Transcript Slide 1

CMB and cluster lensing
Antony Lewis
Institute of Astronomy, Cambridge
http://cosmologist.info/
Lewis & Challinor, Phys. Rept. 2006 : astro-ph/0601594
Lewis & King, PRD 2006 : astro-ph/0512104
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: deflection angle on sky given in terms of lensing potential
Lensed temperature given by
Lewis 2005,
astro-ph/0502469
LensPix sky simulation code:
http://cosmologist.info/lenspix
Lensed temperature Cl
and
linear in lensing potential power spectrum
Analogous results for CMB polarization.
Essentially exact to order of weak lensing – 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:
smoothing of acoustic peaks; small scale power
Full-sky calculation accurate to 0.1%: Fortran code CAMB (http://camb.info)
Polarization lensing: Cx and CE
Important ~ 10% smoothing effect
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 power spectra
Current 95% indirect limits for LCDM given WMAP+2dF+HST
Lewis, Challinor : astro-ph/0601594; Lewis Moriond 2006
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
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
Zaldarriaga astro-ph/9910498, Goldberg&Spergel, etc…
Trispectrum: Connected four-point < T T T T>c
- Depends on deflection angle and temperature power spectra
- ‘Easily’ measurable for accurate ell > 1000 observations
Zaldarriaga astro-ph/9910498; Hu astro-ph/0105117
Other signatures
- correlated hot-spot ellipticities
- Higher n-point functions
- Polarization non-Gaussianity
Confusion with primordial non-Gaussianity?
• 1-point function
- lensing only moves points around, so distribution at a point Gaussian
- But complicated by beam effects
Kesden, Cooray, Kamionkowski: astro-ph/0208325
• 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 if large enough
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??
Cluster CMB lensing
e.g. to constrain cosmology via number counts
Lewis & King, astro-ph/0512104
Following: Seljak, Zaldarriaga, Dodelson, Vale, Holder, 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
Is CMB lensing better than galaxy lensing?
• Assume background galaxy shapes random before lensing
• Measure ellipticity after lensing by cluster
Lensing
•
•
•
•
On average ellipticity measures reduced shear
Shear is γab = ∂<a αb>
Constrain cluster parameters from predicted shear
Assume numerous systematics negligible…
Optimistic Futuristic CMB polarization lensing vs galaxy lensing
Less massive case: M = 2 x 1014 h-1 Msun, c=5
CMB polarization only (0.07 μK arcmin noise)
Galaxies (500 gal/arcmin2)
Summary
• Weak lensing of the CMB very important for precision cosmology
- changes power spectra
- potential confusion with primordial gravitational waves for r <~ 10-3
- Non-Gaussian signal, but well known and probably not main problem
• Cluster lensing of CMB
- Temperature lensing difficult because of confusions
- CMB polarisation lensing needs high sensitivity but potentially useful
at high redshift
- galaxy lensing expected to be much better for low redshift clusters
- CMB lensing has quite different systematics to galaxy lensing
Planck (2007+) parameter constraint simulation
(neglect non-Gaussianity of lensed field)
Lewis 2005,
astro-ph/0502469
Important effect, but using lensed CMB power spectrum gets ‘right’ answer
Parameters can be improved using BB/lensing reconstruction;
non-Gaussianity important in the future; c.f. Wayne Hu’s talk
Full calculation: Lensed temperature depends on deflection angle:
Lensing Potential
Deflection angle on sky given in terms of lensing potential
Toy model: spherically symmetric NFW cluster
A
 (r ) 
2
r (cr  rv )
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