General Relativity is one of the pillars of modern theoretical physics: it describes gravity through geometry. Over the years, several alternative theories have been developed to answer some of the questions left open by Einstein's theory. One of the most fruitful approaches is to separate the causal structure of spacetime from its geodesic structure. This separation is obtained by considering the metric and connection as independent fields. The resulting geometrical framework is the so-called metric-affine approach to gravity. Three main geometrical attributes are associated with a given connection: the curvature, the torsion, and the nonmetricity tensors. They represent the rotation of a vector parallel transported along a closed curve, the failure of the infinitesimal parallelogram formed when two infinitesimal vectors are parallel transported along each other to close, and the change of length of a vector parallel transported along a generic curve, respectively. This thesis focuses on Symmetric Teleparallel Gravity and its generalization called $f(\mathcal{Q})$ gravity. They are alternative theories of gravity in which both curvature and torsion are zero and where only metric and nonmetricity tensors are involved in the description of the gravitational interaction. We investigate both the cosmological and astrophysical aspects of $f(\mathcal{Q})$ gravity to assess the possible improvements brought by this theory compared to the alternative ones that already exist. The main points of our study can be summarized as follows. First, we present a reconstruction algorithm for cosmological models. We specifically focus on Bianchi type-I and Friedmann-Lema\^{\i}tre-Robertson-Walker spacetimes, obtaining exact solutions that might have application in a variety of scenarios such as spontaneous isotropization of Bianchi type-I models, dark energy, and inflation as well as pre-Big Bang cosmologies. After that, using the $1 + 3$ covariant formalism, we investigate the effect of nonmetricity on the universe's dynamics. Then, using the Dynamical System Approach, we analyze the evolution of Bianchi type-I cosmologies. We consider several models of the function $f(\mathcal{Q})$, each manifesting isotropic eras of the universe, whether transitional or not. In one case, in addition to the qualitative analysis provided by the dynamical system method, we also obtain analytical solutions, showing agreement with the previously reconstructed results. We also apply the $1+1+2$ formalism, where preferred directions are chosen for time and space. Thanks to this formalism, we can introduce static and Locally Rotationally Symmetric spacetimes. Moreover, we show how nonmetricity affects all kinematic quantities involved in the covariant $1+1+2$ decomposition. We apply the resulting geometrical framework to study spherically symmetric solutions in the context of $f(\mathcal{Q})$ gravity in vacuum. We obtain explicit solutions and sufficient conditions for the existence of Schwarzschild-de Sitter type spacetimes. Finally, we investigate $f(\mathcal{Q})$ gravity coupled with spinor fields of spin-$1/2$. We present a tetrad-affine approach. After deriving the field equations, the conservation law of the spin density ensures the vanishing of the antisymmetric part of the Einstein-like equations, just as it happens in theories with torsion and metricity. We show that spinors are unaffected by the presence of the nonmetricity. We then focus on Bianchi type-I cosmological models, proposing a general procedure to solve the corresponding field equations in the coincident gauge. We provide analytical solutions in the case of gravitational Lagrangian functions of the kind $f(\mathcal{Q})=\alpha\mathcal{Q}^n$. At late times, such solutions are seen to isotropize, and depending on the value of the exponent $n$, they can undergo an accelerated expansion of the spatial scale factors.
Cosmological and astrophysical aspects in f(Q) gravity
ESPOSITO, FABRIZIO
2024-05-22
Abstract
General Relativity is one of the pillars of modern theoretical physics: it describes gravity through geometry. Over the years, several alternative theories have been developed to answer some of the questions left open by Einstein's theory. One of the most fruitful approaches is to separate the causal structure of spacetime from its geodesic structure. This separation is obtained by considering the metric and connection as independent fields. The resulting geometrical framework is the so-called metric-affine approach to gravity. Three main geometrical attributes are associated with a given connection: the curvature, the torsion, and the nonmetricity tensors. They represent the rotation of a vector parallel transported along a closed curve, the failure of the infinitesimal parallelogram formed when two infinitesimal vectors are parallel transported along each other to close, and the change of length of a vector parallel transported along a generic curve, respectively. This thesis focuses on Symmetric Teleparallel Gravity and its generalization called $f(\mathcal{Q})$ gravity. They are alternative theories of gravity in which both curvature and torsion are zero and where only metric and nonmetricity tensors are involved in the description of the gravitational interaction. We investigate both the cosmological and astrophysical aspects of $f(\mathcal{Q})$ gravity to assess the possible improvements brought by this theory compared to the alternative ones that already exist. The main points of our study can be summarized as follows. First, we present a reconstruction algorithm for cosmological models. We specifically focus on Bianchi type-I and Friedmann-Lema\^{\i}tre-Robertson-Walker spacetimes, obtaining exact solutions that might have application in a variety of scenarios such as spontaneous isotropization of Bianchi type-I models, dark energy, and inflation as well as pre-Big Bang cosmologies. After that, using the $1 + 3$ covariant formalism, we investigate the effect of nonmetricity on the universe's dynamics. Then, using the Dynamical System Approach, we analyze the evolution of Bianchi type-I cosmologies. We consider several models of the function $f(\mathcal{Q})$, each manifesting isotropic eras of the universe, whether transitional or not. In one case, in addition to the qualitative analysis provided by the dynamical system method, we also obtain analytical solutions, showing agreement with the previously reconstructed results. We also apply the $1+1+2$ formalism, where preferred directions are chosen for time and space. Thanks to this formalism, we can introduce static and Locally Rotationally Symmetric spacetimes. Moreover, we show how nonmetricity affects all kinematic quantities involved in the covariant $1+1+2$ decomposition. We apply the resulting geometrical framework to study spherically symmetric solutions in the context of $f(\mathcal{Q})$ gravity in vacuum. We obtain explicit solutions and sufficient conditions for the existence of Schwarzschild-de Sitter type spacetimes. Finally, we investigate $f(\mathcal{Q})$ gravity coupled with spinor fields of spin-$1/2$. We present a tetrad-affine approach. After deriving the field equations, the conservation law of the spin density ensures the vanishing of the antisymmetric part of the Einstein-like equations, just as it happens in theories with torsion and metricity. We show that spinors are unaffected by the presence of the nonmetricity. We then focus on Bianchi type-I cosmological models, proposing a general procedure to solve the corresponding field equations in the coincident gauge. We provide analytical solutions in the case of gravitational Lagrangian functions of the kind $f(\mathcal{Q})=\alpha\mathcal{Q}^n$. At late times, such solutions are seen to isotropize, and depending on the value of the exponent $n$, they can undergo an accelerated expansion of the spatial scale factors.
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