1. Dark energy II : Models of dark energy Shinji Tsujikawa (Tokyo University of Science)

2. What is the origin of dark energy? The simplest candidate: Cosmological constant However this suffers from a fine-tuning problem if it originates from a vacuum energy. Dynamical dark energy models Quintessence, k-essence, chaplygin gas, tachyon, f (R) gravity, scalar-tensor theories, Braneworld, Galileon, …

3. Cosmological constant problem The energy scale of dark energy today is or, Cosmo-illogical constant problem (by Rocky Kolb) If we take the Planck scale as a cut-off scale, the energy scale of the vacuum energy is Problem even before 1998 See my review in 1989. by Steven Weinberg

4. The cosmological constant is (i) sufficiently small to explain the energy scale of dark energy? (ii) or, completely zero? Case (i): Both the cosmological constant and the dark energy problems are solved at the same time. Economical Case (ii): The cosmological constant problem is solved, but the dark energy problem has to be addressed. This possibility remains. `Modified matter’ (such as a scalar field) is introduced, or gravity is modified from Einstein gravity (Dynamical dark energy) ．

5. Example of case (i): de-Sitter vacua in string theory Kachru-Kallosh-Linde-Trivedi (KKLT) scenario Type II string theory compactified on a Calabi Yau manifold with a flux. The KKLT scenario consists of three steps. Potential: where

6. We add uplifting potential generated by anti-D3 brane at the tip of warped throat: uplifting It is possible to explain dark energy if The total potential is AdS dS

7. Example of case (ii) [vanishing cosmological constant] _______________________ K: Kahler potential W: Superpotential In supersymmetric theories the vacuum energy is zero if supersymmetry is unbroken, but in real word supersymmetry is broken. Cancellation is required

8. We can classify the models into two classes ． (i) Modified gravity(ii) Modified matter f(R) gravity, Scalar-tensor theory, Braneworlds, Gauss-Bonnet gravity, Galileon gravity, ….. Quintessence, K-essence, Chaplygin gas, Coupled dark energy, ….. Dynamical dark energy models (Einstein equation)

9. Modified matter models based on scalar fields Quintessence (‘fifth element’): Chiba, Sugiyama, Nakamura (1997) ‘X matter’ Caldwell, Dave, Steinhardt (1998) ‘Quintessence’ K-essence: Accelerated expansion based on the potential energy where Chiba, Okabe, Yamaguchi (1999)‘Kinetically driven quintessence’ Accelerated expansion based on the kinetic energy Armendariz-Picon, Mukhanov, Steinhardt (2000) ‘k-essence’ Piorneering papers were written by Fujii (1982), Wetterich (1988), Ratra and Peebles (1988).

10. Potentials of Quintessence As long as the potential is sufficiently flat, cosmic acceleration can be realized. Energy density: Pressure: Equation of state for Quintessence Quintessence phantom Quintessence can be distinguished from the LCDM.

11. Particle physics models of quintessence (i) Fermion condensate in globally supersymmetric QCD theories (Binetruy) The inverse power-law potential was derived. where (ii) Supergravity models (Brax and Martin, Copeland et al) The field potential in SUGRA theories is

12. (iii) Pseudo-Nambu Goldston Boson (PNGB) models (Friemann et al) The filed starts to evolve only recently.

13. Classification of Quintessence potentials (Caldwell and Linder, 2003) (A) Freezing models: Since the potential tends to be flatter, the evolution of the field slows down. (B) Thawing models: The field has been nearly frozen in the past, but it starts to evolve around today.. Example

14. Quintessence in the (w,w’) plane. LCDM The current observations are not still enough to find the evidence for the variation of the equation of state.

