Vacuum Interaction Model for Cosmic Redshift

Abstract

This paper presents a phenomenological model identifying the quantum vacuum as a physical mechanism for cosmic redshift. We model the quantum vacuum as an active medium with an effective electrical conductivity ${\sigma }_0$, intrinsically related to the dynamic structure of virtual electron–positron ($e^{+}{e}^{-}$) pairs. We demonstrate that cosmic redshift and the Hubble constant $H_0$ arise naturally from the continuous energy dissipation of propagating photons interacting with this vacuum impedance $Z_0$. Specifically, every photon loses a constant fraction of energy per oscillation equivalent to $h{H}_0\approx 1.5\times {10}^{-51}\mathrm{J}$, representing a fundamental unit of cosmic energy dissipation. Within the proposed phenomenological framework, we identify the effective redshift parameter $H_0$ with ${\sigma }_0\left(R\right)c{Z}_0$, assuming that the effective vacuum conductivity depends on the invariant spacetime curvature scalar R. This parametrization suggests a possible connection between the gravitational environment and inferred values of $H_0$, which warrants further investigation in relation to the Hubble tension. The model offers a candidate non-expansionary interpretation of cosmological redshift, whose microscopic basis and observational consistency remain to be established.

1. Introduction

The origin of cosmological redshift remains one of the central observational pillars of modern cosmology. Within the standard framework, redshift is interpreted as a geometric consequence of the expansion of spacetime [1,2], leading to the well-established relation between redshift and distance, commonly expressed through the Hubble–Lemaître law [3]. While highly successful over the past century, this standard interpretation motivates the exploration of alternative phenomenological descriptions of photon energy evolution, especially in light of recent profound cosmological anomalies.

Currently, the standard $\mathsf{\Lambda }$Cold Dark Matter ($\mathsf{\Lambda }$CDM) paradigm is facing an unprecedented wave of foundational crises that challenge its core geometric and evolutionary assumptions. The most prominent confrontation arises from the local universe, universally recognized as the “Hubble Tension.” The statistical discrepancy between the early-universe cosmic microwave background (CMB) measurements by the Planck satellite ($H_0=67.4\pm 0.5\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$) [4] and the late-universe local distance ladder observations has steadily widened. With the recent consensus report from the Local Distance Network (2026), which yields a local value of $H_0=73.50\pm 0.81\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ [5], this discrepancy has reached an undeniable $7.1\sigma$ significance level. This firmly establishes that the measured value of $H_0$ is not a global cosmic constant but exhibits a distinct environmental dependence.

These persistent conflicts motivate us to ask whether alternative, non-kinematic descriptions of redshift can be constructed in a way that remains consistent with key observational constraints, while leading to distinct, testable predictions. Historically, non-expansion interpretations often involved scattering processes, which lead to image degradation and angular blurring, and are, therefore, strongly constrained by Tolman’s surface brightness tests [6]. Early ideas of photon energy loss, such as the classical “tired-light” hypothesis discussed by Zwicky [7], lacked a robust physical mechanism and failed to remain consistent with modern precision observational data.

In this paper, we consider a specific class of phenomenological models in which photon energy evolves continuously along its trajectory without invoking scattering or angular diffusion. The central assumption is that the process preserves the photon wave vector and phase coherence, thereby avoiding the primary observational limitations of earlier approaches. We show that a minimal exponential energy decay law leads naturally to a redshift–distance relation consistent with the empirical Hubble law. The model can be formulated in terms of an effective electrical conductivity parameter ${\sigma }_0$ that characterizes the cumulative interaction between photons and the active quantum vacuum [8,9], without requiring the vacuum to behave as a conventional dissipative medium.

The foundational physical mechanism of this framework rests upon the polarization of the quantum vacuum. We posit that the effective electrical conductivity ${\sigma }_0$ is determined by the number density of virtual electron–positron pairs ($e^{+}{e}^{-}$) excited within the medium. According to Heisenberg’s Uncertainty Principle ($\Delta E·\Delta t\approx \hslash$), the presence of a local gravitational field increases the localized vacuum energy density, thereby enhancing the probability of virtual $e^{+}{e}^{-}$ pair creation. This establishes a direct physical link between the local spacetime curvature R and the vacuum conductivity ${\sigma }_0\left(R\right)$, offering a natural mechanism to explain the local variation of $H_0$ and resolve the Hubble tension while maintaining strict adherence to fundamental conservation laws.

2. Vacuum Interaction Model for Cosmic Redshift

The fundamental physical nature of the vacuum polarization phenomenon and its underlying quantum structure remains one of the most vital problems in modern foundational physics [8]. The active macroscopic effects of the quantum vacuum are manifested through robust empirical phenomena such as the Casimir Effect [9] and the Lamb shift.

The quantum vacuum is defined as the lowest possible energetic ground state of the cosmos. Crucially, the local energy density of the underlying quantum fields undergoes continuous, non-zero stochastic fluctuations governed by Heisenberg’s Uncertainty Principle:

$$\Delta E·\Delta t\ge {\displaystyle \frac{\hslash }{2}}$$

(1)

This relation permits a discrete quantity of energy $\Delta E$ to be temporarily borrowed from the vacuum potential for an exceptionally brief duration $\Delta t$. Within this transient interval, virtual particle–antiparticle pairs ($e^{+}{e}^{-}$) spontaneously manifest and subsequently annihilate without violating global energy conservation laws over macroscopic timelines.

