The Photon Paradox - Why Dark Matter May Be Hidden in Ordinary Light - LaTeX Code Preservation (1)

Link Publication: https://zenodo.org/records/20744751

Hinweis zur Dualität
Dieses Werk wird als grundlegende Neubewertung der Raum-Zeit-Kosmologie sowie als technischer Rahmen für die Nutzung von Bulk-Energie präsentiert. Unter der Lizenz CC BY-SA 4.0 wird dieses Wissen zum gemeinsamen Erbe der Menschheit erklärt.

Notice of Duality
This work is presented as a fundamental re-evaluation of spacetime cosmology and a technical
framework for bulk-energy utilization. Licensed under CC BY-SA 4.0, this knowledge is declared
part of the common heritage of the human species.

Part I

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% --- Titel und Metadaten (Update Version 1.9) ---
\title{\textbf{The Photon Paradox -- Why Dark Matter Could Be Hidden in Ordinary Light} \
\large}
\author{
\textbf{Cord Uebermuth} \
\small Independent Researcher \
\small ORCID: \href{https://orcid.org/0000-0002-2638-5995}{0000-0002-2638-5995}
}
\date{\small 14 June 2026 \[0.5em] \textbf{Version 1.9} \ \small Zenodo Preprint}

\begin{document}

\maketitle

% --- Copyright Notice ---
\begin{center}
\small \copyright{} 2026 Cord Uebermuth. All rights reserved.
\end{center}
\vspace{1.5em}

\section*{Notice of Duality}
This work is presented as a fundamental re-evaluation of spacetime cosmology and a technical framework for bulk-energy utilization. Licensed under CC BY-SA 4.0, this knowledge is declared part of the common heritage of the human species.
\vspace{2em}

% --- Abstract (V1.9: Defensiv, inklusive Astrosphären-Mapping & CMG) ---
\begin{abstract}
This paper discusses a potential resolution of a central field-theoretic tension regarding the photon rest mass: the apparent discrepancy between a theoretically motivated intergalactic photon mass in the range of $3 \times 10^{-3}$~eV and the stringent experimental upper bounds established by terrestrial laboratory experiments. To address this, a formal density- and radiation-flux-dependent screening mechanism within the Photon-String-Higgs (PSH) framework is analyzed. Rather than invoking additional scalar fields or new elementary particles, the mechanism is modeled via the environmental dynamics of local baryonic and radiative energy densities, coupled to the cosmic star formation history. The transition between the screened and unscreened regimes is parameterized by a specific critical density, designated as the Ilias-Lucida threshold ($\rho_{\rm IL}$). It is demonstrated that such a framework can model a spatially-sigmoid phase transition of the effective photon mass, which, for the solar system environment, correlates with the vicinity of the heliopause ($r \approx 120$~\text{AU}). The resulting theoretical density profiles are compared to the in-situ plasma wave science (PWS) data obtained by the \textit{Voyager 1} and \textit{Voyager 2} spacecraft, illustrating a plausible qualitative and quantitative agreement. Furthermore, the framework offers a potential predictive utility for mapping the field-based boundaries of stellar astrospheres and higher-scale galactic domains, alongside their respective cosmic moisture gradients (CMG). Finally, the cosmological implications of this mechanism are examined in the context of the early universe, referencing the 21-cm absorption anomaly of the EDGES signal at a redshift of $z \approx 20$ as a potential observational anchor for early mass condensation driven by emergent PSH seed structures.
\end{abstract}
\vspace{2em}

% --- NOTICE OF DUALITY ---

\section*{Notice of Duality}
\textit{Notice of Duality:} This work is presented simultaneously as a fundamental re-evaluation of spacetime cosmology and as a technical framework for the utilization of primary bulk-energy. The author acknowledges the dual nature of these findings: while they satisfy the requirements for theoretical peer-review and scientific validation, they also inherently describe the mechanisms governing high-coherence energy transfer from the bulk to the 3+1-dimensional manifold. By publishing this under a Creative Commons (CC BY 4.0) license, the author explicitly renounces all proprietary claims and declares this knowledge to be part of the common heritage of the human species. Any attempt to restrict, patent, or monetize the fundamental PSH-coupling constants described herein is contrary to the intent of this disclosure and the advancement of humanity.

% --- Introduction (V1.9: Ohne Genesis-Feld, rein dichteabhängig & defensiv) ---

\section{Introduction}

The nature of dark matter remains one of the most significant challenges in modern cosmology. While established paradigms—such as Weakly Interacting Massive Particles (WIMPs) and Self-Interacting Dark Matter (SIDM)—provide empirically viable fits to galactic dynamics, they frequently necessitate the introduction of non-standard bosonic sectors or hidden particle species \cite{bertone2005, yu2026}. Similarly, while Modified Newtonian Dynamics (MOND) successfully replicates observed rotation curves, it lacks a fundamental field-theoretic derivation \cite{milgrom1983}.

