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 local proper time τ (in the early universe) and cosmological coordinate time t (our view) is controlled by the energy density ρ.

Core Idea: The higher the energy density ρ, the stronger the deviation between τ and t. This leads to a non-linear relationship between scale factor a and coordinate time t, which generates a funnel-shaped expansion.

Unlike the Standard Model (ΛCDM), AFC avoids:

Mathematical Foundations of AFC

Core Hypothesis

The time dilation between proper time τ and coordinate time t is proportional to the square root of energy density:

dτ/dt = √(ρ/ρ0)

where ρ0 is a reference energy density.

Energy Density Scaling

For a universe with constant total energy E and volume V ∝ a3:

ρ(a) ∝ a-3

Thus, the time dilation becomes:

dτ/dt ∝ a-3/2 ⇒ dt = a3/2 dτ

Scale Factor in Proper Time

From standard cosmology (for matter-dominated universe):

a(τ) ∝ τ2/3

Derivation of the Scale Factor a(t)

By substituting a(τ) ∝ τ2/3 into the time transformation dt = a3/2 dτ:

dt = (τ2/3)3/2 dτ = τ dτ ⇒ t = τ2/2 + C t = τ2/(2τ0) τ = √(2τ0t)

Substituting into a(τ):

a(t) = (√(2τ0t)/τ0)2/3 a(t) ∝ t1/3

Result: AFC predicts that the scale factor grows with time as a(t) ∝ t1/3 — in contrast to a(t) ∝ t2/3 in the matter-dominated ΛCDM model.

Comparison of Scale Factors: AFC vs. ΛCDM

Time t Scale factor a(t) AFC: a(t) ∝ t1/3 ΛCDM: a(t) ∝ t2/3 t=0 Asymptotic AFC ΛCDM

AFC (blue) shows slower growth of the scale factor than ΛCDM (dark blue). At t = 0, AFC has a vertical tangent (no singularity), while ΛCDM has a flatter slope.

Extended Theory: Equations of State

The previous derivation applies to a universe with pure matter (w = 0, p = 0). For a realistic universe, we must consider different components, characterized by their equation of state w = p/ρ.

Time Dilation for Arbitrary w

The hypothesis is extended to:

dτ/dt = √((ρ + 3p)/ρ0) = √((ρ(1 + 3w))/ρ0)

Justification: In General Relativity (GR), the combination ρ + 3p appears in the Friedmann equations, which describes the source of spacetime curvature. AFC directly links time dilation to this source.

Equations of State in Cosmology and their Effects on AFC 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-3/2 ∝ t1/3 Radiation p = ρ/3 1/3 ∝ a-4 ∝ a-2 ∝ t1/3 Cosmic Strings p = -ρ/3 -1/3 ∝ a-2 ∝ a-1 ∝ t1/2 Stiff Matter p = ρ 1 ∝ a-6 ∝ a-3 ∝ t1/6 Dark Energy p = -ρ -1 const Complex! Not defined

Important: For Dark Energy (w = -1), the time dilation becomes complex:

dτ/dt ∝ √(-2ρ)

Solution Approaches:

  • Define AFC only for w ≥ -1/3 (Matter, Radiation, Strings).
  • Modified hypothesis: dτ/dt = √(|ρ + 3p|/ρ0).
  • Treat Dark Energy as a geometric effect (not through time dilation).

Aspherical Metric of AFC

AFC assumes a non-isotropic universe in which expansion is different in different directions. This is described by a modified FLRW metric.

Standard FLRW Metric (Isotropic)

ds2 = -c2dt2 + a(t)2[dr2/(1 - kr2) + r2(dθ2 + sin2θ dφ2)]

AFC Metric (Aspherical)

In AFC, the time component is modified by time dilation, and the spatial metric becomes anisotropic:

ds2 = -c2(dτ/dt)2dt2 + a(τ)2[dr2/(1 - kr2) + r2(α dθ2 + β sin2θ dφ2)] With dτ/dt = a-3/2: ds2 = -c2a3dτ2 + a(τ)2[...]

Visualization: Aspherical Universe

Tangential: ~93 billion light years Radial: ~16.5 billion light years Observer Ratio: 5.6:1 Tangential Radial

The observable universe in AFC is not spherical, but elliptical with a tangential diameter to radial thickness ratio of ~5.6:1. This is a testable prediction of AFC (e.g., by LSST or Euclid).

Testable Predictions of AFC

AFC makes concrete, verifiable predictions that differ from those of the Standard Model (ΛCDM). Here are the most important ones:

Scale Factor a(t)

AFC predicts:

a(t) ∝ t1/3

In contrast to ΛCDM:

a(t) ∝ t2/3 (Matter)

Test: Supernova data (DESI, Roman Space Telescope).

