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CHAPTER 4: CLUSTERING OF AXIS AND XMS SOURCES tic X-ray sources cluster forming physical structures of cosmological origin although it is more expensive in terms of resources than the angular correlation function because it needs highly complete surveys (which require hundreds of secure redshift identications obtained via ground-based espectroscopy). Many works have reported spatial clustering using this method with different significancies in the strength if the signal ([242], [84], [161], [45]) at all energy bands. Since the XMS sample has identification completenesses over 95% in the Soft band and about ∼85% in the Hard band, it would allow us to test these results using its sources. This chapter is structured as follows: in sections 4.2 and 4.3 we will study the cosmic variance and the angular clustering of the AXIS sources, while in section 4.4, we will investigate the spatial clustering of XMS sources. Throughout this chapter we will assume H 0 = 70 km s −1 Mpc −1 and a flat Universe framework with Ω M = 0.3 and Ω Λ = 0.7 ([213]). 4.2 Cosmic variance The simplest test for source clustering is to compare the actual number of sources detected in each field N k , with the number of expected sources λ k from the best fit log N − log S (see Chapter 3) and the sky area of each individual field Ω k . This procedure is similar to the traditional counts-in-cells method. In principle, if one field (or a few fields) were looking through strong cosmic structures the number of sources detected should be significantly different from the expected number of sources for a random, uniform distribution as measured by the overall log N − log S. The statistical tools we have used to measure the deviation from such a random uniform distribution are the cumulative Poisson distributions: P λk (≥ N k ) = ∞ ∑ l=N k P λk (l); N k > λ k (4.2.1) P λk (≤ N k ) = N k ∑ P λk (l); N k < λ k (4.2.2) l=0 where P λ (l) is the Poisson probability of detecting l sources when the expected number of sources is λ. The cumulative distributions above give the probability 84
CHAPTER 4: CLUSTERING OF AXIS AND XMS SOURCES of finding ≥ N k (eq. 4.2.1) or ≤ N k (eq. 4.2.2) sources when the expected number from the source counts is λ k . This method is similar to the one used in [37], which uses P λ (l) instead of the cumulative probabilities, being our approach a more conservative estimate to determine the likelihood of finding a number of sources in a field which is different from the expected value. Then, the maximum likelihood for the whole sample is: L ′ = −2 ∑ ln P λk (≥ N k ) − 2 ∑ ln P (≤ N λk ′ k ′) (4.2.3) k k ′ where k runs over the fields for which N k > λ k and k ′ runs over the fields for which N ′ k < λ′ k . We have compared the observed L’ values in each band with 10000 simulated values using the values of λ k and Poisson statistics. The number of simulations with likelihood values above the observed ones were 1388, 4580, 778 and 4958 for the Soft, Hard, XID and ultrahard bands, respectively. As it can be seen, we found nothing significant in both the hard and ultrahard bands. However, in the Soft and XID bands we have found some deviations below or at about the 90% significance level. These results are not conclusive at all but indicate that, at least in the Soft and XID bands, a more detailed study of clustering may be worthwhile. 4.3 Angular correlation function of AXIS sources If cosmic structure is present in all (or most) fields, a test that measures significant deviations from the mean number of sources in each field from some overall average will not provide any significant results, as seen in section 4.2. Instead, we should look for evidence of sources tending to appear together in the sky with respect to an unclustered source distribution. 4.3.1 Method According to [174], the angular two-point correlation function of objects distribution w(θ) determines the joint probability of finding two objects in two small angular regions δΩ 1 and δΩ 2 separated at an angular distance θ with respect to 85
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Universidad de Cantabria Departamen
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A mis padres
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explicar las altas luminosidades de
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Además, una importante fracción (
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ealizar estudios de espectroscopía
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distribución diferencial que, al e
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materia sobre el agujero negro cent
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analizadas proviene de fuentes en t
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Haciendo uso del segundo catálogo
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With magic, you can turn a frog int
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Un capítulo aparte merecen mis com
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Summary Since its discovery more th
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CONTENTS 1.6.1 The K-correction . .
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CONTENTS 5.3.2 Analytical model . .
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LIST OF FIGURES 4.6 Soft 1 − P µ
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Chapter 1 Introduction In the last
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CHAPTER 1: INTRODUCTION have confir
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CHAPTER 1: INTRODUCTION by this mis
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CHAPTER 1: INTRODUCTION Figure 1.2
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CHAPTER 1: INTRODUCTION [161], [242
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CHAPTER 1: INTRODUCTION to-noise ra
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CHAPTER 1: INTRODUCTION Compton up-
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CHAPTER 1: INTRODUCTION ([142]). It
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CHAPTER 1: INTRODUCTION AGN are ess
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CHAPTER 6: SUMMARY OF THE RESULTS F
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Appendix A Sensitivity maps The val
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APPENDIX A: SENSITIVITY MAPS clearl
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REFERENCES [16] Barger, A. J., Cowi
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REFERENCES [61] Ehle, M., Breitfell
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REFERENCES [107] Hawkins, M. R. S.
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REFERENCES [190] Richstone, D., Ahj
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REFERENCES [236] Wisotzky, L., 1998
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