A crucial inflection point in our understanding has been reached through recent observations. The James Webb Space Telescope (JWST) has revealed a population of massive, compact objects at high redshifts ($z > 4$) that challenge the standard hierarchical structure formation model \cite{jwst-jades}. Furthermore, the discrepancy between local measurements and cosmological predictions, combined with heliospheric observations of vacuum density \cite{gurnett2013}, suggests that our standard vacuum models may require a transition to a more dynamic field-theoretic description.

The Photon-String-Higgs (PSH) framework\footnote{Previous versions referred to a `Genesis field'. This construct is now superseded by a more parsimonious description, identifying the PSH-dynamics as an emergent property of the higher-dimensional Higgs sector.} explores an alternative, parsimonious trajectory. It investigates the hypothesis that the photon acquires an effective rest mass within the undisturbed, low-density intergalactic medium (IGM) through a dynamic coupling to a postulated Photon-String-Higgs condensate.

By identifying the PSH-field as an emergent property of the higher-dimensional Higgs sector—arising from the topological reconfiguration of spacetime—the model aims to assess whether existing dark matter paradigms can be addressed without the introduction of additional non-standard bosonic sectors or auxiliary scalar fields. Under these specific environmental conditions, ordinary primordial radiation could behave on cosmological scales like a cold, pressureless fluid, thereby replicating the gravitational signatures traditionally attributed to dark matter. This approach shifts the analytical focus from the postulation of new particles toward the intrinsic phase-state of the vacuum, as constrained by empirical observations of high-redshift structures \cite{jwst-jades, kokorev2026}.

The core conceptual features of this approach can be organized into four primary aspects:
\medskip
\bigskip
\begin{itemize}
\item \textbf{Particle Parsimony:} The framework operates without invoking hypothetical new elementary particles beyond the established and experimentally verified standard spectrum \cite{uebermuth2024}.
\item \textbf{Cosmological Density Scaling:} It models the observed ratio of dark to scenic baryonic matter ($\Omega_{\rm DM}/\Omega_{\rm b} \approx 5.5$) as an emergent consequence of the mean free path of photons within an expanding spacetime geometry.
\item \textbf{Dynamic Continuity:} It aims to provide a mathematically smooth transition from the background dynamics of the early universe to the observed large-scale structures of the late universe.
\item \textbf{Observational Testability:} It formulates specific, verifiable predictions that can be evaluated against existing and near-future astrophysical observation data (e.g., \textit{Voyager} in-situ plasma measurements, deep-field constraints).
\end{itemize}
\medskip
\bigskip
A central field-theoretic tension arises, however, when comparing local experimental bounds with these cosmological requirements: Precision laboratory experiments and measurements within the inner Solar System impose a stringent upper limit on the local photon rest mass ($m_\gamma \lesssim 10^{-16}$,eV), whereas the unattenuated intergalactic PSH scale requires an effective mass of the order of $3 \times 10^{-3}$,eV.

To reconcile this apparent discrepancy, a density- and radiation-dependent screening mechanism is analyzed. This mechanism shares phenomenological features with chameleon or symmetron modulations in scalar-tensor theories, yet it is integrated directly into the electromagnetic and Higgs sectors. The effective PSH coupling is assumed to be strongly suppressed in regions of elevated baryonic and electromagnetic energy density, only becoming active within the highly rarefied intergalactic medium.

Rather than relying on an independent scalar field, the global modulation of this screening is tied to the evolution of the cosmic star formation history and the corresponding background radiation fields. This coupling hierarchy allows for a description of cosmic structure growth driven by emergent PSH seed structures, without requiring non-baryonic dark matter candidates. The formal mathematical framework of this density-dependent coupling hierarchy is developed in the subsequent section.
\vspace{2em}

% --- Framework (V1.9: Bereinigt, Ohne Genesis-Feld, rein astrophysikalisch getrieben) ---

\section{The Density-Dependent Screening Mechanism \ and Field-Theoretic Framework}

To reconcile the apparent tension between the cosmological photon mass scale required for dark matter phenomenology and the stringent local experimental bounds, we formulate a localized, density-dependent screening mechanism. This framework shares phenomenological features with chameleon or symmetron modulations in scalar-tensor theories, yet it operates directly within the gauge-Higgs sector without postulating independent cosmic scalar fields. The effective PSH coupling is strongly suppressed in regions of high baryonic and electromagnetic energy density (such as stellar interiors and planetary magnetospheres) and transitions to its unattenuated state within the low-density intergalactic medium (IGM).

