A renewed search for radio emission from the variable $\gamma$-ray pulsar PSR J2021$+$4026

We undertake the first targeted search at 1.5 GHz for radio emission from the variable $\gamma$-ray pulsar PSR J2021$+$4026. This radio-quiet pulsar assumes one of two stable $\gamma$-ray emission states, between which it transitions on a timescale of years. These transitions, in both $\gamma$-ray flux and pulse profile shape, are accompanied by contemporaneous changes to the pulsar's spin-down rate. A number of radio pulsars are known to exhibit similar correlated variability, which in some cases involves an emission state in which the radio emission ceases to be detectable. In this paper, we perform a search for radio emission from PSR J2021$+$4026, using archival radio observations recorded when the pulsar was in each of its emission/spin-down states. Using improved techniques, we search for periodic radio emission as well as single pulse phenomena such as giant radio pulses and RRAT-like emission. Our search reveals no evidence of radio emission from PSR J2021$+$4026. We estimate that the flux density for periodic emission from PSR J2021$+$4026 does not exceed 0.2 mJy at this frequency. We also estimate single-pulse flux limits for RRAT-like bursts and giant radio pulses to be 0.3 and 100 Jy respectively. We discuss the transitioning behaviour of PSR J2021$+$4026 in the context of pulsar glitches, intermittent pulsars and the increasingly common emission-rotation correlation observed in radio pulsars.