Les méthodes standard de détection du fond gravitationnel stochastique reposent sur l'hypothèse simplificatrice selon laquelle sa distribution ainsi que celle du bruit des détecteurs sont Gaussiennes. Nous proposons dans cette thèse des méthodes améliorées de détection du fond gravitationnel stochastique qui tiennent compte explicitement du caractère non-Gaussien de ces distributions. En utilisant un développement d'Edgeworth, nous obtenons dans un premier temps une expression analytique pour la statistique du rapport de vraisemblance en présence d'une distribution non Gaussienne du fonds gravitationnel stochastique. Cette expression généralise l'expression habituelle lorsque le coefficient de symétrie et le coefficient d'aplatissement de la distribution du fond stochastique sont non nuls. Sur la base de simulations stochastiques pour différentes distributions symétriques présentant des queues plus épaisses que celles de la distribution Gaussienne, nous montrons par ailleurs que le 4eme cumulant peut-être estimé avec une précision acceptable lorsque le ratio signal à bruit est supérieur à 1%, ce qui devrait permettre d'apporter des contraintes supplémentaires intéressantes sur les valeurs de paramètres issus des modèles astrophysiques et cosmologiques. Dans un deuxième temps, nous cherchons à analyser l'impact sur les méthodes de détection du fond gravitationnel stochastique de déviations par rapport à la normalité dans la distribution du bruit des détecteurs. / The new generation of interferometers should allow us to detect stochastic gravitational wave backgrounds that are expected to arise from a large number of random, independent, unresolved events of astrophysical or cosmological origin. Most detection methods for gravitational waves are based upon the assumption of Gaussian gravitational wave stochastic background signals and noise processes. Our main objective is to improve the methods that can be used to detect gravitational backgrounds in the presence of non-Gaussian distributions. We first maintain the assumption of Gaussian noise distributions so as to better focus on the impact of deviations from normality of the signal distribution in the context of the standard cross-correlation detection statistic. Using a 4th-order Edgeworth expansion of the unknown density for the signal and noise distributions, we first derive an explicit expression for the non-Gaussian likelihood ratio statistic, which is obtained as a function of the variance, but also skewness and kurtosis of the unknown signal and noise distributions. We use numerical procedures to generate maximum likelihood estimates for the gravitational wave distribution parameters for a set of symmetric heavy-tailed distributions, and we find that the fourth cumulant can be estimated with reasonable precision when the ratio between the signal and the noise variances is larger than 1%, which should be useful for analyzing the constraints on astrophysical and cosmological models. In a second step, we analyze the efficiency of the standard cross-correlation statistic in situations that also involve non-Gaussian noise distributions.

LISA, the Laser Interferometry Space Antenna, is the 3rd large mission (L3) of the ESA program Cosmic Vision with a junior partnership from NASA planned to be launched around 2034. Space-based gravitational wave observatories such as LISA have been developed for observation of sources that produce gravitational wave (GW) signals with frequencies in the mHz regime. The frequency band is achievable by having a longer-baseline interferometer compared to ground-based detectors. In addition, the significant size of the LISA arms-length guarantees the detection of many astrophysical sources. The absence of Newtonian noise in space, which is the dominant source of noise below few hertz for ground-based detectors, allows LISA to be sensitive to lower frequency compared to the former. Thus, going to space allows studying different sources with respect to the ones of interest for ground-based detectors such as supermassive black holes. Although having very long baselines between the satellites generally increases the sensitivity to gravitational waves, it also implies many technical challenges, such that a balance must be found between scientific performance and technical feasibility.In the actual proposal LISA is designed to be a constellation of three identical spacecraft in a triangular formation with six active laser links connecting the three spacecraft, which are separated by 2.5 million km. To fulfil the observatory program every spacecraft has a minimum requirement of two free-falling test masses, two telescopes, and two lasers. The detector’s center-of-mass follows a circular, heliocentric trajectory, trailing 20 degrees behind the Earth and the plane of the detector is tilted by 60 degrees with respect to the ecliptic.The goal of LISA is to detect GWs which manifest themselves as a tiny fluctuation in the frequency of the laser beam measured at the phase-meter. Thus, to detect GW you need to compete with many sources of disturbance that simulate the effect of a GW frequency modulation. Laser noise is an example of those. Therefore, one key element in the LISA