This paper investigates the potential of the eLISA space-based interferometer to detect the stochastic gravitational wave (GW) background produced by strong first-order cosmological phase transitions. The authors discuss the contributions from bubble collisions, magnetohydrodynamic (MHD) turbulence, and sound waves to the stochastic background and estimate the corresponding GW signals. They compute the projected sensitivity of eLISA to cosmological phase transitions for various detector designs and configurations, demonstrating that eLISA can probe many well-motivated scenarios beyond the Standard Model of particle physics that predict strong first-order cosmological phase transitions in the early universe. The paper is structured into several sections, covering the prediction of the GW signal, the dynamics of the phase transition in different scenarios, and the sensitivity of eLISA to these signals. The authors also provide model-independent projections and discuss specific examples of models predicting strong GW signatures through first-order phase transitions. The results highlight the importance of the experimental configuration on the detectability of the GW signal, with the two six-link configurations showing the best reach.This paper investigates the potential of the eLISA space-based interferometer to detect the stochastic gravitational wave (GW) background produced by strong first-order cosmological phase transitions. The authors discuss the contributions from bubble collisions, magnetohydrodynamic (MHD) turbulence, and sound waves to the stochastic background and estimate the corresponding GW signals. They compute the projected sensitivity of eLISA to cosmological phase transitions for various detector designs and configurations, demonstrating that eLISA can probe many well-motivated scenarios beyond the Standard Model of particle physics that predict strong first-order cosmological phase transitions in the early universe. The paper is structured into several sections, covering the prediction of the GW signal, the dynamics of the phase transition in different scenarios, and the sensitivity of eLISA to these signals. The authors also provide model-independent projections and discuss specific examples of models predicting strong GW signatures through first-order phase transitions. The results highlight the importance of the experimental configuration on the detectability of the GW signal, with the two six-link configurations showing the best reach.