Modern Physics
Vol.07 No.06(2017), Article ID:22803,7 pages
10.12677/MP.2017.76028

Evolution of the Universe in Scalar-Tensor Gravity

Xiaofei Zhang

Institute of Aeronautical Engineering, Binzhou University, Binzhou Shandong

Received: Nov. 4th, 2017; accepted: Nov. 17th, 2017; published: Nov. 27th, 2017

ABSTRACT

In this paper, we aim to discuss the evolvement of the dark energy that is assumed to be in the Scalar-tensor gravity. We derive and analyze the characteristics of this kind of dark energy model which exceed the general relativity framework, and compare the difference between the Scalar-tensor dark energy models and the general dark energy models. At last, with the current observational constraints, we choose a special model from the Scalar-tensor models to get the numerical calculation, and obtain results which is difficult for the general single scalar field model in the general relativity framework.

Keywords:Dark Energy Model, The Equation of State

Scalar-Tensor引力下的宇宙演化

1. 引言

2. Scalar-Tensor暗能量模型的演化推导

$L=\frac{F\left(\varphi \right)R}{16\text{π}G}+\frac{1}{2}k\left(\varphi \right){\nabla }_{\mu }\varphi {\nabla }^{\mu }\varphi -V\left(\varphi \right)+{L}_{m}$ (1)

${\nabla }_{\mu }{\nabla }^{\mu }\varphi =-\frac{1}{k\left(\varphi \right)}\left[\frac{-{F}^{\prime }\left(\varphi \right)R}{16\text{π}G}+\frac{1}{2}{k}^{\prime }\left(\varphi \right){\nabla }_{\mu }\varphi {\nabla }^{\mu }\varphi +{V}^{\prime }\left(\varphi \right)\right]$ (2)

$S={\int }^{\text{​}}\text{ }{\text{d}}^{4}x\sqrt{-g}\left[\frac{F\left(\varphi \right)R}{16\text{π}G}+\frac{1}{2}k\left(\varphi \right){\nabla }_{\mu }\varphi {\nabla }^{\mu }\varphi -V\left(\varphi \right)+{L}_{m}\right]$ (3)

${g}_{\mu \nu }=\left(1,-{a}^{2}\left(t\right),-{a}^{2}\left(t\right),-{a}^{2}\left(t\right)\right)$

$\text{δ}{I}_{g}+\delta {I}_{m}=0$ (4)

$\frac{F\left(\varphi \right)}{16\text{π}G}\left({R}^{\mu \nu }-\frac{1}{2}{g}^{\mu \upsilon }R\right)=-\frac{1}{2}{T}_{m}^{\mu \nu }-\frac{1}{2}{T}_{\varphi }^{\mu \nu }+\frac{1}{16\text{π}G}\left[{\nabla }_{\mu }{\nabla }^{\mu }F\left(\varphi \right){g}^{\mu \upsilon }-F{\left(\varphi \right)}^{;\mu ;\nu }\right]$ (5)

$R=8\text{π}G\frac{-k\left(\varphi \right){\nabla }_{\mu }\varphi {\nabla }^{\mu }\varphi +4V\left(\varphi \right)+{\rho }_{m}-3{P}_{m}-3{\nabla }_{\mu }{\nabla }^{\mu }F\left(\varphi \right)}{F\left(\varphi \right)}$ (6)

${T}_{\varphi }^{\mu \nu }=k\left(\varphi \right)\left({\nabla }^{\mu }\varphi {\nabla }^{\nu }\varphi -\frac{1}{2}{g}^{\mu \nu }{\nabla }_{\mu }\varphi {\nabla }^{\mu }\varphi \right)+V\left(\varphi \right){g}^{\mu \upsilon }$ (7)

${T}_{eff}^{\mu \nu }$ 为宇宙中所有组分的能动量张量，包括了所有可能的构成成分。

${T}_{eff}^{\mu \nu }=\frac{{T}_{m}^{\mu \nu }+{T}_{\varphi }^{\mu \nu }}{F\left(\varphi \right)}+\frac{-{\nabla }_{\mu }{\nabla }^{\mu }F\left(\varphi \right){g}^{\mu \nu }+F{\left(\varphi \right)}^{;\mu ;\nu }}{F\left(\varphi \right)}$ (8)

