Abstract
The anomalous acceleration of Pioneer-10 and Pioneer-11 is known since 1992. These spacecrafts show the unexplainable acceleration near $10^{-7}\;cm/sec^2$ in the direction to the Sun. Later the unknown acceleration of the same order was found in the motion of Ulysses in its motion from the Jupiter to Mercury, and in the motion of Galileo, NEAR, Cassini, Rosetta, and Messenger at the flybys of these spacecrafts near the Earth. The possibility of unexplainable acceleration near $10^{-7}\;cm/sec^2$ was discussed also for stellar globular clusters and for galaxies. We propose the empirical formula for taking into account this acceleration and overview the predictions of this formula at cosmological scales. Several unknown observational effects are found. One of these effects is the anomalous redshifts in the clusters of galaxies. It was known previously only for small groups of galaxies. We show the existence of anomalous redshifts in the clusters of galaxies using the spectral observations of near one million galaxies from the SLOAN 5th data release.