CMB polarization can constrain cosmology better than CMB temperature

We demonstrate that for a cosmic variance limited experiment, cosmic microwave background (CMB) $E$ polarization alone places stronger constraints on cosmological parameters than CMB temperature. For example, we show that $C_{\ell }^{\mathrm{EE}}$ can constrain parameters better than $C_{\ell }^{\mathrm{TT}}$ by up to a factor 2.8 when a multipole range of $\ell =30–2500$ is considered. We expose the physical effects at play behind this remarkable result and study how it depends on the multipole range included in the analysis. In most relevant cases, $C_{\ell }^{\mathrm{TE}}$ or $C_{\ell }^{\mathrm{EE}}$ surpass the $C_{\ell }^{\mathrm{TT}}$-based cosmological constraints. This result is important as the small-scale astrophysical foregrounds are expected to have a much reduced impact on polarization, thus opening the possibility of building cleaner and more stringent constraints of the $\mathrm{\Lambda }\mathrm{CDM}$ model. This is relevant especially for proposed future CMB satellite missions, such as CORE or PRISM, that are designed to be cosmic variance limited in polarization till very large multipoles. We perform the same analysis for a Planck-like experiment, and conclude that even in this case $C_{\ell }^{\mathrm{TE}}$ alone should determine the constraint on ${\mathrm{\Omega }}_c{h}^2$ better than $C_{\ell }^{\mathrm{TT}}$ by $\sim 15\%$, while determining ${\mathrm{\Omega }}_b{h}^2$, $n_s$ and $\theta$ with comparable accuracy. Finally, we explore a few classical extensions of the $\mathrm{\Lambda }\mathrm{CDM}$ model and show again that CMB polarization alone provides more stringent constraints than CMB temperature in the case of a cosmic variance limited experiment.

Figure 1

Derivatives of the $C_{\ell }^{\mathrm{TT}}$ (top panel), $C_{\ell }^{\mathrm{EE}}$ (middle panel) and $C_{\ell }^{\mathrm{TE}}$ (bottom panel) power spectrum with respect to the $\mathrm{\Lambda }\mathrm{CDM}$ parameters. The plot is in logarithmic scale, so we plot the absolute value of the derivatives, showing negative values as dashed lines.

Figure 2

Signal to noise for a Planck-like full mission experiment (solid lines) with $f_{\mathrm{sky}}=0.5$, as detailed in Table 2, for the $C_{\ell }^{\mathrm{TT}}$ (blue), $C_{\ell }^{\mathrm{EE}}$ (red) and $C_{\ell }^{\mathrm{TE}}$ (green) power spectra. Dashed lines show the signal to noise for a CVL experiment with $f_{\mathrm{sky}}=1$.

Figure 3

Standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\max }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\max }=2500$. We consider a CVL experiment with ${\ell }_{\min }=30$ and a prior on $\tau$. We consider here lensed CMB power spectra.

Figure 4

Left: same as Fig. 3, but for unlensed CMB power spectra. Right: same as Fig. 3, but the constraints in this case are calculated as the inverse of the diagonal of the Fisher matrix, i.e. they are not marginalized over degeneracies between parameters. By comparing with Fig. 3, one can determine whether constraints on the parameters are limited by their degeneracies or by the sensitivity of the spectra to each parameter separately.

Figure 5

Standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\min }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\min }=30$. We consider a CVL experiment with ${\ell }_{\max }=2500$ and a prior on $\tau$. We consider here lensed CMB power spectra.

Figure 6

Evolution of the correlations between selected parameters as a function of ${\ell }_{\min }$. We consider a CVL experiment with ${\ell }_{\max }=2500$ and a prior on $\tau$. We consider here lensed CMB power spectra.

Figure 7

Standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\max }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\max }=2500$. We consider a Planck-like full mission experiment with two channels at 143 and 217 GHz with ${\ell }_{\min }=30$ and a prior on $\tau$. We consider here lensed CMB power spectra.

Figure 8

Left: same as Fig. 7, but for unlensed spectra. Right: standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\min }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\min }=30$. We consider a Planck-like full mission experiment ${\ell }_{\max }=2500$ and a prior on $\tau$. We consider here lensed CMB power spectra.

Figure 9

Standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\max }$ without any prior on $\tau$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\max }=2500$. We consider a CVL experiment with ${\ell }_{\min }=30$ (left) and with ${\ell }_{\min }=2$ (right). We consider here lensed CMB power spectra.

Figure 10

Standard deviations on $\mathrm{\Lambda }\mathrm{CDM}$ parameters as a function of ${\ell }_{\max }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\max }=2500$. We consider a Planck-like experiment with ${\ell }_{\min }=30$ (left) and ${\ell }_{\min }=2$ (right), without any prior on $\tau$. We consider here lensed CMB power spectra.

Figure 11

Standard deviations on $\mathrm{\Sigma }{m}_{\nu }$ (first row), $N_{\text{eff}}$ (second row), $Y_p$ (third row) and $n_{\text{run}}$ as a function of ${\ell }_{\max }$, normalized to the standard deviation ${\sigma }^{\mathrm{ref}}$ obtained from $C_{\ell }^{\mathrm{TT}}$ with ${\ell }_{\max }=2500$. We consider a CVL experiment (left) and a Planck-like full mission experiment (right) with ${\ell }_{\min }=30$ and a prior on $\tau$.