Can AGN and clusters explain the CXB fluctuations?

Can AGN and galaxy clusters explain the surface brightness fluctuations of the cosmic X-ray background?

Alexander Kolodzig, Marat Gilfanov, Gert Hütsi, Rashid Sunyaev
Kavli Institute for Astronomy and Astrophysics (KIAA), Peking University, 100871 Beijing, China – KIAA fellow, http://kiaa.pku.edu.cn
Max-Planck-Institut für Astrophysik (MPA), Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany
Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya ul. 84/32, Moscow, 117997 Russia
Tartu Observatory, Tõravere 61602, Estonia
Accepted 2O!6 Xxx XX. Received 2O!6 Xxx XX; in original form 2O!6 Xxx XX
Abstract

Fluctuations of the surface brightness of cosmic X-ray background (CXB) carry unique information about faint and low luminosity source populations, which is inaccessible for conventional large-scale structure (LSS) studies based on resolved sources. We used Chandra data of the XBOOTES field ( deg) to conduct the most accurate measurement to date of the power spectrum of fluctuations of the unresolved CXB on the angular scales of . We find that at sub-arcmin angular scales, the power spectrum is consistent with the AGN shot noise, without much need for any significant contribution from their one-halo term. This is consistent with the theoretical expectation that low-luminosity AGN reside alone in their dark matter halos. However, at larger angular scales we detect a significant LSS signal above the AGN shot noise. Its power spectrum, obtained after subtracting the AGN shot noise, follows a power law with the slope of and its amplitude is much larger than what can be plausibly explained by the two-halo term of AGN. We demonstrate that the detected LSS signal is produced by unresolved clusters and groups of galaxies. For the flux limit of the XBOOTES survey, their flux-weighted mean redshift equals , and the mean temperature of their intracluster medium (ICM),  keV, corresponds to the mass of . The power spectrum of CXB fluctuations carries information about the redshift distribution of these objects and the spatial structure of their ICM on the linear scales of up to Mpc, i.e. of the order of the virial radius.

keywords:
– Galaxies: active – X-rays: galaxies – large-scale structure of Universe – X-rays: diffuse background – Galaxy clusters
pagerange: Can AGN and galaxy clusters explain the surface brightness fluctuations of the cosmic X-ray background?Can AGN and galaxy clusters explain the surface brightness fluctuations of the cosmic X-ray background?pubyear: 2O!6

1 Introduction

Since the discovery of the cosmic X-ray background (CXB) about half a century ago (Giacconi et al., 1962), understanding of its origin has been one of the major drivers for the development of X-ray astronomy and most X-ray space telescopes, such as the currently active missions: XMM-Newton, Chandra, and NuSTAR (e.g. Fabian & Barcons, 1992; Giacconi, 2013; Tanaka, 2013). Thanks to the many, in particular deep X-ray surveys of Chandra (e.g. Brandt & Hasinger, 2005; Alexander et al., 2013; Brandt & Alexander, 2015), we now know for certain that the CXB is dominated by extragalactic discrete sources, with Active Galactic Nuclei (AGN) leading the way (e.g. Comastri et al., 1995; Moretti et al., 2003; Hickox & Markevitch, 2006, 2007; Gilli, Comastri & Hasinger, 2007; Moretti et al., 2012; Lehmer et al., 2012). This makes the CXB the prefect window to study the accretion history of the Universe up to high redshift () (e.g. Hasinger, Miyaji & Schmidt, 2005; Gilli, Comastri & Hasinger, 2007; Aird et al., 2010; Ueda et al., 2014; Miyaji et al., 2015), which is an essential base to understand galaxy evolution (e.g. Hopkins et al., 2006; Hickox et al., 2009; Alexander & Hickox, 2012).

Since the first X-ray surveys, angular correlation studies of the CXB had two major applications. They are used to disentangle the components of the CXB and at the same time to perform large-scale structure (LSS) studies (e.g. Scheuer, 1974; Hamilton & Helfand, 1987; Shafer & Fabian, 1983; Barcons & Fabian, 1988; Soltan & Hasinger, 1994; Vikhlinin & Forman, 1995; Miyaji & Griffiths, 2002). The advantage of such studies is that one can analyze the CXB beyond the survey sensitivity limit, since one does not require any source identification or/and redshift information. Thanks to these studies it has been long known that the CXB must be dominated by point-sources with a redshift distribution similar to optical QSOs but somewhat higher clustering strength.

These results were confirmed in the last two decades by very deep pencil beam surveys (e.g. Hickox & Markevitch, 2006, 2007; Lehmer et al., 2012) and LSS studies with resolved samples of X-ray-selected AGN from wide but more shallow surveys (see reviews of Cappelluti, Allevato & Finoguenov, 2012; Krumpe, Miyaji & Coil, 2014). This became possible thanks to the high-angular resolution of the current generation of X-ray telescopes, complemented with optical spectroscopic redshift surveys of sufficient size and depth. Due to this clustering measurements with resolved AGN developed in the last decade to an important branch of LSS studies in general. It led to major advances in understanding how AGN activity is triggered and how does it depend on its environment, such as the host galaxy and dark matter halo (DMH) properties, and how do supermassive black holes (SMBH) grow and co-evolve with their DMH over cosmic time, which are essential questions in the field of galaxy evolution (e.g. Cappelluti, Allevato & Finoguenov, 2012; Krumpe, Miyaji & Coil, 2014). In the future, it will become possible to use AGN as a cosmological probe via baryon acoustic oscillation measurements (for details see e.g. Kolodzig et al., 2013a; Hütsi et al., 2014) with the  million AGN to be detected in the upcoming SRG/eROSITA all-sky survey (for details see Predehl et al., 2010; Merloni et al., 2012; Kolodzig et al., 2013b).

Due the focus on resolved AGN, the current knowledge of AGN clustering properties and its implications for AGN and galaxy evolution are biased towards objects of , in particular for higher redshifts (), due to the luminosity cut from the AGN identification process and the signal-to-noise ratio (S/N) cut for the spectroscopic redshift (e.g. Allevato et al., 2011, 2012, 2014; Krumpe, Miyaji & Coil, 2010; Krumpe et al., 2012; Miyaji et al., 2011; Krumpe et al., 2015). An important question to ask is if we are able to extrapolate these clustering properties to less luminous AGN, which trace galaxies at an earlier evolutionary stage with a less massive SMBH and/or smaller accretion rate than luminous AGN? A significant step towards answering this question is to study the surface brightness fluctuations of the unresolved CXB measured with the current generation of X-ray telescopes, which allows us to measure angular fluctuations on small scales down to the arc-second regime.

