Research Center for the Early UniverseThe University of Tokyo Jun’ichi Yokoyama

Based on M. Nakashima, R. Nagata & JY, Prog. Theor. Phys. 120(2008)1207M. Nakashima, K. Ichikawa, R. Nagata & JY, JCAP 1001(2010)030

Since Dirac’s large number hypothesis , there have been many theories that allow time variation of physical constants, such as higher-dimensional theories and string theories.

In the framework of these theories, it is very natural that multiple constants vary simultaneously.

In this talk, I consider cosmological constraints on time variation of fundamental constants, mainly the fine-structure constant α,but together with the electron and the proton masses usingCosmic Microwave Background Radiation (CMB) which has beenobserved with high precision by WMAP.

recombination era

Constraint from Oklo natural reactor (e.g. Fujii et al.,2002)

Constraint from spectra of quasars

Constraint from BBN （ e.g. Ichikawa and Kawasaki, 2002)

Constraint from CMB （　　　 ）　　→ Complementary to these observations and has many advantages such as “good understanding of the physics” or “high precision data of WMAP”

2Gyear ago, redshift

A number of observational results at redshifts

1sec 3mint

380kyrt 1088.2 1.2z

Now

３８万年後

Tra

cing b

ack th

e co

smic

histo

ry Big Bang

Expand and Cool

Helium was produced out of protons and neutrons from t=1sec to 3minutes.　　（ Cosmic temperature： 10Billion K　　　　 Size： 1/10Billion today　　　 Size is inversely proportional to Temp.）

WMAP

380ｋｙｒ

Plasma

Decoupling

Cosmic MicrowaveBackground (CMB)

Scale factor Curvature

一様等方宇宙Standard Inflation predicts with high accuracy. 1

Hubble parameter

Density parameter

cosmological constant(dark energy)

階層

1024m

1022m

1020m

1012m

107m

1m

Earth

Solar system

galaxy

cluster

supercluster

grew out of linear perturbations under the gravity

Potential fluctuation Curvature fluctuation

Cosmological ParametersH,

ds t dt a t t d2 2 2 21 2 1 2 ( , ) ( ) ( , )x x xb g b g

Power Spectrumof Initial Fluctuation

Anisotropies in cosmicmicrowave background

Large-Scale Structures

Present Power Spectrum

Angular Power Spectrum

P k t t( , ) | ( )|0 02 k

P k t ti i( , ) | ( )| k2

Cl

Linear perturbation

Three dimensional spatial quantities: Fourier expansion

( , ) ( )x kkxt t e

d kz i

3

23

2bg k k k k( ) ( ) ( , )*t t P k t 3b g ( , ) ( , ) ( , )x y x y k x yt t P k t e

d ki zc h bgb g 3

32

Power Spectrum：

Correlation Function：

Length scale r: rk

Two dimensional angular quantities: Spherical harmonics expansion

T

Ta Ylm lm

m l

l

l

, ,b g b g

0

Angular scaleθ:

l

Angular Power Spectrum：

Angular Correlation Function：

a a Cl m l m l l l m m1 1 2 2 1 1 2 1 2

*

1 1,b g 2 2,b g 12

C 12b g

T

T

T

TC

lC Pl l

l1 1 2 2 12 12

0

2 1

4, , cosb g b g b g b g

Cl

Last Scattering Surface

d

r

Observer

Decoupling

tightly coupled local thermal equilibrium

Free streaming

Plasma

Neutral

Recombination

The Boltzmann equation for photon distribution in a perturbed spacetime

Collision term due to the Thomson scattering

free electron density

ds t dt a t t d2 2 2 21 2 1 2 ( , ) ( ) ( , )x x xb g b gf p x ,c h

Df

Dt

f

x

dx

dt

f

p

dp

dtC f

C f x nme e T T

e

,

8

3

2

2

In the ionized plasma many Thomson scattering occurs and thethermal equilibrium distribution is realized.

As the electrons are recombined with the protons, the collisionterm vanishes and photons propagates freely. The distributionfunction keeps the equilibrium form but with a redshifted

temperature:( )

( )( )dec

dec

a tT t T

a t

The Boltzmann equation for photon distribution in a perturbed spacetime

Collision term due to the Thomson scattering

free electron density

ds t dt a t t d2 2 2 21 2 1 2 ( , ) ( ) ( , )x x xb g b gf p x ,c h

Df

Dt

f

x

dx

dt

f

p

dp

dtC f

C f x nme e T T

e

,

8

3

2

2

0

( , , ) ( ) ( , ) ( ),k i k P

23

0

30

( , )2 1.

