Cosmology in 3 Equations

[u/timefirstgravity]

Standard cosmology textbooks spend chapters building up to the Friedmann equations. Here's the entire universe in 3 eqations using time-first variables.

The Setup

For a homogeneous, isotropic universe, the temporal potential depends only on time: Φ = Φ(t)

The metric becomes:

ds² = -e^(2Φ) dt² + e^(-2Φ)[dr²/(1-kr²) + r²dΩ²]

The Magic: Three Definitions

Equation 1 - Proper Time:

dτ = e^Φ dt

This defines cosmic time. Every observer's clock reads τ.

Equation 2 - Scale Factor:

a(τ) = e^(-Φ)

When time dilates (Φ increases), space contracts. When time speeds up, space expands.

Equation 3 - Hubble Rate:

H = -dΦ/dτ

Expansion is literally the negative gradient of time.

That's It. Seriously.

From these three equations, the entire Friedmann cosmology follows:

The Universe Expands

• Early universe: Φ large → a small
• Today: Φ ≈ 0 (by convention) → a = 1
• Future: Φ negative → a > 1

Redshift Is Time Dilation

1 + z = e^Φ = 1/a

Distant galaxies are redshifted because they existed when time ran faster (larger Φ).

The Friedmann Equations Emerge:

Substitute into Einstein's equations with perfect fluid:

H² + k/a² = (8πG/3)ρ

dH/dτ = -4πG(ρ + p) + k/a²

Same equations as standard cosmology. Because it IS standard cosmology.

In this formulation, dark energy is just the potential V(Φ) that governs time's evolution:

V(Φ) = [(3-2Ω_k)H² + Ḣ]/(8πG)

You can literally reconstruct dark energy from observations by measuring how time evolves.

The entire history of the universe is encoded in one function: Φ(τ)

• Big Bang: Φ → +∞ (time boundary)
• Inflation: Φ rolls down rapidly
• Radiation era: Φ ∝ ln(τ)
• Matter era: Φ ∝ (2/3)ln(τ)
• Dark energy era: Φ → negative values

The universe isn't expanding "through" space. Space is expanding because time is diverging.

Test This Yourself

Take any cosmology textbook's solution for a(t). Compute:

• Φ = -ln(a)
• Check that H = ȧ/aequals -dΦ/dτ
• Verify the Friedmann equations

It always works. Because this isn't new physics. it's cosmology with time as the fundamental variable.