weak lensing of the cmb antony lewis institute of astronomy, cambridge
TRANSCRIPT
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Weak Lensing of the CMBAntony Lewis
Institute of Astronomy, Cambridgehttp://cosmologist.info/
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• From the beginning• Lensing order of magnitudes• Lensed power spectrum• Effect on CMB polarization• Cluster masses from CMB lensing
Outline
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Hu & White, Sci. Am., 290 44 (2004)
Evolution of the universe
Opaque
Transparent
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Source: NASA/WMAP Science Team
O(10-5) perturbations (+galaxy)
Dipole (local motion)
(almost) uniform 2.726K blackbody
Observations:the microwave sky today
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Where do perturbations come from?
Quantum Mechanics“waves in a box” calculation
vacuum state, etc…
Inflationmake >1030 times bigger
After inflationHuge size, amplitude ~ 10-5
New physics Known physics
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Perturbation evolution – what we actually observeCMB 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
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Hu & White, Sci. Am., 290 44 (2004)
CMB temperature power spectrumPrimordial perturbations + later physics
diffusiondampingacoustic oscillations
primordial powerspectrum
finite thickness
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Weak lensing of the CMB
Last scattering surface
Inhomogeneous universe - photons deflected
Observer
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Lensing order of magnitudes
β
Newtonian argument: β = 2 Ψ General Relativity: β = 4 Ψ
Ψ
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
assume uncorrelated
total deflection ~ 501/2 x 10-4
~ 2 arcminutes
(neglects angular factors, correlation, etc.)
(β << 1)
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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
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Full calculation: Lensed temperature depends on deflection angle:
Lensing PotentialDeflection angle on sky given in terms of lensing potential
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Deflections O(10-3), but coherent on degree scales important!
Deflection angle power spectrum
Computed with CAMB: http://camb.info
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LensPix sky simulation code:http://cosmologist.info/lenspixLewis 2005
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Lensing effect on CMB temperature power spectrum
Full-sky calculation accurate to 0.1%: Challinor & Lewis 2005, astro-ph/0502425
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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
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Thomson Scattering Polarization
W Hu
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CMB PolarizationGenerated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution
Hu astro-ph/9706147
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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
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E and B polarization
“gradient” modesE polarization
“curl” modes B polarization
e.g.
B modes only expected from gravitational waves and CMB lensing
e.g. cold spot
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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
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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
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Polarization lensing: Cx and CE
Lewis, Challinor : astro-ph/0601594
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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
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Current 95% indirect limits for LCDM given WMAP+2dF+HST
Polarization power spectra
Lewis, Challinor : astro-ph/0601594
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Cluster CMB lensing
GALAXYCLUSTER
Last scattering surface What we see
Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, etc.
CMB very smooth on small scales: approximately a gradient
Lewis & King, astro-ph/0512104
0.1 degrees
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Toy model: spherically symmetric NFW cluster
)()(
vrcrr
Ar
M200 ~ 1015 h-1 Msun
c ~ 5, z ~ 1 (rv ~ 1.6Mpc)Deflection ~ 0.7 arcmin
(approximate lens as thin, constrain projected density profile)
assume we know where centre is
2
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Unlensed Lensed Difference
RMS gradient ~ 13 μK / arcmindeflection from cluster ~ 1 arcmin Lensing signal ~ 10 μK
BUT: depends on CMB gradient behind a given cluster
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Constraining cluster parameters
Calculate P(c,M200 | observation)
Simulated realisations with noise 0.5 μK arcmin, 0.5 arcmin pixelsSomewhat futuristic: 160x lower noise 14x higher resolution than Planck; few times better than ACT
CMB approximately Gaussian – know likelihood function
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Unlensed T+Q+U Difference after cluster lensing
Add polarization observations?
Less sample variance – but signal ~10x smaller: need 10x lower noise
Plus side: SZ (etc) fractional confusion limit probably about the same as temperature
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0.5 μK arcmin 0.7 μK arcmin 0.07 μK arcmin
Temperature Polarisation Q and U
Noise:
less dispersion in error
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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
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Galaxy lensing comparisonMassive case: M = 1015 h-1 Msun, c=5
CMB temperature only (0.5 μK arcmin noise) Galaxies (100 gal/arcmin2)
(from expected log likelihoods)
Ground (30/arcmin)
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CMB temperature only (0.07 μK arcmin noise)
Optimistic Futuristic CMB polarization vs galaxy lensingLess massive case: M = 2 x 1014 h-1 Msun, c=5
Galaxies (500 gal/arcmin2)
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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
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Rest frame of CMB:
Redshiftedcolder
Blueshiftedhotter
Moving Lenses and Dipole lensing
Homogeneous CMB
Rest frame of lens: Dipole gradient in CMB
Deflected from colderdeflected from hotter
v
T = T0(1+v cos θ)
`Rees-Sciama’(non-linear ISW)
‘dipole lensing’
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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
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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
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Convergence dipole expected ~ 5 x 10-4
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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
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Physics Reports review: astro-ph/0601594
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http://CosmoCoffee.infoarXiv paper filtering, discussion and comments
Currently 420 registered readers
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Calculate Cl by series expansion in deflection angle?
Series expansion only good on large and very small scalesAccurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005No
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arXivJournal.org
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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/0412276Claims 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)