The Role of Energy Density Diffusion in Galactic Dynamics and Cosmic Expansion: A Unified Theory for MOND and Dark Energy

1. Introduction

The discrepancy between observed galactic rotation curves and Newtonian predictions based on visible matter [10] represents one of astronomy’s most persistent challenges. The predominant solution—postulating dark matter halos—remains empirically unverified despite decades of search. Similarly, the accelerated expansion of the universe [9] requires yet another unexplained component: dark energy.

Modified Newtonian Dynamics (MOND) successfully describes rotation curves with a single new constant $a_0$ [8] but has been criticized as phenomenological rather than fundamental. Meanwhile, $\Lambda$CDM cosmology requires two mysterious components (dark matter and dark energy) that together constitute 95% of the universe’s energy content, violating the principle of parsimony.

This paper proposes a fundamental unified mechanism based on a wave-centric paradigm of the universe [20,21]. We demonstrate that both galactic dynamics and cosmic expansion emerge from a single physical process: the conversion of mass to energy and its subsequent incorporation and diffusion through the fabric of space itself. The anomalous dynamics in galaxies arise not from unseen matter but from measurable energy density gradients created by this continuous process of energy injection and diffusion, providing a physical interpretation for both MOND phenomenology and dark energy effects.

The central thesis of this paper is that the process of energy diffusion from astrophysical and primordial sources is the fundamental mechanism underlying these phenomena. We will show that the stationary solution to the diffusion equation for a central source yields an energy density field ${\rho }_{\mathrm{diff}}\left(r\right)\propto 1/r$. This specific profile is not postulated but derived, and it directly leads to the constant orbital velocities observed in galactic rotation curves. This provides a mechanistic, non-phenomenological foundation for MOND. Furthermore, we extend this diffusion paradigm to cosmological scales, arguing that the primordial injection and subsequent diffusion of energy funded the universe’s expansion and gravitational potential energy.

2. Theoretical Framework

2.1. Wave-Based Reality and Emergent Gravity

The core premise of our model is not a new postulate but a physical interpretation of established facts. Quantum field theory dictates that the vacuum possesses a non-zero energy density [22]. The most parsimonious physical interpretation of this energy is that it represents a dynamic, vibrating state of the spatial medium itself. In this framework, matter consists of complex standing wave structures—resonances—within this universal medium. Protons and neutrons, for instance, can be understood as a hierarchical organization of trapped electromagnetic standing waves [6,7].

Forces emerge from wave propagation dynamics in a medium with varying energy density $\rho \left(\mathbf{r}\right)$. Wave-packets (particles) follow trajectories that are geodesics of this energy density landscape. The resulting perceived acceleration derives from the gradient of the energy density field:

$$a\left(\mathbf{r}\right)\propto {\displaystyle \frac{\nabla \rho \left(\mathbf{r}\right)}{\rho \left(\mathbf{r}\right)}}$$

(1)

This constitutive law is not an ad hoc choice but a mechanistic reformulation of the geometric description of General Relativity. In GR, the curvature of spacetime dictates the motion of matter. Here, this curvature finds its physical origin in the gradients of the vibrational energy density. The relative gradient $\nabla \rho /\rho$ naturally emerges as the parameter governing wave deflection, much like the gradient of the refractive index guides light in a medium. This provides a physical explanation for the equivalence principle: both inertial and gravitational mass are manifestations of a wave’s response to the energy density landscape.

2.2. The Diffusion Mechanism: From Energy Injection to the 1/r Profile

The missing element in previous models is a physical process that distributes energy from local sources to create large-scale, long-range fields. We propose this process is diffusion. Once perturbed by a local energy injection, the vibrational medium will tend to redistribute this energy to maximize entropy. This process is universally described by a diffusion equation.

The energy flux $\mathbf{J}$ is proportional to the negative gradient of the energy density (Fick’s law: $\mathbf{J}=-D\nabla \rho$). Combining this with the continuity equation (${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\mathbf{J}=S$), which enforces local conservation of energy, yields the standard diffusion or heat equation for a constant diffusion coefficient D:

$${\displaystyle \frac{\partial \rho }{\partial t}}=D{\nabla }^2\rho +S(\mathbf{r},t)$$

(2)

where $S(\mathbf{r},t)$ is the source term representing the rate of energy injection per unit volume.

