Born-Infeld理论中的光线偏折及暗能量宇宙学模型
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摘要
自1934年波恩(Born)和英菲尔德(Infeld)为克服麦克斯韦理论的内在困难而提出非线性电动力学以来，Born-Infeld(B-I)的非线性理论得到长足的发展，特别是最近在弦理论和D-Brane理论中的发展使得B-I理论重新得到重视，它在宇宙学中的应用正在被广泛研究着。
本论文中我们研究了B-I型引力理论中的光线偏折问题并求出光线偏转角为△(?)≈4M/r_(min)[1-(4M/r_(min))~3 9k~2β/r_(min)~6]。当参数β或k趋向于0时，偏转角回到第一项爱因斯坦理论的结果。在宇宙小尺度上第二项给出的修正非常小，目前的观测精确还不足以区分。
作为暗能量模型的候选者，我们提出了拉格朗日为L_(NLBI)=1/η(1-(?))-u(φ)的非线性B-I(Nonlinear B-I，NLBI)标量场模型。该标量场和正则标量场(Quintessence)一样，违反强能量条件，态参量w演化处于±1之间，可以解释当前的宇宙加速膨胀现象。NLBI标量场可以看成是态参量w为-1的暗能量和压强为0的暗物质的统一模型。对自治系统的稳定性分析得到宇宙演化存在晚期类de Sitter稳定吸引子解的充分条件为势存在正的最小值。
针对态参量w可能小于-1的观测结果，我们提出了拉格朗奇为L_(PNLBI)=1/η(1-(?))-u(φ)的Phantom NLBI标量场模型，此时标量场违反弱能量条件。宇宙演化存在晚期类de Sitter稳定吸引子解的充分条件为势存在正的最大值，此时Phantom模型中的大劈裂可以避免。在势为常数时，宇宙标度因子存在最小值可以避免奇性。在普通物质和辐射存在时宇宙的演化能和当前观测吻合，而且Phantom标量场的密度参量在早期很小以至于不影响原初的核合成。
基于负势对宇宙演化的重要影响和它在循环宇宙模型中的应用，我们研究了负势的NLBI标量场模型，其中的宇宙在经历加速膨胀，减速膨胀后会塌缩到奇点。由于NLBI标量场在φ很大时的行为会和正则标量场明显不同，因此研究宇宙塌缩到奇点附近的行为并和正则标量场模型比较对研究它们的差异和优缺点非常重要。我们详细研究了宇宙演化的几个重要阶段，并和正则标量场的情况进行了类比。在奇点附近，NLBI标量场的演化表现为普通物质而正则标量场则表现为“stiff”物质。我们研究了模型中势参数取不同值对宇宙演化的影响并得出结论：基于人择原理的考虑，负势情况下模型的势参数值必然要存在上限。
在势为线性势和平方势时我们详细考察了NLBI标量场和正则标量场的宇宙演化。在相同初始条件和参数值情况下，NLBI标量场模型中的宇宙年龄更大，暗能量态参量w的值更负，从而比正则标量场更与观测吻合；运用超新星数据对两模型的分析显示NLBI标量场模型要略优于正则标量场模型。不过这一优势在暗能量态参量w趋向-1时变得不明显，因此有必要研究其它观测数据对两模型的约束。
我们研究了Einstein框架下的Brans-Dicke(EBD)理论的自治系统的临界点稳定性问题。该理论下的动力学自治系统等价于耦合正则标量场模型；但只有普通物质和Dilaton标量场耦合，而辐射实质上没有耦合。我们分析了各个临界点的稳定性并给出了它们的相空间演化图。利用太阳系观测给予EBD理论中β参数的限制并结合临界点的分析结果，我们讨论了宇宙未来的演化。
作为NLBI标量场模型的推广，我们研究了一般非正则标量场模型(GeneralNon-canonical Scalar Field Model)。在标量场随时间线性演化的前提下我们精确求出了具有负最小值的平方势，此时宇宙将会经历加速膨胀，减速膨胀并最终塌缩到奇点。我们发现场随时间线性演化的模型有着很高的简并性质：只要不同形式的非正则动能项F(X)在X=X_0处泰勒展开的零阶和一阶系数F_0，F(?)_0一样，则宇宙的演化就完全一样，我们还给出了场随时间线性演化解的稳定性条件。我们研究并发现声速发散的模型实际上是场随时间线性演化模型的一种。对正压指数γ为常数解的研究发现只有γ_0≤1的常数解才是稳定的。对势为常数情况的研究表明此时的非正则标量场可以作为统一暗物质和暗能量的模型。
Nonlinear Born-Infeld theory has won full-grown development since it is firstly proposed by Born and Infeld in order to overcome the inherent problem of classical Maxwell theory in 1934. Especially due to its development in string and D-Brane theory, Nonlinear Born-Infeld theory recovers new emphasis. Its important role in cosmology has been widely studied these years.
