Lecture 2:Observational constraints on dark energy

Shinji Tsujikawa(Tokyo University of Science)

Observational constraints on dark energy

The properties of dark energy can be constrained by a number of observations:

1. Supernovae type Ia (SN Ia)2. Cosmic Microwave Background (CMB) 3. Baryon Acoustic Oscillations (BAO)

4. Large-scale structure (LSS)5. Weak lensing

The cosmic expansion history is constrained.

The evolution of matter perturbations is constrained.Especially important for modified gravity models.

Supernovae Ia constraints

The cosmic expansion history is known by measuring the luminosity distance

(for the flat Universe)

Parametrization to constrain dark energy

Consider Einstein gravity in the presence of non-relativistic matterand dark energy with the continuity equations

In the flat Universe the Friedmann equation gives

where

Joint data analysis of SN Ia, WMAP, and SDSS with the parametrization (Zhao et al, 2007)

Best fit case

Cost for standard two-parameter compressions

The two-parameter compression may not accommodate thecase of rapidly changing equation of state.

Bassett et al (2004) proposed the ‘Kink’ parametrization allowing rapid evolution of w :

DE

Maximizied limits of kink parametrizations

Parmetrization (i)

Parmetrization (iii)

Parmetrization (ii)

Kink

The best-fit kink solutionpasses well outside the limits of all the other parametrizations.

Observational constraints from CMBThe observations of CMB temperature anisotropies can also place constraints on dark energy.

2013? PLANCK

　　 CMB temperature anisotropiesDark energy affects CMB anisotropies in two ways.

1. Shift of the peak position2. Integrated Sachs Wolfe (ISW) effect

ISW effect

Larger

€

ΩDE(0)

Smaller scales

(Important for large scales)

Shift for

Angular diameter distance

The angular diameter distance is

(flat Universe)

This is related with the luminosity distance via

(duality relation)

Causal mechanism for the generation of perturbations

Second horizon crossing

After the perturbations leavethe horizon during inflation, the curvature perturbations remain constant by the second horizon crossing.

Scale-invariant CMB spectra on large scales

After the perturbationsenter the horizon, they start to oscillate as a sound wave.

CMB acoustic peaks

where

Hu Sugiyama

(CMB shift parameter)where

and

The WMAP 5yr bound:

(Komatsu et al, WMAP 5-yr data)

Observational constraints on the dark energy equation of state

Joint data analysis of SN Ia + CMB (for constant w)

The constraints from SN Ia and CMB are almost orthogonal.

　　 ISW effect on CMB anisotropies

　　 Evolution of matter density perturbations

( )

The growing mode solution is

The growing mode solution is

Responsible for LSS

Perturbationsdo not grow.

　　　 Poisson equation

The Poisson equation is given by

(i) During the matter era

(ii) During the dark energy era

(no ISW effect)

Usually the constraint coming from the ISW effect is notstrong compared to that from the CMB shift parameter.(apart from some modified gravity models)

ISW effect

　　　 Baryon Acoustic Oscillations (BAO)

Baryons are tightly coupled to photons before the decoupling.

The oscillations of sound waves should be imprinted in the baryon perturbations as well as the CMB anisotropies.

In 2005 Eisenstein et al founda peak of acoustic oscillations in the large scale correlation function at

　　　 BAO distance measure

The sound horizon at which baryons were released from the Compton drag of photons determines the location of BAO:

We introduce

(orthogonal to the line of light)

(the oscillations along the line of sight)

The spherically averaged spectrum is

We introduce the relative BAO distance

where

The observational constraint by Eisenstein et al is

The case (i) is favored.

Joint data analysis of SN Ia + CMB + BAO (for constant w)