The Astrophysical Journal, 537:L73–L76, 2000 July 10q 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

THE OBSERVED MASS PROFILES OF DARK HALOS AND THE FORMATION EPOCH OF GALAXIESShinji Sato, Fumie Akimoto, Akihiro Furuzawa, Yuzuru Tawara, and Manabu Watanabe
Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya 464-01, JapanandYasuki Kumai
Kumamoto Gakuen University, 2-5-1 Oe, Kumamoto, Japan
Received 2000 April 12; accepted 2000 May 18; published 2000 July 6ABSTRACT
We have determined the mass profiles of dark halos in 83 objects observed by ASCA. The point-spread functionof the X-ray telescope was deconvoled by the Richardson-Lucy algorithm, and the temperature profiles werecalculated to obtain the mass profiles. The derived mass profiles are consistent with the Navarro, Frenk, & Whitemodel in (0.01–1.0)rvirial. We found a good correlation between the scale radius rs and the characteristic massdensity dc, which indicates the self-similarity of dark halos. The spectrum index of primordial density fluctuation,P(k) ∝ k n, was determined from the slope of the rs-dc relation. For M200 = 10 12–10 15 M,, our analysis givesn = 21.2 5 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 dc ∝ (1 1 z f ) 3, wehave determined the epoch when each dark halo was assembled. Our analysis indicates that the field ellipticalgalaxies and groups of galaxies formed approximately at 1 1 z f . 15 and at 1 1 z f . 7–10, respectively.
Subject headings: cosmology: observations — dark matter — galaxies: clusters: general —galaxies: formation — galaxies: halos — X-rays: galaxies

way to determine the formation epochs of galaxies and clustersof galaxies.
Throughout the Letter, we assume Q = 1, L = 0, and H0 =75 km s21 Mpc21.

1. INTRODUCTION

Early on, the density fluctuation grows linearly as the universe expands, dr/r ∝ (1 1 z)21. Once a density enhancementof a spherical region approaches dr/r ∼ 1 , the sphere begins
to turn around from the Hubble flow and collapse rapidly toform a virialized halo. The overdensity of virialized halos
reaches 178r 0 (1 1 z c ) 3, where z c is the collapse redshift andr0 is the critical mass density in the present (Kaiser 1986). As
the universe expands, the virial radius expands gradually bysteady mass accretion. N-body simulations show that the accreting dark matter forms an extended envelope, while the density profile of the original halo remains unchanged. The mass
profile of a dark halo is rearranged only by a major merger
through violent relaxation. A halo formed by a major merger
at z m is characterized by the mass density of 178r 0 (1 1 z m ) 3,and it is preserved until the next major merger takes place(Salvador-Solé, Solanes & Manrique 1998; Raig, GonzálezCasado, & Salvador-Solé 1998; Henriksen & Widrow 1999).
By this means, the density profile of a dark halo can be a good
indicator of the redshift at which the last merger took place.
Navarro, Frenk, & White (1996, hereafter NFW) suggested
that all of the dark halos have the self-similar mass profile
described asr(r) =r 0 dc,(r/rs )(1 1 r/rs ) 2

2. SAMPLE AND ANALYSIS

Our sample consists of rich clusters, poor clusters, groupsof galaxies, and elliptical galaxies. The spectrum analysis wasperformed by the standard method using the XSELECT andthe XSPECK software to determine the H i absorption, themetal abundance, and the luminosity-weighted temperature.
The results of spectrum analysis are described in a forthcoming
paper along with a full description of our sample objects(F. Akimoto, A. Furuzawa, Y. Kumai, S. Sato, Y. Tawara, &M. Watanabe 2000, in preparation).
The point-spread function (PSF) of the ASCA telescope ischaracterized by a sharp central peak (FWHM ∼150) and asignificant scattering tail extending to ∼39. We have deconvolved this scattering tail by the Richardson-Lucy algorithm(Lucy 1974). The observed images of 3C 273 were employed
as the PSFs. The deconvolutions were performed in the threeenergy bands (0.5–1.5, 1.5–3.0, and 3.0–10 keV) separately,
since the PSF has a weak dependence on the photon energy.
All of the observed images went through 100 iterations.
We have calculated the X-ray brightness profiles from thedeconvoled images, masking the bright subpeaks and obvioussubclusters. The brightness profiles obtained in the lowest energy band, kT = 0.5–1.5 keV, were compared to those fromROSAT PSPC in the energy band of kT = 0.5–2.0 keV, andgood agreements were found in these independent observations.
The temperature profiles are calculated from the brightness
ratio of the X-ray profiles in the three energy bands, kT =0.5–1.5, 1.5–3.0, and 3.0–10 keV. We assumed that there wereno radial gradients in metal abundance and H i absorption. The
typical temperature profiles of our sample are shown in Fig-(1)where r0, rs, and dc are the critical mass density of universe,
the scale radius, and the characteristic density, respectively. In
this case, the dc would be a direct indicator of the formation
epochs of dark halos.
In this Letter, we will present the mass profiles of dark halos
over the mass range of 1012–1015 M, using a sample of 83
objects observed with ASCA. This furnishes us with a possible
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Fig. 1.—Measured temperature profiles of ICM. The shaded boxes are ourresults. The height of the boxes indicates the error of 90% confidence level.
The crosses are the results from Fukazawa et al. (1996; NGC 5044), Kikuchiet al. (1999; MKW 3S), and Markevitch et al. (1998; A3571 and A3266). The
vertical lines correspond to the 90% errors. The diamonds are from White(2000). The heights of diamonds indicate the 1 j errors.

