  With an effective chiral model, we investigate how the mesonic cross
couplings $\sigma-\rho$ and $\omega-\rho$ affect the density content of the
symmetry energy and its higher-order slope parameters. Earlier mentioned
cross-couplings are crucial to controlling the density content of symmetry
energy. For this purpose, we did a case study for different values of the
symmetry energy $J_1$, defined at density 0.1 fm$^{-3}$ in the range (23.4 -
25.2) for a fixed value of the slope of the symmetry energy $L_0 = 60$ MeV at
saturation density and investigate its effect on the higher-order coefficients
and their influence on the underlying equation of state. We found that the
model with $J_1= 24.6$ MeV is more favorable with the pure neutron matter (PNM)
constraints obtained from $\chi$EFT calculations. In addition, we show that all
of our models predict a monotonically increasing speed of sound up to four
times the saturation density. The speed of sound decreases/saturates above that
point and approaches the conformal limit approximately $\sqrt{1/3}~c$ at the
center of the maximum mass star.
