论文标题
背景为主的扩展中宇宙学标量场演变的确切通用解决方案
Exact general solutions for cosmological scalar field evolution in a background-dominated expansion
论文作者
论文摘要
我们得出了确切的一般解决方案（与吸引子特定解决方案相反）和相应的第一个积分，用于在具有状态参数$ w_b $方程的背景流体主导的宇宙中标量场$ ϕ $的演变。除了先前审查的线性[$ v（ϕ）= v_0 ϕ $]和二次[$ v（ϕ）= v_0 ϕ^2 $]电位外，我们还表明，Power Law电位$ v（ϕ）= v_0 ϕ^n $具有精确的解决方案，带有$ n = 4（1 + w_b）/（1 + w_b）/（1-W_B） + 2 $ n = 2（1+W_B）/（1-W_B）$。这些对应于电势$ v（ϕ）= v_0 ϕ^6 $和$ v（ϕ）= v_0 ϕ^2 $用于物质主导，$ v（ϕ）= v_0 ϕ^{10} $和$ v（ϕ）= v_0 ϕ^4 $用于辐射主导。 $ ϕ^6 $和$ ϕ^{10} $电势可以产生振荡或非振荡的演化，我们使用第一个积分来确定初始条件如何映射到每种形式的演变上。指数势可以为刚性/智力（$ w_b = 1 $）背景提供精确的解决方案。对于这种情况，我们使用此精确解决方案来得出状态参数方程的演变，以$ w_ϕ $的演变。
We derive exact general solutions (as opposed to attractor particular solutions) and corresponding first integrals for the evolution of a scalar field $ϕ$ in a universe dominated by a background fluid with equation of state parameter $w_B$. In addition to the previously-examined linear [$V(ϕ) = V_0 ϕ$] and quadratic [$V(ϕ) = V_0 ϕ^2$] potentials, we show that exact solutions exist for the power law potential $V(ϕ) = V_0 ϕ^n$ with $n = 4(1+w_B)/(1-w_B) + 2$ and $n = 2(1+w_B)/(1-w_B)$. These correspond to the potentials $V(ϕ) = V_0 ϕ^6$ and $V(ϕ) = V_0 ϕ^2$ for matter domination and $V(ϕ) = V_0 ϕ^{10}$ and $V(ϕ) = V_0 ϕ^4$ for radiation domination. The $ϕ^6$ and $ϕ^{10}$ potentials can yield either oscillatory or non-oscillatory evolution, and we use the first integrals to determine how the initial conditions map onto each form of evolution. The exponential potential yields an exact solution for a stiff/kination ($w_B = 1$) background. We use this exact solution to derive an analytic expression for the evolution of the equation of state parameter, $w_ϕ$, for this case.
