Updated constraint for the gravitational wave background from the gamma-ray Pulsar Timing Array

Fermi LAT observations of gamma-ray pulsars can be used to build a pulsar timing array (PTA) experiment to search for gravitational wave (GW) signals at nanohertz frequencies. At those frequencies, the dominant signal is expected to be a stochastic gravitational wave background (GWB) produced by the incoherent superposition of the quasi-monochromatic GW emissions from a population of supermassive black hole binaries. While the radio PTAs have recently announced compelling evidence for a GWB signal with a power law spectrum of strain amplitude $\approx2-3\times10^{-15}$ (at the frequency of $1 {\rm yr}^{-1}$), in 2022 an analysis of $12.5$ years of Fermi data for 35 pulsars led to an upper limit of $1\times10^{-14}$ for the GWB amplitude. The analysis was carried out on times-of-arrival (TOAs) obtained by folding from six months up to one year of photon observations. A photon-by-photon approach was also tested to infer constraints on the GWB amplitude from individual pulsars, but without accounting for the cross-pulsar correlations that a GWB would induce. Here, we reanalyse the same dataset using a regularized likelihood method that correctly models cross-pulsar correlations directly from the photons, while additionally marginalising over the uncertain pulse profile shape. While the two methods are not expected to have significant differences in sensitivity, we prove through simulations of gamma-ray PTA datasets that the photon-by-photon method for GWB recoveries is, statistically, more robust. The resulting upper limit obtained for the GWB strain amplitude is $1.2\times10^{-14}$, indicating that the improved method yields a consistent result with the previous analyses.