ure 1, together with a comparison with the previous measurements.
3. CALCULATION OF MASS PROFILE

The total mass of a dark halo can be calculated from the gas
density and the temperature profiles of intracluster medium(ICM), assuming the hydrostatic equilibrium. If the observed

Fig. 3.—Upper and lower panels are the measured M200-c and M200-kT relations. The kT is the luminosity weighted temperature determined by theX-ray spectrum. The error bars correspond to the 90% errors. The best-fitrelations with 90% error are log c = (7.2 5 1.8) 2 (0.44 5 0.13) log M200 andlog kT = (26.4 5 1.4) 1 (0.49 5 0.10) log M200.

X-ray surface brightness profile S(v) is described by the bmodel, S(v) = S 0 [1 1 (v/vc ) 2 ]23b10.5, where v, vc, and b are theangular radius, the core radius, and the b parameter, respectively, and the radial gradient of temperatures is relatively small
as compared with that of the gas density, the total mass withinradius r is given byM(r) =Fig. 2.—Composite mass profiles scaled by rs. The width of shaded lines
indicates the 1 j errors. The average scaling factors of emission measure are
kT1/2(1 1 z)9/2 = 0.88, 2.3, 3.6, 6.1, and 9.0 in (a), (b), (c), (d), and (e), respectively. The solid lines are mass profiles scaled by rs and r(rs) /109. The
dashed line is the best-fit NFW model. The arrows indicate the virial radiir200.

3bkT(r)(r/rc ) 2r,Gmm p 1 1 (r/rc ) 2(2)where T, k, mm p, and rc are the gas temperature, Boltzmann’sconstant, the mean molecular weight of the hot gas, and thecore radius, respectively.
We found that rc, b, and kT were not constant in a single
cluster, but changed slowly with radius. The angular profiles of
two parameters, rc(v) and b(v), were calculated by fitting the bmodel to the local brightness profiles of v 5 0.3v. Although theangular profiles rc(v), b(v), and kT(v) are the luminosity-weighted
properties of the hot gas along the line of sight, these are nearly
equal to the radial profiles rc(r), b(r), and kT(r) if the radial gradientof gas density [or S(v)] is much larger than those of rc, b, andkT. In our sample, rc(v), b(v), and kT(v) change by a factor of 2 at
most, while the S(v) changes 2 or 3 orders of magnitude. We
therefore assumed that the observed angular profiles represented
the radial profiles of these parameters. The local gas density
profile was therefore given by n(r) = n 0 {1 1 [r/rc(r) ]2}23b(r) /2 , andthe total mass M(r) was calculated from equation (2). The massdensity r(r10.5dr) = [M(r1dr) 2 M(r) ] /(4pr 2dr) was obtained fromM(r) and M(r1dr), keeping dr = 26 00.
We have constructed the models of gas halos bound by theNFW dark halos to evaluate the systematic errors in ourmethod. The mass profiles were calculated from the model
profiles—which correspond to typical galaxies, poor clusters,
and rich clusters—by applying the same method. The best-fit

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SATO ET AL.

L75Fig. 5.—Finding method for the zero-age clusters. The (1 1 z) is the redshiftof each object, and the dc /(1 1 z)3 indicate the age of dark halo since the lastassembly. The dashed line is the average of five clusters on the lower boundaryof dc /(1 1 z)3 2 (1 1 z) distribution.

Fig. 4.—Measured dc-rs relation and the best-fit model. The error bars correspond to the 90% errors. The residuals are shown in the upper panel. The
formation epoch of dark halos is indicated in the right side. The best-fit relation
with 90% error is log dc = (7.61 5 0.29) 2 (1.44 5 0.14) log rs.