15. K-essence K-essence is described by the action where The models that belong to k-essence is Conformal transformation or

16. Equation of state for k-essence

17. Stability conditions for k-essence

18. Some people tried to solve the coincidence problem of dark energy by considering a specific Lagrangian However it is difficult to construct such models theoretically. Moreover they typically have the superluminal propagation speed. k-essence density parameter Armendariz-Picon, Mukhanov, Steinhardt (2000)

19. Modified gravity models of dark energy This corresponds to large distance modification of gravity. (i) Cosmological scales (large scales) Modification from General Relativity (GR) can be allowed. ??? Beyond GR (ii) Solar system scales (small scales) The models need to be close to GR from solar system experiments. GR+small corrections

20. Concrete modified gravity models or

21. f(R) gravity GR Lagrangian:(R is a Ricci scalar) Extensions to arbitrary function f (R) f(R) gravity The first inflation model (Starobinsky 1980) Starobinsky Inflation is realized by the R term. 2 Favored from CMB observations Spectral index: Tensor to scalar ratio: N: e-foldings

22. f(R) dark energy models More than 700 papers, see the Living Review of De Felice and S.T. (2010). CapozzielloTurner The first dark energy model is Capozziello, Carloni and Troisi (2003) Carroll, Duvvuri, Trodden and Turner (2003) _____________ This term leads to the late-time acceleration. However this model is not valid because of the following reasons. (I) Incompatible with local gravity tests Chiba, Dolgov and Kawasaki, … (II) Instability of cosmological perturbations Hu, Tegmark, Trodden,… (III) Absence of the matter era Amendola, Polarski and S.T,… (n > 0) The main reason why the model does not work is

23. Conditions for the cosmological viability of f(R) dark energy models 1. To avoid ghosts 2. The mass M of a scalar-field degree of freedom needs to be positive for consistency with local gravity constraints (LGC). This condition is also required for the stability of perturbations. 3. For the presence of the matter era and for consistency with LGC. 4. The presence of a stable late-time de Sitter point (R : present cosmological Ricci scalar) 0 To avoid tachyonic instability

24. Viable f(R) dark energy models 1. Hu and Sawicki, 2007 2. Starobinsky, 2007 3. S.T., 2007 Cosmological constant disappears in flat space-time. The models approach the LCDM for. (for the models 1 and 2) The local gravity constraints can be satisfied for (Capozziello and S.T., 2008)

25. Braneworld models of dark energy Dvali, Gabadadze, Porrati (DGP) model 3-brane is embedded in the 5-dimensional bulk Bulk 3-brane (for the flat case) (self acceleration) 5-th dimension On the 3-brane the Friedmann equation is where There is a de Sitter attractor with

26. DGP model is disfavored from observations ． BAO SN Ia Even in the presence of cosmic curvature K, the DGP model is in high tension with observations. Moreover the DGP model contains a ghost mode. The DGP model is disfavored from both theoretical and observational point of view. Theoretical curve

27. Galileon gravity

28. Galileon cosmology : five covariant Galileon Lagrangians (second-order)

29. Cosmological evolution in Galileon cosmology De Felice and S.T., PRL (2010) Tracker solution

30. The most general single-field scalar-tensor theories having second-order equations of motion: Horndeski (1974) Deffayet et al (2011) This action covers most of the dark energy models proposed in literature. Quintessence and K-essence Non-minimal coupling models Scalar-tensor theories (including f(R) gravity, Brans-Dicke theory) Field-derivative coupling models Galileon (Kobayashi, Yamaguchi, Yokoyama, arXiv: 1105.5723)

31. Full background and linear perturbation equations were recently derived in the Horndeski’s most general scalar-tensor theories. A. De Felice, T. Kobayashi, S.T., arXiv:1108.4242 On sub-horizon scales the matter perturbations satsisfies

32. Using our general formula, we can estimate the growth rate of perturbations in each theory. in f(R) gravity at late times

33. Summary of dark energy models (1) Cosmological constant Observationally favored, but theoretically further progress is required. (2) Modified matter models Quintessence, k-essence: these are not distinguished from the LCDM observationally. Chaplygin gas : × Excluded from the observations of large-scale structure. (3) Modified gravity models f(R) gravity, scalar-tensor theories: the models need to be carefully constructed to satisfy all the required constraints. DGP braneworld: × Galileon model: Ruled out from the observations and the ghost problem. Strongly constrained from the LSS and CMB observations.