Timing analysis of the 2022 outburst of the accreting millisecond X-ray pulsar SAX J1808.4\(-\)3658: hints of an orbital shrinking

We present a pulse timing analysis of NICER observations of the accreting millisecond X-ray pulsar SAX J1808.4\(-\)3658 during the outburst that started on 2022 August 19. Similar to previous outbursts, after decaying from a peak luminosity of \(\simeq 1\times10^{36} \, \mathrm{erg \, s^{-1}}\) in about a week, the pulsar entered in a \(\sim 1\) month-long reflaring stage. Comparison of the average pulsar spin frequency during the outburst with those previously measured confirmed the long-term spin derivative of \(\dot{\nu}_{\textrm{SD}}=-(1.15\pm0.06)\times 10^{-15} \, \mathrm{Hz\,s^{-1}}\), compatible with the spin-down torque of a \(\approx 10^{26} \, \mathrm{G \, cm^3}\) rotating magnetic dipole. For the first time in the last twenty years, the orbital phase evolution shows evidence for a decrease of the orbital period. The long-term behaviour of the orbit is dominated by a \(\sim 11 \, \mathrm{s}\) modulation of the orbital phase epoch consistent with a \(\sim 21 \, \mathrm{yr}\) period. We discuss the observed evolution in terms of a coupling between the orbit and variations in the mass quadrupole of the companion star.