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The Astrophysical Journal, 659:29 Y 51, 2007 April 10
# 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

A

CHANDRA MULTIWAVELENGTH PROJECT X-RAY POINT SOURCE NUMBER COUNTS AND THE COSMIC X-RAY BACKGROUND
Minsun Kim,1,2 Belinda J. Wilkes,1 Dong-Woo Kim,1 Paul J. Green,1 Wayne A. Barkhouse, Myung Gyoon Lee,2 John D. Silverman,4 and Harvey D. Tananbaum1
Received 2006 April 24; accepted 2006 November 20
3

ABSTRACT We present the Chandra Multiwavelength Project (ChaMP) X-ray point source number counts and cosmic X-ray background (CXRB) flux densities in multiple energy bands. From the ChaMP X-ray point source catalog, $5500 sources are selected, covering 9.6 deg 2 in sky area. To quantitatively characterize the sensitivity and completeness of the ChaMP sample, we perform extensive simulations. We also include the ChaMP + CDFs (Chandra Deep Fields) number counts to cover large flux ranges from 2 ; 10þ17 to 2:4 ; 10þ12 (0.5 Y 2 keV ) and from 2 ; 10þ16 to 7:1 ; 10þ12 (2 Y 8 keV ) ergs cmþ2 sþ1. The ChaMP and the ChaMP + CDFs differential number counts are well fitted with a broken power law. The best-fit faint and bright power indices are 1:49 ô 0:02 and 2:36 ô 0:05 (0.5 Y 2 keV ), and 1:58 ô 0:01 and 2:59×0::06 (2 Y 8 keV ), respectively. We detect breaks in the differential number counts that appear at þ0 05 different fluxes in different energy bands. Assuming a single power-law model for a source spectrum , we find that the same population(s) of soft X-ray sources causes the break in the differential number counts for all energy bands. We measure the resolved CXRB flux densities from the ChaMP and the ChaMP+CDFs number counts with and without bright target sources. By adding the known unresolved CXRB to the ChaMP+CDF resolved CXRB, we also estimate total CXRB flux densities. The fractions of the resolved CXRB without target sources are 78% ô 1% and 81% ô 2% in the 0.5 Y 2 and 2 Y 8 keV bands, respectively, somewhat lower than but generally consistent with earlier numbers because of their large errors. These fractions increase by $1% when target sources are included. Subject headings: cosmology: observations -- methods: data analysis -- surveys -- X-rays: diffuse background -- g X-rays: general Online material: color figures

1. INTRODUCTION What is the origin and nature of the cosmic X-ray background (CXRB)? Can detected X-ray sources account for the CXRB? The CXRB consists of resolved and unresolved components. The resolved CXRB originates in discrete sources such as point and extended sources, while diffuse components and faint sources that are below current flux limits contribute to the unresolved CXRB. The contribution of discrete X-ray sources to the CXRB can be directly measured from their number counts. Using the deep surveys of ROSAT (Rontgensatellit), Chandra,and XMM-Newton, ¨ the X-ray number counts have been determined down to flux limits of $2:3 ; 10þ17 (0.5 Y 2 keV ), $2:0 ; 10þ16 (2 Y 8 keV ), and $1:2 ; 10þ15 (5 Y 10 keV ) ergs cmþ2 sþ1, and %80% Y 90% of the CXRB is resolved into discrete X-ray sources in the 0.5 Y 2and 2 Y 8 keV bands (see Brandt & Hasinger 2005 for a detailed review). In this study, using the Chandra Multiwavelength Project (ChaMP) and the Chandra Deep Fields (CDFs) data, which include the largest number of sources and cover the widest sky area and flux range from a single satellite (Chandra) to date, we provide statistically robust X-ray number counts and CXRB flux densities without the cross calibration problem that is usually included in data from multiple satellites. We also study the X-ray number counts in multiple energy bands to systematically understand their behavior in each energy band.
Harvard-Smit hsonian Center for Astrophysics, Cambridge , MA. Department of Physics and Astronomy, Astronomy Program , Seoul National University, Seoul, Korea. 3 Department of Astronomy, University of Illinois at Urbana-Champaign , Urbana, IL. 4 Max-Planck-Institut fur extraterrestrische Physik, Garching, Germany. ¨
2 1

