Weak Gravitational Lensing of the Cosmic Microwave Background

 

Yuuki Omori  (U.Chicago/KICP)

Credit: Aman Chokshi (2021 Winter Over)

Cosmic history

1

Cosmic microwave background

CMB

(source z~1100)

2

CMB weak lensing

CMB

(source z~1100)

2

signal peak

z~2

Source galaxies

CMB weak lensing

CMB

(source z~1100)

2

signal peak

z~2

Source galaxies

signal peak

z~0.5

CMB weak lensing

2

CMB

(source z~1100)

signal peak

z~2

Source galaxies

signal peak

z~0.5

CMB weak lensing

 

Undistorted temperature map

3

CMB weak lensing

 

Distorted temperature map

T

 

Undistorted temperature map

3

CMB weak lensing

 

Distorted temperature map

 

Undistorted stokes Q/U map

 

Distorted stokes Q/U map

T

Q

U

 

Undistorted temperature map

3

CMB weak lensing

 

Distorted temperature map

 

Undistorted E/B map

 

Distorted E/B map

T

E

B

Lensing reconstruction

3D gravitational potential

\Psi (\chi\hat{n},\chi)
\Psi

4

3D gravitational potential

\phi(\hat{n})=-2\int d\chi \frac{\chi_{\rm CMB}-\chi}{\chi\chi_{\rm CMB}} \Psi (\chi\hat{n},\chi)

2D potential

\Psi (\chi\hat{n},\chi)
\Psi
\phi

4

Lensing reconstruction

3D gravitational potential

\phi(\hat{n})=-2\int d\chi \frac{\chi_{\rm CMB}-\chi}{\chi\chi_{\rm CMB}} \Psi (\chi\hat{n},\chi)

2D potential

\Psi (\chi\hat{n},\chi)
\Psi
\phi
\vec{\alpha}=\vec{\nabla}\phi(\hat{n})

Deflection

4

{\kappa}=\vec{\nabla}^{2}\phi(\hat{n})

Convergence

Lensing reconstruction

T^{d}(\vec{n})=T^{u}(\vec{n}+\vec{\alpha})\\
=T^{u}(\vec{n})+\vec{\nabla} T^{u}(\vec{n}) \vec{\nabla}\phi(\hat{n})\\
\langle T^{u}_{\ell m} (T^{u}_{\ell' m'})^{*} \rangle_{\ell\neq\ell',m\neq m'}=0
\langle T^{d}_{\ell m} (T^{d}_{\ell' m'})^{*} \rangle_{\ell\neq\ell',m\neq m'}=\sum_{LM}(-1)^{M}\begin{pmatrix} \ell & \ell' & L\\ m & m' & M \end{pmatrix}W^{\phi}_{\ell L \ell'}\phi_{LM}

Two independent modes become correlated through the lensing potential

\vec{\alpha}=\vec{\nabla}\phi(\hat{n})

5

Lensing reconstruction

{\rm Convergence}\ (\kappa=-\vec{\nabla}^{2}\phi)
{\rm Lensing\ potential}\ (\phi)
{\rm Deflection}\ (\vec{\alpha}=\vec{\nabla}\phi)
{\rm Direction\ with\ more\ mass}
{\rm Direction\ with\ less\ mass}

SPT-3G lensing map

(see also: Omori+ 2017, Omori+ 2023)

6

Foregrounds (temperature)

7

SPT-3G temperature map

SPT-3G temperature map

SPT-CLJ0234-5831

7

Foregrounds (temperature)

SPT-3G temperature map

SPT-CLJ0234-5831

7

Foregrounds (temperature)

\langle T^{u}_{\ell m} (T^{u}_{\ell' m'})^{*} \rangle_{\ell\neq\ell',m\neq m}=0

SPT-3G temperature map

PKS 2356-61

SPT-CLJ0234-5831

7

Foregrounds (temperature)

8

Mitigation strategies for temperature biases

  • Vary input          .
     
  • Use deprojection techniques
    • (Madhavacheril&Hill2018, Raghunathan& Omori2023).
       
  • Use hardening techniques (point-source/profile hardening).
     
  • Use shear-based lensing estimator (Schaan & Ferraro 2019).
     
  • Use the combinations of above.
\ell_{\rm max}

Agora simulation (Omori 2022)

Multi-Dark Planck 2

N-body simulation 

(Klypin+ 2017)

9

tSZ

kSZ

CIB

radio

Foregrounds (temperature)

\kappa/\gamma

10

Polarization

T

Galaxy cluster

10

Polarization

  • Fewer astronomical sources are polarized (less prone to astrophysical systematic biases)

Galaxy cluster

T

Q

U

Point source

11

Polarization

  • Fewer astronomical sources are polarized (less prone to astrophysical systematic biases)
  • Less affected by atmospheric noise (lower 1/f noise)

12

Polarization

  • Instrumental noise in polarization is higher (and lensing reconstruction scales as noise  )
  • The survey needs to be sufficiently deep to fully take advantage of polarization

2

14

Lensing noise comparison

2\mu{\rm K}{\text -}{\rm arcmin}\\

Text

Experiment 1:

Experiment 2:

10\mu{\rm K}{\text -}{\rm arcmin}\\

SPT-3G -> P dominated 

SimonsObs -> non-negligible contribution from T

Baryons?

  • For CMB lensing auto-spectrum, the impact of Baryons is small (signal peaks at z~2) 
  • For cross-correlations, the impact depends on the redshift of the other tracer.

15

Current state of CMB lensing

Planck

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Current state of CMB lensing

Atacama Cosmology Telescope

17

Current state of CMB lensing

SPT-3G 2018 Imminent

SPT-3G 2019/2020 in a few months

South Pole Telescope

18

Comparison with galaxy weak lensing

19

Jeffery+ 2021

Map of        from the

Dark Energy Survey 

\kappa_{\rm E}
red border
red border
red border

DES galaxy lensing

map (overlay)

 SPT-3G CMB  lensing map (base)

20

(see also: Omori+2018, Chang+2023)

Comparison with galaxy weak lensing

21

Comparison with galaxy weak lensing

Qu+ 2023

Optimal lensing

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(Millea+ 2021; see also Carron+ 2019)

CMB lensing forward modeling

X^{d}(\vec{n})=X^{u}(\vec{n})+\vec{\nabla} X^{u}(\vec{n}) \vec{\nabla}\phi(\hat{n})\\
\langle X^{d}_{\ell m} (X^{d}_{\ell' m'})^{*} \rangle_{\ell\neq\ell',m\neq m'}=\sum_{LM}(-1)^{M}\begin{pmatrix} \ell & \ell' & L\\ m & m' & M \end{pmatrix}W^{\phi}_{\ell L \ell'}\phi_{LM}
+\mathcal{O}((\vec{\nabla}\phi(\hat{n}))^{2} )\\

+ higher

Credit: Marius Millea

Incoming data

Simons Observatory

South Pole Observatory

23

Incoming data

Simons Observatory

South Pole Observatory

23

Summary

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  • The most common approach to reconstruct a CMB lensing map: Quadratic estimator.
     
  • Temperature maps have astrophysical foregrounds in them, and some treatment has to be made before/after lensing reconstruction (we now have various mitigation techniques).
     
  • Polarization maps are cleaner but require low-noise surveys to produce powerful lensing maps.
     
  • Forward modeling (optimal) approaches are becoming increasingly important as the noise levels in the polarization channel continue to decrease.
     
  • Currently at an exciting time -- we already have good data, but the quality will become orders of magnitude better soon.