6.2. The clustering of Lyman-break galaxies
The high efficiency of the Lyman-break technique and the relatively narrow range of redshifts/cosmic time that it probes make angular clustering a particularly economic means to study large-scale structure at high redshifts, once the redshift distribution N(z) of the galaxy candidates has been measured. Not only is one free from securing a complete spectroscopic follow-up of all the candidates, but the systematics due to selection effects are easier to handle than those that affect studies of spatial clustering using the full redshift information. Knowing N(z), the real-space correlation function can be accurately derived from the angular one through the Limber transform (Peebles 1980; Efstathiou et al. 1991). It turns out that the inversion of the w() is rather insensitive to the still relatively large uncertainties on the angular function, and the spatial correlation length is much more tightly constrained than this (Giavalisco et al. 1998).
The angular correlation function w() is defined in terms of the excess probability over the random (poisson) distribution of finding a companion in an angular shell of size d placed at an angular separation from a selected galaxy, given the surface density of sources N (Peebles 1980):
We used the 5 largest and deepest fields of our survey to produce a weighted average angular correlation function w() of LBGs. Figure 10a shows the data points, their error bars and the fitted power-law model.
We subsequently inverted the angular correlation function using the Limber transform and the observed redshift distribution N(z) to derive the parameters of the spatial correlation function at the epoch of observations, namely the correlation length r0 and the slope = 1 + , under the assumption of the power-law model (r) = (r/r0)-.
To estimate confidence intervals on the parameters and r0, we used montecarlo simulations. Figure 10b shows the distribution of values of r0 and obtained from the simulations. As mentioned above, the correlation length turns out to be much more tightly constrained than the individual parameters of the angular correlation function, with a typical fractional error of 30% at the 1- level. As our fiducial measure and error bar we adopt the median of the distributions of the simulations and its corresponding 68% confidence intervals. These are r0 = 2.1+0.4-0.5 and r0 = 3.3+0.7-0.6 h-1 Mpc (comoving coordinates) for q0 = 0.5 and q0 = 0.1, respectively, for the correlation length, and = 1.98+0.32-0.28 for the slope.
Figure 10. a) Weighted average angular correlation function of LBGs. The filled points are from the PB estimator, the open points from the LS one. The error bars are shown on the top of the figure. The continuous line is the best-fit power law to the PB data points, the dotted line is the fit to the LS. The thick horizontal continuous segment on the x axis marks the angular range over which the we computed the fits. b) The histogram of the correlation length r0 (lower panel) and of the slope (upper panel) from the montecarlo simulations.
We can estimate the bias of these galaxies by comparing their correlation function g to the correlation function of the mass m:
Although (as eq. 3 shows) the bias is in principle a function of scale, our constraint on the power-law exponent is relatively weak and we can only estimate a "typical" value of the bias over the scales of a few Mpc which are probed here. In practice, we use the ratio of the correlation length of the LBGs to that of m(r) predicted by the CDM theory to compute the bias, which is therefore relative to r = 1 h-1 Mpc. Using a CDM power-spectrum with shape parameter * = 0.25, claimed to fit the shape of the local large-scale structure very well (Peacock 1997), and normalization of Eke, Cole, & Frenk (1996), we estimate b ~ 4.5 (1.5) for q0 = 0.5 (0.1). Choosing * = 0.20 results in b ~ 5 (1.5), while adopting the normalization of White, Efstathiou & Frenk (1993) results in b ~ 4 (1).
Very interestingly, the correlation function of the Lyman-break galaxies has a slope that is comparable to or steeper than that measured at intermediate and low redshifts. The evolution of the slope of the correlation function of the mass (or, equivalently, that of the power spectrum at small scales) has a pronounced dependence on . For a CDM-like power spectrum, it depends very weakly on the shape parameter * and, for flat models, on the normalization. As eqn. (3) shows, the slope of g(r) differs from that of m(r) because of the dependence of the bias parameter b(r) with the spatial scale. The form of b(r), its dependence on galaxy properties and how it evolves with redshift are still subjects of discussion (e.g., Mann, Peacock & Heavens 1998; Bagla 1997). If the scale dependence of b(r) for the LBGs over the spatial scales probed by our correlation analysis, namely 1 r 10 h-1 Mpc, is similar to that of the local galaxies, then our measures of are inconsistent with m(r) from the CDM theory if = 1. With our choice of * = 0.25 we found m = 1.25 (over the range 1 < r < 10 h-1 Mpc), independently of the normalization. As mentioned above, the dependence on * is very weak. If * = 0.1 then m = 0.98, while if * = 0.6, then m = 1.14. Thus, the observed slope rules out the steepest CDM slope (m = 1.25) is ruled at the 99.95% confidence level. Open CDM models with the same parameters as above produce slopes in the range 1.6 < m < 2.1 (in open models the slope of m(r) has a more pronounced dependence on the normalization), which are all consistent with our data.
The above computations assume a bias constant with spatial scale. We stress, however, that the evolution of the slope of the correlation function is useful for constraining cosmological models only if the dependence of the bias with the spatial scale and its evolution with redshift are known. The function b(r) also depends on the properties of the halos, which further complicates the interpretation of the data because of the difficulty of establishing an evolutionary sequence between the systems observed at high redshifts and the local galaxies. Bagla's (1997) N-body simulations seem to suggest that the bias will not be strongly scale-dependent - his b(r) for M > 2 x 1012 M halos at z = 0 in standard CDM has a power-law slope of only ~ -0.18 - and if b(r) for Lyman-break galaxies is similarly flat, our conclusions about the slope would not be importantly changed. But until more is known about the scale-dependence of the bias they will remain speculative.