peak amplitude as they move forward along $x^4$, thus not subject to local destabilization as in fixed-point configurations.

Hence, we propose that all ordinary matter—including baryons and Standard Model particles—arises as localized excitations within the membrane’s geometric deformation. This unifies matter and spacetime structure within a single geometric framework, bridging membrane cosmology and particle physics.

Furthermore, this perspective is consistent with the idea of induced matter theory, which proposes that 4D matter arises from the geometry of higher-dimensional spacetime [17] [18] . However, our model goes further by offering a dynamical wave-based geometric mechanism for generating both matter and curvature.

The universe emerges as a wave crest or trough propagating along a higher-dimensional brane embedded in an extended bulk spacetime. All matter and energy in the universe come from the brane deformation, and the curvature of space-time in the universe is equivalent to the brane deformation. Undamped propagation of brane deformation waves ensures the conservation of mass and energy in the universe.

Within the stable 4D slice spacetimes at wave crests and troughs, these deformation knots can act as secondary wave sources, emitting waves that exhibit diffraction and interference phenomena. These effects arise naturally from the wave-like nature of the knots’ wave functions, potentially influencing the distribution of matter and energy in the 4D slices, which may leave observable imprints in the cosmic microwave background (CMB) anisotropies or large-scale structure formation.

The localized wave crests and troughs correspond to regions where Φ attains extreme values, forming self-sustaining solitonic structures that support stable gauge fields. These regions serve as the basis for two types of observable universes:

The fluctuating inter-wave regions remain unstable and are identified with dark energy and dark matter domains.

The total action $S_{total}$ includes contributions from the 6D bulk, 5D brane, Gibbons-Hawking-York boundary terms, and the 4D slices [14] :

$S_{total}=S_{bulk}+S_{brane}+S_{GHY}+{\displaystyle {\int }_{slices}{S}_{slice}}$(6)

The action for the 6D bulk spacetime ( $M^6$) is given by [13] :

$S_{bulk}={\displaystyle {\int }_{M^6}{\text{d}}^{6}x\sqrt{-g^{\left(6\right)}}\left[-{\text{Λ}}_6\right]}$(7)

where:

Thus, the 6D bulk action assumes $g_{MN}^{\left(6\right)}$ as a fixed background geometry, such as Minkowski ( ${\text{Λ}}_6=0$) or AdS ( ${\text{Λ}}_6<0$), with gravitational dynamics confined to the brane.

The action for the 5D brane embedded in the 6D bulk is [14] :

$S_{brane}={\displaystyle {\int }_{M5}{\text{d}}^{5}x\sqrt{-g^{\left(5\right)}}\left(-\sigma +{ℒ}_{deform}^{\left(5\right)}\right)}$(8)

where:

${ℒ}_{deform}^{\left(5\right)}=\frac{1}{2}{g}_{ab}^{\left(5\right)}{\partial }_a\Phi {\partial }_b\Phi -V\left(\Phi \right)$(9)

where Φ is a scalar field representing membrane deformations.

In our model, the quantized deformation “knots” display quantized microscopic characteristics when they exist alone, but when they are gathered together, they display certain macroscopic characteristics. The emergence of gravity in 4D slice spacetimes—formed at the wave crests and troughs of the brane deformation wave—is not postulated but derived from a physical mechanism rooted in the brane’s elastic dynamics. Specifically, as microscopic longitudinal deformations “knots” individual knots induce only small spacetime distortions, since the membrane is connected, this “knot” will also affect other points on the membrane, so their cumulative effect over large regions leads to measurable macroscopic spacetime bending of the brane within each stable slice region. This macroscopic bending, encoded geometrically as the Ricci scalar R of the induced 4D metric, serves as a geometric response balancing the internal tension generated by the cumulative microscopic longitudinal deformations “knots”. So the gravity is not an external field added by hand, but a natural geometric consequence of the membrane’s intrinsic deformation. Specifically, as the deformation wave propagates along the fifth spatial dimension, it accumulates and causes the membrane to bend, resulting in effective 4D curvature.

We propose the following relation as the physical basis for the emergence of gravity (The proof is in Appendix A):

${\epsilon }_{\Phi }\left(x\right)=kR\left(x\right)$(10)

where

This equation shows that the 4D curvature R arises naturally as a geometric response to the internal deformation stress. Unlike traditional brane-world models that derive Einstein equations via Gauss-Codazzi projections from 5D Einstein tensors [14] [19] , our approach ties the curvature directly to the membrane’s physical deformation dynamics.

Importantly, although Equation (10) differs in form from the traditional Einstein field equations (EFE), it encapsulates the same fundamental principle: that spacetime curvature is determined by the local distribution of matter and energy. This proportionality explicitly demonstrates, both mathematically and physically, that the energy density arising from deformation knots induces curvature in the 4D slices. In this sense, Equation (10) provides a concrete realization of the core idea behind the EFE-that matter tells spacetime how to curve-even though it arises here from the specific dynamics of brane deformations rather than being assumed as a geometric postulate.

Moreover, while the Einstein-Hilbert action postulates a Lagrangian density linear in the Ricci scalar R, it does not explain why matter and energy should generate curvature in this way. In contrast, Equation (10) suggests a possible physical interpretation for this proportionality, arising from the dynamics of deformation knots. Our model thus proposes that matter and energy arise as dynamical geometrical deformations of the brane, providing a physical mechanism that naturally links local energy densities to spacetime curvature and offering insight into how matter not only curves spacetime but also maintains stability through propagating deformation waves. Although this does not constitute a rigorous derivation of the Einstein-Hilbert action, it offers a complementary perspective on the physical origins of gravitational curvature.

To consistently encode this geometric response in the variational formulation, the gravitational Lagrangian density ${ℒ}_{grav}$ for each 4D slice must then include a term linear in R. This choice is not an arbitrary borrowing from General Relativity but arises naturally as the unique linear term consistent with (i) the observed curvature-force relation, (ii) general covariance, and (iii) the requirement that the resulting field equations remain second order.

In each localized 4D slice—formed at the crests and troughs of the membrane deformation wave—the total action is composed of a gravitational part and a deformation field part:

Total action;

$S_{slice}=S_{grav}+S_{deform}$(11)

Gravitational action;

$S_{grav}={\displaystyle \int {\text{d}}^{4}x\sqrt{-h}{ℒ}_{grav}}={\displaystyle \int {\text{d}}^{4}x\sqrt{-h}\frac{1}{2{\kappa }_4}R}$(12)

where ${\kappa }_4=\frac{8\pi G}{c^4}$, $h=det\left(h_{\mu \nu }\right)$, and $R=h^{\mu \nu }{R}_{\mu \nu }$ is the 4D Ricci scalar constructed from the induced metric $h_{\mu \nu }$.

This is the Einstein-Hilbert action, indicating that the Einstein Field Equations emerge self-consistently from the wave-induced deformation of the membrane. Thus, gravity has a concrete geometric origin within this model, without relying on extrinsic assumptions or additional fields

Deformation field action:

$S_{deform}={\displaystyle \int {\text{d}}^{4}x\sqrt{-h}{ℒ}_{slice}}$(13)

where:

The deformation Lagrangian density is expressed as Equation (5): The potential term $V\left(A\left(x^{\mu }\right)\right)$ encodes effective interactions or energy density associated with the deformation modes. The Ricci scalar R captures the intrinsic curvature of the 4D slice, which arises as a macroscopic geometric response to the accumulation and coherence of microscopic deformation waves propagating along the fifth spatial direction.

