1. Lecture 2: Observational constraints on dark energy Shinji Tsujikawa (Tokyo University of Science)

2. Observational constraints on dark energy The properties of dark energy can be constrained by a number of observations: 1.Supernovae type Ia (SN Ia) 2. Cosmic Microwave Background (CMB) 3.Baryon Acoustic Oscillations (BAO) 4. Large-scale structure (LSS) 5.Weak lensing The cosmic expansion history is constrained. The evolution of matter perturbations is constrained. Especially important for modified gravity models.

3. Supernovae Ia constraints The cosmic expansion history is known by measuring the luminosity distance (for the flat Universe)

4. Parametrization to constrain dark energy Consider Einstein gravity in the presence of non-relativistic matter and dark energy with the continuity equations In the flat Universe the Friedmann equation gives

5. where

6. Joint data analysis of SN Ia, WMAP, and SDSS with the parametrization (Zhao et al, 2007) Best fit case

7. Cost for standard two-parameter compressions The two-parameter compression may not accommodate the case of rapidly changing equation of state. Bassett et al (2004) proposed the ‘Kink’ parametrization allowing rapid evolution of w : DE

8. Maximizied limits of kink parametrizations Parmetrization (i) Parmetrization (iii) Parmetrization (ii) Kink The best-fit kink solution passes well outside the limits of all the other parametrizations.

9. Observational constraints from CMB The observations of CMB temperature anisotropies can also place constraints on dark energy. 2013? PLANCK

10. CMB temperature anisotropies Dark energy affects CMB anisotropies in two ways. 1. Shift of the peak position 2. Integrated Sachs Wolfe (ISW) effect ISW effect Larger Smaller scales (Important for large scales) Shift for

11. Angular diameter distance The angular diameter distance is (flat Universe) This is related with the luminosity distance via (duality relation)

12. Causal mechanism for the generation of perturbations Second horizon crossing After the perturbations leave the horizon during inflation, the curvature perturbations remain constant by the second horizon crossing. Scale-invariant CMB spectra on large scales After the perturbations enter the horizon, they start to oscillate as a sound wave.

13. CMB acoustic peaks where HuSugiyama

14. (CMB shift parameter) where and

15. The WMAP 5yr bound:

16. (Komatsu et al, WMAP 5-yr data) Observational constraints on the dark energy equation of state

17. Joint data analysis of SN Ia + CMB (for constant w) The constraints from SN Ia and CMB are almost orthogonal.

18. ISW effect on CMB anisotropies

19. Evolution of matter density perturbations ( ) The growing mode solution is Responsible for LSS Perturbations do not grow.

20. Poisson equation The Poisson equation is given by (i) During the matter era (ii) During the dark energy era (no ISW effect)

21. Usually the constraint coming from the ISW effect is not strong compared to that from the CMB shift parameter. (apart from some modified gravity models) ISW effect

22. Baryon Acoustic Oscillations (BAO) Baryons are tightly coupled to photons before the decoupling. The oscillations of sound waves should be imprinted in the baryon perturbations as well as the CMB anisotropies. In 2005 Eisenstein et al found a peak of acoustic oscillations in the large scale correlation function at

23. BAO distance measure The sound horizon at which baryons were released from the Compton drag of photons determines the location of BAO: We introduce (orthogonal to the line of light) (the oscillations along the line of sight) The spherically averaged spectrum is

24. We introduce the relative BAO distance where The observational constraint by Eisenstein et al is The case (i) is favored.

25. Joint data analysis of SN Ia + CMB + BAO (for constant w)