Two-color ($N_c=2$) QCD world is one of the useful testing grounds to delineate cold and dense QCD matter, since lattice QCD simulations are straightforwardly applicable thanks to the disappearance of the sign problem. Motivated by recent lattice results, I am being investigating properties of dense two-color QCD theoretically by constructing a new chiral model, the linear sigma model for $N_c=2$ based on the Pauli-Gursey $SU(4)$ symmetry. In this write-up, I summarize my recent works on the linear sigma model in dense two-color QCD, such as the hadron mass spectrum and appearance of the sound-velocity peak.