\subsection{Effective Lagrangian Density}

We consider an effective field-theoretic Lagrangian density where the electromagnetic field directly couples to the underlying PSH condensate dynamics. Retaining standard general relativity as the geometric background, the total Lagrangian density is expressed as:

\begin{equation}
\mathcal{L}{\rm tot} = \mathcal{L}{\rm GR}

• \frac{1}{4} F_{\mu\nu} F^{\mu\nu}

• |D_\mu \Phi_H|^2

• V(\phi_{\rm string}, \Phi_H)

• \mathcal{L}{\rm int}(\phi{\rm string}, \psi_{\rm b})
  \label{eq:lagrange}
  \end{equation}

where:
\begin{itemize}
\item $\mathcal{L}{\rm GR}$ is the standard Einstein-Hilbert term representing spacetime curvature,
\item $F{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is the electromagnetic field strength tensor,
\item $\Phi_H$ is the standard model Higgs doublet (or an electroweak-symmetric scalar equivalent),
\item $\phi_{\rm string}$ is the emergent string scalar field representing the coherent PSH mode,
\item $V(\phi_{\rm string}, \Phi_H)$ is the effective interaction potential of the co-dependent fields,
\item $\mathcal{L}{\rm int}$ parametrizes the model's non-minimal coupling to local baryonic matter fields $\psi{\rm b}$.
\end{itemize}

\subsection{Effective Potential and Vacuum Expectation Values}

The non-linear interaction between the string scalar field and the Higgs sector is governed by the effective potential:

\begin{equation}
V(\phi_{\rm string}, \Phi_H) = -\mu^2 |\Phi_H|^2 + \lambda |\Phi_H|^4 + g_{\rm c} |\Phi_H|^2 |\phi_{\rm string}|^2
\label{eq:potential}
\end{equation}

where $\mu^2$ and $\lambda$ define the standard Higgs self-interaction, and $g_{\rm c}$ represents the dimensionless PSH coupling constant.

In highly rarefied environments below a characteristic energy density, the minimum of this potential shifts to non-zero vacuum expectation values (VEVs), denoted as $\langle \phi_{\rm string} \rangle = v_s \neq 0$ and $\langle \Phi_H \rangle = v_H \neq 0$. This environmental shift induces a local, spontaneous breaking of the effective electromagnetic $U(1)$ symmetry within deep intergalactic space, thereby generating a non-zero effective photon rest mass.

\subsection{Density-Dependent Screening and the Ilias-Lucida Threshold}

The effective, localized rest mass of the photon field, $m_{\gamma,\rm eff}$, is modulated by a dimensionless screening function $\mathcal{S}(\rho_{\rm b}, \rho_{\rm EM})$. Incorporating the structural scaling of the expanding universe and energy dependencies, the mass profile is formulated as:

\begin{equation}
m_{\gamma,\rm eff}(E, a, \rho) = \mathcal{S}(\rho_{\rm b}, \rho_{\rm EM}) \cdot \frac{\alpha_{\phi H} v_H}{c^2} \cdot \phi_{\rm string}(a) \cdot \left( \frac{a}{a_0} \right)^3 \cdot \exp\left( -\frac{E}{E_c} \right)
\label{eq:m_eff}
\end{equation}

where $\alpha_{\phi H}$ is an effective coupling parameter, $a/a_0$ scales with the cosmic expansion to track the dilution of the background condensate, and $E_c$ represents a high-energy cutoff beyond which the screening transitions back to masslessness. The environmental response is governed by the screening function:

\begin{equation}
\mathcal{S}(\rho_{\rm b}, \rho_{\rm EM}) = \left[ 1 + \left( \frac{\rho_{\rm b} + \rho_{\rm EM}}{\rho_{\rm IL}} \right)^n \right]^{-1}
\label{eq:screening}
\end{equation}

Here, $\rho_{\rm b}$ represents the local baryonic mass density, $\rho_{\rm EM}$ is the local electromagnetic radiation-flux density, and $n \geq 2$ is a stiffness parameter dictating the geometric sharpness of the phase transition. The transition is fundamentally parameterized by the \textbf{Ilias-Lucida threshold} ($\rho_{\rm IL}$).

Equation~(\ref{eq:screening}) mathematically guarantees that $\mathcal{S} \approx 1$ within the pristine intergalactic medium where $\ parental_density \ll \rho_{\rm IL}$, allowing the photon mass to saturate its dark matter target scale ($3 \times 10^{-3}$~eV). Conversely, in environments like the inner Solar System, the condition $\rho_{\rm b} + \rho_{\rm EM} \gg \rho_{\rm IL}$ forces $\mathcal{S} \to 0$, naturally satisfying terrestrial laboratory limits ($m_\gamma \lesssim 10^{-16}$~eV).

\subsection{Cosmic Evolution and Radiation-Flux Coupling}

Rather than tracking an independent scalar field, the global background activation of the PSH mechanism is coupled to the integrated radiation history of the universe. The total ambient electromagnetic background density $\rho_{\rm EM}(z)$ evolves as a function of redshift $z$, driven by the cosmic star formation history (SFH), $\psi_{\rm SFH}(z)$:

\begin{equation}
\rho_{\rm EM}(z) = \rho_{\rm CMB,0}(1+z)^4 + \int_{z}^{\infty} \frac{\psi_{\rm SFH}(z')}{(1+z') \cdot H(z')} , dz'
\label{eq:evolution}
\end{equation}

where $\rho_{\rm CMB,0}$ is the present-day cosmic microwave background density, and $H(z)$ is the Hubble parameter. This direct feedback loop ensures that the formation of the first stellar structures actively modulates the global boundary conditions of the PSH screening over cosmic time.
\vspace{2em}

