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Weak Lensing of the CMB - Antony Lewis

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					Weak Lensing of the CMB
               Antony Lewis
   Institute of Astronomy, Cambridge
          http://cosmologist.info/
                   Outline


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




Opaque




                       Transparent


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




                                                Dipole (local motion)




                                                            O(10-5) perturbations
                                                                       (+galaxy)


Observations:
the microwave
sky today




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




                               Inflation
                         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

                                                                   diffusion
                                           acoustic oscillations   damping




                                                                               primordial power
                                                                               spectrum




                                                     finite thickness
Hu & White, Sci. Am., 290 44 (2004)
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: 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: http://camb.info
LensPix sky simulation code:
http://cosmologist.info/lenspix
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.




         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
                                           statistics).
                                           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




                                         GALAXY
                                         CLUSTER




      0.1 degrees
      Toy model: spherically symmetric NFW cluster

                                           A
                              (r ) 
                                      r (cr  rv )
                                                   2




M200 ~ 1015 h-1 Msun
                                  Deflection
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

                         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



`Rees-Sciama‟
(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)


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
                 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
Convergence dipole expected ~ 5 x 10-4
                        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
Physics Reports review: astro-ph/0601594
http://CosmoCoffee.info
  arXiv paper filtering, discussion and comments




       Currently 420 registered readers
     Calculate Cl by series expansion in deflection angle?




      Series expansion only good on large and very small scales
No
      Accurate calculation uses correlation functions: Seljak 96; Challinor, Lewis 2005
arXivJournal.org
                         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)

				
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