$\begin{array}{l}{\rho }_{eff}=\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}+V\left(\varphi \right)-3H\stackrel{˙}{F}\left(\varphi \right)+{\rho }_{matt}+{\rho }_{rad}\right]\frac{1}{F\left(\varphi \right)}\\ {P}_{eff}=\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}-V\left(\varphi \right)+2H\stackrel{˙}{F}\left(\varphi \right)+\stackrel{¨}{F}\left(\varphi \right)+{P}_{matt}+{P}_{rad}\right]\frac{1}{F\left(\varphi \right)}\end{array}$ (9)

$\begin{array}{l}{\rho }_{DE}=\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}+V\left(\varphi \right)-3H\stackrel{˙}{F}\left(\varphi \right)\right]\frac{1}{F\left(\varphi \right)}\\ {P}_{DE}=\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}-V\left(\varphi \right)+2H\stackrel{˙}{F}\left(\varphi \right)+\stackrel{¨}{F}\left(\varphi \right)\right]\frac{1}{F\left(\varphi \right)}\end{array}$ (10)

$\begin{array}{c}w=\frac{{P}_{DE}}{{\rho }_{DE}}=\frac{\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}+V\left(\varphi \right)-3H\stackrel{˙}{F}\left(\varphi \right)\right]\frac{1}{F\left(\varphi \right)}}{\left[\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}-V\left(\varphi \right)+2H\stackrel{˙}{F}\left(\varphi \right)+\stackrel{¨}{F}\left(\varphi \right)\right]\frac{1}{F\left(\varphi \right)}}\\ =\frac{\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}+V\left(\varphi \right)-3H\stackrel{˙}{F}\left(\varphi \right)}{\frac{k\left(\varphi \right){\stackrel{˙}{\varphi }}^{2}}{2}-V\left(\varphi \right)+2H\stackrel{˙}{F}\left(\varphi \right)+\stackrel{¨}{F}\left(\varphi \right)}\end{array}$ (11)

3. 具体模型的数值分析

$\stackrel{¨}{\varphi }+3H\stackrel{˙}{\varphi }=\frac{{\rho }_{m}-3{P}_{m}}{2{\omega }_{0}+3}+\frac{2}{2{\omega }_{0}+3}\left(2V-\varphi \frac{\text{d}V}{\text{d}\varphi }\right)$ (12)

${w}_{DE}=\frac{{P}_{DE}}{{\rho }_{DE}}=-1+\frac{1}{\varphi }\frac{\frac{2{\omega }_{0}{\stackrel{˙}{\stackrel{˙}{\varphi }}}^{2}}{\varphi }-H\stackrel{˙}{\varphi }+\stackrel{¨}{\varphi }}{\rho }$ (13)

$g\left(\varphi \right)~\frac{2{\omega }_{0}{\stackrel{˙}{\varphi }}^{2}}{\varphi }+\left[\frac{{\rho }_{m}-3{P}_{m}}{2{\omega }_{0}+3}+\frac{2}{2{\omega }_{0}+3}\left(2V-\varphi \frac{\text{d}V}{\text{d}\varphi }\right)-4H\stackrel{˙}{\varphi }\right]$

Figure 1. ${\Omega }_{m}$ as a function of lna

Figure 2. $\varphi$ as a function of lna

Figure 3. ${\varphi }^{\prime }$ as a function of lna

Figure 4. wtot as a function of lna

Figure 5. w-(de) as a function of lna

4. 结论

Evolution of the Universe in Scalar-Tensor Gravity[J]. 现代物理, 2017, 07(06): 242-248. http://dx.doi.org/10.12677/MP.2017.76028

1. 1. Riess, A.G., et al. (1998) Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. As-tronomical Journal, 116, 1009-1038. https://doi.org/10.1086/300499

