8 Test 5 - completeness and confusion

In this section we investigate the confusion and completeness problems for XMM-Newton shallow and deep observations like the first XMM Deep X-ray Survey in the Lockman Hole (Hasinger et al. 2001).

A set of 10 images with exposure times of 10 ks and 100 ks in the energy bands [0.5-2] and [2-10] keV were generated; the fluxes were drawn using the latest $\log N-\log S$ relations from Hasinger et al. (2001) and Giacconi et al. (2000). Detection and analysis were performed with exactly the same parameters for all simulations: detection threshold, analysis threshold, background map size, detection likelihood, etc. (see Sect. 3). Cross-identification was achieved using the input sources above 10 counts and 30 counts for 10 ks and 100 ks exposures respectively. Lowering the count limits yields more cross-identifications but increases considerably the number of spurious detections.

The input image for 100 ks and [0.5-2] keV band is shown in Fig. 12. The inner $10\hbox {$^\prime $ }$ zone where all analysis is performed is indicated, as well as the total XMM-Newton field-of-view. It is informative to compare it with the images for 10 ks in Fig. 8.

Figure 12: Simulated 100 ks XMM-Newton deep field in [0.5-2] keV with the same parameters ( $N_{\rm H},\ \log N - \log S,\ \Gamma$, background) as in the Lockman Hole (Hasinger et al. 2001; Watson et al. 2001). We restrict the analysis to the inner $10\hbox {$^\prime $ }$

In order to estimate the effect of confusion we have generated images with only point-like sources, distributed on a grid such to avoid close neighbours, and with fixed fluxes spanning the interval [10^-16,10^-13] erg/s/cm².

In the following we discuss some important points.

• Confusion and completeness
  The detection rate of the input sources as a function of flux (Fig. 13) indicates that confusion problems are significant for 100 ks exposures in the [0.5-2] keV band. They are marginal for 100 ks exposures in [2-10] keV band and absent for 10 ks in both bands.
  

  Figure 13: Ratio of number of cross-identified objects to the number of input objects, with counts greater than 10 and 30 for 10 ks and 100 ks exposures respectively, as a function of the input flux. The results of 10 generations are shown by crosses, the heavy histogram is the corresponding average; the curve indicates the detection rate if confusion is absent and the dotted line marks 90% completeness

  
  Energy conversion factors and the limiting fluxes below which more than 10% of the input objects are lost are shown in Table 8;
  
   

   

  Table 8: The energy conversion factors (ECF) and the 90% completeness limits for detections. ECF is computed assuming power-low spectrum with photon index $\Gamma =2,\ N_{\rm H}=5$10¹⁹ cm^-2 (Hasinger et al. 2001) and the three EPIC instruments (pn+2MOS) with thin filters

  	[0.5-2] keV	[2-10] keV
  ECF (cts/s per erg/s/cm2)
  	6.70 10-13	3.66 10-12
  Flux limits (erg/s/cm2)
  10 ks	2 10-15	10-14
  100 ks	6 10-16	3 10-15

  
  
• Differential flux distribution
  The differential flux distributions for 100 ks in [0.5-2] keV obtained by MR/1+SE and EMLDETECT are shown in Fig. 14. Spurious detections appears to be numerous with EMLDETECT below 5 10^-16 erg/s/cm² and tend to compensate the loss of sensitivity and confusion. MR/1+SE appears to be less affected and allows us to set a conservative flux limit of 6 10^-16 erg/s/cm² ($\sim$90 photons for 100 ks), below which the incompleteness becomes important - 65% of the input sources are lost between 3 10^-16 and 6 10^-16 erg/s/cm²;
  

  Figure 14: Differential number counts as a function of flux. The continuous histogram is the input distribution, the dots with error bars (Poissonian) are the measured distribution (without any cross-identifications), the dashed histogram indicates the false detections and the continuous line is the corrected distribution excluding the false detections. Left for MR/1+SE, right for EMLDETECT - all averaged from 10 simulations

  
  
• $\log N-\log S$
  The $\log N-\log S$ functions in [0.5-2] keV for 10 ks and 100 ks exposures are shown in Fig. 15. The inferred $\log N-\log S$ (by simple source counting) are in very good agreement with the input ones down to fluxes about 2 10^-15 and 6 10^-16 erg/s/cm² with MR/1+SE and 10^-15 and 3 10^-16 erg/s/cm² for EMLDETECT for 10 ks and 100 ks respectively. Although there are confusion and completeness problems for 100 ks starting at 5-6 10^-16 erg/s/cm² as discussed above, their effect is completely masked in the EMLDETECT integral distribution, which seems to be in very good agreement down to 3 10^-16 erg/s/cm² (the lower flux limit for the Lockman Hole Deep Survey analysis of Hasinger et al. 2001);
  

  Figure 15: The integral number of objects in the inner $10\hbox {$^\prime $ }$ (surface 0.087 sq.deg.) as a function of the flux for [0.5-2] keV and two exposures: 10 ks and 100 ks. The points with error bars (Poissonian) are the detections (without cross-identification) with MR/1+SE boxes are EMLDETECT results, while the histogram is the input $\log N-\log S$ function

  
  
• Photometric accuracy
  The photometric reconstruction is relatively similar for 10 ks and 100 ks in for fluxes greater than 2 10^-15 and 6 10^-16 erg/s/cm² respectively (the 90% completeness limit, Fig. 16). The flux uncertainties are about 25-30% and going down to fainter fluxes leads to very poor photometry.
  

  Figure 16: Photometry reconstruction for all 10 simulated images at 10 ks (left) and 100 ks (right) in the [0.5-2] keV band. The solid line is exact match between detected and input counts while the dashed lines are for two-fold differences. The vertical dashed line marks the 90% completeness limit (see Table 8) and mean and st.dev. (in brackets) above this limit are denoted