= Dark-energy acceleration criterion
{title2=$q_0=\tfrac12\Omega_{m,0}+\tfrac12(1+3w_0)\Omega_{{\rm DE},0}$}
Neglect <radiation in cosmology> and take <pressureless matter> plus a separately conserved <dark energy> component with $p_{\rm DE}=w\rho_{\rm DE}c^2$, where $\rho$ here denotes mass-equivalent <energy density>. The <Friedmann acceleration equation> gives
$$
\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left[\rho_m+\rho_{\rm DE}(1+3w)\right].
$$
Thus $w<-1/3$ for a positive-density component is necessary for that component to accelerate expansion, but the total must also overcome <pressureless matter>. For a <cosmological constant>, $w=-1$ and $q_0=\Omega_{m,0}/2-\Omega_{\Lambda,0}$, so acceleration requires $\Omega_{\Lambda,0}>\Omega_{m,0}/2$. <Spatial curvature of an FLRW universe> enters the distance relation but has no separate term in this acceleration criterion. A <Hubble diagram> alone does not identify a microscopic <dark energy> mechanism.
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