Submitted to The Astrophysical Journal

A Deficiency of High-Redshift, Luminous X-Ray Clusters: Implications for the Fate of the Universe

arXiv:astro-ph/9802153v1 12 Feb 1998

D. E. Reichart1,2 , R. C. Nichol2 , F. J. Castander1 , D. J. Burke3 , A. K. Romer2 , B. P. Holden1 , C. A. Collins3 , M. P. Ulmer4 ABSTRACT From Press-Schechter formalism and the observed X-ray cluster L-T relation, we derive an X-ray cluster luminosity function that can be applied to the growing number of high-redshift, X-ray cluster luminosity catalogs to constrain cosmological parameters. In this paper, we apply this luminosity function to the Einstein Medium Sensitivity Survey (EMSS) to constrain Ω0 . Our results favor high values of this parameter: values that are consistent with unity. In the case of the EMSS, we find a deficiency of high-redshift, luminous X-ray clusters, which demonstrates that the luminosity function appears to have evolved above L⋆ . For the typical value of the slope of CDM-like mass density fluctuation power spectra on cluster scales (n = −1), we find that Ω0 > 0.70 at the 84% confidence level and that Ω0 > 0.33 at the 98% confidence level. We additionally apply a simplified version of this luminosity function to the published luminosity function of the ROSAT Brightest Cluster Survey to better constrain n. In combination with our EMSS results, our fitted value of n again favors high values of Ω0 . We do not assume any particular value of σ8 .

Subject headings: cosmology: theory - cosmology: observation - galaxies: clusters: general - galaxies: luminosity function, mass function - X-rays: galaxies

1.

Introduction

The combination of Press-Schechter formalism (e.g., Lacey & Cole 1993) with present and future X-ray cluster catalogs presents a unique opportunity to constrain the cosmological mass 1

Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637

2

Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213

3

Astrophysics Research Institute, Liverpool John Moores University, Liverpool L3 3AF, England, UK

4

Dearborn Observatory, Northwestern University, 2131 Sheridon Road, Evanston, IL 60208

–2– density parameter, Ω0 . The Press-Schechter approach offers a number of advantages over more traditional methods of determining this parameter. First of all, numerical simulations reproduce Press-Schechter formalism to a high degree of accuracy (Eke et al. 1996; Bryan & Norman 1997; Borgani et al. 1998). Secondly, unlike methods that only probe Ω0 over limited spatial scales - methods that may be insensitive to an underlying, more uniformly distributed component of the dark matter - Press-Schechter formalism probes Ω0 over the largest scales possible. Thirdly, Press-Schechter formalism is relatively insensitive to a cosmological constant (e.g., Henry 1997; Mathiesen & Evrard 1997); consequently, a Press-Schechter determination of Ω0 could be compared to independent determinations of the deceleration parameter, q0 , to constrain this constant. Finally and most importantly, a number of independent, high-redshift, X-ray cluster luminosity catalogs with well-understood selection functions will soon be available (see §4). A potential problem with the Press-Schechter approach is that some X-ray clusters may not correspond to virialized, collapsed dark matter halos. Gravitational lensing measurements of X-ray clusters suggest that this may be a real concern at high redshifts (Clowe et al. 1998); however, with the advent of AXAF temperature profiles, it will become easier to evaluate the degree to which these objects are virialized. The archetypal X-ray cluster catalog is the X-ray cluster subsample of the Einstein Medium Sensitivity Survey, which we refer to here as the EMSS. A complete description of this sample and its selection criteria can be found in Henry et al. (1992) (see also Gioia et al. 1990; Stocke et al. 1991; Gioia & Luppino 1994). The original EMSS consists of 93 X-ray clusters; however, Nichol et al. (1997) updated the z > 0.14 portion of this sample with 21 ROSAT luminosities and optical information from the literature. This revised EMSS consists of 64 X-ray clusters, of which 25 have been updated. The redshift and luminosity ranges of the revised EMSS are 0.140 < z < 0.823 and 7.5 × 1043 < LX < 2.336 × 1045 erg s−1 (0.3 - 3.5 keV). This catalog is of great importance because at present, it is the only X-ray cluster catalog that probes masses above M⋆ , where the Press-Schechter mass function is the most sensitive to Ω0 (see §3). A potential weakness of the EMSS is that it may suffer from optical selection biases, particularly at high redshifts. More recently, X-ray cluster catalogs such as the Serendipitous High-Redshift Archival ROSAT Catalog (SHARC) (Burke et al. 1997) have required that selected X-ray sources also show X-ray extent, in order to curtail these potential biases. Recently, Henry (1997) has measured, for the first time, the X-ray cluster temperature function at a redshift other than zero and has applied Press-Schechter formalism to this temperature function to constrain Ω0 . By comparing the temperature function of ten EMSS clusters in the redshift range 0.3 < z < 0.4 with the low-redshift (z < 0.1) temperature function of Henry & Arnaud (1991), Henry finds that Ω0 = 0.50 ± 0.14 and that Ω0 = 1 is ruled out at a confidence level of greater than 99%. A potential concern with this result is that it appears to be dependent upon the selected redshift range; e.g., had instead the redshift range 0.4 < z < 0.5 been chosen, only three EMSS clusters would have been selected and none of these clusters can have masses above M⋆ , where the Press-Schechter mass function is the most sensitive to Ω0 . In this case,

–3– higher values of Ω0 would certainly have been favored. Assuming that X-ray clusters correspond to virialized, dark matter halos, Press-Schechter formalism describes the evolution of the X-ray cluster mass function with redshift. Since virialization is already assumed, one may convert this mass function to a temperature function with the virial theorem (see §2). To convert this mass function to a luminosity function, one must additionally assume a luminosity-temperature (L-T ) relation. However, since the L-T relation must be determined empirically, Press-Schechter mass and temperature functions are generally preferred. Unfortunately, X-ray cluster mass and temperature catalogs with sufficient breadth to independently constrain Ω0 do not yet exist. However, there are several luminosity catalogs that meet this description, and the number of such catalogs is growing. Consequently, the need for a Press-Schechter luminosity function that incorporates the L-T relation with sufficient rigor is apparent. In §2, we construct just such a luminosity function. In §3, we reanalyze the L-T relation and with the results of this analysis in hand, we apply this luminosity function to the EMSS and, in a simplified way, to the ROSAT Brightest Cluster Survey (BCS). In §4, we draw conclusions and discuss future applications of this luminosity function to the SHARC and the Bright SHARC.

2.

