  We perform a Bayesian search in the latest Pulsar Timing Array (PTA) datasets
for a stochastic gravitational wave (GW) background sourced by curvature
perturbations at scales $10^5~\text{Mpc}^{-1}\lesssim k\lesssim
10^8~\text{Mpc}^{-1}$. These re-enter the Hubble horizon at temperatures around
and below the QCD crossover phase transition in the early Universe. We include
a stochastic background of astrophysical origin in our search and properly
account for constraints on the curvature power spectrum from the overproduction
of primordial black holes (PBHs). We find that the International PTA Data
Release 2 significantly favors the astrophysical model for its reported
common-spectrum process, over the curvature-induced background. On the other
hand, the two interpretations fit the NANOgrav 12.5 years dataset equally well.
We then set new upper limits on the amplitude of the curvature power spectrum
at small scales. These are independent from, and competitive with, indirect
astrophysical bounds from the abundance of PBH dark matter. Upcoming PTA data
releases will provide the strongest probe of the curvature power spectrum
around the QCD epoch.
