Knowledge of the Earth's inner and outer core structure can be considerably improved by the observation of the translational motion of the inner core, the so-called Slichter triplet. The theory for deriving the elastic-gravitational mode frequencies is still controversial and many observational attempts to detect the triplet have so far been unsuccessful. The period of the Slichter modes depends directly of the value of the density jump at the inner core boundary (ICB). We propose a detection method based on the seismological theory for the splitting of the modes. Depending on the density jump at ICB, new theoretical periods for the Slichter modes are computed and the corresponding frequency splittings are used to detect possible candidates. The scanning of the product spectrum obtained after stacking five time-varying gravity records from superconducting gravimeters after the 2001 Peruvian earthquake has confirmed the detection of the 2S1 seismic mode, but the application after stacking one-year datasets has not led to any convincing candidate for the Slichter triplet at this stage.