There have been many similar studies. Using the Chandra survey of SSA13, Mushotzky et al. (2000) presented the X-ray number counts in the 0.5 Y 2 and 2 Y 10 keV bands and suggested that detected hard X-ray sources account for at least 75% of the hard CXRB and that the mean X-ray spectrum of these sources is in good agreement with that of the background. Cowie et al. (2002) presented the 2 Y 8 keV number counts from the Chandra Deep Field Y South (CDF-S) and Chandra Deep Field Y North (CDF-N ) with SSA13/SSA22, and Rosati et al. (2002) presented those of the CDF-S, finding that at most $10% ($15%) of the CXRB is unresolved in the soft ( hard) energy band. Manners et al. (2003) presented the X-ray number counts in the 0.5 Y 2, 2 Y 8, and 0.5 Y 8 keV bands using the ELAIS ( European Large-Area ISO [Infrared Space Observatory] Survey) data. Moretti et al. (2003, hereafter M03) presented the X-ray number counts in the 0.5 Y 2 and 2 Y 10 keV bands from combining data from three different surveys (ROSAT, Chandra, and XMM-Newton). They concluded that 95% and 89% of the soft and hard CXRB, respectively, can be resolved into discrete X-ray sources. Bauer et al. (2004, hereafter B04) combined the CDF-N and CDF-S data and measured the contributions of the faint X-ray source populations to the CXRB. They found that 90% (0.5 Y 2 keV ) and 93% (2 Y 8 keV ) of the total CXRB was resolved into discrete sources. Basilakos et al. (2005) presented the number counts of the XMM-Newton/ Two Degree Field (2dF ) survey in the 0.5 Y 2 and 0.5 Y 8 keV bands, and Chiappetti et al. (2005) presented the number counts of the XMM-Newton Large-Scale Structure ( LSS) survey in the 0.5 Y 2 and 2 Y 10 keV bands. Worsley et al. (2005) found that the resolved fractions of the CXRB are $85% (0.5 Y 2keV ), $80% (2 Y 10 keV ), and $50% at k8 keV. Recently, Hickox & Markevitch (2006, hereafter HM06) directly measured the absolute unresolved CXRB 29


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TABLE 1 Defini ti on of Energy Bands Definition ( keV ) Broad B ............................................. Bc ........................................... Soft S ............................................. Sc............................................ Ssa .......................................... Hard H............................................. Hc ........................................... Heb ......................................... 2.5 Y 8 2Y8 2 Y 10 0.3 Y 2.5 0.5 Y 2 1Y2 0.3 Y 8 0.5 Y 8

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Band

a The Ss (2 Y 10 keV ) band was used only for estimating the CXRB flux density (see x 6). b The He (2 Y 10 keV ) band was not used in this study; however, it is referred to in previous studies.