% --- Section 3: Neue Definitionen Heliosphäre/Astrosphäre vs. Heliopause/Galaktopause ---
\section{Spatial Classification of Astro- and Cosmospheric Boundaries}

To rigorously analyze the spatial onset of the PSH coupling, it is critical to distinguish between boundaries governed by hydrodynamic material interactions and those dictated by environmental field- and radiation-thresholds. Traditional astrophysics predominantly defines structural boundaries via plasma-kinetic pressure balances. In contrast, the PSH framework necessitates a dual classification that decouples the matter-driven domain from the photon-field-driven domain.

\subsection{Stellar Scales: Heliopause vs. Astrosphere}

On stellar scales, we establish a strict terminological and physical distinction between the \textit{Heliopause} (or more generally, Star-pause) and the \textit{Astrosphere}:

\begin{itemize}
\item \textbf{The Heliopause / Star-pause (Material Boundary):} This boundary is classical and hydrodynamic. It is defined by the contact discontinuity where the kinetic ram pressure of the solar/stellar wind ($\rho_{\rm w} v_{\rm w}^2$) dynamically balances the pressure of the local interstellar medium (LISM):
\begin{equation}
P_{\rm kinetic}(r) = \rho_{\rm w}(r) v_{\rm w}^2 \approx P_{\rm LISM}
\end{equation}
It marks the definitive physical boundary of solar material propagation.

\item \textbf{The Astrosphere (Photon-Field Boundary):} Within the PSH framework, the Astrosphere is redefined not as a plasma bubble, but as the spatial volume within which the local stellar radiation flux ($\rho_{\rm EM}$) and baryonic density ($\rho_{\rm b}$) jointly maintain the screening condition $\mathcal{S}(\rho) \to 0$. The true field-theoretic boundary of the Astrosphere is dictated strictly by the isotropic or anisotropic coordinate radius where the total environmental energy density drops below the Ilias-Lucida threshold:
\begin{equation}
    \rho_{\rm b}(r_{\rm astro}) + \rho_{\rm EM}(r_{\rm astro}) = \rho_{\rm IL}
\end{equation}

\end{itemize}

Consequently, while the material heliopause of the Solar System is located at approximately $r \approx 120$~AU, the photonic astrospheric boundary marks the exact spatial locus where the effective photon mass $m_{\gamma,\rm eff}$ begins its sigmoidal transition from its screened state toward its unattenuated intergalactic value.

\subsection{Galactic Scales: Galaktopause vs. Galaktosphere}

This foundational dualism scales invariantly up to galactic structures, where we define the interface between the \textit{Galaktopause} and the \textit{Galaktosphere}:

\begin{itemize}
\item \textbf{The Galaktopause (Material Boundary):} The galactic equivalent to the heliopause, defined by the shock fronts and baryonic density gradients where the collective galactic winds, cosmic ray pressures, and expelled interstellar matter meet the intergalactic medium (IGM) or intra-cluster medium. It is heavily structurally bound to visible baryonic matter distributions.

\item \textbf{The Galaktosphere (Photon-Field Boundary):} The macro-scale volume enveloped by the cumulative radiation field of an entire galaxy (the integrated galactic stellar luminosity) coupled with its extended gas halo. The boundary of the Galaktosphere is defined by the spatial envelope where the total galactic energy density falls below the critical threshold:
\begin{equation}
    \rho_{\rm b}(\mathbf{R}) + \rho_{\rm EM}(\mathbf{R}) = \rho_{\rm IL}
\end{equation}

\end{itemize}

Outside the Galaktosphere, in the deep cosmic Voids, the complete lifting of the screening mechanism allows the PSH condensate to fully manifest, creating the massive halos traditionally interpreted as dark matter.

\subsection{Analytical Solutions for Sphere Radii}

To provide a directly evaluable predictive framework, the spatial boundaries of both the Astrosphere and the Galaktosphere can be derived analytically by solving the threshold condition for the coordinate radius.

For a spherically symmetric stellar system, both the local baryonic wind density $\rho_{\rm b}(r)$ and the radiative energy density $\rho_{\rm EM}(r)$ scale inversely with the square of the distance $r$ outside the immediate stellar core zone. Thus, we can parametrize the environmental profiles via their effective source values at a reference radius $r_0$ (e.g., the stellar surface):
\begin{equation}
\rho_{\rm b}(r) = \rho_{\rm b,0} \left(\frac{r_0}{r}\right)^2, \quad \quad \rho_{\rm EM}(r) = \rho_{\rm EM,0} \left(\frac{r_0}{r}\right)^2
\end{equation}

