Geometric Response Cosmology (GRC)

Geometric Response Cosmology (GRC): A Model Based on Linear Expansion, Emergent Time, and Extended Gravitational Influence

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To explain current observations, the standard cosmological model requires that 95% of the universe's content be unknown (Dark Matter and Dark Energy) and depends on a set of free parameters whose numerical values must be specifically tuned to match the data (fine-tuning). We consider it necessary to explore alternative solutions given current observational tensions; on this basis, we propose a cosmological framework called Geometric Response Cosmology (GRC) based on a global linear expansion at the inertial velocity c.

This expansion is understood as an intrinsic property of space, independent of any energy or matter content. In this framework, time emerges as a magnitude emerging from global expansion rather than as a fundamental dimension. The proper time of matter is the result of the decoupling between mass and said expansion. This approach also proposes an extension of gravity based on the relationship between global dynamics and local gravity. This mechanism derives the critical acceleration scale a₀ = cH₀/(2π) as the angular density of cosmic expansion, and offers a natural explanation for the rotation dynamics in spiral galaxies without resorting to non-baryonic dark matter. The results presented suggest that this work constitutes a contribution towards the construction of an alternative cosmological description.

Geometric Response Cosmology (GRC): Induced Gravity and Emergent Time from the Energy Substrate as a Phenomenological Framework for the Dark Sector

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We propose Geometric Response Cosmology (GRC), a phenomenological effective framework where time emerges as a kinematic residue of a real 3D substrate and gravity is described as an elastic response of the energy substrate through effective couplings with baryonic matter density.

This coupling generates MOND-like dynamics at low accelerations, with the critical scale a₀ ≈ cH₀/(2π) emerging from linear expansion R_h = ct as a natural scale identification. We find consistency with the Radial Acceleration Relation (RAR) through the critical screening density ρ_crit = 10^-22 kg m^-3 (fitted to the SPARC catalog). The “Vacuum Catastrophe” is phenomenologically mitigated through a dynamic cancellation condition, which determines the non-minimal coupling ξ₀ ≃ -2.4 × 10^-3.

Finally, GRC kinematics suggests an extended structural collapse time at high redshifts (z ≈ 10), whose quantitative factor requires calculation of the emergent temporal integral τ(z). The model identifies as its most direct observational signature the absence of primordial B-modes (r ≈ 0). We additionally identify qualitative research directions consistent with the substrate, such as a cumulative gravitational excess in cluster lensing where light deflection follows the same Radial Acceleration Relation as stellar dynamics, without artificial bound on individual amplification. The total cluster excess depends on the three-dimensional galaxy distribution and the photon trajectory. The framework is constructed to be minimal in free parameters, with effective quantities primarily anchored to galactic phenomenology. The determination of exact magnitudes for lensing and BAO constitutes open computational challenges, necessary for future contrasts with CMB-S4, Euclid, and LiteBIRD missions.

This work does not aim to provide a complete replacement for the standard cosmological model, but to establish a minimal phenomenological framework that captures observed scaling relations and provides testable constraints for future observations.

Mechanical Ontology of Black Holes: Singularity Resolution in Geometric Response Cosmology (GRC)

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Within the framework of Geometric Response Cosmology (GRC), we present a mechanistic interpretation of gravitational phenomenology that achieves the empirical success of General Relativity through fundamentally distinct physical principles. We propose that the event horizon is not an internal volume, but a temporal discontinuity arising from the rupture of a scalar substrate. The observed mass manifests as emergent energy from the boundary condition of the substrate's edge, eliminating singularities. The exterior metric emerges from the kinematic restriction imposed by the temporal flow v_t(r) on the substrate.

Planck-scale corrections of the model are suppressed by ~10^-40 for all astrophysical black holes, which explains the century-long precision of GR while providing a physical cutoff. GRC reproduces the predictions of GR where it has been tested, diverging only at Planck scale where quantum gravity is expected.

We demonstrate consistency with the observed masses of V616 Mon, Sgr A* and M87* (error < 22%), topological stability and thermodynamic coherence. The model offers an ontologically consistent alternative that grounds General Relativity in the mechanical dynamics of the substrate.

Kinematic Reconstruction of the Schwarzschild Regime in Geometric Response Cosmology (GRC): From the Weak-Field Limit to the Photon Sphere

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The Schwarzschild metric constitutes the exact solution of Einstein's equations for a non-rotating spherical body. In the weak-field limit (r_s/r ≪ 1), it predicts three observable phenomena: gravitational time dilation (redshift), Shapiro time delay, and angular deflection of light. In the strong-field regime, it predicts the photon sphere (r = 3r_s/2), the unstable circular light orbit observed by EHT as the bright ring of the black hole.

