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 ₀^gw (f) ; or the observables ₀^gw (f₁) and ₀^gw (f₂). Here, r is the so-called ``tensor-to-scalar ratio, '' which is constrained by cosmic-microwave-background experiments; and ₀^gw (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 dln₀^gw (f) /dlnf of the present-day gravitational-wave spectrum. The relationship between r and ₀^gw (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: ^w (f) and ^{n}ₓ (f), where ^{n}ₓ (f) is a certain logarithmic average over nₓ (k) (the primordial tensor spectral index) ; and ^w (f) is a certain logarithmic average over w (a) (the effective equation-of-state parameter in the early universe, after horizon re-entry). Here, the effective equation-of-state parameter w (a) is a combination of the ordinary equation-of-state parameter w (a) and the bulk viscosity (a). Thus, by comparing observational constraints on r and ₀^gw (f), one obtains (remarkably tight) constraints in the ^{w (f), ^{n}ₓ (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 ^w0. 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.