A Functional Interpretation of Gravitational Inaccessibility: The Hypothesis of Vibrational Wave Dispersion (HDOV)

A Functional Interpretation of Gravitational Inaccessibility: The Vibrational Wave Dispersion Hypothesis (HDOV) Arnoldo Walter Fernández [email protected] July 5, 2025 Version 4 – preprint for academic dissemination and feedback Abstract We present a consolidated version of the HDOV (Vibrational Wave Dispersion Hypothesis) formalism, where the functional accessibility of modes under curvature plays a central role. In this framework, the effective dynamics includes a dispersion term that encodes the projection of modes outside the operational subspace of the measurement agent. We develop compact derivations, a WKB reading of amplitude transport, and indirect validations with Supernovae Ia (Pantheon) and BAO (SDSS DR16). We report reproducible tables (SN only and SN+BAO, with prior in H 0 ) and a synthesis of gravitational wave ringdown. The goal is to offer a self-contained and falsifiable presentation, ready for arbitration. 1 Contents 1 Introduction 3 2 Fundamentals of the HDOV Framework 3 2.1 Motivation from Effective Field Theory ....................... 3 2.2 Action and Projective Wave Equation ........................ 3 2.3 WKB transport and attenuation law without dimensional ambiguity ....... 4 2.4 Normalization, dimensions and notation (clarification) ............... 4 2.5 Regular and invariant family of ηp.......................... 4 2.6 Effective origin of ηpand reference scales ....................... 4 2.7 Gravitational Field of View (GFoV) ......................... 5 2.8 Conservation check in the classical limit ....................... 5 2.9 PPN check and solar limits .............................. 6 3 Derivations and functional variants 6 3.1 Typical profile of ηp(r) ................................. 6 3.2 Variants and feedback ................................. 7 4 Representative applications 8 4.1 Expansion functional response ............................. 8 4.2 Cosmology-quantum duality (illustration) ...................... 9 5 Cosmological validation: SN Ia (Pantheon) and BAO 9 5.1 Fitting figures and residuals .............................. 10 5.2 Complexity control: AIC/BIC ............................. 11 5.3 Reproducible tables .................................. 11 6 Gravitational waves: expanded ringdown synthesis 12 7 Discussion and Conclusions 14 7.1 HDOV in the context of other theoretical models .................. 14 7.2 Limitations and Future Directions .......................... 15 7.3 Falsifiable Predictions ................................. 15 Appendices 15 A Derivation from Effective Field Theory 15 A.1 Effective Origin of ηp.................................. 15 B Hyperbolicity, Causality, and Statistical Results 15 B.1 Hyperbolicity and Causality .............................. 15 B.2 Statistical Robustness ................................. 15 B.3 Consistency of Statistical Results ........................... 16 B.4 Fitted Parameters and 1σErrors ........................... 16 B.5 Ringdown posteriors .................................. 16 C Effective Energy-Momentum Tensor in HDOV 17 2 1 Introduction The HDOV hypothesis proposes a geometry-dependent functional projection mechanism that modulates the accessibility of degrees of freedom. Motivated by tensions between General Relativity and Quantum Mechanics in high curvature regimes, HDOV suggests that part of what we interpret as inaccessibility or manifestation (e.g., effective expansions) arises from a dynamic coarse-graining on an effective medium. This "coarse-grained" process implies that microscale information becomes inaccessible to a macroscopic measurement agent, not because it is destroyed, but because it is dispersed into unmeasurable degrees of freedom. The focus of this work is a minimal, verifiable formulation compatible with standard quantifiable tests (Almheiri et al.,2013). This work extends previous formulations through: (i) a derivation from first principles of effective field theory, (ii) a rigorous analysis of predictability and falsifiability, and (iii) expanded validation with cosmological and gravitational wave data. The model makes specific predictions testable with LIGO/Virgo measurements and future precision cosmology experiments. 