This type of clustering measurement offers us a great window to the small-scale clustering regime (). Clustering studies of spatially resolved AGN samples have difficulties to access this regime, because of the low spatial density of AGN in general, and because multiobject spectroscopy surveys are typically limited to an angular separation of (e.g. Blanton et al., 2003; Dawson et al., 2013). Therefore, our best spatially resolved measurement of the small-scale clustering regime comes from the use of dedicated catalogs of close AGN pairs (e.g. SDSS Quasar Lens Search, Kayo & Oguri, 2012) or the direct measurement of the halo occupation distribution (HOD) of AGN from galaxy groups (e.g. Allevato et al., 2012). Both types of measurement require an extensive amount of multi-wavelength survey data. In terms of standard clustering studies, the best results come from studies of optically-selected AGN thanks to the sufficient size of the available survey data (e.g. Hennawi et al., 2006; Kayo & Oguri, 2012; Richardson et al., 2012; Shen et al., 2013). For X-ray-selected AGN, the situation is more difficult due to the so far rather limited survey data (e.g. Allevato et al., 2012; Richardson et al., 2013). Here, non-spatially resolved studies, such as the brightness fluctuations of the unresolved CXB, may offer a true alternative for small-scale clustering measurements, which have not been fully utilized yet.

Due to their scientific focus, the only two existing studies of the brightness fluctuations of the unresolved CXB at these angular scales used very deep surveys (e.g. Cappelluti et al., 2013; Helgason et al., 2014). However, this also implies a very small sky coverage of these surveys ().

In our study, we aim to conduct the most accurate measurement to date of the brightness fluctuations of the unresolved CXB on angular scales below . We are able to achieve this by using the XBOOTES survey (Murray et al., 2005; Kenter et al., 2005, hereafter K05), the currently largest available continuous Chandra survey, with a surface area of . The advantage in comparison to previous studies is that a higher S/N makes any comparison with current clustering models from known source populations much more meaningful and it enables us to do clustering measurements in an energy resolved manner in order to separate different source populations. In this first study we present our measurement of the brightness fluctuations of the unresolved CXB with angular scales up to , and make novel tests for systematic uncertainties such as the brightness fluctuations of the instrumental background.

Covering the angular scales from the arc-second to arc-minute regime may allow us to study the clustering properties of AGN within the same DMH (one-halo-term) and AGN of different DMHs (two-halo-term) of low-luminosity AGN () and redshifts of . This parameter regime is inaccessible for conventional clustering studies of the resolved CXB with current X-ray surveys (e.g. Cappelluti, Allevato & Finoguenov, 2012).

Diffuse emission from the intracluster medium (ICM) of clusters and groups of galaxies and the associated warm-hot intergalactic medium (WHIM) also contributes to the CXB (e.g. Rosati, Borgani & Norman, 2002; Hickox & Markevitch, 2007; Kravtsov & Borgani, 2012; Roncarelli et al., 2012). Since clusters and groups of galaxies are more difficult to detect and an order of magnitude more sparse than AGN, our knowledge about their population, in particular at low fluxes () is less certain (e.g. Finoguenov et al., 2007, 2010, 2015; Clerc et al., 2012; Böhringer, Chon & Collins, 2014). Thanks to cosmological hydrodynamical simulations (e.g. Roncarelli et al., 2006, 2007, 2012; Ursino et al., 2011; Ursino, Galeazzi & Huffenberger, 2014) and analytical studies (e.g. Diego et al., 2003; Cheng, Wu & Cooray, 2004) we have nevertheless some reasonable understanding of their clustering properties. As we will demonstrate in this paper, angular correlation studies of CXB fluctuations can help to dramatically improve the situation from the observational side.

This paper is organized as following: In Sect. 2 we explain our data processing procedure, in Sect. 3 we show the energy spectrum of the unresolved CXB of XBOOTES and estimate the contribution by different components of the CXB, in Sect. 4 we present our measurement of the surface brightness fluctuations of the unresolved CXB, and in Sect. 5 we study the origin of the detected LSS signal at large angular scales. In the Appendixes we present results of tests for various systematic effects and investigate the impact of the instrumental background on our measurements.

For the work we assume a flat CDM cosmology with the following parameters: (), (), , . The values for and were chosen to match the values assumed in the X-ray luminosity function studies, which we use in our calculations (e.g. Sect. 3.3 or 4.3), and and are derived from the cosmic microwave background (CMB) study of WMAP111http://map.gsfc.nasa.gov (Komatsu et al., 2011). We note that the results of this work are not very sensitive to the exact values of the cosmological parameters and if we used the recently published, more precise cosmological parameters of the CMB study by PLANCK222http://www.cosmos.esa.int/web/planck (Planck Collaboration et al., 2015) our results would not change.

2 Data preparation and processing

For our analysis of the surface brightness fluctuations of the unresolved CXB we are using the XBOOTES survey (Murray et al., 2005, K05), which is currently the largest available continuous Chandra survey with a surface area of . It consists of 126 individual, contiguous Chandra ACIS-I observations. In order to avoid unnecessary complication in our analysis, we exclude eight of them. The six observations with the ObsIDs 3601, 3607, 3617, 3625, 3641 & 3657 are excluded because they all show much higher background count rate than the average. The observations with the ObsIDs 4228 and 4224 are excluded because they contain a very bright point- and extended source, respectively. Therefore, when referring to the “XBOOTES survey”, we mean from now on the 118 remaining observations (for a full list of observations see Table 1 of Murray et al., 2005).

The average exposure time of an XBOOTES observation is  ksec and the combined exposure time of 118 observations is almost  Msec. By excluding observations the surface area of XBOOTES reduces from originally to . Those values will further decrease after processing the observations.

For processing the observations of the XBOOTES survey we are using Chandra’s data analysis system CIAO (v4.7, CALDB v4.6.9, Fruscione et al., 2006) and follow their standard analysis threads, unless stated otherwise. Since the observations were performed in the very faint mode (VFAINT), we are able to make use of CIAO’s most strict filtering method333 For details see http://cxc.harvard.edu/cal/Acis/Cal_prods/vfbkgrnd. We activate it in the data processing script chandra_repro with check_vf_pha = yes. of background events in ACIS-I data.

Unless otherwise stated, we use throughout the paper for the Galactic absorption a hydrogen column density of as determined for the XBOOTES survey and we convert the flux of extragalactic sources between different energy bands and between physical and instrumental units assuming an absorbed powerlaw with a photon-index of (K05, Sect.3.3).

Figure 1: Shape of the PSF for Chandra ACIS-I (averaged over 4 azimuthal angles) for different offset angles () for the source flux of (). The vertical dashed line shows the radius of the circular exclusion area for this flux group (Table 1). The horizontal dotted lines show levels corresponding to and of the surface brightness of unresolved AGN (Sect. 3.1.3).

2.1 Exposure map and mask

For the following data processing and analysis, we will need the exposure map [seconds] and mask , which we describe here.

We use the exposure map to convert our count maps [counts] into flux maps [cts s] but also to take the vignetting into account. For creating the exposure map we are using the spectral model and best-fit parameters of our spectral fit of the unresolved CXB (Table 2). Note that the exposure map is not very sensitive to the choice of the spectral model in the band, because we only compute the exposure map in units of seconds and the vignetting is not very energy depended in this energy range.

