In this Section we will describe a χ2 analysis of the redshift evolution of the near- infrared LFs. This test uses the Schechter parametrisation
Ψ(L) = Φ∗ L∗ L L∗ α exp −L L∗ (6.7) of the LF, where L∗is the characteristic luminosity, α the faint-end slope, andΦ∗ the normalisation of the LF (Schechter 1976). The corresponding equation in absolute magnitudes reads
Ψ(M) = 2 5Φ
∗ln 10 100.4(M∗−M)(1+α)exp
−100.4(M∗−M). (6.8) To estimate the rate of evolution of the parameters with redshift, we define evolu- tion parametersµ andν as follows:
Φ∗(z) = Φ∗(0) (1+µz),
M∗(z) = M∗(0) +νz, and (6.9)
α(z) = α(0) ≡ α.
Note that the faint end of the LF cannot be determined very well from our data, thus we leave the faint-end slopeα of the Schechter LF fixed, as we have also done during the fitting of a Schechter function to our data.
To quantify the redshift evolution ofΦ∗ and M∗we now compare our LF data in
all redshift bins with the local Schechter function evolved according to equation (6.9)
to the appropriate redshift. We do this for a grid of values forν andµ , and calculate the value ofχ2for each grid point according to
χ2(ν,µ) = 1 n
∑
i∑
j Φ(Mi,zj)−Ψ(Mi,ν,µ,zj)2 σ2 i j , (6.10)80 CHAPTER6. NEAR-INFRAREDGALAXYLUMINOSITYFUNCTION
whereΦ(Mi,zj) is the measurement of the LF at median redshift zj in the mag- nitude bin centred on Mi, Ψ(Mi,ν,µ,zj)is the local Schechter function evolved ac- cording to the evolution model defined in equation (6.9) to the redshift zj, σi j is the RMS error of the LF value in the appropriate redshift bin, and n is the number of free parameters of the approximation, i.e. the number of data points used minus the number of parameters derived from the fitting.
We want to compare our measurement of the K-band LF with the Schechter ap- proximations to the local determinations. We use the measurements by Loveday (2000) and Kochanek et al. (2001) for the K-band (since the LF parameters derived from local samples are very similar anyway), and the local J-band LF is the one by Cole et al. (2001). The Schechter parameters derived by those authors are shown in Tables 6.5 and 6.6. Choosing a shallower faint-end slope in the K band, similar to the one derived by Cole et al. (2001), changes the result slightly, but – within the errors – not significantly.
To avoid that data points with large completeness correction factors affect the re- sult, we exclude all LF measurements with a total correction factor (photometric in- completeness, spectroscopic incompleteness, and V/Vmaxcorrection) larger than three. The result of the likelihood analysis is shown in Figure 6.7. For the K-band, we compare our measurements at redshifts 0.2, 0.4, and 0.7 to the local measurements by Loveday (2000) and Kochanek et al. (2001), and to the average of their Schechter parameters. We detect a brightening of∆MK∗/∆z' −0.70 magnitudes, and a decline of the number density of objects to redshift one. The decrease ofΦ∗K with redshift is obviously quite strongly dependent on the parameters of the local LF, however, for the average value we derive∆Φ∗K/(Φ∗K∆z)' −0.35. These results also give quantitative estimates of the evolution which can already be seen in the Schechter parameters de- rived from our data, see Table 6.1 for details. Note that Huang et al. (2003) derive a significantly brighter M∗, a slightly larger normalisation, and a steeper faint-end slope, which they ascribe to redshift selection effects. If their measurements are valid, the brightening to redshift one would be smaller, whereas the evolution in number density would be even larger.
Within the errors, the results found in this work agree nicely with the measure- ments of the K-band LF derived from the full MUNICS sample based on photometric redshifts. First results are shown in MUNICS III and show the same trend for the evo- lution of the LF with redshift. A more detailed analysis is presented in MUNICS II, where the evolutionary trend with a brightening of 0.5 to 0.7 mag and a decrease in number density of roughly 25 per cent to redshift one is confirmed.
Very recently, Pozzetti et al. (2003) used about 500 spectroscopic redshifts to de- rive very similar values for the evolution of the K-band LF from the K 20 survey (Cimatti et al. 2002b).
In the case of the J-band (right-hand panel of Figure 6.7), the evolution of the LF is obviously not very well constrained. This is also apparent from the error contours of the Schechter parameters shown in Figure 6.3 (lower left-hand panel), where one can see that the local measurement by Cole et al. (2001) has a characteristic magnitude similar to the one derived in our lowest redshift bin, but a normalisation in between the ones derived from our two lower redshift intervals, thus making any conclusions about
6.8. IMPROVED LOCALLUMINOSITY FUNCTIONS FROM THE6DFGS 81 K −3 −2 −1 0 1 −2 −1 0 1 2 d M* K / d z [mag] d Φ * K / ( Φ * K d z) d M* K / d z [mag] d Φ * K / ( Φ * K d z) d M* K / d z [mag] d Φ * K / ( Φ * K d z) J −3 −2 −1 0 1 −2 −1 0 1 2 dM* J / dz [mag] d Φ * J / ( Φ * J d z)
Figure 6.7: Left-hand panel: Result of the estimation of the redshift evolution of the Schechter parameters MK∗ and Φ∗K between z=0.4 (MUNICS LF) and the local uni- verse from Loveday (2000; dotted line), Kochanek et al. (2001; dashed line), and for an average of these two local measurements (solid line) as derived from aχ2ap- proach (see text for details). The contours correspond to 1σ and 2σ confidence level. The dΦ∗/dz=0 and dM∗/dz=0 lines indicate the non-evolution values. Right-hand
panel: The same for the J-band LF. In this case, the local measurement is taken from
Cole et al. (2001, dotted line), with the appropriate Schechter parameters from Ta- ble 6.6, and the evolution is very badly constrained.
evolution with respect to the local sample difficult. However, we not that the evolution in brightness seems to be similar to the one seen in the K band. We will also show in Section 6.8 that measurements from the 6dFGS indicate that the normalisation of the
J-band LF measured by Cole et al. might be too low.
Furthermore, we note that our J-band LF data seem to confirm the trend seen for the K band, which is also evident from the Kolmogorov-Smirnov tests presented in Section 6.6.