DataSheet1_Carroll Symmetry, Dark Energy and Inflation.zip

Carroll symmetry arises from Poincaré symmetry upon taking the limit of vanishing speed of light. We determine the constraints on the energy-momentum tensor implied by Carroll symmetry and show that for energy-momentum tensors of perfect fluid form, these imply an equation of state E+P=0 for energy density plus pressure. Therefore Carroll symmetry might be relevant for dark energy and inflation. In the Carroll limit, the Hubble radius goes to zero and outside it recessional velocities are naturally large compared to the speed of light. The de Sitter group of isometries, after the limit, becomes the conformal group in Euclidean flat space. We also study the Carroll limit of chaotic inflation, and show that the scalar field is naturally driven to have an equation of state with w = − 1. Finally we show that the freeze-out of scalar perturbations in the two point function at horizon crossing is a consequence of Carroll symmetry. To make the paper self-contained, we include a brief pedagogical review of Carroll symmetry, Carroll particles and Carroll field theories that contains some new material as well. In particular we show, using an expansion around speed of light going to zero, that for scalar and Maxwell type theories one can take two different Carroll limits at the level of the action. In the Maxwell case these correspond to the electric and magnetic limit. For point particles we show that there are two types of Carroll particles: those that cannot move in space and particles that cannot stand still.

卡罗尔对称性（Carroll symmetry）源自庞加莱对称性（Poincaré symmetry）在光速趋于零的极限情形。我们推导了卡罗尔对称性对能量动量张量（energy-momentum tensor）所施加的约束，并证明：对于满足理想流体形式的能量动量张量，该约束给出了能量密度与压强满足的物态方程$E+P=0$。因此，卡罗尔对称性或可与暗能量（dark energy）及宇宙暴胀（inflation）相关联。
在卡罗尔极限下，哈勃半径（Hubble radius）趋于零，其外部的退行速度（recessional velocities）自然远大于光速。取极限后的德西特等距群（de Sitter group of isometries）退化为欧几里得平直空间（Euclidean flat space）中的共形群（conformal group）。我们还研究了混沌暴胀（chaotic inflation）的卡罗尔极限，证明标量场（scalar field）将自然满足物态方程$w=-1$。最后我们证明：两点函数（two point function）中标量扰动在视界穿越（horizon crossing）时的冻结效应，是卡罗尔对称性的必然结果。
为使本文具备自洽性，我们附加了一篇关于卡罗尔对称性、卡罗尔粒子及卡罗尔场论的简要教学性综述（pedagogical review），其中包含若干原创内容。具体而言，通过围绕光速趋于零的展开分析，我们证明：对标量场与麦克斯韦型理论（Maxwell type theories），可在作用量（action）层面取两种不同的卡罗尔极限。在麦克斯韦情形下，这两种极限分别对应电极限与磁极限。对于点粒子（point particles），我们证明存在两类卡罗尔粒子：一类无法在空间中运动，另一类则无法静止。