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 ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}(f)$; or the observables ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}({f}_{1})$ and ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}({f}_{2})$. Here, $r$ is the so-called ``tensor-to-scalar ratio,'' which is constrained by cosmic-microwave-background experiments; and ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}(f)$ 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 $d\mathrm{ln}{\ensuremath{\Omega}}_{0}^{\mathrm{gw}}(f)/d\mathrm{ln}f$ of the present-day gravitational-wave spectrum. The relationship between $r$ and ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}(f)$ 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: $\stackrel{^}{w}(f)$ and ${\stackrel{^}{n}}_{t}(f)$, where ${\stackrel{^}{n}}_{t}(f)$ is a certain logarithmic average over ${n}_{t}(k)$ (the primordial tensor spectral index); and $\stackrel{^}{w}(f)$ is a certain logarithmic average over $\stackrel{\texttildelow{}}{w}(a)$ (the effective equation-of-state parameter in the early universe, after horizon re-entry). Here, the effective equation-of-state parameter $\stackrel{\texttildelow{}}{w}(a)$ is a combination of the ordinary equation-of-state parameter $w(a)$ and the bulk viscosity $\ensuremath{\zeta}(a)$. Thus, by comparing observational constraints on $r$ and ${\ensuremath{\Omega}}_{0}^{\mathrm{gw}}(f)$, one obtains (remarkably tight) constraints in the ${\stackrel{^}{w}(f),{\stackrel{^}{n}}_{t}(f)}$ 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 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 $\stackrel{^}{w}\ensuremath{\lesssim}0.55$ 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.

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