The Press-Schechter Luminosity Function

Assuming that X-ray clusters correspond to virialized, dark-matter halos, we model the comoving number density of X-ray clusters with the Press-Schechter mass function, which is given by (e.g., Lacey & Cole 1993) dnc (M, z) =− dM

r

"

#

2 (z) −δc0 2 ρ¯0 d ln σ0 (M ) δc0 (z) exp , π M 2 d ln M σ0 (M ) 2σ02 (M )

(1)

where nc (M, z) is the comoving number density of X-ray clusters of mass M at redshift z, δc0 (z) is the present linear theory overdensity of perturbations that collapsed and virialized at redshift z, σ0 (M ) is the present linear theory variance of the mass density fluctuation power spectrum filtered on mass scale M , and ρ¯0 is the present mean mass density of the universe. In the case of zero cosmological constant, which we assume throughout this paper, the present overdensity is given by (Lacey & Cole 1993; Peebles 1980)

δc0 (z) =

where D(0) =

   2/3 2π 3   D(0) +1  sinh η−η  2 3

(12π)2/3 (1 + z)

20    2/3    3 2π  D(0) −1 2 η−sin η

  1 +

3 x0

  −1 +

(Ω0 < 1) (Ω0 = 1) , (Ω0 > 1)

√ √ √ 3 1+x0 ln ( 1 + x0 − x0 ) 3/2 x0 √ q 3 1+x0 x0 3 − tan−1 1−x 3/2 x0 0 x0

+

(2)

(Ω0 < 1) (Ω0 > 1)

,

(3)

–4–



x0 = Ω−1 0 − 1 ,

   2  cosh−1 − 1  Ω(z)  η=  cos−1 1 − 2 Ω(z)

and

Ω(z) =

(4) (Ω0 < 1) (Ω0 > 1)

Ω0 (1 + z) . 1 + Ω0 z

,

(5)

(6)

We assume a scale-free mass density fluctuation power spectrum of power-law index n, so the present variance is given by 3+n σ0 (M ) = σ8 M − 6 , (7) where σ8 , the amplitude of the mass density fluctuation power spectrum over spheres of radius 8h−1 Mpc, is a function of n. Since the EMSS and the BCS measure X-ray fluxes and redshifts, which together yield X-ray luminosities given a value of Ω0 , we now convert equation (1) from a mass function to an appropriately defined luminosity function. Following the notation of Mathiesen & Evrard (1997), we begin by assuming that X-ray clusters’ bolometric luminosities scale as power laws in both mass and redshift: Lbol ∝ M p (1 + z)s , (8) although we modify this assumption below to account for intrinsic scatter about this relation. As did Henry et al. (1992) in the case of the EMSS, we find that the fractions of this luminosity that fall into the EMSS band of 0.3 - 3.5 keV and the BCS band of 0.1 - 2.4 keV are well approximated by power laws in X-ray cluster temperature: LX ∝ T −β Lbol ,

(9)

where β = 0.407 ± 0.008 and 0.455 ± 0.008 for the representative temperature ranges of the EMSS (3 < ∼T < ∼ 10 keV) and the BCS (1.5 < ∼T < ∼ 10 keV), respectively. However, at lower temperatures, these approximations quickly fail. Equation (9) is independent of redshift because X-ray luminosities are measured in the source frame. The temperature dependence introduced by equation (9) is removed with the virial theorem: ρ(zf ) T ∝ M (1 + zf ) ρ¯(zf ) 2 3

!1 3

,

(10)

where zf is the redshift at which the X-ray cluster formed, i.e., the redshift at which the corresponding overdensity collapsed and virialized, ρ(zf ) is the mass density of the X-ray cluster at this redshift, and ρ¯(zf ) is the mean mass density of the universe at this redshift. This ratio of mass densities raised to the one-third power is a negligible function of redshift (Lacey & Cole 1993), so it may be grouped with the factor of proportionality and ignored. Together, equations

–5– (8), (9), and (10) yield the following expression that relates an X-ray cluster’s mass to its observed luminosity, LX :   zf − z −β p− 2β s−β LX ∝ M 3 (1 + z) . (11) 1+ 1+z Since X-ray cluster formation redshifts are unknown, we make the following, necessary approximation: 2β LX ∝ M p− 3 (1 + z)s−β .

(12)

This approximation is reasonable for two reasons. First of all, since β is small, the difference between equations (11) and (12) is small for most values of zf and z. Secondly and more importantly, since the most massive X-ray clusters are also the last to form, this approximation is particularly good above M⋆ , where the Press-Schechter mass function is the most sensitive to Ω0 (Henry & Arnaud 1991). Substitution of equation (12) into equation (1) yields the following luminosity function: 3+n −1− 3−n dnc (LX , z) 3p−2β = f (z)LX 2(3p−2β) exp −g(z)LX , dLX



where f (z) = a(1 + z)

(s−β)(3−n) 2(3p−2β)

g(z) = c(1 + z)−



δc0 (z),

(s−β)(3+n) 3p−2β

2 δc0 (z),

(13)

(14) (15)

and a and c depend upon σ8 and the factor of proportionality of equation (12).

2.1.

The L-T Relation Scatter Correction

Equation (13) requires three modifications before it can be fit to real data. First of all, the combination of equations (8) and (10) is not fully justified by the existing data. Together, these two equations imply that 3p 3p Lbol ∝ T 2 (1 + z)s− 2 , (16) which is known as the L-T relation. In §3.1, we fit this equation to an appropriately defined subset of the X-ray cluster temperature measurements of Mushotzky & Scharf (1997) and David et al. (1993) to constrain the parameters p and s. However, we find that the scatter of these measured temperatures about their best-fit L-T relation is too broad to be accounted for by the uncertainties in these measurements (see §3.1). Consequently, there is an intrinsic component to this scatter of which equation (16) describes only the mean. In §3.1, we measure the standard width of this intrinsic component to be σlog T = 0.081 along the temperature axis and this value is independent of Ω0 . Although this width is narrow, it corresponds to a significant uncertainty in luminosity. Equations (9) and (16) imply that LX ∝ T

3p −β 2

3p

(1 + z)s− 2 .

(17)

–6– Consequently, the standard width of this intrinsic distribution along the observed luminosity axis, σlog LX , is given by   3p σlog LX = − β σlog T . (18) 2 The fitted value of p from §3.1 and the values of β stated above imply that σlog LX = 0.27 ± 0.02 in the case of the EMSS and that σlog LX = 0.26 ± 0.02 in the case of the BCS. Hence, given an X-ray cluster’s mass and redshift, the value of its observed luminosity implied by equation (12) can be in error by a factor of two or more. Consequently, the derivation of equation (13) must be revisited. Assuming that the intrinsic distribution is Gaussian in log LX , we replace this luminosity function with its mean weighted over the intrinsic distribution: dnc (LX , z) = dLX where

f (z)

R∞ 0



L−b exp −g(z)Ld − R∞ 0

X

2



L−log LX ) exp − (log2σ dL 2

b=1+ and d= Equation (19) simplifies to



(log L−log LX )2 2 2σlog L

log LX



dL ,

(19)

3−n 2(3p − 2β)

(20)

3+n . 3p − 2β

(21)

"