origin of the break in the differential number counts using the hardness ratio [ HR ¼ (Hc þ Sc)/( Hc × Sc); see Table 1 for energy band definitions] and redshift distribution of the X-ray sources. In addition, we combine the ChaMP and CDFs ( hereafter ChaMP+ CDFs) number counts to cover the full available flux range. From the ChaMP and the ChaMP+CDFs number counts, we estimate the resolved CXRB flux densities in six energy bands. By adding the known unresolved CXRB ( HM06) to the resolved ChaMP+CDFs CXRB flux density, we estimate the total CXRB flux densities in the 0.5 Y 2, 1 Y 2, and 2 Y 8 keV bands. In x 2, we briefly describe the ChaMP data selection. In x 3, we describe the method and results of the ChaMP simulations. In x 4, the ChaMP and the ChaMP+CDFs number counts are presented in six energy bands and are compared with previous studies. In x 5, we study the nature and origin of the break flux in the number counts. In x 6, we estimate the resolved CXRB flux densities in six energy bands and the total CXRB flux densities in three energy bands. In x 7, the summary and conclusions of this study are presented. Throughout this study, quoted errors are for a ô1 confidence level, unless otherwise noted. Although we perform this study in six energy bands (see Table 1), we only present the figures in the 0.5 Y 8, 0.5 Y 2, and 2 Y 8 keV bands for simplicity; however, tables include the results in all energy bands. To compare with previous studies, we assume photon indices of þph ¼ 1:4 and 1.7; however, only figures with þph ¼ 1:4 are provided. 2. THE ChaMP SAMPLE SELECTION The X-ray point source sample is from the ChaMP X-ray point source catalog ( KM07), which consists of $6800 X-ray sources in 149 Chandra archival observations. The ChaMP fields were selected to include ACIS observations at high Galactic latitude, jbj > 20 . Fields containing large extended sources, planetary objects, fields intended by the PI for survey, and Local Group galaxies were excluded ( Kim et al. 2004a). The ChaMP X-ray point source properties were obtained using a ChaMP-specific pipeline, XPIPE, which uses wavdetect5 detections as source positions and extracts source properties within a given aperture appropriate for the local point-spread function ( PSF ) size (a 95% encircled energy radius at 1.5 keV ) using xapphot ( E. Kim et al. 2007, in preparation). The ChaMP X-ray point source catalog is divided into main and supplementary catalogs. Thirty-five ChaMP fields overlap one another, and the supplementary catalog contains sources from the 19 shorter exposure fields among these. To avoid confusion due to duplicated fields, our analysis uses the main ChaMP catalog, which contains 130 ChaMP fields. From the main ChaMP catalog, we selected sources in the I0, I1, I2, and I3 CCD chips for 32 ACIS-I observations and sources in the I2, I3, S2, and S3 CCD chips for 98 ACIS-S observations. These sources are located within an off-axis angle of $150 . In addition, we selected sources with signal-to-noise ratio S/ N > 1:5, corresponding to source counts of C k 5. XPIPE detects sources in the B band (0.3 Y 8keV; see Table 1 for energy band definitions), and for all energy bands, photometry is performed at the source positions determined in the Bband(see x 3 in KM07 ). Therefore, in our sample, it is possible to miss very soft ( hard) sources that might be detected only in the soft ( hard) band but not detected in the B band. For sources with S/ N > 1:5 in the S ( H ) band, the missing percentage of very soft ( hard ) sources is 5% (10%), when we assume matching of all possible counterparts in the B and S ( H ) bands. However, since we perform simulations to correct for the incompleteness and
5

from Chandra Deep Field images after excluding point and extended sources in those fields. They also estimated the resolved X-ray source intensity from the CDFs and from the number counts for brighter sources ( Vikhlinin et al. 1995; M03), and then estimated the total CXRB flux density by combining the two. They found that the resolved fractions of the CXRB are 77% ô 3% (1 Y 2 keV ) and 80% ô 8% (2 Y 8 keV ). Until now, using the ROSAT, XMM-Newton, and Chandra data, these studies have revealed that $80% of the CXRB is resolved into discrete X-ray sources in the 0.5 Y 2 and 2 Y 8 keV bands; however, the resolved fraction of the CXRB significantly decreases at k8 keV. The ChaMP is a serendipitous, wide-area survey covering intermediate and high fluxes using Chandra archival data. Kim et al. (2004a) presented the initial ChaMP catalog, which contains $800 X-ray point sources in the central region of 62 of 149 ChaMP fields. From the initial ChaMP catalog, Kim et al. (2004b, hereafter KD04) presented X-ray number counts in the 0.5 Y 2 and 2 Y 8 keV bands. To avoid the incompleteness of the selected fields, they selected sources having large X-ray source counts (>20) and located close to on-axis (<40000 ). The selected sample covered $1.1 deg 2 in sky area. In the flux range from 10þ15 to 10þ13 ergs cmþ2 sþ1 (0.5 Y 2 keV ), they detected the break in the differential number counts. However, due to the shallow flux limit, they could not detect the break in the 2 Y 8 keV band. In this study, we use the latest ChaMP X-ray point source catalog, which contains $6800 X-ray point sources in 149 ChaMP fields with sky coverage area of $10 deg 2 ( Kim et al. 2007, hereafter KM07 ) to determine the X-ray point source number counts in six energy bands. To correct for incompleteness, Eddington bias, and instrumental effects, and to include large off-axis angles (up to $150 ) and faint (down to $5 source counts) sources, we perform extensive simulations to calculate the sky coverage of the selected sources as a function of flux. Using this large sample and the simulation results, we present the X-ray point source number counts that fully cover the break flux in each energy band with small statistical errors. Due to the wide flux range of the sample, we detect breaks in the differential number counts in all energy bands and investigate what causes the different break flux in different energy bands. We also investigate the nature and the

See http://cxc.harvard.edu /ciao/.