4 (2 ) 2 1

kd kC

We consider temperature fluctuation averaged over photon energy in Fourier and multipole spaces.

direction vector of photon

T

T

T

T ki , , , , , , ,k k k

kc h b g b g :conformal time

Boltzmann equation

collision term

directionally averaged

Baryon (electron) velocity

LNM

OQP ik P i Vb b g 0 2 2

1

10( )

Euler equation for baryons

Va

aV k

RV V R

p pb b bb

b

b

d i,

3

4

Metric perturbation generated during inflation

:Poisson equation , k

a

k

a

H2

2

2

2

23

2

Boltzmann eq. can be transformed to an integral equation.

zb gb g

b gm r b g

, ,

( ) ( ) ( )

0

00

00

k

i V e e e dbik

ax ne e T

conformal time

Boltzmann equation:Interaction Between Radiation and Matter

Euler equation: Hydrodynamics

Einstein equation: Gravitational Evolution of Fluctuations

Optical depth

zb gb g

b gm r b g

, ,

( ) ( ) ( )

0

00

00

k

i V e e e dbik

( ) ( ) z zd ax n de e T

0 0

If we treat the decoupling to occur instantaneously at ,

1

now

Last scattering surface Propagation

e

v e

d

d

( )

( )( ) ( )

b g

b g

, , ,0 0 00

00k kb g b gb gbg b gb g b g zi V e e db d

ik ikd

d

e ( )

d

d 0

manyscattering

no scattering

Visibility function

g

In reality, decoupling requires finite time and the LSS has a finite thickness. Short-wave fluctuations that oscillate many times during itdamped by a factor with corresponding to 0.1deg. e k kDb g2 Mpck hD 10 1

Observable quantity

on Last scattering surface

Integrated Sachs-Wolfe effect

, ,0

1

4

1

30

2 00kb g bg b gb g b g b g

FHG

IKJ zi V e e e db d

ik k k ikd D

d

： Temperature fluctuations

： Doppler effect

： Gravitational Redshift Sachs-Wolfe effect

small scale

Large scale

0

1

4 d

i Vb d bg1

3 dbg

They can be calculated from the Boltzman/Euler/Poisson eqs., if the initial condition of k,tiand cosmological parameters are given.

LSS

d

r

Observer

kdFourier mode with wavenumber k is related to the angular multipole as as depicted in the figure. : distance to the last scattering surface.

14.3Gpcd

2

k

Short wave modes with ( ) s dec

dec

kc t

a t

which is smaller than the sound horizonat decoupling are oscillatory due tosound pressure.

Longer wave modes do not havetime to oscillate yet, and so areconstant, being affected by generalrelativistic effects only.

大スケールでほぼ一定

小スケールで振動

一般相対論

的重力赤方偏

移

流体力学的揺らぎ

Sound horizon at LSS corresponds to about 1 degree,which explains the location ofthe peak

180200

hydorodynamical

Gravitational

大スケールでほぼ一定

小スケールで振動

一般相対論

的重力赤方偏

移

流体力学的揺らぎ

hydorodynamical

Gravitationalsmall scale

Large scale

0

1

4 d

i Vb d bg1

3 dbg

All of them have the same origin, the inflaton fluctuation, in the simplest inflation model, so that its phase can be observed as in the figure by taking the snapshotat the last scattering surface.

The shape of the angular power spectrum depends on

（ spectral index　　 etc） as well as the values of cosmological parameters.（　　　　　 corresponds to the scale-invariant primordial fluctuation.）

42( , ) | ( ) | sni iP k t t Ak k

sn

1sn

Increasing baryon density relatively lowers radiation pressure,which results in higher peak.Decreasing Ω （ open Universe ） makes opening angle smallerso that the multipole l at the peak is shifted to a larger value.Smaller Hubble parameter means more distant LSS with enhanced early ISW effect.Λalso makes LSS more distant, shifting the peak toward right with enhanced Late ISW effect.

Thick line

2

1, 0

1, 0.5

0.01b

n h

h

Old standard CDMmodel.

