Hemispherical Power Asymmetry

郭宗宽

昆明2013.11.2

anomalies in CMB map

• the quadrupole-octopole alignment• power deficit at low-l • hemispherical asymmetry• parity asymmetry• the cold spot• ……

∑𝑚

𝑚2|𝑎𝑙𝑚(�̂�)|2

∆𝑇 (�̂�)𝑇

=∑𝑙 𝑚

𝑎𝑙𝑚𝑌 𝑙𝑚(�̂�)

𝑎𝑙𝑚=∫𝑑�̂�𝑌 𝑙𝑚∗(�̂�)

∆𝑇 (�̂�)𝑇

⟨𝑎𝑙𝑚∗𝑎𝑙′ 𝑚′ ⟩=𝛿𝑙 𝑙′ 𝛿𝑚𝑚′ 𝐶𝑙

𝐶 (𝜃 )=⟨ ∆𝑇 (�̂�1)𝑇

∆𝑇 (�̂�2)𝑇 ⟩

¿14 𝜋∑

𝑙

(2𝑙+1)𝐶𝑙𝑃 𝑙 (cos𝜃)

𝑃+¿ (𝑙 )=∑

𝑛=2

𝑙

cos2 (𝑛 𝜋2 )𝑛(𝑛+1 )2 𝜋

𝐶𝑛 ¿

𝑃− (𝑙 )=∑𝑛=2

𝑙

sin2(𝑛𝜋2 )𝑛(𝑛+1)2𝜋

𝐶𝑛

𝑔 (𝑙 )= 𝑃+¿(𝑙)

𝑃−( 𝑙)¿

data analysis

d (�̂�)= (1+ 𝐴�̂� ∙ �̂�) s (�̂�)+n (�̂�)

the CMB temperature sky maps is modeled as

the likelihood is given by

−2ℒ ( 𝐴 , �̂� ,𝑞 ,𝑛)=d𝑇C−1d+ log|C|

C.Gordon, W.Hu, D.Huterer and T.M.Crawford, arXiv:astro-ph/0509301

H.K.Eriksen, A.J.Banday, K.M.Gorski, F.K.Hansen and P.B.Lilje, arXiv:astro-ph/0701089

C (�̂� ,�̂�)=S (�̂� ,�̂� )+N+F

the full covariance matrix is

S (�̂� ,�̂� )=(1+𝐴�̂� ∙ �̂�)S iso (�̂� ,�̂� )(1+𝐴�̂� ∙�̂�)

Siso (�̂� ,�̂� )= 14𝜋 ∑

𝑙

(2 𝑙+1)𝐶𝑙𝑃 𝑙(�̂� ∙�̂�)

𝐶𝑙=𝑞 ( 𝑙𝑙0 )𝑛

𝐶𝑙❑fid

• WMAP3: (l, b) = (225◦,−27◦), A=0.114

• WMAP5(l<64, 4.5◦): (l, b) = (224◦,−22◦), A=0.0720.022

• Planck:J.Hoftuft, H.K.Eriksen, A.J.Banday, K.M.Gorski, F.K.Hansen and P.B.Lilje, arXiv:0903.1229

P.A.R.Ade et al. [Planck Collaboration], arXiv:1303.5083

preferred dipole directions dependence on smoothing scale

Comment: the dipole anisotropy induced by our velocity

𝑇 (�̂� )= 𝑇 ′ (�̂� ′)𝛾(1−�̂� ∙ 𝛽)

�̂�=�̂�′+[ (𝛾−1 )�̂�′ ∙ �̂�+𝛾𝛽 ] �̂�

𝛾 (1+�̂�′ ∙𝜷)

𝛿 𝐼𝜈 (𝜈 ,�̂�)𝑑 𝐼𝜈/𝑑𝑇

=𝑇0 �̂� ∙𝜷+𝛿𝑇 ′(�̂�−𝛻 (�̂� ∙𝜷))(1+𝑏𝜈 �̂� ∙𝜷)

the temperature and direction in the observed frame

the inferred temperature fluctuations

𝛽≡𝑣𝑐

=1.23×10− 3

(l, b) = (264◦,48◦)

models

• a parameterization [astro-ph/0509301]• our peculiar motion [arXiv:1304.3506]• a gauge field [arXiv:1302.7304]• anisotropic metric [arXiv:1303.6058]• a modulation of a cosmological parameter

