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Gravity theories that modify General Relativity in the slow-motion regime canintroduce nonperturbative corrections to the stochastic gravitational-wavebackground~(SGWB) from supermassive black-hole binaries in the nano-Hertz band,while remaining perturbative in the highly-relativistic regime and satisfyingcurrent post-Newtonian~(PN) constraints. We present a model-agnostic formalismto map such theories into a modified tilt for the SGWB spectrum, showing thatnegative PN corrections (in particular -2PN) can alleviate the tension in therecent pulsar-timing-array data if the detected SGWB is interpreted as arisingfrom supermassive binaries. Despite being preliminary, current data havealready strong constraining power, for example they set a novel (conservative)upper bound on theories with time-varying Newton's constant at least at thelevel of $\dot{G}/G \lesssim 10^{-5} \text{yr}^{-1}$ for redshift$z=[0.1\div1]$. We also show that NANOGrav data are best fitted by a brokenpower-law interpolating between a dominant -2PN or -3PN modification at lowfrequency, and the standard general-relativity scaling at high frequency.Nonetheless, a modified gravity explanation should be confronted with binaryeccentricity, environmental effects, nonastrophysical origins of the signal,and scrutinized against statistical uncertainties. These novel tests of gravitywill soon become more stringent when combining all pulsar-timing-arrayfacilities and when collecting more data.

Novel tests of gravity using nano-Hertz stochastic gravitational-wave background signals