Follow-up procedure for gravitational wave searches from isolated neutron stars using the time-domain $\mathcal{F}$-statistic method

Among promising sources of gravitational waves are long-lived nearly periodic signals produced by rotating, asymmetric neutron stars. Depending on the astrophysical scenario, the sources of asymmetry may have thermal, viscous, elastic and/or magnetic origin. In this work we introduce a follow-up procedure for an all-sky search for gravitational wave signals from rotating neutron stars. The procedure denoted as Followup implements matched-filtering $\mathcal{F}$-statistic method. We describe data analysis methods and algorithms used in the procedure. We present tests of the Followup for artificial signals added to white, Gaussian noise. The tests show a good agreement with the theoretical predictions. The Followup will become part of the Time-Domain $\mathcal{F}$-statistic pipeline that is routinely used for all-sky searches of LIGO and Virgo detector data.