# 物理代写|广义相对论代写General relativity代考|The Early Universe

#### Doug I. Jones

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## 物理代写|广义相对论代写General relativity代考|The Early Universe

In the very young smaller universe, the energy density was dominated by the radiation. At present, the universal CMB presents the most perfect black-body spectrum or Planck distribution. Penzias and Wilson, radio astronomers at Bell Labs, discovered this spectrum in 1964-65. The interesting story of that discovery is described by S. Weinberg (1977). In the Planck distribution, the energy per unit volume in the range of wavelengths between $\lambda$ and $\lambda+d \lambda$ is given by
\begin{aligned} d u & =16 \pi^2 \hbar c \frac{d \lambda}{\lambda^5}\left(\exp \left[\frac{2 \pi \hbar c}{\lambda k_B T_0}\right]-1\right)^{-1}, \ u & =\int_0^{\infty} d u=a T^4 \ & =\frac{8 \pi^5\left(k_B T\right)^4}{15(2 \pi \hbar c)^3}=7.565 \times 10^{-16} T^4 \mathrm{Jm}^{-3} \end{aligned}
The above formula is written in MKS units so it appears as you have learned it in elementary thermodynamics. Problem 8 explores the conversion to natural units. The temperature must be given in Kelvin. At present, $T_0=2.726$ K. The distribution vanishes at both $\lambda=0, \infty$, and reaches a maximum at $\lambda \approx 1.263 \hbar c /\left(k_B T\right)$.

## 物理代写|广义相对论代写General relativity代考|Matter and Dark Energy Domination

For most of the history of the universe, the dominant energies have been a combination of matter and dark energy $\Lambda$. Dark energy is now taking over. However, the period where matter was dominant and dark energy was neglected can be examined,
\begin{aligned} \left(\frac{d Q}{d t}\right)^2 & =H_0^2 Q^2 \frac{\rho_{\mathrm{OM}+\mathrm{DM}}}{\rho_c} \equiv H_0^2 Q^2 \frac{\rho_M}{\rho_c} \ & =H_0^2 Q^2 \frac{\rho_{M 0}}{\rho_c}\left(\frac{Q_0}{Q}\right)^3 \ & =H_0^2 Q_0^3 \Omega_{M 0} Q^{-1} \equiv b^2 Q^{-1} \ Q^{1 / 2} d Q & =b d t, 2 Q^{3 / 2} / 3=b\left(t-t^{\prime \prime}\right), t^{\prime \prime} \approx 0 \ Q & \propto t^{2 / 3}, \frac{d Q}{d t} \propto 2 t^{-1 / 3} / 3 \ H & =\frac{1}{Q} \frac{d Q}{d t}=\frac{2}{3 t}, t=\frac{2}{3} H^{-1} \end{aligned}
Here, $t^{\prime \prime} \approx 0$, compared to the billions of years that the radiation was no longer important. In this case, $t_{H_0}=(2 / 3) H_0^{-1}=9.7$ By. The factor $2 / 3$ due to GR was not included in Hubble’s evaluation of the age of the universe. This was the expectation before dark energy was added to the energy-momentum tensor. The calculation above really jammed up the works for awhile because some of the globular clusters in our galaxy are measured to be older. It takes the cosmological constant or dark energy to make an older universe. If dark energy becomes completely dominant, then you get $H$ (future) $=$ constant. So the critical density will not change in the future.

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|The Early Universe

\begin{aligned} d u & =16 \pi^2 \hbar c \frac{d \lambda}{\lambda^5}\left(\exp \left[\frac{2 \pi \hbar c}{\lambda k_B T_0}\right]-1\right)^{-1}, \ u & =\int_0^{\infty} d u=a T^4 \ & =\frac{8 \pi^5\left(k_B T\right)^4}{15(2 \pi \hbar c)^3}=7.565 \times 10^{-16} T^4 \mathrm{Jm}^{-3} \end{aligned}

## 物理代写|广义相对论代写General relativity代考|Matter and Dark Energy Domination

\begin{aligned} \left(\frac{d Q}{d t}\right)^2 & =H_0^2 Q^2 \frac{\rho_{\mathrm{OM}+\mathrm{DM}}}{\rho_c} \equiv H_0^2 Q^2 \frac{\rho_M}{\rho_c} \ & =H_0^2 Q^2 \frac{\rho_{M 0}}{\rho_c}\left(\frac{Q_0}{Q}\right)^3 \ & =H_0^2 Q_0^3 \Omega_{M 0} Q^{-1} \equiv b^2 Q^{-1} \ Q^{1 / 2} d Q & =b d t, 2 Q^{3 / 2} / 3=b\left(t-t^{\prime \prime}\right), t^{\prime \prime} \approx 0 \ Q & \propto t^{2 / 3}, \frac{d Q}{d t} \propto 2 t^{-1 / 3} / 3 \ H & =\frac{1}{Q} \frac{d Q}{d t}=\frac{2}{3 t}, t=\frac{2}{3} H^{-1} \end{aligned}

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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。

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