dc and rs are then calculated by fitting the NFW model to themass profiles. Comparing the calculated dc and rs with the original values, we have confirmed that the systematic errors of
our method were much smaller than the typical photon noiseof our sample objects.
In our analysis, the brightness profiles within v ! 09. 5 wereexcluded to avoid the contribution of the cooling flow component. The maximum radii were extending up to 109–409 depending on the photon number available. We define r200 as theradius at which the mean interior density becomes 200 times
the critical mass density of the universe and M200 as the total
mass within r200. In this Letter, we consider r200 and M200 as
the virial radius and the virial mass (Cole & Lacey 1996),
respectively.
4. THE OBSERVED MASS PROFILESThe NFW model was fitted to our mass profiles to determine
dc, rs, r200, and M200. The mass profiles of several objects, whichhave similar characteristic radius, are co-added to improve thesignal-to-noise ratio. The composite mass profiles, scaled byrs, are shown in Figure 2 by the shaded lines. The widths oflines correspond to the 51 j errors. The average scale radii
are Ars S = 24, 59, 150, 200, and 460 kpc in Figures 2a, 2b, 2c,and 2d, respectively. The NFW model and the b-model werefitted to these composite profiles. Since the composite mass
profiles drop more quickly at larger radii, the NFW model gives
a better fit to the observations. The x2/degrees of freedom of
the NFW model are 3.0/4, 4.3/5, 5.5/5, and 3.6/5 in Figure 2a,2b, 2c, and 2d, while those of the b-model are 2.0/4, 24/5,23/5, and 14/5. To illustrate a similarity of mass profile, wehave normalized the composite density profiles by r(rs) /10 9 andillustrated in the upper portion of Figure 2 by the solid lines.
It is remarkable that these halos have very similar mass profiles.
The brightness profiles scaled by kT 1/2 (1 1 z) 9/2 can be agood indicator of mass profiles. Ponman, Cannon, & Navarro(1999) illustrated the scaled brightness profiles of 25 clusters,

normalized by the viral radius, and pointed out the systematicchange of profiles with kT 1/2 (1 1 z) 9/2. Our measurements,
however, do not conflict with this result. Since the observed
dark halos have different concentration parameters c = r200 /rs
with M200 as shown in the upper panel of Figure 3, the virial
radius is not an adequate scale to illustrate the similarity ofmass profiles. If our results are scaled by r200, we see the same
systematic trend as found by Ponman et al. (1999).
The correlation between the M200 and the luminosityweighted temperature kT is shown in the lower panel of Fig0.4950.10
ure 3. Our measurements give kT ∝ M200
with a confidence level of 90%. This result supports our assumption that
the ICM of dark halos are in the hydrostatic equilibrium (Evrard, Metzler, & Navarro 1996).
We have found a good correlation between dc and rs in our83 objects as shown in Figure 4. Our result is essentially thesame as the c-Mvir and dc-Mvir correlations found by Wu & Xue(2000), but have larger dynamic ranges and smaller errors. Theupper panel indicates residuals from the best-fit model. Therms scattering of dc shown by the dashed lines (545%) issignificantly larger than the 1 j errors of measurements. Ourresults indicate that the mass profile of a dark halo is describedby a single parameter, such as rs, dc, or M200.
5. THE SPECTRUM INDEX OF PRIMORDIAL FLUCTUATION AND
THE FORMATION EPOCH OF GALAXIESThe amplitude of primordial density fluctuation P(k) ∝ k n ischaracterized by the spectrum index n. In the cold dark matteruniverse, the radius and the mass density of dark halo are connected to n as r ∝ r2(3n19)/(n15) (Peebles 1980). The slope of
rs-dc relation shown in Figure 4 is therefore the direct indicatorof the power spectrum. If the P(k) is described by a single powerin the mass range of our sample M200 = 10 12–10 15 M,, thers-dc relation gives n = 21.2 5 0.3 with a confidence level of90%. The error is calculated from the intrinsic scattering of theobjects. When the objects of M200 ! 2 # 10 14 M, are excluded,
we get n = 20.8 5 0.7 (90%), which is consistent with the resultof Wu & Xue (2000), n = 20.7 5 0.3 (1 j).
We assume that the characteristic density dc is preserved
during the steady accretion phase as N-body simulations suggest. In that case, dc should be proportional to the mean mass

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density of the universe when the halo was assembled, dc =f (1 1 z f ) 3 = f (1 1 z obj ) 3[(1 1 z f )/(1 1 z obj )]3, where f, 1 1 z f ,and 1 1 z obj are a proportional constant, the formation epoch,and the observed redshift of object, respectively. Since 1 1z f ≥ 1 1 z obj, dc /(1 1 z obj ) 3 = f [(1 1 z f )/(1 1 z obj )]3 ≥ f. If oursample includes the zero-age clusters z f = z obj, the dc /(1 1z obj ) 3 of these objects would be equal to f. As shown in Figure
5, we found the clear locus in the lower end of the (1 1 z obj )versus dc /(1 1 z obj ) 3 plot, and five clusters on the locus (A370,A1758, CL 0016116, MS 04516, and RXJ 1347.521145) wereVol. 537

designated to the zero-age clusters. These zero-age clusters givethe proportional constant and the formation epoch as f =1640 5 260 and 1 1 z f = (0.085 5 0.005)dc1/3. Applying thisequation to individual objects, we get 1 1 z f = 13–17 for theformation epochs of three elliptical galaxies in our sample(NGC 1399, NGC 3923, and NGC 4636). Our results are nearly
independent of the cosmological parameters as illustrated inWu & Xue (2000).
We are grateful to the ASCA team for their efforts on thedesign and operation of ASCA hardware and software.

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