Unlike conventional brane-world models where fields are assumed to be confined by postulated mechanisms, in our framework the stability of gauge fields naturally emerges due to the structure of the deformation potential. The gauge field energy density is tightly correlated with the spatial gradient of the deformation potential:

$V_{gauge}=\lambda {\left(\frac{\partial V_{\Phi }}{\partial x^4}\right)}^2$(14)

where:

This formulation ensures that gauge fields are only stable in wave crest and trough regions, while in inter-wave zones, where $\frac{\partial V_{\Phi }}{\partial x^4}\ne 0$, the gauge fields decay, preventing the formation of localized matter.

The specific form of the brane potential $V_{\Phi }$ remains to be determined, with its parameters intended to be constrained by observational data.

In our model, the gauge field excitations acquire an effective mass that depends on the spatial structure of the brane’s deformation potential. Instead of reconstructing a full interaction Lagrangian between the gauge field $A_M$ and the deformation field Φ, we directly postulate an effective mass term based on physical intuition. Specifically, we assume that the squared effective mass is induced by the spatial gradient of the deformation potential $V_{brane}\left(\Phi \right)$ along the propagation direction $x^4$ [20] :

${ℳ}_{eff}^2\left(\Phi \right)=-\lambda {\left(\frac{\partial V_{\Phi }}{\partial x^4}\right)}^2$(15)

where:

This construction captures the core physical mechanism: at wave crests and troughs where the potential reaches extrema, the spatial gradient vanishes and gauge fields become massless, allowing for stable propagation. In contrast, in inter-wave regions where the deformation gradient is significant, the induced negative mass squared leads to tachyonic instabilities. Although this effective mass is not derived from a full five-dimensional action, it reflects a physically motivated structure commonly used in effective field theories and topological soliton models. This formulation allows us to analyze field stability without introducing additional coupling terms in the brane action.

Although the interaction does not appear as an explicit cross term in the Lagrangian, the gauge field mass arises directly from the deformation field profile via Equation (15). This establishes a geometry-induced coupling, where the brane deformation determines gauge field dynamics. Similar mechanisms appear in domain-wall localization and Kaluza-Klein [17] inspired models.

This implies that wave crest and trough regions maintain nearly massless gauge bosons, enabling standard model physics, whereas in inter-wave regions, the effective mass becomes nonzero, inhibiting matter formation.

For simplicity, we assume that all gauge fields share the same effective mass, as their stability is primarily determined by the geometric deformation of the brane. Future work will explore the impact of distinct effective masses for different gauge fields to account for their physical differences.

In this section, we explore how the previously introduced membrane deformation field (Φ) transitions into inflationary dynamics and subsequently leads to the formation of “Universe Factory”—mechanisms that continuously generate new observable universes through deformation wave ripples. This process is delineated through a series of detailed theoretical derivations and physical explanations. We delineate this process into three distinct evolutionary stages, analogous yet distinct from the standard cosmological paradigm: the inflation zone stage, the transition stage, and the membrane oscillation-dominated stage. These stages are driven by the interplay between external energy input and the intrinsic properties of the five-dimensional (5D) brane, offering a novel framework that aligns with key observational constraints, such as the Cosmic Microwave Background (CMB) power spectrum, while extending beyond the conventional ΛCDM model. We provide detailed theoretical derivations, physical explanations, and comparisons with standard cosmology to substantiate this model.

The initial stage of our model, termed the inflation zone stage, corresponds broadly to the radiation-dominated era of standard cosmology but encompasses a significantly extended inflationary period driven by high-energy Gaussian perturbations. These perturbations, arising from random energy impacts within the 6D bulk spacetime, inject a sudden influx of kinetic energy into the 5D brane, initiating a rapid deformation that propagates as a shock wave. The radial extent of the Gaussian perturbation is limited to $R_0={10}^{-12}$ meters and the temporal profile of the energy is modeled as:

$E_t=E_0{\text{e}}^{-\frac{t^2}{2{\tau }^2}}$(16)

where:

Such an extremely energetic perturbation inevitably generates a deformation shock wave. Assuming the medium has a strong restoring force (analogous to a stiff membrane) and the medium’s density $\rho$ has a wide linear response range to external force, we use the Hugoniot equation of linear shock propagation to describe the velocity of the shock front:

$V_s=\frac{\eta }{\eta -1}{u}_{\Phi }$(17)

where:

If the radius of the Gaussian energy disturbance is $R_0$, according to the above formula, the radius of the shock wave caused by it is $\frac{\eta R_0}{\eta -1}$, and the volume of the sphere in the four-dimensional space on the membrane is $\frac{{\pi }^2}{2}{\left(\frac{\eta R_0}{\eta -1}\right)}^4$, Due to the high rigidity of the membrane, it can be assumed that $\rho$ is approximately a constant, and applying Newton’s second law and energy loss formula $-\text{d}E=F\text{d}x$, one arrives at:

$F=m{\stackrel{\dot{}}{u}}_{\Phi }=\rho \left(\frac{{\pi }^2}{2}{\left(\frac{\eta R_0}{\eta -1}\right)}^4\right){\stackrel{\dot{}}{u}}_{\Phi }=-\frac{\frac{\text{d}{E}_t}{\text{d}t}}{\frac{\text{d}x}{\text{d}t}}=-\frac{\frac{\text{d}{E}_t}{\text{d}t}}{u_{\Phi }}$(18)

${\stackrel{\dot{}}{u}}_{\Phi }{u}_{\Phi }=-\frac{2\frac{\text{d}{E}_t}{\text{d}t}}{\rho {\pi }^2{\left(\frac{\eta R_0}{\eta -1}\right)}^4}=\frac{\text{d}{u}_{\Phi }^2}{\text{d}t}$(19)

$u_{\Phi }^2=\frac{4}{\rho {\pi }^2{\left(\frac{\eta R_0}{\eta -1}\right)}^4}\left(E_0-E_t\right)$(20)

$u_{\Phi }={\left(\frac{4}{\rho {\pi }^2{\left(\frac{\eta R_0}{\eta -1}\right)}^4}\left(E_0-E_t\right)\right)}^{\frac{1}{2}}=\frac{2}{\pi {\left(\frac{\eta R_0}{\eta -1}\right)}^2}{\rho }^{-\frac{1}{2}}{\left(E_0-E_t\right)}^{\frac{1}{2}}$(21)

$V_s=\frac{\eta }{\eta -1}{u}_{\Phi }=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\rho }^{-\frac{1}{2}}{\left(E_0-E_t\right)}^{\frac{1}{2}}=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}$(22)

Assuming the radial scale of the inflation zone is $R\left(t\right)$, then:

$\stackrel{\dot{}}{R}\left(t\right)=V_s=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}$(23)

$\ddot{R}\left(t\right)=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}\frac{t{\text{e}}^{-\frac{t^2}{2{\tau }^2}}}{{\tau }^2{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}}$(24)

$R\left(t\right)={\displaystyle {\int }_0^t\stackrel{\dot{}}{R}\left(t^{\prime }\right)\text{d}{t}^{\prime }}=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}{\displaystyle {\int }_0^t{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}\text{d}{t}^{\prime }}$(25)

