Cosmological perturbation theory: meeting the challenges set by current and future observations

在宇宙学标准模型中，小的非均质性是在宇宙早期的空间指数膨胀时期产生的，这一时期被称为暴胀。这些微小的不均匀性，仅在10万分之一的水平上，被印在“原始密度扰动”中，今天主要有三种可观察到的效应：它导致宇宙微波辐射（CMB）辐射具有轻微的各向异性和极化，这可以用微波望远镜（如WMAP和普朗克卫星）观察到。第二个效应是在大爆炸遗留下来的中性氢的分布中也产生各向异性。如今，这些各向异性可以用射电望远镜，如LOFAR，通过氢的21厘米跃迁来绘制。原始波动的第三个作用是为大尺度结构的形成提供了种子，星系、星系团甚至更大的结构在天空中的分布，我们可以在星系调查中测量。这些结构是通过引力坍缩形成的，当物质“落入”原始波动的潜在井中时。这种结构形成模型需要另一种成分——暗物质，它使自己只能通过与正常物质的引力相互作用而被注意到，并增强了原始波动的引力。原始密度扰动是由暴胀期间存在的标量场的真空波动产生的：要么来自负责暴胀本身的场，即“暴胀”，要么来自一个单独的标量场，即“曲率”。然而，根据早期宇宙模型，可能会有更多的场参与其中。为了找到正确的模型，我们必须将理论预测与观测数据进行比较。近年来，可用数据的数量和质量都有了显著提高，特别是WMAP和其他实验以及2dF和SDSS大尺度结构调查的CMB地图。将于2008年发射的普朗克卫星将进一步改善CMB的数据。然而，最近人们意识到，在小尺度上，通过21厘米跃迁来绘制中性氢的地图，可以为早期宇宙提供一个新的、甚至可能更丰富的数据来源。计划于2009年完工的射电望远镜LOFAR已于2007年4月开始收集数据，并计划进行更多的实验。早期宇宙的每个模型都做出了不同的可观测预测，比如宇宙微波背景中热点和冷点的分布和大小，宇宙微波背景各向异性的“光谱”，以及它们的统计特性。如果这些场相互作用，或者与引力场相互作用，这将导致另一个观测结果，即非高斯性。然而，为了计算谱线性（一阶）微扰理论是足够的，因为我们处理的是与场涨落成比例的量，为了处理非高斯性，我们需要二阶微扰理论，因为现在我们必须处理场涨落中的二次量。二阶微扰理论目前仍处于起步阶段，但对于非高斯性和其他高阶效应的计算是必不可少的。因此，我们将计算早期宇宙的不同现实模型的观测预测，并在线性和二阶水平上将它们与数据进行比较。为了能够做到这一点，我们将扩展二阶摄动理论本身。我们将推导出的控制方程组太复杂，以至于无法解析解决，即使在线性顺序下也是如此。这就需要发展数值方法来解这组方程，我们将在微扰中做一阶和二阶。

In the cosmological standard model small inhomogeneities are generated in the early universe during a period of exponential expansion of space, called inflation. These small inhomogeneities, only at the level of 1 in 100000, are imprinted in the 'primordial density perturbation' which has mainly three observable effects today: it causes the Cosmic Microwave Radiation (CMB) radiation to be slightly anisotropic and polarised which can be observed with microwave telescopes such as the WMAP and PLANCK satellites. The second effect is to generate anisotropies also in the distribution of the neutral hydrogen left over from the big bang. These anisotropies can today be mapped with radio telescopes such as LOFAR through the 21cm transition of the hydrogen. The third effect of the primordial fluctuations is to act as seeds for the formation of large scale structure, the distribution of galaxies, galaxy clusters and even bigger structures in the sky, which we can measure in galaxy surveys. These structures form through gravitational collapse, when matter 'falls' into the potential wells of the primordial fluctuations. This model of structure formation needs another ingredient, dark matter, which makes itself only noticed through its gravitational interaction with normal matter, and which enhances the gravitational attraction of the primordial fluctuations. The primordial density perturbation is generated from the vacuum fluctuations of the scalar fields present during inflation: either from the field responsible for inflation itself, the 'inflaton', or from a separate scalar field, the 'curvaton'. However, there might be more fields involved, depending on the early universe model. In order to find the correct model, we have to compare the theoretical predictions to the observational data. In recent years the amount of data available and its quality have improved significantly, in particular with maps of the CMB by WMAP and other experiments and the 2dF and SDSS large scale structure surveys. The PLANCK satellite, to be launched in 2008, will improve the data on the CMB even further. Recently, however, it has been realised that on small scales the mapping of neutral hydrogen via its 21cm transition could provide a new and potentially even richer source of data about the early universe. The radio telescope LOFAR, scheduled for completion in 2009, has begun to take data in April 2007, and more experiments are planned. Each model of the early universe makes different observable predictions, such as the distribution and size of hot and cold spots in the CMB, the 'spectrum' of the CMB anisotropies, and their statistical properties. If the fields interact with each other, or with the gravitational field, this will result in yet another observational consequence, non-gaussianity. Whereas to calculate the spectrum linear (first-order) perturbation theory is sufficient since we are dealing with quantities proportional to the field fluctuations, to get a handle on non-gaussianity we need second-order perturbation theory because now we have to deal with quantities quadratic in the field fluctuations. Second-order perturbation theory is still in its infancy today, but is essential for the calculation of non-gaussianity and other higher-order effects. We will therefore calculate the observational predictions of different realistic models of the early universe and compare them with the data, at the linear and at the second-order level. In order to be able to do that, we will extend second-order perturbation theory itself. The set of governing equations we will derive are too complicated to be solved analytically, even at linear order. This necessitates the development of numerical methods to solve the set of equations, which again we will do at first- and second-order in the perturbations.