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Weak Lensing of the CMB Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/ Outline • From the beginning • Lensing order of magnitudes • Lensed power spectrum • Effect on CMB polarization • Cluster masses from CMB lensing Evolution of the universe Opaque Transparent Hu & White, Sci. Am., 290 44 (2004) (almost) uniform 2.726K blackbody Dipole (local motion) O(10-5) perturbations (+galaxy) Observations: the microwave sky today Source: NASA/WMAP Science Team Where do perturbations come from? New physics Known physics Inflation make >1030 times bigger Quantum Mechanics “waves in a box” calculation vacuum state, etc… After inflation Huge size, amplitude ~ 10-5 Perturbation evolution – what we actually observe CMB monopole source till 380 000 yrs (last scattering), linear in conformal time scale invariant primordial adiabatic scalar spectrum photon/baryon plasma + dark matter, neutrinos Characteristic scales: sound wave travel distance; diffusion damping length CMB temperature power spectrum Primordial perturbations + later physics diffusion acoustic oscillations damping primordial power spectrum finite thickness Hu & White, Sci. Am., 290 44 (2004) Weak lensing of the CMB Last scattering surface Inhomogeneous universe - photons deflected Observer Lensing order of magnitudes Ψ β Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ (β << 1) Potentials linear and approx Gaussian: Ψ ~ 2 x 10-5 β ~ 10-4 Characteristic size from peak of matter power spectrum ~ 300Mpc Comoving distance to last scattering surface ~ 14000 MPc pass through ~50 lumps total deflection ~ 501/2 x 10-4 assume uncorrelated ~ 2 arcminutes (neglects angular factors, correlation, etc.) So why does it matter? • 2arcmin: ell ~ 3000 - on small scales CMB is very smooth so lensing dominates the linear signal • Deflection angles coherent over 300/(14000/2) ~ 2° - comparable to CMB scales - expect 2arcmin/60arcmin ~ 3% effect on main CMB acoustic peaks Full calculation: Lensed temperature depends on deflection angle: Lensing Potential Deflection angle on sky given in terms of lensing potential Deflection angle power spectrum Deflections O(10-3), but coherent on degree scales important! Computed with CAMB: http://camb.info LensPix sky simulation code: http://cosmologist.info/lenspix Lewis 2005 Lensing effect on CMB temperature power spectrum Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/0502425 Planck (2007+) parameter constraint simulation (neglect non-Gaussianity of lensed field) Important effect, but using lensed CMB power spectrum gets „right‟ answer Lewis 2005, astro-ph/0502469 Thomson Scattering Polarization W Hu CMB Polarization Generated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution Hu astro-ph/9706147 Polarization: Stokes‟ Parameters - - Q U Q → -Q, U → -U under 90 degree rotation Q → U, U → -Q under 45 degree rotation Rank 2 trace free symmetric tensor E and B polarization “gradient” modes “curl” modes E polarization B polarization e.g. e.g. cold spot B modes only expected from gravitational waves and CMB lensing Why polarization? • E polarization from scalar, vector and tensor modes (constrain parameters, break degeneracies) • B polarization only from vector and tensor modes (curl grad = 0) + non-linear scalars Polarization lensing: CB Nearly white BB spectrum on large scales Lensing effect can be largely subtracted if only scalar modes + lensing present, but approximate and complicated (especially posterior statistics). Hirata, Seljak : astro-ph/0306354, Okamoto, Hu: astro-ph/0301031 Lewis, Challinor : astro-ph/0601594 Polarization lensing: Cx and CE Lewis, Challinor : astro-ph/0601594 Primordial Gravitational Waves • Well motivated by some inflationary models - Amplitude measures inflaton potential at horizon crossing - distinguish models of inflation • Observation would rule out other models - ekpyrotic scenario predicts exponentially small amplitude - small also in many models of inflation, esp. two field e.g. curvaton • Weakly constrained from CMB temperature anisotropy - significant power only at l<100, cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc) Look at CMB polarization: „B-mode‟ smoking gun Polarization power spectra Current 95% indirect limits for LCDM given WMAP+2dF+HST Lewis, Challinor : astro-ph/0601594 Cluster CMB lensing Lewis & King, astro-ph/0512104 Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, etc. CMB very smooth on small scales: approximately a gradient Last scattering surface What we see GALAXY CLUSTER 0.1 degrees Toy model: spherically symmetric NFW cluster A (r ) r (cr rv ) 2 M200 ~ 1015 h-1 Msun Deflection c ~ 5, z ~ 1 (rv ~ 1.6Mpc) ~ 0.7 arcmin (approximate lens as thin, constrain projected density profile) assume we know where centre is RMS gradient ~ 13 μK / arcmin deflection from cluster ~ 1 arcmin Lensing signal ~ 10 μK BUT: depends on CMB gradient behind a given cluster Unlensed Lensed Difference Constraining cluster parameters