[arXiv:1303.6949]• a superhorizon perturbation

[arXiv:0806.0377]• …… [arXiv:yymm.xxxx]

a superhorizon perturbation

A.L.Erickcek, M.Kamionkowski and S.M.Carroll, arXiv:0806.0377

𝛿𝑇𝑇

≃13

Φ=13ℛ (matter −dominated )the Sachs-Wolfe effect

𝜙(𝑡∗)⟶𝑘(𝑡∗)𝒫ℛ (𝑘)=(𝐻�̇� )2

( 𝐻2𝜋 )2

𝑘=𝑎𝐻

the diploe modulation of curvature perturbation

𝒫ℛ1/2 (𝑘 , 𝒙 )=(1+𝐴 �̂� ∙ 𝒙

𝑥 ls)𝒫ℛ

1/2 (𝑘 )

the asymmetry A is

𝐴=|𝛻𝒫ℛ

1/2|𝒫ℛ1 /2 𝑥 ls=𝑑 ( ln𝒫ℛ

1/2¿ ¿𝑑 ( ln𝑘¿

¿𝑑 ( ln𝑘¿ ¿𝑑𝑡

𝑑𝑡𝑑𝜙

|𝛻𝜙|𝑥 ls

¿ 12

(𝑛ℛ −1 )𝐻 (1−𝜖 ) 1�̇�

(𝑘¿¿𝐿𝛿𝜙𝐿)𝑥 ls=(1−𝜖)(𝑛ℛ −1 )

2(𝑘𝐿𝑥ls )(𝐻�̇� 𝛿𝜙𝐿)¿

¿ (1−𝜖)(𝑛ℛ −1 )

2(𝑘𝐿𝑥 ls)𝒫ℛ ,𝐿

1/2

a single modulated mode𝒫ℛ (𝑘)=𝒫ℛ ,𝐿𝛿(ln𝑘− ln𝑘𝐿)

D.H.Lyth, arXiv:1304.1270

Z.G.Liu, Z.K.Guo and Y.S.Piao, arXiv:1304.6527

𝐶𝑙=4𝜋∫0

∞𝑑𝑘𝑘

𝑇 𝑙2(𝑘)𝒫ℛ (𝑘)

the CMB angular power spectrum

the GZ effect (the Sachs-Wolfe approximation)

𝐶2GZ=4𝜋

25∫0

1/𝑥 ls 𝑑𝑘𝑘 ( 𝑘2 𝑥 ls

2

15 )2𝒫ℛ (𝑘 )=4𝜋

25 (𝑘𝐿2𝑥 ls

2

15 )2𝒫ℛ ,𝐿

√𝐶2GZ≲1.8×10− 5⟹(𝑘¿¿𝐿𝑥 ls)

4𝒫ℛ ,𝐿≲16×10− 8¿

⟹(𝑘𝐿𝑥 ls)𝒫ℛ ,𝐿

12≲ 0.02

observational constraint

• a single-field slow-roll inflation

• a curvaton-type field

• a bounce inflation

𝐴 𝒪(10¿¿−4)¿

𝐴=65𝑓 NL (𝑘𝐿𝑥 ls )𝒫ℛ ,𝐿

1 /2 , 𝑓 𝑁𝐿≳8

(𝑛ℛ −1 ) 3 , ϵ 3 , 𝐴 0.06

A.L.Erickcek, M.Kamionkowski and S.M.Carroll, arXiv:0806.0377

D.H.Lyth, arXiv:1304.1270

Z.G.Liu, Z.K.Guo and Y.S.Piao, arXiv:1304.6527

𝒫ℛ=𝒫ℛinf 2𝜋𝑘|𝐶1−𝐶2|

2

𝒫ℛinf=𝐴inf( 𝑘𝑘0

)𝑛inf −1

primordial power spectrum of curvature perturbations

• semispherical power asymmetry• power deficit on large angular scales• signals in the TE and EE spectra?• scale dependent? A<0.015 (99% CL)

outlook

Thanks!