$\begin{array}{l}ϵ\left(t\right)=1-\frac{\ddot{R}\left(t\right)R\left(t\right)}{\stackrel{\dot{}}{R}{\left(t\right)}^2}\\ =1-\frac{\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}\frac{t{\text{e}}^{-\frac{t^2}{2{\tau }^2}}}{{\tau }^2{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}}}{{\left(\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}\right)}^2}\cdot \frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}{\displaystyle {\int }_0^t{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}\text{d}{t}^{\prime }}\\ =1-\frac{t{\text{e}}^{-\frac{t^2}{2{\tau }^2}}}{{\tau }^2{\left(1-{\text{e}}^{-\frac{t^2}{2{\tau }^2}}\right)}^{\frac{3}{2}}}\cdot {\displaystyle {\int }_0^t{\left(1-{\text{e}}^{-\frac{{t^{\prime }}^2}{2{\tau }^2}}\right)}^{\frac{1}{2}}\text{d}{t}^{\prime }}\end{array}$(26)

where:

Let $C=\frac{\eta -1}{\eta }\frac{2}{\pi R_0^2}{\left(\frac{E_0}{\rho }\right)}^{\frac{1}{2}}$ equal 2.05 × 10⁴⁷, and let the characteristic time scale be $\tau =1.78\times {10}^{-32}$ seconds, according to the calculation (The Python calculation and plotting code is provided in Appendix B): within the interval 0 - 10⁻³² seconds, the function $R\left(t\right)$ increases from 10⁻¹² meters to approximately 3.99 × 10¹⁴ meters. The expansion factor of this inflationary region is 3.99 × 10²⁶. This matches the inflationary expansion factor generally assumed by standard cosmology. During most of this inflationary period, the parameter $ϵ\left(t\right)\ll 1$, which satisfies the slow-roll condition. And it is a single-field inflation, the energy is also high. Therefore, the scalar spectral index: $n_s\left(t\right)\approx 1-2ϵ\left(t\right)$ [23] , remains close to 1 throughout this phase. The average value of $n_s\left(t\right)$ is approximately 0.9607, aligning closely with the scalar spectral index inferred from cosmic microwave background observations.

As the shock wave expands, the energy of the Gaussian perturbation is rapidly depleted. The expansion velocity $\stackrel{\dot{}}{R}\left(t\right)$ and acceleration $\ddot{R}\left(t\right)$ decline sharply. This causes the slow-roll parameter $ϵ\left(t\right)$ to increase rapidly at the end of inflation, leading to a strong negative drift in the scalar spectral index $n_s\left(t\right)$, which can approach −1 after inflation concludes.

If the total perturbation energy is taken as $E_0=1.86\times {10}^{82}$ joules, this exceeds the total energy of the observable universe by a factor of approximately 6.6 × 10¹⁰, this is sufficient to maintain the mass-energy density of the entire 5D propagation region of the deformation wave and generate several peak and trough universes therein. Despite the massive total energy, the peak energy density still remains several orders of magnitude below the Planck limit, so energy conservation is not violated during inflation.

This model thus offers an inflationary scenario in which energy originates from a Gaussian perturbation rather than from a vacuum expectation value. The model represents a “hot inflation” in which no reheating phase is needed. That is, radiation and relativistic particles exist throughout the inflationary process. However, before the peaks and troughs are generated, due to the instability of the gauge field, these radiation and relativistic particles must be quickly stabilized in the stable gauge field of the 4D slice space-time region of the peaks and troughs that appear later, and then they can combine to form protons and neutrons after gaining mass. When the energy density is further reduced, nuclear synthesis also occurs synchronously in this region. More detailed quantitative research will be carried out in the next step.

The inflation zone stage consists of two subphases:

This extended inflation ensures the isotropy observed in our universe, a 4D slice along the tangential direction of the 5D deformation wave. The prolonged expansion smooths out tangential variations, consistent with the CMB’s high degree of uniformity ( $\frac{\text{Δ}T}{T}~{10}^{-5}$) [7] . The standard inflationary phase is thus a small subset of this stage, occurring during the Gaussian perturbation phase, while the subsequent external energy-driven expansion amplifies the spatial and temporal scales of the “universe factory”.

The second stage, the transition stage, corresponds to the matter-dominated era in standard cosmology, where the universe’s expansion decelerates ( $a~t^{\frac{2}{3}}$) due to the dominance of matter density ( ${\rho }_m\propto a^{-3}$) [10] . In our model, this stage marks the gradual conversion of the shock wave energy residual from the inflation zone into the membrane’s intrinsic oscillation energy. The wave speed remains faster than the membrane’s natural propagation speed ( $V_T>C_{brane}$), but it decelerates as the shock wave energy dissipates:

$V_T\left(t\right)=\sqrt{\frac{T+E_{s0}{\text{e}}^{-\gamma t}-V_{gravity}\left(t\right)}{\rho }}$(27)

where:

The deformation wave equation governs this transition:

$-\frac{1}{\sqrt{-g}}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\Phi \right)+\frac{\partial V_{\Phi }}{\partial \Phi }=0$(28)

with the shock wave energy term $E_{S0}{\text{e}}^{-\gamma t}$ diminishing over time.

During this stage, the wave crests and troughs—representing matter and antimatter-dominated universes—continue to form and detach from the inflation zone, while the expansion rate slows. This mirrors the standard model’s matter dominated phase, where gravitational clustering begins, but in our framework, the deceleration reflects the waning influence of external energy rather than a shift in energy density components. The transition stage ends when the external energy is fully converted, and the membrane’s intrinsic oscillations take over.

The third stage, the membrane oscillation-dominated stage, aligns with the dark energy-dominated era of standard cosmology, characterized by accelerated expansion ( $a\propto {\text{e}}^{Ht}$) driven by a cosmological constant or dark energy ( ${\Omega }_{\Lambda }=0.7$) [7] . In our model, this stage begins when the membrane’s intrinsic oscillation energy dominates, with the wave speed $V_m\left(t\right)$ governed by:

$V_m\left(t\right)=\sqrt{\frac{T-V_{gravity}\left(t\right)}{\rho }}$(29)

where is the gravitational potential induced by the deformation’s energy-momentum tensor. Initially, $V_m\left(t\right)<C_{brane}$, due to gravitational binding from the matter density within wave crests and troughs. As the deformation wave propagates and the spatial extent increases, the matter density dilutes ( ${\rho }_m\propto a^{-3}$), reducing $V_{gravity}$, and the wave speed accelerates, approaching:

$C_{brane}=\sqrt{\frac{T}{\rho }}$(30)

So in the third stage, the accelerated expansion typically attributed to dark energy in standard cosmology arises from the dynamics of the deformation wave Φ along the 5D brane. The wave’s intrinsic speed $C_{brane}$ representing its propagation across the entire 5D spacetime (4 spatial dimensions), is a stable property determined by the brane’s bulk modulus $T$ (units: J∙m⁻⁴) and brane bulk density $\rho$ (units: J∙m⁻⁶∙s⁻²). In natural units (c = 1), $T$ units: GeV⁵ and $\rho$ units: GeV⁵.

The brane bulk modulus $T$ and brane bulk density $\rho$ are fundamental parameters of the 5D brane, determining the wave speed $C_{brane}$ that reflects its elastic dynamics.

