2. Physical Constraints on Gauss-Bonnet Dark Energy Cosmologies Ishwaree Neupane University of Canterbury, NZ University of Canterbury, NZ DARK 2007, Sydney September 25, 2007

3. Recently, there has been a renewal of Interest in scenarios that propose alternatives or corrections to Einstein’s gravity. The proposals are of differing origin as well as motivations: some are based on multi -dimensional theories, others on scalar-curvature couplings.

4. Gauss-Bonnet Gravity: Motivations Gauss-Bonnet gravity is motivated by the stability and naturalness of the models, uniqueness of a Lagrangian in higher dimensions, the low-energy effective string actions (heterotic string),

5. Dark energy from stringy gravity One-loop corrected (heterotic) superstring action a Brans-Dicke-like runway dilaton modulus Gauss-Bonnet curvature density In a known example of string compactification No good reason to omit the scalar-curvature couplings apart from complication

6. How can current observations constrain such models?

7. The simplest Example: A fixed modulus & no Gauss-Bonnet coupling  This simplifies the theory a lot Define EOMs sufficiently simple Effective Equation of State x=0 and y=3 is a de Sitter fixed point : Lambda-CDM

8. Too Many choices Quadratic Inverse power-law Axion potential Exponential potential The issue may not be simply to achieve the dark energy equation of state For the model to work the scalar field must relax its potential energy after inflation down to a sufficiently low value: close to the observed of dark energy

9. Gauss-Bonnet driven effective dark energy Number of e-folds primarily depends on the field value GB gravity may be a solution to the dark energy problem, but a large scalar coupling strength is required GB term is topological in 4D, and, if coupled, no Ghost for Minkowski background. Cosmology requires FRW, Inflation  non-constant scalar coupling

10. Crossing the barrier of cosmological constant Equation of state parameter for the potential From top to bottom

11. Dynamics may be well behaved, but

12. An exact solution : Let Ansatz

13. Scalar spectral index

14. CMB + LSS Nature of the dark energy Tegmark et al. 2004 Is Null dominant energy condition : energy doesn’t propagate outside the light cone A model with Gauss-Bonnet corrections: No need to introduce a wrong sign kinetic term preferred?

15. A couple of remarks: 1. 2. does not depend on the equation of state of other fluid components, while definitely does Dark energy or cosmological constant problem is a cosmological problem: Almost every model of scalar gravity behaves as Einstein’s GR for

16. The Simplest Potentials Perhaps too naïve : The slopes of the potentials considered in a post inflation scenario are too large to allow the required number of e-folds of inflation The above choices hold some validity as a post-inflation approximation

17. Dashed lines (SNe IA plus CMBR shift parameter) Shaded regions (including Baryon Acoustic Oscillation scale) Koivisto & Mota hep-th/0609155

18. A non-minimally coupled scalar field Local GR constraints on Q and its derivatives (Damour et al. 1993, Esposito-Farese 2003) Within solar system and laboratories distances: is less than years

19. For the validity of weak equivalence principle Damour et al. gr-qc/0204094 (PRL)

20. Crossing of w = -1? In the absence of GB-scalar coupling, a crossing between non- phantom and phantom cosmology is unlikely.

21. A smooth progression to

22. Ghost and Superluminal modes  One may also consider a metric spacetime under quantum effect: perturbed metric about a FRW background A gauge invariant quantity: so-called a comoving perturbation Speed of propagation No-ghost and stability conditions :

23. Propagation speed of a scalar mode

24. Propagation speed for a tensor mode

25. Observing the effects of a GB coupling is the matter density contrast The growth of matter fluctuations

26. Growth of matter perturbations is With the inputs the observational limit on growth factor implies thaton large cosmological scales

27. Summary  Gauss-Bonnet modification of Einstein’s gravity can easily account for an accelerated expansion with quintessence, cosmological constant or phantom equation-of-state  The scalar-curvature coupling can also trigger onset of a late dark energy domination with  The model to be compatible with astrophysical observations, the GB dark energy density fraction should not exceed 15%.  The solar system constraints, due to a small fractional anisotropic stress can be more stronger