Dark Energy and Extended Gravity theories

Francesca Perrotta

(SISSA, Trieste)

Overview

• The case for Dark Energy and possible approaches;

• “Quintessence’’ models properties;• Extended Gravity Theories: effects on

cosmological background evolution (expansion rate, R-boost) and perturbative effects (CMB, weak lensing, clustering properties);

• Current constraints on G variations

The case for Dark Energy

In the old standard picture, Gravity is an attractive force, decelerating the cosmic expansion by means of the mutual attraction between matter particles and structures.

This scenario has been upset when distant Type 1A Supernovae evidenced an accelerating expansion of the Universe (Riess et al. 1998; Perlmutter et al. 1999). CMB and LSS observations strengthen this view.

Some type of “DARK ENERGY” must drive the acceleration through a REPULSIVE gravitational force.

POSSIBLE APPROACHES:

i) Add a new component with negative equation of state (e.g. Quintessence fields);

ii) Geometrically modify Gravity, e.g. including or terms depending on the curvature;

iii) Add a new fundamental force coupling (Extended theories of Gravity).

The cosmological constant

ikikikik Tc

GgRgR

4

8

2

1

If arising from the “zero point” quantum fluctuations of the known forms of matter (“vacuum energy”), then

Imposing an ultraviolet cutoff at the Planck scale,

4762500 GeV 10/ GcT

vac

while447 GeV 10

discrepancy of 123 orders of magnitude !

“Quintessence’’ models of DE

A classical, minimally-coupled scalar field evolves in a potential Vwhile its energy density and pressure combine to produce a negative equation of state w=p/ .Unlike the cosmological constant, the Quintessence field admits fluctuations .

• Fine-tuning problem: in analogy with the need to tune initial values of to get the observed energy density and equation of state;

• “Coincidence’’ problem: why m ~just today ? Search for ATTRACTOR SOLUTIONS (“tracking fields”)Search for ATTRACTOR SOLUTIONS (“tracking fields”)

e.g. Ratra-Peebles (1988) potential:

4M

V

Tracking solutions are defined in the background (matter or rad.) dominated epoch. In the presentQuintessence-dominated era, the field has already passed the tracking phase. Analyzing the post-tracking regime, good trackers (attractors with large basin of attraction) end up with an e.o.s. too different from –1, ruled outby observations of CMB, LSS, IA Supernovae(Bludmann 2004).

12.098.0 Qw (Spergel et al. 2003)

The fine-tunig problem is resumed (Bludman 2004)

Beyond General Relativity

• Can the Dark Energy be the signature of a modification of Gravity?

• Hints from Quantum Gravity: coupled to R to allow for a mechanics of geometry, equivalent to a new force in the classical limit (modify the gravitational sector of the low energy Lagrangian)

• Prototype: Jordan-Brans-Dicke theory

Deserves further scrutiny as a testing ground of many aspectsof more general NMC theories

Generalized theories of Gravity

Explored classes: RFG

R)(

8

coupling function

2

8

1)(

GF

fluid;; )(

2

1),(

2

1LL

VRf

Perrotta F., Matarrese S., Baccigalupi C., Phys. Rev. D 61 (2000) 023507

(“Extended Quintessence”)

• Modifications of the background evolution:cosmic expansion, R-boost.

• Modifications of the perturbed quantities: CMB, clustering properties, weak lensing…

Baccigalupi, Matarrese, Perrotta 2000;

Background effects

)2(2

12 ,

2,

2 VaRFa H

22

222 3

2

1

3

1

a

FV

aa

F fluid

HH

)(

1

FGeff H

Friedmann equation:

Klein-Gordon:

Changing effective G changes cosmic expansion rate.

The R-term originates a ``boost’’ in the field dynamics

R-boost

Since R diverges as a -3 as a 0 (if non-relativistic species arepresent), an “effective” potential is generated in the KG equation, boosting the dynamics of at early times. (Baccigalupi et al.2000) EQ admits tracking trajectories AND they are good trackers(large basin of attraction).

Approaching without fine-tuning

Matarrese S., Baccigalupi C., Perrotta F., 2004

W0= -0.999, TEQ for different initial K,V

Even if w-1, R boost enlarges the allowed range of initial energydensities

Perturbations effects

• Clustering properties: scalar field perturbations may interact with matter perturbations in EQ models.

• Weak lensing : variations of G induce corrections in distance calculations; perturbations gain a new d.o.f., the anisotropic stress

• CMB effects (ISW, projection, lensing, bispectrum)

Perrotta F., Baccigalupi 2002; Perrotta et al. 2003

Acquaviva V., Baccigalupi C., Perrotta F., 2004

Modifications to the Poisson equation (Perrotta et al. 2004):

Possible effects on collapsed structures?

(Acquaviva, Baccigalupi, Perrotta 2004).

The lensing signal is affected by the Dark Energy both at the background and perturbation level.

BACKGROUND effects: modification of the measures of distances (time-varying G).

PERTURBATIONS: in Generalized theories, the anisotropic stress is non-vanishing, contrarily to the “ordinary Quintessence” models, and is sourced by .

Weak lensing in Generalized Gravity theories

2128/ GPP kk

Projection effects:dec

decHH d

,

Integrated Sachs-Wolfe (ISW) effect:

.)()(1

1

3

2)( constz

zwz

Dark Energy and CMB

Part of the CMB normalization at low multipoles is due to the ISW

Affects the location of acoustic peaks

Enhanced in DE models

ISW and Projection effects on CMB

1w

5.0wmultipoles

CMB and Extended Quintessence

FG

1 Integrated Sachs-Wolfe effect:

For l < 10, 02112

decG

G

C

C

Projection:

F

F

l

l

These corrections will depend on the value and sign of the coupling constant

CMB and Dark Energy• ISW not detectable, because of Cosmic Variance; • Projection effects show degeneracy with

variations of m, H0, K, … but still the basic effect on which CMB constraints on dark energy are based so far;

• EQ: the possibility of testing EQ scenarios is related to the actual value of the coupling constant

HOW BIG IS ?

Constraints on a time-varying G

Jordan-Brans-Dicke parameter:

114 yr10|/||/| FFGG tt

000,40JBD

32

42

)(

18

JBD

JBDeff F

G

2'F

FJBD

Recent solar-system experiments (Cassini spacecraft) give a lower bound:

(Bertotti et al. 2004)

This can be translated, in a model-dependent way, into a constraint on the time variation of G. E.g., In a Brans-Dicke theory in a matter dominated universe,

HOWEVER, a time–dependent G alters the Hubble lengthat matter-radiation equality, which is a scale imprinted on thepower spectrum. CMB and large-scale structure experimentscan provide complementary constraints, on different scales(Liddle, Mazumdar, Barrow 1998). (Acquaviva V., Baccigalupi C., Leach S., Liddle A.R., Perrotta F., 2004)

The coupling parameter 1/ in scalar-tensor theories may be larger than locally is

Solar system experiments probe scales different from theones probed by the CMB: we should expect different constraintson and

Conclusions• Generalized theories of Gravity have advantages

with respect to “ordinary” Quintessence and . Fine-tuning can be alleviated

• Possible effects on structure formation and gravitational collapse

• Possible signatures from CMB and weak lensing• Coupling constants may be larger than expected