When gravitational potential energy becomes negligible, the wave front propagation distance in our model is defined as

$R\left(t\right)=R_{n0}+C_{brane}\left(t-t_{n0}\right)$(31)

where:

The Hubble parameter is:

$H=\frac{\stackrel{\dot{}}{a}}{a}=\frac{{\partial }_{t}R\left(t\right)}{R\left(t\right)}$(32)

This reflects the synchronous expansion of wave crests and troughs with the wavefront distance R, aligning with the expansion behavior of the 4D cosmological scale factor $a\left(t\right)$.

In the standard ΛCDM model, when gravitational potential energy is negligible (i.e., matter and radiation densities approach zero), the Hubble parameter is dominated by dark energy, satisfying:

$H^2=\frac{\Lambda c^2}{3}$(33)

where Λ is the cosmological constant and c is the speed of light. The dark energy density ${\rho }_{DE}$ relates to Λ via ${\rho }_{DE}=\frac{\Lambda c^2}{8\pi G}$. To connect our model with standard cosmology, we assume the initial moment ( $t=t_{n0}$) when gravitational potential energy becomes negligible:

$R\left(t_{n0}\right)=R_{n0}$(34)

${{\partial }_{t}R\left(t\right)|}_{t=t_{n0}}=C_{brane}$(35)

$H=\frac{{\partial }_{t}R\left(t\right)}{R\left(t\right)}=\frac{C_{brane}}{R_0}$(36)

Comparing this with the ΛCDM expression for the dark energy-dominated phase:

$H^2=\frac{\Lambda c^2}{3}=\frac{C_{brane}^2}{R_0^2}$(37)

Solving for Λ:

$\Lambda =\frac{3C_{brane}^2}{c^2{R}_{n0}^2}=\frac{3T}{c^2{R}_{n0}^2\rho }$(38)

${\rho }_{DE}=\frac{\Lambda c^2}{8\pi G}=\frac{3T}{8\pi G{R}_{n0}^2\rho }$(39)

These expressions indicate that both Λ and ${\rho }_{DE}$ depend on the brane’s bulk modulus $T$, brane bulk density $\rho$ and the initial wave front distance when gravitational potential energy becomes negligible $R_{n0}$.

In future work, their values will be constrained to align the projected 4D density with the observed dark energy density ${\rho }_{DE}\approx 7\times {10}^{-27}\text{kg}/{\text{m}}^{\text{3}}$, yielding a pressure $P_{DE}={\rho }_{DE}$ consistent with the equation of state [7] . This framework ties the stable, long-term expansion of our universe to the 5D brane’s intrinsic properties, distinct from short-lived phases, offering a physical basis for dark energy without ad hoc fields.

Our three-stage framework-spanning the inflation zone, transition, and membrane oscillation-dominated stages-offers a coherent alternative to standard cosmology. It aligns with CMB observations, producing a near-scale-invariant spectrum ( $n_s\approx 0.96$) through an inflationary mechanism, while providing a unified explanation for universe formation, deceleration, late-time acceleration and dark energy mechanism within the deformation wave paradigm [14] [24] .

In this section, we rigorously derive the stability conditions for gauge fields localized at the wave crests and troughs of the membrane deformation field Φ. We demonstrate that gauge fields are stable only in regions where the deformation field’s potential energy $V_{brane}\left(\Phi \right)$ reaches extrema (absolute maxima) and the deformation field’s kinetic energy vanishes. Furthermore, wave crests are interpreted as matter-dominated universes, wave troughs as antimatter-dominated universes, and their annihilation is explained geometrically as the cancellation of deformation “knots.”

Gauge Field Settings:

The gauge fields are set as [20] :

$U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)\times U\left(1\right)\text{'}\times B_{\mu \nu }\times C_{d\nu \rho }$(40)

where:

The additional gauge fields are induced by the geometric deformation of the brane:

Total System Action:

$\begin{array}{l}S={\displaystyle \int {\text{d}}^{5}x}\sqrt{-g_{MN}}(-\frac{1}{4}{F}_{MN}{F}^{MN}+\frac{1}{2}{ℳ}_{eff}^2\left(\Phi \right)A_M{A}^M+\frac{1}{2}{\partial }_M\Phi {\partial }^M\Phi \\ -V\left(\Phi \right)-\frac{1}{4}{X}_{MN}{X}^{MN}-\frac{1}{12}{H}_{MNP}{H}^{MNP}-\frac{1}{48}{G}_{MNPQ}{G}^{MNPQ})\end{array}$(41)

where:

Effective Mass Definition as Equation (15):

The negative sign is chosen to induce tachyonic instability in dark sectors enabling exponential growth of perturbations that lead to energy storms, while ensuring stability at crests and troughs.

Gauge Field Potential Function:

The gauge field potential function is:

$V_{gauge}=\lambda {\left(\frac{\partial V_{\Phi }}{\partial x^4}\right)}^2$(42)

which forms a potential well at the wave crests and troughs ( $\frac{\partial V_{\Phi }}{\partial x^4}=0$).

Field Equations:

Gauge Field Equation:

$\frac{1}{\sqrt{-g}}{\partial }_N\left(\sqrt{-g}{F}^{MN}\right)+\frac{1}{2}{ℳ}_{eff}^2\left(\Phi \right)A^M=0$(43)

Deformation Field Equation:

$\frac{1}{\sqrt{-g}}{\partial }_N\left(\sqrt{-g}{F}^{MN}\right)-\frac{\partial V_{brane}}{\partial \Phi }+\frac{1}{2}\frac{\partial {ℳ}_{eff}^2\left(\Phi \right)}{\partial \Phi }{A}_M{A}^M=0$(44)

To derive the behavior of gauge field perturbations in the wave crest and trough regions, we begin with the linearized field equation. Since the deformation potential reaches an extremum in these regions, its spatial gradient vanishes, and thus:

${ℳ}_{eff}^2\left(\Phi \right)=-\lambda {\left(\frac{\partial V_{\varphi }}{\partial x^4}\right)}^2=0$(45)

The gauge field satisfies the free-field equation:

${\nabla }^N{F}_{NM}=0$(46)

Under the Lorenz gauge condition ${\nabla }^M{A}_M=0$, we substitute $F_{NM}={\partial }_N{A}_M-{\partial }_M{A}_N$ and obtain:

$\square A_M=0$(47)

We now solve this wave equation. In locally flat coordinates ( $t$, $x^1$, $x^2$, $x^3$, $x^4$), the d’Alembertian becomes:

$\square A_M={\eta }^{PQ}{\partial }_P{\partial }_Q{A}_M=-{\partial }_t^2{A}_M+\underset{i=1}{\overset{4}{{\displaystyle \sum }}}{\partial }_i^2{A}_M$(48)

We assume a separable plane-wave solution:

$A_M\left(x\right)={ϵ}_M{\text{e}}^{i{k}_N{x}^N}$(49)

Substituting into the wave equation gives:

$A_M=k_N{x}^N{ϵ}_M{\text{e}}^{i{k}_N{x}^N}=0\Rightarrow k_N{x}^N=0$(50)

Thus, the solution describes a massless plane wave satisfying:

$A_M\left(x\right)={ϵ}_M{\text{e}}^{i{k}_N{x}^N},k_N{x}^N=0$(51)

This solution is oscillatory and does not grow with time, indicating the field is linearly stable in wave crest and trough regions.

