% Explaining Dark Energy with the PNP Theory of Gravitation % Fred Nedrock, An M. Rodríguez, Adrien Hale, Leera Vale, Max Freet % August 13, 2025 ## Abstract We show that the PNP Theory of Gravitation — a scalar-field framework already accounting for galaxy-scale dynamics without dark matter — naturally produces a late-time negative-pressure term. This term arises from a dispersive component of the field’s stress–energy tensor and drives cosmic acceleration without introducing a separate dark-energy substance. The effect depends on a single scale parameter $\alpha$ already constrained by halo fits. ## One-Sentence Summary In PNP gravity, the same scalar field explaining dark matter yields a small negative-pressure term that accounts for dark energy. ## Keywords PNP, gravitation, scalar field, stress–energy, negative pressure, dark energy, cosmic acceleration ## 1. Introduction The PNP Theory of Gravitation replaces the Newton–Einstein potential with a covariant scalar-field formulation that matches observed galactic rotation curves without dark matter. Here we show that this same field, without additional assumptions, produces a late-time acceleration term. ## 2. PNP gravitational framework The field $U$ obeys: $$ d(\star dU) = 0 $$ with stress–energy tensor: $$ T_{\mu\nu} = \nabla_\mu U\,\nabla_\nu U - g_{\mu\nu}\,\mathcal{L}(U,\nabla U) $$ Conservation $\nabla_\mu T^{\mu\nu} = 0$ holds by construction. ## 3. Stress–energy split We separate the total energy density $\rho(u)$ into: $$ \rho(u) = \rho_{\text{flow}}(u) + \rho_{\text{disp}}(u) $$ - $\rho_{\text{flow}}$: isotropized kinetic energy from large-scale flows (dominates in high-density regimes). - $\rho_{\text{disp}}$: dispersive term from small-scale phase structure. For a minimal model: $$ \rho_{\text{disp}}(u) = \alpha \ln\!\frac{u}{u_*} $$ where $u$ is the local field energy density, $u_*$ a reference scale, and $\alpha$ the same parameter controlling halo dynamics. ## 4. Negative pressure and acceleration From $\mathcal{L} = -\rho$: $$ p_{\text{disp}}(u) = u\frac{d\rho_{\text{disp}}}{du} - \rho_{\text{disp}}(u) = \alpha - \alpha\ln\!\frac{u}{u_*} $$ In the low-density regime $u \ll u_*$: $$ p_{\text{disp}} \approx -\alpha\,|\ln(u/u_*)| < 0 $$ The Friedmann equation: $$ \frac{\ddot a}{a} = -\frac{4\pi G}{3} \left( \rho + \frac{3p}{c^2} \right) $$ shows that acceleration occurs when $3|p_{\text{disp}}| \gtrsim \rho c^2$. ## 5. Results - The same $\alpha$ that fits galactic dynamics produces late-time acceleration. - No new fields or exotic components are introduced. - The sign of $p_{\text{disp}}$ is fixed by the dispersive form of $\rho_{\text{disp}}$. ## 6. Observational consequences - Fits to supernova and BAO distances can constrain $\alpha$ jointly with halo rotation curves. - Evolution of $H(z)$ can test the predicted link between galaxy-scale and cosmic-scale phenomena. ## 7. Conclusion PNP gravitation explains both dark matter and dark energy as different manifestations of the same scalar-field stress–energy. The acceleration of the universe emerges naturally from a dispersive negative-pressure term, with no additional degrees of freedom. ## Corresponding author(s) An M. Rodríguez: an@preferredframe.com ## References 1. Freet, M., Rodríguez, A. M., Explaining Dark Matter with the Point–Not–Point Framework and a PNP Theory of Gravitation, ResearchGate Preprint, August 2025.