No abstract

We derive a general master equation relating the gravitational-wave observables r and Ω gw 0 (f ); or the observables Ω gw 0 (f 1 ) and Ω gw 0 (f 2 ). Here r is the so-called "tensor-to-scalar ratio," which is constrained by cosmic-microwave-background (CMB) experiments; and Ω gw 0 (f ) is the energy spectrum of primordial gravitational-waves, which is constrained e.g. by pulsar-timing (PT) measurements, laser-interferometer (LI) experiments, and the standard Big Bang Nucleosynthesis (BBN) bound. Differentiating the master equation yields a new expression for the tilt d ln Ω gw 0 (f )/d ln f of the present-day gravitational-wave spectrum. The relationship between r and Ω gw 0 (f ) depends sensitively on the uncertain physics of the early universe, and we show that this uncertainty may be encapsulated (in a model-independent way) by two quantities:ŵ(f ) andn t (f ), wheren t (f ) is a certain logarithmic average over n t (k) (the primordial tensor spectral index); andŵ(f ) is a certain logarithmic average overw(a) (the effective equation-of-state parameter in the early universe, after horizon re-entry). Here the effective equation-of-state parameterw(a) is a combination of the ordinary equation-of-state parameter w(a) and the bulk viscosity ζ(a). Thus, by comparing observational constraints on r and Ω gw 0 (f ), one obtains (remarkably tight) constraints in the {ŵ(f ),n t (f )} plane. In particular, this is the best way to constrain (or detect) the presence of a "stiff" energy component (with w > 1/3) in the early universe, prior to BBN. (The discovery of such a component would be no more surprising than the discovery of a tiny cosmological constant at late times!) Finally, although most of our analysis does not assume inflation, we point out that if CMB experiments detect a non-zero value for r, then we will immediately obtain (as a free by-product) a new upper boundŵ < ∼ 0.55 on the logarithmically averaged effective equation-of-state parameter during the "primordial dark age" between the end of inflation and the start of BBN.