Energy Analysis:

Perturbation energy density:

$\epsilon =\frac{1}{2}\left[{\left|{\partial }_t\delta A_M\right|}^2+{\left|\nabla \delta A_M\right|}^2\right]$(52)

Due to energy conservation, the system is stable against perturbations. The law of energy conservation ensures that even in the presence of perturbations in the wave crest and trough regions, the total energy of the system does not change without limit, thus maintaining the stability of the gauge field.

Effective Mass Analysis:

In regions far from wave crests and troughs, the deformation potential varies significantly, and the induced effective mass becomes negative:

$\left|\frac{\partial V_{brane}}{\partial x^4}\right|\gg 0$(53)

${ℳ}_{eff}^2\left(\Phi \right)\ll 0$(54)

The linearized equation becomes:

$\square \delta A_M=M_{eff}^2\left(\Phi \right)\delta A_M$(55)

Define ${\mu }^2\equiv -M_{eff}^2\left(\Phi \right)>0$, then:

$\square \delta A_M=-{\mu }^2\delta A_M$(56)

In locally flat coordinates, this reads:

$-{\partial }_t^2\delta A_M+{\nabla }^2\delta A_M=-{\mu }^2\delta A_M\Rightarrow {\partial }_t^2\delta A_M={\nabla }^2\delta A_M+{\mu }^2\delta A_M$(57)

Assume a separable solution:

$A_M\left(x\right)={ϵ}_M{\text{e}}^{i{k}_N{x}^N}$(58)

We work in a locally flat coordinate system where $x^N=\left(t,x\right)$, and accordingly decompose the momentum as $k^N=\left(\omega ,k\right)$.

Substituting yields:

${\omega }^2={\mu }^2-{\left|k\right|}^2$(59)

Hence, for low spatial wave number ${\left|k\right|}^2<{\mu }^2$, we obtain exponential growth:

$\delta A_M\left(t\right)~{\text{e}}^{\gamma t},\gamma =\sqrt{{\mu }^2-{\left|k\right|}^2}$(60)

This means that in areas far from the wave crests and troughs, the effective mass of the gauge field becomes a large negative value, in sharp contrast to the zero effective mass at the wave crests and troughs. This large negative effective mass will lead to the instability of the system 20.

In the long-wavelength limit $\left|k\right|\to 0$, we have:

$\gamma \approx \sqrt{\left|M_{eff}^2\left(\Phi \right)\right|}$(61)

Due to large $\left|{ℳ}_{eff}^2\left(\Phi \right)\right|$, tachyonic modes grow rapidly. The exponential growth mode indicates that the perturbation will increase rapidly with time, destroying the stability of the system. The growth time scale $\tau ~1/\gamma$ is very short. This means that the perturbation will change significantly in a very short time, making it difficult for the system to maintain a stable state. It leads to rapid gauge field collapse. Eventually, the gauge field cannot stably exist in the region far from the wave crests and troughs.

Energy Storm:

Violent tachyonic modes lead to:

Effective Mass Asymptotic Behavior:

At distance $\delta x^4$ from wave crest/trough points:

${ℳ}_{eff}^2\left(\Phi \right)=-\lambda {\left({\frac{{\partial }^2{V}_{brane}}{\partial {\left(x^4\right)}^2}|}_{x_0^4}\right)}^2{\left(\delta x^4\right)}^2+O\left({\left(\delta x^4\right)}^3\right)$(62)

This equation gives the asymptotic expression of the effective mass of the gauge field in the region near the wave crest and trough. It shows that the effective mass is related to the distance from the wave crest and trough points and that the absolute value of the effective mass is relatively small in the nearby region.

Slow Region Barrier:

In the nearby regions:

Quantum Tunneling Effect Shielding:

Since $V_{gauge}=\lambda {\left(\frac{\partial V_{\varphi }}{\partial x^4}\right)}^2$ forms a deep potential well at the wave crests and troughs, the tunneling probability approaches zero, ensuring the stability of the gauge field.

For further analysis of the quantum tunneling probability [25] :

$P_{tunnel}\propto \exp \left(-\frac{2}{\hslash }{\displaystyle \int \sqrt{2m\left(E-V_{gauge}\right)}\text{d}x}\right)$(63)

Because the potential well formed by $V_{gauge}=\lambda {\left(\frac{\partial V_{\varphi }}{\partial x^4}\right)}^2$ is extremely deep at the wave crests and troughs, the tunneling probability decays exponentially and is almost zero, strictly confining the gauge field to the wave crest and trough regions.

Localization Proof:

The localization of the gauge field in the wave crest/trough region is protected because:

Topological Irrelevance:

Due to the slow region barrier:

Wave Crest and Trough Regions:

Distant Regions:

Nearby Regions:

The detailed mathematical description of stable deformation knots, including their quantization, wave function forms, and interactions, will be addressed in subsequent studies. Future work will focus on deriving the quantum field theory framework for these knots, their correspondence to gauge fields, and their potential observational signatures in the CMB or large-scale structure.

This section derives the effective gravitational field equations on in the 4D wave crest and trough slice spacetimes within our deformation wave model. In 2.3.4, we propose that gravity emerges directly from a physical mechanism rooted in the brane’s elastic dynamics and the expression of gravitational action is obtained. In the section, starting from the gravitational and the deformation wave’s action, which is a purely geometric construct, we employ variational principles to systematically derive the EFE. The resulting energy-momentum tensor arises solely from the deformation wave’s action, reflecting the brane’s intrinsic dynamics, and incorporates contributions from matter, dark matter, and dark energy. These equations align with cosmological observations, such as the CMB scalar spectral index ( $n_s\approx 0.96$) [7] , galactic dark matter ratios ( $\frac{{\rho }_{DM}}{{\rho }_b}\approx 5$) [10] , and dark energy density ( ${\Omega }_{\Lambda }\approx 0.7$) [11] , while providing a physical mechanism for spacetime curvature, addressing a fundamental gap in General Relativity (GR).

In 2.3.4, we obtain the Gravitational action in Equation (12) and total action in Equation (13).

In our deformation wave model, the gravitational action is mathematically and physically identical to the Einstein-Hilbert action $S_{EH}$ in General Relativity (GR):

$S_{EH}={\displaystyle \int {\text{d}}^{4}x\sqrt{-g}\frac{1}{2{\kappa }_4}R}$(64)

with $h_{\mu \nu }$ replacing $g_{\mu \nu }$. As established in standard GR, the variation of this action with respect to the metric $h_{\mu \nu }$, incorporating the Gibbons-Hawking-York (GHY) boundary term to ensure a well-posed variational principle, yields:

$\begin{array}{c}\delta S_{grav}={\displaystyle \int \frac{1}{2{\kappa }_4}\left(R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R\right)\delta h^{\mu \nu }\sqrt{-h}{\text{d}}^{4}x}\\ ={\displaystyle \int \frac{c^4}{16\pi G}\left(R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R\right)\delta h^{\mu \nu }\sqrt{-h}{\text{d}}^{4}x}\end{array}$(65)

where the boundary term vanishes under standard boundary conditions (e.g. $h_{\mu \nu }\to 0$ at spatial infinity). Given the equivalence to the Einstein model, we adopt this well-known result directly.

