论文标题
通货膨胀的初始条件与通货膨胀时段的大地测量不完整之间的可能关系
Possible relationship between initial conditions for inflation and past geodesic incompleteness of the inflationary spacetime
论文作者
论文摘要
根据Borde-Guth-Vilenkin（BGV）定理的说法，空间的扩展区域不能扩展到某个边界\ Mathcal {B}以外的过去。因此，通货膨胀宇宙必须有某种开始。但是，BGW定理对\ Mathcal {B}甚至其位置的边界条件一无所知。在这里，我们提出了两个测量理论的单尺度场模型，其中非摩曼音量元素$υd^4x $存在于动作中。由于模型动力学，标量场φ类的可允许值的上限φ_0以边界条件的形式设置了\ Mathcal {B}的位置，其形式为spacelikehypersurfaceυ（x）= 0，具有边界条件：∂\至0^asφasφ\ toφ\ to； toφ\ to； toφ\toφ_0^ - 。一项详细的研究已经确定，如果初始动能密度ρ_{kin}^{（in）}占优势，而不是初始梯度能量密度ρ_{grad}^{（in）}，那么初始值φ___________________________________________________________________________________________________________i ρ_{kin}^{（in）}和ρ_{grad}^{（in）}不能超过势能密度，因此满足了通货膨胀发作所需的初始条件。结果表明，在\ Mathcal {b}上几乎所有可能的左侧边界条件下，这就是υ\ \ 0^ - 的地方，爱因斯坦框架中的度量张量在\ Mathcal {b}上具有跳跃不连续性，因此基督距离连接系数在空格式的超级脉冲上未定义。 Thus, if φ_{in}^{(min)}\leq φ_{in}<φ_0 and ρ_{kin}^{(in)}>ρ_{grad}^{(in)}, then there was an inflationary stage in the history of our Universe and the congruence of timelike geodesics cannot be extended to the past beyond the hypersurface Υ=0.
According to the Borde-Guth-Vilenkin (BGV) theorem an expanding region of spacetime cannot be extended to the past beyond some boundary \mathcal{B}. Therefore, the inflationary universe must have had some kind of beginning. However, the BGW theorem says nothing about the boundary conditions on \mathcal{B}, or even about its location. Here we present a single-scalar field model of the Two-Measure Theory, where the non-Riemannian volume element $Υd^4x$ is present in the action. As a result of the model dynamics, an upper bound φ_0 of admissible values of the scalar field φappears, which sets the position of \mathcal{B} in the form of a spacelike hypersurface Υ(x)=0 with a boundary condition: Υ\to 0^+ as φ\toφ_0^-. A detailed study has established that if the initial kinetic energy density ρ_{kin}^{(in)} prevails over initial gradient energy density ρ_{grad}^{(in)} then there is an interval of initial values φ_{in}^{(min)}\leqφ_{in}<φ_0, where ρ_{kin}^{(in)} and ρ_{grad}^{(in)} cannot exceed the potential energy density and hence the initial conditions necessary for the onset of inflation are satisfied. It is shown that under almost all possible left-handed boundary conditions on \mathcal{B}, that is where Υ\to 0^-, the metric tensor in the Einstein frame has a jump discontinuity on \mathcal{B}, so the Christoffel connection coefficients are not defined on the spacelike hypersurface Υ=0. Thus, if φ_{in}^{(min)}\leq φ_{in}<φ_0 and ρ_{kin}^{(in)}>ρ_{grad}^{(in)}, then there was an inflationary stage in the history of our Universe and the congruence of timelike geodesics cannot be extended to the past beyond the hypersurface Υ=0.
