1. Higher Derivative Dark Energy Mingzhe Li Department of Physics, Nanjing University May 22, 2012 IHEP Beijing

2. Outline Dark energy models: classified by w Quintom dark energy model building: difficulties Single scalar field model: higher derivatives Single scalar field model: degenerate higher derivatives Metric-affine formalism of higher derivative dark energy Conclusions

14. Cannot correspond to perfect fluid

25. Metric-Affine Formalism of Higher Derivative Dark Energy Metric approach: Christoffel symbols Torsion free A Priori Metric compatible Einstein equation ML & X.Wang, arXiv:1205.0841

26. Metric-affine formalism: both and the metric are treated independently in the variational principle Covariant derivatives Torsion tensor Non-metricity determined by Einstein-Hilbert

27. A better strategy If matter has no direct interaction with connections, two approaches coincide. Higher derivative dark energy couples to connections directly Differences are expected

28. Metric approach Metric-affine formalism Effective Lagrangian Higher derivative dark energy

29. Contracting Usually not satisfied by the scalar field Inconsistent! Reason: Einstein-Hilbert action has higher symmetry, it is invariant under projective transformation But the scalar-connection coupling is not invariant To avoid inconsistency, we should break the projective symmetry in the gravity sector For example, adding the Lagrange multiplier term Hehl & Kerling, GRG(1978)

30. Then substitute it to

31. Example Metric approach: Shift symmetry Case I Metric-affine: Effective mass Much higher order terms No shift symmetry, approximately corresponds to Case II

32. Case ICase II

33. Conclusions Current data are consistent with the cosmological constant, but mildly favors quintom dark energy. Single field quintom model can be constructed successfully from degenerate higher derivatives. In the higher derivative model, the metric approach and the metric-affine formalism predict different dynamics. Usually these differences are suppressed by the powers of the Planck mass. In some cases, the differences might be significant.

34. Thanks!