The critical insight comes from analyzing the steady-state solution for an isolated, continuous point source. For a central, steady-state source ($S\left(\mathbf{r}\right)=S_0\delta \left(\mathbf{r}\right)$) the time derivative vanishes ($\partial \rho /\partial t=0$). Outside the source location ($\mathbf{r}\ne 0$), the equation reduces to Laplace’s equation:

$${\nabla }^2\rho =0$$

(3)

The spherically symmetric solution to this equation is:

$$\rho \left(r\right)={\displaystyle \frac{A}{r}}+B$$

(4)

Applying the boundary condition that the energy density must vanish at infinity ($\rho \to 0$ as $r\to \infty$) forces the constant $B=0$. This yields the fundamental result:

$${\rho }_{\mathrm{diff}}\left(r\right)={\displaystyle \frac{A}{r}}$$

(5)

This $1/r$ energy density profile is not an assumption but a direct mathematical consequence of energy diffusion from a central source in a homogeneous medium. It is this specific profile that will be shown to generate the MONDian gravitational behavior.

2.3. Energy Sources and Unification of Scales

The continuous injection of energy into the spatial medium is not an additional assumption but the identification of a common physical process behind seemingly disparate phenomena. This process provides a unified physical origin for both dark energy (global, historical energy input) and the galactic dark matter phenomenon (local, ongoing energy input). The ’missing mass’ is not a substance but the energy continuously added to space, which then diffuses according to Eq. (2).

• Primordial annihilation: Matter-antimatter annihilation in the early universe released colossal energy that thermalized the cosmos, establishing the background energy density ${\rho }_0$ and driving the initial expansion. This is the dominant cosmological source.
• Stellar nucleosynthesis: Stars, novae, supernovae, black holes, neutron stars, accretion disks, and other astrophysical systems continuously convert mass to energy through fusion processes [4], acting as local sources $S(\mathbf{r},t)$ that create energy overdensities $\delta \rho$. These are the dominant galactic sources.

2.4. Recent Cosmic Acceleration and Neutrino Condensation

The observed recent acceleration of cosmic expansion [9] finds a natural explanation within our framework. We propose that this acceleration results from the condensation of the vast reservoir of low-energy neutrinos produced throughout cosmic history but unable to condense until now.

As the universe expands, matter becomes increasingly diluted, weakening local gravitational potentials in the vast intergalactic voids. This creates conditions favorable for the condensation of low-energy neutrinos that have been accumulating since the Big Bang. The condensation process follows a threshold mechanism:

$$E_{\nu }<E_{\mathrm{cond}}\propto \left|\varphi \left(\mathbf{r}\right)\right|$$

(6)

where $E_{\nu }$ is the neutrino energy, $\varphi \left(\mathbf{r}\right)$ is the local gravitational potential, and $E_{\mathrm{cond}}$ is the condensation energy threshold. In regions where $\left|\varphi \right(\mathbf{r}\left)\right|$ drops below a critical value (as in modern-era intergalactic space), even the lowest-energy neutrinos can condense.

This process creates a positive feedback loop:

• Neutrino condensation increases the local energy density of space.
• The local metric responds to this increase, since the energy density $\rho \left(\mathbf{r}\right)$ determines the frequency $\nu \left(\mathbf{r}\right)$ of the standing waves that constitute the fabric of space.
  The link between energy density, frequency, and the metric is fundamental. In a wave-based model, the elementary "cells" or "grains" of space are defined by standing wave structures. The energy density $\rho \left(\mathbf{r}\right)$ is a measure of the vibrational energy contained within these cells. A higher energy density directly implies a higher frequency of vibration $\nu \left(\mathbf{r}\right)$ for these fundamental waves, as per the Planck relation $E=h\nu$ applied to the energy content of a spatial cell. Consequently, a spatial region with a higher vibrational frequency will have a greater number of wave nodes and anti-nodes per unit of proper volume. It is this node density—the number of elementary wave events per unit proper volume—that ultimately defines the local metric $g_{\mu \nu }\left(\mathbf{r}\right)$. In this framework, the metric is an emergent property that encodes the underlying density of the wave structure of space.
• A higher frequency corresponds to a greater number of nodes $N\left(\mathbf{r}\right)$ in the standing-wave lattice. The metric can be expressed as a function of this node density:

  $$d{s}^2=g_{\mu \nu }\left(\mathbf{r}\right)d{x}^{\mu }d{x}^{\nu }\mathrm{with}{g}_{\mu \nu }\left(\mathbf{r}\right)\propto f\left(N\left(\mathbf{r}\right)\right).$$