#

∞ X ((ln 10)σlog LX )2 (dm − b)(dm − b + 2) (−g(z)LdX )m dnc (LX , z) = f (z)L−b exp . X dLX m! 2 m=0

(22)

In the event that σlog LX = 0 or is negligible, this equation is equivalent to equation (13). However, even in the event that σlog LX is not negligible, much of its effect upon the luminosity function can be hidden by expanding the argument of the exponential into three terms. The m0 term can be grouped into the parameter a and the m1 term can be grouped into the parameter c. However, the m2 term cannot be hidden in this way: "

#

∞ X dnc (LX , z) (−g(z)LdX )m ((ln 10)σlog LX dm)2 = f (z)L−b exp . X dLX m! 2 m=0

(23)

Unfortunately, this series does not converge. However, it is equivalent to the following double series that does converge: ∞ ∞ X ((ln 10)σlog LX d)2l X dnc (LX , z) (−g(z)LdX )m m2l . = f (z)L−b X dLX 2l l! m! m=0 l=0

(24)

The first (l = 0) term of this series is simply the exponential of equation (13). The effect of the remainder of this series is to stretch equation (13) toward higher luminosities. This effect is particularly pronounced at luminosities above L⋆ , where the luminosity function is the most

–7– sensitive to Ω0 (see §3). Generally, the values of σlog LX d and g(z)LdX are small enough that the outer series of equation (24) converges within a few terms. The inner series converges more slowly, but not unreasonably so. Consequently, this modification can be implemented at only moderate additional computation expense. However, the value of σlog LX d is not generally small enough that this modification can be ignored.

2.2.

Dependences upon Cosmological Parameters

The second modification to equation (13) accounts for dependences upon cosmological parameters in the observed luminosity: since fluxes and not luminosities are measured, luminosities are computed and consequently, LX is a function of H0 and Ω0 . With one exception, all dependences upon H0 can be grouped into the parameters a and c and this exception is noted below. The EMSS and the BCS provide luminosities in their respective X-ray bands that have been computed for H0 = 50 km s−1 Mpc−1 and Ω0 = 1: LΩ0 =1 . The relationship between LX and LΩ0 =1 is given by LX = x(z)LΩ0 =1 , (25) where

 2 fF (dA (Ω0 =1))   dL (Ω0 ) d (Ω =1) f (d (Ω )) x(z) =  L 0 2 F A 0 d (Ω )  0 L  dL (Ω0 =1))

(Einstein)

,

(26)

(ROSAT)

dL (Ω0 ) is luminosity distance, dA (Ω0 ) is angular diameter distance, and fF (dA (Ω0 )) is the fraction of an X-ray cluster’s flux that is detected in the 2′ .4 × 2′ .4 detect cell of the original EMSS. A complete description of this quantity can be found in Henry et al. (1992). Since ROSAT measures total fluxes, fF (dA (Ω0 )) = 1 here. So in the case of the BCS, the latter expression applies. However, the revised EMSS subsample that we fit to in §3.2 is a combination of 42 Einstein luminosities and 17 ROSAT luminosities. Fortunately, 36 of the 42 Einstein clusters have redshifts of z < 0.33 and 4 of the remaining 6 clusters have redshifts of z < 0.47. At these redshifts, the ratio of fractional fluxes in the former expression for x(z) is within a few percent of unity for a wide range of values of Ω0 . Furthermore, ROSAT luminosities are available for 7 of the 8 clusters that carry the majority of the weight in the fits of §3.2 and the remaining Einstein cluster is at a redshift of z = 0.259. Consequently, we also use the latter expression for x(z) in the case of the EMSS. Besides, we show in §3 that the sensitivity of x(z) to Ω0 plays only a tertiary role in the determination of this parameter. Substitution of equation (25) into equation (13) allows the luminosity function to be fitted to LΩ0 =1 data without loss of generality (see equation (29)).

2.3.

The Selection Function

The third modification to equation (13) incorporates the selection function. Let A(LΩ0 =1 , z) be the area of the sky that an X-ray survey surveys at redshift z as a function of luminosity LΩ0 =1 .

–8– In the case of the EMSS, this quantity is given by (Avni & Bachall 1980; Henry et al. 1992; Nichol et al. 1997) A(LΩ0 =1 , z) = A(Flim = F (LΩ0 =1 , z)), (27) where A(Flim ) is the area of the sky that the EMSS surveyed below sensitivity limit Flim (see Henry et al. 1992), fF (dA (z)) h250 LΩ0 =1 , (28) F (LΩ0 =1 , z) = k(z) 4πd2L (z) and k(z) is the k-correction from the observer frame to the source frame for a T = 6 keV X-ray cluster. (The exact temperature dependence is relatively unimportant for the representative temperature range of the EMSS.) For the EMSS, we have computed A(LΩ0 =1 , z) for 41 values of LΩ0 =1 between 1043.5 and 1045.5 erg s−1 in the EMSS band and for Ω0 = 0, 0.5, 1, 3.16, and 10. For intermediate values of LΩ0 =1 and Ω0 , we use linear interpolation between 43.5 < log LΩ0 =1 < 45.5 and between 0 < Ω0 < 1 and we use logarithmic interpolation between 1 < Ω0 < 10. The cases of Ω0 = 0 and 1 are plotted in Figure 1. The dependence of A(LΩ0 =1 , z) upon H0 cannot be grouped into the parameters a and c, unlike all of the other H0 dependences in this analysis (§2.2). Instead of introducing H0 as an additional parameter, we fix H0 = 50 km s−1 Mpc−1 in this paper. However, if others wish to be more general, they need only consider the H0 dependence of this single quantity. The case of the BCS is treated separately in §3.3. As a consequence of these three modifications, the luminosity function is now correctly given by

∞ ∞ X (−g(z)LdΩ0 =1 )m m2l dnc (LΩ0 =1 , z) ((ln 10)σlog LX d)2l X = f (z)L−b , Ω0 =1 dLΩ0 =1 2l l! m! m=0 l=0

where f (z) = a(1 + z)

(s−β)(3−n) 2(3p−2β)

and −

g(z) = c(1 + z)

δc0 (z)x

(s−β)(3+n) 3p−2β

3−n − 2(3p−2β)

(z)

3+n

2 δc0 (z)x 3p−2β (z).

(32)

1

(Ω0 z + (Ω0 − 2)((Ω0 z + 1) 2 − 1))2 4dz dV (z) = 3 4 1 H0 Ω0 (1 + z)3 (1 + Ω0 z) 2

is the comoving volume element.

(30) (31)

The total number of X-ray clusters observed between luminosity and redshift limits Ll < LΩ0 =1 < Lu and zl < z < zu , i.e., the cumulative luminosity function, is given by Z Lu Z zu dnc (LΩ0 =1 , z) N (Ll , Lu ; zl , zu ) = A(LΩ0 =1 , z) dLΩ0 =1 dV (z), dLΩ0 =1 Ll zl where

(29)

(33)

–9– 3.