We considered the deviation of the light path and calculated the deviation valuewas△(?)≈4M/r_(min)[1-(4M/r_(min))~3 9k~2β/r_(min)~6],which would recover the Einstein's result whenβor k equals zero. The second term in the result is so small that can not be detected by current observational apparatus.
As a candidate of dark energy, we proposed a Nonlinear Born-Infeld (NLBI) typescalar field model that was described by lagrangian L_(NLBI)=1/η(1-(?))-u(φ).NLBI scalar field violated the strong energy condition and its state equation w was between -1 and 1, just like the canonical scalar field (i.e. Quintessence). NLBI scalar field can be thought as the sum of dark energy with state equation w=-1 and matter with state equation w=0. Via phase plane and critical point analysis, we get a sufficient condition for an arbitrary potential that admits a later time attractor solution: the potential must have a positive minimum.
As to the observational result that state equation w may be less than -1. we proposed the Phantom NLBI scalar field which was described by lagrangianL_(PNLBI)=1/η(1-(?))-u(φ). This phantom NLBI scalar field violated the weakenergy condition. When the potential is constant, we find that the scale factor has minimum and therefore the universe can avoid the singularity. We find the sufficient condition for the phantom NLBI scalar field that admits a later time attractor solution is the potential must have a positive maximum. In this case, the universe can avoid the "big rip" destiny. We study a specific potential with the form ofu(φ) = u_0(1+φ/φ_0)exp(-φ/φ_0) via phase plane analysis and compute the cosmologicalevolution by numerical analysis in detail. In the presence of radiation and matter, the results show that the phantom field can survive till today(to account for the present observed accelerating expansion) without interfering with the nucleosynthesis of the standard model(the density parameterΩ_φ≈10~(-12) at the equipartition epoch), and alsoavoid the future collapse of the universe. This evolution of universe can fit the observation.
Due to the important effect of negative potentials on cosmological evolution and its role in cyclic universe, we investigate the NLBI scalar field model with negative potentials, in which the universe undergoes accelerating expansion, decelerating expansion and then contract to singularity. We study some important evolutive epochs and compare them with the Quintessence model. A notable characteristic is that NLBI scalar field behaves as ordinary matter near the singularity while the canonical scalar field behaves as stiff matter. We compare the cosmological evolutions with different potential parameters and find that the value of potential parameters must have an upper bound for anthropic consideration.
We study the NLBI scalar field model and the canonical scalar field model with the linear negative potential and the square potential in detail. In the case of same initial conditions and same potential parameters, we find that the age of the universe in NLBI scalar field model is older than the one in canonical scalar field model, and the state parameter w is less than the one in canonical scalar field model. We also use the Gold dataset of 157 SN-Ia to constrain the parameters of the two models. All the results show that the NLBI scalar field is slightly superior to the canonical scalar field model. However, it shows that this superiority is not distinct when the state parameter w approaches -1 and we need compare two models with other observations.
We present the Brans-Dicke theory in Einstein frame (EBD) in which Pauli metric is regarded as the physical space-time metric and study the stability of the critical points of the autonomous system. The EBD theory is essentially equivalent to coupled quintessence model. However, the dilatonic scalar field only couples to ordinary matter and does not couple to radiation. We present the stable condition for all critical points and numerically plot the phase plane. We discuss the future destiny of universe using the results of stable condition of critical points and the bound onβfrom the solar-system gravitational experiments.
We generalize the NLBI scalar field model and investigate the General Non-canonical Scalar Field Model。We find that a special square potential with a negative minimum is needed to drive the linear field solutionφ=φ_0/, in which theuniverse will undergo accelerating expansion, decelerating expansion, accelerating contraction, decelerating contraction and then collapse to singularity. We find that the linear field solution model is highly degenerate since the cosmic evolutions with any F(X) will be equivalent as long as the first two coefficients ( F_0, F_0) of the seriesexpansion of the function F(X) around X = X. are the same, disregarding the higher order terms of F(X). We give the stable condition for the linear field solution. Westudy the special model where the sound speed diverges and find it is actually one type of the linear field solution model. We analyze the case with a constant barotropic indexγand show that this constantγsolution is only stable forγ_0≤1 . When thepotential is taken to be constant, we obtain the first integral of the field equation and find that in this case the scalar field can be a model unified the dark matter and dark energy.