In the standard framework, these phenomena require a four-dimensional pseudo-Riemannian geometry with two independent metric components (g_tt and g_rr). In this work we demonstrate that the complete Schwarzschild regime (weak and strong) emerges mechanically in Geometric Response Cosmology (GRC) from a single hypothesis: the conservation of the velocity-vector norm in a flat, absolute spatial substrate. The effective propagation velocity v_prop(r) = c√(1 − r_s/r), modulated by the local gravitational resistance of the substrate, quantitatively reproduces the three classical weak-field phenomena and the exact photon sphere (b_crit = 3√3/2 r_s) through exact solution of Fermat's principle in the substrate, without invoking geometric curvature.

We discuss a differentiable ontological prediction: in GRC, the bright ring observed by EHT is not light orbiting in curved vacuum, but the optical projection of the matter-to-radiation conversion zone, testable through high-resolution spectroscopy.

Geometric Response Cosmology (GRC): Early Cosmology

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This work extends Geometric Response Cosmology (GRC) to early cosmology. GRC postulates that gravity is the mechanical tension of a substrate field φ, not geometric curvature. Starting from parameters fixed by local observations (v₀, ξ₀, λ, H₀), we mechanically derive the cosmic web scale L_red ≈ 28–31 Mpc through three estimations based on distinct aspects of the model. Three independent routes converge to this scale.

BBN is compatible via reduced effective time within self-organized bubbles. The CMB is reinterpreted as a three-dimensional photograph of the substrate's organization transition, projected from an extended volume rather than a thin surface. The main CMB scale is postulated as a geometric feature ℓ* = 1/(2|ξ₀|) ≈ 208 from a galaxy parameter.

Cosmological Redshift in Geometric Response Cosmology (GRC): Propagation in a Substrate with Growing Horizon

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In Geometric Response Cosmology (GRC), the photon propagates in a homogeneous cosmological background described by a scalar substrate φ at its vacuum value φ = v₀ everywhere. The local propagation velocity is v_prop = c constant, with no spatial gradient of φ. This homogeneity poses a paradox: if the Schwarzschild gravitational redshift arises from the gradient of φ(r), where does the cosmological redshift 1 + z come from?

We resolve this paradox through a geometric propagation mechanism: the cosmological horizon grows as R_h = ct, and the photon, always travelling at local speed c, traverses a substrate whose global scale increases during the journey. The separation between wavefronts stretches as a consequence of the growth of the path, not due to a change in local velocity or a variation in the ticking rate of a clock. This effect is dominant in the early universe, where the emission horizon was more than 1000 times smaller than the present one (z ∼ 1100), and becomes negligible for nearby sources (z ∼ 0.1).

The redshift is a property of the effective geometry of the path, not an intrinsic property of the photon nor a modification of time. This mechanism unifies the propagation principle under a single framework: in local gravity, the path is modified by the gradient of φ; in cosmology, by the growth of the horizon. Both are manifestations of the same principle (propagation at c in a substrate whose effective geometry changes) but with different causes.

Perturbations and Structure Formation in Geometric Response Cosmology (GRC): Four Phases, Organization Transition, and Early JWST Galaxies

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In Geometric Response Cosmology (GRC), the local Newtonian limit exactly recovers Poisson's equation and, consequently, the linear perturbation growth equation of General Relativity. The ontological difference lies not in the mathematics of local growth, but in the global ingredients: no dark matter, no dark energy, and gravity at large scales transitions to the MOND regime when the local substrate tension falls below the scale a₀ = cH₀/(2π).

We propose that the history of perturbations in GRC divides into four phases: (I) pre-organization (z ≫ 1100), where the substrate is disorganized and perturbations remain frozen; (II) organization transition (z ~ 1100), where the substrate φ organizes massively, generating the CMB as a three-dimensional photograph of that transition; (III) autocatalytic collapse by nucleation of organized domains (z < 1100), where the matter-substrate feedback loop enables direct baryonic collapse in ~10⁸ years; and (IV) standard linear growth, where the already organized substrate permits Newtonian growth with initial conditions established by Phase III.

The power spectrum exhibits a preferred scale L_red ~ 28 Mpc (k ~ 0.22 Mpc^-1), distinct from the BAO scale of ΛCDM.

Dynamical Substrate Theory: A Cosmological Extension of Geometric Response Cosmology

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We present an extension of Geometric Response Cosmology (GRC) in which the vacuum expectation value of the scalar substrate φ evolves with redshift. The working hypothesis, motivated by energy conservation in a cosmic volume with a linearly growing horizon, postulates v₀(z) = v₀(0) · (1+z)^(3/4).

This evolution modifies the characteristic scale of the gravitational gradient and the Newton–MOND transition scale at each epoch. We show that, with p = 2 —an exponent consistent with the quadratic structure of emergent time vₜ = c(φ/v₀)² postulated in the GRC framework— the framework consistently reproduces galaxy rotation curves in the local universe, the observations of galaxies with pronounced baryonic dominance at z ~ 2 (Genzel et al.), and the early formation of massive structures at z > 10 (JWST).

