  The European Pulsar Timing Array (EPTA) and Indian Pulsar Timing Array
(InPTA) collaborations have measured a low-frequency common signal in the
combination of their second and first data releases respectively, with the
correlation properties of a gravitational wave background (GWB). Such signal
may have its origin in a number of physical processes including a cosmic
population of inspiralling supermassive black hole binaries (SMBHBs);
inflation, phase transitions, cosmic strings and tensor mode generation by
non-linear evolution of scalar perturbations in the early Universe;
oscillations of the Galactic potential in the presence of ultra-light dark
matter (ULDM). At the current stage of emerging evidence, it is impossible to
discriminate among the different origins. Therefore, in this paper, we consider
each process separately, and investigate the implications of the signal under
the hypothesis that it is generated by that specific process. We find that the
signal is consistent with a cosmic population of inspiralling SMBHBs, and its
relatively high amplitude can be used to place constraints on binary merger
timescales and the SMBH-host galaxy scaling relations. If this origin is
confirmed, this is the first direct evidence that SMBHBs merge in nature,
adding an important observational piece to the puzzle of structure formation
and galaxy evolution. As for early Universe processes, the measurement would
place tight constraints on the cosmic string tension and on the level of
turbulence developed by first-order phase transitions. Other processes would
require non-standard scenarios, such as a blue-tilted inflationary spectrum or
an excess in the primordial spectrum of scalar perturbations at large
wavenumbers. Finally, a ULDM origin of the detected signal is disfavoured,
which leads to direct constraints on the abundance of ULDM in our Galaxy.
