An efficient and robust method to estimate halo concentration

We propose an efficient and robust method to estimate the halo concentration based on the first moment of the density distribution, which is $R_1\equiv {\int }_0^{r_{\mathrm{v}\mathrm{i}\mathrm{r}}}4\pi r^3\rho (r)dr\mathrm{/}{M}_{\mathrm{v}\mathrm{i}\mathrm{r}}\mathrm{/}{r}_{\mathrm{v}\mathrm{i}\mathrm{r}}$. We find that $R_1$ has a monotonic relation with the concentration parameter of the NFW profile, and that a cubic polynomial function can fit the relation with an error $\lesssim 3\mathrm{\%}$. Tests on ideal NFW halos show that the conventional NFW profile fitting method and the $V_{\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{/}{V}_{\mathrm{v}\mathrm{i}\mathrm{r}}$ method produce biased halo concentration estimation by $\approx 10\mathrm{\%}$ and $\approx 30\mathrm{\%}$, respectively, for halos with 100 particles. In contrast, the systematic error for our $R_1$ method is smaller than $0.5\mathrm{\%}$ even for halos containing only 100 particles. Convergence tests on realistic halos in $N$-body simulations show that the NFW profile fitting method underestimates the concentration parameter for halos with $\lesssim 300$ particles by $\gtrsim 20\mathrm{\%}$, while the error for the $R_1$ method is $\lesssim 8\mathrm{\%}$. We also show other applications of $R_1$, including estimating $V_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and the Einasto concentration $c_{\mathrm{e}}\equiv r_{\mathrm{v}\mathrm{i}\mathrm{r}}\mathrm{/}{r}_{-2}$. The calculation of $R_1$ is efficient and robust, and we recommend including it as one of the halo properties in halo catalogs of cosmological simulations.