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CHAPTER 4: CLUSTERING OF AXIS AND XMS SOURCES Combining these equations, Limber’s equation becomes: w(θ) = 2 H 0 c ∫ ∞ 0 ( 1 N ) dN 2 ∫ ∞ E(z)dz ξ(r, z)du. (4.3.15) dz 0 Considering that the physical separation between two sources that project an angle θ in the sky can be written as: r ≃ 1 1 + z (u2 + x 2 θ 2 ) 1/2 (4.3.16) under the small angle approximation assumption, if we now combine equations 4.3.10 and 4.3.15 we obtain: where θ γ−1 0 = H γ ( r γ 0 H 0 c ) ∫ ∞ ( 1 N 0 ) dN 2 E(z)(1 + z) −3−ɛ+γ dz Dc γ−1 (z) dz (4.3.17) with Γ being the gamma function. H γ = Γ( 2 1 γ−1 )Γ( 2 ) Γ( γ 2 ) (4.3.18) We can now invert this equation and obtain r 0 given an angular amplitude θ 0 , but we also need to determine the source redshift distribution dN/dz. Since we have no redshift information of the AXIS sources we can approximately measure dN/dz using an estimate of the AGN luminosity function. We can do that via the equation 4.3.14 writing the selection function (degradation of sampling as a function of the distance in flux-limited surveys) as: φ(D c ) = ∫ ∞ L min (z) Φ(L x , z)dL x (4.3.19) which depends on the evolution of the source luminosity function Φ(L x , z) but is independent of the cosmological model. We have determined the selection function (see Figure 4.8) of the AXIS sources using a luminosity function following a Luminosity-dependent Density evolution model (see Chapter 5). The best-fit model used for both the Soft and Hard bands is that obtained for the luminosity function of the XMS sources along with other deeper and shallower surveys (see Chapter 5 for details). The computed redshift distribution plotted in Figure 4.8 has been calculated using the 102
CHAPTER 4: CLUSTERING OF AXIS AND XMS SOURCES Figure 4.8 The redshift selection function for the AXIS soft (solid line) and hard (dashed line) sources. 103
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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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xxxiv
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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 tant contri
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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 (NLR) and i
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CHAPTER 1: INTRODUCTION optically c
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CHAPTER 1: INTRODUCTION ber of AGN
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CHAPTER 1: INTRODUCTION Figure 1.5
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CHAPTER 1: INTRODUCTION 1.7.3 The X
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CHAPTER 1: INTRODUCTION AGN are ess
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CHAPTER 1: INTRODUCTION subjects ma
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CHAPTER 2: AGN SURVEYS WITH XMM-New
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R R OQP ONM 40 Table 2.1: Continued
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Page 105 and 106: 61 Table 3.1 Maximum likelihood fit
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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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APPENDIX A: SENSITIVITY MAPS 160
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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 [148] Matarrese, S., Col
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REFERENCES [190] Richstone, D., Ahj
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REFERENCES [236] Wisotzky, L., 1998
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