Cosmic Complementarity: Joint Parameter Estimation from CMB Experiments and Redshift Surveys

TL;DR

This study demonstrates that a 13-parameter adiabatic CDM framework exhibits strong parameter degeneracies when constrained by CMB data alone, but combining future CMB measurements (MAP/Planck) with redshift surveys (SDSS/2dF) yields substantially tighter, more robust constraints. The authors formulate a Fisher-matrix forecast, show how baryon-induced features and the sound horizon in the matter power spectrum help break the angular-diameter-distance degeneracy, and quantify improvements across fiducial cosmologies, including effects from massive neutrinos and tensor modes. They also examine the impact of assumptions (foreground removal, lensing, $k_{max}$), the benefits of smaller parameter spaces, and consistency checks with other cosmological data such as SN Ia and $H_0$ measurements. Overall, the results indicate that Planck+SDSS-like data can deliver percent-level precision on key parameters, with neutrino mass and tensor signals becoming accessible under favorable conditions, while emphasizing the crucial role of cross-validation and careful numerical treatment in Fisher analyses.

Abstract

We study the ability of future CMB anisotropy experiments and redshift surveys to constrain a thirteen-dimensional parameterization of the adiabatic cold dark matter model. Each alone is unable to determine all parameters to high accuracy. However, considered together, one data set resolves the difficulties of the other, allowing certain degenerate parameters to be determined with far greater precision. We treat in detail the degeneracies involving the classical cosmological parameters, massive neutrinos, tensor-scalar ratio, bias, and reionization optical depth as well as how redshift surveys can resolve them. We discuss the opportunities for internal and external consistency checks on these measurements. Previous papers on parameter estimation have generally treated smaller parameter spaces; in direct comparisons to these works, we tend to find weaker constraints and suggest numerical explanations for the discrepancies.

Figures (7)

• Figure 1: Our most used model along with 1--$\sigma$ band-power error bars from MAP and Planck. The model is $\Omega_m=0.35$, $h=0.65$, $\Omega_B=0.05$, $\Omega_\Lambda=0.65$, $\Omega_\nu=0.0175$, $\tau=0.05$, $n_S(k_{\rm fid})=1$, and $T/S=\alpha=0$. a) The temperature power spectrum $\Delta T = [\ell (\ell+1)C_{T\ell}/2\pi]^{1/2}$. b) The $E$-channel polarization power spectrum $\Delta T= [\ell (\ell+1)C_{E\ell}/2\pi]^{1/2}$; the panel shows a blow-up of the large-angle feature caused by reionization. MAP errors are the lighter, larger boxes; Planck errors are the darker, smaller boxes. The bands reflect an averaging over many $\ell$; the actual experiments will have finer $\ell$ resolution (and correspondingly larger errors). MAP will be able to average the polarization bands together to get a marginal detection at $\ell\approx150$, but Planck can trace out the full curve. Note that because polarization and temperature are correlated, the significance of detecting a change in parameters using both data sets is not simply given by the combination of errors from each.
• Figure 2: The power spectrum for the model of Figure \ref{['fig:fidmodel']}. SDSS BRG 1--$\sigma$ error bars are superposed. Dashed boxes are for $k>k_{\rm max}=0.1h{\rm\,Mpc}^{-1}$; we neglect information from these scales unless otherwise noted.
• Figure 3: The derivative $d(\ln C_{T\ell})/d\Omega_\nu h^2$ as a function of neutrino fraction $\Omega_\nu/\Omega_m$. Top to bottom (for the first peak): 0.5%, 1%, 2%, 5%, and 20%. The cosmology is the $\Omega_m=0.35$$\Lambda$CDM model, but with $\tau=0.1$, for a variety of $\Omega_\nu$.
• Figure 4: Constraint regions in the $\Omega_m$-$h$ plane from various combinations of data sets. MAP data with polarization yields the ellipse from upper left to lower right; assuming the universe flat yields a small region (short-dashed line). SDSS ($k_{\rm max}=0.1h{\rm\,Mpc}^{-1}$) gives the vertical shaded region; combined with MAP gives the small filled ellipse. A projection of future supernovae Ia results (Teg98a 1998a, middle prediction) gives the solid vertical lines as bounds; combining this with MAP yields the solid ellipse. A direct 10% measurement of $H_0$ gives the long-dashed lines and ellipse. All regions are 68% confidence. The fiducial model is the $\Omega_m=0.35$$\Lambda$CDM model.
• Figure 5: As Figure \ref{['fig:omegahubble']}, but for constraints in the $\Omega_m$-$\Omega_\Lambda$ plane. Lines and shadings are unchanged in meaning. Unlike Figure \ref{['fig:omegahubble']}, assuming the universe flat (short-dashed line) yields a line, not an ellipse. SDSS-only constraints are not shown.