New Cosmic Microwave Background Constraint to Primordial Gravitational Waves

Primordial gravitational waves (GWs) with frequencies $\gtrsim {10}^{-15}\mathrm{Hz}$ contribute to the radiation density of the Universe at the time of decoupling of the cosmic microwave background (CMB). This affects the CMB and matter power spectra in a manner identical to massless neutrinos, unless the initial density perturbation for the GWs is nonadiabatic, as may occur if such GWs are produced during inflation or some post-inflation phase transition. In either case, current observations provide a constraint to the GW amplitude that competes with that from big-bang nucleosynthesis (BBN), although it extends to much lower frequencies ($\sim {10}^{-15}\mathrm{Hz}$ rather than the $\sim {10}^{-10}\mathrm{Hz}$ from BBN): at 95% confidence level, ${\Omega }_{\mathrm{gw}}{h}^2\lesssim 8.4\times {10}^{-6}$ for homogeneous (i.e., nonadiabatic) initial conditions. Future CMB experiments, like Planck and CMBPol, should allow sensitivities to ${\Omega }_{\mathrm{gw}}{h}^2\lesssim 1.4\times {10}^{-6}$ and ${\Omega }_{\mathrm{gw}}{h}^2\lesssim 5\times {10}^{-7}$, respectively.

Figure 1

Adiabatic: the marginalized (unnormalized) likelihoods for the CGWB energy density if perturbations to the CGWB density are adiabatic. The dotted curve is the result obtained using only CMB data. The thick solid curve includes galaxies as well as the Lyman-$\alpha$ forest. In all of the aforementioned curves, the marginalization is over the nonrelativistic-matter density ${\Omega }_m{h}^2$, baryon density ${\Omega }_b{h}^2$, scalar spectral index $n_s$, power-spectrum amplitude $A_s$, the optical depth $\tau$ to the surface of last scatter, and the angle $\theta$ subtended by the first acoustic peak (marginalization over $\theta$ essentially stands in for marginalization over the Hubble constant). We hold the geometry fixed to flat, the number of neutrinos to $N_{\nu }=3.04$, and the neutrino masses fixed to zero. Finally, the dot-dashed curve (to the right) shows current constraints from the $\mathrm{\text{CMB}}+\mathrm{\text{galaxies}}+\mathrm{Ly}\alpha$ if we allow for and marginalize over nonzero neutrino masses as well. The number of equivalent neutrino degrees of freedom ($N_{\mathrm{gw}}$) is shown on the bottom axis. Homogeneous: same as the left panel, except for homogeneous initial conditions for the CGWB. The arrow indicates the 95% C.L. upper limit ${\Omega }_{\mathrm{gw}}{h}^2\le 8.4\times {10}^{-6}$ that we adopt as our central result. This is obtained from the analysis that includes current $\mathrm{\text{CMB}}+\mathrm{\text{galaxy}}+\mathrm{Ly}\alpha +\mathrm{\text{free}}$ $m_{\nu }$.

Figure 2

The gravitational-wave density ${\Omega }_{\mathrm{gw}}{h}^2$ vs frequency. The BBN constraint corresponds to a limit of 1.4 extra neutrino degrees of freedom. We also show our constraints, from current CMB, galaxy, and Lyman-$\alpha$ data, for a CGWB with adiabatic primordial perturbations [“adiabatic (current)”] and for homogeneous initial conditions [“homogeneous (current)”], as well as our forecasts for the sensitivities if current CMB data are replaced by data from CMBPol. These constraints apply to a monochromatic CGWB spectrum. Also shown are the reaches of LIGO and LISA. BBO (not shown) should go deeper, but primarily at frequencies $\sim 1\mathrm{Hz}$. Large-angle CMB fluctuations (also not shown) constrain ${\Omega }_{\mathrm{gw}}{h}^2\lesssim {10}^{-10}$ for a scale-invariant spectrum at frequencies $\lesssim {10}^{-17}\mathrm{Hz}$. The LIGO S3 upper limit is from Ref. 21 and the msec pulsar curve is from Refs. 5, 6.