Abstract
Recently the so-called mimetic gravity approach has been used to obtain corrections to the Friedmann equation of General Relativity similar to the ones present in loop quantum cosmology. In this paper, we propose an alternative way to derive this modified Friedmann equation via the so-called non-polynomial gravity approach, which consists of adding geometric non-polynomial higher derivative terms to Hilbert–Einstein action, which are nonetheless polynomials and lead to a second-order differential equation in Friedmann–Lemaître–Robertson–Walker space-times. Our explicit action turns out to be a realization of the Helling proposal of effective action with an infinite number of terms. The model is also investigated in the presence of a non-vanishing cosmological constant, and a new exact bounce solution is found and studied.
Highlights
Because they indicate a breakdown of the predictivity of the theory under consideration, it is believed that singularities are not part of nature
In principle, it is possible to reproduce the loop quantum cosmology modification of the Friedmann equation—and the bounce that replaces the big bang—via higher-order corrections to Einstein–Hilbert action [32,33]. These corrections must lead to second-order equations of motion, and so are truly geometrical corrections in the sense that unlike a generic modified gravity model, they do not involve additional fields with no direct geometrical meaning compared to the metric, or compared to the scalar field responsible for the local rescaling invariance in some models of conformal gravity, for example
We have found an explicit covariant Lagrangian formulation of the loop quantum cosmology tree level correction [55] to Friedmann equation in terms of an infinite sum of non-polynomial gravity corrections to Einstein–Hilbert action
Summary
Because they indicate a breakdown of the predictivity of the theory under consideration, it is believed that singularities are not part of nature. Galaxies 2017, 5, 51 equation as LQC, with the same bounce This approach was intended to be an effective action formulation of the loop quantization procedure of FLRW space-times. It should be stressed that when dealing with modified gravity models on FLRW space-times, in general other singularities may arise (e.g., [44] and references therein). The NPG approach is intended to mimic a specific sector of a fundamental (i.e., background-independent) effective theory, in which only gravitational metric corrections with no additional derivatives are present. In this way, invariants built making use of non-polynomial terms in the metric become polynomials in the FLRW sector, becoming candidates to build an effective action there.
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