$$f(\mathcal {G})$$f(G) Noether cosmology
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Regular Article - Theoretical Physics
f (G ) Noether cosmology Francesco Bajardi1,2 , Salvatore Capozziello1,2,3,4,a 1
Universitá di Napoli “Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Naples, Italy INFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, 80126 Naples, Italy 3 Gran Sasso Science Institute, viale F. Crispi 7, 67100 L’Aquila, Italy 4 Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia
2
Received: 19 May 2020 / Accepted: 16 July 2020 © The Author(s) 2020
Abstract We develop the n-dimensional cosmology for f (G) gravity, where G is the Gauss–Bonnet topological invariant. Specifically, by the so-called Noether Symmetry Approach, we select f (G) G k power-law models where k is a real number. In particular, the case k = 1/2 for n = 4 results equivalent to General Relativity showing that we do not need to impose the action R + f (G) to reproduce the Einstein theory. As a further result, de Sitter solutions are recovered in the case where f (G) is non-minimally coupled to a scalar field. This means that issues like inflation and dark energy can be addressed in this framework. Finally, we develop the Hamiltonian formalism for the related minisuperspace and discuss the quantum cosmology for this model.
1 Introduction Despite the successes and probes of General Relativity (GR), it presents issues at IR and UV scales pointing out that it is not the final theory of gravity [1,2]. Clearly there are problems with quantization of spacetime geometry (the lack of a final Quantum Gravity Theory) and with large scale structure (the unknown dark side to fit astrophysical and cosmological dynamics). In this context, modified theories of gravity (obtained by extending or changing the Hilbert–Einstein action) could be suitable to fix Dark Energy and Dark Matter issues emerging along the cosmic history. Basically, the philosophy consists in considering extended/modified gravitational Lagrangians where extra-terms in the field equations could play the role of the “Dark” components and explain the expansion of the universe and the large scale structure. Dark Matter and Dark Energy, in fact, represent a controversial problem in cosmology and astrophysics, since they are supposed to cover almost 95% of the universe content but have never been observed at a e-mail:
fundamental scales even if they manifest their effects at large scales. Nevertheless, extending/modifying GR allows to overcome several other issues, see e.g. [3,4]. In particular, they provide new polarization modes for gravitational waves [5], are capable of describing the fundamental plane of galaxies [6,7], can fit Dark Energy dynamics [8–13], can address astrophysical structures through corrections to the Newtonian potential [14]. From another point of view, modified theories may better adapt to the QFT formalism for several reasons: it is possible to extend the action in order to construct super-renormalizable theories [15] or to build up effective theories towards quantu
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