1. Lecture 3: Modified matter 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: Originally introduced by Einstein to realize the static Universe ． 1917 (38 old) 1945 (66 old) ‘Biggest Blunder in my life’ 1998 (119 old:heaven) In 1929 Hubble found the expansion of the Universe. Static Universe Big Bang Cosmology Big Bang cosmology+ cosmic acceleration

4. 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

5. 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) ．

6. 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

7. 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

8. String Landscape We may live in a vacuum with a small energy density (related with anthropic selection). 10 uplifted vacua! 500

9. 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

10. 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, (including mass varying neutrinos) ….. Dynamical dark energy models (Einstein equation)

11. 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’

12. Quintessence: French wine! _____________________________

13. 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.

14. 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

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

16. 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

17. 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.

18. Dynamical system approach to quintessence

19. Dynamical equations The fixed point responsible for the cosmic acceleration is

20. Phase space Attractor (cosmic acceleration) Saddle (matter point)

21. General potentials where (tracking condition) Tracking always occurs.

22. Numerical simulations for

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

24. Equation of state for k-essence

25. Stability condition for k-essence

26. 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)

27. Chaplygin gas model Chaplygin gasGeneralized Chaplygin gas This corresponds to unified dark energy models in which dark matter and dark energy are explained as a single component. (pressureless matter) (dark energy) Continuity equation:

28. Past: Future:

29. Chaplygin gas satisfies observational constrants ? No! Matter power spectrum _____________________ The sound speed term prevents the growth of large-scale structure. Observational constraints This cannot be distinguished from the LCDM.