Transcript Lensing of the CMB

Lensing of the CMB
Antony Lewis
Institute of Astronomy, Cambridge
http://cosmologist.info/
Review ref: Lewis, Challinor , Phys. Rep: astro-ph/0601594
Evolution of the universe
Opaque
Transparent
Hu & White, Sci. Am., 290 44 (2004)
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
acoustic oscillations
diffusion
damping
primordial power
spectrum
finite thickness
Hu & White, Sci. Am., 290 44 (2004)
Temperature anisotropy data: WMAP 3-year + smaller scales
BOOMERANG
Hinshaw et al
+ many more coming up
e.g. Planck (2008)
Weak lensing of the CMB
Last scattering surface
Inhomogeneous universe
- photons deflected
Observer
Not to scale!
All distances are comoving
largest overdensity
~200/14000 ~ degree
Ionized plasma - opaque
Recombination
Neutral gas - transparent
~200Mpc
14 000 Mpc
Good approximation: CMB is single source plane at ~14 000 Mpc
Angular diameter distance well measured by angle of acoustic peaks
~100Mpc
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
In detail, lensed temperature depends on deflection angle:
Lensing Potential
Deflection angle on sky given in terms of lensing potential
Deflection angle power spectrum
Non-linear
Linear
Deflections O(10-3), but coherent on degree scales  important!
Computed with CAMB: http://camb.info
Simulated full sky lensing potential and (magnified) deflection angle fields
Easily simulated assuming Gaussian fields
- just re-map points using Gaussian realisations of CMB and potential
Lensed temperature Cl
- convolution of unlensed Cl
- W is non-linear in lensing potential power
Essentially exact to order of weak lensing by Gaussian field
– very well understood effect on power spectra.
Non-linear Pk 0.2% on TT, ~5% on BB
Lewis, Challinor Phys. Rept. 2006 : astro-ph/0601594
Full-sky fully non-perturbative generalization of method by Seljak 1996
Lensing effect on CMB temperature power spectrum
CAMB’s 0.1% calculation; http://camb.info : Challinor & Lewis 2005, astro-ph/0502425
Lensing important at 500<l<3000
Dominated by SZ on small scales
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
or spin-2 field
- just like shear
E and B polarization
“gradient” modes
E polarization
e.g.
e.g. cold spot
“curl” modes
B polarization
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
B modes only expected from gravitational waves and CMB lensing
Lensing of polarization
• Polarization not rotated w.r.t. parallel transport (vacuum
is not birefringent)
• Q and U Stokes parameters simply re-mapped by the
lensing deflection field
e.g.
Last scattering
Observed
Polarization lensing: Cx and CE
Polarization lensing: CB
Nearly white BB spectrum
on large scales
Polarization power spectra
Current 95% indirect limits for LCDM given WMAP+2dF+HST
Lewis, Challinor : astro-ph/0601594
Non-Gaussianity
• Unlensed CMB expected to be close to Gaussian
• With lensing:
• For a FIXED lensing field, lensed field also Gaussian
• For VARYING lensing field, lensed field is non-Gaussian
• Specific form of non-Gaussianity
- e.g. 1 point still Gaussian, very small 3-point function
- should be able to distinguish from primordial non-Gaussianity
• Modifies covariance of lensed Cl (esp. BB)
• Can be used to learn about lensing potential – reconstruction
methods…
Likelihoods
• Small number of lensing modes: BB Cl correlated between l.
(Smith, Challinor, Rocha 2006)
• Correction to temperature likelihood is small; on full sky usual result
is quite good
Correct BB (and others) using covariance from simulations. Good approx is
Smith, Challinor, Rocha 2006
ASIDE: Also works for cut sky – can use for convergence power spectrum
For multiple redshift bins can generalise for correlated fields:
X= (k11,k22,k12,…)
for details see Hammimeche & Lewis (in prep).
Large scale lensing reconstruction
• As with galaxy lensing, ellipticities of hot and cold spots
could be used to constrain the lensing potential
• But diffuse, know source statistics, can use magnification
- need general method
• Think about fixed lensing potential: lensed CMB is then
Gaussian (T is Gaussian) but not isotropic
- use off-diagonal correlation to constrain lensing
potential
• Can show that ‘optimal’ quadratic estimator is
- simple function of filtered fields
Analogous results for CMB polarization
For more details see Hu astro-ph/0105424 or review; c.f. Metcalf & White 2007
e.g. estimate lensing potential power spectrum
- more information on cosmological parameters
Hu: astro-ph/0108090
(‘ideal’ is limit using non-optimal quadratic estimator)
e.g. reconstruct lensing potential field
• should correlate with other matter tracers
• Constrain large-scale matter distribution to redshift z ~ 6
• De-lens the CMB (remove B-mode lensing contamination to see
primordial B modes)
First claimed detection in cross-correlation (see talk by Olivier Doré)
(http://cosmocoffee.info discussion)
Reconstruction complications
• Limited by cosmic variance on T, other secondaries,
higher order terms
• Quadratic method useful but not optimal
-especially for polarization (Hirata&Seljak papers)
• Requires high resolution: effectively need lots of hot and
cold spots behind each potential
• Reconstruction with polarization is much better: no
cosmic variance in unlensed B
• Polarization reconstruction can in principle be used to
de-lens the CMB
- required to probe tensor amplitudes r <~ 10-4
- requires very high sensitivity and high resolution
Input
astro-ph/0306354
Quadratic (filtered)
Approx max likelihood
Other information in CMB lensing (>> arcminute)
• Lensed CMB power spectra contain essentially two new numbers:
- one from T and E, depends on lensing potential at l<300
- one from lensed BB, wider range of l
astro-ph/0607315
• Can break degeneracies in linear CMB: improve constraints on dark
energy, curvature, etc.
