  We forecast the prospective of detection for a stochastic gravitational wave
background sourced by cosmological first-order phase transitions. We focus on
first-order phase transitions with negligible plasma effects, and consider the
experimental infrastructures built by the end of the LISA mission. We make
manifest the synergy among LISA, pulsar time array experiments, and
ground-based interferometers. For phase transitions above the TeV scale or
below the electroweak scale, LISA can detect the corresponding gravitational
wave signal together with Einstein Telescope, SKA or even aLIGO-aVIRGO-KAGRA.
For phase transitions at the electroweak scale, instead, LISA can be the only
experiment observing the gravitational wave signal. In case of detection, by
using a parameter reconstruction method that we anticipate in this work, we
show that LISA on its own has the potential to determine when the phase
transition occurs and, consequently, the energy scale above which the standard
model of particle physics needs to be modified. The result may likely guide the
collider community in the post-LHC era.