Data Analysis & Model Fits

If properly applied, equation (32) can be an effective probe of Ω0 . In this cumulative luminosity function, δc0 (z), A(LΩ0 =1 , z), x(z), and dV (z) depend upon Ω0 . We now consider how sensitive each of these quantities is to Ω0 . The comoving volume element dV (z), is approximately given by  2.26 z dV (z)  (1+z)3 (Ω0 = 0) (34) ∝  z 1.993 (Ω0 = 1) dz (1+z)

where the indices are for the redshift range 0.14 < z < 0.6. The luminosity conversion expression, x(z), is approximately given by (equation (26)) x(z) ≈ 1 −

1 − Ω0 z 4

(35)

and in equation (29), it is always raised to a power that is closer to zero than it is to ±1. Consequently, dV (z) and x(z) contribute only weak dependences upon Ω0 to equation (32). The present overdensity is a stronger function of Ω0 (equation (2)): δc0 (z) =



1.5 (Ω0 = 0) . 1.69(1 + z) (Ω0 = 1)

(36)

Since this expression appears to the second power in the exponential-like cutoff of equation (29), it contributes a significant dependence upon Ω0 to the cumulative luminosity function. For example, if the luminosity function is observed to cut off prematurely at higher redshifts, i.e., if there is a deficiency of high-redshift, luminous X-ray clusters, then higher values of Ω0 are favored. However, if little or no evolution is manifest in the observed X-ray cluster luminosity function, particularly above L⋆ , then lower values of Ω0 are favored. The surveyed area, A(LΩ0 =1 , z), contributes a different type of dependence upon Ω0 to the cumulative luminosity function. In the case of the EMSS (Figure 1), this dependence is negligible at low luminosities and redshifts. However, at luminosities > ∼ L⋆ , A(LΩ0 =1 , z) is a non-negligible, increasing function of Ω0 at sufficiently high redshifts. For example, in the case of a LΩ0 =1 = 1045 erg s−1 , z = 0.8 EMSS cluster, A(LΩ0 =1 , z) is roughly twice as large in an Ω0 = 1 universe than it is in an Ω0 = 0 universe. Although this effect is suppressed by the exponential-like cutoff of equation (29) above L⋆ , about L⋆ , this effect is amplified by the fact that the luminosity function itself is a non-negligible, increasing function of Ω0 at luminosities < ∼ L⋆ at these high redshifts (see §3.2). Consequently, we find that an overabundance of high-redshift, ∼ L⋆ EMSS clusters favors high values of Ω0 and not low values of this parameter as is generally thought. We return to this idea in §3.2. Consider first the case of X-ray cluster luminosity data that lies within a narrow redshift band of effective redshift zef f . Then, up to a factor of A(LΩ0 =1 , zef f )dV (zef f )/dz, the integrand of equation (32) (equation (29)) is simply a power-law in luminosity with an Ω0 -dependent exponential-like cutoff. This exponential-like cutoff is a function of the parameters σlog T , β, p,

– 10 – n, and gef f = g(zef f ), which itself is a function of Ω0 (see below). We have already constrained the value of β (§2) and we constrain the values of σlog T and p, as well as that of s, with the L-T relation in §3.1. However, there are too few high-luminosity X-ray clusters to simultaneously constrain n and gef f . Fortunately, n is also constrained by the low-luminosity, power-law limit of equation (29) for which data is more plentiful. Consequently, by fitting this luminosity function to data of this type, n and gef f can be jointly constrained. By equation (31), the parameter gef f is a function of zef f and the parameters β, p, s, n, c, and Ω0 . The effective redshift is a given and the parameters β, p, s, and n can be constrained as described above. However, the parameter c is left unconstrained by data of this type and consequently, Ω0 can only be constrained if the value of c is otherwise known, i.e., if the values of H0 , σ8 , and the factor of proportionality of equation (12) are otherwise known (§2). Even in the event that temperature data is used instead of luminosity data, any fitted value of Ω0 will still depend strongly upon the assumed values of H0 , σ8 , and the factor of proportionality of equation (10) for data of this type. However, now consider X-ray cluster luminosity data that spans a breadth of redshifts. Instead of constraining the single parameter gef f , one instead constrains a distribution of such parameters with redshift, i.e., g(z). The normalization of this distribution is c and its shape yields Ω0 since the parameters β, p, s, and n are otherwise constrained. Consequently, by fitting equation (32) to the EMSS, which spans a breadth of luminosities and redshifts, the parameters n, c, and Ω0 can be jointly constrained regardless of the value of σ8 and the factor of proportionality of equation (12) (but not regardless of the value of H0 since A(LΩ0 =1 , z) is a function of this parameter (§2.3)). We do this for the EMSS in §3.2. In §3.3, we further constrain the parameter n with the local (zef f ∼ 0.1) luminosity function of the BCS. First however, we constrain σlog T , p and s with the L-T relation in §3.1.

3.1.

The L-T Relation

The L-T relation (equation (16)) can be restated as Lbol = L0 T

3p 2

3p

(1 + z)s− 2 ,

(37)

where L0 is the constant of proportionality and T is measured in keV. To constrain the parameters σlog T , p and s, we χ2 -fit equation (37) to a combined sample of 124 X-ray clusters for which both Lbol and T have been measured. This sample consists of the 38 0.143 < z < 0.541 X-ray clusters of Mushotzky & Scharf (1997) (see also Henry 1997) and 86 of the 104 low-redshift X-ray clusters of David et al. (1993) (see also Edge & Stewart 1991). Mushotzky & Scharf measure temperatures with ASCA; David et al. measure temperatures with Einstein and they supplement their sample with EXOSAT (Edge 1989) and Ginga (Hatsukade 1989) measurements. From these two samples, we select only those X-ray clusters that have temperatures with bounded 90% confidence intervals. (1 σ confidence intervals are not used here because Mushotzky & Scharf

– 11 – provide only 90% confidence intervals.) This eliminates 17 David et al. clusters. We eliminate an 18th David et al. cluster because it is also in the Mushotzky & Scharf sample. When more than one temperature measurement is available, as is the case for 18 of the David et al. clusters, we select the measurement with the most constrained 90% confidence interval. The combined sample is plotted in Figure 2. Approximate 1 σ confidence intervals are recovered by scaling the 90% confidence intervals by a factor of 0.61. A constant uncertainty of σlog T = 0.081 must be added in quadrature to these 1 σ confidence intervals for equation (37) to fit this sample of data, i.e., for χ2min = ν, where ν is the number of degrees of freedom. This additional uncertainty is the standard width of the intrinsic distribution of temperatures about the best-fit L-T relation (§2.1), assuming that this width is a constant with luminosity. This width is particularly noticeable at the high luminosity end of Figure 2 where uncertainties in the measured temperatures (and luminosities) are much smaller than σlog T . If this width is not taken into account, the confidence intervals of the fitted parameters, L0 , p, and s, will be underestimated (see below). Finally, since Lbol depends upon Ω0 , we fit equation (37) to the combined sample for Ω0 = 0, 0.5, 1, 3.16, and 10. The following equations adequately describe the fitted values of these parameters between 0 < Ω0 < 10: log h250 L0 =

