Research Center for the Early Universe The University of Tokyo Jun’ichi Yokoyama Based on M....
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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 →