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大尺度洛伦兹破缺与空间平坦性

李静 薛迅

李静, 薛迅. 大尺度洛伦兹破缺与空间平坦性[J]. 华东师范大学学报(自然科学版), 2021, (1): 67-81. doi: 10.3969/j.issn.1000-5641.202022004
引用本文: 李静, 薛迅. 大尺度洛伦兹破缺与空间平坦性[J]. 华东师范大学学报(自然科学版), 2021, (1): 67-81. doi: 10.3969/j.issn.1000-5641.202022004
LI Jing, XUE Xun. Spatial flatness and large-scale Lorentz violation[J]. Journal of East China Normal University (Natural Sciences), 2021, (1): 67-81. doi: 10.3969/j.issn.1000-5641.202022004
Citation: LI Jing, XUE Xun. Spatial flatness and large-scale Lorentz violation[J]. Journal of East China Normal University (Natural Sciences), 2021, (1): 67-81. doi: 10.3969/j.issn.1000-5641.202022004

大尺度洛伦兹破缺与空间平坦性

doi: 10.3969/j.issn.1000-5641.202022004
基金项目: 国家自然科学基金(11775080, 11865016)
详细信息
    通讯作者:

    薛 迅, 男, 教授, 博士生导师, 研究方向为粒子物理与场论. E-mail: xxue@phy.ecnu.edu

  • 中图分类号: O413.4

Spatial flatness and large-scale Lorentz violation

  • 摘要: 局域测量得到的哈勃参数($ {H_0}$)与基于$\Lambda {\rm{CDM}}$(Lambda Cold Dark Matter Model)模型从微波背景辐射(Cosmic Microwave Background, CMB)测出来的哈勃参数之间存在不一致性, 促使人们超越标准宇宙学模型去考察新宇宙学模型, 比如空间非平坦的大尺度洛伦兹破缺模型. 由于空间曲率项、 宇宙学常数项和宇宙学扭曲(contortion)对于解释观测数据的贡献存在重叠简并, 这使得现有的模型对于解释观测数据具有存活空间. 通过对光度距离模量红移关系观测与模型预言的对比, 以及计算物质密度和有效宇宙学常数随时间的演化等途径, 将空间曲率密度限制在一定的范围以内, 并在此取值范围内讨论了空间不平坦的大尺度洛伦兹破缺模型的表现.
  • 图  1  $K = + 1,{\varLambda _0} = - 0.02\varLambda$, a0H0=3.5的情况下, 哈勃参数随尺度因子的演化

    Fig.  1  The Hubble constant changes with the scale factor when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$

    图  2  $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$的情况下, 哈勃参数随尺度因子的演化

    Fig.  2  The Hubble constant changes with the scale factor when $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$

    图  3  $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$${a_0}{H_0}=2.5 $的情况下, 哈勃参数随尺度因子的演化

    Fig.  3  The Hubble constant changes with the scale factor when $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$ and $2.5$

    图  4  $K = + 1,{w_0} = - 1,a_0 H_0=3.5,{\varLambda _0}{\rm{ = }} - 0.02\varLambda$的情况下, 光度距离随着红移的演化

    Fig.  4  The luminosity distance changes with redshift when $K = + 1,{w_0} = - 1,a_0 H_0=3.5,{\varLambda _0}{\rm{ = }} - 0.02\Lambda $

    图  5  不同的${a_0}{H_0}$对应的${\varLambda _{\min}}$: a) K=+1; b) K=–1

    Fig.  5  ${\varLambda _{\min }}$ changes with ${a_0}{H_0}$: a) K=+1; b) K=–1

    图  6  K=+1, a0H0=3.5, $\varLambda_0=-0.02\varLambda$情况下, 物质密度${\varOmega _{\rm{M}}}$随尺度因子的演化

    Fig.  6  The matter density ${\varOmega _{\rm{M}}} $ changes with the scale factor when K=+1, a0H0=3.5, $\varLambda_0=-0.02\varLambda$

    图  7  $K=+1, a_0 H_0=3.5$情况下, ${\varLambda _{{\rm{eff}}}}$从单调下降的quintessence类型到出现局域极小类型的转变

    Fig.  7  ${\varLambda _{{\rm{eff}}}}$ transitions from a monotonically decreasing quintessence type to a local minimum type with changes in the scale factor when $K=+1, a_0 H_0=3.5$

    图  8  $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$情况下, 距离模数的模型预言值与测量值的对比

    Fig.  8  Comparison of the measured distance modulus with its expected value when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$

    图  9  $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$情况下, 距离模数的模型预言值与测量值的对比.

