The universe is filled with cosmic microwave radiation left over from the big bang. At each event in spacetime the microwave radiation has a mean rest frame. As seen in that mean rest frame the radiation’s distribution function η is almost precisely isotropic and thermal with zero chemical potential:

Here ν is the frequency of a photon as measured in the mean rest frame.(a) Show that the specific intensity of the radiation as measured in its mean rest frame has the Planck spectrum, Eq. (3.23). Plot this specific intensity as a function of frequency, and from your plot determine the frequency of the intensity peak.(b) Show that η can be rewritten in the frame-independent form

where p(vector) is the photon 4-momentum, and u_o(vector) is the 4-velocity of the mean rest frame.

(c) In actuality, Earth moves relative to the mean rest frame of the microwave background with a speed v of roughly 400 km s−1 toward the Hydra-Centaurus region of the sky. An observer on Earth points his microwave receiver in a direction that makes an angle θ with the direction of that motion, as measured in Earth’s frame. Show that the specific intensity of the radiation received is precisely Planckian in form [Eqs. (3.23)], but with a direction-dependent Doppler-shifted temperature

Note that this Doppler shift of T is precisely the same as the Doppler shift of the frequency of any specific photon [Eq. (2.33)]. Note also that the θ dependence corresponds to an anisotropy of the microwave radiation as seen from Earth. Show that because Earth’s velocity is small compared to the speed of light, the anisotropy is very nearly dipolar in form. Measurements by the WMAP satellite give T_o = 2.725 K and (averaged over a year) an amplitude of 3.346 × 10⁻³ K for the dipolar temperature variations (Bennett et al., 2003). What, precisely, is the value of Earth’s year-averaged speed v?

Transcribed Image Text:

n = 1 ehv/(kB To) - 1 with T=2.725 K. (3.29)