With the homogeneous/isotropic metric (Step 16) and our identifications \(a=e^{-\Phi}\) and \(H=\dot a/a=-\dot\Phi\) (dot = \(d/d\tau\)), the Einstein \(tt\)–equation (equivalently the Hamiltonian constraint) collapses to
\[ \boxed{ \; H^{2} \;+\; \frac{k}{a^{2}} \;=\; \frac{8\pi}{3}\,\rho \; } \qquad (G=c=1) \]
This says the universe’s expansion rate squared is set by its energy density \(\rho\), up to the curvature “tax” \(k/a^{2}\). In time-first language: once you specify the time potential \(\Phi(\tau)\), you’ve fixed \(H=-\dot\Phi\) and \(a=e^{-\Phi}\), so the equation simply balances the slope of time against the stuff in the universe and any curvature of the spatial slice.
Quick intuitions:
If you prefer standard constants, replace the right-hand side by \((8\pi G/3)\rho\) and the curvature term by \(k c^{2}/a^{2}\).
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(H\) | Hubble rate | fractional expansion rate | \(H=\dot a/a=-\dot\Phi\) (1/time) | “Universe’s breathing rate” |
| \(a(\tau)\) | scale factor | spatial ruler on slices | \(a=e^{-\Phi}\) (dimensionless) | “Rubber-sheet stretch” |
| \(\Phi(\tau)\) | time potential | sets lapse & size via \(a=e^{-\Phi}\) | dimensionless | “Altitude of time” controlling expansion |
| \(k\) | curvature index | \(+1,0,-1\) for closed/flat/open | dimensionless; appears as \(k/a^{2}\) | “Built-in bend of the grid” |
| \(\rho\) | energy density | total density (matter+radiation+etc.) | energy/volume | “Mass–energy filling the room” |
| dot \((\dot{ })\) | proper-time derivative | \(d/d\tau\) along comoving clocks | 1/time × (units of variable) | “Rate a comoving watch sees” |