Stability and Hamiltonian formulation of higher derivative theories

TL;DR

This paper investigates the origins of instabilities in higher derivative theories, develops Hamiltonian formulations for such theories including the Starobinsky model, and provides solutions to the Wheeler de Witt equation in this context.

Contribution

It introduces Hamiltonian formulations for higher derivative gravity theories and demonstrates their application to the Starobinsky model, including solving the Wheeler de Witt equation.

Findings

Hamiltonian formulations for fourth and sixth order gravity.

Explicit solution of Wheeler de Witt equation for R^2 gravity.

Clarification on geodesic completeness in flat Friedmann models.

Abstract

We analyze the presumptions which lead to instabilities in theories of order higher than second. That type of fourth order gravity which leads to an inflationary (quasi de Sitter) period of cosmic evolution by inclusion of one curvature squared term (i.e. the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which is necessary for deducing the Wheeler de Witt equation) is found both in the Ostrogradski approach and in another form. As an example, a closed form solution of the Wheeler de Witt equation for a spatially flat Friedmann model and L=R\sp 2 is found. The method proposed by Simon to bring fourth order gravity to second order can be (if suitably generalized) applied to bring sixth order gravity to second order. In the Erratum we show that a spatially flat Friedmann model need not be geodesically complete even if the scale factor a(t) is positive and smooth for all real values of the synchronized time t.