1 0.5 0.30.05

0.03

0.01

0.3

0.5

0.7

0.7

0.3 0

0

cdm

71.0 2.5km/s/Mpc

0.222 0.026

0.0449 0.0028

0.734 0.029B

H

These are obtained using the current values ofthe fundamental constants.

Fundamental Physical Constants affect the angular powerspectrum of CMB temperature anisotropy mainly throughrecombination processes of protons and electrons.

wrong, because ⓔ was combined to at 380kyr for theⓟfirst time in cosmic history.

The collision term in the Boltzmann equation is proportionalto

C f x nme e T T

e

,

8

3

2

2

Thomson crosssectionIonized Electron Fraction

The most sensitive parameters are and , while plays almost the same role as .

empm

B

Fraction of ionized electrons evolves according to Saha eqnin chemical equilibrium

2em

Binding energy

Larger results in earlier and more rapid recombination.2em

10(10 )O The smallness of baryon-to-photon ratio explainswhy recombination occurs at 4000K instead ofT=13.6eV.

The larger values of and lead to 1 Earlier recombination 2 Narrower peaks of the visibility function

vis

ibilit

y f

un

cti

on

conformal time

vis

ibilit

y f

un

cti

on

conformal time

Visibility function

Probability distribution of the time when each photon decoupled (last-scattered).

Past

Past

αが大

Narrower peaks of the visibility function

Small-scale diffusion damping decreases, resulting in larger anisotropy.

Earlier Recombination

Last-scattering surface more distant

Peak shifts to higher multipoleLarger peak amplitude

Larger Δα

Larger Δα

Larger Δme

Larger Δme

： the Position of the First Acoustic Peak

[Hu, Fukugita, Zaldarriaga and Tegmark (2001) ]

Fiducial values are

which yield

h : Hubble parameter in unit of 100km/s/Mpc

: Optical depth of CMB photons due to reionization

sn : Power-law index of primordial fluctuation spectrum2

B Bh 2m mh

The matrix expression,

can be transformed to…

with

DegenerateDirections

We use WMAP ５ yr Data including both temperature anisotropy data as well as E-mode polarization data ( & HST ). Parameter estimations are implemented by Markov-Chain Monte Carlo (MCMC) method　 （ using modified CosmoMC code [ Lewis and Bridle(2002) ] ） We assume the flat-ΛCDM model. Parameters are

&

First we incorporate only time dependence of α.

If we incorporate time dependence on α, the Hubble parametercannot be determined well from CMB alone.

Standard model Time varying α

1D posterior statistical distribution functions

If we incorporate the Hubble-Space-Telescope (HST) result of , the constraints are improved significantly.0 72 8km/s/MpcH

without HST prior with HST prior

1D Posterior Statistical Distribution Functions obtained from MCMC analysis

with HST prior

without HST prior

Based on WMAP 5year observation.They are about 30% more stringent than those obtained based on WMAP 1year data by Ichikawa et al (2006).

95% confidence interval mean value

The relation through 　　　

• is a dilaton f

ield.

• 　　　　　　　

If we adopt a specific theoretical model, physical constantschange in time in a mutually dependent manner. [Olive et al. (1999) , Ichikawa et al. (2006)]

Example : low energy effective action of a string theory in the Einstein frame

In this model, small causes large . 　

In the same model, QCD energy scale can change.　　⇒ From 　 , can also change! 　

One-loop renormalization equation suggests that

⇒

⇒ large factor

, , and e pm m

only

-0.04 -0.02 0 0.02 0.04

0.0083 0.0018

0.028 0.026

95% confidence level

, , and e pm m and em only onlypm

, , and e pm m and em only onlypm

, , and e pm m onlypmand

yield very similar constraints, which impliesthe most dominant constraint comes from in this model.p pm m

Cosmic Microwave Background Radiation provides us with useful information to constrain the time variation of physical constants between now and the recombination epoch, 380kyr after the big bang.

Resultant constraint on at 95%C.L. varies depending on underlying theoretical models as well as on the prior of the value of the Hubble parameter.

Ongoing PLANCK experiment will provide us with even more useful information on the possible time variation of fundamental constants.

-0.0083 < Δα/α < 0.0018

-0.025 < Δα/α < 0.019

-0.028 < Δα/α < 0.026

, , and e pm m

and em

only

These constraints are not so stringent compared with those from other observations, but are very meaningful because the previous works could not have limited in the CMB epoch.

: the proton-to-electron mass ratio → : the dilaton field variation　　　　　　　　→