Quantifying the impact of selection effects on FRB DM-$z$ relation cosmological inference

Fast Radio Bursts (FRBs) have emerged as powerful probes of baryonic matter in the Universe, offering constraints on cosmological and feedback parameters through their extragalactic dispersion measure-redshift (DM$_\mathrm{exgal}$-$z$) relation. However, the observed FRB population is shaped by complex selection effects arising from instrument sensitivity, DM-dependent search efficiency, and FRB source population redshift-evolution. In this work, we quantify the impact of such observational and population selection effects on cosmological inference derived from the conditional distribution $p(\mathrm{DM}_{\mathrm{exgal}}|z)$. Using forward-modeled FRB population simulations, we explore progressively realistic survey scenarios incorporating redshift evolution, luminosity function, and instrument DM selection function. To enable rapid likelihood evaluations, we build a neural-network emulator for the variance in cosmic DM, $\sigma^2[\mathrm{DM}_{\mathrm{cosmic}}(z)]$, trained on $5\times10^4$ baryonification halo-model simulations, achieving $\leq4\%$ accuracy up to $z=4$. We demonstrate that while redshift and DM-dependent selection effects substantially alter the joint distribution $p(\mathrm{DM},z)$, they have a negligible impact on the conditional distribution $p(\mathrm{DM}_{\mathrm{exgal}}|z)$ for current sample sizes. The parameter biases are $\lesssim0.8\sigma$ for $10^2$ FRBs, indicating that conditional analyses are robust for present surveys. However, depending on the survey DM-dependent search efficiency, these biases may exceed $3\sigma$ for $10^4$ FRBs, thus implying that explicit modeling of selection effects will be essential for next-generation samples.