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源自弦景观的有效Quintessence

翟韩豫 ,  申佳音 ,  薛迅

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源自弦景观的有效Quintessence

翟韩豫, 申佳音, 薛迅
物理学报, 2019, 68(13): 139501.
doi: 10.7498/aps.68.20190282

Effective quintessence from string landscape

Zhai Han-Yu, Shen Jia-Yin, Xue Xun
Acta Phys. Sin., 2019, 68(13): 139501.
doi: 10.7498/aps.68.20190282
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  • 摘要

    最近Vafa等人提出了两个弦沼泽地的判据, 并且在研究宇宙学中两个弦沼泽地判据所施加的约束条件时, 发现暴胀模型通常与这两个判据很难相容. 将这两个判据应用于当前时期宇宙的加速膨胀时, 发现特定的quintessence模型可以满足这些限制, 同时满足根据当前观测所施加的约束. 基于大尺度洛伦兹破缺的宇宙学模型, 我们得到了修正的Friedmann方程, 在计算的过程中给出了三种可行的近似, 并定义了一个有效的真空能量密度$ {\varLambda _{{\rm{eff}}}}$, 其行为对于弦景观表现为单调下降的类quintessence势能, 对于大部分具有裸的正真空能量密度的沼泽地, 有效宇宙学常数随时间的演化会出现局域极小. 将光度距离的计算结果与天文观测值做对比, 我们得到负宇宙学常数也可以使宇宙产生加速膨胀, 因而用亚稳的de Sitter真空来解释宇宙后期的加速膨胀是不必要的, 沼泽地猜想所导致的沼泽地模型与后期宇宙加速膨胀不相容性的困难也就不存在了.

    Abstract

    Recently, Vafa et al. proposed two string swampland criteria, and studying the constraints imposed by the two string swampland criteria on cosmology, they found that the inflationary models are generally difficult to be compatible with these two criteria. Applying these two criteria to the accelerated expansion of the universe during the current period, it was found that the specific quintessence model can satisfy these constraints while satisfying the constraints imposed by the current observations. Applying the gravitational theory of large scale Lorentz violation to cosmology, the vacuum energy density is not the only cause of the accelerated expansion of the universe. The large scale Lorentz violation combined with the cosmological constant term results in the observed accelerated expansion of the late universe. The vacuum energy density is a bit like a naked cosmological constant. The equivalent energy density considering the large scale Lorentz violation effect is the effective cosmological constant that determines the evolution of the universe. In this way, we find that the negative cosmological constant in the string landscape can also accelerate the expansion of the universe, and compared with the $ {\varLambda _{{\rm{CDM}}}}$ model, it leads to a cosmological constant as an effective vacuum energy density. Effective vacuum energy density behaves as a monotonically decreasing quintessence potential energy for the string landscape, for most of the naked positive vacuum energy densities in the swampland, the evolution of effective cosmological constant with time will show a local minimum. Comparing the calculated results of the distance modulus withthe astronomical observations, we can obtain that a negative cosmological constant also accelerates the expansion of the universe. Thus, the vacuum energy density derived from the string landscape will give quintessence potential that satisfies the swampland criterion, while the evolution of vacuum energy density given by the swampland model of the metastable dS vacuum is not quintessence potential, so it cannot satisfy the second de Sitter criterion. Therefore, the effective potential leading to the accelerated expansion of the late universe can only come from the string landscape, which is naturally UV completion. Therefore, it gives that the accelerated expansion of the late universe is the feature of early quantum gravity. It is not necessary to use the metastable de Sitter vacuum to explain the accelerated expansion of the late universe. The difficulty of incompatibility between the swampland model and the accelerated expansion of the late universe caused by the swampland conjecture will be eliminated.

    作者及机构信息

    • 1.

      华东师范大学物理系, 上海　200241

    • 2.

