A numerical fit of analytical to simulated density profiles in dark matter haloes

Analytical and geometrical properties of generalized power-law (GPL) density profiles are investigated in detail. In particular, a one-to-one correspondence is found between mathematical parameters (a scaling radius, r_0, a scaling density, rho_0, and three exponents, alpha, beta, gamma), and geometrical parameters (the coordinates of the intersection of the asymptotes, x_C, y_C, and three vertical intercepts, b, b_beta, b_gamma, related to the curve and the asymptotes, respectively): (r_0,rho_0,alpha,beta,gamma) <--> (x_C,y_C,b,b_beta,b_gamma). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and nonlinear least-absolute values and least-squares fits involving the above mentioned five parameters (RFSM5 method) are prescribed. More specifically, the sum of absolute values or squares of absolute logarithmic residuals, R_i= log rhoSDH(r_i)-log rhoGPL(r_i), is evaluated on 10^5 points making a 5- dimension hypergrid, through a few iterations. The size is progressively reduced around a fiducial minimum, and superpositions on nodes of earlier hypergrids are avoided. An application is made to a sample of 17 SDHs on the scale of cluster of galaxies, within a flat LambdaCDM cosmological model (Rasia et al. 2004). In dealing with the mean SDH density profile, a virial radius, rvir, averaged over the whole sample, is assigned, which allows the calculation of the remaining parameters. Using a RFSM5 method provides a better fit with respect to other methods. The geometrical parameters, averaged over the whole sample of best fitting GPL density profiles, yield (alpha,beta,gamma) approx(0.6,3.1,1.0), to be compared with (alpha,beta,gamma)=(1,3,1), i.e. the NFW density profile (Navarro et al. 1995, 1996, 1997), (alpha,beta,gamma)=(1.5,3,1.5) (Moore et al. 1998, 1999), (alpha,beta,gamma)=(1,2.5,1) (Rasia et al. 2004); and, in addition, gamma approx 1.5 (Hiotelis 2003), deduced from the application of a RFSM5 method, but using a different definition of scaled radius, or concentration; and gamma approx 1.2-1.3 deduced from more recent high-resolution simulations (Diemand et al. 2004, Reed et al. 2005). No evident correlation is found between SDH dynamical state (relaxed or merging) and asymptotic inner slope of the fitting logarithmic density profile or (for SDH comparable virial masses) scaled radius. Mean values and standard deviations of some parameters are calculated, and in particular the decimal logarithm of the scaled radius, xivir, reads < log xivir >=0.74 and sigma_s log xivir=0.15-0.17, consistent with previous results related to NFW density profiles. It provides additional support to the idea, that NFW density profiles may be considered as a convenient way to parametrize SDH density profiles, without implying that it necessarily produces the best possible fit (Bullock et al. 2001). A certain degree of degeneracy is found in fitting GPL to SDH density profiles. If it is intrinsic to the RFSM5 method or it could be reduced by the next generation of high-resolution simulations, still remains an open question.