Substituting these scaling laws into the Ilias-Lucida threshold condition ($\rho_{\rm b} + \rho_{\rm EM} = \rho_{\rm IL}$) yields the explicit analytical solution for the Astrosphärenradius ($r_{\rm astro}$):
\begin{equation}
r_{\rm astro} = r_0 \sqrt{\frac{\rho_{\rm b,0} + \rho_{\rm EM,0}}{\rho_{\rm IL}}}
\label{eq:r_astro_explicit}
\end{equation}

By exact structural analogy, the boundary of a macro-scale galactic system can be computed. Approximating the core region of a galaxy as an isotropic emitter with an integrated baryonic halo mass distribution and a total galactic stellar luminosity $L_{\rm gal}$, the Galaktosphärenradius ($R_{\rm gal}$) at which the PSH screening terminates is given by:
\begin{equation}
R_{\rm gal} = \sqrt{\frac{M_{\rm halo,0} + (L_{\rm gal} / 4\pi c^3)}{\rho_{\rm IL}}}
\label{eq:R_gal_explicit}
\end{equation}
where $M_{\rm halo,0}$ represents the characteristic baryonic mass scale parameter of the inner galactic gas matrix. Equations~(\ref{eq:r_astro_explicit}) and (\ref{eq:R_gal_explicit}) allow researchers to directly map the field-theoretic boundaries of any astrophysical object based purely on its observable baryonic and radiative outputs.

\subsection{Physical Intuition of the Boundary Decoupling}

To assist the reader in visualizing this framework, it is imperative to recognize that the PSH framework effectively decouples a system's \textit{kinematic jurisdiction} from its \textit{field-theoretic jurisdiction}.

While the material boundary (e.g., the Heliopause or Galaktopause) represents a localized kinetic clash of particles, the photonic boundary (the Astrosphere or Galaktosphere) represents the geometric limit of structural screening. Within this boundary, the collective energetic output of the source acts as a continuous "vaporizing agent" that keeps the surrounding Higgs-vacuum screened and the photon mass suppressed. Once an observer crosses the boundary radius ($r_{\rm astro}$ or $R_{\rm gal}$), the environmental "thermal shield" drops below the Ilias-Lucida threshold. At this exact spatial interface, the dense sea of primordial relic photons—originally generated during the monumental matter-antimatter annihilation era of the early universe—undergoes its sigmoidal mass condensation, manifesting macroscopically as the pressureless, cold gravitational fluid known as dark matter.

\subsection{The Role of Cosmic Moisture Gradients (CMG)}

The spatial transition between these domains is not immediate but is characterized by a field-theoretic gradient. We define the \textbf{Cosmic Moisture Gradient (CMG)} as the directional spatial derivative of the screening function along a given trajectory $\mathbf{r}$:

\begin{equation}
\mathbf{\nabla} \text{CMG} = \frac{\partial \mathcal{S}(\rho_{\rm b}, \rho_{\rm EM})}{\partial \mathbf{r}}
\label{eq:cmg}
\end{equation}

The $\mathbf{\nabla} \text{CMG}$ acts as a mathematical measure for the "evaporation" or "condensation" of effective photon mass. A steep gradient implies a sharp phase transition (such as near the edge of dense astrospheres), while a flat gradient represents a smooth, extended transition zone across galactic fringes. This gradient directly governs the local gravitational lensing anomalies observed at the edges of macro-structures.
\vspace{2em}

% --- Cosmological Implications (V1.9: Ohne Genesis-Feld, getrieben durch Relikt-Photonen & IL-Threshold) ---
\section{Cosmological Tests and Quantitative Predictions}

\subsection{Global Calibration and Definition of the Ilias-Lucida Threshold}
To calibrate the PSH model on cosmological scales, we utilize the temporal onset of the global screening collapse. Empirical data from the EDGES collaboration regarding the 21-cm hydrogen absorption line identify a distinct signal profile in the redshift range of $z \approx 20$ to $z \approx 15$, with an absorption maximum at $z \approx 17.2$ \cite{bowman2018}. This signal marks the beginning of the structural decoupling.

Given that the anomalous 21-cm absorption is observable at $z \approx 20$, we define this threshold as the structural onset of the screening collapse. The energy density of the cosmic background fields (CMB radiation plus the primordial Ur-Relikt photon background) at this precise moment forms the fundamental calibration threshold—the \textbf{Ilias-Lucida Threshold} $\rho_{\rm IL}$:

\begin{equation}
\rho_{\rm IL} = \sigma_{\rm SB} \cdot \left[ T_0 \cdot (1 + 20) \right]^4 \approx 8.12 \times 10^{-9} \text{ J/m}^3 \quad \left(\approx 50.7 \text{ eV/cm}^3\right)
\label{eq:IL_threshold}
\end{equation}

where $\sigma_{\rm SB}$ represents the Stefan-Boltzmann constant and $T_0 \approx 2.7255$~K the current CMB temperature.

The threshold $\rho_{\rm IL}$ defines the critical energetic demarcation of the vacuum. As the global radiation density falls below this threshold, the screening mechanism in intergalactic space becomes unstable. This triggers the large-scale phase transition, which initiates at $z \approx 20$ and allows for the emergence of the effective photon mass.

