Last scattering surface
Inhomogeneous universe
- photons deflected
Observer
CMB Lensing
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)
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
Deflections O(10-3), but coherent on degree scales important!
Deflection angle power spectrum
Computed with CAMB: http://camb.info
Linear
Non-linear
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
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
- convolution of unlensed Cl
- W is non-linear in lensing potential power
Lensing effect on CMB temperature power spectrum
CAMB’s 0.1% calculation; http://camb.info : Challinor & Lewis 2005, astro-ph/0502425
Current 95% indirect limits for LCDM given WMAP+2dF+HST
Polarization power spectra
Lewis, Challinor : astro-ph/0601594
Bispectrum in ultra-squeezed limit
Lensing potential-temperature correlation Slope of the lensed temperature
power spectrum
Reduced bispectrum
Creminelli & Zaldarriagaa 2004; c.f. Maldacena 2003, Creminelli & Zaldarriaga 2004 for primordial bispectrum
+ similar argument for shear
Why is there a correlation between large-scale
lenses and the temperature?
Overdensity: magnification correlated with positive Integrated Sachs-Wolfe (net blueshift)
Underdensity: demagnification correlated with negative Integrated Sachs-Wolfe (net redshift)
(small-scales: also SZ , Rees-Sciama..)
- Lensing bispectrum depends on power difference: has phase shift
compared to any adiabatic primordial bispectrum (and different scale
dependence)
BUT:
Lensing
CMB temperature lensing
Lensing bispectrum also squeezed triangles but quite distinctive
See Lewis, Challinor & Hanson 2010 (in prep) for details
Lensing field is FIXED:
Anisotropic Gaussian temperature distribution
- Different parts of the sky magnified and demagnified
- Re-construct the actual lensing field
Lensing field is RANDOM:
Non-Gaussian statistically isotropic temperature distribution
- Significant connected 4-point function
- Excess variance to anisotropic-looking realizations
- Lensed temperature power spectrum
We see only one sky - both interpretations can be useful
Trispectrum – or anisotropy?
Anisotropy estimators
Maximum likelihood:
First iteration solution: Quadratic Maximum Likelihood (QML)
Hanson & Lewis, 0908.0963
Hanson, Challinor & Lewis, 0911.0612
Reconstruct statistical anisotropy
in WMAP data?
E.g. primordial power spectrum
anisotropy anomaly?
Look for direction-dependence in primordial power spectrum:
Simple case:
e.g.
Ackerman et.al. astro-ph/0701357
Gumrukcuoglu et al 0707.4179
• Reconstruct g(k).
QML estimator:
Quadrupole primordial
power asymmetry??
Hanson & Lewis, 0908.0963
Very significant evidence for
~ 10% quadrupole angular dependence!
Variance from
simulations
a(k)=1
Dashed: KQ75
Solid: KQ85
WMAP team agree!: Bennet et al. 1001.4758
Fully explained by analytic model of beam asymmetries
can be explained as correlated noise
No detection..
Consistent with Pullen et al 2010
constraint from large-scale structure 1003.0673
Hanson, Lewis & Challinor 1003.0198
Back to CMB lensing..
Use Quadratic Maximum Likelihood estimators to
reconstruct the lensing potential:
.. then estimate lensing potential power spectrum
Hu: astro-ph/0108090 (‘ideal’ is limit using non-optimal quadratic estimator)
Observed CMB temperature power spectrum
Primordial perturbations + known physics with unknown parameters
Observations Constrain theory of early universe
+ evolution parameters and geometry
Nolta et al.
Already will help with Planck
Perotto et al. 2009
(simulation)
Neutrino mass fraction
with and without
lensing (Planck only)
Perotto et al. 2006
Probe 0.5<~ z <~ 6: depends on geometry and matter power spectrum
CMB polarization only (0.07 μK arcmin noise)
Optimistic Futuristic CMB polarization lensing vs galaxy lensing
e.g. M = 2 x 1014 h-1 Msun, c=5
Galaxies (500 gal/arcmin2)
Lewis &
King 2006
CMB cluster lensing. e.g. to measure mass and concentration
Can stack
for constraints
from multiple
clusters
14 May 2009
Planck and the future, 2009+
High sensitivity and resolution
CMB temperature and polarization
Summary • Weak lensing of the CMB very important for precision cosmology
- changes power spectra at several per cent - potential confusion with primordial gravitational wave B-modes for r <~ 10-3 - introduces non-Gaussian signal; important potential confusion for Planck - well understood in theory – accurately modelled with linear theory + small non-linear corrections
• Potential uses - Break parameter degeneracies, improve parameter constraints : dark energy, neutrino mass, curvature - Reconstruction of potential at 0.5 <~ z <~ 7 - cluster lensing at high redshift