2016年IOAA理论第3题-早期宇宙

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英文原题

(T3) Early Universe

Cosmological models indicate that radiation energy density, 𝜌_r , in the Universe is proportional to (1 + 𝑧) ⁴ , and the matter energy density, 𝜌_m, is proportional to (1 + 𝑧) ³ , where 𝑧 is the redshift. The dimensionless density parameter, Ω, is given as Ω = 𝜌/𝜌_c , where 𝜌_c is the critical energy density of the Universe. In the present Universe, the density parameters corresponding to radiation and matter, are Ω_r0 = 10⁻⁴ and Ω_m0 = 0.3, respectively.

(T3.1) Calculate the redshift, 𝑧e , at which radiation and matter energy densities were equal.

(T3.2) Assuming that the radiation from the early Universe has a blackbody spectrum with a temperature of 2.732 K, estimate the temperature, 𝑇_e , of the radiation at redshift 𝑧_e .

(T3.3) Estimate the typical photon energy, 𝐸_ν (in eV), of the radiation as emitted at redshift 𝑧_e .

中文翻译

(T3)早期宇宙

宇宙学模型表明，宇宙中的辐射能量密度$$\rho_r$$与$$(1+z)^4$$成正比，物质能量密度$$\rho_m$$与$$(1+z)^3$$成正比，其中$$z$$是红移。无量纲密度常数$$\Omega $$定义为$$\Omega = \rho /\rho_c$$，其中$$\rho_c$$是宇宙的临界能量密度。在当前宇宙中，与辐射和物质相对应的密度参数分别为$$\Omega_{r0}=10^{-4}$$和$$\Omega_{m0}=0.3$$。

(T3.1) 计算在辐射和物质能量密度相等时的红移$$z_e$$。

(T3.2) 假设来自早期宇宙的辐射是温度的黑体光谱温度为$$2.732\rm{K}$$，估计红移$$z_e$$处的辐射温度$$T_e$$。

(T3.3) 估计红移为$$z_e$$时的典型光子能量$$E_v$$（单位为$$\rm{eV}$$）。