  42.28+0.15(0.28)(0.40) −0.17(0.37)(0.58)

 (42.28

p=

  2.46+0.17(0.36)(0.60)

s=

  (4.00 − 0.62Ω0 )+0.71(1.38)(2.12)

and

(1 < Ω0 < 10)

(0 < Ω0 < 1)

−0.15(0.28)(0.39)

 (2.46 −

(0 < Ω0 < 1)

+0.15(0.27)(0.40) + 0.01 log Ω0 )−0.17(0.37)(0.59)

+0.17(0.36)(0.59) 0.04 log Ω0 )−0.15(0.28)(0.38)

−0.75(1.56)(2.39)

 (3.38 − 2.49 log Ω0 )+0.71(1.39)(2.07) −0.74(1.54)(2.39)

(1 < Ω0 < 10) (0 < Ω0 < 1) (1 < Ω0 < 10)

,

(38)

,

(39)

,

(40)

where the confidence intervals are 1, 2, and 3 σ for one interesting parameter. Borgani et al. (1998) find similar results with different L-T data. The cases of Ω0 = 0 and 1 are also plotted in Figure 2. Only the parameter s is strongly dependent upon Ω0 . Consequently, the degree of evolution in the L-T relation, given by s − 3p/2 by equation (37), is a function of Ω0 . However, due to the moderate extent of the confidence intervals that are associated with this parameter, we find that the L-T relation is consistent with no evolution for values of Ω0 between 0 and ≈ 3.5, which demonstrates agreement with the primary result of Mushotzky & Scharf (1997). We note, however, that we do not assume that the L-T relation does not evolve in this paper. Instead, we use the best-fit values of the parameters p and s in the fits of the following subsections and in §3.2, we estimate the effect that the uncertainties in these parameters have upon the fitted values. We also note that the literature has produced a wide range of values for these parameters, especially for the parameter p. Worse yet, this wide range of values is accompanied by an even wider range of confidence intervals. Sometimes, the reasons for these discrepancies are statistical. For example, occasionally Lbol error bars have been used instead of T error bars. Since Lbol error

– 12 – bars are generally much smaller than their corresponding T error bars, this practice inevitably leads to a poor fit: χ2min ≫ ν. The consequences of this are fitted values that are simultaneously incorrect and over-constrained. A related statistical problem is that often, the intrinsic scatter about the best-fit L-T relation is not taken into account, which again results in a value of χ2min > ν with similar, albeit less severe, consequences. However, not all of the discrepancies in the literature can be attributed to statistic problems. For example, Markevitch (1998) does a statistically sound analysis of a local (z < 0.1) X-ray cluster temperature sample and finds that Lbol ∝ T 2.64 , which at first glance appears to be in contradiction to our results, as well as to the results of David et al. (1993) and Mushotzky & Scharf (1997) (naturally) and others. However, each of these samples probe different luminosity and redshift ranges, which suggests that the L-T relation may not be well-modeled by simple power-laws in luminosity and redshift. This has been suggested on theoretical grounds by Cavaliere, Menci, & Tozzi (1997). If this is the case, we recommend that when applying a Press-Schechter luminosity function to an X-ray cluster sample, one uses an L-T relation that has been derived from data that spans similar ranges in luminosity and redshift as the X-ray cluster sample. Finally, we also recommend that when using high-redshift X-ray cluster samples, one uses an L-T relation that has taken into account the dependence of Lbol upon Ω0 , as we have done with the parameter s. This will become increasingly important in the future as significant numbers of high-redshift X-ray clusters are added to these L-T samples. For example, LΩ0 =0 /LΩ0 =1 ≈ 1.5 for a z = 0.8 X-ray cluster. We now consider the physical implications of our best-fit L-T relation, given by (equations (16), (39), and (40)): ( T 3.69 (1 + z)−0.31 (Ω0 = 0) . (41) Lbol ∝ T 3.69 (1 + z)0.31 (Ω0 = 1) Consider first the simplest case of an X-ray cluster population that behaves self-similarly, as would be the case if gravitational shocks during collapse dominate the heating mechanisms and if cooling mechanisms can be ignored. In this case, the theoretical L-T relation for Ω0 = 1 is given by (Kaiser 1986) 3 Lbol ∝ T 2 (1 + z) 2 . (42) However, it is well-known that this L-T relation differs from even the local observed L-T relation. Consider instead the general case: 3

2 Lbol ∝ fgas gT 2 (1 + z) 2 ,

(43)

where fgas is cluster baryon fraction and g is a function of cluster concentration and cosmology (e.g., Eke, Navarro, & Frenk 1997). The difference between the observed L-T relation and the self-similar L-T relation can consequently be explained by cluster baryon fractions and/or cluster concentrations that are functions of, for example, mass and redshift (e.g., Evrard & Henry 1991; Cavaliere, Menci, & Tozzi 1997; Eke, Navarro, & Frenk 1997). Note however, cosmology alone cannot account for this difference. If cluster concentration and cosmological effects are ignored,

– 13 – we find that our best-fit L-T relation suggests that fgas ∝ M 0.56 , which is consistent with the theoretical results of Cavaliere, Menci, & Tozzi (1997). A prominent physical mechanism that approximately reproduces the observed L-T relation is preheating of the intracluster medium leading to isoentropic core evolution. In this case, the theoretical L-T relation for Ω0 = 1 is given by (Evrard & Henry 1991; Kaiser 1991) Lbol ∝ T 3.2.

11 4

(1 + z)0 .

