$$f(\mathcal {G})$$ Noether cosmology
Francesco Bajardi, Salvatore Capozziello
Abstract
Abstract We develop the n -dimensional cosmology for $$f(\mathcal {G})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> gravity, where $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> is the Gauss–Bonnet topological invariant. Specifically, by the so-called Noether Symmetry Approach, we select $$f(\mathcal {G})\simeq \mathcal {G}^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≃</mml:mo><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math> power-law models where k is a real number. In particular, the case $$k = 1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> for $$n=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math> results equivalent to General Relativity showing that we do not need to impose the action $$R+f(\mathcal {G})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>R</mml:mi><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> to reproduce the Einstein theory. As a further result, de Sitter solutions are recovered in the case where $$f(\mathcal {G})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> 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.