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We derive a general master equation relating the gravitational-waveobservables r and Omega_gw(f). Here r is the tensor-to-scalar ratio,constrained by cosmic-microwave-background (CMB) experiments; and Omega_gw(f)is the energy spectrum of primordial gravitational-waves, constrained e.g. bypulsar-timing measurements, laser-interferometer experiments, and Big BangNucleosynthesis (BBN). Differentiating the master equation yields a newexpression for the tilt d(ln Omega_gw(f))/d(ln f). The relationship between rand Omega_gw(f) depends sensitively on the uncertain physics of the earlyuniverse, and we show that this uncertainty may be encapsulated (in amodel-independent way) by two quantities: w_hat(f) and nt_hat(f), wherent_hat(f) is a certain logarithmic average over nt(k) (the primordial tensorspectral index); and w_hat(f) is a certain logarithmic average over w_tilde(a)(the effective equation-of-state in the early universe, after horizonre-entry). Here the effective equation-of-state parameter w_tilde(a) is acombination of the ordinary equation-of-state parameter w(a) and the bulkviscosity zeta(a). Thus, by comparing constraints on r and Omega_gw(f), one canobtain (remarkably tight) constraints in the [w_hat(f), nt_hat(f)] plane. Inparticular, 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.Finally, although most of our analysis does not assume inflation, we point outthat if CMB experiments detect a non-zero value for r, then we will immediatelyobtain (as a free by-product) a new upper bound w_hat < 0.55 on thelogarithmically averaged effective equation-of-state parameter during the``primordial dark age'' between the end of inflation and the start of BBN.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

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