The shape of the CMB lensing bispectrum

Lensing of the CMB generates a significant bispectrum, which should be detected by the Planck satellite at the 5-sigma level and is potentially a non-negligible source of bias for fNL estimators of local non-Gaussianity. We extend current understanding of the lensing bispectrum in several directions: (1) we perform a non-perturbative calculation of the lensing bispectrum which is ~ 10% more accurate than previous, first-order calculations; (2) we demonstrate how to incorporate the signal variance of the lensing bispectrum into estimates of its amplitude, providing a good analytical explanation for previous Monte-Carlo results; and (3) we discover the existence of a significant lensing bispectrum in polarization, due to a previously-unnoticed correlation between the lensing potential and E-polarization as large as 30% at low multipoles. We use this improved understanding of the lensing bispectra to re-evaluate Fisher-matrix predictions, both for Planck and cosmic variance limited data. We confirm that the non-negligible lensing-induced bias for estimation of local non-Gaussianity should be robustly treatable, and will only inflate fNL error bars by a few percent over predictions where lensing effects are completely ignored (but note that lensing must still be accounted for to obtain unbiased constraints). We also show that the detection significance for the lensing bispectrum itself is ultimately limited to 9 sigma by cosmic variance. The tools that we develop for non-perturbative calculation of the lensing bispectrum are directly relevant to other calculations, and we give an explicit construction of a simple non-perturbative quadratic estimator for the lensing potential and relate its cross-correlation power spectrum to the bispectrum. Our numerical codes are publicly available as part of CAMB and LensPix.

[1]  J. Peacock,et al.  Stable clustering, the halo model and non-linear cosmological power spectra , 2002, astro-ph/0207664.

[2]  Integral solution for the microwave background anisotropies in nonflat universes , 1997, astro-ph/9704265.

[3]  L. Verde,et al.  Non-Gaussianity and the CMB bispectrum: Confusion between primordial and lensing-Rees-Sciama contribution? , 2009, 0906.2317.

[4]  Edward J. Wollack,et al.  FIVE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE OBSERVATIONS: COSMOLOGICAL INTERPRETATION , 2008, 0803.0547.

[5]  Uros Seljak,et al.  Reconstruction of lensing from the cosmic microwave background polarization , 2003 .

[6]  C. Baccigalupi,et al.  Cosmic microwave background constraints on dark energy dynamics : Analysis beyond the power spectrum , 2004, astro-ph/0411702.

[7]  A. Melchiorri,et al.  Searching for integrated Sachs–Wolfe effect beyond temperature anisotropies: CMB E-mode polarization–galaxy cross-correlation , 2005, astro-ph/0511054.

[8]  M. Zaldarriaga,et al.  The Far-Infrared Background Correlation with Cosmic Microwave Background Lensing , 2003 .

[9]  Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background , 2007, astro-ph/0701921.

[10]  R. Sachs.  Gravitational waves in general relativity. VI. The outgoing radiation condition , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  A. Lewis,et al.  Cosmological parameters from CMB and other data: A Monte Carlo approach , 2002, astro-ph/0205436.

[12]  E. Komatsu,et al.  Cosmic Microwave Background-Weak Lensing Correlation: Analytical and Numerical Study of Nonlinearity and Implications for Dark Energy , 2007, 0711.1696.

[13]  F. Bernardeau,et al.  Full-sky lensing shear at second order , 2009, 0911.2244.

[14]  P. Serra,et al.  Impact of Secondary non-Gaussianities on the Search for Primordial Non-Gaussianity with CMB Maps , 2008, 0801.3276.

[15]  F. Bernardeau,et al.  The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations , 2010, 1003.0481.

[16]  Edward J. Wollack,et al.  Evidence for dark energy from the cosmic microwave background alone using the Atacama Cosmology Telescope lensing measurements. , 2011, Physical review letters.

[17]  Dark energy and cosmic microwave background bispectrum , 2001, astro-ph/0108179.

[18]  J. R. Bond,et al.  Cosmic confusion: degeneracies among cosmological parameters derived from measurements of microwave background anisotropies , 1998 .

[19]  A. Lewis,et al.  Weak gravitational lensing of the CMB , 2006, astro-ph/0601594.

[20]  Optimising Boltzmann codes for the PLANCK era , 2009, 0903.0382.

[21]  F. Vernizzi,et al.  The CMB bispectrum in the squeezed limit , 2011, 1109.1822.