2. 2. Perlmutter, S. et al. (1999) Measurements of Omega and Lambda from 42 High Redshift Supernovae. Astronomical Journal, 517, 565-586. https://doi.org/10.1086/307221

3. 3. Tonry, J.L. et al. (2003) Cosmological Results from High-Z Supernovae. Astronomical Journal, 594, 1-24. https://doi.org/10.1086/376865

4. 4. Riess, A.G. et al. (2004) Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. Astronomical Journal, 607, 665-687. https://doi.org/10.1086/383612

5. 5. Clocchiatti, A. et al. (2006) Hubble Space Telescope and Ground-Based Observations of Type Ia Supernovae at Redshift 0.5: Cosmological Implications. Astronomical Journal, 642, 1-21. https://doi.org/10.1086/498491

6. 6. Feng, B. Wang, X. and Zhang, X. (2005) Dark Energy Constraints from the Cosmic Age and Supernova. Physical Letters B, 607, 35-41. https://doi.org/10.1016/j.physletb.2004.12.071

7. 7. Zhao, G.B., Xia, J.Q., Li, H., et al. (2007) Probing for Dynamics of Dark Energy and Curvature of Universe with Latest Cosmological Observations. Physical Letters B, 648, 8-13.

8. 8. Zhang, X. and Wu, F.Q. (2005) Constraints on Holographic Dark Energy from Type Ia supernova Observations. Physical Review D, 72, 043524-043540. https://doi.org/10.1103/PhysRevD.72.043524

9. 9. Chang, Z., Wu, F. Q. and Zhang, X. (2006) Constraints on Holographic Dark Energy from X-Ray Gas Mass Fraction of Galaxy Clusters. Physical Review Letters, 633, 14-18. https://doi.org/10.1016/j.physletb.2005.10.095

10. 10. Di Valentino, E. and Melchiorri, A. (2017) First Cosmological Constraints Combining Planck with the Recent Gravitational-Wave Standard Siren Measurement of the Hubble Constant.

11. 11. Brans, C. and Dicke, R.H. (1961) Mach’s Principle and a Relativistic Theory of Gravitation. Physical Review, 124, 925-935. https://doi.org/10.1103/PhysRev.124.925

12. 12. Hrycyna, O. and Szydlowsk, M. (2013) Dynamical Complexity of the Brans-Dicke Cosmology. Journal of Cosmology and Astroparticle Physics, 1312, 16-49. https://doi.org/10.1088/1475-7516/2013/12/016

13. 13. Emanuele, B., et al. (2015) Testing General Relativity with Present and Future Astrophysical Observations. Classical and Quantum Gravity, 32, 243001-243179. https://doi.org/10.1088/0264-9381/32/24/243001

14. 14. Li, J.X., Wu, F.Q., Li, Y.C., et al. (2014) Cosmological constraint on Brans-Dicke Model. Research in Astronomy and Astrophysics, 15, 2151-2163.

15. 15. Avilez, A. and Skordis, C. (2014) Cosmological Constraints on Brans-Dicke Theory. Physical Review Letters, 113, 011101-011105. https://doi.org/10.1103/PhysRevLett.113.011101

16. 16. Ooba, J., Ichiki, K., Chiba, T., et al. (2017) Cosmological Constraints on Scalar-Tensor Gravity and the Variation of the Gravitational Constant. Progress of Theoretical and Experimental Physics, 4, 043E03-043E18. https://doi.org/10.1093/ptep/ptx046

17. 17. Hrycyna, O., Szydlowski, M. and Kamionka, M. (2014) Dynamics and Cosmological Constraints on Brans-Dicke Cosmology. Physical Review D, 90, 124040-124052. https://doi.org/10.1103/PhysRevD.90.124040

18. 18. Berti, E., Barausse, E., Cardoso, V. et al. (2015) Testing General Relativity with Present and Future Astrophysical Observations. Classical and Quantum Gravity, 32, 243001-243179. https://doi.org/10.1088/0264-9381/32/24/243001