The expansion history of the Universe between the end of inflation and the onset of radiation-domination ( RD ) is currently unknown .
If the equation of state during this period is stiffer than that of radiation , w > 1 / 3 , the gravitational wave ( GW ) background from inflation acquires a blue-tilt { d \log \rho _ { GW } \over d \log f } = { 2 ( w - 1 / 3 ) \over ( w + 1 / 3 ) } > 0 at frequencies f \gg f _ { RD } corresponding to modes re-entering the horizon during the stiff-domination ( SD ) , where f _ { RD } is the frequency today of the horizon scale at the SD-to-RD transition .
We characterized in detail the transfer function of the GW energy density spectrum , considering both ’ instant ’ and smooth modelings of the SD-to-RD transition .
The shape of the spectrum is controlled by w , f _ { RD } , and H _ { inf } ( the Hubble scale of inflation ) .
We determined the parameter space compatible with a detection of this signal by LIGO and LISA , including possible changes in the number of relativistic degrees of freedom , and the presence of a tensor tilt .
Consistency with upper bounds on stochastic GW backgrounds , however , rules out a significant fraction of the observable parameter space .
We find that this renders the signal unobservable by Advanced LIGO , in all cases .
The GW background remains detectable by LISA , though only in a small island of parameter space , corresponding to scenarios with an equation of state in the range 0.46 \lesssim w \lesssim 0.56 and a high inflationary scale H _ { inf } \gtrsim 10 ^ { 13 } GeV , but low reheating temperature 1 ~ { } { MeV } \lesssim T _ { RD } \lesssim 150 MeV ( equivalently , 10 ^ { -11 } ~ { } { Hz } \lesssim f _ { RD } \lesssim 3.6 \cdot 10 ^ { -9 } ~ { } { Hz } ) .
Implications for early Universe scenarios resting upon an SD epoch are briefly discussed .