本论文中我们研究了B-I型引力理论中的光线偏折问题并求出光线偏转角为△(?)≈4M/r_(min)[1-(4M/r_(min))~3 9k~2β/r_(min)~6]。当参数β或k趋向于0时，偏转角回到第一项爱因斯坦理论的结果。在宇宙小尺度上第二项给出的修正非常小，目前的观测精确还不足以区分。
作为暗能量模型的候选者，我们提出了拉格朗日为L_(NLBI)=1/η(1-(?))-u(φ)的非线性B-I(Nonlinear B-I，NLBI)标量场模型。该标量场和正则标量场(Quintessence)一样，违反强能量条件，态参量w演化处于±1之间，可以解释当前的宇宙加速膨胀现象。NLBI标量场可以看成是态参量w为-1的暗能量和压强为0的暗物质的统一模型。对自治系统的稳定性分析得到宇宙演化存在晚期类de Sitter稳定吸引子解的充分条件为势存在正的最小值。
针对态参量w可能小于-1的观测结果，我们提出了拉格朗奇为L_(PNLBI)=1/η(1-(?))-u(φ)的Phantom NLBI标量场模型，此时标量场违反弱能量条件。宇宙演化存在晚期类de Sitter稳定吸引子解的充分条件为势存在正的最大值，此时Phantom模型中的大劈裂可以避免。在势为常数时，宇宙标度因子存在最小值可以避免奇性。在普通物质和辐射存在时宇宙的演化能和当前观测吻合，而且Phantom标量场的密度参量在早期很小以至于不影响原初的核合成。
基于负势对宇宙演化的重要影响和它在循环宇宙模型中的应用，我们研究了负势的NLBI标量场模型，其中的宇宙在经历加速膨胀，减速膨胀后会塌缩到奇点。由于NLBI标量场在φ很大时的行为会和正则标量场明显不同，因此研究宇宙塌缩到奇点附近的行为并和正则标量场模型比较对研究它们的差异和优缺点非常重要。我们详细研究了宇宙演化的几个重要阶段，并和正则标量场的情况进行了类比。在奇点附近，NLBI标量场的演化表现为普通物质而正则标量场则表现为“stiff”物质。我们研究了模型中势参数取不同值对宇宙演化的影响并得出结论：基于人择原理的考虑，负势情况下模型的势参数值必然要存在上限。
在势为线性势和平方势时我们详细考察了NLBI标量场和正则标量场的宇宙演化。在相同初始条件和参数值情况下，NLBI标量场模型中的宇宙年龄更大，暗能量态参量w的值更负，从而比正则标量场更与观测吻合；运用超新星数据对两模型的分析显示NLBI标量场模型要略优于正则标量场模型。不过这一优势在暗能量态参量w趋向-1时变得不明显，因此有必要研究其它观测数据对两模型的约束。
我们研究了Einstein框架下的Brans-Dicke(EBD)理论的自治系统的临界点稳定性问题。该理论下的动力学自治系统等价于耦合正则标量场模型；但只有普通物质和Dilaton标量场耦合，而辐射实质上没有耦合。我们分析了各个临界点的稳定性并给出了它们的相空间演化图。利用太阳系观测给予EBD理论中β参数的限制并结合临界点的分析结果，我们讨论了宇宙未来的演化。
作为NLBI标量场模型的推广，我们研究了一般非正则标量场模型(GeneralNon-canonical Scalar Field Model)。在标量场随时间线性演化的前提下我们精确求出了具有负最小值的平方势，此时宇宙将会经历加速膨胀，减速膨胀并最终塌缩到奇点。我们发现场随时间线性演化的模型有着很高的简并性质：只要不同形式的非正则动能项F(X)在X=X_0处泰勒展开的零阶和一阶系数F_0，F(?)_0一样，则宇宙的演化就完全一样，我们还给出了场随时间线性演化解的稳定性条件。我们研究并发现声速发散的模型实际上是场随时间线性演化模型的一种。对正压指数γ为常数解的研究发现只有γ_0≤1的常数解才是稳定的。对势为常数情况的研究表明此时的非正则标量场可以作为统一暗物质和暗能量的模型。
Nonlinear Born-Infeld theory has won full-grown development since it is firstly proposed by Born and Infeld in order to overcome the inherent problem of classical Maxwell theory in 1934. Especially due to its development in string and D-Brane theory, Nonlinear Born-Infeld theory recovers new emphasis. Its important role in cosmology has been widely studied these years.
