Measuring the cosmic proper distance from fast radio bursts

The cosmic proper distance dP is a fundamental distance in the Universe. Unlike the luminosity and angular diameter distances, which correspond to the angular size, the proper distance is the length of light path from the source to observer. However, the proper distance has not been measured before. The recent redshift measurement of a repeat fast radio burst (FRB) can shed light on the proper distance. We show that the proper distance-redshift relation can indeed be derived from dispersion measures (DMs) of FRBs with measured redshifts. From Monte Carlo simulations, we find that about 500 FRBs with DM and redshift measurements can tightly constrain the proper distance-redshift relation. We also show that the curvature of our Universe can be constrained with a model-independent method using this derived proper distance-redshift relation and the observed angular diameter distances. Owing to the high event rate of FRBs, hundreds of FRBs can be discovered in the future by upcoming instruments. The proper distance will play an important role in investigating the accelerating expansion and the geometry of the Universe