Formulating Weak Lensing from the Boltzmann Equation and Application to Lens-lens Couplings

Abstract

The Planck mission has conclusively detected lensing of the Cosmic Microwave Background (CMB) radiation from foreground sources to an overall significance of greater than $25\sigma$. The high precision of this measurement motivates the development of a more complete formulation of the calculation of this effect. While most effects on the CMB anisotropies are widely studied through direct solutions of the Boltzmann equation, the non-linear effect of CMB lensing is formulated through the solutions of the geodesic equation. In this paper, we present a new formalism to the calculation of the lensing effect by \emph{directly solving the Boltzmann equation}, as we did in the calculation of the CMB anisotropies at recombination. In particular, we developed a diagrammatic approach to efficiently keep track of all the interaction terms and calculate all possible non-trivial correlations to arbitrary high orders. Using this formalism, we explicitly articulate the approximations required to recover the usual remapping approach used in current studies of the weak lensing. In addition, we point out additional unexplored corrections that are manifest in our formalism to which experiments may be sensitive. As an example, we calculate the correction to the CMB temperature power spectrum for the \emph{lens-lens} coupling effects which are neglected in standard calculations. We find that the correction is $\lesssim 0.1\%$ of the CMB temperature power spectrum for $\ell$ up to 3000 and thus is comparable to the cosmic variance.

25 pages, 3 figures, 4 tables, CMB, lensing

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• The intrinsic bispectrum of the Cosmic Microwave Background [PhD thesis]
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• Gauge-Invariant Formalism of Cosmological Weak Lensing
• Second-order gauge-invariant formalism for the cosmological observables: Complete verification of their gauge-invariance
• Geodesic "curve"-of-sight formulae for the cosmic microwave background: a unified treatment of redshift, time delay, and lensing
• A New Solution for the Observed Isotropic Cosmic Birefringence Angle and its Implications for the Anisotropic Counterpart through a Boltzmann Approach
• A new line-of-sight approach to the non-linear Cosmic Microwave Background
• Cosmology using numerical relativity
• Tetrad formalism for exact cosmological observables
• Higher-Order Gravitational Lensing Reconstruction using Feynman Diagrams
• The intrinsic bispectrum of the CMB from isocurvature initial conditions
• Weak lensing "post-Born" effects are equivalent to pure lens-lens couplings
• Living in a Non-Flat Universe: Theoretical Formalism