We use the mask to excluded certain regions of Chandra maps from the analysis. It is set to be large enough ( pixels) to contain the entire ACIS-I FOV of an observation. Pixels of the mask are set to zero, when they are outside the FOV, when they are within the exclusion area of a resolved source (Sect. 2.2), when they have zero exposure time, which takes also bad pixels into account, or when the exposure time falls below of the peak value of . The latter targets low exposed pixels, which are predominantly located in the CCD gaps and edges of the ACIS-I and occur due to the dithering movement of Chandra during an observation. The threshold was chosen to be , because we see a clear break of the pixel distribution of around this value. The average field-of-view (FOV) solid angle of one observation after this filter step but before removing resolved sources is . The solid angle of the mask is computed as

(1)

whereby is the size of a pixel. Since for our analysis we use an image pixel binning of one, the size of a pixel444http://cxc.harvard.edu/proposer/POG/html/chap6.html#tab:acis_char is .

Whenever we convert counts to count-rate, where we can not use the exposure map , we are using the average value of this map:

(2)

This for instance is the case for the energy spectrum in Sect. 3 or for energy bands above , where the effective area of Chandra becomes neglectable.

2.2 Removing resolved sources

Figure 2: Average total counts () per XBOOTES observation (instrumental background not removed) as a function of radius of the circular exclusion area of point sources. We change the radius for each flux group between to of its value in Table 1. The displayed radius is a weighted average based on the number of point sources per flux group. Vertical dotted line: Average radius for our definition in Table 1.
Flux # of (b) Radius Depth ECF (e)
Groups Sources [%] [arcsec] [%] [%] [cts]
1673 30
1328 55
268 80
23 140

(a) In erg cm s ().

(b) Number of sources in fraction of the total number of sources (3293).

(c) The PSF surface brightness in fraction of the surface brightness of unresolved AGN (Sect. 3.1.3) for the brightest sources in the given flux group at the edge of the circular exclusion area.

(d) Enclosed count fraction (based on the integration of our simulated PSF, averaged over all azimuthal and offset angles).

(e) Residual counts per point-source outside the exclusion area for the brightest sources in the given flux group. Computed with the corresponding ECF and an exposure time of  ksec (average value for XBOOTES, Sect. 2.3).

Table 1: Radius of the circular exclusion area of resolved point sources in XBOOTES for different flux groups.

In order to study the unresolved CXB, we need to remove the resolved (point-like and extended) sources to such a level that the residual counts of resolved sources contribute only insignificantly to the surface brightness of the unresolved CXB. For this purpose we are using the two source catalogs of K05 for point and extended sources.

2.2.1 Point sources

The point-source catalog of K05 includes 3293 sources with at least 4 counts in the band. The sky coverage of the XBOOTES survey as a function of the point-source detection sensitivity is shown in Fig. 12 of K05 (for the band). The average flux limit of the survey in the band is . It is defined as the flux level, which gives the same resolved CXB fraction as computed with the sky coverage vs. sensitivity distribution of the survey (e.g. Eq. 9). To estimate the appropriate size of a circular exclusion area of a point source, we simulated the point-spread-function (PSF) shape for 13 offset angles (, in -steps) and for four equally distributed azimuthal angles (from the aimpoint roughly along the diagonal of each CCD) with the Chandra Ray Tracer simulator555http://cxc.harvard.edu/chart (Carter et al., 2003) and the MARX software package666http://space.mit.edu/ASC/MARX (v5.0.0), as shown in Fig. 1 for the average over all azimuthal angles. Based on these simulations we define the circular exclusion area as presented in Table 1, where we split the point sources into different flux groups in order to make source removal more efficient. The shape of the PSF does not depend on the flux of a point source but the normalization of the PSF does. Hence, for each flux group the PSF normalization is defined by the upper-limit of its flux-interval (first column of Table 1). For the first two flux groups, which represent about of all point sources, the radius of the exclusion area is chosen in the way that both groups are removed at a same depth. We quantify this depth with the surface brightness of the PSF at the edge of the exclusion area, which corresponds for both groups to of the surface brightness of unresolved AGN (Sect. 3.1.3). For the other two flux groups we make a compromise in the depth in order to keep the radius of the exclusion area in a reasonable regime. In our simulations we further find that for radii of the PSF shape does not change very significantly as a function of azimuthal and offset angle (Fig. 1). Therefore, we use in the following a PSF averaged over all four azimuthal angles and all offset angles. With this we compute the enclosed count fraction (ECF) and residual counts of a point source for each flux-group, which are displayed as well in Table 1.

In order to test that our definition of the exclusion area of point sources in Table 1 removes sufficiently well the counts of resolved sources, we estimated how the average total counts per XBOOTES observation change as a function of radius of the exclusion area (Fig. 2). We create a list of evaluation steps, where we set the radius from to of its value in Table 1 in each flux group. For presentation purposes (Fig. 2), we compute an average radius per evaluation step of all flux groups, where we weight the radius of each flux group by the corresponding number of point sources. The measured number of total counts per observation is normalized for each evaluation step to the surface area of the observation before sources were removed. To ensure a clean test without any bias due to our choice of removing the extended sources (Sect. 2.2.2), we take here only those observations into account (83 out of 118 observations), which do not contain extended sources.

The result of this test is shown in Fig. 2. We can see for radii of that the total counts do not change significantly. The rise in total counts for indicates that there is still a significant contamination by counts from resolved point sources at these apertures. The average radius of our definition of the exclusion area in Table 1 is (weighted with number of point sources per flux group) and is shown as vertical dotted line in Fig. 2. This figure demonstrates that our definition of the exclusion regions for point sources is rather conservative. In average the FOV is reduce by after removing all resolved point sources with our definition.

To estimate contamination by the residual source counts, we note the following. With our definition of the point-source exclusion area, the ECF averaged over all resolved point sources equals to , i.e. about of the point source counts in the wings of the PSF remain in the image. Using observations without resolved extended sources (83 of 118) we compute the average number of counts in the exclusion regions,  counts per observation in the band. Therefore, there is about residual counts per image left from the resolved point sources. This should be compared with the total number of counts in the unresolved emission, per image, of which are from unresolved CXB and are due to the instrumental background. Thus, residual counts from resolved point sources constitute about of the total unresolved CXB counts, i.e. their contamination can be neglected.

2.2.2 Extended sources

There are 43 extended sources detected with a detection limit of () (K05, Sect. 3.2. & Table 1). The extended sources in the XBOOTES catalog were fitted with a Gaussian model in order to estimate their size. We define the radius of the circular exclusion area as six times this size. We tested circular exclusion areas between four and eight times the size and did not find any significant difference in the remaining source counts after we normalize to the same surface area. Therefore, we believe that this is a reasonable definition. We also note that the total source counts of the resolved extended sources only accounts to the total source counts of all resolved sources, based on the source catalogs of XBOOTES.

2.2.3 Summary

After removing all resolved sources the average FOV area is reduced by down to . The average surface brightness is reduced by from to in the band (after removing the instrumental background, see Sect. 2.3 and 2.4).