    Fig.  9  Comparison of the measured distance modulus with its expected value when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$

    图  10  $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 2.5$情况下, 距离模数的模型预言值与测量值的对比

    Fig.  10  Comparison of the measured distance modulus with its expected value when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 2.5$

    图  11  $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$情况下, 距离模数的模型预言值与测量值的对比

    Fig.  11  Comparison of the measured distance modulus with its expected value when $K = - 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.5$

    图  12  $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$情况下, 距离模数的模型预言值与测量值的对比

    Fig.  12  Comparison of the measured distance modulus with its expected value when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 3.0$

    图  13  $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 2.5$情况下, 距离模数的模型预言值与测量值的对比

    Fig.  13  Comparison of the measured distance modulus with its expected value when $K = + 1,{\varLambda _0} = - 0.02\varLambda ,{a_0}{H_0} = 2.5$

    图  14  $K = +1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02\varLambda$情况下, 3种模型之间的比较

    Fig.  14  Comparison of the three models when $K = +1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02\varLambda$

    图  15  $K = +1 ,{a_0}{H_0} = 2.5 ,{\varLambda _0} = - 0.072\varLambda$情况下, ${\varOmega _{\rm{K}}}$随尺度因子的演化

    Fig.  15  ${\varOmega _{\rm{K}}}$ changes with the scale factor when $K = +1 ,{a_0}{H_0} = 2.5 ,{\varLambda _0} = - 0.072\varLambda$

    图  16  $\ddot a$随尺度因子的演化: a) $K = +1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02 \varLambda$; b) $K = -1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02\varLambda$

    Fig.  16  $\ddot a$ changes with the scale factor: a) $K = +1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02\varLambda$; b) $K = -1 ,{a_0}{H_0} = 3.5 ,{\varLambda _0} = - 0.02\varLambda$

    表  1  $K = + 1$情况下${\varLambda _{\max}}$的值

    Tab.  1  The value of ${\varLambda _{\max}}$ when $K = + 1$

    情形 ${a_0}{H_0}{\rm{ = } }2.0$${a_0}{H_0}{\rm{ = }}2.5$${a_0}{H_0}{\rm{ = 3.0} }$${a_0}{H_0}{\rm{ = 3}}{\rm{.5}}$
    CaseA1 0.0008Λ 0.0007Λ 0.0006Λ 0.0007Λ
    CaseA2 0.0012Λ 0.0014Λ 0.0015Λ 0.0015Λ
    CaseB1 0.0008Λ 0.0007Λ 0.0007Λ 0.0006Λ
    CaseB2 0.0012Λ 0.0014Λ 0.0015Λ 0.0016Λ
    CaseC1(w0 = –1) 0.0008Λ 0.0007Λ 0.0007Λ 0.0006Λ
    CaseC2(w0 = –1) 0.0012Λ 0.0015Λ 0.0015Λ 0.0016Λ
    CaseC1(w0 = $-\frac{8}{9} $) 0.0008Λ 0.0007Λ 0.0007Λ 0.0006Λ
    CaseC2(w0 = $-\frac{8}{9} $) 0.0013Λ 0.0016Λ 0.0017Λ 0.0018Λ
    CaseC1(w0 = $-\frac{7}{9} $) 0.0008Λ 0.0007Λ 0.0007Λ 0.0006Λ
    CaseC2(w0 = $-\frac{7}{9} $) 0.0013Λ 0.0016Λ 0.0018Λ 0.0019Λ
    CaseC1(w0 = $-\frac{2}{3} $) 0.0008Λ 0.0007Λ 0.0006Λ 0.0006Λ
    CaseC2(w0 = $-\frac{2}{3} $) 0.0014Λ 0.0018Λ 0.0020 0.0022Λ
    CaseC1(w0 = $-\frac{1}{3} $) 0.0008Λ 0.0006Λ 0.0006Λ 0.0006Λ
    CaseC2(w0 = $-\frac{1}{3} $) 0.0025Λ 0.0046Λ 0.0056Λ 0.0062Λ
    下载: 导出CSV

    表  2  $K = - 1$情况下${\varLambda _{\max}}$的值

    Tab.  2  The value of ${\varLambda _{\max}}$ when $K = - 1$

    情形 ${a_0}{H_0}{\rm{ = 1}}{\rm{.5}}$${a_0}{H_0}{\rm{ = } }2.0$${a_0}{H_0}{\rm{ = }}2.5$${a_0}{H_0}{\rm{ = 3.0} }$${a_0}{H_0}{\rm{ = 3}}{\rm{.5}}$
    CaseA1 0.0004Λ 0.0005Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseA2 0.0023Λ 0.002Λ 0.0019Λ 0.0018Λ 0.0018Λ
    CaseB1 0.0004Λ 0.0004Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseB2 0.0023Λ 0.002Λ 0.0019Λ 0.0019Λ 0.0018Λ
    CaseC1(w0 = –1) 0.0003Λ 0.0004Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseC2(w0 = –1) 0.015Λ 0.0027Λ 0.0023Λ 0.0021Λ 0.002Λ
    CaseC1(w0 = $-\frac{8}{9} $) 0.0003Λ 0.0004Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseC2(w0 = $-\frac{8}{9} $) 0.0033Λ 0.0033Λ 0.0026Λ 0.0024Λ 0.0023Λ
    CaseC1(w0 = $-\frac{7}{9} $) 0.0003Λ 0.0004Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseC2(w0 = $-\frac{7}{9} $) 0.0032Λ 0.0032Λ 0.0032Λ 0.0028Λ 0.0026Λ
    CaseC1(w0 = $-\frac{2}{3} $) 0.0003Λ 0.0004Λ 0.0004Λ 0.0004Λ 0.0005Λ
    CaseC2(w0 = $-\frac{2}{3} $) 0.0185Λ 0.0091Λ 0.0054Λ 0.0037Λ 0.0031Λ
    CaseC1(w0 = $-\frac{1}{3} $) 0.001Λ 0.0005Λ 0.0005Λ 0.0005Λ 0.0005Λ
    CaseC2(w0 = $-\frac{1}{3} $) 0.0147Λ 0.0119Λ 0.0105Λ 0.0097Λ 0.0092Λ
    下载: 导出CSV

