Consider a source of diffuse background radiation having a continuum volume emissivity, (, t) as a function of wavelength, , and cosmic time t t_{0}. The background intensity due to this source at a given received wavelength, _{0}, results as the integration of the redshifted contributions from all epochs. The background generated during a time interval dt is accumulated over the pathlength cdt where c is the speed of light. If a(t) / a_{0} denotes the ratio of expansion of the Universe between the epoch of emission and today, the received intensity due to emission within dt is therefore
(A1) |
where the emissivity is evaluated at the appropriate blue-shifted wavelength of emission, (t) = _{0}a(t) / a_{0}. The factor in square brackets takes into account the dilution of the intensity due to the expansion of the Universe (as per Liouville's Theorem expressed in the adopted intensity units of photons s^{-1} cm^{-2} sr^{-1} Å^{-1}). Integrating eq. (1) over the age of the Universe gives then for the total background intensity
(A2) |
It is convenient to change variables from cosmic time to redshift, (1 + z) = a_{0} / a(t), using the standard expression for a Friedman Universe
(A3) |
where H_{0} is the Hubble Constant. This leads to
(A4) |
where (z) = _{0} / (1 + z).
Equation (A4) is the general expression for the accumulated background from any continuum emission source. The special case of redshift smeared monochromatic line emission is also of interest. In this case we have for the emissivity
(A5) |
Where _{l} is the rest wavelength of the transition in question. Inserting (A5) into (A4) leads to the following expression for the intensity of the resulting continuum background at wavelengths _{0} _{l}
(A6) |
where = _{0} / _{l} - 1 is the redshift of emission.