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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写宇宙学cosmology方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写宇宙学cosmology代写方面经验极为丰富，各种代写宇宙学cosmology相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|宇宙学代写cosmology代考|The geodesic equation

In Minkowski space, particles travel in straight lines unless they are acted on by a force. Not surprisingly, the paths of particles in more general spacetimes are more complicated. In a curved space, the notion of a straight line gets generalized to a geodesic, the shortest path (or, in general extremal path) between two points. Quite beautifully, general relativity states that this is precisely the path followed by a particle in the absence of any forces apart from gravity. To express this in equations, we must generalize Newton’s law with no forces, $d^2 \boldsymbol{x} / d t^2=0$, to accommodate more general coordinate systems and spacetimes.

The machinery necessary to generalize $d^2 x / d t^2=0$ is perhaps best introduced by starting with a simple example: free particle motion in a Euclidean 2D plane. In that case, the equations of motion in Cartesian coordinates $x^i=(x, y)$ are
$$\frac{d^2 x^i}{d t^2}=0 .$$
However, if we use polar coordinates $x^{\prime i}=(r, \theta)$ instead, the equations for a free particle look significantly different. The fundamental difference between the two coordinate systems is that the basis vectors for polar coordinates $\hat{\boldsymbol{r}}, \hat{\theta}$ vary in the plane. Therefore, the coordinates $r$ and $\theta$ do not satisfy $d^2 x^{\prime i} / d t^2=0$.

## 物理代写|宇宙学代写cosmology代考|Distances

We can anticipate that measuring distance in an expanding universe will be a tricky business. Referring back to the expanding grid of Fig. 1.1, we immediately see two possible ways to measure distance, the comoving distance which remains fixed as the universe expands or the physical distance which grows simply because of the expansion. Frequently,neither of these two measures accurately describes the process of interest. For example, light leaving a distant galaxy at redshift 3 starts its journey towards us when the scale factor was only a quarter of its present value and ends it today when the universe has expanded by a factor of 4 . Which distance do we use in that case to relate, say, the luminosity of the galaxy to the flux we see?

The starting point for the calculation of distances is the comoving (or coordinate) distance which refers to the coordinate grid and is simple to define mathematically. Consider the comoving distance between a distant light source and us. In a small time interval $d t$, light travels a comoving distance $d x=d t / a$ (recall that we are setting $c=1$ ), so the total comoving distance traveled by light that began its journey from an object at time $t$ when the scale factor was equal to $a$ (or redshift $z=1 / a-1$ ) is
$$\chi(t)=\int_t^{t_0} \frac{d t^{\prime}}{a\left(t^{\prime}\right)}=\int_{a(t)}^1 \frac{d a^{\prime}}{a^{\prime 2} H\left(a^{\prime}\right)}=\int_0^z \frac{d z^{\prime}}{H\left(z^{\prime}\right)} .$$
Here we have changed the integration over $t^{\prime}$ to one over $a^{\prime}$, which brings in the additional factor of $\dot{a}=a H$ in the denominator, and finally to $z^{\prime}$. As the final expression makes clear, for small redshifts $z$ we can write the comoving distance as $\chi \approx z / H_0$ (verifying our handwaving discussion of the Hubble diagram at small redshifts in Sect. 1.2). The behavior at larger redshift in the fiducial concordance cosmology is depicted in Fig. 2.3.

Before relating the comoving distance to observables, let us take a quick detour to consider the comoving distance $\eta$ that light could have traveled (in the absence of interactions) since $t=0$,
$$\eta(t) \equiv \int_0^t \frac{d t^{\prime}}{a\left(t^{\prime}\right)}$$

# 宇宙学代考

## 物理代写|宇宙学代写cosmology代考|测地方程

$$\frac{d^2 x^i}{d t^2}=0 .$$

## 物理代写|宇宙学代写cosmology代考|距离

.

$$\chi(t)=\int_t^{t_0} \frac{d t^{\prime}}{a\left(t^{\prime}\right)}=\int_{a(t)}^1 \frac{d a^{\prime}}{a^{\prime 2} H\left(a^{\prime}\right)}=\int_0^z \frac{d z^{\prime}}{H\left(z^{\prime}\right)} .$$这里我们改变了积分 $t^{\prime}$ 到1 / 1 $a^{\prime}$，这就引入了额外的因子 $\dot{a}=a H$ 分母上，最后是 $z^{\prime}$。最后的表达式表明，对于小的红移 $z$ 我们可以把移动距离写成 $\chi \approx z / H_0$ (验证我们在第1.2节中对哈勃图小红移的挥手讨论)。图2.3描述了基准一致宇宙学中较大红移时的行为

$$\eta(t) \equiv \int_0^t \frac{d t^{\prime}}{a\left(t^{\prime}\right)}$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师