data production chain is the post-processing technique called Time Delay Interferometry aimed at suppressing the intense laser frequency noise that would completely cover the astrophysical signal. Data from the six independent inter-satellite links, connecting the three spacecraft, are properly time-shifted and combined to form the final scientific signal. This post-processing technique circumvents the impossibility of physically building in space an equal arm interferometer, which would intrinsically beat the frequency noise by comparing light generated at the same time.The following work is focused on revisiting the Time-Delay-Interferometry (TDI) for LISA and studying the usage of all the possible TDI combinations we can build for the LISA instrument characterisation and science extraction. Many possible TDI combinations that suppress the frequency noise have been identified in the past and this thesis revisits the TDI technique focusing on the physical interpretation of it, that is a virtual interference of photons that have been travelling through the constellation via different paths but performing the same total distance. We illustrate all possible TDI configurations that suppress the laser noise contribution to the level required by the mission to understand how TDI channels can be best used for the diagnostic of the instrument and LISA science. With this philosophy, we develop an algorithm to search for all possible combinations that suppress laser noise at the same level as the classical TDI X, Y, and Z combinations presented in the TDI literature. This algorithm finds new combinations that fulfill the noise suppression requirement as accurately as X, Y, and Z.The LISA mission has been also advertised to probe the early Universe by detecting a stochastic GW background. Once the laser frequency noise has been subtracted, the stochastic signal, both cosmological and astrophysical, is itself going to contribute to the noise curve. Therefore it is necessary to have a good estimate of the noise of the instrument to discriminate between the stochastic background signal and the LISA noise.The strategy that has been suggested in the literature is to use the TDI T, insensitive (up to a certain order) to GW signals to estimate the pure instrumental noise in order to distinguish between the LISA background noise and the GW stochastic signal. Following this idea, as instrument noise is expected to have multiple, independent sources, this thesis explores combinations that could allow discriminating among those sources of noise, and between them and the GW signal, with the purpose of understanding how we can characterise our instrument using TDI. We illustrate special TDI combination signals in LISA, in addition to TDI T, that we call null-channels, which are ideally insensitive to gravitational waves and only carry information about instrumental noise. Studying the noise properties that can be extracted by monitoring these interferometric signals, we state that individual acceleration noise parameters are not well constrained. All null-channels behave as an ideal Sagnac interferometer, sensitive just to a particular linear combination of the six test masses acceleration that resembles a rotational acceleration signal of the entire constellation. Moreover, all null-channels show approximately the same signal to noise ratio remarkably suppressed relative to that of the TDI X. In support and application of our theoretical studies, we also give an introduction on calibrating the LISA instrument by injecting spurious signals in a LISA link and see how these propagates through a TDI channel. Indeed, this will be useful to calibrate the instrument during operations and also to build the basis for the data analysis to discriminate spurious signals from gravitational waves. My contribution to the results we present in this thesis can be summarised as the following. I supported the studies and the realisation of the search TDI algorithm whose results are published in the article. In particular, I took care of cataloging the new TDI combinations and consolidating the results we found. I have updated the TDI combinations reported in the above-mentioned work, the final version of it is reported in this thesis. I worked on the characterisation of these combinations concerning secondary noises such as clock noise, readout noise, residual laser frequency noise, and acceleration noise. In particular, I studied how these noises are transferred through the various TDI and I derive the correspondent analytical models. I then realize a software with Wolfram Mathematica, design to load and combines phase data produced by an external simulator to build the final TDI outputs, besides I also did the noise models’ validation. The basis of this program was then used to implement these TDI combinations in LISANode. Finally, I developed the algorithm to study how disturbances in force, such as glitches, and simple GW signals, such as monochromatic GW binaries, propagate through TDI and null-channels. Moreover, I tested through simulations the validity of these TDI and null-channels to distinguish instrumental artefact from GW signals and to characterise the instrumental noise.