Higgs), if inflation is followed by an SD period. The same mechanism can actually be realized with generic self-interacting scalar fields [@Dimopoulos:2018wfg], other than the SM Higgs. See [@Opferkuch:2019zbd] for a recent re-analysis of the idea.

Models with blue-tilted inflationary GW spectrum due to the presence of an SD epoch have been studied in various contexts [@Giovannini:1998bp; @Giovannini:1999bh; @Giovannini:2008zg; @Giovannini:2009kg; @Tashiro:2003qp; @Sahni:2001qp; @Boyle:2005se; @Boyle:2007zx; @Li:2016mmc]. The resulting GW energy spectrum can be characterized by three quantities: the Hubble rate $H_{\text{inf}}$ during inflation, the equation state parameter $w$ during the stiff epoch, and the Hubble/energy scale at the SD-to-RD transition, parametrized by the redshifted frequency today $f_{\rm RD}$ corresponding to such scale. In general, a blue-tilted GW background can be probed with a variety of experiments, see e.g. [@Zhao:2011bg; @Zhao:2013bba; @Kuroyanagi:2014nba; @Jinno:2014qka; @Lentati:2015qwp; @Lasky:2015lej; @Arzoumanian:2015liz; @Liu:2015psa; @Kuroyanagi:2018csn; @DEramo:2019tit; @Bernal:2019lpc]. The aim of our present work is to assess the ability of the [*Advanced Laser Interferometric Gravitational wave Observatory*]{} (aLIGO), as well as of the (will-be) first generation space-based GW detector, the [*Laser Interferometric Space Antenna*]{} (LISA), to probe the parameter space spanned by $H_{\rm inf}$, $w$, and $f_{\rm RD}$. We assume inflation is well described by a quasi-[*de Sitter*]{} stage, so 