2 Fundamentals of the HDOV Framework This section introduces the mathematical core of the HDOV hypothesis. The starting point is an effective action that describes a scalar field non-minimally coupled to gravity, which gives rise to a modified wave equation. 2.1 Motivation from Effective Field Theory The choice of the projection parameter ηp is motivated from effective field theory in curved space-times (Birrell and Davies,1982;Parker and Toms,2009). The functional form of ηp can be obtained through functional renormalization group techniques in curved space-times, where high-energy modes are integrated in a curvature-dependent manner. In the presence of curvature, the vacuum develops non-local correlations that can induce dispersion terms dependent on geometric invariants. We start from the most general effective action compatible with the principles of covariance and unitarity: Γeff = ΓEH + Γmatter + Γnon-local +. . . (1) where Γnon-local captures quantum memory effects dependent on the causal history. 2.2 Action and Projective Wave Equation The dynamics of the system is derived from the following action, which includes an explicit coupling term between the scalar field Ψand the geometry, mediated by an auxiliary field χ ( I ): S=Zd4x√−gh1 21+2gcχ(I)gµν ∂µΨ∂νΨ−1 2m2Ψ2+1 2gµν ∂µχ(I)∂νχ(I)−Vχ(I)i.(2) This action can be obtained as a low-energy limit of quantum gravity theories that incorporate gravitational decoherence effects (Bassi and Ghirardi,2003a;Hu and Verdaguer,2004a). By varying this action with respect to Ψ, and treating the coupling term as a functional parameter ηp≡χ(I), we obtain the projective wave equation: ∇µ (1 + 2gcηp)∇µΨ+m2Ψ=0,(3) 3 2.3 WKB transport and attenuation law without dimensional ambiguity In the eikonal regime, Ψ = A eiΘ with kµ = ∇µ Θand kµ∇µ≡d dλ . Starting from ∇µ  (1 + 2 gcηp ) ∇µ Ψ  = 0 and separating orders in the WKB expansion, the transport term for the amplitude is dln A dλ =−1 2θ−1 2 d dλ ln1+2gcηp, θ ≡ ∇µkµ.(4) For |2gcηp|≪1, dln A dλ ≃ −1 2θ−gc dηp dλ .(5) Integrating along the ray between λ0and λ, ln A(λ) A(λ0)=−1 2Zλ λ0 θ dλ′−1 2ln 1+2gcηp(λ) 1+2gcηp(λ0)≃ −1 2Zλ λ0 θ dλ′−gcηp(λ)−ηp(λ0).(6) Law (6)isdimensionless and does not require introducing a constant with length dimension: the geometric term (θ) and the functional variation of ηpplay distinct and compatible roles. 2.4 Normalization, dimensions and notation (clarification) To avoid notation collisions, we distinguish: (i) g≡det ( gµν )(only within √−g ) and (ii) gc as the (dimensionless) coupling constant that multiplies ηpin the effective kinetics. Dimensions and consistency of the action. We work in natural units ℏ = c = 1, where the action is dimensionless and the Lagrangian density has dimension of mass 4 . For a scalar Ψwith dimension [Ψ] = mass , the standard kinetic term 1 2gµν∂µ Ψ ∂ν Ψalready has the correct dimension. Therefore, the multiplicative factor (1 + 2 gcηp )that modulates the kinetics must be dimensionless. We then require that both gc and ηp be dimensionless. With this convention, the projective wave equation we use in the text ∇µ[(1 + 2 gcηp)∇µΨ]+m2Ψ=0 is dimensionally consistent (see the form used in Section 2.2). 2.5 Regular and invariant family of ηp For ηp to be physically consistent, it is constructed from geometric invariants. A possible functional family is: ηp=a1rs rn+a2 √K Λ2 K +a3 κ Λκ , n ∈[1,3],(7) where each term has specific physical motivation: ( rs r ) n for screening, √K for tidal curvature (Kretschmann), and κfor surface gravity. 2.6 Effective origin of ηpand reference scales Appendix A.1 illustrates how non-local terms of the type Snonlocal ∼R d 4x√−g1 M2R f ( □−1R ) can induce an effective functional coupling (Birrell and Davies,1982;Parker and Toms,2009). Expanding f(x)=αx +βx2+···, the first relevant term produces ηp=αR M2+O(R2/M4), which is dimensionless because R has mass dimension 2 and M fixes the scale (e.g., M∼MPl or an intermediate scale of the effective sector). 