Gravitational waves (GWs) in the nano-hertz band are great tools for understanding the cosmological evolution of supermassive black holes (SMBHs) in galactic nuclei. We consider SMBH binaries in high-$z$ ultra-luminous infrared galaxies (ULIRGs) as sources of a stochastic GW background (GWB). ULIRGs are likely associated with gas-rich galaxy mergers containing SMBHs that possibly occur at most once in the life of galaxies, unlike multiple dry mergers at low redshift. Adopting a well-established sample of ULIRGs, we study the properties of the GWB due to coalescing binary SMBHs in these galaxies. Since the ULIRG population peaks at $z>1.5$, the amplitude of the GWB is not affected even if BH mergers are delayed by as long as $sim $ 10 Gyrs. Despite the rarity of the high-$z$ ULIRGs, we find a tension with the upper limits from Pulsar Timing Array (PTA) experiments. This result suggests that if a fraction $f_{rm m,gal}$ of ULIRGs are associated with SMBH binaries, then no more than $20 f_{rm m,gal}(lambda_{rm Edd}/0.3)^{5/3}(t_{rm life}/30~{rm Myr})~%$ of the binary SMBHs in ULIRGs can merge within a Hubble time, for plausible values of the Eddington ratio of ULIRGs ($lambda_{rm Edd}$) and their lifetime ($t_{rm life}$).
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