  Recently, the peak structure of the sound velocity was observed in the
lattice simulation of two-color and two-flavor QCD at the finite quark chemical
potential. The comparison with the chiral perturbation theory (ChPT) result was
undertaken, however, the ChPT failed in reproducing the peak structure. In this
study, to extend the ChPT framework, we incorporate contributions of the
$\sigma$ meson, that is identified as the chiral partner of pions, on top of
the low-energy pion dynamics by using the linear sigma model (LSM). Based on
the LSM we derive analytic expressions of the thermodynamic quantities as well
as the sound velocity within a mean-field approximation. As a result, we find
that those quantities are provided by sums of the ChPT results and corrections,
where the latter is characterized by a mass difference between the chiral
partners, the $\sigma$ meson and pion. The chiral partner contributions are
found to yield a peak in the sound velocity successfully. We furthermore show
that the sound velocity peak emerges only when $m_\sigma >\sqrt{3}m_\pi$ and
$\mu_q > m_\pi$, with $m_{\sigma(\pi)}$ and $\mu_q$ being the $\sigma$ meson
(pion) mass and the quark chemical potential, respectively. The correlation
between the sound velocity peak and the sign of the trace anomaly is also
addressed.
