Operational holographic approach to vacuum energy and cosmic acceleration in the HDOV framework: master equation, functional accessibility and cosmological consequences

Operational holographic approach to vacuum energy and cosmic acceleration in the HDOV framework Master equation, functional accessibility and cosmological consequences Arnoldo Walter Fernández [email protected] PREPRINT — November 22, 2025 Abstract The vacuum catastrophe—the discrepancy of ∼10122 orders of magnitude between the zero-point energy density predicted by quantum field theory and the cosmologically inferred value of the effective cosmological constant—is addressed here by means of a strictly operational and holographic approach, consistent with General Relativity and with Bekenstein–Hawking type information bounds [1,2,3,4,5, 6]. The HDOV framework introduces a scalar field of functional accessibility ηp, which parametrizes the fraction of the state space associated with a given physical mode that remains operationally accessible for a macroscopic observer in a specified environment. Incorporating ηpinto a covariant effective action generates an HDOV master equation for the observable field Ψ; in the WKB regime, this equation induces an exponential attenuation A∝exp[−gRχ(I)ηpdλ]of the accessible amplitude. The same notion of accessibility is applied here to high-energy modes of the quantum vacuum, implemented as a smooth spectral weight Wη(k)holographically regulated in momentum space. We show that the accessibility mechanism that explains exponential attenuations in propagation (gravitational waves, plasma signals, femtosecond TRXS, etc.) also regulates the effective vacuum energy 1 without introducing ad hoc cut-offs and in a manner compatible in order of magnitude with the observed cosmological constant. This regulation is formulated as an HDOV Holographic Projection Theorem, which states that, under accessibility and holographic saturation hypotheses, the accessible vacuum density is bounded from below by Λc4/(8πG), independently of the detailed form of Wη(k). Throughout this work, the approach must be understood in a strictly operational and holographic sense: the mismatch is avoided in the accessible sector without attempting to build a complete microphysical theory of the quantum vacuum. Finally, we present explicit numerical examples of smooth spectral weights compared with sharp cut-offs, calibrated with current cosmological parameters. 2 1 Motivation: the vacuum catastrophe and cosmic acceleration The vacuum energy density predicted by the naive sum of zero-point modes in quantum field theory (QFT) is, in natural units, of the order of the Planck density. By contrast, the effective energy density associated with the cosmological constant that drives the accelerated expansion of the universe is ∼10−123 times smaller [1,2]. This mismatch of ∼10122 is known as the “vacuum catastrophe” and constitutes the most severe manifestation of the cosmological constant problem [1,2,3]. On the other hand, precise cosmological observations (CMB, BAO, type Ia supernovae) indicate that the universe is undergoing accelerated expansion and that a Λ-type term dominates the total energy at late times [3]. The ΛCDM model incorporates this term as an a priori cosmological constant fitted to the data. However, from a theoretical point of view, it remains unclear why the quantum contribution of the vacuum takes exactly that value and no other. In this context, HDOV (Vibrational Wave Dispersion Hypothesis) proposes a different approach: instead of summing all possible modes without restrictions, it is postulated that a macroscopic observer has access only to an operational fraction of the field degrees of freedom, encoded in a scalar of functional accessibility ηpand in an environmental gate χ(I)that reflects the interaction with the environment [7,8,9]. Previous sections of the HDOV program have shown that the same accessibility structure explains: (i) exponential amplitude attenuations in the propagation of gravitational waves; (ii) silencing of signals in interplanetary plasmas near the heliopause; (iii) loss of coherence in femtosecond TRXS experiments; (iv) cosmological causal disconnection in an accelerating universe [7,8]. Here we extend that mechanism to the vacuum energy problem. Note. In this work, particular care has been taken regarding formal treatment, references and dimensional consistency, but the HDOV framework applied to vacuum energy and cosmic acceleration remains an exploratory proposal which, at the time of this version, has not yet undergone peer review in specialized journals. We present the arguments and results explicitly so that anyone can examine the assumptions, criticize the methodology, reproduce the basic calculations and, if necessary, refute the conclusions. The related manuscripts in which the HDOV program is developed in other physical regimes [7,8,9] are currently in preparation (2025) and/or available as preprints in open-access repositories, so that referees and interested readers 3 can consult the complete framework. 