2.3 Removing background flares

In order to detect and remove time intervals of an observation, which are contaminated by background flares, we adopt the main concept of Hickox & Markevitch (2006, hereafter H06) and adjust them to the XBOOTES data. We analyze the light curve of each observation in the energy-band . H06 show that this band is the best choice for background flare detection, because of the different energy-spectra of background flares and the quiescent background (see their Fig. 3).

Our de-flaring method consists of three consecutive steps of filtering the light curve:

(a) We run a -clipping with the CIAO tool deflare, which is a standard procedure and removes the most obvious flares. Hereby, we use bins of  sec (10 frames), which is large enough to assume a Gaussian error distribution in each time bin but small enough to not conceal short, strong flares.

(b) We create a light curve with a binning of  sec (80 frames) and remove all bins, which are above the mean count rate of the -clipped light curve from step (a). This step targets weaker and longer lasting flares with a maximum duration of the order of the bin size. In comparison to H06, we only remove positive deviations from the mean.

(c) We compute a light curve in bins of  sec (80 frames) of the ratio between the and band and remove all bins, which are above the mean ratio of all considered XBOOTES observations. This method was introduced by H06 and is best suited for weak flares. It takes advantage of the fact that for a typical flare the flux-ratio of to band will be larger than for the normal instrumental background (alias quiescent background) due to the different energy-spectrum shapes. We use the same threshold for all observations to ensure a constant energy-spectrum shape for all of them.

The major difference between H06 and our filtering arises due to the fact that our observations have exposure times of the order of kiloseconds, whereas H06 use observations with more than one Megasecond. This leads in our case to much smaller bin sizes for the light curves and less restrictive thresholds for removing flare events for step (b) and (c). The light curves of all observations were visually inspected and the thresholds of (b) and (c) were tuned to removed any obvious feature of the light curve, which could be interpret as a background flare.

For a typical observation, our de-flaring method removes on average  sec () of the exposure time. After the de-flaring we have an average exposure time per observation of  ksec and a total exposure time is reduced to  Msec. We note that de-flaring does not significantly affect the power spectrum of the unresolved CXB, but it is necessary for accurate measurement of the absolute CXB flux (Sect. 3).

2.4 Instrumental background and background-subtracted map

We estimate the contribution of the instrumental background with the method presented in H06. They show in their study with the Chandra’s ACIS-I stowed background data777http://cxc.harvard.edu/contrib/maxim/acisbg/ that the shape of the energy spectrum of the instrumental background of ACIS-I from different observations is very stable over the course of five years, which includes the time when the XBOOTES observations were performed. Further, we know that all detected photons in the band are due to the instrumental background because the effective area of Chandra in this energy range is neglectable. With those two facts combined we can estimate the instrumental-background map for an observation with the total-count map in the energy band by scaling the ACIS-I stowed-background map as following:

(3)

With this method we estimate an average background surface brightness of in the band, which is consistent with the value from the spectral fit (see also Table 6). This means that of the total surface brightness of (after removing resolved sources, Sect. 2.2) is due to the instrumental background.

The background-subtracted map is then

(4)

We estimate from the background-subtracted map (after removing resolved sources) the average surface brightness of the the unresolved CXB equal to , which in physical units corresponds to , using our spectral model of the unresolved CXB from Sect. 3. This value is consistent with obtained from the spectral fit in Sect. 3.1.1.

We show in App. D.2 that in the band, surface brightness fluctuations of the instrumental background are much smaller than fluctuations of unresolved CXB. Therefore subtraction of the instrumental background is unnecessary for the calculation of the power spectrum of CXB fluctuations. Accordingly, it is not performed in Sect. 4 where total-count maps () are used for construction of the power spectra. However, accurate account for the instrumental background is necessary for computing the CXB flux and its spectral analysis.

Figure 3: The 0.5–10 keV stacked energy spectrum of 118 XBOOTES observations after removing resolved sources (black crosses, Sect. 2.2). Note that for our brightness fluctuation analysis in Sect. 4 we only use the band. The lower panel shows the ratio of model and data. Green solid curve: Total model (CXB and background model). Dark blue solid curve: Total CXB model (Table 2). Light blue dotted curve: absorbed powerlaw model (phabs(powerlaw)), representing the contribution of extragalactic emission. Orange dotted curve: APEC, representing the contribution of Galactic emission. Pink solid curve: Instrumental background model (App. A). Black dotted curves: components of the instrumental background model.

2.5 Flux and fluctuation maps

The flux map of the energy band is computed as the ratio between the count and exposure map (for each pixel):

(5)

and is only computed in instrumental units [cts s]. The average flux map is defined as:

(6)

These definitions minimizes statistical errors, while taking the vignetting of the exposure map properly into account. Note, that in order to compute the average count map , which also treats vignetting properly, one has to multiply with the exposure map:

(7)

For our analysis in Sect. 4 we need the fluctuation map in different energy bands for each observation. We compute this map for an energy band as following:

(8)

3 Composition of the unresolved CXB

3.1 Energy spectrum

The unresolved CXB888We note that the term “CXB” is used ambiguously in the literature and some may use it exclusively for extragalactic emission. consists of two components: the Galactic and extragalactic emission. We will use spectral analysis to separate their contributions to the CXB. We create the energy spectrum by stacking the energy spectra of all 118 considered XBOOTES observations, after removing resolved sources (Sect. 2.2) and background flares (Sect. 2.3). The stacked energy spectrum has a total exposure time of  Msec and is based on a total surface area of (without taking overlaps into account). We fit999 With the X-Ray spectral fitting package XSPEC (v12.8.2, Arnaud, 1996). it in the energy range of with a model for the unresolved CXB (Sect. 3.1.1, blue curve in Fig. 3) and an instrumental background model (App. A, pink curve in Fig. 3).

3.1.1 Spectral model

Model Component Parameter Value
APEC Temperature ()
Normalization
Surface Brightness
phabs(powerlaw) (fixed)
Photon Index ()
Normalization
Surface Brightness

(a) (); (b) at .

Table 2: Best-fit parameters of our spectral model (APEC + phabs(powerlaw), Sect. 3.1.1) of the unresolved CXB of XBOOTES (Fig. 3).

Our spectral model for the unresolved CXB consists of an absorbed powerlaw (phabs(powerlaw)) with a fixed absorption column of (Kalberla et al., 2005, K05) and of an unabsorbed APEC101010 A collisionally-ionized diffuse gas model, based on the atomic database ATOMDB (v2.0.2), http://www.atomdb.org. Other diffuse gas models, such as RAYMOND or MEKAL are also appropriate. We use the solar abundances of Anders & Grevesse (1989), since it was used in several previous CXB studies. model, the former representing the extragalactic sources and the latter representing the Galactic diffuse emission. The spectrum and model fit are shown in Fig. 3 and the best-fit parameters are listed in Table 2. The CXB model gives a surface brightness of () in the band, which is in good agreement with the value from the flux maps of (Sect. 2.4). The individual components of our CXB model are discussed in the following sections.

