The General Property of Tracking and Thawing Models and Their Observational Constraints
We study the general property of the evolution of a class of scalar fields with tracking and thawing behaviors. For the tracking solutions, we show explicitly with three different potentials that, independent of initial conditions, there exists a general relation between the equation of state <in...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/9/3/146 |
| _version_ | 1797608785959714816 |
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| author | Yujie You Qichao Qiang Qing Gao |
| author_facet | Yujie You Qichao Qiang Qing Gao |
| author_sort | Yujie You |
| collection | DOAJ |
| description | We study the general property of the evolution of a class of scalar fields with tracking and thawing behaviors. For the tracking solutions, we show explicitly with three different potentials that, independent of initial conditions, there exists a general relation between the equation of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>w</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> and the fractional energy density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Ω</mo><mi>ϕ</mi></msub></semantics></math></inline-formula>, so that the scalar field follows the same <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectory during the evolution. The analytical approximations of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectories are derived even though the analytical expression depends upon the particular form of the potential. For thawing solutions, a universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation exists and the relation is independent of both the particular form of the potential and the initial condition of the scalar field. Based on the derived <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation for the thawing models, we derive a tighter upper limit on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>w</mi><mi>ϕ</mi><mo>′</mo></msubsup><mo>=</mo><mi>d</mi><msub><mi>w</mi><mi>ϕ</mi></msub><mo>/</mo><mi>d</mi><mo form="prefix">ln</mo><mi>a</mi></mrow></semantics></math></inline-formula>. The observational data is also used to constrain the thawing potential with the help of the universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation. |
| first_indexed | 2024-03-11T05:48:26Z |
| format | Article |
| id | doaj.art-ae840d569018457abbd06b29112cdb10 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-11T05:48:26Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-ae840d569018457abbd06b29112cdb102023-11-17T14:16:05ZengMDPI AGUniverse2218-19972023-03-019314610.3390/universe9030146The General Property of Tracking and Thawing Models and Their Observational ConstraintsYujie You0Qichao Qiang1Qing Gao2School of Physical Science and Technology, Southwest University, Chongqing 400715, ChinaSchool of Physical Science and Technology, Southwest University, Chongqing 400715, ChinaSchool of Physical Science and Technology, Southwest University, Chongqing 400715, ChinaWe study the general property of the evolution of a class of scalar fields with tracking and thawing behaviors. For the tracking solutions, we show explicitly with three different potentials that, independent of initial conditions, there exists a general relation between the equation of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>w</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> and the fractional energy density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Ω</mo><mi>ϕ</mi></msub></semantics></math></inline-formula>, so that the scalar field follows the same <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectory during the evolution. The analytical approximations of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectories are derived even though the analytical expression depends upon the particular form of the potential. For thawing solutions, a universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation exists and the relation is independent of both the particular form of the potential and the initial condition of the scalar field. Based on the derived <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation for the thawing models, we derive a tighter upper limit on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>w</mi><mi>ϕ</mi><mo>′</mo></msubsup><mo>=</mo><mi>d</mi><msub><mi>w</mi><mi>ϕ</mi></msub><mo>/</mo><mi>d</mi><mo form="prefix">ln</mo><mi>a</mi></mrow></semantics></math></inline-formula>. The observational data is also used to constrain the thawing potential with the help of the universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation.https://www.mdpi.com/2218-1997/9/3/146quintessence fieldtracking solutionthawing solutionobservational constraints |
| spellingShingle | Yujie You Qichao Qiang Qing Gao The General Property of Tracking and Thawing Models and Their Observational Constraints Universe quintessence field tracking solution thawing solution observational constraints |
| title | The General Property of Tracking and Thawing Models and Their Observational Constraints |
| title_full | The General Property of Tracking and Thawing Models and Their Observational Constraints |
| title_fullStr | The General Property of Tracking and Thawing Models and Their Observational Constraints |
| title_full_unstemmed | The General Property of Tracking and Thawing Models and Their Observational Constraints |
| title_short | The General Property of Tracking and Thawing Models and Their Observational Constraints |
| title_sort | general property of tracking and thawing models and their observational constraints |
| topic | quintessence field tracking solution thawing solution observational constraints |
| url | https://www.mdpi.com/2218-1997/9/3/146 |
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