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

在宇宙学标准模型中，早期宇宙在空间指数膨胀期间会产生微小的不均匀现象，称为暴胀。这些微小的不均匀性，仅在100000的水平上，印记在‘原始密度微扰’中，这在今天主要有三个可观察到的效应：它导致宇宙微波辐射轻微的各向异性和偏振，这可以用微波望远镜如wmap和planck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上也产生了各向异性。这些各向异性今天可以用射电望远镜(如LOFAR)通过氢的21厘米跃迁来绘制。原始涨落的第三个效应是为大尺度结构的形成、星系、星系团的分布以及天空中更大的结构提供了种子，我们可以在星系调查中测量到这些结构。这些结构是通过引力坍塌形成的，当物质“坠落”到原始涨落的势井中时。这种结构形成模型需要另一种成分，暗物质，它只通过与正常物质的引力相互作用才能引起人们的注意，从而增强了原始涨落的引力。原始密度微扰是由膨胀过程中存在的标量场的真空涨落产生的：或者来自引起膨胀的场本身，也来自于一个单独的标量场，也就是曲率。然而，根据早期的宇宙模型，可能涉及更多的领域。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，现有数据的数量和质量有了很大改善，特别是通过WMAP和其他实验绘制的CMB地图以及2DF和SDSS大规模结构调查。将于2008年发射的普朗克卫星将进一步改善关于中巴的数据。然而，最近人们已经意识到，在小尺度上，通过21厘米的转变绘制中性氢的图谱，可以提供一个新的、甚至可能更丰富的早期宇宙数据来源。计划于2009年完成的射电望远镜LOFAR已于2007年4月开始采集数据，并计划进行更多实验。早期宇宙的每个模型都做出了不同的可观测预测，如CMB中热点和冷点的分布和大小、CMB各向异性的“谱”及其统计特性。如果两个场相互作用，或与引力场相互作用，这将导致另一个观测结果，即非高斯性。然而，要计算谱线性(一阶)微扰理论就足够了，因为我们处理的是与场涨落成比例的量，而要处理非高斯性，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。二阶微扰理论今天仍处于初级阶段，但对于计算非高斯性和其他高阶效应是必不可少的。因此，我们将计算早期宇宙的不同现实模型的观测预测，并将它们与线性和二阶水平的数据进行比较。为了能够做到这一点，我们将扩展二阶微扰理论本身。我们将推导出的控制方程组太复杂了，即使是线性阶数也无法解析求解。这就需要发展数值方法来求解这组方程，这也是我们将在一阶和二阶摄动中所做的。

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.