= Solution
In a spatially flat <FLRW metric>, a radial <null geodesic> satisfies $d\chi=c\,dt/a(t)$. Normalize $a_0=1$. The <comoving particle horizon> at time $t$ is therefore $r_h=c\int_0^t dt'/a(t')$, using the initial singularity as the time origin. The <cosmological redshift> relation $1+z=1/a$ gives
$$
\frac{dz}{dt}=-(1+z)H(z),\qquad \frac{dt}{a(t)}=-\frac{dz}{H(z)}.
$$
For matter and a <cosmological constant> in a spatially flat model, the <Friedmann equation> gives $H(z)=H_0[\Omega_{m,0}(1+z)^3+\Omega_{\Lambda,0}]^{1/2}$. The initial singularity corresponds to $z\to\infty$, so reversing the integration limits yields
$$
\boxed{r_h(z)=\frac c{H_0}\int_z^\infty\frac{dz'}{[\Omega_{m,0}(1+z')^3+\Omega_{\Lambda,0}]^{1/2}}.}
$$
This is a <comoving particle horizon>, not a distance to the source currently observed at redshift $z$, and not a future-directed <cosmological event horizon>. The corresponding proper horizon radius at that epoch is $r_h(z)/(1+z)$.
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