  Third generation ground-based gravitational wave (GW) detectors, such as
Einstein Telescope and Cosmic Explorer, will operate in the $(\text{few}-10^4)$
Hz frequency band, with a boost in sensitivity providing an unprecedented reach
into primordial cosmology. Working concurrently with pulsar timing arrays in
the nHz band, and LISA in the mHz band, these 3G detectors will be powerful
probes of beyond the standard model particle physics on scales $T\gtrsim
10^{7}$GeV. Here we focus on their ability to probe phase transitions (PTs) in
the early universe. We first overview the landscape of detectors across
frequencies, discuss the relevance of astrophysical foregrounds, and provide
convenient and up-to-date power-law integrated sensitivity curves for these
detectors. We then present the constraints expected from GW observations on
first order PTs and on topological defects (strings and domain walls), which
may be formed when a symmetry is broken irrespective of the order of the phase
transition. These constraints can then be applied to specific models leading to
first order PTs and/or topological defects. In particular we discuss the
implications for axion models, which solve the strong CP problem by introducing
a spontaneously broken Peccei-Quinn (PQ) symmetry. For post-inflationary
breaking, the PQ scale must lie in the $10^{8}-10^{11}$ GeV range, and so the
signal from a first order PQ PT falls within reach of ground based 3G
detectors. A scan in parameter space of signal-to-noise ratio in a
representative model reveals their large potential to probe the nature of the
PQ transition. Additionally, in heavy axion type models domain walls form,
which can lead to a detectable GW background. We discuss their spectrum and
summarise the expected constraints on these models from 3G detectors, together
with SKA and LISA.
