スライド 1

Research Center for the Early Universe
The University of Tokyo
Jun’ichi Yokoyama
Based on
M. Nakashima, R. Nagata & JY, Prog. Theor. Phys. 120(2008)1207
M. 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 using
Cosmic Microwave Background Radiation (CMB) which has been
observed with high precision by WMAP.
recombination era
 Constraint from Oklo natural reactor (e.g. Fujii et al.,2002)
2Gyear ago, redshift
 Constraint from spectra of quasars
A number of observational results at redshifts
 Constraint from BBN（e.g. Ichikawa and Kawasaki, 2002)
t  1sec 3min
Constraint from CMB（ t  380kyr z  1088.2 ）1.2
→Complementary to these observations and has
many advantages such as “good understanding of the
physics” or “high precision data of WMAP”
Tracing back the cosmic history
Cosmic Microwave
Background (CMB)
Big Bang
WMAP
380
ｋｙｒ
Expand
and
Cool
Helium was produced out of protons and
３８万年後
neutrons from t=1sec to 3minutes.
（Cosmic temperature：10Billion K
Size：1/10Billion today
Size is inversely proportional to Temp.）
Plasma
Decoupling
Now
Scale factor
Curvature
Hubble parameter
 Density parameter
cosmological constant
(dark energy)
Standard Inflation predicts
     1 with high accuracy.
一様等方宇宙
cluster
1022m
階層
1020m
1012m
Solar system
galaxy
107m
1024m
supercluster
1m
Earth
grew out of linear perturbations under the gravity
Linear perturbation
b
g
b
g
ds   1  2 ( x , t ) dt  a (t ) 1  2 ( x , t ) dx
2
2
2
Potential fluctuation
Power Spectrum
of Initial Fluctuation
P( k , ti )  | k (ti )|2
Cosmological
Parameters
H, , ,...
2
Curvature fluctuation
Large-Scale Structures
Present Power Spectrum
P( k , t0 )  | k (t0 )|2
Anisotropies in cosmic
microwave background
Angular Power Spectrum
Cl
Three dimensional spatial quantities: Fourier expansion
3

ikx d k
 ( x , t )   k (t )e
Length scale r: r 
2
k
z
bg
3
2
b g
 k (t ) k  * (t )  P( k , t ) 3 k  k 
Power Spectrum：
c hz
d 3k
b
g
Correlation Function：  ( x , t ) ( y, t )   x  y  P( k , t )e
3
ik x  y
b2 g
Two dimensional angular quantities: Spherical harmonics expansion
T
T
bg

l
bg
 ,     almYlm  , 
l  0 m l
Angular scaleθ:
al1m1 al*2m2  Cl1 l1l2  m1m2
Angular Power Spectrum： Cl
bg
Angular Correlation Function： C  12
T
T
1

2
2
12
l
l 0
b , g

b , g
1
2l  1
 , g b
 ,  g Cb
 g

C Pb
cos g
b
T
T
4
1


l
l
12
12
1
2
2
tightly coupled
local thermal
equilibrium
Last Scattering
Surface
Plasma
r
Decoupling
d

Observer
Free
streaming
Recombination
Neutral
c h
b
g


The Boltzmann equation for photon distribution f p , x
in a perturbed spacetime ds2   1  2 ( x, t ) dt 2  a 2 (t ) 1  2 ( x, t ) dx 2
Df
f dx 
f dp 
 
 
 C f Collision term due to
Dt x dt
p dt
the Thomson scattering
b
g
8 2
C f  xe ne T ,  T 
3me2
 free electron density
In the ionized plasma many Thomson scattering occurs and the
thermal equilibrium distribution is realized.
As the electrons are recombined with the protons, the collision
term vanishes and photons propagates freely. The distribution
function keeps the equilibrium form but with a redshifted
temperature: T (t )  Tdec a(tdec )
a(t )
c h
b
g


The Boltzmann equation for photon distribution f p , x
in a perturbed spacetime ds2   1  2 ( x, t ) dt 2  a 2 (t ) 1  2 ( x, t ) dx 2
Df
f dx 
f dp 
 
 
 C f Collision term due to
Dt x dt
p dt
the Thomson scattering
b
g
8 2
C f  xe ne T ,  T 
3me2
 free electron density
We consider temperature fluctuation averaged over photon energy
in Fourier and multipole spaces.
T i
T
k 
 :conformal time
 , , k 
 , , k    , , k ,  
c
T
hTb g b g
k
direction vector of photon
 d k
2 1
C 
 ( , k ,  )   (i)  ( , k ) P (  ),
0 (2 )3
4

0

3
 (0 , k )
2 1
2
.
directionally averaged
Boltzmann equation
L
M
N
b g
   ik    equation:
       0   
Boltzmann
1
 2 P2 (  )  iVb
10
O
P
Q
Interaction Between
Radiation and Matter
collision term    ax n 
e e
conformal time
T
Baryon (electron) velocity
Euler equation for baryons
Euler aequation:
 b    3 b



Vb  Vb  k 
V  Vb , R 

Hydrodynamics
a
R
pb  p
4
d
i
Metric perturbation generated during inflation
k2
k2
3H 2 
   , equation:
  2 
Einstein
:Poisson equation
2
a
a
2 
Gravitational
Evolution
of
Fluctuations
Boltzmann eq. can be transformed to an integral equation.
b  gb, , k g
 zm
    iV  ()e
0
0
0
0
b
 ( )
b
g re
    e
 ( )
b gd
ik   0
b  gb, , k g
 zm
    iV  ()e
0
0
0
 () 
0
b
z
0

