Testing the wavelength dependence of cosmological redshift down to
  $\Delta z \sim 10^{-6}$
  At the core of the standard cosmological model lies the assumption that the
redshift of distant galaxies is independent of photon wavelength. This
invariance of cosmological redshift with wavelength is routinely found in all
galaxy spectra with a precision of $\Delta$z~10$^{-4}$. The combined use of
approximately half a million high-quality galaxy spectra from the Sloan Digital
Sky Survey (SDSS) allows us to explore this invariance down to a nominal
precision in redshift of one part per million (statistical). Our analysis is
performed over the redshift interval 0.02<z<0.25. We use the centroids of
spectral lines over the 3700-6800\AA\ rest-frame optical window. We do not find
any difference in redshift between the blue and red sides down to a precision
of 10$^{-6}$ at z<0.1 and 10$^{-5}$ at 0.1<z<0.25 (i.e. at least an order of
magnitude better than with single galaxy spectra). This is the first time the
wavelength-independence of the (1+z) redshift law is confirmed over a wide
spectral window at this precision level. This result holds independently of the
stellar population of the galaxies and their kinematical properties. This
result is also robust against wavelength calibration issues. The limited
spectral resolution (R~2000) of the SDSS data combined with the asymmetric
wavelength sampling of the spectral features in the observed restframe due to
the (1+z) stretching of the lines prevent our methodology to achieve a
precision higher than 10$^{-5}$, at z>0.1. Future attempts to constrain this
law will require high quality galaxy spectra at higher resolution (R>10,000).