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TABLE 2 Stat is tical P ro perties o f X-Ray Po in t Sources Band (1) Number (2) Min. (3) Counta B ..................... S ..................... H..................... Bc ................... Sc.................... Hc ................... 5515 4864 2575 5229 4554 3078 5.42 5.42 5.41 6.46 5.41 5.42 Fluxb B ..................... S ..................... H..................... Bc ................... Sc.................... Hc ................... 5515 4864 2575 5229 4554 3078 0.63 0.33 1.27 0.69 0.26 1.17 7175.62 3286.49 6690.72 6767.74 2395.21 7112.31 9.09 4.36 8.61 9.04 3.21 8.87 25.97 12.78 21.40 25.38 9.32 21.88 40535.59 38117.52 11604.93 39760.98 36010.96 13624.92 22.57 19.24 12.73 23.46 18.24 13.72 69.53 61.50 28.63 70.52 57.59 31.63 Max. (4) Median (5) Mean (6)

bias in the ChaMP fields using the same detection technique as the ChaMP X-ray point source catalog (see x 3), these very soft ( hard) sources do not introduce an additional error in our number counts. Since the ChaMP is a Chandra archival survey, most ChaMP fields contain target sources selected by the PI, and those targets are likely to be biased toward special X-ray populations such as bright active galactic nuclei (AGNs). Therefore, we excluded target sources to derive less biased X-ray number counts. Our selection results in $5500 sources in the 0.3 Y 8 keV band from the ChaMP X-ray point source catalog. Table 2 lists the number of sources and the statistical properties of the X-ray sources in each energy band. Figure 1 shows the counts and flux distributions of the final X-ray sample. The median value of the distribution is also plotted. 3. THE ChaMP SIMULATIONS To determine accurate number counts, it is necessary to correct for the incompleteness of the sample as well as for instrumental effects such as vignetting and the off-axis degradation of the PSF. There are two major techniques to correct these biases, a semianalytical approach and a Monte Carlo simulation. The semianalytical approach is based on the flux limit map of a given field, which contains the faintest flux corresponding to the assumed significance level of source detection (Johnson et al. 2003; Cappelluti

Notes.--Col. (1): X-ray energy band (see Table 1). Col. (2): Number of sources. Col. (3): Minimum value of the sample. Col. (4): Maximum value of the sample. Col. (5): Median value of the sample. Col. (6): Mean value of the sample. a Source net counts. b Source flux with þph ¼ 1:4 in units of 10þ15 ergs cmþ2 sþ1.

Fig. 1.-- Distributions of source net counts (left) and flux (right) in three energy bands. The vertical dashed line indicates the median of the distribution. The flux was determined assuming a photon index of þph ¼ 1:4.


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Fig. 2.-- Cumulative number counts for the artificial sources in the B band. The solid line represents the number counts for sources whose fluxes were randomly selected from the assumed number counts with a slope of þ1. Due to small number statistics, deviations from the assumed number counts are present in the bright flux regime. Dotted and dashed lines represent number counts for sources generated with MARX and for sources extracted with XPIPE, respectively. The effect of Eddington bias is evident at the faint fluxes (S < 10þ14 ergs cmþ2 sþ1) in the simulated source number counts. [See the electronic edition of the Journal for a color version of this figure.]

Fig. 3.-- Sky coverages for sources with S/ N > 1:5 in six energy bands. The full sky coverage is 9.6 deg 2.

