Integral representation of the cosmic microwave background spectrum

We use an integral representation for nonthermal radiation, which is bounded from below and above, to describe the spectrum of the cosmic microwave background (CMB). The upper bound is given by the Rayleigh-Jeans law with a temperature $T_{\mathrm{RJ}}$ that can be determined by the absorption signal of 21 cm photons, where $T_{\mathrm{RJ}}$ represents the equilibrium temperature of photons in the RJ tail. If $T_{\mathrm{RJ}}>T_{\mathrm{CMB}}$, then the lower bound allows us to conclude that photons, additional to the remnant of the big bang, are needed to explain the present CMB. These constraints are additional to other cosmological or astrophysical constraints in the study of the distortions of the CMB brought about by new physics particles or fields.

Figure 1

Plots of the upper bound given in (6) (green), the lower bound in (4) (red), our example in (13) (blue) and the Planckian blackbody spectrum in (12) (black) at the temperature of the CMB as observed in the present day, viz., $T_{\mathrm{CMB}}=2.725K$. The dot in the green plot at $\omega /T_{\mathrm{CMB}}=0.0251$ represents the upper bound (6) for the 21 cm photons for $T_{\mathrm{RJ}}=1.060T_{\mathrm{CMB}}$.

Figure 2

The blue plot represents our example (13) and the black that of the blackbody radiation in (12). The plots cross at $\omega /T_{\mathrm{CMB}}=0.360$ (not shown) with the higher values of the blue plot over the black plot representing the enhancement of photons over that of the blackbody radiation with temperature $T_{\mathrm{CMB}}$.

Figure 3

Region of maximum difference between our example (13) and the blackbody radiation in inequality (12).