From 2.3.4, deformation field action is in Equation (13):

Define the deformation field’s energy-momentum tensor:

$T_{\mu \nu }=-\frac{2}{\sqrt{-h}}\frac{\delta \left(\sqrt{-h}{ℒ}_{slice}\right)}{\delta h^{\mu \nu }}$(66)

So:

$\begin{array}{c}\delta S_{slice}=\delta S_{grav}+\delta S_{deform}\\ ={\displaystyle \int \frac{c^4}{16\pi G}\left(R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R\right)\delta h^{\mu \nu }\sqrt{-h}{\text{d}}^{4}x}+{\displaystyle \int \frac{\delta \left(\sqrt{-h}{ℒ}_{slice}\right)}{\delta h^{\mu \nu }}\delta h^{\mu \nu }{\text{d}}^{4}x}\\ ={\displaystyle \int \left[\frac{c^4}{16\pi G}\left(R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R\right)-\frac{1}{2}{T}_{\mu \nu }\right]\delta h^{\mu \nu }\sqrt{-h}{\text{d}}^{4}x}=0\end{array}$(67)

Since $\delta h^{\mu \nu }$ is arbitrary, must:

$\frac{c^4}{16\pi G}\left(R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R\right)-\frac{1}{2}{T}_{\mu \nu }=0$(68)

$R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R=\frac{8\pi G}{c^4}{T}_{\mu \nu }$(69)

Therefore, EFE has been derived.

In Section 3.1.3, we established that dark energy in our deformation wave model arises as an intrinsic property of the 5D brane, driven by the membrane’s wave speed $C_{brane}=\sqrt{\frac{T}{\rho }}$, where T is the brane’s bulk modulus (units: J∙m⁻⁴) and $\rho$ is the brane bulk density (units: J∙m⁻⁶∙s⁻²). This wave speed governs the long-term accelerated expansion of the 4D slice spacetimes during the membrane oscillation-dominated stage, analogous to the dark energy-dominated era in standard cosmology. We derived the dark energy density in Equation (39) and the corresponding cosmological constant-like term in Equation (38). This formulation provides a geometric origin for dark energy, eliminating the need for ad hoc scalar fields or vacuum energy assumptions, as required in the standard ΛCDM model. In this section, we incorporate this dark energy term into the effective gravitational field equations derived in Section 3.3.1, ensuring consistency with both the model’s geometric foundation and cosmological observations, such as the dark energy fraction ${\Omega }_{\Lambda }\approx 0.7$.

Incorporation via the Gravitational Action:

To include dark energy in the effective field equations, we modify the gravitational action $S_{grav}$ in the 4D slice to account for the cosmological constant-like term Λ. In standard General Relativity (GR), dark energy is introduced by adding a cosmological constant term to the Einstein-Hilbert action. Following this approach, we modify the gravitational action from Section 2.3.4 in Equation (12) to include the dark energy contribution:

$S_{grav}={\displaystyle \int {\text{d}}^{4}x\sqrt{-h}\frac{1}{2{\kappa }_4}\left(R-2\Lambda \right)}$(70)

Here, the factor $-2\Lambda$ is chosen to ensure that the resulting field equations align with the standard form in GR, where $\Lambda$ appears as a cosmological constant. The total action for the 4D slice $S_{slice}$ remains in Equation (11) with $S_{deform}$ as defined in Section 2.3.4. in Equation (13). We follow the same derivation process as in 3.3.1 to obtain.

$R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R+\Lambda h_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R+\frac{3T}{c^2{R}_{n0}^2\rho }{h}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$(71)

We definite:

$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }R$(72)

Then:

$G_{\mu \nu }+\Lambda h_{\mu \nu }=G_{\mu \nu }+\frac{3T}{c^2{R}_{n0}^2\rho }{h}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$(73)

This is the modified Einstein Field Equation (EFE) in the 4D slice, now incorporating the dark energy term $\Lambda h_{\mu \nu }$, which matches the standard form in GR with a cosmological constant.

Physical Implications and Consistency with Observations:

The inclusion of the dark energy term $\Lambda h_{\mu \nu }$ in the EFE ensures that the 4D slice spacetimes exhibit accelerated expansion during the membrane oscillation-dominated stage, as described in Section 3.1.3. The Hubble parameter in this phase, given in Equation (36), matches the standard $H^2=\Lambda c^2/3$, as shown in Section 3.1.3, confirming that our model reproduces the observed late-time acceleration. Using the observed dark energy density ${\rho }_{DE}\approx 7\times {10}^{-27}$ kg/m³, we can constrain the model parameters $T,\rho$ and $R_{n0}$:

${\rho }_{DE}=\frac{3T}{8\pi G{R}_{n0}^2\rho }=7\times {10}^{-27}\text{kg}/{\text{m}}^3$(74)

Given $G\approx 6.674\times {10}^{-11}{\text{m}}^3\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}$, this implies a relationship between the brane’s bulk modulus, bulk density, and the scale $R_{n0}$, which can be further refined with observational data in future work. The dark energy fraction ${\Omega }_{\Lambda }\approx 0.7$ is naturally achieved by this density, aligning with cosmological observations.

In our model, dark energy is not an ad hoc addition but a direct consequence of the brane’s intrinsic dynamics. The wave speed $C_{brane}$ acts as a fundamental parameter, tying the accelerated expansion to the elastic properties of the 5D brane, offering a geometric explanation for dark energy that contrasts with the vacuum energy or scalar field approaches in standard cosmology.

Impact on the Deformation Wave Dynamics:

The dark energy term $\Lambda h_{\mu \nu }$ introduces a uniform negative pressure across the 4D slice, influencing the propagation of the deformation wave Φ. Specifically, this negative pressure arises from the intrinsic wave speed $C_{brane}$, which drives the expansion of the spatial extent of wave crests and troughs as the deformation wave propagates, mirroring the cosmic expansion observed in our universe. During the membrane oscillation-dominated stage, this term enhances the expansion of the wave crests and troughs, ensuring that matter-dominated universes (crests) and antimatter-dominated universes (troughs) continue to separate and evolve independently.

Comparison with Standard Cosmology:

In the standard ΛCDM model, dark energy is introduced as a cosmological constant Λ, with its origin often attributed to quantum vacuum fluctuations—a hypothesis that faces challenges such as the cosmological constant problem (the discrepancy between theoretical and observed values of Λ). In contrast, our model derives Λ from the brane’s intrinsic properties, providing a physical mechanism for dark energy that avoids fine-tuning issues. The resulting field equations are mathematically equivalent to those in GR with a cosmological constant, ensuring consistency with observational tests of GR, such as the accelerated expansion inferred from Type Ia supernovae and the CMB power spectrum.

Future Directions:

While the current derivation successfully incorporates dark energy into the EFE, further work is needed to:

In summary, the incorporation of dark energy into the effective gravitational field equations provides a unified geometric explanation for late-time cosmic acceleration, aligning with observational data while grounding dark energy in the intrinsic dynamics of the 5D brane.

