Density correlations induced by temperature fluctuations in a photon gas

The impact of angular temperature variations on the thermodynamic variables and real-space correlation functions of black-body radiation are analyzed. In particular, the effect of temperature fluctuations on the number density and energy density correlations of the cosmic microwave background (CMB) is studied. The angular temperature fluctuations are modeled by an isotropic and homogeneous Gaussian random field, whose autocorrelation function is defined on the unit sphere in momentum space. This temperature correlation function admits an angular Fourier transform which determines the density correlations in real space induced by temperature fluctuations. In the case of the CMB radiation, the multipole coefficients of the angular power spectrum defining the temperature correlation function have been measured by the Planck satellite. The fluctuation-induced perturbation of the equilibrium variables (internal energy, entropy, heat capacity and compressibility) can be quantified in terms of the measured multipole coefficients by expanding the partition function around the equilibrium state in powers of the temperature random field. The real-space density correlations can also be extracted from the measured temperature power spectrum. Both the number density and energy density correlations of the electromagnetic field are long-range, admitting power-law decay; in the case of the energy density correlation, the fluctuation-induced correlation overpowers the isotropic equilibrium correlation in the long-distance limit.