Metric-Affine Formalism of Higher Derivative Dark Energy

Higher Derivative Dark Energy
Mingzhe Li
Department of Physics, Nanjing University
May 22, 2012 IHEP Beijing
Dark energy models: classified by w
Quintom dark energy model building: difficulties
Single scalar field model: higher derivatives
Single scalar field model: degenerate higher
• Metric-affine formalism of higher derivative dark
• Conclusions
Cannot correspond to perfect fluid
Metric-Affine Formalism of Higher Derivative
Dark Energy
ML & X.Wang, arXiv:1205.0841
Metric approach:
A Priori
Christoffel symbols
Torsion free
Metric compatible
Einstein equation
Metric-affine formalism:
and the metric are treated independently in the variational principle
Covariant derivatives
Torsion tensor
determined by
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
Higher derivative dark energy
Metric approach
Metric-affine formalism
Effective Lagrangian
Contracting  , 
Usually not satisfied by the scalar field
Hehl & Kerling, GRG(1978)
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
Then substitute it to
Metric approach:
Shift symmetry
Effective mass
Much higher order terms
No shift symmetry, approximately corresponds to Case II
Case I
Case I
Case II
• 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.

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