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 Ω₀ᵍʷ(f); or the observables Ω₀ᵍʷ(f₁) and Ω₀ᵍʷ(f₂). Here, r is the so-called ``tensor-to-scalar ratio,'' which is constrained by cosmic-microwave-background experiments; and Ω₀ᵍʷ(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 dlnΩ₀ᵍʷ(f)/dlnf of the present-day gravitational-wave spectrum. The relationship between r and Ω₀ᵍʷ(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 ^nₜ(f), where ^nₜ(f) is a certain logarithmic average over nₜ(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 ζ(a). Thus, by comparing observational constraints on r and Ω₀ᵍʷ(f), one obtains (remarkably tight) constraints in the ^w(f),^nₜ(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.