This paper investigates the potential of the space-based interferometer eLISA to detect the stochastic gravitational wave (GW) background produced by strong first-order cosmological phase transitions. The study considers contributions from bubble collisions, magnetohydrodynamic (MHD) turbulence, and sound waves to the GW spectrum. The eLISA sensitivity to these signals is analyzed in a model-independent way for various detector configurations. The results are applied to several specific models, demonstrating that eLISA can probe scenarios beyond the Standard Model of particle physics that predict strong first-order phase transitions in the early Universe. The paper discusses the dynamics of phase transitions, focusing on three cases: non-runaway bubbles in a plasma, runaway bubbles in a plasma, and runaway bubbles in vacuum. Each case leads to different contributions to the GW spectrum, depending on the bubble wall velocity, the fraction of latent heat converted into bulk motion, and the presence of plasma effects. The scalar field contribution, sound waves, and MHD turbulence are analyzed in detail, with their respective spectral shapes and dependencies on parameters such as the Hubble parameter, the temperature of the transition, and the bubble wall velocity. The eLISA sensitivity is evaluated for four representative configurations, with the most promising being the six-link configuration (C1), which has a longer arm length and lower noise level. The study shows that adding more laser links increases sensitivity more effectively than increasing the arm length. The detection threshold for GW signals is determined based on the signal-to-noise ratio, with thresholds set for different configurations. The results indicate that eLISA can detect GW signals from first-order phase transitions in the early Universe, particularly in scenarios with high bubble wall velocities and significant latent heat conversion. The paper also provides examples of GW spectra for different phase transition scenarios, showing how the contributions from scalar field gradients, sound waves, and MHD turbulence interact. These examples highlight the importance of understanding the specific dynamics of the phase transition to accurately predict the GW signal. The study concludes that eLISA has the potential to probe a wide range of scenarios beyond the Standard Model, providing valuable insights into the early Universe and the nature of dark matter and other fundamental physics phenomena.This paper investigates the potential of the space-based interferometer eLISA to detect the stochastic gravitational wave (GW) background produced by strong first-order cosmological phase transitions. The study considers contributions from bubble collisions, magnetohydrodynamic (MHD) turbulence, and sound waves to the GW spectrum. The eLISA sensitivity to these signals is analyzed in a model-independent way for various detector configurations. The results are applied to several specific models, demonstrating that eLISA can probe scenarios beyond the Standard Model of particle physics that predict strong first-order phase transitions in the early Universe. The paper discusses the dynamics of phase transitions, focusing on three cases: non-runaway bubbles in a plasma, runaway bubbles in a plasma, and runaway bubbles in vacuum. Each case leads to different contributions to the GW spectrum, depending on the bubble wall velocity, the fraction of latent heat converted into bulk motion, and the presence of plasma effects. The scalar field contribution, sound waves, and MHD turbulence are analyzed in detail, with their respective spectral shapes and dependencies on parameters such as the Hubble parameter, the temperature of the transition, and the bubble wall velocity. The eLISA sensitivity is evaluated for four representative configurations, with the most promising being the six-link configuration (C1), which has a longer arm length and lower noise level. The study shows that adding more laser links increases sensitivity more effectively than increasing the arm length. The detection threshold for GW signals is determined based on the signal-to-noise ratio, with thresholds set for different configurations. The results indicate that eLISA can detect GW signals from first-order phase transitions in the early Universe, particularly in scenarios with high bubble wall velocities and significant latent heat conversion. The paper also provides examples of GW spectra for different phase transition scenarios, showing how the contributions from scalar field gradients, sound waves, and MHD turbulence interact. These examples highlight the importance of understanding the specific dynamics of the phase transition to accurately predict the GW signal. The study concludes that eLISA has the potential to probe a wide range of scenarios beyond the Standard Model, providing valuable insights into the early Universe and the nature of dark matter and other fundamental physics phenomena.