There are two equivalent ways to write the second FRW equation. In “space-first” form:
\[ \boxed{ \ \frac{\ddot a}{a} \;=\; -\,\frac{4\pi}{3}\,(\rho+3p) \ } \qquad (G=c=1) \]
It says positive energy density and positive pressure both decelerate expansion (pressure gravitates, with a factor of 3).
In “time-first” variables, using \(H=\dot a/a=-\dot\Phi\), the same content is usually written as
\[ \boxed{ \ \dot H \;=\; -\,4\pi(\rho+p) \;+\; \frac{k}{a^{2}} \ } \quad\Longleftrightarrow\quad \boxed{ \ \ddot\Phi \;=\; 4\pi(\rho+p)\;-\;\frac{k}{a^{2}} \ } . \]
Read this as: the curvature of the time potential \(\Phi(\tau)\) is driven upward by \((\rho+p)\) and driven downward by positive spatial curvature \(k>0\). Combine with Step 17’s first Friedmann equation and the continuity law \(\dot\rho+3H(\rho+p)=0\) to check consistency.
Quick intuition & checks:
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\ddot a/a\) | acceleration parameter | second derivative of size per size | 1/time\(^2\) | “How quickly the rubber sheet speeds up/slows down” |
| \(H\) | Hubble rate | \(\dot a/a\) | 1/time | “Breathing rate of space” |
| \(\Phi(\tau)\) | time potential | log-lapse; \(a=e^{-\Phi}\) | dimensionless | “Altitude of time” |
| \(\ddot\Phi\) | curvature of time | second proper-time derivative of \(\Phi\) | 1/time\(^2\) | “Concavity of the time landscape” |
| \(\rho\) | energy density | total (matter+radiation+Λ…) | energy/volume | “Stuff per room” |
| \(p\) | pressure | isotropic stress of the contents | energy/volume | “Push of the stuff” |
| \(k\) | curvature index | spatial curvature sign | \(+1,0,-1\) (appears as \(k/a^2\)) | “Built-in bend of the grid” |
| \(w\) | EoS parameter | \(p=w\rho\) | dimensionless | “Material personality: dust 0, rad 1/3, Λ −1” |