Tensor to Scalar Ratio

The tensor-to-scalar ratio \(r\) represents the ratio of the amplitude of the tensor power spectra to the amplitude of the scalar power spectrum in polarization, i.e.: \(r \equiv \frac{A_t}{A_s} \). A non-zero tensor-to-scalar ratio will affect the pattern of fluctuations in the power spectrum of the CMB, specifically in the polarization spectra. Within the context of CMB polarization spectra, there are two distinguished polarization geometries — E-modes and B-modes — which are named as such based on their respective curl and curl-free nature. These are sourced via different mechanisms. For example, when photons scatter off electrons at the surface of last scattering, they become effectively polarized. However, the resulting polarization directions must be either perpendicular or aligned with the photon wavevector. Thus, the scalar temperature fluctuations on the last-scattering surface can only impart a curl-free polarization pattern called E-modes. We are mainly focused on these modes since most of the perturbations we are interested in studying are scalar (e.g. temperature perturbations). Tensor perturbations, on the other hand, are not limited to sourcing only E-mode polarization patterns. These "swirling'' perturbations create both a curl-free E-mode pattern and a curled B-mode pattern. As we increase the tensor-to-scalar ratio, we expect to see the power amplitude of these B-modes to increase, deviating from the general power-law shape on our BB polarization spectrum.

Since these primordial gravitational waves have wavelengths on the order of the size of the observable universe at that time (large angular scales, low multipoles), we expect the amplitude increase to manifest at these scales as well, trailing off at higher multipoles (smaller angular scales) due to Silk damping.

Accurate measurement of the tensor-to-scalar ratio will provide enormous insight into the mechanisms present in the early universe. Since B-mode polarization signatures are unique to tensorial sources like gravitational effects, their search directly targets these perturbations while avoiding the larger E-modes altogether. Some proposed early universe models predict the formation of primordial gravitational waves, which could theoretically have a detectable effect via this B-mode search. Most notably of these early universe models, cosmic inflation hypothesizes that the very early universe went through a period of immense exponential expansion, leaving behind remnant gravitational waves. In other frameworks, such as in bouncing cosmology, there are not yet any well-defined mechanisms that can source tensor perturbations to the degree of inflationary models, although this is still an area of active research. Thus, a discovery of B-modes would differentiate possible cosmic models and hence there are numerous efforts for detection.

There have been numerous experiments aimed at detecting B-modes, as their discovery could provide strong evidence for inflation at large angular scales and help refine the tensor-to-scalar ratio r. Experiments like BICEP and POLARBEAR have pushed the boundaries of B-mode observation. However, detecting these signals is challenging due to several factors, such as the gravitational lensing of large-scale structures, which can convert E-modes into lensing B-modes that obscure the primordial signal. Additionally, foreground emissions such as thermal dust can mimic the B-mode signal, particularly at low multipoles. This was highlighted in the 2014 BICEP2 result, which initially claimed a detection of primordial B-modes but was later attributed to galactic dust. In response, projects like the Simons Observatory and CMB-S4 are working to overcome foreground contamination and reach the sensitivity needed to distinguish a true signal. Despite the multiple challenges, the potential to probe the physics of inflation ensures that the search for B-modes remains a central objective in cosmology.