(44)

The EMSS

As described in §3, the breadth of the luminosity and redshift ranges of the EMSS makes this catalog an ideal sample with which to probe Ω0 (§3). In Figure 3, we plot the LΩ0 =1 -z distribution of the revised EMSS of Nichol et al. (1997) (and the z < 0.14 portion of the original EMSS X-ray cluster subsample). The solid curves are contours of constant sampled differential volume, i.e., A(LΩ0 =1 , z)dV (z)/dz = constant. From left to right, these contours are equally spaced from zero (zero contour not shown). If the luminosity function has not evolved over the redshift range of the EMSS, then at each luminosity, most of the observed X-ray clusters would be where most of the sampled differential volume is. This appears to be the case below LΩ0 =1 ∼ 1045 erg s−1 , which demonstrates agreement with the primary results of Nichol et al. (1997) and Burke et al. (1997), i.e., that the luminosity function evolves only minimally, if at all, below L⋆ . However, above L⋆ , a deficiency of high-redshift X-ray clusters is apparent. This suggests that high values of Ω0 may be favored (§3). High-luminosity and high-redshift X-ray clusters are the key to constraining Ω0 (§3); however, they are also few in number. Consequently, to make the most of the available information, we use the C statistic of Cash (1979). This maximum likelihood statistic is given by C = −2 ln

N tot Y

P (LΩ0 =1,i , zi ),

(45)

i=1

where P (LΩ0 ,i , zi ) is the probability that the ith X-ray cluster fits the model, given values of the model’s parameters. For our model (equation (32)), this probability is given by P (LΩ0 =1 , z) =

A(LΩ0 =1 , z) dV (z) dnc (LΩ0 =1 , z) . N (Ll , Lu ; zl , zu ) dz dLΩ0 =1

(46)

In equation (45), Ntot is the total number of X-ray clusters in the same region of the LΩ0 =1 -z plane as that over which N (Ll , Lu ; zl , zu ) is defined. This region should be as broad as is reasonably possible and it need not be rectangular, as the simple integration limits of equation (32) may −1/2 indicate. Up to a negligible (in this case) term of order Ntot , the C statistic is distributed as the χ2 statistic plus an unknown constant. The primary advantages of the C statistic, aside from the fact that it mimics the familiar χ2 statistic, are (1) that no information is lost to binning

– 14 – and (2) that non-detections also carry weight. The disadvantage of this statistic is that since it only mimics the χ2 statistic up to an unknown additive constant, the absolute probability that a particular model fits the available data cannot be determined, i.e., the Press-Schechter formalism itself cannot be tested in this way. However, assuming that this formalism is indeed correct, as numerical simulations suggest (Eke et al. 1996; Bryan & Norman 1997; Borgani et al. 1998), confidence regions can be determined for the model’s free parameters: the ∆C distribution is conveniently identical to the ∆χ2 distribution. The normalization parameter, a, now drops out of equation (46). This leaves seven parameters: σlog T , β, p, s, c, n, and Ω0 . The parameters σlog T and β are tightly constrained (§3.1 and §2, respectively). The parameters p and s are also constrained, although somewhat less tightly (equations (39) and (40), respectively). We use the best-fit values of these four parameters in the fits of this and the following subsection. However, we also estimate the effect that the uncertainties in the parameters p and s have upon our results at the end of this subsection. Consequently, we are left with three parameters: c, n, and Ω0 , Since we are not interested in the value of c (however, see §3.3), we project over this axis and present confidence contours in the Ω0 -n plane. Consequently, our results are independent of the value of σ8 (§2). We have computed these contours for a broad region of the LΩ0 =1 -z plane. The luminosity bounds of this region are Ll < LΩ0 =1 < Lu = 1045.5 erg s−1 , where Ll is a function of redshift that is given by setting the limiting flux of our subsample to F (Ll , z) = 1.61 x 10−13 erg cm−2 s−1 (the second lowest value of Table 3 of Henry et al. (1992)) for Ω0 = 0. Given this function, higher values of Ω0 yield only more conservative values of F (Ll , z) (Figure 1). This reduces the number of X-ray clusters in our subsample from 64 to 59. The redshift range of this region is 0.14 = zl < z < zu = 0.6. We exclude higher redshifts because (1) optical selection effects become more worrisome at such redshifts and (2) the two highest-redshift EMSS clusters, MS1137.5+6625 (z = 0.782) and MS1054.4-0321 (z = 0.823), are probably not virialized (Clowe et al. 1998); it is unclear whether such objects can legitimately be included in Press-Schechter analyses. Since these two X-ray clusters are additionally excluded by the above flux limit, the total number of X-ray clusters in our subsample remains the same. In Figure 3, this region is marked with a dotted line and solid points are interior to this region. The resultant confidence contours are plotted in Figure 4. The shaded regions are the 1, 2, and 3 σ confidence regions for two interesting parameters (Ω0 and n). For the typical value of the slope of CDM-like mass density power spectra on cluster scales (n = −1), Ω0 > 0.70 at the 84% confidence level and Ω0 > 0.33 at the 98% confidence level. Lower values of Ω0 are favored only if the value of the parameter n is closer to −2; however, even in these cases, high values of Ω0 are not strongly disfavored. Consequently, we find that for reasonable values of the parameter n, the distribution of EMSS clusters in the LΩ0 =1 -z plane favors high values of Ω0 : values that are consistent with unity. In Figure 5, we plot two illustrative examples. In the top figure, we plot the best-fit LΩ0 -z

– 15 – distribution for n = −2 and Ω0 = 0.5. In the bottom figure, we plot that for n = −1 and Ω0 = 1. Both of these distributions fit the data well (Figure 4). From interior to exterior, the solid curves are decreasing contours of constant sampled X-ray cluster number density. Each curve corresponds to one-tenth of the density of its previous curve. Although these distributions are similar, two important differences stand out. The first of these differences is no surprise: higher values of the parameter n produce luminosity functions that cut off at lower luminosities, and this effect is independent of redshift (e.g., equation (13)). The second difference, however, is more interesting: high-redshift, ∼ L⋆ EMSS clusters are more strongly favored in the Ω0 = 1 figure than they are in the Ω0 = 0.5 figure. There are three reasons for this. We have already described the first of these reasons (§3): for this region of the LΩ0 =1 -z plane, the EMSS surveyed roughly twice as much area if Ω0 = 1 than it did if Ω0 = 0 (Figure 1). This is primarily because higher values of Ω0 imply lower luminosity distances, which imply higher fluxes for a given luminosity and consequently, greater surveyed areas by equation (27). This effect can also be seen in Figure 3. The second reason, also mentioned above, is that at luminosities < ∼ L⋆ , the luminosity function itself is a non-negligible, increasing function of Ω0 . At these luminosities, the redshift dependence of the luminosity function is dominated by the function f (z). For the best-fit values of β, p, and s, this function is given by (equations (30), (2), and (26)) f (z) ∝

(

(1 + z)0.29(3−n) (1 + z)1+0.23(3−n)

(Ω0 = 0) (Ω0 = 1)

(47)

Consequently, there is an additional factor of nearly 1 + z in the Ω0 = 1 case. Thirdly, this last effect is solidified by the fact that the L-T relation scatter correction pushes the effective value of L⋆ to even higher luminosities. These three effects suggest that had we not excluded the two high-redshift EMSS clusters from our subsample, which we did for the reasons stated above, they would have favored similar or even higher values of Ω0 , and not lower values as is generally thought (e.g., Gioia 1997). At least, the former two of these three effects might affect the results of Donahue et al. (1997), who found that Ω0 ∼ 0.2 - 0.4 based upon a Press-Schechter temperature analysis of the highest redshift EMSS cluster, MS1054.4-0321. In this analysis, we have used the best-fit values of the parameters σlog T , β, p, and s. Although this is justified in the cases of the former two parameters, we now estimate the effect that the uncertainties in the fitted values of the latter two parameters have upon our results. As we described in §3, the value of the parameter n is primarily constrained by the low-luminosity, power-law limit of equation (29). We have denoted the index of this power law b and it is related to the parameters p and n by equation (20). Assuming that the value of b is reasonably well-defined by the data, one finds that 1 σ variations in the value of the parameter p implies that the value of parameter n may be uncertain by as much as 0.25 to 0.35. Since the value of the parameter n is also constrained in other ways, this uncertainty should be treated as an upper bound. As we also describe in §3, the value of Ω0 is primarily constrained by the distribution g(z) (equation (31)), which is a function of the parameters p, s, and n. Again assuming that this distribution is reasonably well-defined by the data, one finds that 1 σ variations in the values of

