Core Hypothesis
The time dilation between proper time τ and coordinate time t is proportional to the square root of the energy density:
Where ρ₀ is a reference energy density.
Rethinking the Universe
Asymptotic Funnel Cosmology (AFC)
A thought experiment on a geometric-relativistic model of the universe without singularity and dark energy
Introduction: What is AFC?
Asymptotic Funnel Cosmology (AFC) is a new cosmological model that explains the accelerated expansion of the universe and the absence of a Big Bang singularity through the combination of dynamic time dilation and aspherical geometry — without dark energy or modified gravity.
AFC is based on a simple yet profound hypothesis: The time dilation between the local proper time τ (in the early universe) and the cosmological coordinate time t (our view) is controlled by the energy density ρ.
Core Idea: The higher the energy density ρ, the greater the deviation between τ and t. This leads to a non-linear relationship between the scale factor a and coordinate time t, creating a funnel shape of the expansion.
Unlike the standard model (ΛCDM), AFC avoids:
- The Big Bang Singularity: The Big Bang is an asymptotic state, not a point of infinite density.
- Dark Energy: The apparent acceleration arises from geometry and time dilation.
- Ad-hoc Assumptions: All equations are derived from a few physical principles.
Mathematical Basics of AFC
Energy Density Scaling
For a universe with constant total energy E and volume V ∝ a³:
Thus, the time dilation becomes:
Scale Factor in Proper Time
From standard cosmology for a matter-dominated universe:
Derivation of the Scale Factor a(t)
By substituting a(τ) ∝ τ²ᐟ³ into the time transformation dt = a³ᐟ² dτ:
Substituting into a(τ):
Result: AFC predicts that the scale factor grows with time as a(t) ∝ t¹ᐟ³ — in contrast to a(t) ∝ t²ᐟ³ in the matter-dominated ΛCDM model.
Comparison of Scale Factors: AFC vs. ΛCDM
In this model, AFC shows a slower growth of the scale factor compared to the pure matter-dominated ΛCDM model. At t = 0, a(t) ∝ t¹ᐟ³ has a diverging slope.
Extended Theory: Equations of State
The previous derivation applies to a universe with pure matter (w = 0, p = 0). For a realistic universe, different components must be considered, characterized by their equation of state w = p/ρ.
Time Dilation for Arbitrary w
The hypothesis is extended to:
Justification: In General Relativity (GR), the combination ρ + 3p appears in the Friedmann equations, describing the source of spacetime curvature. AFC directly links time dilation to this source.
| Component | Pressure p | w = p/ρ | Energy Density ρ(a) | Time Dilation dτ/dt | Scale Factor a(t) |
|---|---|---|---|---|---|
| Matter (Dust) | p = 0 | 0 | ∝ a⁻³ | ∝ a⁻³ᐟ² | ∝ t¹ᐟ³ |
| Radiation | p = ρ/3 | 1/3 | ∝ a⁻⁴ | ∝ a⁻² | ∝ t¹ᐟ³ |
| Cosmic Strings | p = −ρ/3 | −1/3 | ∝ a⁻² | ∝ a⁻¹ | ∝ t¹ᐟ² |
| Stiff Matter | p = ρ | 1 | ∝ a⁻⁶ | ∝ a⁻³ | ∝ t¹ᐟ⁶ |
| Dark Energy | p = −ρ | −1 | constant | complex! | not defined |
Important: For dark energy (w = −1), the time dilation becomes complex:
Possible Solutions:
- Define AFC only for w ≥ −1/3 (matter, radiation, strings).
- Modified hypothesis:
dτ/dt = √(|ρ + 3p| / ρ₀). - Treat dark energy as a geometric effect.
Aspherical Metric of AFC
AFC assumes a non-isotropic universe where expansion is different in different directions. This is described by a modified FLRW metric.
Standard FLRW Metric (Isotropic)
AFC Metric (Aspherical)
In AFC, the time component is modified by time dilation, and the spatial metric becomes anisotropic:
Visualization: Aspherical Universe
In AFC, the observable universe is not spherical but elliptical, with a ratio of tangential diameter to radial thickness of about 5.6 : 1. This is considered a testable prediction of AFC.
