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Lensed CMB simulation and parameter estimation

Antony Lewis

TL;DR

The paper tackles obtaining robust cosmological parameter constraints in the presence of CMB weak lensing, by developing a fast full-sky lensed-map simulator that uses pixel remapping rather than a slow series expansion in the deflection angle. It demonstrates that, at Planck sensitivity, parameter estimation using a Gaussian likelihood with the lensed power spectra $\widetilde{C}_l$ yields accurate results, though a full non-Gaussian likelihood would be required for more sensitive data. A Planck-like simulation shows that lensing must be modeled consistently to avoid bias, yet the overall parameter constraints remain largely unaffected in width. The authors provide publicly available code for both the lensed simulations and the parameter-estimation pipeline, and discuss limitations related to recombination history and higher-order lensing effects for future, more precise measurements.

Abstract

Modelling of the weak lensing of the CMB will be crucial to obtain correct cosmological parameter constraints from forthcoming precision CMB anisotropy observations. The lensing affects the power spectrum as well as inducing non-Gaussianities. We discuss the simulation of full sky CMB maps in the weak lensing approximation and describe a fast numerical code. The series expansion in the deflection angle cannot be used to simulate accurate CMB maps, so a pixel remapping must be used. For parameter estimation accounting for the change in the power spectrum but assuming Gaussianity is sufficient to obtain accurate results up to Planck sensitivity using current tools. A fuller analysis may be required to obtain accurate error estimates and for more sensitive observations. We demonstrate a simple full sky simulation and subsequent parameter estimation at Planck-like sensitivity. The lensed CMB simulation and parameter estimation codes are publicly available.

Lensed CMB simulation and parameter estimation

TL;DR

The paper tackles obtaining robust cosmological parameter constraints in the presence of CMB weak lensing, by developing a fast full-sky lensed-map simulator that uses pixel remapping rather than a slow series expansion in the deflection angle. It demonstrates that, at Planck sensitivity, parameter estimation using a Gaussian likelihood with the lensed power spectra yields accurate results, though a full non-Gaussian likelihood would be required for more sensitive data. A Planck-like simulation shows that lensing must be modeled consistently to avoid bias, yet the overall parameter constraints remain largely unaffected in width. The authors provide publicly available code for both the lensed simulations and the parameter-estimation pipeline, and discuss limitations related to recombination history and higher-order lensing effects for future, more precise measurements.

Abstract

Modelling of the weak lensing of the CMB will be crucial to obtain correct cosmological parameter constraints from forthcoming precision CMB anisotropy observations. The lensing affects the power spectrum as well as inducing non-Gaussianities. We discuss the simulation of full sky CMB maps in the weak lensing approximation and describe a fast numerical code. The series expansion in the deflection angle cannot be used to simulate accurate CMB maps, so a pixel remapping must be used. For parameter estimation accounting for the change in the power spectrum but assuming Gaussianity is sufficient to obtain accurate results up to Planck sensitivity using current tools. A fuller analysis may be required to obtain accurate error estimates and for more sensitive observations. We demonstrate a simple full sky simulation and subsequent parameter estimation at Planck-like sensitivity. The lensed CMB simulation and parameter estimation codes are publicly available.

Paper Structure

This paper contains 11 sections, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Weak lensing of the CMB
  3. Series expansion
  4. Re-mapping points
  5. Parameter estimation
  6. Gaussian $C_l$ Likelihood function
  7. Sampling from the posterior
  8. WMAP data
  9. Planck-like simulation
  10. Conclusions
  11. Harmonics and map making

Figures (3)

  • Figure 1: The effect of lensing on the CMB power spectra. The top plots show the fractional change in the temperature spectrum $C_l^{TT}$ and the lensing-induced B-polarization spectrum $C_l^{BB}$. The bottom plots show the lensed (grey/red, less peaked) and unlensed (black) T-E cross-correlation $C_l^{TE}$ and E-polarization $C_l^{EE}$ power spectra. All results are for the fiducial model given in the text, and the lensed B-mode power spectrum shown is not very accurate due to the neglect of non-linear evolution in the lensing potential.
  • Figure 2: Power spectra from realizations of the 1st, 2nd, 3rd and 4th order terms in the lensing potential series expansion of the lensed temperature $\widetilde{T}_{lm}$ compared to the full lensed $\widetilde{C}_l$ (top). The spectra may contain some pixelization error.
  • Figure 3: Parameter constraints from a simple Planck-like simulation. Solid lines analyse the lensed sky assuming Gaussianity with the lensed CMB power spectra, dashed lines are for an unlensed sky analysed with the unlensed power spectra, dash-dotted lines show the (inconsistent) result from analysing the lensed sky using the unlensed power spectra. The bottom row shows the dark energy density, age and Hubble constant derived parameters.