– 16 – the parameters p and s and a variation of 0.25 to 0.35 in the value of the parameter n implies that the value of Ω0 may be uncertain by as much as 0.15 to 0.2. If a particular value of the parameter n is assumed, this uncertainty is only ∼ 0.1. Since the value of Ω0 is also constrained in other ways, this uncertainty should be treated as an upper bound. Furthermore, it should be noted that neither of these uncertainty estimates is rigorous, but rather, they are intended to offer a reasonable idea as to what the magnitude of this effect might be.

3.3.

The ROSAT BCS

The ROSAT BCS is a flux-limited sample of 199 bright X-ray clusters. A complete description of this sample and its selection criteria can be found in Ebeling et al. (1997). The redshift range of the BCS is z < 0.3; however, most of the BCS clusters have redshifts of z < 0.2 and the effective redshift of the sample is zef f ∼ 0.1. Consequently, the BCS samples the X-ray cluster population of the local universe. Although such a sample may not have enough redshift-leverage to adequately probe Ω0 , its large size makes it an excellent sample to better constrain the parameters n and c. These constraints can then be used to better constrain Ω0 via the EMSS results of the previous subsection. However, since the BCS is not yet publicly available, we settle here for a simplified analysis based upon the binned, BCS luminosity function of Figure 1 of Ebeling et al. (1997), which we replot in Figure 6. Since the BCS spans only a narrow band of redshifts, we let f (z) = f (zef f ) = fef f and g(z) = g(zef f ) = gef f in equation (29). Also, since Ebeling et al. have already corrected this distribution for sample completeness, we set A(LΩ0 =1 , zef f ) = 1. (Here, we ignore the dependences that this quantity has upon Ω0 , which is a reasonable approximation since zef f < 0.3.) For the best-fit values of σlog T (§3.1), β (§2), and p (equation (39)), the χ2 -fitted value of n = 0.66+0.50 −0.61 (χ2min = 8.78, ν = 9). Unfortunately, this value of n is unreasonably high. 1 σ variations in the parameters p and β do not significantly effect this result (§3.2). Most of the error in this fit is tied to the highest-luminosity bin of Figure 6. If this data point is ignored, the fitted value of n 2 becomes n = −0.60+0.78 −0.74 (χmin = 3.70, ν = 8). Although this value is more reasonable, the value 2 of χmin is now unreasonably low. Both of these fits are plotted in Figure 6. We attribute these problem (1) to the fact that we are fitting to data that has been projected over redshift space and binned in luminosity space and (2) to the simplifying approximations that we must make to fit to such data. When the full BCS becomes available, a more serious analysis can be done. However, if the results of this simple analysis are any indication of what a proper analysis of the BCS may find, then it appears likely that n > ∼ −1. If this is the case, given the results of §3.2 (Figure 4), values of Ω0 that are consistent with unity are preferred even more strongly. Also, we make no attempt here to constrain the parameter c given the fitted value of the parameter gef f . However, once a proper analysis of the BCS is done, this parameter will provide a second avenue by which Ω0 can be further constrained. Blanchard & Bartlett (1997) recently performed a somewhat different, comparative analysis of the EMSS and the BCS and also found

– 17 – that high values of Ω0 are preferred.

4.

Conclusions

In this paper, we derive from Press-Schechter formalism and the observed X-ray cluster L-T relation an X-ray cluster luminosity function that can be applied to the growing number of high-redshift, X-ray cluster luminosity catalogs to constrain cosmological parameters. In particular, we incorporate the redshift dependence and the intrinsic scatter of the L-T relation into our analysis; we also account for all dependences that this luminosity function has upon cosmological parameters. For a fixed value of H0 , we perform a maximum likelihood fit of this luminosity function to an appropriately chosen, broad, subset of the revised EMSS X-ray cluster subsample of Nichol et al. (1997) to constrain Ω0 . For reasonable values of the slope of the mass density power spectrum on cluster scales, n, we find that the distribution of these EMSS clusters in the LΩ0 =1 -z plane favors high values of this parameter: values that are consistent with unity. This result is primarily due to a deficiency in this distribution of high-redshift, luminous X-ray clusters, which demonstrates that the X-ray cluster luminosity function appears to have evolved above L⋆ . Additionally, we show that the overabundance of high-redshift, ∼ L⋆ EMSS clusters may contribute to this result: these X-ray clusters do not favor low values of Ω0 as is generally thought. To better constrain the parameter n, we fit a simplified version of our luminosity function to the projected and binned luminosity function of the ROSAT BCS. We find that the BCS favors high values of this parameter: n > ∼ −1. In combination with our EMSS results, this favors high values of Ω0 even more strongly. However, these results are only as certain as they are reproducible with other X-ray cluster catalogs. Fortunately, the number of X-ray cluster luminosity catalogs is growing rapidly. One such catalog is the Serendipitous High-Redshift Archival ROSAT Catalog (SHARC) (Burke et al. 1997). The redshift and luminosity ranges of the SHARC are z < 0.7 and LΩ0 =1 < 3 × 1044 erg s−1 (0.5 - 2.0 keV). Although the SHARC does not span the luminosity range of the EMSS, it requires that selected X-ray sources show X-ray extent; consequently, its selection function is more reliable. The SHARC will provide a good consistency check of our EMSS results. Our analysis of the SHARC is underway. A similar X-ray cluster catalog that can serve a similar purpose is the ROSAT Deep Cluster Survey (RDCS). It spans the redshift and luminosity ranges z < 0.8 and LΩ0 =1 < 4 × 1044 erg s−1 (0.5 - 2.0 keV). An analysis of the RDCS is also underway (Borgani et al. 1998). The Bright SHARC, a high-luminosity extension of the SHARC, is currently under construction. It will span redshift and luminosity ranges that rival those of the EMSS and consequently, it will provide the first, strong, independent check of our EMSS results (Romer et al. 1998). Finally, local (zef f < ∼ 0.1) X-ray cluster catalogs, such as the ROSAT BCS, are of great

– 18 – importance. Although these samples do not have the redshift leverage to constrain cosmological parameters, their large sizes make them excellent samples to better constrain the parameters n and c. Samples like the EMSS and the Bright SHARC do not have sufficient luminosity leverage to strongly constrain these parameters, which leads to weaker constraints upon the cosmological parameters. However, a simultaneous analysis of a local X-ray cluster catalog with either the EMSS or the Bright SHARC could lead to significantly improved constraints upon all of these parameters. This research has been supported by NASA grant NAG5-6548. We are very grateful to C. A. Metzler and D. Q. Lamb for alerting us to the importance of the L-T relation scatter correction. We are also very grateful to H. Ebeling for providing us with the data for Figure 6. Also, we are grateful to A. Blanchard, S. Borgani, C. Scharf, M. S. Turner, and J. M. Quashnock for valuable discussions. D. E. R. is also grateful to G. P. Holder. D. E. R. is especially grateful to Dr. and Mrs. Bernard Keisler for their hospitality during the summer of 1997.

