  We derive a general master equation relating the gravitational-wave
observables 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. by
pulsar-timing measurements, laser-interferometer experiments, and Big Bang
Nucleosynthesis (BBN). Differentiating the master equation yields a new
expression for the tilt d(ln Omega_gw(f))/d(ln f). The relationship between r
and Omega_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_hat(f) and nt_hat(f), where
nt_hat(f) is a certain logarithmic average over nt(k) (the primordial tensor
spectral index); and w_hat(f) is a certain logarithmic average over w_tilde(a)
(the effective equation-of-state in the early universe, after horizon
re-entry). Here the effective equation-of-state parameter w_tilde(a) is a
combination of the ordinary equation-of-state parameter w(a) and the bulk
viscosity zeta(a). Thus, by comparing constraints on r and Omega_gw(f), one can
obtain (remarkably tight) constraints in the [w_hat(f), nt_hat(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 BBN.
Finally, although most of our analysis does not assume inflation, we point out
that if CMB experiments detect a non-zero value for r, then we will immediately
obtain (as a free by-product) a new upper bound w_hat < 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 BBN.
