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 .