Weak Lensing of the CMB - Antony Lewis

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					Weak Lensing of the CMB
               Antony Lewis
   Institute of Astronomy, Cambridge


•   From the beginning
•   Lensing order of magnitudes
•   Lensed power spectrum
•   Effect on CMB polarization
•   Cluster masses from CMB lensing
                                Evolution of the universe



Hu & White, Sci. Am., 290 44 (2004)
                      (almost) uniform 2.726K blackbody

                                                Dipole (local motion)

                                                            O(10-5) perturbations

the microwave
sky today

Source: NASA/WMAP Science Team
    Where do perturbations come from?
                   New physics                           Known physics

                         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

                                           acoustic oscillations   damping

                                                                               primordial power

                                                     finite thickness
Hu & White, Sci. Am., 290 44 (2004)
Weak lensing of the CMB

       Last scattering surface

                                 Inhomogeneous universe
                                  - photons deflected

          Lensing order of magnitudes

                  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:
LensPix sky simulation code:
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. 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
                                           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


      0.1 degrees
      Toy model: spherically symmetric NFW cluster

                              (r ) 
                                      r (cr  rv )

M200 ~ 1015 h-1 Msun
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


• 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

(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)


   • 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
Convergence dipole expected ~ 5 x 10-4
• 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
  arXiv paper filtering, discussion and comments

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     Calculate Cl by series expansion in deflection angle?

      Series expansion only good on large and very small scales
      Accurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005
                         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)