  (7)
• Since the number of nodes is proportional to the local energy density,

  $$N\left(\mathbf{r}\right)\propto \rho \left(\mathbf{r}\right),$$

  (8)

  an increase in $\rho$ leads directly to an increase in N, which in turn stretches spatial intervals.
• As the number of nodes increases locally, spatial distances expand accordingly:

  $$\Delta l\left(\mathbf{r}\right)\propto N\left(\mathbf{r}\right).$$

  (9)
• This expansion further dilutes matter and weakens gravitational potentials.
• Weaker potentials allow more low-energy neutrinos to condense.

The resulting accelerated expansion will continue until the accumulated reservoir of condensing neutrinos is depleted. This mechanism explains why cosmic acceleration began relatively recently ($z\sim 0.5$–1): it required both sufficient accumulation of low-energy neutrinos and sufficient pre-expansion to create large regions with weak gravitational potentials.

Order-of-magnitude estimates support this mechanism. The total energy available in the cosmic neutrino background is $\sim {10}^{70}$ J, consistent with the energy required to explain the observed acceleration when considering the efficiency of the condensation process.

3. Galactic Dynamics and MOND Emergence

The gravitational acceleration follows from our constitutive law (Eq. (1)):

$$a\left(r\right)\propto {\displaystyle \frac{|\nabla {\rho }_{\mathrm{tot}}\left(r\right)|}{{\rho }_{\mathrm{tot}}\left(r\right)}}$$

(10)

The total energy density ${\rho }_{\mathrm{tot}}\left(r\right)$ is the sum of the density directly associated with matter (${\rho }_m$) and the diffused energy density field (${\rho }_{\mathrm{diff}}$) derived in Section 2.2.

In different regimes: - Newtonian regime (small r): Close to the galactic center, the matter density dominates: ${\rho }_{\mathrm{tot}}\approx {\rho }_m\propto 1/r^2$. This yields $a\left(r\right)\propto |\nabla (1/r^2)|/(1/r^2)\propto (1/r^3)/(1/r^2)\propto 1/r^2$, recovering Newtonian dynamics. - MONDian regime (large r): At large galactocentric distances, the extended diffused energy density profile ${\rho }_{\mathrm{diff}}\propto 1/r$ dominates: ${\rho }_{\mathrm{tot}}\approx {\rho }_{\mathrm{diff}}\propto 1/r$. This yields $a\left(r\right)\propto |\nabla (1/r)|/(1/r)\propto (1/r^2)/(1/r)\propto 1/r$, resulting in a constant circular velocity $v=\sqrt{ar}=\mathrm{constant}$ and a flat rotation curve.

The transition between these regimes occurs at the radius where ${\rho }_m\left(r\right)\sim {\rho }_{\mathrm{diff}}\left(r\right)$. This naturally defines the MOND acceleration constant $a_0$ not as a fundamental constant, but as an emergent property of the spatial medium’s diffusive characteristics (the coefficient D) and the galactic energy output (the source strength $S_0$).

4. Cosmic Energy Budget and Unified Mechanism

Our order-of-magnitude calculations (Appendix A) reveal a profound connection:

$$\begin{array}{cc}\hfill \left|U\right|& \approx 8.36\times {10}^{70}\mathrm{J}(\mathrm{Gravitational}\mathrm{potential}\mathrm{energy})\hfill \\ \hfill E_{\mathrm{thermal}}& \approx 2.6\times {10}^{70}\mathrm{J}(\mathrm{Primordial}\mathrm{annihilation}\mathrm{energy})\hfill \end{array}$$

The remarkable proximity (within a factor of $\sim 3$) suggests that the energy driving cosmic expansion originated from primordial mass conversion. Modern stellar energy release ($\sim {10}^{67}$ J) is negligible cosmologically but dominant locally in galaxies. However, the accumulated energy in the low-energy neutrino background represents a substantial reservoir ($\sim {10}^{70}$ J) that becomes available for driving cosmic acceleration once the gravitational potential in expanding voids drops below the condensation threshold.

