The observed mass profiles of dark halos and the formation epoch of galaxies

We have determined the mass profiles of dark halos in 83 objects observed by ASCA. The point-spread function of the X-ray telescope was deconvoled by the Richardson-Lucy algorithm, and the temperature profiles were calculated to obtain the mass profiles. The derived mass profiles are consistent with the Navarro, Frenk, & White model in (0.01-1.0)r_virial. We found a good correlation between the scale radius r_s and the characteristic mass density δ_c, which indicates the self-similarity of dark halos. The spectrum index of primordial density fluctuation, P(k) ∝ kⁿ, was determined from the slope of the r_s-δ_c relation. For M₂₀₀ = 10¹²-10¹⁵ M_⊙, our analysis gives n = -1.2 ± 0.3 with a confidence level of 90%. The mass density of dark halos is a good indicator of the mean mass density of the universe at the time when the halos were assembled, z = z_f. Assuming δ_c ∝ (1 + z_f)³, we have determined the epoch when each dark halo was assembled. Our analysis indicates that the field elliptical galaxies and groups of galaxies formed approximately at 1 - z_f ≃ 15 and at 1 + z_f ≃ 7-10, respectively.