2014/01/30 KICP seminar, U Chicago

Full-sky formulae forweak lensing power spectra fromtotal angular momentum method
YAMAUCHI, Daisuke
Research Center for the Early Universe (RESCEU),
U. TokyoReferences
Weak lensing generated by vector perturbations and detectability ofcosmic strings
DY, A. Taruya, T. Namikawa, JCAP08(2012)030, arXiv:1205.2139.
Full-sky formulae for weak lensing power spectra from total angularmomentum method
DY, A. Taruya, T. Namikawa, JCAP08(2013)051, arXiv:1305.3348.
Constraining cosmic string parameters with curl mode of CMB lensing
T. Namikawa, DY, A. Taruya, PRD88, 083525, arXiv:1308.6068.

Daisuke Yamauchi (RESCEU)

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INTRODUCTION

Daisuke Yamauchi (RESCEU)

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Gravitational Lensing= method to ``see’’ invisibles

NASA/ESAWEAK LENSING observations can provide a direct evidence forthe intervening ``Scalar/Vector/Tensor modes’’ along a line of sight bymeasuring the spatial patterns of the deformation of the photon path.
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Imaging survey
e.g. HSC, DES, LSST, Euclid…

Weak lensing measurement
CMB lensing
e.g. ACT, SPT, Planck, PolarBear,ACTPol, SPTPol, COrE,…

5Imaging survey
e.g. HSC, DES, LSST, Euclid…

Weak lensing measurement
CMB lensing
e.g. ACT, SPT, Planck, PolarBear,ACTPol, SPTPol, COrE,…

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Cosmic shear
What we can measure in the shear measurement is
the shape of galaxies modified by large scale structure.
Galaxies randomly
distributed

Intrinsic shape ofa background galaxy

WL

z=zS
z=zL

Galaxy shape actuallyseen after WL

z=0

Slight alignment

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Jacobi map
The shape of galaxies modified by lensing is characterizedby the deformation of the two-dim spatial pattern:

2dim→2dim map

φoriginalθ
convergence κshear γ（rotation 8ω）

We should solve the Sachs equation for the Jacobi map:eφeθsymmetric optical tidal matrixn Considering only scalar metric perturbations, the shear(trace-free part of Jacobi map) has the well-known form;
geometry grav. potential

Daisuke Yamauchi (RESCEU)

9E-/B-mode decomposition
The shear field γab can be decomposed into even-paritypart (E-mode) and odd-parity part (B-mode).
ScalarVectorTensorE-modeB-modeThe non-vanishing B-mode shear signal would be a direct
evidence for non-scalar metric perturbations.

10Imaging survey
e.g. HSC, DES, LSST, Euclid…

Weak lensing measurement
CMB lensing
e.g. ACT, SPT, Planck, PolarBear,ACTPol, SPTPol, COrE,…

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CMB lensing
What we observe is a subtly distorted version ofthe primary CMB anisotropy.

Unlensed CMB map

Foreground
perturbations

Daisuke Yamauchi (RESCEU)

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Deflection field
The distortion effect of lensing on the primary CMB isexpressed by a remapping with the deflection angle “Δ”.

+
Deflection field:
Unlensed :Lensed :[figures : Hu+Okamoto(2002)]
Daisuke Yamauchi (RESCEU)

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Deflection field (contd.)
The apparent position modified by the deflection angle(that is what we really see) appears as a solution of thegeodesic equation for the CMB photon.

Δn For scalar metric perturbations,Only gradient-mode!
Daisuke Yamauchi (RESCEU)

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Gradient-/curl-mode decomposition
The two dimensional distortion vector Δ is decomposed
into gradient-mode: ∇φ and curl-mode: (*∇)ω .

ScalarVectorTensorGradient-mode

Daisuke Yamauchi (RESCEU)

Curl-mode

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Question
In usual treatment of lensing, the symmetric trace-free partof the gradient of the deflection Δa (from geodesic eq.) can beused as a proxy for the shear γab (from Sachs eq.).

Is it always true?

?
NO!
Daisuke Yamauchi (RESCEU)

16SHEAR-DEFLECTION RELATION

Daisuke Yamauchi (RESCEU)

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Gauge-invariant metric perturbations

Scalar metric pert.

Vector metric pert.

Tensor metric pert.

Daisuke Yamauchi (RESCEU)

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Deflection angle
with gauge invariant variables
Solving the geodesic equation in the gauge-invariant
manner, we obtain the deflection angle:[see also Yoo (2009),…]with spin-0 part spin-1 part

Note : In the presence of the spin-1 part, that is vector σgand tensor h, there appears the curl-mode.

19Shear fieldwith gauge invariant variables
Solving the Sachs equation for shear field, we havespin-0spin-1spin-2Since the gauge degrees of freedom are completely removed in
the explicit expression for the tidal matrix, the resultant shear
field are manifestly gauge-invariant.
Daisuke Yamauchi (RESCEU)

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Gauge-inv. shear-deflection relation
Spin-2 contributions

 Metric shear [Dodelson+(2003)] / FNC term [Jeong+Schmidt(2012)](based on the geodesic eq)
Cosmic shear measurement via galaxy survey are usually referenced tothe coordinate in which galaxies are statistically isotropic. This is in
general different from our reference coordinate (FLRW). The correction
from the gravitational potential should appear.

