The angular two-point correlation function of galaxies down to B_J ≲ 26 magnitudes on ≲10′ scales

We place constraints on the evolution of galaxy clustering from deep CCD frames in Gunn gri from two widely separated, intermediate Galactic latitude fields (b ≳ 30°), each roughly square in shape and ≃0°.5 in extent. More than 90% of the galaxies brighter than gri ≃ 24.5 mag are detected. For each field in each pass-band, a total of ≳ 23,000 objects are found. Neuschaefer, Windhorst, & Dressler (1991) (the first paper in the series) and Neuschaefer & Windhorst (1994) (the second paper in the series) discuss the details of the photometric and astrometric calibrations, object classification and galaxy counts. The current paper, the third in this series, presents the faint galaxy angular two-point correlation function (as a function of median redshift and apparent magnitude) and assesses the relative importance of effects due to sample completeness, galaxy surface brightness, evolution in galaxy spectra, evolution in clustering, as well as the overall matter density in the universe. Galaxies undergo a moderate bluing trend with increasing sample depth, consistent with previous work. We establish two important trends in the correlation function, w(θ) ∝ θ^δ (with δ ≃ -0.7), as function of increasing survey depth: (1) w(θ)'s amplitude declines moderately with survey depth down to gri ≃ 24.5 mag (where our galaxy catalogs are more than 90% complete); (2) w(θ)'s slope marginally flattens from δ ≃ -0.8 in all-sky surveys to δ -0.5 at g_med = 25 mag in our Four-Shooter data. The decline of the correlation amplitude with fainter magnitude is stronger than expected for an Ω₀ = 1 universe that contains only galaxies with nonevolving spectral energy distributions. Better agreement with the observations is obtained through a combination of the following explanations: (1) w(θ)'s slope δ(z) grows steeper with cosmic time roughly as γ(z) ∝ γ(0)(1 + z)^-0.2±0.2 with γ(0) = 1 - δ(0) = 1.75, confirming predictions of N-body CDM galaxy clustering simulations; (2) galaxy clustering has increased somewhat in amplitude since z ≲ 1, parameterized by a clustering exponent ε ≳ 0; (3) field galaxies have undergone mild spectral evolution out to redshifts z ≲ 1; and/or (4) low surface brightness galaxies are more weakly clustered than the general field galaxy population.