Lecture 1: Basics of dark energy Shinji Tsujikawa (Tokyo University of Science) ``Welcome to the dark side of the world.”

Outline of lectures Letcure 1: Basics of dark energy Letcure 2: Observational constraints on dark energy (SN Ia, CMB, BAO) Lecture 3: Modified matter models of dark energy Lecture 4: Modified gravity models of dark energy

1. E. Copeland, M. Sami, S. Tsujikawa, ``Dynamics of dark energy’’, IJMPD, 1753 (2006), hep-th/ L. Amendola, S. Tsujikawa, ``Dark energy—Theory and observations’’, Cambridge University Press (2010) 3. S. Tsujikawa, ``Modified gravity models of dark energy’’, Lect. Notes, Phys. 800, 99 (2010), [gr-qc] Suggested readings

Dark energy From the observations of SN Ia, CMB, and BAO etc, about 70 % of the energy density of the Universe is dark energy responsible for cosmic acceleration.

The energy components in the present universe 72 %: Dark Energy: Negative pressure 23%: Dark Matter: Pressure-less dust Responsible for cosmic acceleration Responsible for the growth of large-scale structure 4.6%: Atoms (baryons) Responsible for our existence! 0.01 %: Radiation Remnants of black body radiation Today Decoupling epoch

Einstein equations In order to know the expansion history of the Universe, we need to solve the Einstein equation _________ Einstein tensor Energy momentum tensor For a given metric we can evaluate Perfect fluids have only diagonal components.

Homogenous and isotropic background The metric in the homogenous and isotropic background is described by K=0: flat, K>0: closed, K<0: open The non-vanishing components of the Einstein tensors are The energy-momentum tensor for perfect fluid is is the Hubble parameter (energy density) (Pressure)

Friedmann equations In the homogenous and isotropic background we have Eliminating the curvature term, we obtain (negative pressure) Combining the above equations, we also have (continuity equation)

Dark energy: Negative pressure Equation of state : Friedmann equation: Continuity equation: Cosmic acceleration Exponential expansion (Cosmological constant: =const) Negative In the flat Universe (K=0) we have For constant w, the solutions are (matter) (radiation)

Observational constraints on w (flat Universe) For constant w: Constant w However, the large variation of w can be still allowed.

Observational evidence for dark energy 1. Age of the Universe The age of the Universe must be larger than those of globular clusters. 2. Supernovae type Ia (SN Ia): 1998~ Perlmutter et al, Riess et al.,… 3. Cosmic Microwave Background (CMB): 1992~ (WMAP: 2003~) Mather and Smoot (2006, Nobel prize): COBE satellite Spergel et al, Komatsu et al, … : WMAP satellite 4. Baryon Acoustic Oscillations (BAO): 2005~ Eisenstein et al,.. 5. Large-scale structure (LSS): 1999~ (SDSS) Tegmark et al,… 6. ….

Age of the Universe As the matter components of the Universe we consider We introduce the redshift: We assume that the equation of state of dark energy is constant. These are substituted into the Friedmann equation

We introduce the today’s density parameters Then the Friedmann equation can be written as On using the relation the age of the Universe is where

Estimation of the age of the Universe

Dark energy makes the cosmic age larger We require dark energy so that the cosmic age is larger than the ages of the oldest globular clusters. The open Universe without dark energy is insufficient to explain the cosmic age because large cosmic curvature is required. 11 Gyr

SN Ia observations The luminosity distance L s : Absolute lumonisity F : Observed flux is related with the Hubble parameter H, as for the flat Universe (K=0) The absolute magnitude M of SN Ia is related with the observed apparent magnitude m, via

Comoving distance ． SN Ia Observer (z=0) In the flat FLRW background the light travels along the geodesic with The comoving distance to SN Ia is given by where

Luminosity distance in the flat Universe ． SN Ia Observer (z=0) The observed flux is at z=0 is given by The luminosity distance squared is Finally

Luminosity distance with the cosmic curvature For the metric with the cosmic curvature K, the luminosity distance is given by where Expansion around z=0 gives Using the relation we have

Luminosity distance with/without dark energy Flat Universe without dark energy Open Universe without dark energy Flat Universe with dark energy

Perlmutter et al, Riess et al (1998) (Perlmutter et al, 1998) High-z data began to be obtained ． Perlmutter et al showed that the cosmological constant ( ) is present at the 99 % confidence level, with the matter density parameter The rest is dark energy.

Several groups are competing! Brian Schmidt (Head of HSST (Riess et al) group) Saul Perlmutter (Head of SCP group

Observational constraints on the dark energy equation of state for constant w (Kowalski et al, 2008) SN Ia data only