1. Dark Energy and Extended Gravity theories Francesca Perrotta (SISSA, Trieste)

2. 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

3. 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.

4. 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).

5. The cosmological constant  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, while  discrepancy of 123 orders of magnitude !

6. “Quintessence’’ models of DE A classical, minimally-coupled scalar field evolves in a potential V  while 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”)

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

8. Tracking solutions are defined in the background (matter or rad.) dominated epoch. In the present Quintessence-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 out by observations of CMB, LSS, IA Supernovae (Bludmann 2004). (Spergel et al. 2003) The fine-tunig problem is resumed (Bludman 2004)

9. 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 aspects of more general NMC theories

10. Generalized theories of Gravity Explored classes: coupling function 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;

11. Background effects Friedmann equation: Klein-Gordon:  Changing effective G changes cosmic expansion rate. The R-term originates a ``boost’’ in the field dynamics

12. R-boost Since R diverges as a -3 as a  0 (if non-relativistic species are present), 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).

13. Approaching  without fine-tuning Matarrese S., Baccigalupi C., Perrotta F., 2004 W 0 = -0.999, TEQ for different initial K,V Even if w  -1, R boost enlarges the allowed range of initial energy densities

14. 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

15. Modifications to the Poisson equation (Perrotta et al. 2004): Possible effects on collapsed structures?

16. (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

17. Projection effects: Integrated Sachs-Wolfe (ISW) effect: 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

18. ISW and Projection effects on CMB multipoles

19. CMB and Extended Quintessence Integrated Sachs-Wolfe effect: For l < 10, Projection: These corrections will depend on the value and sign of the coupling constant 

20. CMB and Dark Energy ISW not detectable, because of Cosmic Variance; Projection effects show degeneracy with variations of  m, H 0, 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  ?

21. Constraints on a time-varying G Jordan-Brans-Dicke parameter: 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,

22. HOWEVER, a time–dependent G alters the Hubble length at matter-radiation equality, which is a scale imprinted on the power spectrum. CMB and large-scale structure experiments can 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 the ones probed by the CMB: we should expect different constraints on  and 

23. 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