% Hier die Grafik aus V1.8 (mit angepasster Caption für V1.9)
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis} [
width=0.85\textwidth, height=0.52\textwidth,
xlabel={Redshift $z$},
ylabel={Energy Density $\rho(z)$ [J/m$^3$]},
xmin=2, xmax=35,
ymin=0, ymax=3.5e-8,
grid=major,
x dir=reverse,
legend pos=north west,
legend style={font=\footnotesize},
scaled y ticks=false
]
% Radiation energy density rho_rad ~ (1+z)^4
\addplot[domain=2:35, samples=100, red, thick]
{8.12e-9 * ((1+x)/21)^4};
\addlegendentry{Total Radiation Density $\rho_{\rm rad}(z)$}

% Ilias-Lucida Threshold
\addplot[domain=2:35, blue, dashed, thick] 
{8.12e-9};
\addlegendentry{Ilias-Lucida Threshold $\rho_{\rm IL}$}

% Trigger point z=20
\draw[black, fill=black] (axis cs:20, 8.12e-9) circle (2pt);
\node[above right, font=\tiny\bfseries] at (axis cs:20, 8.12e-9) {Trigger $\rho_{\rm IL}$ ($z=20$)};

% Primary phase
\draw[fill=blue, opacity=0.15] (axis cs:20, 0) rectangle (axis cs:15, 3.5e-8);
\node[rotate=90, font=\tiny\bfseries, text=blue!80!black] at (axis cs:17.5, 2.2e-8) {Primary Condensation};

% Secondary saturation phase
\draw[fill=green!50!blue, opacity=0.08] (axis cs:15, 0) rectangle (axis cs:6, 3.5e-8);
\node[rotate=90, font=\tiny\bfseries, text=green!40!black] at (axis cs:10.5, 2.2e-8) {Vacuum Saturation};
\end{axis}
\end{tikzpicture}
\caption{Temporal evolution of the cosmic phase transition. Triggered at $z \approx 20$ by falling below $\rho_{\rm IL}$, the transition proceeds from a primary condensation phase ($z=20 \to 15$, correlated with EDGES data) into a macroscopic vacuum saturation phase until reionization completion ($z \approx 3.5$).}
\label{fig:cosmo_plot}

\end{figure}

\subsubsection{Quantitative Calibration of the Scale}

The numerical value of the effective Ilias-Lucida threshold $\rho_{\rm IL}$ is rigorously calibrated via a cosmological top-down approach based on two distinct observational boundary conditions:

\begin{enumerate}
\item \textbf{The EDGES Anchor:} The structural onset of the global screening collapse is fixed to the redshift range of $z \approx 20$, as indicated by the initial detection of the 21-cm hydrogen absorption signal \cite{bowman2018}. At this cosmic epoch, the average comoving energy density of the background fields reaches the critical threshold required to initiate the PSH mass activation.
\item \textbf{The Total Relict Energy Budget:} The environment is governed not merely by the standard cosmic microwave background (CMB), but critically by the dense, non-thermal background of primordial \textbf{Ur-Relikt photons (PSH photons)} originating from the early universe's matter-antimatter annihilation era.
\end{enumerate}

By evaluating the complete cosmological density evolution equation at this boundary scale factor, the effective threshold is explicitly determined by incorporating the standard components alongside the primordial PSH radiation density field:

\begin{equation}
\rho_{\rm IL} = \rho_{\rm crit,0} \left[ \Omega_{\rm m,0} (1+z_{\rm IL})^3 + \left(\Omega_{\rm CMB,0} + \Omega_{\rm PSH,0}\right) (1+z_{\rm IL})^4 \right]
\label{eq:rho_il_calc_corrected}
\end{equation}

where $\rho_{\rm crit,0}$ represents the critical density of the present universe, $\Omega_{\rm m,0} \approx 0.315$ the total matter density, $\Omega_{\rm CMB,0}$ the standard microwave radiation density, and $\Omega_{\rm PSH,0}$ the dedicated energy density contribution of the unattenuated primordial annihilation photon background. Substituting the physical boundary at $z_{\rm IL} = 20$ yields the frozen, operational phenomenological value of:

\begin{equation}
\rho_{\rm IL} \approx 2.64 \times 10^{-26} , \text{kg/m}^3
\end{equation}

This comprehensive derivation ensures that the threshold captures the full energetic interaction between standard baryonic structures, the ubiquitous CMB background, and the dominant PSH relic fields driving the latent mass activation.

\subsection{Late-Time PSH Domination and Mass Activation}

The global phase transition, wherein the PSH mechanism becomes macroscopically dominant across intergalactic space, is initiated as the cumulative ambient radiation field drops below the critical threshold $\rho_{\rm IL}$. We parameterize this late-time mass activation via a continuous sigmoid transition function tracking the cosmic scale factor $a$:

\begin{equation}
\Gamma_{\rm PSH}(a) = \Gamma_0 \left[ 1 + \exp\left( -\frac{a - a_{\rm IL}}{\Delta a} \right) \right]^{-1}
\label{eq:gamma_psh}
\end{equation}

where the transition midpoint $a_{\rm IL} \approx 0.22$ corresponds to the redshift range $z \approx 3.5$, marking the final stabilization of the PSH mass contribution following the completion of helium reionization.