The extension does not invalidate any previous result of the GRC framework, which is recovered in the limit z → 0. We discuss falsifiable predictions and open problems.

The Fundamental Equation of Geometric Response Cosmology (GRC): Substrate Ontology and Unified Gravitational Dynamics

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We present the fundamental practical equation of Geometric Response Cosmology (GRC): ϕ(r) = v₀ [1 + 2Φ_N(r)/c²]^{1/4}. This algebraic relation translates any distribution of organized baryonic matter, through its standard Newtonian potential Φ_N, into the local profile of the scalar substrate ϕ.

From ϕ one obtains the local clock speed v_t = c(ϕ/v₀)², which generates the effective Painlevé–Gullstrand metric and all weak-field observables. Organized matter extracts energy from the substrate, creating a local deficit (ϕ < v₀). The substrate attempts to compensate this deficit (Newtonian gravity), while the global expansion of the universe (R_h = ct) drags the compensation outward, extending the gravitational range beyond the Newtonian regime.

This mechanical competition gives rise to the transition scale a₀ = cH₀/(2π) of the MOND regime. The equation introduces no free parameters beyond baryonic mass and the Hubble constant.

Polar Currents from the Substrate: An Ontological Mechanism for Jets in Geometric Response Cosmology (GRC)

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In Geometric Response Cosmology (GRC), the scalar substrate φ imposes a kinematic boundary condition on organized matter: near a black hole horizon, the radial degree of freedom is suppressed and matter is forced into non-radial trajectories. When infalling organized matter disorganizes into plasma at the disruption zone, it loses its internal self-tension and ceases to behave as a coherent block.

The resulting plasma flow, dense and collisional, auto-organizes into secondary currents. A fraction of these currents circulates toward the polar regions; if a stable polar current is established, it captures subsequent flow and emerges as a jet. The jet is therefore an emergent, contingent structure—not a universal consequence of accretion, but the result of a density threshold competition between the infalling flow and the local vacuum basal energy.

We propose an initial mathematical framework for this threshold, to be verified by future variational closure. The mechanism connects naturally with the observed thickness of the black hole shadow ring and the recently resolved connection between the ring and the inner jet in M87*.

The Effective Lagrangian of Geometric Response Cosmology

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In Geometric Response Cosmology (GRC), gravity is described as the mechanical response of a scalar substrate φ to organized energy, rather than as curvature of space-time. This work proposes the macroscopic effective Lagrangian of GRC: a scalar field φ in flat space with kinetic term, potential V(φ) = λ(φ² − v₀²)², and coupling to organized energy.

The particles and gauge forces (electromagnetic, weak, strong) are those of known physics; they are not modified. What changes is the description of gravity. This Lagrangian is constructed to recover the Newtonian limit (Poisson), the galactic regime (MOND), the compact environment (Schwarzschild), and the cosmological regime (redshift and perturbations) without dark matter, without dark energy, without inflation, and without postulating time as a fundamental dimension.

Ontological Framework of Black Hole Collisions in Geometric Response Cosmology (GRC)

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In Geometric Response Cosmology (GRC), a black hole is not a gravitational object but a rupture of the substrate field φ → 0, with an edge that grows at c. The edge is a kinematic condition, not a source of active gravity; it is unobservable to any inhabitant of the universe, who only perceives its restrictive effects, never the edge or its interior. When multiple edges coexist in a shared finite substrate, their isotropic growth at c compresses the interstice, raising the gradient of the surrounding substrate until contact becomes mechanically inevitable. We describe the rupture of this interstice upon contact of the edges, and the resulting mechanical perturbation of the surrounding substrate that propagates laterally through reorganization of the medium. We discuss falsifiable differential predictions with respect to General Relativity, including the intrinsically short duration of events, consistent with LIGO/Virgo/KAGRA observations. This work establishes the ontological and mechanical framework; the formulation of boundary equations, the quantitative derivation of the waveform, the estimation of the signal, and the calculation of optical coupling constitute open research fronts.

Multiverses and Hierarchy of Nested Universes under Geometric Response Cosmology (GRC)

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We develop a conceptual framework for a hierarchy of nested universes grounded in Geometric Response Cosmology (GRC). Unlike speculative multiverse models, GRC postulates a mechanical fission ontology: gravitational collapse is identified as the rupture node where the parent universe's substrate terminates, giving birth to a causally closed child domain with its own emergent time.

This architecture integrates the Immersion Principle, recognizing hierarchical blindness as a constitutive property of cosmic structure. The validity of this hierarchy is subjected to a specific falsifiability test: the prediction that black hole mass distribution must exhibit a structure of multiple statistical "modes" or peaks, invariant with redshift.

By dispensing with ad hoc components and mathematical singularities, this work transforms the multiverse concept into a physical process of primordial energy depletion, offering a theory of principles that is sufficiently specific to be refuted by observation.

This work does not aim to provide a complete dynamical model, but to establish a coherent ontological framework from which such a model could be developed.