= Low-redshift acceleration cosmography
{title2=$d_L=(c/H_0)[z+(1-q_0)z^2/2+O(z^3)]$}
A local derivation makes the acceleration sign transparent. Normalize $a(t_0)=1$ and put $\tau=t_0-t_e$. A <Taylor series> gives
$$
a(t_0-\tau)=1-H_0\tau-\frac12q_0H_0^2\tau^2+O(\tau^3),\qquad
z=H_0\tau+(1+q_0/2)H_0^2\tau^2+O(\tau^3).
$$
The <radial null geodesic in FLRW spacetime> gives $D_C=c\int_{t_e}^{t_0}dt/a(t)=c[\tau+H_0\tau^2/2+O(\tau^3)]$. Invert the <redshift> series, use $d_L=(1+z)D_C$ through second order, and obtain the displayed <luminosity distance>. <Spatial curvature of an FLRW universe> first changes this relation at third order. Lower $q_0$ therefore means larger $d_L$ at a fixed sufficiently small positive <redshift> and common $H_0$.
At $z=0.05$, the quadratic approximation gives $222.4353\,\mathrm{Mpc}$ for $q_0=-0.55$, within $0.07\%$ of the full flat-model value $222.2891\,\mathrm{Mpc}$. At $z=0.5$ it gives $2971.1574\,\mathrm{Mpc}$, about $4.9\%$ too large. A useful sign argument at low <redshift> is not a precision replacement for the full distance integral in a distant <supernova> analysis.
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