= Solution
Each separately conserved component obeys the <constant-equation-of-state density scaling> $\rho\propto a^{-3(1+w)}$. For the flat <Friedmann-Lemaître-Robertson-Walker metric>, the <Friedmann equations> therefore give
$$
\frac{H(z)^2}{H_0^2}=\Omega_m(1+z)^3+\Omega_r(1+z)^4+\Omega_X(1+z)^{3(1+w)}.
$$
Write $H(z)/H_0=1+h_1z+O(z^2)$, where $h_1=[3\Omega_m+4\Omega_r+3(1+w)\Omega_X]/2$. Expanding the reciprocal inside the <luminosity distance> integral gives
$$
H_0d_L=(1+z)\left[z-\frac{h_1}2z^2+O(z^3)\right]
=z+\left(1-\frac{h_1}2\right)z^2+O(z^3).
$$
Matching its quadratic coefficient gives the <deceleration parameter> $q_0=h_1-1$. Using flatness in this part,
$$
\boxed{q_0=\frac{\Omega_m}2+\Omega_r+\frac{1+3w}{2}\Omega_X
=\frac12\left(1+\Omega_r+3w\Omega_X\right).}
$$
A negative <deceleration parameter> signifies accelerated expansion; negative pressure must overcome the positive matter and radiation terms. Restoring units replaces $H_0d_L$ by $H_0d_L/c$ in the expansion, leaving $q_0$ unchanged.
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