General Relativity: An Introduction for Physicists · Problem 15.1

Cosmological models — Hobson, Efstathiou & Lasenby

Physics · cmb-photon-number-density

Problem

For blackbody radiation, the number density of photons with frequencies in the range [ν,ν+dν][\nu, \nu + d\nu] is given by n(ν,T) dν=8πν2c3(ehν/kT−1) dν,(E15.1)n(\nu, T)\, d\nu = \frac{8\pi\nu^2}{c^3\left(e^{h\nu/kT} - 1\right)}\, d\nu, \tag{E15.1} where TT is the 'temperature' of the radiation. By conserving the total number of photons, show that the photon energy distribution of the cosmic microwave background (CMB) radiation retains its general blackbody form as the universe expands. Show further that the total number density nn of photons is n(T)=0.244(2πkBThc)3.n(T) = 0.244\left(\frac{2\pi k_\mathrm{B} T}{hc}\right)^3. Hence show that the present-day number density of CMB photons in the universe is n0≈4×108 m−3n_0 \approx 4\times 10^8\ \mathrm{m}^{-3}, and compare this with the present-day number density of protons. How does this ratio vary with cosmic time? Hint: ∫0∞x2ex−1 dx=0.244π2\int_0^\infty \dfrac{x^2}{e^x - 1}\, dx = 0.244\pi^2.

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