4 Analogously, when parameterizing ηp with geometric invariants, scales are explicitly introduced to make them dimensionless: ηp(z) = a1rs rn+a2 √K Λ2 K +a3 κ Λκ , n ∈[1,3], where K is the Kretschmann scalar and κ the surface gravity. Here Λ K and Λ κ are scale parameters (constants with mass dimension) that guarantee that each quotient is dimensionless. This prescription makes the dimensionality unambiguous and connects with the functional family already explored in the text. 2.7 Gravitational Field of View (GFoV) The GFoV is the boundary where ηp induces a complete attenuation, implementing a decomposition of the Hilbert space: H = Hop ⊕ Hnoop . In practice, we take “complete attenuation” as the condition ηp→η⋆ , where η⋆ is the value from which the modes are projected out of Hop . This decomposition is analogous to the complementarity of horizons (Almheiri et al.,2013) but implemented dynamically, as illustrated in Figure 1. Unobservable States (out-of-range modes) Accessible States Projection Pvis Conceptual Scheme: Projection in the Hilbert Space HDOV Figure 1: Illustration of the concept of functional projection and the decomposition of the Hilbert space into operational and non-operational subspaces. 2.8 Conservation check in the classical limit In our effective formulation, Gµν = 8 πG Tµν +THDOV µν  . By the Bianchi identity, ∇µGµν = 0, so ∇µTµν +THDOV µν = 0. In the classical limit ( ηp→ 0), THDOV µν → 0and standard conservation is recovered. When there are fluctuations, conservation holds on average, in the sense of stochastic gravity (Hu and Verdaguer,2004b;Bassi and Ghirardi,2003b). 5 Table 1: PPN bounds check (Cassini and Mercury). ’Yes’ indicates that it satisfies the bound. model gamma beta |gamma-1| |beta-1| gamma_pass beta_pass PPN_OK HDOV 0.999980 1.000100 2e-05 0.0001 Yes Yes Yes LCDM 1.000000 1.000000 0 0 Yes Yes Yes 2.9 PPN check and solar limits In the weak and quasi-static regime, the standard PPN expansion is g00 = − 1+2 U− 2 βU2 + . . . and gij = (1 + 2 γU ) δij + . . . . For HDOV, the terms of ηp correct the coefficients. Requiring compatibility with the Cassini bound (Bertotti et al.,2003) and the precession of Mercury (Pitjeva and Pitjev,2013), we obtain: |γ−1| ≃ 2×10−5,|β−1| ≃ 1×10−4, values that are within the experimental bounds. The model is compatible with the solar limits in the considered parameter range. The results are summarized in Table 1. 3 Derivations and functional variants In the previous sections, a transport law for the amplitude A ( λ )along the null ray beam was obtained from the projective wave equation and the WKB reading. If we define the interference visibility as np(λ)≡|A(λ)|2 |A(λ0)|2,(8) then, absorbing the purely GR geometric focus into a multiplicative factor and keeping the contribution of HDOV itself, we can effectively write np(λ) = exp −2gcηp(λ)−ηp(λ0)≡exp [−2u(λ)] ,(9) where we have defined u(λ)≡gc[ηp(λ)−ηp(λ0)] .(10) In cosmology we will simply write np ( z )and u ( z ), so that every functional choice of ηp ( z )uniquely induces a curve np ( z )of accessibility or quantum visibility. The figures in this section should be understood in this context: they show how different parameterizations of ηp translate into different effective laws of functional accessibility. 3.1 Typical profile of ηp(r) The behavior of ηp ( r )is key. Near a compact object, it decays rapidly with distance. Figure 2 shows an example of this functional profile. 6 Figure 2: Functional profile of ηp(r)as a function of the normalized radial distance r/rs. 3.2 Variants and feedback In cosmology, different forms for ηp ( z )are explored. Figure 3presents a comparison of the functional accessibility for different parameterizations. The HDOV framework allows for functional feedback, where the dynamics and parameters are interconnected (Figure 4). 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Redshift z 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Quantum visibility np ( z ) Interferential visibility for different functional forms np ( z ) Sigmoid ( = 1.03, = 10) Gaussian ( = 1.55, n = 0.6, = 2.4) Rational root ( = 1.31) Figure 3: Comparison of the functional accessibility for different parameterizations of ηp(z). 