Cosmological constraints on Time variation of the Fundamental Constants

2. Cosmological constraints on Time variation of the Fundamental Constants 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

3. Introduction • 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

4. Other constraints on other epochs 2Gyear ago, redshift A number of observational results at redshifts Constraint from CMB（　　　 ） 　　→Complementary to these observations and has many advantages such as “good understanding of the physics” or “high precision data of WMAP” • 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)

5. Big Bang 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.） ３８万年後 Expand and Cool Now Tracing back the cosmic history Cosmic Microwave Background (CMB) WMAP Plasma 380 ｋｙｒ Decoupling

6. 一様等方宇宙 Hubble parameter Density parameter cosmological constant (dark energy) Standard Inflation predicts with high accuracy. The Universe is globally isotropic and homogeneous Scale factor Curvature

7. 1022m 1012m cluster 階層 1020m Solar system galaxy 107m 1024m 1m Earth supercluster Hierarchical Structures of the Universe

8. Large-Scale Structures Present Power Spectrum Power Spectrum of Initial Fluctuation Anisotropies in cosmic microwave background Angular Power Spectrum Hierarchical Structures in the present Universe grew out of linear perturbations under the gravity Linear perturbation Potential fluctuation Curvature fluctuation Cosmological Parameters H, W, L,...

9. Two dimensional angular quantities: Spherical harmonics expansion Angular scaleθ: Angular Power Spectrum： Angular Correlation Function： Three dimensional spatial quantities: Fourier expansion Length scale r: Power Spectrum： Correlation Function：

10. Cocmic Microwave Background Radiation tightly coupled local thermal equilibrium Last Scattering Surface Plasma r Recombination Decoupling d Free streaming Neutral Observer

11. Physics of CMB anisotropy The Boltzmann equation for photon distribution in a perturbed spacetime Collision term due to the Thomson scattering 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:

12. We consider temperature fluctuation averaged over photon energy in Fourier and multipole spaces. h :conformal time direction vector of photon Physics of CMB anisotropy The Boltzmann equation for photon distribution in a perturbed spacetime Collision term due to the Thomson scattering free electron density

13. conformal time Euler equation for baryons Metric perturbation generated during inflation :Poisson equation Boltzmann eq. can be transformed to an integral equation. directionally averaged Boltzmann equation Boltzmann equation: Interaction Between Radiation and Matter collision term Baryon (electron) velocity Euler equation: Hydrodynamics Einstein equation: Gravitational Evolution of Fluctuations

14. If we treat the decoupling to occur instantaneously at , no scattering many scattering 1 Visibility function now Last scattering surface Propagation Optical depth

15. Integrated Sachs- Wolfe effect Observable quantity on Last scattering surface small scale ： Temperature fluctuations ：Doppler effect ：Gravitational Redshift Sachs-Wolfe effect Large scale They can be calculated from the Boltzman/Euler/Poisson eqs., if the initial condition of F (k,ti)and cosmological parameters are given. In reality, decoupling requires finite time and the LSS has a finite thickness. Short-wave fluctuations that oscillate many times during it damped by a factor with corresponding to 0.1deg.

16. r LSS d Observer Fourier mode with wavenumber k is related to the angular multipole as as depicted in the figure. : distance to the last scattering surface. Short wave modes with 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.

17. Angular power spectrum of CMB temperature　anisotropy Sound horizon at LSS corresponds to about 1 degree, which explains the location of the peak 小スケールで振動 Gravitational 一般相対論的 重力赤方偏移 流体力学的揺らぎ 大スケールで ほぼ一定 hydorodynamical

18. small 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. Large scale Angular power spectrum of CMB temperature　anisotropy 小スケールで振動 Gravitational 一般相対論的 重力赤方偏移 流体力学的揺らぎ 大スケールで ほぼ一定 hydorodynamical

19. The shape of the angular power spectrum depends on （spectral indexetc）as well as the values of cosmological parameters. （　　　　　corresponds to the scale- invariant 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.

20. Thick line 0.05 0.03 0.01 1 0.5 0.3 Old standard CDM model. 0.7 0.3 0 0.3 0.5 0.7

21. 7 year WMAP results These are obtained using the current values of the fundamental constants. So much for CMB Cosmology in Standard Physics

22. wrong, because ⓔ was combined to ⓟ at 380kyr for the first time in cosmic history. The most sensitive parameters are and , while plays almost the same role as . The collision term in the Boltzmann equation is proportional to Thomson crosssection Ionized Electron Fraction CMB and Fundamental Constants Fundamental Physical Constants affect the angular power spectrum of CMB temperature anisotropy mainly through recombination processes of protons and electrons.

23. Binding energy Larger results in earlier and more rapid recombination. The smallness of baryon-to-photon ratio explains why recombination occurs at 4000K instead of T=13.6eV. Fraction of ionized electrons evolves according to Saha eqn in chemical equilibrium

24. Visibility function Probability distribution of the time when each photon decoupled (last-scattered). visibility function Past conformal time The larger values of and lead to 1 Earlier recombination 2 Narrower peaks of the visibility function visibility function Past conformal time

25. Narrower peaks of the visibility function Small-scale diffusion damping decreases, resulting in larger anisotropy. αが大 Larger Δα Larger Δα Larger Δme Earlier Recombination Last-scattering surface more distant Peak shifts to higher multipole Larger peak amplitude Larger Δme

26. Parameters that characterize CMB observables ：the Position of the First Acoustic Peak [Hu, Fukugita, Zaldarriaga and Tegmark (2001) ] Fiducial values are which yield

27. How these characterictic parameters change according to the cosmological parameters and/or fundamental constants : Hubble parameter in unit of 100km/s/Mpc : Optical depth of CMB photons due to reionization : Power-law index of primordial fluctuation spectrum

28. Which paremeters can be determined? Which parameters cannot be determined? Singular value decomposition The matrix expression, can be transformed to…

29. Degenerate Directions with

30. Statistical analysis using observatinal data • 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.

31. If we incorporate time dependence on α, the Hubble parameter cannot be determined well from CMB alone. Time varying α Standard model 1D posterior statistical distribution functions

32. If we incorporate the Hubble-Space-Telescope (HST) result of , the constraints are improved significantly. with HST prior without HST prior 1D Posterior Statistical Distribution Functions obtained from MCMC analysis

33. Summary for the case only α is time dependent 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).

34. Simultaneous variations of fundamental constants • 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 • is a dilaton field. The relation through

35. In the same model, QCD energy scale can change. • 　　⇒ From , can also change! • One-loop renormalization equation suggests that ⇒ ⇒ large factor • In this model, small causes large .

36. -0.04 -0.02 0 0.02 0.04 1-Dim posterior statistical distribution 95% confidence level

37. 1-Dim posterior statistical distributions

38. and yield very similar constraints, which implies the most dominant constraint comes from in this model. 1-Dim posterior statistical distributions

39. Conclusion • 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

41. Limit on e-p mass ratio • : 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.