Dark Energy Perturbations 李明哲 南京大学物理学院 2011.10.13 中国科技大学交叉学科理论研究中心 合肥.

Dark Energy Perturbations

李明哲南京大学物理学院

2011.10.13 中国科技大学交叉学科理论研究中心 合肥

Outline

• Importance of cosmological perturbations• Dark energy models• Dark energy perturbations• Dark energy coupling to CMB photons I• Dark energy coupling to CMB photons II• Conclusions

Importance of cosmological perturbations

Inflation quantum fluctuation

Primordial perturbation

dm...,,,,, enp

1965

COBE

WMAP

CMB anisotropy

Tests of perturbation theory

CMB Angular Power Spectrum

Matter Power Spectrum

Dark Energy (DE)

3/1

0)3(/ 34

382

pw

paa

HG

G

Accelerating universe

Negative pressure

GTgRgR 82

1

)8

(82

1

gG

TGRgR

g

GT

8)(

pg

G

8

1w

1, Cosmological Constant (Einstein 1917)

No perturbation

43 )102( eV

Cosmological constant problem

Observation

Zero point energy density

M

v

Mdk

mkk

02

4

2

222

164

12128 10~,10~ vpl eVMM

2, Dynamical dark energy

I, quintessence Peccei, Sola, Wetterich, 1987 Wetterich, 1988 Peebles, Ratra, 1988 Zlatev, Wang, Steinhardt, 1998

)()(2/1 2 VL

)()(2/1 2 VL

1 ,2/1 ,12/1

2/1 22

2

wVV

Vw

II, phantom Caldwell, 1999

12/1

2/12

2

V

Vw

2)1/2(X ),( XKL

III, k-essence Amendariz-Picon, Mukhanov, Steinhardt, 2000

XXppKp 2 ,

Quintessence, phantom are special cases of k-essence

W cannot cross -1

IV, quintom w crosses -1

),()(2/1)(2/1 212

12

1 VL Feng, Wang, Zhang, 2004

)()()(2/1 222

2 VM

cL ML, Feng, Zhang, 2005

22)(1)( VL Cai, ML, Lu, Piao, Qiu, Zhang, 2007

W a crucial parameter to distinguish different models

Data fitting, model independent, parameterization

DE perturbation, only vanished when w=-1 (cosmological constant)Naively switch off DE perturbation is not consistent

Using observational data to search for DE models

Without DE perturbation With DE perturbation

Weller, Lewies, 2003

Constant w and sound speed

))(( 222 jiij dxdxdads

TGG 8

FRW background

Metric perturbation

Equations of perturbations

0 T

Conformal Newtonian Gauge

Dark Energy Perturbations

All matter including DEcontribute to the metric perturbation

: sound speed in the comoving frame

Adiabatic sound speed

Single fluid

Quintessence, phantom

0

K-essence )2/(/2XXXXXXs Xppppc

Equations of dark energy perturbation

The problem of dark energy perturbations

Singular when w_e crosses -1No-Go Theorem J.Xia, Y.Cai, T.Qiu, G.Zhao, X.Zhang (2008) Quintom dark energy B.Feng, X.Wang, X.Zhang (2005)More degrees of freedom

Parameterization, e.g.,

Multi fluids or multi fields need more equations

not applicable

Not convenient in data analysis

Method without new parameters Zhao, Xia, ML, Feng, Zhang, 2005

Quintessence like

Phantom like

Fitting result with and without DE perturbation

Matching conditionML, Cai, Li, Brandenberger, Zhang, 2010

Space-like surface :

The induced 3-metric on and its extrinsic curvature be continuous on both sides

go to the “tilde coordinate system”

Gauge transformation

Matching condition

3-metric

extrinsic curvature

In arbitrary gauge

In conformal Newtonian gauge

Gauge-invariant variables

Initial conditions: adiabatic & isocurvature

Super-horizon scales

Adiabatic perturbation

.constr

Isocurvature perturbationr

Mixture of adiabatic and isocurvature modes

Adiabatic perturbation

Pure dark matter isocurvature perturbationRuled out by experiments

DE isocurvature perturbationLiu, ML, Zhang, 2010

The action integral is gauge invariant.

Geometric Optics Approximation

Dark energy coupling to photons I: Chern-Simons and CPTviolation

M

cp

Stokes parameters

I→ intensity Q&U→ linear polarization V→ circular polarization

ieUQiUQ 222

The polarization angle: Q

Uarctan

2

1

Spin 2

BETXX ,,',

EBl

TBl

TEl

BBl

EEl

TTl CCCCCC ,,,,,Six spectra

CPT violation induced the rotation of the polarization direction

Rotation angle characterizes the CPT-violating effect!

)( f

i

f

iif dxpdkp

)( fiM

c

II obs

)ˆ()2exp()ˆ(

)ˆ())(ˆ(

11

11

11 22,2

2,2

nYinYda

niUQnYda

mllmml

ml

obslm

obslm

Background homogeneous

lmobs

lm aia ,2,2 )2exp(

Without CPT violation, the correlations of TB and EB vanish

Consider the rotation angle as a free parameter

0001.0CMBPol can detect

Simulation result:

Current Status

deg0.40.6 WMAP3+BOOMERanG03

Perturbation, spatial dependent rotation angle

ML, Zhang, 2008

)]([),(4

02

2

2

decldecl kjkPk

dk

M

cC

)( fiM

c

)])(4cos()1([2

)12)(12(

0 2 2

)41)](2(cos)2(sin[

11112

21

21

2

21

222,

BBl

EEl

LBBl

EEll

ll

BBl

EEl

obsBBl

CCCCClllll

CCC

)]()1([

16

)12)(12(

0 2 2

)]1()1()1([

])12)(1(8

41[

~

11112

21

1

1

21

2

2122211

111

2

BBl

EEl

LBBl

EEll

ll

BBll

l

BBl

CCCCC

lllllllllll

CClllll

C

A new method to produce B-mode polarization

CPT violation

Weak gravitational lensing W.Hu 2000

Dark energy coupling to photons II: varying fine structure constant

Wang, ML, 2009

T

T

r: recombination

Linear fluctuation, new long range force

)]()1([

16

)12)(12(

0 2 2

)]1()1()1([

])12)(1(8

41[

~

11112

21

1

1

21

2

2122211

111

2

BBl

EEl

LBBl

EEll

ll

BBll

l

BBl

CCCCC

lllllllllll

CClllll

C

Weak gravitational lensing

Conclusions

• Naively turn off dark energy perturbation is not consistent.

• We need new method to treat the dark energy perturbation in the whole parameter space.

• DE isocurvature perturbation is not strongly constrained by current data, but it is expected to be limited more tightly by CMB-LSS cross correlation.

• In models of dark energy interacting with photons, the perturbation of dark energy has interesting implications.

Thanks!