We considered the deviation of the light path and calculated the deviation valuewas△(?)≈4M/r_(min)[1-(4M/r_(min))~3 9k~2β/r_(min)~6],which would recover the Einstein's result whenβor k equals zero. The second term in the result is so small that can not be detected by current observational apparatus.
As a candidate of dark energy, we proposed a Nonlinear Born-Infeld (NLBI) typescalar field model that was described by lagrangian L_(NLBI)=1/η(1-(?))-u(φ).NLBI scalar field violated the strong energy condition and its state equation w was between -1 and 1, just like the canonical scalar field (i.e. Quintessence). NLBI scalar field can be thought as the sum of dark energy with state equation w=-1 and matter with state equation w=0. Via phase plane and critical point analysis, we get a sufficient condition for an arbitrary potential that admits a later time attractor solution: the potential must have a positive minimum.
As to the observational result that state equation w may be less than -1. we proposed the Phantom NLBI scalar field which was described by lagrangianL_(PNLBI)=1/η(1-(?))-u(φ). This phantom NLBI scalar field violated the weakenergy condition. When the potential is constant, we find that the scale factor has minimum and therefore the universe can avoid the singularity. We find the sufficient condition for the phantom NLBI scalar field that admits a later time attractor solution is the potential must have a positive maximum. In this case, the universe can avoid the "big rip" destiny. We study a specific potential with the form ofu(φ) = u_0(1+φ/φ_0)exp(-φ/φ_0) via phase plane analysis and compute the cosmologicalevolution by numerical analysis in detail. In the presence of radiation and matter, the results show that the phantom field can survive till today(to account for the present observed accelerating expansion) without interfering with the nucleosynthesis of the standard model(the density parameterΩ_φ≈10~(-12) at the equipartition epoch), and alsoavoid the future collapse of the universe. This evolution of universe can fit the observation.
Due to the important effect of negative potentials on cosmological evolution and its role in cyclic universe, we investigate the NLBI scalar field model with negative potentials, in which the universe undergoes accelerating expansion, decelerating expansion and then contract to singularity. We study some important evolutive epochs and compare them with the Quintessence model. A notable characteristic is that NLBI scalar field behaves as ordinary matter near the singularity while the canonical scalar field behaves as stiff matter. We compare the cosmological evolutions with different potential parameters and find that the value of potential parameters must have an upper bound for anthropic consideration.
We study the NLBI scalar field model and the canonical scalar field model with the linear negative potential and the square potential in detail. In the case of same initial conditions and same potential parameters, we find that the age of the universe in NLBI scalar field model is older than the one in canonical scalar field model, and the state parameter w is less than the one in canonical scalar field model. We also use the Gold dataset of 157 SN-Ia to constrain the parameters of the two models. All the results show that the NLBI scalar field is slightly superior to the canonical scalar field model. However, it shows that this superiority is not distinct when the state parameter w approaches -1 and we need compare two models with other observations.
We present the Brans-Dicke theory in Einstein frame (EBD) in which Pauli metric is regarded as the physical space-time metric and study the stability of the critical points of the autonomous system. The EBD theory is essentially equivalent to coupled quintessence model. However, the dilatonic scalar field only couples to ordinary matter and does not couple to radiation. We present the stable condition for all critical points and numerically plot the phase plane. We discuss the future destiny of universe using the results of stable condition of critical points and the bound onβfrom the solar-system gravitational experiments.
We generalize the NLBI scalar field model and investigate the General Non-canonical Scalar Field Model。We find that a special square potential with a negative minimum is needed to drive the linear field solutionφ=φ_0/, in which theuniverse will undergo accelerating expansion, decelerating expansion, accelerating contraction, decelerating contraction and then collapse to singularity. We find that the linear field solution model is highly degenerate since the cosmic evolutions with any F(X) will be equivalent as long as the first two coefficients ( F_0, F_0) of the seriesexpansion of the function F(X) around X = X. are the same, disregarding the higher order terms of F(X). We give the stable condition for the linear field solution. Westudy the special model where the sound speed diverges and find it is actually one type of the linear field solution model. We analyze the case with a constant barotropic indexγand show that this constantγsolution is only stable forγ_0≤1 . When thepotential is taken to be constant, we obtain the first integral of the field equation and find that in this case the scalar field can be a model unified the dark matter and dark energy.
引文
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