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/0603057

2. 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), 1101.0191 [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 needto solve the Einstein equation

_____ ____Einsteintensor

Energy momentumtensor

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)

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ρ

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 satellite4. 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 ageis larger than the ages of the oldest globular clusters.

The open Universe without dark energy is insufficient to explain the cosmic agebecause 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

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dL = Ls /(4πF)

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dL = (1+ z)dz

H(z)0

z

∫ for the flat Universe (K=0)

The absolute magnitude M of SN Ia is related with the observedapparent 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 relationwe have

Luminosity distance with/without dark energy

Flat Universe withoutdark energy

Open Universe without dark energy

Flat Universe withdark energy

Perlmutter et al, Riess et al (1998)

(Perlmutter et al, 1998)

High-z data began to be obtained ．

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mB −M = 5log10(dL /10pc)

Perlmutter et al showed thatthe cosmological constant ( ) is present at the 99 % confidence level, withthe 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