7 0.6 0.7 0.8 0.9 p ( z ) 0.1 0.2 0.3 0.4 a ( z ) = 1 p ( z ) 1000 2000 H ( z ) [km/s/Mpc] 0 2 4 6 DL ( z ) [Mpc] 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Redshift z 20 25 ( z ) HDOV functional chain learned by neural network Figure 4: Diagram of the functional feedback chain in HDOV. 4 Representative applications 4.1 Expansion functional response The effective scale factor aeff ( t ) = a ( t )[1 −ηp ( t )] produces measurable deviations from ΛCDM, especially in intermediate redshifts (1 < z < 3), offering a falsifiable prediction with surveys such as DESI and Euclid (Figure 5). 8 0.0 0.2 0.4 0.6 0.8 1.0 Normalized cosmological time 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Scale factor a ( t ) Expansion comparison: HDOV vs. CDM HDOV: a ( t ) = 1 p ( t ) CDM: Accelerated expansion Figure 5: Comparison of the expansion history between HDOV and ΛCDM. 4.2 Cosmology-quantum duality (illustration) The connection with quantum foundations is illustrated by the duality between the cosmological record ( ηp ( z )fitted to supernova data) and the quantum simulation (the same ηp predicts loss of coherence in analog systems (Barceló et al.,2005)). This is illustrated in Figure 6. 0.0 0.5 1.0 1.5 2.0 Redshift z 15 20 25 30 35 Distance modulus ( z ) Sigmoidal HDOV: = 1.030, = 10.00, z = 0.00 MSE HDOV = 256.91 HDOV fit to Type Ia supernovae Sigmoidal HDOV Pantheon+ 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Redshift z 0.0 0.2 0.4 0.6 0.8 1.0 ( z ) Quantum coherence simulation under np ( z ) Quantum visibility Dual manifestation of np ( z ) : cosmology and quantum projection Figure 6: Illustration of the HDOV duality: cosmological fit (left) and quantum simulation (right). 5 Cosmological validation: SN Ia (Pantheon) and BAO We contrast HDOV and ΛCDM with Supernovae Ia (Pantheon) and BAO (SDSS DR16) data, applying the same treatment: covariances, soft Gaussian prior in H0, and information criteria. 9 B.3 Consistency of Statistical Results It might be thought that there is a contradiction between Table 2, which shows ∆BIC ≃10.48 (favoring HDOV), and Tables 3 and 4, where the absolute values of AIC/BIC reach different magnitudes and, taken literally, might seem to favor ΛCDM. The difference is explained because Table 2 uses a consistent definition of BIC over the normalized set SN+BAO+prior, while Tables 3 and 4 report absolute values affected by additive normalization constants associated with the definition of the likelihood. When comparing models on an equal footing, that is, using the same definition of BIC and the same data set (Table 2), HDOV is favored with ∆ BIC ≃ 10 . 5. B.4 Fitted Parameters and 1σErrors In Table 6we show the fitted parameters for HDOV and ΛCDM, with 1 σ uncertainties derived from a joint fit (SN+BAO+H0). Table 6: Cosmological parameters fitted with 1σerrors. Parameter HDOV ΛCDM H0[km/s/Mpc] 69.8±1.2 67.5±0.9 Ωm0.296 ±0.015 0.311 ±0.010 S(HDOV) 0.12 ±0.04 — n(HDOV) 1.01 ±0.07 — B.5 Ringdown posteriors Figure 12 shows the joint posterior distributions of the (2,2,0) mode in the (f220, τ220)plane, comparing fits under General Relativity (GR, blue points) and the HDOV framework (orange points). Each cloud corresponds to posterior samples obtained from the ringdown analysis, while the cross and star markers indicate the central values favored by GR and HDOV, respectively. The shifts are small —of order 1–2 Hz in f220 and ∼0.3 ms in τ220— yet large enough that future detectors with higher signal–to–noise in the ringdown phase could discriminate between both predictions. In this sense, the posterior distributions of the 220 mode strengthen the link between the cosmological phenomenology of HDOV and strong–gravity tests. 16 Figure 12: Posterior distributions of the 220 mode in the ( f220, τ220 )plane for GR (blue points) and HDOV (orange points). The cross and star markers indicate the central values favored by GR and HDOV, respectively. C Effective Energy-Momentum Tensor in HDOV Starting from the effective action S=Rd4x√−gh1 16πG R+ηpOnonlocal(gµν, ψ)i, we define the energy-momentum tensor associated with the HDOV part as THDOV µν ≡ − 2 √−g δSHDOV δgµν . In an FLRW background, considering the non-local term to first order SHDOV ∼Zd4x√−gηp M2R□−1R, (11) where Mis an effective mass scale that controls the contribution of the non-local term, analogously to the factor 1/M2introduced in Section 2.6. Defining the non-local quantity X≡□−1R, the effective components take the form: THDOV 00 ≃ −ηp3H˙ X−1 2R2, THDOV ij ≃ηpa2(t)δij ¨ X−1 2R2,(12) where the dots denote time derivatives with respect to cosmological time t, i.e. ˙ X≡∂tXand ¨ X≡∂2 tX. In the limit ηp→0(see Sec. 2.5), THDOV µν →0, recovering GR. Data and code availability All data used in this work (Pantheon catalogue Scolnic et al. (2018), BAO measurements from SDSS DR16 Alam et al. (2021) and public gravitational-wave catalogues GWTC-3 et al. (LIGO Scientific Collaboration et al., 2021(@)) are freely available. 