[22]  Matias Zaldarriaga,et al.  Algorithms for bispectra: Forecasting, optimal analysis, and simulation , 2006, astro-ph/0612571.

[23]  C. Hirata,et al.  HyRec: A fast and highly accurate primordial hydrogen and helium recombination code , 2010, 1011.3758.

[24]  R. B. Barreiro,et al.  Planck early results. XVIII. The power spectrum of cosmic infrared background anisotropies , 2011, 1101.2028.

[25]  Wayne Hu,et al.  Mass Reconstruction with CMB Polarization , 2001 .

[26]  J. Chluba,et al.  Precise cosmological parameter estimation using CosmoRec , 2011, 1102.3683.

[27]  Wayne Hu,et al.  Cosmological information from lensed CMB power spectra , 2006 .

[28]  Estimators for local non-Gaussianities , 2006, astro-ph/0606001.

[29]  A. Lewis,et al.  CMB lensing and primordial squeezed non-gaussianity , 2012, 1201.1010.

[30]  R. Stompor,et al.  Gravitational lensing of cosmic microwave background anisotropies and cosmological parameter estimation , 1999 .

[31]  Eiichiro Komatsu,et al.  Hunting for primordial non-Gaussianity in the cosmic microwave background , 2010, 1003.6097.

[32]  Evolution of cosmological dark matter perturbations , 2002, astro-ph/0203507.

[33]  A. Lewis,et al.  Estimators for CMB statistical anisotropy , 2009, 0908.0963.

[34]  Antony Lewis,et al.  Likelihood Analysis of CMB Temperature and Polarization Power Spectra , 2008, 0801.0554.

[35]  A. Lewis.  Cosmological parameters from WMAP 5-year temperature maps , 2008, 0804.3865.

[36]  J. Lesgourgues,et al.  The Cosmic Linear Anisotropy Solving System (CLASS) IV: efficient implementation of non-cold relics , 2011, 1104.2935.

[37]  Forecasting cosmic parameter errors from microwave background anisotropy experiments , 1997, astro-ph/9702100.

[38]  A. Rukhin.  Matrix Variate Distributions , 1999, The Multivariate Normal Distribution.

[39]  Gravitational Lensing Effect on Cosmic Microwave Background Anisotropies: A Power Spectrum Approach , 1995, astro-ph/9505109.

[40]  Matias Zaldarriaga,et al.  Direct signature of an evolving gravitational potential from the cosmic microwave background , 1999 .

[41]  A. Lewis,et al.  Weak lensing of the CMB , 2009, 0911.0612.

[42]  Wayne Hu,et al.  Mass Reconstruction with Cosmic Microwave Background Polarization , 2002 .

[43]  Michael Doran.  CMBEASY: an object oriented code for the cosmic microwave background , 2005 .

[44]  S. Ho,et al.  Correlation of CMB with large-scale structure. II. Weak lensing , 2008, 0801.0644.

[45]  Matias Zaldarriaga,et al.  General solution to the E-B mixing problem , 2007 .

[46]  David N. Spergel,et al.  Measuring Primordial Non-Gaussianity in the Cosmic Microwave Background , 2003, astro-ph/0305189.

[47]  Michael Doran,et al.  Analyse this! A cosmological constraint package for CMBEASY , 2004 .

[48]  A. Lewis.  The real shape of non-Gaussianities , 2011, 1107.5431.

[49]  Antony Lewis,et al.  Lensed CMB simulation and parameter estimation , 2005, astro-ph/0502469.

[50]  K. Gorski,et al.  HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere , 2004, astro-ph/0409513.

[51]  Kendrick M. Smith,et al.  CMB lensing and primordial non-Gaussianity , 2009, 0905.4732.

[52]  A. Lewis,et al.  Efficient computation of CMB anisotropies in closed FRW models , 1999, astro-ph/9911177.

[53]  A. Moss,et al.  How well do we understand cosmological recombination , 2007, 0711.1357.

[54]  M. Zaldarriaga,et al.  Microwave Background Constraints on Cosmological Parameters , 1997, astro-ph/9702157.

[55]  A. Challinor,et al.  CMB temperature lensing power reconstruction , 2010, 1008.4403.

[56]  Comparison of cosmological Boltzmann codes: Are we ready for high precision cosmology? , 2003, astro-ph/0306052.

[57]  M. Zaldarriaga,et al.  Lensing of the CMB: non-Gaussian aspects. , 1999, Annals of the New York Academy of Sciences.

