Imprints of relic gravitational waves in cosmic microwave background radiation

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 cosmic microwave background (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 $C_{\ell }^{X{X}^{\prime }}$ for $X$, $X^{\prime }=T$, $E$, $B$ 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.

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 (27), while the dotted curves are analytical solutions on the scale factor (28).

Figure 2

The present-day spectra for $h_{\mathrm{rms}}(\nu )$ and ${\Omega }_{gw}(\nu )$. The solid lines correspond to the primordial spectral index $\beta =-1.9$, i.e. $\mathrm{n}=1.2$, while the dashed lines are for $\beta =-2$, i.e. $\mathrm{n}=1$.

Figure 3

Function ${\Phi }_n(\eta )$ for different values of $n$. The solid line is exact numerical solution to Eq. (61). The dashed line is the zero-order approximation (73). The dotted line shows the approximation (91) (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 (87), while the dashed line shows the approximation (88). The normalization has been chosen such that $h_n({\eta }_r)=1$.

Figure 6

Relative contributions of an individual Fourier mode $n=100$ to various multipoles $\ell$ in the power spectra $\ell (\ell +1)C_{\ell }$ for temperature and polarization. The normalization has been chosen arbitrarily but same in both the graphs.

Figure 7

The left panel shows (a) the power spectrum of temperature anisotropies $\ell (\ell +1)C_{\ell }^{TT}$ (in $\mu {\mathrm{K}}^2$) generated by (b) the power spectrum of g.w. metric perturbations (13), $\beta =-2$. The right panel shows (c) the power spectra of polarization anisotropies $\ell (\ell +1)C_{\ell }^{BB}$ (solid line) and $\ell (\ell +1)C_{\ell }^{EE}$ (dashed line), panel (d) shows the power spectrum of the first time derivative of the same g.w. field, Eq. (15).

Figure 8

The bottom panel shows the spectrum (96) of gravitational waves, whereas the top panel shows the angular power spectrum $\ell (\ell +1)C_{\ell }^{TE}$ caused by these waves. The negative values of these functions are depicted by broken lines.

Figure 9

The source function ${\Phi }_n(\eta )$ in reionization era for different wave numbers.

Figure 10

The contributions to the angular power spectra from an individual mode $n$. The dashed line is for $\ell (\ell +1)C_{\ell }^{EE}$ and the solid line is for $\ell (\ell +1)C_{\ell }^{BB}$.

Figure 11

The reionization bump in the polarization power spectra. The dashed lines show the spectra when the reionization era is ignored. The solid lines include both eras. The darker lines are for $B$-mode, and the lighter lines for $E$-mode.

Figure 12

The dotted line shows the contribution of density perturbations alone, and the dashed line shows the contribution of gravitational waves alone. The solid line is the sum of these contributions. It is seen from the graph that the inclusion of g.w. makes the total curve to be always below the d.p. curve.

Figure 13

The dashed line shows the best fit $\Lambda \mathrm{CDM}$ model without gravitational waves. The solid line shows a model with spectral index $n=1.2$ and gravitational waves $R=1$.

Figure 14

The summary of CMB temperature and polarization anisotropies due to relic gravitational waves with $n=1.2$ and $R=1$.

Figure 15

The graphs (a), (b), (c) show the quantities $q(\eta )$, $\tau (\eta )$, $g(\eta )$. The graph (d) is a zooming of $g(\eta )$ in the region of recombination. The dashed line shows a model without reionization. The solid line takes reionization into account, with ${\tau }_{\mathrm{reion}}=0.1$.