2 HDOV master equation and functional accessibility In what follows we adopt the (−,+,+,+) signature convention for the metric tensor gµν and units with c= 1 unless explicitly stated otherwise. 2.1 Effective action and field equation In HDOV we postulate that the observable field Ψdoes not live in isolation: it is immersed in an environment (geometry, plasma, quantum vacuum, instrumentation, strong gravity, etc.) that regulates which fraction of the mode is actually accessible. This accessibility is encoded in a scalar ηpand in an environmental gate χ(I)that indicates where the environment couples effectively [7,8]. In this note we treat both ηpand χ(I)as effective quantities that encode, at a macroscopic level, the average response of the environment; we do not assign them an independent action nor their own equations of motion, so that their detailed dynamics is left for future developments of the HDOV framework. In particular, we do not vary the action with respect to ηpor χ(I); both act as effective background fields, so that the exact conservation of Tµν and the back-reaction on the geometry are only discussed in the fixed-background regime and lie beyond the scope of this note. A minimal covariant effective action that incorporates this can be written schematically as S=Zd4x√−gM2 Pl 2R+Lmat + (1 + 2g χ(I)ηp)∇µΨ∇µΨ−m2Ψ2,(1) where gis an accessibility coupling, Ris the Ricci curvature scalar and MPl is the Planck scale. In (1) the accessibility factor multiplies only the kinetic term of the field Ψ, while the rest mass mis kept fixed. This choice guarantees that the effective field equation (2) is derived in the standard way from the variation of the action and reflects the fact that, at this stage of the HDOV program, functional accessibility is interpreted as a modulation of amplitude transport rather than as a microscopic mass generation mechanism. Varying the action (1) with respect to Ψwe obtain the HDOV master equation: ∇µ(1 + 2g χ(I)ηp)∇µΨ+m2Ψ=0.(2) 4 In all regimes explored in this work we explicitly impose the condition 1 + 2g χ(I)ηp>0, so that the effective kinetic term preserves its sign and no ghost modes or linear-level energy instabilities appear. Physical interpretation: the multiplicative factor (1 + 2g χ(I)ηp)does not remove degrees of freedom; it modulates which part of Ψis operationally realizable in that region of space-time. 2.2 WKB decomposition: phase and amplitude We adopt a WKB-type ansatz, Ψ(x)=A(x)eiΘ(x),(3) where A(x)is a slowly varying amplitude and Θ(x)a rapidly varying phase. We define the local wave 4-vector kµ=∇µΘ. For an observer with 4-velocity uµ, the locally observed frequency is ωobs =−kµuµ, i.e. the rate of increase of the phase along their proper time. Inserting (3) into (2) and separating leading-order terms in the phase from lower-order terms in gradients of A(x), one obtains two coupled equations: an eikonal equation for Θ(ray propagation) and a transport equation for A. In particular, the transport equation takes the form dln A dλ =−g χ(I)(λ)ηp(λ)−1 2θ(λ),(4) where λis the affine parameter along the ray and θ(λ)is the geometric expansion of the beam (focusing/defocusing, gravitational lensing, geometric opening, etc.). Integrating (4) we obtain A(λ)∝exp"−gZλχ(I)(λ′)ηp(λ′)dλ′#×exp"−1 2Zλθ(λ′)dλ′#.(5) The first factor, involving χ(I)ηp, is functional accessibility: how much of the mode remains physically supported by the environment. The second factor, involving θ, is purely geometric (focusing or attenuation by opening). Key point: the exponential attenuation in (5) does not say that the mode is destroyed. It says that it ceases to be accessible for the observer integrating along λ. This is exactly what we see in: (a) signals that “switch off” as they approach gravitational horizons; (b) loss of coherence in interplanetary plasmas near the heliopause; (c) loss of visibility in femtosecond TRXS; (d) cosmological causal disconnection [7,8,9]. 