Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes
In the initial part of this paper, we survey (in arbitrary spacetime dimension) the general FLRW cosmologies with non-interacting perfect fluids and with a canonical or phantom scalar field, minimally coupled to gravity and possibly self-interacting; after integrating the evolution equations for the...
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| author | Michela Cimaglia Massimo Gengo Livio Pizzocchero |
| author_facet | Michela Cimaglia Massimo Gengo Livio Pizzocchero |
| author_sort | Michela Cimaglia |
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| description | In the initial part of this paper, we survey (in arbitrary spacetime dimension) the general FLRW cosmologies with non-interacting perfect fluids and with a canonical or phantom scalar field, minimally coupled to gravity and possibly self-interacting; after integrating the evolution equations for the fluids, any model of this kind can be described as a Lagrangian system with two degrees of freedom, where the Lagrange equations determine the evolution of the scale factor and the scalar field as functions of the cosmic time. We analyze specific solvable models, paying special attention to cases with a phantom scalar; the latter favors the emergence of nonsingular cosmologies in which the Big Bang is replaced, e.g., with a Big Bounce or a periodic behavior. As a first example, we consider the case with dust (i.e., pressureless matter), radiation, and a scalar field with a constant self-interaction potential (this is equivalent to a model with dust, radiation, a free scalar field and a cosmological constant in the Einstein equations). In the phantom subcase (say, with nonpositive spatial curvature), this yields a Big Bounce cosmology, which is a non-absurd alternative to the standard (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula>CDM) Big Bang cosmology; this Big Bounce model is analyzed in detail, even from a quantitative viewpoint. We subsequently consider a class of cosmological models with dust and a phantom scalar, whose self-potential has a special trigonometric form. The Lagrange equations for these models are decoupled passing to suitable coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which can be interpreted geometrically as Cartesian coordinates in a Euclidean plane: in this description, the scale factor is a power of the radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></semantics></math></inline-formula>. Each one of the coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></semantics></math></inline-formula> evolves like a harmonic repulsor, a harmonic oscillator, or a free particle (depending on the signs of certain constants in the self-interaction potential of the phantom scalar). In particular, in the case of two harmonic oscillators, the curves in the plane described by the point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> as a function of time are the Lissajous curves, well known in other settings but not so popular in cosmology. A general comparison is performed between the contents of the present work and the previous literature on FLRW cosmological models with scalar fields, to the best of our knowledge. |
| format | Article |
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| institution | Kabale University |
| issn | 2218-1997 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Universe |
| spelling | doaj-art-2ff4ff369b164038baf739efc91c1d7c2024-12-27T14:57:20ZengMDPI AGUniverse2218-19972024-12-01101246710.3390/universe10120467Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular UniversesMichela Cimaglia0Massimo Gengo1Livio Pizzocchero2Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20133 Milano, ItalyDipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20133 Milano, ItalyDipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, I-20133 Milano, ItalyIn the initial part of this paper, we survey (in arbitrary spacetime dimension) the general FLRW cosmologies with non-interacting perfect fluids and with a canonical or phantom scalar field, minimally coupled to gravity and possibly self-interacting; after integrating the evolution equations for the fluids, any model of this kind can be described as a Lagrangian system with two degrees of freedom, where the Lagrange equations determine the evolution of the scale factor and the scalar field as functions of the cosmic time. We analyze specific solvable models, paying special attention to cases with a phantom scalar; the latter favors the emergence of nonsingular cosmologies in which the Big Bang is replaced, e.g., with a Big Bounce or a periodic behavior. As a first example, we consider the case with dust (i.e., pressureless matter), radiation, and a scalar field with a constant self-interaction potential (this is equivalent to a model with dust, radiation, a free scalar field and a cosmological constant in the Einstein equations). In the phantom subcase (say, with nonpositive spatial curvature), this yields a Big Bounce cosmology, which is a non-absurd alternative to the standard (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula>CDM) Big Bang cosmology; this Big Bounce model is analyzed in detail, even from a quantitative viewpoint. We subsequently consider a class of cosmological models with dust and a phantom scalar, whose self-potential has a special trigonometric form. The Lagrange equations for these models are decoupled passing to suitable coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, which can be interpreted geometrically as Cartesian coordinates in a Euclidean plane: in this description, the scale factor is a power of the radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>=</mo><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></msqrt></mrow></semantics></math></inline-formula>. Each one of the coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></semantics></math></inline-formula> evolves like a harmonic repulsor, a harmonic oscillator, or a free particle (depending on the signs of certain constants in the self-interaction potential of the phantom scalar). In particular, in the case of two harmonic oscillators, the curves in the plane described by the point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> as a function of time are the Lissajous curves, well known in other settings but not so popular in cosmology. A general comparison is performed between the contents of the present work and the previous literature on FLRW cosmological models with scalar fields, to the best of our knowledge.https://www.mdpi.com/2218-1997/10/12/467FLRW universesEinstein–scalar cosmologiesphantom scalar fieldsnonsingular universesBig Bounce |
| spellingShingle | Michela Cimaglia Massimo Gengo Livio Pizzocchero Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes Universe FLRW universes Einstein–scalar cosmologies phantom scalar fields nonsingular universes Big Bounce |
| title | Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes |
| title_full | Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes |
| title_fullStr | Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes |
| title_full_unstemmed | Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes |
| title_short | Cosmologies with Perfect Fluids and Scalar Fields in Einstein’s Gravity: Phantom Scalars and Nonsingular Universes |
| title_sort | cosmologies with perfect fluids and scalar fields in einstein s gravity phantom scalars and nonsingular universes |
| topic | FLRW universes Einstein–scalar cosmologies phantom scalar fields nonsingular universes Big Bounce |
| url | https://www.mdpi.com/2218-1997/10/12/467 |
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