 ( )
 ( )d  
b
z
0

g re
    e
 ( )
b gd
ik   0
axe ne T d  Optical depth
If we treat the decoupling to occur instantaneously at
b g
e  ( )      d
Visibility function
e  ( )
1
no scattering
many
scattering
b g
gv ()   ()e  ( )      d
  d ,
d

0
now
b g b gb
gb g
  ,  0 , k    0 , k   0    iVb  d e b g
ik  d  0
Last scattering surface
zb
0
d
g
     eik b  gd
0
Propagation
In reality, decoupling requires finite time and the LSS has a finite
thickness. Short-wave fluctuations
that oscillate many times during it
2
damped by a factor e  bk k D gwith k D  10h1Mpc corresponding to 0.1deg.
Observable quantity
F
1 
b
, , k g
G
H4 
0
0 
Ib g
JK
2
1
ik b
 d  0 g  b
k kD g
 iVb    d e
e

3

on Last scattering surface

Integrated SachsWolfe effect
b



e


b
g
z
0
d
1 
d  ： Temperature fluctuations
4 
ik   0
gd
small scale
b g：Doppler effect
iVb  d
b g：Gravitational Redshift Sachs-Wolfe effect
1
 d
3
Large scale
They can be calculated from the Boltzman/Euler/Poisson eqs., if the initial
condition of  k,ti and cosmological parameters are given.
Fourier mode with wavenumber k is related to the angular multipole
as  kd as depicted in the figure.
d  14.3Gpc : distance to the last scattering surface. 2
r
k
k
 cs tdec
Short wave modes with
LSS
a(tdec )
which is smaller than the sound horizon
at decoupling are oscillatory due to
sound pressure.
Longer wave modes do not have
time to oscillate yet, and so are
constant, being affected by general
relativistic effects only.
d


Observer
Sound horizon at LSS
corresponds to about 1 degree,
which explains the location of
the peak
 180
 
 200


小スケールで振動
Gravitational
重一
力般
赤相
方対
偏論
移的
流体力学的揺らぎ
大スケールで
ほぼ一定
hydorodynamical
1 
0 
d 
4 
小スケールで振動
small scale
bg
iVb  d
bg
1
 d
3
Large scale
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 snapshot
at the last scattering surface.
Gravitational
重一
力般
赤相
方対
偏論
移的
流体力学的揺らぎ
大スケールで
ほぼ一定
hydorodynamical
The shape of the angular power
spectrum depends on
P(k , ti )  |  k (ti ) |2  Ak ns 4
（spectral index ns etc）as well as the
values of cosmological parameters.
（ ns  1 corresponds to the scaleinvariant primordial fluctuation.）
Increasing baryon density relatively lowers radiation pressure,
which results in higher peak.
Decreasing Ω（open Universe）makes opening angle smaller
so 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
  1,   0
n  1, h  0.5
1 0.5 0.3
0.05
0.03
0.01
b h2  0.01
Old standard CDM
model.
0.7
0.3
0.5
0.7
0.3
0
H 0  71.0  2.5km/s/Mpc
 cdm  0.222  0.026
 B  0.0449  0.0028
   0.734  0.029
These are obtained using the current values of
the fundamental constants.
Fundamental Physical Constants affect the angular power
spectrum of CMB temperature anisotropy mainly through
recombination processes of protons and electrons.
wrong, because ⓔ was combined to ⓟ at 380kyr for the
first time in cosmic history.

The most sensitive parameters are
and me ,
while m p plays almost the same role as  B .
The collision term in the Boltzmann equation is proportional
to
2
8
C f  xe ne T ,  T 
3me2
Ionized Electron Fraction
Thomson crosssection
Fraction of ionized electrons evolves according to Saha eqn
in chemical equilibrium
Binding energy
 me
2
Larger me 2 results in earlier and more rapid recombination.
10
 O(10
)
The smallness of baryon-to-photon ratio explains
why recombination occurs at 4000K instead of
T=13.6eV.
visibility function
Visibility function
Probability distribution of the time
when each photon decoupled
(last-scattered).
Past
The larger values of
and
lead to
1 Earlier recombination
2 Narrower peaks of the
visibility function
visibility function
conformal time
conformal time
Past
Larger Δα
α
が
大
Narrower peaks of
the visibility function
Small-scale diffusion damping
decreases, resulting in larger
anisotropy.
Larger Δα
Larger Δme
Earlier Recombination
Last-scattering surface more
distant
Peak shifts to higher multipole
Larger peak amplitude
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
ns : Power-law index of primordial fluctuation spectrum
B  B h2
m  mh2
The matrix expression,
can be transformed to…
with
Degenerate
Directions
 We use WMAP５yr 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 parameter
cannot 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
H0  72  8km/s/Mpc , the constraints are improved significantly.
without HST prior
with HST prior
1D Posterior Statistical Distribution Functions obtained from MCMC analysis
95% confidence interval
mean value
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).
 If
we adopt a specific theoretical model, physical constants
change 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
•
•
The relation through
is a dilaton field.
In the same model, QCD energy scale
can change.
⇒ From
,
can also change!
One-loop renormalization equation suggests that
⇒
⇒
In this model, small
large factor
causes large
.
 , me , and mp
 only
-0.04
-0.02
0
0.0083     0.0018
0.028     0.026
0.02
0.04
95% confidence level
 , me , and mp
 and me
 only
mp only
 , me , and mp
 and me
 only
mp only
 , me , and mp and
mp only
yield very similar constraints, which implies
the most dominant constraint comes from
mp mp in this model.
 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.
-0.0083 < Δα/α < 0.0018
 , me , and mp
-0.025 < Δα/α < 0.019
 and me
-0.028 < Δα/α < 0.026
 only
Ongoing PLANCK experiment will provide us with even more
useful information on the possible time variation of fundamental
constants.


: the proton-to-electron mass ratio →
: the dilaton field variation
→
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.

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