Recovering Pulsar Periodicity from Time-of-arrival Data by Finding the Shortest Vector in a Lattice

The strict periodicity of pulsars is one of the primary ways through which their nature and environment can be studied, and it has also enabled precision tests of general relativity and studies of nanohertz gravitational waves using pulsar timing arrays (PTAs). Identifying such a periodicity from a discrete set of arrival times is a difficult algorithmic problem, In particular when the pulsar is in a binary system. This challenge is especially acute in γ -ray pulsar astronomy, as there are hundreds of unassociated Fermi-LAT sources that may be produced by γ -ray emission from unknown pulsars. Recovering their timing solutions will help reveal their properties and may allow them to be added to PTAs. The same issue arises when attempting to recover a strict periodicity for repeating fast radio bursts (FRBs). Such a detection would be a major breakthrough, providing us with the FRB source’s age, magnetic field, and binary orbit. The problem of recovering a timing solution from sparse time-of-arrival data is currently unsolvable for pulsars in unknown binary systems, and incredibly hard even for isolated pulsars. In this paper, we frame the timing recovery problem as the problem of finding a short vector in a lattice and obtain the solution using off-the-shelf lattice reduction and sieving techniques. As a proof of concept, we solve PSR J0318+0253, a millisecond γ -ray pulsar discovered by FAST in a γ -ray-directed search, in a few CPU minutes. We discuss the assumptions of the standard lattice techniques and quantify their performance and limitations.