Number Count of Peaks in the CMB Map

We investigate the dependence of cosmological parameters on the number count of peaks (local maxima and minima) in the cosmic microwave background (CMB) sky. The peak statistics contains the whole information of acoustic oscillations in the angular power spectrum $C_l$ over $l$-space and thus it can place complementary constraints on the cosmological parameters to those obtained from measurements of $C_l$. Based on the instrumental specifications of Planck, we find that the number count of peaks can provide new constraints on the combination of the matter density ${\mathrm{\Omega }}_{\mathrm{m}}$ and the Hubble parameter $h$ approximately scaled as ${\mathrm{\Omega }}_{\mathrm{m}}{h}^{-4.9}$ for a flat $\mathrm{\Lambda }$CDM model with ${\mathrm{\Omega }}_m=0.3$ and $h=0.7$. Therefore, we suggest that combining it with the constraints from $C_l$ scaled as ${\mathrm{\Omega }}_m{h}^{3.8}$ (or commonly ${\mathrm{\Omega }}_{\mathrm{m}}{h}^2$) can potentially determine ${\mathrm{\Omega }}_{\mathrm{m}}$ or equivalently solve the cosmic degeneracy by the CMB data alone.