– 19 – REFERENCES Avni, Y., & Bachall, J. N. 1980, ApJ, 235, 694 Blanchard, A., & Bartlett, J. G. 1997, A&A (Letters), submitted (astro-ph 9712078) Borgani, S., et al. 1998, ApJ, submitted Bryan, G. L., & Norman, M. L. 1997, ApJ, in press (astro-ph 9710107) Burke, D. J., et al. 1997, ApJ, 488, L83 Cash, A. 1979, ApJ, 228, 939 Cavaliere, A., Menci, N., & Tozzi, P. 1997, ApJ, 484, L21 Clowe, D., et al. 1998, preprint (astro-ph 9801208) David, L. P., et al. 1993, ApJ, 412, 479 Donahue, M., et al. 1997, preprint (astro-ph 970710) Ebeling, H., et al. 1997, ApJ, 479, L101 Edge, A. C. 1989, Ph.D. thesis, Univ. of Leicester Edge, A. C., & Stewart, G. 1991, MNRAS, 252, 414 Eke, V. R., et al. 1996, MNRAS, 282, 263 Eke, V. R., Navarro, J. F., & Frenk, C. S. 1997, ApJ, in press (astro-ph 9708070) Evrard, A. E., & Henry, J. P. 1991, ApJ, 383, 95 Gioia, I. M. 1997, “The Young Universe: Galaxy Formation and Evolution at Intermediate and High Redshift”, in press (astro-ph 9712003) Gioia, I. M., et al. 1990, ApJS, 72, 567 Gioia, I. M., & Luppino, G. A. 1995, ApJS, 94, 583 Hatsukade, I. 1989, Ph.D. thesis, Harvard Univ. Henry, J. P. 1997, ApJ, 489, L1 Henry, J. P., et al. 1992, ApJ, 386, 408 Henry, J. P., & Arnaud, K. A. 1991, ApJ, 372, 410 Kaiser, N. 1986, MNRAS, 222, 323

– 20 – Kaiser, N. 1991, ApJ, 383, 104 Lacey, C., & Cole, S. 1993, MNRAS, 262, 627 Markevitch, M. 1998, ApJ, submitted (astro-ph 9802059) Mathiesen, B., & Evrard, A. E. 1997, MNRAS, submitted (astro-ph 9703176) Mushotzky, R. F., & Scharf, C. A. 1997, ApJ, 482, L13 Nichol, R. C., et al. 1997, ApJ, 481, 644 Peebles, P. J. E. 1980, The Large Scale Structure of the Universe (Princeton: Princeton University Press) Romer, A. K., et al. 1997, AN, 319, 83 Stocke, J. T., et al. 1991, ApJS, 76, 813

This preprint was prepared with the AAS LATEX macros v4.0.

– 21 – Fig. 1.— The area A(LΩ0 =1 , z) of the sky that the EMSS sampled at redshift z as a function of luminosity LΩ0 =1 . The solid curve is for Ω0 = 0 and the dotted curve is for Ω0 = 1. From left to right, the curves correspond to LΩ0 =1 = 1043.5 , 1044 , 1044.5 , 1045 , and 1045.5 erg s−1 in the EMSS band. Fig. 2.— X-ray cluster bolometric luminosity, Lbol , versus X-ray cluster temperature, T , for the combined sample of §3.1. Error bars are 90% confidence intervals. Open circles are for X-ray clusters of redshift z < 0.14 and solid circles are for X-ray clusters of redshift 0.14 < z < 0.55. The solid line is the best fit to equation (37) for z = 0 and the dotted line is that for z = 0.55. The top figure is for Ω0 = 0 and the bottom figure is for Ω0 = 1. Fig. 3.— The LΩ0 =1 -z distribution of the EMSS X-ray clusters (all points). From left to right, the solid curves are increasing, equally spaced (from zero, zero contour not shown) contours of constant sampled differential volume. A deficiency of high-redshift X-ray clusters is apparent above LΩ0 =1 ∼ 1045 erg s−1 . The dotted curve is the region over which equation (32) is fitted in §3.2. Points interior to this region are solid. The top figure is for Ω0 = 0 and the bottom figure is for Ω0 = 1. Fig. 4.— The 1, 2, and 3 σ confidence contours of the fit of §3.2. For the typical value of n = −1, Ω0 > 0.70 at the 84% confidence level and Ω0 > 0.33 at the 98% confidence level. Fig. 5.— Fitted LΩ0 =1 -z distributions for the EMSS X-ray clusters (see Figure 3). From interior to exterior, the solid curves are decreasing contours of constant sampled X-ray cluster number density. Each curve corresponds to one-tenth of the density of its previous curve. The top figure is for n = −2 and Ω0 = 0.5 and the bottom figure is for n = −1 and Ω0 = 1. Higher values of Ω0 are more accommodating to the high-redshift, ∼ L⋆ EMSS clusters. Fig. 6.— The BCS luminosity function of Ebeling et al. (1997). The solid line is the best PressSchechter-fit of §3.3. The dotted line is the best Press-Schechter-fit to all but the highest-luminosity bin. The dashed line is the best Schechter-fit (as opposed to Press-Schechter) of Ebeling et al. These fits favor values of n > −1.

1

.8

.6

– 22 –

.4

.2

0 0

.2

.4

.6 z

.8

1

– 23 –

log T (keV)

1

.5

0

-.5 43.5

44

44.5

45

45.5

46

46.5

43.5

44

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45

45.5

46

46.5

log T (keV)

1

.5

0

-.5

– 24 –

.8

z

.6

.4

.2

0 0

5

10

15

20

25

0

5

10

15

20

25

.8

z

.6

.4

.2

0

– 25 –

– 26 –

.8

z

.6

.4

.2

0 0

5

10

15

20

25

0

5

10

15

20

25

.8

z

.6

.4

.2

0

-5

-6

-7

– 27 –

-8

-9

-10 43.2

43.4

43.6

43.8

44

44.2

44.4

44.6

44.8

45

45.2

45.4