5 From a technical viewpoint, the WKB derivation of (4) and (5) must be understood as a leading-order approximation in smooth gradients of Aand of (1 + 2g χ(I)ηp)along the ray. In situations with sharp variations of ηpor with additional nonlinearities, one expects higher-order corrections to enter, which are not modeled in this note. 3 Application to the quantum vacuum: holographic spectral weight 3.1 Vacuum energy regularized by accessibility The zero-point energy density in QFT, in flat space, can be written schematically as ρvac ∼ℏc 4π2Z∞ 0k3dk. (6) This integral diverges in the ultraviolet: modes with arbitrarily large kcontribute unbounded energy [1,2]. Instead of imposing a hard cut-off by hand, HDOV proposes that functional accessibility ηpinduces a smooth spectral weight over vacuum modes, analogous to the exponential factor in (5). Intuitively: there is a limit to how much energy can be operationally concentrated in a finite region before violating holographic bounds; beyond a certain scale, modes cease to be accessible before being able to sustain themselves without gravitational collapse. We thus introduce a spectral accessibility weight Wη(k): ρeff vac =ℏc 4π2Z∞ 0k3Wη(k)dk, (7) with Wη(k) = exp"− k k0!α#, α > 0.(8) The function Wη(k)is not a hard cut-off ; it is a smooth exponential suppression that says: “modes with k≫k0formally exist in the field, but their accessible part decays exponentially” [9]. From the viewpoint of standard quantum field theory, the expression (7) must be interpreted against the background of the usual vacuum renormalization procedure: we assume that the reference state in flat space has already been subtracted, and that the density ρeff vac corresponds only to the accessible/operational part of residual vacuum fluctuations. The accessibility weight 6 Wη(k)thus acts on the physically operative sector of the vacuum energy. The shape parameter αcontrols how fast the suppression in the ultraviolet is, but as long as α=O(1) the order of magnitude of ρeff vac is dominated by k0, so that the resulting phenomenology is robust against reasonable variations of α. This is the spectral (in k) version of the same exponential mechanism that already appeared in (5) in the context of space-time propagation. It is the same physics: functional accessibility ηpthat attenuates the observable part of a mode [7]. 3.2 Holographic condition and scale fixing The mere introduction of Wη(k)in (7) makes the integral convergent, but leaves the value of k0undetermined. To fix it without arbitrariness, we invoke Bekenstein–Hawking type holographic bounds [4,5,6]. The maximal entropy of a region of radius Rdoes not grow as the volume R3but as the area R2: Smax ∼A 4L2 P∼πR2 L2 P .(9) Translated into an effective energy density, this implies that the accessible energy in a volume cannot exceed the limit imposed by horizon formation. Schematically: ρeff vac L4 P≲1 4π,(10) where in (10) we have rewritten the Bekenstein–Hawking bound in terms of the Planck density, assuming an effective horizon of order H−1; this translation is heuristic and only intended to fix an order-of-magnitude bound for ρeff vac in accelerating cosmological contexts. Equivalently, we can introduce the second spectral moment C≡Z∞ 0k2Wη(k)dk, (11) which controls the effective number of accessible modes in a given volume. The holographic condition then imposes a maximum value (or, in the saturation scenario, a fixed value) for C, determined by the area of the effective horizon. For a late-time de Sitter universe with horizon radius RH=c/HΛ and Λ=3H2 Λ/c2, one has SH∝R2 Hand, consequently, C∝R2 H∝1/Λ. In the next section we show that, for any smooth and monotonically decreasing weight function that saturates this condition, the accessible energy density is controlled by the cosmological term in Einstein’s equations. 