We note that there is a significant emission feature in the energy spectrum around (close to the right wing of the third instrumental line, Fig. 3). We believe that it arises from the instrumental background, since we can see a similar feature in the spectrum of the latter (Fig. 14). This is further supported by the fact that we do not detect any excess continuum associated with this feature. However, we can not entirely exclude the possibility that it may be of astrophysical origin As it is outside the energy range of our fluctuation analysis () we do not investigate it any further here.

3.1.2 Galactic emission

The APEC model of our spectral model encapsulates all the Galactic emission, which is a superposition of various diffuse sources (e.g. Lumb et al., 2002; Hickox & Markevitch, 2006; Henley & Shelton, 2013): the Galactic halo emission and the foreground emission, which is a composite of emission from solar wind charge exchange (SWCX) and the local bubble. All of them have in common that they are anisotropically distributed over the sky. In Fig. 3 we can see that the Galactic emission dominates the soft part of the energy spectrum but above it becomes negligibly small in comparison to the extragalactic component.

The surface brightness of our Galactic emission model is  erg cm s deg for the band (Table 2). This is in reasonable agreement with the measurements of H06 (Table 2, ), which use Chandra Deep Field surveys (hereafter CDFs), and Lumb et al. (2002, Table 3, ), which use several deep XMM-Newton observations.

3.1.3 Extragalactic emission

Figure 4: Comparison of recent surface brightness measurements of the CXB in the band. See also Table 5 and Table 6 (last column) of H06.

The absorbed powerlaw (phabs(powerlaw)) of our spectral model describes the extragalactic emission and its total surface brightness is for the band. Together with the emission of the resolved sources () computed from the removed source counts and converted to physical units with the same spectral model, we obtain for XBOOTES a total extragalactic CXB surface brightness of . The values are summarized in Table 3.

We can see in Fig. 3 that the extragalactic emission dominates the energy spectrum above . For the band it has a surface brightness of , while the Galactic emission has less than , based on our spectral model. Therefore, the contribution by any known Galactic sources can be neglected, which makes the band very suitable to compare our extragalactic CXB measurement with other studies. In Fig. 4 we show a summary of previous CXB measurement taken from H06 (Table 5 and Table 6, last column) together with our measurement. A comparison reveals that our measurement is the highest but still in good agreement with almost all of these studies. Except for the measurement of Gendreau et al. (1995) with ASCA, the differences remains below . We are consistent within one standard deviation with the CDF-North (CDF-N) () and within two standard deviation with the CDF-South (CDF-S) (). Note that since both CDFs are deep pencil-beam surveys with a sky area of  deg each, which is about times smaller than for XBOOTES, cosmic variance needs to be considered. H06 estimate that this adds an additional uncertainty of about to the measurement of the CDFs. Furthermore, the much bigger sky area used in our analysis (by a factor of ) is the main reason of our much smaller statistical uncertainty in comparison to the CDFs.

Unresolved
Resolved
Total

(a) From the spectral fit (Table 2); (b) From the removed source counts of the resolved sources in the background-subtracted maps (Sect. 2.2-2.4).

Table 3: Extragalactic emission of XBOOTES
[].

3.2 Flux budget

The unresolved extragalactic emission is a superposition of contributions of various types of sources of which most important are expected to be: AGN, normal galaxies (no indication of AGN activity), and clusters and groups of galaxies, which we refer to in the following as the major X-ray source populations (e.g. Lumb et al., 2002; De Luca & Molendi, 2004; Hickox & Markevitch, 2006, 2007; Kim et al., 2007; Georgakakis et al., 2008; Lehmer et al., 2012; Rosati, Borgani & Norman, 2002; Finoguenov et al., 2007, 2010, 2015). Here, we estimate the absolute and fractional contribution of each population to the unresolved extragalactic emission, which will be relevant for our fluctuation analysis (Sect. 4 and 5). We compute the surface brightness of the unresolved emission of an X-ray population as follows:

(9)

Hereby, we use differential from the literature and the normalized XBOOTES survey sensitivity curve for each source population.

Lehmer et al. (2012, hereafter L12) present the currently best measurement of the of AGN and normal galaxies, which is based on source samples down to a flux limit of . They demonstrated that above this flux limit AGN dominate the source counts, while below it normal galaxies are becoming the dominant source population (see Fig. 6 of L12). We find a very good agreement between the of L12 and of XBOOTES (K05) down to the flux-limit of XBOOTES (). Based on the of L12, we estimate a surface brightness of and for the unresolved AGN and normal galaxy population in XBOOTES, respectively. For this we used the Eq. (9), the point-source sensitivity distribution of XBOOTES (Fig. 12 of K05, see also Sect. 2.2), and a flux limit of L12 of . If we extrapolate the of L12 by two orders of magnitude down to , the surface brightness of AGN increases by () and of normal galaxies by a factor of ().

Figure 5: The predicted surface brightnesses of unresolved X-ray source populations in XBOOTES () as a function of the lower flux limit of Eq. (9) for point sources. The dotted part of a curve illustrates, where we extrapolate the used . Note that these parts of the curves should be interpreted with some caution as they represent extrapolations over several orders of magnitude. The orange bar shows the contribution by clusters and groups of galaxies. The lower limit of the bar (red line) illustrates the default estimate of Table 4, while for the upper limit the XLF slope of Ebeling et al. (1997) is increased by . Purple horizontal line: The unresolved extragalactic emission of XBOOTES (Table 3). The thickness of the line represents one standard deviation.

This calculation shows that the total flux of AGN is relatively independent of , while for normal galaxies it is very sensitive to , as it is further illustrated in Fig. 5. Such a behavior is caused by the steeper slope of the of normal galaxies in comparison to AGN ( for normal galaxies, for AGN) and makes it difficult to estimate accurately the contribution of normal galaxies to the unresolved CXB emission.

Contribution Absolute Fractional
Energy-Band [keV]
AGN
Normal Galaxies
Galaxy Clusters
Sum

(a) In units of . (b) In respect to the unresolved extragalactic emission of XBOOTES (Table 3). (c) Spectral models for energy band conversion are described in Sect. 5.4.3. (d) Lower flux limit: (). (e) Lower temperature limit: (, for ).

Table 4: Contributions of major X-ray source populations to the unresolved extragalactic emission of XBOOTES.

To estimate the contribution of clusters and groups of galaxies to the unresolved emission of XBOOTES, we are using the X-ray luminosity function (XLF) of Ebeling et al. (1997). It is based on the ROSAT Brightest Cluster sample (BCS, Ebeling et al., 1996), which includes 199 sources with a flux above , and covers redshifts up to and luminosities down to (). Despite the small redshift range of the XLF, it predicts consistent number densities of galaxy clusters down to the currently deepest studies with XMM-Newton having the sensitivity limit for extended sources of (e.g. Rosati, Borgani & Norman, 2002; Finoguenov et al., 2007, 2010, 2015). Further, the XLF does not incorporate any redshift evolution, which is consistent with more recent studies over a larger redshift range up to (e.g. Böhringer, Chon & Collins, 2014; Pacaud et al., 2016). To compute the from this XLF, we integrate the XLF over redshift and luminosity. For the K-correction in this integration, we are assuming an APEC model, where we couple its temperature to the luminosity with the luminosity-temperature scaling relation of Giles et al. (2015, Table 2, bold font) and assume a typical metallicity of of the solar value (Anders & Grevesse, 1989). As a lower temperature limit we use ( K). This limits our integration to luminosities above () and DMH masses above (), using the luminosity-mass relation of Anderson et al. (2015). It also leads to a decline of the differential at fluxes below .