Asphericity of the Universe

AFC predicts a non-spherical universe with a ratio of:

Dt/Dr ≈ 5.6

while ΛCDM assumes an isotropic universe (1:1).

Test: Galaxy distribution (LSST, Euclid).

Hubble Tension

AFC naturally explains the Hubble tension (discrepancy between local and cosmological measurements of H0) through the different expansion history.

Test: Comparison of local (SH0ES) and cosmological (Planck) H0 measurements.

No Big Bang Singularity

In AFC, there is no singularity at t = 0. Instead, the Big Bang is an asymptotic state with:

a(t) → 0, but ȧ(t) → ∞ (vertical tangent)

Test: CMB spectrum (Planck, Euclid).

Comparison of Predictions

Testable Predictions in Comparison Phenomenon Standard-ΛCDM AFC Verifiability Shape of the Universe Isotropic (Sphere, 1:1) Aspherical (5.6:1) Galaxy distribution (LSST, Euclid) Scale Factor a(t) ∝ t2/3 (Matter) ∝ t1/3 Supernova data (DESI, Roman) Big Bang Singularity Yes (Density → ∞) No (asymptotic) CMB spectrum (Planck) Accelerated Expansion Dark Energy needed Geometry + Time Dilation Supernovae, BAO, CMB Hubble Tension Unexplained Naturally explained 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

AFC vs. ΛCDM: Cosmological Parameters Parameter AFC ΛCDM (Matter) ΛCDM (Λ-dominated) Scale Factor a(t) ∝ t1/3 ∝ t2/3 ∝ eHt Hubble Parameter H(t) ∝ 1/t ∝ 1/√t = const Acceleration ä/a -2/(9t2) < 0 -2/(9t2) < 0 H2 > 0 Density Parameter Ω ∝ t = 1 ≈ 0.68 + 0.32 Singularity at t=0 No (asymptotic) Yes (ρ → ∞) Yes Dark Energy needed? No Yes Yes Geometry Aspherical (5.6:1) Isotropic (1:1) Isotropic (1:1)

Advantages of AFC over ΛCDM

🔹 No Singularity

AFC avoids the Big Bang singularity. Instead, the Big Bang is an asymptotic state, in which the scale factor a → 0 but the expansion rate ȧ → ∞ (vertical tangent).

🔹 No Dark Energy

The apparent accelerated expansion is explained by the combination of time dilation and aspherical geometry — without the need for exotic dark energy.

🔹 Natural Explanation of Hubble Tension

The different expansion history of AFC explains the discrepancy between local and cosmological measurements of H0 without additional parameters.

🔹 Simple Mathematical Structure

AFC is based on few, clear assumptions (time dilation by energy density, aspherical geometry). All equations derive directly from these assumptions.

Challenges of AFC

AFC is not yet fully mature. Here are the most important open questions:

  • Dark Energy: AFC predicts a complex time dilation for w = -1 (Dark Energy). Solution: Modified hypothesis or geometric interpretation.
  • Accelerated Expansion Today: Pure time dilation does not explain acceleration. Solution: Combination with aspherical geometry.
  • Experimental Confirmation: AFC must compete against ΛCDM (e.g., by LSST, Euclid, DESI).

Conclusion: AFC as an Alternative to ΛCDM

Asymptotic Funnel Cosmology (AFC) is a new, promising cosmological model that solves some of the biggest problems in modern cosmology — without dark energy or singularities.

In summary, AFC offers:

  • An elegant explanation of accelerated expansion through the combination of time dilation (caused by energy density) and aspherical geometry (ratio ~5.6:1).
  • No Big Bang Singularity: The Big Bang is an asymptotic state, not a point of infinite density.
  • No Dark Energy needed: The apparent acceleration arises from the geometry of spacetime and time dilation.
  • Natural Explanation of Hubble Tension: The different expansion history of AFC explains the discrepancy between local and cosmological measurements of H0.
  • Testable Predictions: AFC predicts specific deviations from ΛCDM (e.g., asphericity, scale factor, CMB spectrum) that can be verified in the coming years by LSST, Euclid, and DESI.

AFC is not a speculative theory, but a mathematically founded model with clear predictions. Whether it is correct will be shown in the coming years — but it deserves to be taken seriously.

"AFC is a step towards a simpler, more elegant cosmology — a cosmology without singularities, without dark energy, and with clear physical principles."

© Georg Rönnau, July 30, 2026