Introduction to Cosmology Ryden · Problem 2.5

Fundamental Observations

Physics — Astrophysics · blackbody-high-energy-fraction

Problem

Consider blackbody radiation at a temperature TT. Show that for an energy threshold E0≫kTE_0 \gg kT, the fraction of the blackbody photons that have energy hf>E0hf > E_0 is n(hf>E0)nγ≈0.42(E0kT)2exp⁡ ⁣(−E0kT).(2.43)\frac{n(hf > E_0)}{n_{\gamma}} \approx 0.42 \left(\frac{E_0}{kT}\right)^2 \exp\!\left(-\frac{E_0}{kT}\right). \tag{2.43} The cosmic background radiation is currently called the ``cosmic microwave background.'' However, photons with λ<1 mm\lambda < 1\,\mathrm{mm} actually lie in the far infrared range of the electromagnetic spectrum. It's time for truth in advertising: what fraction of the photons in today's ``cosmic microwave background'' are actually far infrared photons?

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