In contrast to the previous studies, the metric shear/FNC term
naturally arises in our case from the Sachs equation
Daisuke Yamauchi (RESCEU)

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Why does such correction appear
in Sachs equation?

Since the leading correction of the metric in the FNC is known to
be described by the Riemann curvature, the FNC contribution is
automatically included in the symmetric tidal matrix perturbed
around FLRW.
Daisuke Yamauchi (RESCEU)
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LENSING POWER SPECTRA

Daisuke Yamauchi (RESCEU)

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In deriving lensing power spectrum,…
One complication is that while the weak lensing observables
are defined on a spherical sky, the metric perturbations usuallyappear in the three-dim space.
Even decomposing the perturbations into the plane waves, they
contribute to many multipole due to their angular structure…
E.g.) Bardeen potential

The situation is more complicated for vector/tensor perturbations…
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Total angular momentum wave
Combining the intrinsic angular structure with that of theplane-wave spatial dependence (TAM) substantially simplifies
the derivation of the full-sky formula!

[originally developed by Hu+White (2001)]

It enables us to simultaneously treat the lensing by vector
and tensor modes on an equal footing with those generated
by scalar modes.
Daisuke Yamauchi (RESCEU)

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TAM expansion
 The m=0, +-1, +-2 modes corresponds to the scalar, vector, andtensor metric perturbations.
 We can isolate the total angular dependence by each perturbation.

 Spin-0 component

 Spin-1 component

 Spin-2 component26

A complete set of full-sky formulafor scalar/vector/tensor metric pert.
The m=0, +-1, +-2 modes corresponds to thescalar, vector, tensor metric perturbations.

Auto-power spectrum
for m-mode Transfer functions for X and X’

Daisuke Yamauchi (RESCEU)

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Radial transfer functions for
E-/B-mode cosmic shear

 E-mode shear

N(χS) : source redshift distribution

The m=0, +-1, +-2 modes corresponds to thescalar, vector, tensor metric perturbations.

 B-mode shear

Metric shear/FNC term (from spin-2)

We explicitly show that B-mode is not generated by the
m=0 mode (scalar metric perturbations), as is expected.
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Radial transfer functions forgradient-/curl-mode CMB-lensing gradient mode
The m=0, +-1, +-2 modes corresponds to thescalar, vector, tensor metric perturbations.

 Curl mode

Curl-mode is not generated by the m=0 mode
(scalar metric perturbations), as is expected.
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APPLICATION

Daisuke Yamauchi (RESCEU)

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Primordial gravitational wave
As an application for the curl-/B-mode, let us consider theprimordial gravitational wave as the representative passive
sources for tensor perturbations.

 the tensor metric perturbations ⇔ m=+-2 modes

The contribution from the PGW (m=+-2) are shown to coincide with thosederived by other authors [Dodelson+(2003), Schmidt+Jeong(2012)].
Daisuke Yamauchi (RESCEU)

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Primordial gravitational wave
 Curl-mode of CMB lensing

 B-mode shear

ACTPol+Planck noise
Cosmic Variance limit

w/ FNC
w/o FNC

r=0.2
Multipole lr=0.2
Multipole lComparison between the statistical errors and the predictions
immediately follows that it is challenging to detect PGW via
the weak lensing measurement.
(Full-sky curl-mode estimator [Namikawa+DY+Taruya, 1110.1718])

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Cosmic strings
We consider the cosmic strings as intriguing examples of
the active source for vector and tensor perturbations.
(In this talk, I use the simple model for the string network to calculate analytically.) Line-like topological defects

 generally form during phasetransition in the very early
universe. [Jeannerot+(2003)]
 could be a probe for the earlyphases of the universe before
the CMB epoch[Hiramatsu+Sendouda+Takahashi+DY+Yoo, 1307.0308]

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Cosmic strings
We consider the cosmic strings as intriguing examples of
the active source for vector and tensor perturbations.
(In this talk, I use the simple model for the string network to calculate analytically.) the vector/tensor perturbations ⇔ m=+-1, +-2 modes

Daisuke Yamauchi (RESCEU)

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B-mode shear from cosmic strings[DY+Namikawa+Taruya, 1205.2139, 1305.3348]
Fiducial string parameters : String tension Gμ =10-8, reconnection prob. P= 10-335

(Full-sky curl-mode estimator [Namikawa+DY+Taruya 1110.1718])

Curl-deflection from cosmic strings[DY+Namikawa+Taruya, 1205.2139, 1305.3348]

String tension Gμ =10-8
reconn. prob. P= 10 -3

The curl-mode measurement would provide not only a directprobe of cosmic strings, but also a diagnosis helpful to check thesystematics in the derived constraints from the CMB TT. 36

Constraint on string parametersfrom curl mode for ACT and Planck(95%CL, Planck)

String tension Gμ

Lensing curl mode is moresensitive to small values ofreconnection probability P
compared to the small scaletemperature power spectrum.

[Namikawa+DY+Taruya, 1308.6068]

Reconnection probability P37

Summary
1. Gauge-invariant deflection-shear relation

In contrast previous studies, the metric shear/FNC term
naturally arises in our case from Sachs eq.

2. Lensing power spectra with TAM
Total angular momentum method substantially simplifies
the derivation of the full-sky formula.

Thank you !