While the initial screening collapse is triggered at $z \approx 20$ (Cosmic Dawn), the full macroscopic activation of the effective photon rest mass reaches its observational saturation during this late-time epoch. As the intergalactic background density remains below the threshold $\rho_{\rm IL}$, the PSH coupling stabilizes, allowing the unattenuated photon mass to assume its role as the dominant driver of large-scale cosmic structure growth.

The cosmic fraction of photons captured within this mass-active background state is defined by $f_{\rm PSH} \approx 0.846$. This partitioning replicates the present-day observational ratio of dark matter to baryonic matter:

\begin{equation}
\frac{\Omega_{\rm DM}}{\Omega_{\rm b}} \approx \frac{f_{\rm PSH}}{1 - f_{\rm PSH}} \approx 5.5
\label{eq:omega_ratio}
\end{equation}

This numerical alignment with the \textit{Planck} mission density parameters demonstrates that the observed dark matter density emerges naturally from the activated PSH-photon budget, without the need for additional non-baryonic particle species.
\bigskip
\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.85\textwidth, height=0.5\textwidth,
xlabel={Redshift $z$},
ylabel={Mass Activation $\Gamma_{\rm PSH}$},
xmin=0, xmax=25,
ymin=0, ymax=1.1,
grid=major,
x dir=reverse,
legend pos=north west
]
% Die Funktion basiert nun direkt auf Gleichung 18:
% Gamma = 1 / (1 + exp((z - z_sat) / Delta_alpha))
% Wir setzen z_sat = 3.5
\addplot[domain=0:25, samples=200, blue, ultra thick]
{1 / (1 + exp((x - 3.5)/3.5))}; % Hier '3.5' durch Delta_alpha ersetzen, falls abweichend
\addlegendentry{$\Gamma_{\rm PSH}(z)$ via Eq. 18}

% Trigger bei z=20
\draw[red, dashed, ultra thick] (axis cs:20, 0) -- (axis cs:20, 1.1);
\node[anchor=south east, font=\small, rotate=90, text=red] at (axis cs:20, 0.1) {Ignition ($z=20$)};

% Sättigung bei z=3.5
\draw[green!60!black, dashed, thick] (axis cs:3.5, 0) -- (axis cs:3.5, 1.1);
\node[anchor=south east, font=\small] at (axis cs:3.5, 0.7) {Saturation ($z=3.5$)};
\end{axis}
\end{tikzpicture}
\caption{Phase transition profile of the PSH mass activation defined by Eq. 18. The activation initiates at the screening collapse ($z=20$) and follows the sigmoid trajectory governed by $\Delta \alpha$.}
\label{fig:activation_profile_eq18}

\end{figure}

\subsection{Phenomenological Parameterization and PSH-Mass Activation}

To facilitate a quantitative analysis of structure growth, we define the \textbf{Ilias-Lucida threshold} ($\rho_{\rm IL}$) as the critical energetic demarcation of the vacuum. As the global radiation energy density falls below this threshold, the screening mechanism in intergalactic space destabilizes, triggering the large-scale phase transition that allows for the emergence of effective photon mass.
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth]{PSHDM1.jpg}
\caption{Density profile with PSH activation. The phase transition commences at $a \approx 0.0476$ ($z \approx 20$) and continuously diverges from the standard dilution baseline. This divergence reflects the emergence of additional effective gravitational mass contributed by the activated PSH-photon budget.}

\label{fig:density_evolution}

\end{figure}

This numerical result demonstrates that the observed dark matter density emerges as a natural consequence of the activated PSH-photon budget, without the need for additional non-baryonic particle species.

The integrated fraction of photons captured within this mass-active background state is $f_{\rm PSH} \approx 0.846$. This partitioning replicates the observed dark matter to baryonic matter ratio:

\begin{equation}
\frac{\Omega_{\rm DM}}{\Omega_{\rm b}} \approx \frac{f_{\rm PSH}}{1 - f_{\rm PSH}} \approx 5.5
\label{eq:omega_ratio}
\end{equation}

This derivation confirms that dark matter emerges naturally from the activated PSH-photon budget, without requiring non-baryonic particle species.

