Detecting relic gravitational waves in the CMB: Comparison of different methods

In this paper, we discuss the constraint on the relic gravitational waves by both temperature and polarization anisotropies power spectra of cosmic microwave background radiation. Taking into account the instrumental noises of the Planck satellite, we calculate the signal-to-noise ratio $S/N$ by the simulation and the analytic approximation methods. We find that, compared with the $BB$ channel, the value of $S/N$ is much improved in the case where all the power spectra, $TT$, $TE$, $EE$, and $BB$, are considered. If the noise power spectra of the Planck satellite increase for some reason, the value of $S/N$ in the $BB$ channel is much reduced. However, in the latter case where all the power spectra of cosmic microwave background radiation are considered, the value of $S/N$ is less influenced. We also find that the free parameters $A_s$, $n_s$, and $n_t$ have little influence on the value of $S/N$ in both cases.

Figure 1

The distribution of $r_p$ in the 1000 realization. The black dots (upper left panel) denote the result in the $B$ case, the red dots (upper right panel) are the result in the $CT$ case, the magenta dots (lower left panel) are the result in the $CTE$ case, and the blue dots (lower right panel) denote the result in the $CTEB$ case. In all these panels, we have considered one free parameter $r$ in the likelihood analysis. The input simulated data are up to ${\ell }_{\max }=100$, and the input value is $\widehat{r}=0.3$.

Figure 2

The value of $S/N$ depends on the input value $\widehat{r}$. The black, red, magenta, and blue dots (lines) are the simulation (analytic approximation) results in the $B$ case, $CT$ case, $CTE$ case, and $CTEB$ case, respectively.

Figure 3

The same graph as in Fig. 2, the only difference is that in this figure we have assumed the realistic noise power spectra $N_{\ell }^{XX}$ are 4 times larger than the Planck instrumental noises.

Figure 4

This figure shows the individual signal-to-noise ratio $(S/N{)}_{\ell }^2$ as a function of $\ell$. The data presented in left panel is the result with the Planck instrumental noises, and the data presented in right panel is the result in the case where we assume the realistic noise power spectra $N_{\ell }^{XX}$ are 4 times larger than the Planck instrumental noises.

Figure 5

This figure shows three quantities of $({\widehat{C}}_{\ell }^{X{X}^{\prime }}(gw)/\Delta D_{\ell }^{X{X}^{\prime }}{)}^2$ ($X{X}^{\prime }=TE$, $TT$, $EE$) as a function of $\ell$. As the combination of these three components, the individual signal-to-noise ratios $(S/N{)}_{\ell }^2$ in the $CTE$ case are plotted in solid lines. In the left panel, we have considered the Planck instrumental noises, and in the right panel, we have assumed the realistic noise power spectra are 4 time larger than the Planck instrumental noises.

Figure 6

The locations of the maxima from the 300 realization projected into $n_t-r_p$ (upper panels), and $n_t-r_p^{*}$ (lower panels) planes. Black (red, magenta, blue) dots denote the results in the $B$ ($CT$, $CTE$, $CTEB$) case. In all these graphs, we have considered two free parameters $(r,n_t)$ in the likelihood analysis. The input simulated data are up to ${\ell }_{\max }=100$, and the sign “$+$” denotes the input values of the parameters.

Figure 7

The locations of the maxima from the 300 realization projected into $n_t-r_p$ (upper panels), and $n_t-r_p^{*}$ (lower panels) planes. Red (magenta, blue) dots denote the results in the $CT$ ($CTE$, $CTEB$) case. In all these graphs, we have considered four free parameters $(r,n_t,A_s^{\prime },n_s)$ in the likelihood analysis. The input simulated data are up to ${\ell }_{\max }=500$, and the sign “$+$” denotes the input values of the parameters.