tokens: CP problem in particle physics, as was first proposed by Peccei Quinn ( 1977 ), Wilczek ( 1978 ), and Weinberg ( 1978 ). Ultralight axions are an extrapolation of the QCD axions (with expected masses in the range eV) to much smaller masses eV (Press et al. 1990 ; Hu et al. 2000 ; Peebles 2000 ). At low redshift, they form a Bose–Einstein condensate that oscillates coherently (unlike a stochastic background) on a timescale (e.g., Sikivie Yang 2009 ; Marsh 2016 ; Hui et al. 2017 ; Grin et al. 2019 ; Niemeyer 2019 , and references therein). Constraints on their mass m a have been set using Ly α forest measurements (Armengaud et al. 2017 ; Iršič et al. 2017 ; Kobayashi et al. 2017 ) and cosmic microwave background lensing (Hlozek et al. 2018 ) on megaparsec scales, dwarf spheroidals (Marsh Pop 2015 ; González-Morales et al. 2017 ; Broadhurst et al. 2019 ; Marsh Niemeyer 2019 ; Safarzadeh Spergel 2019 ) and ultradiffuse galaxies (Wasserman et al. 2019 ) on kiloparsec scales, and galactic core observations on (sub)parsec scales (Bar et al. 2019 ; Desjacques Nusser 2019 ; Davies Mocz 2020 ). Pulsar timing offers another avenue to probe the existence of coherent oscillations induced by ultralight scalar fields (Khmelnitsky Rubakov 2014 ; Blas et al. 2017 ; de Martino et al. 2017 ). Upper limits on the amplitude of such an oscillating gravitational potential in the Milky Way halo have already been derived from pulsar timing arrays (Porayko Postnov 2014 ; Porayko et al. 2018 ). Cross-correlation of residuals from different pulsars should improve these constraints (Hellings Downs 1983 ). Axion coherent oscillations also resonate with binary pulsars (Blas et al. 2017 ). While the effect is strongest near resonance, the very small width of the latter (when the coupling is purely gravitational) implies that one shall monitor instantaneous variations (Rozner et al. 2019 ) or the secular drift of orbital elements (Blas et al. 2019 ) away from resonances. Mashhoon ( 1978 ) used Lagrange’s planetary equations to develop an approximate theory of the interaction of a weak gravitational wave with a Keplerian binary. In this paper, we use angle-action variables to investigate the instantaneous variations (that is, not averaged over one orbital time) of a Keplerian system produced by an oscillating background of axion dark matter. We refer the reader to Binney Tremaine ( 1987 ) for an overview of the angle-action formalism. The paper is organized as follows. After a brief presentation of the astrophysical/cosmological context and our numerical implementation in Section 2 , we solve for the time evolution of the perturbed binary system using angle-action variables in Section 3 . We explore the instantaneous variations of the Roemer time delay as a function of orbital parameters, etc., in Section 4 . We conclude in Section 5 . 2. Setup We will use the numerical simulations of Rozner et al. ( 2019 ) to validate our theoretical predictions. We consider a binary
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