This establishes a unified picture powered by the injection and diffusion of energy: - Cosmic scale: Primordial mass-energy conversion and its diffusion drove universal expansion and established the global energy budget. - Galactic scale: Ongoing mass-energy conversion creates local sources. The diffusion of this energy creates the ${\rho }_{\mathrm{diff}}\propto 1/r$ field that explains MOND phenomena. - Recent acceleration: Accumulated low-energy neutrino condensation provides a new energy source, whose incorporation and diffusion into the medium drives the current accelerated expansion.

4.1. Additional Observational Evidence: The Case of Old and Diffuse Galaxies

A key prediction of our diffusion-based model is that the MOND phenomenon, interpreted as the dominance of the diffused energy density profile ${\rho }_{\mathrm{diff}}\propto 1/r$, should be correlated with a galaxy’s historical astrophysical activity. This activity determines the source strength $S(\mathbf{r},t)$ and thus the amplitude of the ${\rho }_{\mathrm{diff}}$ field. Remarkably, observations indicate that certain old, very diffuse, and low-surface-brightness galaxies (such as the galaxy NGC 1052-DF2 [28]) exhibit rotation curves that are consistent with the predictions of standard Newtonian dynamics based on baryonic mass alone, showing no significant need for dark matter or a MOND-like effect. This contrasts sharply with the MONDian behavior observed in more massive and active spiral galaxies. Within our theoretical framework, this disparity finds a natural explanation: these ancient and quiescent galaxies have had a very weak star formation history or effectively became dormant billions of years ago. Consequently, their cumulative energy output from nucleosynthetic processes (supernovae, etc.) is negligible. With the source term S being very weak, the diffusion process has not generated a significant large-scale diffused energy density surplus ${\rho }_{\mathrm{diff}}$. The total energy density ${\rho }_{\mathrm{tot}}$ in these systems is therefore dominated by the baryonic matter density (${\rho }_m$), restoring the dynamics to the standard Newtonian regime ($a\left(r\right)\propto 1/r^2$). This correlation between stellar age, past astrophysical activity, and the presence or absence of rotation curve anomalies provides a strong observational hint and a further falsifiable test for our unified model based on energy diffusion.

5. Discussion and Conclusion

We have presented a unified model where both galactic dynamics and cosmic expansion emerge from a single physical process: mass-energy conversion and its subsequent diffusion through the spatial medium. The model provides a mechanistic foundation for MOND, not by introducing new physics, but by deriving the key $1/r$ density profile from a diffusion process and providing a physical interpretation for the geometric description of GR (energy density gradients instead of spacetime curvature).

The mathematical consistency between the total gravitational potential energy and the primordial thermal energy release is not a mere coincidence but a strong indication of a common origin. This model exemplifies Occam’s razor, solving two major mysteries of modern cosmology—dark matter and dark energy—with a single cause: the transfer and diffusion of energy from matter to the spatial medium.

The MOND acceleration constant $a_0$ is not a fundamental constant but an emergent property. It marks the transition radius where the diffused energy density ${\rho }_{\mathrm{diff}}\propto 1/r$ begins to dominate over the direct matter field ${\rho }_m\propto 1/r^2$. Its value is determined by the medium’s properties—its effective diffusion coefficient D and the source function S—and is therefore a measurable quantity rather than a derivable one, much like the speed of sound in a material.

The recent acceleration of cosmic expansion emerges naturally from our model as the consequence of accumulated low-energy neutrinos reaching condensation conditions in the increasingly rarefied intergalactic medium, initiating a new phase of energy injection and diffusion. This provides a direct physical alternative to the standard postulate of Dark Energy, replacing an enigmatic fundamental constant with a dynamic, testable process tied to the universe’s thermal history and matter distribution.