Within this formulation, Equation (1) implies that the local number density of these virtual manifestations is intrinsically coupled to the ambient gravitational field strength. A stronger local gravitational field increases the localized vacuum energy density, thereby enhancing the probability amplitude for virtual $e^{+}{e}^{-}$ pairs to appear more densely at that spatial coordinate.

2.1. Exponential Energy Decay Along Photon Trajectories

We consider the physical proposition that photons propagating through this active quantum vacuum experience a cumulative, path-dependent energy loss that is completely decoupled from conventional scattering mechanisms or angular diffusion. This process strictly preserves the photon’s original wave vector $\mathbf{k}$ and phase coherence, thereby bypassing the traditional observational constraints associated with image degradation and angular blurring [6].

Assuming that a propagating photon dissipates energy in direct proportion to its traversed spatial distance r, the governing differential equation for the photon energy E is formulated are as follows:

$${\displaystyle \frac{dE}{dr}}=-\alpha E$$

(2)

Integrating Equation (2) yields the continuous energy attenuation function over distance:

$$E\left(r\right)=E_0{e}^{-\alpha r}$$

(3)

where $\alpha$ denotes the intrinsic attenuation coefficient of the quantum vacuum medium.

By employing the standard definition of cosmological redshift, $z=({\lambda }_{\mathrm{obs}}-{\lambda }_{\mathrm{emit}})/{\lambda }_{\mathrm{emit}}$, and invoking the Planck–Einstein relation $E=hc/\lambda$, we establish the exact relation:

$$1+z={\displaystyle \frac{E_0}{E\left(r\right)}}=e^{\alpha r}$$

(4)

For relatively small propagation distances ($r\ll {\alpha }^{-1}$), applying a first-order Taylor expansion ($e^x\approx 1+x$) yields the following:

$$1+z\approx 1+\alpha r\Rightarrow z\approx \alpha r$$

(5)

By matching this linear approximation directly with the empirical Hubble law for local distances ($z\approx {\displaystyle \frac{H_0}{c}}r$), we map the cosmological parameter directly onto the intrinsic vacuum attenuation coefficient:

$$\alpha ={\displaystyle \frac{H_0}{c}}$$

(6)

To provide a concrete electrodynamic basis, we examine the propagation of electromagnetic waves within a medium possessing a finite effective electrical conductivity ${\sigma }_0$. Under the condition of a weak effective conductor (${\sigma }_0\ll \omega {\epsilon }_0$, where ${\epsilon }_0$ represents the vacuum permittivity), the standard Maxwellian amplitude attenuation coefficient is analytically derived as follows:

$${\alpha }_{\mathrm{amp}}={\displaystyle \frac{{\sigma }_0}{2}}\sqrt{{\displaystyle \frac{{\mu }_0}{{\epsilon }_0}}}={\displaystyle \frac{{\sigma }_0{Z}_0}{2}}$$

(7)

where $Z_0\approx 377\mathsf{\Omega }$ represents the characteristic impedance of free space.

Since the total energy density u of an electromagnetic wave is proportional to the square of its electric field amplitude ($u\propto {|\mathbf{E}|}^2$), the energy attenuation coefficient $\alpha$ is precisely twice the amplitude attenuation coefficient ($\alpha =2{\alpha }_{\mathrm{amp}}$):

$$\alpha =2{\alpha }_{\mathrm{amp}}={\sigma }_0{Z}_0$$

(8)

Crucially, Equation (8) demonstrates that the macro-scale attenuation coefficient $\alpha$ remains entirely independent of the photon frequency $\nu$, provided the weak-conduction regime (${\sigma }_0\ll \omega {\epsilon }_0$) holds true.

By unifying Equation (6) with Equation (8), we establish a direct identity bridging cosmology and electrodynamics:

$${\displaystyle \frac{H_0}{c}}={\sigma }_0{Z}_0\Rightarrow {\sigma }_0={\displaystyle \frac{H_0}{c{Z}_0}}$$

(9)

Substituting the standard observational baseline values for the Hubble constant ($H_0\approx 2.27\times {10}^{-18}{\mathrm{s}}^{-1}$), the speed of light c, and space impedance $Z_0$, we compute the baseline effective conductivity of the cosmological vacuum background:

$${\sigma }_0\approx {\displaystyle \frac{2.27\times {10}^{-18}{\mathrm{s}}^{-1}}{(3\times {10}^8\mathrm{m}{\mathrm{s}}^{-1})\times 377\mathsf{\Omega }}}\approx 2\times {10}^{-29}\mathrm{S}{\mathrm{m}}^{-1}$$

(10)

2.2. Dissipation Power and Temporal Interpretation

By modeling the vacuum as a medium with effective conductivity ${\sigma }_0$, we invoke Ohm’s and Joule’s electrodynamic laws. As an electromagnetic wave traverses this medium, the time-averaged power dissipated per unit volume P is given by $P={\sigma }_0{|\mathbf{E}|}^2$. Given the electromagnetic energy density $u={\epsilon }_0{|\mathbf{E}|}^2$, the fractional rate of energy loss per unit time within the vacuum medium is as follows:

$${\displaystyle \frac{P}{u}}={\displaystyle \frac{{\sigma }_0{|\mathbf{E}|}^2}{{\epsilon }_0{|\mathbf{E}|}^2}}={\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}$$

(11)

Transforming the spatial exponential decay law into the time domain via $r=ct$, the fractional energy loss per unit time is expressed as follows:

$${\displaystyle \frac{1}{E}}{\displaystyle \frac{dE}{dt}}=-c\alpha =-H_0$$

(12)

Equation (12) reveals that the characteristic timescale for cosmological photon energy decay is directly governed by $H_0^{-1}$. Substituting ${\sigma }_0$ from Equation (9) into Equation (11) yields the following:

$${\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}={\displaystyle \frac{H_0}{c{Z}_0{\epsilon }_0}}$$

(13)

Recalling that $Z_0={\left(c{\epsilon }_0\right)}^{-1}$, the denominator simplifies identically to unity ($c{Z}_0{\epsilon }_0=1$), leaving:

$${\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}=H_0$$

(14)

This analytical cancellation demonstrates that the continuous rate of photon energy dissipation within this weakly conductive vacuum corresponds identically to the redshift rate governed by $H_0$.

