A necessary condition for best fitting analytical to simulated density profiles in dark matter haloes

Analytical and geometrical properties of generalized power-law (GPL) density profiles are reviewed, and special effort is devoted to the special cases where GPL density profiles reduce to (i) a double power-law (DPL), and (ii) a single power-law (SPL). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and non-linear least-squares fits are prescribed, involving five parameters (a scaling radius, r0, a scaling density, ρ0, and three exponents, α, β, γ), which specify the fitting profile (RFSM5 method). More specifically, the validity of a necessary condition for the occurrence of an extremal point, is related to the existence of an intersection between three surfaces in a three-dimension space. Using the algorithm makes also establish that the extremal point is a fiducial minimum, while the explicit calculation of the Hessian determinant is avoided to gain in simplicity. In absence of a rigorous proof, the fiducial minimum can be considered as nothing but a fiducial absolute minimum. An application is made to a sample of 17 SDHs on the scale of cluster of galaxies, within a flat ΛCDM cosmological model (E. Rasia, G. Tormen, L. Moscardini, MNRAS 351 (2004) 237). In dealing with the averaged SDH density profile (ADP), a virial radius, rvir, equal to the mean over the whole sample, is assigned, which allows the calculation of the remaining parameters. The following results are found. (i) A necessary condition for the occurrence of an extremal point is satisfied for eight sample haloes, and is not for the remaining nine together with ADP. In the former alternative, an extremal minimum point (EMP) may safely exist. In the latter alternative, the occurrence of an EMP cannot be excluded, but only a non-extremal minimum can be determined. (ii) The occurrence of an EMP implies a sum of square residuals which is systematically lower than its counterpart deduced by use of numerical RFSM5 methods. Accordingly, EMPs may safely be thought of as absolute minima. With regard to sample haloes where no EMP is detected, the above result maintains in four cases (including ADP), while the contrary holds for the remaining five cases. (iii) The best fit (with no EMP detected) is provided by DPL density profiles for three sample haloes. In addition, DPL density profiles make a rough, but viable approximation in fitting SDH density profiles. The contrary holds for SPL density profiles. No evident correlation is found between SDH dynamical state (relaxed or merging) and asymptotic inner slope of the 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, ξvir, reads and , the standard deviation exceeding by a factor 3.3–3.4 its counterpart evaluated in an earlier attempt using Navarro et al. (J.F. Navarro, C.S. Frenk, S.D.M. White, MNRAS 275 (1995) 720, J.F. Navarro, C.S. Frenk, S.D.M. White, ApJ 462 (1996) 563, J.F. Navarro, C.S. Frenk, S.D.M. White, ApJ 490 (1997) 493) density profiles (J.S. Bullock, T.S. Kolatt, Y. Sigad, MNRAS 321 (2001) 559). If a large dispersion still maintains for richer samples, in dealing with analytical RSFM5 methods, a low dispersion found in N-body simulations seems to be an artefact, due to the assumption of Navarro et al. (or any equivalent choice) density profile. It provides additional support to the idea, that Navarro et al. density profiles may be considered as a convenient way to parametrize SDH density profiles, without implying that it necessarily produces the best possible fit (J.S. Bullock, T.S. Kolatt, Y. Sigad, MNRAS 321 (2001) 559). With regard to RFSM5 methods formulated in the current paper, the exponents of both the best fitting GPL density profile to ADP, (α, β, γ) ≈ (0.3, 4.5, 1.5), and related averages calculated over the whole halo sample, , are far from their Navarro et al. counterparts, (α, β, γ) = (1, 3, 1). The last result, together with the large value of the standard deviation, , is interpreted as due to a certain degree of degeneracy in fitting GPL to SDH density profiles. If it is a real feature of the problem, or it could be reduced by the next generation of high-resolution simulations, still remains an open question. Values of asymptotic inner slope of fitting logarithmic density profiles, are consistent with results from recent high-resolution simulations (J. Diemand, B. Moore, J. Stadel, MNRAS 353 (2004) 624; D. Reed, F. Governato, L. Verde, et al., MNRAS 357 (2005) 82).