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 delta(c), which indicates the self-similarity of dark halos. The spectrum index of primordial density fluctuation, P(k) proportional to k(n), was determined from the slope of the r(s)-delta(c) relation. For M-200 - 10(12)-10(15) 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 delta(c) proportional to (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) similar or equal to 15 and at 1 + z(f) = 7-10, respectively.