2.3. The Quantum Gravitational Relationship and the Dissipation Unit

To bridge this classical electrodynamic description with quantum mechanics, we transition to discrete photon states, revealing a universal quantum dissipative signature.

2.3.1. Energy Loss per Oscillation Cycle

The intrinsic energy of a photon is governed by $E_p=h\nu$. Utilizing the temporal decay rate in Equations (12) and (14), the instantaneous power dissipated by an individual photon ($P_p$) is as follows:

$$P_p=E_p·H_0=\left(h\nu \right)·H_0$$

(15)

For a single wave period $T={\nu }^{-1}$, the net quantum of energy dissipated ($\Delta E$) over exactly one full oscillation period is analytically evaluated as follows:

$$\Delta E=P_p·T=(h\nu ·H_0)·{\displaystyle \frac{1}{\nu }}=h{H}_0$$

(16)

2.3.2. Alternative Derivation via Instantaneous Power Loss

Alternatively, integrating the instantaneous differential power loss $P={\displaystyle \frac{dE}{dt}}=-H_{0}E=-H_{0}h\nu$ over a single period $T={\nu }^{-1}$ directly recovers the same energy decrement:

$$\Delta E=|P|·T=h{H}_0$$

(17)

2.3.3. Physical Significance of the Quantum Dissipation Constant $h{H}_0$

The exact convergence between Equations (16) and (17) carries profound physical significance:

• Universality: Every individual photon, invariant of its initial frequency or spectral band (from Gamma-rays to Radio waves), loses precisely the exact same quantum of energy, $h{H}_0$, per single oscillation cycle.
• Numerical Scale: Evaluating this cosmic constant using baseline physical parameters yields the following:

  $$h{H}_0\approx (6.626\times {10}^{-34}\mathrm{J}\mathrm{s})\times (2.27\times {10}^{-18}{\mathrm{s}}^{-1})\approx 1.5\times {10}^{-51}\mathrm{J}$$

  (18)
  This invariant value represents the fundamental “minimum energy dissipation unit” of the physical cosmos.
• Continuity: Because this mathematical formulation relies on the effective continuous conductivity ${\sigma }_0$ of the vacuum medium rather than discrete particle scattering events, it inherently guarantees a smooth, continuous spectral evolution free from angular diffusion or image degradation.

Directly mapping the Planck constant onto the vacuum parameters ${\sigma }_0$ and ${\epsilon }_0$ via Equation (14) gives the unified quantum electrodynamic expression:

$$\Delta E=h{\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}$$

(19)

Equation (19) confirms that the active quantum vacuum abstracts energy from propagating electromagnetic waves at an exact rate dictated exclusively by h and the localized material parameters of the vacuum medium.

2.4. Summary

The derivations presented in this section demonstrate that the effective electrical conductivity of the vacuum ${\sigma }_0$ is a foundational material property of the spacetime medium. This framework redefines the Hubble constant from a purely geometric expansion rate into a measurable quantum material property of an active vacuum medium. While ${\epsilon }_0$ and ${\mu }_0$ dictate the constant velocity profile of light ($c={\left[{\mu }_0{\epsilon }_0\right]}^{-1/2}$), the environmental parameter ${\sigma }_0\left(R\right)$ acts as the physical factor governing long-range energy attenuation across the cosmos.

3. Covariant Formulation and Local Gravitational Influence

Based on the unified electrocosmological identity derived in Equation (9), the Hubble parameter is fundamentally governed by the local effective conductivity of the quantum vacuum:

$$H_0\left(R\right)=c{Z}_0·{\sigma }_0\left(R\right)$$

(20)

To ensure full general covariance and coordinate independence, the coupling between the active quantum vacuum and the background spacetime geometry must be formulated using invariant curvature scalars rather than coordinate-dependent 3-accelerations. The local energy-momentum tensor acts as an external energetic source that polarizes the underlying vacuum ground state. This field-induced excitation increases the localized vacuum energy density, thereby enhancing the probability amplitude for the spontaneous manifestation of virtual electron–positron ($e^{+}{e}^{-}$) pairs per unit volume.

To formalize this mechanism in a diffeomorphism-invariant framework, we model the local effective vacuum conductivity ${\sigma }_0\left(R\right)$ as a function of the local Ricci curvature scalar R (or equivalently, through Einstein’s field equations $R=-{\displaystyle \frac{8\pi G}{c^4}}T$, where $T=g^{\mu \nu }{T}_{\mu \nu }$ represents the trace of the local energy-momentum tensor):

$${\sigma }_0\left(R\right)={\sigma }_{0,\mathrm{vac}}\left(1+\xi ·{\mathsf{\ell }}_P^{2}R\right)$$

(21)

where ${\sigma }_{0,\mathrm{vac}}$ represents the fundamental baseline conductivity of the unperturbed, flat cosmic void vacuum ($R\to 0$), ${\mathsf{\ell }}_P=\sqrt{\hslash G/c^3}\approx 1.616\times {10}^{-35}\mathrm{m}$ is the fundamental Planck length, R carries proper dimensions of $\left[{\mathrm{length}}^{-2}\right]$, and $\xi$ is a dimensionless vacuum susceptibility coupling constant.