      新疆大学理论物理中心, 乌鲁木齐　830046

      • 通信作者: 薛迅, xxue@phy.ecnu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11435005, 11775080, 11865016)资助的课题.

    Authors and contacts

    • 1.

      Department of Physics, East China Normal University, Shanghai 200241, China

    • 2.

      Center for theoretical Physics, Xinjiang University, Urumqi 830046, China

      • Corresponding author: Xue Xun, xxue@phy.ecnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11435005, 11775080, 11865016).

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  • 图 1  亚稳态dS的标量势V

    Fig. 1.  The scalar potential V for a metastable dS.

    下载: 全尺寸图片 幻灯片

    图 2  不稳定dS的标量势V

    Fig. 2.  The scalar potential V for an unstable dS.

    下载: 全尺寸图片 幻灯片

    图 3  ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$, ${w_0} = - 0.88$时哈勃常数随时间的演化　(a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 3.  The Hubble constant evolves with the time when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) $ {\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 4  ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$, ${w_0} = - 0.88$时哈勃常数随时间的演化　(a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 4.  The Hubble constant evolves with the time when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) $ {\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 5  ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$, ${w_0} = - 0.88$时${\cal{K}}$随时间的演化　(a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 5.  ${\cal{K}}$ evolves with the time when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 6  ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$, ${w_0} = - 0.88$时${\cal{K}}$随时间的演化　(a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 6.  ${\cal{K}}$ evolves with the time when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 7  ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$, ${w_0} = - 0.88$时光度距离随红移的演化 (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 7.  The luminosity distance evolves with the redshift when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 8  ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$, ${w_0} = - 0.88$时光度距离随红移的演化 (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 8.  the luminosity distance evolves with the redshift when ${w_0} = - 0.88$, ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$ (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $.

    下载: 全尺寸图片 幻灯片

    图 9  ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$, ${w_0} = - 0.88$时距离模数测量值与理论期望值的对比　(a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 9.  Comparison of the measured distance moduluswith its expected value when ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$, ${w_0} = - 0.88$: (a) ${\varLambda _0} = - {\rm{0}}.{\rm{2}}\varLambda $; (b) ${\varLambda _0} = 0$; (c) $ {\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda$

    下载: 全尺寸图片 幻灯片

    图 10  ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$, ${w_0} = - \dfrac{8}{9}$时距离模数实验测量值与理论期望值的对比　(a) ${\varLambda _0} = - 0.2\varLambda $; (b) ${\varLambda _0} = 0$; (c) ${\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda $

    Fig. 10.  Comparison of the measured distance moduluswith its expected value when ${\cal{K}}\left( {{t_0}} \right)\!=\!- {H_0}\left(\!{\sqrt {1\!-\!\dfrac{{\varLambda\!-\!{\varLambda _0}}}{{3{H_0}^2}}}\!+\!1}\!\right)$, ${w_0} = - \dfrac{8}{9}$: (a) ${\varLambda _0} = - 0.2\varLambda $; (b) ${\varLambda _0} = 0$; (c) $ {\varLambda _0} = {\rm{0}}.{\rm{2}}\varLambda$.

    下载: 全尺寸图片 幻灯片

    图 11  ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$时${\varLambda _{{\rm{eff}}}}$从单调的quintessence类型到随时间的演化出现局域极小类型的转变　(a) CaseA1; (b) CaseB1; (c) CaseC1, ${w_0} = - 1$; (d) CaseC1, ${w_0} = - \dfrac{8}{9}$

    Fig. 11.  ${\varLambda _{{\rm{eff}}}}$ shifts from a monotonically decreasing quintessence type to a local minimal type transition with time evolution when ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$ (a) CaseA1; (b) CaseB1; (c) CaseC1, ${w_0} = - 1$; (d) CaseC1, ${w_0} = - \dfrac{8}{9}$.

