Aims . We analyse ESPRESSO data for the stars HD 10700 ( τ Ceti), HD 20794 ( e Eridani), HD 102365, and HD 304636 acquired via its Guaranteed Time Observations (GTO) programme. We characterise the stars’ radial velocity (RV) signals down to a precision of 10 cm/s on timescales ranging from minutes to planetary periods falling within the host’s habitable zone (HZ). We study the RV signature of pulsation, granulation, and stellar activity, inferring the potential presence of planets around these stars. Thus, we outline the population of planets that while undetectable remain compatible with the available data.

Methods . We derived the stellar parameters through different methods for a complete characterisation of the star. We used these parameters to model the effects of stellar pulsations on intra-night RV variations and of stellar activity on nightly averaged values. The RVs were derived both with the cross-correlation method and template matching, as well as over the blue and red ESPRESSO detectors independently to identify colour-dependent parasitic effects of an instrumental or stellar nature. The study of RVs was complemented by an investigation of stellar activity indicators using photospheric information and chromospheric indexes.

Results . A simple model of stellar pulsations successfully reproduced the intra-night RV scatter of HD 10700 down to a few cm/s. For HD 102365 and HD 20794, an additional source of scatter at the level of several 10 cm/s remains necessary to explain the data. A kima analysis was used to evaluate the number of planets supported by the nightly averaged time series of each of these three stars, under the assumption that a quasi-periodic Gaussian process (GP) regression is able to model the activity signal. While a frequency analysis of HD 10700 RVs is able to identify a periodic signal at 20 d, when it is modelled along with the activity signal the signal is formally non-significant. Moreover, its physical origin remains uncertain due to the similarity with the first harmonic of the stellar rotation. ESPRESSO data on their own do not provide conclusive evidence to support the existence of planets around HD 10700, HD 102365, or HD 304636. In addition, the comparison of RVs with the contemporaneous indicators displays a strong correlation for HD 102365. The direct interpretation is that half of the RV variance on this star is directly attributed to activity.

Conclusions . ESPRESSO is shown to reach an on-sky RV precision of better than 10 cm/s on short timescales (<1h) and of 40 cm/s over 3.5 yr. A subdivision of the datasets showcases a precision reaching 20–30 cm/s over one year. These results impose stringent constraints on the impact of granulation mechanisms on RV. In spite of no detections, our analysis of HD 10700 RVs demonstrates a sensitivity to planets with a mass of 1.7 M _⊕ for periods of up to 100 d, and a mass of 2–5 M _⊕ for the star’s HZ.