Equation (21) rigorously satisfies coordinate invariance under general coordinate transformations and preserves exact dimensional consistency, directly resolving the theoretical limitations of classical acceleration formulations. Consequently, the Hubble parameter $H_0\left(R\right)$ is stripped of its status as a universal global constant. Instead, it exhibits structural spatial variation: its value peaks in regions of intense gravitational curvature—such as within or near massive galaxy clusters and cosmic filaments—and falls to a minimum inside immense cosmic voids.

This environmental dependency provides a natural, non-kinematic resolution to the long-standing cosmological conundrum known as the “Hubble Tension”. The observed statistical discrepancy is not an artifact of systemic measurement errors, but a real physical consequence of light traversing distinct local gravitational curvature environments populated by varying virtual particle densities:

• Early/Global Universe Baseline (CMB Measurements): The cosmic microwave background (CMB) photons integrate information over global, highly homogeneous cosmological scales. Over these vast trajectories, the intermediate space is dominated by expansive cosmic voids where the average spacetime curvature is exceptionally weak ($R\to 0$). This yields the baseline unperturbed vacuum conductivity ${\sigma }_{0,\mathrm{vac}}$, corresponding precisely to the lower value reported by the Planck Collaboration [4]:

  $$H_0^{\mathrm{global}}=67.4\pm 0.5\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$$

  (22)
• Late/Local Universe Over-densities (Distance Ladder Measurements): Conversely, localized measurements utilizing Type Ia Supernovae (SNe Ia) and Cepheid variables inherent to the local distance ladder are confined within the dense structures of the local cosmic web. Photons emitted from these local sources must traverse intense gravitational wells generated by nearby massive over-densities, including the Local Supercluster and newly mapped superstructures [10]. This localized environment induces a significant increase in the Ricci curvature scalar R and virtual pair density $n_v\left(R\right)$, inflating the effective conductivity to ${\sigma }_0\left(R\right)$. This manifests empirically as the higher local value confirmed by the Local Distance Network consensus [5]:

  $$H_0^{\mathrm{local}}=73.50\pm 0.81\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$$

  (23)

The divergence between the global baseline of $67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and the localized value of $73.50\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ represents a physical spatial gradient in vacuum attenuation. Recent high-precision observational analyses of local cosmic flows and direct distance catalog cross-calibrations, such as those reported by Stiskalek et al. [11] utilizing the CosmicFlows-4 database, alongside constraints from the Pantheon+ collaboration [12] and the SH0ES team [13], consistently reveal ambient environmental effects in the local expansion profile. These empirical distributions map perfectly onto local matter over-densities, strongly reinforcing the hypothesis that the dissipation rate of photon energy is directly modulated by the localized gravitational curvature configuration of the spacetime medium. Theoretical Status and Tension with Standard QED: It is essential to clarify the precise theoretical status of the effective vacuum conductivity ansatz ${\sigma }_0\left(R\right)={\sigma }_{0,\mathrm{vac}}\left(1+\xi ·{\mathsf{\ell }}_P^{2}R\right)$ presented in Equation (21). While the wavepacket envelope stretching dynamics ($d\overline{\nu }/dt=-H_0\overline{\nu }$) are derived using standard QED operator commutation relations, the curvature-dependent coupling ${\sigma }_0\left(R\right)$ itself remains a purely phenomenological postulate. In standard, flat-spacetime Quantum Electrodynamics, photon-photon interaction via virtual $e^{+}{e}^{-}$ pairs (Euler-Heisenberg Lagrangian) does not yield an effective macroscopic vacuum dissipation rate proportional to spacetime curvature R. Therefore, Equation (21) represents an effective field-theoretic parametrization designed to capture heuristic vacuum polarization in non-zero gravitational curvature, rather than a rigorous result derived from first-principles Standard Model QED.

4. Microscopic QED Derivation: Wavepacket Redshift vs. Particle Absorption

A crucial physical question concerns the transition from classical Maxwellian attenuation to quantum single-photon energy reduction. In classical electrodynamics, propagation through a weakly conductive medium (${\sigma }_0$) leads to amplitude decay. In standard quantum optics, macroscopic absorption typically reduces the total photon number N while surviving photons retain their energy $h\nu$. Here, we demonstrate how continuous photon frequency dissipation arises naturally in a non-trivial quantum vacuum without photon destruction or angular scattering.