Testable Predictions of AFC
AFC makes concrete, testable predictions that differ from those of the standard model (ΛCDM). Here are the most important ones:
Scale Factor a(t)
AFC predicts:
In contrast to ΛCDM:
Test: Supernova data, DESI, and Roman Space Telescope.
Asphericity of the Universe
AFC predicts an aspherical universe with a ratio of:
while ΛCDM assumes an isotropic universe (1 : 1).
Test: Galaxy distribution, LSST, and Euclid.
Hubble Tension
AFC explains the Hubble tension — the discrepancy between local and cosmological measurements of H₀ — through the different expansion history.
Test: Comparison of local and cosmological H₀ measurements.
No Big Bang Singularity
In AFC, there is no singularity at t = 0. Instead, the Big Bang is an asymptotic state with:
Test: CMB spectrum and cosmological observations.
Comparison of Predictions
| Phenomenon | Standard ΛCDM | AFC | Testability |
|---|---|---|---|
| Shape of the Universe | isotropic (1 : 1) | aspherical (5.6 : 1) | Galaxy Distribution |
| Scale Factor a(t) | ∝ t²ᐟ³ (Matter) | ∝ t¹ᐟ³ | Supernova Data |
| Big Bang Singularity | yes (ρ → ∞) | no (asymptotic) | CMB Spectrum |
| Accelerated Expansion | dark energy | Geometry + Time Dilation | Supernovae, BAO, CMB |
| Hubble Tension | open question | explainable by model | local vs. cosmological H₀ measurements |
Comparison with the Standard Model (ΛCDM)
Here are the main differences and similarities between AFC and the standard model of cosmology (ΛCDM).
Cosmological Parameters in Comparison
| Parameter | AFC | ΛCDM (Matter) | ΛCDM (Λ-dominated) |
|---|---|---|---|
| Scale Factor a(t) | ∝ t¹ᐟ³ | ∝ t²ᐟ³ | ∝ eᴴᵗ |
| Hubble Parameter H(t) | ∝ 1/t | ∝ 1/t | ≈ constant |
| Acceleration ä/a | < 0 | < 0 | > 0 |
| Density Parameter | model-dependent | ≈ 1 | Matter + Λ |
| Singularity at t = 0 | no | yes | yes |
| Dark Energy Needed? | no | yes | yes |
| Geometry | aspherical | isotropic | isotropic |
Advantages of AFC Over ΛCDM
No Singularity
AFC avoids the Big Bang singularity. Instead, the Big Bang is an asymptotic state.
No Dark Energy
The apparent accelerated expansion is explained by the combination of time dilation and aspherical geometry.
Hubble Tension
The different expansion history of AFC provides an approach to explaining the discrepancy between local and cosmological measurements of H₀.
Simple Mathematical Structure
AFC is based on a few fundamental assumptions: time dilation through energy density and aspherical geometry.
Challenges of AFC
AFC is not yet fully mature. Here are important open questions:
- Dark Energy: The AFC hypothesis leads to a mathematical problem with w = −1.
- Accelerated Expansion Today: Pure time dilation does not yet provide a complete explanation of the observed expansion.
- Experimental Confirmation: AFC must be quantitatively compared with observational data against ΛCDM.
Conclusion: AFC as an Alternative to ΛCDM
The Asymptotic Funnel Cosmology (AFC) is a new cosmological model that considers some fundamental questions of modern cosmology from a different perspective — without dark energy and without a classical Big Bang singularity.
In summary, AFC offers:
- A possible explanation of expansion through the combination of time dilation and geometry.
- No Big Bang singularity as the starting point of the model.
- A cosmology that does not require dark energy.
- A possible approach to explaining the Hubble tension.
- Testable predictions that can be compared with cosmological observations.
AFC is to be understood as a thought experiment and theoretical model approach. Whether the underlying assumptions correspond to nature must be verified through a complete mathematical development and comparison with observational data.
"AFC is a thought experiment on the way to a simpler cosmology — a cosmology without singularity and without dark energy."
© Georg Rönnau, July 30, 2026