We rescale the computed with XLF of Ebeling et al. (1997) by a factor of to match the observed of extended sources in XBOOTES (K05, Table 1). We note that the scaled is still consistent within one standard deviation with the measurements of Finoguenov et al. (2010, 2015). Based on the shape comparison of the of Ebeling et al. (1997) and the XBOOTES (K05, Table 1), we estimate that the sensitivity limit for extended sources for XBOOTES is around . With this we estimate a surface brightness of for the unresolved clusters and groups of galaxies in XBOOTES.

The computed surface brightness of unresolved clusters and groups of galaxies depends only mildly on the assumed flux limit of XBOOTES for extended sources. For instance, the surface brightness only changes by , if we change the flux limit by or (). Also the exact choice of the lower temperature limit in the XLF integration is not very critical. The surface brightness increases only by , if we decrease the limit to () and it decreases only by , if we increase the limit to (). However, the estimate of the surface brightness of clusters and groups of galaxies is rather sensitive to the assumed shape of their XLF. For example, increasing its slope by , from to , would increase the flux from unresolved clusters and groups of galaxies by . This makes the estimate of their contribution to the unresolved part of the CXB less certain than that of AGN (but still more accurate than the estimate for normal galaxies).

The total flux budget for the unresolved CXB is summarised in Table 4. In computing these numbers we assumed the low flux limit for the integration of AGN and normal galaxies to be equal to (note that as the contribution of clusters and groups of galaxies is computed via their XLF, no explicit low flux limit is needed in this case). With this AGN account for of the unresolved CXB in XBOOTES, normal galaxies for and clusters and groups of galaxies for . All together they account for of the unresolved CXB. About of the unresolved emission remains unaccounted for in this calculation. This is not worrisome however, because of the rather large uncertainty for the contribution of clusters and groups of galaxies and normal galaxies.

We illustrate the dependence of the surface brightness of unresolved AGN and normal galaxies on in Fig. 5. As one can see, the contribution by normal galaxies strongly increases towards small and at can easily explain the remaining part of the unresolved CXB. Although this conclusion is based on a very significant extrapolation of the observed and should be interpreted with caution, it is clear that normal galaxies are an important contributor to the unresolved CXB in XBOOTES. This inference is further supported by the results of Hickox & Markevitch (2007) who showed that faint optical/IR point sources can be associated with of the extragalactic emission below . Combining this result with the of resolved normal galaxies at higher fluxes, we estimate that normal galaxies account for at least of the unresolved extragalactic CXB in XBOOTES, which is close to the value derived above (Table 4).

3.3 Redshift and luminosity distributions of unresolved populations

Figure 6: Flux production rate in the band for different populations in XBOOTES survey versus redshift and luminosity: unresolved AGN (left) and clusters and groups of galaxies (right). For the latter we show the flux production rate for both unresolved (solid curves) and total (dotted curve) populations. The vertical lines show corresponding median values.

In order to characterize the unresolved populations, we calculate their flux production rate per solid angle [erg cm s deg] as a function of redshift and luminosity:

(10)
(11)

where [h Mpc] is the luminosity function of sources, is the normalized survey selection function of for the given type of objects (point-like or extended), is the K-correction, [erg s] is the rest-frame luminosity (), [Mpc h deg] is the co-moving volume element, and [cm] is the luminosity distance (e.g. Hogg, 1999).

3.3.1 Agn

For the AGN luminosity function [h Mpc] we used results of Hasinger, Miyaji & Schmidt (2005) for the band and for the we use is the normalized survey selection function for point sources (K05, Fig. 12). For the XLF we include the exponential redshift cutoff for : , which was proposed by Brusa et al. (2009). For computing the K-correction we assume a powerlaw with the photon-index of , which simplifies the quantity to . The survey selection function is given in the band and we convert it to the band with an absorbed powerlaw with a photon-index of . Note, that the XLF of Hasinger, Miyaji & Schmidt (2005) includes type 1 AGN only, and is defined for the minimum luminosity of and maximum redshift of . Therefore we have to correct the amplitude of the differential flux distributions to match the of L12 (factor of increase). Secondly, one should be aware that the derived flux production rates may be subject to large uncertainties at low luminosity () and large redshift () (see also the discussion of the uncertainties of different XLF of AGN in Sect. 5 of Kolodzig et al., 2013b).

In Fig. 6 (left panels) we show the differential flux distributions of the unresolved AGN for the band. The redshift distribution peaks around with the median value of . About two thirds of unresolved AGN are located between redshift and . The luminosity distribution peaks and has the median value of . About two thirds of unresolved AGN have the luminosity between and . Thus, that with the unresolved CXB of XBOOTES one can study the clustering signal of relatively low-luminosity AGN located around redshift . These objects are largely inaccessible for the conventional clustering studies using resolved AGN (e.g. Cappelluti, Allevato & Finoguenov, 2012; Krumpe, Miyaji & Coil, 2014).

3.3.2 Galaxy clusters & groups

As described in Sect. 3.2, for clusters and groups of galaxies we use the XLF of Ebeling et al. (1997), setting the lower integration limit corresponding to the gas temperature of (Sect. 3.2). The selection function is set to be a step-function with the step at the flux corresponding to the average survey sensitivity for extended source of (Sect. 3.2). We also compute the flux production rate distributions for the entire population of clusters and groups of galaxies, as it will be relevant for the discussion in Sect. 5. In this case we assumed for all fluxes. As with the AGN distributions, result of this calculation is subject to some uncertainty at low luminosities and large redshifts, where the XLF of the objects of interests is poorly constrained.

The obtained distributions are shown in Fig. 6. For the unresolved population, the redshift distribution has a median redshift of and peaks around . The flux-weighted mean redshift equals . About two third of the population are located between redshift and . Their median and peak luminosity is around and , respectively, and about two third have a luminosity between and . Thus, the unresolved population of clusters and groups of galaxies consists mainly of relatively low-luminosity and nearby objects, located around redshift . They are more local and distributed over a more narrow redshift (and luminosity) range than unresolved AGN population.

The total (resolved and unresolved) population of clusters and groups of galaxies is located on average closer than its unresolved part, with a median redshift of (peak at , flux-weighted mean of ) and two third are located between redshift and . The total population of clusters and groups of galaxies is in average more luminous than its unresolved part, with the median luminosity of (peak at ) and two thirds having the luminosity between and . Both results are expect since the resolved fraction of the total population consist of clusters and groups of galaxies, which contribute about to the total surface brightness of clusters and groups of galaxies and are rather close by and more luminous (Vajgel et al., 2014).

Based on the scaling relations at the median redshift and luminosity, we estimate that the unresolved and total population of clusters and groups of galaxies have in average an ICM temperature of ( K) and ( K), and a DMH mass of and , respectively.