To establish this derivation rigorously from Coulomb-gauge Maxwell equations, consider the vector potential operator $\widehat{\mathbf{A}}(\mathbf{r},t)$ propagating in a conductive vacuum (${\sigma }_0\ll \omega {\epsilon }_0$):

$${\nabla }^2\widehat{\mathbf{A}}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2\widehat{\mathbf{A}}}{\partial t^2}}-{\mu }_0{\sigma }_0{\displaystyle \frac{\partial \widehat{\mathbf{A}}}{\partial t}}=0$$

(24)

We represent a single-photon state as a normalized wavepacket state:

$$|1_{\mathsf{\Phi }}\left(t\right)⟩=\int d\omega \mathsf{\Phi }(\omega ,t)a^{†}\left(\omega \right)|0⟩$$

(25)

The total photon number operator $\widehat{N}=\int d\omega a^{†}\left(\omega \right)a\left(\omega \right)$ satisfies strict normalization over time:

$$⟨1_{\mathsf{\Phi }}\left(t\right)|\widehat{N}|1_{\mathsf{\Phi }}{\left(t\right)⟩=\int d\omega |\mathsf{\Phi }(\omega ,t)|}^2=1$$

(26)

The complex dispersion relation resulting from Equation (24) gives $k\left(\omega \right)={\displaystyle \frac{\omega }{c}}\sqrt{1+i{\displaystyle \frac{{\sigma }_0}{\omega {\epsilon }_0}}}\approx {\displaystyle \frac{\omega }{c}}+i{\displaystyle \frac{{\sigma }_0}{2{\epsilon }_{0}c}}$. Unlike discrete atomic absorbers that cause incoherent state collapse, the polarized vacuum acts via forward-coherent phase matching ($k_{\mathrm{scatter}}=k_{\mathrm{incident}}$). Consequently, the interaction Hamiltonian commutes with the photon number operator ($[{\widehat{H}}_{\mathrm{eff}},\widehat{N}]=0$), prohibiting photon destruction or angular scattering out of the beam path.

The continuous rate of EM field energy density attenuation $U\left(t\right)$ matches Equation (14):

$${\displaystyle \frac{dU}{dt}}=-{\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}U\left(t\right)=-H_{0}U\left(t\right)\mathrm{where}{H}_0\equiv {\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}$$

(27)

The total quantum energy expectation value of the single-photon wavepacket is given by the following:

$$⟨\widehat{H}⟩=\hslash \overline{\omega }\left(t\right)\int d\omega {|\mathsf{\Phi }(\omega ,t)|}^2=\hslash \overline{\omega }\left(t\right)$$

(28)

Applying Equation (27) to Equation (28) while maintaining state normalization in Equation (26) forces the temporal evolution of the expectation central frequency $\overline{\omega }\left(t\right)$:

$${\displaystyle \frac{d}{dt}}\left(\hslash \overline{\omega }\left(t\right)\right)=-H_0\left(\hslash \overline{\omega }\left(t\right)\right)\Rightarrow {\displaystyle \frac{d\overline{\nu }}{dt}}=-H_0\overline{\nu }\Rightarrow \overline{\nu }\left(t\right)={\nu }_0{e}^{-H_{0}t}$$

(29)

Equation (29) delivers the explicit microscopic derivation: macroscopic vacuum conductivity ${\sigma }_0$ drives a deterministic adiabatic stretching of the wavepacket envelope $\mathsf{\Phi }(\omega ,t)$, continuously reducing the central quantum frequency $\nu \left(t\right)$ while preserving photon number $N=1$ without scattering or spectral line broadening.

5. Physical Interpretation, Conservation Laws, and Foundational Limitations

5.1. Global Energy Accounting and Comparison with Geometric Redshifts

To address complete global energy conservation, we formulate the coupled Hamiltonian of the single-photon wavepacket and the quantum vacuum medium. We explicitly distinguish this dissipative redshift model from non-dissipative geometric redshifts, such as gravitational redshift analyzed by Vavryčuk [14]. In gravitational redshift, frequency shifts arise purely from metric variations ($\sqrt{-g_{00}}$) without local energy transfer between the photon and its environment. In contrast, our vacuum interaction model represents an explicit energy transfer driven by vacuum polarization in non-zero curvature R.

The global Hamiltonian of the coupled system is strictly conserved:

$${\widehat{H}}_{\mathrm{total}}={\widehat{H}}_{\mathrm{EM}}+{\widehat{H}}_{\mathrm{vac}}+{\widehat{H}}_{\mathrm{int}},\mathrm{with}{\displaystyle \frac{d}{dt}}⟨{\widehat{H}}_{\mathrm{total}}⟩=0$$

(30)

The physical transfer process operates as follows: as the photon wavepacket propagates, its electric field performs work on the vacuum polarization current ${\widehat{\mathbf{j}}}_{\mathrm{vac}}={\sigma }_0\widehat{\mathbf{E}}$ composed of virtual $e^{+}{e}^{-}$ pairs. The rate of energy extraction from the EM field is as follows:

$${\displaystyle \frac{d}{dt}}⟨{\widehat{H}}_{\mathrm{EM}}⟩=-{\int }_V⟨{\widehat{\mathbf{j}}}_{\mathrm{vac}}·\widehat{\mathbf{E}}⟩dV=-{\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}⟨{\widehat{H}}_{\mathrm{EM}}⟩=-H_0⟨{\widehat{H}}_{\mathrm{EM}}⟩$$

(31)

This continuously transferred energy is absorbed into collective low-energy excitations of the quantum vacuum (${\widehat{H}}_{\mathrm{vac}}+{\widehat{H}}_{\mathrm{int}}$). Because the energy loss per wave period is infinitesimally small ($\Delta E=h{H}_0\approx 1.5\times {10}^{-51}\mathrm{J}/\mathrm{cycle}$), it cannot excite real particle pairs ($2m_e{c}^2\approx 1.02\mathrm{MeV}$). Instead, this energy dissipates coherently into the ultra-low-frequency continuum background, ensuring:

$${\displaystyle \frac{d}{dt}}⟨{\widehat{H}}_{\mathrm{vac}}+{\widehat{H}}_{\mathrm{int}}⟩=+H_0⟨{\widehat{H}}_{\mathrm{EM}}⟩\Rightarrow {\displaystyle \frac{d}{dt}}⟨{\widehat{H}}_{\mathrm{total}}⟩=0$$

(32)

Explicit Dynamic Derivation of Per-Cycle Energy Dissipation

To provide a fully explicit dynamic derivation showing why the vacuum absorbs precisely $h{H}_0$ per cycle, we evaluate the work performed by the photon field on the vacuum polarization current ${\mathbf{j}}_{\mathrm{vac}}={\sigma }_0\mathbf{E}$ over one complete electromagnetic cycle $T=1/\nu =2\pi /\omega$.

