The new X-ray Luminosity Function.

Using the new luminosities discussed above the X-ray luminosity function was computed and compared to the EMSS XLF computed using the data from Gioia & Luppino (1994). In the sample of Gioia & Luppino (1994) there are 24 clusters with $z>0.3$ and measured luminosities. In our sample taken from Gioia & Luppino (1994) there are only 21 clusters having these criteria. It is only these 21 clusters which are used in computing the luminosity functions. (The reasons for the differences between the samples are that MS1333.3+1725 has now been identified as a star (Luppino et al. 1999), MS1209.0+3917 as a BL Lac (Rector et al. 1998) and MS1610.4+6616 as a point source at the position of a star (Stocke et al. 1999). Note that MS0354.6-3650 which appears in Henry et al. (1992) does not appear in Gioia & Luppino (1994) and appears to be a soft X-ray source (Nichol et al. 1997).)

The XLF was computed using the $\frac{1}{V_{a}}$ method of Avni & Bahcall (1980) as used in the non-parametric analysis of Henry et al. (1992). This method is summarised here.

For each cluster the luminosity distance and the angular size of the core radius (as measured) were calculated according to the formulae below,

$$D_{L}=\frac{c}{q_{0}^{2}H_{0}}(q_{0}z + (q_{0}-1)(\sqrt{1+2q_{0}z}-1))$$ (19)

$$\theta_{0}=\frac{r_{\textrm{\scriptsize {c}}}(1+z)^2}{D_{L}}$$ (20)

The fraction of counts which fell inside the detect cell of $2.4' \times 2.4'$ was calculated according to,

$$f=\frac{2}{\pi} \arcsin \left\{ \frac{(\frac{\theta_{D}}{\th... ...{\theta_{0}})^2+1)} \right\} (g_{\textrm{\scriptsize {psf}}})$$ (21)

where $\theta_{D}$ is the angular half size of the detect cell and $g_{\textrm{\scriptsize {psf}}} = \frac{f_{\textrm{\scriptsize {psf}}}}{f_{\textrm{\scriptsize {King}}}}$ is a new factor which takes into account the effect of the IPC psf. As figure 3.6 shows, the effect of the IPC psf is constant for a particular cluster above $z = 0.3$, so the inclusion of $g_{\textrm{\scriptsize {psf}}}$ is unimportant here as it will cancel out in equation 3.12 below.

The maximum redshift at which the cluster could have been detected was calculated for each flux limited observation making up the survey. This was found by incrementing $z_{max}$ in the following formula until the statistic, $FIT$ gave the value closest to 1, where

$$FIT=\frac{F_{DET}}{F_{LIM}} \left( \frac{D_{L}(z)}{D_{L}(z_{max})} \right)^{2} \left( \frac{f(z_{max})}{f(z)} \right)$$ (22)

where $F_{DET}$ is the detect cell flux at the observed redshift $z_{obs}$, and $F_{LIM}$ is the limiting survey flux.

The total volume in which each cluster could have been found was then calculated by summing over all the flux limits

$$V_{a} = \sum_{i} \left[ \frac{dV(\Omega_{0},\leq min(z_{u},z_{max,i}))}{d\Omega} \right.$$

$$\left. - \frac{dV(\Omega_{0},\leq z_{l})}{d\Omega} \right] d\Omega_{surv,i}$$ (23)

where $z_{u}$ and $z_{l}$ are the upper and lower redshift limits of the sample and $d\Omega_{surv,i}$ is the solid angle associated with the $i$th flux limit. The mean ratio of the search volumes found here to those assuming no blurring from the IPC psf and a constant core radius of $r_{\textrm{\scriptsize {c}}}=250$kpc (as in Henry et al. 1992) was $0.99 \pm 0.03$.

The values for $\frac{dV}{d\Omega}$ were calculated by numerical integration of the following equation

$$\frac{dV}{d\Omega dz} = 4 \left( \frac{c}{H_{0}} \right)^{3}... ...{0} z})]^{2}} {\Omega_{0}^{4}(1+z)^{3} \sqrt{1+\Omega_{0} z}}$$ (24)

with respect to $z$.

The clusters were binned up into log luminosity bins that are 0.3 wide and then the differential luminosity function was calculated for each bin

$$n(L)=\sum_{j=1}^n \frac {1}{V_{a,j} \Delta L}$$ (25)

where $\Delta L$ is the width of the luminosity bin and $n$ is the number of clusters in that bin.

The XLF is recomputed using the remaining 21 clusters from Gioia & Luppino (1994) and compared to our XLF in two ways. Firstly using our new luminosities for those clusters observed by the PSPC and the luminosities of Gioia & Luppino (1994) for those clusters not observed by the PSPC. This is shown in figure 3.7. Secondly the luminosities of the clusters not observed by the PSPC are corrected by the average increase measured (when not including MS0418.3-3844 and MS0353.6-3642). This is shown in figure 3.8.

Figure 3.7: The revised X-ray Luminosity Function at $z = 0.3$ -- 0.6. Closed circles are from the data of Gioia & Luppino (1994) and open circles are from PSPC data. The dotted line is the local BCS XLF from Ebeling et al. (1997).

Figure 3.8: The revised X-ray Luminosity Function at $z$ = 0.3 -- 0.6. Symbols are the same as in figure 3.7 except that clusters for which there are no PSPC data have had their luminosities corrected by the average correction factor of 1.18.

Figure 3.9: The revised X-ray Luminosity Function at $z$ = 0.3 -- 0.6, compared to the local XLF of Böhringer et al. (2002). The data points are based on the revised luminosities, including an average correction factor for clusters with no PSPC data, but all luminosities have then been decreased by 8%, to be consistent with the approximation used by Böhringer et al. for integrating only to the virial radius.

Superimposed on both figure 3.7 and figure 3.8 is the best fitting Schechter function to the local XLF from Ebeling et al. (1997) which is obtained from the ROSAT Brightest Cluster Sample (BCS). All the luminosities we have used to calculate the XLFs in figures 3.7 and 3.8 are based on an integration of the emission to infinity, as is the BCS XLF. In figure 3.9 we compare the updated XLF with the local XLF of Böhringer et al. (2002), which updates the work of De Grandi et al. (1999) and which was published while this paper was being produced. The Böhringer et al. XLF was also converted from the 0.1-2.4 keV band to the 0.3-3.5 keV band. For cluster temperatures of $T_{\textrm{\scriptsize {X}}}=4$-$10$ keV, a conversion factor of $L_{0.3-3.5}=1.1 L_{0.1-2.4}$ is accurate to within $\approx$8%. Böhringer et al. use luminosities measured by integrating to the virial radius, approximated by 12 times the core radius. As noted by Böhringer et al. (2000), this approximation gives a systematic decrease of 8% compared to integrating to infinity. The same data as in figure 3.8 are plotted, except that all the luminosities were reduced by 8%, to be consistent with the virial radius approximation used by Böhringer et al..

It is clear that whereas previously there existed evidence for a degree of evolution from the local XLF the effect of the increase in luminosity is to bring the high redshift XLF back in line with it. This appears to be the case in figures 3.7, 3.8 and 3.9.