4 Brightness fluctuations of the unresolved CXB

In the following we study the surface brightness fluctuations of the unresolved CXB by analyzing their power spectrum. In this study we focus on the angular scale range between and (angular frequencies of ). This range covers the spatial co-moving scales111111 , using the co-moving distance (e.g. Hogg, 1999, Eq. 16). between and for the redshifts of . Therefore, our measurement is sensitive to the small-scale () clustering regime, where the spatial correlation of galaxies and the ICM within the same DMH (alias the one-halo term, e.g. Cooray & Sheth, 2002) dominates the clustering signal.

For our analysis we combine the power spectra of all considered XBOOTES observations. Since we compute the power spectrum for each observation individually, our maximum angular scale is defined by side size of the ACIS-I FOV121212http://cxc.harvard.edu/proposer/POG/html/chap6.html, which is about . We ignore in our analysis angular scales smaller than () as, at these angular frequencies, the source power spectrum is suppressed by more than a factor of due the PSF-smearing (see Fig. 15). We demonstrate in App. BC and in Fig. 17 that our PSF-smearing models can adequately describe the measured power spectrum up to angular frequencies of (, image pixels).

4.1 Formalism

We study the surface brightness fluctuations of the unresolved CXB via Fourier analysis. The Fourier transform of a density field is defined in our study as:

(12)

We note that due to our choice of having a in the exponent, the angular scale is related to the angular frequency as . Due to our measurement process, the field is transformed from a continuous into a discrete one. This changes Eq. (12) to a 2D discrete Fourier transform:

(13)

where is the position of an image-pixel. The normalization of the Fourier transform was chosen so that the units of the resulting power spectrum (Eq. 14) are per deg. To compute the Fourier transform we use the FFTW library (v.3.3.3, Frigo & Johnson, 2005, http://www.fftw.org). With the assumption of isotropy, we reduce our 2D Fourier transform to an one-dimensional power spectrum as follows:

(14)

Hereby, the ensemble average is replaced with the average over all independent Fourier modes per angular frequency . There are Fourier modes within the interval of the 2D Fourier transform, where is defined by the angular size of the fluctuation map . This size is defined to be equal for both dimensions () of the map and to be large enough to embed the entire FOV of an observation (Sect. 2.1). One can analytically approximate the value of with but we directly count the number of modes for each annulus of the 2D Fourier transform. Since the fluctuations map is a real quantity, half of the 2D Fourier transform is redundant (, where indicates the complex conjugate. Therefore, we only have to average in Eq. (14) over independent Fourier modes. The number and range of independent Fourier frequencies is limited by the pixel size , which sets the maximum angular frequency (or minimum angular scale), known as the Nyquist-Frequency, to , and the angular size of the fluctuation map, which sets the minimum angular frequency (or maximum angular scale) to . In order to obtain the photon-shot-noise-subtracted power spectrum , we subtract from the power spectrum the photon shot-noise estimate , which is explained and discussed in detail in App. C, as following:

(15)

In the following, we will refer to the resulting power spectrum as the measured power spectrum and we will only show it in instrumental units of []. We do not convert it into physical units. Instead, we convert our clustering models into instrumental units.

Based on the assumption that our fluctuations are Gaussian distributed and superimposed by the photon shot noise, we can estimate the statistical uncertainty of as follows:

(16)

Here, one uses the fact that for a given angular frequency the power spectrum follows a distribution (e.g. van der Klis, 1989). For large angular frequencies (small angular scales) the number of modes per frequency-bin becomes large enough () that one can assume a Gaussian distribution for the power spectrum thanks to the central-limit theorem. Due to the fact that we use an averaged power spectrum over more than power spectra, we can assume a Gaussian distribution also for the smallest angular frequencies. This simplifies the error propagation.

In order to directly compare power spectra of different energy bands (as in Fig. 19) we use the flux-normalized power spectrum, which we define as:

(17)

This characterizes the squared fractional amplitude of the fluctuations per unit frequency interval. Since the flux map has units of [], the flux-normalized power spectrum has units of [deg].

4.2 The measured power spectrum

Figure 7: The measured power spectrum of the brightness fluctuations of the unresolved CXB in XBOOTES in the band. Blue curve shows the AGN shot noise (Sect. 4.3) corrected for the PSF-smearing (App. B). Grey areas illustrate its dependence on the photon index assumed for the flux conversion: the dark gray region shows the effect of the photon index varying from , light grey from . The larger photon-index values result in the larger amplitude of the AGN shot noise.

In Fig. 7 we plot the measured (i.e. the photon shot-noise subtracted) power spectrum in the band. In our current study we are interested primarily on the extragalactic part of the CXB. Our spectral analysis in Sect. 3 suggests that fluctuations below  keV can potentially be contaminated by the emission of the Galaxy since it contributes about to the unresolved CXB flux in the band (Table 2). However, as we demonstrate in Sect. 5.4.3, the energy spectrum of CXB fluctuations is much harder than that of Galactic emission. This allows us to place an upper limit of about on the contribution of fluctuations of the Galactic emission to the average power in the band at large angular scales (, Sect. 5.4.3). Furthermore, we compared our results with the those obtained in the band and which is virtually uncontaminated by the emission of the Galaxy and found fully consistent results. We will conduct our analysis in the band for consistency and ease of comparison with previous work.

4.3 Point-source shot noise

Due to their discreteness, point-like X-ray sources (AGN and normal galaxies), give rise to a shot-noise component in the power spectrum131313 For extended sources, the analog of the shot noise has a more complex shape of the power spectrum, carrying information about the spatial structure of their DMHs, and it is usually accounted through the one-halo term, as discussed in the following section. (e.g. Cooray & Sheth, 2002). The shot noise is caused by fluctuations of the number of sources per beam and is generally uncorrelated with the LSS signal itself, i.e. it is added linearly to the power spectrum. It is an analog of the photon shot noise (App. C) and for the definition of the power spectrum used in this paper, is independent of the angular frequency.

The shot noise of unresolved point sources can be computed as:

(18)

where is the differential distribution and is the normalized selection function for point sources (Fig. 12 of K05). Using the distributions from L12, we obtain for the shot noise of AGN a value of in the band. For this calculation we set the lower flux limit in Eq. (18) to , however the result is nearly insensitive to the value of . Although normal galaxies make a comparable contribution to the unresolved CXB flux (Sect. 3.2), due to their steeper in combination with the term in Eq. (18) their shot noise is about times lower than the AGN shot noise and in the following it will be neglected.

In instrumental units the AGN shot noise is , using an absorbed power law with a photon-index of (Sect. 3.1.3) for the conversion from physical to instrumental units. The shot noise level is moderately sensitive to the choice of the photon-index. Varying the latter between and (between and ) results in a variation of the shot noise level by ().

The predicted shot noise of unresolved AGN, corrected for the PSF-smearing (App. B), is shown in Fig. 7, along with the uncertainty due to variations of the photon index used for the units conversion. From Fig. 7 we can see that at small angular scales, below , the measured power spectrum agrees with the theoretical prediction for AGN shot noise quite well, within . However, at larger angular scales, there is a clear LSS signal detected above the shot noise of unresolved AGN. Its origin will be investigated in the next section.

