Model-independent forecasts of CMB angular power spectra for the Planck mission

The Planck mission, designed for making measurements of the cosmic microwave background (CMB) radiation with unprecedented accuracy and angular resolution, is expected to release its entire data in the near future. In this paper, we provide model-independent forecasts for the $TT$, $EE$, and $TE$ angular power spectra for the Planck mission using synthetic data based on the best-fit Lambda cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$) model. The nonparametric function estimation methodology we use here is based on the agnostic viewpoint of allowing the data to speak for themselves rather than letting the models decide what is inferred from the data. Our analysis indicates that the three Planck angular power spectra will be determined sufficiently well for $2\le l\lesssim l_{\max }$, where $l_{\max }=2500\left(TT\right)$, $1377(EE)$, and $1727\text{(}TE\text{)}$ respectively. A key signature of reionization, namely, a bump at low values of $l$, is evident in our forecasts for the $EE$ and $TE$ power spectra. Nonparametric confidence bands in the phase shift (${\varphi }_m$) versus acoustic scale ($l_A$) plane, corresponding to the first eight peaks in the $TT$ power spectrum, show a confluence region for $300\lesssim l_A\lesssim 305$ which is in good agreement with the estimate $l_A=300$ based on the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model. From our results, we expect that the final Planck data should lead to accurate model-independent estimates of CMB angular power spectra using our nonparametric regression formalism.

Figure 1

A realization of simulated $TT$ power spectrum data for the Planck mission, generated using FuturCMB [23]. Black points: data including noise; red points: simulated data after subtracting noise.

Figure 2

A realization of simulated $EE$ power spectrum data for the Planck mission, generated using FuturCMB [23]. Black points: data including noise; red points: simulated data after subtracting noise.

Figure 3

A realization of simulated $TE$ power spectrum data for the Planck mission, generated using FuturCMB [23]. FuturCMB assumes the noise to be zero for the $TE$ data.

Figure 4

$TT$ nonparametric fits. Blue: full-freedom fit ($\mathrm{EDoF}\approx 72$), red: restricted-freedom fit ($\mathrm{EDoF}=27$), black: best-fit $\mathrm{\Lambda }\mathrm{CDM}$ spectrum, grey: simulated data realization.

Figure 5

$EE$ nonparametric fits. Blue: full-freedom fit ($\mathrm{EDoF}\approx 190$), red: restricted-freedom fit ($\mathrm{EDoF}=24$), black: best-fit $\mathrm{\Lambda }\mathrm{CDM}$ spectrum, grey: simulated data realization.

Figure 6

$TE$ nonparametric fits. Blue: full-freedom fit ($\mathrm{EDoF}\approx 95$), red: restricted-freedom fit ($\mathrm{EDoF}=40$), black: best-fit $\mathrm{\Lambda }\mathrm{CDM}$ spectrum, grey: simulated data realization.

Figure 7

The results of a probe of the confidence sets for the $TT$ (red), $EE$ (blue), and $TE$ (green) nonparametric restricted-freedom fits to the synthetic Planck data, to determine how well the fits are expected to be determined by the data alone. The quantity plotted for each data realization is the total vertical variation at each $l$ within the respective 95% ($2\sigma$) confidence set, divided by the absolute value of the fit (assumed nonzero): Values $\ll 1$ indicate that the fit is tightly determined by the data, whereas values $\gtrsim 1$ indicate that the data contain very little information about the height of the angular power spectrum at that $l$. Disregarding the low-$l$ region for the $EE$ fit, and spikes for the $TE$ fit (which arise from nearly zero fitted $C_l$ values), the marked vertical lines indicate the approximate $l$ value at which each curve rises above 1.

Figure 8

95% confidence boxes the locations and heights of peaks and dips in the $TT$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $TT$ data (grey points). The number of acceptable spectral variations sampled from the 95% confidence set is 5000. These uncertainties are tabulated in Table 1.

Figure 9

95% confidence boxes the locations and heights of peaks and dips in the $EE$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $EE$ data (grey points). The number of acceptable spectral variations sampled from the 95% confidence set is 5000. These uncertainties are tabulated in Table 2.

Figure 10

95% confidence boxes the locations and heights of peaks and dips in the $TE$ fit. Black curve is the restricted-freedom monotone fit to the synthetic Planck $TE$ data (grey points). The number of acceptable spectral variations sampled from the 95% confidence set is 5000. These uncertainties are tabulated in Table 3.

Figure 11

The 95% confidence intervals on peak locations in the $EE$ fit (red) plotted against the corresponding confidence intervals in the $TT$ fit (blue). Also plotted are the peak location pairs corresponding to the best-fit $\mathrm{\Lambda }\mathrm{CDM}$ model (black dots), and points (green) based on the theoretical expectation $(m+0.5)/m$. All the plotted quantities are by and large consistent with each other, indicating that the expected behavior of out-of-phase peak locations is indeed vindicated by the data.

Figure 12

Confidence “bands” for the acoustic scale $l_A$ and the shift ${\varphi }_m$ for the $m$th peak, as derived from the 95% confidence intervals on the first eight peak locations of estimated $TT$ power spectrum.

Figure 13

Confidence “bands” for the acoustic scale $l_A$ and the shift ${\varphi }_m$ for the $m$th peak, as derived from the 95% confidence intervals on the first four acoustic peak locations of estimated $EE$ power spectrum.