Understanding the unnaturally small size of the cosmological constant poses one of severest challenges for a theory of gravity. At late times and for large distances, the apparent size of the cosmological constant is constrained to be extremely small in terms of the natural scale for gravity, the Planck mass. In contrast, no observations bound the value of the cosmological constant during the earliest stages of the universe, when corrections to the Einstein-Hilbert action were non-negligible, and its presence can lead to a richer family

of metrics. Among the solutions for a more general gravitational action, the presence of a positive cosmological constant does not inevitably lead to a de Sitter expansion. Such solutions must still yield or evolve into a low energy theory in which the effective cosmological constant is small to be phenomenologically acceptable. If the characteristic scales on which these metrics vary are of extremely high energy or short distance, then it may be possible to integrate out such features to arrive at a slowly varying e®ective theory.

To determine whether an action for gravity, generalized beyond an Einstein-Hilbert term, admits these features | natural coefficients for the terms in the action and a rapid variation | we must ¯rst solve the highly non-linear field equations. This task is difficult even when only the next curvature corrections are added. In 3 + 1 dimensions, Horowitzand Wald and later Starobinsky [3] discovered oscillating solutions for actions that

included quadratic curvature terms but no cosmological constant. Numerical solutions were found in 4 + 1 dimensions in the presence of a cosmological constant and a scalar field, along with the quadratic curvature terms. In this latter scenario, metrics exist that depend periodically on the extra spatial coordinate so that choosing the size of the extra dimension to be the period produces a compact extra dimension without any fine-tunings or singularities. The parameters in the action ¯x the size of the extra dimension uniquely.

However, without an analytical approach it becomes di±cult to generalize these solutions to include an evolution in time. Without this freedom, it is di±cult to understand how a universe starting from a more general state can find itself in one of these configurations.

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# Importance of Cosmological Constant

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