et al. 2005; Chiappetti et al. 2005). This technique is efficient and reliable; however, it is possible to undercorrect the incompleteness of the field because in this method the source detection probability is a function of only the source counts. The actual source detection probability in a Chandra field is a complex function of off-axis angle and source counts: the detection probability decreases as off-axis angle increases and as source counts decrease ( KM07). Therefore, to accurately determine the sky coverage of the ChaMP sample, we performed extensive Monte Carlo simulations to correct incompleteness and biases of the sample fields. 3.1. Method The simulation method is described in detail in KM07 and consists of three parts, (1) generating artificial X-ray sources with MARX,6 (2) adding them to the observed image, and (3) detecting these artificial sources with wavdetect and extracting source properties with xapphot. In step (2), we used the real Chandra observations to accurately reflect the effects of background counts and source confusion in the ChaMP fields. We performed simulations using all selected observations and four CCD chips in each observation (see x 2). We generated 1000 artificial X-ray sources per sample field, which corresponds to $13,000 artificial X-ray sources per square degree. The number of sources in each field depends on the effective exposure time of the observation and the neutral hydrogen column density, NH , toward the observed region of the sky. On average, 11.7% of the 127,178 artificial X-ray sources are detected in our simulations, a total of 14,932 artificial X-ray sources in 130 ChaMP fields. The number of detected artificial sources is 2.5 times the $6000
6

observed sources; this number is statistically sufficient to estimate the properties of the ChaMP sample. The form of the assumed number count distribution is not critical because we use the ratio of input to output number of sources to determine the sensitivity ( Vikhlinin et al. 1995; Kim & Fabbiano 2003). The actual X-ray differential number counts are described by a broken/double power law with a faint slope of $þ1.5 and a bright slope of $þ2.5 ( Yang et al. 2004; Basilakos et al. 2005; Chiappetti et al. 2005) in most energy bands; however, the break flux has not been well determined. Therefore, we assumed a cumulative number count distribution with a single power law with a slope of þ1 corresponding to a slope of þ2in the differential number counts, taking the average of the faint and bright slopes from the literature, in the 0.3 Y 8 keV band. From the assumed number count distribution, we randomly selected the artificial source flux. The artificial source fluxes span from 5 ; 10þ16 to 5 ; 10þ10 ergs cmþ2 sþ1 in the B band, covering the flux range of the observed ChaMP X-ray point sources (6 ; 10þ16 to 6 ; 10þ12 ergs cmþ2 sþ1). The spectrum of the artificial sources was assumed to be a power law (F / þþph ) with a photon index of þph ¼ 1:7, because the photon index þph for the observed ChaMP sources spans þph ¼ 1:5 Y 2 ( KD04; KM07). Tozzi et al. (2006) performed X-ray spectral analysis for 82 X-ray bright sources in the CDF-S and found a weighted mean value for the slope of the power-law spectrum of hþph i ' 1:75 ô 0:02. The flux range of these bright sources in the CDF-S overlaps with the faint flux end of the ChaMP sources; therefore, we assumed that the faint ChaMP sources also have a photon index of þph $ 1:7. We assumed a Galactic absorption, NH (Stark et al. 1992), for each observation; however, we did not include intrinsic absorption for the artificial source spectrum. The spectrum of each X-ray point source was generated using the XSPEC7 package. The artificial source's position was randomly selected in each CCD chip area, but it was rejected if the source area at a given random position had an exposure map value of less than 10% of
7

See http://space.mit.edu /CXC/ MARX / and MARX 4.0 Technical Manual.

See http://xspec.gsfc.nasa.gov.


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Fig. 4.-- Differential (left ) and cumulative (right ) number counts of the ChaMP X-ray point sources in the Bc, Sc, and Hc bands ( from top to bottom, respectively). Solid lines represent the best-fit results with a broken power law. The vertical dashed lines indicate the derived break fluxes. Source fluxes were determined assuming a photon index of þph ¼ 1:4. Since we present the differential number counts brighter than a flux corresponding to 10% of the full sky coverage, the faintest bin still has sufficient sources and shows a small error bar. The error bars in the cumulative number counts are estimated by the error propagation rule using Gehrels (1986) statistics. [See the electronic edition of the Journal for a color version of this figure.]

