Cosmic inflation

Remarks

Equivalent forms: $\ddot a > 0$ holds exactly when the comoving Hubble radius $1/(aH) = 1/\dot a$ decreases, and exactly when the deceleration parameter satisfies $q < 0$. By the second Friedmann equation with $\Lambda = 0$, it holds exactly when $\rho + 3p < 0$, that is, for matter with $\rho > 0$, when $w = p/\rho < -1/3$. Example: a homogeneous scalar field $\phi$ with potential $V$ has $\rho = \tfrac{1}{2}\dot\phi^2 + V(\phi)$ and $p = \tfrac{1}{2}\dot\phi^2 - V(\phi)$, so $\rho + 3p = 2(\dot\phi^2 - V(\phi))$, and it inflates exactly while $\dot\phi^2 < V(\phi)$. Non-example: in the radiation era $a \propto t^{1/2}$, so $\ddot a = -\tfrac{1}{4} t^{-3/2} \cdot \text{const} < 0$, and the expansion decelerates. The amount of inflation is measured by the number of e-folds $N = \ln\left( a(t_{\text{end}}) / a(t_{\text{start}}) \right)$. Alan Guth proposed inflation in 1981 as a solution of the horizon and flatness problems.