= Solution
The <comoving particle horizon> is the maximum radial distance a photon could have travelled since the initial <Big Bang>, expressed in <comoving coordinates>. It describes possible past causal contact, whereas a <cosmological event horizon> concerns photons that can reach an observer in the future.
For a radial <null geodesic> in the <FLRW metric>, $dr/\sqrt{1-kr^2}=c\,dt/a$. Spatial flatness sets $k=0$, so with $a_0=1$,
$$
r_p(t)=c\int_0^t\frac{dt'}{a(t')}=c\int_0^{a(t)}\frac{da'}{a'^2H(a')}.
$$
The given <Hubble parameter> then supplies $a^2H=H_0\sqrt{\Omega_{r,0}+\Omega_{m,0}a+\Omega_{\Lambda,0}a^4}$. Since <cosmological redshift> obeys $a=(1+z)^{-1}$,
$$
\boxed{r_p(z)=\frac c{H_0}\int_0^{(1+z)^{-1}}\frac{da}{\sqrt{\Omega_{r,0}+\Omega_{m,0}a+\Omega_{\Lambda,0}a^4}}.}
$$
The corresponding proper horizon at that epoch is $a(z)r_p(z)$.
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