7 4 HDOV holographic projection theorem The above structure can be condensed into a simple variational result, which makes explicit the independence of the accessible vacuum density from the detailed shape of the spectral weight, provided that accessibility and holographic information constraints are satisfied. Theorem 1 (HDOV holographic projection).Let Wη(k)∈[0,1] be a monotonically decreasing function of the wavenumber modulus k, sufficiently regular for the moments C=Z∞ 0k2Wη(k)dk, I3=Z∞ 0k3Wη(k)dk (12) to exist. Let ρeff vac be the accessible vacuum energy density defined by (7). If the moment Cis fixed by a holographic condition associated with a cosmological horizon of radius RH(late-time de Sitter) and the Bekenstein–Hawking bounds are saturated, then the accessible energy density satisfies the holographic lower bound ρeff vac ≥Λc4 8πG,(13) independently of the detailed form of Wη(k), provided that it respects monotonicity, boundedness and holographic consistency. The equality (or an arbitrarily close numerical approximation) is obtained when Wη(k)approaches the effective cut-off W⋆ η(k) = Θ(k⋆−k). Proof. The effective horizon of radius RHinduces a holographic bound on the number of accessible modes in a volume Vcontained within that horizon. This number is given, in terms of Wη(k), by Neff =V 2π2Z∞ 0k2Wη(k)dk =V 2π2C. (14) The holographic condition identifies Neff with the maximal entropy SHassociated with the horizon, measured in units of an elementary entropy s0=O(1): Neff ≤SH s0 , SH=AH 4L2 P =πR2 H L2 P .(15) In the saturation regime this yields an effective value for C, determined solely by the horizon radius RHand, consequently, by Λin late-time de Sitter (RH∼c/HΛ,Λ=3H2 Λ/c2). Once Cis fixed, the functional I3[Wη] = Z∞ 0k3Wη(k)dk, (16) 8 under the constraints 0≤Wη≤1and Wηmonotonically decreasing, is minimized by decreasing rearrangement (see, for instance, the so-called bathtub principle in rearrangement inequalities) with the effective cut-off W⋆ η(k) = Θ(k⋆−k), k⋆= (3C)1/3,(17) so that Imin 3=Zk⋆ 0k3dk =k4 ⋆ 4=(3C)4/3 4.(18) The minimum accessible energy density compatible with Cis then ρeff,min vac =ℏc 4π2Imin 3=ℏc 16π2(3C)4/3.(19) Consistency with Einstein’s field equations in late-time de Sitter requires that the effective energy density sustaining the horizon be precisely the cosmological term, ρeff,min vac =Λc4 8πG ≡ρΛ.(20) This condition fixes the value of C(and hence of k⋆) uniquely. Any other smooth weight function Wη(k)that satisfies the same holographic condition shares the same Cand produces an accessible density ρeff vac =ℏc 4π2I3[Wη]≥ρeff,min vac =Λc4 8πG,(21) with equality attained by the effective cut-off W⋆ ηand by suitably tuned families of smooth weights. Consequently, the accessible density consistent with the cosmological horizon is bounded from below by (13) and approaches equality when the spectral weight approaches the effective cut-off. Remark 1 (Independence of the shape of Wη(k)).For smooth families such as Wη(k) = exp[−(k/k0)α]or Wη(k) = (1 + (k/k0)α)−1, the scale parameter k0is fixed by imposing that the energy integral R∞ 0k3Wη(k)dk reproduces (or saturates) the bound (13). Within reasonable ranges of α=O(1), the resulting density is robust and approaches the abrupt cut-off case, but with a smooth transition in the ultraviolet, consistent with the operational interpretation of functional accessibility. 5 Numerical implementation and representative figures In this section we illustrate the above construction with explicit numerical examples of smooth spectral weights, compared with effective sharp cut-offs. 9 •Laboratory (TRXS): exponential decays of accessibility, analogous to (5), measured in ultrafast dynamics where the signal decays following a selective accessibility law and not only thermal dissipation [7]. •Plasmas/heliopause: accessibility indicators ηp(or operational equivalents such as κlocal) anticipate regime transitions (for example, heliopause crossing) before standard magnetohydrodynamic diagnostics [7,8]. In all cases, the underlying physics is the same: a fraction of the mode becomes non-operational for the observer and its accessible amplitude decays according to the exponential law already seen in (5). From a quantitative standpoint, these predictions must still be considered exploratory: they identify observational contrast windows for the HDOV framework, but their validation will require detailed calculations, numerical simulations and systematic comparisons with real data in each of these regimes. 