Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity
A scalar–tensor theory of gravity was considered, wherein the gravitational coupling <i>G</i> and the speed of light <i>c</i> were admitted as space–time functions and combined to form the definition of the scalar field <inline-formula><math xmlns="http://www.w3...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/3/709 |
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| author | Rodrigo R. Cuzinatto Rajendra P. Gupta Pedro J. Pompeia |
| author_facet | Rodrigo R. Cuzinatto Rajendra P. Gupta Pedro J. Pompeia |
| author_sort | Rodrigo R. Cuzinatto |
| collection | DOAJ |
| description | A scalar–tensor theory of gravity was considered, wherein the gravitational coupling <i>G</i> and the speed of light <i>c</i> were admitted as space–time functions and combined to form the definition of the scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>. The varying <i>c</i> participates in the definition of the variation of the matter part of the action; it is related to the effective stress–energy tensor, which is a result of the requirement of symmetry under general coordinate transformations in our gravity model. The effect of the cosmological coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula> is accommodated within a possible behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>. We analyzed the dynamics of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> in the phase space, thereby showing the existence of an attractor point for reasonable hypotheses on the potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and no particular assumption on the Hubble function. The phase space analysis was performed both with the linear stability theory and via the more general Lyapunov method. Either method led to the conclusion that the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo>˙</mo></mover><mo>/</mo><mi>G</mi><mo>=</mo><mi>σ</mi><mfenced separators="" open="(" close=")"><mover accent="true"><mi>c</mi><mo>˙</mo></mover><mo>/</mo><mi>c</mi></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> must hold for the rest of cosmic evolution after the system arrives at the globally asymptotically stable fixed point and the dynamics of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> ceases. This result realized our main motivation: to provide a physical foundation for the phenomenological model admitting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mi>G</mi><mo>/</mo><msub><mi>G</mi><mn>0</mn></msub></mfenced><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mi>c</mi><mo>/</mo><msub><mi>c</mi><mn>0</mn></msub></mfenced><mn>3</mn></msup></mrow></semantics></math></inline-formula>, used recently to interpret cosmological and astrophysical data. The thus covarying couplings <i>G</i> and <i>c</i> impact the cosmic evolution after the dynamical system settles to equilibrium. The secondary goal of our work was to investigate how this impact occurs. This was performed by constructing the generalized continuity equation in our scalar–tensor model and considering two possible regimes for the varying speed of light—decreasing <i>c</i> and increasing <i>c</i>—while solving our modified Friedmann equations. The solutions to the latter equations make room for radiation- and matter-dominated eras that progress to a dark-energy-type of accelerated expansion. |
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| language | English |
| last_indexed | 2024-03-11T05:49:53Z |
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| spelling | doaj.art-0d192631420e477fb766427648baed5e2023-11-17T14:09:52ZengMDPI AGSymmetry2073-89942023-03-0115370910.3390/sym15030709Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor GravityRodrigo R. Cuzinatto0Rajendra P. Gupta1Pedro J. Pompeia2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, CanadaDepartment of Physics, University of Ottawa, Ottawa, ON K1N 6N5, CanadaDepartment of Physics, University of Ottawa, Ottawa, ON K1N 6N5, CanadaA scalar–tensor theory of gravity was considered, wherein the gravitational coupling <i>G</i> and the speed of light <i>c</i> were admitted as space–time functions and combined to form the definition of the scalar field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>. The varying <i>c</i> participates in the definition of the variation of the matter part of the action; it is related to the effective stress–energy tensor, which is a result of the requirement of symmetry under general coordinate transformations in our gravity model. The effect of the cosmological coupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Λ</mo></semantics></math></inline-formula> is accommodated within a possible behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula>. We analyzed the dynamics of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> in the phase space, thereby showing the existence of an attractor point for reasonable hypotheses on the potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and no particular assumption on the Hubble function. The phase space analysis was performed both with the linear stability theory and via the more general Lyapunov method. Either method led to the conclusion that the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo>˙</mo></mover><mo>/</mo><mi>G</mi><mo>=</mo><mi>σ</mi><mfenced separators="" open="(" close=")"><mover accent="true"><mi>c</mi><mo>˙</mo></mover><mo>/</mo><mi>c</mi></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> must hold for the rest of cosmic evolution after the system arrives at the globally asymptotically stable fixed point and the dynamics of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϕ</mi></semantics></math></inline-formula> ceases. This result realized our main motivation: to provide a physical foundation for the phenomenological model admitting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="(" close=")"><mi>G</mi><mo>/</mo><msub><mi>G</mi><mn>0</mn></msub></mfenced><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mi>c</mi><mo>/</mo><msub><mi>c</mi><mn>0</mn></msub></mfenced><mn>3</mn></msup></mrow></semantics></math></inline-formula>, used recently to interpret cosmological and astrophysical data. The thus covarying couplings <i>G</i> and <i>c</i> impact the cosmic evolution after the dynamical system settles to equilibrium. The secondary goal of our work was to investigate how this impact occurs. This was performed by constructing the generalized continuity equation in our scalar–tensor model and considering two possible regimes for the varying speed of light—decreasing <i>c</i> and increasing <i>c</i>—while solving our modified Friedmann equations. The solutions to the latter equations make room for radiation- and matter-dominated eras that progress to a dark-energy-type of accelerated expansion.https://www.mdpi.com/2073-8994/15/3/709dynamical analysiscovarying coupling constantsscalar–tensor theorygravitycosmology |
| spellingShingle | Rodrigo R. Cuzinatto Rajendra P. Gupta Pedro J. Pompeia Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity Symmetry dynamical analysis covarying coupling constants scalar–tensor theory gravity cosmology |
| title | Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity |
| title_full | Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity |
| title_fullStr | Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity |
| title_full_unstemmed | Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity |
| title_short | Dynamical Analysis of the Covarying Coupling Constants in Scalar–Tensor Gravity |
| title_sort | dynamical analysis of the covarying coupling constants in scalar tensor gravity |
| topic | dynamical analysis covarying coupling constants scalar–tensor theory gravity cosmology |
| url | https://www.mdpi.com/2073-8994/15/3/709 |
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