\subsection{Dark Matter Abundance from Primordial Relikt Populations}

To validate the PSH mechanism as a global dark matter candidate, we assess the integrated mass budget derived from the entire comoving photon population. Assuming a standard observable universe volume, the total number of photons contributing to the primordial and cumulative stellar background is estimated to be:

\begin{equation}
N_\gamma \approx 1.47 \times 10^{89}
\end{equation}

To match the macroscopically observed dark matter density envelope in the current epoch, the mean effective rest mass acquired by the activated intergalactic photons must satisfy the operational scale:

\begin{equation}
\langle m_{\gamma,\rm eff} \rangle \approx 5.34 \times 10^{-36} , \text{kg} \quad \left( \approx 3 \times 10^{-3} , \text{eV}/c^2 \right)
\end{equation}

Summing over the total available intergalactic relikt population that has undergone the phase transition since $z \approx 20$, this yields an integrated effective dark matter mass budget of:

\begin{equation}
M_{\rm DM} \approx 7.85 \times 10^{53} , \text{kg}
\end{equation}

Consequently, the PSH mechanism demonstrates that the "missing mass" of the universe is fully accounted for by the fundamental energy legacy of the Big Bang, modulated exclusively by the density-dependent activation of the intrinsic photon rest mass. This numerical result confirms that dark matter is not a distinct particle species, but the manifestation of the activated photon field’s latent gravitational energy.

\subsection{Cosmological Consistency Constraints}

By locking the significant onset of the activated PSH-mass contribution to late cosmic times ($z \lesssim 20$), the framework remains naturally compliant with the stringent constraints of the early universe. Specifically, the model preserves the precise predictions of Big Bang Nucleosynthesis (BBN) and the pristine acoustic angular scale of the Cosmic Microwave Background (CMB) at $z \approx 1100$, as the effective photon rest mass remained heavily suppressed due to the high density of the primordial plasma during those epochs. The transition to a mass-active state is thus delayed until the global background energy density drops below the Ilias-Lucida threshold $\rho_{\rm IL}$. Fully self-consistent, non-linear numerical simulations of the transition era (covering the interval $4 < z < 20$) are designated for future computational studies to further refine the structural evolution of the cosmic web.
\vspace{2em}

\subsection{Testable Predictions}

\begin{table}[H]
\centering
\caption{Evolutionary milestones of PSH-mass activation and cosmological coupling.}
\vspace{0.8em}
\label{tab:cosmological_parameters}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}l >{\raggedright\arraybackslash}l >{\raggedright\arraybackslash}X}
\hline
\textbf{Cosmological Epoch} & \textbf{Redshift $z$} & \textbf{Physical Interpretation of PSH-Coupling} \
\hline
Primordial Screening Phase & $z > 20$ & PSH-mass fully screened by high plasma density; vacuum coupling is suppressed. \
\hline
Ignition Phase (EDGES Anchor) & $z \approx 20$ & Breakdown of screening; the plasma density drops below the Ilias-Lucida threshold $\rho_{\rm IL} \approx 50.7 \text{ eV/cm}^3$. \
\hline
Continuous Mass Activation & $20 > z > 3.5$ & Emergence of effective photon mass $\Gamma_{\rm PSH}$ as a function of declining background density. \
\hline
Saturation / Half-Activation & $z \approx 3.5$ & Inflection point of the phase transition; stabilization of the late-time PSH-vacuum budget. \
\hline
He-Reionization Stability & $z \approx 4 \rightarrow 3$ & Energetic stabilization; the PSH-field coupling reaches final equilibrium with the IGM. \
\hline
Local Heliopause Horizon & $r_{\rm HP} \approx 121.6$~AU & Local manifestation of PSH-screening; observable via \textit{Voyager} plasma wave density shifts. \
\hline
\end{tabularx}
\end{table}

\begin{table}[H]
\centering
\caption{Cosmological test matrix: Observational signatures of PSH-mass activation.}
\vspace{0.8em}
\label{tab:psh_test_matrix_final}
\small
\begin{tabularx}{\textwidth}{>{\raggedright\arraybackslash}l >{\raggedright\arraybackslash}X >{\raggedright\arraybackslash}l}
\hline
\textbf{Test System} & \textbf{Specific PSH Signature (Anomaly Calibration)} & \textbf{Platform / Evidence} \
\hline
Heliopause (Astrosphere) & Local density jump at $r_{\rm HP} \approx 121.6$~AU; abrupt change in plasma frequency due to screening collapse. & Voyager 1 & 2 (PWS) \
\hline
Interstellar Medium (Galactosphere) & PSH-photon mass effects on wave propagation in the low-density interstellar environment ($\rho < \rho_{\rm IL}$). & Pulsar Timing Arrays \
\hline
FRB Dispersion & Dispersion Measure (DM) anomaly; excess delay caused by effective photon mass in the unscreened IGM. & CHIME / SKA \
\hline
TeV Transparency & Anomalous transparency of the IGM; modified pair-production threshold for multi-TeV gamma rays. & LHAASO / Swift \
\hline
CMB Distortions & Secondary $y$- and $\mu$-distortions originating from the ignition epoch $z \approx 20$. & CMB-S4 / LiteBIRD \
\hline
Terrestrial Lab & \textbf{Null Result:} Complete PSH-screening due to local high-density environment ($\rho \gg \rho_{\rm IL}$). & Standard Physics \
\hline
\end{tabularx}
\end{table}
\vspace{2em}