This thesis is developed in the framework of the paradigm that seeks for the discovery of an Earth analog. Nowadays, low mass stars, and in particular M dwarf stars, are key targets towards achieving this goal. In this thesis, I focus on the study of the short-time domain of M dwarf stars with the aim of searching for short period planets, but also for the first detection of stellar pulsations on this spectral type. Both science goals are the primary objectives of the "Cool Tiny Beats" (CTB)

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... vey, which has produced most of the data used in this thesis. CTB data consist in high resolution and high-cadence spectroscopic Doppler measurements taken either with HARPS or HARPS-N spectrographs. First of all, a thorough understanding of the spectrographs response in the short time domain was performed to characterize the sources of noise in our range of study. Our first approach to the goals of this thesis consisted in the design of an observational experiment to delve into the HARPS-N sub-night performance. Results unveiled variability of the spectra continuum correlated with instabilities of the spectrograph illumination associated to the airmass. Such distortions, which are wavelength and time dependent, are also present in at least one of the data-products given by the HARPS-N reduction software: the width of the mean-line profiles (i.e. the so-called FWHM index), an index commonly used as a proxy of the stellar activity. As a consequence, we searched for an alternative approach to measure the width index. In particular, we calculated the mean-line profile of the spectrum with a least-squares-deconvolution technique and we obtained the profile indices as the moments of the profile distribution. As part of this study, we also corroborated that the radial velocities calculated with our template matching algorithm TERRA are not affected by the illumination stability. This work unveiled a possible failure of the HARPS-N atmospheric dispersion corrector (or ADC) and outlined two possible solutions: either the FWHM mea [...]