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==英文题目==

'''T5. CMB (10 points)'''

The Cosmic Microwave Background (CMB) is radiation emitted during the early Universe. Its emission spectrum today has a peak at a temperature of approximately $T_{\text{today}} \sim 3K$.

(a) (3 points) What is the redshift ($z$) at which the CMB spectrum has a peak at the infrared wavelength of $\lambda_{\text{IR}} \sim 0.1 \, \text{mm}$?

(b) (7 points) Assuming a spatially flat, matter-dominated Universe, find the age of the Universe corresponding to this redshift.

==中文翻译==

'''T5. 宇宙微波背景（10分）'''

宇宙微波背景（CMB）是早期宇宙发出的辐射，当前光谱峰值温度约$T_{\text{现今}} \sim 3K$。

(a) (3分) 当CMB光谱峰值位于红外波长$\lambda_{\text{红外}} \sim 0.1 \, \text{毫米}$时，对应的红移$z$是多少？

(b) (7分) 假设空间平坦且物质主导的宇宙，求该红移对应的宇宙年龄。

== 官方解答 ==

(a) 应用维恩位移定律：

$$\lambda_{\text{today}} T_{\text{today}} = b \Rightarrow z = \frac{\lambda_{\text{today}} - \lambda_{\text{IR}}}{\lambda_{\text{IR}}} \approx 9$$

(b) 采用物质主导宇宙的年龄公式：

$$t = \frac{2}{3H_0}(1+z)^{-3/2} \approx 0.3 \, \text{Gyr}$$

[[分类:宇宙学]]
[[分类:热学]]
[[分类:由ai生成]]

@@ 第4行： / 第4行： @@
 '''T5. CMB (10 points)'''
-The Cosmic Microwave Background (CMB) is radiation emitted during the early Universe. Its emission spectrum today has a peak at a temperature of approximately \(T_{\text{today}} \sim 3K\).
+The Cosmic Microwave Background (CMB) is radiation emitted during the early Universe. It is reasonably homogeneous and isotropic, and well described by a black-body radiation spectrum. Its emission spectrum today has a peak at a temperature of approximately $$T_{\text{today}} \sim 3\ \text{K}$$, as measured by the COBE satellite FIRAS instrument.
-(a) (3 points) What is the redshift (\(z\)) at which the CMB spectrum has a peak at the infrared wavelength of \(\lambda_{\text{IR}} \sim 0.1 \, \text{mm}\)?
+(a) (3 points) What is the redshift ($$z$$) at which the CMB spectrum has a peak at the infrared wavelength of $$\lambda_{\text{IR}} \sim 0.1\ \text{mm}$$?
-(b) (7 points) Assuming a spatially flat, matter-dominated Universe, find the age of the Universe corresponding to this redshift.
+(b) (7 points) By assuming a spatially flat, matter-dominated Universe, what is the age of the Universe corresponding to the redshift of the previous part?
 ==中文翻译==
@@ 第14行： / 第14行： @@
 '''T5. 宇宙微波背景（10分）'''
-宇宙微波背景（CMB）是早期宇宙发出的辐射，当前光谱峰值温度约\(T_{\text{现今}} \sim 3K\)。
+宇宙微波背景（CMB）是早期宇宙发出的辐射。它具有较好的均匀性和各向同性，并能用黑体辐射谱很好地描述。根据COBE卫星FIRAS仪器的测量，其当前发射谱的峰值温度约为$$T_{\text{today}} \sim 3\ \text{K}$$。
-(a) (3分) 当CMB光谱峰值位于红外波长\(\lambda_{\text{红外}} \sim 0.1 \, \text{毫米}\)时，对应的红移\(z\)是多少？
+(a) (3分) 当CMB光谱的红外波段峰值波长为$$\lambda_{\text{IR}} \sim 0.1\ \text{毫米}$$时，对应的红移$$z$$是多少？
-(b) (7分) 假设空间平坦且物质主导的宇宙，求该红移对应的宇宙年龄。
+(b) (7分) 假设宇宙是空间平坦且物质主导的，计算对应上述红移时宇宙的年龄。
 == 官方解答 ==
-(a) 应用维恩位移定律：
+(a) 解答：
-$$\lambda_{\text{today}} T_{\text{today}} = b \Rightarrow z = \frac{\lambda_{\text{today}} - \lambda_{\text{IR}}}{\lambda_{\text{IR}}} \approx 9$$
+根据维恩位移定律：
-(b) 采用物质主导宇宙的年龄公式：
+$$\lambda_{\text{today}} \cdot T_{\text{today}} = b$$
-$$t = \frac{2}{3H_0}(1+z)^{-3/2} \approx 0.3 \, \text{Gyr}$$
+其中$$b$$为维恩位移常数。已知当前CMB的峰值温度$$T_{\text{today}} \approx 3\ \text{K}$$，对应的波长为：
 [[分类:宇宙学]]
 [[分类:热学]]
 [[分类:由ai生成]]

2024年IOAA理论第5题-宇宙微波背景 - 版本历史

2025年3月6日 (四) 17:11 Quan787

Quan787：创建页面，内容为“{{由ai生成}} ==英文题目== '''T5. CMB (10 points)''' The Cosmic Microwave Background (CMB) is radiation emitted during the early Universe. Its emission spect…”