Imprints of Relic Gravitational Waves in Cosmic Microwave Background Radiation

TL;DR

This work develops a first-principles radiative-transfer framework to predict CMB temperature and polarization from a stochastic relic GW background, reducing the problem to a Volterra-type integral equation for the polarization source. It shows that the low-$\ell$ TE cross-correlation is negative for gravitational waves and positive for density perturbations, making TE a powerful GW diagnostic alongside the anticipated $BB$ signal. The authors connect theory to observations, arguing that the WMAP TE anticorrelation near $\ell\approx30$ supports a significant GW component and present models with $n>1$ and nonzero GW content consistent with $TT$, $TE$, and $EE$, while predicting a potentially detectable $BB$ signal at $\ell\sim90$. These results offer a concrete pathway to identify relic GWs with current and upcoming CMB experiments, providing insights into the early Universe and inflationary alternatives.

Abstract

A strong variable gravitational field of the very early Universe inevitably generates relic gravitational waves by amplifying their zero-point quantum oscillations. We begin our discussion by contrasting the concepts of relic gravitational waves and inflationary `tensor modes'. We explain and summarize the properties of relic gravitational waves that are needed to derive their effects on CMB temperature and polarization anisotropies. The radiation field is characterized by four invariants I, V, E, B. We reduce the radiative transfer equations to a single integral equation of Voltairre type and solve it analytically and numerically. We formulate the correlation functions C^{XX'}_{\ell} for X, X'= T, E, B and derive their amplitudes, shapes and oscillatory features. Although all of our main conclusions are supported by exact numerical calculations, we obtain them, in effect, analytically by developing and using accurate approximations. We show that the TE correlation at lower \ell's must be negative (i.e. an anticorrelation), if it is caused by gravitational waves, and positive if it is caused by density perturbations. This difference in TE correlation may be a signature more valuable observationally than the lack or presence of the BB correlation, since the TE signal is about 100 times stronger than the expected BB signal. We discuss the detection by WMAP of the TE anticorrelation at \ell \approx 30 and show that such an anticorrelation is possible only in the presence of a significant amount of relic gravitational waves (within the framework of all other common assumptions). We propose models containing considerable amounts of relic gravitational waves that are consistent with the measured TT, TE and EE correlations.

Figures (15)

• Figure 1: The g.w. mode functions $h_n(\eta)/h_n(\eta_r)$ in a matter-radiation universe. The solid curves are numerically calculated solutions on the scale factor (\ref{['scalemr']}), while the dotted curves are analytical solutions on the scale factor (\ref{['jointsf']}).
• Figure 2: The present-day spectra for $h_{rms}(\nu)$ and $\Omega_{gw}(\nu)$. The solid lines correspond to the primordial spectral index $\beta =-1.9$, i.e. ${\rm n} =1.2$, while the dashed lines are for $\beta= -2$, i.e. ${\rm n} =1$.
• Figure 3: Function $\Phi_n(\eta)$ for different values of $n$. The solid line is exact numerical solution to Eq. (\ref{['integralequation']}). The dashed line is the zero order approximation (\ref{['zerothorderapproximation1']}). The dotted line shows the approximation (\ref{['zerothorderapproximation2']}) (see below). The g.w. mode functions are normalized such that $h_n(\eta_r)=1$.
• Figure 4: The 'source' functions $H_{n,s}(\eta)$ and $\Phi_{n,s}(\eta)$ ($n=100$) of temperature and polarization anisotropies (the normalization is chosen such that $h_n(\eta_r) = 1$).
• Figure 5: The contributions to the power spectrum $\ell(\ell+1)C_{\ell}^{TT}$ from an individual mode $n$. The solid line shows the exact result calculated according to (\ref{['a_Tapprox1']}), while the dashed line shows the approximation (\ref{['a_Tapprox']}). The normalization has been chosen such that $h_n(\eta_r)=1$.