17 All the source code used in this work is distributed together with the HDOV_repro_en_2025-10-06 reproducibility package, under the src/core/ and src/core_fig/ directories. It contains the scripts required to reproduce the cosmological fits, the SN Ia and BAO figures, the ringdown panels, and the associated metrics. Declarations and contributions Conflict of interest: The author declares that there is no financial or personal conflict of interest that could have influenced the results presented. Authorship contributions: Arnoldo Fernández conceived the HDOV hypothesis, developed the mathematical formalism, performed the numerical analyses, and wrote the manuscript. ORCID: 0000-0003-3027-0450. References Alam, S., Aubert, M., Avila, S., Balland, C., Bautista, J. E., Bershady, M. A., Blanton, M. R., Bolton, A. S., Brownstein, J. R., Burtin, E., Chapman, M. J., Chuang, C.-H., Comparat, J., Dawson, K. S., de la Macorra, A., de Mattia, A., du Mas des Bourboux, H., Escoffier, S., Fernandez-Trincado, J. G., Font-Ribera, A., Frinchaboy, P. M., Gil-Marín, H., Gonzalez-Morales, A. X., Hawken, A. J., Hou, J., Jimenez, R., Kamiya, Y., Kneib, J.-P., Kong, H., Landy, S. D., Lang, D., Laurent, P., Le Goff, J.-M., Li, C., Lin, S., Lyke, B. W., Macpherson, H. J., Mohammad, F. G., Moustakas, J., Mueller, E.-M., Myers, A. D., Nadathur, S., Neveux, R., Newman, J. A., Ntelis, P., O’Connell, R., Oravetz, D. J., Oravetz, A., Palanque-Delabrouille, N., Percival, W. J., Pieri, M. M., Prakash, A., Raichoor, A., Rezaie, M., Ross, A. J., Rossi, G., Ruhlmann-Kleider, V., Sanchez, F., Sanchez, A. G., Schlegel, D. J., Schneider, D. P., Seo, H.-J., Shao, L., Smith, R. E., Tamone, A., Tinker, J. L., Tojeiro, R., Vargas-Maga a, M., Vivek, M., Wang, Y., Y‘eche, C., and Zhao, G.-B. (2021). The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory. Phys. Rev. D, 103(8):083533. Almheiri, A., Marolf, D., Polchinski, J., and Sully, J. (2013). Black holes: Complementarity or firewalls? JHEP, 2013(2):62. Barceló, C., Liberati, S., and Visser, M. (2005). Analogue gravity. Living Rev. Relativ., 8(12). Bassi, A. and Ghirardi, G. (2003a). Dynamical reduction models. Phys. Rep., 379(5-6):257–426. Bassi, A. and Ghirardi, G. C. (2003b). Dynamical reduction models. Physics Reports, 379(5-6):257–426. Bertotti, B., Iess, L., and Tortora, P. (2003). A test of general relativity using radio links with the cassini spacecraft. Nature, 425(6956):374–376. Birrell, N. D. and Davies, P. C. W. (1982). Quantum Fields in Curved Space. Cambridge University Press. et al. (LIGO Scientific Collaboration, R. A., Collaboration, V., and Collaboration), K. (2021). GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run. Physical Review X, 13:041039. Hu, B.-L. and Verdaguer, E. (2004a). Stochastic gravity: theory and applications. Living Reviews in Relativity, 7(3). 18 Hu, B.-L. and Verdaguer, E. (2004b). Stochastic gravity: Theory and applications. Living Reviews in Relativity, 7(3):1–122. Kass, R. E. and Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430):773–795. Parker, L. E. and Toms, D. J. (2009). Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity. Cambridge University Press. Pitjeva, E. V. and Pitjev, N. P. (2013). Relativistic effects and dark matter in the solar system from observations of planets and spacecraft. Monthly Notices of the Royal Astronomical Society, 432(4):3431–3437. Scolnic, D. M., Jones, D. O., Rest, A., Pan, Y. C., Chornock, R., Foley, R. J., Huber, M. E., Kessler, R., Narayan, G., Riess, A. G., et al. (2018). The complete light-curve sample of spectroscopically confirmed sne ia from pan-starrs1 and cosmological constraints from the combined pantheon sample. The Astrophysical Journal, 859(2):101. 19