• May be able to probe neutrino masses ~ 0.04eV (must be there! see
astro-ph/0603494)
Cluster CMB lensing
e.g. to constrain cosmology via number counts
Seljak, Zaldarriaga, Dodelson, Vale, Holder, Lewis, King, Hu. Maturi,. etc.
CMB very smooth on small scales: approximately a gradient
What we see
Last scattering surface
GALAXY
CLUSTER
0.1 degrees
Need sensitive ~ arcminute resolution observations
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
Unlensed CMB unknown, but statistics well understood (background CMB Gaussian) :
can compute likelihood of given lens (e.g. NFW parameters) essentially exactly
Add polarization observations?
Unlensed T+Q+U
Difference after cluster lensing
Less sample variance – but signal ~10x smaller: need 10x lower noise
Note: E and B equally useful on these scales; gradient could be either
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)
- Re-scattered thermal SZ (freq)
- Kinetic SZ (higher order)
- Other lenses
Generally much cleaner
Fitting profiles. e.g. to measure mass and concentration
Optimistic Futuristic CMB polarization lensing vs galaxy lensing
e.g. M = 2 x 1014 h-1 Msun, c=5
Can stack
for constraints
from multiple
clusters
Lewis &
King 2006
CMB polarization only (0.07 μK arcmin noise)
Galaxies (500 gal/arcmin2)
General cluster mass reconstruction
• Can use quadratic reconstruction methods similar to
those on large scales
• Potential problems with bias due to large central
magnifications
- use full likelihood function (e.g. Hirata et al, though prior less clear)
- various ad hoc methods also work (Maturi, Hu..)
• Not competitive with galaxy lensing except possibly for
high redshift
• But systematics very different; may be useful crosscheck
CMB/Galaxy lensing comparison
CMB Lensing
Galaxy lensing
- single source plane, lenses 0.5<~z<~7
- many source planes, lenses <~1.5
- accurate source plane distance
- often only photo-z redshifts
- statistics of source plane well understood
- make no assumption about source
distribution
- systematics: pointing/beam uncertainty,
SZ, foregrounds,…
- systematics: PSF modelling,
source selection, noise bias, ….
- Small corrections from non-linear Pk
- Non-linear Pk crucial
- Smoothes temperature power spectrum
-magnification effect on source number
counts (e.g. smoothes baryon
oscillations; c.f. original Vallinotto talk)
- B modes generated by lensing of E
- Mixing of intrinsic alignment source plane
E and B fields by lensing
Lensing of 21cm
• Very similar to CMB lensing, but 21cm power spectrum much more
small scale power and many source planes/3D information
• Lensed angular power spectrum result simple generalization from
lensed CMB temperature
(Lewis & Challinor 2007
c.f. Mandel & Zaldarriaga 2006)
Cl(z=50,z=50)
Cl(z=50,z=52)
• Can reconstruct potential from lensed 21cm – lots of information in 3D
(Hilbert, Metcalf, White, Zaldarriaga, Zahn, Cooray... see Metcalf poster)
Summary
• Weak lensing of the CMB very important for precision cosmology
- changes power spectra at several percent
- potential confusion with primordial gravitational waves for r <~ 10-3
- introduces non-Gaussian signal
- well understood in theory – accurately modelled with linear theory +
small non-linear corrections
• Potential uses
- Break parameter degeneracies, improve parameter constraints
- Constrain cluster masses to high redshift
- Reconstruction of potential at 0.5 <~ z <~ 7
Correlation with the CMB temperature
very small except on largest scales
Cosmological parameters
Essential to model lensing; but little effect on basic parameter constraints
Planck (2007+) parameter constraint simulation
(neglect non-Gaussianity of lensed field; BB noise dominated so no effect on parameters)
Important effect, but using lensed CMB power spectrum gets ‘right’ answer
Lewis 2005
Moving Lenses and Dipole lensing
Rest frame of CMB:
Homogeneous CMB
‘Rees-Sciama’
(non-linear ISW)
v
Blueshifted
hotter
Rest frame of lens:
Redshifted
colder
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
Non-Gaussianity
(back to CMB temperature)
• Unlensed CMB expected to be close to Gaussian
• With lensing:
• For a FIXED lensing field, lensed field also Gaussian
• For VARYING lensing field, lensed field is non-Gaussian
Three point function: Bispectrum < T T T >
- Zero unless correlation <T Ψ>
• Large scale signal from ISW-induced T- Ψ correlation
• Small scale signal from non-linear SZ – Ψ correlation
Trispectrum: Connected four-point < T T T T>c
- Depends on deflection angle and temperature power spectra
- ‘Easily’ measurable for accurate ell > 1000 observations
Other signatures
- correlated hot-spot ellipticities
- Higher n-point functions
- Polarization non-Gaussianity
Bigger than primordial non-Gaussianity?
• 1-point function
- lensing only moves points around, so distribution at a point Gaussian
- But complicated by beam effects
• Bispectrum
- ISW-lensing correlation only significant on very large scales
- SZ-lensing correlation can dominate on very small scales
- On larger scales oscillatory primordial signal should be
easily distinguishable with Planck
Komatsu: astro-ph/0005036
• Trispectrum (4-point)
Basic inflation:
- most signal
in long thin quadrilaterals
Komatsu: astro-ph/0602099
Lensing:
- broader distribution, less
signal in thin shapes
Hu: astro-ph/0105117
Can only detect inflation signal from cosmic variance if fNL >~ 20
Lensing probably not main problem for flat quadrilaterals if single-field non-Gaussianity
No analysis of relative shape-dependence from e.g. curvaton??
Also non-Gaussianity in polarization…