Cosmic Inhomogeneities and Accelerating Expansion

Ho Le Tuan AnhNational University of Singapore

PAQFT 27-29 Nov 2008

Outline

• Concordance model

• Model with a local void

– Motivation for suggesting model

– Model

– Method to check the model

– Results with Riess 2007 SNe Gold sample

• Conclusion and Discussion

Concordance model

• Homogeneous

• Isotropic

• Nearly flat:

Ωtotal ~ 1

• Dark energy density:

Ωλ ~ 70%

• Use FLRW metric and

Friedmann equations.

Successes in explaining:

• Existence and thermal form of the CMB

radiation.

• Relative abundance of light elements.

• Age of the Universe.

• SNe Ia data with accelerating expansion of the

universe.

Concordance model

Weak points:

• Cosmological constant problem: λ extremely small.

• Cosmic coincidence problem: Ωλ + Ωm ≈ 1

• Mysterious nature of dark energy: What dark energy consists of ?

Whether it is constant or not?

Its equation of state ?

Due to Appearance of Cosmological Constant λ

Concordance model

Solutions of Dark Energy Problems

• Modifying General Relativity Theory at large

distances scales

• Considering systematic uncertainties:

– Intergalactic dust.

– Gravitational lensing.

– Sn progenitors’ evolution.

– Etc…

• Proposals of inhomogeneous models: LTB models,

Stephani models, Swiss-cheese models…

Models with a local void Motivation for suggesting:

– Evidences of local void and the shell (Sloan Great

Wall) from galaxy redshift survey, SDSS, 2dF

redshift survey…

– Systematic deviation of clusters’ motions from

the global Hubble flow.

– Cold spot in the CMB may be associated with a

Big Void in the large-scale structure.

– Etc..

• Consist of 2 homogeneous

and isotropic regions (inner

and outer), separated by a

single, spherical singular

shell.

• Each is FLRW cosmology

with different parameters set.

• Ω0I < Ω0

II ; H0I > H0

II

Model with a local void (Tomita’s model)

SNe and Accelerating expansion

• The homogeneous and isotropic model can

not fit SNe data without dark energy term

accelerating expansion appears.

• Therefore, if dark energy term disappears,

accelerating expansion disappears, too.

This happens in inhomogeneous model.

Distances in Tomita’s model

• Angular Distance:

– General definition:

Where: λ: Affine parameter

θ: Expansion

parameter

• Luminosity Distance:

AA

dDD

d

21L AD z D

• Applying to the model:

– Where:

j: 1, 2 (inner and outer region)

Ω0: Present matter density

parameter

λ0: Present dark energy density

parameter

Distances in Tomita’s model

2

0 02

1 10

2 11 1 3 2 2

1 2

30 1

2

jA j j j j

jj

jA j j

Aj

d Dz z

zd z

d DF F D

dz

2

0 01 1 2j j j j j jF z z z z

Boundary and Initial conditions

• Redshift at the shell are equal:

• For :

• For :

Numerically solving equations (1), we can obtain

angular and luminosity distance.

1 1I IIz z

0

00

0,I

I AA I I

dD cD

dz H

1 100

,II

I II AA A II II

dD cD z D z

dz H

IAD

IIAD

Method to check the model

• Theoretical distance modulus:

• Observed distance modulus:

• Best-fit values are determined by χ2 statistic:

5log 25Ltheory

D

Mpc

observed B Bm M

2

, 0 0 0 0 0 1 ,22 2

, ,

| , , , , ,I II I II IItheory i i observed i

i i mz i

z H H z

Method to check the model

• Relation between σmz and σz :

• Probability distribution function:

• Eliminate nuisance parameters by taking integral:

– y: nuisance parameters set.

– μ0: the set of distance moduli used.

20 0 0 0 0 1 0

1, , , , , | exp

2I II I II IIp H H z

,

5 1

ln10i

Lmz i z

L z z

D

D z

0 0 0 0 0 0, | , , |II II II IIp p y dy

Supernova data and fitting• Apply the model with Riess 2007 Gold sample

• Consider several cases with specific values of

to avoid over-complication.

– z1=0.067, 0.08, 0.1

– = 0.70, 0.082, 0.085, 0.90

– Different matter density profiles:

1 0 0 0, ,I II Iz H H

0 0II IR H H

Profile

A

B

C

D

0I

0 0

0 0 0

0.3 if 0.6

2 if 0.6

I II

I I II

0 0.27I

0 0.20I

0 0.10I

Gold Sample (182 SNe)

Dark Energy density - Matter densityConfidence contours with 68.3% & 95.4% CL (Profile A)

.

Gold Sample

• R increases Ω decreases and λ increases.

• Best-fit values (profile A):

R z1 H0

0.85 0.08 0.5 0.25 - 0.02 63 157.270II

0I 0

II2min

Lam

bda 02

Omega02

.

Comments on results

– The model can fit the SNe data without dark energy.

– Best-fit values are consistent with other measurements

on Hubble constant, local matter density.

– A slightly better fit to the SNe data than ΛCDM model.

– Testing with different matter density profiles A, B, C,

D Confidence contours and are very

insensitive with matter density profiles.

2min

R z1 H0

0.85 0.08 0.5 0.25 - 0.02 63 157.270II

0I 0

II2min

Comparison with Riess 98 SNe sample

– New confidence contours are much more compact

than old ones narrower constraints on

parameters space.

Conclusion and Discussion

– Dark Energy problems can be solved with

inhomogeous models.

– Local void model can consistently account for SNe

data as well as constraints cosmological parameters

values.

– Off-center observer should be considered in the

future.

– Investigating the model with other recent

observations such as WMAP, BAO, ESSENCE…

References1. Alexander, S. a. B., Tirthabir and Notari, Alessio and Vaid, Deepak. 2007, arxiv: astro-ph/0712.0370

2. Alnes, H., Amarzguioui, M., & Gron, O. 2006, Physical Review D, 73

3. Celerier, M.-N. 2007, arxiv: astro-ph/0702416

4. Celerier, M. N. 2000, Astronomy and Astrophysics, 353, 63

5. Liddle, A. 2003, An introduction to modern cosmology (Wiley)

6. Moffat, J. W. 2006, Journal of Cosmology and Astroparticle Physics, arxiv: astro-ph/0505326

7. Peebles, P. J. E. 1993, Principles of physical cosmology (Princeton University Press)

8. Riess, A. G., et al. 1998, Astronomical Journal, 116, 1009

9. ---. 2007, Astrophysical Journal, 659, 98

10. ---. 2004, Astrophysical Journal, 607, 665

11. Roos, M. 2003, Introduction to cosmology (Wiley)

12. Tomita, K. 2000, Astrophysical Journal, 529, 26

13. ---. 2000, Astrophysical Journal, 529, 38

14. ---. 2001, Progress of Theoretical Physics, 106, 929

15. ---. 2001, Monthly Notices of the Royal Astronomical Society, 326, 287

16. Tomita, K., Asada, H., & Hamana, T. 1998. in Workshop on Gravitational Lens Phenomena and High-Redshift Universe, Distances in inhomogeneous cosmological models (Kyoto, Japan: Progress Theoretical Physics Publication Office), 155

17. Wood-Vasey, W. M., et al. 2007, Astrophysical Journal, 666, 694

18. http://www.wikipedia.org.

19. http://braeburn.pha.jhu.edu/~ariess/R06/.

Thank you for your attention