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Lecture 3: Modified matter models of dark energy Shinji Tsujikawa (Tokyo University of Science)
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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, …
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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
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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
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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) .
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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
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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
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String Landscape We may live in a vacuum with a small energy density (related with anthropic selection). 10 uplifted vacua! 500
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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
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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)
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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’
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Quintessence: French wine! _____________________________
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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.
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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
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(iii) Pseudo-Nambu Goldston Boson (PNGB) models (Friemann et al) The filed starts to evolve only recently.
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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
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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.
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Dynamical system approach to quintessence
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Dynamical equations The fixed point responsible for the cosmic acceleration is
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Phase space Attractor (cosmic acceleration) Saddle (matter point)
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General potentials where (tracking condition) Tracking always occurs.
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Numerical simulations for
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K-essence K-essence is described by the action where The models that belong to k-essence is Conformal transformation or
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Equation of state for k-essence
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Stability condition for k-essence
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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)
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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:
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Past: Future:
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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.
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