Integrating the power density ${\mathbf{j}}_{\mathrm{vac}}·\mathbf{E}={\sigma }_0{|\mathbf{E}|}^2$ over the wavepacket volume V and period T gives the following:

$$\Delta E_{\mathrm{cycle}}={\int }_0^T\left({\int }_V{\sigma }_0{|\mathbf{E}(\mathbf{r},t)|}^{2}dV\right)dt$$

(33)

For a single-photon wavepacket, total energy equipartition dictates the cycle-averaged dissipation rate. Substituting ${\sigma }_0/{\epsilon }_0=H_0$ into Equation (33) and integrating over one full cycle $T=2\pi /\omega$ yields the following:

$$\Delta E_{\mathrm{cycle}}=2{\displaystyle \frac{{\sigma }_0}{{\epsilon }_0}}\hslash \omega {\int }_0^{{\displaystyle \frac{2\pi }{\omega }}}{\sin }^2\left(\omega t\right)dt=2H_0\hslash \omega \left({\displaystyle \frac{\pi }{\omega }}\right)=2\pi H_0\hslash$$

(34)

Recalling $\hslash =h/\left(2\pi \right)$, Equation (34) simplifies directly to the following:

$$\Delta E_{\mathrm{cycle}}=2\pi H_0\left({\displaystyle \frac{h}{2\pi }}\right)=h{H}_0$$

(35)

Equation (35) mathematically demonstrates the fundamental origin of the per-cycle energy loss: while higher-frequency photons exert greater field forces ($\propto \hslash \omega$), the duration of one cycle shrinks proportionally ($\propto 1/\omega$). The exact cancellation of $\omega$ in the time integral over T proves that $h{H}_0$ is a universal physical constant of work performed against the characteristic vacuum impedance $Z_0$ in every single oscillation.

5.2. Strict Alignment with Modern Observational Constraints

Any viable alternative to metric expansion must satisfy severe, high-precision observational constraints accumulated over recent decades. Our model is mathematically engineered to naturally satisfy these boundaries without fine-tuning:

• Absence of Angular Image Blurring: Unlike traditional scattering hypotheses where energy loss is accompanied by momentum deflection, the non-scattering, phase-coherent formulation derived in Section 2 rigorously preserves the photon’s original wave vector direction ($\mathbf{k}$). Consequently, the angular resolution and sharp structural boundaries of high-redshift galaxies observed by instruments like the James Webb Space Telescope (JWST) remain perfectly intact over cosmic distances.
• Spectral and Phase Coherence: Because the attenuation is formulated as a smooth, continuous deterministic process governed by global vacuum material constants (${\sigma }_0,{\epsilon }_0$) rather than stochastic discrete particle collisions, it entirely avoids frequency broadening or the degradation of spectral line profiles, maintaining perfect phase coherence along the line of sight.

5.3. Substantive Comparative Analysis with Prior Tired-Light Frameworks

To contextualize this mechanism against earlier non-expansion approaches, we compare our model with notable photon-loss proposals, including Crawford (1979) [15], LaViolette (1986) [16], Brynjolfsson (2004) [17], and Lerner et al. (2014) [18].

Early tired-light hypotheses [7] were disfavored due to predicted angular scattering (image blurring) or failure to account for supernova light-curve time dilation. Subsequent proposals introduced distinct energy loss mechanisms:

• Crawford (1979) [15]: modeled photon decay in curved spacetime, but lacked a gauge-invariant QED interaction framework, leaving open angular deflection.
• LaViolette (1986) [16]: utilized subquantum kinetics to model non-zero energy dissipation, operating outside standard quantum field theory.
• Brynjolfsson (2004) [17]: attributed redshift to plasma redshift in hot intergalactic plasma. However, this depends strongly on electron density $n_e$, struggling to yield a universal Hubble constant across cosmic voids.
• Lerner et al. (2014) [18]: performed empirical Tolman tests on UV galaxy data, serving as observational verification rather than a microscopic field-theoretic derivation.

Our proposed Vacuum Interaction Model overcomes these bottlenecks through three structural features summarized in Table 1: (1) Forward-coherent phase matching prevents angular blurring; (2) General-covariant coupling via Ricci scalar R preserves diffeomorphism invariance; (3) Wavepacket envelope dispersion yields natural supernova light-curve time dilation while local conductivity ${\sigma }_0\left(R\right)$ resolves the $7.1\sigma$$H_0$ tension.

Table 1. Comparative matrix between historic photon-loss proposals and the present Ricci-modulated vacuum interaction model.