In this section, we propose a novel mechanism for the origin of dark matter, rooted in the geometric dynamics of the 5D brane. The aggregation of microscopic deformation “knots” within the 4D slice spacetimes at wave crests and troughs induces macroscopic spacetime curvature. This curvature propagates non-locally through the connectivity of the 5D brane, affecting surrounding regions, including the inter-wave dark sectors. Through a positive feedback mechanism, this process leads to the accumulation of large-scale spacetime curvature, whose gravitational effects we identify as those of dark matter. We demonstrate how this geometric framework naturally produces dark matter, consistent with observational constraints such as the dark matter-to-baryon density ratio ( $\frac{{\rho }_{DM}}{{\rho }_b}\approx 5$).

1) Aggregation of microscopic deformation knots and local spacetime curvature

Within the 4D slice spacetimes at wave crests and troughs (corresponding to matter- or antimatter-dominated universes, as discussed in Section 3.2), the microscopic deformation “knots” represent localized quantum excitations of gauge fields (e.g., photons, gluons; see Section 2.4). The aggregation of these knots within a 4D slice contributes directly to the Einstein Field Equations (EFE) through their energy-momentum tensor $T_{\mu \nu }^{knots}$, inducing local spacetime curvature:

$G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }^{knots}$(75)

where $T_{\mu \nu }^{knots}$ is the energy-momentum tensor of the knots.

When a large number of knots aggregate, their total energy-momentum tensor is $T_{\mu \nu }^{total}={\displaystyle \int T_{\mu \nu }^{knot}}$, producing a local contribution to the Einstein tensor in the m-th slice:

$G_{\mu \nu }^{local,\left(m\right)}=\frac{8\pi G}{c^4}{\displaystyle \sum T_{\mu \nu }^{knots,\left(m\right)}}$(76)

In the non-relativistic approximation, the dominant component of $T_{\mu \nu }^{knots,\left(m\right)}$ is the energy density, ${\displaystyle \sum T_{00}^{knots,\left(m\right)}}\approx {\rho }_{knots,\left(m\right)}{c}^2$, where ${\rho }_{knots,\left(m\right)}$ is the observable mass-energy density of the aggregated knots in the m-th slice. Thus, the corresponding component of the Einstein tensor is:

$G_{00}^{local,\left(m\right)}=\frac{8\pi G}{c^2}{\rho }_{knots,\left(m\right)}$(77)

This local curvature, quantified by $G_{\mu \nu }^{local,\left(m\right)}$ is typically significant on small scales (e.g., stellar scales), but within a single 4D slice, its effects are localized and cannot directly account for the large-scale gravitational effects attributed to dark matter.

2) Non-local propagation of spacetime curvature and positive feedback

Due to the connectivity of the 5D brane (Section 2.1), the macroscopic curvature in the 4D slice spacetimes at wave crests or troughs, quantified by $G_{\mu \nu }^{local,\left(m\right)}$, is not an isolated phenomenon, as microscopic deformation knots. Instead, it affects adjacent regions to cause the corresponding macroscopic curvature, propagates non-locally through the brane’s geometry, including the inter-wave dark sectors. Consequently, the curvature from the crest and troughs slice, initially described by $G_{\mu \nu }^{local,\left(m\right)}$, propagates through the brane to the dark sectors, inducing metric changes and resulting in spacetime curvature within these regions, $G_{\mu \nu }^{dark,\left(m\right)}$. The curvature in the dark sectors exerts a gravitational influence on deformation knots that have not reached extremal values (i.e., where $\partial \Phi /\partial x^4\ne 0$, as discussed in Section 3.2). This gravitational potential well attracts additional knots in the dark sectors to aggregate further, increasing the local energy-momentum tensor $T_{\mu \nu }^{DM,\left(m\right)}$ and thereby enhancing the curvature in the dark sectors and thus the Einstein tensor:

$G_{\mu \nu }^{DM,\left(m\right)}=\frac{8\pi G}{c^4}{T}_{\mu \nu }^{DM,\left(m\right)}$(78)

This process establishes a positive feedback loop:

This positive feedback extends the curvature deeper into the dark sectors and, through the brane’s geometry, propagates to subsequent wave crest and trough 4D slice spacetimes, altering their metric and inducing additional spacetime curvature. So the total Einstein tensor in the subsequent slice, (m + 1)-th slice, $G_{\mu \nu }^{total,\left(m+1\right)}$ is:

$G_{\mu \nu }^{total,\left(m+1\right)}=G_{\mu \nu }^{local,\left(m+1\right)}+G_{\mu \nu }^{DM,\left(m\right)}$(79)

3) Accumulation of curvature effects

The total Einstein tensor in the m-th slice, quantified by $G_{\mu \nu }^{total,\left(m\right)}$, propagates through the brane to subsequent crest and trough slices, leading to a cumulative effect. During propagation, the curvature experiences a net modification due to both the positive feedback in the dark sectors and the geometric decay during propagation across the brane. We denote the net modification factor as ${\beta }_m$ (where ${\beta }_m$ combines the amplification from positive feedback $f_{feedback}$ and the decay from propagation $f_{decayand}$, so ${\beta }_m\equiv f_{feedback}/f_{decay}{}_{and}$ can be greater or less than 1 depending on the relative strengths of these effects). So the curvature $G_{\mu \nu }^{DM,\left(m\right)}$ from the dark sector following the m slice is:

$G_{\mu \nu }^{DM,\left(m\right)}={\beta }_m{G}_{\mu \nu }^{total,\left(m\right)}$(80)

$\begin{array}{c}{G}_{\mu \nu }^{total,\left(m+1\right)}=G_{\mu \nu }^{local,\left(m+1\right)}+G_{\mu \nu }^{DM,\left(m\right)}=G^{local,\left(m+1\right)}+{\beta }_m{G}^{total,\left(m\right)}\\ =G^{local,\left(m+1\right)}+{\beta }_m{G}^{local,\left(m\right)}+{\beta }_m{\beta }_{m-1}{G}^{total,\left(m-1\right)}\\ =G^{local,\left(m+1\right)}+\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right)G^{local,\left(j\right)}\end{array}$(81)

$G_{\mu \nu }^{DM,\left(m\right)}=\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right)G^{local,\left(j\right)}$(82)

The accumulated spacetime curvature produces additional gravitational effects via the EFE, equivalent to an additional energy-momentum tensor $T_{\mu \nu }^{DM}$, This accumulation process is analogous to the non-linear growth of large-scale structures in cosmology (e.g., the Zeldovich approximation), but here it is realized through the geometric propagation across the 5D brane.