[58]  J. Lesgourgues,et al.  Massive neutrinos and cosmology , 2005, astro-ph/0603494.

[59]  Microwave background anisotropies from gravitational waves: The (1+3) covariant approach , 1999, astro-ph/9906474.

[60]  Non-Gaussianity in WMAP data due to the correlation of CMB lensing potential with secondary anisotropies , 2009, 0909.1837.

[61]  Wayne Hu.  Dark Synergy: Gravitational Lensing and the CMB , 2001 .

[62]  Edmund Bertschinger,et al.  Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges , 1995 .

[63]  Sara Seager,et al.  How Exactly Did the Universe Become Neutral? , 1999, astro-ph/9912182.

[64]  J. Maldacena.  Non-Gaussian features of primordial fluctuations in single field inflationary models , 2002, astro-ph/0210603.

[65]  B. Wandelt,et al.  LOCAL NON-GAUSSIANITY IN THE COSMIC MICROWAVE BACKGROUND THE BAYESIAN WAY , 2010, 1010.1254.

[66]  U. Seljak,et al.  A Line of sight integration approach to cosmic microwave background anisotropies , 1996, astro-ph/9603033.

[67]  Martin White,et al.  CMB anisotropies: Total angular momentum method , 1997, astro-ph/9702170.

[68]  J. Lesgourgues,et al.  The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes , 2011, 1104.2933.

[69]  Microwave background bispectrum. II. A probe of the low redshift universe , 1998, astro-ph/9811251.

[70]  CMB 3-point functions generated by nonlinearities at recombination , 2004, astro-ph/0405428.

[71]  B. Gold.  Limits of dark energy measurements from correlations of CMB lensing, the integrated Sachs-Wolfe effect, and galaxy counts , 2005 .

[72]  A. Lewis,et al.  Improving CMB non-Gaussianity estimators using tracers of local structure , 2010, 1009.1549.

[73]  Asantha Cooray,et al.  Secondary non-Gaussianity and cross-correlation analysis , 2009, 0907.3229.

[74]  B. D. Wandelt,et al.  Estimating the impact of recombination uncertainties on the cosmological parameter constraints from cosmic microwave background experiments , 2009, 0910.4383.

[75]  M. Remazeilles,et al.  CMB lensing reconstruction in real space , 2010, 1004.3285.

[76]  B. Wandelt,et al.  More on crinkles in the last scattering surface , 2009, 0903.0871.

[77]  Primordial helium recombination. I. Feedback, line transfer, and continuum opacity , 2007, astro-ph/0702143.

[78]  Berkeley,et al.  Constraining the dark energy dynamics with the cosmic microwave background bispectrum , 2003, astro-ph/0308118.

[79]  Matias Zaldarriaga,et al.  Primordial bispectrum information from CMB polarization , 2004 .

[80]  U. Seljak,et al.  Signature of gravity waves in polarization of the microwave background , 1996, astro-ph/9609169.

[81]  Wayne Hu,et al.  Cosmic microwave background lensing reconstruction on the full sky , 2003 .

[82]  Efficient cosmological parameter estimation from microwave background anisotropies , 2002, astro-ph/0206014.

[83]  Oliver Zahn,et al.  Detection of gravitational lensing in the cosmic microwave background , 2007, 0705.3980.

[84]  Wayne Hu,et al.  Weak lensing of the CMB: A harmonic approach , 2000, astro-ph/0001303.

[85]  A. Lewis.  Harmonic E / B decomposition for CMB polarization maps , 2003, astro-ph/0305545.

[86]  A. Lewis,et al.  Lensed CMB power spectra from all-sky correlation functions , 2005, astro-ph/0502425.

[87]  F. Cyr-Racine,et al.  Photons and baryons before atoms: Improving the tight-coupling approximation , 2010, 1012.0569.

[88]  Matias Zaldarriaga,et al.  Single field consistency relation for the 3-point function , 2004 .

[89]  S. Matarrese,et al.  Non-Gaussianity in the Cosmic Microwave Background Anisotropies at Recombination in the Squeezed limit , 2011, 1109.2043.

[90]  Edward J. Wollack,et al.  SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: POWER SPECTRA AND WMAP-DERIVED PARAMETERS , 2010, 1001.4635.

[91]  Anthony Challinor,et al.  The linear power spectrum of observed source number counts , 2011, 1105.5292.