2 Number Counts

Galaxy differential number counts, giving the number of galaxies per unit sky area per unit magnitude interval as function of total magnitude, have always been a classical tool of observational cosmology. Consequently, a great effort has always been devoted to the extension of the observations to deeper magnitudes, larger sky regions and a wider range of colors. In particular, in the past few years $I$-band counts at high Galactic latitudes have been reliably extended down to $I \simeq 24$, as summarized e.g. by [Shimasaku and Fukugita(1998)]. In our model, counts from three different sources were combined in order to cover as large a magnitude range as possible. At bright magnitudes, i.e. for $I \leq 19$, well-established counts were provided by [Lattanzi(1997)], whereas at fainter magnitudes results from [Glazebrook et al.(1995)] ( $19 < I < 21$) and [Abraham et al.(1996a)] ( $21 < I < 24$) were used. A least-square polynomial fit in $\log N$ vs. $I$ was performed on these data, in order to assess the consistency of the three sources and to obtain a functional form $N=N(I)$ for use in the following. It was thus found that a second degree polynomial was sufficient to obtain a good fit to the data. The number counts and the best-fit parabola are shown in Figure 2, while the best-fit parameters are given in Table 2. According to this approximation, the differential number counts take the following functional form

Figure 2: Galaxy differential number counts in the $I$-band. Data points from [Lattanzi(1997)] (diamonds), [Glazebrook et al.(1995)] (triangles) and [Abraham et al.(1996a)] (squares). The solid line shows the least-square second degree polynomial best-fit.

$$N(I)= \,(a_N+b_N\,I+c_N\,I^2)~~~\mathrm{[number~deg^{-2}~mag^{-1}]}~,$$ (1)

where ``dex'' stands for the exponential function in base ten.

Values of the three parameters contained in Equation 1 are given in Table 2, while counts calculated with this formula are given in Table 4.

Table: Parameters of $\log N$ vs. $I$ least-square second-degree polynomial best-fit. $N$ expressed in $\mathrm{number~deg^{-2}~mag^{-1}}$.

$a_N$	$b_N$	$c_N$
-9.9942	0.90564	-0.011493

The cumulative galaxy number counts, giving the total number of galaxies per unit sky area brighter than a given $I$ magnitude $I_c$, are then given by the definite integral

$$N_c(I) = \int_{-\infty}^{I} N \,\mathrm{d}I'~~~\mathrm{[number~deg^{-2}]}~.$$ (2)

Since the function given by Equation 1 does not have an analytic antiderivative, Romberg numerical integration (see Chapter 4 in [Press et al.(1992)]), was performed. Cumulative galaxy number counts that were thus obtained are listed in Table 4.