7 Discussion The conceptual result is strong and simple: •Nothing “really disappears”. •Under certain geometric, energetic or holographic conditions [4,5,6], parts of the field cease to be accessible to a finite observer before violating physical bounds (horizon formation, information capacity, etc.). •This loss of accessibility is seen from the outside as amplitude attenuation, frequency shifts or as the effect that in standard models is parametrized as “dark energy” in the accelerated expansion of the universe [3,8,9]. HDOV places all of this under a single master equation (2) and a single exponential accessibility law (5). What in different subfields is described as: (a) decoherence, (b) dissipation, (c) loss of contrast, (d) an effective darkenergy-like term in cosmology, appears here as the same phenomenon: a fraction of the state space becomes non-operational for the observer, regulated by ηpand holographic bounds. It is important to emphasize that this approach does not aim to solve the cosmological constant problem at a fundamental level nor to modify the 16 underlying QFT: it simply shows that, once operational limitations on information access (encoded in ηpand the spectral weight Wη(k)) are taken into account together with holographic bounds, the effective vacuum energy entering Einstein’s equations is naturally regulated to the observed order of magnitude. In this sense, HDOV provides a consistent regulation of the accessible sector, without postulating new dark fluids or fundamental cosmological constants fixed by hand. Although the framework presented here is internally consistent, several important issues remain open for future work: (i) constructing an explicit theory for the dynamics of ηpand its coupling to the geometry and material content (in this note we treat ηpand χ(I)as non-dynamical effective fields); (ii) deriving in a more microphysical way the concrete form of Wη(k) and the scale k0, which here are fixed phenomenologically by a global holographic condition; and (iii) developing quantitative and systematic analyses of the cosmological and strong-gravity predictions of HDOV, including detailed comparisons with ΛCDM using public data and standard statistical criteria (BIC, AIC, etc.). These limitations do not affect the formal coherence of the construction, but clearly delimit the still preliminary and operational character of the results presented here. 8 Conclusions We have presented a unified framework (HDOV) in which: •A functional accessibility field ηp, coupled through the environmental gate χ(I), modifies the effective equation of motion for the observable field Ψ, equation (2). •In the WKB regime, this equation induces an exponential accessibility law (5) that explains attenuations observed in contexts as diverse as gravitational waves, interplanetary plasmas and TRXS experiments. •The same mechanism, applied to the spectrum of vacuum modes, generates an effective vacuum density ρeff vac that is finite and compatible in order of magnitude with the observed cosmological constant, in a way consistent with holographic bounds. •Cosmic acceleration can be interpreted as a manifestation of functional inaccessibility (loss of operational causal connectivity) instead of a mysterious dark fluid. 17 The proposal is deliberately operational: it does not aim to derive the entirety of vacuum physics from first principles, but rather to provide a coherent framework in which the accessible part of vacuum energy and cosmological dynamics is regulated through functional accessibility and holographic bounds. Instead of modifying fundamental QFT, we show that operational limits on access to information (encoded in ηpand Wη(k)) naturally regulate the effective vacuum energy, reconciling microscopic predictions with cosmological observations within a self-consistent holographic framework. This approach opens a natural arena to contrast HDOV with ΛCDM and other phenomenological dark energy models in precision cosmology. 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. Author contributions. Arnoldo Fernández conceived the HDOV hypothesis, developed the mathematical formalism, performed the numerical analyses and wrote the manuscript. References [1] A. G. Cohen, D. B. Kaplan, and A. E. Nelson. Effective field theory, black holes, and the cosmological constant. Phys. Rev. Lett., 82:4971, 1999. [2] S. Thomas. Holography stabilizes the vacuum energy. Phys. Rev. Lett., 89:081301, 2002. [3] M. Li. A model of holographic dark energy. Phys. Lett. B, 603:1–5, 2004. [4] J. D. Bekenstein. Black holes and entropy. Phys. Rev. D, 7:2333, 1973. [5] L. Susskind. The world as a hologram. J. Math. Phys., 1995. [6] G. ’t Hooft. The holographic principle, 2000. [7] Arnoldo Walter Fernandez. Una interpretación funcional de la inaccesibilidad gravitacional: La hipótesis de dispersión de onda vibracional (hdov). 2025. [8] Arnoldo Walter Fernandez. Generalized hdov: Functional mass fitting and scalar projection in the standard model. 2025. 18 [9] Chamara A. Perera. Resolution of the vacuum catastrophe from a new quantum scale and the holographic principle. 2025. 19