Once we’ve tied space to time via \(a(\tau)=e^{-\Phi(\tau)}\), the universe’s “breathing rate” is just how quickly \(\Phi\) changes. Differentiate \(a\):
\[ \dot a = -\,\dot\Phi\,e^{-\Phi} \;\Rightarrow\; \frac{\dot a}{a} = -\,\dot\Phi. \]
So the Hubble parameter \(H\) equals the negative slope of the time potential. If \(\dot\Phi<0\), then \(H>0\) (expansion); if \(\dot\Phi>0\), then \(H<0\) (contraction). This makes cosmic expansion a time-first statement: measure how the clock-rate field \(\Phi\) drifts in proper time \(\tau\), and you’ve measured \(H\).
Two quick consequences:
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(H(\tau)\) | Hubble rate | fractional growth rate of spatial scale | \(H=\dot a/a\) (1/time) | “Universe’s breathing rate” |
| \(a(\tau)\) | scale factor | global size of spatial slices | \(a=e^{-\Phi}\) (dimensionless) | “Rubber-sheet stretch factor” |
| \(\dot a\) | scale-factor speed | how fast the scale is changing | \(d a/d\tau\) (1/time × dimensionless) | “How quickly the ruler is stretching” |
| \(\Phi(\tau)\) | time potential | log-lapse controlling clock rate | dimensionless | “Altitude of time” |
| \(\dot\Phi\) | time-slope | how clock-rate field drifts in \(\tau\) | \(d\Phi/d\tau\) (1/time) | “Tilt of the time landscape” |