An example of a Michelson interferometer is the Far Infrared Absolute Spectrophotometer (FIRAS) carried by the Cosmic Background Explorer satellite (COBE). COBE studied the spectrum and anisotropies of the cosmic microwave background radiation (CMB) that emerged from the very early, hot phase of our universe’s expansion. One of the goals of the COBE mission was to see whether the CMB spectrum really has the shape of 2.7 K blackbody (Planckian) radiation, or if it is highly distorted, as some measurements made on rocket flights had suggested. COBE’s spectrophotometer used Fourier transform spectroscopy to meet this goal: it compared accurately the degree of longitudinal coherence γ_∥ of the CMB radiation with that of a calibrated source on board the spacecraft, which was known to be a blackbody at about 2.7K. The comparison was made by alternately feeding radiation from the microwave background and radiation from the calibrated source into the same Michelson interferometer and comparing their fringe spacings. The result (Mather et al., 1994) was that the background radiation has a spectrum that is Planckian with temperature 2.726 ± 0.010 K over the wavelength range 0.5–5.0 mm, in agreement with simple cosmological theory that we shall explore in the last chapter of this book.

(a) Suppose that the CMB had had a Wien spectrum

Show that the visibility of the fringes would have been

where s = cτ is longitudinal distance, and calculate a numerical value for s₀.

(b) Compute the interferogram V (τ) for a Planck function either analytically (perhaps with the help of a computer) or numerically using a fast Fourier transform. Compare graphically the interferogram for the Wien and Planck spectra.

Transcribed Image Text:

Fax w³ exp[-hw/(kBT)].