CMB approximately Gaussian – know likelihood function Calculate P(c,M200 | observation) Simulated realisations with noise 0.5 μK arcmin, 0.5 arcmin pixels Somewhat futuristic: 160x lower noise 14x higher resolution than Planck; few times better than ACT Add polarization observations? Unlensed T+Q+U Difference after cluster lensing Less sample variance – but signal ~10x smaller: need 10x lower noise Plus side: SZ (etc) fractional confusion limit probably about the same as temperature Temperature Polarisation Q and U Noise: 0.5 μK arcmin 0.7 μK arcmin 0.07 μK arcmin less dispersion in error Is it better than galaxy lensing? • Assume galaxy shapes random before lensing • Measure ellipticity after lensing Lensing • On average ellipticity measures reduced shear • Shear is γab = ∂<a αb> • Constrain cluster parameters from predicted shear Galaxy lensing comparison Massive case: M = 1015 h-1 Msun, c=5 (from expected log likelihoods) Ground (30/arcmin) CMB temperature only (0.5 μK arcmin noise) Galaxies (100 gal/arcmin2) Optimistic Futuristic CMB polarization vs galaxy lensing Less massive case: M = 2 x 1014 h-1 Msun, c=5 CMB temperature only (0.07 μK arcmin noise) Galaxies (500 gal/arcmin2) CMB Complications • Temperature - Thermal SZ, dust, etc. (frequency subtractable) - Kinetic SZ (big problem?) - Moving lens effect (velocity Rees-Sciama, dipole-like) - Background Doppler signals - Other lenses • Polarization - Quadrupole scattering (< 0.1μK) - Kinetic SZ (higher order) - Other lenses Generally much cleaner Moving Lenses and Dipole lensing Rest frame of CMB: Homogeneous CMB `Rees-Sciama‟ (non-linear ISW) v Blueshifted Redshifted hotter colder Rest frame of lens: Dipole gradient in CMB T = T0(1+v cos θ) „dipole lensing‟ deflected from hotter Deflected from colder Moving lenses and dipole lensing are equivalent: •Dipole pattern over cluster aligned with transverse cluster velocity – source of confusion for anisotropy lensing signal • NOT equivalent to lensing of the dipole observed by us, - only dipole seen by cluster is lensed (EXCEPT for primordial dipole which is physically distinct from frame-dependent kinematic dipole) Note: • Small local effect on CMB from motion of local structure w.r.t. CMB (Vale 2005, Cooray 2005) • Line of sight velocity gives (v/c) correction to deflection angles from change of frame: generally totally negligible Observable Dipoles • Change of velocity: - Doppler change to total CMB dipole - aberration of observed angles (c.f. dipole convergence) • Can observe: actual CMB dipole: (non-linear) local motion + primordial contribution • Can observe: Dipole aberration (dipole convergence + kinetic aberration) • So: Lensing potential dipole „easily‟ observable to O(10-5) - Can find zero-aberration frame to O(10-5) by using zero total CMB-dipole frame - change of frame corresponds to adding some local kinematic angular aberration to convergence dipole - zero kinematic aberration and zero kinematic CMB dipole frame = Newtonian gauge Convergence dipole expected ~ 5 x 10-4 Summary • Weak lensing of the CMB very important for precision cosmology - changes power spectra - potential confusion with primordial gravitational waves • Cluster lensing of CMB - gravitational lensing so direct probe of mass (not just baryons) - mass constraints independent of galaxy lensing constraints; source redshift known very accurately, should win for high redshifts - galaxy lensing expected to be much better for low redshift clusters - polarisation lensing needs high sensitivity but cleaner and less sample variance than temperature Physics Reports review: astro-ph/0601594 http://CosmoCoffee.info arXiv paper filtering, discussion and comments Currently 420 registered readers Calculate Cl by series expansion in deflection angle? Series expansion only good on large and very small scales No Accurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005 arXivJournal.org Is this right? • Lieu, Mittaz, ApJ L paper: astro-ph/0409048 - Claims shift in CMB peaks inconsistent with observation - ignores effect of matter. c.f. Kibble, Lieu: astro-ph/0412275 • Lieu, Mittaz, ApJ paper:astro-ph/0412276 Claims large dispersion in magnifications, hence peaks washed out - Many lines of sight do get significant magnification - BUT CMB is very smooth, small scale magnification unobservable - BUT deflection angles very small - What matter is magnifications on CMB acoustic scales i.e. deflections from large scale coherent perturbations. This is small. - i.e. also wrong • Large scale potentials < 10-3 : expect rigorous linear argument to be very accurate (esp. with non-linear corrections)

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Dark Energy, Dark Matter, the Dark, cosmological parameters, cosmic shear, power spectrum, photometric redshift, Neutrino mass, energy models, Gravitational Lensing

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posted: | 9/7/2011 |

language: | English |

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