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For $V(\varphi )=\lambda {\varphi }^4$, the slow-roll approximation givesTaking the expanding branch $H>0$ and using $H=\sqrt{8\pi G\lambda /3}{\varphi }^2$ gives the first-order linear differential equationThus the quartic-potential slow-roll solution with the normalization used in this question isIf ${\varphi }_0<0$, the field remains negative and $\dot{\varphi }>0$, so it rolls toward the potential minimum at zero.

For $V=\lambda {\varphi }^4/4$ and positive $\varphi$, the slow-roll Friedmann equation givesThe scalar equation then reduces toWith $\varphi (t_0)={\varphi }_0$,Furthermore,Thus the quartic-potential slow-roll solution is