Model/Framework	Microscopic Mechanism	Angular Blurring?	Covariant R Coupling?	Supernova Time Dilation?	Resolves $7.1\mathsf{\sigma }$ ${\mathit{H}}_0$ Tension?
Crawford (1979) [15]	Curved space decay	Potential	Partial	Unresolved	No
LaViolette (1986) [16]	Subquantum kinetics	No	No	Phenomenological	No
Brynjolfsson (2004) [17]	Plasma scattering	Yes ($n_e$-dependent)	No	No	No
Lerner et al. (2014) [18]	Empirical Tolman test	No	N/A (Empirical)	Phenomenological	No
Present Model	QED Vacuum Polarization ${\sigma }_0\left(R\right)$	Strictly No	Yes (Ricci Scalar R)	Derived ($(1+z)$ stretch)	Yes (Environmental ${\sigma }_0\left(R\right)$)

6. Observational Constraints: Time Dilation and the Tolman Surface Brightness Test

Any viable cosmological framework must demonstrate consistency with precision observational tests, most notably Supernova Type Ia light-curve time dilation and Tolman surface brightness measurements.

6.1. Rigorous Mathematical Derivation of Supernova Light-Curve Time Dilation

Observational studies of distant Type Ia Supernovae (SNe Ia) confirm that observed light-curve durations expand by $(1+z)$. Here we derive this transformation explicitly from QED wavepacket dispersion.

Consider two successive photon wavepacket envelopes emitted at $t_{\mathrm{emit}}^{\left(1\right)}$ and $t_{\mathrm{emit}}^{\left(2\right)}$ across an event duration $\Delta t_{\mathrm{emit}}$ from a source at distance D. The spatial rate of envelope temporal broadening as the wavepacket traverses an increment $dr$ is governed by the dissipation scale $H_0/c$:

$${\displaystyle \frac{d(\Delta t)}{dr}}={\displaystyle \frac{H_0}{c}}\Delta t\left(r\right)$$

(36)

Integrating Equation (36) over the propagation path from $r=0$ to $r=D$ yields the following:

$${\int }_{\Delta t_{\mathrm{emit}}}^{\Delta t_{\mathrm{obs}}}{\displaystyle \frac{d(\Delta t)}{\Delta t}}={\int }_0^D{\displaystyle \frac{H_0}{c}}dr={\displaystyle \frac{H_{0}D}{c}}$$

(37)

Exponentiating both sides of Equation (37) and recalling $1+z=\exp (H_{0}D/c)$ gives the followning:

$${\displaystyle \frac{\Delta t_{\mathrm{obs}}}{\Delta t_{\mathrm{emit}}}}=\exp \left({\displaystyle \frac{H_{0}D}{c}}\right)=1+z\Rightarrow \Delta t_{\mathrm{obs}}=(1+z)\Delta t_{\mathrm{emit}}$$

(38)

Equation (38) provides the rigorous quantitative derivation requested: observed light-curve durations are stretched by exactly $(1+z)$ due to QED envelope dispersion during transit, replicating DES observational data (White et al., 2024 [19]) without spatial expansion.

6.2. Tolman Surface Brightness Scaling and Empirical Alignment with Lubin & Sandage (2001)

We next evaluate the Tolman surface brightness test ($SB=F/\mathsf{\Omega }$). In a static geometry driven by vacuum conductivity, angular area $\mathsf{\Omega }$ is determined by Euclidean geometry, without dynamic metric area stretching. The intrinsic surface brightness attenuation is governed by two fundamental factors:

1.

Single-Photon Energy Dissipation: Central quantum frequency decay reduces photon energy by ${(1+z)}^{-1}$.

2.

Wavepacket Arrival Dilation: QED envelope dispersion in Equation (38) lowers the photon arrival flux rate by ${(1+z)}^{-1}$.

Combining these two intrinsic factors yields the baseline surface brightness law:

$$S{B}_{\mathrm{ideal}}\left(z\right)=S{B}_0\times {(1+z)}^{-1}\times {(1+z)}^{-1}=S{B}_0{(1+z)}^{-2}$$

(39)

Physical Environment Effects and Re-Evaluation of Lubin & Sandage (2001) [20]

Equation (39) establishes the exact theoretical baseline $SB\propto {(1+z)}^{-2}$ in an ideal, unperturbed vacuum, where vacuum polarization continuously stretches the wavepacket envelope without photon destruction. In realistic cosmic environments, propagating light traverses intergalactic dust, neutral gas, and ionized plasma, introducing an additional, path-dependent optical depth extinction $\tau \left(z\right)\propto \ln (1+z)$. Consequently, the observed attenuation exponent n in $SB\propto {(1+z)}^{-n}$ naturally increases modestly above the baseline value of $n=2.0$, varying slightly depending on the localized line-of-sight matter density.

Crucially, this physical interplay provides a compelling explanation for the raw observational data obtained by Lubin and Sandage (2001) [20]. Using Hubble Space Telescope (HST) measurements of cluster galaxies up to $z\approx 0.9$, Lubin and Sandage recorded uncorrected raw exponents of $n\approx 2.6\mathrm{to}3.0$ ($n=2.59\pm 0.17$ in R-band and $n=3.37\pm 0.13$ in I-band). Within our framework, measured values in the range $n\approx 2.6–3.0$ represent the expected physical outcome of the ideal $n=2.0$ baseline supplemented by realistic intergalactic extinction ($n>2.0$).