    下载: 全尺寸图片 幻灯片

    图 12  ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$时${\varLambda _{{\rm{eff}}}}$从单调的quintessence类型到随时间的演化出现局域极小类型的转变　(a) CaseA2; (b) CaseB2; (c) CaseC2, ${w_0} = - 1$; (d) CaseC2, ${w_0} = - \dfrac{8}{9}$

    Fig. 12.  ${\varLambda _{{\rm{eff}}}}$ shifts from a monotonically decreasing quintessence type to a local minimal type transition with time evolution when ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$ (a) CaseA2; (b) CaseB2; (c) CaseC2, ${w_0} = - 1$; (d) CaseC2, ${w_0} = - \dfrac{8}{9}$.

    下载: 全尺寸图片 幻灯片

    表 1  Lorentz破缺宇宙学的几种模型

    Table 1.  Proposed models of Lorentz violation cosmology.

    ${\cal{K}}\left( {{t_0}} \right)$的值${\cal{K}}\left( t \right)$的方程
    CaseA1${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$ $H\left( t \right){\cal{K}}\left( t \right) + {\dot {\cal{K}}}\left( t \right) = \dfrac{1}{3}\left( {{\varLambda _0} - \varLambda } \right);$
    CaseA2${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$H\left( t \right){\cal{K}}\left( t \right) + \dot {\cal{K}}\left( t \right) = \dfrac{1}{3}\left( {{\varLambda _0} - \varLambda } \right);$
    CaseB1${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$\dot {\cal{K}} + 2H{\cal{K}} + \dfrac{1}{2}{{\cal{K}}^2} = \dfrac{1}{2}\left( {{\varLambda _0} - \varLambda } \right);$
    CaseB2${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$\dot {\cal{K}} + 2H{\cal{K}} + \dfrac{1}{2}{{\cal{K}}^2} = \dfrac{1}{2}\left( {{\varLambda _0} - \varLambda } \right);$
    CaseC1${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$\begin{array}{l} \left( {3{w_0} + 1} \right){{\cal{K}}^2} + \left( {6{w_0} + 4} \right)H{\cal{K}} + 2\dot {\cal{K}} = \left( {{w_0} + 1} \right){\varLambda _0}; \\ \end{array} $
    CaseC2${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$\begin{array}{l} \left( {3{w_0} + 1} \right){{\cal{K}}^2} + \left( {6{w_0} + 4} \right)H{\cal{K}} + 2\dot {\cal{K}} = \left( {{w_0} + 1} \right){\varLambda _0}. \\ \end{array} $
    下载: 导出CSV

    表 2  在不同模型下有无${\varLambda _0}$的临界值

    Table 2.  Whether there are critical values of ${\varLambda _0}$ under different models

    ${\cal{K}}\left( {{t_0}} \right)$的值${\varLambda _0}$的临界值
    CaseA${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$ - 0.05\varLambda $
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$ - 0.187\varLambda $
    CaseB${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$ - 0.066\varLambda $
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$ - 0.2144\varLambda $
    CaseC (${w_0} = - 1$) ${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$0.00001\varLambda $
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$0.00001\varLambda $
    CaseC(${w_0} = - {8}/{9}$)${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$$0.119\varLambda $
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$0.075\varLambda $
    CaseC(${w_0} = - {7}/{9}$)${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$无
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$0.152\varLambda $
    CaseC(${w_0} = - {2}/{3}$)${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$ 无
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$0.225\varLambda $
    CaseC(${w_0} = - {1}/{3}$)${\cal{K}}\left( {{t_0}} \right) = {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} - 1} \right)$ 无
    ${\cal{K}}\left( {{t_0}} \right) = - {H_0}\left( {\sqrt {1 - \dfrac{{\varLambda - {\varLambda _0}}}{{3{H_0}^2}}} + 1} \right)$$0.397\varLambda $
    下载: 导出CSV
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目录
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出版历程
  • 收稿日期:  2019-03-01
  • 修回日期:  2019-04-25
  • 上网日期:  2019-07-01
  • 刊出日期:  2019-07-05

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