This framework makes testable predictions:

• Correlation between galactic energy output (e.g., star formation rate) and mass discrepancy severity.
• Specific, measurable energy density profiles ($\sim 1/r$) around galaxies via gravitational lensing or other probes.
• Quantitative relationships between a galaxy’s energy production history and its rotation curve.
• Environmental variations in the effective $a_0$ value.
• Specific spectral distortions in the cosmic neutrino background as low-energy neutrinos condense in large-scale voids.

Future work will explore the detailed mechanisms of energy transfer to the spatial medium and the development of a complete cosmological model based on this diffusion paradigm.

Appendix A.1. Gravitational Potential Energy of the Observable Universe

Calculating the total gravitational potential energy of the universe is a complex problem in general relativity. The Newtonian formula for a homogeneous sphere, while approximate, provides a robust order-of-magnitude estimate and is commonly used in this pedagogical and heuristic context [12,14].

The gravitational potential energy U for a homogeneous sphere of mass M and radius R is given by $U=-{\displaystyle \frac{3}{5}}{\displaystyle \frac{G{M}^2}{R}}$.

Appendix A.1.1. Mean Density of Baryonic Matter

The cosmological parameters from Planck 2018 [24] are

$${\mathsf{\Omega }}_b{h}^2=0.0224,{\mathsf{\Omega }}_m=0.315,H_0=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$$

We deduce

$$h={\displaystyle \frac{H_0}{100}}=0.674$$

$${\mathsf{\Omega }}_b={\displaystyle \frac{{\mathsf{\Omega }}_b{h}^2}{h^2}}={\displaystyle \frac{0.0224}{0.{674}^2}}=0.0493$$

The critical density is

$${\rho }_{\mathrm{crit}}={\displaystyle \frac{3H_0^2}{8\pi G}}=8.53\times {10}^{-27}\mathrm{kg}{\mathrm{m}}^{-3}[25],$$

with $G=6.67430\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$

Thus, the mean baryonic density is

$${\rho }_b={\mathsf{\Omega }}_b{\rho }_{\mathrm{crit}}=4.21\times {10}^{-28}\mathrm{kg}{\mathrm{m}}^{-3}=4.21\times {10}^{-31}\mathrm{g}{\mathrm{cm}}^{-3}.$$

Appendix A.1.2. Gravitational Potential Energy of the Observable Universe

The radius of the observable universe is

$$R=46.5\mathrm{Gly}=4.40\times {10}^{26}\mathrm{m}\left[26\right].$$

The associated volume is

$$V={\displaystyle \frac{4}{3}}\pi R^3$$

The total baryonic mass is

$$M_b={\rho }_{b}V=1.50\times {10}^{53}\mathrm{kg}$$

The total matter mass (baryons + dark matter, ${\mathsf{\Omega }}_m=0.315$) is

$$M_m={\mathsf{\Omega }}_m{\rho }_{\mathrm{crit}}V=9.59\times {10}^{53}\mathrm{kg}$$

For a uniform sphere, the gravitational binding energy is

$$U=-{\displaystyle \frac{3}{5}}{\displaystyle \frac{G{M}^2}{R}}$$

Therefore,

$$U_b=-{\displaystyle \frac{3}{5}}{\displaystyle \frac{G{M}_b^2}{R}}=-2.05\times {10}^{69}\mathrm{J}$$

$$U_m=-{\displaystyle \frac{3}{5}}{\displaystyle \frac{G{M}_m^2}{R}}=-8.36\times {10}^{70}\mathrm{J}$$

• Summary
  • Mean baryonic density: ${\rho }_b=4.21\times {10}^{-28}\mathrm{kg}{\mathrm{m}}^{-3}$.
  • Gravitational potential energy (baryons only): $U_b\approx -2.05\times {10}^{69}\mathrm{J}$.
  • Gravitational potential energy (total matter): $U_m\approx -8.36\times {10}^{70}\mathrm{J}$.

    $$\begin{array}{|c|}\hline |U_m|\approx 8.36\times {10}^{70}\mathrm{J}\\ \hline\end{array}$$

Appendix A.2. Total Primordial Thermal Energy Release

The primordial thermal energy is calculated from current observables of the Cosmic Microwave Background (CMB) and Cosmic Neutrino Background (C$\nu$B), correcting their current energy by the redshift factor $(1+z)$ to estimate their energy at the time of emission/decoupling [3,16].