Conversely, standard expanding $\mathsf{\Lambda }\mathrm{CDM}$ cosmology predicts $n=4.0$ for an ideal expanding universe. When realistic environmental extinction (dust, gas, and plasma) is incorporated into an expanding spacetime, the expected observational exponent must strictly exceed four ($n>4.0$). The empirical finding that raw observational data fall significantly below four ($n\approx 2.6–3.0$) directly contradicts the pure expansion baseline, forcing expanding models to postulate strong, ad hoc luminosity evolution $E\left(z\right)\sim {(1+z)}^{1.5}$ to reconcile theory with observation. In contrast, our Vacuum Interaction model naturally encompasses the raw observational signal without requiring contrived galaxy evolution assumptions.

7. Conclusions

In this paper, we have presented a phenomenological vacuum-interaction model that identifies the effective electrical conductivity of the quantum vacuum, ${\sigma }_0$, as the physical mechanism underlying cosmic redshift. By demonstrating that propagating photons lose an invariant quantum of energy $h{H}_0\approx 1.5\times {10}^{-51}\mathrm{J}$ per oscillation, we established a direct analytical link between electrodynamics, quantum mechanics, and observational cosmology via $H_0={\sigma }_0\left(R\right)c{Z}_0$. We reformulated the vacuum environmental dependency in terms of the general-covariant Ricci curvature scalar R, demonstrating that $H_0\left(R\right)=c{Z}_0{\sigma }_{0,\mathrm{vac}}(1+\xi {\mathsf{\ell }}_P^{2}R)$ provides a fully coordinate-invariant resolution to the $7.1\sigma$ Hubble tension.

7.1. Cosmological Constraints: CMB Acoustic Peaks and Baryon Acoustic Oscillations

While the present Vacuum Interaction Model successfully accounts for cosmic redshift, supernova light-curve time dilation, the Tolman surface brightness test, and provides a localized resolution to the $H_0$ tension, it currently faces important cosmological limitations regarding key precision observables:

• CMB Acoustic Peaks: In the standard $\mathsf{\Lambda }\mathrm{CDM}$ paradigm, the angular power spectrum of the Cosmic Microwave Background (CMB) exhibits characteristic acoustic peaks generated by sound waves in the early photon-baryon fluid prior to recombination. In our non-expanding, static-geometry framework, the CMB is interpreted as the asymptotic thermal degradation state of cosmic radiation. However, a quantitative perturbation theory capable of reproducing the multipole acoustic spectrum ($C_{\mathsf{\ell }}$) without cosmic expansion has not yet been formulated.
• Baryon Acoustic Oscillations (BAO): Similarly, the characteristic preferred clustering scale of galaxies ($\sim 150\mathrm{Mpc}$) observed in BAO surveys is naturally explained in $\mathsf{\Lambda }\mathrm{CDM}$ as a frozen sound horizon. The present model does not yet incorporate a structure-formation perturbation framework to predict BAO feature locations.

Consequently, both CMB acoustic peaks and BAO remain explicitly unaccounted for in the current stage of this model, representing critical avenues for future theoretical development.

7.2. Proposed Falsifiable Empirical Test: Environment-Stratified Supernova Analysis

To decisively distinguish the Vacuum Interaction Model from the standard $\mathsf{\Lambda }\mathrm{CDM}$ expanding paradigm, we propose a sharp, falsifiable observational test based on current Type Ia Supernova (SNe Ia) catalogs (e.g., Pantheon+/CosmicFlows-4):

• Environmental Stratification: SNe Ia at moderate redshifts ($0.01<z<0.15$) should be partitioned into two distinct observational sub-samples based on the line-of-sight matter density environment:
  • Sub-sample A (Void Paths): SNe Ia whose optical propagation paths traverse extensive cosmic voids (low local matter density, $R\to 0$).
  • Sub-sample B (Filament/Cluster Paths): SNe Ia whose sight-lines cross dense galactic filaments, superclusters, or over-dense cosmic web structures ($R>0$).
• Falsifiable Prediction: According to Equation (21), photons traversing void environments experience lower effective conductivity (${\sigma }_{0,\mathrm{vac}}$), whereas those traversing over-densities encounter higher dissipation (${\sigma }_0\left(R\right)$). Therefore, our model strictly predicts that the inferred Hubble parameter from Sub-sample A must be systematically lower than that from Sub-sample B ($H_0^{\mathrm{void}}<H_0^{\mathrm{cluster}}$).

If high-precision cross-matched SNe Ia data reveal no statistically significant difference in $H_0$ between void-dominated and cluster-dominated propagation paths ($H_0^{\mathrm{void}}\approx H_0^{\mathrm{cluster}}$), the core curvature-dependent mechanism of this model would be definitively falsified.

Additionally, extended surface brightness measurements of high-redshift cluster galaxies using modern JWST/HST imaging can serve as a complementary test, where empirical exponents staying in the range $n\approx 2.0-3.0$ without requiring strong, ad hoc luminosity evolution $E\left(z\right)$ would further reinforce the non-expansion vacuum attenuation baseline over pure geometric space expansion.

Furthermore, we provided a microscopic QED wavepacket derivation showing that vacuum energy absorption manifests as adiabatic envelope stretching and central frequency decay $h\Delta \nu$ rather than photon flux destruction or angular scattering. This wavepacket dispersion physics naturally recovers the observed $(1+z)$ supernova light-curve time dilation without spacetime expansion, aligning with precision constraints such as those from the Dark Energy Survey [19], while offering a coherent physical basis for raw Tolman surface brightness scaling [20]. This framework offers a testable and mathematically coherent non-expansion perspective on photon propagation across the cosmos.