5 The origin of the LSS signal

5.1 The (point-source shot-noise subtracted) power spectrum

Figure 8: The power spectrum of the detected LSS signal obtained by subtracting the point-source shot-noise from the measured power spectrum shown in Fig. 7. The green solid line shows the power law with the slope of , best-fit to the power spectrum in the range of angular scales. The blue curves show the two-halo term of unresolved AGN assuming (solid line) and (dashed line). Note that their average slope is . See Sect. 5.5.1 for details.

In order to characterize the LSS signal we subtract the point-source shot noise from the measured power spectrum in order to obtain the point-source shot-noise subtracted power spectrum. Before the subtraction we renormalize the theoretically computed AGN shot noise from Sect. 4.3 to match the observed power in the frequency range (). In this range, we can expect that the power spectrum is dominated by the point-source shot noise. The renormalization of the point-source shot noise is needed because the theoretical calculation in Sect. 4.3 is subject to a number of uncertainties, the main of which are: (i) the conversion from physical to instrumental units; (ii) the accuracy of the AGN ; (iii) conversion of the survey selection function () from band to the band; (iv) neglected contribution from normal galaxies. The shape of the theoretical AGN shot noise is determined by the PSF-smearing model, which has an accuracy of for (App. B). The so computed correction factor is i.e. it is sufficiently close to unity.

The resulting power spectrum of the LSS signal is shown in Fig. 8. A power law fit in the range of angular scales gives the value of the best fit slope of . Note that the power spectrum slope depends on the flux cut. This will be further discussed in Sect. 5.4.2.

5.2 Root-mean-square variation

In order to compute the root-mean-square (RMS) variation, we first compute for each observation the variance of the flux map in the spatial frequency range of interest:

(19)

As before, is the measured power spectrum (Eq. 15). Note that we used Eq. (14) in order to simplify the summation and that the leading coefficient in the above formula depends on the definition of the Fourier transform (Eq. 12).

With we can compute the fractional variance where , and its average over all considered observations. The square root of this value gives the average fractional RMS variation in the spatial frequency range . In order to estimate the uncertainties, we compute the standard deviation of the fractional variance for individual observations and then use error propagation.

For the entire range of the spatial frequencies , we obtain the fractional RMS variation of for the band. The AGN shot noise model corrected for the PSF-smearing predicts , which is fully consistent with the measured value. However, the RMS variation in the full frequency range is dominated by the power at small angular scales (around ), where the product of and the number of modes is the largest, i.e. it is determined by the AGN shot noise and does not characterize the power at large angular scales.

In order to characterize the latter, we compute the RMS variation for the frequency interval corresponding to the angular scales from , where the detected LSS signal dominates the power spectrum. Note that the same frequency interval will be used for our spectral analysis in Sect. 5.4.3. We obtain a RMS variation of while our AGN model predicts . Subtracting quadratically the latter from the former we obtain the RMS variation of for the LSS signal on the angular scales of arcminutes.

5.3 Potential sources of contamination

The angular scales where the detected LSS signal is particularly pronounced () are comparable to the size of the Chandra FOV. There are three main potential sources of contamination at these angular scales: (i) the instrumental background, (ii) large scale spatial non-uniformity of the detector efficiency, and (iii) residual counts in the wings of the PSF from the removed resolved sources.

To investigate significance of the first two factors, we use the instrumental background. In particular, we use both the stowed background data and the XBOOTES data in the band which is entirely dominated by the instrumental background signal. In App. D.2 we compute their power spectra and compare them with each other and with the power spectrum of the unresolved CXB in XBOOTES. We do the comparison both in the units of flux [(cts s) deg] and of the squared fractional RMS [deg] (Fig. 19). The former characterizes the significance of the additive contamination (the instrumental background), while the latter characterizes the role of the multiplicative factor (the non-uniformity of the detector efficiency). In both cases, the power spectrum of the instrumental background is by more than an order of magnitude smaller then the power spectrum of the unresolved CXB at any spatial frequency considered here. This excludes a possibility of any significant contamination of the power spectrum of unresolved CXB due to spatial non-uniformity of the instrumental background or detector efficiency.

Residual counts from the resolved sources on average amount to counts per image in the band (Sect. 2.2.1). This should be compared to the total of counts per image of unresolved emission ( from unresolved CXB and from the instrumental background). This is obviously too small to produce fractional RMS of on the arcminutes angular scales (Sect. 5.2). We also repeated the entire analysis with the point source exclusion radius of and did not find any significant changes in the power spectrum.

Furthermore, the energy spectrum of the LSS signal is much steeper than the energy spectra of the instrumental background and of the resolved sources (Sect. 5.4.3). In the case of the resolved sources the energy spectrum of the residual counts in the wing of the PSF is yet harder than the intrinsic spectrum of sources, due to the energy dependence of the PSF width141414http://cxc.harvard.edu/proposer/POG/html/chap4.html#fg:hrma_ee_pointsource. This adds further confidence in excluding the contamination by the instrumental effects.

Several other less significant systematic effects, such as the mask effect, are investigated in App. D.

5.4 Observational evidences

If the detected LSS signal is caused by one of the known X-ray source populations, it should be present in the resolved part of the CXB as well. Hence, one should expect that the LSS signal is enhanced if (some fraction of) resolved sources are not removed from the analyzed images.

In the course of our data preparation procedure we remove two types of resolved sources: point and extended sources. The point sources are associated predominantly with AGN and a some small contribution from normal galaxies. As the latter can not be always separated from the former, we investigate their effect together, noting that any LSS signal will be by far dominated by AGN. The extended resolved sources are clusters and groups of galaxies. Below we will investigate possible contribution of each source type to the large scale LSS signal.

Figure 9: Top: The measured power spectra of the brightness fluctuations in the band for different flux cuts for point source removal. Black crosses: default case used throughout the paper, with all resolved sources removed. Red crosses: the flux cut of for point sources was applied to the images, retaining of faintest resolved point sources. Bottom: Same as top panel but after subtracting the point-source shot noise (as described in Sect. 5.1). The frequency range used to normalize the point-source shot-noise model is indicated with the two vertical dashed lines in the top panel. The vertical bars are shifted slightly along the x-axis for visualization purposes.

5.4.1 AGN and normal galaxies

In order to investigate the possible role of point sources (AGN and normal galaxies) we construct images keeping of faintest resolved point sources in the images. This corresponds to the flux cut of for the band. With this flux cut, we on average retain 14 resolved point sources per observation, thus approximately doubling the average surface brightness. All other data preparation steps are same as for the default case (Sect. 2).

The power spectra for the two cases are shown in the top panel of Fig. 9. We can see that for the higher flux cut (red crosses) the point-source shot noise noticeably increased, as it should be expected (Eq. 18). In order to study the amplitude of the LSS signal, we subtract the point-source shot noise from the both spectra, as described in Sect.