1.1. History of the Cosmic X-ray background
The discovery of the cosmic X-ray background (CXB) happened at the same time as the detection of the very first extra-solar X-ray source . Both of them marked the beginning of a new era in high-energy astrophysics. Since then, the CXB background has been the object of a lively debate. First about its origin, later about its spectral shape. Indeed, a diffuse isotropic radiation such as the CXB might be produced either by hot gas permeating the Universe, or by millions of point-like X-ray sources or by both. Precise measurements done with the HEAO1-A2 experiment  revealed that the CXB spectral shape is consistent, between 3-50 keV with a bremmstrahlung model with a temperature of 40 keV. This was seen as a natural evidence for the presence of a very hot intergalactic medium. This conclusion was supported by, or was based on, the power-law like emission of Active Galactic Nuclei (AGNs, e.g. ), whose integrated emission (in the case millions of AGNs existed), remains always a power-law like spectrum. The final resolution of the controversy came from the incredibly neat result obtained with the FIRAS instrument on board COBE: the absence of any detectable deviation from a pure black spectrum body of the cosmic microwave background set an upper limit on the contribution of an uniform hot intergalactic gas to the CXB of < 10-4 . Once this issue was solved, it was clear that the CXB emission had to be the unresolved (at that time) emission of millions of AGNs . The deep X-ray surveys [6, 7, 8] have confirmed that indeed a large fraction (80 - 100%) of the CXB can be resolved into point-like sources which can then be identified as AGNs. The discovery that for many AGNs the nuclear radiation is partially obscured by intervening matter led population synthesis models to solve the paradigm of the generation of the CXB [9, 10, 11, 12] by means of AGNs with different amount of absorption. Although the Chandra and XMM-Newton have resolved most of the Cosmic X-ray background below ~ 2 keV, the fraction of resolved CXB emission declines with energy being < 50% above 6 keV . This represents the main evidence for the presence of a population of AGNs which is still currently undetected even in the deepest surveys. The analysis of the unresolved component revealed that it might be consistent with the integrated emission of a population of very absorbed, Compton-thick (τ = NH T ~ 1 and thus NH 1.5 × 1024 atoms cm-2) AGNs. Given the fact that their emission is suppressed below 10 keV, detecting these object is extremely difficult at soft X-rays and until a few years ago only a handful of Compton-thick AGNs were known .
1.2. Constraints from the Cosmic X-ray background
Since the direct detection of Compton-thick objects is difficult in X-rays, the CXB spectrum becomes the final resource to constrain the space density of such objects. Indeed, the CXB represents the integrated emission of the accretion processes onto super-massive black holes (SMBHs) present in the Universe. Integrating the luminosity function of unabsorbed and absorbed AGNs [15, 10], with sensible hypotheses regarding their spectral properties (and their dispersions), allows an estimate of the contribution of these two classes to the total CXB spectrum. The most recent studies [10, 11, 12] show that the contribution of these two classes is not enough to explain the totality of the CXB highlighting a deficit around the CXB peak at 30 keV. Assuming that this deficit emission is due to undetected Compton-thick AGNs, it becomes possible to make an estimate of their space density. With no other constrains left, except the small number of heavily obscured objects known in the local Universe, the absolute normalization of the CXB spectrum (particularly at its peak) represents the main resource to constrain the Compton-thick population. This is why it has been much debated lately. In the 2-10 keV band the CXB measurements of focusing telescopes (as XMM-Newton, Chandra, etc.) lie systematically above the one obtained by non-focusing optics . Moreover, it seems that neither cosmic variance  nor differences in the flux scale calibration of each individual instrument [16, 18] may account for this discrepancy. This led several authors to naturally question the broadband measurement of the CXB performed by HEAO1  and to use CXB spectra renormalized by a factor ~ 1.3 [10, 11, 13]. A change in the normalization of the CXB implies either a change in the number density of AGNs or in their radiative efficiency or in both. This is why the normalization of the CXB has been so much debated in the past. In particular the lack of measurements at the peak of the CXB (~ 30 keV) left the population of Compton-thick AGNs loosely constrained. This situation recently improved with the results shown in the next sections.