the maximum. This requirement is identical to that in the ChaMP X-ray point source catalog reduction procedure. To avoid overcrowding of the artificial sources, $250 artificial sources per CCD were divided into several groups to be added into the observed image: while we did not allow the artificial X-ray point sources to overlap one another, we allowed overlap between artificial and real X-ray sources to provide an estimate of source confusion in each observed field. This resulted in $10 ($20) simulated images per ACIS-I (ACIS-S) CCD, corresponding $9100 CCD images (event files) to run wavdetect (xapphot). Since $11.7% of the artificial sources ($14,900) are detected, on average we added only $1.6 artificial sources to each simulated image. The net counts of the overlapping artificial sources with real sources were corrected following the overlapping source correction methods described in x 3.2.2 of KM07. 3.2. Sk y Coverage Area Using the results of the simulations described in x 3.1, Figure 2 shows the number counts for artificial sources in the B band. The number count for sources whose fluxes were randomly selected from the assumed number counts (solid line) agree well with a slope of þ1. However, there are slight statistical fluctuations at

fluxes brighter than 10þ13 ergs cmþ2 sþ1 due to small number statistics. The random sources were selected per observation (see x 3.1), and 1 Y 2 bright sources out of 1000 sources result in statistical fluctuations in each observation. In addition, since we fixed the flux maximum rather than using infinitely bright flux (see x 3.1) for random sources, the cumulative number of artificial sources drops at $10þ12 ergs cmþ2 sþ1 rather than following a line of slope þ1. Since the aim of our simulations is to correct bias at faint fluxes, we do not require good statistics at bright fluxes. The number counts for artificial sources generated by MARX (dotted line) and that for artificial sources detected by XPIPE (dashed line) are also displayed. The Eddington bias, that sources with counts near the detection threshold will be preferentially detected when they have upward fluctuations (e.g., Kenter & Murray 2003), is evident at faint fluxes (S < 10þ14 ergs cmþ2 sþ1) in the simulated number counts. Near $10þ14 ergs cmþ2 sþ1, the number of detected artificial sources starts to decrease. Figure 3 displays sky coverage for sources with S/ N > 1:5as a function of flux in six energy bands assuming a photon index of þph ¼ 1:4. The sky coverage area is the ratio of the number of detected over input sources at a given flux, multiplied by the total sky area. The full sky area is 9.6 deg 2. The geometrical area of a


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TABLE 3 List of th e Best-Fit Parameters Excl ud ing T ar get Obj ects Band (1) K (2) 1 (3)
ph

Chandra CCD chip is 0.0196 deg 2; however, the net effective area is slightly larger due to the dither. To accurately calculate the effective area, we follow the same method as in xapphot: all pixels in the exposure map were summed, excluding those pixels with an exposure map value less than 10% of the maximum within the corresponding source area. This criterion automatically excludes pixel positions located near the edge of the CCD chip. 4. X-RAY POINT SOURCE NUMBER COUNTS 4.1. The ChaMP Number Counts The cumulative number counts for sources brighter than a given flux S, corrected by the corresponding sky coverage at S, is X1 ; N Ï> S ÷ ¼ S >S i
i

2 (4) = 1.4 2.41 ô 0.05 2.58 ô 0.05 2:44×0::06 þ0 05 2.36 ô 0.05 2.65 ô 0.07 2.48 ô 0.05 = 1.7 2.42 ô 0.05 2.58 ô 0.05 2:45×0::06 þ0 05 2:36×0::05 þ0 04 2.64 ô 0.07 2.48 ô 0.05
ph

Sb (5)

ChaMP Data Set, þ S ........................ H........................ B ........................ Sc....................... Hc ...................... Bc ...................... 769 ô 14 1828×48 þ43 1614×28 þ43 607 ô 12 2040 ô 50 1557×28 þ50 1.57 1.81 1.65 1.54 1.82 1.64 ô ô ô ô ô ô

0.01 0.01 0.01 0.02 0.01 0.01
ph

×0 7 9:9þ1::6 ×0 9 14:2þ1::1 25.0 ô 1.9 6.8 ô 0.5 ×6 3 19:2þ1::8 22.9 ô 1.6

ChaMP Data Set, þ

Ï 1÷

where Si is the flux of the ith X-ray point source and i is the sky coverage that is the maximum solid angle covered by the flux Si . Using the sources selected in x 2 and the corresponding sky coverage derived in x 3.2, we derived the cumulative number counts for the ChaMP point sources. Since the differential number count is a derivative form of the cumulative number count, we derived the differential number counts from the cumulative number counts resulting from equation (1) as follows: dN ¼ þ Ni dS i Si
×1 ×1

S ........................ H........................ B ........................ Sc....................... Hc ...................... Bc ......................

