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Forecasting Cosmic Parameter Errors from Microwave Background Anisotropy Experiments

J. Richard Bond, George Efstathiou, Max Tegmark

TL;DR

The paper forecasts cosmological parameter errors for high-resolution CMB experiments MAP and Planck using a Fisher-matrix approach, with the Fisher matrix F_{ij} = \sum_\ell (\Delta C_\ell)^{-2} (\partial C_\ell/\partial s_i)(\partial C_\ell/\partial s_j) and parameter covariance M = F^{-1}. It adopts an 11-parameter set including matter densities via ω_j = Ω_j h^2, initial-condition amplitudes/tilts, Y_{He}, τ_C, and tensor-scalar parameters (r_{ts}, n_t), optimized for accurate C_l derivatives computed with two Boltzmann codes to better than 1% accuracy. Applying the method to MAP, Planck, and LDB scenarios shows Planck can constrain Ω_b h^2 and h to a few percent or better across target models, while degeneracies such as the angle-distance between Ω_k h^2 and Ω_Λ h^2 persist without priors; priors and external information can mitigate these. The work demonstrates the substantial potential of CMB observations to dramatically sharpen cosmological parameter estimates, while highlighting the need for precise derivative calculations and thoughtful prior choices to obtain robust forecasts.

Abstract

Accurate measurements of the cosmic microwave background (CMB) anisotropies with an angular resolution of a few arcminutes can be used to determine fundamental cosmological parameters such as the densities of baryons, cold and hot dark matter, and certain combinations of the cosmological constant and the curvature of the Universe to percent-level precision. Assuming the true theory is a variant of inflationary cold dark matter cosmologies, we calculate the accuracy with which cosmological parameters can be determined by the next generation of CMB satellites, MAP and Planck. We pay special attention to: (a) the accuracy of the computed derivatives of the CMB power spectrum C_L; (b) the number and choices of parameters; (c) the inclusion of prior knowledge of the values of various parameters.

Forecasting Cosmic Parameter Errors from Microwave Background Anisotropy Experiments

TL;DR

The paper forecasts cosmological parameter errors for high-resolution CMB experiments MAP and Planck using a Fisher-matrix approach, with the Fisher matrix F_{ij} = \sum_\ell (\Delta C_\ell)^{-2} (\partial C_\ell/\partial s_i)(\partial C_\ell/\partial s_j) and parameter covariance M = F^{-1}. It adopts an 11-parameter set including matter densities via ω_j = Ω_j h^2, initial-condition amplitudes/tilts, Y_{He}, τ_C, and tensor-scalar parameters (r_{ts}, n_t), optimized for accurate C_l derivatives computed with two Boltzmann codes to better than 1% accuracy. Applying the method to MAP, Planck, and LDB scenarios shows Planck can constrain Ω_b h^2 and h to a few percent or better across target models, while degeneracies such as the angle-distance between Ω_k h^2 and Ω_Λ h^2 persist without priors; priors and external information can mitigate these. The work demonstrates the substantial potential of CMB observations to dramatically sharpen cosmological parameter estimates, while highlighting the need for precise derivative calculations and thoughtful prior choices to obtain robust forecasts.

Abstract

Accurate measurements of the cosmic microwave background (CMB) anisotropies with an angular resolution of a few arcminutes can be used to determine fundamental cosmological parameters such as the densities of baryons, cold and hot dark matter, and certain combinations of the cosmological constant and the curvature of the Universe to percent-level precision. Assuming the true theory is a variant of inflationary cold dark matter cosmologies, we calculate the accuracy with which cosmological parameters can be determined by the next generation of CMB satellites, MAP and Planck. We pay special attention to: (a) the accuracy of the computed derivatives of the CMB power spectrum C_L; (b) the number and choices of parameters; (c) the inclusion of prior knowledge of the values of various parameters.

Paper Structure

This paper contains 11 sections, 3 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Parameter estimation with priors
  3. The covariance matrix
  4. Choice of variables
  5. Choice of target models
  6. Accuracy of the power spectrum derivatives
  7. Inclusion of prior information on parameters
  8. Cosmological Parameter Errors from MAP and Planck
  9. The CMB power spectrum estimated from MAP and Planck
  10. Accuracy of cosmological parameters
  11. Conclusions

Figures (1)

  • Figure 1: Temperature power spectra for a standard CDM (SCDM) model with ${\rm h}=0.5$, $\Omega_{m}=1$ and $\Omega_{\Lambda}=0$ and an open CDM (OCDM) model with ${\rm h}=0.6$ , $\Omega_{m} = 0.33$ and $\Omega_{\Lambda}=0$. The points show a quadratic power spectrum estimation of ${\rm C}_\ell$ (in bands of width 5% in $\ell$), along with the one sigma error, from simulated CMB skies observed by the MAP and Planck satellites described in Section 3 and Table 1. The errors in the left panels use the MAP$^{+}$ parameters for the three highest frequency channels and two years of observing, while those in the right panels use the four lowest frequency Planck HFI channels. Planck can follow the theoretical curves quite precisely far down the damping tail for both models. In the bottom right panel, a ${\rm C}_\ell$ curve with $\Omega_m=0.15$, $\Omega_{\Lambda}=0.44$ and ${\rm h}=0.9$ almost degenerate with the OCDM model is superposed. The inset, showing these two and also a $\Omega_m=0.10$, $\Omega_{\Lambda}=0.58$ and ${\rm h}=1.1$ model, demonstrate that the angle-distance degeneracy relation (see text) used to define these models is almost exact; these models can be distinguished only at low multipoles no matter how precise the CMB experiment.