Top panel: the cumulative number of sub-halos with mass φ( <M_sh) expected in the observed field for WDM with different thermal relic mass shown in the labels. The thickness of the lines represents the uncertainties in the theoretical predictions related to the uncertainties in the concentration parameter c and in the radius R assuming an uncertainty range 1 ≤ c ≤ 4 and R = 1.5–2 Mpc. The effect of extending the uncertainty ranges of c and R is shown in detail in Fig. 3. The predicted number is compared with the observed values φ_obs at 1-σ and 2-σ confidence levels, represented by the horizontal shaded regions. Bottom panel: for different values of the thermal relic mass m_X, we show the maximum value of the predicted number of DM halos in the observed field, and compare it with the observed number within 1-σ and 2-σ confidence levels.