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

Abstract

We derive a general master equation relating the gravitational-wave observables r and Ω gw 0 (f ); or the observables Ω gw 0 (f 1 ) and Ω gw 0 (f 2 ). Here r is the so-called "tensor-to-scalar ratio," which is constrained by cosmic-microwave-background (CMB) experiments; and Ω gw 0 (f ) is the energy spectrum of primordial gravitational-waves, which is constrained e.g. by pulsar-timing (PT) measurements, laser-interferometer (LI) experiments, and the standard Big Bang Nucleosynthesis (BBN) bound. Differentiating the master equation yields a new expression for the tilt d ln Ω gw 0 (f )/d ln f of the present-day gravitational-wave spectrum. The relationship between r and Ω gw 0 (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:ŵ(f ) andn t (f ), wheren t (f ) is a certain logarithmic average over n t (k) (the primordial tensor spectral index); andŵ(f ) is a certain logarithmic average overw(a) (the effective equation-of-state parameter in the early universe, after horizon re-entry). Here the effective equation-of-state parameterw(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 Ω gw 0 (f ), one obtains (remarkably tight) constraints in the {ŵ(f ),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 BBN. (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 CMB experiments detect a non-zero value for r, then we will immediately obtain (as a free by-product) a new upper boundŵ < ∼ 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.

Sources of Gravitational Waves: Theory and Observations

“…In those science runs, the detectors' sensitivity has passed the BBN bound [501,502] in the frequency band around 100 Hz, Ω GW < 6.9 × 10 −6 , but not yet the CMB bound [503]. The results have started to exclude regions of the parameter space of expected signals from cosmic (super)strings [502,504,505], and have constrained the equation of state of the Universe during the dark age [502,506]. Moreover, pulsar timing observations have set physically meaningful upper limits for the supermassive BH binary background (Ω GW < 1.3 × 10 −9 at 2.8 nHz) [507] and cosmic (super)strings [508].…”

“…In the absence of probes from epochs prior to BBN, it is very hard to infer the equation of state of the Universe between the end of inflation to the epoch when the radiation era started. Thus, it is difficult to predict the spectral slope of the relic GW background from BBN (when the Universe was certainly radiation dominated) to the scales where PTA, space-based and ground-based detectors are sensitive [506]. It is customary and (perhaps) natural to assume that the slope is the same over the huge range of frequency, spanning twenty orders of magnitude, and that it can be determined by CMB observations.…”

Observational constraints on theories with a blue spectrum of tensor modes

“…After completion of this work we became aware of [20] in which a master equation was derived which relates the short wavelength observable Ω gw (f )…”

Blue-tilted tensor spectrum and thermal history of the Universe

“…However, it should be noted that the thermal history of the Universe affects the spectrum of the primordial GWs [42][43][44][45][46][47][48][49][50]. For instance, high-frequency primordial GWs which entered the horizon during a matter-like component dominated epoch, such as that exists during reheating or late-time entropy production, have different frequency dependence compared to those which entered the radiation dominated epoch.…”