Energy-Density Gravity: A Scalar-Potential Toy Model

Energy-Density Gravity: A Scalar-Potential Toy Model with Saturation Radius, Cosmological Lapse, and Tunneling Latency Debra Gavant∗ October 3, 2025 Abstract We present a scalar-potential toy model of gravity sourced by energy density, Energy-Density Gravity (EDG), anchored in the clock law of Dynamic Present Theory (DPΦ). We derive a saturation radius rsat that prevents horizon formation, test the cosmological lapse parameter βagainst DESI DR1 H(z) data, and propose a falsifiable tunneling latency law τCPA derived from Constraint Load (C) in DPΦ. Results are consistent with General Relativity and ΛCDM at the current precision but yield distinct falsification targets for upcoming observations and laboratory tests. 1 Introduction General Relativity (GR) accurately describes gravitational phenomena but encounters difficulties in the strong-field regime (singularities, horizons), cosmological tensions (e.g., the H0discrepancy), and integration with quantum phenomena such as tunneling delays. Energy-Density Gravity (EDG) is a scalar toy model aligned with DPΦ’s postulate that the gravitational field is sourced by energy density ρEthrough a Poisson-like equation, ∇2Φ = 4πG c2ρE,(1) ∗debragavan[email protected] https://orcid.org/0009-0004-5593-713X 1 with Φ entering the clock law dτ dt =r1−2Φ c2.(2) In this note, we present three minimal results: (i) a saturation radius avoiding horizons, (ii) a cosmological lapse parameter βfit to DESI DR1, and (iii) a toy tunneling latency law. Each yields concrete falsification paths. 2 Strong-Field Closure: The Saturation Radius Define ε(r)≡2Φ(r)/c2. As ε→1, GR predicts horizon formation. EDG postulates saturation: the gradient flattens before a horizon forms. Matching the exterior potential Φext =GM/r to a stall threshold, g00(rsat) = ϵ2 ω,(3) with g00 = 1 −2Φ/c2, gives rsat =2GM c2 1 1−ϵ2 ω .(4) For ϵω≪1, this reduces to rsat ≈rs(1 + ϵ2 ω). This flattening modifies nearhorizon observables, such as photon-ring size, testable with GRAVITY+ and the Event Horizon Telescope [1,2]. 2 1234567 r / Rs 0.0 0.2 0.4 0.6 0.8 1.0 Clock-rate factor d / dt General Relativity Energy-Density Gravity Figure 1: Comparison of GR (solid) vs EDG (dashed) time dilation factor dτ/dt near the Schwarzschild radius. EDG flattens at rsat, avoiding horizon divergence. 3 3 Cosmological Lapse Parameter DPΦ introduces a lapse N(a)∝a3β, modifying the Hubble rate by a factor a−3β. We fit βto DESI DR1 H(z) data [3]. Using rd= 147.09 Mpc [4], the derived H(z) values are: z H(z) [km/s/Mpc] 0.51 97.2±2.8 0.71 101.5±3.0 0.93 114.0±2.2 1.32 147.5±4.5 Table 1: DESI DR1 BAO-derived Hubble values. A least-squares fit with Ωm= 0.30 yields β= 0.00 ±0.07 (1σ). Thus EDG is consistent with ΛCDM at the current precision. DR2 will halve these uncertainties. 4 Tunneling Latency and Constraint Load DPΦ relates Constraint Load Cto suppression of the Continuous Present Actualization (CPA) rate: ωCPA(C) = ω0e−γC.(5) The effective tunneling delay is then τCPA =ℏ E∗eγC ,(6) where E∗is an effective barrier energy. This exponential scaling is testable in solid-state systems, such as perovskite tunneling, where the barrier properties can be precisely engineered [5]. 5 Falsification Criteria EDG yields distinct falsifiers: •Strong-field: If GRAVITY+ and EHT find no flattening in timing or photon rings at ϵω>10−3, EDG fails. 4 •Cosmology: If DESI DR2 excludes |β|>0.15 at 99% confidence, EDG lapse is ruled out. •Quantum latency: If tunneling delays do not scale exponentially with constraint load, EDG/DPΦ CPA mapping fails. 6 Conclusion This toy model unifies three domains—strong gravity, cosmology, and quantum latency—under the EDG scalar-potential framework. While consistent with GR and ΛCDM within current uncertainties, EDG is directly falsifiable by near-future observations and laboratory tests. References [1] The GRAVITY Collaboration, R. Abuter, N. Aimar, P. Amaro-Seoane, A. Amorim, M. Baub¨ock, J. P. Berger, H. Bonnet, G. Bourdarot, W. Brandner, V. Cardoso, Y. Cl´enet, R. Davies, P. T. de Zeeuw, J. Dexter, A. Drescher, A. Eckart, F. Eisenhauer, H. Feuchtgruber, G. Finger, N. M. F¨orster Schreiber, A. Foschi, P. Garcia, F. Gao, Z. Gelles, E. Gendron, R. Genzel, S. Gillessen, M. Hartl, X. Haubois, F. Haussmann, G. Heißel, T. Henning, S. Hippler, M. Horrobin, L. Jochum, L. Jocou, A. Kaufer, P. Kervella, S. Lacour, V. Lapeyr`ere, J.-B. Le Bouquin, P. L´ena, D. Lutz, F. Mang, N. More, T. Ott, T. Paumard, K. Perraut, G. Perrin, O. Pfuhl, S. Rabien, D. C. Ribeiro, M. Sadun Bordoni, S. Scheithauer, J. Shangguan, T. Shimizu, J. Stadler, O. Straub, C. Straubmeier, E. Sturm, L. J. Tacconi, F. Vincent, S. von Fellenberg, F. Widmann, M. Wielgus, E. Wieprecht, and J. Woillez. Polarimetry and astrometry of NIR flares as event horizon scale, dynamical probes for the mass of Sgr A*. Astronomy & Astrophysics, 677:A14, 2023. [2] Event Horizon Telescope Collaboration, Kazunori Akiyama, Juan Carlos Algaba, et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. The Astrophysical Journal Letters, 930(2):L12, 2022. [3] DESI Collaboration et al. Data release 1 of the dark energy spectroscopic instrument. Submitted to The Astronomical Journal, March 2025. 5 [4] Planck Collaboration. Planck 2018 results. vi. cosmological parameters. Astronomy and Astrophysics, 641:A6, 2020. [5] Alexander Colsmann et al. Highly Efficient Wide Bandgap Perovskite Solar Cells With 2D/3D Hybrid Bilayer for High Open Circuit Voltage and Enhanced Device Stability. Advanced Materials, 36(43):2402053, 2024. 6