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 $\simeq 1\times {10}^{36}\mathrm{e}\mathrm{r}\mathrm{g}{\mathrm{s}}^{-1}$ 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 ${\dot{\nu }}_{\text{SD}}=-(1.15\pm 0.06)\times {10}^{-15}\mathrm{H}\mathrm{z}{\mathrm{s}}^{-1}$, compatible with the spin-down torque of a $\approx {10}^{26}\mathrm{G}\mathrm{c}{\mathrm{m}}^3$ 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 $\sim 21\mathrm{y}\mathrm{r}$ period. We discuss the observed evolution in terms of a coupling between the orbit and variations in the mass quadrupole of the companion star.