Dark Synergy: Gravitational Lensing and the CMB

TL;DR

This work assesses how weak gravitational lensing of the CMB and cosmic shear from galaxies can break key CMB parameter degeneracies, particularly for the dark-energy equation of state $w$ and dark-energy clustering, and it forecasts the gains from combining primary CMB data with lensing and tomographic cosmic shear using a Fisher-matrix framework. It develops a nine-parameter background/dark-energy model including perturbations with a rest-frame sound speed $c_ ext{eff}$, and provides explicit integral-form expressions for the power spectra and their cross-correlations, including the ISW signal via $W^{\Theta_{\rm ISW}}(D)=-2\dot\Phi/\Phi$. The analysis highlights the complementary strengths and limitations of CMB lensing versus galaxy weak lensing: lensing probes high-redshift structure and can directly test dark-energy clustering but contaminates $B$-modes relevant for tensor constraints, while cosmic shear offers tomographic leverage to tighten $w$, $ au$, and tensor bounds when systematics are controlled. The results show substantial degeneracy-breaking potential from deflection and shear measurements for Planck-like and ideal surveys, and emphasize the need to incorporate realistic systematics in future joint analyses. Overall, the paper articulates a coherent framework for leveraging lensing cross-correlations to enhance our understanding of dark energy and early-universe physics.

Abstract

Power spectra and cross-correlation measurements from the weak gravitational lensing of the cosmic microwave background (CMB) and the cosmic shearing of faint galaxies images will help shed light on quantities hidden from the CMB temperature anisotropies: the dark energy, the end of the dark ages, and the inflationary gravitational wave amplitude. Even with modest surveys, both types of lensing power spectra break CMB degeneracies and they can ultimately improve constraints on the dark energy equation of state w by over an order of magnitude. In its cross correlation with the integrated Sachs-Wolfe effect, CMB lensing offers a unique opportunity for a more direct detection of the dark energy and enables study of its clustering properties. By obtaining source redshifts and cross-correlations with CMB lensing, cosmic shear surveys provide tomographic handles on the evolution of clustering correspondingly better precision on the dark energy equation of state and density. Both can indirectly provide detections of the reionization optical depth and modest improvements in gravitational wave constraints which we compare to more direct constraints. Conversely, polarization B-mode contamination from CMB lensing, like any other residual foreground, darkens the prospects for ultra-high precision on gravitational waves through CMB polarization requiring large areas of sky for statistical subtraction. To evaluate these effects we provide fitting formula for the evolution and transfer function of the Newtonian gravitational potential.

Figures (11)

• Figure 1: CMB power spectra in the fiducial model with $w=-1$ (solid) versus a dark energy model with $w=-2/3$ (dashed) and other parameters chosen to preserve the angular diameter distance and amplitude degeneracies (see text). Boxes represent $1\sigma$ errors on band powers for the Planck experiment and an ideal experiment out to $l=3000$ (see Tab. \ref{['tab:exp']}).
• Figure 2: ISW effect in the $w=-1$ fiducial model compared with models with $w=-2/3$ and sound speeds $c_{\rm eff}=1,1/3$ with other parameters held fixed. The ISW effect is highly sensitive to the equation of state and clustering properties of the dark energy but only becomes a substantial fraction of the total temperature anisotropy power spectrum at the lowest multipoles.
• Figure 3: CMB lensing power spectra for the fiducial $w=-1$ model (solid) and the degenerate $w=-2/3$ model (dashed) of Fig. \ref{['fig:cmb']}. Boxes represent 1$\sigma$ errors on band powers assuming the Planck and ideal experiments of Tab. \ref{['tab:exp']}. Top: deflection power spectra. Bottom: cross correlation of deflection and temperature fields.
• Figure 4: Shear power spectra for three redshift bands $z<1$, $1< z < 1.5$ and $z>1.5$ for the fiducial model (solid) and the degenerate $w=-2/3$ model of Fig. \ref{['fig:cmb']}. Error boxes represent $1\sigma$ errors on band powers appropriate to the survey parameters of Tab. \ref{['tab:wlexp']}, $Z_{25}$ and $Z_{1000}$.
• Figure 5: Cross correlation of cosmic shear with the CMB temperature in three redshift bands for the fiducial model (solid) and the degenerate $w=-2/3$ model of Fig. \ref{['fig:cmb']}. Errors are appropriate for Planck and lensing surveys with $1000$ deg$^{2}$ and all of the $65\%$ of sky covered by Planck.