  We revisit the possibility that the stochastic common-spectrum process
recently detected by the NANOGrav pulsar timing array experiment could be due
to primordial gravitational waves (GWs). A na\"{i}ve extrapolation down to
interferometer scales of the blue GW spectrum required to explain NANOGrav
consistently with Cosmic Microwave Background (CMB) observations would strongly
violate upper limits on the stochastic GW background (SGWB) amplitude from
LIGO/Virgo. In combination with the fact that there are over 19 decades in
frequency between CMB and interferometer scales, this motivates us to move
beyond the commonly adopted approximation of a pure power-law GW spectrum. We
consider a broken power-law parametrization for the SGWB spectrum, which turns
from blue to red above the break frequency: while phenomenological, this choice
maps to various well-motivated early-Universe models, including scenarios
featuring non-instantaneous reheating or a non-standard background expansion
following reheating. After a detailed discussion of the contribution of the
resulting SGWB to the early-Universe radiation energy density, we constrain the
broken power-law model against a wide variety of multi-frequency cosmological
and GW observations. We find that this phenomenological model is able to
explain the NANOGrav signal while remaining in agreement with upper limits on
the tensor-to-scalar ratio on CMB scales, Big Bang Nucleosynthesis constraints
on the early-Universe radiation energy density, and upper limits on the SGWB
amplitude on interferometer scales. We briefly discuss the very bright
prospects for testing this model with next-generation probes across the GW
frequency landscape, which motivate further exploring connections to specific
well-motivated early-Universe models.