Appendix A.2.1. Energy from the Cosmic Microwave Background (Photons)

$$\begin{array}{cc}\hfill T_{\gamma ,0}& =2.7255\mathrm{K}\left[11\right]\hfill \\ \hfill {\rho }_{\gamma ,0}& =a{T}_{\gamma ,0}^4\approx 4.17\times {10}^{-14}\mathrm{J}/{\mathrm{m}}^3(\mathrm{where}a\mathrm{is}\mathrm{the}\mathrm{radiation}\mathrm{constant})\hfill \\ \hfill E_{\gamma ,0}& ={\rho }_{\gamma ,0}·V\approx 1.49\times {10}^{67}\mathrm{J}\hfill \end{array}$$

At the redshift of photon decoupling ($z_{\mathrm{dec}}\approx 1090$ [11]), their energy was higher by a factor of $(1+z)$.

$$E_{\gamma ,\mathrm{initial}}\approx E_{\gamma ,0}·(1+z_{\mathrm{dec}})\approx 1.49\times {10}^{67}\times 1090\approx 1.62\times {10}^{70}\mathrm{J}$$

Appendix A.2.2. Energy from the Cosmic Neutrino Background (Neutrinos)

After neutrino decoupling, their temperature was reduced by a factor of ${(4/11)}^{1/3}$ compared to photons [15].

$$\begin{array}{cc}\hfill T_{\nu ,0}& ={\left({\displaystyle \frac{4}{11}}\right)}^{1/3}{T}_{\gamma ,0}\approx 1.95\mathrm{K}\hfill \\ \hfill {\rho }_{\nu ,0}& =3\times {\displaystyle \frac{7}{8}}\times a{T}_{\nu ,0}^4\approx 2.84\times {10}^{-14}\mathrm{J}/{\mathrm{m}}^3(\mathrm{for}3\mathrm{neutrino}\mathrm{flavors})\hfill \\ \hfill E_{\nu ,0}& ={\rho }_{\nu ,0}·V\approx 1.01\times {10}^{67}\mathrm{J}\hfill \\ \hfill E_{\nu ,\mathrm{initial}}& \approx E_{\nu ,0}·(1+z_{\nu -\mathrm{dec}})\approx 1.01\times {10}^{67}\times {10}^9\approx 1.01\times {10}^{70}\mathrm{J}\hfill \\ \hfill & (\mathrm{Note}\text{:}{z}_{\nu -\mathrm{dec}}\sim {10}^9\mathrm{is}\mathrm{a}\mathrm{standard}\mathrm{approximation})\hfill \end{array}$$

Appendix A.2.3. Total Initial Thermal Energy

$$E_{\mathrm{thermal},\mathrm{initial}}=E_{\gamma ,\mathrm{initial}}+E_{\nu ,\mathrm{initial}}\approx (1.62\times {10}^{70})+(1.01\times {10}^{70})=2.63\times {10}^{70}\mathrm{J}$$

$$\begin{array}{|c|}\hline E_{\mathrm{thermal},\mathrm{initial}}\approx 2.6\times {10}^{70}\mathrm{J}\\ \hline\end{array}$$

Appendix A.3. Note on Subsequent Stellar Energy Production

The total energy radiated by all stars in the universe is estimated from the energy density of the extragalactic background light (EBL) [13].

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{EBL}}& \approx 1\times {10}^{-14}\mathrm{J}/{\mathrm{m}}^3\hfill \\ \hfill E_{\mathrm{stars}}& \approx {\rho }_{\mathrm{EBL}}·V\approx 3.6\times {10}^{66}\mathrm{J}\hfill \end{array}$$

Even when corrected for redshift (factor of $\sim 2-3$), this energy ($\sim {10}^{67}\mathrm{J}$) is three orders of magnitude smaller than the primordial thermal energy and is therefore negligible in this cosmological energy budget.