  Within the Standard Model with non-linearly realised electroweak symmetry,
the LHC Higgs boson may reside in a singlet representation of the gauge group.
Several new interactions are then allowed, including anomalous Higgs
self-couplings, which may drive the electroweak phase transition to be strongly
first-order. In this paper we investigate the cosmological electroweak phase
transition in a simplified model with an anomalous Higgs cubic self- coupling.
We look at the feasibility of detecting gravitational waves produced during
such a transition in the early universe by future space-based experiments. We
find that for the range of relatively large cubic couplings, $111~{\rm GeV}~
\lesssim |\kappa| \lesssim 118~{\rm GeV}$, $\sim $mHz frequency gravitational
waves can be observed by eLISA, while BBO will potentially be able to detect
waves in a wider frequency range, $0.1-10~$mHz.
