Ability of LIGO and LISA to probe the equation of state of the early Universe

Abstract: 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 ${d\log\rho_{\rm GW}\over d\log f} = {2(w-1/3)\over (w+1/3)} > 0$ at frequencies $f \gg f_{\rm RD}$ corresponding to modes re-entering the horizon during the stiff-domination (SD), where $f_{\rm RD}$ 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 $w$, $f_{\rm RD}$, and $H_{\rm inf}$ (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 $H_{\rm inf} \gtrsim 10^{13}~{\rm GeV}$, but low reheating temperature $1~{\rm MeV} \lesssim T_{\rm RD} \lesssim 150~{\rm MeV}$ (equivalently, $10^{-11}~{\rm Hz} \lesssim f_{\rm RD} \lesssim 3.6\cdot10^{-9}~{\rm Hz}$). Implications for early Universe scenarios resting upon an SD epoch are briefly discussed.