 Solution

For a homogeneous canonical scalar, ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and $P_{\varphi }={\dot{\varphi }}^2/2-V$. The acceleration equation requires $V>{\dot{\varphi }}^2$ for cosmic inflation, equivalently ${ϵ}_H=-\dot{H}/H^2<1$ in the flat scalar-dominated limit. Inflation need not be exactly de Sitter.

A standard sufficient regime for a long period is positive potential energy dominating the kinetic energy, physical gradient energy and curvature. Homogeneity over an expanding approximately Hubble-sized patch allows the gradient term to redshift away. In comoving coordinates the gradient energy is $|{\nabla }_{\mathrm{com}}\varphi {|}^2/(2a^2)$; the gradient in the displayed energy formula must be interpreted physically if no $a^{-2}$ is written there. RequireThe slow-roll equations are then $3H\dot{\varphi }\simeq -V^{\prime }$ and $H^2\simeq 8\pi V/(3m_{\mathrm{pl}}^2)$. Sufficient potential conditions areThe potential must remain sufficiently flat over a large enough field interval to produce the required expansion. These slow-roll conditions are stronger than the bare necessary condition for acceleration.