light bending as a probe for geometric dark energy

Light Bending as a probe of Geometric Dark Energy models EDEN, Paris December 8,2005

Light Bending

as a probe for

Geometric Dark Energy

Alessandro Gruppuso (INAF/IASF, Bologna)

In collaboration with

Fabio Finelli & Matteo Galaverni (INAF/IASF, Bologna)

EDEN in Paris: 7th-9th December 2005, LPNHE, Paris France


Plan of the Talk

• Introduction• Light Bending as a preferred general relativistic

test to distinguish between a Cosmological Constant and other Dark Energy models

• Focus on Geometric Dark Energy models• Analytic Results• Applications to Astrophysical sets-up• Conclusions


Introduction

• Supernova and WMAP observations provide clear evidence that the CDM model is not suitable to describe our universe.

• A Cosmological Term Λ is the simplest explanation for the mismatch between theory and observation

• But its value is completely at odd with naive estimate of the vacuum energy of quantum fluctuations! (fine-tuning!)


• 2nd way: modify LHS i.e. changing the Einstein Tensor (DGP model, Weyl Gravity,…)

Introduction

• Are there alternatives to Λ? Or otherwise stated: How modify Einstein Equations in order to be compatible with observations?

• 1st way: modify RHS i.e. considering an additional field such that p/ρ < -1/3 (quintessence models,…)

Dynamical Dark Energy models

Geometric Dark Energy models

GTRgR 82

1

)(82

1 TTGRgR

GTgfRgR 8)(2

1


Introduction

Up to now observational data have not been able to discriminate between ΛCDM model and other Dark Energy models (both geometric and dynamical)

The aim of this talk is to analyze DE effects at astrophysical level (for systems that are decoupled from the cosmological expansion)


Introduction

Light Bending!

The importance of DE effects in Astrophysical tests is already known. In particular the perihelion precession of Mars in DGP model is found to be close to the sensitivity

of the next experiments.

Dvali, Gruzinov & Zaldarriaga PRD (2003)

Lue & Strakman PRD (2003)

Lue astro-ph/0510068 (2005)


Light Bending

0r

)(r

observer sourcemass


Light Bending

Why is it preferred among general relativistic tests? 22222 )()( drdrrAdtrBds

21

3

21)()( r

r

GMrArB

Schwarzschild solution

in presence of

S

r

r

rrrBrrrB

dr

r

rBrAS

01

21200

2sin2

)()(

)()(2

0

S

r

r

rrrBrrrB

dr

r

S

01

21200

2sin2

)()(

12

0

drops out!!!


Light Bending

2

00, 2

8

1524

r

GM

r

GM

does not deflect light!!!

In agreement with Islam PLA (1983)


Focus on Geometric DE models

Consider a metric of the following form

rr

GMrB 2

21)( r

r

GMrA 1

1 21)(

22222 )()( drdrrAdtrBds

A solution of this kind come from gDE

2/1 and2

21 /2/ crGM DGP model Dvali, Gabadadze & Porrati PLB (2000)

In DGP theory there is a maximum scale (Vainshtein scale)

1 andNr/121 Weyl gravity

Mannheim & Kazanas ApJ (1989)


Focus on Geometric DE models

CSrr Therefore at scalesmuch smaller than cosmological ones,

there are some deviation from Schwarzschild results

(Cluster of Galaxies!)

2CSr Cosmological Scale

112/2 GMr Critical Scale

A parametrization of this kind introduces two scales

In DGP model this coincides with Vainshtein radius.


DESCHWII

SCHWI )()(

S

SCHWI

r

r

r

GM 0

0

)( 14

S

SCHWII

r

r

r

GM 0

2

0

)( 28

152

2/120

02210

1

1

rr

rrIr

S

SDE

2

012

1

0

,22

3,

22

1,

2

1

1

1

)2/1(

2/2/1

2 S

S

r

rF

r

rI

Analityc Results

F.Finelli, M.Galaverni & A.G. (2005)


DGP model

2/120

02210

1

1

rr

rrIr

S

SDE

2/120

2/10

22/1

0

1

10

rr

rrr

S

SDE

This is why the deflection of light in this model is not rigorously vanishing!

3/1

CCV r

GMrr

Cr is the crossover scale between 4D and 5D behaviour

In contrast with Lue & Starkman PRD (2003)


DGP model

• Since has to be positive in order to describe at cosmological level the recent acceleration of the universe, then the sign

of is positive.

gr

rr

rrr

S

SDE

2/1022/12

0

2/102/1

02

1

1

2

2

DE

g

0/ rrS


DGP model

• In which astrophysical system the contribution is the largest?

• Since (and is constrained by cosmology to be of the order of 5 Gpc) we found that it is the largest for Cluster of galaxies.

2/12

0

2/102/1

02

1

1

rr

rrr

S

SDE

2

22 2 CrGM

DE

Cr


DGP model

For a Cluster we find

with ( , , ) .

• analytic expression is checked to be good at 0.03%

•

•

2/12

0

2/102/1

02

1

1

rr

rrr

S

SDE

2

%4.0/ SCHWDE

GpcrC 5 Mpcr 10 MpcrS 10

)(210 IISCHWDE

SV rr


DGP model

Total = Schw1 + Schw2 + DE

1st order Schwarzschild

1st + 2nd order Schwarzschild

0/ rMG

0/ rMG



Weyl gravity

2/120

02210

1

1

rr

rrIr

S

SDE

gr

rr

rrr

S

SDE 202/12

0

020

1

10

when is much larger than 1 0/ rrS 0rDE

0/ rrS

g

Edery and Paranjape PRD (1998)


Weyl gravity

• can be positive or negative depending on the sign of .

• As in DGP case the contribution is the largest for the largest astrophysical system (i.e.: cluster of galaxies)

2/12

0

00

1

1

rr

rrr

S

SDE

DE


Weyl gravity

For a cluster we find

With ( , )

2/12

0

00

1

1

rr

rrr

S

SDE

%74/ SCHWDE

Mpcr 10 GpcrN 101 )(410 II

SCHWDE •


Weyl gravity

0/ rMG

0/ rMG

1st order Schwarzschild

1st + 2nd order Schwarzschild

GpcrN 1

GpcrN 5





Conclusions

• Light Bending is studied in a static spherically symmetric space time non asymptotically flat ( ). These terms cannot be parametrized in the usual PPN approach!

• This kind of solution is justified in the context of gDE models but it might be that also dDE models are included in the parametrized metric

• Since Λ does not deflect light, it is a preferred test for the study of DE models in astrophysical context

r


ConclusionsConclusions

• In both cases the DE contribution to light deflection is much greater than Schw one at higher orders

• In both cases the main deviation from Λ is given by Cluster of Galaxies.

• In DGP model we have a small effect with respect to Weyl gravity for two reasons 1) The coefficient is smaller and 2) the geometric factor is smaller.


DGP model

Cr

sun

gal

g.gal

cluster

tot )( ISCHW DE)( II

SCHW

5109.5 5109.5 9107.1 0

6109.1 6109.1 12108.1 10100.4

5107.1 5107.1 10106.1 8103.2

5105.6 5105.6 9103.3 7106.8


Weyl gravity

Cr

sun

gal

g.gal

cluster

tot )( ISCHW DE)( II

SCHW

5109.5 5109.5 9107.1 19106.6

6109.3 6109.3 12108.1 6100.2

5109.3 5109.3 10108.1 5100.2

4109.4 5108.9 9107.4 4100.4

light bending as a probe for geometric dark energy

Documents