783 ô 15 1774×44 þ41 1505×25 þ41 612 ô 12 1932×46 þ48 1407×25 þ48

1.58 1.80 1.65 1.53 1.82 1.64

ô ô ô ô ô ô

0.01 0.01 0.01 0.02 0.01 0.01

10.5 ô 0.8 13.5 ô 0.9 21.9 ô 1.7 6.7 ô 0.5 ×4 4 17:8þ1::7 ×1 3 19:2þ1::4

ChaMP+CDFs Data Set, þ Sc....................... Hc ...................... 571 ô 11 1258 ô 29 1.49 ô 0.02 1.58 ô 0.01

= 1.4 2.36 ô 0.05 2:59×0::06 þ0 05 6.5 ô 0.4 14.4 ô 0.9

þ Ni ; þ Si

Ï 2÷

Notes.-- Col. (1): X-ray energy band (see Table 1). Col. (2): Normalization constant. Col. (3): Faint power-law index of a broken power law. Col. (4): Bright power-law index of a broken power law. Col. (5): Break flux in units of 10þ15 ergs cmþ2 sþ1.

where Ni is the cumulative source number at flux Si . Since the sky coverage rapidly decreases near the faint flux limit, there are large statistical errors for the number counts in the faint flux regime. Thus, for better statistics, we present the number counts brighter than the flux corresponding to 10% of the full sky coverage. For example, in the 0.5 Y 8 keV band , this flux cut corresponds to 2 ; 10þ15 ergs cmþ2 sþ1; 500 sources fainter than this flux are not included in the final number counts. In Figure 4, we display the ChaMP differential number counts (left) and cumulative number counts (right) in three energy bands. Statistical errors on the number counts are assigned following Gehrels (1986). The shape of the cumulative number counts is curved rather than a single power-law feature, and the differential number counts can be fitted by a broken power law ( Baldi et al. 2002; KD04) or by a double power law (Cowie et al. 2002; Harrison et al. 2003, hereafter H03; Yang et al. 2004; Chiappetti et al. 2005). Since errors for the cumulative number counts are not independent ( Murdoch et al. 1973), it is difficult to estimate confidence levels of fitting parameters for the cumulative number counts. Therefore, we fitted the differential number count with a broken power law as follows: ( K ÏS =Sref ÷þ 1 ; S < Sb ; dN ¼ Ï 3÷ Ï 2 þ 1 ÷ þ 2 dS ÏS =Sref ÷ ; S ! Sb ; K ÏSb =Sref ÷ where K is a normalization constant and Sref is a normalization flux. In this study, we set a normalization flux of Sref ¼ 10þ15 ergs cmþ2 sþ1. The parameter Sb is the break flux at which the slope of the differential number count changes, and 1 and 2 are faint and bright power indices. The best-fit parameters for the differential number counts are listed in Table 3 for photon indices of þph ¼ 1:4 and 1.7. The photon index þph hardly affects 1 and 2 , but it shifts Sb somewhat. We display the best-fit results in the left panels of Figure 4. In all energy bands, we detected breaks,

which appear at different fluxes in different energy bands. We discuss the break flux of the differential number count in more detail in x 5. By integrating equation (3), we can derive a formula for the cumulative number count as follows: Z dN dS 0 ; Ï 4÷ N Ï> S ÷ ¼ dS therefore, 8 > 1 > >K > > > 1 þ 1 > > > < N Ï> S ÷ ¼ ×K > 1 > > > > > > 1 >K > : 2 þ 1 Ï1þ 1 ÷ 1 Sb þ 1 þ 2 Sref Ï1þ 1 ÷ 1 S ; Sref þ1 Ï 2 þ 1 ÷ Ï1 Sb S Sref Sref

S < Sb ;
þ 2 ÷

; S ! Sb ; Ï 5÷

where definitions of parameters are the same as in equation (3). Using the best-fit parameters derived from the differential number counts, we also plot the best-fit results for the cumulative number counts in the right panels of Figure 4. 4.2. The ChaMP +CDFs Number Counts To measure the discrete X-ray source contributions to the CXRB, it is important to derive the number counts over a wide range of flux. So far, M03 have presented the widest flu