Relating gravitational wave constraints from primordial nucleosynthesis, pulsar timing, laser interferometers, and the CMB: Implications for the early universe

We derive a general equation relating the gravitational-wave observables r and or the observables Omega(gw)(0)(f(1)) and Omega(gw)(0)(f(2)). Here, r is the so-called tensor-to-scalar ratio, which is constrained by cosmic-microwave-background experiments; and Omega(gw)(0)(f) is the energy spectrum of primordial gravitational waves, which is constrained, e.g., by pulsar-timing measurements, laser-interferometer experiments, and the standard big bang nucleosynthesis bound. Differentiating this equation yields a new expression for the tilt dln Omega(gw)(0)(f)/d lnf of the present-day gravitational-wave specutrum. The relationship between r and Omega(gw)(0)(f) depends sensitively oil the uncertain physics of the early universe, and we show that this uncertainty may be encapsulated (in a model-independent way) by two quantities: (w) over cap (f) and (n) over cap (t)(f), where (n) over cap (t)(f) is a certain logarithmic average over n(t)(k) (the primordial tensor spectral index); and (w) over cap (f) is a certain logarithmic average over (w) over tilde (a) (the effective equation-of-state parameter in the early universe, after horizon re-entry). Here, the effective equation-of-state parameter (w) over tilde (a) is a combination of the ordinary equation-of-state parameter w(a) and the bulk viscosity zeta(a). Thus, by comparing observational constraints on r and Omega(gw)(0)(f), one obtains (remarkably tight) constraints in the {(w) over cap (f), (n) over cap (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 big bang nucleosynthesis. (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 cosmic microwave-background experiments detect a nonzero value for r, then we will immediately obtain (as a free by-product) a new upper bound (w) over cap less than or similar to 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 big bang nucleosynthesis.