We propose an experiment, the Cosmic Accelerometer, designed to yield velocity precision of $\leq 1$ cm/s with measurement stability over years to decades. The first-phase Cosmic Accelerometer, which is at the scale of the Astro2020 Small programs, will be ideal for precision radial velocity measurements of terrestrial exoplanets in the Habitable Zone of Sun-like stars. At the same time, this experiment will serve as the technical pathfinder and facility core for a second-phase larger facility at the Medium scale, which can provide a significant detection of cosmological redshift drift on a 6-year timescale. This larger facility will naturally provide further detection/study of Earth twin planet systems as part of its external calibration process. This experiment is fundamentally enabled by a novel low-cost telescope technology called PolyOculus, which harnesses recent advances in commercial off the shelf equipment (telescopes, CCD cameras, and control computers) combined with a novel optical architecture to produce telescope collecting areas equivalent to standard telescopes with large mirror diameters. Combining a PolyOculus array with an actively-stabilized high-precision radial velocity spectrograph provides a unique facility with novel calibration features to achieve the performance requirements for the Cosmic Accelerometer.