Accelerated expansion iff $w < -1/3$

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Let $G,w\in\mathbb R$ with $G>0$, let $a,\varepsilon,p:\mathbb R\to\mathbb R$ and $t\in\mathbb R$. Assume $a(t)>0$, $\varepsilon(t)>0$, $p(t) = w\,\varepsilon(t)$ and $3\,\dfrac{a^{\prime\prime}(t)}{a(t)} = 0 - 4\pi G(\varepsilon(t)+3p(t))$, where $a^{\prime\prime}(t)$ is the derivative at $t$ of the derivative function of $a$ (with the convention that it is $0$ where not differentiable — no differentiability is assumed). Then $a^{\prime\prime}(t) > 0$ if and only if $w < -\tfrac13$. All hypotheses are at the single time $t$.