    表  3  $K = + 1$Λ0-crit的值

    Tab.  3  The value of Λ0-crit when $K = + 1$

    情形${a_0}{H_0}{\rm{ = }}2.5$${a_0}{H_0}{\rm{ = 3.0} }$${a_0}{H_0}{\rm{ = 3}}.5$$K = 0$
    CaseA1 –0.072Λ –0.064Λ –0.059Λ 0.05Λ
    CaseA2 –0.14Λ –0.159Λ –0.167Λ –0.18Λ
    CaseB1 –0.094Λ –0.083Λ –0.078Λ –0.066Λ
    CaseB2 –0.154Λ –0.176Λ –0.187Λ –0.2144Λ
    CaseC1(w0 = –1) 0Λ 0Λ 0Λ 0Λ
    CaseC2(w0 = –1) 0Λ 0Λ 0Λ 0Λ
    CaseC1(w0 = $-\frac{8}{9} $) 0.162Λ 0.152Λ 0.146Λ 0.119Λ
    CaseC2(w0 = $-\frac{8}{9} $) 0.09Λ 0.086Λ 0.083Λ 0.075Λ
    CaseC1(w0 = $-\frac{7}{9} $)
    CaseC2(w0 = $-\frac{7}{9} $) 0.173Λ 0.164Λ 0.164Λ 0.152Λ
    CaseC1(w0 = $-\frac{2}{3} $)
    CaseC2(w0 = $-\frac{2}{3} $) 0.246Λ 0.235Λ 0.228Λ 0.225Λ
    CaseC1(w0 = $-\frac{1}{3} $)
    CaseC2(w0 = $-\frac{1}{3} $) 0.354Λ 0.368Λ 0.375Λ 0.397Λ
    下载: 导出CSV

    表  4  $K = - 1$Λ0-crit的值

    Tab.  4  The value of Λ0-crit when $K = - 1$

    情形${a_0}{H_0}{\rm{ = 1}}.5$${a_0}{H_0}{\rm{ = } }2.0$${a_0}{H_0}{\rm{ = }}2.5$${a_0}{H_0}{\rm{ = 3.0} }$${a_0}{H_0}{\rm{ = 3}}.5$$K = 0$
    CaseA1 –0.023Λ –0.033Λ –0.038Λ –0.041Λ –0.044Λ –0.05Λ
    CaseA2 –0.214Λ –0.208Λ –0.203Λ –0.198Λ –0.195Λ –0.187Λ
    CaseB1 –0.03Λ –0.042Λ –0.049Λ –0.053Λ –0.056Λ –0.066Λ
    CaseB2 –0.284Λ –0.262Λ –0.248Λ –0.239Λ –0.233Λ –0.214Λ
    CaseC1(w0 = –1) 0Λ 0Λ 0Λ 0Λ 0Λ 0Λ
    CaseC2(w0 = –1) 0Λ 0Λ 0Λ 0Λ 0Λ 0Λ
    CaseC1(w0 = $\frac{8}{9} $) 0.05Λ 0.086Λ 0.102Λ 0.107Λ 0.115Λ 0.119Λ
    CaseC2(w0 = $\frac{8}{9} $) 0.03Λ 0.052Λ 0.059Λ 0.064Λ 0.075Λ
    CaseC1(w0 = $\frac{7}{9} $)
    CaseC2(w0 = $\frac{7}{9} $) 0.079Λ 0.104Λ 0.118Λ 0.131Λ 0.152Λ
    CaseC1(w0 = $\frac{2}{3} $)
    CaseC2(w0 = $\frac{2}{3} $) 0.03Λ 0.15Λ 0.173Λ 0.189Λ 0.198Λ 0.225Λ
    CaseC1(w0 = $\frac{1}{3} $)
    CaseC2(w0 = $\frac{1}{3} $) 0.203Λ 0.25Λ 0.278Λ 0.309Λ 0.343Λ 0.397Λ
    下载: 导出CSV
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  • 收稿日期:  2020-03-09
  • 刊出日期:  2021-01-27

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