Theoretical Framework

Author

Raúl Chiclano

Published

November 30, 2025

The Ontology of the Vacuum

The Dynamic Background Hypothesis posits a radical shift in ontology: the universe is not a container populated by fields, but a condensed matter phase (a superfluid) from which all physical phenomena emerge.

Core Postulate: The vacuum is a complex scalar field \(\Psi\) with a non-trivial ground state. Spacetime geometry is the acoustic metric of this fluid, and matter consists of its topological defects.

1. The Master Equation (EFT)

At low energies, the fundamental substrate behaves as a relativistic superfluid. Its dynamics are governed by a non-linear Klein-Gordon equation with a saturation term:

\[ \Box \Psi + \alpha |\Psi|^2 \Psi + \beta |\Psi|^4 \Psi = 0 \]

Physical Interpretation of Parameters

  • \(\rho = |\Psi|^2\): The local density of the vacuum.
  • \(\alpha < 0\): The “mass” parameter. Its negative sign induces Spontaneous Symmetry Breaking, creating a non-zero vacuum expectation value (\(\rho_0\)).
  • \(\beta > 0\): A non-linear saturation term. It acts as a “hard wall” for energy density, ensuring Vacuum Stability and defining the UV cutoff scale (Planck scale).

2. Emergent Gravity

Gravity is not a fundamental force in this framework. It is an effective description of how excitations (phonons/photons) propagate through the inhomogeneous background.

By linearizing the master equation around a background flow \(v_\mu = \partial_\mu \theta\), we derive the Acoustic Metric:

\[ g_{\mu\nu}^{\text{eff}} = \frac{\rho}{c_s} \left[ \eta_{\mu\nu} + \left( \frac{1}{c_s^2} - 1 \right) v_\mu v_\nu \right] \]

The Equivalence Principle

Fluctuations do not “feel” the flat background \(\eta_{\mu\nu}\); they are forced to follow the geodesics of \(g_{\mu\nu}^{\text{eff}}\). Thus, density gradients (\(\nabla \rho\)) manifest as spacetime curvature.

3. The Newtonian Limit & Criticality

Does this metric reproduce Newton’s Law? In the static limit (\(v=0\)), the derived gravitational potential \(\Phi\) follows a Yukawa interaction:

\[ \Phi(r) \propto \frac{e^{-r/\lambda}}{r} \quad \text{where} \quad \lambda \propto \frac{1}{\sqrt{|\alpha|}} \]

The Fine-Tuning Mechanism (SOC)

Standard gravity (\(1/r\)) requires infinite range (\(\lambda \to \infty\)). This implies that the mass parameter must vanish (\(\alpha \to 0\)). We propose that the universe operates in a state of Self-Organized Criticality (SOC): 1. Cosmic Expansion cools the background, driving \(\alpha \to 0\). 2. Gravitational Collapse (Black Holes) heats the background, increasing \(\alpha\). 3. The balance maintains the universe at the critical point of a phase transition, allowing long-range gravity to persist.

4. Cosmology and Dark Energy

The energy density of the vacuum is derived from the effective potential \(V(\Psi)\):

\[ \rho_{\text{vac}} = V_0 - \frac{\alpha^2}{4\beta} \]

  • Dark Energy: The pressure of the fluid naturally leads to an equation of state \(w \approx -1\), mimicking a Cosmological Constant.
  • The Hierarchy Problem: The SOC mechanism explains why \(\rho_{\text{vac}}\) is small but non-zero: it is dynamically tethered to the critical point of the phase transition.