Dark energy II Models of dark energy Shinji

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

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

Cosmological constant problem or, Cosmo-illogical constant problem (by Rocky Kolb) The energy scale of dark energy today is 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

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)．

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

We add uplifting potential generated by anti-D 3 brane at the tip of warped throat: The total potential is It is possible to explain dark energy if d. S uplifting Ad. S

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

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

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

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

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

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

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

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

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

Equation of state for k-essence

Stability conditions for k-essence

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

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. (ii) Solar system scales (small scales) The models need to be close to GR from solar system experiments. Beyond GR ? ? ? GR+small corrections

Concrete modified gravity models or

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 2 Inflation is realized by the R term. Favored from CMB observations Spectral index: Tensor to scalar ratio: N: e-foldings

Capozziello f(R) dark energy models Turner More than 700 papers, see the Living Review of De Felice and S. T. (2010). The first dark energy model is (n > 0) _______ 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, … The main reason why the model does not work is

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

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)

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

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

Galileon gravity

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

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

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. l Quintessence and K-essence l Non-minimal coupling models Scalar-tensor theories (including f(R) gravity, Brans-Dicke theory) Field-derivative coupling models l Galileon (Kobayashi, Yamaguchi, Yokoyama, ar. Xiv: 1105. 5723)

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. , ar. Xiv: 1108. 4242 On sub-horizon scales the matter perturbations satsisfies

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

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