From Equation (77), we obtain:

$\begin{array}{c}{G}_{00}^{total,\left(m+1\right)}=G_{00}^{local,\left(m+1\right)}+\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right)G_{00}^{local,\left(j\right)}\\ =\frac{8\pi G}{c^2}{\rho }_{knots,\left(m+1\right)}+\frac{8\pi G}{c^2}\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{knots,\left(j\right)}\end{array}$(83)

We definite ${\rho }_{total,\left(m+1\right)}$ as total apparent mass-energy density that:

$G_{00}^{total,\left(m+1\right)}=\frac{8\pi G}{c^2}{\rho }_{total,\left(m+1\right)}$(84)

So:

$G_{00}^{total,\left(m+1\right)}=\frac{8\pi G}{c^2}{\rho }_{total,\left(m+1\right)}=\frac{8\pi G}{c^2}{\rho }_{knots,\left(m+1\right)}+\frac{8\pi G}{c^2}\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{knots,\left(j\right)}$(85)

We obtain:

${\rho }_{total,\left(m+1\right)}={\rho }_{knots,\left(m+1\right)}+\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{knots,\left(j\right)}$(86)

In our model, the baryon density ${\rho }_b\approx {\rho }_{knots,\left(m\right)}$, so:

$G_{00}^{DM,\left(m\right)}=\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right)G_{00}^{local,\left(j\right)}=\frac{8\pi G}{c^2}\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{b,\left(j\right)}$(87)

${\rho }_{total,\left(m+1\right)}\approx {\rho }_{b,\left(m+1\right)}+\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{b,\left(j\right)}$(88)

For dark matter density ${\rho }_{DM,\left(m+1\right)}$, we have:

${\rho }_{total,\left(m+1\right)}\approx {\rho }_{b,\left(m+1\right)}+{\rho }_{DM,\left(m+1\right)}={\rho }_{b,\left(m+1\right)}+\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{b,\left(j\right)}$(89)

${\rho }_{DM,\left(m+1\right)}=\underset{j=1}{\overset{m}{{\displaystyle \sum }}}\left(\underset{i=j}{\overset{m}{{\displaystyle \prod }}}{\beta }_i\right){\rho }_{b,\left(j\right)}$(90)

$\frac{{\rho }_{DM,\left(m+1\right)}}{{\rho }_{b,\left(m+1\right)}}=\frac{{{\displaystyle \sum }}_{j=1}^m\left({{\displaystyle \prod }}_{i=j}^m{\beta }_i\right){\rho }_{b,\left(j\right)}}{{\rho }_{b,\left(m+1\right)}}$(91)

With appropriate values of m (the number of predecessor universes), ${\beta }_i$ (the net modification factor of i-th universe) and ${\rho }_{b,\left(j\right)}$ (the baryon density in j-th universe), this ratio can naturally align with the observed value of $\frac{{\rho }_{DM}}{{\rho }_b}\approx 5$ without requiring precise tuning, thus maintaining a natural and physically motivated

framework. More detailed quantitative studies will be carried out in subsequent work.

In our model, dark matter arises not from unknown particles within our current universe, but as a cumulative geometric memory of preceding universes—wave crests and troughs that formed earlier along the deformation wave. Each former universe, indexed by $j=1,2,\cdots ,m$, contributes residual curvature traces ${\rho }_b^{\left(j\right)}$, corresponding to its baryonic matter density. These residual effects are geometrically projected onto the current wave crest (our universe), but each is modified by a factor ${\beta }_i$, which accounts for curvature dispersion and the damping effect of inter-wave energy storms. The dark matter density equation defined previously in our universe encodes the hierarchical inheritance of geometric deformation traces, A higher number of prior universes (larger m) naturally leads to more accumulated dark matter, while geometric screening suppresses short-wavelength curvature features, resulting in a smooth, non-clustering dark component consistent with observations. The existence of the gravitational effect of dark matter indicates that there were several predecessor universes before our universe.

4) Scale-dependent effects of energy storms

In the inter-wave dark sectors, tachyonic instabilities of the gauge fields (Section 3.2.3) lead to a negative effective mass, ${ℳ}_{eff}^2\left(\Phi \right)=-\lambda {\left(\frac{\partial V_{brane}}{\partial x^4}\right)}^2$, causing exponential growth of perturbations in Equations (60) and (61).

This rapid growth of tachyonic modes triggers violent local energy fluctuations, termed “energy storms.” These energy storms result in:

Specifically, energy storms disrupt the spacetime curvature induced by knot aggregation on small scales (e.g., stellar scales). For instance, on stellar scales, the gravitational potential from knot aggregation is ${\Phi }_{grav}~-\frac{GM}{r}$, but the violent fluctuations from energy storms prevent these small-scale structures from remaining stable, effectively smoothing out the spacetime curvature.

However, on large scales (e.g., galactic scales), the impact of energy storms is limited. The growth timescale of energy storms, $\tau ~1/\gamma$, is short (Section 3.2.3), confining their effects to localized regions. Consequently, large-scale curvature can accumulate before being disrupted by energy storms, resulting in persistent gravitational effects. This scale-dependent behavior aligns with cosmological observations of structure formation: dark matter effects are negligible on small scales (e.g., stellar scales), but dominate on large scales (e.g., galactic scales), as evidenced by galaxy rotation curves.

5) Further consistency with observations

The gravitational effects of accumulated curvature in the dark sectors, quantified by $G_{\mu \nu }^{DM}$, influence the geometry of the 4D slice, enhancing large-scale curvature in a manner consistent with CMB density fluctuations ( $n_s\approx 0.96$, Section 3.1). Additionally, this mechanism is compatible with the holographic principle: geometric changes on the 5D brane project onto the 4D slice, producing additional gravitational effects, akin to non-local gravitational propagation in the AdS/CFT [26] correspondence.

Compared to the standard cold dark matter (CDM) model in cosmology, our framework treats dark matter as a geometric effect rather than a particle species. This geometric origin circumvents challenges in the CDM model, such as the lack of significant signals in direct detection experiments for dark matter particles (e.g., WIMPs). Furthermore, the dynamics of the brane provide a testable framework: the distribution of accumulated curvature may leave observable imprints in CMB anisotropies or large-scale structure, which can be probed by next-generation telescopes such as Euclid and LSST.

6) Comparison with ΛCDM and DGP Models

Compared to the ΛCDM model, which treats dark matter as a cold, non-relativistic particle species and dark energy as a vacuum energy component, our geometric framework provides a unified origin from spacetime deformation [27] . The scalar spectral index $n_s\approx 0.96$, dark matter-to-baryon ratio $\frac{{\rho }_{DM}}{{\rho }_b}\approx 5$, and dark energy density ${\Omega }_{\Lambda }\approx 0.7$ emerge naturally from geometric quantities such as the brane wave speed $C_{brane}$, bulk modulus T, and bulk density $\rho$.

Unlike the Dvali-Gabadadze-Porrati (DGP) model, which modifies gravity via a 4D brane embedded in a 5D Minkowski bulk with a crossover scale $r_c$, our model retains General Relativity on the 4D slice and derives its Einstein equations from brane elasticity. There is no need for ghost-free constraints or infrared modifications of gravity.

This model circumvents key limitations of ΛCDM and DGP by providing physically motivated origins for all cosmic components—gravity, dark matter, and dark energy—within a single geometric brane structure.

7) Future directions

While the current model qualitatively aligns with the observational characteristics of dark matter, further quantitative analysis is required:

8) Summary

This section proposes a geometric origin for dark matter’s gravitational effects through the aggregation of microscopic deformation knots in 4D slice spacetimes at wave crests and troughs, the non-local propagation of spacetime curvature, the scale-dependent effects of energy storms. The spacetime curvature, quantified by the Einstein tensor $G_{\mu \nu }^{DM}$, accumulates through propagation and feedback in the

dark sectors, producing gravitational effects equivalent to dark matter. The resulting density ratio, ${\rho }_{DM}/{\rho }_b$, can naturally align with observations ( $\frac{{\rho }_{DM}}{{\rho }_b}\approx 5$) for appropriate values of the number of predecessor universes, the net modification factor and the baryon density in predecessor universe. This mechanism not only provides a unified geometric framework for explaining the origin of dark matter but also aligns with CMB observations and the holographic principle, offering testable predictions for future observations.